# Testing $\Lambda$CDM cosmology in a binned universe: anomalies in the
deceleration parameter

Erick Pastén  erick.contreras[at]postgrado.uv.cl Víctor H. Cárdenas
victor.cardenas[at]uv.cl Instituto de Física y Astronomía, Universidad de
Valparaíso, Gran Bretaña 1111, Valparaíso, Chile

###### Abstract

We study the reconstructed deceleration parameter splitting the data in
different redshift bins, fitting both a cosmographic luminosity distance and
also assuming a flat $\Lambda$CDM model, using the Pantheon+ sample of type Ia
supernova data (SNIA). We observe tensions $\sim 2\sigma-3\sigma$ for
different redshift and distance indicators if the full sample is used.
However, those tensions disappear when the SNIA at $z<0.008$ are removed. If
the data is splitted in 2 hemispheres according to our movement w.r.t CMB, a
strange $3.8\sigma$ tension appears in one of the samples between particular
redshift bins. Finally, considering posterior distribution as Gaussian,
general linear model prefers a positive slope for $q_{0}$ across redshift bins
opposed to a zero slope expected in a $\Lambda$CDM universe. We discuss
possible explanations for our results and the influence of lowest redshift
SNIA data in cosmological analysis.

##  I Introduction

$\Lambda$CDM is the current standard model in physical cosmology, assuming a
unique set of parameters that governs the evolution of the universe at the
present epoch. According to studies using the last Pantheon\+ SNIA compilation
Scolnic et al. (2022), Brout et al. (2022a), a non zero cosmological constant
$\Lambda$ is still preferred, leading to the scenario of an accelerated
universe with a negative deceleration parameter $q_{0}$ today. However, deeper
studies are necessary to extract information of the actual nature of dark
energy (DE) from the data.

Cosmic acceleration of the universe is one of the strongest and most important
astrophysical discoveries in the last years et al. and Project (1999), et al.
(1998). The use of the FLRW metric to describe the evolution of an isotropic
and homogeneous universe has been extremely successful to match observations
such as cosmic microwave background (CMB), SNIA and Baryon acoustic
oscillations (BAO), indicating a negative value for the “deceleration”
parameter $q_{0}$ explained by adding a cosmological constant $\Lambda$ in the
Einstein equations et al. (2014), et al. (2015), et al. (2016) et al. (2017),
Efstathiou and Lemos (2018), et al. (2018), Collaboration (2020). This
constant is ultimately related with an unknown form of energy that has been
called dark energy (DE) and represent one the most important challenges for
the astrophysics and physics in the XXI century (see Peebles and Ratra (2003),
Durrer (2011) for reviews). Although DE is accepted as the standard
explanation for the observations in cosmology, many theoretical Velten et al.
(2014), Weinberg (2000) and observational Valentino et al. (2021), et al.
(2021), Perivolaropoulos and Skara (2022) tensions remains, and the
questioning on the veracity of $\Lambda$CDM has becoming strong in the recent
years. According to ideas on inhomogeneous Celerier (2006), Alexander et al.
(2009), Cosmai et al. (2019), Pastén and Cárdenas (2022),Clarkson et al.
(2011) and tilted cosmologies Tsagas (2011)Tsagas and Kadiltzoglou (2015), the
late structure of the universe could influence strongly the interpretation of
cosmological data as we are not typical observers as Copernican Principle
state, which is related to being located in a under/over density and/or in the
middle of a bulk flow that is not in rest w.r.t the Hubble Flow.

Observational evidence points to the fact that locally the distribution of
matter alters the measurements, causing the homogeneity scale to be even above
300 Mpc. In the context of the LCDM model, this scale indicates that under
redshift 0.1 we should observe such effects. It was using type Ia supernovae
that the author of Zehavi et al. (1998) suggested the existence of a Hubble
Bubble, the idea of a local void. Therefore it is reasonable not to use data
at very low redshift, as in Kenworthy et al. (2019) for example. However,
Pantheon+ includes data at redshift as low as $z=0.0001$, so we expect to get
unusual behavior.

It is important then to study the cosmological data at different redshift and
for different directions to check the cosmological principle assumptions. A
key previous work is Asvesta et al. (2022) where the authors show that a
tilted Einstein-de Sitter universe can reproduce the recent acceleration of
the universe, without any DE component, just by simply taking into account
linear effects of peculiar motions. In this paper we study the local universe
using a cosmological differential approach, extracting the cosmological
deceleration parameter $q_{0}$ for different redshift bins of SNIA data. We
observe a qualitative and quantitative deviations from what is expected in a
$\Lambda$CDM universe. Therefore, we study possible causes for those
anomalies.

##  II Cosmographic analysis

The well known cosmographic Taylor expansion Visser (2004, 2005)up to the
third term or the luminosity distance is given by:

$D_{L}(z)=\frac{z}{H_{0}}+\frac{(1-q_{0})}{2H_{0}}z^{2}-\frac{(1-q_{0}-3q_{0}^{2}+p_{0})}{6H_{0}}z^{3},$
(1)

with $a(t)$ being the scale factor of the FLRW universe, $q_{0}$ is the
deceleration parameter $q=-\ddot{a}{a}/\dot{a}^{2}$ evaluated today and
$p_{0}=j_{0}-\Omega_{k}$, where $j=\dot{\ddot{a}}/aH^{3}$ is the jerk which is
degenerated with the curvature parameter and cannot be determined separately.
Usually it is argued that $q_{0}\sim-0.5$ from supernovae data Scolnic et al.
(2018) and this is interpreted as an acceleration of the universe due to the
presence of the mysterious DE. However, this interpretation changes if we
consider inhomogeneous or tilted cosmologies, in which case $q_{0}$ is
determined by the inhomogeneties rather than a cosmological constant.

Motivated by ideas on inhomogeneous and tilted cosmologies, we perform a
differential study binning the data for different redshift ranges, to
determine if the best fit of $q_{0}$ changes with redshift. In this case we
use

$D_{L}^{i}(z)=\frac{z}{H_{0i}}+\frac{(1-q_{0i})}{2H_{0i}}z^{2}-\frac{(1-q_{0i}-3q_{0i}^{2}+p_{0i})}{6H_{0i}}z^{3},$
(2)

for the $i$-th bin in the sample. The procedure for binning SNIA data is
explained in the following section. To perform a comparison, we use both the
cosmographic luminosity distance as well as the one for the flat $\Lambda CDM$
model. In this case the $q_{0i}$ parameter is the same for every redshift
given by:

$q_{0i}=\frac{1}{2}\Omega_{mi}-\Omega_{\Lambda i}=\frac{3}{2}\Omega_{mi}-1.$
(3)

The main difference is that the fit of $q_{0}$ from the cosmographic approach
needs to be performed for redshifts below $z\simeq 0.3$ as the Taylor
approximation crashes at higher values, while for the fit of $q_{0}$ using
$\Lambda$CDM we can use all the data for which the quantity of SNIA in those
redshift is significant.

We have used $z\sim 0.8$ as an upper redshift limit for $\Lambda$CDM
cosmology, as the number of data points per redshift bin for values larger
than $z=0.8$ falls rapidly to less than $40$ data points per bin, very low to
perform significant statistical analysis. We hope that this redshift threshold
increase in the future.

It is important to notice that the cosmographic approach is commonly described
as a fully independent model relation, but that is not exactly true. This
relation comes from the assumption that we are co-moving observers with the
so-called CMB rest-frame (which is also assumed to be in rest with respect to
the Hubble Flow). This means that possible effects of our relative motion are
reduced to redshift corrections on observer and the data due to peculiar
velocities.

If the data is corrected due to peculiar velocities, corrections has to been
made also over the luminosity distance et al. (2011). However, as pointed out
in some studies Tsagas (2011); Tsagas and Kadiltzoglou (2015), strong
cosmological implications could be hidden in this relative motion more than
just data corrections. If $z_{\odot}$ is the peculiar movement of our solar
system moving w.r.t. the CMB and $z_{sn}$ is the peculiar movement of each
supernovae, then the luminosity distance should be corrected to:

$\bar{D}_{L}(z,z_{\odot},z_{sn})=(1+z_{sn})^{2}(1+z_{\odot})D_{L}(z),$ (4)

where $\bar{D}_{L}$ is the corrected luminosity distance and $z$ is the
cosmological redshift. This redshift corrections should not modify
significantly the cosmological parameters estimation Davis et al. (2019).
However, as we are doing a redshift binned analysis, those corrections could
be more important at low redshift bins.

###  II.1 SNIA analysis: The Pantheon + Sample

The Pantheon + sample Scolnic et al. (2022), Brout et al. (2022a) is the
latest compilation of high and low redshifts SNIA and has a lot of
improvements on photometric uncertainties and systematic as well as peculiar
velocity corrections based on a velocity field reconstruction, improving the
quantity of data at low redshift. To extract the information on $q_{0}$ from
the data, we use the full Tripp Formulae for distance modulus:

$\mu_{obs}=m_{b}^{*}-M=m_{b}+\alpha x-\beta c+\Delta_{M}-M$ (5)

here, $m_{b}$ corresponds to the peak magnitude at time of B-band maximum, $M$
is the absolute B-band magnitude of a fiducial SNIA, $x$ and $c$ are light
curve shape and color parameters corrections, $\alpha$ is a coefficient of the
relation between luminosity and stretch, $\beta$ is a coefficient of the
relation between luminosity and colour, and $\Delta_{M}$ is a corrections
based on the mass of the host galaxy. This relation is used to compare with
the theoretical expectation of a particular model using standard statistical
analysis where:

$\mu=5\log_{10}{\frac{D_{L}}{10Mpc}}+25.$ (6)

In a $\Lambda$CDM universe we expect to obtain a uniform distribution values
for $q_{0}$ as any part of a homogeneous-isotropic universe should lead to the
same “ today” deceleration parameter. This expectation increases as the
corrected magnitudes in Pantheon + has been performed even allowing for
peculiar velocities corrections and redshift dependent nuisance parameters. As
we are interested just in the $q_{0}$ parameter, we marginalize over the
degenerated parameter involving $H_{0}$ and $M$ according to Conley et al.
(2010).

###  II.2 Binning

We follow two different approaches to study the $q_{0}$ distribution as a
function of redshift, searching for cumulative and local effects in the
Pantheon+ sample.

Figure 1: The 2 approaches used to describe $q0$ distribution using redshift
bins.

####  II.2.1 Redshift spheres

We begin with a sphere centered at $z=0$ of radius $z_{out}=0.01$ and we fit
$q_{0}$ for all supernovae inside this sphere. Then, we increase the radius of
the sphere in steps $\Delta z$ repeating the analysis for supernovae inside
the new sphere until we reach the radius $z_{out}=0.3$. This method allows us
to describe the behaviour of the local deceleration parameter and to search
where DE existence becomes noticeable. Also, this procedure has the advantage
that the number of data always increases from a initial value which is the
number of supernovae with $z\leq 0.01$. A similar approach was used in Colgain
(2019). However, we are not able to describe how $q_{0}$ behaves locally for
each redshift as this study search for cumulative effects.

It is well established the presence of a Hubble Bubble in low $z$ (see e.g.
Kenworthy et al. (2019)) leading to the usual removal of low redshift
supernovae with $z\leq 0.023$ in cosmological analysis as we can expect any
unusual cosmological inference at those redshift.

Because SNIA with low redshift induce a bias in the Hubble diagram we also
perform our study avoiding the use of data with $z<0.023$. Overall, it is
difficult to look for effects of the local structure in cosmology if we do not
consider the redshifts in which those effects are more powerful. Therefore, we
first consider the analysis using all supernovae and then remove the low
redshift ones to look for variations.

####  II.2.2 Redshift layers

The second approach is to perform the analysis in layers of redshift. For
example, the studies performed in Kazantzidis and Perivolaropoulos (2020);
Dainotti et al. (2021, 2022), use this type of layer selection. As the
previous method, we begin with a sphere this time of radius $z_{out}=0.1$.
Then we increase the radius in step of $\Delta z$ but this time we also
increase the inner radius in the same step. It means that the second fit is
performed in a layer of redshift $z\in[\Delta z,0.1+\Delta z]$, the second for
$z\in[2\Delta z,0.1+2\Delta z]$ and so on. This method is very useful to
search for local difference in $q_{0}$ at different redshift.

##  III Results

Figure 2: $q_{0}$ parameter using the layers approach. $z$ here is the mean
redshift of the bin.

###  III.1 Initial Qualitative test

We use the code Emcee Foreman-Mackey et al. (2013) to test the model against
the data: the Pantheon+ SNIA Scolnic et al. (2022). This is a pure python
implementation of the affine invariant ensemble sampler for Markov chain Monte
Carlo proposed by Goodman and Weare Goodman and Weare (2010). The results of
fitting $q_{0}$ using both the cosmographic luminosity and $\Lambda$CDM are
shown in Fig. (2) and (3) for the sphere and layer approaches respectively. To
look for insights in the cosmographic approach, we use a weakly uninformative
uniform prior for $q_{0}\sim U(-4,4)$ not restricting our analysis to any
cosmology. Qualitatively, we can observe a deviation from the uniform
distribution for $q_{0}$ expected from a $\Lambda$CDM universe. Also, it is
interesting to note that without a tight prior, MCMC samples tends to prefer
higher values for $q_{0}$ in the cosmographic Taylor approximation than the
ones allowed in $\Lambda$CDM at low redshifts.

Figure 3: $q_{0}$ parameter using the spheres approach. $z$ here is the upper
redshift of the sphere.

###  III.2 Quantifying possible anomalies in the layer approach

In the context of the $\Lambda$CDM model, we expect $q_{0}$ should take the
same value independent of $z$. To quantify the possible deviation of this
result, we explore by using bins in which qualitatively we see a deviation
from a uniform distribution for $q_{0}$. We have chosen the width of each bin
to show the most appreciable differences in the fit of $q_{0}$. The results
are summarized in table 1.

$z$ Bin | Number of SNIA | $q_{0}$
---|---|---
$0-0.15$ | 826 | $-0.087\begin{subarray}{c}+0.18\\\ -0.17\end{subarray}$
$0.008-0.15$ | 746 | $-0.453\begin{subarray}{c}+0.16\\\ -0.16\end{subarray}$
$0.05-0.2$ | 300 | $-0.457\begin{subarray}{c}0.17\\\ -0.16\end{subarray}$
$0.1-0.25$ | 346 | $-0.501\begin{subarray}{c}0.18\\\ -0.17\end{subarray}$
$0.15-0.3$ | 381 | $-0.61\begin{subarray}{c}0.16\\\ 0.15\end{subarray}$
Table 1: Values for $q_{0}$ fit in wide redshift bins using the cosmographic
Taylor expansion of the luminosity distance. Figure 4: Gaussian distribution
of $q_{0}$ cosmographic parameter for different redshift bins. Between
$[0,0.15]$ (green) and $[0.15,0.3]$ (orange) appears a tension of $2.2\sigma$
considering independent realizations. The tension almost disappear when
supernovae below $z<0.008$ are removed, reaching a value $\sim 0.7\sigma$
(dashed line).

Assuming independent realizations, we observe that $q_{0}$ has a tension of
$\sim 2.2\sigma$ between the best fit values from supernovae at $z<0.15$ with
those at $z\in[0.15,0.3]$. Overall as data in the two analysis are correlated,
this tension could increase. However, if we remove the very low redshift SNIA
data at $z<0.08$ the results change, decreasing this tension to zero with a
value $\sim 0.7\sigma$ (Figure 4).

Figure 5: Best general linear model fits for $q_{0}(z)$ using $\Lambda$CDM
luminosity distance with Gaussian error $\sigma$ estimated as the mean of the
asymmetric errors from MCMC (Left) and estimated as the maximum value between
them (Right).

To check the consistency of the derived $q_{0}$ values with the expectations
in the $\Lambda$CDM model, we also performed a general linear model fit using
the package STATSMODELS Seabold and Perktold (2010). In this case we use:

$q_{0}(z)=Q_{m}z+Q_{0},$ (7)

and any deviation from a zero value for $Q_{m}$ is an indicator of a departure
from the $\Lambda$CDM model. For simplicity we approximate each fit for
$q_{0}(z)$ in the $\Lambda$CDM as being gaussian distributed, estimating the
error $\sigma$ as the average between the asymmetric errors obtained by Emcee
and also estimating $\sigma$ as the maximum between them. We do not consider
the first bin for $z\leq 0.008$ data. We also performed a fit with a zero
slope $Q_{m}=0$ model, in order to compare with the general non zero case.
Results are summarized in Table 2 and the best fit are plotted in figure 5.
According to the statistics, the fit tends to prefer a non-zero slope over a
constant value of $q_{0}$ for all redshifts. Overall, the fit is consistent
with a negative mean $q_{0}$ parameter.

Statistic | Best linear fit of SNIA | Constrained fit with zero slope
---|---|---
$Q_{m}$ |  $0.5341\pm 0.190$ ($0.4532\pm 0.173$) | 0
$Q_{0}$ |  $-0.5736\pm 0.071$ ($-0.5189\pm 0.066$) | $-0.3987\pm 0.037(-0.3699\pm 0.034)$
Log-Likelihood | $-9.4073(-5.0585)$ | $-13.241(-8.4040)$
Pearson Chi-Square | $35.7(26.0)$ | $40.6(29.1)$
Pseudo R-Square | $0.1260(0.1101)$ | $0.0001453(0.0001453)$
AIC | $22.81(14.11)$ | $28.48(18.8)$
BIC | $26.96(18.27)$ | $30.55(20.89)$
Table 2: Best linear fit parameters for $q_{0}$ using $\Lambda$CDM luminosity
distance estimating the Gaussian error $\sigma$ as the average between
asymmetric errors from MCMC. In parenthesis, same values but this time
estimating $\sigma$ as the maximum value between them.

###  III.3 Quantifying possible anomalies in the spheres approach

Using the “spheres” method, and from the results displayed in Fig.(3), we
observe a noticeable difference between the best fit for $q_{0}$ from the
cosmographic expansion and those from the $\Lambda$CDM model for redshift
below $z=0.1$. For example, restricting our analysis to data points below
$z=0.05$, we obtain the best fit values quoted in Table (3). Considering
gaussian distributions, we obtain a tension between the cosmographic distance
fit and the $\Lambda$CDM one of $\sim 2.8$ (Figure 6). Even by decreasing the
parameter space for the prior $q_{0}\sim U(-1,1)$, the cosmographic distance
still prefer a best fit $q_{0}>1$. Interestingly, in both cases the best fit
takes positive values of the deceleration parameter at low redshift $z<0.008$,
but the value changes dramatically when they are removed, making the tension
disappears.

Distance indicator | Number of SNIA | $q_{0}$
---|---|---
Cosmographic | 648(568) | $2.78\begin{subarray}{c}0.98\\\ -0.87\end{subarray}(-0.25\begin{subarray}{c}0.73\\\ -0.68\end{subarray})$
$\Lambda$CDM | 648(568) | $0.35\begin{subarray}{c}0.11\\\ -0.22\end{subarray}(-0.26\begin{subarray}{c}0.47\\\ -0.45\end{subarray})$
Table 3: Best fit values for $q_{0}$ in wide redshift bins using the
cosmographic Taylor expansion of the luminosity distance at $z<0.05$. In
parenthesis we quote the values obtained when we have removed the SNIa data
with $z<0.008$ Figure 6: Gaussian distribution of $q_{0}$ cosmographic
luminosity distance (green) and $\Lambda$CDM (orange) up to redshift $0.05$.
The cosmographic distance prefer higher values for $q_{0}$ even when the prior
is tight, giving a tension of $\sim 2.8\sigma$ between both distance
indicators. This tension disappears when the low redshift $z<0.008$ data is
removed (dashed line).

###  III.4 Local Structure Effects

Figure 7: $q0$ dipolar parameter using the layers approach.

In this section we consider a possible orientation dependence of the best fit
value for $q_{0}$ motivated by anomalies in the CMB dipole direction (see
Aluri et al.  for a review). Similar analysis have been performed in the past
with different data sets. Usually, a positive result of this orientation
dependence is interpreted as a local distribution structure effect on the
data. We perform a layer binned analysis with the cosmographic luminosity
distance for two hemispheres, oriented to the dipole of the CMB, the one
interpreted as our motion respect the CMB rest frame. To look for relations
between our movement w.r.t. the CMB and the $q_{0}$ parameter distribution, we
perform a layer binned analysis with the cosmographic luminosity distance for
two hemispheres separated by our direction of movement. We take the direction
reported in Planck Collaboration et al. (2020) corresponding to
$(263.99,48.26)$ in galactic coordinates to split the data in two groups
according to the two hemispheres of the CMB dipole. We call UP and DOWN
respectively to the blue-shifted and red-shifted hemispheres of the dipole.
Although there are others possible anomalous directions in cosmology, we
selected directly the CMB dipole direction as it is the principal standard
cosmological anisotropy (see Aluri et al. (2022) for a review). Also some
alternative cosmologies as tilted cosmologies predict an anisotropy in the
$q_{0}$ parameter close to this direction (see Tsagas (2011); Tsagas and
Kadiltzoglou (2015)).

The results are shown in Figure 7. Qualitatively, we can see two possible
sources of anomalies. First, a visual discrepancy of $q_{0}$ values between
the hemispheres at low redshift is observed. To analyze this, we fit $q_{0}$
for the two hemispheres at $z\in[0.008,0.075]$. Results are in Table 4. For
Gaussian distribution of parameters, we can see that the tension between
hemispheres is non-existent (Figure 8).

Notice that for $z<0.1$ the UP value for $q_{0}$ is lower than the DOWN one by
a factor of $1$, at more than one sigma. In the range $0.11<z<0.15$ the trend
is the opposite, being the UP value higher than the DOWN one. From $z>0.2$
considering the uncertainties, both hemispheres fit essentially the same value
for $q_{0}$, although with larger uncertainties.

Also only on the UP hemisphere, we note an increase in the $q_{0}$ parameter
comparing the redshift ranges lower than $z\sim 0.075$ and $z\sim 0.11-0.15$.
For example, in particular the bins $[0.0232,0.1232]$ and $[0.0754,0.1754]$
show the best fit parameters summarized in Table 5, giving a tension around
$3.8\sigma$ (Figure 9). However, at $z\sim 0.11-0.15$ there are only 23
supernovas, requiring more data to extract significant conclusions.

Figure 8: Gaussian distribution of $q_{0}$ cosmographic luminosity distance at low redshift $[0.008,0.075]$ for the 2 hemispheres. The tension between the fits doesn’t reach the unity. Figure 9: Gaussian distribution of $q_{0}$ cosmographic luminosity distance at $z\in[0.0232,0.1232]$ and $z\in[0.0754,0.1754]$ for the UP hemispheres. The tension between the posterior paremeter distributions reach 3.8 $\sigma$. $z$ bin | Number of SNIA | $q_{0}$
---|---|---
$0.008-0.075$(UP) | 314 | $-0.59\begin{subarray}{c}0.62\\\ -0.59\end{subarray}$
$0.008-0.075$(DOWN) | 314 | $0.19\begin{subarray}{c}0.62\\\ -0.57\end{subarray}$
Table 4: Values for $q_{0}$ fit at low redshift using the cosmographic Taylor expansion of the luminosity distance for the 2 hemispheres. $z$ bin | Number of SNIA | $q_{0}$
---|---|---
$0.0232-0.1232$(UP) | 214 | $-1.37\begin{subarray}{c}0.41\\\ -0.39\end{subarray}$
$0.0754-0.1754$(UP) | 92 | $1.69\begin{subarray}{c}0.68\\\ -0.67\end{subarray}$
Table 5: Values for $q_{0}$ fit in particular redshift bins using the
cosmographic Taylor expansion of the luminosity distance on the UP hemisphere.

##  IV Discussion

In this work we have found several unexpected behavior for the distribution of
the deceleration parameter $q_{0}$ values using the recent type Ia supernova
sample Pantheon+. First, analyzing the low redshift supernovae ($z<0.3$) we
obtain evidence for a local decelerating universe using the cosmographic
distance and considering the full sample of Pantheon+. However, this behaviour
disappears once the SNIA at redshift lower than $z=0.008$ are removed. A
tension of $\sim 2.2\sigma$ between the cosmographic expansion fit at
different redshift bins are observed when the full sample is considered, and
also a discrepancy of $\sim 2.8\sigma$ between the preferred values for
$q_{0}$ with the cosmographic expansion function and the full $\Lambda$CDM
one. Again, this behaviour disappears once the lowest redshift supernovae
(those with $z<0.008$) are removed.

At this point it is interesting to comment the result of Colgain (2019) where
the author find, using the Pantheon sample, the fit for $\Omega_{m0}$ varies
with the maximum redshift of the data used, finding a very low value between
$z\simeq 0.1$ to $z\simeq 0.15$, a result also found in Camarena and Marra
(2020), and even wiggles have been found Kazantzidis et al. (2020).

Also, assuming the $q_{0}(z)$ is a varying function of redshift, being
described by a linear model, the statistical analysis favours a non-zero slope
according to AIC and BIC criteria, showing a clear departure from the
$\Lambda$CDM universe. Finally, we performed a binned study in the hemispheres
defined by the CMB dipole. We did not found evidence for tensions between the
hemispheres. However, in the hemisphere with the positive direction of our
movement, we found a tension of $3.8\sigma$ between two particular redshift
bins. All results can be summarized as:

  1. 1.

When $z<0.008$ supernovaes are considered, a local decelerated universe in all
fits is observed, and tensions $\sim 2\sigma-3\sigma$ between $q_{0}$
parameters at different redshift and between different distance indicator
emerge. According to Brout et al. (2022b), the very nearby Hubble Diagram
shows a positive bias at $z<0.008$, attributed to measurement errors,
unmodeled peculiar velocities and the volumetric bias they induce. Moreover,
they suggest to avoid the use of data for $z<0.01$ and also in Riess et al.
(2022) the author neglect data below $z<0.023$ to measurements of the Hubble
Flow. Then, is therefore possible that the tensions found in $q_{0}$ comes
from the same origin. However, it is difficult to perform a deep analysis of
local large-structure effects if the lowest SNIA redshift are removed, as it
is possible that strong effects for the overall cosmology are mislead with
this approach. As is presented qualitatively in the spheres binned analysis,
the consideration of the lowest redshift SNIA fit a positive value for $q_{0}$
until $z\sim 0.09$ for the cosmographic luminosity distance. We conclude that
it is important to model the very nearby peculiar velocity field to extract
cosmological useful information of the most local universe with Pantheon+ as
is impossible to do it with the data as it is.

  2. 2.

Linear regression favours a non-zero slope for $q_{0}$ in the layer approach
with $\Lambda$CDM luminosity distance when the posterior distributions for
$q_{0}$ are assumed to be gaussians. This result is important because it is
observed even removing the lowest redshift supernovae, concluding that the
binned values $q_{0}(z)$ are not uniform across redshift bins. The
$\Lambda$CDM model predicts the same value for $q_{0}$ independent of redshift
bin chosen. We consider that a departure from this behaviour can be considered
as an evidence against $\Lambda$CDM. This method would become stronger as the
quantity of data increase, reducing the errors in the parameter extraction
from each redshift binned universe.

  3. 3.

Not evidence for tensions between UP and DOWN hemispheres. As previous
studies, we did not find evidence for anisotropies between the hemispheres
defined by the CMB dipole. In our analysis we used as redshift $z$ the value
$z_{HD}$ provided in the catalogue. However, there is an ongoing debate about
which is the correct reference frame to be used to find anisotropies in the
data. Overall, we do not suggest to use $z_{hel}$ as the frame of reference,
because the current cosmology and distance indicators are constructed for
observers that are in rest w.r.t. the Hubble flow, requiring a new theoretical
approach to analyse the data directly from observed redshift.

  4. 4.

$3.8\sigma$ tension for the UP hemisphere at particular redshift bins.
Intriguingly, in one of the hemispheres appears a high tension for the $q_{0}$
parameter at different redshift bins. If the data is correct, this could be
evidence for a local large structure effects and/or peculiar velocities that
are not considered in the current cosmological analysis.

###  IV.1 Interpretations

We summarized the principal results of our analysis in Pantheon+ survey
showing that $q_{0}$ parameter presents certain anomalies when is compared
with what we expect in a full $\Lambda$CDM universe. But, what this deviations
could mean for a cosmological perspective?

First, there is the possibility that just locally the universe could be
described as a $\Lambda$CDM universe and the global parameters
($\Omega_{M}\sim 0.3$, $\Omega_{\Lambda}\sim 0.7$, $\Omega_{K}\sim 0$) doesn’t
represent the behaviour of this universe in every place. This point of view is
a contradiction: if $\Lambda$CDM model doesn’t provide the same parameters in
every portion of the universe (in which homogeneity scale is valid), then the
universe is not $\Lambda$CDM after all. Different $q_{0}$ parameters in
different places of the universe could indicate that the dynamics of the
universe are not the same in every place. We had to remember that our analysis
is restricted to the Taylor approximation and a flat $\Lambda$CDM universe, so
it is possible that other parameters like curvature $\Omega_{k}$ are not the
same everywhere, biasing the local results on $q_{0}$. Again, this point of
view contradict the cosmological principle, promoting the construction and
development of new models. Similar evidence for evolution in cosmological
parameter with redshift, namely a decreasing trend for $H_{0}$ with an
increasing one in $\Omega_{m0}$, has been observed in several previous works
Dainotti et al. (2021, 2022); Colgain (2019); Wong et al. (2019); Millon et
al. (2020); Krishnan et al. (2020); Colgáin et al. (2022); Colgain et al.
(2022); Camarena and Marra (2020). Those are in agreement with the results
shown in Figure 5 as $\Omega_{m0}$ is directly related with $q_{0}$ by Eq. 3
and $H_{0}$ is anti-correlated with $\Omega_{m0}$ in $\Lambda$CDM.

We cannot neglect the idea that the discrepancies observed in the data are
attributed to systematics due to the combination of different surveys in the
Pantheon sample. According to Kazantzidis et al. (2020) there are also
“wiggles” in best $\Omega_{m0}$ values for Pantheon data. Presumably,
Pantheon+ performs better reducing those systematic anomalies but is possible
that still the data has ”wiggles”. Those systematic effects could potentially
explain the anomalies observed in our work, as we compare different surveys at
different redshift bins. Also it is necessary to perform the same analysis
made in this work over mock data to ensure that the results are not generated
by the binning procedure itself. As our analysis consider multiple binning and
a lot of different statistical test that requires expensive time computation,
we are preparing this analysis for a future work.

In a different approach, if the universe is not $\Lambda$CDM, then is valid to
ask if the $q_{0}$ value extracted from the cosmological luminosity distance
is indeed an acceleration; for example in homogeneous cosmology models as LTB,
$q_{0}$ is product of the radial inhomogeneities of our universe and not a
real cinematic quantity. Moreover, some fits of the cosmological Taylor
approximations predicts values of $q_{0}$ out of the range predicted by
$\Lambda$CDM cosmology ($q_{0}\in[-1,0.5]$), but we need more data to
confidently constrain those values.

Also, the possibility that just the data is not well treated has to be
considered. There has been wide criticism about the treatment of Supernovae IA
data as not considering possible magnitude evolution or the introduction of
redshift dependent nuisance parameters in order to account for correlations
between properties of SNIA and redshift. Those correlations are attributed
just to intrinsic properties of SNIA and not related to cosmology. Finally, we
state that any definitive conclusion is far from being strongly stated, as is
really important to get much more data in order to constrain better the
deviation of $q_{0}$ from what is expected in $\Lambda$CDM cosmology.

##  V Conclusions

We proposed two approaches to extract information about the local deceleration
parameter $q_{0}$ from different redshift binned universe, using the well
known cosmological Taylor expansion of the luminosity distance and also a flat
$\Lambda$CDM universe. The first accounting for a cumulative effect (spheres)
and the second to study differences in cosmological parameter for different
redshift bins (layers). We found that when $z<0.008$ supernovae are added to
cosmological analysis, tensions $\sim 2\sigma-3\sigma$ emerge between
different redshift bins and different luminosity distance indicator. We
suggest not to use lowest Supernovae IA in cosmological analysis, but we also
state the importance to improve the scientific utility of the data at lower
redshift as are necessary to understand the influence of the local large-scale
structure in cosmology. Also, using the cosmological Taylor expansion of the
luminosity distance, we observe a strong tension $\sim 3.9$ between particular
redshift bins, when we analyze the data on the hemisphere that the CMB dipole
points. However, significant anisotropies between the 2 hemispheres were not
observed. Finally, performing a simple general linear model fit against the
redshift binned $\Lambda$CDM parameters, we found that the $q_{0}(z)$ function
prefers a non-zero slope over a model with zero slope expected from a
$\Lambda$CDM universe. Those results has to be took carefully as the quantity
of data is not that big to state definitive conclusions. We proposed this
method as a test for $\Lambda$CDM cosmology, which could be improved as more
data is release.

Finally, we conclude that binned cosmology will be a promising tool to analyze
deeply the structure of our universe without assuming the cosmological
principle. Our analysis is similar to a recent paper Perivolaropoulos and
Skara (2023) – that use the Pantheon + sample – where the authors have shown a
$2$ to $3$ $\sigma$ tension between the low and high redshift best fit for the
SNIA absolute magnitude $M$. However they did not find variations in
$\Omega_{m0}$ and therefore not in $q_{0}$. As the quantity of data increase,
this tool could be performed jointly with anisotropic cosmology in order to
get the best insights of the deviations of our universe from $\Lambda$CDM
model.

## ACKNOWLEDGMENTS

EP acknowledge the support through a graduate scholarship ANID-Subdirección de
Capital Humano/Doctorado Nacional/2021-21210824. We wish thank Leandros
Perivolaropoulos for his comments on the low redshift volumetric bias on SNIA
data.

##  Appendix A On the inhomogeneity of redshift bins

There is the possibility that the inhomogeneity in the quantity of data in
each redshift bin affect our results. To study this concern, we seek for
correlations between $q_{0}$ best values and the quantity of data in each
redshift bin for Cosmographic and $\Lambda$CDM approach. Plots of $q_{0}$
against the quantity of data in each bin $n$ appears in Figure 10 for the 2
approaches. The correlation coefficients are $R=-0.168$ and $R=-0.201$
respectively, which pointed a slightly negative but statistically negligible
correlation. We can address the inhomogeneity in bins further studying $q_{0}$
over redshift bins with $z>0.008$ each one having $300$ supernovaes. Note that
despite of an homogeneous distribution of data across redshift bins, each
redshift space is not homogeneous anymore. To perform a good comparison, we
select the lower limits of the bin to be the same obtained in the layer
approach for cosmographic and $\Lambda$CDM luminosity distance, varying
therefore the upper limit. Results are plotted in Figure 11.

The principal problem with this approach, is that we are not able to study
higher redshift data isolated as those bins are extremely wider. For example
the last bin with 300 SNIA data is $z\in[0.40,1.54]$, covering more than
$\Delta z\sim 1$ space. Indeed, if we try to fit the general linear model over
$q_{0}(z)$ function, we cannot discriminate between zero or non-zero slope.
Results are plotted in Figure 12 and best statistics in Table 6. Finally, we
tried to find tensions at low redshift data with the cosmographic Taylor
expansion approach analogous as the results shown in Table 1 and Figure 4, but
this time using the homogeneous redshift bin. We found a tension of $\sim
1.7\sigma$ between 2 particular redshifts. Results are plotted in Figure 13
and details in Table 7. Note that SNIA with $z<0.008$ are not influencing this
result.

Figure 10: $q_{0}$ values against the number $n$ of data in each redshift bin
for the Cosmographic (left) and the $\Lambda$CDM (right) approach. The
correlation coefficients are $R=-0.168$ and $R=-0.201$ respectively.

Figure 11: $q_{0}$ values against $z$ for cosmographic and $\Lambda$CDM
luminosity distance in the layer approach using redshift bins with exactly
$300$ supernovaes each one. $z$ here is the mean redshift value of the bin.
Note that the bins are not equispaced anymore and the presence of the out-
layer value around $z\sim 0.03$ possible attributed to the influence of lower
redshift supernovaes even when $z<0.008$ are removed.

Figure 12: Best general linear model fits for $q_{0}(z)$ using homogeneous redshift bins, each one with $300$ SNIA. We used $\Lambda$CDM luminosity distance with Gaussian error $\sigma$ estimated as the mean of the asymmetric errors from MCMC (Left) and estimated as the maximum value between them (Right). Statistic | Best linear fit of SNIA | Constrained fit with zero slope
---|---|---
$Q_{m}$ |  $0.1970\pm 0.110$ ($0.1817\pm 0.108$) | 0
$Q_{0}$ |  $-0.5220\pm 0.053$ ($-0.5109\pm 0.052$) | $-0.4429\pm 0.030(-0.4383\pm 0.030)$
Log-Likelihood | $10.061(10.356)$ | $8.4402(8.9112)$
Pearson Chi-Square | $27.4(22.9)$ | $30.1(24.8)$
Pseudo R-Square | $0.08897(0.07949)$ | $0.0004160(0.0004160)$
AIC | $-16.12(-16.71)$ | $-14.88(-15.82)$
BIC | $-13.01(1-13.60)$ | $-13.32(-14.27)$
Table 6:  Best statistics for general linear model fits of $q_{0}$ using $\Lambda$CDM luminosity distance, estimating the Gaussian error $\sigma$ as the average between asymmetric errors from MCMC. In parenthesis, same values but this time estimating $\sigma$ as the maximum value between them. $z$ Bin | Number of SNIA | $q_{0}$
---|---|---
$0.0312-0.1364$ | 300 | $-0.19\begin{subarray}{c}+0.27\\\ -0.29\end{subarray}$
$0.0544-0.20549$ | 300 | $-0.50\begin{subarray}{c}+0.16\\\ -0.17\end{subarray}$
$0.1066-0.23684$ | 300 | $-0.52\begin{subarray}{c}+0.19\\\ -0.21\end{subarray}$
$0.2052-0.3221$ | 300 | $-0.815\begin{subarray}{c}+0.25\\\ -0.20\end{subarray}$
Table 7: Values for $q_{0}$ fit in wide homogeneous redshift bins using the
cosmographic Taylor expansion of the luminosity distance. Figure 13: Gaussian
distribution of $q_{0}$ cosmographic parameter for different redshift
homogeneous bins, each one of 300 SNIA. Between $[0.0312,0.1364]$ (blue) and
$[0.2052,0.3221]$ (orange) appears a tension of $1.78\sigma$ considering
independent realizations.

## References

  * Scolnic et al. (2022) D. Scolnic et al., Astrophys. J. 938, 113 (2022), eprint 2112.03863.
  * Brout et al. (2022a) D. Brout et al., Astrophys. J. 938, 110 (2022a), eprint 2202.04077.
  * et al. and Project (1999) S. P. et al. and T. S. C. Project, The Astrophysical Journal 517, 565 (1999), URL https://doi.org/10.1086/307221.
  * et al. (1998) A. G. R. et al., The Astronomical Journal 116, 1009 (1998), URL https://doi.org/10.1086/300499.
  * et al. (2014) B. M. et al., A&A 568, A22 (2014), URL https://doi.org/10.1051/0004-6361/201423413.
  * et al. (2015) A. et al. (BOSS Collaboration), Phys. Rev. D 92, 123516 (2015), URL https://link.aps.org/doi/10.1103/PhysRevD.92.123516.
  * et al. (2016) B. et al., Monthly Notices of the Royal Astronomical Society 461, 4099 (2016), ISSN 0035-8711, eprint https://academic.oup.com/mnras/article-pdf/461/4/4099/8112347/stw1584.pdf, URL https://doi.org/10.1093/mnras/stw1584.
  * et al. (2017) A. et al., Monthly Notices of the Royal Astronomical Society 470, 2617 (2017), ISSN 0035-8711, eprint https://academic.oup.com/mnras/article-pdf/470/3/2617/18315003/stx721.pdf, URL https://doi.org/10.1093/mnras/stx721.
  * Efstathiou and Lemos (2018) G. Efstathiou and P. Lemos, Monthly Notices of the Royal Astronomical Society 476, 151 (2018), ISSN 0035-8711, eprint https://academic.oup.com/mnras/article-pdf/476/1/151/24196952/sty099.pdf, URL https://doi.org/10.1093/mnras/sty099.
  * et al. (2018) D. M. S. et al., The Astrophysical Journal 859, 101 (2018), URL https://doi.org/10.3847/1538-4357/aab9bb.
  * Collaboration (2020) P. Collaboration, A&A 641, A6 (2020), URL https://doi.org/10.1051/0004-6361/201833910.
  * Peebles and Ratra (2003) P. J. E. Peebles and B. Ratra, Reviews of Modern Physics 75, 559 (2003), URL https://doi.org/10.1103%2Frevmodphys.75.559.
  * Durrer (2011) R. Durrer, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, 5102 (2011), URL https://doi.org/10.1098%2Frsta.2011.0285.
  * Velten et al. (2014) H. E. S. Velten, R. F. vom Marttens, and W. Zimdahl, The European Physical Journal C 74 (2014), URL https://doi.org/10.1140%2Fepjc%2Fs10052-014-3160-4.
  * Weinberg (2000) S. Weinberg, _The cosmological constant problems (talk given at dark matter 2000, february, 2000)_ (2000), URL https://arxiv.org/abs/astro-ph/0005265.
  * Valentino et al. (2021) E. D. Valentino, A. Melchiorri, and J. Silk, The Astrophysical Journal Letters 908, L9 (2021), URL https://dx.doi.org/10.3847/2041-8213/abe1c4.
  * et al. (2021) R. et al., _A comprehensive measurement of the local value of the hubble constant with 1 km/s/mpc uncertainty from the hubble space telescope and the sh0es team_ (2021), URL https://arxiv.org/abs/2112.04510.
  * Perivolaropoulos and Skara (2022) L. Perivolaropoulos and F. Skara, New Astron. Rev. 95, 101659 (2022), eprint 2105.05208.
  * Celerier (2006) M.-N. Celerier (2006).
  * Alexander et al. (2009) S. Alexander, T. Biswas, A. Notari, and D. Vaid, Journal of Cosmology and Astroparticle Physics 2009, 025 (2009), URL https://doi.org/10.1088%2F1475-7516%2F2009%2F09%2F025.
  * Cosmai et al. (2019) L. Cosmai, G. Fanizza, F. S. Labini, L. Pietronero, and L. Tedesco, Classical and Quantum Gravity 36, 045007 (2019), URL https://doi.org/10.1088%2F1361-6382%2Faae8f7.
  * Pastén and Cárdenas (2022) E. Pastén and V. H. Cárdenas, _A fractal ltb model cannot explain dark energy_ (2022), URL https://arxiv.org/abs/2212.04529.
  * Clarkson et al. (2011) C. Clarkson, G. Ellis, J. Larena, and O. Umeh, Reports on Progress in Physics 74, 112901 (2011), URL https://doi.org/10.1088%2F0034-4885%2F74%2F11%2F112901.
  * Tsagas (2011) C. G. Tsagas, Physical Review D 84 (2011), URL https://doi.org/10.1103%2Fphysrevd.84.063503.
  * Tsagas and Kadiltzoglou (2015) C. G. Tsagas and M. I. Kadiltzoglou, Physical Review D 92 (2015), URL https://doi.org/10.1103%2Fphysrevd.92.043515.
  * Zehavi et al. (1998) I. Zehavi, A. G. Riess, R. P. Kirshner, and A. Dekel, Astrophys. J. 503, 483 (1998), eprint astro-ph/9802252.
  * Kenworthy et al. (2019) W. D. Kenworthy, D. Scolnic, and A. Riess, The Astrophysical Journal 875, 145 (2019), URL https://doi.org/10.3847%2F1538-4357%2Fab0ebf.
  * Asvesta et al. (2022) K. Asvesta, L. Kazantzidis, L. Perivolaropoulos, and C. G. Tsagas, Mon. Not. Roy. Astron. Soc. 513, 2394 (2022), eprint 2202.00962.
  * Visser (2004) M. Visser, Classical and Quantum Gravity 21, 2603 (2004), URL https://doi.org/10.1088%2F0264-9381%2F21%2F11%2F006.
  * Visser (2005) M. Visser, General Relativity and Gravitation 37, 1541 (2005), URL https://doi.org/10.1007%2Fs10714-005-0134-8.
  * Scolnic et al. (2018) 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, et al., The Astrophysical Journal 859, 101 (2018), URL https://doi.org/10.3847%2F1538-4357%2Faab9bb.
  * et al. (2011) T. M. D. et al., The Astrophysical Journal 741, 67 (2011), URL https://doi.org/10.1088%2F0004-637x%2F741%2F1%2F67.
  * Davis et al. (2019) T. M. Davis, S. R. Hinton, C. Howlett, and J. Calcino, Monthly Notices of the Royal Astronomical Society 490, 2948 (2019), URL https://doi.org/10.1093%2Fmnras%2Fstz2652.
  * Conley et al. (2010) A. Conley, J. Guy, M. Sullivan, N. Regnault, P. Astier, C. Balland, S. Basa, R. G. Carlberg, D. Fouchez, D. Hardin, et al., The Astrophysical Journal Supplement Series 192, 1 (2010), URL https://doi.org/10.1088%2F0067-0049%2F192%2F1%2F1.
  * Colgain (2019) E. O. Colgain, Journal of Cosmology and Astroparticle Physics 2019, 006 (2019), URL https://doi.org/10.1088%2F1475-7516%2F2019%2F09%2F006.
  * Kazantzidis and Perivolaropoulos (2020) L. Kazantzidis and L. Perivolaropoulos, Phys. Rev. D 102, 023520 (2020), eprint 2004.02155.
  * Dainotti et al. (2021) M. G. Dainotti, B. D. Simone, T. Schiavone, G. Montani, E. Rinaldi, and G. Lambiase, The Astrophysical Journal 912, 150 (2021), URL https://doi.org/10.3847%2F1538-4357%2Fabeb73.
  * Dainotti et al. (2022) M. G. Dainotti, B. De Simone, T. Schiavone, G. Montani, E. Rinaldi, G. Lambiase, M. Bogdan, and S. Ugale, Galaxies 10 (2022), ISSN 2075-4434, URL https://www.mdpi.com/2075-4434/10/1/24.
  * Foreman-Mackey et al. (2013) D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, Publications of the Astronomical Society of the Pacific 125, 306 (2013), eprint 1202.3665.
  * Goodman and Weare (2010) J. Goodman and J. Weare, Communications in Applied Mathematics and Computational Science 5, 65 (2010).
  * Seabold and Perktold (2010) J. Seabold and J. Perktold, in _Proceedings of the 9th Python in Science Conference_ (2010).
  * (42) P. K. Aluri, P. Cea, P. Chingangbam, M.-C. Chu, R. G. Clowes, D. Hutsemékers, J. P. Kochappan, A. Krasiński, A. M. Lopez, L. Liu, et al., URL https://arxiv.org/abs/2207.05765.
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., Arroja, F., Ashdown, M., Aumont, J., Baccigalupi, C., Ballardini, M., Banday, A. J., Barreiro, R. B., et al., A&A 641, A1 (2020), URL https://doi.org/10.1051/0004-6361/201833880.
  * Aluri et al. (2022) P. K. Aluri, P. Cea, P. Chingangbam, M.-C. Chu, R. G. Clowes, D. Hutsemékers, J. P. Kochappan, A. M. Lopez, L. Liu, N. C. M. Martens, et al., _Is the observable universe consistent with the cosmological principle?_ (2022), URL https://arxiv.org/abs/2207.05765.
  * Camarena and Marra (2020) D. Camarena and V. Marra, Physical Review Research 2 (2020), URL https://doi.org/10.1103%2Fphysrevresearch.2.013028.
  * Kazantzidis et al. (2020) L. Kazantzidis, H. Koo, S. Nesseris, L. Perivolaropoulos, and A. Shafieloo, Monthly Notices of the Royal Astronomical Society (2020), URL https://doi.org/10.1093%2Fmnras%2Fstaa3866.
  * Brout et al. (2022b) D. Brout, D. Scolnic, B. Popovic, A. G. Riess, A. Carr, J. Zuntz, R. Kessler, T. M. Davis, S. Hinton, D. Jones, et al., The Astrophysical Journal 938, 110 (2022b), URL https://doi.org/10.3847%2F1538-4357%2Fac8e04.
  * Riess et al. (2022) A. G. Riess, W. Yuan, L. M. Macri, D. Scolnic, D. Brout, S. Casertano, D. O. Jones, Y. Murakami, G. S. Anand, L. Breuval, et al., The Astrophysical Journal Letters 934, L7 (2022), URL https://doi.org/10.3847%2F2041-8213%2Fac5c5b.
  * Wong et al. (2019) K. C. Wong, S. H. Suyu, G. C.-F. Chen, C. E. Rusu, M. Millon, D. Sluse, V. Bonvin, C. D. Fassnacht, S. Taubenberger, M. W. Auger, et al., Monthly Notices of the Royal Astronomical Society 498, 1420 (2019), URL https://doi.org/10.1093%2Fmnras%2Fstz3094.
  * Millon et al. (2020) M. Millon, A. Galan, F. Courbin, T. Treu, S. H. Suyu, X. Ding, S. Birrer, G. C.-F. Chen, A. J. Shajib, D. Sluse, et al., Astronomy & Astrophysics 639, A101 (2020), URL https://doi.org/10.1051%2F0004-6361%2F201937351.
  * Krishnan et al. (2020) C. Krishnan, E. Colgáin, Ruchika, A. Sen, M. Sheikh-Jabbari, and T. Yang, Physical Review D 102 (2020), URL https://doi.org/10.1103%2Fphysrevd.102.103525.
  * Colgáin et al. (2022) E. Ó . Colgáin, M. Sheikh-Jabbari, R. Solomon, G. Bargiacchi, S. Capozziello, M. Dainotti, and D. Stojkovic, Physical Review D 106 (2022), URL https://doi.org/10.1103%2Fphysrevd.106.l041301.
  * Colgain et al. (2022) E. O. Colgain, M. M. Sheikh-Jabbari, R. Solomon, M. G. Dainotti, and D. Stojkovic, _Putting flat lcdm in the (redshift) bin_ (2022), URL https://arxiv.org/abs/2206.11447.
  * Perivolaropoulos and Skara (2023) L. Perivolaropoulos and F. Skara (2023), eprint 2301.01024.