Early Universe Sources for Cosmic Microwave Background Non-Gaussianity

Abstract

In the framework of inflationary models with non-vacuum initial states for cosmological perturbations, we study non-Gaussian signatures on the cosmic microwave background radiation produced by a broken-scale-invariant model which incorporates a feature at a privileged scale in the primordial power spectrum.

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arXiv:astro-ph/0302396v1 19 Feb 2003
February 2, 20083:1WSPC/Guidelines-IJMPAFriedmann˙gangui˙2002
International Journal of Modern Physics A
c ? World Scientific Publishing Company
Early Universe Sources for CMB Non-Gaussianity
ALEJANDRO GANGUI
Instituto de Astronom´ ıa y F´ ısica del Espacio, 1428 Buenos Aires, Argentina, and
Departamento de F´ ısica, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina.
Received (Day Month Year)
Revised (Day Month Year)
In the framework of inflationary models with non-vacuum initial states for cosmo-
logical perturbations, we study non-Gaussian signatures on the cosmic microwave back-
ground (CMB) radiation produced by a broken-scale-invariant model which incorporates
a feature at a privileged scale in the primordial power spectrum.
Keywords: cosmic microwave background; inflation.
1. Introduction
The common belief that the CMB is Gaussian distributed can be directly traced
back to the generic assumption that the quantum fluctuations of the inflaton field
are placed in the vacuum state1. Relaxing this assumption might lead to detectable
signatures in various astrophysical tests, most interestingly in future CMB and
large-scale structure observations. In this note, we study CMB non-Gaussian signa-
tures predicted within inflationary models with non-vacuum initial states for cos-
mological perturbations. The model incorporates a privileged scale, which implies
the existence of a feature in the primordial power spectrum. The model predicts a
vanishing three-point correlation function for the CMB temperature anisotropies2.
We here focus on the first non-vanishing moment, the CMB four-point function at
zero lag, namely the kurtosis, and compute its expected value for different locations
of the primordial feature in the spectrum, as suggested in the literature to conform
to observations of large scale structure3,4.
2. Two-point correlation function for non-vacuum initial states
We consider non-vacuum states for the cosmological perturbations of quantum me-
chanical origin. Let D(σ) be a domain in momentum space, such that if k is between
0 and σ, the domain D(σ) is filled by n quanta, while otherwise D contains nothing.
The state |Ψ1(σ,n)? is defined by
(c†
√n!
p?∈D(σ)
|Ψ1(σ,n)? ≡
?
k∈D(σ)
k)n
|0k?
?
|0p? =
?
k∈D(σ)
|nk?
?
p?∈D(σ)
|0p?.(1)
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Alejandro Gangui
The state |nk? is an n-particle state satisfying, at conformal time η = ηi: ck|nk? =
√n|(n − 1)k? and c†
?Ψ1(σ,n)|Ψ1(σ′,n′)? = δ(σ − σ′)δnn′.
It is clear from the definition of the state |Ψ1? that the transition between the
empty and the filled modes is sharp. In order to “smooth out” the state |Ψ1?, we
consider a state |Ψ2? as a quantum superposition of |Ψ1?. In doing so, we introduce
an, a priori, arbitrary function g(σ;kb) of σ. The definition of the state |Ψ2(n,kb)?
?+∞
0
k|nk? =√n + 1|(n + 1)k?. We have the following property
(2)
|Ψ2(n,kb)? ≡dσg(σ;kb)|Ψ1(σ,n)?, (3)
where g(σ;kb) is a given function which defines the privileged scale kb. We as-
sume that the state is normalized and therefore?+∞
|Ψ1(σ,n)?, for any domain D one has
?Ψ1(σ,n)|cpcq|Ψ1(σ,n)? = ?Ψ1(σ,n)|c†
?Ψ1(σ,n)|cpc†
?Ψ1(σ,n)|c†
In these formulas, δ(q ∈ D) is a function that is equal to 1 if q ∈ D and 0 otherwise.
These relations will be employed in the sequel for the computation of the CMB
temperature anisotropies for the different non-vacuum initial states.
0
g2(σ;kb)dσ = 1. In the state
pc†
q|Ψ1(σ,n)? = 0, (4)
q|Ψ1(σ,n)? = nδ(q ∈ D)δ(p − q) + δ(p − q),
pcq|Ψ1(σ,n)? = nδ(q ∈ D)δ(p − q).
(5)
(6)
2.1. Two-point function of the CMB temperature anisotropy
The spherical harmonic expansion of the cosmic microwavebackgroundtemperature
anisotropy, as a function of angular position, is given by
δT
T(e) =
?
ℓm
aℓmYℓm(e)withaℓm=
?
dΩeδT
T(e)Y∗
ℓm(e).(7)
As we are interested in a non-Gaussian signature of primordial origin we will be
focusing on large angular scales, for which the main contribution to the temperature
anisotropy is given by the Sachs-Wolfe effect, namely, δT/T(e) ≃ (1/3)Φ[ηlss,e(η0−
ηlss)], where Φ(η,x) is the Bardeen potential, while η0and ηlssdenote respectively
the conformal times now and at the last scattering surface. Note that the previous
expression is only valid for the standard Cold Dark Matter model (sCDM). In
general, we might also be interested in the case where a cosmological constant is
present (ΛCDM model) since this seems to be favored by recent observations. Then,
the integrated Sachs-Wolfe effect plays a non-negligible role on large scales and the
expression giving the temperature fluctuations is not as simple as the previous one.
In the theory of cosmological perturbations of quantum mechanical origin, the
Bardeen variable becomes an operator, and its expression can be written as
Φ(η,x) =ℓPl
ℓ0
3
4π
?
dk
?
ck(ηi)fk(η)eik·x+ c†
k(ηi)f∗
k(η)e−ik·x
?
, (8)

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Early Universe Sources for Cosmic Microwave Background Non-Gaussianity
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where ℓPl = (G¯ h)1/2is the Planck length. In the following, we will consider the
class of models of power-law inflation since the power spectrum of the fluctuations
is then explicitly known. In this case, the scale factor reads a(η) = ℓ0|η|1+β, where
β ≤ −2 is a priori a free parameter. However, in order to obtain an almost scale-
invariant spectrum, β should be close to −2. In the previous expression of the scale
factor, the quantity ℓ0 has the dimension of a length and is equal to the Hubble
radius during inflation if β = −2. The parameter ℓ0also appears in Eq. (8). The
mode function fk(η) of the Bardeen operator is related to the mode function µk(η)
of the perturbed inflaton through the perturbed Einstein equations. In the case of
power-law inflation and in the long wavelength limit, the function fk(η) is given
in terms of the amplitude ASand the spectral index ns of the induced density
perturbations by
k3|fk|2= ASkns−1.
Using the Rayleigh equation and the completeness relation for the spherical har-
monics and after some algebra we get
(9)
aℓm=ℓPl
ℓ0eiπℓ/2
?
dk
?
ck(ηi)fk(η) + c†
−k(ηi)f∗
k(η)
?
jℓ[k(η0− ηlss)]Y∗
ℓm(k) .(10)
At this point we need to somehow restrict the shape of the domain D. We assume
that the domain only restricts the modulus of the vectors, while it does not act on
their direction. Then, from Eq. (10), one deduces
?Ψ1(σ,n)|aℓ1m1a∗
ℓ2m2|Ψ1(σ,n)? =ℓ2
?σ
0
Pl
ℓ2
0
?
Cℓ1+ 2nD(1)
ℓ1(σ)
?
δℓ1ℓ2δm1m2, (11)
D(1)
ℓ(σ) ≡
j2
ℓ[k(η0− ηlss)]k3|fk|2dk
k
(12)
Thus, the multipole moments C(1)
2nD(1)
ℓ(σ) , where Cℓis the “standard” angular power spectrum, i.e., the multipole
obtained in the case where the quantum state is the vacuum, i.e., n = 0. Let us
calculate the same quantity in the state |Ψ2?. Performing a similar analysis as the
above one, we find7
ℓ, in the state |Ψ1?, are given by C(1)
ℓ(σ) = Cℓ+
?Ψ2(n,kb)|aℓ1m1a∗
ℓ2m2|Ψ2(n,kb)? =ℓ2
=π
2AS
0
Pl
ℓ2
0
?
Cℓ1+ 2nD(2)
ℓ1
?
δℓ1ℓ2δm1m2,(13)
D(2)
ℓ
?+∞
J2
ℓ+1/2(k)¯h(k)knS−3dk (14)
where, to reach this eqn, we defined g2(σ;kb) ≡ dh/dσ [we will see below that
this function h(kb) cannot be arbitrary] and we integrated by parts, and then we
defined¯h(k) ≡ h(∞)[1 − h(k)/h(∞)]. In this we have not assumed anything on
h(∞) or h(0). We see that the relation g2(k) ≡ dh/dk requires the function h(k) to
be monotonically increasing with k. It is interesting that, already at this stage of
the calculations, very stringent conditions are required on the function h(k) which
is therefore not arbitrary. This implies that the function¯h(k) which appears in the

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Alejandro Gangui
correction to the multipole moments is always positive, vanishes at infinity and is
monotonically decreasing with k. An explicit profile for¯h(k) is given in Fig. 1. The
total power spectrum of the Bardeen potential can be written as
k3|Φk|2∝ ASknS−1
?
1 + 2nh(∞)
?
1 −h(k)
h(∞)
??
= ASknS−1[1 + 2n¯h(k)]. (15)
Observations indicate that nS≃ 1 and for simplicity we will take nS= 1. As we
have seen previously, we can write the multipole moments in the state |Ψ2? as
C(2)
ℓ
definition of D(2)
ℓ
by
= Cℓ+ 2nD(2)
ℓ. Substituting the well-known expression for the Cℓ’s and the
given by Eq. (14), one finds that the coefficients C(2)
ℓ
are given
C(2)
ℓ
= AS
π
2
?
1
23−ns
Γ(3 − ns)Γ[ℓ + (ns− 1)/2]
Γ2[(4 − ns)/2]Γ[ℓ − (ns− 5)/2]+ 2n¯D(2)
ℓ
?
. (16)
As a next step, one has to normalize the spectrum (need to determine the value of
AS). We choose to use the value of Qrms−PS= T0[5C(2)
with T0= 2.7K measured by the COBE satellite. Thus, we compute the quadrupole
and then
2/(4π)]1/2(ℓPl/ℓ0) ∼ 18µK
AS=8
5
Q2
rms−PS
T2
0
ℓ2
ℓ2
Pl
0
?1
6π+ 2n¯D(2)
2
?−1
, (17)
for nS= 1. The band power δTℓgives
δTℓ=Qrms−PS
T0
?
12
5
?
?
?
?1 + 2nπℓ(ℓ + 1)¯D(2)
ℓ
1 + 12nπ¯D(2)
2
.(18)
The n-dependence in the above expression is the correction due to the non-vacuum
initial state. We easily check that if n = 0 the corresponding band powers are
constant at large angular scales.
Finally, we calculate the two-point correlation function at zero lag in the state
|Ψ2?. Using Eqs. (7), (13), the second moment, µ2, of the distribution is given by
??δT
ℓ2
µ2≡
T(e)
?2?
=ℓ2
Pl
0
?
ℓ
2ℓ + 1
4π
C(2)
ℓ
.(19)
Once we have reached this point, an obvious first thing to do is to check that the two-
point correlation function calculated above is consistent with present observations.
2.2. Comparison with observations
Among the available observations that one can use to check the predictions of
theoretical models, two are key in cosmology: the CMB anisotropy and the matter-
density power spectra. We will not study in details all the predictions that can
be done from the two-point correlation function since our main purpose in this
note is to calculate the non-Gaussianity which is a clear specific signature of a non

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Early Universe Sources for Cosmic Microwave Background Non-Gaussianity
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?
?
??
Fig. 1.
the matter power spectrum normalized to COBE for different numbers n of quanta in the initial
state. The cosmological parameters are those corresponding to the sCDM model, namely, h = 0.65,
ΩΛ= 0, Ωb= 0.05, Ωcdm= 0.95 and nS= 1. The parameters describing the non vacuum state
are kphys
b
= 0.052hMpc−1and α = 2.5. The data points represent the power spectrum measured
by the PSCz survey.
The function¯h(k) for different values of α (left panel). On the right panel we show
vacuum state. So, we just compute the matter power spectrum to demonstrate that
it fits reasonably well the available astrophysical observations for some values of the
free parameters. In addition, this illustrates well the fact that, using the available
observations, we can already put some constraints on the free parameters. A simple
ansatz for the function¯h(k) is represented in Fig. 1 and can be expressed as
¯h(k) =1
2
?
1 − tanh
?
αlnk
kb
??
.(20)
With it, the matter power spectrum today, after taking into account the transfer
function T(k) which describes the evolution of the Fourier modes inside the horizon,
can be written
P(k) = T2(k)16π
5H4
0
Q2
rms−PS
T2
0
?1
6π+ 2n¯D(2)
2(kb)
?−1?
1 + 2n¯h(k)
?
kphys. (21)
The sCDM transfer function is given approximatively by the numerical fit5
T(k) =ln(1 + 2.34q)
2.34q
?
1 + 3.89q + (16.1q)2+ (5.46q)3+ (6.71q)4
?−1/4
, (22)
where q and the shape parameter Γ can be written as
q ≡ k/[(hΓ)Mpc−1]Γ ≡ Ω0he−Ωb−√2hΩb/Ω0, (23)
where Ω0 is the total energy density to critical energy density ratio and Ωb rep-
resents the baryon contribution. Or, more explicitly, we take Ω0 = ΩΛ+ Ωm =
ΩΛ+ Ωcdm+ Ωb. We have now normalized the matter power spectrum to COBE.
It is important to realize that the above procedure only works for the sCDM model
since we have used the Sachs-Wolfe equation. The sCDM matter power spectrum
is depicted in Fig. 1. The measured power spectrum of the IRAS Point Source