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
ȧ
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
ρ̇d + 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 ρ̇ + 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
ȧ
=
Gρ
a
3
8π
3
ä
G ρ − ρm
⇒ =
a
3
2
6
∂ρ
∂ρm
For modified polytropic fluid, late time acceleration
ä > 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
Θ(τ, k, n) =
(−i)ℓ (2ℓ + 1)Θℓ (τ, k)Pℓ (k̂ · n)
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.
19