# Constraining dark energy cosmologies with spatial curvature using Supernovae
JWST forecasting

Pablo M. Maldonado Alonso  Celia Escamilla-Rivera  and Rodrigo Sandoval-Orozco

###### Abstract

Recent cosmological tensions, in particular, to infer the local value of the
Hubble constant $H_{0}$, have developed new independent techniques to
constrain cosmological parameters in several cosmologies. Moreover, even when
the concordance Cosmological Constant Cold Dark Matter ($\Lambda$CDM) model
has been well constrained with local observables, its physics has shown
deviations from a flat background. Therefore, to explore a possible deviation
from a flat $\Lambda$CDM model that could explain the $H_{0}$ value in tension
with other techniques, in this paper we study new cosmological constraints in
spatial curvature dark energy models. Additionally, to standard current
Supernovae Type Ia (SNIa) catalogs, we extend the empirical distance ladder
method through an SNIa sample using the capabilities of the James Webb Space
Telescope (JWST) to forecast SNIa up to $z\sim 6$, with information on the
star formation rates at high redshift. Furthermore, we found that our
constraints provide an improvement in the statistics associated with
$\Omega_{m}$ when combining SNIa Pantheon and SNIa Pantheon+ catalogs with JW
forecasting data.

##  1 Introduction

The first direct evidence of the late time cosmic acceleration was obtained
through measurements of Type Ia Supernovae (SNIa) [1, 2]. Over the years,
subsequent observations confirmed this result, such as the cosmic microwave
background (CMB) [3], baryon acoustic oscillations (BAO) [4, 5], and weak
gravitational lensing [6]. However, the capability of supernovae to prove the
accelerating expansion remains invaluable since these objects are bright
enough to be seen at large distances. Furthermore, SNIa are common enough to
be found in large quantities, and their properties make them standardized with
a precision of $\sim 0.1$ mag in brightness or $\sim 5\%$ in distance per
object [7]. Also, the increasing number of SNIa observations has considerably
reduced the associated statistical errors and the uncertainties in estimating
cosmological parameters dominated by them [8, 9].

Nevertheless, the nature of this cosmic acceleration is one of the current
inquiries in precision cosmology since still we do not fully understand the
component with which it is associated, the _dark energy_. However, due to the
well-constrained $\Lambda$-Cold Dark Matter ($\Lambda$CDM) model, this dark
energy could be evidence for a component with a negative Equation–of–State
(EoS) constant value [10, 11, 12] or a dynamical EoS [13, 14, 15].
Furthermore, dark energy can be associated with components that can be derived
from first principles in alternative theories of gravity [16, 17, 18] and
extended theories of gravity [19, 20], showing a late cosmic acceleration.

On the nature of dark energy, several missions have been working to find
better cosmological constraints adjoint with better systematics and increasing
data baselines. Some of them as large-scale structure (LSS) observations with
measurements from the Dark Energy Survey (DES) [21], the Dark Energy
Spectroscopic Instrument (DESI) [22], the Legacy Survey of Space and Time
(LSST) on the Vera Rubin Observatory [23], and Euclid [24], among others, have
extended the concordance cosmological model to include EoS parameters of dark
energy with some shifts within 1$\sigma$.

In particular, the recently launched James Webb Space Telescope (JWST) is a
very interesting experiment that can help to elucidate the nature of dark
energy. JWST is a 6.5-meter primary mirror, space-based observatory operating
in the visible and infrared light equipped with four main science instruments
including a near-infrared camera, spectrograph, and imager slitless
spectrograph; and a mid-infrared camera and spectrograph [25]. It is expected
that it will have an estimated lifespan of 20 years in which the research will
be focused on several astrophysical and cosmology areas such as galaxy
formation in the early universe as [26, 27, 28, 29], exoplanet detection [30,
31, 32], metallicity and chemical exploration [33, 34, 35], and life detection
[36, 37]. All these potential and current observations could allow us to
explore physics further than before as testing dark energy models with
structure formation [38, 39], corroborate the Cepheid calibrations in the
distance ladder [40] and adding more SNIa observations [9], cosmic
chronometers [41], and XA/UV quasars [42, 43] to the constraint analysis of
them.

Recently, numerous studies related to the implications of JWST to cosmology
have been developed. In [9] a JWST simulated sample of SNIa within a redshift
range $2\lesssim z\lesssim 6$, was employed to constrain standard cosmological
parameters. Using combinations of the mock sample and SN Pantheon dataset [44]
it was possible to constrain dark energy models with constant EoS. This
analysis was performed using two different forms of the intrinsic evolution of
the standardized SNIa luminosity distance. On one hand, it is assumed a linear
redshift dependence of magnitude evolution. On the other hand, we can consider
other logarithmic evolutions. Analysing the cases with and without systematic
evolution, it was found that the addition of the simulated SNIa sample would
successfully remove the evolutionary effects. However, even though the SN
Pantheon dataset size was increased by a factor of $\approx$ 16 data points,
it is still not able to constrain the systematic evolution and the
cosmological parameters as effectively as the very high redshift SN data.

A further study about the first galaxies discovered by JWST was carried out in
[45]. According to several works [46, 47, 48, 49, 50, 51, 52], there are some
common aspects within the structure morphology, which indicate that galaxies
discovered by JWST do not have enough time to evolve into what it is observed
today. It is important to notice that this study is within the framework of
the standard cosmological $\Lambda$CDM model. Therefore, the new JWST dataset
includes near and mid-infrared images and near-infrared spectra to perform
analyses based on cosmographic theories of the angular size-redshift
relationship [45]. The $\Lambda$CDM interpretation of JWST observations is
compared with the interpretation based on Zwicky’s static universe model [53],
where the origin of the cosmological redshift can be explained through the
photon-energy loss mechanism. However, the redshifted objects detected by the
JWST are not aligned with such interpretation, but before any final
conclusion, data from this mission should be increased.

As a step forward, using the capabilities of the JWST described, in this work,
we develop the forecasting of SNIa up to $z\sim 6$, with information on the
Star Formation Rates (SFR) at high redshift. Once this data is at hand, we
perform a statistical analysis combined with SN Pantheon [44] and SN
Pantheon+111github.com/PantheonPlusSH0ES/DataRelease to constrain spatial
curvature dark energy cosmologies. We based our cosmological models inspired
by bidimensional EoS parameterisation, which preserves the expanding and
accelerating behaviour at late times. Our goal is to show that a simple
deviation in the spatial curvature of a dark energy EoS model can verify a
well-constrained analysis with SNIa JWST forecasting.

This paper is divided as follows: In Sec. 2 we summarise the theory behind
dark energy bidimensional parameterisations inspired in Taylor series around
the cosmological scale factor $a$. All of these parameterisations are
described through their normalised $E(z)$ Friedmann evolution equation,
including the curvature term. Furthermore, we are going to consider standard
$\Lambda$CDM and $w$CDM models in addition to the dark energy cosmologies to
proceed with comparisons between them. Also, we include the latest constraints
reported in the literature so far. In Sec. 3 we present the methodology
employed for observables. We include the description of current SNIa data
baselines and how we can proceed with their forecasting using JWST
characteristics. In Appendix A we describe the technicalities behind this
forecasting. The results on new constraints for the models described are
developed in Sec. 4. Finally, the conclusions are presented in Sec. 5.

##  2 Standard dark energy parameterisations

The standard cosmological scenario $\Lambda$CDM is a remarkable fit for most
cosmological data. Nonetheless, we are still searching for the nature of
inflation, dark matter, and dark energy. Physical evidence for these
components comes only from astrophysical and cosmological observations [54].
Therefore, an increase in experimental sensitivity can produce deviations from
the standard $\Lambda$CDM scenario that could lead to a deeper understanding
of the gravity theory. If it is not a consequence associated with systematic
errors, the cosmological tensions [55] existing between the different
experimental probes could indicate a failure of the $\Lambda$CDM model, and a
better cosmological model should be able to be found.

In this section, we are going to describe, in addition to the $\Lambda$CDM
model with EoS $w=-1$, five dark energy bidimensional in $z$ parameterisations
of $w(z)$, which can be constant or redshift-dependent. Notice that to
describe dark energy, we need to achieve cosmic acceleration with a negative
pressure at late times [11].

  * •

$\Lambda$CDM model. In this model, the universe is composed of cosmological
fluids with different EoS’s $w$ that contribute to the energy constraint. At
present cosmic times, the non-relativistic matter contribution,
$\Omega_{m}\simeq 0.27$, is the sum of the ordinary baryonic matter term,
$\Omega_{b}\simeq 0.044$, and the (cold) dark matter term, $\Omega_{c}\simeq
0.22$. Dark energy ($\Omega_{\Lambda}\simeq 0.73$) is described by $w=-1$,
associated with a cosmological constant $\Lambda$ or a vacuum energy density
[56]. Radiation represents a negligible contribution, $\Omega_{r}\simeq
9\times 10^{-5}$, but it dominated the early cosmic stages, after the end of
the inflationary stage and before matter-radiation decoupling [57].
Additionally, $\Lambda$CDM can be characterized with a flat geometry, which
corresponds to an energy density parameter,
$\Omega_{\Lambda}=1-\Omega_{m}-\Omega_{r}$, where the only parameter to be
constrained is $\Omega_{m}$. The cosmological evolution for this model can be
expressed as

$E(z)\equiv\dfrac{H(z)}{H_{0}}=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{r}(1+z)^{4}+\Omega_{\Lambda}},$
(2.1)

where $H_{0}$ is the Hubble constant today.

We also consider the non-flat $\Lambda$CDM cosmological model, as an extension
of the $\Lambda$CDM model but with curvature $k\neq 0$, with its constraint
equation as $\Omega_{k}=1-\Omega_{m}-\Omega_{r}-\Omega_{\Lambda}$, where
$\Theta=(\Omega_{m},\Omega_{\Lambda})$ is the vector with free parameters. The
evolution for this case can be written as

$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{r}(1+z)^{4}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}}.$
(2.2)

Then, the model-dependent luminous distance can be calculated according to the
$\Omega_{k}$ value [58]:

$D_{L}(z)=\left\\{\begin{array}[]{ll}\dfrac{c}{H_{0}}(1+z)\dfrac{\sinh{\left(\sqrt{\Omega_{k}}\int_{0}^{z}\dfrac{dz^{\prime}}{E(z^{\prime})}\right)}}{\sqrt{\Omega_{k}}},&\quad\quad\Omega_{k}>0\\\
\dfrac{c}{H_{0}}(1+z)\int_{0}^{z}\dfrac{dz^{\prime}}{E(z^{\prime})},&\quad\quad\Omega_{k}=0\\\
\dfrac{c}{H_{0}}(1+z)\dfrac{\sin{\left(\sqrt{-\Omega_{k}}\int_{0}^{z}\dfrac{dz^{\prime}}{E(z^{\prime})}\right)}}{\sqrt{-\Omega_{k}}},&\quad\quad\Omega_{k}<0,\end{array}\right.$
(2.3)

where $c$ the speed of light and $E(z)$ the background evolution equation of
the cosmological models.

The base test for $\Lambda$CDM is the analysis provided by the Planck
collaboration measuring CMB anisotropies finding the base parameters values
$\Omega_{m}=0.316\pm 0.008$ and $H_{0}=67.27\pm 0.6$ km s-1 Mpc-1 [59] in the
context of a flat $\Lambda$CDM. Using late-time data in [58] was found that
using SNIa, BAO, and a quasar sample the values are $\Omega_{m}=0.300\pm
0.012$, with a fixed $H_{0}=70$ km s-1 Mpc-1, while using a non-flat
$\Lambda$CDM model the results were $\Omega_{m}=0.364\pm 0.021$ and
$\Omega_{\Lambda}=0.829\pm 0.035$, finding a slight deviation from the flat
background. Furthermore, with a Cosmic Chronometers (CC – $H(z)$) sample it
was found that $H_{0}=66.7\pm 5.3$ km s-1 Mpc-1 and
$\Omega_{m}=0.33^{+0.08}_{-0.06}$ for the same flat model [60]. Using only
SNIa Pantheon+ compilation [7] the values for the flat $\Lambda$CDM model are
$H_{0}=73.6\pm 1.1$ km s-1 Mpc-1. The latter assumes a Gaussian prior with
$\Omega_{m}=0.334\pm 0.018$. While for the non-flat $\Lambda$CDM the results
are $\Omega_{m}=0.306\pm 0.057$ and $\Omega_{\Lambda}=0.625\pm 0.084$ in
concordance with the flat counterpart at $2\sigma$.

  * •

$w$CDM model. The simplest extension of the $\Lambda$CDM model is the one in
which $w\neq-1$, yet still constant in time meaning that $w(z)=w_{0}$. From
[9, 58] we express $E(z)$ for this model as,

$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}(1+z)^{3(1+w_{0})}}.$
(2.4)

Non-flat $w$CDM has $\Theta=(\Omega_{m},\Omega_{\Lambda},w_{0})$ as free
parameters. However, under flatness assumption, $\Omega_{k}$ = 0, so the only
free parameters are $\Theta=(\Omega_{m},w_{0})$. This model reduces to
$\Lambda$CDM when $w_{0}=-1$.

Using SNIa, BAO, and quasars the values obtained in [58] were:
$\Omega_{m}=0.369^{+0.022}_{-0.023}$ and $w_{0}=-1.283^{+0.094}_{-0.027}$,
corresponding to a deviation from the $\Lambda$CDM model in more than
$1\sigma$ range with the same fixed $H_{0}$ value. While using a non-flat
$w$CDM results in $\Omega_{m}=0.280^{+0.041}_{-0.037}$,
$\Omega_{\Lambda}=1.662^{+0.041}_{-0.048}$ and
$w_{0}=-0.667^{+0.024}_{-0.027}$, where a difference from a flat model is
reported of more than $3\sigma$ using only SNIa and quasars. Furthermore,
adding BAO to the quasar sample [61] results in $\Omega_{m}=0.31\pm 0.03$ and
$w_{0}=-1.00^{+0.14}_{-0.13}$, that is consistent with $\Lambda$CDM assuming a
Gaussian prior of $H_{0}=67.32\pm 4.7$ km s-1 Mpc-1. When using only SNIa [7]
the flat $w$CDM model gives $\Omega_{m}=0.309^{+0.063}_{-0.069}$,
$H_{0}=73.5\pm 1.1$ km s-1 Mpc-1 and $w_{0}=-0.90\pm 0.14$, returning the
confirmation for a $\Lambda$CDM model.

  * •

Chevallier–Polarski–Linder (CPL) model. One of the most used redshift-
dependent parameterisations corresponds to the Chevallier-Polarski-Linder [62,
63] proposal: $w(z)=w_{0}+w_{a}z/(1+z)$. In which, $w(z)=w_{0}+w_{a}$ at
$z=\infty$ and $w(z)=w_{0}$ at $z=0$, but it diverges in the future for
$z\rightarrow(-1)^{+}$. In this bidimensional model, $w_{0}$ denotes the dark
energy EoS today, and $w_{a}$ describes its evolution. This parameterisation
has several advantages including the well behaviour at high redshift, the
linear feature at low redshift, a simple physical interpretation, and the
accuracy in reconstructing a scalar field EoS [63]. The normalised Hubble
parameter for this model can be written as

$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}(1+z)^{3(1+w_{0}+w_{a})}\exp(\dfrac{-3w_{a}z}{1+z})}.$
(2.5)

For the non-flat and flat cases, we consider as free parameters ($\Omega_{m}$,
$\Omega_{\Lambda}$, $w_{0}$, $w_{a}$) and ($\Omega_{m}$, $w_{0}$, $w_{a}$),
respectively. This model can be reduced to $\Lambda$CDM with $w_{0}=-1$ and
$w_{a}=0$.

Using SNIa, BAO, and quasars in [58] was discussed a deviation from the
$\Lambda$CDM with $\Omega_{m}=0.354^{+0.032}_{-0.030}$, along with
$w_{0}=-1.323^{+0.103}_{0.112}$ and $w_{a}=0.745^{+0.483}_{-0.974}$ for a flat
CPL model, values that correspond to a confirmation of the $\Lambda$CDM model.
Using quasars and $H(z)$ measurements for a non-flat CPL parametrization it
was found that $\Omega_{m}=0.44\pm 0.10$, $\Omega_{k}=-0.36\pm 0.24$,
$H_{0}=71.8^{+4.6}_{-7.7}$, with $w_{0}=-1.2\pm 1.0$, and
$w_{a}=-5.0^{+9-0}_{-2.0}$ [64] showing a clear deviation of more than
$2\sigma$ from the flat $\Lambda$CDM model. Using SNIa [7] for the flat CPL
with a Gaussian prior in $H_{0}$ results in $H_{0}=73.3\pm 1.1$ km s-1 Mpc-1,
with $\Omega_{m}=0.403^{+0.054}_{-0.098}$, $w_{0}=-0.93\pm 0.15$ and
$w_{a}=-0.1^{+0.9}_{-2.0}$. This latter corresponds to a flat $\Lambda$CDM
confirmation. Furthermore, adding BAO, CMB to the previous SNIa sample results
in $H_{0}=67.41^{+0.52}_{-0.82}$ km s-1 Mpc-1,
$\Omega_{m}=0.316^{+0.009}_{-0.005}$ and $w_{0}=1.267^{+0.196}_{-0.191}$ and
$w_{a}=-3.771^{+2.113}_{-2.496}$ [58].

  * •

Jassal-Bagla-Padmanabhan (JBP) model. In [65] the parameterisation
$w(z)=w_{0}+w_{a}z/(1+z)^{2}$, in which $w(0)=w_{0}$, $w^{\prime}(0)=w_{1}$,
and $w(\infty)=w_{0}$ is presented based on trying to explain the accelerated
universe covering both CMB and the SNIa measurements. This model was proposed
to solve the high $z$ issues within the CPL parameterisation [11]. Using the
behaviour of this function allows us to have the same EoS at the present epoch
and high-$z$ with a rapid variation at small redshifts. Considering the
corresponding term related to curvature, the derivative expression for $E(z)$
for this model can be written as

$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}(1+z)^{3(1+w_{0})}\exp[\dfrac{3w_{a}z^{2}}{2(1+z)^{2}}]},$
(2.6)

where $\Theta=(\Omega_{m},\Omega_{\Lambda},w_{0},w_{a})$ for the non-flat JBP
model, and $\Theta=(\Omega_{m},w_{0},w_{a})$ for a flat JBP model.

Using SNIa, BAO, and quasars the JBP model is tested obtaining
$\Omega_{m}=0.354^{+0.032}_{-0.030}$, $w_{0}=-1.371\pm 0.141$ and
$w_{a}=1.127^{+1.293}_{-1.547}$ [58] which is considered a deviation from the
confirmation for $\Lambda$CDM for the $w_{0}$ value. In [66] using SNIa, BAO,
CMB and Gamma-Ray Bursts (GRB) it is obtained $\Omega_{m}=0.27\pm 0.03$ with
$w_{0}=-1.02\pm 0.04$ and $w_{a}=0.22\pm 0.23$ using a flat JBP model in
concordance with a flat $\Lambda$CDM at $1\sigma$.

  * •

Exponential model. In [67] was examined five one-parameter dark energy
parameterisations with several datasets. In particular, data from CMB
observations, Joint light-curve analysis from SNIa observations (JLA), BAO
distance measurements, and $H(z)$. It was concluded that the one-parameter
dark energy model can provide a solution to the $H_{0}$ tension between local
measurements and Planck indirect ones. Besides, it was found which of the five
models is better fitted to the data used. This model, relatively close to
$\Lambda$CDM, is the one with an EoS of the form:
$w(z)=w_{0}\exp[z/(1+z)]/(1+z)$, where $w(0)=w_{0}$ and $w(z)=0$, for both
$z=\infty$ and $z\rightarrow(-1)^{+}$. As a result, the normalised Hubble
parameter can be written as

$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}(1+z)^{3}\exp[3w_{0}\left(\exp(\dfrac{z}{1+z})-1\right)]}.$
(2.7)

For non-flat exponential model we have
$\Theta=(\Omega_{m},\Omega_{\Lambda},w_{0})$, while for flat exponential
model, $\Theta=(\Omega_{m},w_{0})$.

Using SNIa, quasars, and BAO was obtained for the Exponential model
$\Omega_{m}=0.359^{+0.023}_{-0.024}$ and $w_{0}=-1.271^{+0.092}_{-0.107}$ [58]
showing again a deviation from a flat $\Lambda$CDM. In [68] the exponential
model is constrained using CMB, SNIa, BAO, and measurements of the distance
using Hydrogen II galaxies resulting in $H_{0}=70.9\pm 7.0$ km s-1 Mpc-1,
$\Omega_{m}=0.284\pm 0.006$ and $w_{0}=-1.202^{+0.027}_{-0.026}$, imposing a
Gaussian local prior in $H_{0}$.

  * •

Barboza-Alcaniz (BA) model. In [69] was proposed a dark energy
parameterization given by $w(z)=w_{0}+w_{a}z(1+z)/1+z^{2}$. This is a well-
behaved function of redshift throughout the entire cosmic evolution,
$z\in[-1,\infty]$, with $w(z)=w_{0}+w_{a}$ for $z=\infty$ and $w(z)=w_{0}$,
when $z\rightarrow(-1)^{+}$. This smooth function allows to define regions on
the plane $(w_{0}-w_{1})$ associated with several classes of dark energy
models to exclude or confirm the models based on the constraints using
observational data. Thus, it was shown that both quintessence and phantom
behaviors have fully acceptable regimes. The $E(z)$ for non-flat case of this
model can be written as

$E(z)=\sqrt{\Omega_{m}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{\Lambda}(1+z)^{3(1+w_{0})}\left(1+z^{2}\right)^{\frac{3w_{a}}{2}}}.$
(2.8)

The free parameter sets for the non-flat BA model and flat BA model are
$\Theta=(\Omega_{m},\Omega_{\Lambda},w_{0},w_{a})$ and
$\Theta=(\Omega_{m},w_{0},w_{a})$, respectively. The analysis using BAO, SNIa
and quasars in [58] showed deviation from the flat $\Lambda$CDM as
$\Omega_{m}=0.307^{+0.044}_{-0.055}$, $w_{0}=-1.303^{+0.115}_{-0.106}$ and
$w_{a}=1.010^{+0.152}_{-0.466}$, meaning that the quasar sample is responsible
for a deviation from the standard model. In [66] using CMB, BAO, SNIa and GRB
results in $\Omega_{m}=0.28\pm 0.03$, $w_{0}=-1.13\pm 0.04$ and $w_{a}=0.37\pm
0.1$, finding a lower deviation from the $\Lambda$CDM model, but using only
BAO and CMB the standard model is recovered with $\Omega_{m}=0.29\pm 0.04$,
$w_{0}=-1.06\pm 0.11$ and $w_{a}=0.35\pm 0.12$.

##  3 Data treatment: observations and forecastings

In this section, we will perform the statistical analysis for the dark energy
models with and without curvature including three different datasets: SNIa
Pantheon and SNIa Pantheon+ samples, along with the extracted simulated data
from JWST.

  * •

Pantheon (PN) [44]: The Pantheon compilation is a combination of measurements
from SNIa distances combining both low and high redshifts from $z\sim 0.01$ up
to $z=2.26$. This sample has shown an improvement in the photometric
calibrations on the distance ladder through the light curve. This transforms
observable quantities into distances adding up to a total of 1048 data points.

  * •

Pantheon+ (PN+) [70, 71]: Pantheon+ is a collection of 18 different SNIa
samples based on the Pantheon compilation above described and by adding new
data points recollected from different surveys as: Foundation Supernova Survey
[72], the Swift Optical/Ultraviolet Supernova Archive (SOUSA) [73], the Lick
Observatory Supernova Search LOSS1 [74], the second sample LOSS2 [75], and DES
[70]. As a result, Pantheon+ consists of 1701 light curves of 1550 distinct
SNIa along a redshift of $z=$ 0.001 up to 2.26. For $H_{0}$, a value of
$73.30\pm 1.04$ km s-1 Mpc-1 is assumed. This sample is represented in Figure
1 in blue color. Cosmological parameter constraints have been carried out
using the affine-invariant ensemble sampler for Markov Chain Monte Carlo
(MCMC) module emcee222emcee.readthedocs.io/en/stable/ that uses random
generation numbers to explore the parameter space based on the probability
function $P\propto\exp(-\chi^{2}/2)$ to minimize the quantity

$\chi^{2}_{\text{SNIa}}(\Theta)=\Delta\mu^{T}(z,\Theta)\cdot
C^{-1}_{\text{SNIa}}\cdot\Delta\mu(z,\Theta)+\ln{\left(\dfrac{S}{2\pi}\right)},$
(3.1)

where $\Delta\mu(z,\Theta)=\mu(z)_{\text{data}}-\mu(z,\Theta)_{\text{model}}$,
$C_{\text{SNIa}}$ is the covariance matrix of the PN (or PN+) sample, $S$ is
the sum of all the components of $C^{-1}_{\text{SNIa}}$,
$\mu(z)_{\text{data}}$ is the distance modulus of the PN (PN+) data, and
$\mu(z,\Theta)_{\text{model}}$ is the distance modulus for a cosmological
model with a parameter set $\Theta$ [76].

  * •

The extracted JWST simulated SN data (JW) [9]: Recently it was created an SNIa
sample at $z>2$ using a Monte Carlo simulation derived from the capabilities
of JWST and based on the FLARE project [77, 78]. The SNIa redshifts (up to
$z\sim 6$) were derived by extrapolating the local supernovae rates with
information on SFR at high redshift [79]. With this survey, in 6 years we
expect to measure 205 SNIa in a range of $z\sim 0.3$ up to $z\sim 6$. We
represent this sample in Figure 1. The mock sample was created based on a
flat-$\Lambda$CDM model using a fit from Pantheon data. To develop our
analysis, we changed the sample for SN PN+. The full covariance matrix is
constructed by adding the off-diagonal value of $0.002^{2}$ with + and – signs
assigned randomly for all the matrix. The value in the diagonal is the mean
value of the Pantheon covariance matrix [8] and the $0.15^{2}$ value in the
diagonal is the assumed error considering the local determination of the
observation errors extrapolated [9]. The covariance matrix has the following
form

$C_{\text{JW}}=\underbrace{\begin{pmatrix}\ 0.15^{2}&\pm 0.002^{2}&\pm
0.002^{2}&\dots&\pm 0.002^{2}\ \hbox{}\\\ \ \pm 0.002^{2}&0.15^{2}&\pm
0.002^{2}&\dots&\pm 0.002^{2}\ \hbox{}\\\ \ \pm 0.002^{2}&\pm
0.002^{2}&0.15^{2}&\dots&\pm 0.002^{2}\ \hbox{}\\\ \
\vdots&\vdots&\vdots&\ddots&\vdots\ \hbox{}\\\ \ \pm
0.002^{2}&\dots&\dots&\dots&0.15^{2}\ \\\
\end{pmatrix}}_{\begin{subarray}{c}\\\\[1.39998pt] \text{\normalsize$n\times
n$}\end{subarray}},$ (3.2)

where $n\times n$ denotes the dimension of the matrix and ($\pm$) the random
assignment of (+) and (–) signs. In this case,

$\chi^{2}_{\text{JW}}(\Theta)=\Delta\mu^{T}(z,\Theta)\cdot
C^{-1}_{\text{JW}}\cdot\Delta\mu(z,\Theta)+\ln{\left(\dfrac{S}{2\pi}\right)},$
(3.3)

where
$\Delta\mu(z,\Theta)=\mu(z)_{\text{data}}-\mu(z,\Theta)_{\text{model}}-\Gamma_{0}$,
$C_{\text{JW}}$ is the covariance matrix presented in Eq.(3.2), $S$ is the sum
of all the components of $C^{-1}_{\text{JW}}$, $\mu(z)_{\text{data}}$ is the
distance modulus of the JW data, $\mu(z,\Theta)_{\text{model}}$ is the
distance modulus for a cosmological model with parameter set $\Theta$ and
$\Gamma_{0}$ is a magnitude bias added to consider a possible systematic
luminosity difference between the JW and PN (PN+) datasets. Notice that if we
compare our Eq.(3.3) with [9], the analysis does not consider logarithmic
systematics in the forecasting.

For more technical details about the methodology followed here see Appendix A.

Figure 1: Left: Histogram of the Pantheon+ data (blue) and the extracted JW
(red). Right: Hubble Diagram of the Pantheon data (blue) and the extracted
mock JW (red).

##  4 Results: Cosmological constraints

In this section, we discuss the constraints for the dark energy models
previously described. In Tables 1 and 2 are reported the values for the
cosmological parameters involved in each flat and non-flat model,
respectively. Additionally, we used the results from the latest SH0ES
measurement for the Hubble constant [71] in which using the local distance
ladder via Cepheid calibration was obtained $H_{0}=73.04\pm 1.04$, and was
introduced as a fixed value for the Hubble constant in our analyses. Also, an
absolute magnitude $M=-19.263$ was assumed, except for flat $\Lambda$CDM where
$M$ was taken as a free parameter. The optimal constraints on the cosmological
parameters were derived using the emcee code. All Confidence Levels (C.L)
presented in this work correspond to 68.3 and 95 $\%$ i.e., to 1 and 2$\sigma$
respectively. Finally, in the presented results the value of $\Omega_{k}$ is
calculated directly from $\Omega_{k}=1-\Omega_{m}-\Omega_{\Lambda}$, combining
the marginalized distributions of each fractional density using a
getdist333getdist.readthedocs.io modified version.

###  4.1 $\Lambda$CDM model

The constraints for this model are given in Figure 2. As we can notice, using
the Pantheon sample gives a relatively lower value of $\Omega_{m}$ than using
the Pantheon+ sample for the flat-$\Lambda$CDM model. Comparing
$\Omega_{m}=0.290\pm 0.008$ with $\Omega_{m}=0.311^{+0.010}_{-0.009}$, shows a
tendency due to the addition of JW data. Furthermore, considering the non-flat
model results in a higher $\Omega_{m}$ estimation for the Pantheon+ sample.

Figure 2: 1-2$\sigma$ C.L results for the $\Lambda$CDM model using SNIa
Pantheon–PN (red color), SNIa Pantheon+–PN+ (orange color), SNIa Pantheon & JW
mock data – PN+JW (green color) and SNIa Pantheon+ & JW mock data – PN++JW
(purple color) baselines: Left: Flat case. Right: Non-flat case. Notice that
the C.L for $M$ is not associated with $\Gamma_{0}$.

The non-flat $\Lambda$CDM model constrained by JW simulated data tends to
reduce the curvature estimation towards flatness $\Omega_{k}\sim 0$. Let us
keep in mind that the results on curvature constraints are negative for all
four different dataset combinations. There is a deviation from a flat universe
with $\Omega_{k}=-0.0092\pm 0.0091$ using the Pantheon compilation.
Additionally, the flat $\Lambda$CDM model constrains $\Gamma_{0}$ for the JW
mock sample lower than $\Gamma_{0}=0.028\pm 0.018$ mag, which implies that
there is no significant difference expected for the calibrated sample.

###  4.2 $w$CDM model

The constraints for this model are given in Figure 3. As we can notice, there
is a correlation between the $w_{0}$ and $\Omega_{m}$ values present in the
flat model, while in the non-flat version of the model, this vanishes and for
the Pantheon and JW datasets there is a high error determination in the
$w_{0}-\Omega_{m}$ parameter space. For this model, the fractional matter
density has a similar value using Pantheon and Pantheon+ with the introduction
of JW as $\Omega_{m}\sim 0.333$. The curvature estimation is closer to
$\Omega_{k}=0$ as expected using the JW simulated data for the flat model.
Furthermore, notice that we have a $1\sigma$ deviation from a flat model using
only the Pantheon+ sample of $\Omega_{k}=0.27^{+0.17}_{-0.11}$, although both
SN samples prefer a non-flat universe.

Figure 3: 1-2$\sigma$ C.L results for the $w$CDM model using SNIa Pantheon–PN
(orange color), SNIa Pantheon+–PN+ (orange color), SNIa Pantheon & JW mock
data – PN+JW (green color) and SNIa Pantheon+ & JW mock data – PN++JW (purple
color) baselines: Left: Flat case. Right: Non-flat case.

Additionally, $\Gamma_{0}$ is constrained with a value of
$\Gamma_{0}=0.036^{+0.020}_{-0.022}$ mag using Pantheon+ and JW for the flat
model. This means that there is not a significant deviation from the
cosmological fits for the $w$CDM model using the JW mock sample compared to
the observed SN samples.

###  4.3 Chevallier–Polarski–Linder (CPL) model

The constraints for this model are given in Figure 4. It is interesting to
note that systematics improve for this model using Pantheon+ in comparison to
the previous SN catalog, Pantheon. This is expected due to the density of data
points at lower redshifts, where the CPL model can be well-constrained. For
the flat model and using Pantheon data, we recover the $\Lambda$CDM model with
$w_{0}=-1.111^{+0.110}_{-0.124}$ and $w_{a}=-1.252^{+1.354}_{-1.709}$ both at
$1\sigma$. Something similar happened when we included the JW sample. This
trend is confirmed when using Pantheon+ data, e.g. using Pantheon+ and JW
results with $\Omega_{m}=0.323^{+0.022}_{-0.023}$,
$w_{0}=-1.035^{+0.047}_{-0.056}$ and $w_{a}=0.130^{+0.432}_{-0.37}$. However,
in the non-flat model, the estimations change when using solely SN
measurements. The curvature estimation is deviated more than $1\sigma$ as for
Pantheon $\Omega_{k}=0.31^{+0.22}_{-0.13}$, and for Pantheon+
$\Omega_{k}=0.447^{+0.14}_{-0.19}$. These results also deviate from the
$\Lambda$CDM confirmation as both $w_{0}$ and $w_{a}$ do not recover the basic
equations with $w_{0}=-1$ and $w_{a}=0$.

In this model, all the results for $\Gamma_{0}$ constraints are lower than the
simulated 0.15 mag error, and therefore, we do not expect to have any
systematic effects on the cosmological parameters contaminated by the
simulated magnitude in the JW. The larger estimation, in comparison to
previous models, was found using the Pantheon+ sample for which
$\Gamma_{0}=0.039\pm 0.022$ mag.

Figure 4: 1-2$\sigma$ C.L results for the CPL model using SNIa Pantheon–PN
(orange color), SNIa Pantheon+–PN+ (yellow color), SNIa Pantheon & JW mock
data – PN+JW (green color) and SNIa Pantheon+ & JW mock data – PN++JW (purple
color) baselines: Left: Flat case. Right: Non-flat case.

###  4.4 Jassal–Bagla–Padmanabhan (JBP) model

The constraints for this model are given in Figure 5. As we can notice, the
flat version of this model recovers the correlation between the $\Omega_{m}$
and the $w_{0}$ parameters for all the datasets while the $w_{a}$
determination is done with a large error determination. Confirmation of
$\Lambda$CDM occurs for Pantheon and JW combinations as $w_{0}=-1$ and
$w_{a}=0$, while this not happen using Pantheon+, which results in a clear
deviation as $w_{a}=1.403^{+0.650}_{-0.845}$. It is worth mentioning that the
only negative $w_{a}$ value is obtained using the Pantheon dataset as
$w_{a}=-0.721^{+1.787}_{-2.492}$. Using Pantheon, Pantheon+ and the Pantheon
and JW combination has a parameter $w_{a}$ determination error larger than the
average. For the fractional matter density, with Pantheon data, we obtain a
larger estimation than Pantheon+, with $\Omega_{m}>0.3$ for Pantheon, and
$\Omega_{m}<0.3$ for the Pantheon+ and the combination with JW.

For the non-flat model, we obtain different results, as none of the
combinations return a deviation from the flat model. With Pantheon+ data we
obtain $\Omega_{k}=0.48^{+0.20}_{-0.10}$, while the JW mock data brings the
estimation closer to $\Omega_{k}=0$ without reaching flatness. It is worth
noticing that Pantheon combinations result in negative $w_{a}$ values, while
Pantheon+ combinations are opposite at $1\sigma$ with a large uncertainty.

Using the JBP model the larger $\Gamma_{0}$ determination is obtained using
the Pantheon+ and JW for the non-flat version as $\Gamma_{0}=0.040\pm 0.022$
mag, which again discards the systematic error in the magnitude determination
for the JW mock data.

Figure 5: 1-2$\sigma$ C.L results for the JBP model using SNIa Pantheon–PN
(orange color), SNIa Pantheon+–PN+ (yellow color), SNIa Pantheon & JW mock
data – PN+JW (green color) and SNIa Pantheon+ & JW mock data – PN++JW (purple
color) baselines: Left: Flat case. Right: Non-flat case.

###  4.5 Exponential model

The constraints for this model are given in Figure 6. As we can notice, a
correlation is presented in the parameter space between $w_{0}$ and
$\Omega_{m}$ for all dataset combinations. For the flat model, all the
combinations result in $w_{0}\sim-1$ between $1\sigma$ with a $\Omega_{m}\sim
0.3$. For the non-flat version is worth noticing that all the combinations
using Pantheon data result in curvature estimations close to the ones expected
for a flat universe. Being closer to flatness only with Pantheon data. Using
the Pantheon+ dataset alone results in a larger deviation from the flatness as
$\Omega_{k}=0.35^{+0.23}_{-0.11}$. This can be alleviated using the JW mock
data as $\Omega_{k}=0.135^{+0.095}_{-0.072}$ reports a larger deviation than
$1\sigma$ but near to the range of the expected value for a flat universe.
Similar to the flat model, the fractional matter density value is close to
$\Omega_{m}\sim 0.3$, being the lowest one obtained with Pantheon+ with
$\Omega_{m}=0.262\pm 0.055$. In the case of $w_{0}$ the results are slightly
lower than $w_{0}=-1$, being the same Pantheon+ the lower estimation as
$w_{0}=-1.86^{+0.74}_{-0.48}$.

Using the Exponential model results for the determination of
$\Gamma_{0}=0.036\pm 0.021$ in combination with Pantheon+ and JW means that
the determination of the cosmological parameters in this model is not affected
by the systematics in the JW mock data.

Figure 6: 1-2$\sigma$ C.L results for the exponential model using SNIa
Pantheon–PN (orange color), SNIa Pantheon+–PN+ (yellow color), SNIa Pantheon &
JW mock data – PN+JW (blue color) and SNIa Pantheon+ & JW mock data – PN++JW
(purple color) baselines: Left: Flat case. Right: Non-flat case.

###  4.6 Barboza–Alcaniz (BA) model

The constraints for this model are given in Figure 7. As we can notice for the
flat model the lower value for $\Omega_{m}$ is the one obtained with the
Pantheon+ and JW combination resulting in
$\Omega_{m}=0.280^{+0.047}_{-0.081}$. Regarding the values of $w_{0}$ and
$w_{a}$ for the flat models, all the combination results are consistent at
$1\sigma$ with the $\Lambda$CDM model, as all the results fall in the range of
$w_{0}\sim-1$ and $w_{a}=0$. Nevertheless, it is interesting to notice that
the only negative value for $w_{a}$ is the one obtained using the Pantheon
compilation with $w_{a}=0.723^{+1.047}_{-1.531}$. In general, this model
agrees with a flat $\Lambda$CDM at $1\sigma$.

Meanwhile, the non-flat model shows larger deviations from the $\Lambda$CDM
model as the curvature estimation is separated from the flatness in more than
$1\sigma$ for all the combinations, except the Pantheon compilation as
$\Omega_{k}=0.20^{+0.29}_{-0.24}$. Other results prefer $\Omega_{k}>0$.

In this model, the larger $\Gamma_{0}$ estimation is obtained for the non-flat
model with the Pantheon+ and JW mock data combination as $\Gamma_{0}=0.039\pm
0.021$ mag, resulting similar to the previous models where the cosmological
parameter inference is not affected by the error in magnitude for the
simulated dataset.

Figure 7: 1-2$\sigma$ C.L results for the BA model using SNIa Pantheon–PN
(orange color), SNIa Pantheon+–PN+ (yellow color), SNIa Pantheon & JW mock
data – PN+JW (green color) and SNIa Pantheon+ & JW mock data – PN++JW (purple
color) baselines: Left: Flat case. Right: Non-flat case.

##  5 Conclusions

In this paper, we studied new cosmological constraints in spatial curvature
dark energy models. We extend the distance ladder method through an SNIa
sample using the capabilities of JWST to forecast SNIa up to $z\sim 6$,
considering the information on the star formation rates at high $z$. Comparing
the results shown in Tables 1 and 2, notice that flat $\Lambda$CDM, flat
$w$CDM, and flat exponential are the only models in which the value of
$\Omega_{m}$ obtained using the Pantheon sample is less than $\Omega_{m}$ for
Pantheon+, i.e. $\Omega_{m,\text{PN}}<\Omega_{m,\text{PN}^{+}}$.

However, in our analysis when including the JW mock data, the flat
$\Lambda$CDM, non-flat $\Lambda$CDM, non-flat $w$CDM, and non-flat exponential
are the cosmological models in which $\Omega_{m}$ for the combination PN+JW is
less than $\Omega_{m}$ for PN++JW, i.e.
$\Omega_{m,\text{PN+JW}}<\Omega_{m,\text{PN}^{+}+\text{JW}}$. Showing a lower
value of this parameter when we have higher SN at $z<1$.

In regards to JW forecasting, all models have a
$\Gamma_{0,\text{PN}^{+}+\text{JW}}>\Gamma_{0,\text{PN+JW}}$. More SN at $z<1$
(e.g. Pantheon+) seems to raise the value of this $\Gamma_{0}$ parameter
associated with JWST. Therefore, according to our $\Delta\mu$ equation (see
below Eq.(3.3)), it is statistically better to employ Pantheon with JW
forecasting due that the vector uncertainty is lower in comparison to the one
obtained with Pantheon+ catalog. However, notice that the JW sample has been
calibrated with Pantheon, therefore $\Gamma_{0}$ shows a preference for this
SN sample.

We have studied the possibility of non-zero curvature in standard dark energy
models, as demonstrated in [9]. The addition of our JW forecasting leads to an
improvement in the statistics associated with $\Omega_{m}$. It is expected
that the JWST will observe more luminous structures with a well-treated
morphology, which can help us find more robust statistics in dark energy
cosmologies.

Model (flat) | Dataset | $\Omega_{M}$ | $w_{0}$ | $w_{a}$ | $\Gamma_{0}$ |
---|---|---|---|---|---|---
| PN | $0.290\pm 0.008$ |  |  |  |
| PN+JW | $0.295\pm 0.007$ |  |  | $0.005^{+0.018}_{-0.018}$ |
$\Lambda$CDM | PN+ | $0.311^{+0.010}_{-0.009}$ |  |  |  |
| PN++JW | $0.312\pm 0.008$ |  |  | $0.028\pm 0.018$ |
| PN | $0.320^{+0.040}_{-0.045}$ | $-1.073^{+0.100}_{-0.113}$ |  |  |
| PN+JW | $0.327^{+0.023}_{-0.021}$ | $-1.086^{+0.057}_{-0.065}$ |  | $0.025\pm 0.021$ |
$w$CDM | PN+ | $0.334^{+0.036}_{-0.041}$ | $-1.053^{+0.091}_{-0.095}$ |  |  |
| PN++JW | $0.322^{+0.023}_{-0.022}$ | $-1.025^{+0.052}_{-0.062}$ |  | $0.036^{+0.020}_{-0.022}$ |
| PN | $0.380^{+0.046}_{-0.058}$ | $-1.111^{+0.110}_{-0.124}$ | $-1.252^{+1.354}_{-1.709}$ |  |
| PN+JW | $0.336^{+0.021}_{-0.022}$ | $-1.065^{+0.061}_{-0.064}$ | $-0.399^{+0.668}_{-0.798}$ | $0.022^{+0.023}_{-0.022}$ |
CPL | PN+ | $0.343^{+0.054}_{-0.065}$ | $-1.081^{+0.103}_{-0.118}$ | $0.150^{+0.638}_{-0.936}$ |  |
| PN++JW | $0.323^{+0.022}_{-0.023}$ | $-1.035^{+0.047}_{-0.056}$ | $0.130^{+0.432}_{-0.537}$ | $0.039\pm 0.022$ |
| PN | $0.358^{+0.064}_{-0.091}$ | $-1.088^{+0.140}_{-0.162}$ | $-0.721^{+1.787}_{-2.492}$ |  |
| PN+JW | $0.310^{+0.036}_{-0.048}$ | $-1.106^{+0.075}_{-0.079}$ | $0.873^{+1.216}_{-1.491}$ | $0.029^{+0.023}_{-0.022}$ |
JBP | PN+ | $0.268^{+0.095}_{-0.121}$ | $-1.013^{+0.142}_{-0.153}$ | $1.182^{+0.707}_{-1.073}$ |  |
| PN++JW | $0.280^{+0.039}_{-0.051}$ | $-1.045^{+0.062}_{-0.063}$ | $1.403^{+0.650}_{-0.845}$ | $0.038^{+0.022}_{-0.021}$ |
| PN | $0.304^{+0.060}_{-0.066}$ | $-1.044^{+0.128}_{-0.163}$ |  |  |
| PN+JW | $0.312^{+0.026}_{-0.025}$ | $-1.062^{+0.060}_{-0.073}$ |  | $0.025^{+0.022}_{-0.020}$ |
Exp | PN+ | $0.321^{+0.050}_{-0.058}$ | $-1.034^{+0.118}_{-0.129}$ |  |  |
| PN++JW | $0.306^{+0.026}_{-0.025}$ | $-1.001^{+0.058}_{-0.063}$ |  | $0.035^{+0.021}_{-0.023}$ |
| PN | $0.383^{+0.061}_{-0.111}$ | $-1.124\pm 0.159$ | $-0.723^{+1.047}_{-1.531}$ |  |
| PN+JW | $0.301^{+0.049}_{-0.092}$ | $-1.045^{+0.109}_{-0.083}$ | $0.301^{+0.254}_{-0.595}$ | $0.028\pm 0.022$ |
BA | PN+ | $0.323^{+0.076}_{-0.110}$ | $-1.044^{+0.162}_{-0.156}$ | $0.158^{+0.334}_{-0.614}$ |  |
| PN++JW | $0.280^{+0.047}_{-0.081}$ | $-0.980^{+0.099}_{-0.075}$ | $0.343^{+0.164}_{-0.311}$ | $0.037\pm 0.021$ |

Table 1: Best-fit cosmological parameters at $1\sigma$ for the six flat models
obtained by combining the following catalogs: Pantheon (PN) and Pantheon+
(PN+) with the JW simulated sample (JW). Empty cells denote parameters not
defined in the model.

Model (non-flat) | Dataset | $\Omega_{M}$ | $\Omega_{\Lambda}$ | $w_{0}$ | $w_{a}$ | $\Gamma_{0}$ | $\Omega_{k}$
---|---|---|---|---|---|---|---
| PN | $0.332\pm 0.043$ | $0.761\pm 0.049$ |  |  |  | $-0.092\pm 0.091$
| PN+JW | $0.306\pm 0.010$ | $0.725\pm 0.017$ |  |  | $0.018\pm 0.019$ | $-0.031\pm 0.022$
$\Lambda$CDM | PN+ | $0.328\pm 0.037$ | $0.710\pm 0.041$ |  |  |  | $-0.038\pm 0.076$
| PN++JW | $0.312\pm 0.010$ | $0.695\pm 0.017$ |  |  | $0.027\pm 0.019$ | $-0.008\pm 0.023$
| PN | $0.302\pm 0.051$ | $0.59^{+0.11}_{-0.20}$ | $-1.29^{+0.38}_{-0.18}$ |  |  | $0.11^{+0.21}_{-0.16}$
| PN+JW | $0.333^{+0.022}_{-0.019}$ | $0.619^{+0.066}_{-0.073}$ | $-1.19^{+0.16}_{-0.11}$ |  | $0.023\pm 0.021$ | $0.048\pm 0.058$
$w$CDM | PN+ | $0.292\pm 0.045$ | $0.435^{+0.068}_{-0.15}$ | $-1.58^{+0.46}_{-0.29}$ |  |  | $0.27^{+0.17}_{-0.11}$
| PN++JW | $0.334^{+0.022}_{-0.018}$ | $0.595^{+0.065}_{-0.079}$ | $-1.158^{+0.124}_{-0.152}$ |  | $0.032\pm 0.021$ | $0.071\pm 0.061$
| PN | $0.280\pm 0.077$ | $0.412^{+0.060}_{-0.16}$ | $-1.537^{+0.346}_{-0.451}$ | $-3.6^{+5.4}_{-2.9}$ |  | $0.31^{+0.22}_{-0.13}$
| PN+JW | $0.358^{+0.022}_{-0.017}$ | $0.533^{+0.054}_{-0.077}$ | $-1.238^{+0.126}_{-0.143}$ | $-2.0^{+2.4}_{-1.3}$ | $0.025\pm 0.021$ | $0.109^{+0.061}_{-0.055}$
CPL | PN+ | $0.239\pm 0.059$ | $0.314^{+0.038}_{-0.11}$ | $-2.059^{+0.465}_{-0.520}$ | $-1.6^{+5.7}_{-2.7}$ |  | $0.447^{+0.14}_{-0.09}$
| PN++JW | $0.344^{+0.029}_{-0.020}$ | $0.539^{+0.069}_{-0.070}$ | $-1.17^{+0.16}_{-0.11}$ | $-0.34^{+1.5}_{-0.71}$ | $0.037\pm 0.021$ | $0.116\pm 0.057$
| PN | $0.292\pm 0.097$ | $0.50^{+0.12}_{-0.24}$ | $-1.50^{+0.62}_{-0.21}$ | $-2.1^{+4.8}_{-2.6}$ |  | $0.21^{+0.29}_{-0.21}$
| PN+JW | $0.358^{+0.031}_{-0.016}$ | $0.493^{+0.055}_{-0.12}$ | $-1.372^{+0.225}_{-0.281}$ | $-3.0^{+4.9}_{-2.1}$ | $0.031\pm 0.022$ | $0.149^{+0.092}_{-0.061}$
JBP | PN+ | $0.204\pm 0.072$ | $0.315^{+0.044}_{-0.15}$ | $-2.320^{+0.781}_{-1.051}$ | $2.1^{+6.7}_{-4.3}$ |  | $0.48^{+0.20}_{-0.10}$
| PN++JW | $0.328^{+0.056}_{-0.012}$ | $0.517^{+0.076}_{-0.16}$ | $-1.424^{+0.277}_{-0.347}$ | $0.5^{+2.4}_{-1.4}$ | $0.040\pm 0.022$ | $0.155^{+0.10}_{-0.088}$
| PN | $0.289\pm 0.068$ | $0.67^{+0.16}_{-0.32}$ | $-1.24^{+0.50}_{-0.18}$ |  |  | $0.04^{+0.30}_{-0.24}$
| PN+JW | $0.335^{+0.035}_{-0.021}$ | $0.573^{+0.08}_{-0.12}$ | $-1.33^{+0.31}_{-0.19}$ |  | $0.028\pm 0.022$ | $0.092^{+0.086}_{-0.076}$
Exp | PN+ | $0.262\pm 0.055$ | $0.391^{+0.046}_{-0.18}$ | $-1.86^{+0.74}_{-0.48}$ |  |  | $0.35^{+0.23}_{-0.11}$
| PN++JW | $0.340^{+0.032}_{-0.018}$ | $0.526^{+0.073}_{-0.12}$ | $-1.38^{+0.34}_{-0.22}$ |  | $0.036\pm 0.021$ | $0.134^{+0.095}_{-0.072}$
| PN | $0.287^{+0.15}_{-0.086}$ | $0.52^{+0.11}_{-0.23}$ | $-1.328^{+0.324}_{-0.459}$ | $-1.1^{+2.6}_{-1.7}$ |  | $0.20^{+0.29}_{-0.24}$
| PN+JW | $0.363^{+0.029}_{-0.014}$ | $0.465^{+0.044}_{-0.090}$ | $-1.447^{+0.206}_{-0.255}$ | $-2.2^{+2.9}_{-1.3}$ | $0.031\pm 0.022$ | $0.172^{+0.075}_{-0.054}$
BA | PN+ | $0.240\pm 0.069$ | $0.332^{+0.046}_{-0.13}$ | $-1.990^{+0.556}_{-0.658}$ | $-0.7^{+3.2}_{-1.3}$ |  | $0.43^{+0.19}_{-0.10}$
| PN++JW | $0.347^{+0.042}_{-0.012}$ | $0.453^{+0.053}_{-0.10}$ | $-1.522^{+0.254}_{-0.280}$ | $-0.69^{+1.8}_{-0.92}$ | $0.039\pm 0.021$ | $0.200^{+0.081}_{-0.070}$

Table 2: Best-fit cosmological parameters at $1\sigma$ for the six non-flat
cosmological models obtained for the following catalogs: combining Pantheon
(PN) and Pantheon+ (PN+) datasets with the JW simulated sample (JW). Empty
cells denote parameters not defined in the model..

## Acknowledgments

The Authors thank J. Vinko and E. Regős for their insights on using JWST
forecasting data. Also, we would like to acknowledge funding from PAPIIT UNAM
Project TA100122. CE-R acknowledges the Royal Astronomical Society as FRAS
10147. PMA and RS are supported by the CONACyT National Grant. The
computational calculations have been carried out using facilities procured
through the Cosmostatistics National Group project. This article is based upon
work from COST Action CA21136 Addressing observational tensions in cosmology
with systematics and fundamental physics (CosmoVerse) supported by COST
(European Cooperation in Science and Technology).

##  Appendix A SNIa JWST baseline forecasting: data and priors

The simulated data set used is derived from the FLARE project, which has the
goal of searching supernovas from population III at redshift $z\geq 10$ [77]
by using the characteristics of the JWST in an area of 0.05 square degrees.
Furthermore, the project employs four broadband in the NIRCAM filters (F150W,
F200W, F322W2, F444W) with exposure times that can reach 10$\sigma$ limiting
magnitudes of $m\gtrsim 27$ in these filters [9]. By using Monte Carlo methods
it was found that for a specific project of JWST observation, at least 200 SNe
Ia could be observed. Therefore, the mock sample is constructed using a flat
$\Lambda$CDM model with $H_{0}=71.66$ km s-1 Mpc-1 and $\Omega_{m}=0.3$, as
derived using Pantheon data [8]. To ensure consistency between the local and
distant samples, we consider a Gaussian error associated with the supernovae
distance modules of 0.15 mag and extrapolated them to a larger uncertainty
with higher redshift, although this is an oversimplification of the typical
distance measurements [80]. For the FLARE project [78] the simulations were
done using the standard cosmology and a different SFR according to the
redshift in which the simulation is done. Also, it is considered the
calculation of the rate of occurrence to estimate the range of the detection
needed in the telescope to perform the observations.

The used SFR functions are explicitly proposed to have a redshift dependence.
For low $z\lesssim 3$, the function used is [81]:

$\mathrm{SFR}(z)=K\frac{(a+bz)h}{1+(z/c)^{d}},$ (A.1)

where $h=H_{0}/(100\mathrm{km}\mathrm{s}^{-1}\mathrm{Mpc}^{-1})$, $a=0.017$,
$b=0.13$, $c=3.3$, and $d=5.3$. These quantities constrain $K$ with the SN
rates observed. For higher redshifts between $3<z\leq 8$, the function used
is:

$\mathrm{SFR}(z)\propto(1+z)^{-3.6},$ (A.2)

and for $z>8$:

$\mathrm{SFR}(z)\propto(1+z)^{-10.4}.$ (A.3)

Additionally, other SFRs are proposed for the whole interval [82]:

$\mathrm{SFR}(z)=0.015\frac{(1+z)^{2.7}}{1+[(1+z)/2.9]^{5.6}},$ (A.4)

or [83]:

$\mathrm{SFR}(z)=\frac{C}{10^{A(z-z_{0})}+10^{B(z-z_{0})}},$ (A.5)

using the constants $C=0.18$, $A=-0.997$, $B=0.241$, and $z_{0}=1.243$. All
the parameterisations studied in Sec.2 for the SFR give similar redshift
dependence with a peak in star formation in $z\sim 2$. The volume rate was
calculated using the observed SN rate per redshift bin as [78]:

$\dot{N}_{\mathrm{SN}}(z)=\frac{\dot{n}_{\mathrm{SLSN}}(z)}{1+z}\derivative{V}{z},$
(A.6)

where the comoving volume can be rewritten as:

$\dot{n}_{\mathrm{SLSN}}(z)=\varepsilon(z)\mathrm{SFR}(z),$ (A.7)

where explicitly is expressed the comoving rate of SNe. $\varepsilon(z)$ is a
factor of the efficiency taking into account the metallicity dependence. Even
though, $\varepsilon(z)$ can be studied through GRBs [78]. So, the expected
number of SN can be calculated as:

$N_{\mathrm{SN}}=\Omega T\int_{z}^{z+\Delta
z}\frac{\dot{n}_{\mathrm{SLSN}}(z)}{1+z}\derivative{V}{z}\mathrm{d}z,$ (A.8)

resulting in a number per unit of redshift interval in the survey area.
$\Omega$ is the survey area and $T$ is the dedicated time of observation. The
redshift dependence also has to consider the number of SNIa progenitors that
occur (as SNIa needs a White Dwarf) and the delay for the stellar evolution in
such binary systems.

For the simulations of mock data sets the code developed by [84] is used to
create light curves at different redshifts taking into account the calculated
rate and detection of the observatory [78]. So, for the simulation the
redshift $z$, the luminosity distance $D_{L}$, maximum light moment
$t_{\mathrm{max}}$, V absolute magnitude $M_{\mathrm{V}}$, stretch and color
are calculated for every SN. It is important to mention that the assumptions
of the FLARE project imply that the JSWT will take deep observations for at
least three years with a 90-day cadence allowing us to discover between 5 and
20 supernovae events meaning at least fifty in redshift $1<z<4$ [9].
Additionally, the simulation takes into account that the SNIa ideal
observation time has to be from 2 weeks before the maximum up to one month
after in which spectrum would have the ideal quality [85]. So, for the
simulations, the results are the Hubble diagram for the apparent magnitude
assuming detection with the NIRCam of the telescope.

## References

  * [1] Supernova Cosmology Project collaboration, _Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae_, _Astrophys. J._ 517 (1999) 565 [astro-ph/9812133].
  * [2] Supernova Search Team collaboration, _Observational evidence from supernovae for an accelerating universe and a cosmological constant_ , _Astron. J._ 116 (1998) 1009 [astro-ph/9805201].
  * [3] Boomerang collaboration, _Cosmology from MAXIMA-1, BOOMERANG and COBE / DMR CMB observations_ , _Phys. Rev. Lett._ 86 (2001) 3475 [astro-ph/0007333].
  * [4] SDSS collaboration, _Detection of the Baryon Acoustic Peak in the Large-Scale Correlation Function of SDSS Luminous Red Galaxies_ , _Astrophys. J._ 633 (2005) 560 [astro-ph/0501171].
  * [5] BOSS collaboration, _The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: baryon acoustic oscillations in the Data Releases 10 and 11 Galaxy samples_ , _Mon. Not. Roy. Astron. Soc._ 441 (2014) 24 [1312.4877].
  * [6] DES collaboration, _Dark Energy Survey year 1 results: Cosmological constraints from galaxy clustering and weak lensing_ , _Phys. Rev. D_ 98 (2018) 043526 [1708.01530].
  * [7] D. Brout et al., _The Pantheon+ Analysis: Cosmological Constraints_ , _Astrophys. J._ 938 (2022) 110 [2202.04077].
  * [8] D. Scolnic et al., _The Next Generation of Cosmological Measurements with Type Ia Supernovae_ , 1903.05128.
  * [9] J. Lu, L. Wang, X. Chen, D. Rubin, S. Perlmutter, D. Baade et al., _Constraints on Cosmological Parameters with a Sample of Type Ia Supernovae from JWST_ , _Astrophys. J._ 941 (2022) 71 [2210.00746].
  * [10] C. Escamilla-Rivera and S. Capozziello, _Unveiling cosmography from the dark energy equation of state_ , _Int. J. Mod. Phys. D_ 28 (2019) 1950154 [1905.04602].
  * [11] C. Escamilla-Rivera, _Status on bidimensional dark energy parameterizations using SNe Ia JLA and BAO datasets_ , _Galaxies_ 4 (2016) 8 [1605.02702].
  * [12] L.A. Escamilla, W. Giarè, E. Di Valentino, R.C. Nunes and S. Vagnozzi, _The state of the dark energy equation of state circa 2023_ , 2307.14802.
  * [13] C. Escamilla-Rivera and A. Nájera, _Dynamical dark energy models in the light of gravitational-wave transient catalogues_ , _JCAP_ 03 (2022) 060 [2103.02097].
  * [14] C. Escamilla-Rivera, M.A.C. Quintero and S. Capozziello, _A deep learning approach to cosmological dark energy models_ , _JCAP_ 03 (2020) 008 [1910.02788].
  * [15] H.-C. Zhang, _Dynamical dark energy can amplify the expansion rate of the Universe_ , _Phys. Rev. D_ 107 (2023) 103529 [2305.16586].
  * [16] L.G. Jaime, M. Jaber and C. Escamilla-Rivera, _New parametrized equation of state for dark energy surveys_ , _Phys. Rev. D_ 98 (2018) 083530 [1804.04284].
  * [17] T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, _Modified Gravity and Cosmology_ , _Phys. Rept._ 513 (2012) 1 [1106.2476].
  * [18] L. Amendola, R. Gannouji, D. Polarski and S. Tsujikawa, _Conditions for the cosmological viability of f(R) dark energy models_ , _Phys. Rev. D_ 75 (2007) 083504 [gr-qc/0612180].
  * [19] S. Bahamonde, K.F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry et al., _Teleparallel gravity: from theory to cosmology_ , _Rept. Prog. Phys._ 86 (2023) 026901 [2106.13793].
  * [20] Y.-F. Cai, S. Capozziello, M. De Laurentis and E.N. Saridakis, _f(T) teleparallel gravity and cosmology_ , _Rept. Prog. Phys._ 79 (2016) 106901 [1511.07586].
  * [21] DES collaboration, _The Dark Energy Survey: more than dark energy – an overview_ , _Mon. Not. Roy. Astron. Soc._ 460 (2016) 1270 [1601.00329].
  * [22] DESI collaboration, _The DESI Experiment Part I: Science,Targeting, and Survey Design_ , 1611.00036.
  * [23] LSST Dark Energy Science collaboration, _The LSST Dark Energy Science Collaboration (DESC) Science Requirements Document_ , 1809.01669.
  * [24] L. Amendola et al., _Cosmology and fundamental physics with the Euclid satellite_ , _Living Rev. Rel._ 21 (2018) 2 [1606.00180].
  * [25] J.P. Gardner et al., _The James Webb Space Telescope_ , _Space Sci. Rev._ 123 (2006) 485 [astro-ph/0606175].
  * [26] Y. Asada, M. Sawicki, G. Desprez, R. Abraham, M. Bradač, G. Brammer et al., _Jwst catches the assembly of a $z\sim 5$ ultra-low-mass galaxy_, _arXiv preprint arXiv:2212.07540_ (2022) .
  * [27] I. Yoon, C.L. Carilli, S. Fujimoto, M. Castellano, E. Merlin, P. Santini et al., _Alma observation of a $z\sim 10$ galaxy candidate discovered with jwst_, _arXiv preprint arXiv:2210.08413_ (2022) .
  * [28] B.E. Robertson et al., _Identification and properties of intense star-forming galaxies at redshifts z $>$ 10_, _Nature Astron._ 7 (2023) 611 [2212.04480].
  * [29] Y. Harikane, M. Ouchi, M. Oguri, Y. Ono, K. Nakajima, Y. Isobe et al., _A comprehensive study of galaxies at z 9–16 found in the early jwst data: Ultraviolet luminosity functions and cosmic star formation history at the pre-reionization epoch_ , _The Astrophysical Journal Supplement Series_ 265 (2023) 5.
  * [30] K. Luhman, P. Tremblin, S. Birkmann, E. Manjavacas, J. Valenti, C.A. de Oliveira et al., _Jwst/nirspec observations of the planetary mass companion twa 27b_ , _The Astrophysical Journal Letters_ 949 (2023) L36.
  * [31] J.L. Bean, Q. Xue, P.C. August, J. Lunine, M. Zhang, D. Thorngren et al., _High atmospheric metal enrichment for a saturn-mass planet_ , _Nature_ 618 (2023) 43.
  * [32] T.P. Greene, T.J. Bell, E. Ducrot, A. Dyrek, P.-O. Lagage and J.J. Fortney, _Thermal emission from the earth-sized exoplanet trappist-1 b using jwst_ , _Nature_ 618 (2023) 39.
  * [33] T.S.-Y. Lai, L. Armus, T. Diaz-Santos, K.L. Larson, A. Evans, M.A. Malkan et al., _Goals-jwst: Tracing agn feedback on the star-forming ism in ngc 7469_ , _arXiv preprint arXiv:2209.06741_ (2022) .
  * [34] E.W. Schwieterman, S.L. Olson, D. Pidhorodetska, C.T. Reinhard, A. Ganti, T.J. Fauchez et al., _Evaluating the plausible range of n2o biosignatures on exo-earths: An integrated biogeochemical, photochemical, and spectral modeling approach_ , _The Astrophysical Journal_ 937 (2022) 109.
  * [35] S. Mukherjee, J.J. Fortney, N.E. Batalha, T. Karalidi, M.S. Marley, C. Visscher et al., _Probing the extent of vertical mixing in brown dwarf atmospheres with disequilibrium chemistry_ , _The Astrophysical Journal_ 938 (2022) 107.
  * [36] M. Leung, E.W. Schwieterman, M.N. Parenteau and T.J. Fauchez, _Alternative methylated biosignatures. i. methyl bromide, a capstone biosignature_ , _The Astrophysical Journal_ 938 (2022) 6.
  * [37] T. Mikal-Evans, _Detecting the proposed ch4–co2 biosignature pair with the james webb space telescope: Trappist-1e and the effect of cloud/haze_ , _Monthly Notices of the Royal Astronomical Society_ 510 (2022) 980.
  * [38] P. Wang, B.-Y. Su, L. Zu, Y. Yang and L. Feng, _Exploring the Dark Energy Equation of State with JWST_ , 2307.11374.
  * [39] S.A. Adil, U. Mukhopadhyay, A.A. Sen and S. Vagnozzi, _Dark energy in light of the early JWST observations: case for a negative cosmological constant?_ , 2307.12763.
  * [40] A.G. Riess, G.S. Anand, W. Yuan, S. Casertano, A. Dolphin, L.M. Macri et al., _Crowded No More: The Accuracy of the Hubble Constant Tested with High Resolution Observations of Cepheids by JWST_ , 2307.15806.
  * [41] F. Cogato, M. Moresco, L. Amati and A. Cimatti, _An analytical late-Universe approach to the weaving of modern cosmology_ , 2309.01375.
  * [42] M.G. Dainotti, G. Bargiacchi, A.L. Lenart, S. Nagataki and S. Capozziello, _Quasars: Standard Candles up to z = 7.5 with the Precision of Supernovae Ia_ , _Astrophys. J._ 950 (2023) 45 [2305.19668].
  * [43] R. Sandoval-Orozco, C. Escamilla-Rivera, R. Briffa and J. Levi Said, _$f(T)$ cosmology in the regime of quasar observations_, 2309.03675.
  * [44] Pan-STARRS1 collaboration, _The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample_ , _Astrophys. J._ 859 (2018) 101 [1710.00845].
  * [45] N. Lovyagin, A. Raikov, V. Yershov and Y. Lovyagin, _Cosmological Model Tests with JWST_ , _Galaxies_ 10 (2022) 108 [2212.06575].
  * [46] H. Atek, M. Shuntov, L.J. Furtak, J. Richard, J.-P. Kneib, G. Mahler et al., _Revealing galaxy candidates out to z 16 with JWST observations of the lensing cluster SMACS0723_ , _MNRAS_ 519 (2023) 1201 [2207.12338].
  * [47] C. Jacobs, K. Glazebrook, A. Calabrò, T. Treu, T. Nannayakkara, T. Jones et al., _Early Results from GLASS-JWST. XVIII. A First Morphological Atlas of the 1 ¡ z ¡ 5 Universe in the Rest-frame Optical_ , _ApJ_ 948 (2023) L13 [2208.06516].
  * [48] H. Yan, Z. Ma, C. Ling, C. Cheng and J.-S. Huang, _First Batch of z $\approx$ 11-20 Candidate Objects Revealed by the James Webb Space Telescope Early Release Observations on SMACS 0723-73_, _ApJ_ 942 (2023) L9 [2207.11558].
  * [49] M. Castellano, A. Fontana, T. Treu, P. Santini, E. Merlin, N. Leethochawalit et al., _Early Results from GLASS-JWST. III. Galaxy Candidates at z 9-15_ , _ApJ_ 938 (2022) L15 [2207.09436].
  * [50] D. Schaerer, R. Marques-Chaves, L. Barrufet, P. Oesch, Y.I. Izotov, R. Naidu et al., _First look with JWST spectroscopy: Resemblance among z $\sim$ 8 galaxies and local analogs_, _A &A_ 665 (2022) L4 [2207.10034].
  * [51] M.A. Marshall, S. Wilkins, T. Di Matteo, W.J. Roper, A.P. Vijayan, Y. Ni et al., _The impact of dust on the sizes of galaxies in the Epoch of Reionization_ , _MNRAS_ 511 (2022) 5475 [2110.12075].
  * [52] K.A. Suess, R. Bezanson, E.J. Nelson, D.J. Setton, S.H. Price, P. van Dokkum et al., _Rest-frame Near-infrared Sizes of Galaxies at Cosmic Noon: Objects in JWST’s Mirror Are Smaller than They Appeared_ , _ApJ_ 937 (2022) L33 [2207.10655].
  * [53] F. Zwicky, _On the Red Shift of Spectral Lines through Interstellar Space_ , _Proc. Nat. Acad. Sci._ 15 (1929) 773.
  * [54] E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri et al., _In the realm of the Hubble tension—a review of solutions_ , _Class. Quant. Grav._ 38 (2021) 153001 [2103.01183].
  * [55] E. Abdalla et al., _Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies_ , _JHEAp_ 34 (2022) 49 [2203.06142].
  * [56] L. Amendola and S. Tsujikawa, _Dark Energy: Theory and Observations_ , Cambridge University Press (2010), 10.1017/CBO9780511750823.
  * [57] P.D. Bari, _Cosmology and the Early Universe_ , Taylor & Francis (2018).
  * [58] G. Bargiacchi, M. Benetti, S. Capozziello, E. Lusso, G. Risaliti and M. Signorini, _Quasar cosmology: dark energy evolution and spatial curvature_ , _Mon. Not. Roy. Astron. Soc._ 515 (2022) 1795 [2111.02420].
  * [59] Planck collaboration, _Planck 2018 results. VI. Cosmological parameters_ , _Astron. Astrophys._ 641 (2020) A6 [1807.06209].
  * [60] M. Moresco, _Addressing the Hubble tension with cosmic chronometers_ , 2307.09501.
  * [61] X. Zheng, S. Cao, M. Biesiada, X. Li, T. Liu and Y. Liu, _Multiple measurements of quasars acting as standard probes: model independent calibration and exploring the Dark Energy Equation of States_ , _Sci. China Phys. Mech. Astron._ 64 (2021) 259511 [2103.07139].
  * [62] M. Chevallier and D. Polarski, _Accelerating universes with scaling dark matter_ , _Int. J. Mod. Phys. D_ 10 (2001) 213 [gr-qc/0009008].
  * [63] E.V. Linder, _Exploring the expansion history of the universe_ , _Phys. Rev. Lett._ 90 (2003) 091301 [astro-ph/0208512].
  * [64] B.R. Dinda, H. Singirikonda and S. Majumdar, _Constraints on cosmic curvature from cosmic chronometer and quasar observations_ , 2303.15401.
  * [65] H.K. Jassal, J.S. Bagla and T. Padmanabhan, _WMAP constraints on low redshift evolution of dark energy_ , _Mon. Not. Roy. Astron. Soc._ 356 (2005) L11 [astro-ph/0404378].
  * [66] D. Staicova, _DE Models with Combined H 0 · rd from BAO and CMB Dataset and Friends_, _Universe_ 8 (2022) 631 [2211.08139].
  * [67] W. Yang, S. Pan, E. Di Valentino, E.N. Saridakis and S. Chakraborty, _Observational constraints on one-parameter dynamical dark-energy parametrizations and the $H_{0}$ tension_, _Phys. Rev. D_ 99 (2019) 043543 [1810.05141].
  * [68] M.N. Castillo-Santos, A. Hernández-Almada, M.A. García-Aspeitia and J. Magaña, _An exponential equation of state of dark energy in the light of 2018 CMB Planck data_ , _Phys. Dark Univ._ 40 (2023) 101225 [2212.01974].
  * [69] E.M. Barboza, Jr. and J.S. Alcaniz, _A parametric model for dark energy_ , _Phys. Lett. B_ 666 (2008) 415 [0805.1713].
  * [70] D. Brout, M. Sako, D. Scolnic, R. Kessler, C.B. D’Andrea, T.M. Davis et al., _First cosmology results using type ia supernovae from the dark energy survey: Photometric pipeline and light-curve data release_ , _The Astrophysical Journal_ 874 (2019) 106.
  * [71] A.G. Riess 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_ , _Astrophys. J. Lett._ 934 (2022) L7 [2112.04510].
  * [72] R.J. Foley et al., _The Foundation Supernova Survey: Motivation, Design, Implementation, and First Data Release_ , _Mon. Not. Roy. Astron. Soc._ 475 (2018) 193 [1711.02474].
  * [73] P.J. Brown, A.A. Breeveld, S. Holland, P. Kuin and T. Pritchard, _SOUSA: the Swift Optical/Ultraviolet Supernova Archive_ , _Astrophys. Space Sci._ 354 (2014) 89 [1407.3808].
  * [74] M. Ganeshalingam, W. Li, A.V. Filippenko, C. Anderson, G. Foster, E.L. Gates et al., _Results of the lick observatory supernova search follow-up photometry program: Bvri light curves of 165 type ia supernovae_ , _The Astrophysical Journal Supplement Series_ 190 (2010) 418.
  * [75] B.E. Stahl et al., _Lick Observatory Supernova Search Follow-Up Program: Photometry Data Release of 93 Type Ia Supernovae_ , _Mon. Not. Roy. Astron. Soc._ 490 (2019) 3882 [1909.11140].
  * [76] R. Briffa, C. Escamilla-Rivera, J. Said Levi, J. Mifsud and N.L. Pullicino, _Impact of $H_{0}$ priors on $f(T)$ late time cosmology_, _Eur. Phys. J. Plus_ 137 (2022) 532 [2108.03853].
  * [77] L. Wang et al., _A First Transients Survey with JWST: the FLARE project_ , 1710.07005.
  * [78] E. Regos and J. Vinko, _Detection and Classification of Supernovae Beyond z $\sim$ 2 Redshift with the James Webb Space Telescope_, _Astrophys. J._ 874 (2019) 158 [1811.00891].
  * [79] X. Chen, L. Hu and L. Wang, _Constraining Type Ia Supernova Delay Time with Spatially Resolved Star Formation Histories_ , _Astrophys. J._ 922 (2021) 15 [2101.06242].
  * [80] M.M. Phillips, _The Absolute Magnitudes of Type IA Supernovae_ , .
  * [81] A.M. Hopkins and J.F. Beacom, _On the normalization of the cosmic star formation history_ , _The Astrophysical Journal_ 651 (2006) 142.
  * [82] P. Madau and M. Dickinson, _Cosmic Star Formation History_ , _Ann. Rev. Astron. Astrophys._ 52 (2014) 415 [1403.0007].
  * [83] P.S. Behroozi, R.H. Wechsler and C. Conroy, _The average star formation histories of galaxies in dark matter halos from z= 0–8_ , _The Astrophysical Journal_ 770 (2013) 57.
  * [84] K. Barbary, T. Barclay, R. Biswas, M. Craig, U. Feindt, B. Friesen et al., “SNCosmo: Python library for supernova cosmology.” Astrophysics Source Code Library, record ascl:1611.017, Nov., 2016.
  * [85] L. Hu, X. Chen and L. Wang, _Spectroscopic Studies of Type Ia Supernovae Using LSTM Neural Networks_ , _Astrophys. J._ 930 (2022) 70 [2202.02498].


proposed in Cho:2022kzf . We leave these for future work.

## Acknowledgements

We would like to thank Gil Young Cho, Zhihao Duan, Hee-Cheol Kim, Sungjay Lee,
Linfeng Li, Minyoung You and Haowu Wang for useful discussions. KL and KS are
supported by KIAS Grants PG006904 and QP081001 respectively. KL is also
supported in part by the National Research Foundation of Korea (NRF) Grant
funded by the Korea government (MSIT) (No.2017R1D1A1B06034369).

## References

  * (1) J. B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis, “Fermionic rational conformal field theories and modular linear differential equations,” PTEP 2021, no.8, 08B104 (2021).
  * (2) J. B. Bae, Z. Duan, K. Lee, S. Lee and M. Sarkis, “Bootstrapping fermionic rational CFTs with three characters,” JHEP 01, 089 (2022).
  * (3) Z. Duan, K. Lee, S. Lee and L. Li, “On Classification of Fermionic Rational Conformal Field Theories,” [arXiv:2210.06805 [hep-th]].
  * (4) T. Lan, L. Kong and X. G. Wen, “Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries,” Phys. Rev. B 94, no.15, 155113 (2016).
  * (5) P. Bruillard, C. Galindo, T. Hagge, S. H. Ng, J. Y. Plavnik, E. C. Rowell and Z. Wang, “Fermionic Modular Categories and the 16-fold Way,” J. Math. Phys. 58 (2017) no.4, 041704.
  * (6) P. Bonderson, E. Rowell, Z. Wang and Q. Zhang, “Congruence subgroups and super-modular categories,” Pacific Journal of Mathematics 296.2 (2018): 257-270.
  * (7) P. J. Bruillard, J. Y. Plavnik, E. C. Rowell and Q. Zhang, “Classification of super-modular categories,” arXiv preprint arXiv:1909.09843 (2019).
  * (8) P. J. Bruillard, C. Galindo, S. H. Ng, J. Y. Plavnik, E. C. Rowell and Z. Wang, “Classification of Super-Modular Categories by Rank,” Algebras and Representation Theory 23 (2019): 795-809.
  * (9) G. Y. Cho, H. c. Kim, D. Seo and M. You, “Classification of Fermionic Topological Orders from Congruence Representations,” [arXiv:2210.03681 [cond-mat.str-el]].
  * (10) J. Duncan, “Super-moonshine for Conway’s largest sporadic group,” Duke Mathematical Journal 139.2 (2007): 255-315.
  * (11) T. Creutzig, J. F. R. Duncan and W. Riedler, “Self-Dual Vertex Operator Superalgebras and Superconformal Field Theory,” J. Phys. A 51 (2018) no.3, 034001.
  * (12) T. Johnson-Freyd, “Supersymmetry and the Suzuki chain,” Tunisian Journal of Mathematics, 2020, 3(2): 309-359.
  * (13) S. M. Harrison, N. M. Paquette, D. Persson and R. Volpato, “Fun with $F_{24}$,” JHEP 02 (2021), 039.
  * (14) J. Albert, J. Kaidi and Y. H. Lin, “Topological modularity of Supermoonshine,” [arXiv:2210.14923 [hep-th]].
  * (15) Y. Tachikawa, “Topological phases and relativistic QFTs,” Notes of the lectures given in the CERN winter school, February 2018.
  * (16) A. Karch, D. Tong and C. Turner, “A Web of 2d Dualities: ${\bf Z}_{2}$ Gauge Fields and Arf Invariants,” SciPost Phys. 7 (2019), 007.
  * (17) I. Runkel and G. M. T. Watts, “Fermionic CFTs and classifying algebras,” JHEP 06 (2020), 025.
  * (18) C. T. Hsieh, Y. Nakayama and Y. Tachikawa, “Fermionic minimal models,” Phys. Rev. Lett. 126 (2021) no.19, 195701.
  * (19) J. Kulp, “Two More Fermionic Minimal Models,” JHEP 03 (2021), 124.
  * (20) J. B. Bae and S. Lee, “Emergent supersymmetry on the edges,” SciPost Phys. 11 (2021) no.5, 091.
  * (21) K. Kikuchi, J. Chen, F. Xu and C. M. Chang, “Emergent SUSY in Two Dimensions,” [arXiv:2204.03247 [hep-th]].
  * (22) J. A. Harvey and Y. Wu, “Hecke Relations in Rational Conformal Field Theory,” JHEP 09, 032 (2018).
  * (23) J. A. Harvey, Y. Hu and Y. Wu, “Galois Symmetry Induced by Hecke Relations in Rational Conformal Field Theory and Associated Modular Tensor Categories,” J. Phys. A 53 (2020) no.33, 334003.
  * (24) Y. Wu, “Hecke Operators and Galois Symmetry in Rational Conformal Field Theory,” (Doctoral dissertation, The University of Chicago, 2020).
  * (25) Z. Duan, K. Lee and K. Sun, “Hecke relations, cosets and the classification of 2d RCFTs,” JHEP 09 (2022), 202.
  * (26) G. Höhn, “Selbstdual Vertesoperatorsuperalgebren und das Babymonster,” Ph.D. thesis, Bonn University, 1995.
  * (27) M. R. Gaberdiel, H. R. Hampapura and S. Mukhi, “Cosets of Meromorphic CFTs and Modular Differential Equations,” JHEP 04, 156 (2016).
  * (28) A. N. Schellekens, “Meromorphic $c=24$ conformal field theories,” Commun. Math. Phys. 153, 159-186 (1993).
  * (29) _SageMath, the Sage Mathematics Software System (Version 8.8)_ , The Sage Developers, 2019, https://www.sagemath.org.
  * (30) E. P. Verlinde, “Fusion Rules and Modular Transformations in 2D Conformal Field Theory,” Nucl. Phys. B 300 (1988), 360-376.
  * (31) D. Friedan, Z. a. Qiu and S. H. Shenker, “Conformal Invariance, Unitarity and Two-Dimensional Critical Exponents,” Phys. Rev. Lett. 52, 1575-1578 (1984).
  * (32) D. Friedan, Z. a. Qiu and S. H. Shenker, “Superconformal Invariance in Two-Dimensions and the Tricritical Ising Model,” Phys. Lett. B 151 (1985), 37-43.
  * (33) M. A. Bershadsky, V. G. Knizhnik and M. G. Teitelman, “Superconformal Symmetry in Two-Dimensions,” Phys. Lett. B 151 (1985), 31-36.
  * (34) P. Goddard, A. Kent and D. I. Olive, “Unitary Representations of the Virasoro and Supervirasoro Algebras,” Commun. Math. Phys. 103 (1986), 105-119.
  * (35) T. Gannon, “Comments on nonunitary conformal field theories,” Nucl. Phys. B 670 (2003), 335-358.
  * (36) K. Schoutens, “Supersymmetry and Factorizable Scattering,” Nucl. Phys. B 344 (1990), 665-695.
  * (37) E. Melzer, “Supersymmetric analogs of the Gordon-Andrews identities, and related TBA systems,” [arXiv:hep-th/9412154 [hep-th]].
  * (38) A. Berkovich, B. M. McCoy and W. P. Orrick, “Polynomial identities, indices, and duality for the N=1 superconformal model $SM(2,4\nu)$,” J. Statist. Phys. 83 (1996), 795.
  * (39) A. Berkovich and B. M. McCoy, “Generalizations of the Andrews-Bressoud identities for the N=1 superconformal model $SM(2,4\nu)$,” Mathematical and Computer Modelling, 1997, 26(8-10): 37-49.
  * (40) D. Kastor, “Modular Invariance in Superconformal Models,” Nucl. Phys. B 280 (1987), 304-320.
  * (41) Y. Matsuo and S. Yahikozawa, “Superconformal Field Theory With Modular Invariance on a Torus,” Phys. Lett. B 178 (1986), 211-218.
  * (42) A. Cappelli, “Modular Invariant Partition Functions of Superconformal Theories,” Phys. Lett. B 185 (1987), 82-88.
  * (43) P. Di Vecchia, J. L. Petersen, M. Yu and H. B. Zheng, “Explicit Construction of Unitary Representations of the N=2 Superconformal Algebra,” Phys. Lett. B 174 (1986), 280-284.
  * (44) Y. Kazama and H. Suzuki, “New N=2 Superconformal Field Theories and Superstring Compactification,” Nucl. Phys. B 321 (1989), 232-268.
  * (45) S. D. Mathur, S. Mukhi and A. Sen, “On the Classification of Rational Conformal Field Theories,” Phys. Lett. B 213 (1988), 303-308.
  * (46) M. Dittmann and H. Wang. “Theta blocks related to root systems,” Mathematische Annalen (2021): 1-24.
  * (47) N. R. Scheithauer, “The Fake monster superalgebra,” Advances in Mathematics 151.2 (2000): 226-269.
  * (48) S. M. Harrison, N. M. Paquette and R. Volpato, “A Borcherds–Kac–Moody Superalgebra with Conway Symmetry,” Commun. Math. Phys. 370 (2019) no.2, 539-590.
  * (49) K. Sun, H. Wang and B. Williams, “On the hyperbolization of affine Kac–Moody algebras,” to appear.
  * (50) P. Bantay, “The Kernel of the modular representation and the Galois action in RCFT,” Commun. Math. Phys. 233 (2003), 423-438.
  * (51) J. Sturm, “On the Congruence of Modular Forms,” in Number Theory, Lecture Notes in Mathematics 1240, 275-280, Springer-Verlag Berlin, 1987.
  * (52) J. B. Bae, K. Lee and S. Lee, “Monster Anatomy,” JHEP 07 (2019), 026.
  * (53) J. B. Bae, J. A. Harvey, K. Lee, S. Lee and B. C. Rayhaun, “Conformal Field Theories with Sporadic Group Symmetry,” Commun. Math. Phys. 388, no.1, 1-105 (2021).