# Constraining Bianchi-I universe with Pantheon SNIa data

Anshul Verma1, Sanjeet K. Patel1, Pavan K<EMAIL_ADDRESS>Sukanta Panda2, David F. Mota3

###### Abstract

We study the Bianchi-I cosmological model motivated by signals of statistical
isotropy violation seen in cosmic microwave background (CMB) observations and
others. To that end, we consider various kinds of anisotropic matter that
source anisotropy in our model, specifically cosmic strings, magnetic fields,
domain walls and Lorentz violation generated magnetic fields. These
anisotropic matter sources, taking one at a time, are studied for their co-
evolution with standard model (isotropic) sources viz., dust-like
(dark/normal) matter and dark energy modelled as cosmological constant. We
constrain the Hubble parameter, density fractions of anisotropic matter,
isotropic cold dark matter (CDM) and dark energy ($\Lambda$) in a Bianchi-I
universe with planar symmetry i.e., with a global ellipsoidal geometry, and
try to find signatures of a cosmic preferred axis if any. Pantheon Type Ia
Supernova (SNIa) data is used in our study to obtain constraints on
cosmological parameters of this model. In our analysis, we found a cosmic axis
of anisotropy using SN1e data. Interestingly this preferred axis lies in the
same direction, broadly, as some other cosmic anisotropy axes reported in the
literature such as the CMB hemispherical power asymmetry, etc., with some of
anisotropic sources considered. We also found a non-zero evidence for cosmic
shear and eccentricity for our universe with those sources. To be more
conclusive, we require more SNIa host galaxy data for tighter constraints on
distance and absolute magnitude calibration which we hope will be available
from the future JWST observations and other surveys.

1Department of Physics, Indian Institute of Technology (BHU), Varanasi -
221005, India

2Department of Physics, Indian Institute of Science Education and Research,
Bhopal - 462066, India

3Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029
Blindern, N-0315 Oslo, Norway

##  1 Introduction

Observations of Cosmic Microwave Background (CMB) from COBE, WMAP, and Planck
missions [1, 2, 3] lead to the current era of ‘Precision Cosmology’. This
naturally paved the way in-turn for testing the fundamental hypothesis of
current standard model of cosmology itself, specifically homogeneity and
isotropy of our universe. It resulted in finding many instances of isotropy
violation in CMB large angular scales indicating a preferred direction for our
universe [4, 5, 6]. Such violations of statistical isotropy were also seen in
other data sets such as Type Ia Supernovae [7, 8, 9, 10, 11, 12], alignment of
optical polarization vectors of distant Quasars [13, 14, 15, 16], radio
polarization vector alignments [17, 18, 19], large scale velocity flows being
larger than those predicted in $\Lambda$CDM cosmology [20], etc. A host of
other observational challenges to the current standard model of cosmology are
also present. See, for example, Ref. [21, 22, 23, 24] for a review. Unexpected
alignments of anisotropy axes seen in such diverse astronomical and
cosmological data sets were initially discussed in Ref. [25]. Although, we are
in an era of unprecedented precision in terms of cosmological data that is
available, and the standard cosmological model based on ‘Cosmological
principle’ explaining a variety of observational data, we still have these
challenges or anomalies mentioned above that needs to be resolved for a better
understanding of our universe.

Many of these anomalies suggest re-examining the notion of ‘isotropy’ via the
Cosmological principle. Such anisotropy axis may be indicative of different
expansion rates in different directions of our universe. In order to explain
such anisotropy, we consider a global geometry using some metric that is non-
trivial and yet has minimal modifications to the Friedmann-Lemaître-Robertson-
Walker (FLRW) background. There exists such anisotropic, but homogeneous
models called Bianchi-type cosmological models [26]. The simplest among them
is the Bianchi-I model whose invariant space-time line element is given by

$ds^{2}=c^{2}dt^{2}-a(t)^{2}dx^{2}-b(t)^{2}dy^{2}-c(t)^{2}dz^{2}\,,$ (1)

where ‘$a(t)$’, ‘$b(t)$’ and ‘$c(t)$’ are the scale factors characterizing
expansion of the universe along the three comoving coordinates $x$, $y$ and
$z$ respectively. Further ‘$c$’ is the speed of light in vacuum and $t$ is the
cosmic time coordinate. Hereafter, we work in units of $c=1$. This difference
in cosmic expansion in different directions leads to an axis of anisotropy for
the universe. Given the nature of physical sources which will be studied in
the present work that can induce anisotropy, we consider Eq. (1) with a
residual planar symmetry taking $a(t)=b(t)$ and the $z$-axis normal to the
plane serving as universe’s preferred direction, which will be constrained
from observational data later in this paper.

While we change the left hand side of Einstein’s field equations by assuming a
different metric for our background space-time, we also need to take care of
the right hand side by choosing appropriate source terms via the energy-
momentum tensor. The question is what kind of matter can be added as a source
of anisotropy (if there is any) in the universe? During inflation, the
anisotropic shear in the universe decreases and becomes isotropically
negligible at last scattering surface [27]. To address any non trivial shear
in present times, there has to be some source of anisotropy after that period.
This may happen if we have an anisotropic dark energy or any other form of
anisotropic matter source. In this paper, we investigate the latter case with
four types of anisotropic sources : Cosmic Strings (CS), Magnetic Fields (MF),
Domain Walls (DW), Lorentz Violation generated Magnetic Fields (LVMF) [28,
29]. It was shown in Ref. [30] that power suppression at low multipoles can be
explained by assuming magnetic fields. Large scale magnetic fields could have
been produced during cosmic inflation due to a Lorentz-violating term in the
photon sector [31]. Anisotropic dark energy as a possible source was
discussed, for example, in Ref. [32].

Rest of the paper is organized as follows. In the next section we briefly
discuss Einstein’s field equations in cosmic time and also the distance
modulus - redshift relation for a Bianchi-I space-time is presented along with
the fractional density and Hubble parameter evolution equations to be solved
for obtaining constriants later on. In section 3, the observational data used
and the likelihood analysis to constrain the model parameters are described.
Then, in section 4, we present our results and analysis i.e., constraints
obtained on various fractional energy densities of (an)isotropic sources
considered for this work and the presence of any cosmic preferred axis. These
are followed by section 5 where we discuss and summarize our findings.

##  2 Bianchi-I Universe

###  2.1 Planar Bianchi-I space-time and Sources of anisotropy

Given the hints for statistical anisotropy from diverse astronomical data, we
should consider a metric that generalizes flat FLRW metric and may explain the
cosmological skewness [28, 29, 30]. To that end, we study the Bianchi-I space-
time given by Eq. (1) but with residual planar symmetry described by the line
element,

$ds^{2}=dt^{2}-a(t)^{2}(dx^{2}+dy^{2})-b(t)^{2}dz^{2}\,,$ (2)

where ‘$a(t)$’ and ‘$b(t)$’ are now the scale factors perpendicular to and
along the axis of anisotropy. By virtue of its geometry, it is also called
Eccentric or Ellipsoidal Universe [29, 30].

The diagonal anisotropic energy-momentum tensor for this metric is then taken
to be,

$T^{\mu}_{\mspace{12.0mu}\nu}={\rm diag}(\rho,-p_{a},-p_{a},-p_{b})\,,$ (3)

where, $p_{a}$ = $w_{a}\rho$ and $p_{b}$ = $w_{b}\rho$ are the anisotropic
pressure terms in the plane normal to and along the anisotropy axis. As
mentioned earlier, we consider four types of anisotropic matter sources in
this paper. First we consider two types of topological defects viz., cosmic
strings (CS) and domain walls (DW) as a form of anisotropic matter. Then we
consider a magnetic field generated due to a Lorentz-violating term in the
photon sector, abbreviated as LVMF, as another anisotropic source. Finally the
well known configuration of a uniform magnetic field is also considered. Their
anisotropic equations of state are summarized in Table 1 [29, 33]. Note that
because of the symmetry one assumes in Eq. (1) i.e., depending on our choice
of geometry being prolate or oblate ($a(t)=b(t)$ or $b(t)=c(t)$), the equation
of state parameters, $w_{a}$ and $w_{b}$, may get interchanged accordingly for
all anisotropic sources. Also, the convention used to define the energy-
momentum tensor in Eq. (3) or the signature of the metric in Eq. (2), will
further result in a change of sign for $w_{a}$ and $w_{b}$ listed in Table 1.

Among these, the cosmic string and domain wall distributions can grow
gradually anisotropic with time [34]. The origin of all these sources is
primordial in the sense that all of these were produced through broken
discrete symmetries during phase transitions in the early universe [35]. We
study the anisotropic co-evolution of these sources with dust-like
(normal+dark) matter and dark energy represented by a cosmological constant,
$\Lambda$. Corresponding to our choice of energy-momentum tensor, we have two
different equations of state $w_{a}$ for any direction within the plane of
symmetry and $w_{b}$ along the normal to that plane.

The necessary mathematical relations following Einstein’s field equations,
that are required to constraint the Ellipsoidal model being studied were
earlier presented in Ref. [32, 33]. However, we briefly summarize them
consistently for the form of planar Bianchi-I metric shown in Eq. (2).

Anisotropic Source | $w_{a}$ | $w_{b}$
---|---|---
Cosmic Strings | 0 | -1
Domain Walls | -1 | 0
LVMF | 0 | 1
MF | 1 | -1
Table 1: Anisotropic sources used in this paper and their anisotropic equation
of state parameter values are given here.

###  2.2 Einstein’s Field Equations

The Einstein’s field equation following our choice of metric given by Eq. (2)
and anisotropic stress-energy tensor given by Eq. (3) can be found to be,

$\displaystyle H_{a}^{2}+2H_{a}H_{b}$ $\displaystyle=$ $\displaystyle 8\pi
G\rho\,,$
$\displaystyle\dot{H_{a}}+\dot{H_{b}}+H_{a}^{2}+H_{b}^{2}+H_{a}H_{b}$
$\displaystyle=$ $\displaystyle-8\pi Gp_{a}\,,$ (4) $\displaystyle
2\dot{H_{a}}+3H_{a}^{2}$ $\displaystyle=$ $\displaystyle-8\pi Gp_{b}\,,$

where $H_{a}=\dot{a}/a$ and $H_{b}=\dot{b}/b$ are the rates of expansion or
Hubble parameters in and normal to the residual plane of symmetry. Here,
$\dot{a}=da/dt$ and $\dot{b}=db/dt$. The continuity equation,
$\nabla_{\mu}T^{\mu}_{\mspace{12.0mu}\nu}=0$, leads to finding,

$\dot{\rho}+(2H_{a}+H_{b})\rho+2H_{a}p_{a}+H_{b}p_{b}=0\,.$ (5)

We set $8\pi G=1$ for the rest of this paper. The expansion rates $H_{a}$ and
$H_{b}$ can be more conveniently expressed in terms of average Hubble
parameter, $H=(2H_{a}+H_{b})/3$, and differential expansion rate, also called
the shear parameter, $\Sigma$ = ($H_{a}-H_{b})/\sqrt{3}$, respectively. In
terms of these new variables, Eq. (2.2) and (5) can be rewritten with respect
to cosmic time ‘$t$’ as:

$\displaystyle\frac{dH}{dt}$ $\displaystyle=$
$\displaystyle-H^{2}-{\frac{2}{3}}\Sigma^{2}-{\frac{1}{6}}(\rho+2p_{a}+p_{b})\,,$
$\displaystyle\frac{d\Sigma}{dt}$ $\displaystyle=$
$\displaystyle-3H\Sigma+\frac{1}{\sqrt{3}}(p_{a}-p_{b})\,,$ (6)
$\displaystyle\frac{d\rho}{dt}$ $\displaystyle=$
$\displaystyle-3H(\rho+\frac{2p_{a}+p_{b}}{3})-\frac{2\Sigma}{\sqrt{3}}(p_{a}-p_{b})\,,$

along with a constraint equation given by,

$H^{2}=\frac{\rho}{3}+\frac{\Sigma^{2}}{3}\,.$ (7)

Note that the isotropic non-interacting sources viz., dust-like matter and a
cosmological constant are automatically described by the anisotropic energy-
momentum tensor given in Eq. (3), where $p_{a}=p_{b}$ or equivalently
$w_{a}=w_{b}$.

###  2.3 Evolution equations and luminosity distance

In order to constraint various parameters characterizing our Bianchi-I model,
we employ latest data corresponding to Type Ia Supernova (SN1a) that will be
described in the next section. Here we present the theoretical luminosity
distance ($d_{L}$) versus redshift ($\mathcal{z}$) relation that is dependent
on various cosmological parameters of our model and the evolution equations of
these parameters to be solved in conjugation. First we define the mean scale
factor and the eccentricity parameter of Bianchi-I model as,

$A=(a^{2}b)^{1/3}\quad{\rm and}\quad e^{2}=1-\frac{b^{2}}{a^{2}}\,.$ (8)

This definition of mean scale factor can be seen to be easily reconciled with
our definition of mean Hubble parameter, $H$, introduced earlier as,

$H=\frac{\dot{A}}{A}=\frac{2H_{a}+H_{b}}{3}\,.$ (9)

Further, it is convenient to define some dimensionless variables as follows.
Using the constraint equation, Eq. (7), we define dimensionless energy density
parameter $\Omega=\rho/(3H^{2})$ (in units of $c=8\pi G=1$) and expansion
normalized dimensionless shear parameter $\sigma=\Sigma/(\sqrt{3}H)$.

Here we note that the energy density $\rho$ in Eq. (7), and consequently
$\Omega$, is comprised of both isotropic sources such as (ordinary+dark)
matter denoted by ‘$\Omega_{IM}$’ and dark energy ‘$\Omega_{\Lambda}$’
represented by a cosmological constant, and the anisotropic sources denoted by
$\Omega_{AM}$. As listed in Table 1, we consider four types of anisotropic
matter viz., cosmic strings, magnetic fields, domain walls, and Lorentz
violation generated magnetic fields, along with isotropic matter. By defining
critical energy density as $\rho_{c,0}$ or simply $\rho_{0}=3H^{2}_{0}$, where
$H_{0}$ is the value of Hubble parameter at current times, we can write the
constraint equation at present as,

$\Omega_{AM,0}+\Omega_{IM,0}+\Omega_{\Lambda,0}+\sigma^{2}_{0}=1\,,$ (10)

where

$\Omega_{AM,0}=\rho_{AM,0}/\rho_{0}\,,\quad\Omega_{IM,0}=\rho_{IM,0}/\rho_{0}\,,\quad{\rm
and}\quad\Omega_{\Lambda,0}=\rho_{\Lambda}/\rho_{0}\,,$ (11)

are the fractional energy densities of various sources at current time,
$t_{0}$, and $\sigma_{0}$ is the current value of cosmic shear in the
universe. Hereafter we drop the subscript ‘$0$’ in the fractional energy
densities for brevity, and simply denote them as $\Omega_{AM}$, $\Omega_{IM}$,
$\Omega_{\Lambda}$. Nevertheless their usage should be clear from the context,
whether they denote energy densities at current time having fixed values or
time dependent variables. Finally, a dimensionless time variable is defined as
$\tau=\log(A)$. Thus we have $H=d\tau/dt$.

The direction dependent redshift ($\mathcal{z}$) and luminosity distance
($d_{L}$) of an SN1a object, observed in the direction $\hat{n}$ = ($\theta$,
$\phi$) in a Bianchi-I universe are given by [32],

$\displaystyle 1+\mathcal{z}(\hat{n})$ $\displaystyle=$
$\displaystyle\frac{1}{A}\frac{(1-e^{2}\sin^{2}\alpha)^{\frac{1}{2}}}{(1-e^{2})^{\frac{1}{3}}}\,,$
(12) $\displaystyle d_{L}(\hat{n})$ $\displaystyle=$ $\displaystyle
c(1+\mathcal{z})\int_{A(\mathcal{z})}^{1}{\frac{dA}{{A^{2}}H}}{\frac{(1-e^{2})^{\frac{1}{6}}}{(1-e^{2}\cos^{2}\alpha)^{\frac{1}{2}}}}\,,$
(13)

where $c$ is the speed of light, $\alpha=\cos^{-1}(\hat{\lambda}\cdot\hat{n})$
is the angle between cosmic preferred axis ‘$\hat{\lambda}$’ (i.e., the
$z$-axis of Eq. 2). and the location of an SN1a at ‘$\hat{n}$’ in the sky. To
solve for the integral in the above expression for ‘$d_{L}$’, the complete set
of evolution equations (following Eq. (2.2) and Eq (8)) in terms of
dimensionless variables, just defined, can be found to be given by [33],

$\displaystyle\frac{h^{\prime}}{h}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}(1+\sigma^{2}+\bar{w}\Omega_{AM}-\Omega_{\Lambda})\,,$
$\displaystyle(e^{2})^{\prime}$ $\displaystyle=$ $\displaystyle
6\sigma(1-e^{2})\,,$ $\displaystyle\sigma^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{3}{2}[\sigma(1-\sigma^{2})-(\frac{2}{3}\delta_{w}+\bar{w}\sigma)\Omega_{AM}+\sigma\Omega_{\Lambda}]\,,$
(14) $\displaystyle\Omega_{AM}^{\prime}$ $\displaystyle=$
$\displaystyle-3\Omega_{AM}(\bar{w}+\frac{2}{3}{\delta_{w}}\sigma-\bar{w}\Omega_{AM}+\Omega_{\Lambda}-\sigma^{2})\,,$
$\displaystyle\Omega_{IM}^{\prime}$ $\displaystyle=$ $\displaystyle
3\Omega_{IM}(\bar{w}\Omega_{AM}-\Omega_{\Lambda}+\sigma^{2})\,,$
$\displaystyle\Omega_{\Lambda}^{\prime}$ $\displaystyle=$ $\displaystyle
3\Omega_{\Lambda}(1+\bar{w}\Omega_{AM}-\Omega_{\Lambda}+\sigma^{2})\,,$

where, for some parameter ‘$X$’, its prime denotes a derivative with respect
to the new time variable, $\tau$, i.e., $X^{\prime}=dX/d\tau$222We note that
there was a typographical error in Eq. 7.9 of Ref. [33] for $e^{2}$ \- an
extra factor of $1/A$ \- that is correctly presented here.,
$\bar{w}=(2w_{a}+w_{b})/3$ and $\delta_{w}=w_{a}-w_{b}$. Here we don’t need to
solve for, say, $\Omega_{\Lambda}$ which is determined by the constraint
equation $\Omega_{\Lambda}=1-\Omega_{IM}-\Omega_{AM}-\sigma^{2}$ at any time.
Further, $h=H/(100~{}km.s^{-1}/Mpc)$ in the preceding set of equations is the
dimensionless Hubble parameter.

Since we are provided with redshift ($\mathcal{z}$) of an object, these
evolution equations can be solved for any of the cosmological parameters ‘$X$’
with respect to $\mathcal{z}$ as,

$\frac{dX}{d\mathcal{z}}=\frac{dA}{d\mathcal{z}}\frac{d\tau}{dA}\frac{dX}{d\tau}=\frac{1}{A}\frac{dA}{d\mathcal{z}}X^{\prime}\,,$
(15)

where we used the relation $d\tau/dA=1/A$. Also, $dA/d\mathcal{z}$ can be
found to be given by,

$\displaystyle\frac{dA}{d\mathcal{z}}$ $\displaystyle=$
$\displaystyle\frac{-f_{1}}{(1+\mathcal{z})(1+\mathcal{z}-{f_{1}}{f_{2}}\frac{\sigma}{A})}\,,$
$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{(1-e^{2}\sin^{2}\alpha)^{\frac{1}{2}}}{(1-e^{2})^{\frac{1}{3}}}\,,$
(16) $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\frac{2+(e^{2}-3)\sin^{2}{\alpha}}{1-e^{2}\sin^{2}{\alpha}}\,.$

Let ‘$\mathcal{z}^{*}$’ be the observed redshift of an SN1a. The evolution
equations are solved from $\mathcal{z}=0$ (corresponding to the initial
condition $A_{0}=1$ at current time $t=t_{0}$) up to
$\mathcal{z}=\mathcal{z}^{*}$ in order to obtain constraints on our model
parameters. The set of cosmological parameters to be constrained in the
anisotropic Bianchi-I model are
$\\{h_{0},e^{2}_{0},\sigma_{0},\Omega_{AM},\Omega_{IM},\Omega_{\Lambda},\hat{\lambda}=(l_{a},b_{a})\\}$,
where we consider the anisotropic matter sources (given in Table 1) taken one
at a time, along with the isotropic sources. The cosmic anisotropy axis in our
model denoted by $\hat{\lambda}=(l_{a},b_{a})$ will be estimated in galactic
coordinates.

Constraints thus obtained on our anisotropic model are compared with the
standard flat $\Lambda$CDM model, for which the luminosity distance is given
by,

$d_{L}=\frac{c(1+\mathcal{z}^{*})}{H_{0}}\int_{0}^{\mathcal{z}^{*}}{\frac{d\mathcal{z}}{\sqrt{\Omega_{IM}(1+\mathcal{z})^{3}+\Omega_{\Lambda}}}},$
(17)

where $\Omega_{\Lambda}=1-\Omega_{IM}=1-\Omega_{m,0}$. Thus we have two
cosmological parameters viz., $H_{0}$ (or equivalently $h_{0}$) and
$\Omega_{IM}$ (or $\Omega_{\Lambda}$), to be constrained in the standard
concordance model, which will be our reference model to compare with.

##  3 Data sets used and Methodology

###  3.1 The Pantheon SNIa data

In order to obtain constraints on our model parameters, in this work we will
be using the _Pantheon_ Type Ia Supernovae (SN1e)
data333https://github.com/dscolnic/Pantheon compiled from several surveys,
consists of 1048 spectroscopically confirmed SN1a [36], that is much larger a
sample size than any of the previously compiled SN1e data. It includes PS1
survey (Pan-STARRS, 279 SNe [37]), CSP (Carnegie Supernova Project, 26 SNe)
[38], CFA1-CFA4 (146 SNe) [39, 40, 41], SDSS (Sloan Digital Sky Survey, 335
SNe) [42], SNLS (SuperNova Legacy Survey, 236 SNe) [43]. All of these contain
SNe with mean redshift $\bar{\mathcal{z}}<1$. Few surveys with
$\bar{\mathcal{z}}>1$ are SCP (Supernova Cosmology Project, 3 SNe) [44], GOODS
(Great Observatories Origins Deep Survey, 15 SNe) [45], CANDELS (6 SNe) [46]
and CLASH (2 SNe) [47, 48] surveys.

Only the best fitting Supernovae to _Type Ia_ which includes class
identification, measurement time, convergence of the fit and other sample
cuts, were included in Pantheon data set. For example, the PS1 survey includes
three level quality cuts namely Initial criteria, First-pass criteria and the
Final criteria [37]. Similarly, SNe from other surveys were also subjected to
several quality checks to confirm that they unambiguously fall under ‘Type Ia’
category as physics of them is presumably well understood, although there
exists debate in literature on their progenitors [49, 50].

In the _left_ panel of Fig. [1], we exhibit the redshift distribution of the
Pantheon compilation of Type Ia Supernaovae that we used in our present study.
In the _right_ panel of same figure, we have shown the sky distribution of
1048 SN1a that make up the Pantheon sample and the accompanying colour bar
denotes redshift of each SN1a in that sample. As is obvious from Fig. [1], the
distribution of SN1a in Pantheon sample is very inhomogeneous with respect to
redshift ($\mathcal{z}$) as well as spatial location ($\hat{n}$).

Figure 1: _Left :_ Number count histogram of SNIa in Pantheon data with
respect to redshift in 50 bins. _Right :_ Sky distribution of Pantheon SNIe in
galactic coordinates, with clourbar indicating their redshift.

###  3.2 Likelihood analysis

In this section, we describe the likelihood function ($\mathcal{L}$) that is
dependent on various SN1a fit parameters including cosmological parameters of
our interest. The likelihood function is minimized by sampling various
parameters that it depends on using an MCMC method.

From the data, one is provided with, among others, the redshift
‘$\mathcal{z}^{*}$’ and rest-frame peak $B$-band magnitude ‘$m_{B}^{\rm obs}$’
for each observed SN1a. Supernovae cosmology relies on relating the luminosity
distance, $d_{L}$, which is dependent on cosmological parameters, and the
observed apparent magnitude, $m_{B}^{\rm obs}$, via the _distance modulus_ as

$\mu^{\rm
th}=5\log\left(\frac{d_{L}({\boldsymbol{\theta}},\mathcal{z})}{10~{}pc}\right)\quad{\rm
and}\quad\mu^{\rm obs}=m_{B}^{\rm corr}-M_{0}=m_{B}^{\rm obs}+\alpha
x_{1}-\beta\mathsf{c}-M_{0}={\bf V}^{T}{\bf X}-M_{0}\,,$ (18)

where
${\boldsymbol{\theta}}=\\{h_{0},\Omega_{IM},\Omega_{\Lambda},e^{2}_{0},\sigma_{0},\Omega_{AM},(l_{a},b_{a})\\}$
are the cosmological parameters of the model being fitted, ‘$M_{0}$’ is the
absolute magnitude of an SN1a as measured from a reference distance of
$10~{}pc$ (parsec) from it, $m_{B}^{\rm corr}=m_{B}+\alpha
x_{1}-\beta\mathsf{c}$ is corrected $B$-band magnitude of SN1a, ‘$x_{1}$’ and
‘$\mathsf{c}$’ are SN1a light curve fit parameters corresponding to shape and
colour, and finally ‘$\alpha$’ and ‘$\beta$’ are global parameters
characterizing SN1a population corresponding to $x_{1}$ and $\mathsf{c}$. For
brevity, we also defined the vectors ${\bf V}=(1,~{}\alpha,~{}-\beta)^{T}$ and
${\bf X}=(m_{B},~{}x_{1},~{}\mathsf{c})^{T}$. All Type Ia Supernovae are
assumed to have the same $M_{0}$, $\alpha$, and $\beta$. So they are
constrained globally along with the cosmological parameters of our interest.

Thus, in order to obtain constrains, we minimize the $\chi^{2}$
($=-\ln(\mathcal{L})$) function defined as,

$\chi^{2}({\boldsymbol{\Theta}})=\Delta{\bf m}_{B}^{T}{\bf C}^{-1}\Delta{\bf
m}_{B}\,,$ (19)

where ${\boldsymbol{\Theta}}=\\{{\boldsymbol{\theta}},\alpha,\beta\\}$ are all
the fit parameters including cosmological and nuisance parameters, and
$\Delta{\bf m}_{B}={\bf m}_{B}^{\rm obs}-{\bf m}_{B}^{\rm th}$ is a vector
whose elements are given by the difference $m_{B,i}^{\rm obs}-m_{B,i}^{\rm
th}$ for each Supernova ‘$i$’ in the data sample. Then, ${\bf C}$ is the
covariance matrix containing both diagonal and non-diagonal terms which will
be described in detail in the next section. The theoretical $B$-band
magnitude, $m_{B,i}^{\rm th}$ of a Supernova ‘$i$’ in the data is defined
following Eq. (18) as,

$\displaystyle m_{B,i}^{\rm th}$ $\displaystyle=$ $\displaystyle\mu^{\rm
th}_{i}+M_{0}-\alpha x_{1,i}+\beta\mathsf{c}_{i}\,,$ (20) $\displaystyle=$
$\displaystyle
5\log\left(\frac{d_{L}({\boldsymbol{\theta}},\mathcal{z}^{*}_{i})}{10~{}pc}\right)+M_{0}-\alpha
x_{1,i}+\beta\mathsf{c}_{i}\,,$ $\displaystyle=$ $\displaystyle
5\log(\tilde{d}_{L}({\boldsymbol{\theta}},\mathcal{z}^{*}_{i}))+\mu_{0}+M_{0}-\alpha
x_{1,i}+\beta\mathsf{c}_{i}$

where $\tilde{d}_{L}$ is now dimensionless and the constant offset term,

$\mu_{0}=5\log\left(\frac{c}{100~{}km.s^{-1}/Mpc}~{}\frac{1}{10~{}pc}\right)\approx
42.384\,,$ (21)

arises as a result of dimensionful parameters in Eq. (13) or Eq. (17) viz.,
the speed of light ‘$c$’ (in $km.s^{-1}$) and the dimensionful part of Hubble
parameter $H=h\times 100~{}km.s^{-1}/Mpc$, and the $10~{}pc$ term in the
denominator of Eq. (18).

Therefore, in a Bianchi-I universe, the dimensionless luminosity distance is
given by,

$\tilde{d}_{L}(\hat{n})=(1+\mathcal{z}^{*})\int_{A(\mathcal{z}^{*})}^{1}{\frac{dA}{{A^{2}}h}}{\frac{(1-e^{2})^{\frac{1}{6}}}{(1-e^{2}\cos^{2}\alpha)^{\frac{1}{2}}}}\,,$
(22)

following Eq. (13). For a flat $\Lambda$CDM cosmological model, it is given
by,

$\tilde{d}_{L}=\frac{(1+\mathcal{z}^{*})}{h_{0}}\int_{0}^{\mathcal{z}^{*}}{\frac{d\mathcal{z}}{\sqrt{\Omega_{IM}(1+\mathcal{z})^{3}+\Omega_{\Lambda}}}}\,,$
(23)

following Eq. (17).

An interesting point to note here is that, in the standard flat $\Lambda$CDM
model, the Hubble parameter ($H_{0}$ or $h_{0}$) and the absolute SN1a
magnitude ($M_{0}$) are degenerate with each other. One can see from Eq. (20)
and (23) that both parameters form an effective one parameter term given by
“$-5\log(h_{0})+M_{0}$”. Hence they cannot be constrained simultaneously using
Supernova data alone. But, following Eq. (22), we see that there is no such
degeneracy between (mean) Hubble parameter and the absolute magnitude of
Supernovae in a Bianchi-I universe, at least in principle. So they may be
constrained independently.

However, while trying to obtain constraints on the Bianchi-I model treating
$M_{0}$ and $h_{0}$ as free parameters, we find that there exists some
degeneracy between them. This arises as a result of anisotropic matter
density, $\Omega_{AM}$, being low at current times as estimated. For this
reason, we fix $M_{0}$ using the latest precise SN1a absolute magnitude
estimate viz., $M_{0}=-19.253\pm 0.027$ [51]. This is also based on the
assumption that $M_{0}$ i.e., apparent magnitude of a Supernova measured at a
fixed distance of $10~{}pc$ (that is much smaller than the cosmological
distance scales) is same for all Supernova which is reasonable, and may not
effect Bianchi-I model parameter constraints.

###  3.3 Covariance matrix

One can form the $\chi^{2}$ function following the quadrature approach by
defining it as,

$\chi^{2}({\boldsymbol{\Theta}})=\sum_{i={\rm SN1e}}\frac{(m_{B,i}^{\rm
obs}-m_{B,i}^{\rm
th}({\boldsymbol{\Theta}},\mathcal{z}_{i}^{*},M_{0},x_{1,i},\mathsf{c}_{i}))^{2}}{\delta{m_{B,i}}^{2}},$
(24)

when there are no correlations between Supernovae. But, there are correlations
between SN1e in the data that are calibrated simultaneously due to both
statistical and systematic uncertainties in SN1e light curve fitting - for a
given Supernova as well as among various SN1e. A discussion on this can be
found, for example, in Ref. [52]. Therefore, we use the full covariance matrix
as given in Ref. [53] for the Pantheon SN1e sample which has a diagonal and a
non-diagonal part. So the covariance matrix in Eq. (19) can be represented as,

${\bf C}={\bf C}_{\rm stat}+{\bf C}_{\rm sys}\,,$ (25)

where ${\bf C}_{\rm stat}(=C_{{\rm stat},i}\delta_{ij})$ is the diagonal part
and ${\bf C}_{\rm sys}$ is the non-diagonal part of the full covariance matrix
${\bf C}$.

For a particular Supernova, diagonal uncertainties are given by,

$C_{{\rm stat},i}=\sigma_{{\rm int},i}^{2}+\sigma_{{\rm
ext},i}^{2}+\sigma_{{\rm
lens},i}^{2}+\left(\frac{5(1+\mathcal{z}_{i})}{\mathcal{z}_{i}(1+\mathcal{z}_{i}/2)\ln(10)}\right)^{2}\sigma_{\mathcal{z},i}^{2}+{\bf
V}^{T}{\bf C}_{X}{\bf V}\,.$ (26)

Here $\sigma_{\rm int}$ is the uncertainty due to intrinsic scatter of SN1e
magnitudes, and we take $\sigma_{\rm int}=0.1$ which is same as that used for
the Pantheon sample based on ‘C11’ scatter model [36]. We note that this value
depends on the type of dispersion model assumed. For example, here we use
$\sigma_{\rm int}$ following the prescribed ‘C11’ model. However, $\sigma_{\rm
int}=0.08$ for the ‘G10’ intrinsic dispersion model. Next, $\sigma_{\rm ext}$
is the uncertainty for host galaxy extinction or precisely due to uncorrelated
dust along the line of sight and is taken to be $\sigma_{\rm
ext}=0.013\mathcal{z}$ for $\mathcal{z}<0.2$ [54]. Uncertainty due to
stochastic gravitational lensing effects $\sigma_{\rm lens}=0.055\mathcal{z}$
is used as given in Ref. [55]. Following Ref. [56], we use
$c\sigma_{\mathcal{z}}=\pm 250$ km/s for the peculiar velocity uncertainty.
Lastly, ${\bf C}_{X}$ is a $3\times 3$ covariance matrix corresponding to the
derived parameters $m_{B}$, $x_{1}$ and $\mathsf{c}$ for individual SN1a from
lightcurve fitting. Systematic covariance matrix, ${\bf C}_{\rm sys}$,
involving all 1048 Type Ia Supernovae is used as provided by Pantheon
collaboration [43].

##  4 Results

###  4.1 Cosmological constraints

In this section, we present the results from fitting cosmological parameters
of our Bianchi-I model for all four anisotropic sources listed in Table 1 at
the current epoch. We do four separate MCMC likelihood analysis for the four
anisotropic sources considered along with the usual isotropic sources of the
standard cosmological model. Therefore, the set of cosmological parameters to
be constrained are
$\\{h_{0},\Omega_{IM},\Omega_{\Lambda},e^{2}_{0},\sigma_{0},\Omega_{AM},(l_{a},b_{a})\\}$
along with the nuisance parameters $\alpha$ and $\beta$. Due to improved SNIa
calibration techniques (quality) and a much larger data sample (quantity)
compared to previous releases [57, 58, 54, 59, 44, 60], we get good
constraints on all the parameters.

Two dimensional contour plots with $1\sigma$ and $2\sigma$ confidence levels
(CL) for all the cosmological parameters with one anisotropic source included
at a time viz., $\\{h_{0},\Omega_{IM},e^{2}_{0},\sigma_{0},\Omega_{AM}\\}$ are
shown in Fig. [2]. Following the constraint equation, Eq. (10), we have chosen
to sample all the parameters except for the cosmological constant term,
$\Omega_{\Lambda}$. The posterior distributions for ‘$\Omega_{\Lambda}$’
following this constraint equation for different choices of anisotropic matter
sources are shown in _left_ panel of Fig. [3]. Constraints on the two
additional (global nuisance) parameters, namely $\alpha$ and $\beta$,
corresponding to color and stretch parameters of SN1a light curves
respectively, are depicted in the _right_ panel of Fig. [3].

We also obtained constraints on the two parameter standard cosmological model
i.e., the flat $\Lambda$CDM model which is our base model to compare with.
Contour plots with $68\%$ and $95\%$ CL for the two cosmological parameters of
the flat $\Lambda$CDM model viz., {$h_{0},\Omega_{m,0}$} are shown in Fig. [4]
to the _left_. In the same figure, the posterior distribution of dark energy
denisty fraction ($\Omega_{\Lambda}=1-\Omega_{IM}=1-\Omega_{m,0}$) is shown to
the _right_. All the cosmological model parameters for both anisotropic
Bianchi-I universe and the standard flat $\Lambda$CDM model are listed in
Table 2.

Figure 2: 2D contour plots of all Cosmological parameters in our Bianchi-I
model assuming Cosmic Strings (CS), Domain Walls (DW), Lorentz Violation
generated Magnetic Fields (LVMF), and Magnetic Fields (MF) as sources of
anisotropic matter along with isotropic standard model sources.

Figure 3: _Left :_ Posterior distribution of dark energy density fraction
($\Omega_{\Lambda}$) that is fixed by the constraint equation
$\Omega_{\Lambda}=1-\Omega_{IM}-\Omega_{AM}-\sigma^{2}_{0}$, where the others
are sampled as free parameters. _Right :_ $1\sigma$ contour plot between the
nuisance parameters $\alpha$ and $\beta$ when considering different
anisotropic matter sources are shown.

Figure 4: 2D confidence contour plot for the flat $\Lambda$CDM model
parameters $\\{h_{0},\Omega_{m,0}\\}$ is shown in the _left_ panel. Posterior
distribution of dark energy density fraction as fixed by the constraint
equation $\Omega_{\Lambda}=1-\Omega_{IM}=1-\Omega_{m,0}$, where $\Omega_{m,0}$
is the current cold dark matter density fraction, is depicted in the _right_
panel.

Parameter | CS | DW | LVMF | MF |  $\Lambda$CDM
---|---|---|---|---|---
$h_{0}$ | $0.7263\pm 0.0043$ | $0.7228\pm 0.0044$ | $0.7238\pm 0.0042$ | $0.7263\pm 0.0045$ | $0.7239\pm 0.0036$
$\sigma_{0}$ | $-0.0073^{+0.014}_{-0.0060}$ | $-0.034^{+0.019}_{-0.014}$ | $-0.032^{+0.019}_{-0.012}$ | $-0.0073^{+0.014}_{-0.0060}$ | -
$e^{2}_{0}$ | $<0.0578$ | $0.080^{+0.038}_{-0.061}$ | $0.075^{0.031}_{0.064}$ | $<0.0578$ | -
$\Omega_{AM}$ | $<0.0195$ | $0.057^{+0.028}_{-0.035}$ | $0.028^{+0.013}_{-0.0318}$ | $<0.00653$ | -
$\Omega_{\Lambda}$ | $0.676\pm 0.028$ | $0.642\pm 0.037$ | $0.707\pm 0.031$ | $0.686\pm 0.028$ | $0.674\pm 0.024$
$\Omega_{IM}$ | $0.308\pm 0.029$ | $0.299\pm 0.029$ | $0.264^{+0.042}_{-0.035}$ | $0.308\pm 0.029$ | $0.326\pm 0.024$
$\hat{\lambda}=(l_{a},b_{a})$ | $(230^{+22}_{-26},45\pm 10)$ | $(241\pm 24,-52^{+17}_{-15})$ | $(240\pm 24,-49^{+21}_{-14})$ | $(227^{+21}_{-24},43^{+20}_{-10})$ | -
$\chi^{2}$ | 1094.6 | 1093.2 | 1093.2 | 1095.1 | 1098.6
$\Delta\chi^{2}$ | 4 | 5.4 | 5.4 | 3.5 | -
${\rm AIC}_{\rm rel}$ | 0.0498 | 0.1003 | 0.1003 | 0.0388 | 1
Table 2: Best fit values of all the cosmological parameters of anisotropic
Bianchi-I model with cosmic strings (CS), domain walls (DW), Lorentz violation
generated magnetic fields (LVMF) and magnetic fields (MF) taken a sources of
anisotropy, including the axes of anisotropy in galactic coordinates, and the
goodness-of-parameters are tabulated here.

Observing the likelihood density contour plots, one can see that although most
of the cosmological parameters agree with those from standard cosmological
model, we find few interesting results for cosmic shear, $\sigma_{0}$, and the
eccentricity parameter, $e^{2}_{0}$. Shear, at present times, is maximally
_negative_ at 95% confidence level in the case of domain walls and LVMF as
anisotropic sources while being essentially zero (within error bars) for the
other two anisotropic sources viz., cosmic strings and magnetic fields. Same
is found to be the case for eccentricity parameter, at current epoch, where it
prefers a non-zero value when domain walls and Lorentz violation generated
magnetic fields are the anisotropic matter sources, and is consistent with
zero for the other two anisotropic sources. Likewise, the energy density
fractions, $\Omega_{AM}$, of cosmic strings and magnetic fields, when
considered as sources of anisotropic matter, turn out to be consistent with
zero whereas it prefers a non-zero value at $1\sigma$ level for domain walls
and LVMF. So there is an obvious correlation, expectedly, between the
parameters $\\{e^{2}_{0},\sigma_{0},\Omega_{AM}\\}$. Note that the estimated
cosmic shear whether consistent with zero or has a non-zero value it has a
preference for a negative value for all choices of anisotropic matter sources
we considered. However, given the fit for non-zero cosmic shear from SNIe data
in case of domain walls and LVMF, we can infer that there may have existed a
small fraction of energy density from these sources in early universe which
still remains inducing observed anisotropy at current times.

Another interesting observation is the preferred anisotropy axis that was
found for cosmic strings and mangetic fields as sources of anisotropy. Their
axis of anisotropy turns out to be $(l_{a},b_{a})=(230^{\circ},45^{\circ})$
and $=(227^{\circ},43^{\circ})$ for cosmic strings and magneticfields,
respectively, in galactic coordinates. These two axes consistent with each
other are broadly closer to the CMB dipole ($\ell=1$) direction [3], with
which CMB quadrupole ($\ell=2$) and octupole ($\ell=3$) axes [61] are also
aligned, along with other such dipole directions seen in radio data from NRAO
VLA Sky Survey (NVSS) [62], TIFR GMRT Sky Survey (TGSS) [62]. Note that the
bounds on current energy density fraction due to these two sources viz.,
cosmic strings and magnetic fields are consistent with zero.

An important result is the broad clustering of anisotropy axis found in case
of domain walls and LVMF, when considered as sources of anisotropy, with other
cosmic preferred axes seen in CMB data. The axes of anisotropy when studying
domain walls and LVMF were found to be
$(l_{a},b_{a})=(241^{\circ},-52^{\circ}$) and $=(240^{\circ},-49^{\circ}$),
respectively, in galactic coordinates. Note that in a previous analysis [33],
the axis of anisotropy for cosmic strings and magnetic fields were also
aligned broadly with those of domain walls and LVMF mentioned in the
preceeding paragraph. These preferred axes with domain walls and LVMF by and
large span the region containing hemispherical power asymmetry axis, CMB cold
spot, and mirror parity asymmetry axis, which are all some of the significant
anomalous anisotropy axes or features seen in CMB data [63], among others. We
further note that the constraints on energy density fractions of domain walls
and LVMF prefer a non-zero value at $1\sigma$ level.

Figure 5: Mollweide view projection of the cosmic preferred axis for all four
anisotropic sources considered in our work are shown with $1\sigma$ confidence
contour. Also shown are similar axis of anisotropy found in various
astronomical/cosmological data listed in Table 3.

All the four anisotropy axis found in each case (anisotropic matter sources
viz., cosmic strings, magnetic fields, domain walls, and LVMF) are shown in
Fig. [5] with $1\sigma$ contour, along with other known axes of anisotropy
previously reported in the literature as listed in Table 3. We can see that
there are two sets of cosmic axes broadly spanning the upper right and lower
right quadrants in Fig. 5. Those that are pointing in the CMB dipole direction
(i.e., broadly in its vicinity) may possibly be arising due to an unknown
systematic or data processing artifact, if not a cosmic preferred axis. It
would be reasonable to say so given that we were only able to obtain upper
bounds on the energy density fractions in case of cosmic strings and magnetic
fields as anisotropic matter sources. However the anisotropy axes seen when
assessing domain walls and LVMF as sources of anisotropy could be an
indication for a real cosmic preferred axis for our Universe, in whose case we
obtain a definite preference for non-zero energy density fractions. This
reasoning is further reinforced by the fact that other anisotropic axis in
their proximity are all found to be significant large scale CMB anomalies
among others.

Axis | $(l_{a},b_{a})$ | Ref.
---|---|---
CMB dipole | (263.99,48.26) | [3]
CMB quadrupole | (224.2,69.2) | [61]
CMB octupole | (239,64.3) | [61]
CMB Even mirror parity axis | (260.4,48.1) | [63]
CMB Odd mirror parity axis | (264.4,-17) | [63]
CMB Hemispherical Power Asymmetry axis | (218,-21) | [63]
CMB Dipole Modulation axis | (228,-18) | [63]
CMB Local Variance estimator | (212,-13) | [64]
Radio dipole NVSS | (253.12,27.28) | [62]
Radio dipole TGSS | (243,45) | [62]
Fine structure constant ($\alpha$) dipole | (320.5,-11.7) | [65]
SNIa dipole | (316,-5) | [66]
Table 3: Other cosmic preferred axes (in galactic coordinates), depicted in
Fig. [5], as found in various astronomical/cosmological data are listed here.

Finally, we see that the nuisance parameters $\alpha=0.1414,\beta=3.028$
corresponding to the dependence of SN1e light curves’ stretch and color
parameters with luminosity is in close agreement with those originally
reported by Pantheon data [36].

###  4.2 Goodness of fit

While performing any kind of ‘goodness of fit’ test, two different models used
to fit the same data having different number of fit parameters may seem
difficult to compare, both qualitatively and quantitatively. One approach
would be to simply check the difference in the minima of the $\chi^{2}$ for
the two models wherein one of the models is taken as base model and also
calculating the $\chi^{2}$ per degree of freedom. For the various models
considered in the present work, $\Delta\chi^{2}=\chi^{2}_{\rm
ref}-\chi^{2}_{\rm model}$ with respect to the reference flat $\Lambda$CDM
model are listed in the second row from the bottom of Table 2.

Introducing more parameters to describe observed data almost always leads to a
better fit i.e., it returns a lower $\chi^{2}$ minimum value. So a better
procedure in identifying the right model that describes the observed data
should factor in the additional parameters introduced into model fitting with
respect to a _base_ model. One such model selection criterion that is
ubiquitously used comes from information theory known as Akaike Information
Criterion (AIC) [67]. Along with the minimum of the
$\chi^{2}=-\ln(\mathcal{L})$, where $\mathcal{L}$ is the likelihood function
of the model being studied, the formula for AIC includes an additional penalty
term that depends on the number of parameters ‘$k$’ used to fit the data. It
is given by,

${\rm AIC}=2k-2\ln(\mathcal{L})=2k+\chi^{2}\,.$ (27)

Here $\chi^{2}$ is its minimum value ($\chi^{2}_{\rm min}$) obtained in the
space of model parameters after optimization of Eq. (19).

Now, we can calculate the relative AIC for the candidate models being studied
using one of the models (the one with the least AIC value among the candidate
models) to get an estimate of the probability of the other models in
describing the observed data relative to this _reference_ (base) model. It is
given by,

${\rm AIC}_{\rm rel}=\exp\left(\frac{{\rm AIC}_{\Lambda CDM}-{\rm
AIC}_{Bianchi-I}}{2}\right)\,,$ (28)

where we have chosen the flat $\Lambda$CDM model as our base model to compare
anisotropic Bianchi-I model with different types of anisotropic sources. These
${\rm AIC}_{\rm rel}$ values relative to the standard concordance model are
also listed at the end of Table 2. We find that the Bianchi-I universe with
Lorentz violating magnetic fields and domain walls as anisotropic matter
sources is preferred over that with cosmic strings and magnetic fields.
However, the relative AIC values indicate that a Bianchi-I model with LVMF and
domain walls themselves are relatively less probable that the standard model.
However their relative probability is not small enough to negate these
sources. Further, given that the ${\rm AIC}_{\rm rel}$ found for domain walls
and LVMF being same, we don’t get a clear indication as to which anisotropic
source is a better fit to the data between these two.

##  5 Conclusions

Our goal here is to estimate the level of anisotropy in the ‘observed’
universe (cosmic shear and eccentricity of the universe), constrain the
density fractions of (an)isotropic matter sources and dark energy modeled as
cosmological constant ($\Lambda$), and identify any cosmic preferred axis
assuming a Bianchi-I universe. We considered various anisotropic matter
sources viz., cosmic strings, magnetic fields, domain walls and Lorentz
violating magnetic fields (LVMF) in the source term of Einstein’s equation,
taken one at a time, along with the standard model (isotropic) sources viz.,
(dark+visible) matter and dark energy ($\Lambda$). Bianchi-I model that we
studied here is an anisotropic cosmological model with a planar symmetry in
spatial part of it’s metric (as defined in Eq. 2). The evolution equations
corresponding to various cosmological parameters of this model in the form of
coupled differential equations (Eq. 2.3) were (re)derived. To obtain
constraints on the model parameters, we made use of the latest compilation of
Type Ia supernovae (SN1e) data from the _Pantheon_ collaboration [36]. With
this SN1e data, we performed an MCMC likelihood analysis of the Bianchi-I
model solving these evolution equations in conjugation with the corresponding
luminosity distance versus redshift relation (Eq. 13) by defining a $\chi^{2}$
function (Eq. 19).

The current work is an extension of the analysis presented in Ref. [33] where
constraints on a Bianchi-I universe with various kinds of anisotropic matter
sources were obtained, but using Union2 SN1e data [59] to probe for the
presence of any isotropy violation in the universe. With nearly double the
data size in Pantheon compilation compared to Union2 sample, we get improved
constraints on the parameters of the anisotropic Bianchi-I model.

We found a $2\sigma$ level evidence for a negative cosmic shear, non-zero
anisotropic matter sources’ energy density fractions at $1\sigma$ confidence
level at current times, and a cosmic preferred axis for our universe. These
finding are in line with those reported in Ref. [33]. However, with improved
data set in quality and quantity, we found such non-zero values for the
corresponding cosmological parameters when considering domain walls and LVMF,
only, as sources of anisotropy. Constraints on cosmic shear and anisotropic
matter density fractions are consistent with zero when considering cosmic
strings and magnetic fields. The cosmic anisotropy axis found here is also
consistent with that reported in Ref. [33], but only in case of domain walls
and LVMF as anisotropic sources.

Interestingly, in case of cosmic strings and magnetic fields, the axis of
anisotropy, by and large, points in the CMB dipole ($\ell=1$) direction, among
others. We suspect that this may be due to some systematic in the data, which
is inducing to pick up the a direction closer to CMB dipole direction as axis
of anisotropy via the anisotropic equation of state parametrization, though
the constraints on the corresponding anisotropic sources themselves viz.,
cosmic strings and magnetic fields are consistent with zero. In case of domain
walls and LVMF, the axis of anisotropy broadly span the same region in which
anomalous preferred directions/features seen in CMB data and others fall.
Therefore, we find that the two sets of axis of anisotropy reported here are
aligned with other axis of isotropy violation seen in various
astronomical/cosmological data such as CMB, radio observations, etc.

Although the degeneracy between $h_{0}$ and $M_{0}$ is absent, in principle,
in a Bianchi-I model (Eq. 20, 22 and 23), we found that the data couldn’t yet
break that degeneracy. Hence we used the cepheid based absolute magnitude
measurement [51]. The (mean) Hubble parameter value ($h_{0}$) that we obtained
in our Bianchi-I model analysis with various anisotropic sources doesn’t
elleviate the ‘Hubble tension’ [68, 69, 70]. Thus, we may say that the ‘Hubble
tension’ still remains within this model.

Despite finding non-zero values (at 1-2$\sigma$ level) for the cosmological
parameters with some of the anisotropic matter sources suggesting their
presence at current times, by employing a model selection criteria such as
Akaike information criterion (AIC), the standard cosmological model - a flat
$\Lambda$CDM model - is relatively still the preferred model with a lower AIC
value. But the likeliness of a Bianchi-I model to “expalin” the observed data
(SN1e) with domain walls and LVMF as anisotropic sources cannot be dismissed
as their AIC values are not negligibly low. All these results suggest that a
mild anisotropy is present in our universe now (as revealed through SN1e data
used here), or if there is any isotropy violation in the universe at present,
the current data set is not sufficient to constraint it conclusively but is
only suggestive of it.

As the present analysis was underway, a larger sample of Type Ia supernovae
data, the _Pantheon+SH0ES_ compilation444https://pantheonplussh0es.github.io/
[71], was released that contains $\sim 1700$ SNIe. We expect to get better
constraints on our model by putting it under the lens again with much denser
and more homogeneous sky distribution of SNIe as in the Pantheon+SH0ES sample.
We plan to follow up this work with that larger data set in the near future.

## Acknowledgements

AV acknowledges the financial support received through research fellowship
awarded by Council of Scientific & Industrial Research (CSIR), India (file no.
09/1217(0079)/2020-EMR-I) during this project. DFM thanks the Research Council
of Norway for their support and the resources provided by UNINETT Sigma2 - the
National Infrastructure for High Performance Computing and Data Storage in
Norway. We would to thank David Jones for a helpful communication about using
the Pantheon data set. PKA would like to thank Pankaj Jain for a useful
communication. We acknowledge using Cobaya MCMC
Sampler555https://cobaya.readthedocs.io/en/stable/samplermcmc.html [72], a
code for Bayesian analysis in Cosmology, and
GetDist666https://getdist.readthedocs.io/ [73], a user friendly GUI interface
for the analysis and plotting MCMC samples. We also acknowledge the use of
SciPy777https://scipy.org/ [74], NumPy888https://numpy.org/ [75],
Astropy999http://www.astropy.org [76, 77, 78], and
Matplotlib101010https://matplotlib.org/ [79].

## References

  * [1] C. L. Bennett, A. J. Banday, K. M. Gorski, G. Hinshaw, P. Jackson, P. Keegstra, A. Kogut, G. F. Smoot, D. T. Wilkinson, and E. L. Wright.  Four-Year COBE DMR Cosmic Microwave Background Observations: Maps and Basic Results.  ApJ, 464:L1, June 1996.
  * [2] C. L. Bennett, D. Larson, J. L. Weiland, N. Jarosik, G. Hinshaw, N. Odegard, K. M. Smith, R. S. Hill, B. Gold, M. Halpern, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, J. Dunkley, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, and E. L. Wright.  Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results.  ApJS, 208(2):20, October 2013.
  * [3] Planck Collaboration, N. Aghanim, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, R. Battye, K. Benabed, J. P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J. F. Cardoso, J. Carron, B. Casaponsa, A. Challinor, H. C. Chiang, L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, F. Cuttaia, P. de Bernardis, G. de Zotti, J. Delabrouille, J. M. Delouis, F. X. Désert, E. Di Valentino, C. Dickinson, J. M. Diego, S. Donzelli, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, E. Falgarone, Y. Fantaye, J. Fergusson, R. Fernandez-Cobos, F. Finelli, F. Forastieri, M. Frailis, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Génova-Santos, M. Gerbino, T. Ghosh, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley, F. K. Hansen, G. Helou, D. Herranz, S. R. Hildebrandt, E. Hivon, Z. Huang, A. H. Jaffe, W. C. Jones, A. Karakci, E. Keihänen, R. Keskitalo, K. Kiiveri, J. Kim, T. S. Kisner, L. Knox, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J. M. Lamarre, M. Langer, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Jeune, J. P. Leahy, J. Lesgourgues, F. Levrier, A. Lewis, M. Liguori, P. B. Lilje, M. Lilley, V. Lindholm, M. López-Caniego, P. M. Lubin, Y. Z. Ma, J. F. Macías-Pérez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero, M. Maris, P. G. Martin, M. Martinelli, E. Martínez-González, S. Matarrese, N. Mauri, J. D. McEwen, P. D. Meerburg, P. R. Meinhold, A. Melchiorri, A. Mennella, M. Migliaccio, M. Millea, S. Mitra, M. A. Miville-Deschênes, D. Molinari, A. Moneti, L. Montier, G. Morgante, A. Moss, S. Mottet, M. Münchmeyer, P. Natoli, H. U. Nørgaard-Nielsen, C. A. Oxborrow, L. Pagano, D. Paoletti, B. Partridge, G. Patanchon, T. J. Pearson, M. Peel, H. V. Peiris, F. Perrotta, V. Pettorino, F. Piacentini, L. Polastri, G. Polenta, J. L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubiño-Martín, B. Ruiz-Granados, L. Salvati, M. Sandri, M. Savelainen, D. Scott, E. P. S. Shellard, M. Shiraishi, C. Sirignano, G. Sirri, L. D. Spencer, R. Sunyaev, A. S. Suur-Uski, J. A. Tauber, D. Tavagnacco, M. Tenti, L. Terenzi, L. Toffolatti, M. Tomasi, T. Trombetti, J. Valiviita, B. Van Tent, L. Vibert, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K. Wehus, M. White, S. D. M. White, A. Zacchei, and A. Zonca.  Planck 2018 results. I. Overview and the cosmological legacy of Planck.  A&A, 641:A1, September 2020.
  * [4] C. L. Bennett, R. S. Hill, G. Hinshaw, D. Larson, K. M. Smith, J. Dunkley, B. Gold, M. Halpern, N. Jarosik, A. Kogut, E. Komatsu, M. Limon, S. S. Meyer, M. R. Nolta, N. Odegard, L. Page, D. N. Spergel, G. S. Tucker, J. L. Weiland, E. Wollack, and E. L. Wright.  Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Are There Cosmic Microwave Background Anomalies?  ApJS, 192(2):17, February 2011.
  * [5] Planck Collaboration, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, K. Benabed, M. Bersanelli, P. Bielewicz, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J. F. Cardoso, B. Casaponsa, H. C. Chiang, L. P. L. Colombo, C. Combet, D. Contreras, B. P. Crill, P. de Bernardis, G. de Zotti, J. Delabrouille, J. M. Delouis, E. Di Valentino, J. M. Diego, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, Y. Fantaye, R. Fernandez-Cobos, F. Finelli, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frolov, S. Galeotta, S. Galli, K. Ganga, R. T. Génova-Santos, M. Gerbino, T. Ghosh, J. González-Nuevo, K. M. Górski, A. Gruppuso, J. E. Gudmundsson, J. Hamann, W. Handley, F. K. Hansen, D. Herranz, E. Hivon, Z. Huang, A. H. Jaffe, W. C. Jones, E. Keihänen, R. Keskitalo, K. Kiiveri, J. Kim, N. Krachmalnicoff, M. Kunz, H. Kurki-Suonio, G. Lagache, J. M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, M. Le Jeune, F. Levrier, M. Liguori, P. B. Lilje, V. Lindholm, M. López-Caniego, Y. Z. Ma, J. F. Macías-Pérez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, A. Marcos-Caballero, M. Maris, P. G. Martin, E. Martínez-González, S. Matarrese, N. Mauri, J. D. McEwen, P. R. Meinhold, A. Mennella, M. Migliaccio, M. A. Miville-Deschênes, D. Molinari, A. Moneti, L. Montier, G. Morgante, A. Moss, P. Natoli, L. Pagano, D. Paoletti, B. Partridge, F. Perrotta, V. Pettorino, F. Piacentini, G. Polenta, J. L. Puget, J. P. Rachen, M. Reinecke, M. Remazeilles, A. Renzi, G. Rocha, C. Rosset, G. Roudier, J. A. Rubiño-Martín, B. Ruiz-Granados, L. Salvati, M. Savelainen, D. Scott, E. P. S. Shellard, C. Sirignano, R. Sunyaev, A. S. Suur-Uski, J. A. Tauber, D. Tavagnacco, M. Tenti, L. Toffolatti, M. Tomasi, T. Trombetti, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, N. Vittorio, B. D. Wandelt, I. K. Wehus, A. Zacchei, J. P. Zibin, and A. Zonca.  Planck 2018 results. VII. Isotropy and statistics of the CMB.  A&A, 641:A7, September 2020.
  * [6] Dominik J. Schwarz, Craig J. Copi, Dragan Huterer, and Glenn D. Starkman.  CMB anomalies after Planck.  Classical and Quantum Gravity, 33(18):184001, September 2016.
  * [7] D. J. Schwarz and B. Weinhorst.  (An)isotropy of the Hubble diagram: comparing hemispheres.  A&A, 474(3):717–729, November 2007.
  * [8] I. Antoniou and L. Perivolaropoulos.  Searching for a cosmological preferred axis: Union2 data analysis and comparison with other probes.  J. Cosmology Astropart. Phys, 2010(12):012, December 2010.
  * [9] Jacques Colin, Roya Mohayaee, Subir Sarkar, and Arman Shafieloo.  Probing the anisotropic local Universe and beyond with SNe Ia data.  MNRAS, 414(1):264–271, June 2011.
  * [10] David L. Wiltshire, Peter R. Smale, Teppo Mattsson, and Richard Watkins.  Hubble flow variance and the cosmic rest frame.  Phys. Rev. D, 88(8):083529, October 2013.
  * [11] Stephen Appleby, Arman Shafieloo, and Andrew Johnson.  Probing Bulk Flow with Nearby SNe Ia Data.  ApJ, 801(2):76, March 2015.
  * [12] Zhe Chang, Dong Zhao, and Yong Zhou.  Constraining the anisotropy of the Universe with the Pantheon supernovae sample.  Chinese Physics C, 43(12):125102, December 2019.
  * [13] D. Hutsemekers.  Evidence for very large-scale coherent orientations of quasar polarization vectors.  A&A, 332:410–428, April 1998.
  * [14] D. Hutsemékers and H. Lamy.  Confirmation of the existence of coherent orientations of quasar polarization vectors on cosmological scales.  A&A, 367:381–387, February 2001.
  * [15] Pankaj Jain, Gaurav Narain, and S. Sarala.  Large-scale alignment of optical polarizations from distant QSOs using coordinate-invariant statistics.  MNRAS, 347(2):394–402, January 2004.
  * [16] D. Hutsemékers, L. Braibant, V. Pelgrims, and D. Sluse.  Alignment of quasar polarizations with large-scale structures.  A&A, 572:A18, December 2014.
  * [17] P. Birch.  Is the Universe rotating?  Nature, 298(5873):451–454, July 1982.
  * [18] Pankaj Jain and John P. Ralston.  Anisotropy in the Propagation of Radio Polarizations from Cosmologically Distant Galaxies.  Modern Physics Letters A, 14(6):417–432, January 1999.
  * [19] Prabhakar Tiwari and Pankaj Jain.  Polarization Alignment in Jvas/class Flat Spectrum Radio Surveys.  International Journal of Modern Physics D, 22(12):1350089, October 2013.
  * [20] Richard Watkins, Hume A. Feldman, and Michael J. Hudson.  Consistently large cosmic flows on scales of 100h-1Mpc: a challenge for the standard $\Lambda$CDM cosmology.  MNRAS, 392(2):743–756, January 2009.
  * [21] Philip Bull, Yashar Akrami, Julian Adamek, Tessa Baker, Emilio Bellini, Jose Beltrán Jiménez, Eloisa Bentivegna, Stefano Camera, Sébastien Clesse, Jonathan H. Davis, Enea Di Dio, Jonas Enander, Alan Heavens, Lavinia Heisenberg, Bin Hu, Claudio Llinares, Roy Maartens, Edvard Mörtsell, Seshadri Nadathur, Johannes Noller, Roman Pasechnik, Marcel S. Pawlowski, Thiago S. Pereira, Miguel Quartin, Angelo Ricciardone, Signe Riemer-Sørensen, Massimiliano Rinaldi, Jeremy Sakstein, Ippocratis D. Saltas, Vincenzo Salzano, Ignacy Sawicki, Adam R. Solomon, Douglas Spolyar, Glenn D. Starkman, Danièle Steer, Ismael Tereno, Licia Verde, Francisco Villaescusa-Navarro, Mikael von Strauss, and Hans A. Winther.  Beyond $\Lambda$ CDM: Problems, solutions, and the road ahead.  Physics of the Dark Universe, 12:56–99, June 2016.
  * [22] Elcio Abdalla, Guillermo Franco Abellán, Amin Aboubrahim, Adriano Agnello, Özgür Akarsu, Yashar Akrami, George Alestas, Daniel Aloni, Luca Amendola, Luis A. Anchordoqui, Richard I. Anderson, Nikki Arendse, Marika Asgari, Mario Ballardini, Vernon Barger, Spyros Basilakos, Ronaldo C. Batista, Elia S. Battistelli, Richard Battye, Micol Benetti, David Benisty, Asher Berlin, Paolo de Bernardis, Emanuele Berti, Bohdan Bidenko, Simon Birrer, John P. Blakeslee, Kimberly K. Boddy, Clecio R. Bom, Alexander Bonilla, Nicola Borghi, François R. Bouchet, Matteo Braglia, Thomas Buchert, Elizabeth Buckley-Geer, Erminia Calabrese, Robert R. Caldwell, David Camarena, Salvatore Capozziello, Stefano Casertano, Geoff C. F. Chen, Jens Chluba, Angela Chen, Hsin-Yu Chen, Anton Chudaykin, Michele Cicoli, Craig J. Copi, Fred Courbin, Francis-Yan Cyr-Racine, Bożena Czerny, Maria Dainotti, Guido D’Amico, Anne-Christine Davis, Javier de Cruz Pérez, Jaume de Haro, Jacques Delabrouille, Peter B. Denton, Suhail Dhawan, Keith R. Dienes, Eleonora Di Valentino, Pu Du, Dominique Eckert, Celia Escamilla-Rivera, Agnès Ferté, Fabio Finelli, Pablo Fosalba, Wendy L. Freedman, Noemi Frusciante, Enrique Gaztañaga, William Giarè, Elena Giusarma, Adrià Gómez-Valent, Will Handley, Ian Harrison, Luke Hart, Dhiraj Kumar Hazra, Alan Heavens, Asta Heinesen, Hendrik Hildebrandt, J. Colin Hill, Natalie B. Hogg, Daniel E. Holz, Deanna C. Hooper, Nikoo Hosseininejad, Dragan Huterer, Mustapha Ishak, Mikhail M. Ivanov, Andrew H. Jaffe, In Sung Jang, Karsten Jedamzik, Raul Jimenez, Melissa Joseph, Shahab Joudaki, Marc Kamionkowski, Tanvi Karwal, Lavrentios Kazantzidis, Ryan E. Keeley, Michael Klasen, Eiichiro Komatsu, Léon V. E. Koopmans, Suresh Kumar, Luca Lamagna, Ruth Lazkoz, Chung-Chi Lee, Julien Lesgourgues, Jackson Levi Said, Tiffany R. Lewis, Benjamin L’Huillier, Matteo Lucca, Roy Maartens, Lucas M. Macri, Danny Marfatia, Valerio Marra, Carlos J. A. P. Martins, Silvia Masi, Sabino Matarrese, Arindam Mazumdar, Alessandro Melchiorri, Olga Mena, Laura Mersini-Houghton, James Mertens, Dinko Milaković, Yuto Minami, Vivian Miranda, Cristian Moreno-Pulido, Michele Moresco, David F. Mota, Emil Mottola, Simone Mozzon, Jessica Muir, Ankan Mukherjee, Suvodip Mukherjee, Pavel Naselsky, Pran Nath, Savvas Nesseris, Florian Niedermann, Alessio Notari, Rafael C. Nunes, Eoin Ó Colgáin, Kayla A. Owens, Emre Özülker, Francesco Pace, Andronikos Paliathanasis, Antonella Palmese, Supriya Pan, Daniela Paoletti, Santiago E. Perez Bergliaffa, Leandros Perivolaropoulos, Dominic W. Pesce, Valeria Pettorino, Oliver H. E. Philcox, Levon Pogosian, Vivian Poulin, Gaspard Poulot, Marco Raveri, Mark J. Reid, Fabrizio Renzi, Adam G. Riess, Vivian I. Sabla, Paolo Salucci, Vincenzo Salzano, Emmanuel N. Saridakis, Bangalore S. Sathyaprakash, Martin Schmaltz, Nils Schöneberg, Dan Scolnic, Anjan A. Sen, Neelima Sehgal, Arman Shafieloo, M. M. Sheikh-Jabbari, Joseph Silk, Alessandra Silvestri, Foteini Skara, Martin S. Sloth, Marcelle Soares-Santos, Joan Solà Peracaula, Yu-Yang Songsheng, Jorge F. Soriano, Denitsa Staicova, Glenn D. Starkman, István Szapudi, Elsa M. Teixeira, Brooks Thomas, Tommaso Treu, Emery Trott, Carsten van de Bruck, J. Alberto Vazquez, Licia Verde, Luca Visinelli, Deng Wang, Jian-Min Wang, Shao-Jiang Wang, Richard Watkins, Scott Watson, John K. Webb, Neal Weiner, Amanda Weltman, Samuel J. Witte, Radosław Wojtak, Anil Kumar Yadav, Weiqiang Yang, Gong-Bo Zhao, and Miguel Zumalacárregui.  Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies.  Journal of High Energy Astrophysics, 34:49–211, June 2022.
  * [23] L. Perivolaropoulos and F. Skara.  Challenges for $\Lambda$CDM: An update.  New A Rev., 95:101659, December 2022.
  * [24] Pavan Kumar Aluri, Paolo Cea, Pravabati Chingangbam, Ming-Chung Chu, Roger G. Clowes, Damien Hutsemékers, Joby P. Kochappan, Alexia M. Lopez, Lang Liu, Niels C. M. Martens, C. J. A. P. Martins, Konstantinos Migkas, Eoin Ó Colgáin, Pratyush Pranav, Lior Shamir, Ashok K. Singal, M. M. Sheikh-Jabbari, Jenny Wagner, Shao-Jiang Wang, David L. Wiltshire, Shek Yeung, Lu Yin, and Wen Zhao.  Is the observable Universe consistent with the cosmological principle?  Classical and Quantum Gravity, 40(9):094001, May 2023.
  * [25] John P. Ralston and Pankaj Jain.  The Virgo Alignment Puzzle in Propagation of Radiation on Cosmological Scales.  International Journal of Modern Physics D, 13(9):1857–1877, January 2004.
  * [26] M. A. H. MacCallum and G. F. R. Ellis.  A class of homogeneous cosmological models: II. Observations.  Communications in Mathematical Physics, 19(1):31–64, March 1970\.
  * [27] A. Emir Gümrükçüoglu, Carlo R. Contaldi, and Marco Peloso.  Inflationary perturbations in anisotropic backgrounds and their imprint on the cosmic microwave background.  J. Cosmology Astropart. Phys, 2007(11):005, November 2007.
  * [28] John D. Barrow.  Cosmological limits on slightly skew stresses.  Phys. Rev. D, 55(12):7451–7460, June 1997.
  * [29] Arjun Berera, Roman V. Buniy, and Thomas W. Kephart.  The eccentric universe.  J. Cosmology Astropart. Phys, 2004(10):016, October 2004.
  * [30] L. Campanelli, P. Cea, and L. Tedesco.  Ellipsoidal Universe Can Solve the Cosmic Microwave Background Quadrupole Problem.  Phys. Rev. Lett., 97(13):131302, September 2006.
  * [31] Leonardo Campanelli.  Model of universe anisotropization.  Phys. Rev. D, 80(6):063006, September 2009.
  * [32] Tomi Koivisto and David F. Mota.  Anisotropic dark energy: dynamics of the background and perturbations.  J. Cosmology Astropart. Phys, 2008(6):018, June 2008.
  * [33] Pavan K. Aluri, Sukanta Panda, Manabendra Sharma, and Snigdha Thakur.  Anisotropic universe with anisotropic sources.  J. Cosmology Astropart. Phys, 2013(12):003, December 2013.
  * [34] A. Vilenkin.  Cosmic Strings and Domain Walls.  In Edward W. Kolb and Michael S. Turner, editors, The Early Universe: Reprints, page 431. 1988.
  * [35] Edward W Kolb and Michael S Turner.  The early universe.  Frontiers in physics. Westview Press, Boulder, CO, 1990.
  * [36] D. M. Scolnic, D. O. Jones, A. Rest, Y. C. Pan, R. Chornock, R. J. Foley, M. E. Huber, R. Kessler, G. Narayan, A. G. Riess, S. Rodney, E. Berger, D. J. Brout, P. J. Challis, M. Drout, D. Finkbeiner, R. Lunnan, R. P. Kirshner, N. E. Sanders, E. Schlafly, S. Smartt, C. W. Stubbs, J. Tonry, W. M. Wood-Vasey, M. Foley, J. Hand, E. Johnson, W. S. Burgett, K. C. Chambers, P. W. Draper, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, F. Bresolin, E. Gall, R. Kotak, M. McCrum, and K. W. Smith.  The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample.  ApJ, 859(2):101, June 2018.
  * [37] A. Rest, D. Scolnic, R. J. Foley, M. E. Huber, R. Chornock, G. Narayan, J. L. Tonry, E. Berger, A. M. Soderberg, C. W. Stubbs, A. Riess, R. P. Kirshner, S. J. Smartt, E. Schlafly, S. Rodney, M. T. Botticella, D. Brout, P. Challis, I. Czekala, M. Drout, M. J. Hudson, R. Kotak, C. Leibler, R. Lunnan, G. H. Marion, M. McCrum, D. Milisavljevic, A. Pastorello, N. E. Sanders, K. Smith, E. Stafford, D. Thilker, S. Valenti, W. M. Wood-Vasey, Z. Zheng, W. S. Burgett, K. C. Chambers, L. Denneau, P. W. Draper, H. Flewelling, K. W. Hodapp, N. Kaiser, R. P. Kudritzki, E. A. Magnier, N. Metcalfe, P. A. Price, W. Sweeney, R. Wainscoat, and C. Waters.  Cosmological Constraints from Measurements of Type Ia Supernovae Discovered during the First 1.5 yr of the Pan-STARRS1 Survey.  ApJ, 795(1):44, November 2014.
  * [38] Carlos Contreras, Mario Hamuy, M. M. Phillips, Gastón Folatelli, Nicholas B. Suntzeff, S. E. Persson, Maximilian Stritzinger, Luis Boldt, Sergio González, Wojtek Krzeminski, Nidia Morrell, Miguel Roth, Francisco Salgado, María José Maureira, Christopher R. Burns, W. L. Freedman, Barry F. Madore, David Murphy, Pamela Wyatt, Weidong Li, and Alexei V. Filippenko.  The Carnegie Supernova Project: First Photometry Data Release of Low-Redshift Type Ia Supernovae.  AJ, 139(2):519–539, February 2010.
  * [39] Adam G. Riess, Robert P. Kirshner, Brian P. Schmidt, Saurabh Jha, Peter Challis, Peter M. Garnavich, Ann A. Esin, Chris Carpenter, Randy Grashius, Rudolph E. Schild, Perry L. Berlind, John P. Huchra, Charles F. Prosser, Emilio E. Falco, Priscilla J. Benson, César Briceño, Warren R. Brown, Nelson Caldwell, Ian P. dell’Antonio, Alexei V. Filippenko, Alyssa A. Goodman, Norman A. Grogin, Ted Groner, John P. Hughes, Paul J. Green, Rolf A. Jansen, Jan T. Kleyna, Jane X. Luu, Lucas M. Macri, Brian A. McLeod, Kim K. McLeod, Brian R. McNamara, Brian McLean, Alejandra A. E. Milone, Joseph J. Mohr, Dan Moraru, Chien Peng, Jim Peters, Andrea H. Prestwich, Krzysztof Z. Stanek, Andy Szentgyorgyi, and Ping Zhao.  BVRI Light Curves for 22 Type IA Supernovae.  AJ, 117(2):707–724, February 1999.
  * [40] Saurabh Jha, Robert P. Kirshner, Peter Challis, Peter M. Garnavich, Thomas Matheson, Alicia M. Soderberg, Genevieve J. M. Graves, Malcolm Hicken, João F. Alves, Héctor G. Arce, Zoltan Balog, Pauline Barmby, Elizabeth J. Barton, Perry Berlind, Ann E. Bragg, César Briceño, Warren R. Brown, James H. Buckley, Nelson Caldwell, Michael L. Calkins, Barbara J. Carter, Kristi Dendy Concannon, R. Hank Donnelly, Kristoffer A. Eriksen, Daniel G. Fabricant, Emilio E. Falco, Fabrizio Fiore, Michael R. Garcia, Mercedes Gómez, Norman A. Grogin, Ted Groner, Paul J. Groot, Jr. Haisch, Karl E., Lee Hartmann, Carl W. Hergenrother, Matthew J. Holman, John P. Huchra, Ray Jayawardhana, Diab Jerius, Sheila J. Kannappan, Dong-Woo Kim, Jan T. Kleyna, Christopher S. Kochanek, Daniel M. Koranyi, Martin Krockenberger, Charles J. Lada, Kevin L. Luhman, Jane X. Luu, Lucas M. Macri, Jeff A. Mader, Andisheh Mahdavi, Massimo Marengo, Brian G. Marsden, Brian A. McLeod, Brian R. McNamara, S. Thomas Megeath, Dan Moraru, Amy E. Mossman, August A. Muench, Jose A. Muñoz, James Muzerolle, Orlando Naranjo, Kristin Nelson-Patel, Michael A. Pahre, Brian M. Patten, James Peters, Wayne Peters, John C. Raymond, Kenneth Rines, Rudolph E. Schild, Gregory J. Sobczak, Timothy B. Spahr, John R. Stauffer, Robert P. Stefanik, Andrew H. Szentgyorgyi, Eric V. Tollestrup, Petri Väisänen, Alexey Vikhlinin, Zhong Wang, S. P. Willner, Scott J. Wolk, Joseph M. Zajac, Ping Zhao, and Krzysztof Z. Stanek.  UBVRI Light Curves of 44 Type Ia Supernovae.  AJ, 131(1):527–554, January 2006.
  * [41] Malcolm Hicken, Peter Challis, Robert P. Kirshner, Armin Rest, Claire E. Cramer, W. Michael Wood-Vasey, Gaspar Bakos, Perry Berlind, Warren R. Brown, Nelson Caldwell, Mike Calkins, Thayne Currie, Kathy de Kleer, Gil Esquerdo, Mark Everett, Emilio Falco, Jose Fernandez, Andrew S. Friedman, Ted Groner, Joel Hartman, Matthew J. Holman, Robert Hutchins, Sonia Keys, David Kipping, Dave Latham, George H. Marion, Gautham Narayan, Michael Pahre, Andras Pal, Wayne Peters, Gopakumar Perumpilly, Ben Ripman, Brigitta Sipocz, Andrew Szentgyorgyi, Sumin Tang, Manuel A. P. Torres, Amali Vaz, Scott Wolk, and Andreas Zezas.  CfA4: Light Curves for 94 Type Ia Supernovae.  ApJS, 200(2):12, June 2012.
  * [42] Joshua A. Frieman, Bruce Bassett, Andrew Becker, Changsu Choi, David Cinabro, Fritz DeJongh, Darren L. Depoy, Ben Dilday, Mamoru Doi, Peter M. Garnavich, Craig J. Hogan, Jon Holtzman, Myungshin Im, Saurabh Jha, Richard Kessler, Kohki Konishi, Hubert Lampeitl, John Marriner, Jennifer L. Marshall, David McGinnis, Gajus Miknaitis, Robert C. Nichol, Jose Luis Prieto, Adam G. Riess, Michael W. Richmond, Roger Romani, Masao Sako, Donald P. Schneider, Mathew Smith, Naohiro Takanashi, Kouichi Tokita, Kurt van der Heyden, Naoki Yasuda, Chen Zheng, Jennifer Adelman-McCarthy, James Annis, Roberto J. Assef, John Barentine, Ralf Bender, Roger D. Blandford, William N. Boroski, Malcolm Bremer, Howard Brewington, Chris A. Collins, Arlin Crotts, Jack Dembicky, Jason Eastman, Alastair Edge, Edmond Edmondson, Edward Elson, Michael E. Eyler, Alexei V. Filippenko, Ryan J. Foley, Stephan Frank, Ariel Goobar, Tina Gueth, James E. Gunn, Michael Harvanek, Ulrich Hopp, Yutaka Ihara, Želko Ivezić, Steven Kahn, Jared Kaplan, Stephen Kent, William Ketzeback, Scott J. Kleinman, Wolfram Kollatschny, Richard G. Kron, Jurek Krzesiński, Dennis Lamenti, Giorgos Leloudas, Huan Lin, Daniel C. Long, John Lucey, Robert H. Lupton, Elena Malanushenko, Viktor Malanushenko, Russet J. McMillan, Javier Mendez, Christopher W. Morgan, Tomoki Morokuma, Atsuko Nitta, Linda Ostman, Kaike Pan, Constance M. Rockosi, A. Kathy Romer, Pilar Ruiz-Lapuente, Gabrelle Saurage, Katie Schlesinger, Stephanie A. Snedden, Jesper Sollerman, Chris Stoughton, Maximilian Stritzinger, Mark Subba Rao, Douglas Tucker, Petri Vaisanen, Linda C. Watson, Shannon Watters, J. Craig Wheeler, Brian Yanny, and Donald York.  The Sloan Digital Sky Survey-II Supernova Survey: Technical Summary.  AJ, 135(1):338–347, January 2008.
  * [43] A. Conley, J. Guy, M. Sullivan, N. Regnault, P. Astier, C. Balland, S. Basa, R. G. Carlberg, D. Fouchez, D. Hardin, I. M. Hook, D. A. Howell, R. Pain, N. Palanque-Delabrouille, K. M. Perrett, C. J. Pritchet, J. Rich, V. Ruhlmann-Kleider, D. Balam, S. Baumont, R. S. Ellis, S. Fabbro, H. K. Fakhouri, N. Fourmanoit, S. González-Gaitán, M. L. Graham, M. J. Hudson, E. Hsiao, T. Kronborg, C. Lidman, A. M. Mourao, J. D. Neill, S. Perlmutter, P. Ripoche, N. Suzuki, and E. S. Walker.  Supernova Constraints and Systematic Uncertainties from the First Three Years of the Supernova Legacy Survey.  ApJS, 192(1):1, January 2011.
  * [44] N. Suzuki, D. Rubin, C. Lidman, G. Aldering, R. Amanullah, K. Barbary, L. F. Barrientos, J. Botyanszki, M. Brodwin, N. Connolly, K. S. Dawson, A. Dey, M. Doi, M. Donahue, S. Deustua, P. Eisenhardt, E. Ellingson, L. Faccioli, V. Fadeyev, H. K. Fakhouri, A. S. Fruchter, D. G. Gilbank, M. D. Gladders, G. Goldhaber, A. H. Gonzalez, A. Goobar, A. Gude, T. Hattori, H. Hoekstra, E. Hsiao, X. Huang, Y. Ihara, M. J. Jee, D. Johnston, N. Kashikawa, B. Koester, K. Konishi, M. Kowalski, E. V. Linder, L. Lubin, J. Melbourne, J. Meyers, T. Morokuma, F. Munshi, C. Mullis, T. Oda, N. Panagia, S. Perlmutter, M. Postman, T. Pritchard, J. Rhodes, P. Ripoche, P. Rosati, D. J. Schlegel, A. Spadafora, S. A. Stanford, V. Stanishev, D. Stern, M. Strovink, N. Takanashi, K. Tokita, M. Wagner, L. Wang, N. Yasuda, H. K. C. Yee, and The Supernova Cosmology Project.  The Hubble Space Telescope Cluster Supernova Survey. V. Improving the Dark-energy Constraints above $z>1$ and Building an Early-type-hosted Supernova Sample.  ApJ, 746(1):85, February 2012.
  * [45] Adam G. Riess, Louis-Gregory Strolger, Stefano Casertano, Henry C. Ferguson, Bahram Mobasher, Ben Gold, Peter J. Challis, Alexei V. Filippenko, Saurabh Jha, Weidong Li, John Tonry, Ryan Foley, Robert P. Kirshner, Mark Dickinson, Emily MacDonald, Daniel Eisenstein, Mario Livio, Josh Younger, Chun Xu, Tomas Dahlén, and Daniel Stern.  New Hubble Space Telescope Discoveries of Type Ia Supernovae at $z>=1$: Narrowing Constraints on the Early Behavior of Dark Energy.  ApJ, 659(1):98–121, April 2007.
  * [46] O. Graur, S. A. Rodney, D. Maoz, A. G. Riess, S. W. Jha, M. Postman, T. Dahlen, T. W. S. Holoien, C. McCully, B. Patel, L. G. Strolger, N. Benítez, D. Coe, S. Jouvel, E. Medezinski, A. Molino, M. Nonino, L. Bradley, A. Koekemoer, I. Balestra, S. B. Cenko, K. I. Clubb, M. E. Dickinson, A. V. Filippenko, T. F. Frederiksen, P. Garnavich, J. Hjorth, D. O. Jones, B. Leibundgut, T. Matheson, B. Mobasher, P. Rosati, J. M. Silverman, V. U, K. Jedruszczuk, C. Li, K. Lin, M. Mirmelstein, J. Neustadt, A. Ovadia, and E. H. Rogers.  Type-Ia Supernova Rates to Redshift 2.4 from CLASH: The Cluster Lensing And Supernova Survey with Hubble.  ApJ, 783(1):28, March 2014.
  * [47] Steven A. Rodney, Adam G. Riess, Louis-Gregory Strolger, Tomas Dahlen, Or Graur, Stefano Casertano, Mark E. Dickinson, Henry C. Ferguson, Peter Garnavich, Brian Hayden, Saurabh W. Jha, David O. Jones, Robert P. Kirshner, Anton M. Koekemoer, Curtis McCully, Bahram Mobasher, Brandon Patel, Benjamin J. Weiner, S. Bradley Cenko, Kelsey I. Clubb, Michael Cooper, Alexei V. Filippenko, Teddy F. Frederiksen, Jens Hjorth, Bruno Leibundgut, Thomas Matheson, Hooshang Nayyeri, Kyle Penner, Jonathan Trump, Jeffrey M. Silverman, Vivian U, K. Azalee Bostroem, Peter Challis, Abhijith Rajan, Schuyler Wolff, S. M. Faber, Norman A. Grogin, and Dale Kocevski.  Type Ia Supernova Rate Measurements to Redshift 2.5 from CANDELS: Searching for Prompt Explosions in the Early Universe.  AJ, 148(1):13, July 2014.
  * [48] Adam G. Riess, Steven A. Rodney, Daniel M. Scolnic, Daniel L. Shafer, Louis-Gregory Strolger, Henry C. Ferguson, Marc Postman, Or Graur, Dan Maoz, Saurabh W. Jha, Bahram Mobasher, Stefano Casertano, Brian Hayden, Alberto Molino, Jens Hjorth, Peter M. Garnavich, David O. Jones, Robert P. Kirshner, Anton M. Koekemoer, Norman A. Grogin, Gabriel Brammer, Shoubaneh Hemmati, Mark Dickinson, Peter M. Challis, Schuyler Wolff, Kelsey I. Clubb, Alexei V. Filippenko, Hooshang Nayyeri, Vivian U, David C. Koo, Sandra M. Faber, Dale Kocevski, Larry Bradley, and Dan Coe.  Type Ia Supernova Distances at Redshift $>1.5$ from the Hubble Space Telescope Multi-cycle Treasury Programs: The Early Expansion Rate.  ApJ, 853(2):126, February 2018.
  * [49] Keiichi Maeda and Yukikatsu Terada.  Progenitors of type Ia supernovae.  In Massimo Bianchi, Robert T. Jansen, and Remo Ruffini, editors, Fourteenth Marcel Grossmann Meeting - MG14, pages 503–524, January 2018.
  * [50] Ken Shen.  The progenitors of Type Ia supernovae.  In The Extragalactic Explosive Universe: the New Era of Transient Surveys and Data-Driven Discovery, page 46, October 2019.
  * [51] Adam G. Riess, Wenlong Yuan, Lucas M. Macri, Dan Scolnic, Dillon Brout, Stefano Casertano, David O. Jones, Yukei Murakami, Gagandeep S. Anand, Louise Breuval, Thomas G. Brink, Alexei V. Filippenko, Samantha Hoffmann, Saurabh W. Jha, W. D’arcy Kenworthy, John Mackenty, Benjamin E. Stahl, and WeiKang Zheng.  A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s-1 Mpc-1 Uncertainty from the Hubble Space Telescope and the SH0ES Team.  ApJ, 934(1):L7, July 2022.
  * [52] Roger Barlow.  Asymmetric Systematic Errors.  arXiv e-prints, page physics/0306138, June 2003.
  * [53] D. O. Jones, D. M. Scolnic, A. G. Riess, A. Rest, R. P. Kirshner, E. Berger, R. Kessler, Y. C. Pan, R. J. Foley, R. Chornock, C. A. Ortega, P. J. Challis, W. S. Burgett, K. C. Chambers, P. W. Draper, H. Flewelling, M. E. Huber, N. Kaiser, R. P. Kudritzki, N. Metcalfe, J. Tonry, R. J. Wainscoat, C. Waters, E. E. E. Gall, R. Kotak, M. McCrum, S. J. Smartt, and K. W. Smith.  Measuring Dark Energy Properties with Photometrically Classified Pan-STARRS Supernovae. II. Cosmological Parameters.  ApJ, 857(1):51, April 2018.
  * [54] M. Kowalski, D. Rubin, G. Aldering, R. J. Agostinho, A. Amadon, R. Amanullah, C. Balland, K. Barbary, G. Blanc, P. J. Challis, A. Conley, N. V. Connolly, R. Covarrubias, K. S. Dawson, S. E. Deustua, R. Ellis, S. Fabbro, V. Fadeyev, X. Fan, B. Farris, G. Folatelli, B. L. Frye, G. Garavini, E. L. Gates, L. Germany, G. Goldhaber, B. Goldman, A. Goobar, D. E. Groom, J. Haissinski, D. Hardin, I. Hook, S. Kent, A. G. Kim, R. A. Knop, C. Lidman, E. V. Linder, J. Mendez, J. Meyers, G. J. Miller, M. Moniez, A. M. Mourão, H. Newberg, S. Nobili, P. E. Nugent, R. Pain, O. Perdereau, S. Perlmutter, M. M. Phillips, V. Prasad, R. Quimby, N. Regnault, J. Rich, E. P. Rubenstein, P. Ruiz-Lapuente, F. D. Santos, B. E. Schaefer, R. A. Schommer, R. C. Smith, A. M. Soderberg, A. L. Spadafora, L. G. Strolger, M. Strovink, N. B. Suntzeff, N. Suzuki, R. C. Thomas, N. A. Walton, L. Wang, W. M. Wood-Vasey, and J. L. Yun.  Improved Cosmological Constraints from New, Old, and Combined Supernova Data Sets.  ApJ, 686(2):749–778, October 2008.
  * [55] J. Jönsson, M. Sullivan, I. Hook, S. Basa, R. Carlberg, A. Conley, D. Fouchez, D. A. Howell, K. Perrett, and C. Pritchet.  Constraining dark matter halo properties using lensed Supernova Legacy Survey supernovae.  MNRAS, 405(1):535–544, June 2010.
  * [56] Edward F. Schlafly and Douglas P. Finkbeiner.  Measuring Reddening with Sloan Digital Sky Survey Stellar Spectra and Recalibrating SFD.  ApJ, 737(2):103, August 2011.
  * [57] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, S. Fabbro, A. Goobar, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C. R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon, P. Ruiz-Lapuente, N. Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M. Newberg, W. J. Couch, and The Supernova Cosmology Project.  Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae.  ApJ, 517(2):565–586, June 1999.
  * [58] Adam G. Riess, Louis-Gregory Strolger, John Tonry, Stefano Casertano, Henry C. Ferguson, Bahram Mobasher, Peter Challis, Alexei V. Filippenko, Saurabh Jha, Weidong Li, Ryan Chornock, Robert P. Kirshner, Bruno Leibundgut, Mark Dickinson, Mario Livio, Mauro Giavalisco, Charles C. Steidel, Txitxo Benítez, and Zlatan Tsvetanov.  Type Ia Supernova Discoveries at $z>1$ from the Hubble Space Telescope: Evidence for Past Deceleration and Constraints on Dark Energy Evolution.  ApJ, 607(2):665–687, June 2004.
  * [59] R. Amanullah, C. Lidman, D. Rubin, G. Aldering, P. Astier, K. Barbary, M. S. Burns, A. Conley, K. S. Dawson, S. E. Deustua, M. Doi, S. Fabbro, L. Faccioli, H. K. Fakhouri, G. Folatelli, A. S. Fruchter, H. Furusawa, G. Garavini, G. Goldhaber, A. Goobar, D. E. Groom, I. Hook, D. A. Howell, N. Kashikawa, A. G. Kim, R. A. Knop, M. Kowalski, E. Linder, J. Meyers, T. Morokuma, S. Nobili, J. Nordin, P. E. Nugent, L. Östman, R. Pain, N. Panagia, S. Perlmutter, J. Raux, P. Ruiz-Lapuente, A. L. Spadafora, M. Strovink, N. Suzuki, L. Wang, W. M. Wood-Vasey, N. Yasuda, and The Supernova Cosmology Project.  Spectra and Hubble Space Telescope Light Curves of Six Type Ia Supernovae at $0.511<z<1.12$ and the Union2 Compilation.  ApJ, 716(1):712–738, June 2010.
  * [60] M. Betoule, R. Kessler, J. Guy, J. Mosher, D. Hardin, R. Biswas, P. Astier, P. El-Hage, M. Konig, S. Kuhlmann, J. Marriner, R. Pain, N. Regnault, C. Balland, B. A. Bassett, P. J. Brown, H. Campbell, R. G. Carlberg, F. Cellier-Holzem, D. Cinabro, A. Conley, C. B. D’Andrea, D. L. DePoy, M. Doi, R. S. Ellis, S. Fabbro, A. V. Filippenko, R. J. Foley, J. A. Frieman, D. Fouchez, L. Galbany, A. Goobar, R. R. Gupta, G. J. Hill, R. Hlozek, C. J. Hogan, I. M. Hook, D. A. Howell, S. W. Jha, L. Le Guillou, G. Leloudas, C. Lidman, J. L. Marshall, A. Möller, A. M. Mourão, J. Neveu, R. Nichol, M. D. Olmstead, N. Palanque-Delabrouille, S. Perlmutter, J. L. Prieto, C. J. Pritchet, M. Richmond, A. G. Riess, V. Ruhlmann-Kleider, M. Sako, K. Schahmaneche, D. P. Schneider, M. Smith, J. Sollerman, M. Sullivan, N. A. Walton, and C. J. Wheeler.  Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples.  A&A, 568:A22, August 2014.
  * [61] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, N. Bartolo, E. Battaner, R. Battye, K. Benabed, A. Benoît, A. Benoit-Lévy, J. P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bridges, M. Bucher, C. Burigana, R. C. Butler, J. F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, R. R. Chary, H. C. Chiang, L. Y. Chiang, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, F. Couchot, A. Coulais, B. P. Crill, M. Cruz, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J. M. Delouis, F. X. Désert, J. M. Diego, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, Y. Fantaye, J. Fergusson, F. Finelli, O. Forni, M. Frailis, E. Franceschi, M. Frommert, S. Galeotta, K. Ganga, M. Giard, G. Giardino, Y. Giraud-Héraud, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gregorio, A. Gruppuso, F. K. Hansen, M. Hansen, D. Hanson, D. L. Harrison, G. Helou, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, K. M. Huffenberger, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, J. Kim, T. S. Kisner, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lähteenmäki, J. M. Lamarre, A. Lasenby, R. J. Laureijs, C. R. Lawrence, J. P. Leahy, R. Leonardi, C. Leroy, J. Lesgourgues, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. López-Caniego, P. M. Lubin, J. F. Macías-Pérez, B. Maffei, D. Maino, N. Mandolesi, A. Mangilli, D. Marinucci, M. Maris, D. J. Marshall, P. G. Martin, E. Martínez-González, S. Masi, M. Massardi, S. Matarrese, F. Matthai, P. Mazzotta, J. D. McEwen, P. R. Meinhold, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, K. Mikkelsen, S. Mitra, M. A. Miville-Deschênes, D. Molinari, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, H. U. Nørgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, S. Osborne, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, G. Patanchon, H. V. Peiris, O. Perdereau, L. Perotto, F. Perrotta, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, D. Pogosyan, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, T. Poutanen, G. W. Pratt, G. Prézeau, S. Prunet, J. L. Puget, J. P. Rachen, B. Racine, C. Räth, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, S. Ricciardi, T. Riller, I. Ristorcelli, G. Rocha, C. Rosset, A. Rotti, G. Roudier, J. A. Rubiño-Martín, B. Ruiz-Granados, B. Rusholme, M. Sandri, D. Santos, G. Savini, D. Scott, M. D. Seiffert, E. P. S. Shellard, T. Souradeep, L. D. Spencer, J. L. Starck, V. Stolyarov, R. Stompor, R. Sudiwala, F. Sureau, P. Sutter, D. Sutton, A. S. Suur-Uski, J. F. Sygnet, J. A. Tauber, D. Tavagnacco, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, M. Türler, L. Valenziano, J. Valiviita, B. Van Tent, J. Varis, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, I. K. Wehus, M. White, A. Wilkinson, D. Yvon, A. Zacchei, and A. Zonca.  Planck 2013 results. XXIII. Isotropy and statistics of the CMB.  A&A, 571:A23, November 2014.
  * [62] Carlos A. P. Bengaly, Roy Maartens, and Mario G. Santos.  Probing the Cosmological Principle in the counts of radio galaxies at different frequencies.  J. Cosmology Astropart. Phys, 2018(4):031, April 2018.
  * [63] Planck Collaboration, P. A. R. Ade, N. Aghanim, Y. Akrami, P. K. Aluri, M. Arnaud, M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, N. Bartolo, S. Basak, E. Battaner, K. Benabed, A. Benoît, A. Benoit-Lévy, J. P. Bernard, M. Bersanelli, P. Bielewicz, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, F. Boulanger, M. Bucher, C. Burigana, R. C. Butler, E. Calabrese, J. F. Cardoso, B. Casaponsa, A. Catalano, A. Challinor, A. Chamballu, H. C. Chiang, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, C. Combet, D. Contreras, F. Couchot, A. Coulais, B. P. Crill, M. Cruz, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, F. X. Désert, J. M. Diego, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, Y. Fantaye, J. Fergusson, R. Fernandez-Cobos, F. Finelli, O. Forni, M. Frailis, A. A. Fraisse, E. Franceschi, A. Frejsel, A. Frolov, S. Galeotta, S. Galli, K. Ganga, C. Gauthier, T. Ghosh, M. Giard, Y. Giraud-Héraud, E. Gjerløw, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gregorio, A. Gruppuso, J. E. Gudmundsson, F. K. Hansen, D. Hanson, D. L. Harrison, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, Z. Huang, K. M. Huffenberger, G. Hurier, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, J. Kim, T. S. Kisner, J. Knoche, M. Kunz, H. Kurki-Suonio, G. Lagache, A. Lähteenmäki, J. M. Lamarre, A. Lasenby, M. Lattanzi, C. R. Lawrence, R. Leonardi, J. Lesgourgues, F. Levrier, M. Liguori, P. B. Lilje, M. Linden-Vørnle, H. Liu, M. López-Caniego, P. M. Lubin, J. F. Macías-Pérez, G. Maggio, D. Maino, N. Mandolesi, A. Mangilli, D. Marinucci, M. Maris, P. G. Martin, E. Martínez-González, S. Masi, S. Matarrese, P. McGehee, P. R. Meinhold, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, K. Mikkelsen, S. Mitra, M. A. Miville-Deschênes, D. Molinari, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, F. Nati, P. Natoli, C. B. Netterfield, H. U. Nørgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, N. Pant, D. Paoletti, F. Pasian, G. Patanchon, T. J. Pearson, O. Perdereau, L. Perotto, F. Perrotta, V. Pettorino, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, L. Popa, G. W. Pratt, G. Prézeau, S. Prunet, J. L. Puget, J. P. Rachen, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, I. Ristorcelli, G. Rocha, C. Rosset, M. Rossetti, A. Rotti, G. Roudier, J. A. Rubiño-Martín, B. Rusholme, M. Sandri, D. Santos, M. Savelainen, G. Savini, D. Scott, M. D. Seiffert, E. P. S. Shellard, T. Souradeep, L. D. Spencer, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, D. Sutton, A. S. Suur-Uski, J. F. Sygnet, J. A. Tauber, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, T. Trombetti, M. Tucci, J. Tuovinen, L. Valenziano, J. Valiviita, B. Van Tent, P. Vielva, F. Villa, L. A. Wade, B. D. Wandelt, I. K. Wehus, D. Yvon, A. Zacchei, J. P. Zibin, and A. Zonca.  Planck 2015 results. XVI. Isotropy and statistics of the CMB.  A&A, 594:A16, September 2016.
  * [64] Y. Akrami, Y. Fantaye, A. Shafieloo, H. K. Eriksen, F. K. Hansen, A. J. Banday, and K. M. Górski.  Power Asymmetry in WMAP and Planck Temperature Sky Maps as Measured by a Local Variance Estimator.  ApJ, 784(2):L42, April 2014.
  * [65] Antonio Mariano and Leandros Perivolaropoulos.  Is there correlation between fine structure and dark energy cosmic dipoles?  Phys. Rev. D, 86(8):083517, October 2012.
  * [66] Hai-Nan Lin, Sai Wang, Zhe Chang, and Xin Li.  Testing the isotropy of the Universe by using the JLA compilation of Type Ia supernovae.  MNRAS, 456(2):1881–1885, February 2016.
  * [67] H. Akaike.  A New Look at the Statistical Model Identification.  IEEE Transactions on Automatic Control, 19:716–723, January 1974\.
  * [68] G. Efstathiou.  A Lockdown Perspective on the Hubble Tension (with comments from the SH0ES team).  arXiv e-prints, page arXiv:2007.10716, July 2020.
  * [69] Eleonora Di Valentino, Olga Mena, Supriya Pan, Luca Visinelli, Weiqiang Yang, Alessandro Melchiorri, David F. Mota, Adam G. Riess, and Joseph Silk.  In the realm of the Hubble tension-a review of solutions.  Classical and Quantum Gravity, 38(15):153001, July 2021.
  * [70] Wendy L. Freedman.  Measurements of the Hubble Constant: Tensions in Perspective.  ApJ, 919(1):16, September 2021.
  * [71] Dan Scolnic, Dillon Brout, Anthony Carr, Adam G. Riess, Tamara M. Davis, Arianna Dwomoh, David O. Jones, Noor Ali, Pranav Charvu, Rebecca Chen, Erik R. Peterson, Brodie Popovic, Benjamin M. Rose, Charlotte M. Wood, Peter J. Brown, Ken Chambers, David A. Coulter, Kyle G. Dettman, Georgios Dimitriadis, Alexei V. Filippenko, Ryan J. Foley, Saurabh W. Jha, Charles D. Kilpatrick, Robert P. Kirshner, Yen-Chen Pan, Armin Rest, Cesar Rojas-Bravo, Matthew R. Siebert, Benjamin E. Stahl, and WeiKang Zheng.  The Pantheon+ Analysis: The Full Data Set and Light-curve Release.  ApJ, 938(2):113, October 2022.
  * [72] Jesús Torrado and Antony Lewis.  Cobaya: code for Bayesian analysis of hierarchical physical models.  J. Cosmology Astropart. Phys, 2021(5):057, May 2021.
  * [73] Antony Lewis.  GetDist: a Python package for analysing Monte Carlo samples.  2019\.
  * [74] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern, Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas, Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero, Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa, Paul van Mulbregt, and SciPy 1.0 Contributors.  SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python.  Nature Methods, 17:261–272, 2020.
  * [75] Charles R. Harris, K. Jarrod Millman, Stéfan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jaime Fernández del Río, Mark Wiebe, Pearu Peterson, Pierre Gérard-Marchant, Kevin Sheppard, Tyler Reddy, Warren Weckesser, Hameer Abbasi, Christoph Gohlke, and Travis E. Oliphant.  Array programming with NumPy.  Nature, 585(7825):357–362, September 2020.
  * [76] Astropy Collaboration, T. P. Robitaille, E. J. Tollerud, P. Greenfield, M. Droettboom, E. Bray, T. Aldcroft, M. Davis, A. Ginsburg, A. M. Price-Whelan, W. E. Kerzendorf, A. Conley, N. Crighton, K. Barbary, D. Muna, H. Ferguson, F. Grollier, M. M. Parikh, P. H. Nair, H. M. Unther, C. Deil, J. Woillez, S. Conseil, R. Kramer, J. E. H. Turner, L. Singer, R. Fox, B. A. Weaver, V. Zabalza, Z. I. Edwards, K. Azalee Bostroem, D. J. Burke, A. R. Casey, S. M. Crawford, N. Dencheva, J. Ely, T. Jenness, K. Labrie, P. L. Lim, F. Pierfederici, A. Pontzen, A. Ptak, B. Refsdal, M. Servillat, and O. Streicher.  Astropy: A community Python package for astronomy.  A&A, 558:A33, October 2013.
  * [77] Astropy Collaboration, A. M. Price-Whelan, B. M. Sipőcz, H. M. Günther, P. L. Lim, S. M. Crawford, S. Conseil, D. L. Shupe, M. W. Craig, N. Dencheva, A. Ginsburg, J. T. Vand erPlas, L. D. Bradley, D. Pérez-Suárez, M. de Val-Borro, T. L. Aldcroft, K. L. Cruz, T. P. Robitaille, E. J. Tollerud, C. Ardelean, T. Babej, Y. P. Bach, M. Bachetti, A. V. Bakanov, S. P. Bamford, G. Barentsen, P. Barmby, A. Baumbach, K. L. Berry, F. Biscani, M. Boquien, K. A. Bostroem, L. G. Bouma, G. B. Brammer, E. M. Bray, H. Breytenbach, H. Buddelmeijer, D. J. Burke, G. Calderone, J. L. Cano Rodríguez, M. Cara, J. V. M. Cardoso, S. Cheedella, Y. Copin, L. Corrales, D. Crichton, D. D’Avella, C. Deil, É. Depagne, J. P. Dietrich, A. Donath, M. Droettboom, N. Earl, T. Erben, S. Fabbro, L. A. Ferreira, T. Finethy, R. T. Fox, L. H. Garrison, S. L. J. Gibbons, D. A. Goldstein, R. Gommers, J. P. Greco, P. Greenfield, A. M. Groener, F. Grollier, A. Hagen, P. Hirst, D. Homeier, A. J. Horton, G. Hosseinzadeh, L. Hu, J. S. Hunkeler, Ž. Ivezić, A. Jain, T. Jenness, G. Kanarek, S. Kendrew, N. S. Kern, W. E. Kerzendorf, A. Khvalko, J. King, D. Kirkby, A. M. Kulkarni, A. Kumar, A. Lee, D. Lenz, S. P. Littlefair, Z. Ma, D. M. Macleod, M. Mastropietro, C. McCully, S. Montagnac, B. M. Morris, M. Mueller, S. J. Mumford, D. Muna, N. A. Murphy, S. Nelson, G. H. Nguyen, J. P. Ninan, M. Nöthe, S. Ogaz, S. Oh, J. K. Parejko, N. Parley, S. Pascual, R. Patil, A. A. Patil, A. L. Plunkett, J. X. Prochaska, T. Rastogi, V. Reddy Janga, J. Sabater, P. Sakurikar, M. Seifert, L. E. Sherbert, H. Sherwood-Taylor, A. Y. Shih, J. Sick, M. T. Silbiger, S. Singanamalla, L. P. Singer, P. H. Sladen, K. A. Sooley, S. Sornarajah, O. Streicher, P. Teuben, S. W. Thomas, G. R. Tremblay, J. E. H. Turner, V. Terrón, M. H. van Kerkwijk, A. de la Vega, L. L. Watkins, B. A. Weaver, J. B. Whitmore, J. Woillez, V. Zabalza, and Astropy Contributors.  The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package.  AJ, 156(3):123, September 2018.
  * [78] Astropy Collaboration, Adrian M. Price-Whelan, Pey Lian Lim, Nicholas Earl, Nathaniel Starkman, Larry Bradley, David L. Shupe, Aarya A. Patil, Lia Corrales, C. E. Brasseur, Maximilian N”othe, Axel Donath, Erik Tollerud, Brett M. Morris, Adam Ginsburg, Eero Vaher, Benjamin A. Weaver, James Tocknell, William Jamieson, Marten H. van Kerkwijk, Thomas P. Robitaille, Bruce Merry, Matteo Bachetti, H. Moritz G”unther, Thomas L. Aldcroft, Jaime A. Alvarado-Montes, Anne M. Archibald, Attila B’odi, Shreyas Bapat, Geert Barentsen, Juanjo Baz’an, Manish Biswas, M’ed’eric Boquien, D. J. Burke, Daria Cara, Mihai Cara, Kyle E. Conroy, Simon Conseil, Matthew W. Craig, Robert M. Cross, Kelle L. Cruz, Francesco D’Eugenio, Nadia Dencheva, Hadrien A. R. Devillepoix, J”org P. Dietrich, Arthur Davis Eigenbrot, Thomas Erben, Leonardo Ferreira, Daniel Foreman-Mackey, Ryan Fox, Nabil Freij, Suyog Garg, Robel Geda, Lauren Glattly, Yash Gondhalekar, Karl D. Gordon, David Grant, Perry Greenfield, Austen M. Groener, Steve Guest, Sebastian Gurovich, Rasmus Handberg, Akeem Hart, Zac Hatfield-Dodds, Derek Homeier, Griffin Hosseinzadeh, Tim Jenness, Craig K. Jones, Prajwel Joseph, J. Bryce Kalmbach, Emir Karamehmetoglu, Mikolaj Kaluszy’nski, Michael S. P. Kelley, Nicholas Kern, Wolfgang E. Kerzendorf, Eric W. Koch, Shankar Kulumani, Antony Lee, Chun Ly, Zhiyuan Ma, Conor MacBride, Jakob M. Maljaars, Demitri Muna, N. A. Murphy, Henrik Norman, Richard O’Steen, Kyle A. Oman, Camilla Pacifici, Sergio Pascual, J. Pascual-Granado, Rohit R. Patil, Gabriel I. Perren, Timothy E. Pickering, Tanuj Rastogi, Benjamin R. Roulston, Daniel F. Ryan, Eli S. Rykoff, Jose Sabater, Parikshit Sakurikar, Jes’us Salgado, Aniket Sanghi, Nicholas Saunders, Volodymyr Savchenko, Ludwig Schwardt, Michael Seifert-Eckert, Albert Y. Shih, Anany Shrey Jain, Gyanendra Shukla, Jonathan Sick, Chris Simpson, Sudheesh Singanamalla, Leo P. Singer, Jaladh Singhal, Manodeep Sinha, Brigitta M. SipHocz, Lee R. Spitler, David Stansby, Ole Streicher, Jani Šumak, John D. Swinbank, Dan S. Taranu, Nikita Tewary, Grant R. Tremblay, Miguel de Val-Borro, Samuel J. Van Kooten, Zlatan Vasovi’c, Shresth Verma, Jos’e Vin’icius de Miranda Cardoso, Peter K. G. Williams, Tom J. Wilson, Benjamin Winkel, W. M. Wood-Vasey, Rui Xue, Peter Yoachim, Chen Zhang, Andrea Zonca, and Astropy Project Contributors.  The Astropy Project: Sustaining and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.0) of the Core Package.  apj, 935(2):167, August 2022.
  * [79] J. D. Hunter.  Matplotlib: A 2d graphics environment.  Computing in Science & Engineering, 9(3):90–95, 2007.