2022

1]Institute of Space Sciences and Astronomy, University of Malta, Malta, MSD
2080 2]Department of Physics, University of Malta, Malta 3]Instituto de
Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior
C.U., A.P. 70-543, México D.F. 04510, México

# Impact of $H_{0}$ priors on $f(T)$ late time cosmology

Rebecca Briffa<EMAIL_ADDRESS>Celia Escamilla-Rivera
<EMAIL_ADDRESS>Jackson Levi Said<EMAIL_ADDRESS>Jurgen Mifsud<EMAIL_ADDRESS>and Nathan Lee Pullicino
<EMAIL_ADDRESS>[  [  [

###### Abstract

We present a detailed analysis of the impact of $H_{0}$ priors from recent
surveys in the literature on the late time cosmology of five $f(T)$
cosmological models using cosmic chronometers, the Pantheon data set, and
baryonic acoustic oscillation data. In this work, we use three recently
reported values of $H_{0}$ that have contributed to the recent $H_{0}$ tension
problem. We find that these priors have a strong response in these analyses in
terms of all the cosmological parameters. In general, our analyses gives much
higher values of $H_{0}$ when considered against equivalent analyses without
priors while, by and large, giving lower values of the matter density
parameter. We close with a cross-analysis of each of our model, data set and
prior combination choices.

###### keywords:

Cosmology, Dark Energy

##  1 Introduction

The growing pressure from the so-called $H_{0}$ tension Bernal:2016gxb ;
DiValentino:2020zio ; DiValentino:2021izs  has prompted a reconsideration of
novel approaches to formulating a consistent cosmological model. On the other
hand, the standard model of cosmology, the $\Lambda$CDM cosmological model,
rests on overwhelming consistency with observational cosmology Planck:2018vyg
. This is only possible with the inclusion of matter beyond the standard model
of particle physics in terms of cold dark matter (CDM) which stabilizes
galactic structures Baudis:2016qwx ; Bertone:2004pz  while on larger scales
dark energy, through the cosmological constant ($\Lambda$) Peebles:2002gy ;
Copeland:2006wr , produces the acceleration observed in this regime
SupernovaSearchTeam:1998fmf ; SupernovaCosmologyProject:1998vns . While great
efforts have been put into the detection of these exotic forms of matter, they
remain elusive to direct observations Gaitskell:2004gd , and continue to be
plagued by foundational issues Weinberg:1988cp .

The prospect of a possible observational disparity in $\Lambda$CDM has
prompted renewed efforts to determine the current value of the cosmological
expansion in order to better assess the degree of inconsistency that the
concordance model may be expressing. This parameter characterizes the
discrepancy between $\Lambda$CDM independent measurements of the Hubble
parameter at current times Riess:2019cxk ; Wong:2019kwg  and its $\Lambda$CDM-
based predicted value from early time observations Planck:2018vyg ;
DES:2021wwk . While tip of the red giant branch (TRGB, Carnegie-Chicago Hubble
Program) point to a lower $H_{0}$ tension, the $H_{0}$ tension has been
growing for some time and may only be resolved either by novel observations
such as using using gravitational astronomy Graef:2018fzu ; Abbott:2017xzu ;
Baker:2019nia ; 2017arXiv170200786A , or possibly by considering other
gravitational contributions to our cosmological model.

There exist many possible modifications of general relativity (GR) that may be
further developed into an observationally viable gravitational base for a
concordance model for cosmology Sotiriou:2008rp ; Clifton:2011jh ;
CANTATA:2021ktz . These are largely built on correction terms to the Einstein-
Hilbert action Faraoni:2008mf ; Capozziello:2011et . Here, gravitational
interactions continue to be based on the curvature associated with the Levi-
Civita connection, which is the source of curvature in GR
misner1973gravitation ; nakahara2003geometry . However, there is a growing
body of work in which torsion rather than curvature is considered as the form
in which gravitation is expressed Bahamonde:2021gfp ; Aldrovandi:2013wha ;
Cai:2015emx ; Krssak:2018ywd , which has produced interesting cosmological
models. Teleparallel gravity (TG) incorporates the theories in which the
teleparallel connection Weitzenbock1923 ; Bahamonde:2021gfp  is used, which
expresses this torsion in geometry. The teleparallel connection is curvature-
less and satisfies metricity, and so all measures of curvature identically
vanish irrespective of the components of the metric. A consequence of this is
that the regular Ricci scalar $\accentset{\circ}{R}$ (over-circles represent
quantities calculated with the Levi-Civita connection) will be zero when
calculated with the teleparallel connection, i.e. $R=0$. By relating both
forms of the Ricci scalar, TG produces a torsion scalar $T$ which is
dynamically equivalent to GR, called the Teleparallel equivalent of General
Relativity (TEGR), which differs from the Einstein-Hilbert action by a
boundary term $B$. The division between the torsion scalar and boundary terms
means that a much larger range of theories that are second-order in
derivatives can be formed giving a weaker form of the generalized Lovelock
theory Lovelock:1971yv ; Gonzalez:2015sha ; Bahamonde:2019shr .

Using the same rationale as in other modified theories of gravity, such as
$f(\accentset{\circ}{R})$ gravity Sotiriou:2008rp ; Faraoni:2008mf ;
Capozziello:2011et , TEGR can be directly generalized to $f(T)$ gravity
Ferraro:2006jd ; Ferraro:2008ey ; Bengochea:2008gz ; Linder:2010py ;
Chen:2010va ; Bahamonde:2019zea . $f(T)$ gravity is a second-order
gravitational theory that has shown promise at meeting the observational
challenges that are becoming all the more pressing Cai:2015emx ;
Farrugia:2016qqe ; Finch:2018gkh ; Farrugia:2016xcw ; Iorio:2012cm ;
Deng:2018ncg . In Ref. Nesseris:2013jea  both expansion and growth data are
used to constrain several prominent models of $f(T)$ gravity, resulting in
model parameters that are within 1$\sigma$ of their corresponding $\Lambda$CDM
values where specific extended models are considered. The three most promising
extended models are then reconsidered in Ref. Anagnostopoulos:2019miu  where
the most recent observations are considered, and which shows consistency with
$\Lambda$CDM. $f(T)$ gravity has also been analyzed in terms of its impact on
the CMB power spectrum from gravitational waves where in Ref. Nunes:2018evm
the power-law model was explored in this regime. More recently, Ref.
Benetti:2020hxp  include data from big bang nucleosynthesis to constrain
further these three models.

In recent years there have been a number of cosmology independent measurements
of the current value of Hubble parameter, which is the main driver of the
growing tension between local, and early-Universe predicted values of $H_{0}$.
In this work, we explore the impact of these priors on five core models in
$f(T)$ cosmology. The literature contains several additional works where
observational data is used to constrain TG models with various uses of priors
in these models. To explore the impact of these values of measurements of
$H_{0}$ we perform several background studies on these models using these
various settings in order to better discern the impacts of priors and $f(T)$
models.

TG also produces other interesting theories such as $f(T,B)$ gravity
Bahamonde:2015zma ; Bahamonde:2016grb ; Paliathanasis:2017flf ;
Farrugia:2018gyz ; Bahamonde:2016cul ; Wright:2016ayu ; Farrugia:2020fcu ;
Capozziello:2019msc ; Farrugia:2018gyz ; Escamilla-Rivera:2019ulu  where the
boundary term plays a more active role, as well as $f(T,T_{G})$ where $T_{G}$
is the Gauss-Bonnet invariant Kofinas:2014owa ; Kofinas:2014daa ; delaCruz-
Dombriz:2017lvj ; delaCruz-Dombriz:2018nvt , and numerous scalar-tensor
theories such as Bahamonde:2019shr ; Bahamonde:2019ipm ; Bahamonde:2020cfv .
We first describe the technical details of TG and its modification to $f(T)$
gravity in Sec. 2, while in Sec. 3 we describe the data sets we use in our
Markov chain Monte Carlo (MCMC) implementation. Our core results are presented
in Sec. 4 where we also introduce the five models under consideration. We
close with a summary of our results and their place against similar studies in
the literature in Sec. 5.

##  2 $f(T)$ FLRW Cosmology

TG recasts the curvature of GR and its modifications Clifton:2011jh  with a
torsional geometric framework Hohmann:2019nat  which is built on the exchange
of the Levi-Civita connection with the teleparallel connection. This then
envelopes into a gravitational theory that embodies torsion from its
foundations Bahamonde:2021gfp .

The source of curvature in GR is the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu}$ (over-circles
are used throughout to denote quantities determined using the Levi-Civita
connection) rather than the metric tensor, which acts as the dynamical
variable of the theory but actually only quantifies the amount of geometric
deformation. TG characterizes gravitation as torsion through its associated
teleparallel connection $\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu}$ which
satisfies metricity but which is curvature-less Hayashi:1979qx ;
Aldrovandi:2013wha . This realization means that all the curvature-based
quantities will identically vanish for the teleparallel connection such as the
teleparallel Riemann tensor (the regular Levi-Civita connection Riemann tensor
naturally does not vanish). In this background, TG requires an entirely new
formulation of gravitational tensors on which to build theories (see reviews
in Refs. Krssak:2018ywd ; Cai:2015emx ; Aldrovandi:2013wha ).

The most direct way of formulating teleparallel theories of gravity is through
the tetrad $e^{A}_{\phantom{A}\mu}$ (and its inverses
$E_{A}^{\phantom{A}\mu}$) which replaces the metric tensor as the fundamental
dynamical variable in TG theories through

$\displaystyle
g_{\mu\nu}=e^{A}_{\phantom{A}\mu}e^{B}_{\phantom{B}\nu}\eta_{AB}\,,$
$\displaystyle\eta_{AB}=E_{A}^{\phantom{A}\mu}E_{B}^{\phantom{B}\nu}g_{\mu\nu}\,,$
(1)

where Latin indices represent coordinates on the tangent space while Greek
indices represent coordinates on the general manifold Cai:2015emx . In GR,
tetrads are largely suppressed in their usage, as an example, they are used to
describe spinors Chandrasekhar:1984siy . Naturally, tetrads have to satisfy
orthogonality conditions

$\displaystyle e^{A}_{\phantom{A}\mu}E_{B}^{\phantom{B}\mu}=\delta^{A}_{B}\,,$
$\displaystyle
e^{A}_{\phantom{A}\mu}E_{A}^{\phantom{A}\nu}=\delta^{\nu}_{\mu}\,,$ (2)

for internal consistency.

The teleparallel connection can be defined as Weitzenbock1923 ; Krssak:2018ywd

$\Gamma^{\sigma}_{\phantom{\sigma}\nu\mu}:=E_{A}^{\phantom{A}\sigma}\left(\partial_{\mu}e^{A}_{\phantom{A}\nu}+\omega^{A}_{\phantom{A}B\mu}e^{B}_{\phantom{B}\nu}\right)\,,$
(3)

where $\omega^{A}_{\phantom{A}B\mu}$ is a flat spin connection which is
responsible for incorporating the local Lorentz transformation invariance of
the theory (which arises explicitly due to the appearance of the tangent space
indices). This can be contrasted with GR where the spin connections
(associated with their tetrads) are not flat misner1973gravitation . In TG,
tetrad-spin connection pairs represent gravitational and local degrees of
freedom respectively and both contribute to a system’s equations of motion.
Similar to the way that the Levi-Civita connection builds up to the Riemann
tensor, the teleparallel connection can be used to describe a torsion tensor
Hayashi:1979qx

$T^{\sigma}_{\phantom{\sigma}\mu\nu}:=2\Gamma^{\sigma}_{\phantom{\sigma}[\nu\mu]}\,,$
(4)

where square brackets denote an antisymmetric operator, and where
$T^{\sigma}_{\phantom{\sigma}\mu\nu}$ represents the gauge field strength of
gravity in TG Aldrovandi:2013wha . The torsion tensor is covariant under both
diffeomorphisms and local Lorentz transformations. By taking suitable
contractions of the torsion tensor, a torsion scalar can be written down as
Krssak:2018ywd ; Cai:2015emx ; Aldrovandi:2013wha ; Bahamonde:2021gfp

$T:=\frac{1}{4}T^{\alpha}_{\phantom{\alpha}\mu\nu}T_{\alpha}^{\phantom{\alpha}\mu\nu}+\frac{1}{2}T^{\alpha}_{\phantom{\alpha}\mu\nu}T^{\nu\mu}_{\phantom{\nu\mu}\alpha}-T^{\alpha}_{\phantom{\alpha}\mu\alpha}T^{\beta\mu}_{\phantom{\beta\mu}\beta}\,,$
(5)

which can be arrived at either by demanding that $T$ be equivalent to the
Ricci scalar (up to a boundary term), or by interpreting TG as a gauge theory
of translations, which then naturally leads to this form of the torsion
scalar. The Ricci scalar is dependent only on the Levi-Civita connection, and
similarly, the torsion scalar is entirely dependent on the teleparallel
connection.

Theories based on the Levi-Civita connection produce the Ricci scalar
$\accentset{\circ}{R}$ among other measures of curvature. By exchanging this
with the teleparallel connection, this form of the Ricci scalar will
identically vanish, meaning $R\equiv 0$ (where we emphasize that
$R=R(\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu})$ and
$\accentset{\circ}{R}=\accentset{\circ}{R}(\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu})$).
In terms of gravitational scalars, the torsion and Ricci scalar are equivalent
up to a boundary term $B$, which can be represented through the relation
Bahamonde:2015zma ; Farrugia:2016qqe

$R=\accentset{\circ}{R}+T-B=0\,.$ (6)

This is the base of the TEGR action, which is thus guaranteed to feature
identical equations of motion as the Einstein-Hilbert action.

Following the same reasoning as the many extensions to GR, such as
$f(\accentset{\circ}{R})$ gravity DeFelice:2010aj ; Capozziello:2011et , TEGR
can be arbitrarily extended to $f(T)=-T+\mathcal{F}(T)$ gravity by raising the
TEGR action to Ferraro:2006jd ; Ferraro:2008ey ; Bengochea:2008gz ;
Linder:2010py ; Chen:2010va ; RezaeiAkbarieh:2018ijw

$\mathcal{S}_{\mathcal{F}(T)}=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{4}x\;e\left(-T+\mathcal{F}(T)\right)+\int\mathrm{d}^{4}x\;e\mathcal{L}_{\text{m}}\,,$
(7)

where $\kappa^{2}=8\pi G$, $\mathcal{L}_{\text{m}}$ is the matter Lagrangian,
and $e=\det\left(e^{a}_{\phantom{a}\mu}\right)=\sqrt{-g}$ is the tetrad
determinant. The TEGR limit will then be described by
$\mathcal{F}(T)\rightarrow 0$, while $\Lambda$CDM is recovered when the
arbitrary Lagrangian tends to a constant $\Lambda$ value. The most important
difference, in terms of the dynamical equations, between
$\mathcal{F}(\accentset{\circ}{R})$ and $\mathcal{F}(T)$ is that the total
divergence term $B$ (6) is no longer a boundary term in
$\mathcal{F}(\accentset{\circ}{R})=\mathcal{F}(-T+B)$ gravity. This is the
reason why $\mathcal{F}(\accentset{\circ}{R})$ equations of motion become
fourth-order while $f(T)$ remains second-order, as in TEGR. This is
advantageous for several reasons such as being naturally Gauss-Ostrogadsky
ghost free Aldrovandi:2013wha  and being more amenable to numerical
approaches.

The field equations for $f(T)$ gravity can then be written as

$\displaystyle W_{a}^{\phantom{a}\mu}:=e^{-1}$
$\displaystyle\partial_{\nu}\left(eE_{a}^{\phantom{a}\rho}S_{\rho}^{\phantom{\rho}\mu\nu}\right)\left(-1+\mathcal{F}_{T}\right)-E_{a}^{\phantom{a}\lambda}T^{\rho}_{\phantom{\rho}\nu\lambda}S_{\rho}^{\phantom{\rho}\nu\mu}\left(-1+\mathcal{F}_{T}\right)+\frac{1}{4}E_{a}^{\phantom{a}\mu}\left(-T+\mathcal{F}(T)\right)$
$\displaystyle+E_{a}^{\phantom{a}\rho}S_{\rho}^{\phantom{\rho}\mu\nu}\partial_{\nu}\left(T\right)\mathcal{F}_{TT}+E_{b}^{\phantom{b}\lambda}\omega^{b}_{\phantom{b}a\nu}S_{\lambda}^{\phantom{\lambda}\nu\mu}\left(-1+\mathcal{F}_{T}\right)=\kappa^{2}E_{a}^{\phantom{a}\rho}\Theta_{\rho}^{\phantom{\rho}\mu}\,,$
(8)

where subscripts denote derivatives
($\mathcal{F}_{T}=\partial\mathcal{F}/\partial T$ and
$\mathcal{F}_{TT}=\partial^{2}\mathcal{F}/\partial T^{2}$), and
$\Theta_{\rho}^{\phantom{\rho}\nu}$ is the regular energy-momentum tensor. The
separate tetrad and spin connection variations produce the field equations
Bahamonde:2021gfp

$W_{(\mu\nu)}=\kappa^{2}\Theta_{\mu\nu}\,,\quad\text{and}\quad
W_{[\mu\nu]}=0\,,$ (9)

which represent the degrees of freedom associated with the tetrad and spin
connection, respectively. For any metric ansatz a unique tetrad-spin
connection pair exists such that the local frame is compatible with all the
spin connection components vanishing, which is called the Weitzenböck gauge
Krssak:2018ywd ; Bahamonde:2021gfp . In this setting, $W_{[\mu\nu]}$ vanishes
taking a select choice of tetrad components, while still satisfying the metric
equations in Eq. (1).

In this work we explore the cosmology of a flat homogeneous and isotropic
Universe which can be represented by the tetrad Krssak:2015oua ;
Tamanini:2012hg

$e^{A}_{\phantom{A}\mu}=\text{diag}\left(1,\,a(t),\,a(t),\,a(t)\right)\,,$
(10)

where $a(t)$ is the scale factor in cosmic time $t$, and which was shown to
universally satisfy the Weitzenböck gauge conditions in Ref. Hohmann:2019nat .
This reproduces, through Eq. (1), the regular flat
Friedmann–Lemaître–Robertson–Walker (FLRW) metric misner1973gravitation

$\mathrm{d}s^{2}=\mathrm{d}t^{2}-a^{2}(t)\left(\mathrm{d}x^{2}+\mathrm{d}y^{2}+\mathrm{d}z^{2}\right)\,.$
(11)

Taking the standard definition of Hubble parameter as $H=\dot{a}/a$ where
over-dots refer to derivatives with respect to cosmic time, the equivalence in
Eq. (6). In this setting, we find $T=-6H^{2}$ and
$B=-6\left(3H^{2}+\dot{H}\right)$, which produces the regular Ricci scalar
through

$\accentset{\circ}{R}=-T+B=-6\left(2H^{2}+\dot{H}\right)\,.$ (12)

The Friedmann equations can then be written down as Bahamonde:2021gfp

$\displaystyle H^{2}+\frac{T}{3}\mathcal{F}_{T}-\frac{\mathcal{F}}{6}$
$\displaystyle=\frac{\kappa^{2}}{3}\rho\,,$ (13)
$\displaystyle\dot{H}\left(1-\mathcal{F}_{T}-2T\mathcal{F}_{TT}\right)$
$\displaystyle=-\frac{\kappa^{2}}{2}\left(\rho+p\right)\,,$ (14)

where we denote the energy density and pressure of the total matter sector by
$\rho$ and $p$, respectively.

##  3 Observational Data

We here present the observational data sets which will be considered in the
below analyses. For our baseline data set, we consider $H(z)$ data along with
a Supernovae type Ia (SNIa) compilation data set.

  * •

For the Hubble parameter data, we adopt thirty–one data points which were
inferred via the cosmic chronometers (CC) technique. Such a technique enables
us to directly derive information about the Hubble function at several
redshifts, up to $z\lesssim 2$. Since the adopted CC data is primarily based
on measurements of the age difference between two passively–evolving galaxies
that formed at the same time but are separated by a small redshift interval
(from which one can compute $\Delta z/\Delta t$), CC were found to be more
reliable than any other method based on an absolute age determination for
galaxies Jimenez:2001gg . Our adopted CC data points were compiled from Refs.
2014RAA….14.1221Z ; Jimenez:2003iv ; Moresco:2016mzx ; Simon:2004tf ;
2012JCAP…08..006M ; 2010JCAP…02..008S ; Moresco:2015cya , which are
independent of the Cepheid distance scale and from any cosmological model,
although they are dependent on the modelling of stellar ages, which is based
on robust stellar population synthesis techniques (see, for instance, Refs.
Gomez-Valent:2018hwc ; Lopez-Corredoira:2017zfl ; Lopez-Corredoira:2018tmn ;
Verde:2014qea ; 2012JCAP…08..006M ; Moresco:2016mzx  for analyses related to
CC systematics). The corresponding $\chi^{2}_{H}$ estimator is given by

$\chi^{2}_{H}(\Theta)=\sum_{i=1}^{31}\frac{\left(H(z_{i},\,\Theta)-H_{\mathrm{obs}}(z_{i})\right)^{2}}{\sigma_{H}^{2}(z_{i})}\,,$
(15)

where $H(z_{i},\Theta)$ are the theoretical Hubble parameter values at
redshift $z_{i}$ with model parameters $\Theta$, $H_{\mathrm{obs}}(z_{i})$ are
the corresponding measured values of the Hubble parameter at $z_{i}$ with
observational error of $\sigma_{H}(z_{i})$.

  * •

The other component of our baseline data set consists of the Pantheon
compilation of 1048 SNIa relative luminosity distance measurements spanning
the redshift range of $0.01<z<2.3$ Scolnic:2017caz . Henceforth, we will be
denoting the Pantheon SNIa compilation by SN. The publicly available release
of the SN catalog provides SNIa magnitudes corrected for systematic effects,
including the stretch of the light–curve, the color at maximum brightness and
the stellar mass of the host galaxy. Since the apparent magnitude of each SNIa
needs to be calibrated via an arbitrary fiducial absolute magnitude $M$, we
will be considering $M$ as a nuisance parameter in our MCMC analyses. This can
be implemented through the use of the theoretical values of the distance
moduli

$\mu(z_{i},\,\Theta)=5\log_{10}\left[D_{L}(z_{i},\,\Theta)\right]+M\,,$ (16)

at redshift $z_{i}$ via the corresponding computation of the luminosity
distance

$D_{L}(z_{i},\Theta)=c\,(1+z_{i})\int_{0}^{z_{i}}{\frac{\mathrm{d}z^{\prime}}{H(z^{\prime},\,\Theta)}}\,,$
(17)

where $c$ is the speed of light, and the nuisance parameter $M$ encodes the
Hubble constant which has to be marginalized over in the MCMC analyses. The
associated $\chi^{2}_{\mathrm{SN}}$ is specified by 2011ApJS..192….1C

$\chi_{\text{SN}}^{2}(\Theta)=\left(\Delta\mu(z_{i},\,\Theta)\right)^{T}C_{\text{SN}}^{-1}\,\Delta\mu(z_{i},\,\Theta)+\ln\left({\frac{S}{2\pi}}\right)-\frac{k^{2}(\Theta)}{S}\,,$
(18)

where $C_{\text{SN}}$ is the total covariance matrix, $S$ is the sum of all
the components of $C_{SN}^{-1}$, while $k$ is given by

$k(\Theta)={\left(\Delta\mu(z_{i},\,\Theta)\right)^{T}\cdotp
C_{\text{SN}}^{-1}}\,,$ (19)

with $\Delta\mu(z_{i},\,\Theta)=\mu(z_{i},\,\Theta)-\mu_{\text{obs}}(z_{i})$.

  * •

We further consider a joint baryon acoustic oscillation (BAO) data set
composed of independent data points. The BAO data set incorporates the SDSS
Main Galaxy Sample measurement at $z_{\mathrm{eff}}=0.15$ Ross:2014qpa , the
six–degree Field Galaxy Survey measurement at $z_{\mathrm{eff}}=0.106$
2011MNRAS.416.3017B , and the BOSS DR11 quasar Lyman–$\alpha$ measurement at
$z_{\mathrm{eff}}=2.4$ Bourboux:2017cbm . We further consider the angular
diameter distances and $H(z)$ measurements of SDSS–IV eBOSS DR14 quasar survey
at $z_{\mathrm{eff}}=\\{0.98,\,1.23,\,1.52,\,1.94\\}$ Zhao:2018gvb , along
with the SDSS–III BOSS DR12 consensus BAO measurements of the Hubble parameter
and the corresponding comoving angular diameter distances at
$z_{\mathrm{eff}}=\\{0.38,\,0.51,\,0.61\\}$ Alam:2016hwk , where in these two
BAO data sets we consider the full covariance matrix in our MCMC analyses. For
the considered BAO data sets we computed the Hubble distance $D_{H}(z)$,
comoving angular diameter distance $D_{M}(z)$, and volume–average distance
$D_{V}(z)$, which are respectively specified by

$D_{H}(z)=\frac{c}{H(z)}\,,\;\;D_{M}(z)=(1+z)D_{A}(z)\,,\;\;D_{V}(z)=\left[(1+z)^{2}D_{A}^{2}(z)\frac{z}{H(z)}\right]^{1/3}\,,$
(20)

with $D_{A}(z)=(1+z)^{-2}D_{L}(z)$ being the angular diameter distance. In
order to use the reported BAO results in our MCMC analyses, we had to consider
the corresponding combination of parameters
$\mathcal{G}(z_{i})=\\{D_{V}(z_{i})/r_{s}(z_{d}),\allowbreak\,r_{s}(z_{d})/D_{V}(z_{i}),\allowbreak\,D_{H}(z_{i}),\allowbreak\,D_{M}(z_{i})(r_{s,\mathrm{fid}}(z_{d})/r_{s}(z_{d})),\allowbreak\,H(z_{i})(r_{s}(z_{d})/r_{s,\mathrm{fid}}(z_{d})),\allowbreak\,D_{A}(z_{i})(r_{s,\mathrm{fid}}(z_{d})/r_{s}(z_{d}))\\}$,
for which we had to compute the comoving sound horizon $r_{s}(z)$ at the end
of the baryon drag epoch at redshift $z_{d}\approx 1059.94$ Planck:2018vyg ,
such that

$r_{s}(z)=\int_{z}^{\infty}\frac{c_{s}(\tilde{z})}{H(\tilde{z})}\,\mathrm{d}z=\frac{1}{\sqrt{3}}\int_{0}^{1/(1+z)}\frac{\mathrm{d}a}{a^{2}H(a)\sqrt{1+\left[3\Omega_{b,0}/(4\Omega_{\gamma,0})\right]a}}\,,$
(21)

where we have adopted $\Omega_{b,0}=0.02242$ Planck:2018vyg ,
$T_{0}=2.7255\,\mathrm{K}$ 2009ApJ…707..916F , and a fiducial value of
$r_{s,\mathrm{fid}}(z_{d})=147.78\,\mathrm{Mpc}$. The corresponding
$\chi^{2}_{\mathrm{BAO}}(\Theta)$ is specified by

$\chi^{2}_{\mathrm{BAO}}(\Theta)=(\Delta\mathcal{G}(z_{i},\Theta))^{T}\,C^{-1}_{\mathrm{BAO}}\,\Delta\mathcal{G}(z_{i},\Theta)\,,$
(22)

where
$\Delta\mathcal{G}(z_{i},\Theta)=\mathcal{G}(z_{i},\Theta)-\mathcal{G}_{\mathrm{obs}}(z_{i})$,
and $C_{\mathrm{BAO}}$ is the covariance matrix of all the considered BAO
observations.

Further to the above data sets, we will also be analysing the impact of an
$H_{0}$ prior value on our $f(T)$ model parameter constraints. We will be
considering the latest SH0ES local estimate Riess:2019cxk  of $H_{0}=74.22\pm
1.82\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$ (R19) based on SN in the Hubble flow,
the H0LiCOW Collaboration’s Wong:2019kwg  measurement which relies on strong
lensing from quasars and has a value of $73.3^{+1.7}_{-1.8}\,{\rm
km\,s}^{-1}{\rm Mpc}^{-1}$ (HW), and finally the measurement using the tip of
the red giant branch (TRGB) as a standard candle with $H_{0}=69.8\pm 1.9\,{\rm
km\,s}^{-1}{\rm Mpc}^{-1}$ Freedman:2019jwv . While other measurements exist
(see, for instance, Ref. Abbott:2017xzu ), the aforementioned measurements are
the most representative model–independent local values which have been
exhaustively studied in the literature.

##  4 Constraints of $f(T)$ Cosmological Models

We take five models within the $f(T)$ gravity framework in which to probe the
impact of recent cosmology model independent measurements of $H_{0}$. These
have gained prominence in the literature and are frequently studied since they
are reported to mirror very well our cosmological history. We analyze the
impact that these priors have on the resulting parameter values for the data
sets described in Sec. 3. In this context, we try to separate the impacts
between the choice of prior on $H_{0}$ and the $f(T)$ model under
consideration.

In order to better evaluate the performance of each $f_{i}$CDM model against
$\Lambda$CDM, we compute the Akaike information criterion (AIC) Akaike:1974
to assess which model ultimately best supports the data. The AIC is defined as

$\text{AIC}=-2\ln L_{\rm max}+2n\,,$ (23)

where $L_{\rm max}$ is the maximum value of the likelihood function for each
model and data set together with prior combination, and $n$ is the number of
parameters involved in the estimation routine. In practice, lower values of
AIC indicate better performance against observational data set combinations.
On the other hand, we also consider the Bayesian information criterion (BIC)
10.1214/aos/1176344136  which is closely related to AIC and defined as

$\text{BIC}=-2\ln L_{\rm max}+n\ln m\,,$ (24)

where $m$ is the sample size of the observational data combination. This
variant of the AIC also imposes a penalty for models having a large number of
parameters to estimate, which is more impactful in BIC. The explicit
appearance of the number of points in a data set also makes the BIC an
arguably better measure of the performance of AIC against observational data.

###  4.1 $f_{1}(T)$CDM Model

The power-law model Bengochea:2008gz  was first explored due to its ability to
produce an accelerated late-time Universe. This model is expressed as

$\mathcal{F}_{1}(T)=\alpha_{1}\left(-T\right)^{b_{1}}\,,$ (25)

where $\alpha_{1}$ and $b_{1}$ are constants. By evaluating the Friedmann
equation (13) at current times, we can obtain

$\alpha_{1}=\left(6H_{0}^{2}\right)^{1-b_{1}}\frac{1-\Omega_{M_{0}}}{1-2b_{1}}\,,$
(26)

where $\Omega_{M_{0}}=\Omega_{m_{0}}+\Omega_{r_{0}}$, and where
$\Omega_{m_{0}}$ and $\Omega_{r_{0}}$ are the density parameters for matter
and radiation at current times, respectively. This makes $b_{1}$ the only new
model parameter for $f_{1}$CDM rather than the superficial addition of two
parameters shown in Eq. (25).

By defining the normalised Hubble parameter $E(z):=H(z)/H_{0}$, we can write
the Friedmann equation 13 for this model as

$E^{2}(z)=\Omega_{m_{0}}\left(1+z\right)^{3}+\Omega_{r_{0}}\left(1+z\right)^{4}+\left(1-\Omega_{m_{0}}-\Omega_{r_{0}}\right)E^{2b_{1}}(z)\,,$
(27)

which reduces to the $\Lambda$CDM model for $b_{1}=0$, and to the Dvali,
Gabadadze and Porrati (DGP) Dvali:2000hr ; Barcenas-Enriquez:2018ili  model
for $b_{1}=1/2$. For $b_{1}=1$, the additional component in the Friedmann
equation produces a rescaled gravitational constant term in the density
parameters, i.e. this is the pure GR limit. This gives an upper bound such
that $b_{1}<1$ for an accelerating Universe.

Figure 1: Left: Confidence levels (C.L) and posteriors for the $f_{1}$CDM (Power Law) model (25) using CC+SN data along the $H_{0}$ priors: R19 (green color), HW (red color) and TRGB (purple color), are respectively shown. Right: C.L and posteriors for the Power Law model (25) using CC+SN+BAO data are shown using the same prior colors. Table 1: Results for the $f_{1}$CDM (Power Law) model (25). First column: data sets used to constrain the model, including the $H_{0}$ priors. Second column: $H_{0}$ values derived from the analysis. Third column: Constrained $\Omega_{m,0}$. Fourth column: Best fit $b_{1}$ values. Fifth column: Nuisance parameter, $M$. Sixth column: $\chi^{2}_{\text{min}}$. From seventh up to tenth column: AIC and BIC with their respective differences with $\Lambda$CDM within the corresponding data set and prior combination (as shown in Appendix A). Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $b_{1}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $68.5\pm 1.8$ |  $0.350^{+0.045}_{-0.064}$ |  $-0.22^{+0.41}_{-0.48}$ |  $-19.390^{+0.053}_{-0.055}$ |  1040.94 |  1048.94 |  1068.88 |  1.45 |  6.43
$\mathrm{CC+SN+R19}$ |  $71.3^{+1.3}_{-1.4}$ |  $0.326^{+0.045}_{-0.065}$ |  $-0.13^{+0.40}_{-0.50}$ |  $-19.314^{+0.039}_{-0.038}$ |  1045.83 |  1053.83 |  1073.77 |  1.51 |  6.50
$\mathrm{CC+SN+HW}$ |  $71.0\pm 1.3$ |  $0.329^{+0.045}_{-0.062}$ |  $-0.16^{+0.41}_{-0.48}$ |  $-19.324^{+0.038}_{-0.037}$ |  1044.50 |  1052.50 |  1072.44 |  1.51 |  6.50
$\mathrm{CC+SN+TRGB}$ |  $69.1^{+1.4}_{-1.3}$ |  $0.344^{+0.045}_{-0.063}$ |  $-0.20^{+0.42}_{-0.47}$ |  $-19.375\pm 0.040$ |  1041.55 |  1049.55 |  1069.49 |  1.87 |  6.85
CC + SN + BAO |  $67.1\pm 1.6$ |  $0.294\pm 0.015$ |  $0.06\pm 0.13$ |  $-19.435\pm 0.047$ |  1057.13 |  1065.13 |  1085.13 |  1.68 |  6.68
$\mathrm{CC+SN+BAO+R19}$ |  $69.9\pm 1.2$ |  $0.305^{+0.014}_{-0.013}$ |  $-0.14^{+0.12}_{-0.13}$ |  $-19.359^{+0.035}_{-0.034}$ |  1066.87 |  1074.87 |  1094.87 |  0.56 |  5.56
$\mathrm{CC+SN+BAO+HW}$ |  $69.7\pm 1.2$ |  $0.304^{+0.014}_{-0.012}$ |  $-0.12^{+0.12}_{-0.13}$ |  $-19.366^{+0.035}_{-0.033}$ |  1064.92 |  1072.92 |  1086.92 |  0.89 |  5.89
$\mathrm{CC+SN+BAO+TRGB}$ |  $68.1\pm 1.2$ |  $0.298\pm 0.014$ |  $-0.01^{+0.11}_{-0.12}$ |  $-19.407\pm 0.036$ |  1058.56 |  1066.56 |  1086.56 |  2.00 |  7.00

In Fig. 1, the posteriors and confidence regions are shown for both CC+SN and
CC+SN+BAO data set combinations. In these plots, we also show the results for
each of the priors on $H_{0}$ which are described in Sec. 3. Immediately, one
notices that the biggest impact of having a prior is to raise the value of
$H_{0}$ in the results from the scenario in which there is no prior. A similar
effect occurs for the $\Omega_{\rm m,0}$ parameter but this is less
pronounced.

To see the precision values, we give specific results in Table 1 where it
becomes clear that the highest value of the Hubble constant is achieved for
the CC+SN data set with an R19 prior
($H_{0}=71.3^{+1.3}_{-1.4}$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$)
which occurs due to early Universe impacts on BAO data having the effect of
reducing the value of the $H_{0}$ parameter while R19 being the highest prior
produces a shift to higher values in the MCMC runs. In tandem, the value of
$\Omega_{\rm m,0}$ reaches a minimum for the CC+SN data set for the R19 prior,
which tallies with having a large value of $H_{0}$ since most of the energy in
this scenario will appear as an effective dark energy. Following a similar
reasoning, the lowest value of the Hubble constant appears for the no prior
($H_{0}=67.1\pm
1.6$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$)
scenario in which CC+SN+BAO data set is selected. Here, the combination of BAO
data and the impact of not assuming a prior on $H_{0}$ brings the value of the
Hubble constant to its lowest point. Conversely, this raises the value of
$\Omega_{\rm m,0}$ to its highest value for the CC+SN+BAO data set.

In all cases, the value of the $b_{1}$ parameter is found to be within
1$\sigma$ of their corresponding $\Lambda$CDM value except for the CC+SN+BAO
with an R19 prior run where this moves to being within 2$\sigma$ of the
corresponding $\Lambda$CDM value. Coincidentally, from the array of all the
MCMC runs, this run also turns out to give the lowest values of both
$\Delta$AIC and $\Delta$BIC in comparison to the respective $\Lambda$CDM run
which is interesting since this becomes the favoured of model of the group of
runs. On the other hand, the $b_{4}$ parameter may effect cosmic perturbations
giving an intriguing division between background and perturbation behaviour in
terms of model parameters.

These results are within the confidence regions of previous studies in the
literature Nesseris:2013jea ; 2018ApJ…855…89X ; Benetti:2020hxp ; Wang:2020zfv
for the closest prior value. We nuance these results in the literature with
more in-depth analysis of the impact of other priors on the eventual
cosmological parameter values.

###  4.2 $f_{2}(T)$CDM Model

The Linder model Linder:2010py  was designed to produce late-time accelerated
expansion and is described by

$\mathcal{F}_{2}(T)=\alpha_{2}T_{0}\left(1-\text{Exp}\left[-b_{2}\sqrt{T/T_{0}}\right]\right)\,,$
(28)

where $\alpha_{2}$ and $b_{2}$ are constants, and where
$T_{0}=T|_{t=t_{0}}=6H_{0}^{2}$. Evaluating at current times, the Friedmann
equation relates these constants through

$\alpha_{2}=\frac{1-\Omega_{M_{0}}}{\left(1+b_{2}\right)e^{-b_{2}}-1}\,,$ (29)

which makes $b_{2}$ the new parameter for the $f_{2}$CDM model. The
corresponding Friedmann equation can then be written as

$E^{2}\left(z\right)=\Omega_{m_{0}}\left(1+z\right)^{3}+\Omega_{r_{0}}\left(1+z\right)^{4}+\frac{1-\Omega_{m_{0}}-\Omega_{r_{0}}}{(b_{2}+1)e^{-b_{2}}-1}\left[\left(1+b_{2}E(z)\right)\text{Exp}\left[-b_{2}E(z)\right]-1\right]\,,$
(30)

which reduces to $\Lambda$CDM as $b_{2}\rightarrow+\infty$. We perform our
analysis for $1/b_{2}$ so that this limit becomes $1/b_{2}\rightarrow 0^{+}$,
which also makes the numerical approach more stable.

Figure 2: Left: C.L and posteriors for the $f_{2}$CDM (Linder) model (28) using CC+SN data along the $H_{0}$ priors: R19 (green color), HW (red color) and TRGB (purple color), respectively. Right: C.L and posteriors for the Linder model (28) using CC+SN+BAO data along the same priors denoted. Table 2: Results for the $f_{2}$CDM (Linder) model (28). First column: data sets used to constrain the model, including the $H_{0}$ priors. Second column: $H_{0}$ values derived from the analysis. Third column: Constrained $\Omega_{m,0}$. Fourth column: Best fit $1/b_{2}$ values. Fifth column: Nuisance parameters, $M$. Sixth column: $\chi^{2}_{\text{min}}$. From seventh up to tenth column: AIC and BIC with their respective differences with $\Lambda$CDM within the corresponding data set and prior combination (as shown in Appendix A). Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $\frac{1}{b_{2}}$ |  M |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $68.7^{+1.8}_{-1.7}$ |  $0.298^{+0.031}_{-0.035}$ |  $0.101^{+0.227}_{-0.098}$ |  $-19.43^{+0.57}_{-0.47}$ |  1041.49 |  1049.49 |  1069.43 |  2.00 |  6.98
$\mathrm{CC+SN+R19}$ |  $71.4\pm 1.3$ |  $0.283^{+0.027}_{-0.036}$ |  $0.088^{+0.252}_{-0.086}$ |  $-19.28^{+0.50}_{-0.51}$ |  1046.32 |  1054.32 |  1074.25 |  2.00 |  6.99
$\mathrm{CC+SN+HW}$ |  $71.0^{+1.3}_{-1.2}$ |  $0.285^{+0.027}_{-0.036}$ |  $0.096^{+0.245}_{-0.093}$ |  $-19.37^{+0.45}_{-0.34}$ |  1044.99 |  1052.99 |  1072.93 |  2.00 |  6.99
$\mathrm{CC+SN+TRGB}$ |  $69.2\pm 1.3$ |  $0.296^{+0.028}_{-0.035}$ |  $0.088^{+0.239}_{-0.085}$ |  $-19.36^{+0.36}_{-0.37}$ |  1041.69 |  1049.69 |  1069.62 |  2.00 |  6.99
CC + SN + BAO |  $66.9^{+1.5}_{-1.6}$ |  $0.294\pm 0.016$ |  $0.22^{+0.12}_{-0.15}$ |  $-19.38^{+0.22}_{-0.35}$ |  1056.52 |  1064.62 |  1084.52 |  1.06 |  6.06
$\mathrm{CC+SN+BAO+R19}$ |  $68.71^{+0.88}_{-0.96}$ |  $0.300\pm 0.014$ |  $0.079^{+0.098}_{-0.064}$ |  $-19.35^{+0.19}_{-0.24}$ |  1068.31 |  1076.31 |  1096.31 |  2.00 |  7.00
$\mathrm{CC+SN+BAO+HW}$ |  $68.58^{+0.89}_{-0.92}$ |  $0.300^{+0.013}_{-0.014}$ |  $0.076^{+0.105}_{-0.060}$ |  $-19.389^{+0.045}_{-0.047}$ |  1066.03 |  1074.03 |  1094.03 |  2.00 |  7.00
$\mathrm{CC+SN+BAO+TRGB}$ |  $67.7\pm 1.0$ |  $0.297\pm 0.014$ |  $0.128^{+0.111}_{-0.099}$ |  $-19.46^{+0.37}_{-0.26}$ |  1058.47 |  1066.47 |  1086.47 |  1.90 |  6.90

For this case, we show the posteriors and confidence regions of all the output
MCMC runs in Fig. 2. Some similar patterns to the $f_{1}$CDM model emerge such
as priors on $H_{0}$ pushing the value of the Hubble constant from the MCMC
runs to higher values, and some anti-correlation between $\Omega_{\rm m,0}$
and the model parameter $b_{i}$. However, this is a different model and has
different properties. By design, this naturally leads to an accelerating late
time Universe Linder:2010py , which we do confirm in this case and, in fact we
get slightly higher values for the expansion rate when compared with the
$f_{1}$CDM power-law model.

In Table. 2, we show the precision outputs for each of the MCMC runs together
with their settings for priors on $H_{0}$. While slightly higher, we again
find that the largest $H_{0}$ is achieved for the CC+SN data set with an R19
prior
($H_{0}=71.4\pm^{1}.3$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$)
which is an expected feature as in $f_{1}$CDM . In tandem, we then find the
lowest value of $\Omega_{\rm m,0}$ for this MCMC run in this data set.
Analogously, the CC+SN+BAO MCMC run with no prior
($H_{0}=66.9^{+1.5}_{-1.6}$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$)
gives the least expanding cosmology. Given how close the R19 and HW priors are
in value, in all the MCMC runs, they always gave comparatively similar results
for the ensuing cosmological parameters.

For numerical stability we take $1/b_{2}$ to be the active parameter in our
MCMC runs, which means that the $\Lambda$CDM limit would be represented by
$1/b_{2}\rightarrow 0^{+}$. Dissimilar to $f_{1}$CDM, most models fall within
a 2$\sigma$ rather than 1$\sigma$ making them slightly further away from
favouring $\Lambda$CDM directly. Another curious feature of the results is
that despite each of the $\Delta$AIC and $\Delta$BIC being calculated for each
corresponding data set and prior combination for $\Lambda$CDM, it turns out
that most $\Delta$AIC and $\Delta$BIC values tend to about 2.0 and 6.99
respectively, with the closest to $\Lambda$CDM being the CC+SN+BAO with no
prior setting. Given that BAO depends on the early Universe to some extend,
this is not an unexpected result for the model.

While still consistent with the literature in Refs. Nesseris:2013jea ;
2018ApJ…855…89X ; Benetti:2020hxp ; Wang:2020zfv , the various priors do add a
lot more detail to the impact of the priors on $H_{0}$ and their effect on the
MCMC runs and the resulting model parameter values. For instance, the priors
by and large reduce the uncertainties in the resulting parameters which is
interesting for purposes of comparison with $\Lambda$CDM.

###  4.3 $f_{3}(T)$CDM Model

Motivated by works in $f(\accentset{\circ}{R})$ gravity Linder:2009jz , Ref.
Nesseris:2013jea  proposes the following variant of the Linder model

$\mathcal{F}_{3}(T)=\alpha_{3}T_{0}\left(1-\text{Exp}\left[-b_{3}T/T_{0}\right]\right)\,,$
(31)

where $\alpha_{3}$ and $b_{3}$ are constants, which can be related through

$\alpha_{3}=\frac{1-\Omega_{M_{0}}}{\left(1+2b_{3}\right)e^{-b_{3}}-1}\,,$
(32)

where the Friedmann equation (13) was evaluated at current times. Thus, the
Friedmann equation can be written as

$E^{2}\left(z\right)=\Omega_{m_{0}}\left(1+z\right)^{3}+\Omega_{r_{0}}\left(1+z\right)^{4}+\frac{1-\Omega_{m_{0}}-\Omega_{r_{0}}}{(1+2b_{3})e^{-b_{3}}-1}\left[\left(1+2b_{3}E^{2}(z)\right)\text{Exp}\left[-b_{3}E^{2}(z)\right]-1\right]\,,$
(33)

which tends to $\Lambda$CDM as $b_{3}\rightarrow+\infty$ similar to
$f_{2}$CDM. As in $f_{2}$CDM, we again perform our analysis for $1/b_{3}$ so
that the analysis is more stable. In this case, the $\Lambda$CDM limits comes
as $1/b_{3}\rightarrow 0^{+}$.

Figure 3: Left: C.L and posteriors for the $f_{3}$CDM model (31) using CC+SN data along the $H_{0}$ priors: R19 (green color), HW (red color) and TRGB (purple color), respectively. Right: C.L and posteriors for the model (31) using CC+SN+BAO data along the same priors denoted. Table 3: Results for the $f_{3}$CDM model (31). First column: data sets used to constrain the model, including the $H_{0}$ priors. Second column: $H_{0}$ values derived from the analysis. Third column: Constrained $\Omega_{m,0}$. Fourth column: Best fit $1/b_{3}$ values. Fifth column: Nuisance parameter, $M$. Sixth column: $\chi^{2}_{\text{min}}$. From seventh up to tenth column: AIC and BIC with their respective differences with $\Lambda$CDM within the corresponding data set and prior combination (as shown in Appendix A). Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $\frac{1}{b_{3}}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $69.6^{+1.8}_{-1.9}$ |  $0.286^{+0.021}_{-0.023}$ |  $0.067^{+0.078}_{-0.054}$ |  $-19.367^{+0.053}_{-0.056}$ |  1045.04 |  1053.04 |  1072.97 |  5.55 |  10.53
$\mathrm{CC+SN+R19}$ |  $72.0^{+1.3}_{-1.4}$ |  $0.273\pm 0.020$ |  $0.042^{+0.099}_{-0.032}$ |  $-19.302^{+0.037}_{-0.039}$ |  1048.16 |  1056.16 |  1076.10 |  3.84 |  8.82
$\mathrm{CC+SN+HW}$ |  $71.5\pm 1.4$ |  $0.275^{+0.019}_{-0.020}$ |  $0.070^{+0.072}_{-0.058}$ |  $-19.317^{+0.039}_{-0.038}$ |  1047.06 |  1055.07 |  1075.01 |  4.08 |  9.06
$\mathrm{CC+SN+TRGB}$ |  $69.7^{+1.3}_{-1.4}$ |  $0.285^{+0.020}_{-0.021}$ |  $0.048^{+0.094}_{-0.037}$ |  $-19.366\pm 0.040$ |  1045.04 |  1053.04 |  1072.98 |  5.36 |  10.34
CC + SN + BAO |  $67.35^{+0.94}_{-0.97}$ |  $0.289\pm 0.013$ |  $0.043^{+0.101}_{-0.026}$ |  $-19.441^{+0.032}_{-0.031}$ |  1060.55 |  1068.55 |  1088.55 |  5.09 |  10.09
$\mathrm{CC+SN+BAO+R19}$ |  $68.70^{+0.84}_{-0.85}$ |  $0.293^{+0.013}_{-0.012}$ |  $0.059^{+0.056}_{-0.047}$ |  $-19.397^{+0.029}_{-0.028}$ |  1071.71 |  1079.71 |  1099.71 |  5.41 |  10.41
$\mathrm{CC+SN+BAO+HW}$ |  $68.52^{+0.85}_{-0.82}$ |  $0.295^{+0.011}_{-0.014}$ |  $0.034^{+0.089}_{-0.024}$ |  $-19.401^{+0.028}_{-0.029}$ |  1069.03 |  1077.03 |  1097.03 |  4.99 |  9.10
$\mathrm{CC+SN+BAO+TRGB}$ |  $67.79\pm 0.85$ |  $0.292^{+0.012}_{-0.014}$ |  $0.074^{+0.057}_{-0.059}$ |  $-19.425^{+0.027}_{-0.030}$ |  1061.78 |  1069.78 |  1089.78 |  5.21 |  10.21

The posterior and confidence regions for the $f_{3}$CDM model are shown in
Fig. 3. This is an interesting model since in removing the square root in the
exponential index there is a clear impact on the confidence regions on
$\Omega_{\rm m,0}$ which now has tighter confidence regions, while $H_{0}$ is
largely left unaffected. Saying that, we again see margin increases in the
value of $H_{0}$ for some data set and prior settings.

The precision values shown in Table. 3 make clearer the stricter confidence
levels in the density parameter values which certainly helps understand the
predictions of the model better. As in the previous models, we again see the
impact of priors producing a shift of the value of $H_{0}$ to higher values,
as would be expected. Also, we observe a mildly higher maximum value of the
Hubble constant
($H_{0}=72.0^{+1.3}_{-1.4}$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$)
with almost Gaussian errors, while the lowest value is also slightly increased
with this model
($H_{0}=67.35^{+0.94}_{-0.97}$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$).

Now, along the same lines of reasoning as in the $f_{2}$CDM model, we take
$1/b_{3}$ as our parameter in the MCMC runs for the numerical stability of
this model. While the mean values are closer to their $\Lambda$CDM limits, the
confidence regions still point to a largely 2$\sigma$ distance between the
$f_{3}$CDM and $\Lambda$CDM models. On the other hand, the $\Delta$AIC and
$\Delta$BIC classifiers now give a range of values for the different MCMC
settings. However, in this case, the runs favour the CC+SN MCMC run with an
R19 prior.

The results presented in Fig. 3 and Table. 3 are consistent with those
reported in Refs. Nesseris:2013jea ; 2018ApJ…855…89X ; Benetti:2020hxp ;
Wang:2020zfv . However, the addition of other priors shows how the value of
$H_{0}$ can also be smaller.

###  4.4 $f_{4}(T)$CDM Model

The logarithmic model, proposed in Ref. Bamba:2010wb , is described by

$\mathcal{F}_{4}(T)=\alpha_{4}T_{0}\sqrt{\frac{T}{b_{4}T_{0}}}\log\left[\frac{b_{4}T_{0}}{T}\right]\,,$
(34)

where $\alpha_{4}$ and $b_{4}$ are constants. Evaluating the Friedmann
equation at current times gives

$\alpha_{4}=-\frac{\left(1-\Omega_{M_{0}}\right)\sqrt{b_{4}}}{2}\,,$ (35)

which reduces the Friedmann equation (13) to the relatively simple form

$E^{2}\left(z\right)=\Omega_{m_{0}}\left(1+z\right)^{3}+\Omega_{r_{0}}\left(1+z\right)^{4}+\left(1-\Omega_{m_{0}}-\Omega_{r_{0}}\right)E(z)\,,$
(36)

which interestingly does not feature $b_{4}$, meaning that background data
cannot constrain this parameter. Another important point to highlight is that
no choice of parameter values can reproduce $\Lambda$CDM for $f_{4}$CDM. On
the other hand, the background behavior of this model does coincide with that
of a spatially at self-accelerating branch of the DGP braneworld model
Dvali:2000hr ; Deffayet:2000uy . Models of this kind are intriguing because
they cannot feature confirmation bias with $\Lambda$CDM.

Figure 4: Left: C.L and posteriors for the $f_{4}$CDM model (34) using CC+SN data along the $H_{0}$ priors: R19 (green color), HW (red color) and TRGB (purple color), respectively. Right: C.L and posteriors for the model (34) using CC+SN+BAO data along the same priors denoted.  Table 4: Results for the Logarithmic model (34). First column: data sets used to constrain the model, including the $H_{0}$ priors. Second column: $H_{0}$ values derived from the analysis. Third column: Constrained $\Omega_{m,0}$. Fourth column: Nuisance parameter, $M$. Fifth column: $\chi^{2}_{\text{min}}$. From sixth up to ninth column: AIC and BIC with their respective differences with $\Lambda$CDM within the corresponding data set and prior combination (as shown in Appendix A). Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---
CC+ SN |  $68.8^{+1.8}_{-1.7}$ |  $0.207^{+0.018}_{-0.017}$ |  $-19.371^{+0.051}_{-0.052}$ |  1043.46 |  1042.46 |  1064.41 |  1.97 |  1.97
$\mathrm{CC+SN+R19}$ |  $71.4\pm 1.3$ |  $0.194^{+0.016}_{-0.015}$ |  $-19.301^{+0.038}_{-0.035}$ |  1047.98 |  1053.98 |  1068.94 |  1.67 |  1.67
$\mathrm{CC+SN+HW}$ |  $71.0^{+1.3}_{-1.2}$ |  $0.195^{+0.016}_{-0.015}$ |  $-19.309\pm 0.036$ |  1046.70 |  1052.70 |  1067.65 |  1.70 |  1.70
$\mathrm{CC+SN+TRGB}$ |  $69.2\pm 1.3$ |  $0.205\pm 0.016$ |  $-19.358^{+0.037}_{-0.040}$ |  1043.60 |  1049.60 |  1064.55 |  1.91 |  1.91
CC + SN + BAO |  $60.89^{+0.75}_{-0.71}$ |  $0.252^{+0.014}_{-0.013}$ |  $-19.611^{+0.027}_{-0.031}$ |  1078.52 |  1084.52 |  1099.52 |  21.07 |  21.07
$\mathrm{CC+SN+BAO+R19}$ |  $62.77^{+0.72}_{-0.76}$ |  $0.261^{+0.013}_{-0.014}$ |  $-19.543^{+0.028}_{-0.029}$ |  1124.87 |  1130.87 |  1145.87 |  56.56 |  56.56
$\mathrm{CC+SN+BAO+HW}$ |  $62.77^{+0.70}_{-0.76}$ |  $0.260^{+0.014}_{-0.013}$ |  $-19.544^{+0.029}_{-0.028}$ |  1121.49 |  1127.49 |  1142.49 |  55.47 |  55.46
$\mathrm{CC+SN+BAO+TRGB}$ |  $62.02^{+0.72}_{-0.73}$ |  $0.257^{+0.013}_{-0.014}$ |  $-19.567^{+0.026}_{-0.033}$ |  1097.80 |  1103.80 |  1118.80 |  39.23 |  39.23

For this model, the posteriors and confidence regions are shown in Fig. 4.
This is quite an interesting model since it has no $\Lambda$CDM limit and so
there is no a priori expected values for the extra model parameter. Another
crucial aspect of the $f_{4}$CDM model is that at background it does not
contain any additional parameters in comparison to $\Lambda$CDM.

In Table. 4, the precision results are presented for each of the data set and
prior combinations. Here a significant difference appears depending on whether
BAO data is included or not. For the cases of simply taking CC+SN data, we
find reasonable values of $H_{0}$ with very low values of $\Omega_{\rm m,0}$
giving a maximum Hubble constant for the R19 prior ($71.4\pm
1.3$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$), as
expected. On the other hand, when the BAO data set is included both the Hubble
constant and density parameter readings turn out to be very low with a minimum
$H_{0}=60.89^{+0.75}_{-0.71}$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$.
In all cases, the uncertainties are tightly constrained due to the small
number of parameters.

Nothing can be said about the $b_{4}$ parameter from these MCMC runs since the
parameter does not appear in the background equations for a flat FLRW
cosmology. On the other hand, the statistical indicators tell a different
story. For the CC+SN data set, irrespective of whether a prior is put on
$H_{0}$ or not, the $\Delta$AIC and $\Delta$BIC both turn out to be low in
comparison with the respective $\Lambda$CDM data set and prior combinations.
However, once BAO data is, its not simply the cosmological parameters that
start to veer away from their higher confidence observational regions, but
also the statistical indicators. Indeed the $\Delta$AIC and $\Delta$BIC in
comparison with $\Lambda$CDM both become extremely large in this scenario
bringing the model into question.

In line with previous studies, Nesseris:2013jea ; 2018ApJ…855…89X  we find
that logarithmic models do not fair well with observational data, but the
addition of BAO data totally removes any possibility of the $f_{4}$CDM model
having any significance against observational data.

###  4.5 $f_{5}(T)$CDM Model

Our last model under consideration is the hyperbolic tangent model given by
Wu:2010av

$\mathcal{F}_{5}(T)=\alpha_{5}(-T)^{b_{5}}\text{Tanh}\left(\frac{T_{0}}{T}\right)\,,$
(37)

where $\alpha_{5}$ and $b_{5}$ are constants. Using Eq. (13) at current times
results in the relation

$\alpha_{5}=\frac{\left(6H_{0}^{2}\right)^{1-b_{5}}\left(1-\Omega_{M_{0}}\right)}{(1-2b_{5})\text{tanh}(1)+2\text{Sech}^{2}(1)}\,.$
(38)

The Friedmann equation can then be written as

$\displaystyle E^{2}\left(z\right)$
$\displaystyle=\Omega_{m_{0}}\left(1+z\right)^{3}+\Omega_{r_{0}}\left(1+z\right)^{4}+$
$\displaystyle\frac{1-\Omega_{m_{0}}-\Omega_{r_{0}}}{(2b_{5}-1)\text{tanh}(1)-2\text{Sech}^{2}(1)}E^{2(b_{5}-1)}(z)\left[(2b_{5}-1)E^{2}(z)\text{Tanh}(E^{-2}(z))-2\text{Sech}^{2}(E^{-2}(z))\right]\,.$
(39)

As in the $f_{4}$CDM model, we do not recover $\Lambda$CDM in any limit here
so that $f_{5}$CDM does not represent a deviation from $\Lambda$CDM in so much
as it is an alternative to it.

Figure 5: Left: C.L and posteriors for the $f_{5}$CDM model (37) using CC+SN data along the $H_{0}$ priors: R19 (green color), HW (red color) and TRGB (purple color), respectively. Right: C.L and posteriors for the model (37) using CC+SN+BAO data along the same priors denoted. Table 5: Results for the hyperbolic tangent model (34). First column: data sets used to constrain the model, including the $H_{0}$ priors. Second column: $H_{0}$ values derived from the analysis. Third column: Constrained $\Omega_{m,0}$. Fourth column: $b_{5}$ best fits. Fifth column: Nuisance parameter, $M$. Sixth column: $\chi^{2}_{\text{min}}$. From seventh up to tenth column: AIC and BIC with their respective differences with $\Lambda$CDM within the corresponding data set and prior combination (as shown in Appendix A). Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $b_{5}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $69.2^{+1.9}_{-2.0}$ |  $0.369^{+0.031}_{-0.036}$ |  $-0.36^{+0.29}_{-0.38}$ |  $-19.390^{+0.056}_{-0.058}$ |  1044.44 |  1052.44 |  1072.38 |  4.95 |  9.93
$\mathrm{CC+SN+R19}$ |  $71.8\pm 1.3$ |  $0.349^{+0.029}_{-0.035}$ |  $-0.28^{+0.28}_{-0.38}$ |  $-19.314^{+0.038}_{-0.041}$ |  1048.13 |  1056.13 |  1076.06 |  3.81 |  8.79
$\mathrm{CC+SN+HW}$ |  $71.3\pm 1.3$ |  $0.353^{+0.029}_{-0.036}$ |  $-0.29^{+0.28}_{-0.38}$ |  $-19.329\pm 0.040$ |  1046.94 |  1054.94 |  1074.87 |  3.94 |  8.93
$\mathrm{CC+SN+TRGB}$ |  $69.5^{+1.3}_{-1.4}$ |  $0.366^{+0.030}_{-0.034}$ |  $-0.35^{+0.29}_{-0.37}$ |  $-19.381^{+0.041}_{-0.042}$ |  1044.50 |  1052.50 |  1072.44 |  4.81 |  9.80
CC + SN + BAO |  $68.4^{+1.5}_{-1.4}$ |  $0.302^{+0.014}_{-0.012}$ |  $0.144^{+0.087}_{-0.107}$ |  $-19.400^{+0.039}_{-0.037}$ |  1062.88 |  1070.88 |  1090.88 |  7.43 |  12.43
$\mathrm{CC+SN+BAO+R19}$ |  $70.6^{+1.1}_{-1.2}$ |  $0.308\pm 0.013$ |  $0.079^{+0.098}_{-0.064}$ |  $-19.342^{+0.031}_{-0.033}$ |  1069.77 |  1077.77 |  1097.77 |  3.47 |  8.47
$\mathrm{CC+SN+BAO+HW}$ |  $70.2\pm 1.1$ |  $0.308^{+0.012}_{-0.013}$ |  $0.039^{+0.095}_{-0.112}$ |  $-19.351^{+0.032}_{-0.031}$ |  1067.90 |  1075.90 |  1095.90 |  3.88 |  8.88
$\mathrm{CC+SN+BAO+TRGB}$ |  $68.9^{+1.1}_{-1.2}$ |  $0.304\pm 0.013$ |  $0.115^{+0.082}_{-0.100}$ |  $-19.386^{+0.032}_{-0.031}$ |  1063.34 |  1071.34 |  1091.34 |  6.78 |  11.78

In Fig. 5, the posterior and confidence regions are shown for the $f_{5}$CDM
model. As with the $f_{4}$CDM model, our interest in $f_{5}$CDM is in its lack
of a $\Lambda$CDM limit which removes any preference biases. In contrast to
the $f_{4}$CDM model, the model parameter, namely $b_{5}$, does contribute at
background level in the $f_{5}$CDM model.

Table 5 shows the precision results for the $f_{5}$CDM model. The situation is
now drastically different to that of the $f_{4}$CDM model in that the spread
of $H_{0}$ values is wider and the density parameter at current times prefers
a much larger value. The maximum value of the Hubble constant which is
obtained for the CC+SN data set with an R19 prior ($71.8\pm
1.3$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$) occurs
between the $f_{2}$CDM and $f_{3}$CDM models, while the lowest value is again
obtained for the CC+SN+BAO data set with no prior
($68.4^{+1.5}_{-1.4}$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$$)
on $H_{0}$. Another issue to point out is that the uncertainties in this
scenario are slightly larger than in the cases with a clear $\Lambda$CDM
limit.

In this scenario the extra model parameter does play a role for the background
cosmology and so has an input in the MCMC runs. By and large, the $b_{5}$
parameter is within the 2$\sigma$ region of being zero. However, a significant
distinction occurs for the CC+SN and CC+SN+BAO data sets in that the value of
$b_{4}$ turns out to either be negative or positive respectively. Moreover,
the confidence regions are much stricter for the latter case. Another crucial
difference between the $f_{4}$CDM and this one is that the statistical
indicators are much more realistic in this scenario. Saying that, the
$\Delta$AIC and $\Delta$BIC continue to feature a significant distance from
$\Lambda$CDM. This calls into question whether hyperbolic tangent models are
competitive scenarios for cosmology.

While adding more details with the various prior settings under consideration
here, we are in agreement with the literature where this model has been
investigated Nesseris:2013jea ; 2018ApJ…855…89X .

##  5 Conclusion

Figure 6: Whisker plot for $H_{0}$, $\Omega_{m0}$ and $b_{i}$, respectively.
The parameter $b_{i}$ corresponds to: $f_{1}$CDM as $b_{1}$, $f_{2}$CDM as
$1/b_{2}$, $f_{3}$CDM as $1/b_{3}$, $f_{4}$CDM as $b_{4}$ and $f_{5}$CDM as
$b_{5}$. The best fits are reported for CC+SN and CC+SN+BAO as: without prior
(blue color), with R19 prior (green color), with HW prior (red color) and with
TRGB prior (purple color). The rectangles in colors denotes the 1-$\sigma$
measured value for each project. The dashed yellow line in $b_{i}$ denotes the
$\Lambda$CDM model.

Our study probes the impact of $H_{0}$ priors in $f(T)$ cosmology using five
specific models together with CC, SN and BAO data sets, and 3 particularly
relevant cosmology-independent priors from literature. Our interest was in
assessing the way in which the cosmological parameters are altered in these
various forms of $f(T)$ gravity as well as how they compare statistically with
$\Lambda$CDM. To make the cross-analysis between the various models, data sets
and prior choices more straightforward, we show each of the cosmological
parameters against each other in the whisker plot in Fig. 6. Here we also show
the value of each prior in shaded regions where the direct impact they have on
the cosmological parameters for each model becomes much clearer.

In this work, the first three models we explore in $f_{1}$CDM, $f_{2}$CDM and
$f_{3}$CDM are modifications of $\Lambda$CDM with a clear-cut limit in this
case. We also run our MCMC analysis for $\Lambda$CDM which appears in Appendix
A which we use for our statistical analysis. These models produce higher
values of $H_{0}$ and slightly lower values of $\Omega_{\rm m,0}$ as compared
with $\Lambda$CDM at the expanse of an additional model parameter which takes
the form of $b_{i}$ ($i=1,2,3$). This additional freedom in our cosmological
model means that we can better approximate the data but produce AIC and BIC
values which are higher than in $\Lambda$CDM. In our analyses, we find
cosmological parameter values that are in agreement with previous works in the
literature Nesseris:2013jea ; 2018ApJ…855…89X ; Benetti:2020hxp ; Wang:2020zfv
and within 1-2$\sigma$ of $\Lambda$CDM.

On the other hand, we also probe the $f_{4}$CDM and $f_{5}$CDM models which
are very interesting to explore since they do not have an associated
$\Lambda$CDM limit, meaning that no set of model parameter values returns an
identical $\Lambda$CDM behaviour. In fact, in $f_{4}$CDM, the model parameter
$b_{4}$ does not appear at background level meaning that it has the same
number of parameters as $\Lambda$CDM in this regime. This also means that we
have one less parameter in our MCMC analyses which means that the number of
parameters in the statistical analysis will not be harshly effected by this
modified form of gravity. In some case this turns out to be an advantage such
as in the CC+SN runs for $f_{4}$CDM which have AIC and BIC values very close
to $\Lambda$CDM, while for the CC+SN+BAO scenario of $\Lambda$CDM give
excessively large values for these statistical quantifiers.

The work presented here probes the impact of priors on the Hubble data in the
late Universe. It would be interesting to further study this effect using
forecast data from future surveys at higher redshift in order to better
understand how the cosmological parameters of these models may be effected in
comparison to the present analysis. Furthermore, another interesting direction
would be to include a perturbative analysis in which growth and gravitational
wave data may be included to better probe how these priors effect those
associated cosmological parameters. This will be reported elsewhere.

##  Appendix A $\Lambda$CDM Model

In all $f_{i}$CDM models we provide comparisons with each respective
$\Lambda$CDM MCMC run. To this end, we here present the results for
$\Lambda$CDM for transparency of our results. Firstly in Fig. 7, we show the
MCMC posteriors and confidence regions for the various priors for the CC+SN
and CC+SN+BAO data sets. As expected, the convergence for each data set and
prior combination occurs very fast giving nearly Gaussian uncertainties in
every case.

Figure 7: C.L for the $\Lambda$CDM model. Left: C.L and posteriors for the model (34) using CC+SN data along the $H_{0}$ priors: R19 (green color), HW (red color) and TRGB (purple color), respectively. Right: C.L and posteriors for the model (34) using CC+SN+BAO data along the same priors denoted.  Table 6: Results for the LCDM model. First column: data sets used to constrain the model, including the $H_{0}$ priors. Second column: $H_{0}$ values derived from the analysis. Third column: Constrained $\Omega_{m,0}$. Fourth column: $\chi^{2}_{\text{min}}$. From fifth-sixth column: AIC and BIC results, respectively. Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{s}^{-1}{\mathrm{M}\mathrm{pc}}^{-1}$] |  $\Omega_{m,0}$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC
---|---|---|---|---|---
CC+ SN |  $68.7\pm 1.8$ |  $0.306^{+0.022}_{-0.021}$ |  1041.49 |  1047.49 |  1062.44
$\mathrm{CC+SN+R19}$ |  $71.3^{+1.4}_{-1.3}$ |  $0.290^{+0.019}_{-0.020}$ |  1046.32 |  1052.32 |  1067.27
$\mathrm{CC+SN+HW}$ |  $71.0\pm 1.3$ |  $0.291^{+0.020}_{-0.019}$ |  1044.99 |  1050.99 |  1065.94
$\mathrm{CC+SN+TRGB}$ |  $69.2\pm 1.4$ |  $0.303^{+0.021}_{-0.020}$ |  1041.69 |  1047.69 |  1062.64
CC + SN + BAO |  $67.63\pm 0.90$ |  $0.297\pm 0.013$ |  1057.46 |  1063.46 |  1078.45
$\mathrm{CC+SN+BAO+R19}$ |  $68.81^{+0.82}_{-0.84}$ |  $0.300\pm 0.013$ |  1068.30 |  1074.30 |  1089.30
$\mathrm{CC+SN+BAO+HW}$ |  $68.70\pm 0.83$ |  $0.300^{+0.014}_{-0.013}$ |  1066.03 |  1072.03 |  1087.03
$\mathrm{CC+SN+BAO+TRGB}$ |  $67.98^{+0.85}_{-081}$ |  $0.298\pm 0.013$ |  1058.56 |  1064.56 |  1079.56

Table 6 shows the precision results for each of the MCMC runs with the
specific reference values for the AIC and BIC statistical indicators. these
are the values used through the work in each of the comparison entries.

*Acknowledgments The authors would like to acknowledge networking support by the COST Action CA18108 and funding support from Cosmology@MALTA which is supported by the University of Malta. This research has been carried out using computational facilities procured through the European Regional Development Fund, Project No. ERDF-080 “A supercomputing laboratory for the University of Malta”. The authors would also like to acknowledge funding from “The Malta Council for Science and Technology” in project IPAS-2020-007. CE-R acknowledges the Royal Astronomical Society as FRAS 10147 and by DGAPA-PAPIIT-UNAM Project IA100220.

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