# Observational Constraints on $f(T)$ Gravity from Model-Independent Data

F. B. M. dos Santos  J. E. Gonzalez  R. Silva

###### Abstract

We establish new constraints on $f(T)$ gravity models by using cosmological
data. In particular, we investigate the restrictions given by the gas mass
fraction measurements of galaxy clusters and transversal BAO data. Both data
sets are regarded as weakly dependent on a fiducial cosmology. In addition, we
also include a CMB measurement of the temperature power spectrum first peak,
along with $H(z)$ values from cosmic chronometers and supernovae data from the
Pantheon data set. We also perform a forecast for future constraints on the
deviation of $f(T)$ models from the $\Lambda$CDM scenario by following the
specifications of the J-PAS and Euclid surveys and find significant
improvements on the constraints of the $b$-parameter, when compared to the
results of the statistical analysis.

##  1 Introduction

The $\Lambda$CDM model has become the standard model to describe the evolution
of the Universe at large scales since the discovery of its late-time cosmic
acceleration [1, 2]. This model is based on the existence of a fluid with
negative pressure described by a cosmological constant ($\Lambda$) added to
the Einstein field equations, the so-called dark energy. The model also
assumes the presence of pressureless non-baryonic matter comprises most of
galaxies composition denominated cold dark matter (CDM). This model is the one
that best describes data from type Ia supernovae and other astrophysical
objects to the cosmic microwave background (CMB) temperature power spectrum
[3]. However, the increasing measurements of the late-time Universe might
suggest that the framework described by general relativity (GR) is not the
most general for describing gravity, motivating the idea of extensions of the
standard model in a variety of ways, such as a modified theory of gravity [4,
5].

One of the most discussed questions in the literature recently regarding the
predictions of the standard model is the tension between data at different
eras of the Universe, especially the values of $H_{0}$ (see [6] for a review)
from the Planck satellite [3] and the local measurements from astrophysical
objects [7, 8, 9]. Therefore, several modifications to GR were considered in
recent years as an attempt to explain why such tensions appear and to answer
other questions that the $\Lambda$CDM model does not address. One of the most
studied theories involves a function of the Ricci scalar $f(R)$ [10, 11, 12]
in the field equations that could account for phenomena not explained by the
cosmological standard model. More profound modifications from the standard
picture lead to alternative options. Suppose one considers that instead of the
metric, the gravitational field is described by tetrads, where the Riemann
tensor, the main ingredient for the dynamics of GR, is replaced by a non-zero
torsion. A change of this kind allows us to an alternative description of
gravity so that the resulting field equations could lead to relevant effects
in the observables. Developments towards this way led to the Teleparallel
Equivalent General Relativity (TEGR), in which the torsion scalar $T$ is
related with the Ricci scalar as $\bar{R}=-T+B$, with $B$ being a boundary
term, meaning that it is equivalent to GR at the level of field equations. A
natural extension of this picture can be realized when we look at how $f(R)$
theories are constructed, giving rise to $f(T)$ theories [13, 14, 15, 16, 17],
with a function of $T$ being added to the gravitational action.

A number of forms for the $f(T)$ function were proposed in the literature [13,
18, 19] where they have proved to be viable models. In Ref. [20, 21, 22, 23,
24, 25, 26], statistical analyses were performed to find constraints on
cosmological parameters when the models are confronted with various data sets.
In Ref. [27], it was shown that a power-law dependence on $T$ can greatly
alleviate the Hubble tension by increasing the associate error in the
parameter, and in [24], an exponential $f(T)$ form was particularly preferred
over the standard $\Lambda$CDM model. More recently, the impact on $f(T)$
constraints by different $H_{0}$ priors was investigated in [28]. In Ref.
[29], a $f(T)$ model with exponential form was introduced as an infrared (IR)
correction to GR. A complete analysis was done in Ref. [30, 31], at the
background and linear perturbation level, where it was found that the Hubble
tension can also be alleviated. In a recent work, [26], the study of
concordance with the Big Bang Nucleosynthesis helped constrain the free
parameters of $f(T)$ models with great precision, with a deviation from the
standard model at 3$\sigma$ confidence level. It is also interesting to note
that an extension that is being widely explored is the $f(T,B)$ class of
models [32, 33, 34, 35, 36]. Here, the boundary term $B$ contribute to the
equations of motion as being part of an arbitrary function. It makes the
construction of many different models possible, and the equations of motion
are generally more complicated.

In this work, we seek to constrain some of these models with data that is
generally regarded as being model-independent. In particular, we want to
verify the impact of including measurements of the gas mass fraction of galaxy
clusters, which is the ratio between the baryonic and total mass of a given
cluster. Recently, these data sets have been used to constrain cosmological
scenarios [37, 38], being another way of confirming cosmic acceleration.
Moreover, they are a cosmology-independent way of determining cosmological
parameters, especially the matter density parameter $\Omega_{m0}$. In
particular, we use the data set in Ref. [39], which consists of 40 points at
low and intermediate redshift range of $0.078\leq z\leq 1.063$. Another data
set that we expect to impact our analysis and produce strong constraints is
the 2D baryonic acoustic oscillations (BAO) data due to its cosmological model
independence. [40, 41, 42] Additionally, we also use $H(z)$ values from the
Cosmic Chronometers method compiled in [43, 44], the position of the first
peak of the CMB temperature power spectrum $l_{1}$ [45, 46], and the latest
Type Ia supernovae measurements of the Pantheon compilation [47].

This work is organized in the following manner. In Sec. 2, we review the
$f(T)$ formalism, focusing on the background dynamics. In Sec. 3, we briefly
review the $f(T)$ models that will be investigated. Sec. 4 describes the data
and methodology used in the analysis. In Sec. 5, we discuss the results of the
statistical analysis, while in Sec. 6 we perform a forecast on the models for
future experiments. Finally, in Sec 7, we present our considerations.

##  2 Teleparallel Gravity

In this section we briefly introduces the teleparallel formalism, the
generalization for a $f(T)$ function, and its cosmological consequences.

###  2.1 Formalism

Teleparallel gravity is a way of describing gravity in which the fundamental
object is the tetrad $e_{\mu}^{A}$ instead of the usual metric tensor
$g_{\mu\nu}$ for GR. Gravity is then described by a non-zero torsion, while
the Riemann tensor, along with the non-metricity tensor, are both zero (in the
teleparallel picture). A consequence of this approach is that when deriving
the field equations, the Levi-Civita connection
($\bar{\Gamma}^{\lambda}_{\mu\nu}$) is substituted by the teleparallel
connection ${\Gamma}_{\mu\nu}^{\lambda}$ [48]. The metric tensor of GR is
related with the tetrad as [14, 16]

$\displaystyle g_{\mu\nu}=\eta_{AB}e^{A}_{\mu}e^{B}_{\nu},$ (2.1)

with capital latin letters corresponding to the tangent space, while Greek
letters correspond to space-time coordinates on the manifold. The teleparallel
connection is written as

$\displaystyle\Gamma^{\lambda}_{\mu\nu}=E_{A}^{\lambda}\
\partial_{\mu}e^{A}_{\nu},$ (2.2)

with $E_{A}^{\lambda}$ being the inverse tetrad. In the teleparallel picture,
the connection ${\Gamma}_{\mu\nu}^{\lambda}$ is related to the Riemannian one
as

$\displaystyle{\Gamma}_{\;\;\mu\nu}^{\lambda}=\bar{{\Gamma}}_{\;\;\mu\nu}^{\lambda}+K_{\;\;\mu\nu}^{\lambda}$
(2.3)

with

$\displaystyle
K^{\lambda}_{\;\;\mu\nu}\equiv-\frac{1}{2}\left(T_{\;\;\mu\nu}^{\lambda}-T^{\lambda}_{\;\;\mu\nu}-T_{\;\;\nu\mu}^{\lambda}\right),$
(2.4)

being the contortion tensor that is defined in terms of the torsion tensor
$T_{\mu\nu}^{\lambda}$, which has an equivalent role as the Riemann tensor in
GR

$\displaystyle T_{\mu\nu}^{\lambda}=2\Gamma^{\lambda}_{[\mu\nu]}.$ (2.5)

Contraction of the torsion tensor leads to

$\displaystyle
T=\frac{1}{4}T^{\rho\mu\nu}T_{\rho\mu\nu}+\frac{1}{2}T^{\rho\mu\nu}T_{\nu\mu\rho}-T^{\;\;\;\;\rho}_{\rho\mu}T^{\nu\mu}_{\;\;\;\;\nu},$
(2.6)

which can be related to the Levi-Civita Ricci scalar $\bar{R}$ as

$\displaystyle R$
$\displaystyle=\bar{R}+\frac{2}{e}\partial_{\rho}\left(eT^{\mu\rho}_{\mu}\right)+T$
$\displaystyle=\bar{R}-B+T=0,$ (2.7)

where $B=-\frac{2}{e}\partial_{\rho}\left(eT^{\mu\rho}_{\mu}\right)$ is a
boundary term, showing an equivalence between GR and TEGR at the level of
field equations. This allows us to write a similar gravitational action as the
Einstein-Hilbert one. This way, we can write the field equations that can be
shown to be equivalent to GR by providing the same equations of motion,
according to (2.6). Furthermore, in an analogous way to $f(R)$ gravity, one
can generalize the action by introducing a function $f(T)$ to the
gravitational Lagrangian, so that the action becomes

$\displaystyle\mathcal{S}=\frac{1}{16\pi G}\int
d^{4}xe\Big{(}T+f(T)+\mathcal{L}_{m}\Big{)},$ (2.8)

where $\mathcal{L}_{m}$ is the matter Lagrangian, and $e$ can be identified as
$e=\operatorname{det}\left(e^{a}_{\mu}\right)=\sqrt{-g}$. Just as in
$f(\bar{R})$ gravity, $f(T)$ will be responsible for deviation from GR, where
for instance, if the function is taken as a constant, we reproduce the
$\Lambda$CDM model.

We can vary this action concerning the tetrad to obtain the field equations as

$\displaystyle
e^{-1}\partial_{\mu}(ee^{\rho}_{A}S_{\rho}^{\mu\nu})(1+f_{T})+e^{\rho}_{A}S_{\rho}^{\mu\nu}\partial_{\mu}(T)f_{TT}-(1+f_{T})e_{A}^{\lambda}T^{\rho}_{\mu\lambda}S_{\rho}^{\nu\mu}+\frac{1}{4}e_{A}^{\nu}(1+f(T))=4\pi
Ge^{\rho}_{A}\mathcal{T}_{\rho}^{v},$ (2.9)

where the susbscript T denotes derivatives with respect to the torsion scalar,
and $\mathcal{T}^{\nu}_{\rho}$ is the energy-momentum tensor, and
$S_{\rho}^{\;\;\mu\nu}\equiv\frac{1}{2}\left(K^{\mu\nu}_{\;\;\;\;\rho}+\delta^{\mu}_{\rho}T^{\sigma\nu}_{\;\;\;\;\sigma}-\delta^{\nu}_{\rho}T^{\sigma\mu}_{\;\;\;\;\sigma}\right)$
is a superpotential that can be used to obtain the tensor scalar as
$T=S_{\rho}^{\;\;\mu\nu}T^{\rho}_{\;\;\mu\nu}$.

###  2.2 Background Dynamics

To study the cosmological implications of $f(T)$ gravity in the context of a
homogeneous, isotropic, and spatially flat universe, characterized by
$e_{\mu}^{A}=\operatorname{diag}(1,a,a,a)$, we see that this corresponds to
the FLRW geometry characterized by the line element

$\displaystyle ds^{2}=dt^{2}-a^{2}(t)\delta_{ij}dx^{i}dx^{j},$ (2.10)

so that the Friedmann equations become, from (2.9) are [14, 16]

$\displaystyle 3H^{2}=8\pi G(\rho_{m}+\rho_{r})-\frac{f}{2}+Tf_{T}$ (2.11)

and

$\displaystyle\dot{H}=-\frac{4\pi
G(\rho_{m}+P_{m}+\rho_{r}+P_{r})}{1+f_{T}+2Tf_{TT}},$ (2.12)

with $H\equiv\frac{\dot{a}}{a}$ being the Hubble parameter, and $\rho$, $P$
being the energy density and pressure that come from the total energy-momentum
tensor, respectively. We note that if $f=0$, the $f(T)$ formulation is
equivalent to GR, while the dynamics can be modified entirely by assuming a
different $f(T)$ function since we in the FLRW geometry have that $T=-6H^{2}$.

Before solving the equations, we should define some quantities. Since, in
general we can interpret the r.h.s. of Eq. (2.11) as corresponding to the
contribution of all matter components, while the Eq. (2.12) contains the
contributions from the pressures$+$densities of the fluids, it is possible to
make the following definitions for the dark energy density and pressure,
respectively:

$\displaystyle\rho_{DE}\equiv\frac{1}{16\pi
G}\left[2Tf_{T}-f\right]\quad\text{and}\quad P_{DE}\equiv\frac{1}{16\pi
G}\left[\frac{f-f_{T}T+2T^{2}f_{TT}}{1+f_{T}+2Tf_{TT}}\right].$ (2.13)

Then, the dark energy equation of state can be written as

$\displaystyle
w_{DE}\equiv\frac{P_{DE}}{\rho_{DE}}=\frac{f-f_{T}T+2T^{2}f_{TT}}{(2Tf_{T}-f)(1+f_{T}+2Tf_{TT})}.$
(2.14)

The cosmological fluids considered will have their evolution dictated by the
conservation of the energy-momentum tensor

$\displaystyle\dot{\rho}_{m}+3H\rho_{m}(1+w_{m})=0\quad\text{and}\quad\dot{\rho}_{r}+3H\rho_{r}(1+w_{r})=0,$
(2.15)

with $w_{m}$ and $w_{r}$ being the the equation of state parameters of matter
and radiation, respectively; and we can find that the defined dark energy
density will also follow the same conservation equation:

$\displaystyle\dot{\rho}_{DE}+3H\rho_{DE}(1+w_{DE})=0,$ (2.16)

with $\rho_{DE}$ and $P_{DE}$ defined by (2.13). Since $T=-6H^{2}$, the
normalized Hubble parameter $E(z)$ can be written as
$E^{2}(z)\equiv\frac{H^{2}(z)}{H^{2}_{0}}=\frac{T(z)}{T_{0}}$, with $H_{0}$ is
the present value of the Hubble parameter, and $T_{0}=-6H_{0}^{2}$. Also,
assuming that $\rho_{m}$ is pressureless dust, so $w_{m}=0$, and that
radiation follows $w_{r}=1/3$, we can write the Friedmann equation (2.11) as

$\displaystyle
E^{2}(z,r)=\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}+\Omega_{dark0}y(z,r),$
(2.17)

with $y(z,r)$ being

$\displaystyle
y(z,r)\equiv\frac{1}{6H_{0}^{2}\Omega_{dark0}}\left[2Tf_{T}-f\right],$ (2.18)

and $\Omega_{dark0}$ being the dark energy density parameter today,

$\displaystyle\Omega_{dark0}=1-\Omega_{m0}-\Omega_{r0},$ (2.19)

produced by the modifying $f(T)$ term. Note the distortion function $y(z,r)$
that controls the effect from the modified dynamics of teleparallel gravity,
where $r$ corresponds to the free parameters of the specific model considered.
The main characteristics of this function are that GR must (preferentially) be
reproduced for some limit of parameter, while at the cosmological level, the
concordance model $\Lambda$CDM can also be achieved (when $y=1$). Numerical
analysis of the main $f(T)$ models indicate that deviations from the standard
model are generally small [24], when the model in question can reproduce the
$\Lambda$CDM one, showing that different $f(T)$ scenarios are concordant with
the standard model, and might even compete with it.

##  3 $f(T)$ Models

We present the $f(T)$ models investigated in this work. The three selected
functions are well studied in the literature, and previous numerical analyses
have shown that they are among the best ones preferred by data when compared
to the $\Lambda$CDM model. We will see how different data can affect the
predictions for each model while verifying the consistency with previous
works.

  * •

Power-law model

Currently, one of the most favored by data $f(T)$ models is the power-law form
given by [13]

$\displaystyle f_{1}(T)=\alpha(-T)^{b},$ (3.1)

where $\alpha$ and $b$ are the two free parameters that can be related through

$\displaystyle\alpha=\frac{\Omega_{dark0}(6H_{0}^{2})^{1-b}}{2b-1},$ (3.2)

by taking $z=0,H(z=0)=H_{0}$ in Eq. (2.20). The distortion factor becomes
simply

$\displaystyle y(z,b)=E^{2b}(z,b),$ (3.3)

and then the Friedmann equation is

$\displaystyle
E^{2}(z,b)=\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}+\Omega_{dark0}E^{2b}(z,b),$
(3.4)

We can easily see that $b=0$ reproduces the $\Lambda$CDM cosmology. This model
gives a de-Sitter limit for $z=-1$, and deviations from the standard model are
more evident for higher $|b|$. However, these deviations are generally small,
as verified by numerical analyses performed in past years [20, 21, 24]. Also,
the power-law model is capable of alleviating the Hubble tension [27]. The
parameter $b$ is anti-correlated with $H_{0}$, meaning that a larger $H_{0}$
is achieved for small $b$, a feature that does not happen with the other
models investigated due to a strong degeneracy between parameters.

  * •

Exponential model

Another model investigated is inspired by $f(R)$ gravity, where an exponential
dependence exists, and the $f(T)$ function takes the form [20]

$\displaystyle f_{2}(T)=\alpha T_{0}\left(1-e^{-pT/T_{0}}\right),$ (3.5)

where, again, $\alpha$ and $p$ are dimensionless parameters that can be
related though the Friedmann equation as

$\displaystyle\alpha=\frac{\Omega_{dark0}}{1-(1+2p)e^{-p}},$ (3.6)

so the distortion term is

$\displaystyle
y(z,b)=\frac{1-\left(1+\frac{2E^{2}}{b}\right)e^{-\frac{E^{2}}{b}}}{1-\left(1+\frac{2}{b}\right)e^{-\frac{1}{b}}}.$
(3.7)

Consequently, the Friedmann equation for this model becomes

$\displaystyle
E^{2}(z,b)=\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}+\Omega_{dark0}\frac{1-\left(1+\frac{2E^{2}}{b}\right)e^{-\frac{E^{2}}{b}}}{1-\left(1+\frac{2}{b}\right)e^{-\frac{1}{b}}},$
(3.8)

where we can define $b\equiv 1/p$, so the $\Lambda$CDM model is recovered for
$b\rightarrow 0^{+}$, while the GR limit is achieved for
$b\rightarrow+\infty$.

  * •

The square-root exponential model

The last $f(T)$ model we consider here is the exponential form studied in
[18], with functional form

$\displaystyle f_{3}(T)=\alpha T_{0}\left(1-e^{-p\sqrt{T/T_{0}}}\right),$
(3.9)

where the $\alpha$ and $b$ parameters are related as

$\displaystyle\alpha=\frac{\Omega_{dark0}}{1-(1+p)e^{-p}},$ (3.10)

and the distortion factor becomes

$\displaystyle
y(z,b)=\frac{1-\left(1+\frac{E}{b}\right)e^{-\frac{E}{b}}}{1-(1+\frac{1}{b})e^{-\frac{1}{b}}}.$
(3.11)

with $p=1/b$. In a similar manner to the $f_{2}$ model, one can see that the
limit $b\rightarrow 0^{+}$ reproduces the $\Lambda$CDM model, while
$b\rightarrow+\infty$ corresponds to the pure GR limit.

##  4 Observational data and methods

To check the viability of these models, we will perform a statistical analysis
using the Monte Carlo Markov Chain (MCMC) method, where we compare the
predictions with different data sets of the cosmological observables.

###  4.1 Data Sets

  * •

Gas mass fraction data

The first data used in this work is the cluster mass gas fraction
$f_{gas}\equiv\frac{M_{gas}}{M_{total}}$ [49, 50, 51, 52, 53, 54] (one can
check [39] for further references). Since these clusters can be assumed as
containing a good part of the total content of non-relativistic matter in the
Universe, we can initially, approximate $f_{gas}$ as

$\displaystyle f_{gas}=b_{gas}(z)\frac{\Omega_{b}}{\Omega_{m}},$ (4.1)

where $\Omega_{b}$ is the total fraction of baryonic matter, while
$b_{gas}(z)$ is some function that expresses how different the cluster mass
gas fraction is from the cosmic one. As $f_{gas}\propto d_{L}d_{A}^{1/2}$ and
following the Ref. [55, 39], we can use the cosmic distance duality relation
$d_{A}=d_{L}/(1+z)^{2}$ to write $f_{gas}$ in terms of the angular diameter
distance

$\displaystyle f_{gas}\equiv
A(z)K(z)\gamma(z)\frac{\Omega_{b0}}{\Omega_{m0}}\left(\frac{D_{A}^{fid}(z)}{D_{A}(z)}\right)^{3/2},$
(4.2)

where $D_{A}(z)$ is the angular diameter distance for a given model,
normalized by a fiducial model that is taken as a $\Lambda$CDM one with
$\Omega_{m0}=0.3$ and $H_{0}=70$ km/s/Mpc for the data we are using. The
$A(z)$ factor is the angular correction between two models, which is usually
close to one, but can be modeled as

$\displaystyle
A(z)=\left(\frac{H(z)D_{A}(z)}{H^{fid}(z)D_{A}^{fid}(z)}\right)^{\eta},$ (4.3)

where $\eta$ is estimated as $\eta=0.442\pm 0.035$ [39]. $K(z)$ and
$\gamma(z)$ are respectively the calibration bias and the depletion factor,
where the former takes into account instrumental inaccuracies as well as
astrophysical effects in the cluster mass, while the latter measures the
depletion of hot gas in the cluster relative to the baryon cosmic fraction.
Some works in the literature have investigated the possible variation of these
quantities with redshift (in partcular $\gamma$) [56, 57], but in this
analysis, we take them as constants, as estimated by hydrodynamical
simulations; therefore, we use the values $\gamma=0.848\pm 0.085$ [58], and
$K=0.96\pm 0.09\pm 0.09$ [59]. By using these three parameters
($\eta,K,\gamma$ as discussed, the $\chi^{2}$ function for gas mass fraction
measurements is given by

$\displaystyle\chi^{2}_{fgas}=\sum_{i=1}^{40}\left(\frac{f_{gas}(z_{i})-f_{gas}^{obs}}{\sigma_{fgas,i}}\right)^{2}.$
(4.4)

In this expression, $f_{gas}(z_{i})$ represent the theoretical predictions
given by Eqs. (4.2-4.3), $f_{gas}^{obs}$ are the observational values; the
uncertainties $\sigma_{f_{gas,i}}$ have the effective form

$\displaystyle\sigma_{fgas,i}^{2}=\sigma_{obs,i}^{2}+\left[f^{th}_{gas}(z_{i})\right]^{2}\Bigg{[}\left(\frac{\sigma_{K}}{K}\right)^{2}+\left(\frac{\sigma_{\gamma}}{\gamma}\right)^{2}+\ln^{2}\left(\frac{H(z_{i})D_{A}(z_{i})}{H^{fid}(z_{i})D_{A}^{fid}(z_{i})}\right)\sigma_{\eta}^{2}\Bigg{]},$
(4.5)

where $\sigma_{obs,i}^{2}$ are the uncertainties associated with the data. We
use the following data set for the analysis. In [39], the fraction was derived
for 40 cluster measurements at the radius $r_{2500}$ 111$r_{2500}$ refers to
the radius of spherical shells in which the matter of the cluster is
contained. For these data, it means that the mean density inside is 2500
higher than the cosmic critical density at the cluster’s redshift. improving
the previous work done in Ref. [55]. These points cover the redshift interval
of $0.078\leq z\leq 1.063$. A recent application of these data points in
constraining cosmological parameters is described in [38] for the $\Lambda$CDM
and $w$CDM models, in a way that we can compare our results with theirs,
especially in the determination of the Hubble parameter $H_{0}$. It is good to
mention that other measurements of $f_{gas}$ are available in the literature
[60, 61, 62, 63], from lower to higher redshifts ($0.0473\leq z\leq 1.235$),
but measured in the radius $r_{500}$, which will not be used in the present
work.

  * •

BAO2D data

As in the case of gas mass fraction data, the BAO data we will use in this
analysis is computed in a way that can be regarded as almost model-
independent. The method, presented in Refs. [42, 40, 41], involves the 2-point
correlation function for a distribution of galaxies, where only the angular
separation is considered in redshift shells of the order $\delta z\simeq
10^{-2}$. This allows one to obtain information on the BAO transversal signal
without the effect of a fiducial cosmology222To obtain the BAO-2D signal it is
used a fiducial cosmology, however, the final angular distance estimates are
weakly model dependent [40]., and can be used to test different cosmological
scenarios. The expression gives the BAO angular scale $\theta_{BAO}$

$\displaystyle\theta_{BAO}=\frac{r_{s}}{(1+z)D_{A}(z)},$ (4.6)

where $r_{s}$ is the comoving sound horizon, obtained as 333To compute the
integral in Eq. (4.7), we have used the expression for $E(z)$ in (2.17) for
each model, while approximating an universe with matter and radiation at the
right-hand side of said equation, allowing us to account for the effect of the
$f(T)$ function at high redshifts.

$\displaystyle
r_{d}(z)=\int_{z_{d}}^{\infty}\left(3+\frac{9\Omega_{b0}a^{\prime}}{4\Omega_{\gamma
0}}\right)^{-1/2}\frac{da^{\prime}}{a^{{}^{\prime}2}H(a^{\prime})},$ (4.7)

where $\Omega_{b0}$ is the present baryon density parameter, and
$\Omega_{\gamma 0}$ is the present photon density parameter. The redshift at
the drag epoch $z_{d}$ is estimated by the fitting formula [64]

$\displaystyle
z_{d}=\frac{1291(\Omega_{m0}h^{2})^{0.251}}{1+0.659(\Omega_{m0}h^{2})^{0.828}}\left[1+b_{1}(\Omega_{b}h^{2})^{b_{2}}\right],$
(4.8)

with

$\displaystyle
b_{1}=0.313(\Omega_{m0}h^{2})^{-0.419}\left[1+0.607(\Omega_{m}h^{2})^{0.674}\right]\text{
and}$ $\displaystyle b_{2}=0.238(\Omega_{m0}h^{2})^{0.223}.$ (4.9)

The total BAO $\chi^{2}$ function ($\chi^{2}_{BAO}$) is then

$\displaystyle\chi^{2}_{BAO}=\sum_{i=1}^{14}\left[\frac{\theta_{BAO}^{th}(z_{i})-\theta_{BAO}^{obs}(z_{i})}{\sigma_{\theta,i}}\right]^{2}.$
(4.10)

These data points have been used previously in different investigations. For
instance, in [65], the $\Lambda$CDM and CPL models were analyzed along with
Planck data. It was shown that a dynamical dark energy scenario in this
context provides a value for $H_{0}$ that is compatible with local
measurements. Following this work, the same data was used with H0LiCOW data
[66] to obtain constraints on the $H_{0}-r_{d}$ plane independently of CMB
data and investigate the impact on spatial curvature. In [67] cosmological
constraints were obtained by imposing observational and thermodynamics limits
on interacting dynamical dark energy models.

  * •

Type Ia Supernovae (SNe) data

We also use in this analysis the SNIa Pantheon compilation [47]. In
particular, we consider the binned version where the 1048 points are compacted
to 40, which span the redshift interval $0.01\leq z\leq 1.6$. The $\chi^{2}$
function is given as

$\displaystyle\chi^{2}_{SNe}=\Delta\mu\mathcal{C}^{-1}_{SNe}\Delta\mu^{T},$
(4.11)

where $\mathcal{C}^{-1}_{SNe}$ corresponds to the inverse covariance matrix of
the data, and $\Delta\mu=\mu_{i}-\mu_{i,th}$ is a vector with the difference
between the observational and theoretical distance modulus. The distance
modulus is defined as $\mu=m_{B}-\mathcal{M}$, where $\mu_{B}$ is the observed
apparent magnitude at a given redshift, while $\mathcal{M}$ is the absolute
magnitude which is treated as a nuisance parameter in the statistical
analysis. This is compared with the theoretical form calculated via

$\displaystyle\mu_{th}=5\log\frac{D_{L}(z)}{Mpc}+25,$ (4.12)

where $D_{L}(z)=(1+z)\int_{0}^{z}\frac{dz^{\prime}}{H(z^{\prime})}$ is the
luminosity distance.

  * •

$H(z)$ data

We use measurements of the Hubble parameter obtained from the differential age
method, also known as cosmic chronometer (CC) data. This method of measuring
the differential age of galaxies allows us to determine the Hubble parameter
at a certain redshift without assuming a specific model. Here, we will
consider 31 points cataloged in [43], and compiled in Table 1 of [44] spanning
the redhsift range of $0.07\leq z\leq 2$. The $\chi^{2}$ function is
constructed as

$\displaystyle\chi^{2}_{CC}=\sum_{i=1}^{31}\left(\frac{H(z_{i})-H^{obs}(z_{i})}{\sigma_{H,i}(z_{i})}\right)^{2}.$
(4.13)

  * •

CMB data

The last data set used in this work is the Planck CMB data encoded on the
first peak of the temperature power spectrum, indicated by $l_{1}$, expressed
as [45]

$\displaystyle
l_{1}=l_{A}\left[1-0.267\left(\frac{\rho_{r}(z_{d})}{0.3(\rho_{b}(z_{d})+\rho_{c}(z_{d}))}\right)^{0.1}\right],$
(4.14)

with $l_{A}=\pi(1+z_{d})\frac{d_{A}(z_{d})}{r_{dec}}$ being the acoustic sound
scale, and all quantities are evaluated at the decoupling redshift $z_{dec}$
[68]. The measured value of the peak we use is $l_{1}=220.0\pm 0.5$ [46]

Therefore, to analyse the impact of the $f_{gas}$ and BAO data, we consider
four total $\chi^{2}$ functions:
$\chi^{2}=\chi^{2}_{\text{Base}}+\chi^{2}_{f_{gas}}$,
$\chi^{2}=\chi^{2}_{\text{Base}}+\chi^{2}_{BAO}$,
$\chi^{2}=\chi^{2}_{\text{Base}}+\chi^{2}_{f_{gas}}+\chi^{2}_{BAO}$ along with
all data sets combined, and where
$\chi^{2}_{\text{Base}}\equiv\chi^{2}_{SNe}+\chi^{2}_{CC}$. We assume uniform
priors on $H_{0}$, $w$, $b$ and $\Omega_{m0}$ and a Gaussian prior on the
baryon parameter density of $\Omega_{b}h^{2}\equiv\omega_{b}=0.0226\pm
0.00034$ [69]. To perform the MCMC analysis, we use the emcee sampler[70], and
the GetDist [71] Python module to plot the results.

Model |  $H_{0}$[Km/s/Mpc] | $\Omega_{m0}$ | $b$ | $\omega_{b}$ | $\mathcal{M}_{B}$
---|---|---|---|---|---
|  |  | Base + $f_{gas}$ |  |
$\Lambda$CDM | $68.962^{+1.688}_{-1.741}$ | $0.301^{+0.012}_{-0.011}$ | $-$ | $0.0225^{+0.00033}_{-0.00031}$ | $-19.382^{+0.049}_{-0.052}$
$f_{1}(T)$ | $69.046^{+1.701}_{-1.652}$ | $0.302^{+0.012}_{-0.011}$ | $-0.0672^{+0.128}_{-0.144}$ | $0.0226^{+0.00035}_{-0.00035}$ | $-19.384^{+0.050}_{-0.049}$
$f_{2}(T)$ | $68.83661^{+1.604}_{-1.650}$ | $0.302^{+0.012}_{-0.011}$ | $0.0982^{+0.0581}_{-0.0639}$ | $0.0226^{+0.00032}_{-0.00032}$ | $-19.381^{+0.047}_{-0.049}$
$f_{3}(T)$ | $68.808^{+1.625}_{-1.574}$ | $0.301^{+0.011}_{-0.012}$ | $0.111^{+0.091}_{-0.076}$ | $0.0225^{+0.00034}_{-0.00035}$ | $-19.384^{+0.047}_{-0.047}$
|  |  | Base + BAO2D |  |
$\Lambda$CDM | $70.735^{+1.149}_{-1.140}$ | $0.280^{+0.013}_{-0.013}$ | $-$ | $0.0225^{+0.00033}_{-0.00036}$ | $-19.337^{+0.036}_{-0.036}$
$f_{1}(T)$ | $69.602^{+1.618}_{-1.626}$ | $0.268^{+0.015}_{-0.015}$ | $0.159^{+0.125}_{-0.140}$ | $0.0226^{+0.00033}_{-0.00035}$ | $-19.364^{+0.045}_{-0.044}$
$f_{2}(T)$ | $70.406^{+1.349}_{-1.498}$ | $0.277^{+0.014}_{-0.012}$ | $0.131^{+0.061}_{-0.090}$ | $0.0226^{+0.00036}_{-0.00034}$ | $-19.341^{+0.043}_{-0.040}$
$f_{3}(T)$ | $70.042^{+1.335}_{-1.483}$ | $0.274^{+0.015}_{-0.014}$ | $0.223^{+0.117}_{-0.143}$ | $0.0226^{+0.00033}_{-0.00035}$ | $-19.351^{+0.040}_{-0.041}$
|  |  | Base + $f_{gas}$ \+ BAO2D |  |
$\Lambda$CDM | $70.990^{+1.137}_{-1.189}$ | $0.288^{+0.0092}_{-0.0087}$ | $-$ | $0.0224^{+0.00031}_{-0.00031}$ | $-19.325^{+0.034}_{-0.036}$
$f_{1}(T)$ | $70.917^{+1.450}_{-1.323}$ | $0.288^{+0.0088}_{-0.0082}$ | $0.0137^{+0.108}_{-0.119}$ | $0.0224^{+0.00032}_{-0.00032}$ | $-19.326^{+0.040}_{-0.037}$
$f_{2}(T)$ | $70.832^{+1.114}_{-1.217}$ | $0.290^{+0.008}_{-0.008}$ | $0.104^{+0.064}_{-0.071}$ | $0.0224^{+0.00035}_{-0.00033}$ | $-19.323^{+0.034}_{-0.033}$
$f_{3}(T)$ | $70.679^{+1.200}_{-1.221}$ | $0.289^{+0.008}_{-0.008}$ | $0.156^{+0.095}_{-0.098}$ | $0.0224^{+0.00033}_{-0.00034}$ | $-19.330^{+0.035}_{-0.035}$
|  |  | Base + $f_{gas}$ \+ BAO2D \+ CMB |  |
$\Lambda$CDM | $70.712^{+0.592}_{-0.603}$ | $0.288^{+0.008}_{-0.008}$ | $-$ | $0.02248^{+0.00030}_{-0.00033}$ | $-19.334^{+0.016}_{-0.016}$
$f_{1}(T)$ | $70.509^{+1.137}_{-1.098}$ | $0.287^{+0.008}_{-0.008}$ | $0.0019^{+0.104}_{-0.104}$ | $0.0224^{+0.00033}_{-0.00032}$ | $-19.340^{+0.025}_{-0.025}$
$f_{2}(T)$ | $70.562^{+0.664}_{-0.815}$ | $0.289^{+0.009}_{-0.008}$ | $0.120^{+0.059}_{-0.070}$ | $0.0224^{+0.00035}_{-0.00033}$ | $-19.333^{+0.017}_{-0.017}$
$f_{3}(T)$ | $70.255^{+0.693}_{-0.757}$ | $0.288^{+0.008}_{-0.008}$ | $0.156^{+0.097}_{-0.103}$ | $0.0224^{+0.00033}_{-0.00032}$ | $-19.344^{+0.017}_{-0.017}$
Table 1: Cosmological constraints for all models investigated with $1\sigma$
uncertainties. We divide the results into Base+$f_{gas}$, Base+BAO2D,
Base+$f_{gas}$+BAO2D and Base+$f_{gas}$+BAO2D+CMB data sets.

###  4.2 Model Selection

After determining the parameters posteriors distributions for each model, we
must use a way to compare them, which will help us to determine which model is
more favored by the data used. The most robust estimator used in cosmology for
statistical comparison is the Bayes factor, the ratio between the Bayesian
evidence of a model of interest and a reference model. We also compute the
value of the Akaike Information Criteria (AIC) [72], which, under the
assumption of at least near gaussianity of the posterior distribution, it is
given as [73]

$\displaystyle AIC\equiv-2\ln\mathcal{L}_{max}+\frac{2k(k+1)}{N-k-1}.$ (4.15)

In (4.15), $\mathcal{L}_{max}$ is the value of the maximum likelihood for a
given model. At the same time, $k$ and $N$ are the number of free parameters
of the model and the total number of data used in the analysis. The criterion
for model comparison is as follows: Smaller AIC corresponds to a better model,
and a larger number of free parameters $k$ penalizes the model, resulting in a
larger criterion value. For two competing models, one can define the
difference $\Delta IC\equiv IC_{model}-IC_{ref}$, where $IC_{ref}$ represents
the AIC of the reference model. We use the same classification as [24], where
$\Delta IC\leq 2$ corresponds to statistical compatibility between models,
$2<\Delta IC<6$ represents a tension between them, while $\Delta IC\geq 10$
represents robust evidence against the model we want to compare with the
reference one.

We also use the Bayes’ factor as an evidence-based statistical estimator for
model selection. This quantity considers not only the best-fit point (the
minimum $\chi^{2}$ parameters values) but also the entire probability
distribution. The definition of the Bayes’ factor, $B_{01}$, is the evidence
ratio between two models:

$B_{0i}=\frac{E_{0}}{E_{i}},$ (4.16)

being $E_{i}$ the evidence in the Bayes’ theorem for the $i-$model and $E_{0}$
the evidence for a reference one. As in the AIC criterion, the comparison is
performed with a reference model and a qualitative inference is interpreted by
the Jefreys’ logarithmic scale [74]. In this scale, the logarithm of the
Bayes’ factor determines the preference for a model with the higher Bayesian
evidence. The characteristic values of the scale are: $|\ln B_{0i}|<1$,
$1<|\ln B_{0i}|<2.5$, $2.5<|\ln B_{0i}|<5$, and $|\ln B_{0i}|>5$ for
inconclusive, weak, moderate and strong evidence, respectively.

##  5 Results

The results of the statistical analysis for all models considered are
displayed in Figures 1, 2, 3 and 4 and Tables 1 and 2, where the values for
the statistical criteria in Table 2 correspond to the combination of all data
sets. For the standard $\Lambda$CDM model, we realize the following: When
using the combination Base+$f_{gas}$ we have the lowest value for $H_{0}$, of
$68.962^{+1.688}_{-1.741}$, compatible with the value of Planck [3] of
$67.36\pm 0.54$ Km/s/Mpc at $1\sigma$ confidence. This value is also
considerably higher than the one obtained in [38]. The present matter density,
on the other hand, is the highest, of $\Omega_{m0}=0.301^{+0.012}_{-0.011}$,
as suggested by the anti-correlation of the parameters in Figure 1 (grey
contour). When we consider Base+BAO2D, we note a considerable increase in
$H_{0}$ and a decrease in $\Omega_{m0}$, where $H_{0}$ goes more towards the
R19 value [8], and there is an improvement in the error bars, when considering
BAO2D data. For the third combination, Base+$f_{gas}$+BAO2D, we note a slight
increase in $H_{0}$, but with the uncertainties essentially preserved.
However, there is now a significant improvement in the uncertainties of
$\Omega_{m0}$. Finally, for all data sets combined, there is a great decrease
in the error bars for $H_{0}$, with similar results as in [24], but in our
case, the values of parameters remain almost the same.

We note similar behaviors in the constraints of the $f(T)$ models, where the
lowest value of $H_{0}$ is always obtained when we consider the combination
Base+$f_{gas}$, while the highest is achieved for Base+$f_{gas}$+BAO2D. For
the $f_{1}$ power-law model, we have $H_{0}=69.602^{+1.618}_{-1.626}$ (for the
Base+BAO2D combination) and $H_{0}=70.990^{+1.114}_{-1.217}$ (for the
Base+$f_{gas}$+BAO2D combination), being quite different from the results in
Ref. [28], where the ’Base’ data set was used, but with another BAO data, and
closer to Ref. [24], where Base+$f\sigma_{8}$ was used. A good improvement in
the $H_{0}$ constraint was obtained by adding CMB data. As for the parameter
$b$, the combination Base+$f_{gas}$+BAO2D leads to $1\sigma$ concordance with
the $\Lambda$CDM scenario. Such a conclusion was also obtained from previous
analyses in the literature. One interesting feature of the power-law model is
the ability to greatly alleviate the $H_{0}$ tension due to its anti-
correlation with $b$. In our results (Figure 2), this anti-correlation is
preserved, while there is an inversion of correlations in the
$H_{0}-\Omega_{m0}$ plane, when we consider Base+$f_{gas}$ and Base+BAO2D data
sets. For all data sets combined, we note that this specific correlation is
not as evident, so the value of $\Omega_{m0}$ is better determined. In
addition, the value of $b$ agrees even more with the standard model, where
this time, we obtain a small positive value.

As shown in Figures 3 and 4 , the results are also similar for both
exponential $f_{2}$ and $f_{3}$ models. For all data sets combined, $H_{0}$ is
better constrained when compared with the power-law model, while the parameter
$b$ does not have the $\Lambda$CDM limit at $1\sigma$ level. These general
results are also present in the literature and are part of the explanation as
to why the power-law model can solve the $H_{0}$ tension, while the $f_{2}$
and $f_{3}$ models cannot [27]. We then see that using these data for
constraining $f(T)$ models leads to results consistent with recent previous
studies. We also achieve a similar level of restriction as other data sets
available, despite the larger associated error bars, as it is in the case of
$f_{gas}$ and BAO2D data.

Model | $\chi^{2}_{min}$ | AIC |  $\Delta$AIC | $\ln B_{0i}$
---|---|---|---|---
$\Lambda$CDM | 88.660 | 96.991 | - | -
$f_{1}$ | 88.946 | 99.446 | 2.455 | 2.11
$f_{2}$ | 89.747 | 100.247 | 3.256 | 0.92
$f_{3}$ | 88.984 | 99.484 | 2.493 | 0.52
Table 2: Values of $\chi^{2}_{min}$, AIC and the logarithm of the Bayes’
factor for the analysis performed with the Base+$f_{gas}$+BAO2D+CMB data set.
Figure 1: $1\sigma$ and $2\sigma$ confidence contours and posterior
distributions for the $\Lambda$CDM model. The grey contours represent the
analysis with SNe+CC+$f_{gas}$ data, the purple contours correspond to
SNe+CC+BAO2D data, the green contours correspond to SNe+CC+$f_{gas}$+BAO2D,
and the blue contours are referent to all data combined,
SNe+CC+BAO2D+$f_{gas}$+CMB.

To conclude, we discuss the statistical results from the Bayes factor and AIC.
The AIC criterion indicates the $\Lambda$CDM model as the best one, followed
by $f_{1}$, $f_{3}$ and $f_{2}$ models, which is consistent with the previous
results in the literature, where the power-law model is the best one. From the
scale described in the previous section, all models are in mild tension with
$\Lambda$CDM for these data. Also, as pointed out in Ref. [24], a slight
difference of $\Delta$IC between models makes it difficult to establish which
of the competing models is the best. Hence, we also have a statistical
equivalence between the $f(T)$ models. As for the Bayes’ factor, we have
different results. Although the $\Lambda$CDM is still preferred, we note a
significant preference for the $f_{3}$ model, when we look only at the $f(T)$
models; the $f_{1}$ models seem to be performing the worst in the light of
this criterion.

##  6 Forecast

The next generation of surveys mapping the Large Scale Structure of the
Universe will obtain tighter constraints on cosmological models. These surveys
will be important to distinguish between modified gravity theories and dark
energy models by considering precise data in a wide range of cosmic history.
In this section we follow the specifications of the J-PAS and Euclid surveys
to quantify the future constraints on $f(T)$-gravity that will be obtained
from the radial BAO signal [75, 76, 77, 78, 79, 80]444 A brief description of
these data can be found in Ref. [81].. For this purpose, we use the expected
$H(z)$ relative errors to simulate Hubble parameter data considering the
fiducial cosmology given by the results presented in Table 1 (Base + $f_{gas}$
\+ BAO2D \+ CMB). We replace the BAO2D real data with the simulated $H(z)$
data in the statistical analysis to avoid double counting of the same
observable and maintain the rest of the data sets. It is worth mentioning that
this approach considers the constraints obtained from the distribution of
galaxies and their effect in conjunction with the other observables used in
this work in their current state.

In Table 3, we present the results of the $f(T)$-gravity parameter constraints
using the J-PAS-like and Euclid-like $H(z)$ estimates. As is shown, the most
essential improvements on the $b$-parameter constraints occur for the $f_{1}$
and $f_{3}$ models in comparison with the results presented in Table 1. Such
results are particularly relevant for the $f_{3}$ model because it would be
possible to measure deviance of the $\Lambda$CDM model in $\sim 2\sigma$.
Another important point is that the constraints for these two surveys are
similar, being the Euclid $H(z)$ estimates are more precise while the J-PAS
estimates cover a wider redshift range.

Figure 2: Same as Figure 1, but for the $f_{1}$ power-law teleparallel model. Model | $\sigma_{b}$
---|---
| J-PAS | Euclid
$f_{1}$ | 0.056 | 0.049
$f_{2}$ | 0.054 | 0.053
$f_{3}$ | 0.079 | 0.071
Table 3: Results of the future constraints for each $f(T)$-gravity parameter
model using the J-PAS-like and Euclid-like $H(z)$ estimates and real data.

##  7 Conclusions

This work has explored the statistical viability of some $f(T)$ gravity models
with the essential cosmological datasets. The analysis considered the $H(z)$
from Cosmic Chronometers, Type Ia supernovae from the Pantheon set, and the
first peak of the CMB temperature power spectrum, with the addition of other
data regarded as being model-independent, in our case, the gas mass fraction
of galaxy clusters and radial BAO data. Our main goal was to study the impact
of these data in constraining the cosmological parameters, especially the $b$
parameter present in the $f(T)$ functions that control the deviation from the
standard $\Lambda$CDM scenario. We have found that the free parameters can be
constrained with similar accuracy as previous works. Although the associated
error bars in the $f_{gas}$ and $BAO_{2D}$ data are considerably large, it
would be interesting then to observe the impact of using such data in the
constraints of other modified gravity models. From a statistical point of
view, we have seen that the AIC/Bayes’ factor criteria prefer the standard
scenario, so the $\Lambda$CDM remains the best model. However, among the
$f(T)$ models, the AIC indicates a statistical equivalence, especially between
the $f_{1}$ and $f_{3}$ models, while the Bayes’ factor shows the $f_{3}$
model as the best one.

Figure 3: Same as Figure 1, but for the $f_{2}$ exponential teleparallel
model.

Finally, we have performed a forecast for the statistical analysis using two
next-generation galaxy surveys: J-PAS and Euclid, to predict the improvement
in the measurements of the $b$-parameter of the three $f(T)$ models. Compared
with the results of Table 1, we have found a significant improvement in the
error bars of $b$, especially for the $f_{1}$ model. We also note that for
both J-PAS and Euclid, similar constraints with the simulated $H(z)$
measurements are found, and in the context of the $f_{3}$ model, a $\sim
2\sigma$ deviance from the standard $\Lambda$CDM model is observed.

## Acknowledgements

F.B.M. dos Santos is supported by Coordenação de Aperfeiçoamento de Pessoal de
Nível Superior (CAPES). JEG acknowledges financial support from the Conselho
Nacional de Desenvolvimento Científico e Tecnológico CNPq (Grants no.
165468/2020-3). R. Silva acknowledges financial support from CNPq (Grant No.
307620/2019-0).

Figure 4: Same as Figure 1, but for the $f_{3}$ square-root exponential
teleparallel model.

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