Second Order Gravitational Effects on CMB Temperature Anisotropy in Lambda dominated flat universes

Abstract

We study second order gravitational effects of local inhomogeneities on the cosmic microwave background radiation in flat universes with matter and a cosmological constant $\Lambda$. We find that the general relativistic correction to the Newtonian approximation is negligible at second order provided that the size of the inhomogeneous region is sufficiently smaller than the horizon scale. For a spherically symmetric top-hat type quasi-linear perturbation, the first order temperature fluctuation corresponding to the linear integrated Sachs-Wolfe (ISW) effect is enhanced(suppressed) by the second order one for a compensated void(lump). As a function of redshift of the local inhomogeneity, the second order temperature fluctuations due to evolution of the gravitational potential have a peak before the matter-$\Lambda$ equality epoch for a fixed comoving size and a density contrast. The second order gravitational effects from local quasi-linear inhomogeneities at a redshift z~1 may significantly affect the cosmic microwave background.

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arXiv:0712.1291v2 [astro-ph] 15 May 2008
Second order gravitational effects on CMB temperature anisotropy
in Λ dominated flat universes
Kenji Tomita
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Kaiki Taro Inoue
Department of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502, Japan
(Dated: May 15, 2008)
We study second order gravitational effects of local inhomogeneities on the cosmic microwave
background radiation in flat universes with matter and a cosmological constant Λ. We find that
the general relativistic correction to the Newtonian approximation is negligible at second order
provided that the size of the inhomogeneous region is sufficiently smaller than the horizon scale.
For a spherically symmetric top-hat type quasi-linear perturbation, the first order temperature
fluctuation corresponding to the linear integrated Sachs-Wolfe (ISW) effect is enhanced(suppressed)
by the second order one for a compensated void(lump).
inhomogeneity, the second order temperature fluctuations due to evolution of the gravitational
potential have a peak before the matter-Λ equality epoch for a fixed comoving size and a density
contrast. The second order gravitational effects from local quasi-linear inhomogeneities at a redshift
z ∼ 1 may significantly affect the cosmic microwave background.
As a function of redshift of the local
PACS numbers: 98.80.-k, 98.70.Vc, 04.25.Nx
I.INTRODUCTION
Generation of temperature anisotropy in the Cosmic Microwave Background (CMB) due to linear pertubatons
of gravitational potentials is called the Sachs-Wolfe (SW) effect[1]. There are two contribution to the SW effect.
The first one called the non-integrated Sachs-Wolfe effect is produced by fluctuations of gravitational potentials
at the surface of last scattering. The second one called the integrated Sachs-Wolfe (ISW) effect is generated by
time-varying gravitational potentials as the CMB photons pass through them from the recombination epoch to
the present epoch.
Recently, much attention has been paid to the non-linear version of the ISW effect called the Rees-Sciama
(RS) effect[2] on cosmological scales, since the RS effect of local inhomogeneities at quasi-linear regime may
explain the origin of observed anomalies [3, 4, 5, 6, 7, 8] in the large-angle CMB anisotriopy [9, 10]. It is argued
that the anisotropy for compensated asymptotically expanding local voids can be larger because the second
order effect enhances the ISW effect [11]. For compensated local lumps, the nature of the second order effect
at the Λ dominated epoch remains unknown.
So far, two types of treatment have been used for deriving the RS effect of local inhomogeneities. One is the
general relativistic treatment [12, 13, 14], in which a non-linear version of the SW effect at second order has
been studied. Another one is the simplified Newtonian treatment [15, 16, 17, 18, 19, 20, 21] in which influences
of nonlinear halo clustering upon the CMB temperature fluctuations and gravitational lensing phenomena have
been explored.
In this paper, we first study the RS effect comparatively using the general relativistic second order pertur-
bation theory and the Newtonian approximation and we show the consistency between the two approaches in
the perturbative regime. Next, we consider the RS effect of a compensating quasi-linear void/lump modelled
by a spherically symmetric top-hat type density perturbation at a redshift z and we investigate the temporal
change in the linear and second order temperature fluctuations as functions of z. In §2, we derive the solutions
of Einstein equations for general relativistic perturbations at second order in the Poisson gauge (generalized
longitudinal gauge) and consider the limit of κ (∼ the scale of inhomogeneities / Hubble radius ) ≪ 1. In §3, we
study the perturbations in cosmological Newtonian approximations and their relation to the general relativistic
perturbations up to second order. In §4, we study the temperature anisotropy owing to first and second order
perturbations based on relativistic perturbation theory [22]. In §5, we investigate the correlations between
first order and second order temperature fluctuations and their temporal behavior for a spherical top-hat type
density perturbation. §6 is dedicated to concluding remarks. In Appendices A and B, the main components of
Einstein equations and the derivation of their solutions are shown. In Appendix C, the integrations of metric
perturbations for spherically symmetric top-hat type perturbations along the light paths are shown.

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II.GENERAL RELATIVISTIC SECOND ORDER PERTURBATIONS
In what follows, we use the units of 8πG = c = 1, the Greek and Latin letters denote 0,1,2,3 and 1,2,3,
respectively. Index “B” represents the value for the unperturbed background spacetime. δij(= δi
Kronecker delta, and subscripts (n) correspond to n-th order quantities.
As a function of conformal time ˜ η and Cartesian coordinates ˜ xi, the metric of spatially flat Friedmann-
Robertson-Walker (FRW) universes with first order and second order perturbations ψ(n),φ(n),z(n)
is described by
j= δij) are the
i
,χ(n)
ij,n = 1,2
ds2≡ gµνd˜ xµd˜ xν= ˜ a2(˜ η)
?
− (1 + 2ψ(1)+ ψ(2))d˜ η2+
?
z(1)
i
+1
2z(2)
i
?
d˜ ηd˜ xi
+
?
(1 − 2φ(1)− φ(2))δij+ χ(1)
ij+1
2χ(2)
ij
?
d˜ xid˜ xj?
, (2.1)
where ˜ a(˜ η) is the scale factor, ˜ η (= ˜ x0) is related to the cosmic time t by dt = ˜ a(˜ η)d˜ η, and χ(n)
χ(n)l
l
xi= ˜ xi/H−1
0
factors at present are defined as ˜ a0= 1 and a0= H−1
while η and r do not have dimensions in our notation.
Energy density and 4-velocity of dust matter are written in terms of the background quantities and pertur-
bations as
ij
satisfy
≡ δlmχ(n)
lm= 0. In what follows, we use the normalized scale factor a = ˜ aH−1
0, and the conformal time η = ˜ η/H−1
0, the comoving coordinate
where H−1
0. Note that the scale fatcor a has a dimension of length
0
is the Hubble radius at present η0. The scale
ρ = ρB(η) + δ(1)ρ +1
2δ(2)ρ,(2.2)
uµ=1
a
?
δµ
0+ v(1)µ+1
2v(2)µ?
, (2.3)
where videnotes the 3-velocity of the dust matter. From the condition gµνuµuν= −1, we obtain the relations
v(1)0= −ψ(1),(2.4)
v(2)0= −ψ(2)+ 3(ψ(1))2+ [2z(1)
i
+ v(1)
i]v(1)i.(2.5)
Einstein equations for the unperturbed background with dust matter and a cosmolological constant Λ are
3(a′/a)2= (ρB+ ρΛ)a2,(2.6)
6(a′/a)′= −(ρB− 2ρΛ)a2
(2.7)
where ρΛis the energy density of the cosmological constant Λ and a prime denotes derivative with respect to η.
To fix the gauge freedom of perturbations, we adopt the Poisson gauge defined by
δlmz(n)
l,m= 0andδlmχ(n)
kl,m= 0(2.8)
for n = 1 and 2. This gauge is a generalized version of the longitudinal gauge which is defined by z(n)
and χ(n)
ij
= 0 [23], and gives us a metric expression convenient for a Newtonian interpretation, as well as the
longitudinal gauge [24].
In what follows, we consider only scalar-type perturabtions at linear order. Then, we have
i
= 0
z(1)
l
= 0andχ(1)
kl= 0 (2.9)
The Ricci tensor, the Einstein tensor and the energy-momentum tensor for dust matter are shown in Appendix
A. Solving the Einstein equation we obtain the expression of first order scalar-type perturbations in the growing

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mode in terms of functions P(η) and F(x) as1
ψ(1)= φ(1)= −1
2
?
1 −a′
aP′?
F,(2.10)
z(1)
i
= 0,χ(1)
ij= 0,(2.11)
δρ(1)/ρB=
1
ρBa2[(a′/a)P′− 1]∆F +3
2(a′/a)P′F,(2.12)
v(1)0= −1
2[(a′/a)P′− 1]F,v(1)i=1
2P′F,i,(2.13)
where F,iis ∂F/∂xi, ∆ is the Laplacian ∂2/∂xi∂xi, and
P(η) = −
2
3Ωm0˜ a−3/2[Ωm0+ ΩΛ0˜ a3]1/2
?˜ a
0
?˜ a
0
d˜ a˜ a3/2[Ωm0+ ΩΛ0˜ a3]−1/2+
2
3Ωm0˜ a,
η =d˜ a˜ a−1/2[Ωm0+ ΩΛ0˜ a3]−1/2,(2.14)
where Ωm0and ΩΛ0are the density parameters for matter and the cosmological constant at present. P(η) is
the solution of the growing mode in equation
P′′+2a′
aP′− 1 = 0.(2.15)
Note that the potential function F(x) is related to the first order matter density contrast ǫmin comoving slices
(defined in the comoving synchronous gauge) as
ǫm=
1
ρBa2[(a′/a)P′− 1]∆F. (2.16)
Next we consider relativistic second order perturbations corresponding to the first order perturbations in the
growing mode. From Eqs. (B9) - (B12) in Appendix B, which are derived by solving the Einstein equations
δ2Gj
i=δ2Tj
i, we have
φ(2)= ζ1F,lF,l+ ζ2· 100Ψ0+ ζ3F2+ ζ4· 100Θ0, (2.17)
ψ(2)= ξ1F,lF,l+ ξ2· 100Ψ0+ ξ3F2+ ξ4· 100Θ0,(2.18)
where
ζ1 =
1
4P
1
4P′?a′
?
1 −a′
aP′?
?
,ζ2=
?a′
a
?1
P′?
21
a′
a
?
PP′−1
6Q′?
−
1
18
?
P +1
2(P′)2??
aP′?
,
ζ3 =
a+−a′′
a
+
?2?
,ζ4= −1
3
a′
aP′?
1 −a′
,(2.19)
ξ1 = ζ1,ξ2= ζ2,
ξ3 =
1
4
?
4 − 7a′
aP′+
?
−a′′
a
+ 5
?a′
a
?2?
(P′)2?
,ξ4=1
6
?
2 −6a′
aP′+
?
−2a′′
a
+ 8
?a′
a
?2?
(P′)2?
,(2.20)
1In paper [12] (referred as Paper[12]), the first order perturbations in the Poisson gauge were derived by transforming the solution
in the comoving synchronous gauge to that in the Poisson gauge[23, 25]. After the publication several misprints were found in
Eqs. (4.6) - (4.8) of Paper[12], which should be taken into account for deriving expressions (2.10) - (2.13). Perturbations in the
decaying mode were derived in Paper [12], but they are omitted here.

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z(2)
i
= P′(1 + P′′)Ci, (2.21)
and
χ(2)
ij=
?
P +1
2(P′)2?
Dij+3
7P2∆Dij+ δχij,(2.22)
where Cisatisfies
∆Ci=
?
−200
9
Ψ0,i+1
2(F,lF,l),i− F,i∆F
?
,(2.23)
and δχij in Eq.(2.22) satisfies
?∂2
∂η2+2a′
a
∂
∂η− ∆
?
δχij=3
7P2∆2Dij+1
7
?
P −5
2(P′)2?
∆Dij.(2.24)
Ψ0and Θ0are defined as
∆Ψ0≡
9
200
?
F,klF,kl− (∆F)2?
, ∆Θ0≡ Ψ0−
3
100F,lF,l,(2.25)
and Q(η) satisfies
Q′′+2a′
aQ′= −
?
P −5
2(P′)2?
. (2.26)
The above second order solutions are consistent with those shown in Eqs.(4.12) - (4.15) in Paper[12], which was
derived using a transformation from the comoving synchronous gauge to the Poisson gauge. Note that ∆Dij
and ∆2Dij correspond to˜Gj
of Paper[12] with misprints must be replaced by the above correct equation (2.26).
In the above solutions, the ratios of terms including Θ0to terms including F,lF,land Ψ0are of the order of
κ2, where κ ≡ |x|/η, and |x| and η are the characteristic spatial scale of local perturbation and the horizon
size. Therefore, the terms including Θ0are negligible for local perturbations that are sufficiently smaller than
the horizon size. Thus for κ ≪ 1, we have
iand Gj
iin Eq.(2.25) of Paper[12], as ∆Dij= −˜Gj
iand ∆2Dij= −Gj
i. Eq.(2.17)
φ(2)= ψ(2)= ζ1F,lF,l+ ζ2100Ψ0,(2.27)
z(2)
i
= P′(1 + P′′)Ci,(2.28)
and
χ(2)
ij=3
7P2∆Dij.(2.29)
III.COSMOLOGICAL NEWTONIAN APPROXIMATION
In the cosmological Newtonian approximation, we assume that ǫ ≡ a|v|/c ≪ 1 and κ ≡ |x|/η ≪ 1 but
χ ≡ (ρ/ρB− 1)1/2is arbitrary, and consider only ψ and φ(= ψ) as the metric perturbations[26]. Then, from
the difference between the perturbed equation R0
0= T0
0−1
2Tµ
µand the background counterpart, we obtain
∆ψ =1
2a2(ρ − ρB).(3.1)
From the conservation equation Tµν
p ) Tµν= (ρ + p)uµuν+ pgµν, we obtain the equation of continuity
;ν= 0 and the energy-momentum tensor (for the perfect fluid with pressure
ρ′+3a′
aρ + (ρvi),i= 0(3.2)
and the equation of motion
vi′+ vi
,jvj+a′
avi− ψ,i+ p,i/ρ = 0,(3.3)

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where vi= aui. By solving these equations (3.1) -(3.3), we have ρ,viand ψ, which can determine the lowest-
order RS effect in the form of spatial integration of ψ,ialong a light path.
In what follows, we only consider the perturbative case with ǫ ≪ 1, κ ≪ 1, and χ ≪ 1. First, expressing the
perturbations as
ψ = ψ(1)+1
2ψ(2),ρ = ρB+ δρ(1)+1
2δρ(2),vi= v(1)i+1
2v(2)i,(3.4)
the first order equations for these perturbations in the pressureless case are given by
∆ψ(1)−
1
2a2δρ(1)= 0,
3a′
aδρ(1)+ ρBv(1)i
a′
av(1)i+ ψ(1)
δρ(1)′
+
,i= 0,
v(1)i′
+
,i
= 0,(3.5)
and the first order solutions are
δρ(1)/ρB=
1
ρBa2[(a′/a)P′− 1]∆F,(3.6)
v(1)i=1
2P′F,i, (3.7)
and
ψ(1)= −1
2
?
1 −a′
aP′?
F, (3.8)
where the above δρ(1),v(1)iand ψ(1)are equal to Eqs.(2.12), (2.13), and (2.10) in the limit of κ ≪ 1, respectively.
Next the corresponding Newtonian second order equations are
∆ψ(2)−
1
2a2δρ(2)= 0,
3a′
aδρ(2)+ 2[δρ(1)v(1)i],i+ ρBv(2)i
+ 2 v(1)i
av(2)i+ ψ(2)
δρ(2)′
+
,i= 0,
v(2)i′
,jv(1)j+a′
,i
= 0.(3.9)
By substituting the above first order solutions Eqs. (3.6), (3.7), and (3.8) to Eq.(3.9) and eliminating ψ(2)and
v(2)i, we obtain
(δρ(2)/ρB)′′+a′
a(δρ(2)/ρB)′−1
2a2δρ(2)=1
4(P′)2∆(F,lF,l) −
1
ρBa4[a2P′(a′
aP′− 1)]′(∆FF,i),i. (3.10)
Assuming that ψ(2)is given by Eq.(2.27) and using the first line of Eq.(3.9), we can express δρ(2)as
δρ(2)/ρB=
2
ρBa2{ζ1∆(F,lF,l) +9
2ζ2[F,ijF,ij− (∆F)2]}.(3.11)
By substituting this δρ(2)/ρB to Eq.(3.10), we find that it is a solution of Eq.(3.10), so that ψ(2)given by
Eq.(2.27) is also a solution of Eq.(3.9). Thus we proved that in the cosmological Newtonian limit with ǫ ≪ 1,
κ ≪ 1, and χ ≪ 1, the general relativistic second order solution is consistent with the Newtonian one.
IV.TEMPERATURE ANISOTROPY
In this section we consider the observed temperature of the CMB radiation which was emitted at the recom-
bination epoch and received at the present epoch. The relation between the emitted and received temperatures
TeTois
To= (ωo/ωe)Te, (4.1)