8π. 3. Gρ where energy density ρ = ρr + ρb + ρd with dark component ρd = ρm +
ρv. Equations of state p = wρ where for matter wm = 0 and vacuum wv = −1. 2 ...
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, ds2 = dt2 − a2 (t)dx2
Einstein, Eµν = 8πGTµν where Eµν = Rµν − 12 gµν R. Bianchi, ∇µ Eµν = 0
∇µ Tµν = 0.
⇒
Friedmann, 2 a˙ 8π 2 H = = Gρ a 3 where energy density ρ = ρr + ρb + ρd with dark component ρd = ρm + ρv Equations of state p = wρ where for matter wm = 0 and vacuum wv = −1.
2
Unified dark energy: ρd = ρd (ρm ) Energy-momentum conservation ρ˙ d + 3H(ρd + pd ) = 0 of dark component gives pressure of dark fluid ∂ρd − ρd pd = ρm ∂ρm Effective equation of state 0, a → 0 (early) pd → wd = −1, a → 1 (today) ρd should result.
3
Chaplygin gases Aerodynamics (Chaplygin, Moscow, p=−
1904):
A ρ
Used with cosmological conservation ρ˙ + 3H(ρ + p) = 0 √ 1/2 B/a3 , a → 0 B → ⇒ρ= A+ 6 A + B, a → 1 a
Generalized Chaplygin gas (Bento, gr-qc/0202064): p=−
A , α ρ
⇒ρ= A+
Bertolami and Sen,
0≤α≤1 B a3(1+α)
4
1/(1+α)
Late times, 1/(1+α)
ρ→A
B 1 1 + 1 + α Aα/(1+α) a3(1+α)
p → −A1/(1+α) +
1 α B 1 + α Aα/(1+α) a3(1+α)
Last parts describe ’matter’ with pm = αρm , 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 ρd = ρ(ρm ) with ρm ∝ 1/a3 . ρ = ρm + Bρ2m ,
brane world
= ρm + Bρpm , p < 1,
power
1/q = ρm (1 + Bρ−q ) , m
polytropic
1/q = ρm (1 + Bρ−qν , m )
modified polytropic
At late times ρm → 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, 1/q
ρ = (B + ρqm ) i.e. with q = 1 + α.
Friedmann evolution: 2 8π a˙ = Gρ a 3
8π 3 a ¨ G ρ − ρm ⇒ = a 3 2 6
∂ρ ∂ρm
For modified polytropic fluid, late time acceleration a ¨ > 0 when (3ν − 1)Bρ−qν >1 m i.e. ν > 1/3. Cardassian pressure (1−q)/q p = −νBρ−qν+1 (1 + Bρ−qν m m )
and equation of state
0, −νBρ−qν p m w= = −qν → ρ 1 + Bρm −ν,
SN Ia and age of universe (Savage, astro-ph/0403196):
ρm → ∞
ρm → 0
Sugiyama and Freese,
ν 1
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1/3
1
10
7
100
q
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 Z 3 d k ik·x Θ(τ, x, n) = Θ(τ, k, n)e (2π)3 and polar expansion ∞ X ˆ · n) Θ(τ, k, n) = (−i)ℓ (2ℓ + 1)Θℓ (τ, k)Pℓ (k ℓ=0
Observed temperature correlation function C(β)
= h Θ(τ0 , x, n)Θ(τ0 , x, n′ ) i ∞ 1 X (2ℓ + 1)Cℓ Pℓ (cos β) = 4π ℓ=0
where cos β = n · n′ and power spectrum Z 3 d k 2 Cℓ = 4π h |Θ (τ , k)| i ℓ 0 3 (2π)
8
Newtonian or longitudinal gauge, ds2 = a2 (τ )[(1 + 2Φ)dτ 2 − (1 − 2Ψ)dx2 ] Line-of-sight integration gives amplitudes, Θℓ (τ0 , k) = ΘSW (τ0 , k) + ΘISW (τ0 , k) ℓ ℓ where Sachs-Wolfe contribution ΘSW (τ0 , k) = [Θ0 (τ∗ , k) + Ψ(τ∗ , k)]jℓ (kτ0 − kτ∗ ) ℓ (with only Θ0 on LSS) and Integer Sachs-Wolfe effect Z τ0 dτ e−κ(τ ) [Φ′ (τ, k) + Ψ′ (τ, k)]jℓ (kτ0 − kτ ) ΘISW (τ0 , k) = ℓ τ∗
0.4 0.35
0.25
l
l(l+1)C /(2π)
0.3
0.2 0.15 0.1 0.05 0
1
2
10
10
l 9
3
10
Adiabatic perturbations in gravitational potential 3H(Ψ′ + HΦ) + k 2 Ψ = −4πGa2 δρ with Ψ′ = dΨ/dτ and H = a′ /a etc. and densities δ ≡ δρ/ρ δ ′ = (1 + w)(−V + 3Ψ′ ) + 3H(w − c2s )δ and velocity potential w′ k 2 c2s V = (3w − 1)HV − V + δ + k 2 Φ. 1+w 1+w ′
No anisotropic shear stress: Φ = Ψ.
10
Demise of unified models Chaplygin gas with equation of state
Ωm −3(1+α) −1 w =− 1+ a 1 − Ωm and speed of sound c2s =
∂p ∂ρ
= −αw
is negative when α < 0. CDM power spectrum (Sandvik, Tegmark astr-ph/0212114)
11
and Zaldarriaga,
Constraints on α:
0.5 H 0 T 0 = 0.79 95
Ω∗ m
0.4
68 Allowed region from our analysis (actually 1000 times narrower than this line)
0.3
0.2
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, Waga, astro-ph/0209486)
12
de Oliveira and
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, astro-ph/0409163)
Kurki-Suonio and F.R.,
−qν −qν νBρ [(ν − 1)Bρ m m + qν − 1] 2 cs = 2 ρ−2qν 1 + (2 − ν)Bρqν + (1 − ν)B m m
At late times ρs → 0 and c2s → −ν. Avoided only by taking ν = 1 ⇒ Chaplygin gas: ⇒ c2s =
q−1 . q 1 + ρm /B
c2s > 0 ⇒ q > 1 and at late times c2s < 1 ⇒ q < 2
14
0.4 0.35
0.25
l
l(l+1)C /(2π)
0.3
0.2 0.15 0.1 0.05 0
1
2
10
3
10
10
l
3
10
2
10
1
10
δ
0
10
K −1
10
−2
10
−3
10
−4
10
−4
10
−3
10
−2
10
a
15
−1
10
0
10
CDM power spectra in Cardassian model (Amarzguioui, Elgaroy and Multamaki, astro-ph/0410408) 6
10
5
10
4
10
P(k) (h−3 Mpc3)
3
10
2
10
1
10
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
0
10
−1
10
−2
10
−2
−1
10
10 −1 k (h Mpc )
0
10
6
10
5
10
4
10
P(k) (h−3 Mpc3)
3
10
2
10
1
10
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
0
10
−1
10
−2
10
−2
10
−1
10 k (h Mpc−1)
16
0
10
Modified gravity: A resurrection? Einstein-Hilbert action Z 1 4 √ 2 SEH = d x −g − MP R + Lm 2 ⇒ Eµν = Bianchi: ∇µ Eµν = 0
1 Tµν 2 MP ∇µ Tµν = 0.
⇒
Modified gravity: R → f (R) = R + µ4 /R + . . . ⇒ modified Einstein equation: 1 M b Eµν = 2 Tµν MP
Generalized Bianchi identity bµν = 0 ∇µ E
M ∇µ Tµν =0
⇒
If now can write
X bµν = Eµν − 1 Tµν E MP2
then Eµν
1 M C X = 2 Tµν + Tµν ≡ Tµν MP
M C Both Tµν and Tµν conserved! No pressure fluctuations in CDM but anisotropic shear: Φ 6= Ψ. 17
0 −0.1
Ψ, Φ
−0.2 −0.3 −0.4 −0.5 −0.6
−6
−4
10
−2
10
10
0
10
a
0.4 0.35
0.25
l
l(l+1)C /(2π)
0.3
0.2 0.15 0.1 0.05 0
1
2
10
10
l
18
3
10
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.
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