Dark Forces and the ISW Effect Finn Ravndal, University of Oslo Introduction Chaplygin gases Cardassian fluids The integrated Sachs-Wolfe effect Demise of unified dark models Modified gravity: A resurrection? Conclusion *) with T. Koivisto and H. Kurki-Suonio, University of Helsinki, astro-ph/0409163 1
Introduction Flat universe, ds 2 = dt 2 a 2 (t)dx 2 Einstein, E µν = 8πGT µν where E µν = R µν 1 2 g µνr. Bianchi, µ E µν = 0 µ T µν = 0. Friedmann, H 2 = (ȧ a ) 2 = 8π 3 Gρ where energy density ρ = ρ r + ρ b + ρ d with dark component ρ d = ρ m + ρ v Equations of state p = wρ where for matter w m = 0 and vacuum w v = 1. 2
Unified dark energy: ρ d = ρ d (ρ m ) Energy-momentum conservation ρ d + 3H(ρ d + p d ) = 0 of dark component gives pressure of dark fluid ( ) ρd p d = ρ m ρ d ρ m Effective equation of state w d = p d 0, a 0 (early) ρ d 1, a 1 (today) should result. 3
Chaplygin gases Aerodynamics (Chaplygin, Moscow, 1904): p = A ρ Used with cosmological conservation ρ + 3H(ρ + p) = 0 ( ρ = A + B ) 1/2 B/a 3, a 0 a 6 A + B, a 1 Generalized Chaplygin gas (Bento, Bertolami and Sen, gr-qc/0202064): p = A ρ α, 0 α 1 ρ = ( A + B ) 1/(1+α) a 3(1+α) 4
Late times, ρ A 1/(1+α) + 1 1 + α p A 1/(1+α) + α 1 + α B A α/(1+α) 1 a 3(1+α) B 1 A α/(1+α) a 3(1+α) Last parts describe matter with p m = αρ m, i.e. positive pressure! Can made to fit SN Ia data and CMB peaks with 0.2 < α < 0.6 (Bento, Bertolami and Sen, gr-qc/0303538) 5
Cardassian fluids Unified dark energy ρ d = ρ(ρ m ) with ρ m 1/a 3. ρ = ρ m + Bρ 2 m, brane world = ρ m + Bρ p m, p < 1, power = ρ m (1 + Bρ q m ) 1/q, polytropic = ρ m (1 + Bρ qν m ) 1/q, modified polytropic At late times ρ m 0 and last term dominates, simulating dark energy (Kathrine Freese et al, astro-ph/0201229, 0209322). Polytropic gas, i.e. ν = 1 is generalized Chaplygin gas, i.e. with q = 1 + α. Friedmann evolution: (ȧ ρ = (B + ρ q m) 1/q a ) 2 = 8π 3 Gρ ä a = 8π [ 3 G ρ 3 2 ρ m 6 ( ρ ρ m )]
For modified polytropic fluid, late time acceleration ä > 0 when i.e. ν > 1/3. Cardassian pressure p = νbρ qν+1 m (3ν 1)Bρ qν m > 1 (1 + Bρ qν m ) (1 q)/q and equation of state w = p ρ = νbρ qν m 1 + Bρ qν m 0, ρ m ν, ρ m 0 SN Ia and age of universe (Savage, Sugiyama and Freese, astro-ph/0403196): ν 1 000 111 000000000 111111111 0000000000 1111111111 00000000 11111111 00000000 11111111 0000000 1111111 00 11 00000 11111 000000000 111111111 0000 1111 000000000000 111111111111 0000 1111 1/3 1 10 100 q 7
CMB spectrum and ISW effect CMB temperature at point x at conformal time τ observed in direction n, T(τ,x,n) = T(τ)[1 + Θ(τ,x,n)] where Θ(τ,x,n) is fluctuation. Fourier Θ(τ,x,n) = d 3 k (2π) 3 Θ(τ,k,n)eik x and polar expansion Θ(τ,k,n) = ( i) l (2l + 1)Θ l (τ, k)p l (ˆk n) l=0 Observed temperature correlation function C(β) = Θ(τ 0,x,n)Θ(τ 0,x,n ) = 1 (2l + 1)C l P l (cosβ) 4π l=0 where cosβ = n n and power spectrum C l = 4π d 3 k (2π) 3 Θ l(τ 0, k) 2 8
Newtonian or longitudinal gauge, ds 2 = a 2 (τ)[(1 + 2Φ)dτ 2 (1 2Ψ)dx 2 ] Line-of-sight integration gives amplitudes, Θ l (τ 0, k) = Θ SW l where Sachs-Wolfe contribution (τ 0, k) + Θ ISW l (τ 0, k) Θ SW l (τ 0, k) = [Θ 0 (τ, k) + Ψ(τ, k)]j l (kτ 0 kτ ) (with only Θ 0 on LSS) and Integer Sachs-Wolfe effect Θ ISW l (τ 0, k) = τ0 τ dτ e κ(τ) [Φ (τ, k) + Ψ (τ, k)]j l (kτ 0 kτ) 0.4 0.35 0.3 l(l+1)c l /(2π) 0.25 0.2 0.15 0.1 0.05 0 10 1 10 2 10 3 l 9
Adiabatic perturbations in gravitational potential 3H(Ψ + HΦ) + k 2 Ψ = 4πGa 2 δρ with Ψ = dψ/dτ and H = a /a etc. and densities δ δρ/ρ δ = (1 + w)( V + 3Ψ ) + 3H(w c 2 s)δ and velocity potential V = (3w 1)HV w 1 + w V + k2 c 2 s 1 + w δ + k2 Φ. No anisotropic shear stress: Φ = Ψ. 10
Demise of unified models Chaplygin gas with equation of state w = [ 1 + and speed of sound c 2 s = Ω m 1 Ω m a 3(1+α)] 1 ( ) p ρ = αw is negative when α < 0. CDM power spectrum (Sandvik, Tegmark and Zaldarriaga, astr-ph/0212114) 11
Constraints on α: 0.5 H 0 T 0 = 0.79 0.4 95 Ω m 68 0.3 0.2 Allowed region from our analysis (actually 1000 times narrower than this line) H 0 T 0 = 1.27-0.9-0.6-0.3 0 0.3 0.6 0.9 α Fits to background evolution (Makler, de Oliveira and Waga, astro-ph/0209486) 12
6000 5000 4000 3000 2000 1000 0-1000 1 10 10 2 10 3 Amendola, Finelli, Burigana and Carturan, astro-ph/0304325 13
Cardassian fluid (Koivisto, Kurki-Suonio and F.R., astro-ph/0409163) c 2 s = νbρ qν m [(ν 1)Bρ qν m + qν 1] 1 + (2 ν)bρ qν m + (1 ν)b 2 ρ 2qν At late times ρ s 0 and c 2 s ν. Avoided only by taking ν = 1 Chaplygin gas: c 2 s = q 1 1 + ρ q m/b. c 2 s > 0 q > 1 and at late times c 2 s < 1 q < 2 m 14
0.4 0.35 0.3 l(l+1)c l /(2π) 0.25 0.2 0.15 0.1 0.05 0 10 1 10 2 10 3 l 10 3 10 2 10 1 δ K 10 0 10 1 10 2 10 3 10 4 10 4 10 3 10 2 10 1 10 0 a 15
CDM power spectra in Cardassian model (Amarzguioui, Elgaroy and Multamaki, astro-ph/0410408) 10 6 10 5 10 4 P(k) (h 3 Mpc 3 ) 10 3 10 2 10 1 10 0 10 1 n=0.00001, q=1.0 n=0.0001, q=1.0 n= 0.00001, q=1.0 n= 0.0001, q=1.0 ΛCDM 10 2 10 2 10 1 10 0 k (h Mpc 1 ) 10 6 10 5 10 4 P(k) (h 3 Mpc 3 ) 10 3 10 2 10 1 10 0 10 1 n=0.0, q=1.00001 n=0.0, q=1.0001 n=0.0, q=0.99999 n=0.0, q=0.9999 ΛCDM 10 2 10 2 10 1 10 0 k (h Mpc 1 ) 16
Modified gravity: A resurrection? Einstein-Hilbert action S EH = d 4 x ( g 1 ) 2 M2 PR + L m E µν = 1 M 2 P T µν Bianchi: µ E µν = 0 µ T µν = 0. Modified gravity: R f(r) = R + µ 4 /R +... modified Einstein equation: Ê µν = 1 M 2 P Generalized Bianchi identity If now can write T M µν µ Ê µν = 0 µ T M µν = 0 Ê µν = E µν 1 M 2 P T X µν then E µν = 1 M 2 P ( ) Tµν M + Tµν X T C µν Both T M µν and T C µν conserved! No pressure fluctuations in CDM but anisotropic shear: Φ Ψ. 17
0 0.1 0.2 Ψ, Φ 0.3 0.4 0.5 0.6 10 6 10 4 10 2 10 0 a 0.4 0.35 0.3 l(l+1)c l /(2π) 0.25 0.2 0.15 0.1 0.05 0 10 1 10 2 10 3 l 18
Conclusion Both ISW effect and CDM power spectrum rule out Chaplygin gas except in ΛCDM limit α 0. Both ISW effect and CDM power spectrum rule out Cardassian fluid except in ΛCDM limit ν = 1 and q 1. Modified gravity may save the CDM power spectrum but again ruled out by the ISW effect. 19