Origins and Natures of Inflation, Dark Matter and Dark Energy

Figure 1
<p>Concept of a three-beam stimulated resonant photon collider (<math display="inline"><semantics><mrow><msup><mrow/><mi mathvariant="normal">t</mi></msup><mi>SRPC</mi></mrow></semantics></math>) with focused coherent beams [<a href="#B21-universe-09-00355" class="html-bibr">21</a>]. The two focused creation laser beams (green) at the incident angle <math display="inline"><semantics><msub><mi>θ</mi><mi>c</mi></msub></semantics></math> produces an ALP resonance state and the focused inducing laser beam (red) stimulates its decay. The creation photons have different energies <math display="inline"><semantics><msub><mi>ω</mi><mn>1</mn></msub></semantics></math> and <math display="inline"><semantics><msub><mi>ω</mi><mn>2</mn></msub></semantics></math> from the central value of <math display="inline"><semantics><msub><mi>ω</mi><mi>c</mi></msub></semantics></math> and different incident angles <math display="inline"><semantics><msub><mi>ϑ</mi><mn>1</mn></msub></semantics></math> and <math display="inline"><semantics><msub><mi>ϑ</mi><mn>2</mn></msub></semantics></math> from <math display="inline"><semantics><msub><mi>θ</mi><mi>c</mi></msub></semantics></math>, respectively. Similarly, the inducing laser (red) with a central wavelength of <math display="inline"><semantics><msub><mi>ω</mi><mi>i</mi></msub></semantics></math> has part of the beam with <math display="inline"><semantics><msub><mi>ω</mi><mn>4</mn></msub></semantics></math>, increasing the emission probability of the signal photon of <math display="inline"><semantics><msub><mi>ω</mi><mn>3</mn></msub></semantics></math> (blue) via energy-momentum conservation.</p> Full article ">Figure 2
<p>Two proposals for variable angle mechanisms. The green beams are creation lasers, the red beam is an inducing laser, and the blue beam indicates signal photons. <b>Left</b>: parabolic mirror type where the collision angle is changeable by changing the incident position of lasers. <b>Right</b>: rotating stage type where the collision angle is changeable by assembling a beam focusing system on multiple rotary stages.</p> Full article ">Figure 3
<p>Variable angle mechanism using a rotary stage. The incident angle is varied by rotating a stage assembling a beam-focusing system with a periscope (PS) and a parabolic mirror (PM). By using a periscope (PS), the angle is changeable only by rotating the periscope (PS) and the mirror (M) on the x-axis rail stage in front of PS. By setting the parabolic mirror’s focal point at the center of the rotary stage (RS), the collision point remains fixed even when the incident angle is varied. The focal spots can be checked using a monitoring camera.</p> Full article ">Figure 4
<p>Side views of designed variable-angle three-beam stimulated resonant photon colliders for large-angle (<b>left</b>) and narrow-angle (<b>right</b>) setups. Detailed explanations are found in the main text.</p> Full article ">Figure 5
<p>Collisional geometries viewed from the top. (<b>a</b>) Large-angle setup from <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>24.8</mn></mrow></semantics></math> deg (<b>left</b>) to <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>47.9</mn></mrow></semantics></math> deg (<b>right</b>) and (<b>b</b>) narrow-angle setup from <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>9.3</mn></mrow></semantics></math> deg (<b>left</b>) to <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>24.8</mn></mrow></semantics></math> deg (<b>right</b>). The details can be found in the main text.</p> Full article ">Figure 6
<p><b>Left</b>: top view of the camera system on the top layer of the rotary stages to monitor focal spots of all the three lasers c1, c2, and i. <b>Right</b>: picture assembling all the components for the large-angle setup. The details can be found in the main text.</p> Full article ">Figure 7
<p>Picture of large-angle setup (<b>left</b>) and focal images of three individual beams (<b>right</b>) when lasers with a common beam diameter of 5 mm are focused into IP at <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>24.8</mn></mrow></semantics></math> degree. In the picture, the optical paths of the three beams, consisting of the creation beam (c1), the creation beam (c2), and the inducing beam (i), as well as the signal photon line (s) are drawn. The middle column shows the images of three individual lasers when they hit the crossed point between two thin target wires of a 10 <math display="inline"><semantics><mi mathvariant="sans-serif">μ</mi></semantics></math>m diameter.</p> Full article ">Figure 8
<p>Picture of large-angle setup (<b>left</b>) and focal images of three individual beams (<b>right</b>) when lasers with a common beam diameter of 5 mm are focused into IP at <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>35.5</mn></mrow></semantics></math> degree. The other details are the same as in <a href="#universe-09-00355-f007" class="html-fig">Figure 7</a>.</p> Full article ">Figure 9
<p>Picture of large-angle setup (<b>left</b>) and focal images of three individual beams (<b>right</b>) when lasers with a common beam diameter of 5 mm are focused into IP at <math display="inline"><semantics><mrow><msub><mi>θ</mi><mi>c</mi></msub><mo>=</mo><mn>47.9</mn></mrow></semantics></math> degree. The other details are the same as in <a href="#universe-09-00355-f007" class="html-fig">Figure 7</a>.</p> Full article ">Figure 10
<p>Incident angles of the inducing beam <math display="inline"><semantics><msub><mi>θ</mi><mi>i</mi></msub></semantics></math> as a function of ALP masses <math display="inline"><semantics><msub><mi>m</mi><mi>a</mi></msub></semantics></math> which are determined by incident angles of creation beams <math display="inline"><semantics><msub><mi>θ</mi><mi>c</mi></msub></semantics></math>. Three combinations of laser wavelengths for fundamental, second harmonic, and third harmonic cases. Namely, beginning with 800 nm (Ti:Sapphire) for two creation beams and 1064 nm (Nd:YAG) for an inducing beam expressed as <math display="inline"><semantics><mrow><mn>800</mn><mo>×</mo><mn>2</mn><mo>+</mo><mn>1064</mn></mrow></semantics></math> nm (red), we extend the search to those with <math display="inline"><semantics><mrow><mn>400</mn><mo>×</mo><mn>2</mn><mo>+</mo><mn>532</mn></mrow></semantics></math> nm (blue) and <math display="inline"><semantics><mrow><mn>267</mn><mo>×</mo><mn>2</mn><mo>+</mo><mn>355</mn></mrow></semantics></math> nm (magenta). Depending on the combinations between the two angle setups and laser wavelengths, accessible mass ranges are different. This figure shows projections to cover from 0.5 to 6.9 eV.</p> Full article ">Figure 11
<p>Sensitivity projections based on realistic parameters in <a href="#universe-09-00355-t002" class="html-table">Table 2</a>. NA<math display="inline"><semantics><mi>ω</mi></semantics></math>, NA<math display="inline"><semantics><mrow><mn>2</mn><mi>ω</mi></mrow></semantics></math>, LA<math display="inline"><semantics><mrow><mn>2</mn><mi>ω</mi></mrow></semantics></math>, and LA<math display="inline"><semantics><mrow><mn>3</mn><mi>ω</mi></mrow></semantics></math> are sensitivities corresponding to the four arrows in <a href="#universe-09-00355-f010" class="html-fig">Figure 10</a> specifying individual mass ranges. The other details can be found in the main text.</p> Full article ">

Figure 1
<p>Concept of a three-beam stimulated resonant photon collider (<math display="inline"><semantics> <msup> <mrow/> <mi mathvariant="normal">t</mi> </msup> </semantics></math>SRPC) with focused coherent beams.</p> Full article ">Figure 2
<p>Schematic drawing of the search setup.</p> Full article ">Figure 3
<p>Photographs of the search setup (<b>left</b>) and the three focused laser spots (<b>right</b>) at a common thin cross-wire target.</p> Full article ">Figure 4
<p>Photograph of oscilloscope waveforms from the four photodetectors in <a href="#universe-09-00123-f002" class="html-fig">Figure 2</a>. Four-wave mixing (FWM) photons were clearly observed when a thin BBO crystal was positioned at IP.</p> Full article ">Figure 5
<p>Observed number of FWM photons as a function of stage position in the delay line for a fine timing tuning between two creation laser pulses when a thin BBO crystal was positioned at IP.</p> Full article ">Figure 6
<p>Four patterns of the beam combination between the two laser pulses where the green and red pulses are respectively creation and inducing laser pulses. The classifications are: S for two laser pulses, C for only the creation laser pulses, I for only the inducing laser pulses, and P for pedestals without laser pulses.</p> Full article ">Figure 7
<p>Arrival time distribution of FWM photons via the atomic process when a BBO crystal was placed at IP and space–time synchronization was ensured by PMT1. The red lines thus provide the expected time window for FWM photons via ALP-exchange to arrive.</p> Full article ">Figure 8
<p>Arrival time distributions of photons with no target state (air) at IP. The histograms in the upper left, upper right, lower left, and lower right correspond to S, C, I, and P patterns of beam combinations, respectively. The interval between the two red lines in the S-pattern indicates the expected time windows for FWM photons via ALP-exchange to arrive.</p> Full article ">Figure 9
<p>Arrival time distribution of reconstructed photons after subtraction between the four patterns based on Equation (<a href="#FD5-universe-09-00123" class="html-disp-formula">5</a>) over the entire time range in <a href="#universe-09-00123-f008" class="html-fig">Figure 8</a>. The interval between the two red lines indicates the expected time windows for FWM photons via ALP-exchange to arrive.</p> Full article ">Figure 10
<p>Upper limit in the parameter space of the coupling-mass relation (region enclosed by the red solid curve) evaluated at a 95% confidence level for the pseudoscalar field exchange achieved by the three-beam stimulated resonant photon collider (<math display="inline"><semantics> <msup> <mrow/> <mi mathvariant="normal">t</mi> </msup> </semantics></math>SRPC00). The magenta area indicates the excluded range based on SRPC in quasi-parallel collision geometry (SAPPHIRES01) [<a href="#B23-universe-09-00123" class="html-bibr">23</a>]. The purple areas are excluded regions by the Light Shining through a Wall (LSW) experiments (ALPS [<a href="#B27-universe-09-00123" class="html-bibr">27</a>] and OSQAR [<a href="#B28-universe-09-00123" class="html-bibr">28</a>]). The gray area shows the excluded region by the vacuum magnetic birefringence (VMB) experiment (PVLAS [<a href="#B29-universe-09-00123" class="html-bibr">29</a>]). The light-cyan horizontal solid line indicates the upper limit from the search for eV (pseudo)scalar penetrating particles in the SPS neutrino beam (NOMAD) [<a href="#B30-universe-09-00123" class="html-bibr">30</a>]. The horizontal dotted line indicates the upper limit from the Horizontal Branch observation [<a href="#B31-universe-09-00123" class="html-bibr">31</a>]. The blue areas indicate the excluded regions from the optical MUSE-faint survey [<a href="#B32-universe-09-00123" class="html-bibr">32</a>]. The green area is the excluded region by the helioscope experiment CAST [<a href="#B33-universe-09-00123" class="html-bibr">33</a>,<a href="#B34-universe-09-00123" class="html-bibr">34</a>,<a href="#B35-universe-09-00123" class="html-bibr">35</a>,<a href="#B36-universe-09-00123" class="html-bibr">36</a>]. The yellow band and the upper solid brown line are the predictions of QCD axion by the KSVZ model [<a href="#B3-universe-09-00123" class="html-bibr">3</a>,<a href="#B4-universe-09-00123" class="html-bibr">4</a>] with <math display="inline"><semantics> <mrow> <mn>0.07</mn> <mo>&lt;</mo> <mfenced separators="" open="|" close="|"> <mi>E</mi> <mo>/</mo> <mi>N</mi> <mo>−</mo> <mn>1.95</mn> </mfenced> <mo>&lt;</mo> <mn>7</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>/</mo> <mi>N</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, respectively. The bottom dashed brown line is the prediction from the DFSZ model [<a href="#B13-universe-09-00123" class="html-bibr">13</a>,<a href="#B14-universe-09-00123" class="html-bibr">14</a>] with <math display="inline"><semantics> <mrow> <mi>E</mi> <mo>/</mo> <mi>N</mi> <mo>=</mo> <mn>8</mn> <mo>/</mo> <mn>3</mn> </mrow> </semantics></math>. The cyan lines are the predictions from the ALP <span class="html-italic">miracle</span> model [<a href="#B12-universe-09-00123" class="html-bibr">12</a>] with the intrinsic parameter values <math display="inline"><semantics> <mrow> <msub> <mi>c</mi> <mi>γ</mi> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>0.1</mn> <mo>,</mo> <mn>0.01</mn> </mrow> </semantics></math>, respectively.</p> Full article ">Figure 11
<p>Expected incident angles of creation and inducing lasers, <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>c</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>θ</mi> <mi>i</mi> </msub> </semantics></math>, respectively, as a function of ALP mass when two wavelengths of creation (808 nm) and inducing lasers (1064 nm) are assumed, resulting in the fixed wavelength of FWM signals, 651 nm, in vacuum.</p> Full article ">Figure A1
<p>Schematic view of the setup to evaluate two angle parameters to define Jones vectors for the two creation lasers based on Stokes parameters. A polarizer (POL) was placed between a periscope (PS) and a lens (L) in each beamline. Stokes parameters were obtained by measuring transmitted laser intensity through POL at four different rotation angles by using individual cameras C<math display="inline"><semantics> <msub> <mrow/> <mrow> <mi>c</mi> <mn>1</mn> </mrow> </msub> </semantics></math> and C<math display="inline"><semantics> <msub> <mrow/> <mrow> <mi>c</mi> <mn>2</mn> </mrow> </msub> </semantics></math> assigned for the two creation lasers.</p> Full article ">

Figure 1
<p>Schematic view of a three-beam stimulated photon collider.</p> Full article ">Figure 2
<p>Relation between theoretical coordinates with the primed symbol and laboratory coordinates to which laser beams are physically mapped. The <math display="inline"><semantics> <msup> <mi>z</mi> <mo>′</mo> </msup> </semantics></math>-axis is theoretically obtainable so that stochastically selected two incident photons satisfying the resonance condition have zero pair transverse momentum (<math display="inline"><semantics> <msub> <mi>p</mi> <mi>T</mi> </msub> </semantics></math>) with respect to <math display="inline"><semantics> <msup> <mi>z</mi> <mo>′</mo> </msup> </semantics></math>. The Lorentz invariant scattering amplitude is calculated on the primed coordinates where rotation symmetries of the initial and final state reaction planes around <math display="inline"><semantics> <msup> <mi>z</mi> <mo>′</mo> </msup> </semantics></math> are maintained. Definitions of four-momentum vectors <math display="inline"><semantics> <msubsup> <mi>p</mi> <mi>i</mi> <mo>′</mo> </msubsup> </semantics></math> and four-polarization vectors <math display="inline"><semantics> <mrow> <msup> <mi>e</mi> <mo>′</mo> </msup> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <msubsup> <mi>p</mi> <mi>i</mi> <mo>′</mo> </msubsup> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> with polarization states <math display="inline"><semantics> <msub> <mi>λ</mi> <msubsup> <mi>p</mi> <mi>i</mi> <mo>′</mo> </msubsup> </msub> </semantics></math> for the initial state (<math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </semantics></math>) and final state (<math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> </mrow> </semantics></math>) plane waves are given. This figure is extracted from [<a href="#B14-universe-09-00020" class="html-bibr">14</a>].</p> Full article ">Figure 3
<p>Collision geometry between three short pulsed laser beams to define the spacetime overlapping factor <math display="inline"><semantics> <msub> <mi mathvariant="script">D</mi> <mrow> <mi>t</mi> <mi>h</mi> <mi>r</mi> <mi>e</mi> <mi>e</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 4
<p>Flow of the numerical calculation. The left figure depicts the initial state of two scattering photons with incidence of two creation beams (green) and an inducing beam (red), while the right figure indicates the final state photons, that is, the inducing beam photons and signal photons (blue) in the laboratory coordinates by omitting the outgoing two creation beams. The top figure is to remind of the scattering amplitude calculation in the primed coordinates. Probability distribution functions in momentum space <math display="inline"><semantics> <msub> <mi>G</mi> <mi>p</mi> </msub> </semantics></math> as a function of polar angles <math display="inline"><semantics> <msub> <mo>Θ</mo> <mi>i</mi> </msub> </semantics></math> and azimuthal angles <math display="inline"><semantics> <msub> <mo>Φ</mo> <mi>i</mi> </msub> </semantics></math> in the laboratory coordinates and those in energy <math display="inline"><semantics> <mrow> <msub> <mi>G</mi> <mi>E</mi> </msub> <mrow> <mo>(</mo> <msub> <mi>ω</mi> <mi>i</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics></math> for individual photons <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>4</mn> </mrow> </semantics></math> are assigned to individual focused beams by denoting the normalized Gaussian distributions as <span class="html-italic">G</span>.</p> Full article ">Figure 5
<p>Expected sensitivities in the coupling–mass relation for the pseudoscalar field exchange at a 95% confidence level by a three-beam stimulated resonant photon collider (<math display="inline"><semantics> <msup> <mrow/> <mi mathvariant="normal">t</mi> </msup> </semantics></math>SRPC) with focused short-pulsed lasers based on the beam parameters in <a href="#universe-09-00020-t001" class="html-table">Table 1</a>.</p> Full article ">

Figure 1
<p>Holographic Inflationary Cosmology in Conformal Time: Equal Time Surfaces are Hyperbolae Interpolating Between Diamonds. There are additional green spacelike surfaces that we have omitted for clarity, interpolating between those shown, such that the proper time between consecutive surfaces along the central geodesic is always the Planck time. When a non-central geodesic penetrates the causal diamond of the central geodesic, consistency with the fact that that geodesic is still experiencing inflation implies that it must behave like an isolated quantum system with finite entropy given by the area law.</p> Full article ">

Figure 1
<p>Plot (<b>a</b>) shows the behavior of Equation (<a href="#FD28-universe-08-00510" class="html-disp-formula">28</a>) viz <span class="html-italic">r</span> for BH (<a href="#FD19-universe-08-00510" class="html-disp-formula">19</a>). The behavior of the metric potential <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> </mrow> </semantics></math>, which characterizes the horizons by putting <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>(</mo> <mi>r</mi> <mo>)</mo> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>: (<b>b</b>) for linear Maxwell Rastall gravity theory; (<b>c</b>) for the nonlinear electrodynamics Rastall’s theory. The values of <span class="html-italic">m</span> for the linear case are 1.3; 0.99; 0.8 and q = 1, while for the nonlinear case m = 1.3; 1.1 and 0.9, q = 1 and <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Λ</mi> <mrow> <mi>e</mi> <mi>f</mi> <mi>f</mi> <mo>.</mo> </mrow> </msub> </semantics></math>= 0.3.</p> Full article ">Figure 2
<p>Plots of thermodynamical quantities of BHs. (<b>a</b>) The mass-radius relation, which determines the minimal mass. (<b>b</b>) The hawking temperature, which vanishes at <math display="inline"><semantics> <msub> <mi>r</mi> <mi>h</mi> </msub> </semantics></math>. (<b>c</b>) The heat capacity. Moreover, the linear case investigates a second-order phase transition. All the figures are plotted for <math display="inline"><semantics> <mrow> <msub> <mi>m</mi> <mi>h</mi> </msub> <mo>=</mo> <mi>q</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>.</p> Full article ">

Figure 1
<p>(<b>top</b>): Aerial view and geometry of H10T instrument with location of 10 detectors, adapted from Beznosko et al., 2019 [<a href="#B4-universe-07-00384" class="html-bibr">4</a>]; (<b>bottom</b>): A typical MME event recorded on 6 April 2018 by H10T instrument at point <math display="inline"><semantics> <mrow> <mo>#</mo> <mn>9</mn> </mrow> </semantics></math>, adapted from Beznosko et al., 2019 [<a href="#B4-universe-07-00384" class="html-bibr">4</a>]. All pulses are recorded at a single detection point. Delay times, the width of each peak <math display="inline"><semantics> <msub> <mi>τ</mi> <mrow> <mn>0.8</mn> </mrow> </msub> </semantics></math> in ns, and the particle density <math display="inline"><semantics> <msub> <mi>ρ</mi> <mrow> <mn>0.8</mn> </mrow> </msub> </semantics></math> per <math display="inline"><semantics> <msup> <mi mathvariant="normal">m</mi> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </semantics></math> within <math display="inline"><semantics> <msub> <mi>τ</mi> <mrow> <mn>0.8</mn> </mrow> </msub> </semantics></math> are also shown in the table.</p> Full article ">Figure 2
<p>Solid lines: the particle density distribution <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>(</mo> <mi>R</mi> <mo>)</mo> </mrow> </semantics></math> in simulated EAS disk versus distance from axis for different energies shown by different colours, depending on energy of the CR. Blue dots: cumulative particle density for each bimodal pulse vs. distance to EAS axis, adapted from Beznosko et al., 2019 [<a href="#B4-universe-07-00384" class="html-bibr">4</a>].</p> Full article ">Figure 3
<p>Solid lines: simulated EAS disk width versus distance from axis for different energies shown by different colours. Pulse width <math display="inline"><semantics> <msub> <mi>τ</mi> <mrow> <mn>0.8</mn> </mrow> </msub> </semantics></math> in ns for the first pulse (in blue) and second pulse (in red) for the bimodal pulses versus distance to the EAS axis, adapted from Beznosko et al., 2019 [<a href="#B4-universe-07-00384" class="html-bibr">4</a>].</p> Full article ">

Figure 1
<p>The evolution of the Hubble parameter (blue curve) is shown for the best-fit values of model parameters, as given in <a href="#universe-07-00362-t001" class="html-table">Table 1</a>, arising from the joint analysis of <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> + BAO + CMB + BBN + SNIa (Pantheon) dataset (<b>left</b> panel) and <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> + BAO + CMB + BBN + SNIa (Union 2.1) dataset (<b>right</b> panel). Here, the red curve represents the corresponding evolution of <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> in a standard <math display="inline"><semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics></math>CDM model with <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>c</mi> <mi>d</mi> <mi>m</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.315</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>67.4</mn> </mrow> </semantics></math> km/s/Mpc [<a href="#B40-universe-07-00362" class="html-bibr">40</a>] (<b>left</b> panel) and <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>c</mi> <mi>d</mi> <mi>m</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.2675</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>71.3</mn> </mrow> </semantics></math> km/s/Mpc [<a href="#B37-universe-07-00362" class="html-bibr">37</a>] (<b>right</b> panel). In this plot, the green dots correspond to the 51 <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> data points in the redshift range <math display="inline"><semantics> <mrow> <mn>0.07</mn> <mo>≤</mo> <mi>z</mi> <mo>≤</mo> <mn>2.36</mn> </mrow> </semantics></math>, obtained from different surveys and the corresponding <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> values are given in [<a href="#B44-universe-07-00362" class="html-bibr">44</a>]. In the inset diagram, the corresponding relative difference, <math display="inline"><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mo>%</mo> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>100</mn> <mo>×</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mrow> <mi>N</mi> <mi>G</mi> <mi>C</mi> <mi>G</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>−</mo> <msub> <mi>H</mi> <mrow> <mi mathvariant="sans-serif">Λ</mi> <mi>C</mi> <mi>D</mi> <mi>M</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>/</mo> <msub> <mi>H</mi> <mrow> <mi mathvariant="sans-serif">Λ</mi> <mi>C</mi> <mi>D</mi> <mi>M</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, is shown for the best-fit model.</p> Full article ">Figure 2
<p>The evolution of <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> is shown for the best-fit values of model parameters, as given in <a href="#universe-07-00362-t001" class="html-table">Table 1</a>, arising from the joint analysis of <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> + BAO + CMB + BBN + SNIa (Union 2.1) dataset (blue curve). The <math display="inline"><semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics></math>CDM model (<math display="inline"><semantics> <mrow> <msub> <mi mathvariant="sans-serif">Ω</mi> <mrow> <mi>c</mi> <mi>d</mi> <mi>m</mi> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>0.2675</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>H</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>71.3</mn> </mrow> </semantics></math> km/s/Mpc [<a href="#B37-universe-07-00362" class="html-bibr">37</a>]) is also shown in the red line for model comparison. Here, <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> represents the distance modulus, which is the difference between the apparent magnitude and the absolute magnitude of the observed supernova, is given by [<a href="#B3-universe-07-00362" class="html-bibr">3</a>] <math display="inline"><semantics> <mrow> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>25</mn> <mo>+</mo> <mn>5</mn> <msub> <mi>log</mi> <mn>10</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mi>L</mi> </msub> <mo>/</mo> <mi>M</mi> <mi>p</mi> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math>, where <math display="inline"><semantics> <msub> <mi>d</mi> <mi>L</mi> </msub> </semantics></math> is the luminosity distance. In this plot, the black dots correspond to the Error bar plot of 580 points of Union 2.1 compilation Supernovae Type Ia data sets [<a href="#B41-universe-07-00362" class="html-bibr">41</a>].</p> Full article ">Figure 3
<p>Plot of <span class="html-italic">q</span> as a function of <span class="html-italic">z</span> is shown by considering the values of model parameters, as given in <a href="#universe-07-00362-t001" class="html-table">Table 1</a>, arising from the joint analysis of <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> + BAO + CMB + BBN + SNIa (Pantheon) dataset (blue curve). Here, the red, green, and dotted (purple) curves represent the corresponding evolution of <span class="html-italic">q</span> in a standard <math display="inline"><semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics></math>CDM, GCG, and CG models, respectively.</p> Full article ">Figure 4
<p>The time evolutions of the statefinder pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>left</b> panel) and the pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>right</b> panel) for this model are shown using the <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> + BAO + CMB + BBN + SNIa (Pantheon) dataset, as indicated in each panel. The red point <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> in the left panel corresponds to the <math display="inline"><semantics> <mi mathvariant="sans-serif">Λ</mi> </semantics></math>CDM model, while, in the right panel, the green point <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>=</mo> <mn>0.5</mn> <mo>,</mo> <mi>r</mi> <mo>=</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </semantics></math> represents the matter dominated Universe (SCDM). In addition, the black dots on the curves show present values <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>s</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>left</b> panel) and <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <msub> <mi>q</mi> <mn>0</mn> </msub> <mo>,</mo> <msub> <mi>r</mi> <mn>0</mn> </msub> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>right</b> panel) for the NGCG model.</p> Full article ">Figure 5
<p>The time evolutions of the statefinder pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>s</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>left</b> panel) and the pair <math display="inline"><semantics> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>,</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> (<b>right</b> panel) for different models are shown using the <math display="inline"><semantics> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> + BAO + CMB + BBN + SNIa (Pantheon) dataset. Here, the blue, green, and dotted (purple) curves are for the NGCG, GCG, and CG models, respectively.</p> Full article ">Figure 6
<p>Evolution of <math display="inline"><semantics> <mrow> <msub> <mi>O</mi> <mi>m</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics></math> is shown for different models, as indicated in the panel.</p> Full article ">

Figure 1
<p>The integral of the MQN number density from minimum detectable MQN mass <span class="html-italic">M_MQN</span> to infinity times 230 km/s, the velocity of the solar system through the galactic halo, gives the predicted number flux for mass ≥<span class="html-italic">M_MQN</span> plotted on the <span class="html-italic">x</span>-axis for range of currently allowed values of <span class="html-italic">B_o</span>. For <span class="html-italic">B_o</span> = 1.65 × 10¹² T and atmospheric mass density (~10⁻³ kg/m³) at the 50 km altitude to be probed for nuclearites by JEM-EUSO, the nuclearite mass <span class="html-italic">M_nuc</span> with the same interaction cross section as <span class="html-italic">M_MQN</span> is plotted above the graph.</p> Full article ">Figure 2
<p>Representative density maps are shown for times when radial expansion of the channel in the clay-sand layer is approaching its maximum: (<b>a</b>) 9 MJ/m at t = 20 ms, (<b>b</b>) 27 MJ/m at t = 25 ms, (<b>c</b>) 81 MJ/m at t = 40 ms, and (<b>d</b>) 243 MJ/m at t = 55 ms. The red material represents three layers separated by black lines. From the bottom up, the layers are 0.3 m of granite, 1.0 m of clay-sand, and 0.7 m of peat with initial density of 1.12 × 10³ kg/m³. The blue area is atmospheric air. The white spaces are voids at shear planes. Movies of pressure, density, and temperature are available online (accessed on 14 February 2021: <a href="https://datadryad.org/stash/share/Lt7dMvxEAUWNnkfKt2xPg2l5TuWz7Bbec67iY4Kvazg" target="_blank">https://datadryad.org/stash/share/Lt7dMvxEAUWNnkfKt2xPg2l5TuWz7Bbec67iY4Kvazg</a>.</p> Full article ">Figure 3
<p>Solid lines show crater diameter in granite (red), clay-sand (blue), and peat (black) as a function of the energy/length from CTH simulations. The data points show the diameter of fractured granite from reference [<a href="#B37-universe-07-00116" class="html-bibr">37</a>].</p> Full article ">Figure 4
<p>Mass density profiles of peat at <span class="html-italic">t</span> = 10 ms after the start of a simulated MQN interaction depositing 30 MJ/m on a trajectory inclined at (<b>a</b>) 0°, (<b>b</b>) 15°, (<b>c</b>) 30°, and (<b>d</b>) 45° from the vertical.</p> Full article ">Figure 5
<p>(<b>a</b>) Photograph of the crater soon after the event in 1985 illustrate good agreement in the actual and simulated profile of crater sides. (<b>b</b>) Photograph of the crater in March 2005 showing the full diameter and circularity before excavation.</p> Full article ">Figure 6
<p>Cross-sectional view of the three excavations: 2017 (black line), 2018 (red line), and 2019 (blue line), and the three layers: peat (gray), clay-sand (blue), and granite (brown). Brown ellipsoids represent the two granite boulders found to be distributed within the clay-sand volume of the 2018 excavation and the ten found in the 2019 excavation. The brown rectangle shows the location of the only ensemble of fractured rock found in the excavations.</p> Full article ">Figure 7
<p>Two of the ten boulders found within the excavated volume are shown. Their diameters are approximately 0.4 m and 0.6 m, and they were distributed throughout the 633 m³ of the 2019 excavation, but similar boulders were not observed on the surface of the peat bog.</p> Full article ">Figure 8
<p>Granite shards from the volume below the grouping of fractured granite at 4.7 ± 0.1 m are shown. The rocks are covered with the fines from the clay-sand mixture which distorts their natural colors. Although rocks that are similar to the larger samples in <a href="#universe-07-00116-f008" class="html-fig">Figure 8</a> were found on the surface of the peat, no collection of rocks similar to the single shattered boulder was found on the surface.</p> Full article ">Figure 9
<p>Pressure-washed face of the excavation’s west side, adjacent to the grouping of shards at the 4.7 ± 0.1 m depth found in the 2018 excavation.</p> Full article ">Figure 10
<p>The layout of granite rocks (brown) and sediment walls (gray) at depth of ~ 6.3 to 6.5 m. The scale is in meters and the angles are between the vertical and the normal to the largest-area surface. The origin is directly below the center of the original impact crater on the surface with an estimated accuracy of +/− 0.3 m. The dotted ellipse shows the approximate projection of the shattered rock found in the 2018 excavation at 4.7-m depth to the 6.4-m depth shown here.</p> Full article ">Figure A1
<p>View of the 4-m diameter crater from 1985, drained with a channel shown to the down sloping terrain on the right.</p> Full article ">Figure A2
<p>Excavation in 2019 to depth of 5.7 m and showing access ramp, submersible water pump and hose, and exit ladder. Examining the bottom and recovering rock fragments had to be done by feel at this stage.</p> Full article ">Figure A3
<p>Photo of the collection of granite rocks found at approximately 6.3-m depth to the southeast of the center of the crater. The normal to the slab on the right is inclined at 30° to the vertical.</p> Full article ">Figure A4
<p>Wooden stake and three survey lines mark the center of the original impact crater for subsequent expeditions to extend the investigation into the granite bedrock below 6.5-m depth.</p> Full article ">

Figure 1
<p>Three MQN trajectories are shown as red arrows, along with their angle of impact <span class="html-italic">θ</span> with respect to the normal surface of an idealized, not to scale, Earth (blue). The trajectory with impact angel <span class="html-italic">θ₁</span> ≪ 90° is a more common radial impact. Nearly tangential trajectories that transit through Earth are represented by trajectories with impact angles between <span class="html-italic">θ</span>₂ and <span class="html-italic">θ</span>_3. MQNs on a <span class="html-italic">θ</span>₂ trajectory emerge from Earth with negligible velocity after transiting a distance <span class="html-italic">x_max</span>, the maximum range of an MQN. MQNs on a <span class="html-italic">θ</span>₃ trajectory emerge from Earth with considerable velocity.</p> Full article ">Figure 2
<p>Estimated number of events per year somewhere on Earth as a function of <span class="html-italic">B_o</span> for MQNs with 10⁵ kg ≤ <span class="html-italic">m</span> ≤ 10¹⁰ kg. The four solid-line curves correspond to event rates based on interstellar dark-matter density [<a href="#B3-universe-07-00035" class="html-bibr">3</a>,<a href="#B42-universe-07-00035" class="html-bibr">42</a>] of ~7 × 10⁻²² kg/m³: MQNs transiting through water and emerging with any velocity <span class="html-italic">v_exit</span> (blue); MQNs transiting through water and emerging with velocity 10 m/s ≤ <span class="html-italic">v_exit</span> ≤ 100 m/s (gray); MQNs transiting through granite and emerging with any velocity <span class="html-italic">v_exit</span> (red); MQNs transiting through granite and emerging with velocity 10 m/s ≤ <span class="html-italic">v_exit</span> ≤ 100 m/s (black).</p> Full article ">Figure 3
<p>Cross sectional view of the magnetopause is shown (black line) between an MQN (blue circle) with magnetic moment (purple vector) at an angle of 60° to the velocity of the plasma (yellow arrows) flowing into the rest frame of the MQN. The plasma flow produces a net force (red arrow) centered at the top of the MQN and a corresponding torque vector into the page. The magnetopause is the locus of points at which the plasma pressure (on the left in <a href="#universe-07-00035-f003" class="html-fig">Figure 3</a>) is balanced by the magnetic pressure of the compressed magnetic field on the right. The complex shape of the magnetopause and resulting torque have been computed by Papagiannis [<a href="#B43-universe-07-00035" class="html-bibr">43</a>] for Earth, illustrated in <a href="#universe-07-00035-f003" class="html-fig">Figure 3</a>, and extended to the case of MQNs. The effect can be understood by considering that the mean distance between the magnetopause and the MQN on the top half of <a href="#universe-07-00035-f003" class="html-fig">Figure 3</a> is less than on the bottom half, which means that the magnetic field is compressed more on the top than on the bottom. Since force is transmitted by the compressed magnetic field, the net force is a push on the top, as shown by the red arrow.</p> Full article ">Figure 4
<p>Estimated angular velocity in the first 10⁻⁴ s for 10⁶ kg quark nugget with velocity <span class="html-italic">v</span> = 220 km/s, initial angle <span class="html-italic">χ</span> = 0.61 rad, initial angular velocity <span class="html-italic">ω</span> = 0, and passing through matter with mass density of 2300 kg/m³. Note the initial oscillation about 0 until a full rotation occurs, after which the angular velocity increases rapidly.</p> Full article ">Figure 5
<p>Geometry of simulation of rotating magnetized sphere above a highly conducting material. Magnetized, rotating sphere (blue) is shown above conducting material (gray). Arrows inside sphere indicate rotation and arrows in conducting material indicate force on the material. Arrows in air (white) indicate magnetic field lines at one moment in time. The axis of rotation is the <span class="html-italic">y</span>-axis, out of the plane of the figure. The magnetic axis of the magnetized sphere is initially in the <span class="html-italic">x</span>-direction and remains in the <span class="html-italic">xz</span>-plane.</p> Full article ">Figure 6
<p>Components of the time-averaged force between the simulated quark nugget and conducting slab as a function of the electromagnetic skin depth <span class="html-italic">λ</span>. The negative (&lt;0) force <span class="html-italic">F_z</span> (blue) opposes gravity and levitates the rotating magnetized sphere for <span class="html-italic">λ</span> &lt; 0.5 m, with the most negative value for <span class="html-italic">λ</span> &lt; 0.03 m. The force generated by the magnetic field traveling through the deformable conductor in the <span class="html-italic">x</span>-direction, as the magnetized sphere rotates about the <span class="html-italic">y</span>-axis, generates a propulsive force <span class="html-italic">F_x</span> (red). <span class="html-italic">F_x</span> is much less than <span class="html-italic">F_z</span> for small skin depths. The much smaller <span class="html-italic">F_y</span> (black) illustrates ±5% error in the calculation, since symmetry requires <span class="html-italic">F_y</span> = 0.</p> Full article ">Figure 7
<p>Contours of the hole formed by the rotating the magnetic field of the magnetized sphere in (<b>a</b>) the <span class="html-italic">x</span>-direction and (<b>b</b>) the <span class="html-italic">y</span>-direction for times 2.5 ms (blue), 10 ms (gold), 20 ms (red), and 30 ms (black). The magnetic field sweeps through plastically deformable conducting material, and the displacement of the bottom of the hole is approximately −0.25 m in the <span class="html-italic">x</span>-direction. The same contours for the <span class="html-italic">y</span>-direction, which is along the axis of rotation, show the deformation is symmetric about <span class="html-italic">y</span> = 0, as expected. In both cases, the vertical axis has a different scale from the horizontal axis.</p> Full article ">Figure A1
<p><b>Geometry of simulation of a rotating quark nugget.</b> (<b>a</b>) The overall geometry of the simulation shows the 20 m radius conducting boundary with the cylindrical rotating coordinate system centered on the axis and the 4 × 4 × 2 m thick slab of simulated peat. (<b>b</b>) Close up of the peat slab with its top surface located 0.3 m below the center of the 0.1 m radius, spherical magnet simulating the quark nugget, which is inside the cylindrical 0.2 m radius rotating coordinate system.</p> Full article ">Figure A2
<p>Amplitude of the magnetic induction <span class="html-italic">B</span> in the rotating and magnetized sphere required to produce a time-averaged force of 10⁷ N on the peat as a function of the radius of the magnetized sphere. The radius of the magnetized sphere was varied between 5 and 150 mm. The mesh size was too large for the calculation to converge for radii less than 5 mm.</p> Full article ">Figure A3
<p><b>Extract from the 1863 Ordnance Survey map.</b> The locations of the “square hole” (a), the most prominent trench (b), triangular channel (c), and cave (d), and the reported path of the “globe of fire” (dotted line) between features are shown. The field with the “20-foot-square” hole (a) has been drained and is lower than the field with the trench (b) and the triangular channel (c). Therefore, we do not know the relative elevation of the hole (a) and the trench (b) in 1868.</p> Full article ">Figure A4
<p>The location consistent with Fitzgerald’s “20-foot-square hole,” at which the ball of light first disappeared into the peat. (<b>a</b>) Photo of the “square hole” with dimensions and 0.5 m deep contour (dashed blue line). (<b>b</b>) Contours at three depths are shown: 0 m (solid black line), 0.3 m (dashed black line), and 0.5 m (dashed blue line). The natural feeder drainage flows approximately from the lower right to the upper left. Orientation to north is approximate.</p> Full article ">Figure A5
<p>Photograph looking along the seventh trench.</p> Full article ">Figure A6
<p>Fitzgerald’s “25 m long diversion of the stream” and the current path of the stream, which continues to the left and right of the contour map. (<b>a</b>) Photo of the site as seen from the western end; the channel made by the “globe of fire” is on the right and the stream that was recut by the Council is on the left. (<b>b</b>) Contour map of the site constructed from the survey and field notes. Solid lines: surface level. Long dashes: the bottom of the channel at −1.2 ± 0.25 m level. Short dashes: the bottom of the stream, as cut by the County Council in the 1980s, at the −1.9 ± 0.2 m level. Orientation to the north is approximate.</p> Full article ">Figure A7
<p>Cave at the end of the semi-circular channel.</p> Full article ">

Figure 1
<p>The 1<inline-formula><mml:math id="mm1371"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1372"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on <inline-formula><mml:math id="mm1373"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1374"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mo>Λ</mml:mo></mml:msub></mml:semantics></mml:math></inline-formula> parameters. (Left panel) In the standard spatially flat <inline-formula><mml:math id="mm1375"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model from the SNe Ia Pantheon dataset, as well as from the combined BAO peak length scale and Planck datasets. (The figure is adapted from [<xref ref-type="bibr" rid="B210-universe-10-00122">210</xref>]). (Right panel) In the spatially non-flat <italic>o</italic>CDM model using discovery sample of Riess et al. [<xref ref-type="bibr" rid="B211-universe-10-00122">211</xref>] and the full Pantheon sample of Scolnic et al. [<xref ref-type="bibr" rid="B212-universe-10-00122">212</xref>]. Pantheon constraints with systematic uncertainties are presented in red, while only statistical uncertainties are denoted in gray. (The figure is adapted from [<xref ref-type="bibr" rid="B212-universe-10-00122">212</xref>]).</p> Full article ">Figure 2
<p>(Left panel) The location of thawing and freezing scalar fields in the <inline-formula><mml:math id="mm1376"><mml:semantics><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>ϕ</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>d</mml:mi><mml:msub><mml:mi>w</mml:mi><mml:mi>ϕ</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:mi>d</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mi>a</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> plane. (The figure is adapted from [<xref ref-type="bibr" rid="B135-universe-10-00122">135</xref>]). (Right panel) Regimes of the quick rolling down and slow rolling down of the scalar field to the minimum of its potential.</p> Full article ">Figure 3
<p>(Left panel) The 1<inline-formula><mml:math id="mm1377"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1378"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on cosmological parameters <inline-formula><mml:math id="mm1379"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1380"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> in the <italic>w</italic>CDM model from various datasets: SNe Ia apparent magnitude + CMB temperature anisotropy + BAO peak length scale + HST (yellow), BAO peak length scale + CMB temperature anisotropy (blue), SNe Ia apparent magnitude + CMB temperature anisotropy (red). (Right panel) The 1<inline-formula><mml:math id="mm1381"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1382"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours constraints on cosmological parameters <inline-formula><mml:math id="mm1383"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <italic>w</italic> in the <italic>w</italic>CDM model from various datasets: SNe Ia apparent magnitude + CMB temperature anisotropy (black), CMB temperature anisotropy (blue), SNe Ia apparent magnitude (red) (with systematic uncertainties), SNe Ia apparent magnitude (gray line) (with only statistical uncertainties). The figure is adapted from [<xref ref-type="bibr" rid="B212-universe-10-00122">212</xref>].</p> Full article ">Figure 4
<p>The 1<inline-formula><mml:math id="mm1384"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1385"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1386"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the scalar field <inline-formula><mml:math id="mm1387"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. (Left panel) (<bold>a</bold>) For all R98 SNe Ia apparent magnitude data, (<bold>b</bold>) for R98 SNe Ia apparent magnitude data excluding the <inline-formula><mml:math id="mm1388"><mml:semantics><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0.97</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> measurement, (<bold>c</bold>) for P99 Fit C SNe Ia apparent magnitude dataset, (<bold>d</bold>) for three datasets: all R98 SNe Ia apparent magnitude (long-dashed lines) data, R98 SNe Ia apparent magnitude data excluding the <inline-formula><mml:math id="mm1389"><mml:semantics><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0.97</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> measurement (short-dashed lines), and P99 Fit C SNe Ia apparent magnitude data (dotted lines). (Right panel) (<bold>a</bold>) For all the R98 SNe Ia apparent magnitude data, (<bold>b</bold>) for R98 SNe Ia apparent magnitude data excluding the <inline-formula><mml:math id="mm1390"><mml:semantics><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0.97</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> measurement, (<bold>c</bold>) for P99 Fit C SNe Ia apparent magnitude data, (<bold>d</bold>) for the <inline-formula><mml:math id="mm1391"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1392"><mml:semantics><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> constraints used in conjunction with all R98 SNe Ia (long-dashed lines) apparent magnitude data, R98 SNe Ia apparent magnitude data excluding the <inline-formula><mml:math id="mm1393"><mml:semantics><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0.97</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> measurement (short-dashed lines), and the P99 Fit C SNe Ia apparent magnitude data (dotted lines). The figure is adapted from [<xref ref-type="bibr" rid="B241-universe-10-00122">241</xref>].</p> Full article ">Figure 5
<p>(Left panel) The 1<inline-formula><mml:math id="mm1394"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1395"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1396"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on <inline-formula><mml:math id="mm1397"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1398"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> parameters by using a sample of 176 SNe Ia apparent magnitude data. (Upper left sub-panel) For the ordinary quintessence with the inverse power-law RP potential, (upper right sub-panel) for ordinary quintessence with the Sugra potential, (lower left sub-panel) for extended quintessence with the inverse power-law RP potential, (lower right sub-panel) for extended quintessence with the inverse power-law RP potential when upper limits on the time variation of the gravitational constant are satisfied. (Right panel) The 1<inline-formula><mml:math id="mm1399"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1400"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1401"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on <inline-formula><mml:math id="mm1402"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1403"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> parameters for the ordinary <inline-formula><mml:math id="mm1404"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM quintessence model with the inverse power-law RP potential by using SNAP sample data. (Upper left sub-panel) corresponds to constraints obtained by assuming the exact EoS parameter, (upper right sub-panel) corresponds to the linear approximation of the EoS parameter, (lower left sub-panel) corresponds to the constant approximation of the EoS parameter, (lower right sub-panel) corresponds to the superposition of above-mentioned three cases. The figure is adapted from [<xref ref-type="bibr" rid="B214-universe-10-00122">214</xref>].</p> Full article ">Figure 6
<p>(Left panel) As <xref ref-type="fig" rid="universe-10-00122-f005">Figure 5</xref>, but for the ordinary <inline-formula><mml:math id="mm1405"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM quintessence model with the Sugra potential. (Right panel) As <xref ref-type="fig" rid="universe-10-00122-f005">Figure 5</xref>, but for the extended quintessence model with the inverse power-law RP potential. The results are obtained by imposing the upper bound on the time variation of the gravitational constant. The figure is adapted from [<xref ref-type="bibr" rid="B214-universe-10-00122">214</xref>].</p> Full article ">Figure 7
<p>Constraints on parameters <inline-formula><mml:math id="mm1406"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi>e</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1407"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>. The left picture depicts the distribution from WMAP + CBI + VSA + SDSS + HST data and the right picture is that of SNe Ia apparent magnitude versus redshift data alone. The regions of <inline-formula><mml:math id="mm1408"><mml:semantics><mml:mrow><mml:mn>1</mml:mn><mml:mi>σ</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> (<inline-formula><mml:math id="mm1409"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mi>σ</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>) confidence level are enclosed by a white (black) line. The figure is adapted from [<xref ref-type="bibr" rid="B247-universe-10-00122">247</xref>].</p> Full article ">Figure 8
<p>SNe Ia apparent magnitude versus redshift data [<xref ref-type="bibr" rid="B211-universe-10-00122">211</xref>] as data points with thin (brown) error bars. The authors plotted the logarithm of the luminosity distance minus a fiducial model for which <inline-formula><mml:math id="mm1410"><mml:semantics><mml:mrow><mml:msub><mml:mi>d</mml:mi><mml:mi>L</mml:mi></mml:msub><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo form="prefix">ln</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. The solid (black) line is for the spatially flat <inline-formula><mml:math id="mm1411"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model, the dotted (blue) line is for <inline-formula><mml:math id="mm1412"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi>e</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, and the dashed (red) line is for <inline-formula><mml:math id="mm1413"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi>e</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msup><mml:mn>10</mml:mn><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. For all models, <inline-formula><mml:math id="mm1414"><mml:semantics><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B247-universe-10-00122">247</xref>].</p> Full article ">Figure 9
<p>(Left panel) The 2<inline-formula><mml:math id="mm1415"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> contours of the fixed time parameter <inline-formula><mml:math id="mm1416"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> as a function of values of the matter density parameter at the present epoch <inline-formula><mml:math id="mm1417"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> and space curvature density parameter at the present epoch <inline-formula><mml:math id="mm1418"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, as well as the model parameter <inline-formula><mml:math id="mm1419"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> in the scalar field <inline-formula><mml:math id="mm1420"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The results obtained for larger values of free parameters (<inline-formula><mml:math id="mm1421"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>) and for <inline-formula><mml:math id="mm1422"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>[</mml:mo><mml:mn>0.7</mml:mn><mml:mo>,</mml:mo><mml:mn>0.75</mml:mn><mml:mo>,</mml:mo><mml:mn>0.8</mml:mn><mml:mo>,</mml:mo><mml:mn>0.85</mml:mn><mml:mo>,</mml:mo><mml:mn>0.95</mml:mn><mml:mo>,</mml:mo><mml:mn>1.05</mml:mn><mml:mo>,</mml:mo><mml:mn>1.15</mml:mn><mml:mo>]</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right panel) The 2<inline-formula><mml:math id="mm1423"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> contours of the factor by which the growth of matter perturbations falls lower than in the Einstein–de Sitter model. The cosmological test parameter values <inline-formula><mml:math id="mm1424"><mml:semantics><mml:mrow><mml:mo>▵</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> obtained for the larger values of free parameters (<inline-formula><mml:math id="mm1425"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>). The figure is adapted from [<xref ref-type="bibr" rid="B252-universe-10-00122">252</xref>].</p> Full article ">Figure 10
<p>The 1<inline-formula><mml:math id="mm1426"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1427"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the matter density parameter at the present epoch <inline-formula><mml:math id="mm1428"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> and the dark energy density (quintessence) parameter at the present epoch <inline-formula><mml:math id="mm1429"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">Q</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> for scalar field <inline-formula><mml:math id="mm1430"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models. (Left panel) With the inverse power-law PR potential. (Right panel) With Sugra potential <inline-formula><mml:math id="mm1431"><mml:semantics><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∝</mml:mo><mml:msup><mml:mi>ϕ</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msup><mml:mo form="prefix">exp</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>4</mml:mn><mml:mi>π</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. The blue lines accord to the flat universe. The figure is adapted from [<xref ref-type="bibr" rid="B253-universe-10-00122">253</xref>].</p> Full article ">Figure 11
<p>The 1<inline-formula><mml:math id="mm1432"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1433"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1434"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the spatially non-flat scalar field <inline-formula><mml:math id="mm1435"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential from compilations of data: <inline-formula><mml:math id="mm1436"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + SNe Ia apparent magnitude (first row), <inline-formula><mml:math id="mm1437"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (second row), and BAO peak length scale + SNe Ia apparent magnitude (third row). Filled circles denote best-fit points. The dot–dashed lines in the first column panels are 1<inline-formula><mml:math id="mm1438"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1439"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1440"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours obtained by Farooq et al. [<xref ref-type="bibr" rid="B260-universe-10-00122">260</xref>] for the spatially flat <inline-formula><mml:math id="mm1441"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model (open circles denote best-fit points). Dotted lines separate the accelerating and decelerating models (at zero space curvature). The horizontal axis with <inline-formula><mml:math id="mm1442"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1443"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. First, second, and third columns correspond to marginalizing over <inline-formula><mml:math id="mm1444"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1445"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm1446"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, respectively. The figure is adapted from [<xref ref-type="bibr" rid="B255-universe-10-00122">255</xref>].</p> Full article ">Figure 12
<p>The 1<inline-formula><mml:math id="mm1447"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1448"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on <inline-formula><mml:math id="mm1449"><mml:semantics><mml:msub><mml:mi>M</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1450"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> values for <inline-formula><mml:math id="mm1451"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM (left panel), <italic>w</italic>CDM (middle panel), and IDE (right panel) models, obtained from compilations of Pantheon + BAO + BBN and Pantheon + BAO + BBN + <inline-formula><mml:math id="mm1452"><mml:semantics><mml:msub><mml:mi>M</mml:mi><mml:mi>B</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> datasets. The figure is adapted from [<xref ref-type="bibr" rid="B158-universe-10-00122">158</xref>].</p> Full article ">Figure 13
<p>The CMB temperature anisotropy spectrum for different quintessence scalar field <inline-formula><mml:math id="mm1453"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models: with the leaping kinetic term (model A), with the inverse power-law RP potential (model B) (here the dark energy density parameter at the present epoch <inline-formula><mml:math id="mm1454"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.6</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>), and for the <inline-formula><mml:math id="mm1455"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model (model C). Data points from the BOOMERANG [<xref ref-type="bibr" rid="B266-universe-10-00122">266</xref>] and MAXIMA [<xref ref-type="bibr" rid="B267-universe-10-00122">267</xref>] experiments are shown for reference. The figure is adapted from Ref. [<xref ref-type="bibr" rid="B265-universe-10-00122">265</xref>].</p> Full article ">Figure 14
<p>(Left panel) Polarization (TE) and temperature (TT) as functions of the multipole <italic>l</italic>. Two quintessential inflation models with <inline-formula><mml:math id="mm1456"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.99</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1457"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1.05</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> are presented with WMAP data from [<xref ref-type="bibr" rid="B270-universe-10-00122">270</xref>,<xref ref-type="bibr" rid="B271-universe-10-00122">271</xref>]. WMAP-normalized spectra for the best fit for the <inline-formula><mml:math id="mm1458"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model (no <inline-formula><mml:math id="mm1459"><mml:semantics><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> data) with the constant spectral index <inline-formula><mml:math id="mm1460"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>0.97</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> [<xref ref-type="bibr" rid="B6-universe-10-00122">6</xref>] and the <inline-formula><mml:math id="mm1461"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model with the running spectral index <inline-formula><mml:math id="mm1462"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.93</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1463"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mi>k</mml:mi><mml:mo>=</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>−<inline-formula><mml:math id="mm1464"><mml:semantics><mml:mrow><mml:mn>0.031</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> are shown for comparison. For large <italic>l</italic>, CBI data and ACBAR data are used. (Right panel) The CDM power spectrum at the present epoch as a function of <inline-formula><mml:math id="mm1465"><mml:semantics><mml:mrow><mml:mi>k</mml:mi><mml:mo>/</mml:mo><mml:mi>h</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>. The linear spectrum for two quintessential inflation models with spectral indices <inline-formula><mml:math id="mm1466"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.99</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1467"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1.05</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> are plotted. The best fit for the <inline-formula><mml:math id="mm1468"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model with running spectral index <inline-formula><mml:math id="mm1469"><mml:semantics><mml:mrow><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.93</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1470"><mml:semantics><mml:mrow><mml:mi mathvariant="normal">d</mml:mi><mml:msub><mml:mi>n</mml:mi><mml:mi>s</mml:mi></mml:msub><mml:mo>/</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mo form="prefix">ln</mml:mo><mml:mi>k</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>0.031</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> [<xref ref-type="bibr" rid="B6-universe-10-00122">6</xref>]), normalized to WMAP data (no <inline-formula><mml:math id="mm1471"><mml:semantics><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> data), is shown. 2dFGRS measurements and <inline-formula><mml:math id="mm1472"><mml:semantics><mml:msub><mml:mi>L</mml:mi><mml:mrow><mml:mi>y</mml:mi><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> data are converted to <inline-formula><mml:math id="mm1473"><mml:semantics><mml:mrow><mml:mi>z</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B269-universe-10-00122">269</xref>].</p> Full article ">Figure 15
<p>The spectra are in arbitrary units, normalized to unity at the first acoustic peak. (Left panel) CMB angular total intensity power spectra for the scalar field <inline-formula><mml:math id="mm1474"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential (dotted), quadratic (dashed) and exponential coupling extended quintessence (solid) with <inline-formula><mml:math id="mm1475"><mml:semantics><mml:mrow><mml:msub><mml:mi>ω</mml:mi><mml:mrow><mml:mi>JBD</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>50</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right panel) CMB angular polarization power spectra for the <inline-formula><mml:math id="mm1476"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model (dotted), quadratic (dashed), and exponential coupling extended quintessence (solid) with <inline-formula><mml:math id="mm1477"><mml:semantics><mml:msub><mml:mi>ω</mml:mi><mml:mrow><mml:mi>JBD</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B215-universe-10-00122">215</xref>].</p> Full article ">Figure 16
<p>The model with <inline-formula><mml:math id="mm1478"><mml:semantics><mml:mrow><mml:msub><mml:mi>t</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>13</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> Gyr and <inline-formula><mml:math id="mm1479"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.014</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Likelihood functions <inline-formula><mml:math id="mm1480"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>m</mml:mi><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1481"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1482"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> (arbitrarily normalized to unity at the highest peak). (Left panel) Derived from a simultaneous analysis of DMR 53 and 90 GHz galactic-frame data. The faint high-latitude foreground galactic emission is corrected and the quadrupole moment in the analysis is included: (<bold>a</bold>) for <inline-formula><mml:math id="mm1483"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and (<bold>b</bold>) for <inline-formula><mml:math id="mm1484"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right panel) Derived by marginalizing <inline-formula><mml:math id="mm1485"><mml:semantics><mml:mrow><mml:mi>L</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>m</mml:mi><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1486"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1487"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> over <inline-formula><mml:math id="mm1488"><mml:semantics><mml:msub><mml:mi>Q</mml:mi><mml:mrow><mml:mi>r</mml:mi><mml:mi>m</mml:mi><mml:mi>s</mml:mi><mml:mo>−</mml:mo><mml:mi>P</mml:mi><mml:mi>S</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> with a uniform prior: (<bold>a</bold>) the correction for the faint high-latitude foreground galactic emission is ignored and the quadrupole moment from the analysis is excluded, (<bold>b</bold>) for the faint high latitude foreground galactic emission is corrected and the quadrupole moment in the analysis is included. The figure is adapted from [<xref ref-type="bibr" rid="B280-universe-10-00122">280</xref>].</p> Full article ">Figure 17
<p>The 1 <inline-formula><mml:math id="mm1489"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1490"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1491"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the scalar field <inline-formula><mml:math id="mm1492"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential using cluster gas mass fraction data. Solid lines correspond to WMAP prior while dashed lines correspond to the alternate prior. The cross matches the best fit at <inline-formula><mml:math id="mm1493"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.27</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1494"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The circle denotes the best fit at <inline-formula><mml:math id="mm1495"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.26</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1496"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The horizontal axis for which <inline-formula><mml:math id="mm1497"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the spatially flat <inline-formula><mml:math id="mm1498"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B283-universe-10-00122">283</xref>].</p> Full article ">Figure 18
<p>The 1<inline-formula><mml:math id="mm1499"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1500"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the spatially flat and spatially non-flat scalar field <inline-formula><mml:math id="mm1501"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law potential from a joint analysis using the HST <inline-formula><mml:math id="mm1502"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> prior in the scenario with three species of degenerate massive neutrinos. (Upper left, upper right, and lower left panels) Contours are presented in the <inline-formula><mml:math id="mm1503"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mo>∑</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1504"><mml:semantics><mml:mrow><mml:msub><mml:mi>σ</mml:mi><mml:mn>8</mml:mn></mml:msub><mml:mo>−</mml:mo><mml:mo>∑</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm1505"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>−</mml:mo><mml:mo>∑</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> planes. The thin blue (thick red) lines correspond to constraints in the spatially flat (spatially non-flat) universe. The “+” (“x”) denotes the mean values of the pair in the spatially flat (spatially non-flat) universe. (Lower right panel) Contours are in the <inline-formula><mml:math id="mm1506"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">k</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mo>∑</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> plane for the spatially non-flat universe. The “x” denotes the mean values of the (<inline-formula><mml:math id="mm1507"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">k</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mo>∑</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>) pair. The figure is adapted from [<xref ref-type="bibr" rid="B288-universe-10-00122">288</xref>].</p> Full article ">Figure 19
<p>(Upper panels) Evolution of the EoS parameter <inline-formula><mml:math id="mm1508"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mi>ϕ</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and dark energy density parameter <inline-formula><mml:math id="mm1509"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi>ϕ</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> in the tilted spatially flat <inline-formula><mml:math id="mm1510"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model for the range of values of <inline-formula><mml:math id="mm1511"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> parameter <inline-formula><mml:math id="mm1512"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>. The black solid curve accords to the <inline-formula><mml:math id="mm1513"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model, which corresponds to reduced <inline-formula><mml:math id="mm1514"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with <inline-formula><mml:math id="mm1515"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Middle panels) Theoretical predictions for matter density and CMB temperature anisotropy angular power spectra for the <inline-formula><mml:math id="mm1516"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model depending on parameter <inline-formula><mml:math id="mm1517"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula>. (Lower panels) Ratios of the <inline-formula><mml:math id="mm1518"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model power spectra relative to the <inline-formula><mml:math id="mm1519"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B291-universe-10-00122">291</xref>].</p> Full article ">Figure 20
<p>The 1<inline-formula><mml:math id="mm1520"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1521"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours. (Left panel) In the <inline-formula><mml:math id="mm1522"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> plane for the tilted spatially flat scalar field <inline-formula><mml:math id="mm1523"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model. (Right panel) In the <inline-formula><mml:math id="mm1524"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">k</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> plane for the untilted spatially non-flat scalar field <inline-formula><mml:math id="mm1525"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model. Constraints are derived from Planck CMB TT + lowP + lensing and non-CMB datasets. The horizontal dashed line indicates the spatially flat curvature with <inline-formula><mml:math id="mm1526"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. For the spatially non-flat <inline-formula><mml:math id="mm1527"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model constrained with TT + lowP + lensing, the <inline-formula><mml:math id="mm1528"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>&gt;</mml:mo><mml:mn>0.45</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> prior has been used. The figure is adapted from [<xref ref-type="bibr" rid="B291-universe-10-00122">291</xref>].</p> Full article ">Figure 21
<p>(Left panels (<bold>a</bold>,<bold>c</bold>,<bold>e</bold>)) The comparison of the spatially flat tilted <inline-formula><mml:math id="mm1529"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model (gray solid line) with the best-fit <inline-formula><mml:math id="mm1530"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula>’s for the XCDM model and the <inline-formula><mml:math id="mm1531"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model. (Right panels (<bold>b</bold>,<bold>d</bold>,<bold>f</bold>)) The comparison of the spatially flat tilted <inline-formula><mml:math id="mm1532"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model (gray solid line) with the best-fit <inline-formula><mml:math id="mm1533"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula>’s for the <inline-formula><mml:math id="mm1534"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model. The all-<italic>l</italic> region is shown on top panels. The low-<italic>l</italic> region <inline-formula><mml:math id="mm1535"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and residuals are represented on middle panels. The high-<italic>l</italic> region <inline-formula><mml:math id="mm1536"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and residuals are demonstrated on bottom panels. The figure is adapted from [<xref ref-type="bibr" rid="B294-universe-10-00122">294</xref>].</p> Full article ">Figure 22
<p>Constraints on various quantities related to reionization obtained from the MCMC analysis for the untilted spatially non-flat <inline-formula><mml:math id="mm1537"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM, XCDM, and <inline-formula><mml:math id="mm1538"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM quintessential inflation models that fit best the dataset: Planck 2015 TT + lowP + lensing and SNe Ia apparent magnitude, BAO peak length scale, <inline-formula><mml:math id="mm1539"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and LSS growth rate data. The thick central lines along with surrounding shaded regions correspond to best−fit models and their <inline-formula><mml:math id="mm1540"><mml:semantics><mml:mrow><mml:mn>2</mml:mn><mml:mi>σ</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> uncertainty ranges. (Upper panels) (<bold>a</bold>) A number of ionizing photons in the IGM per baryon in stars, (<bold>b</bold>) photoionization rates for hydrogen along with observational data from Wyithe and Bolton [<xref ref-type="bibr" rid="B298-universe-10-00122">298</xref>] and Becker and Bolton [<xref ref-type="bibr" rid="B299-universe-10-00122">299</xref>], (<bold>c</bold>) a specific number of Lyman−limit systems with data points from Songaila and Cowie [<xref ref-type="bibr" rid="B300-universe-10-00122">300</xref>] and Prochaska et al. [<xref ref-type="bibr" rid="B301-universe-10-00122">301</xref>]. (Lower panel) (<bold>d</bold>) Electron scattering optical depths along with their values from Park and Ratra [<xref ref-type="bibr" rid="B291-universe-10-00122">291</xref>], (<bold>e</bold>) volume filling factor of ionized regions, (<bold>f</bold>) global neutral hydrogen fraction with different present observational limits. The figure is adapted from [<xref ref-type="bibr" rid="B297-universe-10-00122">297</xref>].</p> Full article ">Figure 23
<p>The 1<inline-formula><mml:math id="mm1541"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1542"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours on parameters of the spatially non-flat IDE model, presented in (<inline-formula><mml:math id="mm1543"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mi>ξ</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>) plane (left panel) and in (<inline-formula><mml:math id="mm1544"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mo>Λ</mml:mo></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi>m</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>) plane (right panel). Here <inline-formula><mml:math id="mm1545"><mml:semantics><mml:mi>ξ</mml:mi></mml:semantics></mml:math></inline-formula> is the dimensionless coupling parameter which characterizes the strength of the interaction between the dark sectors. The figure is adapted from [<xref ref-type="bibr" rid="B121-universe-10-00122">121</xref>].</p> Full article ">Figure 24
<p>(Left upper panel) The 1<inline-formula><mml:math id="mm1546"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters <inline-formula><mml:math id="mm1547"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1548"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> of the <italic>w</italic>CDM model. (Right upper panel) The 1<inline-formula><mml:math id="mm1549"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters <inline-formula><mml:math id="mm1550"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">k</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1551"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> of the <italic>w</italic>CDM model. (Left lower panel) The 1<inline-formula><mml:math id="mm1552"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters <inline-formula><mml:math id="mm1553"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1554"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> of the scalar field spatially flat <inline-formula><mml:math id="mm1555"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential. (Right lower panel) The 1<inline-formula><mml:math id="mm1556"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the normalized Hubble constant <italic>h</italic> and the parameter <inline-formula><mml:math id="mm1557"><mml:semantics><mml:mi>γ</mml:mi></mml:semantics></mml:math></inline-formula> describing deviations from general relativity for various dark energy models. The figure is adapted from [<xref ref-type="bibr" rid="B306-universe-10-00122">306</xref>].</p> Full article ">Figure 25
<p>The 1<inline-formula><mml:math id="mm1558"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1559"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1560"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the spatially flat scalar field <inline-formula><mml:math id="mm1561"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential from LSS growth rate measurements (blue dashed lines with blue filled circle at best fit <inline-formula><mml:math id="mm1562"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.28</mml:mn><mml:mo>,</mml:mo><mml:mn>0.052</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1563"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>/</mml:mo><mml:mi>dof</mml:mi><mml:mo>=</mml:mo><mml:mn>8.62</mml:mn><mml:mo>/</mml:mo><mml:mn>12</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>); SNe Ia apparent magnitude + <inline-formula><mml:math id="mm1564"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data (red dot–dashed lines with red filled circle at best fit <inline-formula><mml:math id="mm1565"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.26</mml:mn><mml:mo>,</mml:mo><mml:mn>0.302</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1566"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>/</mml:mo><mml:mi>dof</mml:mi><mml:mo>=</mml:mo><mml:mn>562</mml:mn><mml:mo>/</mml:mo><mml:mn>598</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>); and a combination of all datasets (black solid lines and black filled circle at the best fit <inline-formula><mml:math id="mm1567"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.27</mml:mn><mml:mo>,</mml:mo><mml:mn>0.300</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1568"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>/</mml:mo><mml:mi>dof</mml:mi><mml:mo>=</mml:mo><mml:mn>570</mml:mn><mml:mo>/</mml:mo><mml:mn>612</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>). The horizontal axis with <inline-formula><mml:math id="mm1569"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1570"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model and the curved dotted line denotes zero-acceleration models. The figure is adapted from [<xref ref-type="bibr" rid="B315-universe-10-00122">315</xref>].</p> Full article ">Figure 26
<p>The 1<inline-formula><mml:math id="mm1571"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1572"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters <inline-formula><mml:math id="mm1573"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1574"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> in the scalar field <inline-formula><mml:math id="mm1575"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential. (Left panel) Constraints are obtained from the LSS growth rate data [<xref ref-type="bibr" rid="B318-universe-10-00122">318</xref>]. (Right panel) Constraints are obtained from the data compilation of Gupta et al. (2012) [<xref ref-type="bibr" rid="B318-universe-10-00122">318</xref>] and Giostri et al. (2012) [<xref ref-type="bibr" rid="B319-universe-10-00122">319</xref>]. The figure is adapted from [<xref ref-type="bibr" rid="B317-universe-10-00122">317</xref>].</p> Full article ">Figure 27
<p>The 1<inline-formula><mml:math id="mm1576"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1577"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour plots for various pairs of free parameters (<inline-formula><mml:math id="mm1578"><mml:semantics><mml:msub><mml:mi>V</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1579"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <italic>h</italic>, <inline-formula><mml:math id="mm1580"><mml:semantics><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm1581"><mml:semantics><mml:mover accent="true"><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>˙</mml:mo></mml:mover></mml:semantics></mml:math></inline-formula>), for which the spatially flat <inline-formula><mml:math id="mm1582"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the Zlatev–Wang–Steinhardt potential is in the best fit with the standard spatially flat <inline-formula><mml:math id="mm1583"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B320-universe-10-00122">320</xref>].</p> Full article ">Figure 28
<p>The 1<inline-formula><mml:math id="mm1584"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1585"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour plots for various pairs of free parameters (<italic>k</italic>, <inline-formula><mml:math id="mm1586"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <italic>h</italic>, <inline-formula><mml:math id="mm1587"><mml:semantics><mml:msub><mml:mi>V</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1588"><mml:semantics><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm1589"><mml:semantics><mml:mover accent="true"><mml:msub><mml:mi>ϕ</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>˙</mml:mo></mml:mover></mml:semantics></mml:math></inline-formula>), for which the spatially flat <inline-formula><mml:math id="mm1590"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the phantom PNGB potential is in the best fit with the standard spatially flat <inline-formula><mml:math id="mm1591"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B320-universe-10-00122">320</xref>].</p> Full article ">Figure 29
<p>The 1<inline-formula><mml:math id="mm1592"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1593"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour plots for various pairs of free parameters (<inline-formula><mml:math id="mm1594"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1595"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, <italic>h</italic>), for which the spatially flat <inline-formula><mml:math id="mm1596"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential is the best fit with the standard spatially flat <inline-formula><mml:math id="mm1597"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B320-universe-10-00122">320</xref>].</p> Full article ">Figure 30
<p>(Left panel) The comparison of the possible <inline-formula><mml:math id="mm1598"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1599"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> values for quintessence dark energy potentials in the spatially flat scalar field <inline-formula><mml:math id="mm1600"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the CPL-<inline-formula><mml:math id="mm1601"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM 1<inline-formula><mml:math id="mm1602"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1603"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1604"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours. (Right panel) The comparison of possible <inline-formula><mml:math id="mm1605"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1606"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> values for phantom dark energy potentials in the spatially flat scalar field <inline-formula><mml:math id="mm1607"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the CPL-<inline-formula><mml:math id="mm1608"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM 1<inline-formula><mml:math id="mm1609"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1610"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1611"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours. The figure is adapted from [<xref ref-type="bibr" rid="B320-universe-10-00122">320</xref>].</p> Full article ">Figure 31
<p>The 1<inline-formula><mml:math id="mm1612"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1613"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1614"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the spatially flat scalar field <inline-formula><mml:math id="mm1615"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential using different combinations of datasets and the compressed Planck 2018 data [<xref ref-type="bibr" rid="B13-universe-10-00122">13</xref>]. The results obtained from DS2/BSP dataset CMB temperature anisotropy + BAO peak length scale + LSS growth rate (gray contours), DS1/BSP dataset: SNe Ia apparent magnitude + <inline-formula><mml:math id="mm1616"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale + LSS growth rate + CMB temperature anisotropy (dashed red contours), and DS1/BSP dataset without LSS growth rate data CMB temperature anisotropy + BAO peak length scale + SNe Ia apparent magnitude + <inline-formula><mml:math id="mm1617"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> (solid green contours). The results are presented in the <inline-formula><mml:math id="mm1618"><mml:semantics><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>cdm</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula> plane (left panel) and in the <inline-formula><mml:math id="mm1619"><mml:semantics><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>cdm</mml:mi></mml:msub><mml:mo>−</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> plane (right panel), where <inline-formula><mml:math id="mm1620"><mml:semantics><mml:mrow><mml:msub><mml:mi>w</mml:mi><mml:mi>cdm</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> is a physical matter density parameter. The figure is adapted from [<xref ref-type="bibr" rid="B321-universe-10-00122">321</xref>].</p> Full article ">Figure 32
<p>The 1<inline-formula><mml:math id="mm1621"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1622"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1623"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the spatially flat scalar field <inline-formula><mml:math id="mm1624"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential from DS1/SP, DS1/BSP, and DS2/BSP datasets. The figure is adapted from [<xref ref-type="bibr" rid="B321-universe-10-00122">321</xref>].</p> Full article ">Figure 33
<p>The 1<inline-formula><mml:math id="mm1625"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1626"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour for the tilted spatially flat XCDM model (left panel) and for the untilted spatially non-flat XCDM model (right panel), constrained by Planck CMB TT + lowP + lensing and non-CMB datasets. The horizontal and vertical dashed lines indicate the standard spatially flat <inline-formula><mml:math id="mm1627"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model (with <inline-formula><mml:math id="mm1628"><mml:semantics><mml:mrow><mml:mi>w</mml:mi><mml:mo>=</mml:mo><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1629"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">k</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>). The figure is adapted from [<xref ref-type="bibr" rid="B326-universe-10-00122">326</xref>].</p> Full article ">Figure 34
<p>Best-fit CMB temperature anisotropy power spectra of (<bold>a</bold>) the tilted spatially flat XCDM model (top five panels) and (<bold>b</bold>) the untilted spatially non-flat XCDM model (bottom five panels) constrained by Plank CMB TT + lowP data (excluding lensing data) together with data on SNe Ia apparent magnitude, BAO peak length scale, <inline-formula><mml:math id="mm1630"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, and LSS growth rate. The best-fit power spectra of the tilted spatially flat <inline-formula><mml:math id="mm1631"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model are shown as black curves. The residual <inline-formula><mml:math id="mm1632"><mml:semantics><mml:mrow><mml:mi>δ</mml:mi><mml:msub><mml:mi>D</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> of the TT power spectra are shown with respect to the spatially flat <inline-formula><mml:math id="mm1633"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM power spectrum that best fits the TT + lowP data. The high-<italic>l</italic> region <inline-formula><mml:math id="mm1634"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and residuals are shown on the bottom panels. The figure is adapted from [<xref ref-type="bibr" rid="B326-universe-10-00122">326</xref>].</p> Full article ">Figure 35
<p>The 1<inline-formula><mml:math id="mm1635"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1636"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1637"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the scalar field <inline-formula><mml:math id="mm1638"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP scalar field potential. The <inline-formula><mml:math id="mm1639"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> axis corresponds to the standard spatially flat <inline-formula><mml:math id="mm1640"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left panel) Solid lines are constraints derived by Percival et al. (2007) [<xref ref-type="bibr" rid="B17-universe-10-00122">17</xref>] using BAO peak length scale data in conjunction with WMAP data on acoustic horizon angle. Dashed lines are constraints obtained by Eisenstein et al. (2005) [<xref ref-type="bibr" rid="B20-universe-10-00122">20</xref>] from BAO peak length scale data. The circle denotes the best-fit value. Two sets of dotted lines are constraints obtained from galaxy cluster gas mass fraction measurements of Samushia and Ratra (2008) [<xref ref-type="bibr" rid="B327-universe-10-00122">327</xref>]; thick dotted lines are derived using WMAP priors for <italic>h</italic> and <inline-formula><mml:math id="mm1641"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> while thin dotted lines are obtained for alternate priors. The cross denotes the best-fit value. (Right panel) Solid lines are joint constraints obtained by Percival et al. (2007) from BAO peak length scale data in conjunction with WMAP data on acoustic horizon angle and galaxy cluster gas mass fraction measurements. The circle denotes the best-fit value with a suitable <inline-formula><mml:math id="mm1642"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≃</mml:mo><mml:mn>58</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 42 degrees of freedom; dashed lines are joint constraints derived by Eisenstein et al. (2005) [<xref ref-type="bibr" rid="B20-universe-10-00122">20</xref>] using BAO peak length scale data. The cross denotes the best-fit value with a suitable <inline-formula><mml:math id="mm1643"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>≃</mml:mo><mml:mn>52</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 41 degrees of freedom. Thick lines are derived using the WMAP priors for <italic>h</italic> and <inline-formula><mml:math id="mm1644"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, and thin lines are for alternate priors. Joint best-fit values for two prior sets overlap. Here, <inline-formula><mml:math id="mm1645"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1646"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> ranges are smaller than those shown on the left panel. The figure is adapted from [<xref ref-type="bibr" rid="B327-universe-10-00122">327</xref>].</p> Full article ">Figure 36
<p>The 1<inline-formula><mml:math id="mm1647"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1648"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1649"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the scalar field <inline-formula><mml:math id="mm1650"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the inverse power-law RP potential. The horizontal axis with <inline-formula><mml:math id="mm1651"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1652"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left panel) Dotted lines obtained from the lookback time data and measurements of the age of the universe. The cross denotes the best-fit parameters <inline-formula><mml:math id="mm1653"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.04</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1654"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>10</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1655"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>22</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, for <inline-formula><mml:math id="mm1656"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1657"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>359</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 346 degrees of freedom derived using measurements of the lookback time, the age of the universe, SNe Ia apparent magnitude, and BAO peak length scale, while solid lines are derived using only SNe Ia apparent magnitude measurements and BAO peak length scale data. The cross denotes the best-fit point at <inline-formula><mml:math id="mm1658"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.22</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1659"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1660"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>329</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 307 degrees of freedom. The figure is adapted from [<xref ref-type="bibr" rid="B328-universe-10-00122">328</xref>].</p> Full article ">Figure 37
<p>The 1<inline-formula><mml:math id="mm1661"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1662"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours of one- and two-dimensional distributions of <inline-formula><mml:math id="mm1663"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi>b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1664"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1665"><mml:semantics><mml:mrow><mml:mo>∑</mml:mo><mml:msub><mml:mi>m</mml:mi><mml:mi>ν</mml:mi></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm1666"><mml:semantics><mml:msub><mml:mi>σ</mml:mi><mml:mn>8</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> for the quintessential inflation model with the exponential potential <inline-formula><mml:math id="mm1667"><mml:semantics><mml:mrow><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>ϕ</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>∝</mml:mo><mml:mo form="prefix">exp</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:mi>λ</mml:mi><mml:msup><mml:mi>ϕ</mml:mi><mml:mi>n</mml:mi></mml:msup><mml:mo>/</mml:mo><mml:msubsup><mml:mi>M</mml:mi><mml:mrow><mml:mi>pl</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msubsup><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1668"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> (orange line) and <inline-formula><mml:math id="mm1669"><mml:semantics><mml:mrow><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> (blue line). The figure is adapted from [<xref ref-type="bibr" rid="B177-universe-10-00122">177</xref>].</p> Full article ">Figure 38
<p>The <inline-formula><mml:math id="mm1670"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> for the best-fit spatially non-flat <inline-formula><mml:math id="mm1671"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM, spatially non−flat <inline-formula><mml:math id="mm1672"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM, and spatially flat tilted <inline-formula><mml:math id="mm1673"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM (gray solid line) models. (Left panels) (<bold>a</bold>,<bold>c</bold>,<bold>e</bold>) Results obtained from only CMB temperature anisotropy data. (Right panels) (<bold>b</bold>,<bold>d</bold>,<bold>f</bold>) results obtained only from CMB temperature anisotropy + BAO peak length scale data. All-<italic>l</italic> regions are demonstrated in top panels. The low-<italic>l</italic> region <inline-formula><mml:math id="mm1674"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and residuals are shown in the middle panels. The high-<italic>l</italic> region <inline-formula><mml:math id="mm1675"><mml:semantics><mml:msub><mml:mi>C</mml:mi><mml:mi>l</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> and residuals are presented in the bottom panels. The figure is adapted from [<xref ref-type="bibr" rid="B330-universe-10-00122">330</xref>].</p> Full article ">Figure 39
<p>The 1<inline-formula><mml:math id="mm1676"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1677"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1678"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the spatially non-flat <inline-formula><mml:math id="mm1679"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. Solid (dashed) contours correspond to <inline-formula><mml:math id="mm1680"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>2.8</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula><inline-formula><mml:math id="mm1681"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mn>73.24</mml:mn><mml:mo>±</mml:mo><mml:mn>1.74</mml:mn><mml:mo>)</mml:mo></mml:mrow><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> prior; the red dots indicate the location of the best-fit point in each prior case. The horizontal axis with <inline-formula><mml:math id="mm1682"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1683"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left panel) The results obtained for the <inline-formula><mml:math id="mm1684"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> marginalization. (Center panel) The results obtained for the <inline-formula><mml:math id="mm1685"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> marginalization. (Right panel) The results obtained for the parameter <inline-formula><mml:math id="mm1686"><mml:semantics><mml:mi>α</mml:mi></mml:semantics></mml:math></inline-formula> marginalization. The figure is adapted from [<xref ref-type="bibr" rid="B331-universe-10-00122">331</xref>].</p> Full article ">Figure 40
<p>The 1<inline-formula><mml:math id="mm1687"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1688"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level 2D contour constraints on the coupling parameter <inline-formula><mml:math id="mm1689"><mml:semantics><mml:mi>ξ</mml:mi></mml:semantics></mml:math></inline-formula> in the IDE model and 1D posteriors for the cases without lensing. The grey vertical stripe refers to the value of <inline-formula><mml:math id="mm1690"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> measured by the SH0ES team (<inline-formula><mml:math id="mm1691"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.04</mml:mn><mml:mo>±</mml:mo><mml:mn>1.04</mml:mn><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> at 1<inline-formula><mml:math id="mm1692"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level). The figure is adapted from [<xref ref-type="bibr" rid="B67-universe-10-00122">67</xref>].</p> Full article ">Figure 41
<p>The 1<inline-formula><mml:math id="mm1693"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1694"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1695"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the <inline-formula><mml:math id="mm1696"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. Solid lines correspond to <inline-formula><mml:math id="mm1697"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73</mml:mn><mml:mo>±</mml:mo><mml:mn>3</mml:mn><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, while dashed lines correspond to <inline-formula><mml:math id="mm1698"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>4</mml:mn><mml:mspace width="3.33333pt"/><mml:mrow><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. The plus sign denotes the maximum likelihood at <inline-formula><mml:math id="mm1699"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.32</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1700"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.15</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with reduced <inline-formula><mml:math id="mm1701"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1.8</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The cross denotes the maximum likelihood at <inline-formula><mml:math id="mm1702"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.19</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1703"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>4.37</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with reduced <inline-formula><mml:math id="mm1704"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1.89</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The horizontal axis for which <inline-formula><mml:math id="mm1705"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the spatially flat <inline-formula><mml:math id="mm1706"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B335-universe-10-00122">335</xref>].</p> Full article ">Figure 42
<p>The 1<inline-formula><mml:math id="mm1707"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1708"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1709"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters of the <inline-formula><mml:math id="mm1710"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The horizontal axis with <inline-formula><mml:math id="mm1711"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1712"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left panel) Contours obtained from <inline-formula><mml:math id="mm1713"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. The star denotes the best-fit pair <inline-formula><mml:math id="mm1714"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.28</mml:mn><mml:mo>,</mml:mo><mml:mn>0.46</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1715"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>10.1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right panel) Contours were obtained from a joint analysis of the BAO peak length scale and SNe Ia apparent magnitude data (with systematic errors), with (and without) <inline-formula><mml:math id="mm1716"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. The cross denotes the best-fit point determined from the joint sample with <inline-formula><mml:math id="mm1717"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data at <inline-formula><mml:math id="mm1718"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.28</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1719"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, with <inline-formula><mml:math id="mm1720"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>531</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The diamond denotes the best-fit point obtained from the joint sample with <inline-formula><mml:math id="mm1721"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data at <inline-formula><mml:math id="mm1722"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.28</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1723"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1724"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>541</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B338-universe-10-00122">338</xref>].</p> Full article ">Figure 43
<p>Thick solid lines are 1<inline-formula><mml:math id="mm1725"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1726"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1727"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm1728"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential, for the prior <inline-formula><mml:math id="mm1729"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.8</mml:mn><mml:mo>±</mml:mo><mml:mn>2.4</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>. The horizontal axis with <inline-formula><mml:math id="mm1730"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1731"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left upper panel) Contours obtained from <inline-formula><mml:math id="mm1732"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. Thin dot–dashed lines are 1<inline-formula><mml:math id="mm1733"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1734"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1735"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours reproduced from [<xref ref-type="bibr" rid="B338-universe-10-00122">338</xref>], where the prior is <inline-formula><mml:math id="mm1736"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>3.5</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula>; the empty circle corresponds to the best-fit point. The curved dotted lines denote zero-acceleration models. The filled black circles correspond to best-fit points. (Right upper panel) Contours obtained from only SNe Ia apparent magnitude data with (without) systematic errors. Filled (open) circles denote likelihood maxima for the case of data with (without) systematic errors. (Left lower panel) Contours were obtained from only the BAO peak length scale data. Filled circles denote likelihood maxima. (Right lower panel) Contours obtained from data on the BAO peak length scale and SNe Ia apparent magnitude (with systematic errors), with (without) <inline-formula><mml:math id="mm1737"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. The full (empty) circle denotes the best-fit point determined from a joint analysis with (without) <inline-formula><mml:math id="mm1738"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. The figure is adapted from [<xref ref-type="bibr" rid="B339-universe-10-00122">339</xref>].</p> Full article ">Figure 44
<p>Thick solid (thin dot–dashed) lines are 1<inline-formula><mml:math id="mm1739"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1740"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1741"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm1742"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential from the new <inline-formula><mml:math id="mm1743"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data (old <inline-formula><mml:math id="mm1744"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data were used in [<xref ref-type="bibr" rid="B339-universe-10-00122">339</xref>]). The filled (empty) circle is the best-fit point from new (old) <inline-formula><mml:math id="mm1745"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> measurements. The horizontal axis with <inline-formula><mml:math id="mm1746"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1747"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The curved dotted lines denote zero-acceleration models. (Left upper panel) Contours obtained for the <inline-formula><mml:math id="mm1748"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>3.5</mml:mn><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>kms</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> prior. The filled circles correspond to the best-fit pair <inline-formula><mml:math id="mm1749"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.36</mml:mn><mml:mo>,</mml:mo><mml:mn>0.70</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1750"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>15.2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The empty circles correspond to the best-fit pair <inline-formula><mml:math id="mm1751"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.30</mml:mn><mml:mo>,</mml:mo><mml:mn>0.25</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1752"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>14.6</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right upper panel) Contours obtained for the <inline-formula><mml:math id="mm1753"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.8</mml:mn><mml:mo>±</mml:mo><mml:mn>2.4</mml:mn><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>kms</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> prior. The filled circles correspond to the best-fit pair <inline-formula><mml:math id="mm1754"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.25</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1755"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>16.1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Empty circles correspond to the best-fit pair <inline-formula><mml:math id="mm1756"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.27</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1757"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>15.6</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Left lower panel) Contours obtained from joint analysis with SNe Ia apparent magnitude data (with systematic errors) and BAO peak length scale data, with (without) <inline-formula><mml:math id="mm1758"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. The full (empty) circle marks the best-fit point determined from a joint analysis with (without) <inline-formula><mml:math id="mm1759"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data. Contours obtained for <inline-formula><mml:math id="mm1760"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>3.5</mml:mn><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>kms</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> prior. The full circle indicates the best-fit pair <inline-formula><mml:math id="mm1761"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.29</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1762"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>567</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> while the empty circle corresponds to the best-fit pair <inline-formula><mml:math id="mm1763"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.30</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1764"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>551</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right lower panel) Contours obtained for the <inline-formula><mml:math id="mm1765"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.8</mml:mn><mml:mo>±</mml:mo><mml:mn>2.4</mml:mn><mml:mspace width="3.33333pt"/><mml:msup><mml:mi>kms</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> prior. The empty circle denotes the best-fit pair <inline-formula><mml:math id="mm1766"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.30</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1767"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>551</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> while the full circle denotes the best-fit pair <inline-formula><mml:math id="mm1768"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.27</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1769"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>569</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B340-universe-10-00122">340</xref>].</p> Full article ">Figure 45
<p>Thick solid and thin dot–dashed lines are 1<inline-formula><mml:math id="mm1770"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1771"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1772"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the scalar field <inline-formula><mml:math id="mm1773"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential from the compilation of <inline-formula><mml:math id="mm1774"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data for <inline-formula><mml:math id="mm1775"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>3.5</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1776"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.8</mml:mn><mml:mo>±</mml:mo><mml:mn>2.4</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> priors, respectively. The horizontal axis with <inline-formula><mml:math id="mm1777"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1778"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model and the curved dotted line denotes zero-acceleration models. Filled and empty circles are best-fit points for which <inline-formula><mml:math id="mm1779"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.29</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1780"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>18.24</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1781"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.25</mml:mn><mml:mo>,</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1782"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>20.64</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, respectively. The figure is adapted from [<xref ref-type="bibr" rid="B342-universe-10-00122">342</xref>].</p> Full article ">Figure 46
<p>Thick solid and thin dot–dashed lines are 1<inline-formula><mml:math id="mm1783"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1784"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1785"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the <inline-formula><mml:math id="mm1786"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential, the XCDM model, and the <inline-formula><mml:math id="mm1787"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model from 7 or 9 measurements per bin data. In these three rows, the first two plots include red weighted-mean constraints while the second two include red median statistics. Filled red and empty blue circles correspond to the best-fit points. Dashed diagonal lines denote spatially flat models and dotted lines show zero-acceleration models. The figure is adapted from [<xref ref-type="bibr" rid="B260-universe-10-00122">260</xref>].</p> Full article ">Figure 47
<p>The <inline-formula><mml:math id="mm1788"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo><mml:mo>/</mml:mo><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>+</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data binned with 7 or 9 measurements per bin, as well as 5 higher measurements of redshift, and Farooq and Ratra [<xref ref-type="bibr" rid="B260-universe-10-00122">260</xref>] best-fit model predictions. Dashed and dotted lines correspond to <inline-formula><mml:math id="mm1789"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>68</mml:mn><mml:mo>±</mml:mo><mml:mn>3.5</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1790"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.8</mml:mn><mml:mo>±</mml:mo><mml:mn>2.4</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> priors, respectively. The figure is adapted from [<xref ref-type="bibr" rid="B342-universe-10-00122">342</xref>].</p> Full article ">Figure 48
<p>(Left panel) The 1<inline-formula><mml:math id="mm1791"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm1792"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the <inline-formula><mml:math id="mm1793"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. (Right panel) Best-fit model curves from the 28 <inline-formula><mml:math id="mm1794"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> data points for the spatially flat <inline-formula><mml:math id="mm1795"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model, <italic>w</italic>CDM model, and the spatially flat and spatially non-flat <inline-formula><mml:math id="mm1796"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B343-universe-10-00122">343</xref>].</p> Full article ">Figure 49
<p>The 1<inline-formula><mml:math id="mm1797"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1798"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1799"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours constraints on the parameters of the spatially non-flat <inline-formula><mml:math id="mm1800"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. Red (blue) solid lines are for the lower (higher) <inline-formula><mml:math id="mm1801"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> prior. (Left, center, and right panels) The results obtained correspond to the marginalization over <inline-formula><mml:math id="mm1802"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi></mml:mrow></mml:semantics></mml:math></inline-formula>, and <inline-formula><mml:math id="mm1803"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula>, respectively. Red (blue) solid circles are the best-fit points for the lower (higher) <inline-formula><mml:math id="mm1804"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> prior. Red (blue) dot–dashed lines in the left panel are 1<inline-formula><mml:math id="mm1805"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1806"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1807"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> for the lower (higher) <inline-formula><mml:math id="mm1808"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> prior in the spatially flat <inline-formula><mml:math id="mm1809"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B192-universe-10-00122">192</xref>].</p> Full article ">Figure 50
<p>The 1<inline-formula><mml:math id="mm1810"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1811"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1812"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially non-flat <inline-formula><mml:math id="mm1813"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential from data on <inline-formula><mml:math id="mm1814"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, QSO, and BAO peak length scale. (Upper and middle panels) The vertical green dashed line in the upper center panel, and the horizontal green dashed lines in the middle left and middle center panels separate spatially closed models (with <inline-formula><mml:math id="mm1815"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>&lt;</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>) from spatially open models (with <inline-formula><mml:math id="mm1816"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mrow><mml:mo>&gt;</mml:mo><mml:mn>0</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>. The horizontal line with <inline-formula><mml:math id="mm1817"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> in the upper panels corresponds to the spatially non-flat <inline-formula><mml:math id="mm1818"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Lower panel) One-dimensional likelihoods for <inline-formula><mml:math id="mm1819"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B193-universe-10-00122">193</xref>].</p> Full article ">Figure 51
<p>The 1<inline-formula><mml:math id="mm1820"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1821"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1822"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm1823"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential (left panel). The black dotted line splits the parameter space into accelerated and decelerated regions. The axis with <inline-formula><mml:math id="mm1824"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1825"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. Constraints for the spatially non-flat <inline-formula><mml:math id="mm1826"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential are depicted in the right panel. The figure is adapted from [<xref ref-type="bibr" rid="B347-universe-10-00122">347</xref>].</p> Full article ">Figure 52
<p>The 1<inline-formula><mml:math id="mm1827"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1828"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1829"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the untilted spatially non-flat <inline-formula><mml:math id="mm1830"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with RP potential using the combination of datasets: QSO (gray line), <inline-formula><mml:math id="mm1831"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (red line), and QSO + <inline-formula><mml:math id="mm1832"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (blue line). (Left panel) Contours and one-dimensional likelihoods for all free parameters. The red dotted curved lines denote zero-acceleration lines. (Right panel) Plots for <inline-formula><mml:math id="mm1833"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> cosmological parameters, without constraints from QSO data. These plots are for <inline-formula><mml:math id="mm1834"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>73.24</mml:mn><mml:mo>±</mml:mo><mml:mn>1.74</mml:mn><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> as a prior. The black dashed straight lines denote the flat hypersurface with <inline-formula><mml:math id="mm1835"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B195-universe-10-00122">195</xref>].</p> Full article ">Figure 53
<p>The 1<inline-formula><mml:math id="mm1836"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1837"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1838"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours constraints on the parameters of the untilted spatially non-flat <inline-formula><mml:math id="mm1839"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential using the combination of datasets: QSO (gray line), <inline-formula><mml:math id="mm1840"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (red line), and QSO + <inline-formula><mml:math id="mm1841"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (blue line). (Left panel) Contours and one-dimensional likelihoods for all free parameters. (Right panel) Plots for only <inline-formula><mml:math id="mm1842"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>,</mml:mo><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> cosmological parameters, without constraints only from QSO data. These plots are for <inline-formula><mml:math id="mm1843"><mml:semantics><mml:mrow><mml:msub><mml:mi>H</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mrow><mml:mo>=</mml:mo><mml:mn>73.24</mml:mn><mml:mo>±</mml:mo><mml:mn>1.74</mml:mn><mml:mo>)</mml:mo><mml:mspace width="3.33333pt"/><mml:mi>km</mml:mi><mml:mspace width="3.33333pt"/></mml:mrow><mml:msup><mml:mi mathvariant="normal">s</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:msup><mml:mi>Mpc</mml:mi><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> as a prior. Black dashed straight lines denote the spatially flat hypersurface with <inline-formula><mml:math id="mm1844"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B196-universe-10-00122">196</xref>].</p> Full article ">Figure 54
<p>The 1<inline-formula><mml:math id="mm1845"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1846"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1847"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1848"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential from various datasets. The black dotted line splits the parameter space into the regions of the currently decelerating and accelerating cosmological expansion. The axis with <inline-formula><mml:math id="mm1849"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1850"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B352-universe-10-00122">352</xref>].</p> Full article ">Figure 55
<p>The 1<inline-formula><mml:math id="mm1851"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1852"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1853"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and non-flat (right panel) <inline-formula><mml:math id="mm1854"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential, using QSO (blue) and the BAO peak length scale + <inline-formula><mml:math id="mm1855"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> (red) datasets. The black dotted line in the <inline-formula><mml:math id="mm1856"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>−</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> sub-panels is the line of zero acceleration, under which the accelerated cosmological expansion occurs. The axis with <inline-formula><mml:math id="mm1857"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1858"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B197-universe-10-00122">197</xref>].</p> Full article ">Figure 56
<p>The 1<inline-formula><mml:math id="mm1859"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1860"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1861"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially non-flat <inline-formula><mml:math id="mm1862"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The zero-acceleration line splits the parameter space into regions of the currently decelerating and accelerating cosmological expansion. The cyan dash–dot lines show the spatially flat <inline-formula><mml:math id="mm1863"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model; regions with spatially closed geometry are located either below or to the left. The axis with <inline-formula><mml:math id="mm1864"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1865"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B201-universe-10-00122">201</xref>].</p> Full article ">Figure 57
<p>The 1<inline-formula><mml:math id="mm1866"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1867"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1868"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1869"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential, using Chandra + XXL + High-<italic>z</italic> + Newton-3 + Newton-4 (blue) and BAO peak length scale + <inline-formula><mml:math id="mm1870"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> (red) datasets. In all plots, black dotted lines are zero-acceleration lines, which split the parameter space into the regions of current acceleration and deceleration. Black dashed line corresponds to <inline-formula><mml:math id="mm1871"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The axis with <inline-formula><mml:math id="mm1872"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1873"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B199-universe-10-00122">199</xref>].</p> Full article ">Figure 58
<p>The 1<inline-formula><mml:math id="mm1874"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1875"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1876"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1877"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential from 3-parameter <inline-formula><mml:math id="mm1878"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> high-<inline-formula><mml:math id="mm1879"><mml:semantics><mml:msub><mml:mo>ℜ</mml:mo><mml:mrow><mml:mi>F</mml:mi><mml:mi>e</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> (blue), 2-parameter <inline-formula><mml:math id="mm1880"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:msup><mml:mi>β</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:semantics></mml:math></inline-formula> high-<inline-formula><mml:math id="mm1881"><mml:semantics><mml:msub><mml:mo>ℜ</mml:mo><mml:mrow><mml:mi>F</mml:mi><mml:mi>e</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> (green), and <inline-formula><mml:math id="mm1882"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (red) measurements. Black dotted lines correspond to zero-acceleration lines. Black dashed lines represent <inline-formula><mml:math id="mm1883"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B200-universe-10-00122">200</xref>].</p> Full article ">Figure 59
<p>The 1<inline-formula><mml:math id="mm1884"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1885"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1886"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1887"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential from the measurements of <inline-formula><mml:math id="mm1888"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:msup><mml:mi>I</mml:mi><mml:mo>′</mml:mo></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> high-<inline-formula><mml:math id="mm1889"><mml:semantics><mml:msub><mml:mo>ℜ</mml:mo><mml:mrow><mml:mi>F</mml:mi><mml:mi>e</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> (blue), <inline-formula><mml:math id="mm1890"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:msup><mml:mi>I</mml:mi><mml:mrow><mml:mo>″</mml:mo></mml:mrow></mml:msup></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> high-<inline-formula><mml:math id="mm1891"><mml:semantics><mml:msub><mml:mo>ℜ</mml:mo><mml:mrow><mml:mi>F</mml:mi><mml:mi>e</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> (green), <inline-formula><mml:math id="mm1892"><mml:semantics><mml:mrow><mml:mi>M</mml:mi><mml:msub><mml:mi>g</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula> high-<inline-formula><mml:math id="mm1893"><mml:semantics><mml:msub><mml:mo>ℜ</mml:mo><mml:mrow><mml:mi>F</mml:mi><mml:mi>e</mml:mi><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> (gray), and BAO peak length scale + <inline-formula><mml:math id="mm1894"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>. Black dotted lines correspond to zero-acceleration lines. Black dashed lines represent <inline-formula><mml:math id="mm1895"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B362-universe-10-00122">362</xref>].</p> Full article ">Figure 60
<p>The 1<inline-formula><mml:math id="mm1896"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1897"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1898"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1899"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential from various combinations of datasets. The axis with <inline-formula><mml:math id="mm1900"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1901"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The black dash–dotted lines denote spatially flat hypersurfaces <inline-formula><mml:math id="mm1902"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>; closed spatial hypersurfaces are located either below or to the left. The black dotted lines correspond to the lines of zero acceleration and split the parameter space into currently accelerating (bottom left) and decelerating (top right) regions. The figure is adapted from [<xref ref-type="bibr" rid="B203-universe-10-00122">203</xref>].</p> Full article ">Figure 61
<p>The 1<inline-formula><mml:math id="mm1903"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1904"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1905"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours constraints on parameters of the <inline-formula><mml:math id="mm1906"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The horizontal axis with <inline-formula><mml:math id="mm1907"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1908"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left upper panel) Contours are obtained by Wang’s (2008) [<xref ref-type="bibr" rid="B368-universe-10-00122">368</xref>] method. The circle indicates the best-fit parameter values <inline-formula><mml:math id="mm1909"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1910"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>10.2</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1911"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>1.39</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 4 degrees of freedom. (Right upper panel) Contours are derived using the GRB data by Wang’s (2008) [<xref ref-type="bibr" rid="B368-universe-10-00122">368</xref>] method, the SNe Ia Union apparent magnitude data, and the BAO peak length scale measurements, while dotted lines (here the cross denotes the best-fit point) are derived using only the SNe Ia apparent magnitude data and the BAO peak length scale measurements. The best-fit parameters in both cases are <inline-formula><mml:math id="mm1912"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.24</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1913"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1914"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>326</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 313 degrees of freedom (solid lines) and <inline-formula><mml:math id="mm1915"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>321</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 307 degrees of freedom. (Left lower panel) Contours are obtained using GRB data by Schaefer’s (2007) method (here the cross indicates the best-fit parameter values): <inline-formula><mml:math id="mm1916"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1917"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>4.5</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1918"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>77.8</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 67 degrees of freedom. (Right lower panel) Contours are obtained using Schaefer’s (2007) [<xref ref-type="bibr" rid="B367-universe-10-00122">367</xref>] method, the SNe Ia Union apparent magnitude data, and the BAO peak length scale measurements, while dotted lines are obtained using the SNe Ia apparent magnitude data and the BAO peak length scale measurements only. Solid lines (circle denotes best-fit point) are derived using GRB data, here the cross denotes the best-fit point. The best-fit matter density parameters are <inline-formula><mml:math id="mm1919"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.24</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1920"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.30</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1921"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>401</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 376 degrees of freedom (solid lines), and <inline-formula><mml:math id="mm1922"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">m</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mn>0.25</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm1923"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm1924"><mml:semantics><mml:mrow><mml:msup><mml:mi>χ</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>321</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> for 307 degrees of freedom (dotted lines). The figure is adapted from [<xref ref-type="bibr" rid="B365-universe-10-00122">365</xref>].</p> Full article ">Figure 62
<p>The 1<inline-formula><mml:math id="mm1925"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1926"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1927"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1928"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential, using the combination of datasets: GRB (gray line), <inline-formula><mml:math id="mm1929"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (red line), and GRB + <inline-formula><mml:math id="mm1930"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (blue line). The black dotted line splits the parameter space into accelerating and decelerating regions. The axis with <inline-formula><mml:math id="mm1931"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1932"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B369-universe-10-00122">369</xref>].</p> Full article ">Figure 63
<p>The 1<inline-formula><mml:math id="mm1933"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1934"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1935"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1936"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential from MD-SGRB (gray), GWLGRB (green), MD-SGRB + GW-LGRB (orange), and <inline-formula><mml:math id="mm1937"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> + BAO peak length scale (red) data. Black dashed lines denote zero-acceleration lines, which split the parameter space into two regions of current acceleration and deceleration. Dash–dotted crimson lines correspond to spatially flat hypersurfaces with spatially closed hypersurfaces either below or to the left. The magenta lines correspond to the <inline-formula><mml:math id="mm1938"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model; the closed spatial geometry are either below or to the left. The axis with <inline-formula><mml:math id="mm1939"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1940"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B202-universe-10-00122">202</xref>].</p> Full article ">Figure 64
<p>The 1<inline-formula><mml:math id="mm1941"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1942"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1943"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours on parameters of the spatially flat (left panel) and spatially non-flat (right panel) <inline-formula><mml:math id="mm1944"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential, using various combinations of GRB datasets. The axis with <inline-formula><mml:math id="mm1945"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> denotes the spatially flat <inline-formula><mml:math id="mm1946"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The black dotted lines correspond to lines of zero acceleration and split the parameter space into currently accelerating (bottom left) and currently decelerating (top right) regions. The crimson dash–dot lines denote spatially flat hypersurfaces <inline-formula><mml:math id="mm1947"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">k</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>; closed spatial hypersurfaces are located either below or to the left. The figure is adapted from [<xref ref-type="bibr" rid="B205-universe-10-00122">205</xref>].</p> Full article ">Figure 65
<p>The 1<inline-formula><mml:math id="mm1948"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1949"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1950"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm1951"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The horizontal axis with <inline-formula><mml:math id="mm1952"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm1953"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left panel) Contours obtained from <inline-formula><mml:math id="mm1954"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> galaxy data. The best-fit point with <inline-formula><mml:math id="mm1955"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>53.3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> is indicated by the solid black circle at <inline-formula><mml:math id="mm1956"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.17</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1957"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.39</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right panel) Contours obtained from joint <inline-formula><mml:math id="mm1958"><mml:semantics><mml:msub><mml:mi>H</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> galaxy and BAO peak length scale data (solid lines) and BAO peak length scale data only (dashed lines). The best-fit point with <inline-formula><mml:math id="mm1959"><mml:semantics><mml:mrow><mml:mo>−</mml:mo><mml:mn>2</mml:mn><mml:mo form="prefix">log</mml:mo><mml:mo>(</mml:mo><mml:msub><mml:mi>L</mml:mi><mml:mi>max</mml:mi></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>55.6</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> is indicated by the solid black circle at <inline-formula><mml:math id="mm1960"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.27</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1961"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B381-universe-10-00122">381</xref>].</p> Full article ">Figure 66
<p>The 1<inline-formula><mml:math id="mm1962"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1963"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1964"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on parameters. (Left panel) For the spatially flat <inline-formula><mml:math id="mm1965"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential and non-relativistic CDM. Continuous lines are obtained for <inline-formula><mml:math id="mm1966"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.72</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1967"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.0214</mml:mn><mml:mo>±</mml:mo><mml:mn>0.002</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> while dotted lines match <inline-formula><mml:math id="mm1968"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.68</mml:mn><mml:mo>±</mml:mo><mml:mn>0.04</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1969"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.014</mml:mn><mml:mo>±</mml:mo><mml:mn>0.004</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Middle panel) For the spatially flat <inline-formula><mml:math id="mm1970"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. Continuous lines are obtained for <inline-formula><mml:math id="mm1971"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.72</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1972"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.0214</mml:mn><mml:mo>±</mml:mo><mml:mn>0.002</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> while dotted lines obtained for <inline-formula><mml:math id="mm1973"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.68</mml:mn><mml:mo>±</mml:mo><mml:mn>0.04</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1974"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.014</mml:mn><mml:mo>±</mml:mo><mml:mn>0.004</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The diagonal dash–dotted line delimits spatially flat models. (Right panel) For the XCDM model. Continuous lines are obtained for <inline-formula><mml:math id="mm1975"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.72</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1976"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.0214</mml:mn><mml:mo>±</mml:mo><mml:mn>0.002</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> while dotted lines are derived for <inline-formula><mml:math id="mm1977"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.68</mml:mn><mml:mo>±</mml:mo><mml:mn>0.04</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1978"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.014</mml:mn><mml:mo>±</mml:mo><mml:mn>0.004</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. In all pictures, two dots indicate the place of maximum probability. The figure is adapted from [<xref ref-type="bibr" rid="B336-universe-10-00122">336</xref>].</p> Full article ">Figure 67
<p>The 1<inline-formula><mml:math id="mm1979"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1980"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1981"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the <inline-formula><mml:math id="mm1982"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. (Left panel) For the R04 gold SNe Ia apparent magnitude sample. The dot indicates the maximum likelihood for which <inline-formula><mml:math id="mm1983"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.30</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1984"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right panel) For the joint R04 gold SNe Ia apparent magnitude sample and galaxy cluster gas mass fraction data. The solid gray lines are computed for <inline-formula><mml:math id="mm1985"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.72</mml:mn><mml:mo>±</mml:mo><mml:mn>0.08</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1986"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.0214</mml:mn><mml:mo>±</mml:mo><mml:mn>0.002</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> with maximum likelihood at <inline-formula><mml:math id="mm1987"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.28</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1988"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn><mml:mo>.</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula> The black dotted lines are computed for <inline-formula><mml:math id="mm1989"><mml:semantics><mml:mrow><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:mn>0.68</mml:mn><mml:mo>±</mml:mo><mml:mn>0.04</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1990"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mi mathvariant="normal">b</mml:mi></mml:msub><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>=</mml:mo><mml:mn>0.014</mml:mn><mml:mo>±</mml:mo><mml:mn>0.004</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, with maximum likelihood at <inline-formula><mml:math id="mm1991"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.22</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> and <inline-formula><mml:math id="mm1992"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0.45</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is derived from constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm1993"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM, the XCDM, and the <inline-formula><mml:math id="mm1994"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM models with the RP potential adapted from [<xref ref-type="bibr" rid="B337-universe-10-00122">337</xref>].</p> Full article ">Figure 68
<p>The 1<inline-formula><mml:math id="mm1995"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm1996"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm1997"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours in the <inline-formula><mml:math id="mm1998"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula>-<inline-formula><mml:math id="mm1999"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mi>a</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> plane from the data analysis of the galaxy cluster number count, Hubble parameter <inline-formula><mml:math id="mm2000"><mml:semantics><mml:mrow><mml:mi>H</mml:mi><mml:mo>(</mml:mo><mml:mi>z</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:semantics></mml:math></inline-formula>, CMB temperature anisotropy, BAO peak length scale, and the SNe Ia apparent magnitude. The shaded areas represent various types of dynamical dark energy models. The figure is adapted from [<xref ref-type="bibr" rid="B390-universe-10-00122">390</xref>].</p> Full article ">Figure 69
<p>The 1<inline-formula><mml:math id="mm2001"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm2002"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm2003"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the <inline-formula><mml:math id="mm2004"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The horizontal axis with <inline-formula><mml:math id="mm2005"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the standard spatially flat <inline-formula><mml:math id="mm2006"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. (Left upper panel) Contours obtained from angular diameter distance <inline-formula><mml:math id="mm2007"><mml:semantics><mml:msub><mml:mi>d</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> data. The star denotes the best-fit point <inline-formula><mml:math id="mm2008"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.54</mml:mn><mml:mo>,</mml:mo><mml:mn>5</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm2009"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>37.3</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Right upper panel) Contours obtained using BAO peak length scale data. The star marks the best-fit point <inline-formula><mml:math id="mm2010"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.32</mml:mn><mml:mo>,</mml:mo><mml:mn>2.01</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm2011"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>0.169</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. (Left lower panel) Contours obtained from SNe Ia apparent magnitude data. Thin solid lines (best-fit pair at <inline-formula><mml:math id="mm2012"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.27</mml:mn><mml:mo>,</mml:mo><mml:mn>0.00</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm2013"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>543</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, marked by cross “x”) exclude systematic errors, while thick solid lines (best-fit pair at <inline-formula><mml:math id="mm2014"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.27</mml:mn><mml:mo>,</mml:mo><mml:mn>0.00</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula>, <inline-formula><mml:math id="mm2015"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>531</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>, marked by diamond “◊”) calculated for systematics. (Right lower panel) Thick (thin) solid lines are contours obtained from a joint analysis of BAO peak length scale and SNe Ia apparent magnitude (with systematic errors) data, with (and without) angular diameter distance <inline-formula><mml:math id="mm2016"><mml:semantics><mml:msub><mml:mi>d</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula>. The cross “x” means the best-fit point defined from the joint sample without the <inline-formula><mml:math id="mm2017"><mml:semantics><mml:msub><mml:mi>d</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> data at <inline-formula><mml:math id="mm2018"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.28</mml:mn><mml:mo>,</mml:mo><mml:mn>0.00</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm2019"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>531</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The diamond “◊” denotes the best-fit point determined from the joint sample with the <inline-formula><mml:math id="mm2020"><mml:semantics><mml:msub><mml:mi>d</mml:mi><mml:mi>A</mml:mi></mml:msub></mml:semantics></mml:math></inline-formula> data at <inline-formula><mml:math id="mm2021"><mml:semantics><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>,</mml:mo><mml:mi>α</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mn>0.28</mml:mn><mml:mo>,</mml:mo><mml:mn>0.01</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:semantics></mml:math></inline-formula> with <inline-formula><mml:math id="mm2022"><mml:semantics><mml:mrow><mml:msubsup><mml:mi>χ</mml:mi><mml:mi>min</mml:mi><mml:mn>2</mml:mn></mml:msubsup><mml:mo>=</mml:mo><mml:mn>572</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B391-universe-10-00122">391</xref>].</p> Full article ">Figure 70
<p>The 1<inline-formula><mml:math id="mm2023"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> and 2<inline-formula><mml:math id="mm2024"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours arising from lensing statistics and SNe Ia apparent magnitude versus redshift data (solid curves). (Left panel) The <inline-formula><mml:math id="mm2025"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the pNGb potential. Solid curves correspond to constraints from SNe Ia apparent magnitude data. Contours of the constant matter energy parameter at the present epoch <inline-formula><mml:math id="mm2026"><mml:semantics><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub></mml:semantics></mml:math></inline-formula> and the limit for the acceleration parameter at the present epoch <inline-formula><mml:math id="mm2027"><mml:semantics><mml:mrow><mml:msub><mml:mi>q</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> are depicted. (Right panel) The <inline-formula><mml:math id="mm2028"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. The lower bound of <inline-formula><mml:math id="mm2029"><mml:semantics><mml:mrow><mml:msub><mml:mo>Ω</mml:mo><mml:mrow><mml:mi mathvariant="normal">m</mml:mi><mml:mn>0</mml:mn></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>0.15</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> from clusters and curves for the constant value of the EoS parameter at the present epoch <inline-formula><mml:math id="mm2030"><mml:semantics><mml:msub><mml:mi>w</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:semantics></mml:math></inline-formula> are shown. The figure is adapted from [<xref ref-type="bibr" rid="B386-universe-10-00122">386</xref>].</p> Full article ">Figure 71
<p>The 68%, 90%, 95%, and 99% confidence level contour constraints on the parameters of the <inline-formula><mml:math id="mm2031"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential from the strong gravitational lensing data. The thin line represents the 68% confidence level derived from the SNe Ia apparent magnitude versus redshift data by Chen and Ratra [<xref ref-type="bibr" rid="B387-universe-10-00122">387</xref>]. The horizontal axis for which <inline-formula><mml:math id="mm2032"><mml:semantics><mml:mrow><mml:mi>α</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula> corresponds to the spatially flat <inline-formula><mml:math id="mm2033"><mml:semantics><mml:mo>Λ</mml:mo></mml:semantics></mml:math></inline-formula>CDM model. The figure is adapted from [<xref ref-type="bibr" rid="B389-universe-10-00122">389</xref>].</p> Full article ">Figure 72
<p>The 1<inline-formula><mml:math id="mm2034"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm2035"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm2036"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contours constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm2037"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. Solid lines are contours computed for the uniform prior <inline-formula><mml:math id="mm2038"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mo>)</mml:mo><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:semantics></mml:math></inline-formula>. Short dashed lines are obtained for the logarithmic prior <inline-formula><mml:math id="mm2039"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mn>0</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>/</mml:mo><mml:msub><mml:mo>Ω</mml:mo><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:semantics></mml:math></inline-formula>. The figure is adapted from [<xref ref-type="bibr" rid="B387-universe-10-00122">387</xref>].</p> Full article ">Figure 73
<p>The 1<inline-formula><mml:math id="mm2040"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, 2<inline-formula><mml:math id="mm2041"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula>, and 3<inline-formula><mml:math id="mm2042"><mml:semantics><mml:mi>σ</mml:mi></mml:semantics></mml:math></inline-formula> confidence level contour constraints on the parameters of the spatially flat <inline-formula><mml:math id="mm2043"><mml:semantics><mml:mi>ϕ</mml:mi></mml:semantics></mml:math></inline-formula>CDM model with the RP potential. (Left panel) Constraints were obtained using all twenty radio galaxies (including 3C 427.1). (Right panel) Constraints were obtained using only nineteen radio galaxies (excluding 3C 427.1). The figure is adapted from [<xref ref-type="bibr" rid="B388-universe-10-00122">388</xref>].</p> Full article ">