11institutetext: Univ Lyon, ENS de Lyon, Univ Lyon1, CNRS, Centre de Recherche
Astrophysique de Lyon UMR5574, F–69007, Lyon, France 22institutetext:
Department of Physics and Materials Science, University of Luxembourg, 162 A,
Avenue de la Faïencerie, L-1511 Luxembourg City, Luxembourg

# Homology reveals significant anisotropy in the cosmic microwave background

Pratyush Pranav 1122<EMAIL_ADDRESS>Thomas Buchert 11

We test the tenet of statistical isotropy of the standard cosmological model
via a homology analysis of the cosmic microwave background temperature maps.
Examining small sectors of the normalized maps, we find that the results
exhibit a dependence on whether we compute the mean and variance locally from
the masked patch, or from the full masked sky. Assigning local mean and
variance for normalization, we find the maximum discrepancy between the data
and model in the galactic northern hemisphere at more than $3.5$ s.d. for the
PR4 dataset at degree-scale. For the PR3 dataset, the C-R and SMICA maps
exhibit higher significance than the PR4 dataset at $\sim 4$ and $4.1$ s.d.
respectively, however the NILC and SEVEM maps exhibit lower significance at
$\sim 3.4$ s.d. The southern hemisphere exhibits high degree of consistency
between the data and the model for both the PR4 and PR3 datasets. Assigning
the mean and variance of the full masked sky decreases the significance for
the northern hemisphere, the tails in particular. However the tails in the
southern hemisphere are strongly discrepant at more than $4$ standard
deviations at approximately $5$ degrees. The $p$-values obtained from the
$\chi^{2}$-statistic exhibit commensurate significance in both the
experiments. Examining the quadrants of the sphere, we find the first quadrant
to be the major source of the discrepancy. Prima-facie, the results indicate a
breakdown of statistical isotropy in the CMB maps, however more work is needed
to ascertain the source of the anomaly. Regardless, these map characteristics
may have serious consequences for downstream computations such as parameter
estimation, and the related Hubble tension.

###### Key Words.:

Cosmology – Cosmic Microwave Background (CMB) radiation – primordial non-
Gaussianity – topology – relative homology – topological data analysis

##  1 Introduction

The standard Lambda Cold Dark Matter (LCDM) paradigm of cosmology encapsulates
and arises from the cosmological principle (CP), which posits that, on large
enough scales, the Universe is isotropic and homogeneous (Peebles, 1980, 1993;
Durrer, 2015). Though supported by strong mathematical, philosophical and
historical foundations, the veracity of the fundamental tenets of CP has not
yet been comprehensively and conclusively established, motivating theoretical
and observational tests (Secrest et al., 2021; Oayda & Lewis, 2023; Dam et
al., 2023). The recent focus of cosmology towards data gathering and analysis
presents us with an unprecedented opportunity to test the postulates of the
cosmological principle, and the ensuing standard model of cosmology. The data
gathered are from both the early and late epochs in the evolutionary timeline
of the Universe, and have thrown up a number of surprises. Among others, this
includes the famous discrepancy in the inference of the Hubble parameter
between early- and late-Universe data (Di Valentino et al., 2021;
Perivolaropoulos & Skara, 2022; Aluri et al., 2023).

The Cosmic Microwave Background (CMB) radiation is one of the more important
probes of the properties of the early Universe. Emitted at the epoch of
recombination, when the Universe was merely 380,000 years old, the fluctuation
characteristics of the CMB trace the fluctuation characteristics of the matter
distribution in the early Universe (Durrer, 2015; Jones, 2017), and offer the
largest and the oldest canvas on which to test the postulates of CP.
Therefore, studying the CMB fluctuations is essential towards understanding
the properties of the stochastic matter field in the early Universe. The two
components of the CMB radiation – temperature and polarization – present
independent probes in to the properties of the primordial fluctuations
(Durrer, 1999; Seljak & Zaldarriaga, 1997).

The general consensus is that the stochastic fluctuation field of the CMB is
an instance of an isotropic and homogeneous Gaussian random field (Harrison,
1970; Guth & Tye, 1980; Starobinsky, 1982; Guth & Pi, 1982; Komatsu, 2010);
see also Buchert et al. (2017) and references therein for more recent
investigations of Gaussianity. However, CMB data has thrown up a number of
surprises in terms of anomalous features since the launch of the Cosmic
Background Explorer (COBE) satellite (Fixsen et al., 1994). Due to its low
resolution of approximately $7$ degrees, the COBE satellite first discovered
the truly large-scale anomalous lack of correlation in the CMB at 60 degrees
and more. Subsequently, the Wilkinson Microwave Anisotropy Probe (WMAP)
(Bennett et al., 2013) satellite with a higher resolution, discovered a number
of other anomalies at smaller scales, that have persisted in the CMB
measurements by the latest Planck satellite (Planck Collaboration et al.,
2014a, 2016, 2020c). These anomalies which have been detected in both the real
and the harmonic space, seem to be at odds with the postulates of the standard
cosmological model, and perhaps with the more fundamental CP itself.

Representative examples of anomalies in the harmonic space consist of the
_hemispherical power asymmetry_ (HPA) (Eriksen et al., 2004a; Hansen et al.,
2009; Paci et al., 2010; Planck Collaboration et al., 2014b) or the _cosmic
hemispherical asymmetry_ (CHA) (Mukherjee & Souradeep, 2016), the alignment of
low multipoles (Schwarz et al., 2016), as well as the parity anomaly (Land &
Magueijo, 2005; Finelli et al., 2012; Planck Collaboration et al., 2014b). In
particular, the power spectrum has been studied at large scales, for
$\ell=2,40$ (Eriksen et al., 2004a; Mukherjee & Souradeep, 2016), which has
later been extended to smaller scales $\ell\sim 600$ (Hansen et al., 2009;
Paci et al., 2010; Planck Collaboration et al., 2014b), and the analysis
presents the evidence for HPA/CHA at all scales. Important to note is that the
assumption of cosmological isotropy is challenged in other datasets as well
(Bouchet et al., 2001; Colin et al., 2019; Secrest et al., 2021, 2022; Oayda &
Lewis, 2023; Dam et al., 2023). In particular, the analysis of galaxy survey
datasets also points to a hemispherical asymmetry, as the northern and the
southern galactic hemispheres appear to have different topo-geometrical
characteristics (Kerscher et al., 1997, 1998, 2001; Appleby et al., 2022).

The anomalies in the real space have consisted of the discovery of the _cold-
spot_ (Cruz et al., 2005), as well as the unusual behavior of descriptors
emerging from topo-geometrical considerations which involve the integral-
geometric Minkowski functionals (MFs), particularly the genus statistic as a
member of the MFs, first performed on WMAP data (Park, 2004; Eriksen et al.,
2004b), and later extended to Planck data (Planck Collaboration et al., 2014b,
2016, 2020c; Pranav et al., 2019a; Pranav, 2021a, 2022). While Park (2004)
perform their analysis on small sub-degree scales, Eriksen et al. (2004b)
perform a multi-scale analysis spanning a range of sub- and super-degree
scales. In both the cases, there are reported asymmetry between the CMB
hemispheres, where the small scale analysis in Park (2004) reports anomalous
behavior to the tune of $2\sigma$, while (Eriksen et al., 2004b) report a more
than $3\sigma$ deviation in the genus statistic at scales of approximately $5$
degrees for negative thresholds. In this context, it is important to note that
the purely geometric entities of the MFs, such as the area, contour length and
skeleton length have consistently shown a congruence between the data and the
model (Planck Collaboration et al., 2016; Buchert et al., 2017), while the
topological entities such as the genus and the Euler characteristic have shown
deviations between the data and the model (Eriksen et al., 2004b; Park, 2004;
Pranav et al., 2019a; Pranav, 2021a, 2022).

Extending these studies, in this paper, we examine specific sectors of the CMB
temperature maps with a view to test the tenet of statistical isotropy, via
tools that find basis in purely topological notions arising from homology
(Munkres, 1984; Edelsbrunner & Harer, 2010; Pranav, 2015; Pranav et al., 2017)
and its hierarchical extension persistent homology (Edelsbrunner et al., 2002;
Edelsbrunner & Harer, 2010; Pranav, 2015; Pranav et al., 2017; Pranav, 2021),
which form the basis of the recently emerging field of topological data
analysis (TDA) (Carlsson, 2009; Porter et al., 2023). Using these
methodologies, which form the basis for the developed computational pipeline
tailored to examining the CMB datasets (Pranav, 2022), the central
contribution of this paper is the uncovering of a number of anomalous
signatures, which point to different behavior of galactic hemispheres, that
have evidence in the documented literature, but have been overlooked, and not
given due attention in the past.

The advent of TDA holds important place in view of the recent surge in data
acquisition in cosmology and astronomy, which demand increasingly more
sophisticated tools to condense meaningful information from these large and
growing datasets. In view of these observations, the tools and methodologies
presented here particularly stand out as promising candidates to reveal novel
features in the big cosmological datasets, including the completed, on going,
and upcoming CMB observations as evidenced in this paper, as well as galaxy
surveys. Even though a recent development, TDA has already witnessed a strong
presence in a wide range of astrophysical research from studying the large
scale structure of the Universe (van de Weygaert et al., 2011; Sousbie, 2011;
Shivashankar et al., 2016; Xu et al., 2018; Cisewski-Kehe et al., 2018;
Feldbrugge et al., 2019; Biagetti et al., 2021; Kono et al., 2020; Wilding et
al., 2021; Cisewski-Kehe et al., 2022; Ouellette et al., 2023), and the study
of of stochastic properties of astrophysical and cosmological fields in
general (Park et al., 2013; Adler et al., 2017; Makarenko et al., 2018; Pranav
et al., 2019b; Heydenreich et al., 2021; Pranav, 2021b), including the CMB
fluctuation field (Adler et al., 2017; Pranav et al., 2019a; Pranav, 2021a,
2022).

Section 2 presents the data and methods, while Section 3 presents the main
results. We discuss the results and conclude in Section 4.

(a)

(b)

Figure 1: A visualization of the CMB temperature fluctuation field in the
northern hemisphere in two different views. The masked area covers the whole
southern hemisphere, and the relevant parts of the northern hemisphere,
dictated by the PR3 temperature common mask. The visualization is based on the
PR3 observed map, cleaned using the SMICA component separation pipeline,
degraded at ${N}=128$ and smoothed with a Gaussian of $FWHM=80^{\prime}$.

(a)

(b)

(c)

(d)

Figure 2: An illustration of the analyzed map surface in the different
quadrants of the sphere. The first, second, third and fourth quadrants are
presented from left to right, and this convention is followed for quoting the
results.

##  2 Data and Methods

In this section, we briefly describe the data and methods employed at arriving
the results. The computational pipeline, specifically tailored to the CMB
data, but useful for the analysis of other scalar functions on
${{\mathbb{S}}}^{2},$ is a recent development, and a detailed account may be
found in Pranav et al. (2019a) and Pranav (2022).

###  2.1 Data

The data that we investigate are the temperature maps from the latest two data
releases by the Planck team – Planck Data Release 3 (DR3), and the fourth and
final, Planck Data Release 4 (DR4). These data releases represent a natural
evolution of the Planck data processing pipeline, where the final data release
incorporates the best strategies for both the LFI and HFI instruments,
commensurate with an overall reduction in noise and systematics (Planck
Collaboration et al., 2020b). The PR3 and the PR4 data sets are accompanied by
$600$ and $300$ simulations respectively, that originate from the standard
LCDM paradigm, which posits the CMB field to be an instance of an isotropic
and homogeneous Gaussian random field. The PR3 dataset consists of
observational maps obtained via four different component separation methods,
namely C-R, NILC, SEVEM and SMICA, (c.f. Planck Collaboration et al. (2020a)).
We analyze all the four maps in only one experiment, to assess the overall
trend and congruence of results between the different component separation
methods. This choice is dictated by significant computational overhead,
especially for higher resolutions.

###  2.2 Methods

(a)

(b)

(c)

(d)

Figure 3: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the temperature maps for
the NPIPE dataset for the northern (top two rows) and the southern hemisphere
(bottom two rows). The mean and variance are computed for each hemisphere
locally from the unmasked pixels in that hemisphere. The graphs present the
normalized differences, and each panel presents the graphs for a range of
degradation and smoothing scales. The mask used is the PR3 temperature common
mask.

(a)

(b)

(c)

(d)

Figure 4: A visualization of the structure of the superlevel set of the
temperature field at the threshold $\nu=0.5$ for the northern hemisphere. This
is the threshold at which we detect a statistically significant deviation
between the observation and simulations in the number of isolated components.
Clock-wise from the top-left panel, we present the visualization of the
observed CMB map from the FFP10 pipeline smoothed with a Gaussian beam profile
of $FWHM=80^{\prime}$. The rest of the panels present the visualization for
the randomly selected simulation sample numbering 42, and its two higher
multiples 84 and 126. Visually, it is evident that the observational maps have
more fragmented structure compared to the simulated maps.

(a)

(b)

(c)

(d)

Figure 5: Same as Figure 3, however in this case, the mean and variance are
computed from the full sky from the unmasked pixels.

The methods employed in this paper emerge from algebraic and computational
topology, at the level of homology (Munkres, 1984; Edelsbrunner & Harer, 2010)
and persistent homology (Edelsbrunner et al., 2002; Edelsbrunner & Harer,
2010). On ${{\mathbb{S}}}^{2}$, only the 0- and 1D homology groups are of
interest, which are associated with isolated components and holes of the
excursion sets of ${{\mathbb{S}}}^{2}$. These methodologies are complementary
in nature to the integral-geometric Minkowski functionals (Eriksen et al.,
2004b; Schmalzing & Buchert, 1997; Schmalzing & Gorski, 1998; Matsubara, 2010;
Ducout et al., 2013; Buchert et al., 2017; Appleby et al., 2018; Chingangbam
et al., 2017; Telschow et al., 2019), and together they represent a
comprehensive topo-geometrical characterization of fields.

As the CMB data is unreliable in certain regions of the sky, we mask these
regions. We present our results in terms of the homology of the excursion sets
relative to the mask, denoting the number of components relative to the mask
for a given normalized threshold $\nu$ as ${{b}_{0}}(\nu)$. Similar definition
is adopted for the number of loops, ${{b}_{1}}(\nu)$ (Pranav et al., 2019a;
Pranav, 2022).

In continuation of the experiments detailed in Pranav et al. (2019a), Pranav
(2021a), and Pranav (2022), in this paper we perform our investigations on the
hemispheres and quads of the CMB sky, where these regions are defined with
respect to the galactic coordinates. When analyzing the northern hemisphere,
the whole southern hemisphere and the relevant parts of northern hemisphere
are masked, and vice-versa. Similar masking procedure is adopted when
examining the quads. Figure 1 presents a visualization of the temperature
fluctuation field in the northern hemisphere, smoothed with a Gaussian beam
profile of $FWHM=80^{\prime}$. Figure 2 presents a visualization of the quads
of the CMB sky in Molleweide projection view.

We perform a multi-scale analysis by smoothing the original maps given at
$FWHM=5^{\prime}$ and ${N}=2048$, to a range of scales defined by Gaussian
beam profile of
$FWHM=10^{\prime},20^{\prime},40^{\prime},80^{\prime},160^{\prime},320^{\prime}$
and $640^{\prime}$. In order to facilitate faster computations, the maps are
also degraded to ${N}=1024,512,256,128,64,32$ and $16$ in the HealPix format,
prior to the smoothing operation. The mask is subjected to an identical
degrading and smoothing procedure. This results in a non binary mask, which is
re-binarized by setting all pixels above and equal to $0.9$ to $1$, and all
pixels with smaller values to $0$. The mask is applied to the simulations and
observations, and they are transformed to zero-mean and unit-variance fields
by subtracting the mean and re-scaling by the standard deviation. Denoting
$\delta T(\theta,\phi)$ as the fluctuation field at $(\theta,\phi)$ on
${{\mathbb{S}}}^{2}$, $\mu_{\delta T}$ as its mean, and $\sigma_{\delta T}$ as
the standard deviation, computed over the relevant pixels, we examine the
properties of the normalized field:

$\nu(\theta,\phi)=\frac{\delta T(\theta,\phi)-\mu_{\delta T}}{\sigma_{\delta
T}}.$ (1)

In all the experiments we restrict ourselves to $\nu\in[0:3]$ while examining
${{b}_{0}}$, and $\nu\in[-3:0]$ while examining ${{b}_{1}}$, commensurate with
the fact that components are the dominant topological entities at positive
thresholds, while loops are dominant for the negative thresholds.

After the pre-processing steps, performed with the aid of HealPix software
(Górski et al., 2005), the data is subject to the topology computation
pipeline, which briefly involves tessellating the points on the sphere,
computing the upper-star filtration of this tessellation, constructing the
boundary matrix of the filtration, and reducing the boundary matrix to obtain
the $0$\- and $1$-dimensional persistence diagrams. The Betti numbers,
relative to the mask are condensed from the persistence diagrams.

##  3 Results: Topological characteristics of subsets of ${{\mathbb{S}}}^{2}$

In this section, we present the results of the analysis of various sectors of
the CMB sky, with a view to test statistical isotropy. We begin by presenting
the results of examining the hemispheres, followed by an examination of the
quads.

###  3.1 Hemispheres

For the hemispherical analysis, we present the results for two different
experiments, which differ in the regions adopted for computing the mean and
variance for normalizing the maps (c.f. (1)). In the first experiment, we
compute the mean and variance for each hemisphere locally from the unmasked
pixels. For the second experiment, we compute the mean and variance from the
unmasked pixels from the full sky. In all the experiments, we present our
results in terms of the graphs of the Betti numbers relative to the mask. The
graphs are normalized at each threshold to reflect the significance of
deviation. At each threshold, we compute the mean $\mu_{{{b}_{i,sim}}}$, and
the standard deviation $\sigma_{{{b}_{i,sim}}}$ of the Betti numbers for
$i=0,1$, corresponding to the components and the holes. Then, the significance
of difference between the observation and simulations is given by

$\nu_{{{b}_{i}}}=\frac{{{b}_{i,obs}}-\mu_{{{b}_{i,sim}}}}{\sigma_{{{b}_{i,sim}}}},$
(2)

which is the quantity depicted in the graphs.

Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
|  | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.4083 | 0.0733 | 0.1483 | 0.9917 | 0.7617 | 0.8867
512 | 20’ | 0.4633 | 0.0850 | 0.1650 | 0.9933 | 0.8183 | 0.9933
256 | 40’ | 0.0600 | 0.3350 | 0.4767 | 0.3500 | 0.8167 | 0.7283
128 | 80’ | 0.0367 | 0.0367 | 0.0350 | 0.6617 | 0.9017 | 0.9150
64 | 160’ | 0.0433 | 0.0583 | 0.0733 | 0.2833 | 0.7217 | 0.8500
32 | 320’ | 0.0200 | 0.2150 | 0.1017 | 0.5967 | 0.1033 | 0.1717
16 | 640’ | 0.5600 | 0.2900 | 0.4700 | 0.4283 | 0.1783 | 0.2683
summary | N/A | 0.0233 | 0.0533 | 0.0167 | 0.5767 | 0.8767 | 0.9250
(a)

Relative homology – $\chi^{2}$ (empirical) – Gloabal Variance
---
|  | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.4600 | 0.1467 | 0.2750 | 0.5167 | 0.2567 | 0.3150
512 | 20’ | 0.1917 | 0.1450 | 0.1250 | 0.3783 | 0.1117 | 0.1733
256 | 40’ | 0.1817 | 0.3400 | 0.5217 | 0.4533 | 0.0833 | 0.3233
128 | 80’ | 0.0750 | 0.2383 | 0.3567 | 0.1217 | 0.0400 | 0.1367
64 | 160’ | 0.3383 | 0.1250 | 0.2867 | 0.5150 | 0.1583 | 0.5817
32 | 320’ | 0.0383 | 0.2967 | 0.2617 | 0.0650 | 0.0017 | 0.0000
16 | 640’ | 0.8800 | 0.9617 | 0.9000 | 0.3317 | 0.0167 | 0.0150
summary | N/A | 0.1667 | 0.6383 | 0.5283 | 0.0783 | 0.0183 | 0.0000
(b)

Table 1: Table displaying the two-tailed $p$-values for relative homology
obtained from the empirical Mahalanobis distance or $\chi^{2}$ test, computed
from the sample covariance matrices, for different resolutions and smoothing
scales for the NPIPE dataset. Panel (a) presents the $p$-values for
experiments where the variance is computed for each hemisphere separately, and
panel (b) presents results for experiments where the hemispheres are assigned
the variance of full sky. The last entry is the $p$-value for the summary
statistic computed across all resolutions. Marked in boldface are $p$-values
$0.05$ or smaller.

####  3.1.1 Local normalization

For these experiments, the temperature at each pixel is mean-subtracted, and
re-scaled by the standard deviation, where these quantities are computed
locally from the unmasked pixels in each hemisphere. The topological
quantities are computed as a function of the normalized temperature threshold,
$\nu$, at steps of $0.5$. Figure 3 presents the graphs for the significance of
difference between the simulations and the observations, for the number of
components, ${{b}_{0}}$, and the number of loops, ${{b}_{1}}$ for the PR4
dataset. The red curve represents the observed sky, while the gray curves
represent the individual simulations treated as observation. The top two rows
present the graphs for ${{b}_{0}}$ and ${{b}_{1}}$ for the northern
hemisphere, while the bottom two rows present the same for the southern
hemisphere. Similar results for the PR3 dataset are presented in Figure 9 in
the appendix. For this case, we analyze all the four observational maps
obtained from the different component separation pipeline.

We notice a few important things in the graphs. First, that both the PR4 and
PR3 datasets present largely identical results. Second, that the significance
of deviation in the northern hemisphere is in general flared compared to the
southern hemisphere. While the southern hemisphere significance is within the
$2\sigma$ band in general, the northern hemisphere shows significance of
$2\sigma$ or more for most of the scales. The third important thing that we
notice is the very significant deviation in the number of components between
the observations and simulations at the threshold $\nu=0.5$, at
$FWHM=80^{\prime},{N}=128$, which is at degree scale. The significance of
difference for the PR4 datasets stands at approximately $3.5\sigma$. For the
PR3 dataset, two of the maps, namely C-R and SMICA, exhibit higher
significance at approximately $4\sigma$ and $4.1\sigma$ respectively, while
the two other maps, namely NILC and SEVEM, exhibit a $3.4\sigma$ deviation.
Figure 8 in the appendix presents the distribution of Betti numbers at this
threshold, which maybe approximated as a Gaussian, which justifies ascribing a
$\sigma$-significance to the differences. Figure 4 presents the visualization
of the structure of the temperature field at $\nu=0.5$ for the northern
hemisphere. Clock-wise from top-left, we present the field for the observed
CMB map, as well as three randomly selected simulated samples. It is evident
that the observed maps exhibit more fragmented characteristics than any of the
visualized simulations, which engender the significant deviation in the number
of components between the observation and the model.

For the loops, The highest deviation recorded is at $N=64,FWHM=160^{\prime}$
for the northern hemisphere. The PR4 dataset exhibits a $3.4\sigma$ deviation,
while the PR3 dataset shows a $3.2\sigma$ deviation at this scale. At
${N}=128,FWHM=80^{\prime}$, the PR4 dataset exhibits a maximum of $3\sigma$
deviation, while the PR3 dataset exhibits a $2.6\sigma$ deviation.

####  3.1.2 Global normalization

Figure 5 presents the graphs similar to Figure 3 for the PR4 dataset, where
the maps are normalized by the mean and variance computed from unmasked pixels
from the full sky. Similar results for the PR3 dataset are presented in Figure
10. As in the case of local normalization, we notice an agreement in the
feature of the graphs between both the datasets. However, we also note
important differences between observations and simulations. For the northern
hemisphere, the significance of difference is slightly suppressed in general,
and there is an evident change in the behavior of the tail, which is more in
sync with the model. However, in this experiment, there are significant
differences between the observation and the model in the southern hemisphere.
While the number of components, ${{b}_{0}}$, is generally within the $2\sigma$
band, the number of loops exhibits strong differences between the data and
model in the tail. This difference peaks at more than $5$ standard deviations
at scales of approximately $5$ degrees and more. We note that the distribution
of the Betti numbers is manifestly non Gaussian at these scales and thresholds
(c.f. (Pranav, 2022)).

####  3.1.3 Significance of combined thresholds and resolutions

Table 1 presents the $p$-values from the $\chi^{2}$ statistics for
${{b}_{0}}$, ${{b}_{1}}$, and ${\sf EC_{\rm rel}}$ for the PR4 dataset. ${\sf
EC_{\rm rel}}$ is the alternating sum of the Betti numbers, and indicates the
Euler characteristic of the excursion set relative to the mask. The $p$-values
for the PR3 dataset based on SMICA maps are presented in the Appendix in Table
3. The first seven rows in the tables present the $p$-values combining
different thresholds for a given resolution, accounting for multiple testing
at different thresholds. For both the datasets, the $p$-values are consistent
with the graphs, broadly indicating similar properties. For experiments where
the variance is computed separately for the hemispheres, there are significant
deviation between data and model around a degree at $FWHM=80^{\prime}$ for all
the topological quantities. In addition, ${{b}_{0}}$ exhibits differences for
a range of scales $FWHM=80^{\prime},160^{\prime},320^{\prime}$, with
${{b}_{1}}$ also exhibiting mild differences. In contrast the experiments with
common variance exhibit departure between the data and model in the southern
hemisphere at scales of $5$ degrees and larger. The differences are most
evident in the number of loops, which also affects the ${\sf EC_{\rm rel}}$.
However, we also note the deviant behavior of ${{b}_{0}}$ at
$FWHM=320^{\prime}$ in the northern hemisphere.

To account for multiple testing at various resolutions, the last entry in the
tables present the summary $p$-values combining all the tested thresholds and
resolutions. For the experiments with the variance computed separately for the
different hemispheres, the summary $p$-values present significant evidence for
non-random deviation for all the topological descriptors in the northern
hemisphere. Similarly for the experiments with a common variance computed from
the full sky, the southern hemisphere indicates non-random discrepancy between
the data and the model for all the topological descriptors, which is mild for
${{b}_{0}}$, but significant for ${{b}_{1}}$ and ${\sf EC_{\rm rel}}$.

####  3.1.4 The effect of normalization

Normalization is an order preserving transformation topologically. This
ensures that there is a bijection between the original and the normalized
maps, and specifically a correspondence in the dictionary of critical points
as well as the order in which they appear in the filtration to form and
destroy the topological cycles. This entails that, at the most, an erroneous
estimation of mean and variance would induce an offset between the level-sets
of the compared maps, without affecting the deeper topological structure of
the field. Combined with the fact that there are stark differences between the
data and model irrespective of the recipe for normalization, this points to a
difference in the topological structure of the observed field with respect to
the simulations at a level deeper than normalization. Appendix A.1 presents a
short account of the properties of the distribution of mean and variance for
the full sky and the galactic hemispheres. At the level of mean, both the
northern and the southern hemispheres are consistent between the data and
model, which is reflected in the behavior of the mean in the full sky as well.
In contrast, the variance of the northern hemisphere shows stark deviation
between observation and simulations, while the variance in the southern
hemisphere is consistent between the data and the model. This engenders a
variance in the full sky case, which exhibits a mild deviation between the
observation and simulations. Due to the evident discrepancy in variance, it is
also prudent to treat both the hemispheres as arising from different models,
and consequently treat the results from this experimental procedure as more
meaningful.

###  3.2 Quadrants of the sphere

(a)

(b)

(c)

(a)

(b)

(c)

(d)

(b)

Figure 6: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the different quadrants of
the sphere for a range of smoothing scales with Gaussian
$FWHM=20^{\prime},40^{\prime},80^{\prime},160^{\prime},320^{\prime}$. Panel
(a) presents the graphs for ${{b}_{0}}$ while panel (b) presents the graphs
for ${{b}_{1}}$. The values for the different quadrants are presented from top
to bottom, while the scale increases from left to right.

Following the results of the previous section where we detect an anomalous
behavior in the hemispheres, in this section, we compare the behavior of the
observations and simulations in the different quadrants of the sphere for the
PR4 dataset, illustrated in Figure 2, with a view to determine the zone of
discrepancy more accurately. In these experiments, we compute the mean and the
variance locally from the quadrants for map-normalization. We restrict
ourselves to the analysis of scales represented by $FWHM=20,40,80,160,320$.
Our choice of scales is determined by the fact that, on the larger scales,
statistics on smaller sections of the sphere may not be reliable due to the
low numbers involved, while the smaller scales have significant computational
overhead.

Figure 6 presents the graphs for the Betti numbers while Table 2 presents the
$p$-values computed from the empirical $\chi^{2}$ test for the different
quadrants, based on $600$ simulations. Examining the graphs, the first
quadrant stands out due to the maximum deviation for ${{b}_{1}}$ at
$FWHM=160^{\prime}$, with a significance of more than $3.8\sigma$. The
$p$-values presented in Table 2 corroborate the fact that the northern
hemisphere exhibits anomalous behavior with respect to the simulations, where
the source of the discrepancy is clearly associated with the first quadrant.
The summary $p$-values combining all thresholds and resolutions indicate a
strong non-random deviation between data and model in the first quadrant,
which is the strongest for ${{b}_{0}}$. Interesting to note is that the
deviations between the data and model for the quadrants have shifted to a
slightly higher resolution of $FWHM=160^{\prime}$, where also ${{b}_{0}}$
exhibits the strongest deviations. In comparison, the quadrants of the
southern hemisphere exhibit no difference with respect to the model,
consistent with the observation that the southern Galactic hemisphere is
congruent with the standard model, when the maps are normalized by local mean
and variance.

Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
|  | Quad 1 | Quad 2
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
512 | 20’ | 0.1800 | 0.3433 | 0.2917 | 0.5617 | 0.0283 | 0.0567
256 | 40’ | 0.0250 | 0.3650 | 0.2233 | 0.5483 | 0.5533 | 0.7467
128 | 80’ | 0.4117 | 0.3267 | 0.5183 | 0.4350 | 0.4767 | 0.5667
64 | 160’ | 0.0050 | 0.0150 | 0.0033 | 0.2867 | 0.4700 | 0.6450
32 | 320’ | 0.4283 | 0.1517 | 0.4067 | 0.1817 | 0.4083 | 0.6050
summary | N/A | 0.0017 | 0.0383 | 0.0100 | 0.2767 | 0.3200 | 0.2183
(a)

Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
|  | Quad 3 | Quad 4
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
512 | 20’ | 0.9767 | 0.6133 | 0.8833 | 0.9283 | 0.6967 | 0.9700
256 | 40’ | 0.8100 | 0.5233 | 0.7183 | 0.2333 | 0.7300 | 0.6200
128 | 80’ | 0.7983 | 0.6400 | 0.8350 | 0.7683 | 0.6383 | 0.7500
64 | 160’ | 0.5533 | 0.4483 | 0.7083 | 0.5933 | 0.6100 | 0.6267
32 | 320’ | 0.1233 | 0.5200 | 0.1433 | 0.2100 | 0.3617 | 0.3517
summary | N/A | 0.7433 | 0.6200 | 0.4683 | 0.4717 | 0.6917 | 0.5533
(b)

Table 2: Table displaying the two-tailed $p$-values for relative homology
obtained from the empirical Mahalanobis distance or $\chi^{2}$ test, computed
from the sample covariance matrices, for different resolutions and smoothing
scales for the maps in different quadrants of the sphere for the NPIPE
dataset. Panel (a) presents the $p$-values for experiments where the variance
is computed for each hemisphere separately, and panel (b) presents results for
experiments where the hemispheres are assigned the variance of full sky. The
last entry is the $p$-value for the summary statistic computed across all
resolutions. Marked in boldface are $p$-values $0.05$ or smaller.

###  3.3 Comparison with earlier results in literature

##### Local hemisphere variance:

In the experiments with variance computed locally from the hemispheres, the
most striking feature is the significant deviation at around a degree in the
northern hemisphere. We have noticed hints of this degree-scale deviation in
the full sky analysis reported in Pranav (2022), where we report a
$2.96\sigma$ deviation in the PR3 dataset at $FWHM=80^{\prime}$. The PR4
dataset at this scale exhibits a $2.2\sigma$ deviation. Due to a weak display
of anomaly in the PR4 dataset, we rejected it in view of the larger scales
exhibiting stronger anomalies statistically.

We find a similar phenomenon in the reported investigations on the WMAP data
in literature. Park (2004) and Eriksen et al. (2004b) pioneered the
investigation of genus statistics of WMAP CMB data. While Park (2004)
restricted to small sub-degree scales, Eriksen et al. (2004b) performed a
multi-scale topo-geometrical investigation, spanning a range of sub-degree
small scales to super-degree large scales. Due to its multi-scale analysis, as
well as the fact that it is performed on WMAP data, and represents independent
evidence, we find Eriksen et al. (2004b) an excellent source for comparison
with our results. Figure 4 of Eriksen et al. (2004b) presents the Minkowski
functional curves for the WMAP data smoothed at $FWHM=1.28\deg$, where the
genus in the positive threshold range is deviant from the simulations at more
than $2\sigma$. This is more evident in the Figure 5 of Eriksen et al.
(2004b), where the MFs and the skeleton length are presented for a range of
smoothing scales. We notice a discrepancy in the genus between observations
and simulations at more than $2\sigma$ for positive thresholds for a range of
scales, most prominently around $1.28\deg$ and $1.70\deg$. The genus for the
negative thresholds shows no anomaly at these scales. This also results in the
suppression of the signal from the positive threshold anomaly in the
$\chi^{2}$-statistic for genus. Simultaneously, for the larger scales of
approximately $5$ degrees, the genus at negative thresholds is deviant by more
than $3\sigma$, and therefore, like in our case in Pranav (2022), Eriksen et
al. (2004b) also do not deem the degree scale deviation as significant in view
of the anomaly at larger scales. In the same paper, Figure 10 shows that the
asymmetry parameter between negative and positive thresholds for the genus is
weakly correlated. This is experimental evidence in support of the theoretical
fact that the different Betti numbers, which dominate the genus at different
thresholds, are independent, which further motivates the case of examining the
Betti numbers separately, in addition to their linear combination reflected in
the genus. Our experiments further support this observation, as the Betti
numbers of the hemispheres reveal a difference in their topological
properties, which is the source of weak deviation observed in the full sky
analysis for the Betti numbers as well as the associated genus statistics in
both the Planck and WMAP data, which has been overlooked and not given due
attention before.

##### Global variance:

Assigning the global variance to the hemispheres gives rise to discrepant
behavior between observations and simulations in the southern hemisphere,
specifically for the loops at scales of roughly $5$ degrees and larger. The
observed discrepancy stands at approximately $5$ standard deviations. We have
observed the hints of this phenomenon in the full sky analysis as well, where
we report a difference of $3.9$ standard deviation in the number of loops
between the observations and simulations at this scale (Pranav, 2022). The
source of this deviation is linked to the strongly deviant behavior of loops
in the southern hemisphere. Moreover, this phenomenon is observed in the WMAP
data also, where the genus at negative thresholds exhibits large deviations
from the simulations at these scales (Eriksen et al., 2004b).

##  4 Discussions and Conclusion

In this paper, we presented a multi-scale analysis of the topological
properties of the CMB temperature maps in small sectors of the sky, including
hemispheres and quads, with an aim to investigate the veracity of the
postulate of statistical isotropy. This is in continuation of the experiments
performed in Pranav et al. (2019a), Pranav (2022), and Pranav (2021a), where
we report on the full sky properties of the temperature fluctuation maps. We
have employed tools emanating from homology and its hierarchical extension
persistent homology, which form the the foundations of computational topology
(Edelsbrunner et al., 2002; Edelsbrunner & Harer, 2010; Pranav et al., 2017).

We have found various anomalous signatures in the topology of the temperature
fluctuations in the normalized maps in the different hemispheres defined in
the Galactic coordinates. The discovered anomalous signatures in the different
hemispheres depend on the section of the sky adopted for computing the mean
and variance for normalizing the maps. For experiments where the hemispheres
are assigned local mean and variance for normalization, we find that the
northern hemisphere exhibits significant deviations between the data and the
model, most prominently at scales of roughly a degree, in which case the
coincidence between the scale of the anomaly and the horizon at the epoch of
CMB is worth noting. The anomalous signatures in this case are more prominent
for topological components, however the topological holes also exhibit
anomalous behavior. In contrast, the southern hemisphere exhibits remarkable
consistency with the standard model simulations. For the experiments where the
hemispheres are assigned a global mean and variance computed from the masked
full sky, the southern hemisphere exhibits strongly anomalous behavior with
respect to the standard model simulations at scales of roughly 5 degrees and
more.

Noting that the variance of the northern hemisphere is starkly different from
the simulations, while the variance of the southern hemisphere is strongly
consistent with the model, it maybe prudent to treat the hemispheres as
arising from different models at the level of normalization. Consequently, the
results from the experiments where the variance is computed locally from the
hemispheres may be a more accurate reflection of reality than the experiments
where the hemispheres are assigned a global variance for normalization. In
view of this, the degree-scale anomalies in the behavior of the topological
components in the northern hemisphere maybe a stronger and fairer indicator of
the ground-truth, compared to the larger scale anomalies in the topological
loops in the southern hemisphere, which may possibly be an instance of an
artifact of data treatment. Taking hints from the experiments on the
hemispheres, we further tested the quadrants of the sphere assigning local
variance for normalization. In this case, we find that the first quadrant
exhibits significantly anomalous behavior with respect to the simulations.

Despite possible offsets due to an erroneous estimation of mean and variance,
considering the fact that normalization is an order preserving transformation,
and that the anomalies persist irrespective of the recipe adopted for
normalization, this points to deeper differences in the stochastic structure
of the observed and simulated CMB fields, at a level beyond normalization and
offset effects. The fact that the deviations persist in the $\chi^{2}$-tests,
which takes into consideration all the thresholds and resolutions, is another
compelling argument against the deviations being engendered by offset effects.
However, to avoid and mitigate any such effects, in future research we will
present a comparison of topology directly in the space of persistence
diagrams, which encode consolidated information about all level-sets.
Preliminary inroads in this direction have been been made in terms of modeling
persistence diagrams directly, with an analysis of the Galactic hemispheres in
Adler et al. (2017).

An agnostic interpretation of these data characteristics points to a departure
from statistical isotropy in the CMB maps, however further work is required to
convincingly ascribe the source of the anomalies to a genuinely cosmological
effect, unaccounted foreground effect, or merely systematic effects. Regarding
noise and systematics, there is an important point to consider from first
principles. The PR4 dataset purportedly has lower level of systematics and
noise compared to the PR3 dataset. If this is the ground truth, and if the
anomalous signals we have discovered are real, their significance should
increase from PR3 to PR4 datasets. We notice this trend for the NILC and SEVEM
maps, but the opposite trend for C-R and SMICA maps, which makes any
assessment about the origin of the signals inconclusive. Also important to
note is that the slightly differing results for the different maps in PR3
indicate the sensitivity of our methodology to the details of the component
separation pipeline, and maybe considered as a framework for bench-marking. In
the context of foregrounds, the recently discovered foreground effect by
Hansen et al. (2023) also deserves a specific mention. They find that there is
an effect of deepening of CMB temperature profile extending to a few degrees
around nearby large spiral galaxies. While they posit that this foreground
effect may provide a possible common explanation for a number of anomalies of
different kinds in the CMB, restricting to the topological anomalies, such an
effect may generate spurious holes in addition to deepening their temperature
profiles. However, we find it difficult to conclude that the aforementioned
foreground may account for the novel anomalous signatures presented in this
paper, for various reasons. First, the evidence presented in this paper points
to strongly anomalous signatures in the hot-spot regions as well, which cannot
be explained by deepening of temperature profiles, or creation of spurious
cold-spots. Second, the fact that the anomalous number of cold-spots appear in
the southern hemisphere when assigning a global variance for normalization,
and disappear when normalizing the hemispheres by local variance, points to
the fact that the anomalous behavior of the cold-spots in the southern
hemisphere may in fact be misleading, and engendered as an artifact of data
processing. The third compelling reason is that we find the deviations to be
the most significant in the first quad in galactic coordinates, and the
foreground map presented in Hansen et al. (2023) appears to be lacking
significant foreground contamination in this region. Regardless of the finer
details, a crucial point to note from the evidence in this paper is that the
disagreement between the data and the model may engender spurious results for
all subsequent downstream calculations such as cosmological parameter
estimation, as pointed out in Fosalba & Gaztañ aga (2021); Yeung & Chu (2022),
and may have consequences for the Hubble tension through the related mis-
estimation of cosmological parameters.

## Acknowledgements

We are indebted to Herbert Edelsbrunner, Rien van de Weygaert, and Robert
Adler for discussions and comments that have helped shape the draft. We are
grateful to Hans Kristian Eriksen and Geraint F. Lewis for important comments
on the draft. PP would also like to acknowledge the important interactions
with Reijo Keskitalo and Julian Borill, their consistent help with questions,
and incisive comments on the draft. This work is part of a project that has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement ERC
advanced grant 740021 - ARTHUS, PI: TB). P.P. is currently supported by the
Physics of Living Matter Group at the University of Luxembourg, and by the
Luxembourg National Research Fund’s grants awarded to the PI of the Physics of
Living Matter Group (ATTRACT Investigator Grant no. A17/MS/11572821/MBRACE,
and CORE Grant no. C19/MS/13719464/TOPOFLUME/Sengupta). We gratefully
acknowledge the support of PSMN (Pôle Scientifique de Modélisation Numérique)
of the ENS de Lyon, and the Department of Energy’s National Energy Research
Scientific Computing Center (NERSC) at Lawrence Berkeley National Laboratory,
operated under Contract No. DE-AC02-05CH11231, for the use of computing
resources.

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## Appendix A

###  A.1 Distribution characteristics of mean, variance and Betti numbers

In the results presented in the previous sections, we have noticed features in
the topological characteristics that exhibit weak to strong dependence on the
recipe for computing mean and variance for normalizing the maps. Specifically,
computing the mean and variance locally from the hemispheres points to a
difference between the data and model in the northern hemisphere, as opposed
to the southern hemisphere that shows remarkable consistency with the model.
In contrast, computing the variance from the full sky results in a deviation
between the data and model for both the hemispheres for some scales.

As the anomalies presented in the previous sections exhibit a dependence on
the recipe for computing mean and variance, we examine the histograms of mean
and variance of the hemispheres and the full sky at ${N}=512$ in Figure 7.
From the figure, we notice the variance of the observation in the northern
hemisphere to be less than the variance from all the $600$ simulated maps,
while the southern sky is consistent with the simulations. As a result, the
full sky exhibits mildly anomalous characteristics with respect to the
simulations. When examining the mean, it is evident that both the northern and
the southern skies are consistent with the simulations. It has also been noted
in the literature that at all scales the northern hemisphere exhibits
anomalous variance with respect to the simulations, in contrast with the
southern hemisphere, which exhibits no deviation Planck Collaboration et al.
(2014b). It may therefore be prudent to treat the hemispheres as arising from
different models, when computing the mean and variance for normalization.

Figure 8 presents the histogram of ${{b}_{0}}$ from the simulations for this
threshold and resolution, with the observed value indicated by a red vertical
line. It is evident from the histogram that the distribution maybe
approximated by a Gaussian distribution.

###  A.2 PR3 dataset

In this section, we present the graphs and table of $p$-values for the PR3
dataset. Figure 9 presents the graphs of the Betti numbers for the maps that
are normalized with respect to local mean and variance, while Figure 10
presents the graphs for normalization where the mean and the variance ae
computed from the masked full sky. The graphs display consistent
characteristics with that of the PR4 dataset presented in the main body of the
paper, albeit with higher significance of difference between the data and the
model. Table 3 presents the $p$-values computed from the $\chi^{2}$-statistic.
The top seven rows present the $\chi^{2}$-statistic computed from the combined
level sets for a given resolution, while the bottom row presents the same for
the combination of all thresholds at all resolutions. Commensurate with the
graphs, the $p$-values for the PR3 dataset presents higher significance of
difference between the data and the model.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7: Histogram of variance for the north, south and full sky, with the
appropriate regions masked. The histograms are computed for the map at
$N=512$. (a)
Figure 8: Histogram of ${{b}_{0}}$ values at $\nu=0.5$ for the PR3  data set
at $FWHM=80^{\prime}$. The distribution from the simulations indicates a
tendency towards a symmetric Gaussian distribution.

(a)

(b)

(c)

(d)

Figure 9: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the temperature maps for
the FFP10 dataset for the northern (top two rows) and the southern hemisphere
(bottom two rows). The variance is computed for each hemisphere separately
from the unmasked pixels in that hemisphere. The graphs present the normalized
differences, and each panel presents the graphs for a range of degradation and
smoothing scales. The mask used is the PR3 temperature common mask.

(a)

(b)

(c)

(d)

Figure 10: Graphs of ${{b}_{0}}$ and ${{b}_{1}}$ for the temperature maps for
the FFP10 dataset for the northern (top two rows) and the southern hemispheres
(bottom two rows). The graphs present the normalized differences, and each
panel presents the graphs for a range of degradation and smoothing scales. The
variance is computed from the full sky from the unmasked pixels. The mask used
is the PR3 temperature common mask.

Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
|  | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.2400 | 0.1100 | 0.1333 | 0.9667 | 0.4567 | 0.4667
512 | 20’ | 0.4333 | 0.2367 | 0.2867 | 1.0000 | 0.8567 | 0.9667
256 | 40’ | 0.0367 | 0.3267 | 0.2733 | 0.0900 | 0.6767 | 0.2467
128 | 80’ | 0.0000 | 0.0467 | 0.0133 | 0.6533 | 0.6233 | 0.8467
64 | 160’ | 0.0633 | 0.0700 | 0.2333 | 0.2867 | 0.2600 | 0.5067
32 | 320’ | 0.1567 | 0.2467 | 0.3433 | 0.8533 | 0.2467 | 0.4600
16 | 640’ | 0.8167 | 0.5400 | 0.6933 | 0.8733 | 0.0233 | 0.0367
summary | N/A | 0.0100 | 0.0867 | 0.0000 | 0.5433 | 0.3967 | 0.0133
(a)

Relative homology – $\chi^{2}$ (empirical) – Separate Variance
---
|  | North | South
Res | FWHM | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$ | ${{b}_{0}}$ | ${{b}_{1}}$ | ${\sf EC_{\rm rel}}$
threshold = 0.90
1024 | 10’ | 0.3400 | 0.1400 | 0.3100 | 0.3600 | 0.1767 | 0.2333
512 | 20’ | 0.4433 | 0.3233 | 0.5467 | 0.2033 | 0.1533 | 0.1767
256 | 40’ | 0.2067 | 0.2167 | 0.5433 | 0.3333 | 0.0833 | 0.2567
128 | 80’ | 0.0167 | 0.3033 | 0.0867 | 0.2833 | 0.0733 | 0.1867
64 | 160’ | 0.2500 | 0.3033 | 0.3000 | 0.5733 | 0.0933 | 0.5100
32 | 320’ | 0.1500 | 0.5467 | 0.5633 | 0.5067 | 0.0033 | 0.0033
16 | 640’ | 0.6400 | 0.7167 | 0.6500 | 0.3733 | 0.0000 | 0.0000
summary | N/A | 0.2200 | 0.4500 | 0.4133 | 0.0667 | 0.0000 | 0.0000
(b)

Table 3: Table displaying the two-tailed $p$-values for relative homology
obtained from the empirical Mahalanobis distance or $\chi^{2}$ test, computed
from the sample covariance matrices, for different resolutions and smoothing
scales for the FFP10 dataset. Panel (a) presents the $p$-values for
experiments where the variance is computed for each hemisphere separately, and
panel (b) presents results for experiments where the hemispheres are assigned
the variance of full sky. The last entry is the $p$-value for the summary
statistic computed across all resolutions. Marked in boldface are $p$-values
$0.05$ or smaller.