The Haves and Have-nots of Brane Cosmology Supratik Pal Indian Statistical Institute Kolkata and BCTP, Universität Bonn The Haves Dvali, Arkani-Hamed, Antoniadis, Randall, Sundrum, Gabadedge, Brandenberger, Henry-Tye, Poratti, Hawking, Maartens, Barrow, Wands, Bassett, Rubakov, Clarkson, Seahra, Nilles, Starobinski, Sahni, Shtanov, Sasaki, Koyama, Gregory, Chen, Gergely, Brux, Trodden, Tsujikawa, Cooray, Copeland, Sami, de Rahm, Burgess, Himemoto, Kobayashi, Maeda, Binetruy, Ellwanger, Deffayet, Langlois, Kar, Papantonopoulos, Panotopoulos, Seery, Sakharov, Park, Gogberashvili, Shiromizu, Garriga, Tanaka, Dimopoulos, Csaki, Stojkovic, Perivolaropoulos, van de Bruck, Germani ... 1 The Haves Dvali, Arkani-Hamed, Antoniadis, Randall, Sundrum, Gabadedge, Brandenberger, Henry-Tye, Poratti, Hawking, Maartens, Barrow, Wands, Bassett, Rubakov, Clarkson, Seahra, Nilles, Starobinski, Sahni, Shtanov, Sasaki, Koyama, Gregory, Chen, Gergely, Brux, Trodden, Tsujikawa, Cooray, Copeland, Sami, de Rahm, Burgess, Himemoto, Kobayashi, Maeda, Binetruy, Ellwanger, Deffayet, Langlois, Kar, Papantonopoulos, Panotopoulos, Seery, Sakharov, Park, Gogberashvili, Shiromizu, Garriga, Tanaka, Dimopoulos, Csaki, Stojkovic, Perivolaropoulos, van de Bruck, Germani ... The Have-nots Supratik Pal :) :) :) 1-a A cosmologist’s wishlist To explain... 2 A cosmologist’s wishlist To explain... ...in an unambiguous way ! 2-a A brief history of “adventure” . Inflation −→ Acceleration ↓ Radiation-dominated era ↓ Matter-dominated era ց Deceleration ր ↓ Present era 3 −→ Acceleration A brief history of “adventure” . −→ Acceleration Inflation ↓ Radiation-dominated era ↓ ց Deceleration Matter-dominated era ր ↓ Friedmann equations: ä a Present era
ȧ 2 = 8πG a 3 ρ− −→ Acceleration k a2 = − 4πG 3 (ρ + 3p) Ordinary matter satisfy Strong Energy Condition (ρ + 3p) ≥ 0 Who gives this “anti-gravity” ? 3-a Cosmological Constant
ȧ 2 + Λ 3 = − 4πG 3 (ρ + 3p) + Λ 3 a ä a = 8πG 3 ρ M = 43 πa3 (ρ + 3p) =⇒ ↓ − k a2 ä = − GM a2 + ւ Effective mass Attraction Λ 3a ց Repulsion Λ > 0 guarantees repulsion, hence accelerated expansion Inflationary solution Late time solution ⇓ a(t) ∝ Exp[ q  Λi 3 t] a(t) ∝ sinh( 23 4 ⇓ q Λ0 3 t) 2/3 • Extreme fine-tuning to match present value ρΛ0 ∼ 10−47 GeV 4 . How to generate this from a large inflationary value? • Gives ever-accelerating universe!!! How to produce intermediate decelerating phases (RDE and MDE)? • How to explain formation of matter aka (p)reheating? • How to get correct power spectra and several other aspects of CMB observations? • Signatures of dynamical dark energy? • Is cosmic coincidence a solution or a problem? Mukhanov; Dodelson; Weinberg; Carroll; Turner; Smoot... Several fundamental puzzles confronting modern cosmology ! 5 Dynamical models ä a
ȧ 2 a = = 8πG 3 − 4πG 3 P P i i ρi − k a2 (ρi + 3pi ) pi < −ρi /3 ⇒ wi < −1/3 leads to acceleration ⇐⇒ violates SEC Guth; Linde; Starobinsky; Liddle; Lyth; Sahni... Too many candidates with wide range of initial conditions 6 Dynamical models ä a
ȧ 2 a = = 8πG 3 − 4πG 3 P P i i ρi − k a2 (ρi + 3pi ) pi < −ρi /3 ⇒ wi < −1/3 leads to acceleration ⇐⇒ violates SEC Scalar field models Lagrangian density Lφ = 21 φ̇2 − V (φ) EM tensor components ρφ = 21 φ̇2 + V (φ) ; pφ = 21 φ̇2 − V (φ) ′′ Choose the potential to be sufficiently steep so that V V /V ′ 2 ≥1 Scalar field rolls down the potential : “Tracker potential” Includes Quintessence, Kessence, Chaplygin gas... Guth; Linde; Starobinsky; Liddle; Lyth; Sahni... Too many candidates with wide range of initial conditions 6-a Modified gravity models ∗ Einstein’s theory is not directly tested in cosmological scales. ∗ Modify the gravity sector rather than the matter sector. • Brans-Dicke theory: G as a VEV of a geometric field, need vastly different values for solar system and cosmic scales • Gauss-Bonnet theory: Higher derivative corrections to Einstein’s gravity • f(R) gravity: R in the action replaced by f(R), [e.g. R − µ R ], • Phenomenological models: MOND, MOG, Scalar-Tensor... Brans; Lovelock; Odintsov; Polarski; Tsujikawa; Bekenstein... Tools to discriminate =⇒ Observational constraints 7 Brane cosmology ADD, RS, DGP... depending on bulk, embedding, hierarchy issue... No unique candidate to address all the points 8 Have-Not No.1 Brane cosmology: How to visualize? FRW ⇒ Brane ; 5D Static, spherically symmetric metric ⇒ Bulk ds25 = −f (r)dt2 + 1 2 dr f (r) + r 2 dΩ23 Embedding mechanism ⇒ Induced metric on the brane ds24 = −dτ 2 + r 2 (τ )dΩ23 Identify r(τ ) with the scale factor a(τ ) =⇒ FRW ! Visser, PLB(2000); Sahni, JCAP(2003); Maartens, LRR(2004) SP, PRD(2006),(2006),(2008); Mukherji, SP, MPLA(2010) 9 Brane cosmology: How to visualize? FRW ⇒ Brane ; 5D Static, spherically symmetric metric ⇒ Bulk ds25 = −f (r)dt2 + 1 2 dr f (r) + r 2 dΩ23 Embedding mechanism ⇒ Induced metric on the brane ds24 = −dτ 2 + r 2 (τ )dΩ23 Identify r(τ ) with the scale factor a(τ ) =⇒ FRW ! Expanding 4D universe ≡ Moving brane in the bulk ⇑ ⇑ Brane-based observer Bulk-based observer Accelerated universe is the manifestation of geodesic motion Visser, PLB(2000); Sahni, JCAP(2003); Maartens, LRR(2004) SP, PRD(2006),(2006),(2008); Mukherji, SP, MPLA(2010) 9-a RS-type brane: effective field equations Gµν = −Λgµν + κ24 Tµν + κ45 Sµν − Eµν + Fµν | {z } | {z } |{z} |{z} ⇓ 4D GR ⇓ ⇓ ⇓ Quadratic Weyl Bulk Tµν term matter Langlois, PRL(2002); Maartens, LRR(2004) 10 RS-type brane: effective field equations Gµν = −Λgµν + κ24 Tµν + κ45 Sµν − Eµν + Fµν | {z } | {z } |{z} |{z} ⇓ ⇓ 4D GR ⇓ ⇓ Quadratic Weyl Bulk Tµν term matter Friedmann equations H2 = Ḣ = κ24 −2 κ24 3 h ρ+ ρ2 2λb i + ρ∗ + h  (ρ + p) 1 + ρ λb  + Λ 3 − 4 ∗ 3ρ k a2 i + k a2 Langlois, PRL(2002); Maartens, LRR(2004) 10-a Quadratic term Early time: ρ2 ≫ λb > (100GeV )4 results in non-standard evolution during inflation Maartens, PRD(2001); PRL(2001); Sasaki, PRD(2001) 11 Quadratic term Early time: ρ2 ≫ λb > (100GeV )4 results in non-standard evolution during inflation Maartens, PRD(2001); PRL(2001); Sasaki, PRD(2001) Weyl term For empty bulk: ρ∗ = C a4 =⇒ dark radiation ≤ .03% of radiation density (Nucleosynthesis data) For radiative bulk: ρ∗ = C(τ ) a4 may be significant at late time as well Barrow, PLB(2002); Langlois, PRL(2002); SP, PRD(2006),(2006),(2008) 11-a Brane inflation: governing equations  8π 2 H = 3M 2 V 1 + V 2λ P  φ̈ + 3H φ̇ + V ′ (φ) = 0 Slow roll parameters  ′ 2 2 1+ Vλ MP V ǫV = 16π V ≪ 1 ; ηV = (1+ V )2 2λ ξV = 4 MP (8π)2  ′ ′′′ V V V2  1 V 2 (1+ 2λ ) 2 MP 8π  ′′ V V ′ ; σV =  6 MP (V )2 V (8π)3 V3 1 V (1+ 2λ ) ′′′′ Number of e-foldings N= 8π 2 MP R φi  V  φf V′ 1+ 12 V 2λ
dφ ≈ 56 − 70 ≪1 1 V 3 (1+ 2λ ) Chaotic inflation on the brane N≃ 2π M42 V = 21 m2 φ2
π2 m2 4
2 2 4 φi − φf + 3M 6 φi − φf 5 ρ ≫ λ ≪ 1016 GeV ⇒ M5 < 1017 GeV ⇒
M5 2 5 4 φend ≃ 4π2 m M54 NCOBE ≈ 55 ⇒ m ≈ 5 × 10−5 M5 , φcobe ≈ 3 × 102 M5 φcobe < M4 ⇒ η-problem softened by quadratic term Have No.1 Maartens(2000), Sasaki(2001), (2002), Bertolami(2003) Likewise calculate observable quantities 13 will come to this point But... The 4-D field equations are now “derived field equations”. Effective inflaton potential has to be generated from bulk... Have-not No.2 Parameter estimation.... Have-not No.3 Effects on reheating phenomenology, leptogenesis.... Have-not No.4 14 Brane inflation from bulk SUGRA: Schematics Choudhury, SP, PRD(2012), NPB(2012)  R 4 R +πR √  3 P 1 5 S = 2 d x −πR dy g5 M5 (R5 − 2Λ5 ) + LSU GRA + i δ(y − yi )L4i L5SU GRA = ∗ N = 2, D = 5 SUGRA e5 5 I − R2 + 2i Ψ̄im̃ Γm̃ñq̃ ∇ñ Ψiq̃ −SIJ Fm̃ñ F I m̃ñ − 12 gαβ Dm̃ φµ D m̃ φν +F +CS ∗ Radion fields: χ = −ψ52 ; q   T = √12 e5̇5 − i 23 A05 ∗ Kähler function: H(G) =  † 
∂W ∂G W ∂W exp MG2 (Gnm )−1 ∂φ n + ∂φm + ∂φm M 2 ∗ Z2 symmetry ∗ Compactification around a circle S 1 15 ∂G W ∂φn M 2  2 | − 3 |W 2 M  ∗ Dimensional reduction ds25 2A(y) =e ds24 2 2 + R β dy ∗ VD = 0 ⇔ U (1) gauge interaction is absent   1 P † α  P ∂W V = VF = exp M 2 α φα φ β ∂φβ 2 2
2 | − 3 |W 2 M  N = 2, D = 5 bulk SUGRA ⇒ N = 1, D = 4 brane SUGRA 16 ∗ Dimensional reduction ds25 2A(y) =e ds24 2 2 + R β dy ∗ VD = 0 ⇔ U (1) gauge interaction is absent   1 P † α  P ∂W V = VF = exp M 2 α φα φ β ∂φβ 2 2
2 | − 3 |W 2 M  N = 2, D = 5 bulk SUGRA ⇒ N = 1, D = 4 brane SUGRA ∗ Include one-loop corrections ∗ Choose M5 ≤ 1017 GeV (softening η-problem ⇒ fine-tuning!!)      4  φ φ V (φ) = V0 1 + D4 + K4 ln M M Coleman Weinberg potential 16-a ∗ Dimensional reduction ds25 2A(y) =e ds24 2 2 + R β dy ∗ VD = 0 ⇔ U (1) gauge interaction is absent   1 P † α  P ∂W V = VF = exp M 2 α φα φ β ∂φβ 2 2
2 | − 3 |W 2 M  N = 2, D = 5 bulk SUGRA ⇒ N = 1, D = 4 brane SUGRA ∗ Include one-loop corrections ∗ Choose M5 ≤ 1017 GeV (softening η-problem ⇒ fine-tuning!!)      4  φ φ V (φ) = V0 1 + D4 + K4 ln M M N= 2 M U h 1 2 1+ 
α 2 1 φ2f Coleman Weinberg potential i  2 αD D − φ12 + 2M44 (1 + α)(φ2i − φ2f ) + 12M48 (φ6i − φ6f ) i 1/4 V0 ≃ 2 × 1015 GeV 16-b Have No.2 Cosmological perturbations Quantum fluctuations of inflaton are transformed to macroscopic cosmological perturbations Perturbations in the metric ⇓ δGµν = 8πGδTµν ւ Scalar perturbation ↓ Vector perturbation ց Tensor perturbation ⇓ ⇓ Matter Gravitational waves 17 Observable quantities 18 Observable quantities • Amplitude of scalar perturbation i h 3   3 512π V ∆2s ≃ 75M 6 (VV′ )2 1 + 2λ P k=aH 18-a ⇒ ∆2s |k=aH ∼ 2 × 10−9 Observable quantities • Amplitude of scalar perturbation i h 3   3 512π V ∆2s ≃ 75M 6 (VV′ )2 1 + 2λ k=aH P ⇒ ∆2s |k=aH ∼ 2 × 10−9 • Amplitude of tensor perturbation  ∆2t = 32 4 75MP q   V V [1+ 2λ ] V 2V V −1 1+ 2V λ (1+ 2λ )− λ (1+ 2λ ) sinh 18-b 1 r 2V λ V (1+ 2λ )   k=aH Observable quantities • Amplitude of scalar perturbation i h 3   3 512π V ∆2s ≃ 75M 6 (VV′ )2 1 + 2λ k=aH P ⇒ ∆2s |k=aH ∼ 2 × 10−9 • Amplitude of tensor perturbation  ∆2t = 32 4 75MP q   V V [1+ 2λ ] V 2V V −1 1+ 2V λ (1+ 2λ )− λ (1+ 2λ ) sinh 1 r 2V λ V (1+ 2λ )   k=aH • Ratio of tensor to scalar amplitudes r= ∆2t 16 ∆2 s < 0.36 : WMAP7 ∼ 10−2 : what PLANCK can probe 18-c • Scalar spectral index ns − 1 = d(ln(∆2s )) d(ln(k)) ≃ 2ηV⋆ − 6ǫ⋆V 19 ⇒ 0.948 < ns < 1 • Scalar spectral index ns − 1 = d(ln(∆2s )) d(ln(k)) ≃ 2ηV⋆ − 6ǫ⋆V ⇒ 0.948 < ns < 1 • Tensor spectral index nt = d(ln(∆2t )) d(ln(k)) 19-a ≃ −3ǫ⋆V • Scalar spectral index ns − 1 = d(ln(∆2s )) d(ln(k)) ≃ 2ηV⋆ − 6ǫ⋆V ⇒ 0.948 < ns < 1 • Tensor spectral index nt = d(ln(∆2t )) d(ln(k)) ≃ −3ǫ⋆V • Running of scalar spectral index αs = dns |k=aH = 16ηǫ − 18ǫ2 − 2ξ d ln k 19-b ∼ −10−3 • Scalar spectral index ns − 1 = d(ln(∆2s )) d(ln(k)) ≃ 2ηV⋆ − 6ǫ⋆V ⇒ 0.948 < ns < 1 • Tensor spectral index nt = d(ln(∆2t )) d(ln(k)) ≃ −3ǫ⋆V • Running of scalar spectral index αs = dns |k=aH = 16ηǫ − 18ǫ2 − 2ξ d ln k • Running of tensor spectral index dnt |k=aH = 6ǫη − 9ǫ2 αt = d ln k 19-c ∼ −10−3 ln(∆s ) versus ln(|αs |) - 6.4 lnH Ds ¤L - 6.6 - 6.8 - 7.0 - 7.2 - 10.3 - 10.2 - 10.1 - 10.0 - 9.9 lnH Αs ¤L 20 - 9.8 - 9.7 - 9.6 C4 ∆2s ∆2t ≃ D4 ×10−9 ×10−14 nt r αs αt ×10−5 ×10−5 ×10−3 ×10−6 0.951 -4.352 2.176 -0.798 -2.125 0.941 -7.412 3.706 -1.142 -4.323 1.440 0.936 -9.447 4.723 -1.345 -5.975 2.902 0.951 -4.352 2.176 -0.798 -2.125 0.941 -7.412 3.706 -1.142 -4.323 1.327 0.936 -9.447 4.723 -1.345 -5.975 2.679 0.951 -4.352 2.176 -0.798 -2.125 0.941 -7.412 3.706 -1.142 -4.323 0.936 -9.447 4.723 -1.345 -5.975 3.126 -0.70 -0.65 -0.60 1.835 1.704 1.573 1.234 6.803 6.317 5.831 ns 21 Parameter estimation with CAMB Input parameters in CAMB H0 τReion Ωb h2 Ωc h2 TCM B km/sec/MPc 71.0 K 0.09 0.0226 0.1119 2.725 Output parameters from CAMB t0 zReion Ωm ΩΛ Ωk Gyr 13.707 10.704 0.2670 0.7329 22 0.0 ηRec η0 Mpc Mpc 285.10 14345.1 CMB TT Angular Power Spectrum 6000 Best fit WMAP data set 5000 2 l(l+1)Cl/2π[µK ] 4000 3000 2000 1000 0 1 10 100 1000 l 1st peak at l ≈ 241 confirms Ωk ≈ 0 2nd and 3rd peaks at l ≈ 533, 791 confirm adiabatic perturbation Peak positions and heights confirm Ωb ≈ 0.04, ΩM ≈ 0.3, ΩDE ≈ 0.7 Have No.3 23 CMB TE Angular Power Spectrum CMB EE Angular Power Spectrum 150 50 Best fit WMAP data set 40 50 30 l(l+1)Cl/2π[µK2] 100 0 20 -50 10 -100 0 -150 -10 0 100 200 300 400 500 600 700 800 0 200 400 l 600 l CMB TT Angular Power Spectrum 700 Best fit 600 500 l(l+1)Cl/2π[µK2] l(l+1)Cl/2π[µK2] Best fit WMAP data set 400 300 200 100 0 1 10 100 l 24 1000 800 1000 Late time evolutions Quadratic term Late time: ρ2 ≪ λb > (100GeV )4 ⇒ negligible Weyl term For empty bulk: ρ∗ = C a4 =⇒ dark radiation ≤ .03% of radiation density (Nucleosynthesis data) ⇒ negligible Results in standard evolution at late time How to get dark energy? Have-not No.5 25 How to get dark energy? ⇓ Choose different bulk and/or embedding ւ RS, radiative bulk ↓ Generalized dynamics ⇓ ρ∗ = ց DGP braneworlds ⇓ C(τ ) a4 Modifications at large scale (SP, Maartens, Langlois...) (Maartens, Koyama, Chen...) ⇓ New Avatar Galileon (Burgess, Brux, de Rahm, Tsujikawa, Troddeen, Copeland...) 26 Dark energy from generalized bulk dynamics Das, Ghosh, van Holten, SP, JHEP(2009) The general bulk action h R 1 gµν ẋµ ẋν − S = m dτ 2e e 2 µ ν − λgµν ξ ẋ + eλ2 µ ν g ξ ξ µν 2 + eβλ2 2 i The action has been derived by Kaluza-Klein decomposition 27 Dark energy from generalized bulk dynamics Das, Ghosh, van Holten, SP, JHEP(2009) The general bulk action h R 1 gµν ẋµ ẋν − S = m dτ 2e e 2 µ ν − λgµν ξ ẋ + eλ2 µ ν g ξ ξ µν 2 + eβλ2 2 i τ = worldline evolution parameter ξ µ = Killing vectors associated with symmetry of spacetime e(τ ) = worldline einbein to maintain reparametrization-invariance λ(τ ) = auxiliary worldline scalar variable β = a nonzero parameter of the theory The action has been derived by Kaluza-Klein decomposition 27-a Study geodesics in the background
2 2M 2 2 ds5 = − k − r2 + Λ5 r dt + dr 2 +Λ5 r 2 k− 2M r2 + r 2 dΩ23 Using the Killing vectors, radial geodesics look ṙ 2 + Veff (r) = ε2 Effective potential : Veff = k − 2M r2 28 + Λ5 r 2
 1+ l2 r2 + l2 β  Study geodesics in the background
2 2M 2 2 ds5 = − k − r2 + Λ5 r dt + dr 2 +Λ5 r 2 k− 2M r2 + r 2 dΩ23 Using the Killing vectors, radial geodesics look ṙ 2 + Veff (r) = ε2 Effective potential : Veff = k − 2M r2 + Λ5 r 2 8000
 1+ l2 r2 + l2 β  6000 Β=10 Veff 4000 Β=100 2000 Β=100000 Β=-50 0 -2000 -4000 0 2 4 6 8 10 12 14 r Repulsive force is generated for β < 0 ⇒ Dark Energy? 28-a Brane Friedmann equations
ȧ 2 a ä a = = + 2M a4 h + α − Λ5 + l2 (2M/a2 −Λ5 a2 ) β(a2 +l2 ) + 3p) − 2M a4 h + α − Λ5 − l2 (Λ5 a4 +2M +2Λ5 l2 a2 ) β(a2 +l2 )2 8πG4 3 ρ 4 − 4πG 3 (ρ i i Research Highlight in Nature(2009) 29 Brane Friedmann equations
ȧ 2 a ä a = = + 2M a4 h + α − Λ5 + l2 (2M/a2 −Λ5 a2 ) β(a2 +l2 ) + 3p) − 2M a4 h + α − Λ5 − l2 (Λ5 a4 +2M +2Λ5 l2 a2 ) β(a2 +l2 )2 8πG4 3 ρ 4 − 4πG 3 (ρ i i Approximate solution h √ i √ a(t) ≈ M + 12 et/ −β + M 2 e−t/ −β =⇒ Accelerating solution ! Behaves pretty close to ΛCDM, with β −1 ≈ Λ Research Highlight in Nature(2009) 29-a Observable quantities Stringent constraints on dark energy models 30 Observable quantities • Luminosity distance dL (z) = c(1 + z) Rz dz ′ 0 H(z ′ ) ⇒ ΩDE ≈ 0.7 Stringent constraints on dark energy models 30-a Observable quantities Rz ′ dz • Luminosity distance dL (z) = c(1 + z) 0 H(z ′ ) ⇒ ΩDE ≈ 0.7 R∞ ′ • Age of the universe t(z) = z (1+zdz ′ )H(z ′ ) ≈ 13.7 Bn Yrs Stringent constraints on dark energy models 30-b Observable quantities Rz ′ dz • Luminosity distance dL (z) = c(1 + z) 0 H(z ′ ) ⇒ ΩDE ≈ 0.7 R∞ ′ • Age of the universe t(z) = z (1+zdz ′ )H(z ′ ) ≈ 13.7 Bn Yrs • Deceleration parameter q(z) = −ä/a (ȧ/a)2 = H ′ (z) H(z) (1 + z) − 1 < 0 Onset of recent acceleration z ≈ 0.6 Stringent constraints on dark energy models 30-c Observable quantities Rz ′ dz • Luminosity distance dL (z) = c(1 + z) 0 H(z ′ ) ⇒ ΩDE ≈ 0.7 R∞ ′ • Age of the universe t(z) = z (1+zdz ′ )H(z ′ ) ≈ 13.7 Bn Yrs • Deceleration parameter q(z) = −ä/a (ȧ/a)2 = H ′ (z) H(z) (1 + z) − 1 < 0 Onset of recent acceleration z ≈ 0.6 • Effective equation of state w(z) = 2q(z)−1 3[1−Ωm (z)] < −1 at 2σ Stringent constraints on dark energy models 30-d Observable quantities Rz ′ dz • Luminosity distance dL (z) = c(1 + z) 0 H(z ′ ) ⇒ ΩDE ≈ 0.7 R∞ ′ • Age of the universe t(z) = z (1+zdz ′ )H(z ′ ) ≈ 13.7 Bn Yrs • Deceleration parameter q(z) = −ä/a (ȧ/a)2 = H ′ (z) H(z) (1 + z) − 1 < 0 Onset of recent acceleration z ≈ 0.6 • Effective equation of state w(z) = 2q(z)−1 3[1−Ωm (z)] < −1 at 2σ • Statefinder parameters {r, s} ⇒ dynamical dark energy vs Λ     2 ... a /a H ′′ H′ H′ 2 r = (ȧ/a)3 = 1 + H + H (1 + z) − 2 H (1 + z) s= r−1 3(q−1/2) Stringent constraints on dark energy models 30-e Observable quantities Rz ′ dz • Luminosity distance dL (z) = c(1 + z) 0 H(z ′ ) ⇒ ΩDE ≈ 0.7 R∞ ′ • Age of the universe t(z) = z (1+zdz ′ )H(z ′ ) ≈ 13.7 Bn Yrs • Deceleration parameter q(z) = −ä/a (ȧ/a)2 = H ′ (z) H(z) (1 + z) − 1 < 0 Onset of recent acceleration z ≈ 0.6 • Effective equation of state w(z) = 2q(z)−1 3[1−Ωm (z)] < −1 at 2σ • Statefinder parameters {r, s} ⇒ dynamical dark energy vs Λ     2 ... a /a H ′′ H′ H′ 2 r = (ȧ/a)3 = 1 + H + H (1 + z) − 2 H (1 + z) s= r−1 3(q−1/2) • Averaging over entire redshift Om(z) and q̄(z) ⇒ test for Λ Stringent constraints on dark energy models 30-f Observational aspects of generalized dynamics Das, Ghosh, van Holten, SP, IJMPD(2011) Express Friedmann equations in terms of redshift s.t. a ∝ Neglect all terms ≥ (1 + z)4 Hubble parameter boils down to  
H 2 = H02 ΩX 1 + b(1 + z)2 + ΩM (1 + z)3 |{z} |{z} Dark Energy Matter Sector 31 1 1+z Observational aspects of generalized dynamics Das, Ghosh, van Holten, SP, IJMPD(2011) Express Friedmann equations in terms of redshift s.t. a ∝ 1 1+z Neglect all terms ≥ (1 + z)4 Hubble parameter boils down to  
H 2 = H02 ΩX 1 + b(1 + z)2 + ΩM (1 + z)3 |{z} |{z} Dark Energy Matter Sector • H0 = 74.2 ± 3.6 km/s/Mpc : WMAP5/SHOES • ΩM = 8πG4 ρ/3H02 = 0.28 ± 0.08, 95% CL ? : CMB/LSST • ΩX = (α − Λ5 µ2 )/H02 = 0.726 ± 0.015, 95% CL ? : WMAP5/SNIa • b = Λ5 β(µ2 − 1)2 /(α − Λ5 µ2 ) = ? 31-a Luminosity distance: SNIa: ΩX ≈ 0.7; CMB+LSST: ΩM = 0.3 R z dz ′ dL (z) = (1 + z) 0 H(z ′ ) (1+z) R z dz ′ √ = H0 0 ′ 2 ′ 3 ΩX (1+b(1+z ) )+ΩM (1+z ) 4 W X =0.4 dL 3 W X =1 W X =0.7 W X =0 2 1 0 0.0 0.5 1.0 1.5 2.0 z Observational data from all of the SNIa fall on green line. ΩM = 0.3 ; ΩX = 0.7 ; −0.07 ≤ b < 0 Matches observations for Dark Energy and Matter density 32 Age of the universe: Latest accepted value 13.7 ± 0.02 Gyr Correct Dark Energy density results in correct calculation of age R∞ dz ′ t(z) = z (1+z ′ )H(z ′ ) R∞ 1 dz ′ √ = H0 z ′ ′ 2 ′ 3 (1+z ) ΩX (1+b(1+z ) )+ΩM (1+z ) 0.70 0.69 W X =0.95 Ht W X =0.7 0.68 0.67 W X =0.35 W X =0 0.66 0 2 4 6 8 z ΩX = 0.7 matches the plot obtained from observational data 33 Deceleration parameter: q < 0, onset of acceleration z = 0.6 q(z) = = −ä/a ȧ2 /a2 = H ′ (z) H(z) (1 + z) − 1 ΩM (1+z)3 −2ΩX 2[ΩX (1+b(1+z)2 )+ΩM (1+z)3 ] 0.4 0.2 q 0.0 -0.2 -0.4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 z Plot for ΩM = 0.3 ; ΩX = 0.7 ; b = −0.05 ∗ q < 0 at present ∗ Onset of recent acceleration z ≈ 0.6 confirmed 34 Equation of state: SNIa Gold dataset: −1.11 < wX < −1 at 2σ wX (z) = 2q(z)−1 3[1−ΩM (z)] ≈ −1 + W X =0.95 -1.00 W X =0.7 -1.02 w 2b(1+z)2 3 W X =0.4 -1.04 -1.06 W X =0.2 -1.08 -1.10 0.0 0.2 0.4 0.6 0.8 1.0 z ∗ Shows phantom behavior without any phantom field −1.11 < wX < −1 ⇒ −0.15 < b < 0 ; dL ⇒ −0.07 ≤ b < 0 ∗ Will fit well with more precise observational data too 35 Statefinder parameters: Dynamical models vs Λ     ... 2 a /a H′ H ′′ H′ 2 r = (ȧ/a)3 = 1 + H + H (1 + z) − 2 H (1 + z) ; s = ; s= 2 2 b(1+z) 3 [3+b(1+z)2 ] 1.10 W X =0.85 1.08 1.06 W X =0.7 r r =1− bΩX (1+z)2 ΩX (1+b(1+z)2 )+ΩM (1+z)3 2 r−1 3 2q−1 1.04 W X =0.5 1.02 W X =0.1 1.00 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 s Dynamical model assured Have No.4 36 A cosmologist’s checklist • Inflation? Yes • SUGRA origin? Yes • String 7→ Brane cosmology? No • Reheating and leptogenesis? Yes (?) • Parameter estimation with CAMB? Yes • Detailed CMB physics with Monte-Carlo? No • Dark energy from RS? Not possible • Dark energy from modified bulk? Yes • Maximum likelihood analysis? No • Dark matter? Not possible (?) • Effects on reionization? No 37 38