# Local patch analysis for testing statistical isotropy of the Planck
convergence map

Priya Goyal,11footnotetext: Corresponding author Pravabati Chingangbam

###### Abstract

The small but measurable effect of weak gravitational lensing on the cosmic
microwave background radiation provide information about the large-scale
distribution of matter in the universe. We use the all sky distribution of
matter, as represented by the convergence map that is inferred from CMB
lensing measurement by Planck survey, to test the fundamental assumption of
Statistical Isotropy (SI) of the universe. For the analysis we use the
$\alpha$ statistic that is devised from the contour Minkowski tensor, a
tensorial generalization of the scalar Minkowski functional, the contour
length. In essence, the $\alpha$ statistic captures the ellipticity of iso-
field contours at any chosen threshold value of a smooth random field and
provides a measure of anisotropy. The SI of the observed convergence map is
tested against the suite of realistic simulations of the convergence map
provided by the Planck collaboration. We first carry out a global analysis
using the full sky data after applying the galactic and point sources mask. We
find that the observed data is consistent with SI. Further we carry out a
local search for departure from SI in small patches of the sky using $\alpha$.
This analysis reveals several sky patches which exhibit deviations from
simulations with statistical significance higher than 95% confidence level
(CL). Our analysis indicates that the source of the anomalous behaviour of
most of the outlier patches is inaccurate estimation of noise. We identify two
outlier patches which exhibit anomalous behaviour originating from departure
from SI at higher than 95% CL. Most of the anomalous patches are found to be
located roughly along the ecliptic plane or in proximity to the ecliptic
poles.

##  1 Introduction

Observations of primary and secondary anisotropies of the cosmic microwave
background radiation (CMB henceforth) have unveiled many mysteries about the
early universe. CMB photons have been free streaming towards us since the
decoupling epoch, where they were last scattered. The temperature and
polarization fluctuations of the CMB carry imprints of various critical
physical processes that occur in our universe after decoupling, such as the
scattering of CMB photons from the free electrons generated during the epoch
of reionization, inverse Compton scattering of CMB photons off the high energy
electrons in galaxy clusters, and deflection of photon paths by the large
scale structure in the universe (i.e. gravitational lensing) [1].

Out of these sources of secondary anisotropies, gravitational lensing of CMB
is of particular interest, as it is a complementary tracer of matter
distribution (baryonic and dark matter) in the universe. Furthermore, it has
become an important tool in cosmology for studying the growth of structure in
the dark matter distribution as well as cosmic acceleration. CMB lensing is an
integrated phenomenon that provides a measure of the entire mass distribution,
or equivalently, the lensing potential along the line of sight up to the last
scattering surface (LSS). As such it is very sensitive to late universe
parameters, such as the sum of neutrino masses, the dark energy equation of
state and spatial curvature. Lensing remaps the photon distribution in the
sky, which leads to the smoothening of the sharp acoustic peaks in the CMB
temperature power spectrum [2, 3, 4], and generates CMB B-mode polarization at
small scales [5]. CMB lensing was first confirmed in the WMAP data by cross-
correlating CMB observations [6, 7] with the WISE [8] galaxy data. The Planck
2015 [9] and 2018 [10] results have provided the most significant $(\sim
40\sigma)$ detection of gravitational lensing of CMB so far. The data is
further used to reconstruct full-sky lensing potential map and its
corresponding angular power spectrum. From the lensing potential one can
define the convergence field, which is related to the lensing potential via
the 2-dimensional (2D) Poisson equation. It is interpreted as the sky
projection of the surface density of the matter structures that CMB photons
encounter in their paths between the LSS and us. The CMB lensing potential
provides a new observable, in addition to CMB temperature and polarization
anisotropies, to investigate the properties of matter distribution, and hence
another way to test our models of cosmological structure formation. The
lensing potential estimated from the Planck data maps the mass distribution in
the universe over 70% of the sky up to (z$\sim 1100$) in contrast to the
typical surveys which map the distribution of galaxies, quasars which are
concentrated in a limited region of the sky, over limited redshift range.
However, observations from different telescopes/surveys are complementary to
each other. Ultimately, they all observe the same sky, i.e., the same
underlying matter density field. This fact has lead to various cross-
correlation studies between CMB lensing and the large scale structure field.
For example, there have been joint statistical analyses of the CMB lensing
field with the KIDS shear field, Boss galaxies distribution, DESI galaxies
[11, 12, 13, 14, 15] etc. The noise and systematics of CMB lensing
observations are quite different from the ones associated with the
observations of galaxy density, galaxy lensing or galaxy count, and hence
their joint analysis helps in breaking parameter degeneracies and thus improve
constrains on some of the cosmological parameters.

The lensing potential or the convergence field, reconstructed from observed
lensed CMB, can also be utilized to test the fundamental assumptions of the
standard ($\Lambda CDM$) model of cosmology, i.e., the Statistical Isotropy
(SI) of the universe on large scales. This assumption of the universe has been
directly tested employing different kinds of statistics and cosmological data,
for instance, using CMB fields [16, 17, 18], X-ray background and radio
sources [19, 20, 21, 22, 23], clustering properties of galaxies in large scale
structure surveys such as Wiggle-Z [24] and SDSS [25, 26].

In this paper we test the Statistical Isotropy (SI) of the CMB lensing
convergence field using the Tensor Minkowski functionals or Minkowski Tensors
(henceforth MTs). MTs are tensor generalizations of scalar Minkowski
functionals, defined for excursion sets of smooth random fields.
Mathematically they are defined for structures in two- or three- dimensional
Euclidean space in [27]. A subset of MTs which are translationally invariant
in 2D, have been defined for smooth random fields and also generalized to
curved spaces in [28]. MTs carry additional information about the Statistical
Isotropy (SI) of the field (interpreted from the orientation/alignment of the
structures) and the shape of the structures for each excursion set of a field.
One of the translation-invariant rank-2 MTs, the contour Minkowski Tensor
(CMT), has been used to test the SI of random fields [29, 30, 31]. This
approach to test the SI of the CMB fields using real space statistics
complements harmonic space-based statistics such as BiPolar Spherical
Harmonics (BiPoSH) [32, 33, 34].

This analysis is a follow-up of our previous paper [35], where we carried out
theoretical study of the morphology of lensed CMB fields using the CMT on
ideal simulations without taking into account the real data complications like
masking, noise and residual foregrounds. We quantified the distortions caused
by weak lensing in the CMB hotspots and coldspots using the $\beta$ parameter
derived from the CMT, which is corresponding to shear. Here we test the SI of
the reconstructed convergence field from the Planck 2018 data release. In
effect, we are probing the SI of the large scale distribution of matter that
the CMB photons have interacted with. The assumption of SI of the CMB
convergence field should be observationally verified since the detection of
violation of SI could have profound implications for cosmology. Recently,
Marques et al in [36] investigated SI of the Planck 2015 lensing convergence
field using the local variance estimator, and identified sky directions or
outlier regions where the weak lensing imprints anomalous signatures to the
variance estimator revealed through a $\chi^{2}$ analyses at a statistically
significant level. The goal of this paper is to carry out similar analysis,
but using a different methodology. We carry out the global and the local
analysis of the convergence field, using $\alpha$ estimator derived from the
CMT. We obtained $\alpha$ as a function of the threshold level of the Planck
convergence field, and compare with the corresponding $\alpha$ obtained from
Planck simulated convergence map. We identify the thresholds where we find
more than $2~{}\sigma$ disagreement between observed data and simulations. We
further probe those anomalous thresholds by carrying out our local analysis,
by which we mean, the $\alpha$ distribution in non-overlapping patches in the
sky.

This paper is organized as follows. In section 2, we briefly review the method
of the estimation of the convergence field followed by the Planck
collaboration. In section 3, we describe the data and the simulations used in
our analysis. We also review the definition of the contour Minkowski Tensor
and the shape and alignment parameters obtained from it. In section 4, we
present our analysis and the main results. The paper ends with concluding
remarks on Section 5.

##  2 CMB lensing Physics

Weak gravitational lensing (WL) is the physical process by which CMB photons
originating from the LSS are deflected by the gravitational potential of the
inhomogeneous distribution of matter along the line of sight. WL of the CMB is
an achromatic effect which means that lensing by transverse potential
gradients does not change the frequency of the CMB photons. Hence it does not
change the frequency distribution in a given direction, instead it produces an
apparent shift in the angular position of the source.

Let $f$ denote a CMB field which can be either temperature, $T$, or the
Stoke’s parameters $Q,U$ of polarization. Let the deflection caused by lensing
in each sky direction, $\hat{n}$, be denoted by deflection angle
$\vec{d}(\hat{n})$. Then the observed CMB field value in some direction
$\hat{n}$ can be expressed in terms of the corresponding unlensed field value
as,

$f^{\rm L}(\hat{n})=f^{\rm UL}(\hat{n}+\vec{d}).$ (2.1)

For small perturbations in the linear regime, assuming the Born approximation,
the deflection field is given by the gradient of the lensing potential,
$\Psi$, as $\vec{d}=\vec{\nabla}_{\hat{n}}\Psi$. The lensing potential is the
projection along the line of sight of the space and time dependent
gravitational potential field $\psi$, and is expressed as,

$\Psi(\hat{n})=-2\int_{0}^{\chi^{*}}{\rm
d}\chi\left(\frac{\chi^{*}-\chi}{\chi\chi^{*}}\right)\psi\left(\chi\hat{n};\eta_{0}-\chi\right),$
(2.2)

where $\chi$ represents the comoving distance, $\chi^{*}$ is the comoving
distance to the LSS, $\psi$ denotes the 3-dimensional gravitational potential
at conformal distance $\chi$ along the direction $\hat{n}$, and
$\eta_{0}-\chi$ is the conformal time at which the photon was at position
$\chi\hat{n}$. Let the convergence field be denoted by $\kappa$. It is related
to the lensing potential via the 2D Poisson’s equation as,

$\kappa=-\frac{1}{2}{\vec{\nabla}_{\hat{n}}}^{2}\Psi.$ (2.3)

Gravitational lensing by large scale structure introduces non-Gaussianity into
the CMB and provides a new observable (lensing potential), which can be used
as a cosmological probe. Quadratic estimators [37, 38] have been employed to
extract this gravitational lensing signal using CMB modes in harmonic space,
incorporating both temperature and polarization data. Configuration space
based methods like maximum likelihood method have been proposed as well in
[39], where authors analyzed the convergence field by using temperature
anisotropy in real space. This lensing potential reconstruction is possible
due to higher-order correlations between the multipole moments of CMB fields
produced by lensing [40, 41]. The convergence field is then obtained from the
reconstructed lensing potential. The $\kappa$ field captures the magnification
or demagnification of CMB hotspots and coldspots caused by gravitational
lensing. Another measurable consequence of lensing on CMB is the shearing
effect, (area-preserving distortions) on the CMB which may be observable
through changes to ellipticity distribution of the hot and cold spot as shown
in [40]. Unlike the galaxy lensing scenario, where shear is the most crucial
lensing observable, for the CMB we can gain information from both shear and
magnifications.

##  3 Data and Methodology

In this section, we describe the set of observed data and simulations that we
have used for our analysis, followed by the explaination of the statistical
method used to test for SI of the convergence field.

###  3.1 The Planck convergence map: Observed data and Simulations

The Planck 2018 data release [10] provides the most significant measurement of
the lensing potential and its power spectrum over about $70$ percent of the
sky.

Observed data: To reconstruct the lensing potential, the method [42] employed
by the Planck team are based on quadratic estimators that use the features
induced by the lensing process such as the diverse correlations of the CMB
temperature (T) and polarization (E and B) modes. The combination of these
estimators in a minimum-variance (MV) estimator is used to reconstruct the CMB
lensing potential, $\psi^{MV}$. The Planck 2015 lensing estimate was based on
CMB temperature and polarization multipoles i.e.,
$\psi^{TT},\psi^{EE},\psi^{TE},\psi^{TB},\psi^{EB}$ and combination of all
these five multipoles in the minimum variance quadratic estimator. However,
the Planck 2018 estimate of the lensing potential differs from the previous
release because it also includes the contribution from filtered B-modes,
$B^{WF}$. The CMB data set used as input to the MV lensing estimator was the
foreground cleaned map obtained by passing the raw Planck 2018 full mission
frequency maps through the SMICA pipeline.

The reconstructed lensing potential estimate have much red power spectrum,
with most of its power on large angular scales. Cutting the maps with red
power spectrum in small portions can cause leakage issues. For this reason, we
use in our analysis the convergence map, $\kappa^{WF}$, instead of the lensing
potential map. The lensing convergence and its corresponding reconstruction
noise, have a much whiter power spectrum specially on large angular scales
[43].

Simulations: Next, the simulations we use to perform our analysis are the set
of 300 realizations of the convergence field, which constitutes the Planck
Full Focal Plane (FFP10) simulations [44]. These Monte Carlo realizations
comprise of a set of maps that incorporate the dominant instrumental effects
(detector beam, bandpass and correlated noise properties), scanning (pointing
and flags) and data analysis (map-making algorithm and implementation)
effects.

Both the observed and simulation data sets of the convergence field are
provided in the form of multipole expansion coefficients, $\kappa_{\ell
m}^{MV}$, upto $\ell_{max}=4096$ in HEALPix [45]
222http://healpix.sourceforge.net/ FITS format. These are related to harmonic
coefficients of the lensing potential by,

$\kappa_{\ell m}^{MV}=\frac{\ell(\ell+1)}{2}\Psi_{\ell m}^{MV}.$ (3.1)

The $\kappa$ field reconstructed from observed data is highly noise dominated
especially at small angular scales. The signal-to-noise ratio is $S/N\approx
1$ at $\ell=60$. For this reason, we will construct convergence maps by
choosing map resolutions corresponding to the Healpix parameter $N_{\rm
side}\leq 512$. The largest multipole value is then chosen to be $\ell_{\rm
max}=2N_{\rm side}$. The maps that are constructed (both observed data and
simulations) are then further processed following the steps given below.

  * •

Bandpass filtering: We filter out $\ell\leq 8$ modes as done in the Planck
2015 and 2018 lensing analysis. The reason is that the low $\ell$ modes have
high sensitivity to the mean-field subtraction. The form of the filter,
$F_{\ell}$, is given by the following expression,

$\centering F_{\ell}=\frac{1}{2}\left\\{1+{\rm
tanh}\left(\frac{\ell-\ell_{0}}{\Delta}\right)\right\\},\@add@centering$ (3.2)

where $\ell_{0}$ denotes the center of a chosen band of $\ell$, and $\Delta$
denotes the bandwidth. We choose $\ell_{0}=8$ and $\Delta=3$ to carry out our
analysis. The bandpass filtered multipoles are then generated by multiplying
${\kappa}_{\ell m}$s of the input map with $F_{\ell}$.

  * •

Wiener filtering: To mitigate the effect of noise, we apply Wiener filter to
the spherical harmonic coefficients [46], as given by the following
expression,

$\kappa_{\ell
m}^{WF}=\frac{C_{\ell}^{\kappa,fid}}{C_{\ell}^{\kappa,fid}+N_{\ell}^{\kappa}}\kappa_{\ell
m}^{MV}.$ (3.3)

Here $C_{\ell}^{\kappa,fid}$ is the convergence power spectrum in the fiducial
cosmological model, and $N_{\ell}^{\kappa}$ is the reconstruction noise power
spectrum.

  * •

Masking: We further mask the convergence map using the lens reconstruction
analysis mask (lens mask henceforth) provided among the lensing products,
which has $f_{\rm sky}=0.67$. This is an improved mask over the 2015 weak
lensing mask due to reduced point source contamination for the same sky
fraction. We downgrade the lens mask to the lower resolution that is
compatible with the resolution of the convergence map.

###  3.2 Contour Minkowski tensor for smooth random fields and its numerical
computation

The contour Minkowski tensor (henceforth CMT) is defined for a smooth closed
curve, $C$, on a general smooth two-dimensional manifold as,

$\displaystyle{\mathcal{W}}_{1}$ $\displaystyle\equiv$
$\displaystyle\int_{C}\,\hat{T}\otimes\hat{T}\,{\rm d}s,$ (3.4)

where ${\rm d}s$ is the infinitesimal arc length, and $\hat{T}$ is the unit
tangent vector at each point of the curve with the direction chosen to be one
of the two possibilities. The symbol $\otimes$ represents the symmetric tensor
product of two vectors.

Let $f$ denote a random field on the sphere for which we want to compute
${\cal W}_{1}$. We will work with the mean subtracted field, $u\equiv f-\mu$,
where $\mu$ is the mean of $f$. We have taken field threshold values denoted
by $\nu$, and then the set of all points on the manifold where the field has a
value greater than or equal to $\nu$ form an excursion set,
$\mathcal{Q}_{\nu}$. It consists of connected regions and holes. The
boundaries, denoted by $\partial\mathcal{Q}_{\nu}$, of these connected regions
and holes form closed curves. Therefore, each threshold gives a set of curves.

Let $\nabla u=\left(u_{;1},u_{;2}\right)$ denote the covariant derivative of
$u$. The components of the unit tangent vector, $\hat{T}$, can be expressed in
terms of the field derivatives as,
$\hat{T}_{i}=\epsilon_{ij}\,u_{;j}/\left|\nabla u\right|$, where
$\epsilon_{ij}$ is the Levi-Civita tensor in two dimensions. Then,
${\mathcal{W}}_{1}$ for the $k^{th}$ curve in $\partial\mathcal{Q}_{\nu}$ can
be expressed as,

${\mathcal{W}}_{1}(\nu,k)=\int_{C_{k}}{\rm d}s\,\frac{1}{|\nabla u|^{2}}\
{\mathcal{M}},$ (3.5)

where the matrix $\mathcal{M}$ is given by,

$\mathcal{M}=\left(\begin{array}[]{cc}u_{;2}^{2}&-u_{;1}\,u_{;2}\\\
-u_{;1}\,u_{;2}&u_{;1}^{2}\end{array}\right).$ (3.6)

The sum of ${\mathcal{W}}_{1}$ (denoted with an overbar) over all curves in
$\partial\mathcal{Q}_{\nu}$ is then

${\overline{\mathcal{W}}}_{1}=\int_{\partial{\mathcal{Q}}_{\nu}}{\rm
d}s\,\frac{1}{|\nabla u|^{2}}\ {\mathcal{M}}.$ (3.7)

Note that by dividing the above equation by the total area of the manifold
under consideration, we can obtain ${\overline{\mathcal{W}}}_{1}$ per unit
area, which is how the scalar MFs are usually expressed. The line integral
over the boundary of $\mathcal{Q}_{\nu}$ in eq. 3.7 can be further transformed
into an area integral by introducing a Jacobian to give

$\displaystyle\overline{\mathcal{W}}_{1}$ $\displaystyle=$
$\displaystyle\int_{{\mathcal{S}}^{2}}{\rm d}a\,\,\delta(u-\nu)\
\frac{1}{|\nabla u|}\ {\mathcal{M}},$ (3.8)

where ${\rm d}a$ is the infinitesimal area element on ${\mathcal{S}}^{2}$ and
$\delta(u-\nu)$ is the Dirac delta function.

###  3.3 Definition and physical meaning of alignment parameter $\alpha$

Let the eigenvalues of $\overline{\mathcal{W}}_{1}$ be denoted by
$\Lambda_{1}$ and $\Lambda_{2}$. Then $\alpha$ is defined to be the ratio,

$\alpha\equiv\frac{\Lambda_{1}}{\Lambda_{2}},$ (3.9)

Note that by definition the value of $\alpha$ lies between $0$ and $1$. For a
single curve $\alpha$ gives a measure of the ellipticity or anisotropy. Any
curve which has $m$-fold symmetry, with $m\geq 3$, has $\alpha=1$ (see [28]
for detailed explanation). For a distribution of curves the sum of all the
$\mathcal{W}_{1}$ matrices defines one overlap curve, whose anisotropy is
measured by $\alpha$. Therefore, $\alpha$ measures the relative alignment or
the deviation from rotational symmetry in the distribution of curves, and its
departure from one gives the degree of anisotropy.

When we work with finite resolution maps in the space of compact extent, such
as the surface of a sphere, even for a field which is given to be
statistically isotropic, we do not obtain $\alpha$ to be precisely equal to
one at each threshold even for a field which is known to be statistically
isotropic. The $\alpha=1$ is recovered only in the limit of the total
perimeter tending to infinity. At threshold values close to zero, where the
perimeter of $\mathcal{Q}_{\nu}$ is the largest, $\alpha$ is usually closest
to one. For a random distribution of a few curves, the probability that they
will be isotropically distributed is small. As a result, at higher $|\nu|$,
the values of $\alpha$ decrease from unity. Therefore, it is essential to take
into account the threshold dependence of $\alpha$ when probing statistical
isotropy of finite resolution fields.

####  3.3.1 Estimator for $\overline{\mathcal{W}}_{1}$ and $\alpha$

In order to numerically calculate Eq. (3.8) we require, first, threshold bins,
and secondly we need to express the integral as a sum over pixels on the
sphere. Let $\Delta\nu$ denote threshold bin size. Then the $\delta-$function
can be approximated as ( [47, 28]),

$\delta\left(u-\nu\right)=\left\\{\begin{array}[]{l}\frac{1}{\Delta\nu},\quad{\rm
if}\ u\in\left(\nu-\frac{\Delta\nu}{2},\nu+\frac{\Delta\nu}{2}\right)\\\
0,\quad{\rm otherwise}.\end{array}\right.$ (3.10)

For the pixellized sky the total number of pixels is given by $N_{\rm
pix}=12N_{\rm side}^{2}$. The estimator for $\overline{\mathcal{W}}_{1}$
(given by eq. 3.8) for masked pixellized sky can then be expressed as

$\displaystyle\overline{\mathcal{W}}_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{i=0}^{N_{\rm pix}}\,\frac{w_{i}}{\Delta\nu}\
{\mathcal{I}}_{i},$ (3.11)

where ${\mathcal{I}}_{i}\equiv{\mathcal{M}}/|\nabla u|$, with the right hand
side evaluated at pixel $i$, and ${\mathcal{M}}$ is given by eq. 3.6. The
variable $w_{i}$ is the pixel weight which has value zero at pixels that are
masked, and one otherwise. $N$ is the count of all pixels having $w_{i}=1$.

By diagonalizing $\overline{\mathcal{W}}_{1}$ we get its eigenvalues and
obtain $\alpha$ by taking their ratio. This method has inherent numerical
error coming from the discrete approximation of the delta function [48].
However, it was shown in [35] that the error mostly cancels out when computing
$\alpha$. and hence is well suited for applications where $\alpha$ is used to
extract physical information.

##  4 Analysis and results

In this section we present our analysis of SI of the Planck convergence map
using the $\alpha$ statistic. In order to keep the terminology clear we will
refer to the convergence map obtained from Planck data as the observed
convergence map. We first rescale the convergence field by its variance
$\sigma_{0}$. This makes the rescaled field threshold range to be of order
one. Then we compute $\overline{\mathcal{W}}_{1}$, and from it $\alpha$, for
33 equally spaced threshold bins, of width $\Delta\nu=0.25$, ranging from
$-4.0\leq\nu\leq 4.0$.

Minimizing numerical error due to mask boundaries: In order to minimize
numerical error near sharp mask boundaries that can arise due to harmonic
transforms, we first mask the field using downgraded lens mask which is
apodized using the same smoothing kernel as the convergence maps. Then, we
include only those pixels which are sufficiently far away from the boundary in
the calculation of $\alpha$. This is done using a parameter $s_{m}$, whose
value lie between zero and one, and only pixels of the field for which the
corresponding smoothed lens mask has values greater than $s_{m}$ are included.
Since, as $s_{m}$ increases towards one, the fraction of included pixels
decreases, and hence the statistical significance of the results will
decrease. Therefore, it is best to select an optimum value of $s_{m}$ such
that the numerical error is minimized and the statistical significance is
maximized. A rough estimate shows that a smoothed mask pixel value greater
than $0.89$ roughly corresponds to $>2\theta_{s}$ distance from the mask
boundary, $\theta_{s}$ being the smoothing scale. Hence we use $s_{m}=0.89$
for our calculations.

###  4.1 Understanding the effects of bandpass filter and masking on $\alpha$
using ideal simulations

Before analysing the observed data set and looking for subtle signatures of
departure from SI in the universe, it is important to have a clear
understanding of the effects of masking and bandpass filtering on $\alpha$
using ideal Gaussian isotropic maps of the convergence field. This step is
important to isolate any physical phenomenon from the systematic effects. For
this purpose we simulate 300 Gaussian isotropic realizations of $\kappa$ and
then compute $\alpha$ using the method outlined in section 3.2. The input
lensing potential or convergence power spectrum is obtained from CAMB
[49]333https://camb.info/. The values of the input $\Lambda$CDM parameters
are: $\Omega_{b}h^{2}=0.02216$, $\Omega_{cdm}h^{2}=0.1203$, $H_{0}=67\,{\rm
km\,s^{-1}\,Mpc^{-1}}$, $\tau=0.06$, $n_{s}=0.964$, $A_{s}=2.119\times
10^{-9}$, where the symbols have their usual meanings. These are the same
fiducial cosmological parameters which are used as input for the Planck
convergence simulations.

Figure 1: Mean values of $\alpha$ and standard deviation obtained from
isotropic Gaussian simulations of the convergence map for different
resolutions $N_{\rm side}=64,\ 128,\ 256$ and 512. Different colors here
represent $\alpha$ calculated for full sky (red), masked (green), bandpass
filtered (yellow), and masked plus band pass filtered (blue).

We compare $\alpha$ as a function of the threshold for the full sky case with
the corresponding $\alpha$ obtained after (a) applying lens mask with
$s_{m}=0.98$, (b) no masking but band bass filtering, and (c) band pass
filtering and masking. In figure 1 we have shown $\alpha$ for these three
different cases. The four panels correspond to four different resolutions
given by $N_{\rm side}=64,\ 128,\,256$ and $512$. The red curves correspond to
the full sky case, while the green curves represent the masked case, yellow
curves are for the bandpass filtered case with cut-off scale $\ell_{0}=8$, and
the blue curves represents the alpha for band pass filtered and masked maps.
For the ideal case, we expect $\alpha$ to be close to one at all thresholds
considered here. This is what we find that the ideal convergence field is
isotropic as $\alpha$ is close to one, which is as expected from the standard
model of cosmology. Applying lens mask to the ideal $\kappa$ field, reduces
the number of structures in the field which leads to a drop in the value of
$\alpha$ at large thresholds. This effect is more prominent at lower
resolutions i.e. $N_{side}=64,128$ where already the ideal field has few
structures. We also note that the bandpass filtering does not have much effect
on the value of $\alpha$.

###  4.2 Quantifying the statistical significance of $\alpha$ and error bars

The $\alpha$ statistic follows the Beta distribution given by the following
expression,

$P(\alpha)=\frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\alpha^{a-1}(1-\alpha)^{b-1},$

where $a>0,\ b>0$ are parameters that depend on the cosmological model [50].
Hence the standard deviation of $\alpha$, which we denote by
$\sigma_{\alpha}$, will not be equivalent to the usual 68% confidence
interval. So, we first reconstruct the probability density function (PDF) of
$\alpha$ using the 300 FFP10 realizations and using it determine the
confidence intervals as described below. Let $\widetilde{\alpha}^{\,\rm
sim}(\nu)$ denote the median value of $\alpha$ at each threshold obtained from
the simulations. Let the ranges of $\alpha$ be given by
$\alpha_{a}=\left(\widetilde{\alpha}^{\,\rm sim}-\delta^{(j)}_{-}\right)$ to
$\alpha_{b}=\left(\widetilde{\alpha}^{\,\rm sim}+\delta^{(j)}_{+}\right)$,
which denotes the confidence intervals such that $j=1,2,3,\ldots$ correspond
to 68%, 95% and 99%, and so on, respectively. Then we determine
$\delta^{(j)}_{-}$ and $\delta^{(j)}_{+}$ from the condition

$\int_{\widetilde{\alpha}^{\,\rm
sim}-\delta^{(j)}_{-}}^{\widetilde{\alpha}^{\,\rm sim}}{\rm
d}\alpha\,P(\alpha)=\int_{\widetilde{\alpha}^{\,\rm
sim}}^{\widetilde{\alpha}^{\,\rm sim}+\delta^{(j)}_{+}}{\rm
d}\alpha\,P(\alpha)=p/2,$ (4.1)

where $p=0.68,0.95,0.99$ for $j=1,2,3$. We find that the difference between
the mean and median values of $\alpha$ is actually quite small in all cases.
They differ at most by 1% at $|\nu|\sim 3$ and this difference gets smaller
towards $|\nu|\sim 0$.

Figure 2: Left: $P(\alpha)$ for some selected values of $\nu$ for $N_{\rm
side}=512$. Right: $2\sigma_{\alpha}$ (red triangle), $\delta^{(2)}_{-}$
(green square) and $\delta^{(2)}_{+}$ (blue circle), for the same $N_{\rm
side}$ as the left panel.

To demonstrate, the left panel of figure 2 shows the reconstructed $P(\alpha)$
for some selected $\nu$ values, for $N_{\rm side}=512$. We observe that the
maxima of the PDF for $\nu=0$ occurs at smaller value of $\alpha$ compared to
$\nu=2$. Moreover, there is no symmetry between positive and negative $\nu$.
These are due to the presence of instrumental noise, foreground residuals and
other systematic effects. The panel on the right shows comparison of
$2\sigma_{\alpha}$ with $\delta^{(2)}_{-}$ and $\delta^{(2)}_{+}$, calculated
using $p=0.95$. The values of the three quantities do not differ much. There
is mild mismatch at higher $|\nu|$ values where $2\sigma_{\alpha}$ slightly
underestimates $\delta^{(2)}_{-}$ and overestimates $\delta^{(2)}_{+}$. We
find similar behaviour for different values of $N_{\rm side}$.

Let $\alpha^{\rm obs}$ denote the value of $\alpha$ for the observed data, and
let $\widetilde{\alpha}^{\rm sim}$ mean/median $\alpha$ value from 300 FFP10
simulations, then their difference $\Delta\alpha$ is given by,

$\Delta\alpha\equiv\alpha^{\rm obs}-\widetilde{\alpha}^{\rm sim}.$ (4.2)

Then in order to quantify the statistical significance of $\Delta\alpha$ at
each threshold we define the variable $\widetilde{\chi}^{(j)}$ as
follows444$|\widetilde{\chi}^{(1)}|$ reduces to the square root of the
standard chi-squared statistic for Gaussian distribution.:

$\widetilde{\chi}^{(j)}=\left\\{\begin{array}[]{c}\Delta\alpha/\delta^{(j)}_{-},\quad{\rm
if\ \Delta\alpha<0},\\\ \Delta\alpha/\delta^{(j)}_{+},\quad{\rm if\
\Delta\alpha>0}.\end{array}\right.$ (4.3)

If $|\widetilde{\chi}^{(j)}|>1$ then it implies that $\alpha^{\rm obs}$ is
outside the confidence interval corresponding to $j$. Since there can be
unknown systematic or physical effects at different threshold levels we choose
to interpret $\widetilde{\chi}^{(j)}$ as a function of threshold, rather than
condense the information of the threshold variation into a single value. It
can be shown that the values of $\alpha$ at neighbouring thresholds are
uncorrelated if the threshold bin size is sufficiently large. Our choice of
$\Delta\nu=0.25$ is sufficiently large and hence $\delta^{(j)}_{-}$ and
$\delta^{(j)}_{+}$ in the denominator of $\widetilde{\chi}^{(j)}$ captures the
full covariance information.

An important point to keep in mind when we interpret our results is the
following. Since lower value of $\alpha$ means higher degree of alignment of
structures, a significantly negative value of $\Delta\alpha$, and consequently
of $\widetilde{\chi}^{(j)}$, implies that the observed data has higher level
of anisotropy in comparison to the expectation from the median value of
$\widetilde{\alpha}^{\rm sim}$. On the other hand, as seen in section 4.1,
$\alpha$ value for a generic field depends on the number of structures and is
larger for field with higher number of structures. Therefore, a significantly
large positive value of $\Delta\alpha$ indicates anomalous behaviour, but does
not indicate higher level of anisotropy.

Equipped with the understanding of the effects of masking and bandpass
filtering on the ideal convergence field, and the statistics of $\alpha$, we
now focus on our stated goal of testing SI of the observed convergence map and
the corresponding FFP10 simulations. In the following we perform our analysis
using two complementary approaches: (1) global analysis, where we calculate
$\alpha$ for the masked full sky convergence maps, and (2) local analysis,
where we calculate $\alpha$ for non-overlapping small patches of the sky.

###  4.3 Global analysis

For the global analysis we first construct the maps of the observed and FFP10
simulated convergence field following the steps outlined in section 3.1. For
the masking we use the conservative value of $s_{m}=0.98$. The sky fraction
for the full resolution $(N_{\rm side}=2048$) mask is $f_{sky}=0.67$. After
apodizing the lens mask and then applying the $s_{m}$ condition the sky
fractions become $f_{sky}=$ 0.48, 0.49, 0.53 and 0.60 for $N_{\rm side}$ = 64,
128, 256 and 512, respectively. Then $\alpha$ is calculated from each of the
maps constructed as above. We will discuss the results for the cases with and
without Wiener filtering.

Figure 3: Top row: Upper panels show $\alpha$ while lower panels show
$\widetilde{\chi}^{(2)}$ for the Planck convergence map (blue stars) and FFP10
simulations (red triangles), without Wiener filtering, for different values of
$N_{\rm side}$. Bottom row: Same as first row but for Wiener filtered maps.

We first discuss the results for the case without Wiener filtering. The
$\alpha$ and corresponding $\widetilde{\chi}^{(2)}$ values as functions of the
field threshold are presented in the top row of figure. 3. The four panels
correspond to $N_{\rm side}=$ 64, 128, 256 and 512, which are the same as in
figure 1 with the same smoothing FWHM values. The $\alpha$ values computed
from the observed $\kappa$ map are represented by blue stars, while the median
value and error bars given by $\delta^{(2)}_{-}$ (lower) and
$\delta^{(2)}_{+}$ (upper) are shown in red. The black curves in the lower
panels show $\widetilde{\chi}^{(2)}$. We observe the following points:

  * •

We find that $\alpha$ values differ strongly in terms of amplitude as well as
shape of the threshold dependence, between the FFP10 simulations and the ideal
cases shown in figure 1. We observe a dip in the $\alpha$ curve at around
$\nu=0$. Also, we note that $\alpha$ is asymmetric about $\nu=0$, unlike the
case of ideal simulations. We find that the curve slopes down towards negative
threshold values for $N_{\rm side}=64$, while for higher resolution cases the
downward slope is towards positive thresholds. This difference can be
attributed to the complex noise (arising from both instrument and lensing
reconstruction) properties contributing in the observed data and the
corresponding simulations.

  * •

From the values of $\widetilde{\chi}^{(2)}$ we infer that there is good
agreement between the observed data and FFP10 simulations for all the four
cases except for some threshold values, particularly for $N_{\rm side}=64$.

Next we discuss the results for the case with Wiener filtering. In the bottom
row panels of figure 3 we show $\alpha$ for the same observed data (blue
stars) and simulations (red triangles) as in the top row. The superscript ‘WF’
refers to Wiener filter. Wiener filtering optimally weighs for the noise and
hence, is expected to suppress the modes with large noise contribution. We
observe some changes in the results after applying Wiener filter. For $N_{\rm
side}=64$ we observe that the $\alpha$ curve now is tilted toward the positive
threshold values in contrast to the without Wiener filtering case. While for
$N_{\rm side}=256,512$ we find that $\alpha$ becomes relatively more symmetric
about $\nu=0$, as the (large $\ell$ values) modes with high noise contribution
have been suppressed by the Wiener filter. Even though, the shape of the
$\alpha$ curve obtained here also do not mimic the $\alpha$ curve in the case
of ideal Gaussian isotropic $\kappa$ simulation, the observed $\kappa$ is
consistent with Planck isotropic simulations within $2\sigma$. Agreement
between the corresponding values computed from observations and simulations
imply that the observed maps are statistically isotropic. We also observe that
$\widetilde{\chi}^{(2)}$ has positive values at most of the thresholds for
almost all the $N_{\rm side}$ values considered here, in both with and without
Wiener filtering case. positive values of $\widetilde{\chi}^{(2)}$ indicate
presence of higher number of structures in the observed $\kappa$ field than
the corresponding simulated field.

We are guided by two factors to complement the global analysis by further
analysis after restricting to smaller sky patches. The first is that any
anomaly in localized sky regions can get washed out when $\alpha$ is computed
over larger regions. Secondly, we do see some hints of anomaly in the global
analysis, and these can get sharpened when analyzed over smaller regions.
Therefore, we expect the statistical significance of the positive values of
$\widetilde{\chi}^{(2)}$ to get enhanced.

###  4.4 Local patch analysis

We now focus on analysis of small patches of the sky. We first pixellize the
sky using a low value of the Healpix resolution parameter which we denote by
$N_{\rm side}^{\rm local}$. Each pixel determined by $N_{\rm side}^{\rm
local}$ is referred to as a patch. We will work with $N_{\rm side}^{\rm
local}$ = 2 and 4. For $N_{\rm side}^{\rm local}$ = 2 we get 48 patches, each
patch having angular size $(29.3^{\circ})^{2}$. For $N_{\rm side}^{\rm local}$
= 4 there are 192 patches and each patch has angular size
$(14.7^{\circ})^{2}$. We identify each patch by its pixel number in the ‘ring’
pixel numbering format in Healpix. Next, we pixellize the sky using a higher
$N_{\rm side}$ value which we denote with a superscript ‘global’, by $N_{\rm
side}^{\rm global}$. The number of pixels in each patch defined above is then
given by $N_{\rm pix}^{p}=(N_{\rm side}^{\rm global}/N_{\rm side}^{\rm
local})^{2}$. We work with $N_{\rm side}^{\rm global}=128$ and 256. In each
case we use the maximum multipole value of $\kappa$ given by $\ell_{\rm
max}\sim 2N_{\rm side}^{\rm global}$. Therefore, the physical information
encoded in the maps constructed with $N_{\rm side}^{\rm global}=128$ is
contained in maps having $N_{\rm side}^{\rm global}=256$, but the converse is
not true. The values of $N_{\rm side}^{\rm local}$ and $N_{\rm side}^{\rm
global}$ have been chosen keeping in mind that the SNR of the observed
convergence map becomes increasingly smaller at high multipoles, and that the
patches should have good enough resolution for statistical analysis using
morphological properties. Table 1 summarizes the values of $N_{\rm side}^{\rm
local}$ and $N_{\rm side}^{\rm global}$ considered in our analysis, along with
the relevant numbers of patches and pixels, and the angular sizes of patches.

$N_{\rm side}^{\rm local}$ |  Total no of patches |  Angular size of a patch | $N_{\rm side}^{\rm global}$ |  No of pixels in each patch
---|---|---|---|---
2 |  48 |  $(29.3^{\circ})^{2}$ | 128 |  4096
256 |  16384
4 |  192 |  $(14.7^{\circ})^{2}$ | 128 |  1024
256 |  4096
Table 1: Table showing the numbers of patches and their angular size. and
number of pixels in each patch, for the values of $N_{\rm side}^{\rm global}$
and $N_{\rm side}^{\rm local}$ considered in our local analysis.

The estimator of $\overline{\mathcal{W}}_{1}$, which for the global analysis
is given by eqn. 3.11, is modified for each patch indexed by $p$, to the
following form,

$\displaystyle\overline{\mathcal{W}}_{1,p}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{p}}\sum_{i=0}^{N_{\rm
pix}}\,\frac{w_{p,i}}{\Delta\nu}\ {\mathcal{I}}_{i},$ (4.4)

where the weight $w_{p,i}$ has one if the pixel belongs to the $p^{\rm th}$
patch and if the apodized lens mask at that pixel has value $>s_{m}$, else it
has value zero. The normalization $N_{p}$ is now the count of pixels in each
patch having $w_{p,i}=1$. We also exclude patches that have high percentage of
masked regions. This is done by calculating the ratio, $p_{\rm frac}$, of the
number of pixels that are not masked (valid pixels) to the total number of
pixels in each patch. We choose $p_{\rm frac}=0.6$. Figure 4 visually
demonstrates the local analysis. The left panels shows the $\kappa$ field in
one patch. The right panel shows a map of $\alpha$ values for each patch
computed from the observed convergence map, at threshold $\nu=-0.5$ for the
case of $N_{\rm side}^{\rm local}=4$ and $N_{\rm side}^{\rm global}=128$.

Figure 4: Left: An example of a patch of the $\kappa$ map. Right: Map showing
values of $\alpha$ for sky patches at $\nu=-0.5$, with each patch being one of
the 192 pixels given by $N_{\rm side}^{\rm local}=4$. The value of $N_{\rm
side}^{\rm global}$ is 256 and the valid pixel fraction is $p_{\rm frac}=0.6$.

Identification of outlier patches: We calculate $\Delta\alpha$ for each patch
at each threshold. Then we compute $\widetilde{\chi}^{(j)}(\nu)$,
$j=1,2,3,...$. At high thresholds the excursion sets for a typical patch
consist of few structures and hence the analysis of statistical significance
can become unreliable. Therefore, we restrict the threshold range for
identifying outlier patches to the conservative range $-2\leq\nu\leq 2$. If
$|\widetilde{\chi}^{(2)}(\nu)|>1$ (equivalent to higher than $2\sigma$) we
consider the patch to be anomalous at that threshold. A patch can exhibit
anomaly at one or more threshold values. To identify a patch as an outlier for
each value of $N_{\rm side}^{\rm local}$ we demand that it satisfies either of
the following two conditions:

  1. 1.

It must show anomaly for both $N_{\rm side}^{\rm global}$ = 128 and 256. This
condition ensures that anomalous behaviour for $N_{\rm side}^{\rm global}$ =
128 is also manifested in the higher resolution case, and that the anomaly is
robust against variation of resolution.

  2. 2.

For patches that are not common as above they must be anomalous at three or
more threshold values. This condition ensures that the anomaly is not just a
statistical fluctuation.

The outlier patches that have been identified are listed in Table 2. Patches
shown in magenta have positive value of $\widetilde{\chi}^{(2)}(\nu)$, while
those in blue have negative value of $\widetilde{\chi}^{(2)}(\nu)$ at all
anomalous thresholds. The patches that are not common between $N_{\rm
side}^{\rm global}$=128 and 256 are highlighted by the black boxes.

$N_{\rm side}^{\rm local}$ | $N_{\rm side}^{\rm global}$ |  No of valid patches |  No of outlier patches |  Patch ids of outlier patches
---|---|---|---|---
2 | 128 |  28 |  8 |  11, 17, 36, 37, 41, 42, 43, 45
256 |  29 |  11 |  11, 17, 36, 37, 38, 40, 41, 42, 43, 45, 46
4 | 128 |  123 |  22 |  0, 3, 29, 30, 31, 44, 45, 53, 55, 65, 67, 134, 140, 142, 156, 161, 162, 163, 165, 172, 179, 184
256 |  121 |  23 |  0, 3, 29, 30, 31, 44, 45, 53, 55, 65, 125, 140, 142, 152 156, 158, 160, 161, 162, 163, 165, 172, 184
Table 2: Table showing the numbers of anomalous patches, and their
identification numbers. Patches shown in pink have positive value of
$\widetilde{\chi}^{(2)}(\nu)$, while those in blue have negative values of
$\widetilde{\chi}^{(2)}(\nu)$ at all anomalous thresholds. The patches that
are not common between $N_{\rm side}^{\rm global}$=128 and 256 are highlighted
by the black boxes. Figure 5: $\widetilde{\chi}$ values for outlier patches
versus threshold.

Degree of anomaly: We define a new statistic, $\widetilde{\chi}(\nu)$, to
measure the degree of statistical significance of $\Delta\alpha$ for every
outlier patch. It is defined to have value between 2 and 3 if
$\widetilde{\chi}^{(2)}>1$ and $\widetilde{\chi}^{(3)}<1$, between 3 and 4 if
$\widetilde{\chi}^{(3)}>1$ and $\widetilde{\chi}^{(4)}<1$, and so on. Figure 5
shows plots of $\widetilde{\chi}(\nu)$ for all the anomalous patches on the
two dimensional space spanned by $\nu$ and the patch identification numbers.
The four panels correspond to the four combinations of $N_{\rm side}^{\rm
local}$ and $N_{\rm side}^{\rm global}$. Blue circles represent negative,
while pink ones represent positive values of $\widetilde{\chi}$. Larger sized
circles denote larger values of $|\widetilde{\chi}|$. There are no points
having values $3<|\widetilde{\chi}|<4$ for any of the patches. There are no
points in the central regions of the panels because these pixels lie in the
masked region of the Galactic plane.

We first discuss our findings for the case of $N_{\rm side}^{\rm local}=2$.
From table 2 and the top panels of figure 5 we observe the following points:

  * •

The number of outlier patches for $N_{\rm side}^{\rm global}=128$ are 8. All
of them have $\widetilde{\chi}$ value between 2 to 3 at one threshold values
each, except patch id 42 which is anomalous at two threshold values.

  * •

For $N_{\rm side}^{\rm global}=256$ there are 11 outlier patches. All the 8
patches identified in the case of $N_{\rm side}^{\rm global}=128$ form a
subset of these. All the common patches have $\widetilde{\chi}$ value between
2 to 3. Moreover, most of them are found to be anomalous at multiple threshold
values (mostly on the negative side), as can be seen from a comparison of the
top panels of figure 5. This confirms that the anomalous behaviour of the
patches is not erased by variation of resolution, and that the anomaly is more
evident in the case of the higher resolution. Note that the same value of the
threshold will not correspond to the same actual field values for $N_{\rm
side}^{\rm global}=128$ and 256, due to the field rescaling by the respective
standard deviations. A patch exhibiting anomalous behaviour at some threshold
will show similar behaviour at a threshold value that is shifted with respect
to the former. We can see this, for example, for patch id 11 which is
anomalous at only one threshold, namely, $\nu=0.75$, for $N_{\rm side}^{\rm
global}=128$, while for $N_{\rm side}^{\rm global}=256$ it is anomalous at
$\nu=-1.75$. Moreover, the threshold shift need not be the same for all
anomalous patches since the fields at the two different resolutions are
different due to the inclusion of additional multipoles for the higher
resolution.

  * •

The 3 additional outlier patches found in the higher resolution case indicate
new information regarding disagreement of the values of $\kappa$ for
multipoles higher than 256 between observed data and FFP10 simulations. Patch
ids 40 and 46 have $\widetilde{\chi}$ values between 2 to 3, while patch id 38
is strongly anomalous with $\widetilde{\chi}>4$.

Next, we discuss $N_{\rm side}^{\rm local}=4$. In this case what we are
effectively doing is dividing each patch of $N_{\rm side}^{\rm local}=2$ into
four equal parts and treating each part as a new patch. There is no new
information compared to $N_{\rm side}^{\rm local}=2$ in terms of multipoles of
$\kappa$. However, the information of anomalous nature of smaller sized
regions can get washed out as components of larger regions since in that case
$\alpha$ will capture the mean alignment of a larger number of structures. The
analysis for $N_{\rm side}^{\rm local}=4$ can isolate such smaller anomalous
regions. From table 2 and the bottom panels of figure 5 we observe the
following points:

  * •

The number of outlier patches for $N_{\rm side}^{\rm global}=128$ are 22, of
which 5 have negative $\widetilde{\chi}$. All the outliers have
$\widetilde{\chi}$ values between 2 and 3, while 55 is strongly anomalous with
$\widetilde{\chi}>4$.

  * •

For $N_{\rm side}^{\rm global}=256$ we get 23 outliers, out of which 2 have
negative $\widetilde{\chi}$ values. All the outliers have $\widetilde{\chi}$
values between 2 and 3, while 45, 152 and 158 are strongly anomalous with
$\widetilde{\chi}>4$.

  * •

There are 19 common patches. Out of these, we find that patch ids 0, 142 and
161 are anomalous at one threshold each, and the sign of $\widetilde{\chi}$ is
different for different $N_{\rm side}^{\rm global}$. For this reason we
exclude them from the set of outlier patches.

Sky locations of outlier patches: Having identified the outlier patches for
each value of $N_{\rm side}^{\rm local}$, we next examine these patches in
further detail by visualising their location in the sky. The top row of figure
6 shows maps of outlier patches, with the different colours indicating the
values of $\widetilde{\chi}$, for $N_{\rm side}^{\rm local}=2$ for $N_{\rm
side}^{\rm global}=128$ (left) and $N_{\rm side}^{\rm global}=256$ (right).
The bottom row shows the same for $N_{\rm side}^{\rm local}=4$. Dark blue
corresponds to $-3<\widetilde{\chi}<-2$, yellow corresponds to
$2<\widetilde{\chi}<3$ and maroon to $\widetilde{\chi}>4$. Patches ids in
Healpix ring format is also indicated. All maps are shown using Molleweide
projection in Galactic coordinates.

As seen in figure 6 all the outlier patches for $N_{\rm side}^{\rm local}=2$
and most of them for $N_{\rm side}^{\rm local}=4$ have positive
$\widetilde{\chi}$. As discussed in section 4.2, positive $\widetilde{\chi}$
is not an indication of higher level of alignment between the structures.
Rather, it indicates that the observed $\kappa$ field in these patches have
higher number of structures in comparison to the simulations. Further, we
observe a general trend of increase of statistical significance of anomaly for
the higher resolution case for both $N_{\rm side}^{\rm local}=2$ and 4. For
$N_{\rm side}^{\rm local}=2$ the higher resolution case has three additional
outlier patches, of which one (patch id 38) is very highly anomalous. For
$N_{\rm side}^{\rm local}=4$ the higher resolution case has two additional
outlier patches (patch id 152 and 158) which are highly anomalous, while patch
id 45 shows increase of statistical significance of anomaly. It is useful to
mention here that addition of spatially uncorrelated noise to a given physical
random field having some coherence length, in general, results in increase of
the number of small scale structures of the combined field. Hence, positive
values of $\widetilde{\chi}$ and the general trend for it to be higher for
higher resolution correlate with the fact that the SNR of the observed
convergence map decreases with increasing multipole. Therefore, while we
cannot conclusively rule out other possibilities, our results strongly suggest
that the anomalous behaviour of the outlier patches having positive
$\widetilde{\chi}$ are caused by inadequate estimation of the instrumental
noise.

We find two patches that have negative $\widetilde{\chi}$ (patch ids 29 and
179 in the bottom left panel of figure6). These are patches where the
structures in the observed $\kappa$ map exhibit relative alignment that is
significantly higher (lower value of $\alpha$) than is expected from the
simulations. Therefore we conclude that these two patches show departure from
SI at higher than 95% confidence level. Of these, patch id 29 lies close to
the ecliptic north pole, while 179 is located close to the ecliptic plane.

We can visually see that several of the outlier patches obtained for $N_{\rm
side}^{\rm local}=4$ are contained within the outlier patches of $N_{\rm
side}^{\rm local}=2$. Thus, as expected, by subdividing the patches of $N_{\rm
side}^{\rm local}=2$ into smaller parts we can further isolate or localize the
anomalous regions. The regions corresponding to patch ids 29, 30, 31, 44 and
25 for $N_{\rm side}^{\rm local}=4$ are conspicuously absent for $N_{\rm
side}^{\rm local}=2$. This indicates that the alignment information has gotten
washed out when $\alpha$ is calculated over larger patch size.

Lastly, it is interesting to note that most of the outlier patches are located
close to either the ecliptic north or south poles, or the ecliptic plane.
Patch ids 11 and 17 for $N_{\rm side}^{\rm local}=2$, and correspondingly 53,
55 and 65 for for $N_{\rm side}^{\rm local}=4$, lie close to the masked
regions (11 and 55 are close to the north Galactic spur) and are possibly
contaminated by residual foreground.

Figure 6: Top: Maps of showing outlier patches for $N_{\rm side}^{\rm
local}=2$ for $N_{\rm side}^{\rm global}=128$ (left) and $N_{\rm side}^{\rm
global}=256$ (right). The colors indicate the corresponding values of
$\widetilde{\chi}$. Dark blue corresponds to $-3<\widetilde{\chi}<-2$, yellow
corresponds to $2<\widetilde{\chi}<3$ and maroon to $\widetilde{\chi}>4$.
Patches are numbered in Healpix ring format. Bottom: Same as the top row but
for $N_{\rm side}^{\rm local}=4$. All maps are shown using Mollweide
projection in Galactic coordinates.

##  5 Conclusion

In this paper we have carried out statistical isotropy test using the observed
Planck convergence map by comparing it with the FFP10 simulations provided by
the Planck team. For the test we use the $\alpha$ statistic which measures
anisotropy of excursion sets of smooth random fields. We carry out the test
using the global (masked) sky and for small sky patches. In order to focus on
multipoles where the SNR of the observed convergence map is not too small we
restrict our analysis to $N_{\rm side}=512$ in the global analysis and 256 in
the local analysis. The local analysis is complementary to, but more effective
than, the global one for detecting presence of anisotropy or any anomalous
behaviour of the field in localised sky regions. This is because in carrying
out average over a larger set of structures the anomalous information can get
washed out. From the global analysis we find that the observed data and
simulations show good agreement but hints at the presence of some anomaly.
Then using the local patch analysis we identify several anomalous patches
where the observed and simulated data show disagreement of the values of
$\alpha$ at statistical significance higher than 95% CL.

From the positive sign of the statistic $\widetilde{\chi}$ we infer that the
source of the anomalous behaviour of most of the outlier patches is inaccurate
estimation of noise. The locations of a majority of the anomalous patches that
we have identified are close to either the ecliptic plane or the ecliptic
poles. Further we identify two outlier patches, which exhibit anomalous
behaviour originating from departure from SI at higher than 95% CL. One of
these is located near the north ecliptic pole and the other is located close
to the ecliptic plane. Though beyond the scope of this paper it will be
interesting to examine these patches further and cross-correlate them with
large scale structure surveys.

Our local patch analysis is similar in spirit to the analysis carried out in
Marques et al. [36], though we use a different methodology. Their method uses
the variance, and hence direct field values, computed from sky patches, while
$\alpha$ is constructed using the first derivative of the field and can be
directly interpreted as the anisotropy of a curve. Therefore, the anomalous
patches identified by our analysis need not be the same as those identified
using the variance. Comparison of figure 6 with figs. 9 and 10 of [36] shows
that there are some patches common, and both our analysis and theirs indicate
that anomalous regions are more likely near the ecliptic poles and ecliptic
plane.

Our results broadly confirm that the universe is statistically isotropic on
large scales by using a method that is different from the ones that have
previously been used in the literature. We have argued that most of the
anomalous regions we have identified can are well explained by inaccurate
noise estimation. Hence our method and results will be useful for improved
understanding of noise. Each of the anomalous regions, and in particular the
two regions that exhibit statistically significant deviation from SI, are
interesting for further probes and cross-correlation with large scale
structure surveys.

## Acknowledgments

We acknowledge use of the NOVA HPC clusters at the Indian Institute of
Astrophysics. Some of the results in this paper have been obtained by using
the CAMB [49] and HEALPIX [45] packages. We have used Matplotlib [51] library
for generating most of the plots in our work. The work of PC is supported by
the Science and Engineering Research Board of the Department of Science and
Technology, India, under the MATRICS scheme, bearing project reference no
MTR/2018/000896. PC would also like to acknowledge the hospitality of the
Physics department, Indian Institute for Science Education and Research,
Mohali, during the concluding phase of this work.

## References

  * [1] A. Blanchard and J. Schneider, _Gravitational lensing effect on the fluctuations of the cosmic background radiation_ , _A &A_ 184 (1987) 1.
  * [2] U. Seljak, _Gravitational Lensing Effect on Cosmic Microwave Background Anisotropies: A Power Spectrum Approach_ , _ApJ_ 463 (1996) 1 [astro-ph/9505109].
  * [3] R.B. Metcalf and J. Silk, _Gravitational Magnification of the Cosmic Microwave Background_ , _ApJ_ 489 (1997) 1 [astro-ph/9708059].
  * [4] A. Lewis and A. Challinor, _Weak gravitational lensing of the CMB_ , _Phys. Rept._ 429 (2006) 1 [astro-ph/0601594].
  * [5] M. Zaldarriaga and U. Seljak, _Gravitational lensing effect on cosmic microwave background polarization_ , _Phys. Rev. D_ 58 (1998) 023003 [astro-ph/9803150].
  * [6] G. Hinshaw, M.R. Nolta, C.L. Bennett, R. Bean, O. Doré, M.R. Greason et al., _Three-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Temperature Analysis_ , _ApJS_ 170 (2007) 288 [astro-ph/0603451].
  * [7] K.M. Smith, O. Zahn and O. Doré, _Detection of gravitational lensing in the cosmic microwave background_ , _Phys. Rev. D_ 76 (2007) 043510 [0705.3980].
  * [8] J.J. Condon, W.D. Cotton, E.W. Greisen, Q.F. Yin, R.A. Perley, G.B. Taylor et al., _The NRAO VLA Sky Survey_ , _AJ_ 115 (1998) 1693.
  * [9] Planck Collaboration, P.A.R. Ade, N. Aghanim, M. Arnaud, M. Ashdown, J. Aumont et al., _Planck 2015 results. XV. Gravitational lensing_ , _A &A_ 594 (2016) A15 [1502.01591].
  * [10] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., _Planck 2018 results. VIII. Gravitational lensing_ , _A &A_ 641 (2020) A8 [1807.06210].
  * [11] S. Ho, C. Hirata, N. Padmanabhan, U. Seljak and N. Bahcall, _Correlation of cmb with large-scale structure. i. integrated sachs-wolfe tomography and cosmological implications_ , _Phys. Rev. D_ 78 (2008) 043519.
  * [12] B.D. Sherwin, S. Das, A. Hajian, G. Addison, J.R. Bond, D. Crichton et al., _The atacama cosmology telescope: Cross-correlation of cosmic microwave background lensing and quasars_ , _Phys. Rev. D_ 86 (2012) 083006.
  * [13] A.R. Pullen, S. Alam, S. He and S. Ho, _Constraining gravity at the largest scales through CMB lensing and galaxy velocities_ , _Monthly Notices of the Royal Astronomical Society_ 460 (2016) 4098 [https://academic.oup.com/mnras/article-pdf/460/4/4098/8117760/stw1249.pdf].
  * [14] S. Singh, R. Mandelbaum and J.R. Brownstein, _Cross-correlating Planck CMB lensing with SDSS: lensing–lensing and galaxy–lensing cross-correlations_ , _Monthly Notices of the Royal Astronomical Society_ 464 (2016) 2120 [https://academic.oup.com/mnras/article-pdf/464/2/2120/18518180/stw2482.pdf].
  * [15] D. Kirk, Y. Omori, A. Benoit-Lévy, R. Cawthon, C. Chang, P. Larsen et al., _Cross-correlation of gravitational lensing from DES Science Verification data with SPT and Planck lensing_ , _Monthly Notices of the Royal Astronomical Society_ 459 (2016) 21 [https://academic.oup.com/mnras/article-pdf/459/1/21/8113774/stw570.pdf].
  * [16] D.J. Schwarz, C.J. Copi, D. Huterer and G.D. Starkman, _CMB anomalies after Planck_ , _Classical and Quantum Gravity_ 33 (2016) 184001 [1510.07929].
  * [17] Planck Collaboration, P.A.R. Ade, N. Aghanim, Y. Akrami, P.K. Aluri, M. Arnaud et al., _Planck 2015 results. XVI. Isotropy and statistics of the CMB_ , _A &A_ 594 (2016) A16 [1506.07135].
  * [18] S. Ghosh, P. Jain, G. Kashyap, R. Kothari, S. Nadkarni-Ghosh and P. Tiwari, _Probing statistical isotropy of cosmological radio sources using square kilometre array_ , _Journal of Astrophysics and Astronomy_ 37 (2016) 25.
  * [19] D. Alonso, A.I. Salvador, F.J. Sánchez, M. Bilicki, J. García-Bellido and E. Sánchez, _Homogeneity and isotropy in the Two Micron All Sky Survey Photometric Redshift catalogue_ , _MNRAS_ 449 (2015) 670 [1412.5151].
  * [20] J. Colin, R. Mohayaee, M. Rameez and S. Sarkar, _High-redshift radio galaxies and divergence from the CMB dipole_ , _MNRAS_ 471 (2017) 1045 [1703.09376].
  * [21] C.A.P. Bengaly, R. Maartens and M.G. Santos, _Probing the Cosmological Principle in the counts of radio galaxies at different frequencies_ , _J. Cosmology Astropart. Phys._ 2018 (2018) 031 [1710.08804].
  * [22] C.A.P. Bengaly, C.P. Novaes, H.S. Xavier, M. Bilicki, A. Bernui and J.S. Alcaniz, _The dipole anisotropy of WISE × SuperCOSMOS number counts_ , _MNRAS_ 475 (2018) L106 [1707.08091].
  * [23] M. Rameez, R. Mohayaee, S. Sarkar and J. Colin, _The dipole anisotropy of AllWISE galaxies_ , _MNRAS_ 477 (2018) 1772 [1712.03444].
  * [24] M.I. Scrimgeour, T. Davis, C. Blake, J.B. James, G.B. Poole, L. Staveley-Smith et al., _The WiggleZ Dark Energy Survey: the transition to large-scale cosmic homogeneity_ , _MNRAS_ 425 (2012) 116 [1205.6812].
  * [25] P. Ntelis et al., _Exploring cosmic homogeneity with the BOSS DR12 galaxy sample_ , _JCAP_ 06 (2017) 019 [1702.02159].
  * [26] S. Sarkar, B. Pandey and R. Khatri, _Testing isotropy in the Universe using photometric and spectroscopic data from the SDSS_ , _Mon. Not. Roy. Astron. Soc._ 483 (2019) 2453 [1810.07410].
  * [27] S. Alesker, _Description of continuous isometry covariant valuations on convex sets_ , _Geometriae Dedicata_ 74 (1999) 241.
  * [28] P. Chingangbam, K.P. Yogendran, P.K. Joby, V. Ganesan, S. Appleby and C. Park, _Tensor Minkowski Functionals for random fields on the sphere_ , _J. Cosmology Astropart. Phys._ 2017 (2017) 023 [1707.04386].
  * [29] V. Ganesan and P. Chingangbam, _Tensor Minkowski Functionals: first application to the CMB_ , _J. Cosmology Astropart. Phys._ 2017 (2017) 023 [1608.07452].
  * [30] P.K. Joby, P. Chingangbam, T. Ghosh, V. Ganesan and C.D. Ravikumar, _Search for anomalous alignments of structures in Planck data using Minkowski Tensors_ , _J. Cosmology Astropart. Phys._ 2019 (2019) 009 [1807.01306].
  * [31] J.P. Kochappan, A. Sen, T. Ghosh, P. Chingangbam and S. Basak, _Statistical Isotropy of the CMB E-mode signal_ , _arXiv e-prints_ (2020) arXiv:2006.15038 [2006.15038].
  * [32] A. Hajian, T. Souradeep and N. Cornish, _Statistical isotropy of the wilkinson microwave anisotropy probe data: A bipolar power spectrum analysis_ , _The Astrophysical Journal_ 618 (2004) L63.
  * [33] S. Basak, A. Hajian and T. Souradeep, _Statistical isotropy of CMB polarization maps_ , _Phys. Rev. D_ 74 (2006) 021301 [astro-ph/0603406].
  * [34] T. Ghosh, A. Hajian and T. Souradeep, _Unveiling hidden patterns in CMB anisotropy maps_ , _Phys. Rev. D_ 75 (2007) 083007 [astro-ph/0604279].
  * [35] P. Goyal, P. Chingangbam and S. Appleby, _Morphology of CMB fields—effect of weak gravitational lensing_ , _J. Cosmology Astropart. Phys._ 2020 (2020) 020 [1911.04740].
  * [36] G.A. Marques, C.P. Novaes, A. Bernui and I.S. Ferreira, _Isotropy analyses of the Planck convergence map_ , _MNRAS_ 473 (2018) 165 [1708.09793].
  * [37] W. Hu and T. Okamoto, _Mass Reconstruction with Cosmic Microwave Background Polarization_ , _ApJ_ 574 (2002) 566 [astro-ph/0111606].
  * [38] T. Okamoto and W. Hu, _Cosmic microwave background lensing reconstruction on the full sky_ , _Phys. Rev. D_ 67 (2003) 083002 [astro-ph/0301031].
  * [39] C.M. Hirata and U.c.v. Seljak, _Reconstruction of lensing from the cosmic microwave background polarization_ , _Phys. Rev. D_ 68 (2003) 083002.
  * [40] F. Bernardeau, _Weak lensing detection in CMB maps._ , _A &A_ 324 (1997) 15 [astro-ph/9611012].
  * [41] F. Bernardeau, _The effects of source clustering on weak lensing statistics_ , _A &A_ 338 (1998) 375 [astro-ph/9712115].
  * [42] J. Carron and A. Lewis, _Maximum a posteriori CMB lensing reconstruction_ , _Phys. Rev. D_ 96 (2017) 063510 [1704.08230].
  * [43] M. Bucher, C.S. Carvalho, K. Moodley and M. Remazeilles, _CMB lensing reconstruction in real space_ , _Phys. Rev. D_ 85 (2012) 043016 [1004.3285].
  * [44] Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi et al., _Planck 2018 results. III. High Frequency Instrument data processing and frequency maps_ , _A &A_ 641 (2020) A3 [1807.06207].
  * [45] K.M. Górski, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke et al., _HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere_ , _ApJ_ 622 (2005) 759 [astro-ph/0409513].
  * [46] J. Bobin, J.L. Starck, F. Sureau and S. Basak, _Sparse component separation for accurate cosmic microwave background estimation_ , _A &A_ 550 (2013) A73 [1206.1773].
  * [47] J. Schmalzing and K.M. Gorski, _Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps_ , _Mon. Not. Roy. Astron. Soc._ 297 (1998) 355 [astro-ph/9710185].
  * [48] E.A. Lim and D. Simon, _Can we detect Hot or Cold spots in the CMB with Minkowski Functionals?_ , _JCAP_ 01 (2012) 048 [1103.4300].
  * [49] A. Lewis, A. Challinor and A. Lasenby, _Efficient Computation of Cosmic Microwave Background Anisotropies in Closed Friedmann-Robertson-Walker Models_ , _ApJ_ 538 (2000) 473 [astro-ph/9911177].
  * [50] P. Chingangbam, K.P. Yogendran, P. Goyal and S. Appleby.  _Geometric meaning of statistical isotropy of smooth random fields: application to cosmological fields_ , in prep.
  * [51] J.D. Hunter, _Matplotlib: A 2d graphics environment_ , _Computing in Science & Engineering_ 9 (2007) 90.


# A generalized Montel theorem for a class of first order elliptic equations
with measurable coefficients

Erik Duse

###### Abstract

In this paper we prove a generalization of Montel’s theorem for a class of
first order elliptic equations with measurable coefficients involving
Hodge–Dirac operators. We then apply this result to sequences of solutions of
second order uniformly elliptic equations with measurable coefficients on
divergence form and show that this results in a precompactness result for such
sequences.

## Introduction

We begin by recalling a version of Montel’s theorem from classical complex
analysis.

######  Theorem 0.1.

Let $\Omega\subset{\mathbb{C}}$ be a domain and let $\mathscr{F}$ be a family
of bounded holomorphic functions on $\Omega$. The $\mathscr{F}$ contains a
subsequence $\\{f_{n}\\}\subset\mathscr{F}$ that converge compactly to some
$f\in\mathscr{F}$.

Here converge compactly means that $f_{n}\to f$ uniformly on any compact
subset $K\subset\Omega$ as $n\to\infty$. Holomorphic functions are of course
defined by the condition that

$\displaystyle\overline{\partial}f(z)=0,$

and $\overline{\partial}$ is the _elliptic_ Cauchy-Riemann operator. Montel’s
theorem can be generalized to solutions of a large class of partial
differential operators with smooth coefficients acting on sections over
manifolds. For simplicity we assume that we are on euclidean domains and we
have vector valued fields rather than sections. Following [2] we define:

######  Definition 0.1 (Weak hypoellipticity).

Let $\Omega\subset{\mathbb{R}}^{n}$ be a domain and let $E$ and $F$ be
euclidean vector spaces and let $\mathcal{P}$ be a partial differential
operator with smooth coefficients such that

$\displaystyle\mathcal{P}:C^{\infty}(\Omega,E)\to C^{\infty}(\Omega,F).$

We say that $\mathcal{P}$ is _weakly_ hypoelliptic if all solutions of
$\mathcal{P}u=0$ are smooth. Furthermore, we set
$\mathscr{H}(\Omega,\mathcal{P})=\text{ker}(\mathcal{P})=\\{u\in
C^{\infty}(\Omega,E):\mathcal{P}u=0\\}$.

In [2] the following Theorem 4 was proven.

######  Theorem 0.2.

Let $\mathcal{P}$ be a weakly hypoelliptic operator. Then any locally
$L^{1}$-bounded sequence $\\{u_{j}\\}_{j}\in\mathscr{H}(\Omega,\mathcal{P})$
contains a subsequence which converge in the $C^{\infty}$-topology to some
$u\in\mathscr{H}(\Omega,\mathcal{P})$.

The proof of Theorem 4 relies on the estimate

$\displaystyle\|u\|_{C^{j}(K)}\leq C\|u\|_{L^{1}(U)}$

valid for weakly hypoelliptic operators $\mathcal{P}$ and any
$u\in\mathscr{H}(\Omega,\mathcal{P})$ and any compact set $K\subset\Omega$ and
open set $U\Subset\Omega$ containing $K$, see Lemma 2 in [2].

In another direction, let $\Omega\subset{\mathbb{C}}$ and consider the
Beltrami equation

$\displaystyle\overline{\partial}f(z)=\mu(z)\partial f(z)$ (0.1)

where $\mu\in L^{\infty}(\Omega,{\mathbb{C}})$ such that $|\mu|_{\infty}<1$.
This is an elliptic system with non-smooth coefficients for which the notion
of hypoellipticity is meaningless. Yet, all solution $f\in
W^{1,2}_{loc}(\Omega,{\mathbb{C}})$ of the Beltrami equation admits a
factorization

$\displaystyle f(z)=h(g(z))$

where $h$ is any holomorphic function and $g\in W^{1,1}_{loc}(\Omega)$ is a
_fixed homeomorphic solution_ to (0.1). This is the famous Stoilow
factorization theorem, see [1, Thm. 5.5.1]. Using the Stoilow factorization
theorem one can prove a generalization of Montel’s theorem for solutions of
the Beltrami equation by reducing it to Montel’s theorem for holomorphic
functions. Namely, if $\\{f_{j}\\}_{j}\subset
W^{1,2}_{loc}(\Omega,{\mathbb{C}})$ is a family of solutions of the Beltrami
equation (0.1) that is locally bounded in $L^{1}$, then there exists a fixed
homeomorphic solution $g\in W^{1,1}_{loc}(\Omega)$ of (0.1) and a sequence of
holomorphic functions $\\{h_{j}\\}_{j}$ locally bounded in $L^{1}$ such that
$f_{j}=h_{j}\circ g$. By Montel’s theorem for holomorphic functions there
exists a subsequence $\\{h_{j_{k}}\\}_{k}$ that converge locally uniformly to
a holomorphic function $h$. But then $f=h\circ g$ solves (0.1). Our aim now is
to generalize this result to a class of first order elliptic systems in higher
dimensions with merely measurable coefficients. This result will require
slightly stronger assumptions and the result is also slightly less general in
the sense that our limit function may depend on the choice of open subset
$U\Subset\Omega$ considered. This is proven in Theorem 0.3. A consequence of
this result is a precompactness result for solutions of second order uniformly
elliptic equations on divergence form.

### Acknowledgements

Erik Duse was supported by the Knut and Alice Wallenberg Foundation KAW grant
2016.0416. The author thanks Kari Astala for useful suggestions improving the
presentation of the paper.

## Dirac–Beltrami equation

Let $\Lambda{\mathbb{R}}^{n}$ denote the exterior algebra equipped with the
induced euclidean inner product form ${\mathbb{R}}^{n}$. Let $d$ denote the
exterior derivative and $\delta=-d^{\ast}$ the interior derivative, the
negative of the formal adjoint of $d$. Define the Hodge-Dirac operators

$\displaystyle\mathcal{D}^{\pm}=d\pm\delta.$

Then

$\displaystyle(\mathcal{D}^{\tiny+})^{2}=-(\mathcal{D}^{\tiny-})^{2}=\Delta,\quad\mathcal{D}^{\tiny-}\mathcal{D}^{\tiny+}=-\mathcal{D}^{\tiny+}\mathcal{D}^{\tiny-},$

and $\mathcal{D}^{\tiny+}$ is formally skew-adjoint and $\mathcal{D}^{\tiny-}$
is formally self-adjoint. For a more comprehensive discussion of the
Hodge–Dirac operators and their relation to Clifford algebras we refer the
reader to [3]. Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{n}$ and
$\mathcal{M}\in L^{\infty}(\Omega,\mathcal{L}(\Lambda{\mathbb{R}}^{n}))$
satisfy

$\displaystyle M=\|\|\mathcal{M}(x)\|\|_{L^{\infty}}<1.$

Consider the _Dirac–Beltrami equation_

$\displaystyle\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x)$
(0.2)

for $F\in
W^{1,2}_{loc}(\Omega,\Lambda):=W^{1,2}_{loc}(\Omega,\Lambda{\mathbb{R}}^{n})$.
The condition that $M<1$ ensures that this system is elliptic. This can be
seen as a generalization of the classical Beltrami equation in the plane and
solutions of it has many features in common with solutions to the Beltrami
equation, especially in the case when
$F:\Omega\to\Lambda^{ev}{\mathbb{R}}^{n}=\oplus_{j=0}\Lambda^{2j}{\mathbb{R}}^{n}$.
In the same time, it is also a very different equation in the sense that
$\text{dim}(\Lambda{\mathbb{R}}^{n})=2^{n}>n=\text{dim}({\mathbb{R}}^{n})$. In
particular, there is no Stoilow factorization theorem for solutions of (0.2).
This fact necessitates the need to develop new methods to study solutions and
prove analogous theorems in higher dimension. In [3] the following
representation formula for solutions of (0.2) was proven for $F\in
W^{1,2}(\Omega,\Lambda)$ and where $\Omega$ is a $C^{2}$-domain.

$\displaystyle
F(x)=\mathcal{C}_{T}^{+}\circ(I-\mathcal{S}_{T}\mathcal{M})^{-1}\mathcal{S}_{T}\mathcal{M}(\mathcal{D}^{\tiny+}H)+H,$

where $H$ is a solution to the boundary value problem

$\displaystyle\mathcal{D}^{\tiny-}H(x)=0\,\,\,x\in\Omega,\,\,\,\gamma_{T}H=\gamma_{T}F.$
(0.3)

Here $\gamma_{T}$ denotes the _tangential trace_ and $\mathcal{C}_{T}^{+}$ is
the _tangential Cauchy transform_ and $\mathcal{S}_{T}$ is the _tangential
Beurling transform_ with respect to $\Omega$, provided $\Omega$ is at least a
Lipschitz domain. We will omit the definitions of these integral operators but
instead the refer the reader to [3]. In fact the representation formula also
holds for strongly Lipschitz domains but then $F$ belongs to the _partial
Sobolev space_

$\displaystyle
W^{1,2}_{d,\delta}(\Omega,\Lambda)=\\{f\in\mathscr{D}^{\prime}(\Omega,\Lambda):F,dF,\delta
F\in L^{2}(\Omega,\Lambda)\\}.$ (0.4)

instead.

The important point is that $\mathcal{C}_{T}^{+}$ is a weakly singular
integral operator and $\mathcal{S}_{T}$ is a singular integral operator. The
condition $M<1$ now enters and ensures that the Neumann series

$\displaystyle(I-\mathcal{S}_{T}\mathcal{M})^{-1}G=\sum_{j=0}^{\infty}\mathcal{S}_{T}\mathcal{M}G$

converges for any $G\in L^{2}(\Omega,\Lambda)$. If we define the operator

$\displaystyle\mathcal{T}=\mathcal{C}_{T}^{+}\circ(I-\mathcal{S}_{T}\mathcal{M})^{-1}\mathcal{S}_{T}\mathcal{M}\circ\mathcal{D}^{\tiny+}$

the representation formula can be written

$\boxed{F=(I+\mathcal{T})(H).}$ (0.5)

We note that $H$ is unique up to an element in the relative Hodge-De Rham
space $\mathcal{H}_{T}(\Omega)=\\{\omega\in
W^{1,2}(\Omega,\Lambda):d\omega=\delta\omega=0,\,\,\,\gamma_{T}\omega=0\\}$.
Note that whenever $\Omega$ is finitely connected we have
$\text{dim}(\mathcal{H}_{T}(\Omega))<+\infty$. Moreover,
$\mathcal{D}^{\tiny-}$ can be viewed as a generalization of
$\overline{\partial}$, and solutions of $\mathcal{D}^{\tiny-}H=0$ have many
properties in common with holomorphic functions. Just as the Stoilow theorem,
this sets a up a bijective (up to $\mathcal{H}_{T}(\Omega)$) correspondence
between solutions of (0.2) and solutions of $\mathcal{D}^{\tiny-}H=0$. In
particular any $H\in W^{1,2}(\Omega,\Lambda)$ solving
$\mathcal{D}^{\tiny-}H=0$ give rise to a solution $F$ of (0.2) through
$F=(I+\mathcal{T})(H)$. Moreover $\mathcal{T}:W^{1,2}(\Omega,\Lambda)\to
W^{1,2}(\Omega,\Lambda)$ is a bounded operator, which if $M<<1$ can be viewed
as a small perturbation of $H$ in the $W^{1,2}$-topology. The purpose of the
present work is to exploit this correspondence and show how properties of
$H\in\mathscr{H}(\Omega,\mathcal{D}^{\tiny-})$ can be transfered to properties
of $F$.

We first the recall the following Caccioppoli inequality that is proven in
[3]. Let $\Omega\subset{\mathbb{R}}^{n}$ be a domain and let $F\in
W^{1,2}_{loc}(\Omega)$ be a solution of
$\mathcal{D}^{\tiny-}F=\mathcal{M}\mathcal{D}^{\tiny+}F$ on $\Omega$. Then for
any $\varphi\in C^{\infty}_{0}(\Omega)$ there exists a constant $C$ depending
only on $n$ such that

$\displaystyle\int_{{\mathbb{R}}^{n}}|\varphi(x)|^{2}|\nabla\otimes
F(x)|^{2}dx\leq C\int_{{\mathbb{R}}^{n}}|\nabla\varphi(x)|^{2}|F(x)|^{2}dx.$
(0.6)

Let $U\Subset V\Subset\Omega$ be open sets such that
$\text{supp}(\varphi)\subset\overline{V}$ and $\varphi$ is chosen so that
$\varphi(x)=1$ for all $x\in\overline{U}$. Then (0.6) shows that there exists
a constant $C^{\prime}$ depending on $n$, $U$ and $V$ so that

$\displaystyle\|F\|_{W^{1,2}(U,\Lambda)}\leq
C^{\prime}\|F\|_{L^{2}(V,\Lambda)}.$ (0.7)

######  Proposition 0.1.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded $C^{2}$-domain. Let
$\\{F_{j}\\}_{j}\subset W^{1,2}(\Omega,\Lambda)$ be a sequence of solutions of

$\displaystyle\mathcal{D}^{\tiny-}F_{j}(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F_{j}(x)$

for some $\mathcal{M}\in L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$
such that $M:=\|\|\mathcal{M}(x)\|\|_{\infty}<1$. If

$\displaystyle\sup_{j}\|F_{j}\|_{W^{1,2}}<+\infty,$

then there exists a subsequence $\\{F_{j_{k}}\\}_{k}\in
W^{1,2}(\Omega,\Lambda)$ that converges to some $F\in W^{1,2}(\Omega,\Lambda)$
and solves

$\displaystyle\mathcal{D}^{\tiny-}F(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F(x).$

###### Proof.

Since $\Omega$ is a $C^{2}$-domain the trace map
$\gamma:W^{1,2}(\Omega,\Lambda)\to W^{1/2,2}(\Omega,\Lambda)$ is bounded. For
each $j$ define $H_{j}\in W^{1,2}(\Omega,\Lambda)$ through (0.4). In
particular we have

$\displaystyle\|H_{j}\|_{W^{1,2}(\Omega)}\leq C^{\prime}\|\gamma
H_{j}\|_{W^{1/2,2}(\partial\Omega)}=C^{\prime}\|\gamma
F_{j}\|_{W^{1/2,2}(\partial\Omega)}\leq
C^{\prime\prime}\|F_{j}\|_{W^{1,2}(\Omega)}.$

Thus

$\displaystyle\sup_{j}\|H_{j}\|_{W^{1,2}(\Omega)}<+\infty.$

But this implies that $\sup_{j}\|H_{j}\|_{L^{1}(\Omega)}<+\infty$. Since the
Hodge–Dirac operator $\mathcal{D}^{\tiny-}$ is elliptic, it is in particular
weakly hypoelliptic. By Theorem 0.2, $\\{H_{j}\\}_{j}$ contains a weakly
converging subsequence $\\{H_{k}\\}_{k}$ that converge locally uniformly in
$C^{\infty}$ to some $H\in C^{\infty}(\Omega,\Lambda)$ which solves
$\mathcal{D}^{\tiny-}H=0$ in $\Omega$. But this implies that $H_{k}\to H$ in
$W^{1,2}(\Omega,\Lambda)$. Define

$\displaystyle F:=(I+\mathcal{T})(H).$

Then $F$ solves (0.2) and satisfy $\gamma_{T}F=\gamma_{T}H$. ∎

######  Theorem 0.3.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a bounded domain. Let
$\\{F_{n}\\}_{n}\subset W^{1,2}_{loc}(\Omega,\Lambda)$ be a sequence of
solutions of (0.2) for some $\mathcal{M}\in
L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$ such that
$M:=\|\|\mathcal{M}(x)\|\|_{\infty}<1$. Assume that $\\{F_{j}\\}_{j}$ is
locally bounded in $L^{2}$. Then for any smooth relatively compact subset
$U\Subset\Omega$ there exists a subsequence $\\{F_{n_{k}}\\}_{k}$ that
converge in $W^{1,2}$ to some $F=F_{U}\in W^{1,2}(\Omega)$ that solves (0.2).

###### Proof.

By the Caccioppoli inequality (0.7), for any $U\Subset\Omega$ there exists an
open set $V$ such that $U\Subset V\Subset\Omega$ such that any solution $F\in
W^{1,2}_{loc}(\Omega)$ of (0.2) satisfy

$\displaystyle\|F\|_{W^{1,2}(U)}\leq\|F\|_{L^{2}(V)}.$

Hence

$\displaystyle\sup_{n}\|F_{n}\|_{W^{1,2}(U)}\leq+\infty.$

By Proposition 0.1 there exists a subsequence $\\{F_{n_{j}}\\}_{j}$ that
converge in $W^{1,2}$ to some $F_{U}\in W^{1,2}(U,\Lambda)$ and solves (0.2).
∎

######  Remark 0.1.

It is not known to the author whether solutions to (0.2) are Hölder continuous
or not. In a special case this is proven in Theorem 11.4 in [3].

## Applications

######  Proposition 0.2.

Let $\Omega\subset{\mathbb{R}}^{n}$ be a domain and assume that
$\\{u_{n}\\}_{n}\in W^{1,2}_{loc}(\Omega)$ is locally uniformly bounded in
$L^{2}$ and solve (0.8)

$\displaystyle\text{div}\,A(x)\nabla u_{j}(x)=0$ (0.8)

on $\Omega$ for every $j$. Here $A\in
L^{\infty}(\Omega,\mathcal{L}({\mathbb{R}}^{n}))$ is normal, i.e.,
$[A^{\ast}(x),A(x)]=0$ and satisfy the ellipticity bounds

$\displaystyle\lambda|v|^{2}\leq\langle A(x)v,v\rangle,\quad\|A(x)\|<\Lambda$

for a.e. $x\in\Omega$ and some $0<\lambda\leq\Lambda<\infty$. Then on any
smooth simply connected $U\Subset\Omega$ there exists a subsequence
$\\{u_{n_{k}}\\}_{k}$ that converge to some $u\in W^{1,2}(U)$ and solves (0.8)
on $U$.

###### Proof.

Let

$\displaystyle\mathcal{M}(x)=(I-A(x))\circ(I+A(x))^{-1}$

denote the Cayley transform of $A$. By Lemma A.3 in [3] there exists an
$M=M(\lambda,\Lambda)<1$ such that

$\displaystyle M=\|\|\mathcal{M}(x)\|\|_{L^{\infty}}.$

Moreover, on any smooth simply connected open $U\Subset\Omega$ the equations
(0.8) are equivalent to the equations

$\displaystyle\mathcal{D}^{\tiny-}F_{j}(x)=\mathcal{M}(x)\mathcal{D}^{\tiny+}F_{j}(x)$

for some $F_{j}=u_{j}+v_{j}$, where $v_{j}\in
W^{1,2}(\Omega,\Lambda^{2}{\mathbb{R}}^{n})$. Applying Theorem 0.3 to the
sequence $\\{F_{j}\\}$ and using Theorem 6.2 in [3] proves the desired result.
∎

## References

  * [1] K. Astala, T. Iwaniec and G. Martin Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane _Princeton University Press_ (2009) NJ.
  * [2] C. Bär Some properties of solutions of weakly hypoelliptic equations _Int. J. Differ. Equ._ , Art. ID 526390 Band: 2013
  * [3] E. Duse Second Order Linear Elliptic Equations and Hodge-Dirac Operators _arXiv:2010.15699_

Erik Duse, Department of Mathematics and Statistics, KTH, Stockholm, Sweden
<EMAIL_ADDRESS>