# A Non-Parametric Test of Variability of Type Ia Supernovae Luminosity and
CDDR

Darshan Kumar  , Akshay Rana  , Deepak Jain  , Shobhit Mahajan  , Amitabha
Mukherjee  , and R. F. L. Holanda

###### Abstract

The first observational evidence for cosmic acceleration appeared from Type Ia
supernovae (SNe Type Ia) Hubble diagram from two different groups. However,
the empirical treatment of SNe Type Ia and their ability to show cosmic
acceleration have been the subject of some debate in the literature. In this
work we probe the assumption of redshift-independent absolute magnitude
$(M_{\mathrm{B}})$ of SNe along with its correlation with spatial curvature
($\Omega_{k0}$) and cosmic distance duality relation (CDDR) parameter
($\eta(z)$). This work is divided into two parts. Firstly, we check the
validity of CDDR which relates the luminosity distance ($d_{L}$) and angular
diameter distance ($d_{A}$) via redshift. We use the Pantheon SNe Ia dataset
combined with the $H(z)$ measurements derived from the cosmic chronometers.
Further, four different redshift-dependent parametrizations of the distance
duality parameter $(\eta(z))$ are used. The CDDR is fairly consistent for
almost every parametrization within a $2\sigma$ confidence level in both flat
and a non-flat universe. In the second part, we assume the validity of CDDR
and emphasize on the variability of $M_{\mathrm{B}}$ and its correlation with
$\Omega_{k0}$. We choose four different redshift-dependent parametrizations of
$M_{\mathrm{B}}$. The results indicate no evolution of $M_{\mathrm{B}}$ within
$2\sigma$ confidence level. For all parametrizations, the best fit value of
$\Omega_{k0}$ indicates a flat universe at $2\sigma$ confidence level. However
a mild inclination towards a non flat universe is also observed. We have also
examined the dependence of the results on the choice of different priors for
$H_{0}$.
Keywords: supernova type Ia - standard candles, cosmological parameters from
LSS, dark energy theory, cosmology of theories beyond the SM.

##  1 Introduction

The first observational evidence for the cosmic acceleration appeared from
Type Ia supernovae (SNe Type Ia) observations performed by two different
research groups [1, 2]. Over the decades, the number of SNe Type Ia catalogs
have increased significantly. Even now, SNe Type Ia observations provide the
most direct evidence for the current cosmic acceleration. In the context of
Einstein’s General Theory of Relativity (GTR), SNe Type Ia observations
support the existence of a mysterious form of energy called dark energy, that
is either constant ($\Lambda$CDM) or slowly varying with time and space. (See
reviews in Refs. [3, 4, 5]).

Type Ia supernovae (SNe Type Ia) are extremely luminous explosions and these
are observationally identified by the absence of hydrogen and presence of
silicon (Si II) spectral lines in their spectra [6]. The use of SNe Type Ia as
a reliable cosmological probe relies on two fundamental assumptions:

a) The first assumption is that the shape of the light curve of all type Ia
supernovae (SNe Type Ia) is similar; hence these can be standardized [7, 8].
In practice, this task of standardizing light curve of SNe Type Ia is achieved
by using several empirically derived light curve fitters like, for instance,
MLCS/MLCS2k2 [9], SALT [10], SALT2 [10], SiFTO [11] etc. As stated, the
generalized functional forms of the light curves obtained in these fitters are
purely empirical and based on plausible physical explanations to explain the
light curves of SNe Type Ia from the time of explosion to a few weeks after
peak brightness[12].

b) The second basic assumption behind the SNe Type Ia analysis is that the
intrinsic luminosity of a supernova is independent of the redshift and the
host galaxy environment. In other words, it is theoretically assumed that two
different SNe Type Ia in two different host galaxies have the same intrinsic
luminosity, independent of masses and redshifts of the host galaxies. However,
in recent years several dedicated studies have concluded that the light curve
fitting analysis of SNe Type Ia depends on their host galaxy masses [13].
Further, it is seen that SNe Type Ia events occurring in massive early-type,
passive galaxies are brighter than those in late-type star forming galaxies
[14, 15, 16, 17].

The two main progenitor models of SNe Type Ia are the “single degenerate” and
the “double degenerate” models. In the single degenerate model, a white dwarf
accreting from a binary companion star is pushed over the Chandrasekhar mass
limit, while in the double degenerate model of SNe Type Ia explosions, an
orbiting pair of binary white dwarfs merge together and their mass eventually
exceeds the Chandrasekhar mass limit. There is no established formalism to
exactly differentiate between these two channels of SNe Type Ia explosion.
Hence the observed supernova population catalog may have SNe Type Ia
contributions coming from both these channels which will have a direct impact
on the first and second assumptions used in SNe Type Ia analysis [18].

Further, the statistical treatment of SNe Type Ia, their dimming and their
ability to prove the cosmic acceleration is still a topic of debate in the
literature. This is because the process of obtaining the SNe Type Ia
observations is not trivial but requires some priors for interpretations and
corrections to convert an observed-frame magnitude to a rest-frame magnitude.
Examples discussed in the literature are for instance: possible evolutionary
effects in SNe Type Ia events [19, 20], local Hubble bubble [21, 22], modified
gravity [23, 24, 25], unclustetered sources of light attenuation [26, 27, 28,
29] and the existence of Axion-Like-Particles (ALPs) arising in a wide range
of well-motivated high-energy physics scenarios. All these could also lead to
the dimming of SNe Type Ia brightness [30, 31] which would eventually affect
the second assumption used in the SNe Type Ia analysis.

Given the above mentioned limitations, Tutusaus et al. (2019) [32] relaxed the
standard assumption that SNe Type Ia intrinsic luminosity is independent of
the redshift and examined its impact on the cosmic acceleration. The authors
reconstructed the expansion rate of the Universe by fitting the SNe Type Ia
observations with a cubic spline interpolation. They showed that a non-
accelerated expansion rate of the Universe is able to fit all the main
background cosmological probes. In addition, Tutusaus et al. (2017) [33] found
that when SNe Type Ia intrinsic luminosity is not assumed to be redshift
independent, a non-accelerated low-redshift power law model is able to fit the
low-redshift background data as well as the $\Lambda$CDM model. It has also
been found that a significant correlation exists between SNe Type Ia
luminosity (after the standardization) and the stellar population age at a
99.5% confidence level [34] indicating that the light-curve fitters used by
the SNe Type Ia community are not quite capable of correcting for the
population age effect. More recently, Valentino et al. (2020) [35] have also
raised the issue whether intrinsic SNe Type Ia luminosities might evolve with
redshift. They analyse the impact of the latter on the inferred properties of
the dark energy component responsible for cosmic acceleration. However, they
find the evidence for cosmic acceleration to be robust to possible
systematics. Along the same lines, Sapone et al. (2020) [36] also analyse the
cosmological implications of an absolute luminosity of SNe Type Ia which could
vary with respect to the redshift. Further, they study the impact of the
latter on modified gravity models and non-homogeneous models.

These statements seem to offer enough plausible reasons to investigate the
effects of variability of absolute luminosity of SNe Type Ia. However, these
are not the sole reasons of concern about the variability of $M_{B}$. The
cosmic acceleration rate and the cosmological parameters determined by the SNe
Type Ia measurements are highly dependent on the possible dimming effect as
well. Vavrycuk et al. (2019) [37] revived a debate about an origin of Type Ia
supernova (SN Ia) dimming and showed that the standard $\Lambda$CDM model and
the opaque universe model (caused by light extinction by intergalactic dust)
fit the SN Ia measurements at redshifts $z<1.4$ fairly well. Then, there are
still some possible loopholes in the current SNe Type Ia observations and
alternative mechanisms contributing to the acceleration evidence or even
mimicking the dark energy behaviour have been proposed. It is important to
point out that a constant value of the absolute luminosity of SNe Type Ia at
the peak of its light curve is not sensitive to Hubble constant ($H_{0}$).
Nevertheless,a variable absolute magnitude will be sensitive to $H_{0}$.

These recent development motivate us to carry out a model independent study of
finding any correlation between variable luminosity of SNe Type Ia and
observed SNe Type Ia dimming effect due to opacity of the environment. The
most general methodology of testing the cosmic opacity of SNe Type Ia is based
on the cosmic distance duality relation (CDDR) which connects the luminosity
distance $d_{L}$ and angular diameter distance (ADD) $d_{A}$ at the same
redshift and is defined as, $d_{L}(1+z)^{-2}/d_{A}=\eta(z)=1$. In order to
look for the presence of some unknown physics phenomenon beyond the standard
model or an inconsistency between cosmological data we check if $\eta(z)\neq
1$, that is CDDR is violated. This relation holds for general metric theories
of gravity in any background, in which photons travel along null geodesics and
the number of photons is conserved during cosmic evolution [38]. Briefly, the
SNe Type Ia observations have been confronted with several other cosmological
probes (strong gravitation lens systems, angular diameter distances, gas mass
fractions, baryon acoustic oscillations, cosmic microwave background, radio
sources, gravitational waves, $H(z)$ measurements, gamma ray bursts etc.) in
order to put limits on the redshift dependence of $\eta$, that is on $\eta(z)$
[39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 63, 64, 65]. All these works conclude that CDDR is valid
within a $2\sigma$ confidence level. However, it is worth stressing that
current analysis could not distinguish which functional form of $\eta(z)$ best
describes the data (see details in Ref.[66]).

Another cosmological parameter which can be crucial to the variability of
absolute luminosity of SNe Type Ia and its dimming effect is the cosmic
curvature. Cosmic curvature ($\Omega_{k0}$) is a fundamental geometric
quantity of the Universe. It plays a crucial role in the evolution and
dynamics of the universe. As, $\Omega_{k0}$ is directly related to
cosmological distances, a flat or non-flat space-time would obviously impact
the path travelled by the photon and eventually the absolute magnitude of SNe
Type Ia. Hence, in this paper, we also probe the variation of the absolute
luminosity of SNe Type Ia and its correlation with the CDDR parameter
($\eta(z)$) and cosmic curvature ($\Omega_{k0}$).

For clarity of exposition, this paper is divided into two parts. In the first
part, we test the validity of CDDR. We propose a new cosmological non-
parametric test for CDDR by using SNe Type Ia observations and $H(z)$
measurements from cosmic chronometers. The basic procedure is as follows: we
obtain the angular diameter distances at SNe Type Ia redshifts by applying
Gaussian Process integration method on cosmic chronometer $H(z)$ data. By
using a deformed CDDR of the form $d_{L}=\eta(z)d_{A}(1+z)^{2}$ and
considering a flat universe, we impose limits on the SNe Type Ia absolute
magnitude ($M_{B}$), and $\eta(z)$ functions, namely:
$\eta(z)=\eta_{0},\eta(z)=\eta_{0}+\eta_{1}z$,
$\eta(z)=\eta_{0}+\eta_{1}z/(1+z)$ and
$\eta(z)=\eta_{0}+\eta_{1}\text{ln}(1+z)$.

After testing the validity of the CDDR parameter $\eta$ in the first part, in
the second part, we assume the validity of CDDR and put limits on the possible
evolution of SNe Type Ia absolute magnitude by assuming $M_{B}(z)=M_{B0}$,
$M_{B}(z)=M_{B0}+M_{B1}z$, $M_{B}(z)=M_{B0}+M_{B1}z/(1+z)$ and
$M_{B}(z)=M_{B0}+M_{B1}\text{ln}(1+z)$ parametrizations in flat and non flat
cosmologies.

The structure of the paper is as follows: In Section 2, we discuss the
cosmological probes and the data sets used in the analysis along with their
theoretical construction. In Section 3, we outline the methodology used. In
Section 4, the emphasis is on the results of both parts where in the first
part we test the validity of CDDR and in the second part we test the
variability of absolute luminosity of SNe Type Ia. Finally, in Section 5, we
discuss the final outcomes of both parts and also explore the impact of
different $H_{0}$ priors on our analysis.

##  2 Cosmological Probes and Data sets

In this section we present the cosmological probes and data sets used in the
analysis.

###  2.1 Type Ia Supernovae measurement and Pantheon Sample

Type Ia supernovae are considered to be standard candles. Observations of
these are at the core of establishing the validity of cosmic acceleration. The
standard observable quantity in SNe Type Ia analysis is the distance modulus
which is the difference between the apparent and absolute magnitude of the SNe
Type Ia. The observational measurement of this quantity is given by the
relation

$\mu_{\mathrm{SN}}=m_{\mathrm{B}}^{\mathrm{obs}}(z)+\alpha\cdot
X_{1}-\beta\cdot\mathcal{C}-M_{\mathrm{B}}.$ (2.1)

where, $m_{\mathrm{B}}$ is the rest frame B-band observed peak magnitude,
$X_{1}$ and $\mathcal{C}$ are time stretching of the light curve and SNe Type
Ia color at maximum brightness respectively and $M_{\mathrm{B}}$ is the
absolute B-band magnitude. This relation indicates that the variability of the
distance modulus is governed by two additional parameters $X_{1}$ and
$\mathcal{C}$. It must be noted here that we do have two nuisance parameters
$\alpha$ and $\beta$ as well. In this paper we use the recent and largest
database of SNe Type Ia known as the Pantheon data set. In this data set,
these two nuisance parameters have been marginalized and eventually calibrated
to be zero. This data set has $1048$ SNe Type Ia measurements in the redshift
range $0.01\leq z\leq 2.26$ [13]. Hence, for the Pantheon data set, the
observed distance modulus relaxes to the form
$\mu_{\mathrm{SN}}=m_{\mathrm{B}}^{\mathrm{obs}}-M_{\mathrm{B}}$. Once we know
the distance modulus, we can easily define the luminosity distance and
uncertainty in observed luminosity distance as

$d_{\mathrm{L}}^{\mathrm{th}}(z;M_{\mathrm{B}})=10^{\left(m_{\mathrm{B}}^{\mathrm{th}}-M_{\mathrm{B}}-25\right)/5}(\mathrm{Mpc}),$
(2.2)

From Eq.(2.2), it can be easily seen that once we estimate the value of
$M_{\mathrm{B}}$, we can find the luminosity distance at a given redshift.
However, in this work, our aim is to study the variation of the absolute
magnitude ($\mathrm{M}_{B}$) of SNe Type Ia, so if we can get a model-
independent estimate of the luminosity distance from other observational
probes then we can constrain the variability of $\mathrm{M}_{B}$.

We use the cosmic distance duality relation (CDDR) to reconstruct the
luminosity distance theoretically which is given by

$d_{\mathrm{L}}^{\mathrm{th}}(z;\eta,\Omega_{k0})=\eta(z)d_{\mathrm{A}}(z;\Omega_{k0})(1+z)^{2}.$
(2.3)

Using Eq. (2.3) we can define, from Eq. (2.2), $m_{\mathrm{B}}^{\mathrm{th}}$
as

$m_{\mathrm{B}}^{\mathrm{th}}(z;\eta,M_{\mathrm{B}},\Omega_{k0})=5\log\left(\eta(z)d_{A}(z;\Omega_{k0})(1+z)^{2}\right)+M_{\mathrm{B}}+25,$
(2.4)

Here, $\eta$ is the cosmic distance duality parameter which is a measure of
the deviation from CDDR. CDDR holds for $\eta(z)=1$. Here
$d_{\mathrm{L}}^{\mathrm{th}}$ is the theoretical luminosity distance defined
in the terms of angular diameter distance $d_{\mathrm{A}}$ and $\eta(z)$. The
angular diameter distance is defined as

$d_{A}(z;H_{0},\Omega_{k0})=\begin{cases}\dfrac{d_{H}}{(1+z)\sqrt{\Omega_{k0}}}\sinh\left[\sqrt{\Omega_{k0}}\dfrac{d_{C}}{d_{H}}\right]&\mbox{for
$\Omega_{k0}>$0 }.\\\ \dfrac{d_{C}}{(1+z)}&\mbox{for $\Omega_{k0}=$0}.\\\
\dfrac{d_{H}}{(1+z)\sqrt{|\Omega_{k0}}|}\sin\left[\sqrt{|\Omega_{k0}|}\dfrac{d_{C}}{d_{H}}\right]&\mbox{for
$\Omega_{k0}<$0}.\\\ \end{cases}$ (2.5)

Here $\Omega_{k0}$ is the cosmic curvature, where $\Omega_{k0}$ is greater,
equal and less than zero for open, flat and closed universe respectively.
$d_{C}$ is the comoving distance and $d_{H}=c/H_{0}$ is known as the Hubble
distance where $c$ is the speed of light and $H_{0}$ is the Hubble constant.

###  2.2  Constraining Angular Diameter Distance using $H(z)$ measurements

As is evident from Eq. 2.5, if we can obtain a model-independent estimate of
$d_{C}/d_{H}$ then we can easily estimate angular diameter distance
$d_{\mathrm{A}}(z;\Omega_{k0})$ as a function of $\Omega_{k0}$ and
subsequently theoretical luminosity distance
$d_{\mathrm{L}}^{\mathrm{th}}(z;\eta,\Omega_{k0})$ as a function of $\eta$ and
$\Omega_{k0}$. In this section we will first discuss the data sets and the
methodology to obtain angular diameter distances from them.

####  2.2.1 Hubble data set

In cosmology, the Hubble parameter $H(z)$ is a crucial measured quantity which
describes the dynamical properties of the universe such as the expansion rate
and evolution history of the universe. It is also helpful to explore the
nature of the dark energy. The most recent data compilation of Hubble
parameter measurements [67] has $31$ measurements of $H(z)$ which are obtained
by using the differential ages of passively evolving galaxies. We now outline
the steps needed to obtain the angular diameter distance that we need.

  1. (i)

Differential ages of passively evolving galaxies: The Hubble parameter $H(z)$
can be expressed in the terms of the rate of change of cosmic time with the
redshift, given by

$H(z)=-\dfrac{1}{(1+z)}\dfrac{\Delta z}{\Delta t}.$ (2.6)

The change of cosmic time with redshift can be estimated from the ageing of
the stellar population in the galaxies. However, one needs to be extremely
careful in selecting the galaxies while calculating the $H(z)$ values using
this method. In young evolving galaxies, the stars are being born continuously
and the emission spectra will be dominated by the young stellar population.
Hence to estimate accurately the differential ageing of the universe,
passively evolving red galaxies are used as their light is mostly dominated by
the old stellar population [68]. To find $H(z)$ at a given redshift, the ages
of the early type passively evolving galaxies with similar metallicity and
very small redshift interval is calculated. The redshift difference $\Delta z$
between two galaxies can be measured by using spectroscopic observations. For
the estimation of $\Delta t$, [69] suggested the use of a direct spectroscopic
observable (the 4000 Å break) which is known to be linearly related to the age
of the stellar population of a galaxy at fixed metallicity. As the measure of
$H(z)$ is estimated purely by using spectroscopic observations, it is
independent of the cosmological model and has proved to be a strong probe to
constrain cosmological models and assumptions. This method of calculating
$H(z)$ is usually known as the “Cosmic Chronometers” and data points are
generally referred to as CC $H(z)$. We have $31$ data points estimated using
this differential ages of passively evolving galaxies technique [70, 71, 72,
69, 73, 74].

Figure 1:  In the left plot, 31 CC $H(z)$ vs $z$ datapoints are shown. In the
right plot, $H(z)$ is estimated at all intermediate redshifts in the range
$0<z<2$ using a non-parametric smoothening technique, namely Gaussian Process.
In the analysis, the Hubble constant value is taken to be $H_{0}=67.66\pm
0.42\mathrm{\leavevmode\nobreak\ km}\mathrm{\leavevmode\nobreak\
s}^{-1}\mathrm{Mpc}^{-1}$ estimated from CMB measurement [75]. The impact of
other choices of $H_{0}$ values on analysis has been discussed in Section
5(C).

####  2.2.2 Comoving distance using Gaussian Process

In this analysis, we require a model-independent estimate of the angular
diameter distance. For this, we first calculate the comoving distance $d_{C}$
using the $H(z)$ measurement of cosmic chronometers as

$\dfrac{d_{C}}{d_{H}}=\int\limits_{0}^{z}\dfrac{dz^{\prime}}{E(z^{\prime})}.$
(2.7)

Figure 2:  This plot represents the reconstructed values of $d_{C}/d_{H}$ in
the redshift in the range $0<z<2$. The solid black line represents the best
fit line while blue lines includes the $1\sigma$ confidence region. The solid
red line is the theoretical curve for the flat $\Lambda CDM$ model with
$\Omega_{m}=0.3$. The value of $H_{0}$ used in reconstructing this plot is
$H_{0}=67.66\pm 0.42\mathrm{\leavevmode\nobreak\
km}\mathrm{\leavevmode\nobreak\ s}^{-1}\mathrm{Mpc}^{-1}$.

Here $E(z)=H(z)/H_{0}$. If we have the functional form of $E(z)$ then we can
easily integrate Eq. 2.7 by using any numerical integration method. However,
in our case we have only $H(z)$ estimates at certain redshifts. In our
analysis, to obtain continuous smooth values of $H(z)$ we use the Gaussian
Process (GP), a well known hyper-parametric regression method [76]. GP has
been widely used in the literature to reconstruct the shapes of physical
functions and is very useful for such functional reconstructions due to its
flexibility and simplicity. In this method, the complicated parametric
relationship is replaced by parametrizing a probability model over the data.
In mathematical terms, it is a distribution over functions, characterized by a
mean function and covariance function, given by

$K\left(z,z^{\prime}\right)=\left\langle(H(z)-\mu(z))\left(H\left(z^{\prime}\right)-\mu\left(z^{\prime}\right)\right)\right\rangle$
(2.8)

where $\mu(z)$ is the prior mean. In order to avoid model dependence appearing
through the choice of the prior mean function, we have chosen it to be zero.
This is a common choice of mean function as one can always normalize the data
so it has zero mean. We have also checked by taking different values of prior
mean, $\mu(z)$ and observed that the result is independent of the choice of
the prior mean function. This method, however, comes with a few inherent
underlying assumptions- it is assumed that each observation is an outcome of
an independent Gaussian distribution belonging to the same population and the
outcomes of observations at any two redshifts are correlated with the strength
of correlation depending on their nearness to each other. We reconstructed our
data by using the square exponential kernel function, given by

$K(z,z^{\prime})=\sigma_{f}^{2}\exp{\left(-\dfrac{(|z-z^{\prime}|)^{2}}{2\ell^{2}}\right)}$
(2.9)

Here, $\sigma_{f}$ and $l$ are two hyperparameters which control the amplitude
and length-scale of the prior covariance. The value of hyperparameters is
calculated by maximizing the corresponding marginal log-likelihood probability
function of the distribution. For maximization, we use flat priors for
$\sigma_{f}$ and $l$ for all the choices of kernel function. In order to check
the sensitivity of our analysis to the choice of the kernel function, we
repeated our analysis with the Matérn-(3/2, 5/2, 7/2 & 9/2) kernel functions.
Though the values of hyperparameters vary according to the choice of kernel
function, the reconstructed curves do not show any significant deviation from
the curve obtained from square exponential curve. Hence we choose to work with
the square exponential kernel function only.

Once we obtain the reconstructed $H(z)$ at all possible redshifts in the range
$0<z<2$ as shown in Figure 1, we divide it by $H_{0}=67.66\pm
0.42\mathrm{\leavevmode\nobreak\ km}\mathrm{\leavevmode\nobreak\
s}^{-1}\mathrm{Mpc}^{-1}$ value to obtain $E(z)$. We use the Simpson $3/8$
method for numerically integrating Eq. 2.7 and obtain continuous values of
$d_{C}/d_{H}$ at all redshifts in the range $0<z<2$ which is shown in Figure
2. Further, we can use Eq. 2.5 to obtain the angular diameter distance as a
function of cosmic curvature $\Omega_{k0}$. The uncertainty in $d_{A}$ i.e
$\sigma_{d_{A}}$ is estimated by propagating the error obtained in
$d_{C}/d_{H}$ using Gaussian Process.

##  3 Analysis

Generally, the Pantheon dataset comes in terms of SNe Type Ia apparent
magnitudes and with a full covariance
matrix111http://github.com/dscolnic/Pantheon, $C_{\text{sys}}$ correlating the
apparent magnitudes at various redshifts. This covariance matrix is a non-
diagonal matrix of systematic uncertainties that come from the bias
corrections method[13, 77].

In this analysis, we have to fit simultaneously three parameters i.e.
$M_{\mathrm{B}}$, $\eta$ and $\Omega_{k0}$. These parameters are determined by
maximizing the likelihood $\mathcal{L}\sim\exp\left(-\chi^{2}/2\right)$, where
chi-square $(\chi^{2})$ is a quantity summed over all the Pantheon SNe Type Ia
Sample redshifts, and is defined as

$\chi_{\mathrm{Pan}}^{2}=\Delta m^{T}\cdot C^{-1}\cdot\Delta m$ (3.1)

where, $C=D_{\text{stat}}+C_{\text{sys}}$. Here $D_{\text{stat}}$ is the
diagonal covariance matrix of the statistical uncertainties and $\Delta
m=m_{B}^{\mathrm{obs}}(z_{i})-m_{B}^{\mathrm{th}}(z_{i};\eta,M_{\mathrm{B}},\Omega_{k0})$
which is given in Eqs.(2.1,2.4).

In this analysis, we have three unknown parameters, namely, the absolute
magnitude of SNe Type Ia, $M_{B}$, the cosmic distance duality parameter,
$\eta(z)$, and the cosmic curvature, $\Omega_{k0}$. In order to investigate
the variability of $M_{B}$ we have to analyse its correlation with the
remaining two parameters. In order to do so, we divide the work into two
parts:

### Part I: Test of CDDR

In this part, we consider the distance duality relation parameter $(\eta)$ to
test CDDR and put constraints on $\eta(z)$ simultaneously with the other two
parameters i.e. $M_{\mathrm{B}}$ and $\Omega_{k0}$. We take into account four
parametrizations of $\eta(z)$ which are as follows

  * •

P1: $\eta(z)=\eta_{0}$.

  * •

P2: $\eta(z)=\eta_{0}+\eta_{1}z$.

  * •

P3: $\eta(z)=\eta_{0}+\eta_{1}\dfrac{z}{1+z}$.

  * •

P4: $\eta(z)=\eta_{0}+\eta_{1}\text{ln}(1+z)$.

We included these to study the impact of different characterizations on the
analysis. The motivation for this form of parametrizations comes from the
commonly used parametrizations for the equation of state parameter, viz. the
Chevallier-Polarski-Linder (CPL) Parametrization, Jassal-Bagla-Padmanabhan
(JBP) Parametrization etc [78, 79, 80].

In the first parametrization, we are choosing $\eta$ to be redshift
independent. In second parametrization, it is a simple Taylor series expansion
around $z=0$ but is not well behaved at higher z values. The third and fourth
parametrizations are taken as these are well behaved even at high redshifts
and are somewhat more slowly varying as compared to the second one.

Further in each parametrization of $\eta(z)$, we discuss two cases. In the
first case we choose a flat universe by considering $\Omega_{k0}=0$ and then
put constraints on $M_{\mathrm{B}}$ and $\eta$. In the second case, we
consider $\Omega_{k0}$ as a free parameter corresponding to a non-flat
universe.

### Part II: Test of variability of SNe Type Ia absolute luminosity

After testing the validity of CDDR in Part I, we solely focus on the
variability of absolute magnitude of SNe Type Ia and use the following
parametrizations of $M_{B}$

  * •

M1: $M_{B}(z)=M_{B0}$.

  * •

M2: $M_{B}(z)=M_{B0}+M_{B1}z$.

  * •

M3: $M_{B}(z)=M_{B0}+M_{B1}\dfrac{z}{1+z}$.

  * •

M4: $M_{B}(z)=M_{B0}+M_{B1}\text{ln}(1+z)$.

In each parametrization of $M_{B}(z)$, we discuss two cases. In the first case
we choose a flat universe and then put constraints on $M_{\mathrm{B}}$ and in
the second case, we consider a non-flat universe.

We use emcee, a Python based package, to perform the Markov Chain Monte Carlo
(MCMC) analysis [81]. We find the best fit of all parameters and once the MCMC
method is performed, the confidence level with their $1\sigma$, $2\sigma$ and
$3\sigma$ uncertainties are computed with the Python package corner[82].

In this work, we assume a broad flat prior for all parameters (Table 1) . We
use 100 walkers which take 10000 steps for exploring the parameter space with
MCMC chains. The first $10\%$ of the steps are discarded as burn-in period and
the posterior distributions are analysed based on the remaining samples. To
ensure that the chains are converging, an auto-correlation study is performed.
For this we compute the integrated auto-correlation time $\tau_{f}$ using the
autocorr.integrated_time function of the emcee package. For more details,
please see ref.[81] .

Parameter | Prior Range
---|---
$M_{\mathrm{B}}$ | U[-21,-17]
$\eta_{0}$ | U[-3,3]
$\eta_{1}$ | U[-2,2]
$\Omega_{k0}$ | U[-2.5,2.5]
Table 1:  The prior range of $M_{\mathrm{B}},\leavevmode\nobreak\ \eta_{0}$,
$\eta_{1}$ and $\Omega_{k0}$.

##  4 Results

In this paper our aim is to probe the variation of the absolute luminosity
$(M_{B})$ of SNe Type Ia. This paper is divided into two parts. In the first
part, we propose a new (cosmological) model-independent test for CDDR by using
SNe Type Ia observations and $H(z)$ measurements from cosmic chronometers. In
the second part we assume that CDDR is valid and check the dependency of
absolute magnitude on redshift.

###  4.1 Test of CDDR

In this part, we consider the distance duality parameter $(\eta)$ and take
into account four parametrizations of $\eta(z)$. In each parametrization, we
discuss two cases- flat and non-flat universe.

#### P1. $\eta(z)=\eta_{0}$.

Taking $\eta(z)$ to be a constant, the best fit values of $M_{\mathrm{B}}$,
$\eta_{0}$ and $\Omega_{k0}$ parameters are given in Table 2.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B}}$ | $-19.300^{+0.812}_{-0.874}$ | $-19.653^{+0.587}_{-0.431}$
$\eta_{0}$ | $0.960_{-0.476}^{+0.300}$ | $1.124^{+0.250}_{-0.265}$
$\Omega_{k0}$ | — | $0.076^{+0.110}_{-0.106}$
Table 2:  The best fit values of $M_{\mathrm{B}},\leavevmode\nobreak\
\eta_{0}$ and $\Omega_{k0}$ with $1\sigma$ confidence level for P1
parametrization.

The best fit value of $\eta(z)$ shown in Table 2 for both flat as well as non-
flat universe, indicates that the cosmic distance duality relation
$(\eta(z)=1)$ holds at $1\sigma$ confidence level. In both flat and non-flat
case, the value of $M_{\mathrm{B}}$ remains same within $1\sigma$ confidence
level which reflects that $M_{\mathrm{B}}$ doesn’t have any strong dependence
on the curvature parameter. The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\eta_{0}$ and $\Omega_{k0}$ with $1\sigma$, $2\sigma$ and
$3\sigma$ confidence levels for flat and non-flat universe are shown in Fig.3.
The contour plots shown in Fig.3 indicate a negative correlation between
$M_{B}$ and $\eta_{0}$. Further, this parametrization supports a flat universe
within $1\sigma$ confidence level.

(a) Flat Universe

(b) Non-Flat Universe

Figure 3: The 1D and 2D posterior distributions of $M_{\mathrm{B}},\eta_{0}$
and $\Omega_{k0}$ for P1 parametrization.

#### P2. $\eta(z)=\eta_{0}+\eta_{1}z$.

In this parametrization, we consider $\eta(z)$ as a function of redshift. The
best fit values of $M_{\mathrm{B}},\leavevmode\nobreak\
\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ are given in Table
3.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B}}$ | $-19.254^{+0.616}_{-0.778}$ | $-19.574^{+0.518}_{-0.495}$
$\eta_{0}$ | $0.938_{-0.232}^{+0.405}$ | $1.097_{-0.234}^{+0.279}$
$\eta_{1}$ | $0.002^{+0.010}_{-0.009}$ | $-0.098_{-0.049}^{+0.047}$
$\Omega_{k0}$ | — | $1.108_{-0.531}^{+0.503}$
Table 3:  The best fit values of $M_{\mathrm{B}},\leavevmode\nobreak\
\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ with $1\sigma$
confidence level for P2 parametrization.

In case of a flat universe, the best fit value of $\eta_{1}$, suggests that
the cosmic distance duality parameter holds within $1\sigma$ confidence level
. Similarly in the case of a non-flat universe, there is no violation of
cosmic distance duality relation at $3\sigma$ confidence level. The obtained
value of $\Omega_{k0}$ can accommodate a flat universe at $3\sigma$ confidence
level. The best fit values of $M_{B}$ in both a flat universe and a non-flat
universe remain the same within $1\sigma$ confidence level which indicates
that there is no strong impact of $\Omega_{k0}$ on $M_{B}$.

The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ in
both flat and non-flat universes are shown in Fig. 4. The 2D posterior plots
for two cases (flat and non-flat) show a correlation between $M_{B}$ and
distance duality parameters.

(a) Flat Universe

(b) Non-Flat Universe

Figure 4: The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ for
P2 parametrization.

#### P3. $\eta(z)=\eta_{0}+\eta_{1}\dfrac{z}{1+z}$.

In this parametrization, we choose $\eta$ as a function of redshift which
converges at high redshift. The best fit values of
$M_{\mathrm{B}},\leavevmode\nobreak\ \eta_{0},\leavevmode\nobreak\ \eta_{1}$
and $\Omega_{k0}$ are given in Table 4.

In a flat universe, the CDDR holds within $1\sigma$ confidence level. Even,
for the non-flat universe, results indicate that CDDR holds true at $2\sigma$
confidence level. The best fit value of $\Omega_{k0}$ prefer a flat universe
at $2\sigma$ confidence level. The best fit values of $M_{B}$ in both a flat
universe and a non-flat universe at $1\sigma$ confidence level indicates that
there is no strong correlation between $\Omega_{k0}$ and $M_{B}$.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B}}$ | $-19.342^{+0.489}_{-0.369}$ | $-19.368^{+0.432}_{-0.335}$
$\eta_{0}$ | $0.976^{+0.179}_{-0.197}$ | $0.993^{+0.166}_{-0.178}$
$\eta_{1}$ | $0.003^{+0.015}_{-0.015}$ | $-0.047^{+0.036}_{-0.031}$
$\Omega_{k0}$ | — | $0.312^{+0.194}_{-0.204}$
Table 4:  The best fit values of $M_{\mathrm{B}},\leavevmode\nobreak\
\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ with $1\sigma$
confidence level for P3 parametrization.

The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ for
P3 parametrization of Part I are shown in Fig. 5. This figure shows strong
correlation between $M_{B}$ and $\eta_{0}$, and $\Omega_{k0}$ and $\eta_{1}$.

(a) Flat Universe

(b) Non-Flat Universe

Figure 5: The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\leavevmode\nobreak\ \eta_{0},\leavevmode\nobreak\ \eta_{1}$
and $\Omega_{k0}$ for P3 parametrization.

#### P4. $\eta(z)=\eta_{0}+\eta_{1}\text{ln}(1+z)$.

In this parametrization, we choose $\eta$ as a function of redshift which
varies with redshift logarithmic. The best fit value of
$M_{\mathrm{B}},\leavevmode\nobreak\ \eta_{0},\leavevmode\nobreak\ \eta_{1}$
and $\Omega_{k0}$ are given in Table 5.

In a flat universe, the best fit value of $\eta(z)$ supports the validity of
CDDR at $1\sigma$ confidence level. Similarly, the results in a non-flat
universe indicate that CDDR does hold true at $2\sigma$ confidence level. We
find the best fit value of $\Omega_{k0}$ a flat universe at $2\sigma$
confidence level. The best fit values of $M_{B}$ in both a flat universe and a
non-flat universe indicate that there is no strong correlation between
$\Omega_{k0}$ and $M_{B}$ at $1\sigma$ confidence level.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B}}$ | $-19.202^{+0.469}_{-0.505}$ | $-20.000^{+0.582}_{-0.402}$
$\eta_{0}$ | $0.916^{+0.241}_{-0.178}$ | $1.327^{+0.265}_{-0.306}$
$\eta_{1}$ | $0.003^{+0.014}_{-0.012}$ | $-0.046^{+0.041}_{-0.032}$
$\Omega_{k0}$ | — | $0.333^{+0.215}_{-0.243}$
Table 5:  The best fit values of $M_{\mathrm{B}},\leavevmode\nobreak\
\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ with $1\sigma$
confidence level for P4 parametrization.

The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ for
P4 parametrization are shown in Fig. 6. This figure shows strong correlation
between $M_{B}$ and $\eta_{0}$, and $\Omega_{k0}$ and $\eta_{1}$.

(a) Flat Universe

(b) Non-Flat Universe

Figure 6: The 1D and 2D posterior distributions of
$M_{\mathrm{B}},\leavevmode\nobreak\ \eta_{0},\leavevmode\nobreak\ \eta_{1}$
and $\Omega_{k0}$ for P4 parametrization.

###  4.2 Test of variability of SNe Type Ia absolute luminosity

In the first part, we find that the cosmic distance duality relation is not
violated at $2\sigma$ confidence level in four parametrizations of $\eta(z)$
(P1, P2, P3 and P4).

Therefore, in this part we consider that the CDDR is valid and to check the
dependency of absolute magnitude on redshift, we vary $M_{B}$ with redshift in
four different ways in both a flat and a non-flat universe.

#### M1: ${M_{\mathrm{B}}(z)=M_{\mathrm{B0}}}$.

In the first parametrization of $M_{\mathrm{B}}$ we consider it as a constant
parameter. The best fit values of $M_{\mathrm{B}}$ and $\Omega_{k0}$
parameters are given in Table 6.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B0}}$ | $-19.390^{+0.015}_{-0.015}$ | $-19.393^{+0.015}_{-0.015}$
$\Omega_{k0}$ | — | $0.075^{+0.104}_{-0.103}$
Table 6:  The best fit values of $M_{\mathrm{B0}}$ and $\Omega_{k0}$ with
$1\sigma$ confidence level for M1 parametrization.

From table 6, we find that the best fit value of $M_{\mathrm{B}}$ is the same
in both flat and non-flat universes within $1\sigma$ confidence level.
Further, the best fit value of $\Omega_{k0}=0.075^{+0.104}_{-0.103}$ suggests
a flat universe at $1\sigma$ confidence level.

The 1D and 2D posterior distributions of $M_{\mathrm{B}}$ and $\Omega_{k0}$
for M1 parametrization are shown in Fig. 7. This figure shows that there is no
correlation between $M_{B}$ and $\Omega_{k0}$.

(a) Flat Universe

(b) Non-Flat Universe

Figure 7: The 1D and 2D posterior distributions of $M_{\mathrm{B0}}$ and
$\Omega_{k0}$ for M1 parametrization.

#### M2: ${M_{\mathrm{B}}(z)=M_{\mathrm{B0}}+M_{\mathrm{B1}}z}$.

In this parametrization, we consider $M_{\mathrm{B}}$ as a function of
redshift. The best fit values of $M_{\mathrm{B0}},\leavevmode\nobreak\
M_{\mathrm{B1}}$ and $\Omega_{k0}$ are given in Table 7.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B0}}$ | $-19.391^{+0.016}_{-0.016}$ | $-19.376^{+0.018}_{-0.019}$
$M_{\mathrm{B1}}$ | $0.005^{+0.021}_{-0.021}$ | $-0.152^{+0.089}_{-0.091}$
$\Omega_{k0}$ | — | $0.823^{+0.471}_{-0.450}$
Table 7:  The best fit values of $M_{\mathrm{B0}},\leavevmode\nobreak\
M_{\mathrm{B1}}$ and $\Omega_{k0}$ with $1\sigma$ confidence level for M2
parametrization.

In case of a flat universe, we don’t find any signal of the redshift evolution
of absolute magnitude with $1\sigma$ confidence level. Similarly, in a non-
flat universe, there is no redshift dependence of the absolute magnitude at
$2\sigma$ confidence level. Further, the result supports a flat universe at
$2\sigma$ confidence level.

Figure 8 illustrates the 1D and 2D posterior distributions of
$M_{\mathrm{B0}},M_{\mathrm{B1}}$ and $\Omega_{k0}$ for M2 parametrization. In
this figure, the plot for a non-flat universe indicates a mild correlation
between absolute magnitude and cosmic curvature parameter.

(a) Flat Universe

(b) Non-Flat Universe

Figure 8: The 1D and 2D posterior distributions of $M_{\mathrm{B0}}$,
$M_{\mathrm{B1}}$ and $\Omega_{k0}$ for M2 parametrization.

#### M3: ${M_{\mathrm{B}}(z)=M_{\mathrm{B0}}+M_{\mathrm{B1}}\dfrac{z}{1+z}}$.

In this parametrization, we choose $M_{\mathrm{B}}$ as a function of redshift
which converges at high redshift. The best fit values of
$M_{\mathrm{B0}},\leavevmode\nobreak\ M_{\mathrm{B1}}$ and $\Omega_{k0}$ are
given in Table 8.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B0}}$ | $-19.390^{+0.017}_{-0.017}$ | $-19.380^{+0.018}_{-0.018}$
$M_{\mathrm{B1}}$ | $0.001^{+0.038}_{-0.040}$ | $-0.111^{+0.082}_{-0.083}$
$\Omega_{k0}$ | — | $0.343^{+0.214}_{-0.225}$
Table 8:  The best fit values of $M_{\mathrm{B0}},\leavevmode\nobreak\
M_{\mathrm{B1}}$ and $\Omega_{k0}$ with $1\sigma$ confidence level for M3
parametrization.

In this parametrization, the best fit value of $M_{\mathrm{B1}}$ in both a
flat and a non-flat universe suggests that the absolute magnitude does not
evolve with redshift at $2\sigma$ confidence level. Furthermore, there is no
indication of deviation from flat universe at $2\sigma$ confidence level yet
it mildly supports a non flat universe.

(a) Flat Universe

(b) Non-Flat Universe

Figure 9: The 1D and 2D posterior distributions of $M_{\mathrm{B0}}$,
$M_{\mathrm{B1}}$ and $\Omega_{k0}$ for M3 parametrization.

In both flat and non-flat universes, the 1D and 2D posterior distributions of
$M_{\mathrm{B0}}$, $M_{\mathrm{B1}}$ and $\Omega_{k0}$ are illustrated in
Figure 9. The plot for non-flat universe indicates a correlation between
absolute magnitude and cosmic curvature parameter.

#### M4: ${M_{\mathrm{B}}(z)=M_{\mathrm{B0}}+M_{\mathrm{B1}}\text{ln}(1+z)}$.

In this last parametrization, we choose $M_{\mathrm{B}}$ as a logarithmic
function of redshift. The best fit values of
$M_{\mathrm{B0}},\leavevmode\nobreak\ M_{\mathrm{B1}}$ and $\Omega_{k0}$ are
given in Table 9.

Parameter | Flat Universe | Non-Flat Universe
---|---|---
$M_{\mathrm{B0}}$ | $-19.391^{+0.017}_{-0.016}$ | $-19.380^{+0.017}_{-0.018}$
$M_{\mathrm{B1}}$ | $0.005^{+0.030}_{-0.029}$ | $-0.110^{+0.079}_{-0.078}$
$\Omega_{k0}$ | — | $0.442^{+0.282}_{-0.287}$
Table 9:  The best fit values of $M_{\mathrm{B0}},\leavevmode\nobreak\
M_{\mathrm{B1}}$ and $\Omega_{k0}$ with $1\sigma$ confidence level for M4
parametrization.

In this parametrization, for both a flat and a non-flat universe, the absolute
magnitude does not evolve with redshift at $2\sigma$ confidence level.
Furthermore, the best fit value of $\Omega_{k0}$ supports a flat universe at
$2\sigma$ confidence level.

(a) Flat Universe

(b) Non-Flat Universe

Figure 10: The 1D and 2D posterior distributions of $M_{\mathrm{B0}}$,
$M_{\mathrm{B1}}$ and $\Omega_{k0}$ for M4 parametrization.

In both flat and non-flat universes, the 1D and 2D posterior distributions of
$M_{\mathrm{B0}}$, $M_{\mathrm{B1}}$ and $\Omega_{k0}$ are illustrated in
Figure 10. The plot for non-flat universe indicates a correlation between
absolute magnitude and cosmic curvature parameter.

##  5 Discussion and Conclusions

In this work, we test the validity of the cosmic distance duality relation
which relates the luminosity distance to angular diameter distance via
redshift. We use the recent database of $1048$ Type Ia supernovae named
Pantheon for the luminosity distance. For the angular diameter distance, we
first reconstruct the Hubble parameter, $H(z)$ database of 31 datapoints from
Cosmic Chronometers in a model-independent way using the Gaussian Process.
Then by adopting the Planck prior on $H_{0}=67.66\pm
0.42\mathrm{\leavevmode\nobreak\ km}\mathrm{\leavevmode\nobreak\
s}^{-1}\mathrm{\leavevmode\nobreak\ Mpc}^{-1}$, the angular diameter distance
is obtained corresponding to the reconstructed $H(z)$. Further, in the
luminosity distance we have a free parameter, i.e. absolute magnitude of Type
Ia supernovae $(M_{\mathrm{B}})$, and we put constraints on it simultaneously
with other cosmological parameters i.e. $\Omega_{k0}$ and $\eta(z)$. Finally,
we allow $M_{\mathrm{B}}$ and $\eta(z)$ to evolve with redshift to check
whether these are constant quantities or evolve with cosmic time.
We divide this work into two parts as follows.

###  (A) Part I: Test of CDDR

In this part, we do not assume that cosmic distance duality is valid and to
test this, we consider a free parameter i.e. $\eta(z)$. We fit this parameter
simultaneously with $M_{\mathrm{B}}$ and $\Omega_{k0}$. To check the
dependency of the distance duality parameter on redshift, we assume four
parametrizations of $\eta(z)$. A brief summary of the results is as follows:

  * •

In the case of a flat universe, all parametrizations suggest that the cosmic
distance duality relation holds at $1\sigma$ confidence level.

  * •

For the non-flat universe, P1, supports the $\eta(z)=1$ at $1\sigma$
confidence level, P3 & P4 parametrizations at $2\sigma$ confidence level and,
P2 parametrization at $3\sigma$ confidence level. For this case, this
variation in the confidence level for the validity of $\eta(z)=1$ directly
indicates the correlation between the $\eta(z)$ and $\Omega_{k0}$. This
analysis also highlights that the results are sensitive to the choice of
parametrization and hence, it justifies our use of multiple parametrizations.

  * •

Consistently, for flat and non-flat cases, all the 2D contours between
$\eta_{0}$ and $M_{\mathrm{B}}$ in Part I of the analysis indicate a very
strong negative correlation. This points to the need of considering the
validity of cosmic distance duality relation in Part II to independently probe
the variability of the absolute magnitude $M_{\mathrm{B}}$.

  * •

In non-flat universe case, the parametrization, P1, strongly suggests a flat
universe within $1\sigma$ confidence level while parametrization P2, P3 and P4
also supports the flat universe within $2\sigma$ confidence level. Though the
best fit value of $\Omega_{k0}$ mildly prefer a non-flat universe.

###  (B) Part II: Test of variability of SNe Type Ia absolute luminosity

In the second part we assume that the cosmic distance duality relation is
valid, i.e. $\eta$=1. We are thus left with two parameters that we have to fit
simultaneously. These parameters are $M_{\mathrm{B}}$ and $\Omega_{k0}$. To
test the dependency of $M_{\mathrm{B}}$ on redshift, we consider four
parametrizations of $M_{\mathrm{B}}$. All four parametrizations have two model
parameters $M_{B0}$ and, $M_{B1}$. While analysing we choose $M_{B0}$ to be
constrained in the uniform prior range $U[-21,-17]$ and, $M_{B1}$ in the
uniform prior range $U[-2,+2]$. In each parametrization, we discuss two cases.
In the first case we consider a flat universe i.e. $\Omega_{k0}=0$ and in the
second case we choose a non-flat universe.

Our main conclusions of this part are listed below:

  * •

In the flat universe case, the parametrization $M1$, $M2$ and, $M4$, support
no evolution of absolute magnitude with redshift with $1\sigma$ confidence
level. Even for the parametrization $M3$, our analysis does not show any
signal of redshift evolution of absolute magnitude at $2\sigma$ confidence
level.

  * •

Similarly, in the non-flat universe case, for all parametrizations, no
redshift evolution has been found in our analysis in absolute luminosity
$M_{\mathrm{B}}$ at $2\sigma$ confidence level. Hence from our analysis in
Part I and Part II, we don’t find any indication towards a statistically
significant variability of absolute magnitude $(M_{\mathrm{B}})$ of type Ia
supernova.

  * •

For all parametrizations of the absolute magnitude $M_{\mathrm{B}}$, the best
fit value of $\Omega_{k0}$ suggests a flat universe at $2\sigma$ confidence
level. However, in the parametrizations $M2$,$M3$ and M4, the best fit value
of $\Omega_{k0}$ show mild preference for a non-flat universe. Further, from
the $1D$ and $2D$ contours of all four parametrizations of $M_{B}(z)$ for non-
flat case, we observed a negative correlation between the absolute magnitude
and cosmic curvature which should be analysed further.

It is important to note that in Part I, we fit the distance duality parameter
$\eta(z)$ along with $M_{B}$ and $\Omega_{k0}$. On the other hand, in Part II
we assume the validity of the cosmic distance duality relation i.e;
$\eta(z)=1$ and are thus left with only $M_{B}\leavevmode\nobreak\
\&\leavevmode\nobreak\ \Omega_{k0}$ as free parameters. Thus, the number of
free parameters to be probed decreases from four in Part I to three in Part
II. In Part I, our analysis suggests a strong correlation between the
$\eta(z)$ and $M_{B}$ values, which seems to have an impact on the $1\sigma$
error bars of $M_{B}$. However, in Part II, the $\eta(z)$ parameter has been
excluded, hence it results in tighter $1\sigma$ constraints on $M_{B}$. This
correlation between $\eta(z)$ and $M_{B}$ has also been highlighted using BAO
and Cluster observations as well [83].

###  (C) Impact of different prior values of Hubble Parameter ($H_{0}$)

In this analysis, we consider the Planck prior and discuss our results in two
parts. In the first part we test the CDDR and in the second part we test the
evolution of $M_{B}$. However, it is important to check whether our results
are sensitive to the chosen prior of $H_{0}$ or not. To check this sensitivity
on the chosen prior for $H_{0}$, we make two more prior choices of $H_{0}$
while reconstructing the angular diameter distance. These two priors are

  * •

No Prior on $H_{0}$. The reconstructed $H_{0}$ value using GP is $67.64\pm
4.79\mathrm{\leavevmode\nobreak\ km}\mathrm{\leavevmode\nobreak\
s}^{-1}\mathrm{Mpc}^{-1}.$

  * •

Planck Prior on $H_{0}=67.66\pm 0.42\mathrm{\leavevmode\nobreak\
km}\mathrm{\leavevmode\nobreak\ s}^{-1}\mathrm{Mpc}^{-1}$

  * •

SH0ES Prior on $H_{0}=73.20\pm 1.30\mathrm{\leavevmode\nobreak\
km}\mathrm{\leavevmode\nobreak\ s}^{-1}\mathrm{Mpc}^{-1}$.

To check the impact of the prior in our analysis, we repeat the whole analysis
for all parametrizations of $\eta(z)$ and $M_{B}$ with the three priors of
$H_{0}$ i.e. No prior, Planck prior and SH0ES prior [84]. In Table 10 and
Table 11, we show the results only for the third parametrization of $\eta(z)$
(i.e $P3$) and $M_{B}(z)$ ( i.e. $M3$) just to show their behaviour with
different priors . The remaining parametrizations in all three priors of
$H_{0}$ give the same conclusions as we find here in Table 10 and Table 11.

Parameter | No Prior | Planck Prior | SH0ES Prior
---|---|---|---
Case.1 Flat Universe
$M_{\mathrm{B}}$ | $-19.161^{+0.694}_{-0.590}$ | $-19.342^{+0.489}_{-0.369}$ | $-19.038^{+0.514}_{-0.454}$
$\eta_{0}$ | $0.927^{+0.265}_{-0.254}$ | $0.976^{+0.179}_{-0.197}$ | $0.910^{+0.206}_{-0.193}$
$\eta_{1}$ | $-0.016^{+0.008}_{-0.003}$ | $0.003^{+0.015}_{-0.015}$ | $-0.105^{+0.022}_{-0.031}$
Case.2 Non-Flat Universe
$M_{\mathrm{B}}$ | $-19.493^{+0.433}_{-0.439}$ | $-19.368^{+0.432}_{-0.335}$ | $-19.133^{+0.563}_{-0.484}$
$\eta_{0}$ | $1.085^{+0.256}_{-0.196}$ | $0.993^{+0.166}_{-0.178}$ | $0.948^{+0.226}_{-0.217}$
$\eta_{1}$ | $-0.077^{+0.024}_{-0.016}$ | $-0.047^{+0.036}_{-0.031}$ | $-0.089^{+0.017}_{-0.008}$
$\Omega_{k0}$ | $0.132^{+0.152}_{-0.149}$ | $0.312^{+0.194}_{-0.204}$ | $-0.029^{+0.135}_{-0.135}$
Table 10:  The best fit values of $M_{\mathrm{B}},\leavevmode\nobreak\ \eta_{0},\leavevmode\nobreak\ \eta_{1}$ and $\Omega_{k0}$ with $1\sigma$ confidence level for three $H_{0}$ priors for P3 parametrization of Part I. Parameter | No Prior | Planck Prior | SH0ES Prior
---|---|---|---
Case.1 Flat Universe
$M_{\mathrm{B0}}$ | $-19.300^{+0.150}_{-0.155}$ | $-19.390^{+0.017}_{-0.017}$ | $-19.241^{+0.039}_{-0.039}$
$M_{\mathrm{B1}}$ | $-0.130^{+0.040}_{-0.040}$ | $0.001^{+0.038}_{-0.040}$ | $-0.268^{+0.040}_{-0.040}$
Case.2 Non-Flat Universe
$M_{\mathrm{B0}}$ | $-19.286^{+0.147}_{-0.167}$ | $-19.380^{+0.018}_{-0.018}$ | $-19.229^{+0.039}_{-0.039}$
$M_{\mathrm{B1}}$ | $-0.238^{+0.079}_{-0.078}$ | $-0.111^{+0.082}_{-0.083}$ | $-0.407^{+0.079}_{-0.080}$
$\Omega_{k0}$ | $0.342^{+0.213}_{-0.213}$ | $0.343^{+0.214}_{-0.225}$ | $0.382^{+0.192}_{-0.183}$
Table 11:  The best fit values of $M_{\mathrm{B0}},\leavevmode\nobreak\
M_{\mathrm{B1}}$ and $\Omega_{k0}$ with $1\sigma$ confidence level for three
$H_{0}$ priors for M3 parametrization of Part II.

Through careful analysis, we find that while reconstructing the $H(z)$ data,
the choice of the prior value of $H_{0}$ does not make a significant impact on
correlations among the parameters.
We also notice that the $1\sigma$ error bars of the $M_{B}$ also get affected
by the uncertainties of the $H_{0}$ prior chosen for the analysis. For
example, the uncertainty in the value of $H_{0}$ from CMB is $0.42$ while that
from SH0ES is $1.3$. We find that in the case of using the CMB prior on
$H_{0}$, the error bars of $M_{B}$ are much smaller as compared to those
obtained by using the SH0ES prior. It seems that the uncertainty in the prior
propagates in $M_{B}$ during the fitting analysis and affects the error bars
of $M_{B}$. Similar to cosmic probes like Cosmic Chronometers and Supernovae
Type Ia, other independent probes like strong gravitational lensing ( Time
Delay angular distance measure) have also seen that the $M_{B}$ values and
error bars are impacted by the associated uncertainties in the input priors,
data and parameter values [85]. Given the above mentioned observations, we can
conclude that along with the consideration of the $\eta(z)$ parameter, the
uncertainties in the prior value of $H_{0}$ also impacts the $1\sigma$ bounds
on the $M_{B}$.

Finally, we observe that even if we adopt different priors for $H_{0}$, the
absolute magnitude of SNe Type Ia does not show any redshift dependence in
both flat and non-flat universes. This is consistent with the expected non-
variability of $M_{B}$ up to $3\sigma$ confidence level and shows no deviation
from the observationally expected assumptions. Though our analysis supports a
flat universe but we observed the mild preference of best fit value of
$\Omega_{k0}$ towards a non flat universe. It should be noted that there is a
strong correlation between the absolute magnitude, the distance duality
parameter. In Part II, we observed mild correlation between absolute magnitude
and cosmic curvature as well. Any signal of deviation in one of these
parameters will impact the rest of the parameters and the underlying
assumptions. We expect that one can resolve these correlations among different
parameters by performing a similar analysis with a larger data set of
Supernovae and other observational probes. Upcoming surveys such as the Large
Synoptic Survey, the Wide Field Infrared Survey and survey from Large Synoptic
Survey Telescope (LSST) may assist us in detecting any deviations from the
standard supernova type Ia light-curve modeling as well as deviation from the
standard cosmological model, i.e. the $\Lambda$CDM model [86, 87, 88, 89].

## Acknowledgements

We are grateful to the anonymous reviewer for her/his very enlightening
remarks which have helped improve the paper. Darshan Kumar is supported by an
INSPIRE Fellowship under the reference number: IF180293, DST India. A.R. and
Darshan Kumar acknowledges facilities provided by the IUCAA Centre for
Astronomy Research and Development (ICARD), University of Delhi. In this work
some of the figures were created with  numpy [90] and matplotlib [91] Python
modules.

## References

  * [1] A. G. Riess _et al._ _Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant_.  AJ, 116, 1009, (1998).
  * [2] S. Perlmutter _et al._ _Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae_.  ApJ, 517, 565, (1999).
  * [3] P. J. Peebles and B. Ratra.  _The cosmological constant and dark energy_.  Rev. Mod. Phys., 75, 559, (2003).
  * [4] R. R. Caldwell and M. Kamionkowski.  _The Physics of Cosmic Acceleration_.  Annu. Rev. Nucl. Part. Sci., 59, 397, (2009).
  * [5] D. H. Weinberg, M. J. Mortonson, D. J. Eisenstein, C. Hirata, A. G. Riess and E. Rozo.  _Observational probes of cosmic acceleration_.  Phys. Rep., 530, 87, (2013).
  * [6] W. Hillebrandt and J. C. Niemeyer.  _Type Ia Supernova Explosion Models_.  Annu. Rev. Astron. Astrophys., 38, 191, (2000).
  * [7] M. M. Phillips.  _The Absolute Magnitudes of Type IA Supernovae_.  ApJ, 413, L105, (1993).
  * [8] R. Tripp.  _A two-parameter luminosity correction for Type IA supernovae_.  A&A, 331, 815, (1998).
  * [9] S. Jha, A. G. Riess and R. P. Kirshner.  _Improved Distances to Type Ia Supernovae with Multicolor Light-Curve Shapes: MLCS2k2_.  ApJ, 659, 122, (2007).
  * [10] J. Guy _et al._ _SALT2: using distant supernovae to improve the use of type Ia supernovae as distance indicators_.  A&A, 466, 11, (2007).
  * [11] A. Conley _et al._ _SiFTO: An Empirical Method for Fitting SN Ia Light Curves_.  ApJ, 681, 482–498, (2008).
  * [12] W. Zheng and A. V. Filippenko.  _An Empirical Fitting Method for Type Ia Supernova Light Curves: A Case Study of SN 2011fe_.  ApJ, 838, L4, (2017).
  * [13] D. M. Scolnic et al.  _The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample_.  ApJ, 859, 101, (2018).
  * [14] M. Childress _et al._ _Host Galaxy Properties and Hubble Residuals of Type Ia Supernovae from the Nearby Supernova Factory_.  ApJ, 770, 108, (2013).
  * [15] Y.-L. Kim, Y. Kang and Y.-W. Lee.  _Environmental Dependence of Type Ia Supernova Luminosities from the YONSEI Supernova Catalog_.  J. Korean Astron. Soc., 52, 181, (2019).
  * [16] J. S. Gallagher, P. M. Garnavich, N. Caldwell, R. P. Kirshner, S. W. Jha, W. Li, M. Ganeshalingam and A. V. Filippenko.  _Supernovae in Early-Type Galaxies: Directly Connecting Age and Metallicity with Type Ia Luminosity_.  ApJ, 685, 752, (2008).
  * [17] M. Hamuy, S. C. Trager, P. A. Pinto, M. M. Phillips, R. A. Schommer, V. Ivanov and N. B. Suntzeff.  _A Search for Environmental Effects on Type IA Supernovae_.  AJ, 120, 1479, (2000).
  * [18] B. S. Wright and B. Li.  _Type Ia supernovae, standardizable candles, and gravity_.  Phys. Rev. D, 97, 083505, (2018).
  * [19] P. S. Drell, T. J. Loredo and I. Wasserman.  _Type Ia Supernovae, Evolution, and the Cosmological Constant_.  ApJ, 530, 593, (2000).
  * [20] F. Combes.  _Properties of SN-host galaxies_.  New Astron. Rev., 48, 583, (2004).
  * [21] I. Zehavi, A. G. Riess, R. P. Kirshner and A. Dekel.  _A Local Hubble Bubble from Type Ia Supernovae?_ ApJ, 503, 483, (1998).
  * [22] A. Conley, R. G. Carlberg, J. Guy, D. A. Howell, S. Jha, A. G. Riess and M. Sullivan.  _Is There Evidence for a Hubble Bubble? The Nature of Type Ia Supernova Colors and Dust in External Galaxies_.  ApJ, 664, L13, (2007).
  * [23] M. Ishak, A. Upadhye and D. N. Spergel.  _Probing cosmic acceleration beyond the equation of state: Distinguishing between dark energy and modified gravity models_.  Phys. Rev. D, 74, 043513, (2006).
  * [24] M. Kunz and D. Sapone.  _Dark Energy versus Modified Gravity_.  Phys. Rev. Lett., 98, 121301, (2007).
  * [25] E. Bertschinger and P. Zukin.  _Distinguishing modified gravity from dark energy_.  Phys. Rev. D, 78, 024015, (2008).
  * [26] A. Aguirre.  _Intergalactic Dust and Observations of Type Ia Supernovae_.  ApJ, 525, 583, (1999).
  * [27] M. Rowan-Robinson.  _Do Type Ia supernovae prove $\Lambda$$>$0?_ Mon. Not. Roy. Astron. Soc., 332, 352, (2002).
  * [28] A. Goobar, L. Bergström and E. Mörtsell.  _Measuring the properties of extragalactic dust and implications for the Hubble diagram_.  A&A, 384, 1, (2002).
  * [29] A. Goobar, S. Dhawan and D. Scolnic.  _The cosmic transparency measured with Type Ia supernovae: implications for intergalactic dust_.  Mon. Not. Roy. Astron. Soc., 477, L75, (2018).
  * [30] A. Avgoustidis, L. Verde and R. Jimenez.  _Consistency among distance measurements: transparency, BAO scale and accelerated expansion_.  JCAP, 06, 012, (2009).
  * [31] A. Avgoustidis, C. Burrage, J. Redondo, L. Verde and R. Jimenez.  _Constraints on cosmic opacity and beyond the standard model physics from cosmological distance measurements_.  JCAP, 08, 024, (2010).
  * [32] I. Tutusaus, B. Lamine and A. Blanchard.  _Model-independent cosmic acceleration and redshift-dependent intrinsic luminosity in type-Ia supernovae_.  A&A, 625, A15, (2019).
  * [33] I. Tutusaus, B. Lamine, A. Dupays and A. Blanchard.  _Is cosmic acceleration proven by local cosmological probes?_ A&A, 602, A73, (2017).
  * [34] Y. Kang, Y.-W. Lee, Y.-L. Kim, C. Chung and C. H. Ree.  _Early-type Host Galaxies of Type Ia Supernovae. II. Evidence for Luminosity Evolution in Supernova Cosmology_.  ApJ, 889, 8, (2020).
  * [35] E. D. Valentino, S. Gariazzo, O. Mena and S. Vagnozzi.  _Soundness of dark energy properties_.  JCAP, 07, 045, (2020).
  * [36] D. Sapone, S. Nesseris and C. A. P. Bengaly.  _Is there any measurable redshift dependence on the SN Ia absolute magnitude?_ Phys. Dark Universe, 32, 100814, (2021).
  * [37] V. Vavryčuk.  _Universe opacity and Type Ia supernova dimming_.  Mon. Not. Roy. Astron. Soc., 489, L63, (2019).
  * [38] G. F. R. Ellis.  _On the definition of distance in general relativity: I. M. H. Etherington (Philosophical Magazine ser. 7, vol. 15, 761 (1933))_.  Gen. Relativ. Gravit., 39, 1047, (2007).
  * [39] R. F. L. Holanda, J. A. S. Lima and M. B. Ribeiro.  _Testing the distance–duality relation with galaxy clusters and type Ia supernovae_.  ApJ, 722, L233, (2010).
  * [40] J. A. S. Lima, J. V. Cunha and V. T. Zanchin.  _Deformed Distance Duality Relations and Supernovae Dimming_.  ApJ, 742, L26, (2011).
  * [41] Z. Li, P. Wu and H. Yu.  _Cosmological-model-independent tests for the distance-duality relation from Galaxy Clusters and Type Ia Supernova_.  ApJ, 729, L14, (2011).
  * [42] R. S. Gonçalves, R. F. L. Holanda and J. S. Alcaniz.  _Testing the cosmic distance duality with X-ray gas mass fraction and supernovae data_.  Mon. Not. Roy. Astron. Soc., 420, L43, (2011).
  * [43] X.-L. Meng, T.-J. Zhang, H. Zhan and X. Wang.  _Morphology of Galaxy Clusters: A Cosmological Model-Independent Test of the Cosmic Distance-Duality Relation_.  ApJ, 745, 98, (2012).
  * [44] R. F. L. Holanda, R. S. Gonçalves and J. S. Alcaniz.  _A test for cosmic distance duality_.  JCAP, 06, 022, (2012).
  * [45] X. Yang, H.-R. Yu, Z.-S. Zhang and T.-J. Zhang.  _An improved method to test the distance-duality relation_.  ApJ, 777, L24, (2013).
  * [46] N. Liang, Z. Li, P. Wu, S. Cao, K. Liao and Z.-H. Zhu.  _A consistent test of the distance–duality relation with galaxy clusters and Type Ia Supernovae_.  Mon. Not. Roy. Astron. Soc., 436, 1017, (2013).
  * [47] A. Shafieloo, S. Majumdar, V. Sahni and A. A. Starobinsky.  _Searching for systematics in SNIa and galaxy cluster data using the cosmic duality relation_.  JCAP, 04, 042, (2013).
  * [48] Y. Zhang.  _Reconstruct the Distance Duality Relation by Gaussian Process_.  arXiv:1408.3897, (2014).
  * [49] S. S. da Costa, V. C. Busti and R. F. L. Holanda.  _Two new tests to the distance duality relation with galaxy clusters_.  JCAP, 10, 061, (2015).
  * [50] R. Nair, S. Jhingan and D. Jain.  _Observational cosmology and the cosmic distance duality relation_.  JCAP, 05, 023, (2011).
  * [51] Z. Chen, B. Zhou and X. Fu.  _Testing the Distance-Duality Relation from Hubble, Galaxy Clusters and Type Ia Supernovae Data with Model Independent Methods_.  Int. J. Theor. Phys., 55, 1229, (2015).
  * [52] R. F. L. Holanda, V. C. Busti and J. S. Alcaniz.  _Probing the cosmic distance duality with strong gravitational lensing and supernovae Ia data_.  JCAP, 02, 054, (2016).
  * [53] A. Rana, D. Jain, S. Mahajan and A. Mukherjee.  _Revisiting the Distance Duality Relation using a non-parametric regression method_.  JCAP, 07, 026, (2016).
  * [54] K. Liao, Z. Li, S. Cao, M. Biesiada, X. Zheng and Z.-H. Zhu.  _The Distance Duality Relation from Strong Gravitational Lensing_.  ApJ, 822, 74, (2016).
  * [55] R. F. L. Holanda and K. N. N. O. Barros.  _Searching for cosmological signatures of the Einstein equivalence principle breaking_.  Phys. Rev. D, 94, 023524, (2016).
  * [56] R. F. L. Holanda, S. H. Pereira and S. S. da Costa.  _Searching for deviations from the general relativity theory with gas mass fraction of galaxy clusters and complementary probes_.  Phys. Rev. D, 95, 084006, (2017).
  * [57] A. Rana, D. Jain, S. Mahajan, A. Mukherjee and R. F. L. Holanda.  _Probing the cosmic distance duality relation using time delay lenses_.  JCAP, 07, 010, (2017).
  * [58] H.-N. Lin, M.-H. Li and X. Li.  _New constraints on the distance duality relation from the local data_.  Mon. Not. Roy. Astron. Soc., 480, 3117, (2018).
  * [59] X. Fu, L. Zhou and J. Chen.  _Testing the cosmic distance-duality relation from future gravitational wave standard sirens_.  Phys. Rev. D, 99, 083523, (2019).
  * [60] C.-Z. Ruan, F. Melia and T.-J. Zhang.  _Model-independent Test of the Cosmic Distance Duality Relation_.  ApJ, 866, 31, (2018).
  * [61] R. F. L. Holanda, L. R. Colaço, S. H. Pereira and R. Silva.  _Galaxy cluster Sunyaev-Zel'dovich effect scaling-relation and type Ia supernova observations as a test for the cosmic distance duality relation_.  JCAP, 06, 008, (2019).
  * [62] J. Chen.  _Testing the distance-duality relation with the baryon acoustic oscillations data and type Ia supernovae data_.  Commun. Theor. Phys., 72, 045401, (2020).
  * [63] X. Zheng, K. Liao, M. Biesiada, S. Cao, T.-H. Liu and Z.-H. Zhu.  _Multiple Measurements of Quasars Acting as Standard Probes: Exploring the Cosmic Distance Duality Relation at Higher Redshift_.  ApJ, 892, 103, (2020).
  * [64] D. Kumar, D. Jain, S. Mahajan, A. Mukherjee and N. Rani.  _Constraining cosmological and galaxy parameters using strong gravitational lensing systems_.  Phys. Rev. D, 103, 063511, (2021).
  * [65] J. Hu and F. Y. Wang.  _Testing the distance–duality relation in the Rh = ct universe_.  Mon. Not. Roy. Astron. Soc., 477, 5064, (2018).
  * [66] W. J. C. da Silva, R. F. L. Holanda and R. Silva.  _Bayesian comparison of the cosmic duality scenarios_.  Phys. Rev. D, 102, 063513, (2020).
  * [67] J. Magaña, M. H. Amante, M. A. Garcia-Aspeitia and V. Motta.  _The Cardassian expansion revisited: constraints from updated Hubble parameter measurements and type Ia supernova data_.  Mon. Not. Roy. Astron. Soc., 476, 1036, (2018).
  * [68] R. Jimenez and A. Loeb.  _Constraining Cosmological Parameters Based on Relative Galaxy Ages_.  ApJ, 573, 37, (2002).
  * [69] M. Moresco _et al._ _Improved constraints on the expansion rate of the Universe up to z $\sim$ 1.1 from the spectroscopic evolution of cosmic chronometers_.  JCAP, 08, 006, (2012).
  * [70] C. Zhang, H. Zhang, S. Yuan, S. Liu, T.-J. Zhang and Y.-C. Sun.  _Four new observational H(z) data from luminous red galaxies in the Sloan Digital Sky Survey data release seven_.  Res. Astron. Astrophys., 14, 1221, (2014).
  * [71] J. Simon, L. Verde and R. Jimenez.  _Constraints on the redshift dependence of the dark energy potential_.  Phys. Rev. D, 71, 123001, (2005).
  * [72] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S. A. Stanford.  _Cosmic chronometers: constraining the equation of state of dark energy. I:H(z) measurements_.  JCAP, 02, 008, (2010).
  * [73] M. Moresco.  _Raising the bar: new constraints on the Hubble parameter with cosmic chronometers at z $\sim$ 2_.  Mon. Not. Roy. Astron. Soc., 450, L16, (2015).
  * [74] M. Moresco _et al._ _A 6% measurement of the Hubble parameter at z $\sim$0.45: direct evidence of the epoch of cosmic re-acceleration_.  JCAP, 05, 014, (2016).
  * [75] Planck Collaboration _et al._ _Planck 2018 results. VI. Cosmological parameters_.  A&A, 641, A6, (2020).
  * [76] C. E. Rasmussen.  _Gaussian processes in machine learning_.  Summer school on machine learning, Springer, (2003).
  * [77] L. Kazantzidis and L. Perivolaropoulos.  _Hints of a local matter underdensity or modified gravity in the low z Pantheon data_.  Phys. Rev. D, 102, 023520, (2020).
  * [78] M. Chevallier and D. Polarski.  _Accelerating Universes with Scaling Dark Matter_.  Int. J. Mod. Phys. D, 10, 213, (2001).
  * [79] E. V. Linder.  _Exploring the Expansion History of the Universe_.  Phys. Rev. Lett., 90, 091301, (2003).
  * [80] H. K. Jassal, J. S. Bagla and T. Padmanabhan.  _WMAP constraints on low redshift evolution of dark energy_.  Mon. Not. Roy. Astron. Soc., 356, L11, (2005).
  * [81] D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman.  _emcee: The MCMC Hammer_.  Publ. Astron. Soc. Pac., 125, 306, (2013).
  * [82] D. Foreman-Mackey.  _corner.py: Scatterplot matrices in Python_.  J. Open Source Softw., 1, 24, (2016).
  * [83] H.-N. Lin, M.-H. Li and X. Li.  _New constraints on the distance duality relation from the local data_.  Monthly Notices of the Royal Astronomical Society, 480, 3117–3122, (2018).
  * [84] A. G. Riess, S. Casertano, W. Yuan, J. B. Bowers, L. Macri, J. C. Zinn and D. Scolnic.  _Cosmic Distances Calibrated to 1% Precision with Gaia EDR3 Parallaxes and Hubble Space Telescope Photometry of 75 Milky Way Cepheids Confirm Tension with $\Lambda$CDM_.  ApJ, 908, L6, (2021).
  * [85] X. Wen and K. Liao.  _Calibrating the standard candles with strong lensing_.  The European Physical Journal C, 80, (2020).
  * [86] LSST Science Collaboration _et al._ _LSST Science Book, Version 2.0_.  arXiv e-prints, arXiv:0912.0201, (2009).
  * [87] Ž. Ivezić _et al._ _LSST: From Science Drivers to Reference Design and Anticipated Data Products_.  ApJ, 873, 111, (2019).
  * [88] H. Zhan and J. A. Tyson.  _Cosmology with the Large Synoptic Survey Telescope: an overview_.  Rep. Prog. Phys., 81, 066901, (2018).
  * [89] T. Eifler _et al._ _Cosmology with the Roman Space Telescope - multiprobe strategies_.  Mon. Not. Roy. Astron. Soc., 507, 1746, (2021).
  * [90] T. E. Oliphant.  _A guide to NumPy_ , vol. 1, (Trelgol Publishing USA2006).
  * [91] J. D. Hunter.  _Matplotlib: A 2D graphics environment_.  Comput. Sci. Eng., 9, 90, (2007).