Effect of radiation-like solid on CMB anisotropies
arXiv:1402.4434v1 [gr-qc] 18 Feb 2014
Vladimír Balek∗ and Matej Škovran
†
Department of Theoretical Physics, Comenius University, Bratislava, Slovakia
February 19, 2014
Abstract
The form of Sachs-Wolfe plateau in the presence of solid matter with w = 1/3 is investigated.
In order that the plateau fits the flat spectrum within cosmic variance, the shear modulus to
energy density ratio ξ must be close enough to zero, |ξ| . 10−5 .
1
Introduction
The observed acceleration of the universe is usually regarded as an effect of dark energy, identified
either with null oscillations of quantum fields or with some kind of scalar field. An alternative
explanation is that what accelerates the universe is a solid with negative parameter w (pressure
to energy density ratio). The idea appeared shortly after the effect was discovered [1, 2], and the
underlying theory was extensively studied afterwards [3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
Given the attention paid to scenarios in which the solid has negative w, it is natural to ask what
difference would it make if the value of w of the solid was positive. This question was addressed
in [13]. One of the two scenarios explored there was that in a universe filled with radiation there
appeared a radiation-like solid, which had, in addition to w = 1/3, a constant shear modulus to
energy ratio ξ. A possible materialization of such solid could be a Coulomb crystal with relativistic
Fermi gas of moving charges; another, more speculative possibility could be a network of ‘springlike’ strings with energy inversely proportional to length. In the latter case ξ would be negative,
making the vector perturbations unstable. However, this would not necessarily make the theory
useless, since in the process of inflation typically no vector perturbations are created.
In order to obtain an observable effect of the solid on long-wavelength perturbations, one must
assume that it was created with flat internal geometry and nonzero shear stress acting in it. Such
∗
†
e-mail address: [email protected]
e-mail address: [email protected]
1
solidification cannot take place in pure radiation, whose particles move freely and hence the concept
of shear deformation has no sense for them. It must be linked to some other kind of particles
present in the universe, distributed anisotropically from the very start of Friedmann expansion.
Such component of cosmic medium could perhaps be formed in solid inflation [9, 10, 11, 12].
In the paper we investigate how the radiation-like solid, if present in our universe, would manifest itself in the large-scale CMB anisotropies. In section 2 we describe how the long-wavelength
perturbations evolve in the presence of the solid and determine their size at the time of recombination; in section 3 we derive a formula for temperature fluctuations observed at present on Earth
and calculate the CMB power spectrum for small multipole moments; and in section 4 we discuss
the results. A system of units is used in which c = 16πG = 1.
2
Evolution of perturbations
2.1
Pure solid
Consider a flat FRWL universe filled with an isotropic elastic matter and add a small perturbation
of spacetime metric and distribution of matter to it. Following [14], we will use proper-time
comoving gauge for the description of perturbations.; thus, we will assume that φ (the correction
to the (00)-component of the metric tensor) as well as ξ (the shift vector of the matter) is zero.
Denote the scale parameter by a, the mass density and pressure of the matter by ρ and p, the
compressional modulus of the matter by K and the shear modulus of the matter by µ. The
condition φ = 0 means that the scalar part of the metric is
ds(S)2 = a2 [dη 2 + 2B,i dηdxi − (δij − 2ψδij − 2E,ij )dxi dxj ],
(1)
and from the condition ξ = 0 it follows that the scalar part of the energy-momentum tensor is
T0 0 = ρ + ρ+ (3ψ + E),
T
,
Ti j(S) = −pδij − K(3ψ + E)δij − 2µE,ij
Ti 0(S) = ρ+ B,i ,
(2)
T
T
where ρ+ = ρ + p, E = △E and E,ij
is the traceless part of the tensor E,ij , E,ij
= E,ij − Eδij /3.
In the expressions for T0 0 and Ti j(S) we assume that the perturbations are adiabatic, that is, the
entropy per particle S is constant throughout the space.
The proper-time comoving gauge allows for a residual transformation δη = a−1 δt(x), where
δt(x) is a local shift of the moment at which the time count has started. The function E is invariant
under such transformation and the functions B and ψ can be written as
B = B + χ,
ψ = −Hχ,
where B is invariant and χ transforms as χ → χ + δη.
2
(3)
Suppose the solid has flat internal geometry and consider a perturbation of the form of plane
wave with the comoving wave vector k. For B and E as functions of η we have two coupled
differential equations of first order, coming from the equations T0 µ ;µ = 0 and 2G00 = T00 ,
E ′ = −(k 2 + 3αH2 )B − αHE,
B ′ = (3c2S0 + α − 1)HB + c2Sk E,
(4)
where the prime denotes differentiation with respect to η, H = a′ /a is the Hubble parameter with
respect to η, and the functions α, c2S0 (auxiliary sound speed squared) and c2Sk (longitudinal sound
speed squared) are defined as α = (3/2)ρ+ /ρ, c2S0 = K/ρ+ and c2Sk = c2S0 + (4/3)µ/ρ+ . These
equations must be supplemented by the dynamical equations for an unperturbed universe,
1/2
1
, ρ′ = −3Hρ+ ,
ρa4
a′ =
6
(5)
and an expression for K,
K = ρ+
dp dρ
S
(6)
.
If the matter has more than one component, it holds ρ =
X
ρi and K =
equation (5) as well as equation (6) hold for each component separately.
X
Ki , and the second
Consider a one-component universe in which the parameters w = p/ρ and ξ = µ/ρ are constant.
By combining the two equations (5) we obtain that a is a power-like function of η, a ∝ η 2/(1+3w) ,
and from equation (6) we find that the parameter K = K/ρ is constant, too, K = ww+ with
w+ = w + 1. Equations for B and E then combine into an equation for B whose solution are Bessel
functions multiplied by a certain power of η. In case w = 1/3 the equations for B and E are
B ′ = 2η −1 B + c2Sk E,
E ′ = −(k 2 + 6η −2 )B − 2η −1 E,
(7)
where c2Sk = 1/3 + ξ; and after excluding E we obtain an equation for B,
B ′′ + (c2Sk k 2 + 6ξη −2 )B = 0,
(8)
with the solution
√
z(aJ Jn + aY Yn ),
(9)
p
where z = cSk kη, n = 1/4 − 6ξ and Jn and Yn are Bessel functions of first and second kind of
B=
the argument z.
When describing CMB anisotropies it is convenient to pass to Newtonian gauge, in which the
functions B and E are traded for the Newtonian potential φ and the scalar part of the shift vector
ξ (S) . Denote the potentials φ and ψ and the perturbation to the mass density δρ in the Newtonian
gauge by Φ, Ψ and δρ. For the last two functions we will use expressions coming from the gauge
transformation (equations (7.19) and (7.20) in [15]), and for the first function we will write down
an expression following from Einstein equations (equation (7.40) in [15]),
1
Φ = Ψ + a2 τ (2) ,
2
Ψ = ψ + H(B − E ′ ),
3
δρ = δρ − ρ′ (B − E ′ ),
where τ (2) is the longitudinal part of the perturbation to Ti j(S) and δρ is the perturbation to T0 0 .
By inserting here from equations (2) and (3) we find
Φ = Ψ − µa2 E,
Ψ = H(B − E ′ ),
δρ = ρ+ (3Ψ + E),
(10)
and after we insert into the first equation from the first equation in (5) and into the second equation
from the second equation in (4), we obtain
Φ = Ψ + 6ξk −2 H2 E,
Ψ = −αk −2 H2 (3HB + E).
(11)
For w = 1/3 this yields
ΦA ≡ (Φ, Ψ) = −2(kη)−2 (3η −1 B + βA E),
(12)
where βA = (1 − 3ξ, 1). By inserting for E from the first equation in (4), we can rewrite this in
terms of B only,
i
h dB
− (β̃A − 3ξ)z −1 B ,
ΦA ∝ −z −2 βA
dz
(13)
where β̃A = (1 − 6ξ, 1). With B in the form (9), this is an expression for the potentials Φ and Ψ
in terms of Bessel functions and their derivatives.
Let us find the asymptotic of Ψ for perturbations whose wavelength exceeds significantly the
size of the sound horizon. It holds z = cSk rh /λ-, where rh is the radius of the particle horizon
and λ- = λ/(2π) is the reduced wavelength. (We have used the fact that the radius of horizon is
proportional to the conformal time, rh = aη.) Thus, we are interested in the behavior of Ψ in the
limit z ≪ 1. From the approximate formulas
z2 ,
Jn ∝ z n 1 −
4n+
Yn ∝ −z −n ,
where n+ = n + 1, we obtain
Ψ = AJ
where
m
M
p
q
z −m + z −2−m − AY z −2−M ,
(14)
= 1/2 ∓ n and the constants p and q are defined as
p=
1
(2 − m + 3ξ),
4n+
q = m − 3ξ.
(15)
.
.
For |ξ| ≪ 1 it holds p = 1 and q = 3ξ, therefore |p/q| ≫ 1 and the term proportional to z −m can
dominate the term proportional to z −2−m while z is still small.
Consider now a universe filled with an ideal fluid and suppose that the fluid turned into a solid
with the same w at some moment ηs . As before, we put w = 1/3. Suppose the solidification was
anisotropic, producing a solid with flat internal geometry, and consider perturbations that were
long-wavelength at the moment ηs ; in other words, suppose zs ≪ 1. The first condition means that
4
equations (4) are valid without modifications at η > ηs and the second condition implies that Ψ
can be written in the form (14) in some interval η > ηs . In a fluid, the potentials Φ and Ψ coincide
and, if we restrict ourselves to their non-decaying part, they are constant for long-wavelength
perturbations. Denote this constant by Φ0 . If |ξ| is not too small, |ξ| ≫ zs2 , the first term in the
brackets in (14) can be neglected at the moment ηs and the matching conditions reduce to
ÃJ − ÃY = Φ0 ,
(2 + m)ÃJ − (2 + M )ÃY = −ηs [Ψ′ ],
(16)
where ÃJ = AJ zs−2−m and ÃY = AY zs−2−M and the square brackets denote the jump of the
function at η = ηs . To determine [Ψ′ ], we observe that the density contrast δ = δρ/ρ equals −2Φ0
for long-wavelength perturbations in a universe filled with fluid [15]. By using δ = w+ (3Φ0 + E),
see the last equation in (10), and putting w = 1/3, we find E = −9Φ0 /2 and
ηs [Ψ′ ] = −6(kηs )−2 [B ′ ] = −6c2Sk zs−2 ξEs = 27c2Sk ξzs−2 Φ0 .
Our assumption about the values of ξ guarantees that this is in absolute value much greater than
|Φ0 |, so that we can neglect the right hand side of the first equation in (16). The solution then is
ÃJ = ÃY = rzs−2 Φ0 ,
r=
27 2
c ξ.
2n Sk
(17)
After inserting this into (14) and introducing a rescaled time η̃ normalized to 1 at the moment ηs ,
η̃ = η/ηs = z/zs , we have
Ψ = P η̃ −m + Qzs−2 (η̃ −2−m − η̃ −2−M ),
(18)
where P = (p/q)rΦ0 and Q = rΦ0 .
2.2
Adding the solid to the rest of matter
Consider a universe filled with nonrelativistic matter (dust) with density ρm and pressure pm = 0
and radiation with density ρr and pressure pr = ρr /3, and suppose that radiation consists of
ordinary radiation (photons and neutrinos) with density ρrI and zero shear modulus, and of
radiation-like solid with density ρrII and shear modulus µ proportional to ρrII . Both ordinary
radiation and radiation-like solid have the same dependence of p on ρ and hence of ρ on a; thus,
the densities ρrI and ρrII are at any given moment equal to the same fractions of ρr and µ is
proportional to ρr , µ = ξρr with constant ξ.
Let us first write down the solution of equations (5) for an unperturbed universe. Denote
√
ζ = η/η∗ with η∗ = ηeq /( 2 − 1), where the index ‘eq’ refers to the moment when the densities
ρr and ρm are equal. From the second equation (5), written separately for radiation and matter,
we have
ρr ∝ a−4 ,
ρm ∝ a−3 ,
5
and from the first equation (5) with ρ = ρr + ρm on the right hand side we obtain
a = aeq ζ(2 + ζ).
2
Now we turn to the perturbed universe. Denote ζ+ = 1 + ζ, X = ζ(2 + ζ), X+ = 1 + X = ζ+
and X̂+ = 1 + (3/4)X. From the formulas for ρr and ρm we have ρr /ρm = aeq /a = 1/X, so that
ρr /ρ = 1/X+ , ρr+ /ρ+ = 1/X̂+ and ρ+ /ρ = (4/3)X̂+ /X+ . The function α equals (3/2)ρ+ /ρ as
before and the functions c2S0 and c2Sk now equal (1/3)ρr+ /ρ+ (because K = Kr = ρr+ /3) and
(1/3 + ξ)ρr+ /ρ+ ; thus, α = 2X̂+ /X+ , c2S0 = (1/3)/X̂+ and c2Sk = (1/3 + ξ)/X̂+ . Finally, by
differentiating a ∝ X with respect to η we obtain H = (2ζ+ /X)η∗−1 . With these expressions
equations (4) become
1
1 + 21 X 2ζ+
+ξ
dB̃ 1
+
E,
B̃ + 3
=
dζ
X+
X
X̂+
X̂+
dE
4X̂+
24X̂+ B̃ −
= − κ2 +
E,
2
dζ
X
ζ+ X
where B̃ = η∗−1 B and κ = kη∗ ; and equations (11) can be written as
X̂+ 6ζ+
B r + b A Er ,
ΦA = 2
X
X
(19)
(20)
where Br = −8κ−2 B̃, Er = −8κ−2 E and bA = (1, 1 − 3ξ/X̂+ ).
A straightforward way to compute the functions Φ and Ψ would be by solving an equation
of second order for Br , similar as in the theory with one-component matter. However, if we are
interested in small values of κ only, we can also derive an equation of second order directly for Ψ
and solve it perturbatively. As we will see, the equation becomes especially simple if we factor out
of Ψ a certain function of ζ.
By differentiating equation (20) with A = 2 and using equations (19) we obtain
X̂+
2X̂+ 2ζ+
ζ+
dΨ
=− 1+
Ψ + 6ξ 3 Er − κ2 2 Br .
dζ
X
X
X
X
(To simplify the algebra, it is useful to express the coefficient in front of B̃ in the first equation (19)
as logarithmic derivative of the function X 2 /(ζ+ X̂+ ).) The coefficient in front of Ψ is logarithmic
derivative of the function ζ+ /X 3 , therefore if we write Ψ as (ζ+ /X 3 )F , we eliminate the term
with Ψ from the right hand side,
dF
X X̂+
Br .
= 6ξEr − κ2
dζ
ζ+
Besides that, we have an equation for Er and an expression for Br in terms of Er and F ,
X 1 F
4F
dEr
−
Er .
= − 2 − κ2 B r , B r =
dζ
X
6 X̂+
ζ+
(21)
(22)
By differentiating the first equation and inserting into it from the other two equations we obtain
h
1
X i
F
24ξ
d2 F
2
(23)
(X
+
ξ)
+
+
F
=
−κ
Er ,
−
X
dζ 2
X2
3
ζ+
X̂+
where X = [1 + (2/3)X]X̂+/X+ . This becomes an equation of second order for F if Er is expressed
in terms of F and dF/dζ by using equation (21), with Br inserted from the second equation (22).
6
2.3
Long-wave limit
Consider a perturbation that is stretched far beyond horizon at equality, κ ≈ kηeq ≪ 1. We will
solve equation (23) perturbatively up to the first order in κ2 ; thus, we will find F (0) from equation
(0)
(0)
(23) without the right hand side, compute Er , insert F (0) and Er
on the right hand side of (23)
and find ∆F .
Suppose that the radiation-like solid appeared in the universe at some moment ηs in the
radiation dominated era, ηs ≪ ηeq . For ηs ≤ η ≪ ηeq our Ψ must coincide within a good accuracy
with Ψ of the theory with one-component matter that has w = 1/3. Since we have assumed
kηeq ≪ 1, it surely holds zs ≈ kηs ≪ 1, too, hence at η ≪ ηeq our Ψ must be given by equation
(0)
(0)
(18). By using η̃ = ζ/ζs and zs = cSk κζs , where (cSk )2 = 1/3 + ξ, we obtain
(24)
Ψ = ζsm [P ζ −m + κ−2 Q̃(ζ −2−m − ζs2n ζ −2−M )],
(0)
where Q̃ = (cSk )−2 Q. Our aim is to extend this solution to η ∼ ηeq and further. Clearly, the
second part of (24) will be then replaced by Ψ(0) and the first part of (24) will be replaced by ∆Ψ.
For F (0) we have a homogeneous equation
24ξF (0)
d2 F (0)
+ 2
= 0,
2
dζ
ζ (2 + ζ)2
(25)
whose solution is
F (0) = cm fm + cM fM ,
fm = ζ m (2 + ζ)M
fM = ζ M (2 + ζ)m
(26)
.
The asymptotics of F (0) and Ψ(0) = (ζ+ /X 3 )F (0) in the limit ζ ≪ 1 are
F (0) = ĉm ζ m + ĉM ζ M ,
Ψ(0) =
1
(ĉm ζ −2−M + ĉM ζ −2−m ),
8
m
where the constants ĉm and ĉM are defined as ĉm =2M
cm and ĉM =
2 cM , and we have rewritten
m
M
the powers of ζ in Ψ(0) with the help of the identity
= 1−
. The approximate expression
M
m
for Ψ(0) coincides with the second part of (24) if ĉM = −ζs−2n ĉm = 8κ−2 ζsm Q̃.
The ratio of the two terms in the asymptotic expression for F (0) equals in absolute value 1 at
ζ = ζs , and then decreases as ζ −2n . Thus, except for the first period after ηs , the second term can
be neglected and the function F in the zeroth order in κ can be written as F (0) = cM fM .
(0)
Having found F (0) we can compute ∆F . From equation (21) we have Er
= (6ξ)−1 dF (0) /dζ
and if we keep only the term proportional to fM in F (0) , we obtain
Er(0) = (6ξ)−1
(0)
After inserting F (0) and Er
ζ + 2M (0)
F .
X
(27)
on the right hand side of (23), we arrive at the equation
24ξ∆F
d2 ∆F
+ 2
= −CM F ,
dζ 2
ζ (2 + ζ)2
7
(28)
where CM = (6ξ)−1 κ2 cM and
1
h
ζ + 2M i
6ξ
+
F = (X + ξ)
fM .
−X
3
ζ+
X̂+
(29)
The Wronskian of the functions fm and fM is 4n, hence the solution is
1
∆F = − CM fM
4n
Z
fm F dζ − m ↔ M .
(30)
In order that ∆F is small in comparison with F (0) , it must hold |CM /cM | ≈ |ξ|−1 κ2 ≪ 1. We
are interested predominantly in solids with small dimensionless shear modulus, |ξ| ≪ 1, for which
p
the requirement κ ≪ |ξ| is a stronger constraint on κ than just κ ≪ 1. However, it turns out
p
that the theory holds also for κ ≫ |ξ|, as long as the condition κ ≪ 1 is observed. As a result,
p
we can use it for κ ∼ |ξ|, too, regarding it as an interpolating theory between the two opposite
asymptotic regions.
To prove the claim that our theory is valid for κ ≫
p
|ξ|, we must show that in the limit ξ → 0
our F coincides with the function Fid obtained for long-wavelength perturbations in the theory
with an ideal fluid. By putting ξ = 0 in the definitions of fm , fM and F we obtain fm = 2 + ζ,
fM = ζ and F = (1/3 − X )ζ(ζ + 2)/ζ+ . We also notice that the constants cm and cM must go to
zero like ξ in order that the constant CM ∝ ξ −1 cM stays finite. As a result, F (0) vanishes and the
arbitrary constants in the integrals in ∆F must be chosen in such a way that neither a constant
term nor a term proportional to ζ appears there; otherwise F (0) will be restored. Furthermore,
eliminating the constant Cm ∝ ξ −1 cm , which normally means ignoring the decaying part of F (0)
when computing the right hand side of (28), is not necessary here since the function that enters
the right hand side of (28) multiplied by Cm coincides with F . If we denote C = Cm + CM , we
have
3
13
1 1
.
,
Cζ 3 ζ 2 + 3ζ +
+
F = ∆F =
24
5
3
ζ+
(31)
and the function Φ = (ζ+ /X 3 )F is identical with the non-decaying part of Φid , as given in equation
(7.71) in [15]. Thus, F is identical with Fid .
The function Φ for an ideal fluid is constant both at ζ ≪ 1 (radiation dominated era) and ζ ≫ 1
(matter dominated era), and the ratio of its values at ζ ≫ 1 and ζ ≪ 1 is ω0 = 9/10. However,
we are not really interested in the value of Φ at infinite time; instead, we want to know Φ at the
moment of recombination. Denote the quantities referring to that moment by the index ‘re’. It
.
.
holds are = aeq ζre (2 + ζre ) = 3aeq , therefore the value of ζ at the time of recombination is ζre = 1
and the ratio of the values of Φ at recombination and before equality is ω = 253/270 = 0.94.
Let us find the asymptotics of ∆F in case the medium has nonzero shear stress. At ζ ≪ 1 we
have fm = 2M ζ m , fM = 2m ζ M and F = −4s0 fM /3 with s0 = M − 9ξ(1 + ξ)/2, while at ζ ≫ 1
we have fm = ζ + 2M , fM = ζ + 2m and F = −ζ 3 /2. After inserting these expressions into (30)
8
we obtain
∆F =
2
2+M
9 sĈM ζ
1
5
40 CM ζ
if ζ ≪ 1
if ζ ≫ 1
(32)
,
where ĈM = 2m CM and s = 3s0 /(2n+ ).
Now we are ready to write down the asymptotic formulas for the leading term in Ψ and the
first correction to it at ζ ≪ 1 and ζ ≫ 1. From the definitions
ζ+ (0)
F ,
X3
Ψ(0) =
∆Ψ =
ζ+
∆F,
X3
(33)
we obtain
Ψ
(0)
=
1
−2−m
8 ĉM ζ
cM ζ
−4
if ζ ≪ 1
if ζ ≫ 1
, ∆Ψ =
1
−m
36 sĈM ζ
if ζ ≪ 1
if ζ ≫ 1
1
40 CM
(34)
,
One easily verifies that the asymptotics of Ψ(0) and ∆Ψ at ζ ≪ 1 are consistent with the approx-
imate formula (24). Indeed, from the identity (m − 3ξ)(M − 3ξ) = 3ξ(1 + 3ξ) we obtain
q=
3ξ(1 + 3ξ)
,
M − 3ξ
p=
1
s0
1
(2 − q) =
,
4n+
2n+ M − 3ξ
so that the ratio of the coefficients in front of ζ −m and ζ −2−m in (24) is
P
κ−2 Q̃
=
κ2 (1 + 3ξ)p
κ2 s0
κ2 s
=
=
=
3q
18n+ ξ
27ξ
1
36 sĈM
1
8 ĉm
.
According to equation (20), Φ can be expressed in terms of Ψ and Er as Φ = Ψ − (3ξ/X 2 )Er .
(0)
The function Er
is given in (27) and the function ∆Er is
∆Er = (6ξ)−1
h d∆F
dζ
X X̂+ F (0)
X (0) i
E
.
−
6ζ+ X̂+
ζ+ r
(35)
+ m (0)
F
X3
(36)
+ κ2
This yields
Φ(0) =
1
2ζ
and
h 1
1
1
fM
∆Φ = ∆Ψ + CM
2
4
X3
Z
fm F dζ + m ↔ M −
X̂+ 6ξ
ζ + 2M i
−
fM .
3ζ+ X X̂+
ζ+
(37)
The asymptotics of the leading term in Φ and the first correction to it are
Φ
(0)
=
1
−2−m
8 mĉM ζ
1
−4
2 cM ζ
if ζ ≪ 1
if ζ ≫ 1
, ∆Φ =
1
−m
36 σ ĈM ζ
1
40 CM
−
if ζ ≪ 1
1
−2
2 ξCM ζ
if ζ ≫ 1
,
(38)
where σ = 3M − 9ξ − (1 + M )s. In the expression for ∆Φ at ζ ≫ 1 we have skipped the terms
of order ζ −1 and ζ −2 that are the same as in ∆Ψ, and kept only the first term in the expansion
in ζ −1 by which ∆Φ differs from ∆Ψ. A test of the above formulas in the limit ζ ≪ 1 would be
to compare them to an expression for Φ analogical to (24), valid for a universe filled with pure
radiation-like solid. Equivalently, one can compare the ratios of the coefficients in the upper lines
9
of equations (38) and (34) to the ratios of the corresponding coefficients in Φ and Ψ in the case
with pure radiation-like solid. After deriving an expression for Φ analogical to (18), one can check
by a straightforward computation that the ratios are the same.
Let us determine the values of the potentials Φ and Ψ at recombination. We will express them
in terms of their value Φ0 at the initial moment, which can be identified as the end of inflation
and beginning of Friedmann expansion of the universe. To calculate Φre and Ψre we need to know
the constants cM and CM . For the former constant we have
cM = 2−m ĉM = 8κ−2 (ζs /2)m Q̃,
(0)
Q̃ = (cSk )−2 rΦ0 =
so that
cM = 216κ−2 ξ Φ̂0 ,
Φ̂0 =
27
ξΦ0 ,
2n
(ζs /2)m
Φ0 ,
2n
(39)
.
and from the definition of the latter constant we then obtain CM = 36Φ̂0 . Note that Φ̂0 = Φ0 if
|ξ| ≪ 1. The value of Φ and Ψ at ζ → ∞ is Φ∞ = CM /40 = ω0 Φ̂0 , hence the only effect of the
solid on Φ∞ is that Φ0 is replaced by Φ̂0 . Suppose the solid does not contribute to the total mass
density substantially, so that its presence does not affect the temperature of radiation at equality
.
and ζre = 1 as in the case with pure fluid. By inserting this value into the first equation in (33)
(0)
(0)
and into equation (36) we obtain Φre = (2/27)3mcM and Ψre = [(1/2 + m)/27]3m cM , or
(0)
ΦAre = mA κ−2 ξ Φ̂0 ,
mA = (2, 1/2 + m)8 · 3m .
(40)
The corrections to these values are of the form
∆ΦAre = nA Φ̂0 ,
Zζre
.
(fm or fM )F dζ = ω if |ξ| ≪ 1.
nA = expressions composed of
(41)
Thus, the functions Φ and Ψ evaluated at recombination can be written as
ΦAre = (mA κ−2 ξ + nA )Φ̂0 .
(42)
Equation (42) with Φ̂0 inserted from the second equation (39) is the desired expression of Φre
and Ψre in terms of Φ0 . However, when constructing the graphs of Φre and Ψre as functions of
κ, we replace the parameter Φ0 by a more descriptive parameter Φid,re = ωΦ0 . The graphs are
shown in fig. 1. The value of the dimensionless shear modulus is ξ = 1/600 in the left panel and
ξ = −1/600 in the right panel. (The values are the same as in fig. 1 in [13], where the parameter
b = 6ξ is used instead of ξ.) For such ξ the deviation of nA from ω is several thousandths
only and does not need to be taken into account. Heavy curves are computed for ζs = 10−5 ,
which corresponds to shear stress appearing on the scale of nucleosynthesis, and light curves are
computed for ζs = 10−24 , which corresponds to shear stress appearing on the GUT scale. The
curves cross zero in case ξ < 0, therefore we have combined ordinary and logarithmic scale on the
vertical axis in the right panel.
10
100
2
1
0
0.01
-1
Ψ Φid
10
0.1
1
κ
Φ Φid
Φ Φid
1
-10
Ψ Φid
0.1
0.01
0.1
1
κ
-100
Fig. 1: Scalar perturbations at recombination as functions of the wave number
for positive (left) and negative (right) shear stress
.
.
For the conformal time η∗ we have η∗ = ηre , hence κ = kηre = (rh )re /λ-re and the condition
κ ≪ 1 we have used when deriving the expressions for Φre and Ψre means that the perturbation
is stretched far beyond horizon not just at equality, but also at recombination. In the figure we
have extrapolated the curves up to κ = 0.5, where the error is of order 25%. At such κ Φre
and Ψre decrease in the exact theory due to the fact that the inhomogeneities are suppressed
inside horizon during the first quarter-period of acoustic oscillations. In our approximation the
functions are saturated close to κ = 0.5, where the right hand side of (42) is dominated by the
second (constant) term. The first term is small because of small |ξ|. As κ decreases, the first term
becomes dominant and both functions start to rise or fall steeply depending on the sign of ξ.
3
3.1
Large-scale CMB anisotropies
Temperature fluctuations
To compute CMB anisotropies we must know the relative deviation of the temperature from its
mean value ∆T = δT /T at the time of recombination. Since ργ ∝ T 4 , it holds
∆T =
1
δγ ,
4
(43)
where δγ is the density contrast of radiation, δγ = δργ /ργ . Density perturbations of individual
components of matter, if independent from each other, are given by a formula analogical to that
for the total density perturbation, δρi = ρi+ (3Ψ + E). For the density contrast δγ this yields
4
δγ = 4Ψ + E.
3
11
(44)
The quantity ∆T is one of the two terms in the effective fluctuation of temperature ∆Tef f ,
measured by a local observer that is at rest with respect to the unperturbed matter and observes
the radiation arriving in the same direction in which it then propagates to the observer on Earth.
Denote the local velocity of matter by v and the unit vector pointing towards the place from which
the radiation comes to the observer on Earth by n. The function ∆Tef f differs from ∆T by the
term ∆TD = −v · n coming from Doppler effect. Suppose the velocity reduces to its scalar part,
v = v(S) , and denote the unit vector in the direction of k by m. By performing the transformation
from the proper-time comoving gauge to the Newtonian gauge we find v = −ikE ′ = imκ−1 dE/dζ
and
∆TD = −im · nκ−1
dE
.
dζ
(45)
Note that the expression for v can be obtained also from the energy conservation law,
4
(δγ − 4Ψ)′ + ∇ · v = 0,
3
if one inserts into it from (44). The law is the same as in the theory with an ideal fluid, see
equation (7.110) in [15], except that Φ is replaced by Ψ.
CMB anisotropies are calculated from the temperature fluctuations ∆T0 , seen at present on
Earth. (The index 0 denotes the present moment, while previously it denoted the beginning of
Friedmann expansion. The notation is standard and its ambiguity is not considered a problem,
since the meaning of the index can be easily identified from the context.) To compute ∆T0 we
make use of the fact that the temperature fluctuations of the radiation propagating freely from
the surface of last scattering satisfy
d(∆T + Φ)
∂(Φ + Ψ)
=
.
dη
∂η
Again, this is the same equation as in the theory with an ideal fluid, except that one Φ on
the right hand side is replaced by Ψ, see equation (9.20) in [15]. The contribution of the right
hand side to ∆T0 , the integrated Sachs-Wolfe effect, is presumably between 10 and 20% as in
the theory with an ideal fluid [15]; therefore the right hand side can be neglected in the first
approximation. If we put the value of Φ measured on Earth at present equal to zero, the equation
yields ∆T0 = (∆Tef f + Φ)re ; and by inserting here from equations (43), (44) and (45) we obtain
dE 1
.
∆T0 = Φ + Ψ + E − im · nκ−1
3
dζ re
(46)
We are interested in this quantity for long-wavelength perturbations (κ ≪ 1).
We already know the values of Φ and Ψ at recombination, so that all we need to do is to
calculate the values assumed at recombination by the function E and its derivative. Let us first
discuss the behavior of E for an ideal fluid. We have mentioned earlier that for a one-component
12
fluid E is constant and equal to −9Φ0 /2. This stays valid also for a two-component fluid composed
of radiation and matter, if we regard Φ0 as the value of Φ deep in the radiation dominated era.
Indeed, the second equation (22) combined with equation (21) with ξ = 0 yields
3X+ dF
. 6ζ+
B̃ = −
,
E=−
X
4X 2 X̂+ dζ
and since the function F defined in (31) satisfies the identity
X+
dF
1
= CX 2 X̂+ ,
dζ
6
E is constant and equal to −C/8. (We have skipped the term proportional to F in the formula
relating E to B̃ because it is of order κ2 .) The meaning of C can be established by computing
the function Φ = (ζ+ /X 3 )F at ζ ≪ 1 and putting the result equal to Φ0 . In this way we find
C = 36Φ0 , which yields the desired value of E.
Consider now the function E for a solid. Its leading term, the function E (0) , is obtained by
(0)
multiplying the function Er
from equation (27) by −(1/8)κ2 . After inserting for F (0) the product
cM fM with the constant cM given in (39), we find
E (0) = −
This yields
9(ζ + 2M )
fM Φ̂0 .
2X
(47)
18mM
dE (0)
fM Φ̂0 ,
=
dζ
X2
and
dE (0) 3
(0)
Ere
= − (1 + 2M )3m Φ̂0 .
2
dζ
re
(48)
=
2
mM · 3m Φ̂0 .
3
(49)
The first quantity is of order Φ0 , just like ∆Φre and ∆Ψre , and the second quantity is of order
ξΦ0 . Obviously, the first quantity must be included into ∆T0 ; however, the second quantity can
be skipped, because its contribution to ∆T0 is of order κ−1 ξΦ0 and hence is suppressed by the
(0)
(0)
factor κ with respect to the contributions of Φre and Ψre .
(0)
The function ∆E can be calculated in a similar way as Er
starting from equation (35), but
if we are interested only in its order of magnitude at recombination, we can use its expression in
terms of ∆Φ and ∆Φ instead,
∆E =
1
(6ξ)−1 κ2 X 2 (∆Φ − ∆Ψ).
4
(50)
In the limit ξ → 0 both ∆Φ and ∆Ψ equal Φid , hence the Taylor expansion of their difference
begins by a term proportional ξ and hence the function ∆E is of order κ2 Φ0 . As a result, the
contributions of both ∆Ere and (∆E/dζ)re to ∆T0 are negligible, being suppressed by factors κ2
and κ respectively when compared to the contributions of ∆Φre and ∆Ψre .
The estimate of ∆E implies that the function E is well approximated by E (0) . This yields
p
E = −9Φ0 /2 for |ξ| ≪ 1, proving once more that the theory works well in the limit κ ≫ |ξ|.
13
We have seen that in ∆T0 one can leave out the contribution of ∆E to the term E/3 and the
whole Doppler term proportional to dE/dζ. What remains is
∆T0 = (ακ−2 ξ + β)Φ̂0 ,
(51)
where
α=
5
2
1
β = n1 + n2 − (1 + 2M )3m .
2
+ m 8 · 3m ,
(52)
If |ξ| ≪ 1, we can put ξ = 0 in α and β as well as in Φ̂0 , so that the only place where ξ is left is
the product κ−2 ξ in the first term. The expression for ∆T0 then becomes
∆T0 = (20κ−2 ξ + b)Φ̂0 ,
(53)
where b = 2ω − 3/2 = 101/270 = 0.37. For an ideal fluid the first term vanishes and the formula
reduces to ∆T0 = bΦ0 . In approximate calculations, one replaces b by b0 obtained in the limit
ζre ≫ 1, b0 = 2ω0 − 3/2 = 3/10, see par. 9.5 in [15].
3.2
Power spectrum
The power spectrum of CMB anisotropies are the coefficients Cl in the formula
D
E
1 X
(2l + 1)Cl Pl (cos θ),
C(θ) ≡ ∆T0 (n1 )∆T0 (n2 )n1 ·n2 =cos θ =
4π
(54)
where the angle brackets denote averaging over an ensemble of universes and Pl are Legendre
polynomials. (We will use this definition, although when presenting observational data one usually
regards Cl as dimensional quantities, defined in terms of δT0 rather than ∆T0 .) To compute Cl for
small l, we need to know the temperature fluctuations at present ∆T0 for small κ. This is, however,
not what appears in equation (51). The quantity there is Fourier coefficient of the deviation of the
temperature from its mean value, and the quantity Φ0 on the right hand side is Fourier coefficient
of the Newtonian potential at the beginning of Friedmann expansion. Thus, the equation must be
read as
∆T0k = (20κ−2 ξ + b)Φ0k .
(55)
From inflation one obtains a spectrum of Φ0 which is, within a good precision,
flat; that is,
Z
−3
′
−3/2
hΦ0k Φ0k′ i = Bk δ(k − k ) with constant B. By inserting ∆T0 (n) = (2π)
∆T0k e−iη0 k·n d3 k
into (54) and using the expression for hΦ0k Φ0k′ i, we find
2
Cl = B
π
Z∞
dk
(20κ−2 ξ + b)2 jl2 (kη0 ) ,
k
0
where jl is the spherical Bessel function. Next we pass from k to s = kη0 to obtain
2
Cl = Bb2
π
Z∞
ds
(ξ∗ s−2 + 1)2 jl2 (s) ,
s
0
14
(56)
where ξ∗ = (20/b)(η0 /η∗ )2 ξ. The integral can be computed with the help of the formula
Z∞
0
We find
Cl =
where
s−n jl2
ds
π Γ(2 + n) Γ(l − n2 )
.
=
s
8 · 2n Γ2 ( 3+n
Γ(l + 2 + n2 )
2 )
i
h8
ξ∗2
ξ∗
4
+
+ 1 Cl,id ,
15 (l + 3)(l + 2)(l − 1)(l − 2) 3 (l + 2)(l − 1)
Cl,id =
1
Bb2
.
l(l + 1) π
(57)
(58)
After b is replaced by b0 , the expression for Cl,id coincides with that in equation (9.44) in [15].
To complete the theory, we will find the numerical value of the coefficient of proportionality
between ξ∗ and ξ. For a flat universe with the density parameters Ωm and ΩΛ we have
1 ΩΛ −1/6
η0
=
η∗
3 Ωm
Zτ0
0
a 1/2
dτ
0
,
2/3
aeq
(sinh τ )
where
τ0 = sinh−1
Ω 1/2
Λ
Ωm
.
By using Ωm = 0.32 and ΩΛ = 0.68 (Planck results) and taking for the ratio a0 /aeq the value 3560
(computed for the cited Ωm and h = 0.67, which is also a Planck result), we obtain η0 /η∗ = 54.4
and
ξ∗ = 1.58 × 105 ξ.
(59)
The coefficients C0 , C1 and C2 are infinite, first of them even in case the cosmic medium is
p
fluid. (This is true for C0 unless ξ∗ = (15/2)(1 ± 3/5) and for C1 unless ξ∗ = 10. However,
as seen from the following discussion, there is no need to explore these singular cases in detail.)
The coefficients become finite if we take into account that the spectrum of perturbations is cut
off at some wave number kmin given by the duration of inflation; and the coefficient C2 , as well
as the coefficient C0 in case ξ = 0, become finite also if we introduce a small negative tilt of the
primordial spectrum, replacing B by Bsǫ with ǫ > 0. However, of the three coefficients we need
to consider C2 (quadrupole) only. Recall why the other two coefficients (monopole and dipole) do
not enter the theory. An individual observer has no clue how much the mean temperature Tobs he
has measured differs from the true mean temperature T , neither can he tell how much his velocity
with respect to CMB Vobs , which he determines by averaging the product ∆Tobs cos θ, differs from
his true velocity V . He identifies T with Tobs and V with Vobs ; and if we write the coefficients Cl
as
Z
E
D
Cl = ∆T0 (n1 ) ∆T0 (n2 )Pl dΩ2 ,
we can see that such identification means that the coefficients C0 and C1 are put equal to zero.
15
If we write the renormalized coefficient C2 as
1
C2r = uξ∗2 + ξ∗ + 1 C2id ,
3
(60)
we obtain by a straightforward calculation
u=
4
4 −1
ǫ (1 − sǫmin ) =
×
75
75
ǫ−1 ≡ uI if ǫ log(1/smin ) ≫ 1
log(1/smin ) ≡ uII if ǫ log(1/smin ) ≪ 1
.
(61)
The parameter ǫ is the deviation of the scalar spectral index nS from 1, ǫ = 1−nS , so that ǫ = 0.04
and uI = 4/3 for the observational mean value of nS , which is 0.96 (once again a Planck result).
The value of the parameter smin depends on the inflationary scenario. It holds
smin = kmin η0 =
robs,F
Nmin rh,inf
Nmin
rh0
=
=
≈
,
λ-max0
λ-max,F
N H −1
N
where the index ‘F’ denotes the beginning of Friedmann expansion, robs is the radius of the part
of the universe which is observable today, the index ‘inf’ denotes the beginning of inflation, N is
the number of e-foldings during inflation, Nmin is minimum N and H is Hubble constant during
inflation. For inflation on GUT and Planck scale (new and chaotic) Nmin is about 60 and 70 and
N is typically about 2000 and 107 ÷ 1011 respectively. This yields uII = 0.18 and 0.63 ÷ 1.12,
hence uII ≪ uI for inflation on the GUT scale and uII . uI for inflation on the Planck scale.
According to (61) the value of u in both asymptotic regimes is given by the less of uI and uII .
Thus, in approximate calculations we can use u = uII ; in other words, we can ignore the tilt of
the primordial spectrum and take into account only its cutoff.
The observed values of Cl must coincide with the theoretical ones within cosmic variance,
r
2
.
(62)
Cl,obs ∈ h1 − δl , 1 + δl iCl , δl =
2l + 1
Denote the relative deviation of our Cl from Cl,id by ∆l . After identifying Cl,id with Cl,obs we find
that ξ∗ must assume values between −4.15 and 2.30 (a consequence of 1/(1 + ∆2r ) ≥ 1 − δ2 , if one
inserts for u the value uII computed for inflation on GUT scale). However, the identification fails
for the quadrupole, whose observational value lies approximately at the lower limit of the interval
.
allowed by cosmic variance, C2,obs = (1 − δ2 )C2,id . The theory with a solid can provide for that
only for ξ∗ between −1.85 and 0 (a consequence of ∆2r ≤ 0).
It was suggested that the lack of power in the quadrupole is not an effect of cosmic variance, but
comes from a cutoff of the primordial spectrum on the scale smin ∼ 1 [16, 17, 18]. Such cutoff can
be caused either by short duration of inflation of by a jump-like variation of inflationary potential
near the end of inflation. Suppose the integral in the expression (56) is cut off at smin = 2 and
write the power spectrum as Cl = (pl ξ∗2 + 2ql ξ∗ + rl )Cl,id , where Cl,id is the power spectrum in
the theory with pure fluid in which no cutoff occurs. After imposing the condition that the ratio
16
Cl /Cl,id equals 1 − δ2 for l = 2 and 1 for l > 2 within cosmic variance, we find that the maximum
ξ∗ is 0.81 (a consequence of p2 ξ∗2 + 2q2 ξ∗ + r2 = 1) and the minimum ξ∗ is −3.21 (a consequence
of p3 ξ∗2 + 2q3 ξ∗ + r3 = 1/(1 + δ3 )).
A dominant contribution to Cl comes from the part of the integration region around s = l.
.
Thus, relevant values of κ for the given l are κ ∼ (η∗ /η0 )l = l/50. On the other hand, the
magnitude of the parameter ξ∗ cannot exceed values |ξ∗ | ∼ 1, which corresponds to |ξ| ∼ 10−5 and
p
|ξ| ∼ 1/300. If we take into account that the CMB anisotropies can be viewed as large-scale at
most up to l ∼ 100, we can see that the values of κ that give a substantial contribution to the
p
CMB power spectrum for extremal values of ξ are κ ∼ (0.1 ÷ 5) |ξ|.
The ratio Cl /Cl,id as a function of l, computed for extremal ξ∗ in the theory with cutoff at
smin ∼ 1, is depicted in fig. 2. The full and dashed lines represent theoretical and observational
Cl
2
Cl,id
0.51 10
Cl
1.5
Cl,id
5
1.5
5
2.03 10
1
1
0.5
0.5
0
10
2
0
l
2
10
l
Fig. 2: Large-scale power spectrum, computed for extremal values of shear modulus consistent with the observed spectrum
values respectively, the shaded strips represent cosmic variance and the numbers are the values
of ξ. On the left panel, the quadrupole is located on the borderline of the region allowed by
cosmic variance, so that in the theory with solid the cutoff serves to make positive shear modulus
consistent with observations rather than explain the lack of power in the quadrupole.
4
Conclusion
We have calculated the large-scale part of the CMB power spectrum, the Sachs-Wolfe plateau, in
a universe containing radiation-like solid with constant shear modulus to energy ratio ξ. For that
purpose, we had to extend the theory developed in [13] to the case when there is also nonrelativistic matter in the universe. It turned out that in case |ξ| ≪ 1 the theory of long-wavelength
17
perturbations, which was formulated for the parameter κ ≈ kηeq much less than
p
|ξ|, holds also
in the opposite limit. If we sought exact formulas covering both asymptotic regions of κ but still
(0)
restricted to κ ≪ 1, the theory would become quite complicated. (In the expression (27) for Er
(0)
we would have to replace ξ by ξ + κ2 X 2 X̂+ /(36X+ ), which would yield an equation for Fr
that
cannot be solved exactly.) Instead, we have used the original theory to interpolate between the two
p
regimes, which allowed us to consider values of κ from the intermediate region, κ ∼ (0.1 ÷ 5) |ξ|,
and confirm the conclusion of [13] that in order to accommodate observations, the parameter ξ
must satisfy |ξ| . 10−5 . We have restricted ourselves to the Sachs-Wolfe plateau and did not
investigate the position and width of acoustic peaks. Obviously, for such small |ξ| they would look
practically the same as in a universe filled with pure fluid. Of course, |ξ| could be greater if the
effect of the solid was compensated by the tilt of the primordial spectrum, but to assume such
canceling of two independent effects does not seem reasonable.
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18