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Entropy
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Modified Gravity: From Black Holes Entropy to Current Cosmology
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Modified Gravity: From Black Holes Entropy to Current Cosmology

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Astrophysics, Cosmology, and Black Holes".

Deadline for manuscript submissions: closed (20 September 2012) | Viewed by 110155

Special Issue Editors

Dr. Kazuharu Bamba

E-Mail Website
Guest Editor
Faculty of Symbiotic Systems Science, Fukushima University, Fukushima 960-1296, Japan
Interests: particle cosmology; gravity theory; inflation; late time acceleration
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Sergei D. Odintsov

grade E-Mail Website
Guest Editor
1. Institució Catalana de Recerca i Estudis Avançats (ICREA), Passeig Luis Companys, 23, 08010 Barcelona, Spain
2. Institute of Space Sciences (ICE-CSIC), C. Can Magrans s/n, 08193 Barcelona, Spain
Interests: cosmology; dark energy and inflation; quantum gravity; modified gravity and beyond general relativity; quantum fields at external fields
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Recent cosmological observations strongly support that the current expansion of the universe is accelerating. The origin of such a cosmic acceleration mechanism is one of the most significant problems in modern cosmology. Indeed, this is shown by the fact that the Nobel Prize in Physics 2011 was presented to the finding of the current cosmic acceleration by means of the observations of the Type Ia supernovae.

There are two representative approaches to explain the current accelerated expansion of the universe. One is to introduce “dark energy” in the framework of general relativity. The other is to modify a gravitational theory, such as F(R) gravity, so that we can obtain so-called geometrical dark energy. It is believed that a modified gravitational theory must pass cosmological bounds and solar system tests because it corresponds to an alternative theory of gravitation to general relativity. As another meaningful touchstone of modified gravity, it is important to examine whether the second law of thermodynamics can be satisfied in the models of modified gravity.

The fundamental connection between gravitation and thermodynamics has been suggested by the discovery of black hole thermodynamics with black hole entropy and Hawking temperature. In addition, it was shown that the Einstein equation can be derived from the proportionality of the entropy to the horizon area together with the Clausius relation in thermodynamics. This consequence has been applied to various cosmological settings as well as modified gravitational theories. In particular, the connections between thermodynamics and modified gravity have recently been discussed extensively.

In this special issue, we discuss the application of thermodynamics to the test of a successful alternative gravitational theory to general relativity. Through this procedure, we can obtain a clue to resolve the dark energy problem “geometrically”. It is considered that any successful modified gravity theory should obey the second law of thermodynamics. If the second law is violated in certain universes in a model, it is more likely to be due to an incorrect generalization of the second law or some inherent inconsistency of the model itself. For the latter case, the model should be abandoned. It is strongly expected that the considerations of this special issue can produce our new physical understanding on entropy in the context of the relation between thermodynamics and gravitation and shed light on novel ingredients as well as insights on modern cosmology, in particular new properties of dark energy.

Prof. Dr. Sergei D. Odintsov
Dr. Kazuharu Bamba
Guest Editors

 

Keywords

  • Quantum aspects of black holes, evaporation, thermodynamics
  • Black hole entropy
  • Modified theories of gravity
  • Dark energy
  • Cosmology

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Published Papers (17 papers)

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147 KiB  
Article
The Thermal Entropy Density of Spacetime
by Rongjia Yang
Entropy 2013, 15(1), 156-161; https://doi.org/10.3390/e15010156 - 8 Jan 2013
Cited by 9 | Viewed by 7136
Abstract
Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results [...] Read more.
Introducing the notion of thermal entropy density via the first law of thermodynamics and assuming the Einstein equation as an equation of thermal state, we obtain the thermal entropy density of any arbitrary spacetime without assuming a temperature or a horizon. The results confirm that there is a profound connection between gravity and thermodynamics. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
369 KiB  
Article
Entropy and Gravity
by Øyvind Grøn
Entropy 2012, 14(12), 2456-2477; https://doi.org/10.3390/e14122456 - 4 Dec 2012
Cited by 15 | Viewed by 8980
Abstract
The effect of gravity upon changes of the entropy of a gravity-dominated system is discussed. In a universe dominated by vacuum energy, gravity is repulsive, and there is accelerated expansion. Furthermore, inhomogeneities are inflated and the universe approaches a state of thermal equilibrium. [...] Read more.
The effect of gravity upon changes of the entropy of a gravity-dominated system is discussed. In a universe dominated by vacuum energy, gravity is repulsive, and there is accelerated expansion. Furthermore, inhomogeneities are inflated and the universe approaches a state of thermal equilibrium. The difference between the evolution of the cosmic entropy in a co-moving volume in an inflationary era with repulsive gravity and a matter-dominated era with attractive gravity is discussed. The significance of conversion of gravitational energy to thermal energy in a process with gravitational clumping, in order that the entropy of the universe shall increase, is made clear. Entropy of black holes and cosmic horizons are considered. The contribution to the gravitational entropy according to the Weyl curvature hypothesis is discussed. The entropy history of the Universe is reviewed. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
►▼ Show Figures

Figure 1

Figure 1
<p>Thermal entropy (modulus an undetermined constant and in units of (3/2)Nk) as a function of <span class="html-italic">R</span> for a gas cloud contracting or expanding under the action of its own gravity. (a) Contraction. Here <span class="html-italic">R &lt; R</span><sub>0</sub>. Then the entropy first increases towards a maximum at <span class="html-italic">R</span> = <span class="html-italic">R</span><sub>0</sub>/2, and then decreases. (b) Expansion. Here <span class="html-italic">R</span> &gt; <span class="html-italic">R</span><sub>0</sub>. The entropy of the gas increases.</p> Full article ">Figure 2
<p>Evolution of a system (a) in which gravity may be neglected, and (b) in a self gravitating system where the box is much larger than the Jean’s length of the gas it contains.</p> Full article ">Figure 3
<p>Time-variation of the candidate gravitational entropies <math display="inline"> <mrow> <msub> <mi>S</mi> <mrow> <mi>G</mi> <mn>1</mn> </mrow> </msub> </mrow> </math> (upper curve) and <math display="inline"> <mrow> <msub> <mi>S</mi> <mrow> <mi>G</mi> <mn>2</mn> </mrow> </msub> </mrow> </math> in a comoving volume during the beginning of the inflationary era.</p> Full article ">
195 KiB  
Article
Periodic Cosmological Evolutions of Equation of State for Dark Energy
by Kazuharu Bamba, Ujjal Debnath, Kuralay Yesmakhanova, Petr Tsyba, Gulgasyl Nugmanova and Ratbay Myrzakulov
Entropy 2012, 14(11), 2351-2374; https://doi.org/10.3390/e14112351 - 20 Nov 2012
Cited by 15 | Viewed by 6185
Abstract
We demonstrate two periodic or quasi-periodic generalizations of the Chaplygin gas (CG) type models to explain the origins of dark energy as well as dark matter by using the Weierstrass ξ(t), σ(t) and ζ (t) functions with two periods being infinite. If the [...] Read more.
We demonstrate two periodic or quasi-periodic generalizations of the Chaplygin gas (CG) type models to explain the origins of dark energy as well as dark matter by using the Weierstrass ξ(t), σ(t) and ζ (t) functions with two periods being infinite. If the universe can evolve periodically, a non-singular universe can be realized. Furthermore, we examine the cosmological evolution and nature of the equation of state (EoS) of dark energy in the Friedmann–Lemaître–Robertson–Walker cosmology. It is explicitly illustrated that there exist three type models in which the universe always stays in the non-phantom (quintessence) phase, whereas it always evolves in the phantom phase, or the crossing of the phantom divide can be realized. The scalar fields and the corresponding potentials are also analyzed for different types of models. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
►▼ Show Figures

Figure 1

Figure 1
<p>The EoS <span class="html-italic">ω</span> in Equation (<a href="#FD29-entropy-14-02351" class="html-disp-formula">29</a>) as a function of <span class="html-italic">t</span> for <math display="inline"> <mrow> <mi>℘</mi> <mfenced separators="" open="(" close=")"> <mi>t</mi> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mfenced> </mrow> </math>, <span class="html-italic">i.e</span>., the model parameters of the Weierstrass invariants of <math display="inline"> <mrow> <msub> <mi>g</mi> <mn>2</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mi>g</mi> <mn>3</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math>. The line of <math display="inline"> <mrow> <mi>ω</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </math> is also plotted.</p> Full article ">Figure 2
<p>The EoS <span class="html-italic">ω</span> in Equation (<a href="#FD41-entropy-14-02351" class="html-disp-formula">41</a>) as a function of <span class="html-italic">t</span>. The legend is the same as in <a href="#entropy-14-02351-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 3
<p>The EoS <span class="html-italic">ω</span> in Equation (<a href="#FD48-entropy-14-02351" class="html-disp-formula">48</a>) as a function of <span class="html-italic">t</span>. The legend is the same as in <a href="#entropy-14-02351-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 4
<p>The EoS <span class="html-italic">ω</span> in Equation (<a href="#FD55-entropy-14-02351" class="html-disp-formula">55</a>) as a function of <span class="html-italic">t</span>. The legend is the same as in <a href="#entropy-14-02351-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 5
<p>The EoS <span class="html-italic">ω</span> in Equation (<a href="#FD62-entropy-14-02351" class="html-disp-formula">62</a>) as a function of <span class="html-italic">t</span>. The legend is the same as in <a href="#entropy-14-02351-f001" class="html-fig">Figure 1</a>.</p> Full article ">Figure 6
<p>The EoS <span class="html-italic">ω</span> in Equation (<a href="#FD69-entropy-14-02351" class="html-disp-formula">69</a>) as a function of <span class="html-italic">t</span>. The legend is the same as in <a href="#entropy-14-02351-f001" class="html-fig">Figure 1</a>.</p> Full article ">
193 KiB  
Article
Viscosity in Modified Gravity
by Iver Brevik
Entropy 2012, 14(11), 2302-2310; https://doi.org/10.3390/e14112302 - 12 Nov 2012
Cited by 14 | Viewed by 5095
Abstract
A bulk viscosity is introduced in the formalism of modified gravity. It is shownthat, based on a natural scaling law for the viscosity, a simple solution can be found forquantities such as the Hubble parameter and the energy density. These solutions mayincorporate a [...] Read more.
A bulk viscosity is introduced in the formalism of modified gravity. It is shownthat, based on a natural scaling law for the viscosity, a simple solution can be found forquantities such as the Hubble parameter and the energy density. These solutions mayincorporate a viscosity-induced Big Rip singularity. By introducing a phase transition inthe cosmic fluid, the future singularity can nevertheless in principle be avoided. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
201 KiB  
Article
Dark Energy Problem, Physics of Early Universe and Some New Approaches in Gravity
by Alexander Shalyt-Margolin
Entropy 2012, 14(11), 2143-2156; https://doi.org/10.3390/e14112143 - 2 Nov 2012
Cited by 2 | Viewed by 5465
Abstract
The dark energy problem is studied based on the approach associated with the cosmological term in General Relativity that is considered as a dynamic quantity. It is shown that a quantum field theory of the Early Universe (Planck scales) and its limiting transition [...] Read more.
The dark energy problem is studied based on the approach associated with the cosmological term in General Relativity that is considered as a dynamic quantity. It is shown that a quantum field theory of the Early Universe (Planck scales) and its limiting transition at low energy play a significant role. Connection of this problem with Verlinde’s new (entropic) approach to gravity is revealed within the frame of such statement as well as the Generalized Uncertainty Principle (GUP) and Extended Uncertainty Principle (EUP). The implications from the obtained results are presented, and a more rigorous statement of the Concordance Problem in cosmology is treated. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
587 KiB  
Article
Accelerating Universe and the Scalar-Tensor Theory
by Yasunori Fujii
Entropy 2012, 14(10), 1997-2035; https://doi.org/10.3390/e14101997 - 19 Oct 2012
Cited by 3 | Viewed by 7118
Abstract
To understand the accelerating universe discovered observationally in 1998, we develop the scalar-tensor theory of gravitation originally due to Jordan, extended only minimally. The unique role of the conformal transformation and frames is discussed particularly from a physical point of view. We show [...] Read more.
To understand the accelerating universe discovered observationally in 1998, we develop the scalar-tensor theory of gravitation originally due to Jordan, extended only minimally. The unique role of the conformal transformation and frames is discussed particularly from a physical point of view. We show the theory to provide us with a simple and natural way of understanding the core of the measurements, Λobs ∼ t0−2 for the observed values of the cosmological constant and today’s age of the universe both expressed in the Planckian units. According to this scenario of a decaying cosmological constant, Λobs is this small only because we are old, not because we fine-tune the parameters. It also follows that the scalar field is simply the pseudo Nambu–Goldstone boson of broken global scale invariance, based on the way astronomers and astrophysicists measure the expansion of the universe in reference to the microscopic length units. A rather phenomenological trapping mechanism is assumed for the scalar field around the epoch of mini-inflation as observed, still maintaining the unmistakable behavior of the scenario stated above. Experimental searches for the scalar field, as light as ∼ 10−9 eV, as part of the dark energy, are also discussed. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
►▼ Show Figures

Figure 1

Figure 1
<p><math display="inline"> <msup> <mi>ζ</mi> <mn>2</mn> </msup> </math> given by (<a href="#FD13-entropy-14-01997" class="html-disp-formula">13</a>) restricted to be positive is plotted against <math display="inline"> <mrow> <mi>ξ</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math>, taken from Figure 1 of [<a href="#B25-entropy-14-01997" class="html-bibr">25</a>]. There are two branches for <math display="inline"> <mrow> <mi>ϵ</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mrow> </math> bounded by the two (dotted) straight lines <math display="inline"> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>6</mn> </mrow> </math> and <math display="inline"> <mrow> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>6</mn> </mrow> </math>. The results from the solar-system experiments correspond to the points, like the one marked by +, converging toward the origin <math display="inline"> <mrow> <mi>ξ</mi> <mo>=</mo> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math> with <math display="inline"> <mrow> <mi>ϵ</mi> <mo>=</mo> <mo>+</mo> <mn>1</mn> </mrow> </math>. The symbol ) also marked with <span class="html-italic">r</span> at the point <math display="inline"> <mrow> <mo>(</mo> <mi>ξ</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> <mo>,</mo> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>25</mn> <mo>)</mo> </mrow> </math> naturally with <math display="inline"> <mrow> <mi>ϵ</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math> provides with the boundary of selecting the portion of the curve to the upper-left, arising from the positive radiation-dominated matter energy both in J frame, (<a href="#FD26-entropy-14-01997" class="html-disp-formula">26</a>) and (<a href="#FD27-entropy-14-01997" class="html-disp-formula">27</a>), and E frame, (<a href="#FD38-entropy-14-01997" class="html-disp-formula">38</a>). The same <math display="inline"> <mrow> <mo>(</mo> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>5</mn> <mo>,</mo> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>16</mn> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1875</mn> <mo>)</mo> </mrow> </math> marked by <span class="html-italic">d</span> is for the dust-dominated universe. The symbol × shows a prediction of string theory in higher-dimensional spacetime, <math display="inline"> <mrow> <mi>ϵ</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>4</mn> </mrow> </math> thus <math display="inline"> <mrow> <mi>ω</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math>, and <math display="inline"> <mrow> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </math>, based on (4.1) of [<a href="#B25-entropy-14-01997" class="html-bibr">25</a>], taken originally from (3.4.58) of [<a href="#B26-entropy-14-01997" class="html-bibr">26</a>].</p> Full article ">Figure 2
<p>An example of a numerical integration, corresponding to the initial values <math display="inline"> <mrow> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>0</mn> <mo>.</mo> <mn>25</mn> <mo>,</mo> <mover accent="true"> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> <mo>˙</mo> </mover> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>ρ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>1</mn> </mrow> </math> at the initial time <math display="inline"> <mrow> <mo form="prefix">ln</mo> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math>, taken from Figure 4.1 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>], originally from [<a href="#B27-entropy-14-01997" class="html-bibr">27</a>].</p> Full article ">Figure 3
<p>An example of the phase-diagrams in E frame taken from Figure 3 of [<a href="#B28-entropy-14-01997" class="html-bibr">28</a>]. The evolution variable is chosen to be <math display="inline"> <mrow> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>=</mo> <msqrt> <mrow> <mi>V</mi> <mo>(</mo> <mi>σ</mi> <mo>)</mo> </mrow> </msqrt> <mi>d</mi> <msub> <mi>t</mi> <mo>*</mo> </msub> </mrow> </math>, while the coordinates are defined by <math display="inline"> <mrow> <mi>x</mi> <mrow> <mo>(</mo> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>d</mi> <mi>σ</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>τ</mi> <mo>*</mo> </msub> </mrow> </math> and <math display="inline"> <mrow> <mi>y</mi> <mrow> <mo>(</mo> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>ζ</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo>(</mo> <mi>d</mi> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo>/</mo> <mi>d</mi> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>)</mo> </mrow> <mo>/</mo> <msub> <mi>a</mi> <mo>*</mo> </msub> </mrow> </math>, which satisfy the self-autonomous equations (3.15) and (3.16) of [<a href="#B28-entropy-14-01997" class="html-bibr">28</a>]. The solid and dashed curves in (<b>a</b>) are for the null curves of <math display="inline"> <mrow> <mi>d</mi> <mi>x</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>d</mi> <mi>y</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math>, respectively, bounding the area of <math display="inline"> <mrow> <mi>d</mi> <mi>x</mi> <mo>/</mo> <mi>d</mi> <msub> <mi>τ</mi> <mo>*</mo> </msub> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> marked by <math display="inline"> <msub> <mo>+</mo> <mi>x</mi> </msub> </math>, for example. The fixed points are <math display="inline"> <mrow> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>x</mi> <mo>=</mo> <mi>y</mi> <mo>=</mo> <mo>-</mo> <mn>1</mn> </mrow> </math> for an attractor and repeller, respectively, as shown in the close-up views in (<b>b</b>) and (<b>c</b>). The trajectory shown by a dotted curve enters the frame of (<b>b</b>), with <math display="inline"> <mrow> <mi>ξ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>4</mn> </mrow> </math> thus <math display="inline"> <mrow> <msup> <mi>ζ</mi> <mn>2</mn> </msup> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </math>, near the lower-left corner, going out across the right edge, re-entering again at the top, spiraling finally into the attractor at <math display="inline"> <mrow> <mi>x</mi> <mo>=</mo> <mn>1</mn> <mo>=</mo> <mn>1</mn> </mrow> </math>. No such trajectory is shown naturally in (<b>c</b>).</p> Full article ">Figure 4
<p>The simple Yukawa interaction with the coefficient <math display="inline"> <mrow> <mn>2</mn> <mo>-</mo> <mi>d</mi> </mrow> </math> as in (<b>a</b>), but now with a non-gravitational radiative correction included, like in (<b>b</b>), where the dashed curve is for a non-gravitational field with the associated coupling constant <math display="inline"> <msub> <mi>g</mi> <mi>c</mi> </msub> </math>. Heavy dotted lines drawn vertically are for <span class="html-italic">σ</span>.</p> Full article ">Figure 5
<p>A loop diagram generating a mass of the field <span class="html-italic">σ</span> (heavy dotted lines), while solid lines inside a loop represent quarks or leptons, with the coupling strength proportional to their own masses divided by <math display="inline"> <msub> <mi>M</mi> <mi mathvariant="normal">P</mi> </msub> </math>. We also assume the integral to be cut off roughly around <math display="inline"> <msub> <mi>M</mi> <mi>ssb</mi> </msub> </math>, the mass scale of supersymmetry breaking.</p> Full article ">Figure 6
<p>(<b>a</b>) 1-loop photon self-energy part with <span class="html-italic">σ</span> (heavy dotted line) attached to two of the vertices. (<b>b</b>) The same but one of the photon lines (thin dotted lines) attached to another charged field (vertical solid line), with <span class="html-italic">σ</span> attached to three of the vertices.</p> Full article ">Figure 7
<p>An example of hesitation behavior, taken from Figure 5.6 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>]. The solid curve in the upper-half of the plot shows <math display="inline"> <mrow> <mn>2</mn> <mo form="prefix">ln</mo> <msub> <mi>a</mi> <mo>*</mo> </msub> </mrow> </math>, while the dashed curve represents <math display="inline"> <mrow> <mn>2</mn> <mi>σ</mi> </mrow> </math>. In the lower-half of the plot, the dashed and the solid curves are for <math display="inline"> <mrow> <msub> <mo form="prefix">log</mo> <mn>10</mn> </msub> <msub> <mi>ρ</mi> <mo>*</mo> </msub> </mrow> </math> and <math display="inline"> <mrow> <msub> <mo form="prefix">log</mo> <mn>10</mn> </msub> <msub> <mi>ρ</mi> <mi>σ</mi> </msub> </mrow> </math>, respectively. We chose <math display="inline"> <mrow> <mi>ζ</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>5823</mn> </mrow> </math>, the same as will be used in the next subsection. The initial values at <math display="inline"> <mrow> <mo form="prefix">log</mo> <msub> <mi>t</mi> <mrow> <mo>*</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>10</mn> </mrow> </math> is given by <math display="inline"> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>75442</mn> <mo>,</mo> <msub> <mover accent="true"> <mi>σ</mi> <mo>˙</mo> </mover> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math>, while the matter density assumed to be radiation-dominated is <math display="inline"> <mrow> <mn>3</mn> <mo>.</mo> <mn>7352</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>23</mn> </mrow> </msup> </mrow> </math>.</p> Full article ">Figure 8
<p>The potential <math display="inline"> <mrow> <mi>V</mi> <mo>(</mo> <mi>σ</mi> <mo>,</mo> <mi>χ</mi> <mo>)</mo> </mrow> </math> given by (<a href="#FD98-entropy-14-01997" class="html-disp-formula">98</a>), taken from Figure 5.7 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>]. Along the central valley with <math display="inline"> <mrow> <mi>χ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math>, the potential reduces back to the simpler behavior <math display="inline"> <mrow> <mo>Λ</mo> <msup> <mi>e</mi> <mrow> <mo>-</mo> <mn>4</mn> <mi>ζ</mi> <mi>σ</mi> </mrow> </msup> </mrow> </math>, but with <math display="inline"> <mrow> <mi>χ</mi> <mo>≠</mo> <mn>0</mn> </mrow> </math>, it shows an oscillation in the <span class="html-italic">σ</span> direction. The configuration of <span class="html-italic">σ</span> and <span class="html-italic">χ</span> is represented by a point, which is trapped in one of the valleys in the <span class="html-italic">χ</span> direction stays there, hence contributing a lasting <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>σ</mi> <mi>χ</mi> </mrow> </msub> </math> that acts like a cosmological “constant”. As the time elapses, however, the force in the <span class="html-italic">χ</span> direction towards the central valley becomes strong, because of the increase of <math display="inline"> <msubsup> <mi>t</mi> <mo>*</mo> <mn>2</mn> </msubsup> </math> in the last term on LHS of (<a href="#FD101-entropy-14-01997" class="html-disp-formula">101</a>), eventually releasing the point in the positive <span class="html-italic">σ</span> direction, the end of the mini-inflation. For more details, see also Figure 5.14 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>].</p> Full article ">Figure 9
<p>An example of the solution, taken from Figure 5.8 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>]. In accordance with the argument following (<a href="#FD102-entropy-14-01997" class="html-disp-formula">102</a>), we chose <math display="inline"> <mrow> <msub> <mi>ζ</mi> <mi>dm</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math>, without much affecting the result around today. Upper diagram: <math display="inline"> <mrow> <mi>b</mi> <mo>=</mo> <mo form="prefix">ln</mo> <msub> <mi>a</mi> <mo>*</mo> </msub> </mrow> </math> (dotted), <span class="html-italic">σ</span> (solid) and <math display="inline"> <mrow> <mn>2</mn> <mi>χ</mi> </mrow> </math> (dashed) are plotted against <math display="inline"> <mrow> <msub> <mi>τ</mi> <mn>10</mn> </msub> <mo>=</mo> <mo form="prefix">log</mo> <msub> <mi>t</mi> <mo>*</mo> </msub> </mrow> </math>. The present epoch corresponds to <math display="inline"> <mrow> <msub> <mi>τ</mi> <mn>10</mn> </msub> <mo>=</mo> <mn>60</mn> <mo>.</mo> <mn>1</mn> <mo>-</mo> <mn>60</mn> <mo>.</mo> <mn>2</mn> </mrow> </math>, while the primordial nucleosynthesis must have taken place at <math display="inline"> <mrow> <msub> <mi>τ</mi> <mn>10</mn> </msub> <mo>∼</mo> <mn>45</mn> </mrow> </math>. The parameters are <math display="inline"> <mrow> <mo>Λ</mo> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mi>ζ</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>5823</mn> <mo>,</mo> <mi>m</mi> <mo>=</mo> <mn>4</mn> <mo>.</mo> <mn>75</mn> <mo>,</mo> <mi>γ</mi> <mspace width="3.33333pt"/> <mo>=</mo> <mspace width="3.33333pt"/> <mn>0</mn> <mo>.</mo> <mn>8</mn> <mo>,</mo> <mi>κ</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>. The initial values at <math display="inline"> <mrow> <msub> <mi>τ</mi> <mn>10</mn> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>10</mn> </msup> </mrow> </math> are <math display="inline"> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>7544</mn> <mo>,</mo> <msubsup> <mi>σ</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math> (a prime implies differentiation with respect to <math display="inline"> <mrow> <mi>τ</mi> <mo>=</mo> <mo form="prefix">ln</mo> <msub> <mi>t</mi> <mo>*</mo> </msub> </mrow> </math>), <math display="inline"> <mrow> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>21</mn> <mo>,</mo> <msubsup> <mi>χ</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mo>=</mo> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>005</mn> <mo>,</mo> <msub> <mi>ρ</mi> <mrow> <mn>1</mn> <mi>rad</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> <mo>.</mo> <mn>7352</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>23</mn> </mrow> </msup> <mo>,</mo> <msub> <mi>ρ</mi> <mrow> <mn>1</mn> <mi>dust</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo>.</mo> <mn>0</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>45</mn> </mrow> </msup> </mrow> </math>. The dashed-dotted straight line represents the asymptote of <span class="html-italic">σ</span> given by <math display="inline"> <mrow> <mi>τ</mi> <mo>/</mo> <mo>(</mo> <mn>2</mn> <mi>ζ</mi> <mo>)</mo> </mrow> </math>. Notice long plateaus of <span class="html-italic">σ</span> and <span class="html-italic">χ</span>, and their rapid changes during relatively “short” periods. Middle diagram: <math display="inline"> <mrow> <msub> <mi>p</mi> <mo>*</mo> </msub> <mo>=</mo> <msup> <mi>b</mi> <mo>′</mo> </msup> <mo>=</mo> <msub> <mi>t</mi> <mo>*</mo> </msub> <msub> <mi>H</mi> <mo>*</mo> </msub> </mrow> </math> for an effective exponent in the local power-law expansion <math display="inline"> <mrow> <msub> <mi>a</mi> <mo>*</mo> </msub> <mo>∼</mo> <msubsup> <mi>t</mi> <mo>*</mo> <msub> <mi>p</mi> <mo>*</mo> </msub> </msubsup> </mrow> </math> of the universe. Notable leveling-offs can be seen at 0.333, 0.5 and 0.667 corresponding to the epochs dominated by the kinetic terms of the scalar fields, the radiation matter and the dust matter, respectively. Lower diagram: <math display="inline"> <mrow> <mo form="prefix">log</mo> <msub> <mi>ρ</mi> <mrow> <mi>σ</mi> <mi>χ</mi> </mrow> </msub> </mrow> </math> (solid), the total energy density of the <span class="html-italic">σ</span>-<span class="html-italic">χ</span> system, and <math display="inline"> <mrow> <mo form="prefix">log</mo> <msub> <mi>ρ</mi> <mo>*</mo> </msub> </mrow> </math> (dashed), the matter energy density. Notice an “interlacing" pattern of <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>σ</mi> <mi>χ</mi> </mrow> </msub> </math> and <math display="inline"> <msub> <mi>ρ</mi> <mo>*</mo> </msub> </math>, still obeying <math display="inline"> <mrow> <mo>∼</mo> <msubsup> <mi>t</mi> <mo>*</mo> <mrow> <mo>-</mo> <mn>2</mn> </mrow> </msubsup> </mrow> </math> as an overall behavior. Nearly flat plateaus of <math display="inline"> <msub> <mi>ρ</mi> <mrow> <mi>σ</mi> <mi>χ</mi> </mrow> </msub> </math> precede before it overtakes <math display="inline"> <msub> <mi>ρ</mi> <mo>*</mo> </msub> </math>, hence with <math display="inline"> <msub> <mo>Ω</mo> <mo>Λ</mo> </msub> </math> passing through 0.5.</p> Full article ">Figure 10
<p>An example of the solution, taken from Figure 5.11 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>], showing no mini-inflation around the present epoch, though another mini-inflation at <math display="inline"> <mrow> <msub> <mi>τ</mi> <mn>10</mn> </msub> <mo>∼</mo> <mn>27</mn> </mrow> </math> is still present. Symbols and initial values are the same as explained in <a href="#entropy-14-01997-f009" class="html-fig">Figure 9</a>, except for <math display="inline"> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>761</mn> </mrow> </math>, which is different form 6.7544 in <a href="#entropy-14-01997-f009" class="html-fig">Figure 9</a> only slightly. This indicates how sensitively the result might depend on the choice of some of the parameters.</p> Full article ">Figure 11
<p>Magnified view of <span class="html-italic">σ</span> (solid) and <math display="inline"> <mrow> <mn>0</mn> <mo>.</mo> <mn>02</mn> <mi>χ</mi> <mo>+</mo> <mn>44</mn> <mo>.</mo> <mn>25</mn> </mrow> </math> (dashed) in the upper panel of <a href="#entropy-14-01997-f009" class="html-fig">Figure 9</a>, taken from Figure 5.10 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>]. Note that the vertical scale has been expanded by approximately 330 times as large compared with <a href="#entropy-14-01997-f009" class="html-fig">Figure 9</a>.</p> Full article ">Figure 12
<p>Typical plots of the theoretical curves for <math display="inline"> <mrow> <mrow> <mo>(</mo> <mo>Δ</mo> <mi>α</mi> <mo>/</mo> <mi>α</mi> <mo>)</mo> </mrow> <mo>×</mo> <msup> <mn>10</mn> <mn>6</mn> </msup> </mrow> </math> as a function of redshift <span class="html-italic">z</span>, translated from the same of <math display="inline"> <msub> <mi>t</mi> <mo>*</mo> </msub> </math>, taken from Figure 1 of [<a href="#B40-entropy-14-01997" class="html-bibr">40</a>]. See also [<a href="#B41-entropy-14-01997" class="html-bibr">41</a>,<a href="#B42-entropy-14-01997" class="html-bibr">42</a>,<a href="#B43-entropy-14-01997" class="html-bibr">43</a>,<a href="#B44-entropy-14-01997" class="html-bibr">44</a>]. The Oklo phenomenon having occurred <math display="inline"> <mrow> <mo>≈</mo> <mn>1</mn> <mo>.</mo> <mn>95</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>9</mn> </msup> <mi mathvariant="normal">y</mi> </mrow> </math> ago corresponds to <math display="inline"> <mrow> <mi>z</mi> <mo>∼</mo> <mn>0</mn> <mo>.</mo> <mn>15</mn> </mrow> </math>[<a href="#B45-entropy-14-01997" class="html-bibr">45</a>], while two QSO data [<a href="#B46-entropy-14-01997" class="html-bibr">46</a>,<a href="#B47-entropy-14-01997" class="html-bibr">47</a>] are shown; <math display="inline"> <mrow> <mo>-</mo> <mn>0</mn> <mo>.</mo> <mn>12</mn> <mo>±</mo> <mn>1</mn> <mo>.</mo> <mn>79</mn> </mrow> </math> and <math display="inline"> <mrow> <mn>5</mn> <mo>.</mo> <mn>66</mn> <mo>±</mo> <mn>2</mn> <mo>.</mo> <mn>67</mn> </mrow> </math> for <math display="inline"> <mrow> <mi>z</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>15</mn> </mrow> </math> and <math display="inline"> <mrow> <mn>1</mn> <mo>.</mo> <mn>84</mn> </mrow> </math>, also for the fractional look-back time 0.59 and 0.73, respectively. We commonly choose the initial values at <math display="inline"> <mrow> <msub> <mi>t</mi> <mn>1</mn> </msub> <mo>=</mo> <msup> <mn>10</mn> <mn>10</mn> </msup> </mrow> </math> in the reduced Planckian unit system, as in Figure 5.8 of [<a href="#B11-entropy-14-01997" class="html-bibr">11</a>]; <math display="inline"> <mrow> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>6</mn> <mo>.</mo> <mn>77341501</mn> <mo>,</mo> <msubsup> <mi>σ</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mo>=</mo> <mn>0</mn> <mo>,</mo> <msub> <mi>χ</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>21</mn> <mo>,</mo> <msubsup> <mi>χ</mi> <mn>1</mn> <mo>′</mo> </msubsup> <mo>=</mo> <mn>0</mn> </mrow> </math>, where the prime is for the derivative with respect to <math display="inline"> <mrow> <mi>τ</mi> <mo>=</mo> <mo form="prefix">ln</mo> <mi>t</mi> </mrow> </math>.</p> Full article ">Figure 13
<p><span class="html-italic">σ</span>-dominated tree diagrams for the photon-photon scattering process, taken from Figure 3 of [<a href="#B18-entropy-14-01997" class="html-bibr">18</a>]. Solid lines are for the photons with the attached momenta <span class="html-italic">p</span>’s while the dashed lines for <span class="html-italic">σ</span>, in the <span class="html-italic">s</span>-, <span class="html-italic">t</span>-, and <span class="html-italic">u</span>-channels, respectively.</p> Full article ">Figure 14
<p>A single Gaussian laser beam focused by an ideal lens where a scalar field exchange entails a frequency-upshifted photon in the forward direction, taken from Figure 5 of [<a href="#B18-entropy-14-01997" class="html-bibr">18</a>]. The frequency of the incident laser beam is assumed to be within a narrow band, while the incident angle varies largely including the value <math display="inline"> <mrow> <mo>∼</mo> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>9</mn> </mrow> </msup> </mrow> </math>.</p> Full article ">Figure 15
<p>Definitions of kinematical variables, taken from Figure 1 of [<a href="#B18-entropy-14-01997" class="html-bibr">18</a>], in the Quasi-Parallel-Frame.</p> Full article ">Figure 16
<p>Fumitaka Sato’s image of Unification in 1983. His original caption in Japanese goes like <span class="html-italic">“Understanding microscopic world now provides us with a powerful tool to understand the hyper-macroscopic world”</span>. In his own drawing, a guy is looking into a microscope instead of a telescope, yelling “Look, I got the universe!”.</p> Full article ">
538 KiB  
Article
Exact Solution and Exotic Fluid in Cosmology
by Seyen Kouwn, Taeyoon Moon and Phillial Oh
Entropy 2012, 14(9), 1771-1783; https://doi.org/10.3390/e14091771 - 20 Sep 2012
Cited by 2 | Viewed by 5120
Abstract
We investigate cosmological consequences of nonlinear sigma model coupled with a cosmological fluid which satisfies the continuity equation. The target space action is of the de Sitter type and is composed of four scalar fields. The potential which is a function of only [...] Read more.
We investigate cosmological consequences of nonlinear sigma model coupled with a cosmological fluid which satisfies the continuity equation. The target space action is of the de Sitter type and is composed of four scalar fields. The potential which is a function of only one of the scalar fields is also introduced. We perform a general analysis of the ensuing cosmological equations and give various critical points and their properties. Then, we show that the model exhibits an exact cosmological solution which yields a transition from matter domination into dark energy epoch and compare it with the Λ-CDM behavior. Especially, we calculate the age of the Universe and show that it is consistent with the observational value if the equation of the state ωf of the cosmological fluid is within the range of 0.13 < ωf < 0.22. Some implication of this result is also discussed. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
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Figure 1

Figure 1
<p>The plot of <math display="inline"> <mrow> <msub> <mo>Ω</mo> <mrow> <mo>Λ</mo> <mo>,</mo> <mn>0</mn> </mrow> </msub> <mrow> <mo>(</mo> <mi>b</mi> <mi>l</mi> <mi>u</mi> <mi>e</mi> <mo>)</mo> </mrow> <mo>≃</mo> <mn>0.726</mn> <mo>±</mo> <mn>0.015</mn> <mo>,</mo> <mspace width="3.33333pt"/> <msub> <mi>H</mi> <mn>0</mn> </msub> <mrow> <mo>(</mo> <mi>g</mi> <mi>r</mi> <mi>e</mi> <mi>e</mi> <mi>n</mi> <mo>)</mo> </mrow> <mo>≃</mo> <mn>2.28</mn> <mo>±</mo> <mn>0.04</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow> <mo>−</mo> <mn>18</mn> </mrow> </msup> <msup> <mi>s</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </math> on the (<span class="html-italic">B</span>, <span class="html-italic">t</span>)-plane for <math display="inline"> <mrow> <msub> <mi>ω</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.13</mn> </mrow> </math>(left), <math display="inline"> <mrow> <msub> <mi>ω</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.18</mn> </mrow> </math>(right), <math display="inline"> <mrow> <msub> <mi>ω</mi> <mi>f</mi> </msub> <mo>=</mo> <mn>0.22</mn> </mrow> </math>(lower). The red band is the current age with uncertainty <math display="inline"> <mrow> <msub> <mi>t</mi> <mn>0</mn> </msub> <mo>≃</mo> <mn>4.33</mn> <mo>±</mo> <mn>0.04</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>17</mn> </msup> <mi>s</mi> </mrow> </math>.</p> Full article ">
1458 KiB  
Article
Black Holes, Cosmological Solutions, Future Singularities, and Their Thermodynamical Properties in Modified Gravity Theories
by Alvaro De la Cruz-Dombriz and Diego Sáez-Gómez
Entropy 2012, 14(9), 1717-1770; https://doi.org/10.3390/e14091717 - 18 Sep 2012
Cited by 306 | Viewed by 8880
Abstract
Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann–Lemaˆıtre–Robertson–Walker metrics. The thermodynamical properties of fourth order gravity theories are also [...] Read more.
Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann–Lemaˆıtre–Robertson–Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f(R) models. In addition, we revise some attempts to extend the Cardy–Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a deep study on cosmological singularities, which appear as a real possibility for some kind of modified gravity theories, is performed, and the validity of the entropy bounds is studied. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
►▼ Show Figures

Figure 1

Figure 1
<p>Graphics showing horizons positions as solutions of the equation <math display="inline"> <mrow> <msub> <mo>Δ</mo> <mi>r</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </math> taken from [<a href="#B120-entropy-14-01717" class="html-bibr">120</a>,<a href="#B121-entropy-14-01717" class="html-bibr">121</a>]. On the left panel (<math display="inline"> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </math>) the presented cases are: <math display="inline"> <mrow> <mi>h</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> (<b>I</b>), BH with well-defined horizons, dashed with dots), <math display="inline"> <mrow> <mi>h</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> (<b>II</b>), <span class="html-italic">extremal</span> BH, continuous line) and <math display="inline"> <mrow> <mi>h</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math> (<b>III</b>), <span class="html-italic">naked singularity</span>, dashed). On the right panel (<math display="inline"> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>&gt;</mo> <mn>0</mn> </mrow> </math>) the represented cases are: <math display="inline"> <mrow> <mi>h</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math> (<b>I</b>, BH with well-defined horizons, dashed with dots), <math display="inline"> <mrow> <mi>h</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> (<b>II</b>, <math display="inline"> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>r</mi> <mi>e</mi> <mi>m</mi> <mi>a</mi> <mi>l</mi> </mrow> </math> BH and <b>III</b>, <math display="inline"> <mrow> <mi>e</mi> <mi>x</mi> <mi>t</mi> <mi>r</mi> <mi>e</mi> <mi>m</mi> <mi>a</mi> <mi>l</mi> </mrow> </math> <math display="inline"> <mrow> <mi>m</mi> <mi>a</mi> <mi>r</mi> <mi>g</mi> <mi>i</mi> <mi>n</mi> <mi>a</mi> <mi>l</mi> </mrow> </math> BH, continuous line), and <math display="inline"> <mrow> <mi>h</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> (<b>IV</b>, <span class="html-italic">naked singularity</span> and <b>V</b>, <span class="html-italic">naked marginal singularity</span>, dashed).</p> Full article ">Figure 2
<p>The shaded regions, delimited by the upper <math display="inline"> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </math> and lower <math display="inline"> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </math> curves, represent the values of <math display="inline"> <mrow> <mi>a</mi> <mo>/</mo> <mi>M</mi> </mrow> </math> for which the existence of BH is possible once <math display="inline"> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mspace width="0.166667em"/> <msup> <mi>M</mi> <mn>2</mn> </msup> </mrow> </math> value is fixed. Panels show for <math display="inline"> <mrow> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mspace width="0.166667em"/> <mo>/</mo> <mspace width="0.166667em"/> <mi>M</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> (left) and <math display="inline"> <mrow> <mover accent="true"> <mi>Q</mi> <mo>¯</mo> </mover> <mspace width="0.166667em"/> <mo>/</mo> <mspace width="0.166667em"/> <mi>M</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>75</mn> </mrow> </math> (right) on the left and right panels respectively. Note that <math display="inline"> <msub> <mi>R</mi> <mn>0</mn> </msub> </math> has dimensions of [length]<math display="inline"> <msup> <mrow/> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </msup> </math> when normalizing. Original plots at [<a href="#B120-entropy-14-01717" class="html-bibr">120</a>,<a href="#B121-entropy-14-01717" class="html-bibr">121</a>].</p> Full article ">Figure 3
<p>For <math display="inline"> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>=</mo> <mo>−</mo> <mn>0</mn> <mo>.</mo> <mn>2</mn> </mrow> </math>, we graphically display temperature (left) and heat capacity (right) of a BH as functions of the mass parameter <span class="html-italic">M</span> for the cases: (<b>I</b>) <math display="inline"> <mrow> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mo>.</mo> <mn>5</mn> </mrow> </math> y <math display="inline"> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> : “slow” BH that shows a local maximum temperature <math display="inline"> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </math> and a local minimum temperature <math display="inline"> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </math> at the points where the heat capacity diverges, taking the latter negative values between <math display="inline"> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </math> y <math display="inline"> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </math>; (<b>II</b>) <math display="inline"> <mrow> <mi>a</mi> <mo>≈</mo> <mn>0</mn> <mo>.</mo> <mn>965</mn> </mrow> </math> y <math display="inline"> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> : limit case where <math display="inline"> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </math> and <math display="inline"> <msub> <mi>T</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </math> merge, hence resulting in an inflection point in the temperature and an always positive heat capacity; (<b>III</b>) <math display="inline"> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> <mo>.</mo> <mn>5</mn> </mrow> </math> y <math display="inline"> <mrow> <mi>Q</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> : “fast” BH with both temperature and heat capacity monotonously growing (always positive too).</p> Full article ">Figure 4
<p>Thermodynamical regions in the <math display="inline"> <mrow> <mo>(</mo> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>,</mo> <mo>|</mo> <mi>β</mi> <mo>|</mo> <mo>)</mo> </mrow> </math> plane for Model I in <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>4</mn> </mrow> </math> (left) and <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>5</mn> </mrow> </math> (right).</p> Full article ">Figure 5
<p>Thermodynamical regions in the <math display="inline"> <mrow> <mo>(</mo> <mo>|</mo> <mi>α</mi> <mo>|</mo> <mo>,</mo> <mo>|</mo> <mi>β</mi> <mo>|</mo> <mo>)</mo> </mrow> </math> plane for Model I in <math display="inline"> <mrow> <mi>D</mi> <mo>=</mo> <mn>10</mn> </mrow> </math>.</p> Full article ">Figure 6
<p><b>Model I</b>: Region 3: <math display="inline"> <mfenced separators="" open="{" close="}"> <mi>α</mi> <mo>&lt;</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>&gt;</mo> <mn>2</mn> </mfenced> </math>, and Region 4: <math display="inline"> <mfenced separators="" open="{" close="}"> <mi>α</mi> <mo>&gt;</mo> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>&lt;</mo> <mn>1</mn> </mfenced> </math>. BH with a well defined horizon structure will only exist if they have a spin parameter below the upper surface <math display="inline"> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </math>. For the presented regions 3 and 4, the surface <math display="inline"> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </math> does not exist.</p> Full article ">Figure 7
<p>Thermodynamical regions for Model I. We distinguish between three different regions: (<b>i</b>) <math display="inline"> <mrow> <mi>C</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>F</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math>, in black; (<b>ii</b>) <math display="inline"> <mrow> <mi>C</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>F</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math>, in gray; (<b>iii</b>) <math display="inline"> <mrow> <mi>C</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>F</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math>, in white.</p> Full article ">Figure 8
<p><b>Model II</b>. Region 1: <math display="inline"> <mfenced separators="" open="{" close="}"> <mi>κ</mi> <mo>&gt;</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>γ</mi> <mo>&gt;</mo> <mn>0</mn> </mfenced> </math>, and Region 2: <math display="inline"> <mfenced separators="" open="{" close="}"> <mi>κ</mi> <mo>&gt;</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>γ</mi> <mo>&lt;</mo> <mn>0</mn> </mfenced> </math>. BH with a well-defined horizon structure will only exist if they have a spin parameter below the upper surface <math display="inline"> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>a</mi> <mi>x</mi> </mrow> </msub> </math>, and above a second surface <math display="inline"> <msub> <mi>a</mi> <mrow> <mi>m</mi> <mi>i</mi> <mi>n</mi> </mrow> </msub> </math> (in case it exists, only in region 1 for this model) for certain values of <span class="html-italic">κ</span> and <span class="html-italic">γ</span>.</p> Full article ">Figure 9
<p>Thermodynamical regions with negative scalar curvature <math display="inline"> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>&lt;</mo> <mn>0</mn> </mrow> </math> of model I. We distinguish between three different regions: (<b>i</b>) <math display="inline"> <mrow> <mi>C</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>F</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math>, in black; (<b>ii</b>) <math display="inline"> <mrow> <mi>C</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>F</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math>, in gray; (<b>iii</b>) <math display="inline"> <mrow> <mi>C</mi> <mo>&gt;</mo> <mn>0</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>F</mi> <mo>&lt;</mo> <mn>0</mn> </mrow> </math>, in white.</p> Full article ">
353 KiB  
Article
Cosmology of F(T) Gravity and k-Essence
by Ratbay Myrzakulov
Entropy 2012, 14(9), 1627-1651; https://doi.org/10.3390/e14091627 - 4 Sep 2012
Cited by 40 | Viewed by 5977
Abstract
This a brief review on F(T) gravity and its relation with k-essence. Modified teleparallel gravity theory with the torsion scalar has recently gained a lot of attention as a possible explanation of dark energy. We perform a thorough reconstruction analysis on the so-called [...] Read more.
This a brief review on F(T) gravity and its relation with k-essence. Modified teleparallel gravity theory with the torsion scalar has recently gained a lot of attention as a possible explanation of dark energy. We perform a thorough reconstruction analysis on the so-called F(T) models, where F(T) is some general function of the torsion term, and deduce the required conditions for the equivalence between of F(T) models with pure kinetic k-essence models. We present a new class of models of F(T)-gravity and k-essence. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
314 KiB  
Article
Stability of Accelerating Cosmology in Two Scalar-Tensor Theory: Little Rip versus de Sitter
by Yusaku Ito, Shin’ichi Nojiri and Sergei D. Odintsov
Entropy 2012, 14(8), 1578-1605; https://doi.org/10.3390/e14081578 - 23 Aug 2012
Cited by 43 | Viewed by 5945
Abstract
We develop the general reconstruction scheme in two scalar model. The quintom-like theory which may describe (different) non-singular Little Rip or de Sitter cosmology is reconstructed. The number of scalar phantom dark energy models (with Little Rip cosmology or asymptotically de Sitter evolution) [...] Read more.
We develop the general reconstruction scheme in two scalar model. The quintom-like theory which may describe (different) non-singular Little Rip or de Sitter cosmology is reconstructed. The number of scalar phantom dark energy models (with Little Rip cosmology or asymptotically de Sitter evolution) is presented. Stability issue of such dark energy cosmologies as well as the flow to fixed points are studied. The stability of Little Rip universe which leads to dissolution of bound objects sometime in future indicates that no classical transition to de Sitter space occurs. The possibility of unification of inflation with Little Rip dark energy in two scalar theory is briefly mentioned. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
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Figure 1

Figure 1
<p>Each vector denotes <math display="inline"> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>50</mn> <mo>,</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>50</mn> <mo>)</mo> </mrow> </math>, which is independent of <span class="html-italic">Z</span> and <span class="html-italic">W</span>. The parameters are <math display="inline"> <mrow> <mover accent="true"> <mi>λ</mi> <mo>˜</mo> </mover> <mo>=</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>α</mi> <mo>˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math>. The point A is located in <math display="inline"> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>, where the EoS parameter is <math display="inline"> <mrow> <mo>−</mo> <mn>2</mn> </mrow> </math>. The point B is located in <math display="inline"> <mrow> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>2</mn> <mo>)</mo> </mrow> </math>, where the EoS parameter is <math display="inline"> <mrow> <mo>−</mo> <mn>3</mn> </mrow> </math>.</p> Full article ">Figure 2
<p>Each vector denotes <math display="inline"> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>,</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>)</mo> </mrow> </math>, which is independent of <span class="html-italic">Z</span> and <span class="html-italic">W</span>. The parameters are <math display="inline"> <mrow> <mover accent="true"> <mi>λ</mi> <mo>˜</mo> </mover> <mo>=</mo> <mo>−</mo> <mn>3</mn> <mo>/</mo> <mn>2</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>α</mi> <mo>˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math>. The point A is located in <math display="inline"> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>, where the EoS parameter is 0.</p> Full article ">Figure 3
<p>Each vector denotes <math display="inline"> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>50</mn> <mo>,</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>50</mn> <mo>)</mo> </mrow> </math> with <math display="inline"> <mrow> <mi>Z</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>W</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> (<math display="inline"> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>χ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mi>κ</mi> </mrow> </math>). The parameters are <math display="inline"> <mrow> <mi>γ</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>α</mi> <mo>˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math>. The point A is located in <math display="inline"> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>, which corresponds to the Little Rip universe.</p> Full article ">Figure 4
<p>Each vector denotes <math display="inline"> <mrow> <mo>(</mo> <msup> <mi>X</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>,</mo> <msup> <mi>Y</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>)</mo> </mrow> </math> with <math display="inline"> <mrow> <mi>Z</mi> <mo>=</mo> <mn>1</mn> </mrow> </math> and <math display="inline"> <mrow> <mi>W</mi> <mo>=</mo> <mn>0</mn> </mrow> </math> (<math display="inline"> <mrow> <mi>ϕ</mi> <mo>=</mo> <mi>χ</mi> <mo>=</mo> <mn>1</mn> <mo>/</mo> <mn>2</mn> <mi>κ</mi> </mrow> </math>). The parameters are <math display="inline"> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mrow> </math> and <math display="inline"> <mrow> <msub> <mover accent="true"> <mi>α</mi> <mo>˜</mo> </mover> <mn>0</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math>. The point A is located in <math display="inline"> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </math>. The point B is located in <math display="inline"> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </math>, which corresponds to the de Sitter universe.</p> Full article ">Figure 5
<p>Each vector denotes <math display="inline"> <mrow> <mo>(</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>,</mo> <msup> <mi>W</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>)</mo> </mrow> </math>, which is independent of the form of <math display="inline"> <mrow> <mi>ω</mi> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </math>, <math display="inline"> <mrow> <mi>η</mi> <mo>(</mo> <mi>χ</mi> <mo>)</mo> </mrow> </math> and <math display="inline"> <mrow> <mi>V</mi> <mo>(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>χ</mi> <mo>)</mo> </mrow> </math>. The dynamics of <span class="html-italic">Z</span> and <span class="html-italic">W</span> are classified into four types according to the values of <span class="html-italic">X</span> and <span class="html-italic">Y</span> as (<b>a</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>Y</mi> </mrow> </math>, <math display="inline"> <mrow> <mi>X</mi> <mo>&lt;</mo> <mi>Y</mi> </mrow> </math>; (<b>b</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>Y</mi> <mo>&lt;</mo> <mi>X</mi> </mrow> </math>; (<b>c</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&gt;</mo> <mi>Y</mi> </mrow> </math>, <math display="inline"> <mrow> <mi>X</mi> <mo>&gt;</mo> <mi>Y</mi> </mrow> </math> and (<b>d</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&gt;</mo> <mi>Y</mi> <mo>&gt;</mo> <mi>X</mi> </mrow> </math>. The fixed points are located in <math display="inline"> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </math>.</p> Full article ">Figure 5 Cont.
<p>Each vector denotes <math display="inline"> <mrow> <mo>(</mo> <msup> <mi>Z</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>,</mo> <msup> <mi>W</mi> <mo>′</mo> </msup> <mo>/</mo> <mn>20</mn> <mo>)</mo> </mrow> </math>, which is independent of the form of <math display="inline"> <mrow> <mi>ω</mi> <mo>(</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </math>, <math display="inline"> <mrow> <mi>η</mi> <mo>(</mo> <mi>χ</mi> <mo>)</mo> </mrow> </math> and <math display="inline"> <mrow> <mi>V</mi> <mo>(</mo> <mi>ϕ</mi> <mo>,</mo> <mi>χ</mi> <mo>)</mo> </mrow> </math>. The dynamics of <span class="html-italic">Z</span> and <span class="html-italic">W</span> are classified into four types according to the values of <span class="html-italic">X</span> and <span class="html-italic">Y</span> as (<b>a</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>Y</mi> </mrow> </math>, <math display="inline"> <mrow> <mi>X</mi> <mo>&lt;</mo> <mi>Y</mi> </mrow> </math>; (<b>b</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>Y</mi> <mo>&lt;</mo> <mi>X</mi> </mrow> </math>; (<b>c</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&gt;</mo> <mi>Y</mi> </mrow> </math>, <math display="inline"> <mrow> <mi>X</mi> <mo>&gt;</mo> <mi>Y</mi> </mrow> </math> and (<b>d</b>) <math display="inline"> <mrow> <mn>0</mn> <mo>&gt;</mo> <mi>Y</mi> <mo>&gt;</mo> <mi>X</mi> </mrow> </math>. The fixed points are located in <math display="inline"> <mrow> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0</mn> <mo>)</mo> </mrow> </math>.</p> Full article ">
112 KiB  
Article
A Model of Nonsingular Universe
by Changjun Gao
Entropy 2012, 14(7), 1296-1305; https://doi.org/10.3390/e14071296 - 23 Jul 2012
Cited by 11 | Viewed by 5701
Abstract
In the background of Friedmann–Robertson–Walker Universe, there exists Hawking radiation which comes from the cosmic apparent horizon due to quantum effect. Although the Hawking radiation on the late time evolution of the universe could be safely neglected, it plays an important role in [...] Read more.
In the background of Friedmann–Robertson–Walker Universe, there exists Hawking radiation which comes from the cosmic apparent horizon due to quantum effect. Although the Hawking radiation on the late time evolution of the universe could be safely neglected, it plays an important role in the very early stage of the universe. In view of this point, we identify the temperature in the scalar field potential with the Hawking temperature of cosmic apparent horizon. Then we find a nonsingular universe sourced by the temperature-dependent scalar field. We find that the universe could be created from a de Sitter phase which has the Planck energy density. Thus the Big-Bang singularity is avoided. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
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Figure 1

Figure 1
<p>The evolution of the scale factor <span class="html-italic">a</span> with respect to the cosmic time <span class="html-italic">t</span>. It shows that the scale factor approaches zero when <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mo>∞</mo> </mrow> </math>. The unite of time is the Planck time <math display="inline"> <msub> <mi>t</mi> <mi>p</mi> </msub> </math>.</p> Full article ">Figure 2
<p>The evolution of the Hubble radius <span class="html-italic">s</span> with respect to the cosmic time <span class="html-italic">t</span>. It shows that the Hubble horizon approaches a finite constant when <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mo>∞</mo> </mrow> </math>. In other words, the physical size of the universe is bounded below. The unit of time and length are the Planck time <math display="inline"> <msub> <mi>t</mi> <mi>p</mi> </msub> </math> and Planck length <math display="inline"> <msub> <mi>l</mi> <mi>p</mi> </msub> </math>, respectively.</p> Full article ">Figure 3
<p>The evolution of energy density <span class="html-italic">ρ</span> with respect to the cosmic time <span class="html-italic">t</span>. It shows that the energy density approaches the Planck density when <math display="inline"> <mrow> <mi>t</mi> <mo>=</mo> <mo>−</mo> <mo>∞</mo> </mrow> </math>. The unit of time and energy density are the Planck time <math display="inline"> <msub> <mi>t</mi> <mi>p</mi> </msub> </math> and the Planck density <math display="inline"> <msub> <mi>ρ</mi> <mi>p</mi> </msub> </math>, respectively.</p> Full article ">
309 KiB  
Article
The Dark Energy Properties of the Dirac–Born–Infeld Action
by Xinyou Zhang, Qing Zhang and Yongchang Huang
Entropy 2012, 14(7), 1203-1220; https://doi.org/10.3390/e14071203 - 9 Jul 2012
Cited by 2 | Viewed by 5819
Abstract
Introducing a new potential, we deduce a general Lagrangian for Dirac–Born– Infeld (DBI) inflation, in which the determinant of the induced metric naturally includes the kinetic energy and the potential energy. In particular, the potential energy and kinetic energy can convert into each [...] Read more.
Introducing a new potential, we deduce a general Lagrangian for Dirac–Born– Infeld (DBI) inflation, in which the determinant of the induced metric naturally includes the kinetic energy and the potential energy. In particular, the potential energy and kinetic energy can convert into each other at any same order, which is in agreement with the limit of classical physics. We also present a general sound speed in the evolutions of the universe, and the exact expressions of energy-momentum tensor, pressure and density. Furthermore, from the results we obtain the new equation of states. The analytic form of the action that is consistent with data turns out to be surprisingly simple and easy to categorize. Finally, we examine properties of the dark energy and introduce a novel mechanism for realizing either quintessence or phantom dark energy dominated phases within a string theoretical context. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
214 KiB  
Article
On Chirality of the Vorticity of the Universe
by Davor Palle
Entropy 2012, 14(5), 958-965; https://doi.org/10.3390/e14050958 - 16 May 2012
Cited by 14 | Viewed by 5544
Abstract
The presence of dark energy in the Universe challenges the Einstein’s theory of gravity at cosmic scales. It motivates the inclusion of rotational degrees of freedom in the Einstein–Cartan gravity, representing the minimal and the most natural extension of the General Relativity. One [...] Read more.
The presence of dark energy in the Universe challenges the Einstein’s theory of gravity at cosmic scales. It motivates the inclusion of rotational degrees of freedom in the Einstein–Cartan gravity, representing the minimal and the most natural extension of the General Relativity. One can, consequently, expect the violation of the cosmic isotropy by the rotating Universe. We study chirality of the vorticity of the Universe within the Einstein–Cartan cosmology. The role of the spin of fermion species during the evolution of the Universe is studied by averaged spin densities and Einstein–Cartan equations. It is shown that spin density of the light Majorana neutrinos acts as a seed for vorticity at early stages of the evolution of the Universe. Its chirality can be evaluated in the vicinity of the spacelike infinity. It turns out that vorticity of the Universe has right-handed chirality. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
108 KiB  
Article
A Phase Space Diagram for Gravity
by Xavier Hernandez
Entropy 2012, 14(5), 848-855; https://doi.org/10.3390/e14050848 - 4 May 2012
Cited by 7 | Viewed by 7664
Abstract
In modified theories of gravity including a critical acceleration scale a0, a critical length scale rM = (GM/a0)1/2 will naturally arise with the transition from the Newtonian to the dark matter mimicking regime occurring for systems larger [...] Read more.
In modified theories of gravity including a critical acceleration scale a0, a critical length scale rM = (GM/a0)1/2 will naturally arise with the transition from the Newtonian to the dark matter mimicking regime occurring for systems larger than rM. This adds a second critical scale to gravity, in addition to the one introduced by the criterion v < c of the Schwarzschild radius, rS = 2GM/c2. The distinct dependencies of the two above length scales give rise to non-trivial phenomenology in the (mass, length) plane for astrophysical structures, which we explore here. Surprisingly, extrapolation to atomic scales suggests gravity should be at the dark matter mimicking regime there. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
►▼ Show Figures

Figure 1

Figure 1
<p>Phase space diagram for self-gravitating equilibrium configurations. The labelled solid lines give the mass dependant scale radii resulting from the two limit conditions <math display="inline"> <mrow> <mi>v</mi> <mo>=</mo> <mi>c</mi> </mrow> </math> and <math display="inline"> <mrow> <mi>a</mi> <mo>=</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> </mrow> </math>, <math display="inline"> <mrow> <msub> <mi>r</mi> <mi>S</mi> </msub> <mo>=</mo> <mn>2</mn> <mi>G</mi> <mi>M</mi> <mo>/</mo> <msup> <mi>c</mi> <mn>2</mn> </msup> </mrow> </math> and <math display="inline"> <mrow> <msub> <mi>r</mi> <mi>M</mi> </msub> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mi>G</mi> <mi>M</mi> <mo>/</mo> <msub> <mi>a</mi> <mn>0</mn> </msub> <mo>)</mo> </mrow> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </math>. The approach to the former from below signals the relativistic region, whilst the approach to latter from the left denotes the transition from the Newtonian to the dark matter mimicking regime. The labels identify the regions occupied by different astrophysical objects; the solar system, SS, stars, S, wide binaries, WB, globular clusters, GC, dwarf spheroidal galaxies, dSph, elliptical galaxies, E, spiral galaxies, S Gal and galaxy clusters, GaC. Distinct regions of the diagram are labelled; black holes, BH, appearance of relativistic effects, GR, the Newtonian region, N, the modified gravity regime, M, and the critical density of the universe, or the dark energy density, coinciding with the critical point b = 1 where <math display="inline"> <mrow> <msub> <mi>r</mi> <mi>S</mi> </msub> <mo>=</mo> <msub> <mi>r</mi> <mi>M</mi> </msub> </mrow> </math>.</p> Full article ">

Review

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216 KiB  
Review
Conformal Gravity: Dark Matter and Dark Energy
by Robert K. Nesbet
Entropy 2013, 15(1), 162-176; https://doi.org/10.3390/e15010162 - 9 Jan 2013
Cited by 24 | Viewed by 6495
Abstract
This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion and [...] Read more.
This short review examines recent progress in understanding dark matter, dark energy, and galactic halos using theory that departs minimally from standard particle physics and cosmology. Strict conformal symmetry (local Weyl scaling covariance), postulated for all elementary massless fields, retains standard fermion and gauge boson theory but modifies Einstein–Hilbert general relativity and the Higgs scalar field model, with no new physical fields. Subgalactic phenomenology is retained. Without invoking dark matter, conformal gravity and a conformal Higgs model fit empirical data on galactic rotational velocities, galactic halos, and Hubble expansion including dark energy. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
316 KiB  
Review
Conformal Relativity versus Brans–Dicke and Superstring Theories
by David B. Blaschke and Mariusz P. Dąbrowski
Entropy 2012, 14(10), 1978-1996; https://doi.org/10.3390/e14101978 - 18 Oct 2012
Cited by 8 | Viewed by 6303
Abstract
We show how conformal relativity is related to Brans–Dicke theory and to low-energy-effective superstring theory. Conformal relativity or the Hoyle–Narlikar theory is invariant with respect to conformal transformations of the metric. We show that the conformal relativity action is equivalent to the transformed [...] Read more.
We show how conformal relativity is related to Brans–Dicke theory and to low-energy-effective superstring theory. Conformal relativity or the Hoyle–Narlikar theory is invariant with respect to conformal transformations of the metric. We show that the conformal relativity action is equivalent to the transformed Brans–Dicke action for ω = -3/2 (which is the border between standard scalar field and ghost) in contrast to the reduced (graviton-dilaton) low-energy-effective superstring action which corresponds to the Brans–Dicke action with ω = -1. We show that like in ekpyrotic/cyclic models, the transition through the singularity in conformal cosmology in the string frame takes place in the weak coupling regime. We also find interesting self-duality and duality relations for the graviton-dilaton actions. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
267 KiB  
Review
Exact Solutions in Modified Gravity Models
by Andrey N. Makarenko and Valery V. Obukhov
Entropy 2012, 14(7), 1140-1153; https://doi.org/10.3390/e14071140 - 25 Jun 2012
Cited by 6 | Viewed by 5458
Abstract
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper [...] Read more.
We review the exact solutions in modified gravity. It is one of the main problems of mathematical physics for the gravity theory. One can obtain an exact solution if the field equations reduce to a system of ordinary differential equations. In this paper we consider a number of exact solutions obtained by the method of separation of variables. Some applications to Cosmology and BH entropy are briefly mentioned. Full article
(This article belongs to the Special Issue Modified Gravity: From Black Holes Entropy to Current Cosmology)
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