TESTING THE GAUSSIANITY OF THE CMB WITH
SCALAR STATISTICS
C. Monteserín, R.B. Barreiro, E.M. Martínez-González, J.L. Sanz
Abstract
A method to compute several scalar quantities on the sphere of Cosmic Microwave Background maps is presented. We consider in this article only two of
them, namely the normalized Laplacian and the shape index. Both quantities are
obtained directly from the spherical harmonic coefficients of the map, using
the derivatives of the field on the sphere. We also study the probability density
function of these scalars for the case of a homogeneous and isotropic Gaussian
field and compare the theoretical results with simulations.The power of these
quantities to detect non-Gaussianity is also tested using non-Gaussian simulations. The simulations are generated at the resolution of the 30 GHz Planck
channel.
Introduction
The Cosmic Microwave Background (CMB) contains very valuable information about the primitive universe and its study offers us a unique chance to
solve many of the current enigmas
of
the universe and in addition, it constitutes
an important proof of the
theory. A particularly interesting subject
is whether the CMB temperature fluctuations follow or not a Gaussian distribution. While the standard theory predicts Gaussianity, other alternative models
introduce non-Gaussian features in the CMB. So any non-Gaussian detection
in the CMB could have far reaching consequences and could discard many
of the cosmological models supported nowadays. As the CMB data are contaminated by several non-Gaussian components (emissions from our galaxy,
Sunyaev-Zeldovich effect from galaxy clusters, etc...), we must carefully study
the origin of any possible non-Gaussianity detected.
Several Gaussianity studies have been recently carried out using the multifrequency all-sky CMB data provided by the WMAP satellite of the NASA
(Bennet et al. 2003), finding, in some cases, hints of non-Gaussianity. While
some authors have found that the WMAP data are consistent with Gaussianity using different methods for the analysis (e.g. Komatsu et al. 2003, Colley
& Gott 2003), other studies have detected non-Gaussianity and/or north-south
asymmetries (e.g. Eriksen et al. 2004a,b, Vielva et al. 2004, Hansen et al.
2
2004, Cruz et al. 2004). Although foreground residuals could explain some of
these results, in other cases the origin of the detection has not been established
and a primordial origin can not be discarded (see e.g. Vielva et al.2004 and
Cruz et al 2004). These results motivate even more the development of novel
techniques to perform further Gaussianity analysis of the CMB.
In this article we have focused on the study of statistical properties of several scalar quantities constructed from the derivatives of the CMB field on the
sphere. This type of analysis is particularly well suited for all-sky experiments
such as WMAP or the future Planck mission of ESA, to be launched in 2007.
Planck will provide multifrecuency observations of the microwave sky at unprecedent resolution and sensitivity.
Other interesting studies on statistical properties of scalars have already
been done. For example Barreiro et al. (1997) studied the mean number of
maxima and the probability distribution of the Gaussian curvature and the eccentricity of the CMB peaks for different power spectra. The power to detect
non-Gaussianity in the CMB of the number, eccentricity and Gaussian curvature of excursion sets above (and below) a threshold, was tested by Barreiro,
Martínez-González & Sanz (2001) using Gaussian and non-Gaussian simulations, finding that the Gaussian curvature was the best discriminator. Doré,
Colombi & Bouchet (2003) tested the power of a technique based on the proportion of hill, lake and saddle points (which are defined attending to their local
curvature) on flat patches of the sky. Also, the length of the skeleton (a quantity
obtained from the derivatives of the field)) has been applied to study the Gaussianity of the WMAP data, finding evidence of non-Gaussianity and asymmetry between the northern and southern hemispheres (Eriksen et al. 2004b).
This article is organized as follows. In 1 we present the scalars of a 2D
field, focusing on two of them, namely the normalized Laplacian and shape index, describing how to calculate them from the covariant derivatives of the field
and, in particular, for the spherical coordinate system. We also include the theoretical probability distribution function of these scalars, for a homogeneous,
isotropic and Gaussian random field (HIGRF) on the sphere. In 2 we show
how to construct our Gaussian and non-Gaussian simulations using the Edgeworth expansion and test the power of the method to detect non-Gaussianity.
Finally, in 3 we present our conclusions and outline future applications of this
work.
1.
Derivatives and scalars on the sphere
Let us consider a 2-dimensional field T . Using the field derivatives,
several quantities that are scalars under a change of the coordinate system,
can be constructed.
(i. e. under regular general transformation : "!#$%&' ).
3
Testing the Gaussianity of the CMB with scalar statistics
For a given coordinate system, the covariant derivatives (*) +-, , of T .
are related to the ordinary derivatives (0/ +1, trough the Christoffel’s symbols 2+13 ,
as follows:
(1)
(4) +1,5!6(4/ +1,8792 +13 , ( /
3
To construct linear scalars we need to contract the indices of these covariant
tensorial quantities.
The scalars that are constructed with second derivatives, even though they
can be expressed as functions of the covariant derivatives, are usually defined
using the eigenvalues :.;:< , of the negative Hessian matrix = of the field
T % :
=>!@?A7B(4) +-,DCE
(2)
that is, :F and :G are the (negative) second derivatives along the two principal
directions. In the following we will assume :48H6:G .
Attending to the values of :. , :G , we can distinguish among three types of
points (e. g. Dor«e et al. 2003): hill (where both eigenvalues are positive),
lake (where both are negative) and saddle (where : HJI and : LK I ). For
saddle points we will also distinguish between saddle M where N-:FON K N-:<N and
saddle P where N-:FQN$HRN-:GN (the sign +,- refers to the sign of the Laplacian, see
below).
Hereinafter we will work on the sphere of unity radius, so we consider
spherical coordinates TSU;V , which give the following non-zero Christoffel’s
symbols:
S
!>2XW ![ZD\^
_a` ] S
WY
YW
]
2XW
2 Y
WOW
_c`
!b7
S
]
ZD\^]
Sed
(3)
On the sphere, and under certain continuity properties, a common procedure is
to write the temperature field as a series of harmonic functions:
h
(TS<;V'0!gfh
/i
h
h
i5j i TSU;Vk
(4)
where i are the coefficients of this expansion. With these coefficients we
can obtain the power spectrum of the field:
h
l h
!
nOo*m p
f
hGr
i
B
X
q
P
m
h
i
th s
(5)
iEu and define the moments of the field:
v !
+
h
f h
nOo*p
l h
w^x
o okp
m ? m
yC
+
d
(6)
Using the i coefficients it is possible to obtain the harmonic coefficients of
the first and second derivatives of the initial field (for details see appendix A of
Monteserín et al. (2004))
4
We have classified all the scalars of a field in four families, attending to the
physical meaning of these quantities: The first family is formed by the Hessian
matrix scalars that include algebraical quantities related to the Hessian matrix
as its eigenvalues, trace (negative Laplacian) and determinant. The distortion
scalars conform the second family, we consider here quantities related with
powers of the difference of the eigenvalues ( :47:G ) as the shear, the ellipticity
or the shape index. All of them give us information about the distortion of
the initial field. The third family is composed by the gradient related scalars,
where we include the squared modulus of the gradient field, and also another
quantity related to its derivative. The curvature scalars are the fourth family
and they give us information about the geometry of the initial field. Gaussian
and extrinsic curvature belong to this family.
In this article we will consider only two of these scalars, the Laplacian and
the shape index, (for a description of the other scalars, see Monteserín et al.
2004). In the following, we present the considered scalars as a function of the
covariant derivatives and also how to rewrite them in terms of the eigenvalues. Finally we show the probability distribution function of the scalars for an
HIGRF, obtained by Monteserín et al. 2004.
The Laplacian
The Laplacian z T is defined as the trace of the Hessian matrix. It can be
expressed as a function of the eigenvalues as follows:
(7)
z(>!b7E:k7{:G8d
Therefore negative values of the Laplacian correspond to hill or saddle P points
of the field, while lake and saddle M points correspond to positive Laplacian
values. The Laplacian can also be written as a function of the field covariant
derivatives:
+
z(|!6( ) + d
(8)
If the initial temperature field (}TS<;V' is a HIRGF with zero mean, then its
Laplacian will also be Gaussian with zero mean and dispersion v (defined in
equation 6), which only depends on the power spectrum of the initial field.
For convenience we will consider the normalized Laplacian ~!X& , which
simply follows a Gaussian distribution of zero mean and unit dispersion:
"!
m n x
P
d
(9)
As an example, we show in Fig. 1 a CMB Gaussian simulation (top panel) and
its corresponding Laplacian (bottom left). Note that those regions with low
values of z( correspond to regions with high positive curvature in the initial
temperature field, and analogously regions with high values of z( correspond
to regions with high negative curvature in ( .
5
Testing the Gaussianity of the CMB with scalar statistics
The shape Index
This quantity is defined (Koenderink 1990) in terms of the eigenvalues of
the negative Hessian matrix as follows:
n
x
~
p
` : :
(10)
!
d
ZD
:GB7{:^
8
7 . Hill points correspond
By definition, the shape index is bounded
9
m m to values of
7 &7 , lake points to
, saddle P points to
m
m
0 0
7 I and finally saddle M points are in the range
It .
As a function of the covariant derivatives of the field, the shape index can
be rewritten as
n
x
~
!
ZD
`}
n
+
?( )+ C 7
( )+
+
(11)
d
,
,
+
+
? ( ) + ( ) , 7( ) + ( ) , C
If the initial field is a HIGRF with zero mean then the following distribution
function can be deduced for the shape index.
v'¡Uv x
w
k!
§¦
¢
¢ ZD_c\^`U]*
¨ £¥¤ ¦ ¢
¢
¢
£ ¤
¢]
¢
©
v ¡ p
v
ZD\
x
n
P0«
kª
(12)
¢
v
v is a constant that only depends on the power spectrum
where v'¡ !
7
of the initial field (see equation 6).
The structure of the shape index for a Gaussian CMB simulation is shown
in the bottom right panel of Fig. 1. High values of correspond to lake points
in the original temperature field while low values of correspond to hill points.
2.
Simulations
There are many ways in which physically motivated non-Gaussian features
can affect the CMB temperature distribution. For instance, non-standard inflation or topological defects can produce non-Gaussian signatures in the CMB.
The recent non-Gaussian detections in the WMAP data reveal the importance
of carrying out Gaussianity analyses intended to detect different kinds of nonGaussianities.
A method that introduces small deviations from Gaussianity is used here
to generate non-Gaussian simulations and to test the performance of the considered scalars. This method is based on the Edgeworth expansion and characterizes the deviation from normality considering the third and fourth order
moments: skewness and kurtosis.
6
Figure 1. A Gaussian CMB simulation (in units of K, top panel) smoothed with a Gaussian
beam of FWHM= ¬D , and its corresponding Laplacian (bottom left) and shape index (bottom
right) are shown.
The Edgeworth expansion of a density function ®"F can be expressed in
terms of ¯° (the Hermite polynomials) as:
®±²<k!g³´T²<Xµ
²
°
p·¶
n
"¹ n
f
¯ °
°
º
m
¸
°q ¨
p¼»
°
¸
°O½ O¾
(13)
¸
where ° are the cumulants of the expansion and ³±²< represent the Gaussian
distribution
(see e.g. Martínez-González et al. (2002) and references therein).
¸
Keeping only the first terms in the corresponding Hermite polynomials and
considering only the skewness and kurtosis perturbations, we aproximate the
distribution function of our non-Gaussian simulations by these two equations.
® ¡ ²<k!
®É{²<k!
P0¿
n xgÀ
P ¿
n x
pÂÁ Ã6Ä
©
m
m
pËÊ
n w
²
²
Ã
²$Ì87
(14)
7ÆÅ $ÇtÈ
²
p
Å
ª
(15)
where S and K denote skewness and kurtosis, respectively. These distribution
functions are not always well defined, because they can become negative even
7
Testing the Gaussianity of the CMB with scalar statistics
for relatively small values of S or K. To avoid this problem we fix the function to zero when it becomes negative and then we normalise the distribution
function. Note that for values of S, K Í 1, values of ² where the zeros of these
functions appear are always in the tails of the distribution so the renormalization value is close to 1.
Following Martínez-González et al (2002), we have generated non-Gaussian
simulations using the previous pdf’s in the following way. First we simulate
white noise realizations distributed according to equations (14) and (15)Ã for difnÓ
. These
ferent values of S and K using HEALPix pixelization at ÎÐÏ +AÑÒ !
maps are then convolved in real space with a Gaussian beam of FWHM=23
arcmin. Finally we renormalize the power spectrum of the resulting field to the
desired CMB power spectrum using the HEALPix package. In particular, we
have used the power spectrum given by the best-fit model found by the WMAP
team (Spergel et al. 2003) filtered with a gaussian beam of FWHM=33 arcmin.
Note that this corresponds to thel resolution
of the 30 GHz channel of the Planck
h
low frecuency instrument. The ’s for this model were generated using CMBFast (Seljak & Zaldarriaga 1996). As a consequence of the beam convolution
and the introduction of correlations in the temperature maps the original levels
of skewness and kurtosis injected trough the Edgeworth expansion are reduced.
This reduction has been quantified by averaging over simulations of different
levels of injected skewness and kurtosis. Table 1 gives the mean and standard
deviation of the resultant skewness and kurtosis of 1000 non-Gaussian simulations for different injected values of S and K. Note that Gaussian simulations
have been simulated using values of S and K equal to zero.
Table 1.
Mean and standard deviation of skewness and kurtosis
injected S
injected K
0.0
0.3
0.5
0.0
0.0
0.0
0.0
0.0
0.5
1.0
ÔÖÕ×ÙØTÚXÛ
ã.ä&å ä;æBçéèäê
ðå äDñBçéèä ê
æ&å ñíïBçéèä ê
ä&å1èñB
ë çéèä ê
ã.ä&åóè çéèä ê
Ü^Ý;ÞTßà
ë ë
ë å ë èkçéèäê
ë å ë èkçéèä ê
ë å äBçéèä ê
ë å ðë èkçéèä ê
å äBçéèä ê
ÔÖá×ÙØTÚ4Û
ã.ä&å ìíìîçéèäê
ã.ä&å ëï¥ðçéèä ê
ã.ä&å æBçéèä ê
ä&å òDïîçéèä ê
ñå ðOèkçéèä ê
Üâ<ÞTßà
ë
ë å ï&è0çéèäê
ë å æíæçéèä ê
ë å æë è0çéèä ê
ë å ï çéèä ê
å ï&è0çéèä ê
These numbers have been obtained averaging over 1000 simulations of each family. The reduction in the
moments S and K mentioned in the text is shown. This decrease is greater for the injected kurtosis than for
the skewness.
Test and results
We have studied five different families of simulations: one Gaussian (S=K=0)
and four non-Gaussian with certain levels of injected skewness or kurtosis. In
particular, we consider two families with injected Skewness (K=0 either S=0.3
8
Figure 2. Histograms of the Kolmogorov-Smirnov distances corresponding to the temperature maps obtained from 1000 simulations for each family. Left panel: Gaussian family (solid
line), skewness 0.3 (dashed), and skewness 0.5 (dotted). Right panel: Gaussian family (solid
line), kurtosis 0.5 (dashed), and kurtosis 1.0 (dotted).
or S=0.5) and two more with injected kurtosis (S=0 either K=0.5 or K=1.0)
(see table 1). We have generated a total of 1000 simulations for each family. In order to quantify the deviation from Gaussianity of these simulations,
we have compared the empirical distribution function of the temperature maps
obtained from the five families of simulations with the expected for the Gaussian case distribution using the well-know Kolmogorov-Smirnov (KS) test (e.g.
von Mises 1964). For each family of simulations we have constructed the histogram of KS distances (this is given in Figure 2). We see that the histograms
for the Gaussian and non-Gaussian simulations clearly overlap, which indicates that the temperature distribution can not discriminate between these type
of simulations.
With the aim of studying the power of the distribution function of the considered scalars to discriminate between different initial fields, we have also compared the theoretical prediction for a HIGRF initial temperature field, given in
1, with the ones obtained from the Gaussian and non-Gaussian simulations.
This comparison is given in Figures 3 and 4 for the normalized Laplacian
and shape index, respectively. Note that, as expected, the agreement between
the theoretical pdf’s and the ones obtained from Gaussian simulations is very
good. Conversely, we find certain deviations between the pdf’s computed from
the non-Gaussian simulations and the theoretical distribution for the Gaussian
case. To quantify these differences, we have also constructed the histogram of
the KS distances for the two considered scalars and for each family of simulations.
We present in Figures 5 and 6 the KS distance histograms of the normalized
Laplacian and shape index. As expected, increasing the values of the injected
Testing the Gaussianity of the CMB with scalar statistics
9
Figure 3.
For all the panels the solid line represents the theoretical pdf of the normalized
Laplacian for the Gaussian case. The crosses and error bars have been obtained as the average
and dispersion of the distribution of ô from 100 simulations with different levels of injected
skewness or kurtosis. The different panels correspond to: Gaussian simulations (top), simulations with S=0.3 (middle left), S=0.5 (middle right), K=0.5 (bottom left) and K=1.0 (bottom
right).
10
Figure 4. This figure has been constructed analogously to the Fig 3. For all the panels the
solid line represents the theoretical pdf of the shape index for the Gaussian case. The crosses
and error bars have been obtained as the average and dispersion of the distribution of õ from
100 simulations with different levels of injected skewness or kurtosis. The different panels
correspond to: Gaussian simulations (top), simulations with S=0.3 (middle left), S=0.5 (middle
right), K=0.5 (bottom left) and K=1.0 (bottom right).
Testing the Gaussianity of the CMB with scalar statistics
11
Figure 5. Histograms of the Kolmogorov-Smirnov distance corresponding to the normalized
Laplacian obtained from 1000 simulations for each family. In the left panel, the lines correspond
to: Gaussian family (solid), skewness 0.3 (dashed), and skewness 0.5 (dotted). In the right panel
the histograms are given for: Gaussian family (solid line), kurtosis 0.5 (dashed), and kurtosis
1.0 (dotted).
S or K leads to histograms peaked at larger distances from the Gaussian case
and,therefore, to a better separation. Using both, the normalized Laplacian and
shape index, it is possible to completely discriminate between the Gaussian
simulations and those with skewness since the corresponding histograms do
not overlap. In fact, the discriminating power of both scalars to detect Skewness is very similar (the curves appear completely separated and peaked in
similar positions for both cases). With regards to the simulations with kurtosis, we find that both scalars can distinguish between the Gaussian simulations
and the simulations with K=1. However, there is a certain overlap between
the distribution of KS distances for the Gaussian case and the simulations with
K=0.5, which indicates that some of these non-Gaussian simulations can not be
discriminated from the Gaussian ones using this method. Note that the shape
index is slightly better than the normalised Laplacian for detecting the kurtosis
of the simulations since at 0.01 significance level the power of the shape index
test is 82.4 ö while the power of the normalised Laplacian test is 74.2 ö . We
can not discriminate between injected K=0.5 and Gaussianity but we can distinguish between injected S=0.3 and Gaussianity. This is not due to a failure
of the method detecting kurtosis, the real reason is that the injected kurtosis is
more reduced than the skewness when we construct the simulations (see table
1). We would like to remark that the discriminating power of the distribution
of the scalars is clearly superior to that of the temperature map.
12
Figure 6. Histograms of the Kolmogorov-Smirnov distance corresponding to the shape index
obtained from a 1000 simulations for each family. In the left panel, the lines correspond to:
Gaussian family (solid), skewness 0.3 (dashed), and skewness 0.5 (dotted). In the right panel
the histograms are given for: Gaussian family (solid line), kurtosis 0.5 (dashed), and kurtosis
1.0 (dotted).
3.
Conclusions and future work
In this article we have introduced a new method to detect non-Gaussianity in
the CMB using the statistical properties of two scalars, the normalized Laplacian and the shape index. The power of this technique has been tested using Gaussian and non-Gaussian simulations generated through the Edgeworth
expansion. In order to quantify the power of the method we have used the
Kolmogorov-Smirnov test. We find that both scalars are very sensitive to
the presence of skewness or kurtosis in the simulations. In particular, nonGaussian simulations containing skewness (constructed with S=0.3 or S=0.5)
can be clearly distinguished using the normalised Laplacian or the shape index.
These quantities can also distinguish well between the non-Gaussian simulations with K=1 and the Gaussian case. However, some overlapping is found
in the histograms of the KS distances corresponding to the non-Gaussian simulations with K=0.5 and the Gaussian case, which indicates that both kinds of
simulations can not be completely discriminated using this method. We also
find that the shape index performs slightly better than the normalised Laplacian with regards to the simulations with kurtosis whereas the performance of
both scalars is very similar to discriminate between Gaussian simulations and
non-Gaussian ones with skewness. In any case the scalars amplify the nonGaussianities present in the temperature maps.
We are currently studying the discriminating power of other scalars (mentioned in ) and aim to apply this technique to both physically motivated nonm
Gaussian models and to the available CMB data (such as the WMAP data).
Finally, we would like to point out that the study of the scalars can be adapted,
Testing the Gaussianity of the CMB with scalar statistics
13
for instance, to work with regions or extrema above (or below) different thresholds. This will also be the subject of a future work.
Acknowledgments
CM thanks Marcos Cruz, Marcos López-Caniego and Patricio Vielva for
useful comments. CM acknowledges the Spanish Ministerio de Ciencia y Tecnología (MCyT) for a predoctoral FPI fellowship. RBB thanks UC and the
MCyT for a Ramón y Cajal contract. We acknowledge partial financial support from the Spanish MCyT project ESP2002- 04141-C03-01. This work has
used the software package HEALPix (Hierarchical, Equal Area and iso-latitude
pixelization of the sphere, http://www.eso.org/science/healpix), developed by
K.M. Górski, E.F. Hivon, B.D. Wandelt, A.J. Banday, F.K. Hansen and M.
Barthelmann. We acknowledge the use of the software package CMBFAST
(http://www.cmbfast.org) developed by U. Seljak and M. Zaldarriaga.
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