Beyond the power spectrum: primordial and secondary non-Gaussianity in the microwave background

Abstract

Cosmic microwave background observations are most commonly analyzed by
estimating the power spectrum. In the limit where the CMB statistics are
perfectly Gaussian, this extracts all the information, but the CMB also
contains detectable non-Gaussian contributions from secondary, and possibly
primordial, sources. We review possible sources of CMB non-Gaussianity and
describe statistical techniques which are optimized for measuring them,
complementing the power spectrum analysis. The machinery of $N$-point
correlation functions provides a unifying framework for optimal estimation of
primordial non-Gaussian signals or gravitational lensing. We review recent
results from applying these estimators to data from the WMAP satellite mission.

Full text

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arXiv:1111.1783v1 [astro-ph.CO] 8 Nov 2011
Beyond the power spectrum: Primordial and secondary non-Gaussianity in the microwave background
ASP Conference Series, Vol. 432, 2009
Edited by L. M. Stanford, J. D. Green, L. Hai, and Y. Mao
Beyond the power spectrum: primordial and secondary
non-Gaussianity in the microwave background
Kendrick M. Smith
Department of Astrophysical Sciences, Princeton University
Abstract.
analyzed by estimating the power spectrum. In the limit where the CMB statis-
tics are perfectly Gaussian, this extracts all the information, but the CMB also
contains detectable non-Gaussian contributions from secondary, and possibly
primordial, sources. We review possible sources of CMB non-Gaussianity and
describe statistical techniques which are optimized for measuring them, comple-
menting the power spectrum analysis. The machinery of N-point correlation
functions provides a unifying framework for optimal estimation of primordial
non-Gaussian signals or gravitational lensing. We review recent results from
applying these estimators to data from the WMAP satellite mission.
Cosmic microwave background observations are most commonly
1.Introduction
Observations of the cosmic microwave background have had an enormous im-
pact on our understanding of cosmology and the level of precision with which
cosmological parameters can be constrained. Beginning with first detection of
anisotropy on large angular scales by COBE (Smoot et al. 1992), successive
generations of experiments have provided increasingly precise measurements
of CMB temperature flucutations on smaller scales (e.g. Hanany et al. 2000;
Netterfield et al. 2002; Hinshaw et al. 2003; Kuo et al. 2007; Ade et al. 2008;
Lueker et al. 2009; Fowler et al. 2010). In particular, the WMAP satellite has
measured CMB fluctuations on the full sky with high signal-to-noise for all an-
gular scales larger than ≈20 arcmin.
It is convenient to represent the CMB in harmonic space. If T(n) denotes
the value of the temperature in line-of-sight direction n, then we expand in
spherical harmonics:
∞
?
to define the multipoles aℓm. In this representation, the most general rotationally
invariant two-point correlation function is:
T(n) =
ℓ=2
ℓ
?
m=−ℓ
aℓmYℓm(n) (1)
?aℓ1m1aℓ2m2? = (−1)m1Cℓ1δℓ1ℓ2δm1,−m2
(2)
where this equation defines the power spectrum Cℓ.
CMB observations are typically analyzed via the power spectrum: the main
result of the analysis is a measurement of Cℓand an estimate of the statistical
uncertainty, which can be compared with theoretical predictions for the power
spectrum. This approach is powerful because the CMB is a Gaussian field to a
1

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Kendrick M. Smith
good approximation. Gaussianity implies that the full probability distribution
for the CMB map (i.e. the multivariate PDF for the variables aℓm) is determined
by the two-point correlation function, so that the power spectrum contains all
the information. However, there are weak sources of non-Gaussianity which
contain qualitatively new cosmological information, and require statistical tech-
niques which complement the power spectrum. In this article, we will review
sources of non-Gaussianity either in the early universe during inflation, or in
the late universe via gravitational lensing. Our emphasis will be on statistical
techniques for extracting these non-Gaussian signals, and we will present results
from applying these techniques to WMAP data when possible.
2. Primordial non-Gaussianity
2.1. Three-point signals from inflation
Consider the simplest model of inflation: single-field slow-roll inflation with
standard kinetic term,
L =
M2
2
pl
R −1
2(∂µφ)(∂µφ) − V (φ) (3)
In such a model, the adiabatic curvature perturbation ζ(k) generated during
inflation is a Gaussian field; in particular the three-point correlation function is
zero.1However, there are inflationary models which can generate a three-point
function which is observationally distinguishable from zero. In this section we
will review the phenomenology of such models.
Let us first make some general mathematical comments.
attention to models which satisfy statistical translation and rotation invariance,
then the three-point function of the curvature perturbation ζ must be of the
form
?ζ(k1)ζ(k2)ζ(k3)? = F(k1,k2,k3)(2π)3δ3(k1+ k2+ k3)
If we restrict
(4)
where the function F is called the bispectrum, and only depends on the lengths ki
of the Fourier wavenumbers kiwhich form a closed triangle. For the inflationary
models we will consider in this section, the bispectrum will have a simple de-
pendence on overall scale (roughly F(αk1,αk2,αk3) ≈ α−6+2(1−ns)F(k1,k2,k3))
so that we can think of the bispectrum as a function of a 2-parameter family of
triangle shapes without regard to overall scale. We will use the term “squeezed”
for a triangle shape with k1≪ k2(which implies k1≪ k3), “flattened” for a
shape with k1≈ (k2+ k3), and “equilateral” for a shape with k1≈ k2≈ k3.
The curvaton model is a multifield model of inflation in which the source
of primordial curvature fluctuations is not the inflaton, but a second field σ
which does not dominate the energy density during inflation, but decays after
the inflaton, when the curvaton is oscillating near the minimum of its potential
(Linde & Mukhanov 1997; Moroi & Takahashi 2001; Lyth, Ungarelli & Wands
1This statement is an approximation, but a good one: the three-point function in such models
corresponds roughly to floc
2003; Maldacena 2003).
NL≈ (5/12)(1 − ns), far too small to be detectable (Acquaviva et al.

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Non-Gaussianity in the CMB
3
2003).
(δσ/σ) and the fraction f = ρσ/ρtot of the energy density due to the curva-
ton (both quantities evaluated at curvaton decay) are free parameters of the
model. The perturbation to the curvaton energy density is given by (δρσ/ρσ) =
2(δσ/σ) + (δσ/σ)2, and can therefore be a non-Gaussian field if the quadratic
term is non-negligible compared to the linear term. There is a region of pa-
rameter space (i.e. f ≪ 1 with Pδσ/σ= O(10−9f−2)) in which the primordial
curvature fluctuation (after curvaton decay) is significantly non-Gaussian and
has a power spectrum which is consistent with CMB observations. The non-
Gaussian curvature fluctuation in the curvaton model can be written:
ζ(x) = ζG(x) −3
The power spectrum P(δσ/σ)of the fractional curvaton perturbation
5flocal
NLζG(x)2
(5)
where ζG is a Gaussian field, and flocal
is purely conventional.) If we write this equation in Fourier space2, we get the
following form of the bispectrum F:
NL
is a free parameter. (The factor 3/5
F(k1,k2,k3) = flocal
NLFlocal(k1,k2,k3) (6)
where
Flocal(k1,k2,k3) = −6∆2
ζ
5
?
1
1k3
k3
2
+
1
2k3
k3
3
+
1
1k3
k3
3
?
(7)
The local bispectrum is largest for squeezed triangles. Conversely, there is a theo-
rem, the single-field consistency relation (Maldacena 2003; Creminelli & Zaldarriaga
2004), which states that the bispectrum is always small (i.e. O(10−2)) in squeezed
triangles, provided that inflation is single-field (but allowing for aribtrary self-
interactions of the inflaton). Thus the presence of a second field in the curvaton
model is actually a necessary ingredient for any model which generates flocal
larger than O(10−2).
For single-field inflation, there is also a theorem (Senatore, Smith & Zaldarriaga
2010) which shows that the most general primordial bispectrum is:
NL
F(k1,k2,k3) = fequil
NLFequil(k1,k2,k3) + forthog
NL
Forthog(k1,k2,k3) (8)
where the “equilateral” and “orthogonal” bispectra are defined by:3
Fequil(k1,k2,k3)=
−18∆2
ζ
5
(k1+ k2− k3)(k1− k2+ k3)(−k1+ k2+ k3)
k3
3Fequil(k1,k2,k3) +36∆2
5
k2
1k3
1
1k2
2k3
3
Forthog(k1,k2,k3)=
ζ
2k2
3
(9)
2Note that in Eqs. (7) and (9), we have assumed scale invariance, so that Pζ(k) = ∆ζk−3.
See Senatore, Smith & Zaldarriaga (2010) for expressions for the primordial bispectra with a
power law spectrum Pζ(k) = ∆ζk−3(k/k0)ns−1.
3The equilateral and orthogonal bispectra defined in Eq. (9) are actually approximations to
more precise expressions for the bispectra generated during inflation. The approximation is
made so that F(k1,k2,k3) will be a sum of a small number of terms which are factorizable
in the form f(k1)g(k2)h(k3). As will be discussed in the next subsection, this factorizability
condition is necessary in order to make the data analysis computationally tractable.

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Kendrick M. Smith
The equilateral shape is largest for equilateral triangles (as the name suggests)
while the orthogonal shape changes sign between equilateral and flattened tri-
angles. Both single-field shapes vanish in the squeezed limit, as required by the
consistency relation.
For a particular single-field model, such as DBI inflation (Alishahiha, Silverstein & Tong
2004) or ghost inflation (Arkani-Hamed et al. 2004), the coefficients fequil
forthog
NL
in Eq. (8) can be calculated in terms of fundamental parameters of the
model. In this article, our emphasis will be on data analysis. We simply remark
that the forms of the bispectra Flocal, Fequil, Forthoggiven above provide a point
of contact between theory and data. Given data from a CMB experiment like
WMAP, our job is to determine observational limits on the three fNLparam-
eters with statistical errors which are as small as possible. Once this has been
done, any inflationary model can be compared with observations by calculating
the fNLparameters.
Primordial non-Gaussianity is a particularly interesting probe of inflation
because it can rule out qualitative classes of models. A robust detection of
flocal
NL?= 0 would rule out all single-field models. Detection of any nonzero fNL
parameter would rule out the simplest inflationary model, single-field inflation
with standard kinetic term and slow-roll potential V (φ). In the next two sub-
sections we will turn to data analysis: we will construct optimal estimators for
constraining the fNLparameters from data, and present results from WMAP.
NL
and
2.2.Estimators for primordial non-Gaussianity
Let us begin by making some general statements about how three-point signals
are estimated from data, before specializing to the case of the three inflationary
shapes from the preceding subsection.
We will make the approximation that the evolution from the 3D intitial
curvature fluctuation ζ(k) to the 2D CMB aℓmis a linear operation, so that the
three-point function of ζ translates linearly into the three-point function of the
CMB:
?aℓ1m1aℓ2m2aℓ3m3? = fX
NLBX
ℓ1m1,ℓ2m2,ℓ3m3
(10)
where BX
When we measure the CMB in a real experiment, we measure the aℓm’s
plus some instrumental noise:
ℓ1m1,ℓ2m2,ℓ3m3is different for each X ∈ {local,equil,orthog}.
aobs
ℓm= atrue
ℓm+ anoise
ℓm
(11)
We will assume that the instrumental noise is Gaussian, and that the total
covariance matrix of the observed aℓm’s is given by the sum of a signal term
(which is diagonal in ℓ,m) and a noise term (which is non-diagonal):
?aℓ1m1aℓ2m2? = (−1)m1Cℓ1δℓ1ℓ2δm1,−m2+ Nℓ1m1,ℓ2m2
(12)
The detailed form of the noise covariance matrix Nℓ1m1,ℓ2m2encapsulates details
of the experiment such as spatial gradients in instrumental noise level (repre-
sented by assigning different noise variance to different pixels), the sky mask
imposed to remove regions of high foreground emission (represented by assign-
ing infinite noise variance to pixels which are masked), and the shapes of the

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Non-Gaussianity in the CMB
5
beams. We will denote the total covariance matrix on the RHS of Eq. (12) by
Cℓ1m1,ℓ2m2and its inverse by (C−1)ℓ1m1,ℓ2m2. The matrix C represents all the
properties of the experiment that we will need to know about in order to write
down an estimator for fNL.
Now we can ask the following general question: Given BX
Cℓ1m1,ℓ2m2, how should we estimate the value of the coefficient fX
from the noisy CMB aℓm? The optimal (i.e. mininum variance) estimator was
found in (Creminelli et al. 2006):
ℓ1m1,ℓ2m2,ℓ3m3and
NLin Eq. (10)
ˆfX
NL=1
F
?
T[C−1a] − (C−1a)ℓm?
∂ℓmT[C−1a′]
?
a′
?
(13)
where the quantities T and ∂T are defined by (Smith & Zaldarriaga 2006):
T[C−1a]=
1
6
?
?
ℓimi
Bℓ1m1,ℓ2m2,ℓ3m3(C−1a)ℓ1m1(C−1a)ℓ2m2(C−1a)ℓ3m3
∂ℓmT[C−1a]=
1
2
ℓimi
Bℓm,ℓ1m1,ℓ2m2(C−1a)ℓ1m1(C−1a)ℓ2m2
(14)
The normalization constant F, which is included so thatˆfX
estimator of fX
estimatorˆfX
NLin Eq. (13) is a sum of a three-point term and a one-point “coun-
terterm” which reduces the estimator variance in the presence of inhomogeneous
noise. The one-point term vanishes if the noise is homogeneous, since rotation
invariance implies ?∂ℓmT[C−1a′]? = 0 for ℓ > 0. We will see more examples
of this estimator structure, with an N-point leading term plus counterterms of
lower order, in §3.3.
To evaluate the estimatorˆfX
NL, we need algorithms for computing aℓm→
(C−1a)ℓmand a → T[a]. The first algorithm depends on details of the exper-
iment being analyzed (via C−1) but not the form of the three-point function
Bℓ1m1,ℓ2m2,ℓ3m3; we will discuss its implementation in WMAP in the next sub-
section. The rest of this subsection is devoted to briefly describing the second
algorithm, which depends only on the three-point function.
For a completely general three-point function Bℓ1m1,ℓ2m2,ℓ3m3, there is no
algorithm for the operation a → T[a] which is faster than the O(ℓ5
space sum in Eq. (14). For current CMB experiments (with ℓmax= O(103)) this
is computationally prohibitive. However, if the three-point function is a sum of
N factorizable terms, in the sense that
NLwill be an unbiased
NL(i.e. ?ˆfX
NL? = fX
NL) can be determined by Monte Carlo. The
max) harmonic-
Bℓ1m1,ℓ2m2,ℓ3m3
=
N
?
i=1
α(i)
ℓ1ℓ2ℓ3β(i)
ℓ1ℓ2ℓ3γ(i)
ℓ1ℓ2ℓ3
(15)
×
?
(2ℓ1+ 1)(2ℓ2+ 1)(2ℓ3+ 1)
4π
?
ℓ1
m1
ℓ2
m2
ℓ3
m3
?
and N is not too large, then there is a fast (i.e. O(ℓ3
a → T[a] operation (Komatsu, Spergel & Wandelt 2005; Creminelli et al. 2006;
maxN)) algorithm for the