#  $f(T,B)$ Gravity in the late Universe

Rebecca Briffa<EMAIL_ADDRESS>Institute of Space Sciences and
Astronomy, University of Malta, Msida, Malta  Department of Physics,
University of Malta, Msida, Malta  Celia Escamilla-Rivera
<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad
Nacional Autónoma de México, Circuito Exterior C.U., A.P. 70-543, México D.F.
04510, México  Jackson Levi Said<EMAIL_ADDRESS>Institute of Space
Sciences and Astronomy, University of Malta, Msida, Malta  Department of
Physics, University of Malta, Msida, Malta  Jurgen Mifsud
<EMAIL_ADDRESS>Institute of Space Sciences and Astronomy, University
of Malta, Msida, Malta  Department of Physics, University of Malta, Msida,
Malta

###### Abstract

We explore the viability of three models in $f(T,B)$ gravity using data from
recent surveys based on cosmic chronometers, the Pantheon data set, and
baryonic acoustic oscillation data. We also assess the consistency of these
models and data set combinations with two important priors on the Hubble
constant coming from the SH0ES Team and measurements using the tip of the red
giant branch respectively. These give the highest and lowest values of the
Hubble constant coming from cosmology independent studies. In general, our
analysis does provide a more consistent fit for the late time data being
analyzed. However, each model does include an additional model parameter in
comparison with the concordance model. We close the analysis with a
comparative analysis in which each model, data set and Hubble constant prior
combination are cross-analyzed against each other.

##  I Introduction

The recent indication of a statistical tension Bernal:2016gxb ;
DiValentino:2020zio ; DiValentino:2021izs ; Abdalla:2022yfr  in value of the
Hubble constant $H_{0}$ has prompted a reassessment of possible modifications
beyond standard vanilla $\Lambda$CDM. While the $H_{0}$ problem may be the
result of systematics, the mismatch in reported values of the Hubble constant
has been reported by a number of surveys indicating a more serious nature to
the issue. On the other hand, this standard model of cosmology can produce a
consistent evolution of the Universe Planck:2018vyg . However, this is only
possible with the inclusion of exotic particles physics using cold dark matter
(CDM) Baudis:2016qwx ; Bertone:2004pz  which govern the dynamics of galaxies,
and a cosmological constant Peebles:2002gy ; Copeland:2006wr , which is the
source of the late time accelerating Universe SupernovaSearchTeam:1998fmf ;
SupernovaCosmologyProject:1998vns . Despite great efforts to directly detect
CDM Gaitskell:2004gd  and to formulate a deeper explanation for the
cosmological constant Weinberg:1988cp , these prospects remain elusive.

The $H_{0}$ tension problem has led to increased efforts to determine the
current value of the Hubble parameter in order to better understand the degree
of disparity between measurements that are independent of cosmological models
and those that require a cosmological model to make a Hubble constant
estimate. The most prominent example of the former is the latest announced
result of the SH0ES Team Scolnic:2021amr ; Brout:2021mpj ; Riess:2021jrx
which is based on observations of supernovae type Ia (SNe Ia) with calibration
coming from Cepheid measurements. This gives a Hubble constant of $73.3\pm
1.04\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$ which is the result of data points taken
below $z=1$. A relatively high value of the Hubble constant is also supported
by the H0LiCOW Collaboration Wong:2019kwg  which is based on six strong
lensing measurements and gives a value of $73.3^{+1.7}_{-1.8}\,{\rm
km\,s}^{-1}{\rm Mpc}^{-1}$. Other model independent measurements do point to
lower values of the Hubble constant such as measurements of the tip of the red
giant branch who give a value of $69.8\pm 1.7\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$
Freedman:2019jwv ; Freedman:2021ahq , and measurements from novel approaches
of gravitational wave standard sirens LIGOScientific:2017adf  which gives a
higher uncertainty result of $70.0^{+12}_{-8}\,{\rm km\,s}^{-1}{\rm
Mpc}^{-1}$. On the other hand, model dependent measurements based on early
Universe measurements give a drastically lower value of the Hubble constant.
The most prominent of these measurements is the Planck Collaboration result of
$67.4\pm 0.5\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$ and the Atacama Cosmology
Telescope (ACT) Hubble constant $67.9\pm 1.5\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$
ACT:2020gnv .

One direction that has received heightened community effort in the literature
is that of modifications to general relativity (GR) where numerous proposals
have been made to better meet the observational constraints of recent surveys
Sotiriou:2008rp ; Clifton:2011jh ; CANTATA:2021ktz . By and large, most of
these proposed gravitational models build on GR and extend the Einstein-
Hilbert action to meet specific theoretical or phenomenological goals
Faraoni:2008mf ; Capozziello:2011et . However, these expressions of gravity
are mostly based on the same curvature-based geometric description of gravity
as GR, which is sourced by the Levi-Civita connection misner1973gravitation ;
nakahara2003geometry . Interestingly, other descriptions of geometric gravity
are possible BeltranJimenez:2019esp . One class of theories that has garnered
growing interest in the literature is that of teleparallel gravity (TG)
Bahamonde:2021gfp ; Aldrovandi:2013wha ; Cai:2015emx ; Krssak:2018ywd . This
is based on the exchange of the curvature associated with the Levi-Civita
connection with its teleparallel connection analogue which is curvature-less
and gives an expression for torsion in gravity. Naturally, this implies that
all measures of curvature will vanish so we must consider new tensors to
define gravitational theories. One critical scalar that is central to TG is
the torsion scalar $T$ which turns out to be equivalent to the regular
curvature scalar $\accentset{\circ}{R}$ (over-circles represent quantities
calculated with the Levi-Civita connection), up to a total divergence term.
This means that we can define a Teleparallel equivalent of General Relativity
(TEGR) where the Einstein-Hilbert action is replaced with one based solely on
the torsion scalar. The total divergence difference between the curvature and
torsion scalars is denoted by $B$ and will be important for the work that
follows. Indeed, it embodies the fourth order aspects of the curvature scalar
which produce fourth order terms in many modifications of GR Lovelock:1971yv ;
Gonzalez:2015sha ; Bahamonde:2019shr .

In the TEGR context, the curvature scalar of the Einstein-Hilbert action is
divided into the active torsion scalar $T$ and the total divergence
contribution from $B$. Taking the same approach of many of the extended
theories of gravity in curvature-based geometries, such as
$f(\accentset{\circ}{R})$ gravity Sotiriou:2008rp ; Faraoni:2008mf ;
Capozziello:2011et , we can write an $f(T,B)$ action for the gravitational
sector Bahamonde:2015zma ; Bahamonde:2016grb ; Paliathanasis:2017flf ;
Farrugia:2018gyz ; Bahamonde:2016cul ; Wright:2016ayu ; Farrugia:2020fcu ;
Capozziello:2019msc ; Farrugia:2018gyz ; Escamilla-Rivera:2019ulu . Models in
$f(T,B)$ gravity have been explored at all levels of phenomenology, it hosts
many models that are consistent with solar system level observations
Farrugia:2020fcu . In terms of gravitational waves, it predicts generically
that they propagate at the speed of light but it does predict extra
polarizations in comparison with GR Capozziello:2019msc ; Farrugia:2018gyz .
Another important development was the introduction of exact and perturbed
black hole solutions in Ref. Bahamonde:2021srr . In terms of cosmology, there
have been a number of studies that probe both the background cosmological
evolution of models in the theory Bahamonde:2016grb ; Paliathanasis:2017flf ;
Bahamonde:2016cul ; Escamilla-Rivera:2019ulu ; Escamilla-Rivera:2021xql ;
Franco:2020lxx ; Caruana:2020szx , as well as its perturbative structure
Bahamonde:2020lsm . There has also been significant foundational work on the
structure of the theory Wright:2016ayu  such as its exact relationship to
$f(\accentset{\circ}{R})$ gravity Bahamonde:2015zma .

The surge in surveys reporting values of the Hubble constant is the main
driver of current efforts in modified gravity to define favoured models that
may meet the observational challenges of the coming decade. In this work, we
explore three models in $f(T,B)$ cosmology which have garnered popularity in
the community. We also consider the impact that prior values on the Hubble
constant may have on the resulting cosmological model parameter values. To do
this, we first give an introduction to TG and its $f(T,B)$ formalism in Sec.
II. We also describe its cosmology here, and then move onto information about
the observational data we use and the $H_{0}$ priors in Ref. III. We give our
main results in Sec. IV where we introduce the three models together with the
statistical outputs of our analysis. Finally, we summarize our results in Sec.
V.

##  II $f(T,B)$ FLRW Cosmology

TG is based on the exchange of the curvature associated with the Levi-Civita
connection Clifton:2011jh  with the torsion sourced through the teleparallel
connection where torsion replaces curvature as the mediator of gravitation
Bahamonde:2021gfp . Curvature is a produce not of the metric tensor but of 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), and so identical metric components can continue to result from a
TG description of gravity. However, TG is different in that geometric torsion
emerges from the teleparallel connection teleparallel connection
$\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu}$ which continues to satisfy
metricity Hayashi:1979qx ; Aldrovandi:2013wha . As a note on the
characteristic measures in gravitation, the regular Riemann tensor obviously
does not vanish
($\accentset{\circ}{R}^{\beta}_{\phantom{\beta}\mu\nu\alpha}\neq 0$), but if
the Levi-Civita connections are replaced with their teleparallel connection
analogues then indeed curvature vanishes identically
($R^{\beta}_{\phantom{\beta}\mu\nu\alpha}=0$) (see reviews in Refs.
Krssak:2018ywd ; Cai:2015emx ; Aldrovandi:2013wha ), which points to the need
for a different description of gravitational interactions.

Torsion is more conveniently expressed in terms of the tetrad
$e^{A}_{\phantom{A}\mu}$ (and its inverses $E_{A}^{\phantom{A}\mu}$) whereby
the metric becomes a derived quantity through the expressions

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

where Latin indices represent coordinates on the tangent space while Greek
indices continue to represent indices on the general manifold Cai:2015emx .
Tetrads are largely suppressed in GR but they do appear in some circumstances
Chandrasekhar:1984siy . Naturally, these tetrads must 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 consistency.

Given the tetrad description, the teleparallel connection can directly be
written as Weitzenbock1923 ; Krssak:2015oua

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

where the freedom of the theory is spread over the tetrad and the spin
connection $\omega^{A}_{\phantom{A}B\mu}$ which is a flat connection, and thus
represents the local Lorentz freedom, which is a result of the explicit
appearance of the local indices in this description. Thus, the tetrad-spin
connection pair embody the gravitational and Lorentz degrees of freedom of the
theory which then appear in the equations of motion. Analogous to the Riemann
tensor dependence on the Levi-Civita connection, we can write a torsion tensor
as Hayashi:1979qx

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

where square brackets denote the anti-symmetry operator, and where torsion is
the result of anti-symmetry Aldrovandi:2013wha . The torsion tensor is
covariant under both diffeomorphisms and local Lorentz transformations. The
torsion tensor can be contracted in such a way to produce the torsion scalar
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 is the result of demanding that an action based only on $T$ gives the
same equations of motion as the regular Einstein-Hilbert action with
$\accentset{\circ}{R}$ (up to a boundary term). Analogous to the curvature
scalar being solely dependent on the Levi-Civita connection, the torsion
scalar ultimately is dependent only on the teleparallel connection.

The curvature scalar will organically vanish when calculated using the
teleparallel connection since this is a measure of curvature and the
teleparallel connection has none, i.e. $R\equiv 0$ (where we emphasize that
$R=R(\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu})$ and
$\accentset{\circ}{R}=\accentset{\circ}{R}(\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu})\neq
0$). In this background, we can write an expression relating the curvature and
torsion scalars with each other such that Bahamonde:2015zma ; Farrugia:2016qqe

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

where $B$ represents a total divergence term and is defined as

$B=\frac{2}{e}\partial_{\rho}\left(eT^{\mu\phantom{\mu}\rho}_{\phantom{\mu}\mu\phantom{\rho}}\right)\,,$
(7)

where $e=\det\left(e^{a}_{\phantom{a}\mu}\right)=\sqrt{-g}$ is the determinant
of the tetrad. These relations guarantee the equivalence of the GR with TEGR,
where TEGR has an action based on the torsion scalar analogous to the
Einstein-Hilbert being based on the curvature scalar.

As in GR, we can consider extensions to the TEGR action where TEGR is
complemented by additional terms which may embody possibly dynamical features
of dark energy. In this work, we explore the generalization of the TEGR action
to an arbitrary function of both the torsion scalar and the boundary term,
which gives Bahamonde:2015zma ; Capozziello:2018qcp ; Bahamonde:2016grb ;
Paliathanasis:2017flf ; Farrugia:2018gyz ; Bahamonde:2016cul ; Wright:2016ayu

$\mathcal{S}_{f(T,B)}=\frac{1}{2\kappa^{2}}\int\mathrm{d}^{4}x\;e\,f(T,B)+\int\mathrm{d}^{4}x\;e\mathcal{L}_{\text{m}}\,,$
(8)

where $\kappa^{2}=8\pi G$, and $\mathcal{L}_{\text{m}}$ is the matter
Lagrangian in the Jordan frame. TEGR is recovered for the limit where
$f(T,B)\rightarrow-T$. In this description the second and fourth order
contributions to the equations of motion are decoupled with the respective
torsion and boundary term scalars. In this context, there is a generalization
of the fourth-order $f(\accentset{\circ}{R})$ gravity Sotiriou:2008rp ;
Capozziello:2011et  which now is recovered for the limit $f(T,B)\rightarrow
f(-T+B)=f(\accentset{\circ}{R})$ (on comparison with Eq. (6)). Taking a
variation with respect to the tetrad the produces the field equations
Bahamonde:2015zma ; Farrugia:2018gyz

$\displaystyle
W^{\nu}_{\phantom{\nu}\lambda}:=2\delta_{\nu}^{\lambda}\mathring{\Box}f_{B}-2\mathring{\nabla}^{\lambda}\mathring{\nabla}_{\nu}f_{B}+Bf_{B}\delta_{\nu}^{\lambda}+4\Big{[}(\partial_{\mu}f_{B})+(\partial_{\mu}f_{T})\Big{]}S_{\nu}{}^{\mu\lambda}$
$\displaystyle+4e^{-1}e^{A}{}_{\nu}\partial_{\mu}(eS_{A}{}^{\mu\lambda})f_{T}-4f_{T}T^{\sigma}{}_{\mu\nu}S_{\sigma}{}^{\lambda\mu}-f\delta_{\nu}^{\lambda}$
$\displaystyle=$ $\displaystyle
2\kappa^{2}\Theta^{\nu}_{\phantom{\nu}\lambda}\,,$ (9)

where subscripts denote derivatives, and $\Theta^{\nu}_{\phantom{\nu}\lambda}$
is the regular energy-momentum tensor for matter.

At this juncture, its important to point out that the tetrad and spin
connection are independent variables in the theory and so produce independent
field equations. The tetrad produces the ten metrical field equations while
the spin connection produces the six local Lorentz field equations. However,
TG has a very useful property whereby the tetrad variation is connected to the
spin connection field equations through
$W_{a}^{\phantom{a}\mu}=\delta\mathcal{S}_{\mathcal{F}(T)}/\delta
e^{A}_{\phantom{A}\mu}$ Hohmann:2021fpr . The property states that the tetrad
and spin connection field equations are respectively the result of the
symmetric and anti-symmetric operator on $W_{\mu\nu}$. Due to the symmetry of
the energy-momentum tensor, this means that these field equations can be
written as Bahamonde:2021gfp

$W_{(\mu\nu)}=\kappa^{2}\Theta_{\mu\nu}\,,\quad\text{and}\quad
W_{[\mu\nu]}=0\,,$ (10)

which holds for any formalism based on the teleparallel connection. For
particular choices of tetrads, the anti-symmetric field equations can be
satisfied for vanishing spin connection components. This is called the
Weitzenböck gauge and can be helpful for making calculations.

In this work, we consider a flat Friedmann–Lemaître–Robertson–Walker (FLRW)
cosmology where the tetrad can be written as Escamilla-Rivera:2019ulu

$e^{A}_{\phantom{A}\mu}=\textrm{diag}(1,a(t),a(t),a(t))\,,$ (11)

where $a(t)$ is the scale factor, and which produces the flat homogeneous and
isotropic metric

$ds^{2}=dt^{2}-a(t)^{2}(dx^{2}+dy^{2}+dz^{2})\,,$ (12)

through Eq. (1). This diagonal form of the tetrad is compatible with the
Weitzenböck gauge, i.e. $\omega^{A}_{\phantom{A}B\mu}=0$ Krssak:2015oua ;
Tamanini:2012hg . Using the definitions in Eqs. (5,7), we can write

$T=-6H^{2}\,,\quad B=-6(3H^{2}+\dot{H})\,,$ (13)

which together reproduce the regular curvature-based curvature scalar
$\accentset{\circ}{R}=-T+B=-6(\dot{H}+2H^{2})$. The Friedmann equations can
also be determined giving

$\displaystyle 3H(\dot{f}_{B}-2Hf_{T})+\frac{1}{2}(Bf_{B}-f)$ $\displaystyle=$
$\displaystyle\kappa^{2}\rho\,,$ (14)
$\displaystyle-\ddot{f}_{B}+2f_{T}\dot{H}+2H(3Hf_{T}+\dot{f}_{T})+\frac{1}{2}(f-Bf_{B})$
$\displaystyle=$ $\displaystyle\kappa^{2}P\,,$ (15)

where overdots refer to derivatives with respect to cosmic time $t$, and where
$\rho$ and $P$ respectively represent the energy density and pressure of
matter.

For background cosmology, we can rewrite these equations with the extension to
the TEGR Lagrangian term as an effective fluid which embodies the additional
terms that are sourced from the extra torsion contributions. We first consider
the mapping $f(T,B)\rightarrow-T+\mathcal{F}(T,B)$. Then, the equations of
motion in Eqs. (14,15) take the form

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\kappa^{2}\left(\rho+\rho_{\text{eff}}\right)\,,$ (16)
$\displaystyle 3H^{2}+2\dot{H}$ $\displaystyle=$
$\displaystyle-\kappa^{2}\left(P+P_{\text{eff}}\right)\,,$ (17)

where the effective fluid is defined through

$\displaystyle\kappa^{2}\rho_{\text{eff}}$ $\displaystyle:=$ $\displaystyle
3H^{2}\left(3\mathcal{F}_{B}+2\mathcal{F}_{T}\right)-3H\dot{\mathcal{F}}_{B}+3\dot{H}\mathcal{F}_{B}+\frac{1}{2}\mathcal{F}\,,$
(18) $\displaystyle\kappa^{2}P_{\text{eff}}$ $\displaystyle:=$
$\displaystyle-\frac{1}{2}\mathcal{F}-\left(3H^{2}+\dot{H}\right)\left(3\mathcal{F}_{B}+2\mathcal{F}_{T}\right)-2H\dot{\mathcal{F}}_{T}+\ddot{\mathcal{F}}_{B}\,.$
(19)

This effective fluid for the $\mathcal{F}(T,B)$ gravity extension satisfies
the fluid equation Bahamonde:2016grb

$\dot{\rho}_{\text{eff}}+3H\left(\rho_{\text{eff}}+P_{\text{eff}}\right)=0\,,$
(20)

and can be used to define an effective equation of state (EoS)

$\displaystyle\omega_{\text{eff}}$ $\displaystyle:=$
$\displaystyle\frac{P_{\text{eff}}}{\rho_{\text{eff}}}$ (21) $\displaystyle=$
$\displaystyle-1+\frac{\ddot{\mathcal{F}}_{B}-3H\dot{\mathcal{F}}_{B}-2\dot{H}\mathcal{F}_{T}-2H\dot{\mathcal{F}}_{T}}{3H^{2}\left(3\mathcal{F}_{B}+2\mathcal{F}_{T}\right)-3H\dot{\mathcal{F}}_{B}+3\dot{H}\mathcal{F}_{B}-\frac{1}{2}\mathcal{F}}\,.$
(22)

In the $\Lambda$CDM limit, as expected this EoS parameter limits to
$\omega_{\text{eff}}\rightarrow-1$.

##  III Observational Data

We here present and briefly discuss the observational data sets which will be
considered in the below analyses. For our baseline data set, we consider
Hubble expansion data along with an SNe Ia compilation data set.

  * •

For the Hubble parameter data, we adopt thirty–one cosmic chronometer data
points 2014RAA….14.1221Z ; Jimenez:2003iv ; Moresco:2016mzx ; Simon:2004tf ;
2012JCAP…08..006M ; 2010JCAP…02..008S ; Moresco:2015cya . It is well-known
that 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, CC were found to be more reliable than any other
method based on an absolute age determination for galaxies Jimenez:2001gg . We
should also remark that our CC data set is independent of the Cepheid distance
scale and also considered to be model independent, although CC data points are
known to be 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).

  * •

We shall also be considering the Pantheon SNe Ia data compilation consisting
of 1048 SNe Ia relative luminosity distance measurements spanning the redshift
range of $0.01<z<2.3$ Scolnic:2017caz . Henceforth, we will be denoting the
Pantheon SNe Ia compilation by SN. We should remark that since the apparent
magnitude of each SNe Ia needs to be calibrated via an arbitrary fiducial
absolute magnitude $M$, we will be considering $M$ as a nuisance parameter in
our MCMC analyses.

  * •

Furthermore, we shall also be considering a joint baryon acoustic oscillation
(BAO) data set. The considered BAO data set is composed of 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
each set of parameters in the MCMC analysis, we computed the comoving sound
horizon $r_{s}(z)$ at the end of the baryon drag epoch at redshift
$z_{d}\approx 1059.94$ Planck:2018vyg  via an accurate expression which has
been inferred from genetic algorithms Aizpuru:2021vhd , in which we have
adopted a baryon content of $\Omega_{b,0}=0.02166\,h^{-2}$ with $h=H_{0}/100$.

We shall also be analysing the impact of $H_{0}$ prior values on our inferred
$f(T,B)$ model parameter constraints. We will be considering the latest SH0ES
local estimate Riess:2021jrx  of $H_{0}=73.30\pm 1.04\,{\rm km\,s}^{-1}{\rm
Mpc}^{-1}$ (R21) based on SN in the Hubble flow, and the measurement using the
tip of the red giant branch (TRGB) as a standard candle Freedman:2021ahq  with
$H_{0}=69.8\pm 1.7\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$ (F21).

##  IV Constraints of $f(T,B)$ Cosmological Models

In this work, we consider three $f(T,B)$ models together with two recent
cosmological model independent $H_{0}$ priors which will test the consistency
of these models with the priors. There exists several measurements of $H_{0}$
in literature as studies have shown how well they can reproduce our
cosmological history. We examine the effect that these priors have on the
model parameter values for the data sets mentioned in Sec. III. In this paper,
we implement an MCMC algorithm using a combination of two choices. The first
being the data sets chosen, i.e CC+SN or CC+SN+BAO and the second being the
selection from having no prior, or one of the $H_{0}$ prior values (either R21
or F21).

In addition, we make a comparison of each $f_{i}$CDM model against the
$\Lambda$CDM model by calculating the Akaike information criteria (AIC)
Akaike:1974  and the Bayesian information criteria (BIC)
10.1214/aos/1176344136 . This allows us to evaluate which model best fits the
observational data. The AIC is defined as

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

where $L_{\text{max}}$ is the maximum value of the likelihood of each model,
which includes likelihood of the data set and prior combination, and $n$ is
the number of parameters involved in the MCMC algorithm. The AIC reward
goodness of fit through the maximum likelihood, however, the additional term
of the AIC serves as a penalty for models which have a large number of
parameters. The BIC is defined as

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

where $m$ is the number of data points used in the fit. From Eq. (24), it is
evident that the penalty for BIC is higher than that of AIC. Moreover, the
involvement of the number of data points in the BIC criterion, makes it a
potentially better measure of performance than the AIC criterion. In general,
the model with lower AIC and BIC values corresponds to the model that best
supports the data.

For our model comparison, we calculate the $\Delta$AIC and $\Delta$BIC, where
we compare each model data set and prior combination with the corresponding
$\Lambda$CDM (See Appendix A). The difference between the two models can be
defined as $\Delta\text{AIC}=\Delta\chi^{2}_{\text{min}}+2\Delta n$ and
$\Delta\text{BIC}=\Delta\chi^{2}_{\text{min}}+\Delta n\ln m$ for AIC and BIC,
respectively. In this context, the small values of $\Delta$AIC and $\Delta$BIC
correspond to the model with the chosen data set and prior combination being
closer to $\Lambda$CDM.

###  IV.1 Power Law Model – $f_{1}(T,B)$CDM Model

The power law model was first considered by Bengochea and Ferraro in
Bengochea:2008gz  because of its ability to reproduce the late-time cosmic
acceleration in $f(T)$ gravity. The $f(T,B)$ model is given by

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

where $\alpha_{1}$ and $p_{1}$ are constants. From the modified Friedmann
equation 14, we can obtain $\alpha_{1}$ by evaluating at current time

$\alpha_{1}=\frac{6H_{0}^{2}(\Omega_{m_{0}}+\Omega_{r_{0}}-1)}{(p_{1}-1)(-B_{0})^{p_{1}}+6H_{0}\left(p_{1}(p_{1}-1)(-B_{0})^{p_{1}-2}\dot{B}|_{t=t_{0}}\right)}\,,$
(26)

where $\Omega_{m_{0}}$ and $\Omega_{r_{0}}$ are the matter and radiation
density parameters at current times, respectively. Moreover, $B_{0}$ is the
boundary term value at current times, i.e $B_{0}=B|_{t=t_{0}}$ whilst
$\dot{B}|_{t=t_{0}}$ is its derivative also evaluated at current times. Thus,
$p_{1}$ is the only new model parameter for $f_{1}(T,B)$CDM model.

Thus, the Friedmann equation in Eq. 14 for this model can be written as

$\displaystyle H^{\prime\prime}$
$\displaystyle=\frac{-1}{36\alpha_{1}(1+z)^{2}H^{3}p_{1}(p_{1}-1)(-B)^{p_{1}-2}}$
$\displaystyle+\Big{[}6H_{0}^{2}\Big{(}\Omega_{m_{0}}(1+z)^{3}+\Omega_{r_{0}}(1+z)^{4}\Big{)}-6H_{0}^{2}+\alpha_{1}(1-p_{1})(-B)^{p_{1}}-216\alpha_{1}p_{1}(p_{1}-1)H^{3}H^{\prime}(1+z)(-B)^{p_{1}-2}\Big{]}$
(27) $\displaystyle-\frac{H^{\prime}}{1+z}-\frac{H^{\prime 2}}{H}\,,$

where we have transformed to redshift space, and where primes refer to
derivatives with respect to redshift $z$. The above model reduces to the
$\Lambda$CDM model when $p_{1}=0$. When $p_{1}=1$, we obtain the GR limit,
which gives an upper bound, $p_{1}<1$, such that we have an accelerating
Universe.

Table 1: Power Law Model Caption Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $p_{1}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $68.5^{+2.1}_{-2.2}$ |  $0.281^{+0.027}_{-0.023}$ |  $0.140^{+0.090}_{-0.109}$ |  $-19.381^{+0.063}_{-0.064}$ |  1040.94 |  1048.94 |  1068.88 |  $1.21$ |  $6.19$
$\mathrm{CC+SN+R21}$ |  $72.0\pm 1.1$ |  $0.269^{+0.021}_{-0.016}$ |  $0.080^{+0.108}_{-0.068}$ |  $-19.27^{+0.19}_{-0.22}$ |  1040.50 |  1053.50 |  1076.10 |  $1.04$ |  $6.02$
$\mathrm{CC+SN+F21}$ |  $69.1^{+1.5}_{-1.4}$ |  $0.279^{+0.025}_{-0.022}$ |  $0.143^{+0.079}_{-0.113}$ |  $-19.362^{+0.075}_{-0.078}$ |  1040.98 |  1048.98 |  1068.92 |  $1.10$ |  $6.08$
CC + SN + BAO |  $67.5\pm 1.4$ |  $0.305\pm 0.017$ |  $0.058\pm 0.053$ |  $-19.396^{+0.090}_{-0.136}$ |  1047.97 |  1055.97 |  1075.97 |  $-0.04$ |  $4.96$
$\mathrm{CC+SN+BAO+R21}$ |  $70.32^{+0.95}_{-0.94}$ |  $0.314\pm 0.016$ |  $0.012^{+0.027}_{-0.0094}$ |  $-19.327^{+0.029}_{-0.034}$ |  1062.23 |  1070.23 |  1090.24 |  0.79 |  5.79
$\mathrm{CC+SN+BAO+F21}$ |  $68.1\pm 1.2$ |  $0.308\pm 0.017$ |  $0.046^{+0.045}_{-0.042}$ |  $-19.35^{+0.69}_{-0.78}$ |  1048.81 |  1056.81 |  1076.81 |  0.13 |  5.12

Figure 1: Confidence level (C.L) for the Power Law model Eq. (25). The regions
in color denotes: CC+SN sample (blue), CC+SN+R21 (green) and CC+SN+F21 (red).
The priors R21 and F21 indicates the values quoted in Sec. III.

For this model, we use the MCMC algorithm to plot the posteriors and the
confidence regions for CC+SN and CC+SN+BAO data sets shown in Fig. 1. In
addition, we also plot the results for each prior of $H_{0}$ mentioned in Sec.
III. The most apparent feature of the results is the impact that these priors
have on $H_{0}$ values as one immediately notices that these priors tend to
increase the Hubble constant value when compared to the no prior situation.
Similarly, although to a lesser extent, $\Omega_{m_{0}}$ has a lower value
when priors are added to the chosen data sets, which is to be expected since
the priors shift the value of $H_{0}$. There is also some anti-correlation
between the $\Omega_{m,0}$ and $p_{1}$ parameters for the CC+SN data sets. An
interesting feature of this model is that for all cases of the CC+SN+BAO data
sets there are two possible valid sets for $p_{1}$ for $H_{0}$.
Correspondingly, in a Hubble diagram of $H(z)$ versus redshift, $z$, the two
values of the parameter $p_{1}$ are changing the tilt of the graph such that
an $H_{0}$ within the same range is created. Similarly, this feature is also
attributed to the two parameters $\Omega_{m,0}$ and $p_{1}$, where for a set
of value of $\Omega_{m,0}$, $p_{1}$ can have two valid sets.

The exact values for the model parameters obtained from the MCMC runs are
found in Table.1, together with the values of $\chi^{2}_{\text{min}}$, AIC,
BIC and $\Delta$AIC and $\Delta$BIC. The first column confirms the fact that
the priors tend to raise the value of $H_{0}$ and, in fact the highest value
of the Hubble constant is obtained for the CC+SN with the R21 prior
($H_{0}=72.0\pm 1.1\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$). This, however, is to be
expected as the R21 prior has a higher value than the F21 prior and thus, it
tends to shift the value of $H_{0}$ to a higher value. Consequently, the
$\Omega_{m,0}$ parameter for CC+SN+R21 has the lowest value, which agrees with
having a large value of the Hubble constant. On the other hand, the minimum
value of $H_{0}$ is reached for the CC+SN+BAO data set with no prior
($H_{0}=67.5\pm 1.4\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$), which is also
foreseeable since the BAO data is dependent on the early Universe. In turn,
this raises the value of $\Omega_{m,0}$ to the highest value.

For all cases, the $p_{1}$ parameter shows slight deviation from the
$\Lambda$CDM model as the values for such parameter fall within 2$\sigma$ of
the standard model of cosmology. An interesting feature in this model is that
the $\Delta$AIC for CC+SN+BAO is slightly negative, meaning that the maximum
likelihood of this model with the CC+SN+BAO with no prior is marginally higher
than that of the $\Lambda$CDM. In consequence, one can say that this model
with this particular data set is slightly preferred over the $\Lambda$CDM.
Coincidentally, the BIC for such data set also has the lowest value.

###  IV.2 Exponential Model – $f_{2}(T,B)$ Model

The exponential model Linder:2010py  can be described by

$\mathcal{F}_{2}(B)=\alpha_{2}\,\text{Exp}\left[-p_{2}\sqrt{\frac{B}{B_{0}}}\right]\,,$
(28)

where $\alpha_{2}$ and $p_{2}$ are constants. These constants can be related
by evaluating the Friedmann equation at current time such that

$\alpha_{2}=\frac{e^{p_{2}}6H_{0}^{2}(\Omega_{m_{o}}+\Omega_{r_{0}}-1)}{-1-\frac{p_{2}}{2}+\frac{6H_{0}p_{2}}{4B_{0}^{2}}(1+p_{2})\dot{B}|_{t=t_{0}}}\,,$
(29)

making $p_{2}$ the new model parameter for this model. Thus, the Friedmann
equation can then be written as

$\displaystyle H^{\prime\prime}$
$\displaystyle=\frac{-BB_{0}\text{Exp}\left[p_{2}\sqrt{\frac{B}{B_{0}}}\right]}{9\alpha_{2}p_{2}(1+z)^{2}H^{3}\left(p_{2}\sqrt{\frac{B_{0}}{B}}\right)}$
$\displaystyle\Bigg{[}6H_{0}^{2}\Big{(}\Omega_{m_{0}}(1+z)^{3}+\Omega_{r_{0}}(1+z)^{4}\Big{)}-6H_{0}^{2}+\alpha_{2}\text{Exp}\left[p_{2}\sqrt{\frac{B}{B_{0}}}\right]\left(1-\frac{54p_{2}(1+z)H^{3}H^{\prime}}{BB_{0}}\left(p_{2}+\sqrt{\frac{B_{0}}{B}}\right)\right)\Bigg{]}$
(30) $\displaystyle-\frac{H^{\prime}}{1+z}-\frac{H^{\prime 2}}{H}\,.$

which reduces to $\Lambda$CDM when $p_{2}=0$

The confidence regions together with the posteriors are shown in Fig. 2 for
all data sets. Similar to the power law model, the priors tend to shift the
value of the Hubble constant to a higher value compared to when no priors are
implemented, whilst $\Omega_{m,0}$, is shifted towards the lower values. The
highest value of $H_{0}$ is in agreement with the previous model and is also
reached for the CC+SN with the R21 prior ($H_{0}=72.1^{+1.1}_{-1.0}\,{\rm
km\,s}^{-1}{\rm Mpc}^{-1}$ ). In addition, the value of $\Omega_{m,0}$ is the
lowest in this case at a value of $\Omega_{m,0}=0.307\pm 0.016$. Indeed, the
anti-correlation feature between the parameters $H_{0}$ and $\Omega_{m,0}$ is
more evident in this model especially for the CC+SN data sets. On the other
hand, the lowest value, is again obtained for the CC+SN+BAO
($H_{0}=68.0^{+1.1}_{-1.2}\,{\rm km\,s}^{-1}{\rm Mpc}^{-1}$).

The $p_{2}$ values in this model are even lower than $f_{1}$CDM and closer to
zero. However, unlike $f_{1}$CDM, the values are within 1$\sigma$ of
$\Lambda$CDM, and thus, this model seems to be closer to the $\Lambda$CDM than
the power law model. The statistical indicators, however, are very similar to
the previous model and again in this case the smallest values obtained for
$\Delta$AIC and $\Delta$BC are for the data set CC+SN+BAO with no priors.
Despite that, the values for such statistical indicators for the CC+SN+BAO
with the F21 prior are very similar to the CC+SN+BAO with no prior and in
fact, the values of the Hubble constant and the density parameter of both
these data sets are very close to each other.

Table 2: Exponential Model Caption Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $p_{2}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $68.7^{+2.1}_{-2.0}$ |  $0.302^{+0.024}_{-0.023}$ |  $-0.049^{+0.045}_{-0.095}$ |  $-19.398^{+0.115}_{-0.051}$ |  1040.70 |  1048.70 |  1068.64 |  $0.96$ |  $5.95$
$\mathrm{CC+SN+R21}$ |  $72.1^{+1.1}_{-1.0}$ |  $0.282\pm 0.020$ |  $-0.045^{+0.041}_{-0.082}$ |  $-19.288^{+0.030}_{-0.031}$ |  1045.69 |  1053.69 |  1073.63 |  $1.23$ |  $6.22$
$\mathrm{CC+SN+F21}$ |  $69.3\pm 1.5$ |  $0.299\pm 0.021$ |  $-0.048^{+0.044}_{-0.090}$ |  $-19.368^{+0.043}_{-0.044}$ |  1040.89 |  1048.89 |  1068.83 |  $1.01$ |  $6.00$
CC + SN + BAO |  $68.0^{+1.1}_{-1.2}$ |  $0.307\pm 0.016$ |  $-0.050^{+0.046}_{-0.084}$ |  $-18.95^{+0.25}_{-0.00}$ |  1048.31 |  1056.31 |  1076.31 |  0.30 |  5.30
$\mathrm{CC+SN+BAO+R21}$ |  $70.38^{+0.85}_{-0.87}$ |  $0.314\pm 0.016$ |  $-0.037^{+0.033}_{-0.051}$ |  $-19.329^{+0.053}_{-0.052}$ |  1062.46 |  1070.46 |  1090.46 |  1.02 |  6.02
$\mathrm{CC+SN+BAO+F21}$ |  $68.4^{+1.1}_{-1.2}$ |  $0.307^{+0.016}_{-0.014}$ |  $-0.044^{+0.046}_{-0.071}$ |  $-18.7^{+7.7}_{-0.0}$ |  1049.22 |  1057.23 |  1077.23 |  0.54 |  5.54

Figure 2: C.L for the Exponential model Eq. (28). The regions in color
denotes: CC+SN sample (blue), CC+SN+R21 (green) and CC+SN+F21 (red). The
priors R21 and F21 indicates the values quoted in Sec. III.

###  IV.3 Logarithmic Model – $f_{3}(T,B)$ Model

The logarithmic model Bamba:2010wb  is given by

$\mathcal{F}_{3}=\alpha_{3}\ln\left[p_{3}\frac{B_{0}}{B}\right]\,,$ (31)

where $\alpha_{3}$ and $p_{3}$ are constants. By evaluating the Friedmann
equation at current times we get

$\alpha_{3}=\frac{6H_{0}^{2}\left(\Omega{m_{0}}+\Omega{r_{0}}-1\right)}{\frac{6H_{0}}{B_{0}^{2}}\dot{B}|_{t=t_{0}}-1-\ln
p_{3}}\,.$ (32)

The Friedmann equation, thus, can be written as

$\displaystyle H^{\prime\prime}$
$\displaystyle=\frac{-B^{2}}{36\alpha_{3}(1+z)^{2}H^{3}}\Bigg{[}6H_{0}^{2}\Big{(}\Omega_{m_{0}}(1+z)^{3}+\Omega_{r_{0}}(1+z)^{4}\Big{)}-6H_{0}^{2}+\alpha_{3}\ln\left[p_{3}\frac{B_{0}}{B}\right]+\alpha_{3}-\frac{216\alpha_{3}}{B^{2}}H^{3}H^{\prime}(1+z)\Bigg{]}$
(33) $\displaystyle-\frac{H^{\prime}}{1+z}-\frac{H^{\prime 2}}{H}\,.$

An interesting feature in this model is that no choice of parameter values can
recover the $\Lambda$CDM. However, it is this lack of $\Lambda$CDM limit that
provides an intriguing characteristic to this model as there are no expected
values for $p_{3}$.

Fig.3 shows the posterior and confidence regions obtained through MCMC for all
data sets. The precision results of such outputs are shown in Table. 3, in
which one can notice that even though this model has no $\Lambda$CDM limit,
there is no significant difference in the values of the Hubble constant
compared to previous models. Indeed, the highest value obtained for $H_{0}$ is
similar to that of the two preceding models and is again for the case of CC+SN
with a R21 prior ($H_{0}=72.09^{+0.97}_{-0.98}\,{\rm km\,s}^{-1}{\rm
Mpc}^{-1}$). In addition, the density parameter for CC+SN data prefers low
values, irrespective of whether a prior was included or not. On the other
hand, even though the lowest value of $H_{0}$ still appears for CC+SN+BAO, its
magnitude turns out to be quite low in comparison to $f_{1}$CDM and
$f_{2}$CDM, with $H_{0}=66.0^{+1.3}_{-1.4}{\rm km\,s}^{-1}{\rm Mpc}^{-1}$.

Another noteworthy characteristic is that the values of $p_{3}$ for all
choices of the data set is very small, close to zero. The statistical
indicators also provide a slightly different scenario than preceding models,
especially for the CC+SN data sets. In fact, the $\Delta$AIC and the
$\Delta$BIC for such data sets have relatively higher values. Moreover, for
CC+SN the values of $\Delta$AIC and the $\Delta$BIC are very similar
regardless if a prior was included or not. The minimum value of $\Delta$AIC
and the $\Delta$BIC also changes slightly, as in this case it is CC+SN+F21
which has the lowest value. Another crucial difference from previous models is
the exceptionally high value of $\Delta$AIC and $\Delta$BIC for CC+SN+BAO+F21.
These considerable differences from $\Lambda$CDM model challenges the
suitability of this model for cosmology.

Table 3: Logarithmic Model Caption Data Sets |  $H_{0}$[$\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}\mathrm{/}\mathrm{M}\mathrm{pc}$] |  $\Omega_{m,0}$ |  $p_{3}/10^{-3}$ |  $M$ |  $\chi^{2}_{\mathrm{min}}$ |  AIC |  BIC |  $\Delta$AIC |  $\Delta$BIC
---|---|---|---|---|---|---|---|---|---
CC + SN |  $68.8\pm 1.9$ |  $0.258^{+0.025}_{-0.024}$ |  $5.9^{+10.4}_{-4.0}$ |  $-19.370^{+0.059}_{-0.056}$ |  1042.08 |  1050.08 |  1070.02 |  2.35 |  7.33
$\mathrm{CC+SN+R21}$ |  $72.09^{+0.97}_{-0.98}$ |  $0.243^{+0.021}_{-0.022}$ |  $5.5^{+10.8}_{-3.4}$ |  $-19.274\pm 0.028$ |  1046.93 |  1054.93 |  1074.87 |  2.48 |  7.46
$\mathrm{CC+SN+F21}$ |  $69.3\pm 1.3$ |  $0.257\pm 0.022$ |  $5.3^{+9.4}_{-3.2}$ |  $-19.355^{+0.039}_{-0.036}$ |  1042.23 |  1050.23 |  1070.17 |  2.35 |  7.33
CC + SN + BAO |  $66.0^{+1.3}_{-1.4}$ |  $0.300\pm 0.014$ |  $1.53^{+5.13}_{-1.00}$ |  $-19.442^{+0.035}_{-0.049}$ |  1049.21 |  1057.21 |  1077.20 |  1.20 |  6.20
$\mathrm{CC+SN+BAO+R21}$ |  $69.64^{+0.73}_{-0.76}$ |  $0.310\pm 0.013$ |  $0.32^{+0.19}_{-0.0}$ |  $-19.340^{+0.023}_{-0.024}$ |  1066.77 |  1074.77 |  1094.77 |  5.33 |  10.33
$\mathrm{CC+SN+BAO+F21}$ |  $67.17\pm 0.94$ |  $0.303^{+0.013}_{-0.012}$ |  $0.49^{+1.7}_{-0.12}$ |  $-19.429^{+0.047}_{-0.019}$ |  1049.78 |  1057.78 |  1077.78 |  1.09 |  6.09

Figure 3: C.L for the Logarithmic model Eq. (31). The regions in color
denotes: CC+SN sample (blue), CC+SN+R21 (green) and CC+SN+F21 (red). The
priors R21 and F21 indicates the values quoted in Sec. III.

##  V Conclusion

In our study, we probe the performance of three prominent models in $f(T,B)$
gravity against cosmological observations of expansion. We also survey how
priors on the Hubble constant effect the posterior outcomes for the individual
cases. This is done in the context of CC, SN and BAO data which are
particularly relevant for the $H_{0}$ problem in the late Universe. Our main
interest was to determine the impact that these combinations of analyses would
have on the output cosmological parameters that each model produces, as
compared with $\Lambda$CDM. In order to highlight the contrasts between these
analyses, we show the whisker plot in Fig. 4. Here, each prior on the Hubble
constant is shown as a shaded region with its uncertainties, thus showing how
each of the three models is individually impacted. Together with this, the
other model parameters are shown, which do show some variance across the
models. The appearance of the $H_{0}^{\rm F21}$ appears to be consistent with
the models, in particular the CC+SN combination results in less than 1-sigma
difference between the priors and posterior outputs, which rises slightly in
the case that BAO data is included. On the other hand, the very high
$H_{0}^{\rm R21}$ prior on the Hubble constant results in a mostly 2-sigma
difference between these values, which is somewhat expected due to this being
one of the highest literature values of $H_{0}$.

The first two models we explore in this work, $f_{1}$CDM and $f_{2}$CDM, host
a continuous $\Lambda$CDM in which a specific setting of the additional model
parameter recovers a constant cosmological constant contribution. In this way,
we are able to assess whether the analysis prefers a nonzero evolution for the
additional Lagrangian terms in Eqs. (25,28). For comparison purposes, we also
run the same analyses for the $\Lambda$CDM model putting the respective
results in Appendix A, which is useful for statistical comparisons. By and
large, these models are consistent with the $\Lambda$CDM model. However, the
$f_{1}$CDM model consistently allows for a lower value of the $\Omega_{m,0}$
parameter while the $f_{2}$CDM model increases this value, except for the case
where BAO data is included where both models produce very similar values for
the matter density parameter. On the other hand, the additional model
parameters for these models at times goes marginally out of the 1$\sigma$
uncertainties but still being consistent with $\Lambda$CDM. This motivates
further study using more data sets in cosmology or astrophysics. These results
are also consistent with Ref. Escamilla-Rivera:2019ulu  which also performed a
background analysis on similar $f(T,B)$ models.

On the other hand, the $f_{3}$CDM model does not have a $\Lambda$CDM limit
since no value of the $p_{3}$ parameter in Eq. (31) will make this term a
constant. These models can be interesting since they do not alter $\Lambda$CDM
in a marginal way but offer a genuine alternative to it. Interestingly, this
model points to a lower value of Hubble parameter for most data set
combinations, and similarly for the matter density parameter. Interestingly,
the additional parameter for the $f_{3}$CDM model gives very small
uncertainties for some of the data samples even at background level. This may
indicate further analysis may yield decisive information about the model.

The work presented here explores the viability of literature models in
$f(T,B)$ gravity together with their consistency with priors on the Hubble
constant. It would be interesting to now extend this work to forecast data of
future surveys, and also to consider CMB data from recent surveys such as the
Planck Mission. This would however require the use of perturbations such as in
Ref. Bahamonde:2020lsm .

Figure 4: Whisker plot for model parameters $H_{0}$, $\Omega_{0}$ and $p_{i}$
respectively. The additional parameter $p_{i}$ corresponds to: $p_{1}$ for
$f_{1}$CDM, $p_{2}$ for $f_{2}$CDM and $p_{3}$ for $f_{3}$CDM. The best fits
are reported for CC+SN and CC+SN+BAO as: without prior (green), with R21 prior
(blue), and with F21 prior (red). The rectangles in colors denote the
1$\sigma$ uncertainties for each prior for the Hubble constant. The dashed
orange line for the $p_{i}$ parameter represents the $\Lambda$CDM value for
each mode (except the $f_{3}$CDM model).

##  Appendix A LCDM Model

In this appendix we present, for comparison purposes, the statistical results
using the same analyses for the $\Lambda$CDM model (see Figure 5) and its
statistical comparison between data samples in Table 4.

Figure 5: C.L for the $\Lambda$CDM model. The regions in color denotes: CC+SN sample (blue), CC+SN+R21 (green) and CC+SN+F21 (red). The priors R21 and F21 indicates the values quoted in Sec. III. Table 4: LCDM Model Caption 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
---|---|---|---|---|---|---
CC + SN |  $68.8\pm 1.8$ |  $0.300^{+0.023}_{-0.019}$ |  $-19.381^{+0.051}_{-0.054}$ |  1041.74 |  1047.74 |  1062.83
$\mathrm{CC+SN+R21}$ |  $72.05^{+0.99}_{-0.90}$ |  $0.282^{+0.018}_{-0.017}$ |  $-19.292^{+0.028}_{-0.027}$ |  1046.45 |  1052.45 |  1067.41
$\mathrm{CC+SN+F21}$ |  $69.3\pm 1.3$ |  $0.299^{+0.020}_{-0.019}$ |  $-19.370\pm 0.038$ |  1041.88 |  1047.88 |  1062.83
CC + SN + BAO |  $68.2\pm 1.0$ |  $0.305\pm 0.015$ |  $-19.403^{+0.037}_{-0.034}$ |  1050.01 |  1056.01 |  1071.00
$\mathrm{CC+SN+BAO+R21}$ |  $70.46^{+0.80}_{-0.77}$ |  $0.313\pm 0.014$ |  $-19.328^{+0.027}_{-0.026}$ |  1063.44 |  1069.44 |  1084.45
$\mathrm{CC+SN+BAO+F21}$ |  $68.54\pm 0.90$ |  $0.306^{+0.015}_{-0.014}$ |  $-19.389^{+0.030}_{-0.032}$ |  1050.68 |  1056.68 |  1077.69

###### Acknowledgements.

CE-R acknowledges the Royal Astronomical Society as FRAS 10147 and supported
by PAPIIT Project TA100122. This work is part of the Cosmostatistics National
Group (CosmoNag) project. 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.

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where $r$ varies between $0$ and $\Lambda$ and $\tau$ goes from $0$ to
$\infty$. Notice that at $r=\Lambda$ the metric is not singular but there is a
horizon.

### Open slicings

Picking the coordinates,

$\displaystyle X_{0}$ $\displaystyle=$
$\displaystyle\Lambda\sin(\xi/\Lambda)\sinh(\chi)\,,$ (A.7) $\displaystyle
X_{1}$ $\displaystyle=$ $\displaystyle\Lambda\cos(\xi/\Lambda)\,,$
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle\Lambda\sin(\xi/\Lambda)\cosh(\chi)\sin\theta\cos\phi\,,$
$\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle\Lambda\sin(\xi/\Lambda)\cosh(\chi)\sin\theta\sin\phi\,,$
$\displaystyle X_{4}$ $\displaystyle=$
$\displaystyle\Lambda\sin(\xi/\Lambda)\cosh(\chi)\cos\theta\,,$

the metric in Eq. (A.2) becomes,

$ds^{2}=d\xi^{2}+\Lambda^{2}\sin^{2}(\xi/\Lambda)(-d\chi^{2}+\cosh^{2}\chi^{2}d\Omega_{2}^{2})\,.$
(A.8)

This foliation does not describe the full dS but only the causal patch of an
observer at the centre of the hyperboloid. The region covered is depicted in
red in Fig. 14. Note that the analytical continuation $\chi\to i\chi$ leads to

$ds_{E}^{2}=d\xi^{2}+\Lambda^{2}\sin^{2}(\xi/\Lambda)(d\chi^{2}+\cos^{2}\chi^{2}d\Omega_{2}^{2})\,,$
(A.9)

which is the euclidean metric of a 4 sphere. This continuation was equivalent
to make $X_{0}\to iX_{0}$.

Figure 14: Left panel: The green part is the region covered by the static
patch of an observer at the south pole. The horizontal dotted lines are the
constant time slices in the global coordinates. Right panel: Penrose diagram
for the open slicing metrics in Eq. (A.7) and Eq. (A.10). The timelike lines
are hypersurfaces of constant $\xi$ and spacelike lines are hypersurfaces of
constant $\chi$

Another useful foliation is given by,

$\displaystyle X_{0}$ $\displaystyle=$
$\displaystyle\Lambda\cosh(\xi/\Lambda)\,,$ $\displaystyle X_{1}$
$\displaystyle=$ $\displaystyle\Lambda\sinh(\xi/\Lambda)\sinh(\chi)\,,$ (A.10)
$\displaystyle X_{2}$ $\displaystyle=$
$\displaystyle\Lambda\sinh(\xi/\Lambda)\cosh(\chi)\sin\theta\cos\phi\,,$
$\displaystyle X_{3}$ $\displaystyle=$
$\displaystyle\Lambda\sinh(\xi/\Lambda)\cosh(\chi)\sin\theta\sin\phi\,,$
$\displaystyle X_{4}$ $\displaystyle=$
$\displaystyle\Lambda\sinh(\xi/\Lambda)\cosh(\chi)\cos\theta\,,$

which foliates dS with spacelike hyperboloids. This foliation covers the
region that lies between the outside region of the causal patch and the
asymptotic future. It intersects the region covered by Eq. (A.7), at $\xi=0$
which are the null surfaces for the observer on the hemisphere . The metric in
Eq. (A.2) becomes,

$ds^{2}=-d\xi^{2}+\Lambda^{2}\sinh^{2}(\xi/\Lambda)\left(d\chi^{2}+\sinh^{2}\chi
d\Omega_{2}^{2}\right)\,,$ (A.11)

which is also pictured as the region in blue in Fig. 14. To get the Euclidean
continuation we need to do the transformations, $\chi\to i\chi$ and $\xi\to
i\xi$, which lead to Eq. (A.9). finally note that this transformation are
equivalent to $X_{0}\to iX_{0}$ .

###  A.2 Analytic Continuation in CDL

Given two vacua $\phi_{F}$ and $\phi_{T}$ such that
$V(\phi_{F})>V(\phi_{T})\geq 0$, according to [3] to calculate the scalar
field that interpolates between the two minima first we need to consider the
Euclidean solution that extremises the action in order to derive the remaining
geometry inside and outside the bubble. This was done in several steps:

  * •

In field theory, the Euclidean bounce solution is $O(4)$ invariant in such a
way that the scalar field depends on the Euclidean distance
$\xi^{2}=|x|^{2}+\tau^{2}$. Analytic continuation changes this to a $O(3,1)$
and $\xi^{2}\rightarrow|x|^{2}-t^{2}$. Once gravity is included the
corresponding line element is assumed to share that symmetry.

Figure 15: Penrose diagram for the dS to Minkowski transition mediated by a
CdL instanton. The red causal diagram is obtained after analytically continue
the $O(4)$ instanton to $O(3,1)$ whose metric is given in Eq. (A.14). The
dotted line is the wall which is also a constant $\sigma$ slice.The blue
region is open FRW with $V(\phi_{t})=0$ obtained after analyticaly continue
Eq. (A.14) past the lightcone. In this region we also draw the constant radius
hypersurfaces.

  * •

Starting with the $O(4)$ symmetric Euclidean dS metric,

$ds^{2}=d\xi^{2}+\rho^{2}(\xi)(d\psi^{2}+\sin^{2}\psi d\Omega_{2}^{2})$ (A.12)

and writing $\psi\to\pi/2+i\sigma$ we get the $O(3,1)$ invariant metric,

$ds^{2}=d\xi^{2}+\rho^{2}(\xi)(-d\sigma^{2}+\cosh^{2}\sigma d\Omega_{2}^{2})$
(A.13)

where $\sigma$ now runs from $-\infty$ to $\infty$. This spacetime is dS
foliated by timelike hyperboloids. If we fix the angles and write
$d\xi^{2}=\rho^{2}(\xi)(dy^{2})$ we get,

$ds^{2}=\rho^{2}(y)\,(-d\sigma^{2}+dy^{2})$ (A.14)

whose geometry is represented by the causal diamond in Fig. 15. The wall lies
within this region as indicated by the dotted timelike hypersurface in Fig.
15. An observer in the hemisphere will see the wall moving at a speed
approaching the speed of light.

  * •

This space is not geodesically complete, because for timelike geodesics it is
possible to go past the lightlike hypersurfaces. To complete the description
we can make a further analytical continuation $\sigma\to i\pi/2+\chi$ and a
rotation $\xi\to it$ which leads to,

$ds^{2}=-dt^{2}+\rho^{2}(-i\xi)(d\chi^{2}+\sinh^{2}\chi d\Omega_{2}^{2}).$
(A.15)

This region describes an FRW open space, and covers the blue patch in Fig. 14.
Note that the analytical continuation $\sigma\to-i\pi/2+\chi$ describes the
upper right region of the diagram. Constant time hypersurfaces end on
$\mathscr{I}_{+}$ at $\xi=\pi/2$

  * •

To interpret the diagram in terms of dynamics of the scalar field, let us
assume that the tunneling points where $\phi_{b}$ and $\phi_{a}$, as indicated
in Fig. 13. Both points are hemispheres of dS where $\rho$ vanishes. The wall
separating the two regions is inside the causal diamond as indicated in Fig.
15 by the dotted line. The left region outside the causal diamond describes
the dynamics after the tunneling while the right region describes the false
vacuum dynamics. Spacelike hypersurfaces in these region represent constant
field surfaces. Then after the tunneling the field rolls down to the true
vacuum $V(\phi_{t})$ at ${\mathcal{I}}_{+}$. After analytic continuation the
surfaces of constant field values correspond to constant values of
$|x|^{2}-t^{2}$ which are hyperbolae. This defines the natural foliation of
the spacetime.

In Fig. 15 we have assumed that the true vacuum is Minkowski. In the case of
dS this only extends up to the horizontal line at $T=\pi/2$.

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