Unication models of dark matter and dark energy

Transcription

1 Unication models of dark matter and dark energy Neven ƒaplar March 14, 2012 Neven ƒaplar () Unication models March 14, / 25

2 Index of topics Some basic cosmology Unication models Chaplygin gas Generalized Chaplygin gas Generalized unied model Comparison with observational data Large Scale Structure Supernova data Hubble parameter data Conclusion Neven ƒaplar () Unication models March 14, / 25

3 Some basic cosmology Big Bang Theory Standard model of cosmology General theory of relativity Quantum eld theory Crucial observations Expansion of the Universe Existence of Cosmic microwave background radiation Ratio of light elements in the Universe Einstein equation Rµν 1 2 g µν R Λgµν = 8πGTµν Neven ƒaplar () Unication models March 14, / 25

4 Some basic cosmology Friedmann equations Isotropy and homogeneity ( ) ds 2 = dt 2 a 2 dr 2 (t) 1 + kr 2 r 2 dθ 2 + r 2 sinθ 2 dφ 2 First Friedmann equation Second Friedmann equation ( ȧ a ) 2 + k = 8πG a 2 3 ρ tot 2ä a + ( ȧ a ) 2 + k a 2 = 8πGp Neven ƒaplar () Unication models March 14, / 25

5 Some basic cosmology Equation of state Relation that connects pressure and density p = α ρ General solution ρ = const a 3(1+α) non relativistic matter α = 0 ρ a 3 radiation α = 1/3 ρ a 4 cosmological constant α = 1 ρ const Neven ƒaplar () Unication models March 14, / 25

6 Some basic cosmology Hubble parameter Hubble parameter H(t) ȧ a Hubble parameter v = H 0 d Hubble diagram with data from ve independent measurements. Lower diagram shows H 0 as a function of distance, while horizontal line shows best t, 72 km sec 1 Mpc 1. Neven ƒaplar () Unication models March 14, / 25

7 Some basic cosmology Denition of Hubble parameter can be written ( H 2 ȧ a ) 2 = 8πG 3 (ρ m + ρ vac ) k a 2 Equation above written with help of critical density Ω M + Ω Λ + Ω K = 1 With denition of cosmological redshift Following expression follows t 0 t 1 = H 1 0 z z a(t obs) a(t emit ) dz (1 + z)[ω m (1 + z) 3 + Ω k (1 + z) 2 + Ω Λ ] 1/2 Neven ƒaplar () Unication models March 14, / 25

8 Some basic cosmology Some cosmological parameters (WMAP 5-year data) CMB temperatura T = ± K Energy density of photons Ω γ0 = h 2 0 Energy density of baryons ± Ω b0 = Energy density of dark matter ± Ω c0 =. h 2 0 h 2 0 Neven ƒaplar () Unication models March 14, / 25

9 Some basic cosmology Deducing cosmological parameters Temperature derived from the WMAP 5-year data. Data are represented as points. All lines are for ΛCDM model with dierent parameters. Blue corresponds to Ωm = 0.27, Ωk = 0. Dark gray is made with Ωm = 0.27, Ωk = 0.1, while light curve corresponds to choice of Ωm = 0.17, Ωk = 0.1 Cosmological parameter bounds, calculated from dierent observations. Lines represent 99.7%, 95.4% and 68.3% condence. Neven ƒaplar () Unication models March 14, / 25

10 Some basic cosmology Age of Universe, t 0 in units of H 1 0 in relation to Ωm0. Full line represents at Universe in which Λ is present, so relation Ωm0 + Ω Λ0 = 1. Dashed line corresponds to Universe without cosmological constant. Full horizontal line bounds minimal age of Universe, as suggested by observations ( 11 Gyr). Age deduced by WMAP-5 measurements is also drawn (h 0 =0.7). Flat universe with Λ is in agreement with WMAP measurement for < Ωm < Neven ƒaplar () Unication models March 14, / 25

11 Unication models Chaplygin gas [1] Equation of state p = A ρ, From Friedman equations ρ = A + B a 6, When expanding ρ A + p A + B 4A a 6, B 4A a 6. Neven ƒaplar () Unication models March 14, / 25

12 Unication models Chaplygin gas Lagrangian density Energy density L = 1 φ 2 V (φ), 2 ρ φ = 1 2 φ 2 + V (φ) = A + B a 6 Uz k = 0, te φ 8πG/3φ V (φ) = A 2 ( cosh3φ + Speed of sound is bound and positive (ω(a) p/ρ) c 2 s p ρ = A ρ 2 = ω ) 1 cosh3φ Neven ƒaplar () Unication models March 14, / 25

13 Unication models Generalized Chaplygin gas Equation of state Energy density Speed of sound ( ρ = p = A ρ α, A + B a 3(α+1) ) 1 α+1 c 2 s p ρ = α ω Neven ƒaplar () Unication models March 14, / 25

14 Unication models Generalized unied model Neven ƒaplar () Unication models March 14, / 25

15 Observational constraints Theoretical basics of perturbation growth Density contrast δ(t, x) ρ(t, x) ρ bg (t), ρ bg (t) where ρ bg is background energy density. Power spectrum P(k, t) = δ k (t) 2 Fluctuation of δ k with wave vector k evolves as δ k + [2 + ξ 3(2ω c s 2 )]δ k [ ( 3 = 2 (1 6c s 2 + 8ω 3ω 2 ) kc s ah ) 2 ] δ k, with d/dlna and ξ (H2 ) 2H 2. Neven ƒaplar () Unication models March 14, / 25

16 Observational constraints Procedure Inital conditions CAMB evolution until z = 100 for δ CAMB evolution until z = 100 for δ Evolution of δ k (a) until today with Mathematica program package χ 2 analysis Sloan Digital Sky Survey 2004 [2] For every choice of (α, γ) we use normalization which gives best t Only linear regime, k < 0.15 h/mpc Neven ƒaplar () Unication models March 14, / 25

17 Observational constraints From top to bottom, thick lines were made for generalized Chaplygin gas model with α = 10 4 (dashed-dotted), α = 10 5 (dotted), α = 10 5 (dashed), α = 10 4 (full line). Three thin lines in middle show evolution of ΛCDM model. From top to bottom they show our simulation, CAMB simulation and WMAP measurement. Compare with gure 1. from [3] Neven ƒaplar () Unication models March 14, / 25

18 Observational constraints Likelihood function for generalized Chaplygin gas. Horizontal line on correspond to the value χ 2 = 1. Allowed values of parameter α < α < Neven ƒaplar () Unication models March 14, / 25

19 Observational constraints Nonrelativistic perturbation equation δ k + 2H δ k + c 2 s k 2 a 2 δ k = 4πG N ρδ k. Small dierences in the area of parametric space allowed by observations Power spectrum of perturbations as function of k. Full and dashed line have been calculated using relativistic equation and correspond to α = 10 4 for full line and α = 10 5 for dotted, respectively. Dashed-dotted and dotted line have been calculated using nonrelativistic equation for α = 10 4 and α = 10 5, respectively. Neven ƒaplar () Unication models March 14, / 25

20 Observational constraints Supernova observations Luminosity distance D L Distance module µ ( ) µ = 5log DL Mpc Supernova Ia data from 1998, used in rst works on the problem of the acceleration of the Universe. This sample has been extensively studied in many cosmological works. Neven ƒaplar () Unication models March 14, / 25

21 Observational constraints Supernova observations [4] Analysis of ΛCDM Universe. Darker region denote the area where agreement with observations is better. Preferred parameters are Ωm 0.3 and h Neven ƒaplar () Unication models March 14, / 25

22 Observational constraints Hubble parameter data Relatively new method with great potential in dark sector Has already been used for parameter estimation in Chaplygin gas models [6] Neven ƒaplar () Unication models March 14, / 25

23 Conclusion Conclusion With adding additional parameter it is possible to solve ne tunning problem Future observations Theoretical motivation Model Speed of sound Allowed parametric space CG -ω - GCG -α ω < α < Neven ƒaplar () Unication models March 14, / 25

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25 Links to articles (Arxiv) Links to articles (Arxiv) (First idea and main article describing Chaplygin gas model) (Data from SDSS 2004 that was analyzed) (Analysis that served as basis for numerical calculations of perturbations) (Data for Supernova analysis) (Data for Hubble parameter analysis) (Analysis of Chaplygin gas model with Hubble technique) Neven ƒaplar () Unication models March 14, / 25