Limits on Primordial Non-Gaussianity from Minkowski Functionals of the WMAP Temperature Anisotropies

Abstract

We present an analysis of the Minkowski Functionals (MFs) describing the WMAP three-year temperature maps to place limits on possible levels of primordial non-Gaussianity. In particular, we apply perturbative formulae for the MFs to give constraints on the usual non-linear coupling constant fNL. The theoretical predictions are found to agree with the MFs of simulated CMB maps including the full effects of radiative transfer. The agreement is also very good even when the simulation maps include various observational artifacts, including the pixel window function, beam smearing, inhomogeneous noise and the survey mask. We find accordingly that these analytical formulae can be applied directly to observational measurements of fNL without relying on non-Gaussian simulations. Considering the bin-to-bin covariance of the MFs in WMAP in a chi-square analysis, we find that the primordial non-Gaussianity parameter is constrained to lie in the range -70<fNL<91 at 95% C.L. using the Q+V+W co-added maps. Comment: 9 pages, 4 figures, accpeted for publication in MNRAS

Full text

Page 1

arXiv:0802.3677v2 [astro-ph] 3 Jul 2008
Mon. Not. R. Astron. Soc. 000, 000–000 (0000)Printed 3 July 2008(MN LATEX style file v2.2)
Limits on Primordial Non-Gaussianity from Minkowski
Functionals of the WMAP Temperature Anisotropies
C. Hikage1,2⋆, T. Matsubara3, P. Coles1, M. Liguori4, F. K. Hansen5, S. Matarrese6,7
1School of Physics and Astronomy, Cardiff University, Queens Buildings, 5, The Parade, Cardiff CF24 3AA
2School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD
3Department of Physics and Astrophysics, Nagoya University, Chikusa, Nagoya 464-8602, Japan
4Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge,
Wilberfoce Road, Cambridge, CB3 0WA
5Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, 0315 Oslo, Norway
6Dipartimento di Fisica “G. Galilei” universit´ a di Padova, INFN Sezione di Padova, via Marzolo 8, 1-35131 Padova, Italy
7INFN, Sezione di Padova, via Marzolo 8, I-35131, Padova, Italy
Submitted 2008 February 25; Accpeted 2008 July 2
ABSTRACT
We present an analysis of the Minkowski Functionals (MFs) describing the WMAP
three-year temperature maps to place limits on possible levels of primordial non-
Gaussianity. In particular, we apply perturbative formulae for the MFs to give con-
straints on the usual non-linear coupling constant fNL. The theoretical predictions are
found to agree with the MFs of simulated CMB maps including the full effects of radia-
tive transfer. The agreement is also very good even when the simulation maps include
various observational artifacts, including the pixel window function, beam smearing,
inhomogeneous noise and the survey mask. We find accordingly that these analytical
formulae can be applied directly to observational measurements of fNLwithout rely-
ing on non-Gaussian simulations. Considering the bin-to-bin covariance of the MFs in
WMAP in a chi-square analysis, we find that the primordial non-Gaussianity param-
eter is constrained to lie in the range −70 < fNL< 91 (95% C.L.) using the Q+V+W
co-added maps.
Key words: Cosmology: early Universe – cosmic microwave background – methods:
statistical – analytical
1INTRODUCTION
The existence of non-Gaussianity in primordial density fields
has the potential to provide a unique observational probe
that will enable discrimination among wide variety of in-
flationary models of the early Universe. Versions of the in-
flation scenario based on the idea of a single slow-rolling
scalar field predict levels of non-Gaussianity too small to
be observed. On the other hand, multi-field inflation models
and models with a non-standard kinetic term for the infla-
ton may yield larger non-Gaussian effects which could in
principle be detected in current or next-generation obser-
vations (e.g. Bartolo et al. 2002; Bernardeau & Uzan 2002;
Lyth et al. 2003; Dvali et al. 2004; Arkami-Hamed et al.
2004; Alishahiha et al. 2004; Bartolo et al. 2004; Chen et al.
2007; Battefeld & Battefeld 2007; Koyama et al. 2007).
In this paper we focus on the local parametrisation of
primordial non-Gaussianity by including quadratic correc-
⋆chiaki.hikage@astro.cf.ac.uk
tions to the curvature perturbation during the matter era
(e.g. Komatsu & Spergel 2001):
Φ = φ + fNL(φ2− ?φ2?),
where φ represents an auxiliary random-Gaussian field and
fNL characterizes the amplitude of the non-linear contribu-
tion to the overall perturbation. This local form is motivated
by the simple slow-rolling single scalar inflation scenario and
other models, including curvaton models; for an alternative
parameterization of fNL, see (Creminelli et al. 2007a). Cur-
rent observations are not sufficiently sensitive to detect the
wavelength dependence of fNL so a constant fNL provides a
reazonable parameterization of the level of non-Gaussianity.
Analysis of the angular bispectrum for the WMAP
3-year data provides a constraint on fNL to lie between
−54 and 114 at the 95% C.L. (Komatsu et al. 2003;
Spergel et al. 2007). Creminelli et al. (2007a) obtains more
stringent constraint −36 < fNL < 100. On the other
hand, Yadav & Wandelt (2008) recently reported a detec-
tion of primordial non-Gaussianity at greater than 99.5%
significance. Further detailed analyses of non-Gaussianity is
(1)

Page 2

2
Hikage et al.
clearly necessary in order to reconcile and understand the
various constraints and claimed detections.
Different approaches to the study of non-Gaussianity
exploit different statistical properties and will be sensitive
to different aspects of the behaviour of the pattern being
tested. In general, there is no unique statistic to describe the
non-Gaussian nature of a sample in a complete manner. A
given method may have strong discriminatory power for one
particular form of non-Gaussianity, but this is not necessar-
ily the case for all possible alternative distributions. Testing
non-Gaussianity therefore requires a battery of complemen-
tary techniques rather than a single approach. It is particu-
larly important to use different statistical approaches in the
context of primordial non-Gaussianity, because the physical
mechanism responsible remains unknown. Furthermore, in
the real world, issues including survey masks and inhomo-
geneous noise have to be taken into consideration. Different
statistics may be sensitive to different systematics and fore-
ground artefacts, so that complementary analysis using dif-
ferent statistics are essential for a robust detection. Analyses
using different statistical methods are useful to validate or
refute basic theoretical models and constrain model parame-
ters more accurately. The most commonly used statistic for
non-Gaussian analysis is the bispectrum (or even trispec-
trum) which focuses on information contained within three-
point (or four-point) correlations. Other approaches repre-
sented by Minkowski Functionals (MFs) and genus statistics
(one of MFs) utilize information concerning the integrated
morphology and topology of the density structure, and are
dependent on all order of correlation functions. Their robust-
ness and generality therefore makes them ideal complements
to standard correlation analyses.
In this analysis, we focus on the local model of primor-
dial non-Gaussianity characterized by fNL in Equation (1).
Creminelli et al. (2007b) show that the bispectrum is the
optimal statistics in the estimation of fNL and then other
statistics (e.g., trispectrum) are useless even if there are dif-
ferent foreground contaminations. This is, however, only the
case when the local model exactly describes the real uni-
verse. Other forms of non-Gaussianity, different from the
one purely characterized by fNL, which may exist in a real
observation, could make the fit of the theoretical estima-
tion as a function of fNL to the observation worse and also
influence the estimation of fNL among different statistics.
Different statistical approaches are, therefore, still useful to
test the assumed model of primordial non-Gaussianity and
also possible observational systematics by checking if they
have a reasonable goodness of the fit to observations and
thus give a consistent limit on fNL compared to that that
from the bispectrum.
In this paper, we present a measurement of primor-
dial non-Gaussianity from the MFs of the WMAP three-
year temperature maps. We apply perturbative formulae
recently derived by Hikage et al. (2006) to do the com-
parison with observations; previous analyses rely on non-
Gaussian simulations (Komatsu et al. 2003; Spergel et al.
2007). The agreement of the theoretical predictions with
non-Gaussian simulations has already been established in
Sachs-Wolfe limit (Hikage et al. 2006). In this paper, we ap-
ply non-Gaussian simulations based on full radiative trans-
fer computations and then demonstrate that the analyti-
cal predictions accurately reproduce the simulation results.
Gott et al. (2007) already derived an analytical formula for
the genus statistic in the Sachs-Wolfe approximation to com-
pare with WMAP data. Our analysis takes more detailed
physics into account and is consequently more accurately
applicable to a wider range of scales.
Observational effects (including antenna beam pattern,
inhomogeneous noise and the survey mask) could be other
sources of confusion. From a comparison with simulations
including these observational issues, we find that the obser-
vational systematics are negligible to estimate the primor-
dial non-Gaussianity from WMAP data directly using our
method.
The organization of this paper is as follows. In §2 the
WMAP three-year data studied here are briefly introduced.
In §3 we test whether the perturbative formulae well describe
the MFs for the non-Gaussian simulation maps even includ-
ing the various observational effects mentioned above. In §4,
we show the MFs for WMAP three-year temperature maps
compared with theoretical formulae and give constraints on
fNL. § 5 is devoted to a summary and the conclusions.
2WMAP THREE-YEAR DATA
The CMB temperature maps derived from the WMAP ob-
servation are pixelized in HEALPix format with the total
number of pixels npix = 12N2
our analysis, we use the maps for Q, V and W frequency
bands with Nside = 512. The linearly co-added maps are
constructed using an inverse weight of the pixel-noise vari-
ance σ2
ferential assembly (DA) given in Bennett et al. (2003b) and
¯ Nobs represents the full-sky average of the effective number
of observations per each pixel. We adopt two maps with dif-
ferent combinations of frequency bands: V and W (written as
“V+W”) and Q, V and W (written as “Q+V+W”). The co-
added maps are masked with the Kp0 galaxy mask includ-
ing point-source mask provided by Bennett et al. (2003b),
which leaves 76.8% of the sky available for the data analy-
sis.
In comparison with WMAP observations to give con-
straint on fNLin §4, a ΛCDM cosmology is assumed with the
cosmological parameters at the maximum likelihood peak
from the WMAP three-year data only fit (Spergel et al.
2007): Ωb = 0.04309, Ωcdm = 0.211, ΩΛ = 0.74591, H0 =
71.227 km s−1Mpc−1, τ = 0.08982, and ns = 0.95537. The
amplitude of the primordial fluctuations has been normal-
ized by the first acoustic peak of the temperature power
spectrum, l(l + 1)Cl/(2π) = 5617.05(µK)2at l = 220
(Hinshaw et al. 2007).
side(G´ orski et al. 2005). In
0/¯ Nobs, where σ0 denotes the pixel noise for each dif-
3PERTURBATIVE FORMULAE VERSUS
NON-GAUSSIAN SIMULATIONS
3.1Perturbative Formulae of MFs for CMB with
Primordial Non-Gaussianity
The topology of random fluctuation fields is generally stud-
ied using their excursion sets, i.e. regions where the field
exceeds some threshold level. In a two-dimensional ran-
dom field such as a CMB temperature map, three MFs

Page 3

Primordial Non-Gaussianity from WMAP
3
are defined: the fraction of area V0 exceeding the thresh-
old, the total circumference V1 of all the entire excursion
set, and the corresponding Euler Characteristic V2 (Coles
1988). We measure MFs for CMB temperature maps as a
function of the threshold density ν, defined as the tempera-
ture fluctuation ∆T/T normalized by its standard deviation
σ0 ≡ ?(∆T/T)2?1/2. Based on the general formalism of per-
turbation theory for MFs (Matsubara 2003), Hikage et al.
(2006) derived perturbative formulae of the MFs as a func-
tion of the non-linear coupling parameter fNL (eq.[1]).
The MFs are separately written with the amplitude and
the function of ν as follows.
Vk(ν) = Akvk(ν).(2)
The amplitude Ak, which is determined only by the angular
power spectrum Cl, is given by
Ak=
1
(2π)(k+1)/2
ω2
ω2−kωk
?
σ1
√2σ0
?k
,(3)
σ2
j≡
1
4π
?
l
(2l + 1)[l(l + 1)]jClW2
l,(4)
where ωk ≡ πk/2/Γ(k/2 + 1) gives ω0 = 1, ω1 = 2, ω2 =
π and Wl represents the smoothing kernel determined by
the pixel and beam window functions and any additional
smoothing (e.g. a Gaussian kernel). In weakly non-Gaussian
fields, the function vk(ν) can be divided into the Gaussian
term v(G)
k
and the non-Gaussian term at lowest order ∆vk:
vk(ν) = v(G)
k
(ν) + ∆vk(ν,fNL).(5)
Each term has the following form
v(G)
k
= e−ν2/2Hk−1(ν),(6)
∆vk(ν,fNL)=e−ν2/2??1
k(k − 1)
6
6S(0)Hk+2(ν) +k
3S(1)Hk(ν)
+
S(2)Hk−2(ν)
?
σ0+ O(σ2
0)
?
,(7)
where Hn(ν) represent the n-th Hermite polynomials and
the skewness parameters S(k)are given in Equations [27-
29] of Hikage et al. (2006). The amplitude Ak (eq. [3]) is
not directly relevant to non-Gaussianity but is dependent
on the shape of Cl. We therefore concentrate on the non-
Gaussian term ∆vkhereafter. The quantity ∆vkis the same
as the relative difference of MFs, which are plotted in Fig. 2
in Hikage et al. (2006), except for its normalization factor;
in this paper the difference of MFs is normalized by Ak (3),
while the maximum value of MFs for Gaussian fields is used
in Hikage et al. (2006).
3.2 Comparison with Non-Gaussian Simulations
The above analytical formulae have already been found to
match accurately the MFs for non-Gaussian maps in Sachs-
Wolfe limit (Appendix C in Hikage et al. 2006). Here we test
them against non-Gaussian simulations including the full ra-
diative transfer function (Liguori et al. 2003, 2007). As we
mentioned in the introduction, actual observations of CMB
also involve different effects which may produce other confu-
sions: the pixel window function, beam smearing, the inho-
mogeneous noise, survey mask and so on. We include these
observational effects into the simulations to check whether
they could have a systematic effect on our topological mea-
sures.
The cosmology in the non-Gaussian simulations is
based on Lambda CDM, but the cosmological param-
eters have slightly different values from WMAP three-
year best-fit; Ωb = 0.05,Ωcdm = 0.25,ΩΛ = 0.7,H0 =
65 km s−1Mpc−1,τ = 0, and ns = 1. The amplitude of pri-
mordial fluctuations is, however, set to be same as WMAP
three-year best-fit value l(l + 1)Cl/(2π) = 5617.05(µK)2.
Observational effects related to WMAP data are in-
cluded as follows. First we convolve the original simulation
maps with the Q+V+W co-added beam transfer function
with inverse weight of the full-sky averaged pixel-noise vari-
ance in each DA. Next we add independent Gaussian noise
realizations following the noise pattern co-added with the
same weight. The simulation map is then masked with the
Kp0 Galaxy mask. Finally we smooth the simulation maps
using a Gaussian filter with a smoothing scale of θs,
Wl= exp
?
−1
2l(l + 1)θ2
s
?
.(8)
The MFs are sensitive to the resolution (or smoothing)
scale of a density field and thereby we can obtain a vari-
ety of information from density fields using different lev-
els of smoothing. The information extracted from varying
smoothing scales is nevertheless limited because they are all
derived from the same original field; the smoothed fields are
not independent. Here we focus on the field smoothed by
three different smoothing scales 10’, 20’ and 40’, where the
limit on fNL is sufficiently converged. To remove the effect
of the survey mask near the boundary of the mask, we only
use the pixels more than 2θs away from the boundary. The
sky fraction used in the analysis for each smoothing scale is
41% for θs = 40′, 62% for θs = 20′and 73% for θs = 10′.
The MFsforthemeasured
anisotropy are computed from the integral of the curvature
of iso-temperature contour lengths (the details are described
in Appendix A.1. of Hikage et al. 2006). The binning range
of ν is set to be −3.6 to 3.6 with 18 equally spaced bins of
ν per each MF. This binning way produces well converged
results irrespective of other choices of the range of ν and the
number of bins.
We obtain the normalized MFs (eq. [3]) with the am-
plitude Ak computed from Cl of each realization. Then the
residuals of the normalized MFs from Gaussian predictions
∆˜ vα are calculated at each bin of α, which denote a thresh-
old value ν, a kind of MF k, and a smoothing scale parame-
terized with θs or Nside. Even when the MFs of Gaussian re-
alizations are computed, however, the function ∆˜ vα are not
exactly equal to 0 due to the effect of pixelization, survey
mask and other numerical artifacts. We therefore measure
the deviations from the average of the measurements over
Gaussian realizations and subtract them as
CMBtemperature
∆vα = ∆˜ vα− [∆˜ vα]Gaussian,mean.
In Fig. 1, we compare the analytical predictions of vari-
ance, skewness, and MFs with the measurements from the
simulations for fNL = 100. The simulated results are the av-
erage over 200 realizations and the error-bars represent the
error for the average (the sample variance divided by the
square-root-of 200). The averaged measurements for Gaus-
(9)

Page 4

4
Hikage et al.
sian CMB maps are subtracted from those for non-Gaussian
maps in the simulated plots including variance and skewness
as well as MFs (see eq. [9]). In the plots, we adopt the Gaus-
sian maps which are generated from the same realizations
of linear potential fields (φ in eq. [1]) as the non-Gaussian
CMB maps. The sample variances, represented by the error-
bars, are cancelled very well in such plots so one can focus
on the systematic effect of primordial enon-Gaussianity. The
analytical formulae are found to agree with the simulations
extremely well even including all observational effects. This
indicates that we can measure fNL from direct comparison
of the analytical formulae with observations without hav-
ing to worry excessively about the presence of such sys-
tematics. We also check that both the artificial systematics
[∆˜ vα]Gaussian,mean and covariance matrix are not strongly
dependent on the details of cosmology. These results are en-
couraging, but not unexpected: being based on integrated
properties, the Minkowski Functionals are expected to be
robust to such effects.
4 CONSTRAINTS ON PRIMORDIAL
NON-GAUSSIANITY FROM WMAP
THREE-YEAR DATA
4.1Covariance Matrix for MFs
We have adopted a maximum likelihood method to esti-
mate the best-fit value of fNL and its associated uncer-
tainty. In ‘nearly Gaussian’ fields, the distribution functions
of ∆vα are well described as multivariate Gaussians. The
likelihood function of fNL is, therefore, simply proportional
to exp(−χ2(fNL)/2) where χ2(fNL) is computed using the
theoretical formulae (eq. [7]) as
χ2(fNL)=
?
[∆v(obs)
α′
αα′
[∆v(obs)
α
− ∆v(theory)
α
(fNL)]Σ−1
αα′
×− ∆v(theory)
α′
(fNL)], (10)
where α and α′denote the binning number of threshold val-
ues ν, different kinds of MF k, and smoothing scale param-
eterized with θs or Nside. The full covariance matrix Σαα′ is
required because MFs are strongly correlated between dif-
ferent ν, different kinds of MF, and also different Nsideor θs.
We estimate the covariance matrix of MFs from 1000 Gaus-
sian simulation maps including the pixel and beam window
function, Kp0 survey mask, and inhomogeneous noise for
WMAP three-year maps.
The MFs contain information about fluctuations at dif-
ferent scales, so the results depend on the choice of window
function. Here, two different types of window functions are
adopted (in addition to the beam window functions). One is
a Gaussian window function with the scale characterized by
θs (which is chosen to be sufficiently large compared with
the pixel size). The other is just the pixel window function
in HEALPix format with a scale characterized by Nside. The
multipole components with l higher than 2Nside are cut be-
cause they suffer from serious aliasing effects.
Before applying these ideas to the observational data,
we check if our method based on χ2analysis is valid using
simulations with primordial non-Gaussianity. We apply the
non-Gaussian simulations to check that the likelihood func-
tion using the equation (10) reproduces a valid probability
distribution of the true value f(true)
the likelihood function of f(true)
a Gaussian distribution around the best-fit value f(best)
NL
. Here we consider that
in each realization follows
NL
NL
as
P(f(true)
NL
|f(best)
NL
)=
1
√2πσfNL
exp
?
−1
2
?f(true)
NL
− f(best)
σfNL
NL
?2?
(11)
σfNL
=
??
αα′
∂Vα
∂fNL(Σ−1)αα′∂Vα′
∂fNL
?−1/2
(12)
where the binning number α represents a threshold ν for k-th
MF at a given scale Nside(or θsif the Gaussian smoothing is
added) and the covariance matrix Σαα′ is numerically esti-
mated from Gaussian simulations. The function ∂Vα/∂fNL
is independent of fNL (see equation [7]) and thus the un-
certainty σfNLis independent of fNL. According to Bayes’
theorem, f(best)
NL
should distribute around f(true)
way:
NL
in the same
P(f(best)
NL
|f(true)
NL
) = P(f(true)
NL
|f(best)
NL
)(13)
We estimate the distribution function of f(best)
non-Gaussian CMB simulated maps at a given f(true)
then compare with the equation (11). The simulated maps
include observational effects represented by pixel and beam
window functions, noise, and Kp0 survey cut for WMAP
three-year data. Fig. 2 shows the theoretical predictions of
P(f(best)
NL
|f(true)
from the MFs for the combined maps at Gaussian smoothing
scales θs = 20′and 10′where the uncertainty is σfNL= 44.
The histograms show the distribution of f(best)
(non-)Gaussian realizations. The averages of the best fit val-
ues of fNL from the simulations are respectively 0 ± 44 (for
f(true)
NL
= 0) and 101 ± 46 (for f(true)
tions reproduce the theoretical predictions of the likelihood
function very well. Our method is thus well established to
give constraints on fNL from WMAP three-year map.
NL
from 200
NL
and
NL
) at f(true)
NL
= 0 (solid) and 100 (dotted).
NL
from 200
NL
= 100). The simula-
4.2Constraints on fNL from Minkowski
Functionals for WMAP Three-Year
Temperature Anisotropy
The three MFs for the CMB temperature maps from WMAP
three-year data are respectively plotted with symbols in each
column of Fig. 3 (left three columns for the “Q+V+W” map
and right three columns for the “V+W” map). In each col-
umn, the top panel shows the MF Vk at a representative
scale (θs = 20′). and the lower three panels illustrate ∆vk
at different θs = 10′,20′and 40′. The perturbative formulae
with the best-fit value of fNL to each observed MF are plot-
ted with lines. The best-fit values and 1σ uncertainty are
written in the left-bottom side of each panel. In top pan-
els, all of the amplitude of observed MFs are found to be
smaller than the theoretical estimations. This comes from
the deficit of the observed power at low l which generates
the larger amplitude of MFs determined by σ1/σ0 (eq.[3]),
as pointed out by Gott et al. (2007). Fig. 4 shows the same
plot but for the MFs with the pixel window function only.
The top panel shows the MF at Nside = 128 and the lower
three panels illustrate ∆vk for Nside=256, 128, and 64. It

Page 5

Primordial Non-Gaussianity from WMAP
5
Figure 1. Comparison between the analytical predictions (lines) and the numerical estimations averaged over 200 realizations of non-
Gaussian simulation maps (symbols) for fNL = 100; Upper-left: variances σ0 and σ1 (eq. [4]) Upper-right: skewness parameters S(a)
(a = 0,1, and 2) in the equation (7), Middle: MFs for non-Gaussian fields, Vk (eq. [2]), Lower: the difference ratio of MFs ∆vk
(eq. [7]). CMB maps are smoothed with a Gaussian kernel Wl= exp[−l(l + 1)θ2
radiative transfer function is considered for both the theoretical predictions and the simulations. The simulations also include the various
observational effects for WMAP three-year Q+V+W coadded map; pixel window function, beam smearing, inhomogeneous noise pattern,
and Kp0 cut. The error-bars represent the errors for the averaged simulation results over 200 realizations (the sample variance divided
by the square-root-of 200).
s/2] where θs denotes the smoothing scale. The fully

Page 6

6
Hikage et al.
Figure 2. The distribution function of the best-fit value of fNLusing WMAP three-year mock simulation maps (histogram). We use 200
realizations of Gaussian simulations (solid) and non-Gaussian simulations with f(true)
obtained by fitting the analytical formulae (eq. [7]) to all of the MFs for the simulations at θs= 10′and 20′combined. For comparison,
we plot the likelihood function of fNLat f(true)
NL
= 0 and 100 with σfNL= 44 (eq. [11]), which is the expected uncertainty of fNLfrom
all of the MFs for CMB maps at θs= 10′and 20′combined.
NL
= 100 (dotted) respectively. The best-fit values are
is interesting that all MFs at Nside=64 have large positive
values of fNL, though the significance is less than 2σ.
Table 1 lists the best-fit values and the 1σ uncertainty
of fNL for each MF and their combined values at differ-
ent sets of Gaussian smoothing scales θs. The 1σ uncer-
tainty of fNL is estimated from the range of fNL with
∆χ2= χ2− χ2
and the goodness-of-fit Pχ2>χ2
results for the pixel window function only are shown in Ta-
ble 2. The goodness-of-fit values are reasonable for all the
fits, which means that the simple form of the primordial
non-Gaussianity (equation [1]) well describes the behaviour
of the observed MFs. In other words, present observations
are too uncertain to allow the extraction of any further in-
formation about primordial non-Gaussianity (e.g. scale de-
pendence of fNL). The constraint −70 < fNL < 91 at 95%
C.L. is obtained from all MFs for the Q+V+W co-added
map at combined different Gaussian smoothing scales of 10,
20 and 40 arcmin. A similar constraint is obtained from the
MFs with pixel-window only as −84 < fNL < 105. The re-
sults from Q+V+W co-added map are consistent with the
previous ones (Spergel et al. 2007; Creminelli et al. 2007a).
There is some friction (but not disagreement) between
our results and those by Yadav & Wandelt (2008); our V+W
analysis finds fNL = −22 ± 43 whereas they find fNL =
87±30. Moreover our averaged fNLdecreases from Q+V+W
to V+W whereas their fNL increases. This is very interest-
ing because there is the possibility that e.g., foregrounds
and point sources might be biasing one of the two results.
Yadav & Wandelt (2008) show in their analysis that these
effects do not seem to contaminate the primordial bispec-
trum measurement significantly. It will be then important
to check their effect on the MFs statistics in order to verify
if this can explain the differencies among the two results.
However the observed discrepancies show already how ana-
min≤ 1. The minimum of chi-square χmin
minare listed for each fit. The
lyzing non-Gaussianity using different statistics can provide
additional interesting information.
5SUMMARY AND CONCLUSIONS
We have presented an analysis of MFs for WMAP the
three-year temperature maps to limit the primordial non-
Gaussianity characterized by the nonlinear coupling param-
eter fNL. To do this we compared perturbative formulae for
MFs of weakly non-Gaussian fields directly with the obser-
vations. The analytical formulae are found to be in excel-
lent agreement with results from non-Gaussian simulations
of CMB maps including full radiative transfer effects. The
agreement is still very good when including systematic ob-
servational effects including the Kp0 survey mask, pixel and
beam window functions, and inhomogeneous noise distribu-
tion for WMAP three-year data.
We have performed a χ2analysis to the comparison of
the analytical formulae with WMAP three-year data. The
fits of the analytical formulae to the observations are ac-
ceptable and we thus obtain a robust constraint of −70 <
fNL < 91 at 95% C.L. from the Q+V+W coadded maps with
Gaussian filter at different scales 10’, 20’ and 40’ combined.
The result is consistent with previous results (Spergel et al.
2007; Creminelli et al. 2007a; Yadav & Wandelt 2008).
The behaviour of the results for the V+W maps raises
some interesting issues; our constraint is negatively shifted
−108 < fNL < 64 while Yadav & Wandelt (2008) find a
more positive range 27 < fNL < 147. The difference between
the two results should be clearer in the near future survey
represented by Planck. It is worth investigating this result
in further detail through a careful analysis of foregrounds
and point source effects. This will be the subject of future
work.

Page 7

Primordial Non-Gaussianity from WMAP
7
Figure 3. Comparison between MFs for WMAP three-year temperature maps (symbols) and the analytical formulae with the best-fit
value of fNLfor each MF (lines). The MFs are calculated from the Q+V+W co-added map (Left) and the V+W map (Right). Top panels
show the MFs Vk(k=0,1, and 2) at θs=20 arcmin, and other panels illustrate ∆vk(eq. [7]) at θs=10, 20 and 40 arcmin respectively.
Error-bars denote the standard deviation of MFs at each bin of ν computed from 1000 Gaussian realizations including the WMAP
three-year noise distribution, Kp0 mask and pixel and beam window function. The systematics due to the pixelization effect is estimated
from the Gaussian realizations and is subtracted from the observed MFs (see eq. [9]).
Figure 4. Same as Fig. 3 but for different Nside= 256,128 and 64 without Gaussian smoothing.
ACKNOWLEDGMENTS
We thank the anonymous referee for providing very use-
ful comments and suggestions. We deeply appreciate Ei-
ichiro Komatsu who originally proceeded with the project
together. C. H. acknowledges support from the Particle
Physics and Astronomy Research Council grant number
PP/C501692/1. C.H. also acknowledges support from a
JSPS (Japan Society for the Promotion of Science) fellow-
ship. T. M. acknowledges the support from the Ministry
of Education, Culture, Sports, Science, and Technology,
Grant-in-Aid for Scientific Research (C), 18540260, 2006,
and Grant-in-Aid for Scientific Research on Priority Areas
No. 467 “Probing the Dark Energy through an Extremely
Wide & Deep Survey with Subaru Telescope”. S. M. ac-

Page 8

8
Hikage et al.
Table 1. The constraints on fNLfrom MFs for WMAP three-year co-added maps, Q+V+W and V+W, at different Gaussian smoothing
scales θs [arcmin] of 40, 20 and 10 and their combination. For the calculation of their constraints, the maximum likelihood method are
employed with the analytical formulae (eq. [7]) and the covariance matrix of MFs estimated from 1000 Gaussian realizations. The range
of ν is set from −3.6 to 3.6 with the binning number per each MF of 18. The goodness-of-fit of the analytical formulae is represented by
the minimum values of χ2value, χ2
min, and the probability with χ2larger than χ2
min.
Q+V+W V+W
θs [arcmin] MFd.o.f.
χ2
min(Pχ2>χ2
min)fNL
χ2
min(Pχ2>χ2
min)fNL
40
40
40
40
V0
V1
V2
All
17
17
17
53
19.2 (0.32)
12.9 (0.75)
8.2 (0.96)
46.8 (0.71)
429 ± 310
−20 ± 82
−50 ± 109
7 ± 76
23.7 (0.13)
16.7 (0.48)
7.6 (0.97)
48.8 (0.64)
465 ± 305
−28 ± 83
−83 ± 109
−14 ± 77
20
20
20
20
V0
V1
V2
All
17
17
17
53
20.8 (0.24)
14.0 (0.66)
13.9 (0.67)
47.9 (0.67)
93 ± 198
−29 ± 62
9 ± 62
−32 ± 48
20.7 (0.24)
12.6 (0.76)
10.9 (0.86)
49.4 (0.61)
105 ± 191
−42 ± 63
−4 ± 62
−53 ± 49
10
10
10
10
V0
V1
V2
All
17
17
17
53
9.0 (0.94)
11.8 (0.81)
9.9 (0.91)
42.4 (0.85)
98 ± 154
−7 ± 61
0 ± 57
−10 ± 46
9.4 (0.93)
11.3 (0.84)
7.9 (0.97)
52.7 (0.48)
86 ± 149
5 ± 67
−31 ± 61
−25 ± 52
10, 20 & 40
10, 20 & 40
10, 20 & 40
10, 20 & 40
V0
V1
V2
All
53
53
53
161
49.0 (0.63)
41.4 (0.88)
34.8 (0.98)
148.4 (0.75)
8 ± 74
−20 ± 55
5 ± 52
11 ± 40
55.5 (0.38)
44.2 (0.80)
28.3 (1.00)
173.2 (0.24)
−20 ± 76
−23 ± 57
−19 ± 53
−22 ± 43
Table 2. Same as Table 1 but for different Nside= 64,128 and 256 without Gaussian smoothing
Q+V+WV+W
Nside
MFd.o.f.
χ2
min(Pχ2>χ2
min)fNL
χ2
min(Pχ2>χ2
min)fNL
64
64
64
64
V0
V1
V2
All
17
17
17
53
17.5 (0.42)
18.7 (0.34)
12.2 (0.79)
46.9 (0.71)
357 ± 260
133 ± 131
342 ± 186
137 ± 122
18.7 (0.34)
19.2 (0.32)
8.3 (0.96)
44.1 (0.80)
400 ± 257
141 ± 132
344 ± 187
104 ± 123
128
128
128
128
V0
V1
V2
All
17
17
17
53
25.0 (0.09)
25.5 (0.08)
14.1 (0.66)
55.8 (0.37)
130 ± 165
−24 ± 73
−58 ± 76
−10 ± 60
15.5 (0.56)
19.7 (0.29)
16.5 (0.49)
41.6 (0.87)
92 ± 160
−44 ± 76
−62 ± 77
−45 ± 62
256
256
256
256
V0
V1
V2
All
17
17
17
53
7.2 (0.98)
8.7 (0.95)
10.1 (0.90)
38.1 (0.94)
95 ± 139
32 ± 81
40 ± 71
26 ± 60
13.1 (0.73)
9.4 (0.93)
9.4 (0.93)
41.0 (0.89)
87 ± 118
91 ± 98
−30 ± 85
−13 ± 69
256, 128 & 64
256, 128 & 64
256, 128 & 64
256, 128 & 64
V0
V1
V2
All
53
53
53
161
55.9 (0.37)
61.4 (0.20)
41.5 (0.87)
165.3 (0.39)
−44 ± 109
−15 ± 63
−7 ± 60
11 ± 47
54.5 (0.42)
55.2 (0.39)
41.2 (0.88)
152.4 (0.67)
−21 ± 99
−20 ± 67
−51 ± 63
−48 ± 48
knowledges ASI contract Planck LFI Activity of Phase E2,
for partial financial support.
REFERENCES
Alishahiha M., Silverstein E., Tong D., 2004, Phys. Rev.
D., 70, 123505
Arkani-Hamed N., Creminelli P., Mukohyama S., Zaldar-
riaga M., 2004, JCAP, 4, 1
Bartolo N., Matarrese S., Riotto, A., 2002 Phys. Rev. D
65, 103505
Bartolo, N., Komatsu, E., Matarrese, S., & Riotto, A.,
2004, Phys. Rept., 402, 103
Battefeld D., Battefeld T., 2007, JCAP, 5, 1
Bennett, C. L. et al., 2003b, ApJS, 148, 97
Bernardeau F., Uzan, J.-P., 2002, Phys. Rev. D., 66, 103506
Chen X., Richard E, Eugene A. L., 2007, JCAP, 6, 23
Coles P., 1988, MNRAS, 234, 509

Page 9

Primordial Non-Gaussianity from WMAP
9
Creminelli, P., Senatore, L., Zaldarriaga, M., & Tegmark,
M., 2006, JCAP, 03, 005
Creminelli, P., Senatore, L., & Zaldarriaga, M., 2007,
JCAP, 03, 019
Dvali G., Gruzinov A., Zaldarriaga M., 2004, Phys. Rev.
D., 69, 083505
Hikage, C., Komatsu, E. & Matsubara, T., 2006, ApJ, 653,
11
G´ orski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D.,
Hansen, F. K., Reinecke, M., & Bartelman, M., 2005, ApJ,
622, 759
Gott III, J. R., Colley, W. N., Park, C. G., Park, C., &
Mugnolo, C., 2007, MNRAS, 377, 1668
Hinshaw, et al., 2007, ApJS, 170, 288
Komatsu, E. & Spergel, D. N., 2001, Phys. Rev. D, 63,
63002
Komatsu, E. et al., 2003, ApJS, 148, 119
Koyama, K., Mizuno, S., Vernizzi, F., Wands, D., 2007,
JCAP, 11, 24
Liguori, M., Matarrese, S., Moscardini, L., 2003, ApJ, 597,
57
Liguori, M., Hansen, F. K., Komatsu, E., Matarrese, S., &
Riotto, A., 2006, Phys. Rev. D, 73, 043505
Liguori, M., Yadav, A., Hansen, F. K., Komatsu, E., Matar-
rese, S., Wandelt, B., 2007, Phys. Rev. D, 76, 105016
Lyth D.H., Ungarelli C., Wands D., 2003, Phys. Rev. D.,
67, 23503
Matsubara, T., 2003, ApJ, 584, 1
Spergel D.N. et al., 2007, ApJS, 170, 377
Yadav, A. P. S., & Wandelt, B., D., 2008, Phys. Rev. Lett.,
100, 181301