To ensure that only the influence of perspective deformation is being
considered ($\sigma^{2}_{\varphi}=\sigma^{2}_{\varepsilon}$), we validate our
model for planar photometric patches in the synthetic ICL-NUIM dataset
[handa2014benchmark], which provides RGB-D sequences and the ground truth
values for the camera and scene parameters.

As in [molton2004locally], we consider patches as locally planar regions on 3D
world surface instead as 2D templates in image space or more complex surfaces,
since our formulation allows patches to be placed on any parametric surface
(see Section LABEL:subsec:geoCov). Although salient points often appear at
discontinuities, it is commonly assumed that it is possible to find a locally
dominant plane for their representation (see Figure LABEL:fig:intro).

We gather a massive number of high gradient pixel projections between all
image pairs within the same sequence, and extract their photometric errors,
intensity gradients and predicted deformations. Figures 4 and 4 show
histograms illustrating the number of data points, the deformation range and
photometric errors in the dataset.

We group photometric errors according to their deformation and normalize them
with the intensity gradient. Finally, we compute the covariance of the errors
within each cluster obtaining the value for $\sigma^{2}_{x}$ in Equation
eq:photoCov

$\sigma^{2}_{x}=\frac{\sigma_{r}^{2}}{G^{2}}=\frac{\sigma_{N}^{2}}{G^{2}}+\sigma_{\varepsilon}^{2}(\varepsilon^{2}).$
(1)

Figure 4 shows a representative sample of the results (using a pattern of $9$
pixels), clearly confirming a relation between perspective deformation and
visual covariances and how our model captures it accurately. We highlight
three significant outcomes: 1) expressing compressions as its homologous
tractions ($\varepsilon_{t}^{2}=\varepsilon_{c}^{-2}$, see Section
LABEL:sec:perspDef) unifies the behaviour of both covariance responses
($\sigma_{t}^{2}\sim\sigma_{c}^{2}$). 2) Following Ockham’s razor, we define
$\sigma_{t}^{2}(\varepsilon^{2}-1)$ and $\sigma_{c}^{2}(\varepsilon^{-2}-1)$
simply with constant values. 3) From the minimum covariance value in the
absence of perspective deformation ($\sigma_{\varepsilon}^{2}=1$) we can
derive with Equation eq:patchNoise the photometric noise in the images
$\sigma^{2}_{I}$. Note how as expected the use of bilinear interpolation for
intensities reduces the image noise to around $\frac{4}{9}$.

[b].45  [b].45

Figure 1: Deformation data. Figure 2: Normalized photo. errors.

[b].9

Figure 3: Photometric covariance model. Results from the validation in the
ICL-NUIM dataset [handa2014benchmark] show that the covariance for photometric
patches increases along with the perspective deformation according to Equation
eq:geoCovSplit. Figure 4: Photometric model validation.

Patch patterns. Table LABEL:table:patchApproach evaluates our model for
photometric residuals in different patch patterns, as proposed in
[engel2017direct]. The most relevant outcomes are: 1) Bigger patches act as
photometric filters, reducing the image noise $\sigma_{I}$. 2) Bigger patches
experience a faster degradation of the performance under traction, but perform
better than smaller patches under compression. And vice versa, smaller patches
deteriorate faster under compression and handle traction better. This effect
is easily recognizable observing the changes of the response function
parameters ($\sigma_{t}^{2}$, $\sigma_{c}^{2}$) or the range of deformation
$\varepsilon^{2}$ in Table LABEL:table:patchApproach .

The last columns of Table LABEL:table:patchApproach show results assuming the
surface is perpendicular to the optical axis ($S_{0}$) or the backprojected
ray ($S_{\perp}$). We observe that now traction effects are mostly dominated
by the surface assumption, _i.e._ , as the camera moves towards the points,
depth inconsistencies arise. Yet, bigger patches still work better under
compression.

modelFittingTable.tex

[t]0.12  [t]0.17  [t]0.17  [t]0.12  [t]0.17  [t]0.17

Figure 5: Fr1. Figure 6: Photo. Model Figure 7: ORB. Model Figure 8: Fr2.
Figure 9: Photo. Model Figure 10: ORB. Model

[t]0.12  [t]0.17  [t]0.17  [t]0.12  [t]0.17  [t]0.17

Figure 11: Fr3. Figure 12: Photo. Model Figure 13: ORB. Model Figure 14: realsense Figure 15: Photo. Model Figure 16: ORB. Model Figure 17: Model validation with real data. We perform the model fitting of Equation eq:geoCovSplit, described in sections LABEL:subsec:photoPatches and LABEL:subsec:featMethods, with data from three cameras of the sequences of the RGB-D TUM dataset [sturm12iros] and with data recorded with a realsense D435i depth camera. It can be seen how our covariance model for photometric patches and ORB features fits the noise distribution of the four cameras. camera | $\sigma_{I}$ | $\sigma^{2}_{t}$ | $\sigma^{2}_{c}$ | $\varepsilon^{2}\in$ | $\mathtt{R}^{2}$ | $\frac{\sigma^{2}_{t}}{\sigma^{2}_{c}}$
---|---|---|---|---|---|---
fr1 | 47.2 | 1.80 | 1.34 | 0.6-1.4 | 0.95 | 1.3
fr2 | 22.0 | 0.88 | 0.89 | 0.6-1.8 | 0.97 | 1.0
fr3 | 34.4 | 0.79 | 0.90 | 0.7-1.4 | 0.98 | 0.9
realSense | 30.8 | 0.63 | 0.62 | 0.6-1.9 | 0.97 | 1.0
Table 1: Model validation for photometric 9-pixel patches with real data. The
table shows the parameters of Equation eq:geoCovSplit estimated for the
cameras of the RGBD-TUM dataset (fr) and a realsense D435i depth camera. Note
how the 9-pixel patch behaves similarly under traction and compression
deformation.

Real Data. We repeat the experiment in Figure 4 with real data from three
different cameras of the public RGB-D TUM dataset [sturm12iros]. Figure 17
shows how again, our model accurately captures the visual covariance produced
by perspective deformation of photometric planar patches. An important outcome
from the real data with different cameras and sequences is the validation of
the assumption of considering patches as locally planar regions rather than
more complex surfaces. However, it also leads to saturation of the model under
strong changes in perspective deformation. Finally, to have a cleaner
experiment with data from a depth sensor but trying to minimize any source of
noise, we validated the model with data from an intel realsense D435i depth
camera facing a checkerboard pattern. Figure 17 shows how our model fits more
reliably in this validation performed under more suitable conditions. Table 1
collects the coefficients of the model validation that we will use in the
experiments of Section LABEL:sec:experiments.


#  $f(T,B)$ cosmography for high redshifts

Celia Escamilla-Rivera  0000-0002-8929-250X<EMAIL_ADDRESS>Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México,
Circuito Exterior C.U., A.P. 70-543, México D.F. 04510, México.  Geovanny A.
Rave-Franco  0000-0003-0638-8344<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
0000-0002-7835-4365<EMAIL_ADDRESS>Institute of Space Sciences and
Astronomy, University of Malta, Msida, Malta  Department of Physics,
University of Malta, Msida, Malta

###### Abstract

In light of the statistical performance of cosmological observations, in this
work we present the cosmography in $f(T,B)$ gravity. In this scenario we found
a cosmological viable standard case that allows to reduce the degeneracy
between several $f(T,B)$ models already proposed in the literature.
Furthermore, we constrain this model using Pantheon SNeIa compilation, Cosmic
Chronometers and a newly GRB calibrated data sample. We found that with an
appropriate strategy for including the cosmographic parameter, we do produce a
viable cosmology with our model within $f(T,B)$ gravity.

##  I Introduction

General relativity (GR) has gone through decades of success as the fundamental
theory of gravity of the concordance model of cosmology with observational
consistency across all scales where measurements can be taken
dodelson2003modern ; Weinberg:1972kfs . In cosmology, the combination of cold
dark matter (CDM) together with a cosmological constant $\Lambda$ has led to
the $\Lambda$CDM model of the Universe which is precisely compatible with
observations related to the late-time accelerated expansion of the Universe
Riess:1998cb ; Perlmutter:1998np  as well as those from galactic dynamics
alma991010731099705251 . Despite these accomplishments, $\Lambda$CDM is now
facing a growing challenge from new observations related to the specific value
of the Hubble constant Bernal:2016gxb ; DiValentino:2020zio ;
DiValentino:2021izs , as well as a noticeable tension in the growth of large
scale structure Aghanim:2018eyx ; DiValentino:2020vvd  over time. In addition,
the theoretical problems within $\Lambda$CDM have been ever present during
this period with the arbitrariness of the value of the cosmological constant
being a serious issue in modern cosmology Weinberg:1988cp ; Appleby:2018yci ,
as well as the lack of direct measurements of dark matter in particle physics
experiments despite enormous efforts Gaitskell:2004gd ; Baudis:2016qwx ;
Bertone:2004pz .

The growing observational and theoretical problems in $\Lambda$CDM have led to
several efforts to modify the theory to better confront these developing
challenges. While efforts to continue to allow for more exotic forms of matter
continue Barboza:2008rh ; Jassal:2005qc ; Efstathiou:1999tm ; Linder:2002et ;
Astier:2000as ; Cooray:1999da ; Escamilla-Rivera:2021boq ; Colgain:2021pmf ,
the possibility of modified gravity providing a possible resource to this
problem has started to receive much more serious attention in the literature
Clifton:2011jh ; Capozziello:2011et ; CANTATA:2021ktz . There now exists an
abundance of theories beyond GR in which potentially viable cosmological
histories are possible. However, it is not enough for these gravitational
models to produce fine-tuned models that agree with the numerical value of the
late time accelerating Universe. In these generalized models, it is crucial
that each component of the theory be connected with observational
measurements. Cosmography Bamba:2012cp ; Yang:2019vgk  opens that possibility
where rather than repeatedly solving the Friedmann equations for slightly
different assumptions and testing each of these choices, cosmographic
parameters can be used since they are simply related to the derivatives of the
scale factor Weinberg:1972kfs . Thus, by just assuming a flat, homogeneous and
isotropic cosmology, the distance–redshift relation can be used to produce
successive derivative components of different generalized theories of gravity.

The weight of observational tensions in late time cosmology as well as the
growing problems of dark matter particle physics and the foundational problems
related to the structure of the theory has prompted several radical proposals
to revisiting the origins of gravitation in order to meet these growing
challenges. One of these possible descriptions of gravity is teleparallel
gravity (TG) which embodies those theories in which the curvature associated
with the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\alpha}_{\phantom{\alpha}\mu\nu}$ (over-circles
represent quantities calculated with the Levi-Civita connection) is replaced
with torsion Bahamonde:2021gfp ; Aldrovandi:2013wha ; Cai:2015emx ;
Krssak:2018ywd  through the teleparallel connection
$\Gamma^{\alpha}_{\phantom{\alpha}\mu\nu}$, which is curvature-less but
continues to satisfy metricity. In this regime, all quantities that act as a
measure of curvature will identically vanish such as the Riemann tensor
$R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\equiv 0$ Clifton:2011jh . One
consequential effect of this property is that the teleparallel Ricci scalar
will be identically zero Aldrovandi:2013wha , $R\equiv 0$. Thus, teleparallel
theories must be based on a new geometric setting through the so-called
torsion tensor Bahamonde:2021gfp  $T^{\alpha}_{\phantom{\alpha}\mu\nu}$. An
interesting feature of the torsion tensor is that by taking an appropriate
series of contractions leads to a torsion scalar $T$ which produces a
teleparallel equivalent of general relativity (TEGR) Aldrovandi:2013wha . TEGR
and GR differ by a boundary term $B$ in their Lagrangians but produce
dynamically equivalent field equations.

The boundary term is the source of the largely fourth order nature of most
modified theories of gravity that involve direct generalizations of the
scalars based on the Riemann tensor, which is encapsulated in the Lovelock
theorem Lovelock:1971yv  through its stringent conditions on the possible
generalizations of GR that produce second order theories. Thus, in TG the
class of theories which produce second order models immediately becomes
drastically larger Gonzalez:2015sha ; Bahamonde:2019shr  due to the decoupling
of the boundary term in TEGR. Similar to GR, TEGR can be directly generalized
through the many popular mechanisms by which modified theories of gravity can
be formed. For instance, similar to $f(\accentset{\circ}{R})$ Sotiriou:2008rp
; Faraoni:2008mf ; Capozziello:2011et , TEGR can be straightforwardly used to
write a more arbitrary action based on an $f(T)$ gravity Lagrangian
Ferraro:2006jd ; Ferraro:2008ey ; Bengochea:2008gz ; Linder:2010py ;
Chen:2010va ; Bahamonde:2019zea . The decoupling of the Ricci scalar into the
torsion scalar and boundary terms means that a more representative analogue to
$f(\accentset{\circ}{R})$ gravity would be through $f(T,B)$ gravity Escamilla-
Rivera:2019ulu ; Bahamonde:2015zma ; Capozziello:2018qcp ; Bahamonde:2016grb ;
Paliathanasis:2017flf ; Farrugia:2018gyz ; Bahamonde:2016cul ; Wright:2016ayu
. This limits to $f(\accentset{\circ}{R})$ gravity for the special limit in
which $f(T,B)=f(-T+B)=f(\accentset{\circ}{R})$. $f(T,B)$ models have been
explored in several regimes ranging from weak field solar system tests
Farrugia:2020fcu ; Capozziello:2019msc ; Farrugia:2018gyz ; Bahamonde:2020bbc
, to theoretical predictions from cosmology Bahamonde:2015zma ;
Bahamonde:2016grb ; Bahamonde:2016cul ; Bahamonde:2015zma , as well as
observational cosmology Escamilla-Rivera:2019ulu . To this end, it is timely
that further studies be performed to complement the work being done on
determining which models of $f(T,B)$ gravity are more viable than others in
terms of their confrontation with observations at all scales.

In this work, we explore the possibility of using cosmography as a tool to
link observations on state parameters with the derivative term contributions
to the $f(T,B)$ Lagrangian in the hope of further probing models which are
observationally preferred. To this end, we first briefly introduce $f(T,B)$
gravity in Sec. II where we describe the nuances of the theory. In Sec. III,
we show how the different components of the $f(T,B)$ model can connect to the
different cosmographic parameters at play. Through this analysis, we explore
the extent to which this analysis can be used to formulate a cosmologically
viable model in this scheme. Finally, we close with a brief summary of our
work in Sec. V. We also have included two appendices to better explain the
details of this work. In Appendix A we discuss the theoretical background of
the model analysis we explore in this work, while in Appendix B we explore
some technical details related to how phenomenology.

##  II $f(T,B)$ gravity on basis

GR is based on the curvature associated with the Levi-Civita connection
$\accentset{\circ}{\Gamma}^{\sigma}_{\phantom{\sigma}\mu\nu}$ (recalling that
over-circles refer to any quantities based on the Levi-Civita connection)
while TG is built on its exchange with the teleparallel connection
$\Gamma^{\sigma}_{\phantom{\sigma}\mu\nu}$ Aldrovandi:2013wha ;
Bahamonde:2021gfp ; Cai:2015emx ; Krssak:2018ywd . This alternative approach
to building the foundations of gravitation is arrived at by first replacing
the metric tensor $g_{\mu\nu}$ as the fundamental dynamical variable with the
tetrad $e^{A}_{\phantom{A}\mu}$ and spin connection
$\omega^{B}_{\phantom{B}C\nu}$. In all instances, Greek indices refer to
coordinates on the general manifold while Latin ones refer to coordinates on
the local Minkowski space. Thus, tetrads connects both spaces and can raise
and lower indices between the different spaces RevModPhys.48.393

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

where orthogonality conditions produce the conditions

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

However, there exists an infinite number of solutions of this equation which
occurs due to the freedom in choosing the local Lorentz frame. This freedom is
embodied in the TG flat spin connection.

The tetrad and flat spin connection pair can then be used to define the
teleparallel connection as Cai:2015emx ; Krssak:2018ywd

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

which is curvature-less and satisfies metricity Hohmann:2021fpr . For the
gauge choice where the spin connection vanishes, this is called the
Weitzenböck gauge Weitzenbock1923 . Thus, the tetrad spin connection pair
represent the fundamental variables of the theory and so produce independent
field equations.

Analogous to GR, TG is build on the concept of tensor objects defined on a
Riemann manifold. However, the Riemann tensor
$\accentset{\circ}{R}^{\alpha}_{\phantom{\alpha}\beta\mu\nu}$ is a measure of
curvature, and so if the Levi-Civita connection is replaced with the
teleparallel connection then its components will organically vanish
$R^{\alpha}_{\phantom{\alpha}\beta\mu\nu}\equiv 0$. Thus, a torsion tensor is
defined to give a measure of torsion Aldrovandi:2013wha ; ortin2004gravity

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

where the brackets denote the antisymmetric operator, and this acts as a
measure of the field strength of gravity in TG Bahamonde:2021gfp ;
Aldrovandi:2013wha . Taking a suitably chosen addition of contractions leads
to 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 arrived at by demanding that $T$ is equivalent to the Ricci scalar
(up to a total divergence term). Analogous to the Ricci scalar being dependent
only on the Levi-Civita connection, the torsion scalar is only dependent on
the teleparallel connection. The teleparallel Ricci scalar will identically
vanish due to it being torsion-less. However, this relation does show the
equivalence of the regular Ricci scalar and the torsion scalar through
Bahamonde:2015zma

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

where $B$ is a boundary term defined by

$B:=2\mathring{\nabla}_{\mu}\left(T^{\mu}\right)\,,$ (7)

and where $e=\text{det}\left(e^{A}_{\phantom{A}\mu}\right)=\sqrt{-g}$ is the
tetrad determinant. The torsion scalar is the Lagrangian of TEGR and through
this relation, we have a guarantee that the equations of GR and TEGR will be
dynamically equivalent to each other.

In regular curvature-based gravity, one of the most popular approaches to
modifying GR is that of $f(\accentset{\circ}{R})$ gravity Sotiriou:2008rp ;
Capozziello:2011et  in which the Einstein-Hilbert action Ricci scalar is
generalized to an arbitrary function thereof. The decoupling of the second and
fourth order contributions to the Ricci scalar means that the analogous setup
would be contained in a broader $f(T,B)$ generalization Bahamonde:2015zma ;
Capozziello:2018qcp ; Bahamonde:2016grb ; Paliathanasis:2017flf ;
Farrugia:2018gyz ; Bahamonde:2016cul ; Bahamonde:2016cul ; Wright:2016ayu ;
Zubair:2021gve ; Zubair:2020wyu ; Zubair:2018wyy . The action for this theory
can then be written as

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

where the $f(\accentset{\circ}{R})$ limits occurs for
$f(T,B)=f(-T+B)=f(\accentset{\circ}{R})$ and TEGR is found when $f(T,B)=-T$.

For a flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric in $f(T,B)$
gravity we can take the tetrad

$e^{A}_{\phantom{A}\mu}=\textrm{diag}(1,a(t),a(t),a(t))\,,$ (9)

which is in the Weitzenböck gauge Krssak:2015oua ; Tamanini:2012hg , and where
$a(t)$ is the scale factor. This tetrad reproduces the FLRW metric to give

$ds^{2}=dt^{2}-a(t)^{2}(dx^{2}+dy^{2}+dz^{2})\,,$ (10)

through Eq. (1). The corresponding torsion scalar turns out to be

$T=6H^{2}\,,$ (11)

while the boundary term is given by

$B=6(3H^{2}+\dot{H})\,,$ (12)

which together produce the regular Ricci scalar, namely
$\accentset{\circ}{R}=-T+B=6(\dot{H}+2H^{2})$. The analogous Friedmann
equations for this tetrad are given by

$\displaystyle-3H^{2}(3f_{B}+2f_{T})+3H\dot{f}_{B}-3\dot{H}f_{B}+\frac{1}{2}f$
$\displaystyle=\kappa\rho,$ (13)
$\displaystyle-\left(3H^{2}+\dot{H}\right)(3f_{B}+2f_{T})-2H\dot{f}_{T}+\ddot{f}_{B}+\frac{1}{2}f$
$\displaystyle=-\kappa P.$ (14)

where overdots represent time derivatives, while $\rho$ and $P$ are
respectively the energy density and pressure of a perfect fluid, and where
underscore denotes partial derivatives (such as $f_{T}=\partial f/\partial T$
and $f_{B}=\partial f/\partial B$).

##  III Extended cosmography for $f(T,B)$ gravity

Cosmography offers a way to parameterize the evolution of the Universe through
its state parameters which can be used to infer progressively higher order
derivatives of our gravitational Lagrangian, namely $f(T,B)$. This provides a
directly observationally-driven approach to attempting to resolve the open
questions surrounding possible modifications to the Einstein-Hilbert action.
To this end, we employ the standard setting of defining the cosmographic
parameters up to the seventh order in terms of derivatives of the scalar
factor, namely Capozziello:2011hj ; Aviles:2012ay

$\displaystyle H=\frac{1}{a}\frac{da}{dt}\,,\quad
q=-\frac{1}{a}\frac{d^{2}a}{dt^{2}}H^{-2}\,,\quad
j=\frac{1}{a}\frac{d^{3}a}{dt^{3}}H^{-3}\,,$ (15) $\displaystyle
s=\frac{1}{a}\frac{d^{4}a}{dt^{4}}H^{-4}\,,\quad
l=\frac{1}{a}\frac{d^{5}a}{dt^{5}}H^{-5}\,,\quad\tilde{m}=\frac{1}{a}\frac{d^{6}a}{dt^{6}}H^{-6}\,,\quad
n=\frac{1}{a}\frac{d^{7}a}{dt^{7}}H^{-7}\,,$ (16)

which all describe the kinematic evolution of the Universe. By considering the
Friedmann equations of $f(T,B)$ Franco:2020lxx ; Escamilla-Rivera:2019ulu ;
Caruana:2020szx  we can relate the derivatives of the Lagrangian with the
evolution of the Universe. On a practical level, this occurs due to the
relations between the contributing torsion scalar and boundary term, and the
derivatives of the Hubble parameter. We thus write down these calculations for
clarity for the work that follows

$\displaystyle\dot{T}=12H\dot{H}\,,$
$\displaystyle\dot{B}=3\dot{T}+6\ddot{H}\,,$ (17)
$\displaystyle\ddot{T}=12\left[\dot{H}^{2}+H\ddot{H}\right]\,,$
$\displaystyle\ddot{B}=3\ddot{T}+6\dddot{H}\,,$ (18)
$\displaystyle\dddot{T}=12\left[3\dot{H}\ddot{H}+H\ddot{H}\right]\,,$
$\displaystyle\dddot{B}=3\dddot{T}+6H^{\text{(iv)}}\,,$ (19) $\displaystyle
T^{\text{(iv)}}=12\left[3\ddot{H}^{2}+4\dot{H}\dddot{H}+HH^{\text{(iv)}}\right]\,,$
$\displaystyle B^{\text{(iv)}}=3T^{\text{(iv)}}+6H^{\text{(v)}}\,,$ (20)
$\displaystyle
T^{\text{(v)}}=12[10\ddot{H}\dddot{H}+5\dot{H}H^{\text{(iv)}}+HH^{\text{(v)}}]\,,$
$\displaystyle B^{\text{(v)}}=3T^{\text{(v)}}+6H^{\text{(vi)}}\,,$ (21)

which can then be written in terms of the cosmographic parameters (15)-(16) as

$\begin{aligned} T&=6H^{2}\,,\\\ \dot{T}&=-12H^{3}\left(q+1\right)\,,\\\
\ddot{T}&=12H^{4}\left[j+q\left(q+5\right)+3\right]\,,\\\
\dddot{T}&=12H^{5}\left[-j\left(3q+7\right)\right.\\\ &\text{\hskip 17.07182pt
}\left.-3q\left(4q+9\right)+s-12\right]\,,\\\
T^{\text{(iv)}}&=12H^{6}\left[j\left(3j+44q+48\right)\right.\\\ &\text{\hskip
17.07182pt }\left.+l-\left(4q+9\right)s\right.\\\ &\text{\hskip 17.07182pt
}\left.+3q\left(4q^{2}+39q+56\right)+60\right]\,,\\\
T^{\text{(v)}}&=-12H^{7}\left(50j^{2}+10j(q(8q+51)-s+36)\right.\\\
&\left.+l(5q+11)-\tilde{m}-5(14q+15)s\right.\\\
&\left.+30q(q+2)(9q+20)+360\right)\,,\end{aligned}\quad\begin{aligned}
B&=-6H^{2}\left(q-2\right)\,,\\\ \dot{B}&=6H^{3}\left(j-3q-4\right)\,,\\\
\ddot{B}&=6H^{4}\left[2j+3q,\left(q+6\right)+s+12\right]\,,\\\
\dddot{B}&=6H^{5}\left[-2j\left(4q+11\right)+l\right.\\\ &\text{\hskip
11.38092pt }\left.-6\left(q\left[7q+17\right)+8\right]+s\right]\,,\\\
B^{\text{(iv)}}&=6H^{6}\left[24j\left(6q+7\right)+8j^{2}\right.\\\
&\text{\hskip 11.38092pt }\left.+\tilde{m}-3\left(3q+8\right)s\right.\\\
&\text{\hskip 11.38092pt
}\left.+6q\left(q\left[7q+72\right]+108\right)+240\right]\,,\\\
B^{\text{(v)}}&=-6H^{7}(160j^{2}+\tilde{m}-n+3l(8+3q)\\\
&+30(q(156+3q(48+11q)-7s)-8(-6+s))\\\
&+5j(264+18q(20+3q)-5s))\,.\end{aligned}$ (22)

Here, we have the gravitational scalars directly related with the cosmographic
parameters which makes it possible to infer their values through observational
data. At this point, we consider a simple model $f(T,B)=-T+\tilde{f}(T,B)$ as
a power law model $\tilde{f}(T,B)=\tilde{t}_{0}T^{m}+\tilde{b}_{0}B^{k}$.
Power law models are interesting ways to probe new physics in these settings,
but this model also exhibits the property that any cross derivatives will
vanish. the remainder of the contributing derivatives are shown in Appendix A.
In $f(T,B)$ gravity, the power law model is motivated not only for its ease of
purpose in the cosmographic setting but also for its observational and
theoretical advantages. In Ref. Escamilla-Rivera:2019ulu , this model is shown
to model cosmic chronometer, Pantheon and baryonic acoustic oscillation data
in the late Universe. Moreover, in Ref. Caruana:2020szx  this model seems to
be a genetic solution in many instances some early Universe physics, while the
power law model also seems to have some interesting features for an
astrophysics setting Bahamonde:2021srr ; Farrugia:2020fcu .

The Friedmann equations (13) and (14) can be now be rewritten as

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\kappa^{2}\rho+3H^{2}\left(3f_{B}+2f_{T}\right)-3H\dot{f}_{B}+3\dot{H}f_{B}-\frac{1}{2}f\,,$
(23) $\displaystyle 2\dot{H}$ $\displaystyle=$
$\displaystyle-\kappa^{2}\rho(1+\omega)+2\dot{H}f_{T}+2H\dot{f}_{T}+3H\dot{f}_{B}-\ddot{f}_{B}\,,$
(24)

where $\omega=p/\rho$ is the equation of state for matter. If we compute
higher order derivatives of $H$ in a dust model $\omega=0$, we obtain

$\displaystyle 2\ddot{H}$ $\displaystyle=$ $\displaystyle
3H\kappa^{2}\rho+2\ddot{H}f_{T}+2H\ddot{f}_{T}+4\dot{H}\dot{f}_{T}+3\dot{H}\dot{f}_{B}+3H\ddot{f}_{B}-\dddot{f}_{B}\,,$
(25) $\displaystyle 2\dddot{H}$ $\displaystyle=$ $\displaystyle
3\dot{H}\kappa^{2}\rho-9H^{2}\kappa^{2}\rho+2\dddot{H}f_{T}+2H\dddot{f}_{T}+6\ddot{H}\dot{f}_{T}+6\dot{H}\ddot{f}_{T}+3\ddot{H}\dot{f}_{B}$
$\displaystyle+6\dot{H}\ddot{f}_{B}+3H\dddot{f}_{B}-f^{\text{(iv)}}_{B},$
$\displaystyle 2H^{\text{(iv)}}$ $\displaystyle=$ $\displaystyle
3\ddot{H}\kappa^{2}\rho-27H\dot{H}\kappa^{2}\rho+27H^{3}\kappa^{2}\rho+2H^{\text{(iv)}}f^{\text{(iv)}}_{T}+8\dddot{H}\dot{f}_{T}+8\dot{H}\ddot{f}_{T}+12\ddot{H}\ddot{f}_{T}$
$\displaystyle+9\ddot{H}\ddot{f}_{B}+3\dddot{H}\dot{f}_{B}+9\dot{H}\dddot{f}_{B}+3Hf^{\text{(iv)}}_{B}-f^{\text{(v)}}_{B}\,.$

These expressions allow us to write the cosmographic parameters in terms of
the arbitrary $f(T,B)$ Lagrangian and its derivatives, explicitly, they can be
written as

$\displaystyle
q=-1+\frac{1}{3f_{B}+2f_{T}-2}\left[-\frac{\ddot{f}_{B}}{H^{2}}+9f_{B}-\frac{f}{2H^{2}}+\frac{2\dot{f}_{T}}{H}+6f_{T}-3\right],$
(28)
$\displaystyle\Omega_{\text{m}}=1-3f_{B}-2f_{T}+\frac{1}{H}\dot{f}_{B}+(1+q)f_{B}+\frac{1}{6H^{2}}f\,,$
(29) $\displaystyle
j=-2-3q+\frac{1}{2(1-f_{T})}\Bigg{[}-\frac{\dddot{f}_{B}}{H^{3}}+\frac{3\ddot{f}_{B}+2\ddot{f}_{T}}{H^{2}}-\frac{(q+1)\left(3\dot{f}_{B}+4\dot{f}_{T}\right)}{H}+9\Omega_{\text{m}}\Bigg{]}\,,$
(30) $\displaystyle
s=4j+3q(q+4)+6+\frac{1}{2(1-f_{T})}\Bigg{[}-\frac{f^{\text{(iv)}}_{B}}{H^{4}}+\frac{3\dddot{f}_{B}+2\dddot{f}_{T}}{H^{3}}-\frac{6(q+1)\left(\ddot{f_{B}}+\ddot{f_{T}}\right)}{H^{2}}$
$\displaystyle+\frac{(j+3q+2)\left(3\dot{f}_{B}+6\dot{f}_{T}\right)}{H}-(9(q+1)+27)\Omega_{\text{m}}\Bigg{]}\,,$
(31) $\displaystyle
l=5s-10(q+2)j-30(q+2)q-24+\frac{1}{2(1-f_{T})}\Bigg{[}-\frac{f^{\text{(v)}}_{B}}{H^{5}}+\frac{3f^{\text{(iv)}}_{B}+2f^{\text{(iv)}}_{T}}{H^{4}}$
$\displaystyle-\frac{(q+1)\left(9\dddot{f}_{B}+8\dddot{f}_{T}\right)}{H^{3}}+\frac{(j+3q+2)\left(9\ddot{f}_{B}+12\ddot{f}_{T}\right)}{H^{2}}+\frac{\left(3\dot{f}_{B}+8\dot{f}_{T}\right)(-4j-3q(q+4)+s-6)}{H}$
$\displaystyle+\Omega_{\text{m}}(9(j+3q+2)+81(q+1)+81)\Bigg{]}\,,$ (32)

where $\Omega_{\rm m}$ is the density parameter for matter.

However, when substituting the derivatives given by the Eqs.(63)–(73) into the
above expression, we obtain a system of nonlinear relations that makes it
impossible to isolate the cosmographic parameters in a general model for
arbitrary $k$ and $m$. To highlight this issue, we can see that the
deceleration parameter when substituting the Eqs.(63)–(73), becomes

$\displaystyle q$
$\displaystyle=-1+3\left(\frac{\tilde{b}_{0}6^{k-1}(k-1)k\left(-H^{2}(q-2)\right)^{k}\left(j^{2}(k-2)-2jk(3q+4)+10j(q+2)+k(3q+4)^{2}-3q^{3}-30q^{2}-qs-24q+2s-8\right)}{H^{2}(q-2)^{3}}\right.$
(33)
$\displaystyle\left.-\frac{\tilde{b}_{0}6^{k}\left(-H^{2}(q-2)\right)^{k}+6^{m}\tilde{t}_{0}\left(H^{2}\right)^{m}}{2H^{2}}+9\tilde{b}_{0}k\left(18H^{2}-6H^{2}(q+1)\right)^{k-1}-2^{m+1}3^{m-1}(m-1)m(q+1)\tilde{t}_{0}\left(H^{2}\right)^{m-1}\right.$
$\displaystyle\left.+6^{m}m\tilde{t}_{0}\left(H^{2}\right)^{m-1}-3\right)\div\Bigg{(}9\tilde{b}_{0}k\left(18H^{2}-6H^{2}(q+1)\right)^{k-1}+6^{m}m\tilde{t}_{0}\left(H^{2}\right)^{m-1}-6\Bigg{)},$

which can not be isolated for arbitrary values of $k$ and $m$. The same issue
holds for the other cosmographic parameters. Thus, the cosmographic parameters
are inherently interrelated in this sense.

###  III.1 Standard Case

As an example of a standard case, it is possible to isolate the cosmographic
parameters in the simplest case with $m=1$ and $k=2$. In this context, the
equation (33) simplifies to

$\displaystyle
q=\frac{6\tilde{b}_{0}H^{2}(4j+9q(q+4)+2(s+6))-\tilde{t}_{0}+1}{36\tilde{b}_{0}H^{2}(q-2)-2\tilde{t}_{0}+2}\,,$
(34)

where it is possible to solve for $q$ as

$\displaystyle q_{1}$
$\displaystyle=\frac{\left(\tilde{t}_{0}-1\right)\left(\sqrt{1-\frac{18\tilde{b}_{0}H^{2}\left(12\tilde{b}_{0}H^{2}(2j+s-90)-17\left(\tilde{t}_{0}-1\right)\right)}{\left(\tilde{t}_{0}-1\right){}^{2}}}-1\right)}{18\tilde{b}_{0}H^{2}}-8\,,$
(35) $\displaystyle q_{2}$
$\displaystyle=-\frac{\left(\tilde{t}_{0}-1\right)\left(\sqrt{1-\frac{18\tilde{b}_{0}H^{2}\left(12\tilde{b}_{0}H^{2}(2j+s-90)-17\left(\tilde{t}_{0}-1\right)\right)}{\left(\tilde{t}_{0}-1\right){}^{2}}}+1\right)}{18\tilde{b}_{0}H^{2}}-8\,.$
(36)

Under the same assumption, is is also possible to isolate the other
cosmographic parameters which are given by

$\displaystyle\Omega_{\text{m}}$
$\displaystyle=\frac{2\tilde{b}_{0}\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)}{H}+\frac{\tilde{b}_{0}\left(18H^{2}-6H^{2}(q+1)\right)^{2}+6H^{2}\tilde{t}_{0}}{6H^{2}}$
$\displaystyle+2\tilde{b}_{0}(q+1)\left(18H^{2}-6H^{2}(q+1)\right)-6\tilde{b}_{0}\left(18H^{2}-6H^{2}(q+1)\right)-2\tilde{t}_{0}+1$
(37) $\displaystyle j$
$\displaystyle=\frac{-\frac{6\tilde{b}_{0}H^{2}l}{\tilde{t}_{0}-1}+\frac{333\tilde{b}_{0}H^{2}q^{2}}{\tilde{t}_{0}-1}+\frac{1008\tilde{b}_{0}H^{2}q}{\tilde{t}_{0}-1}+\frac{12\tilde{b}_{0}H^{2}s}{\tilde{t}_{0}-1}+\frac{252\tilde{b}_{0}H^{2}}{\tilde{t}_{0}-1}+3q-\frac{5}{2}}{-\frac{6\tilde{b}_{0}H^{2}(5q+34)}{\tilde{t}_{0}-1}-1}$
(38) $\displaystyle s$
$\displaystyle=\Bigg{(}-\frac{30\tilde{b}_{0}H^{2}j^{2}}{\tilde{t}_{0}-1}-\frac{54\tilde{b}_{0}H^{2}j(21q+32)}{\tilde{t}_{0}-1}+\frac{18\tilde{b}_{0}H^{2}l}{\tilde{t}_{0}-1}-\frac{6\tilde{b}_{0}H^{2}m}{\tilde{t}_{0}-1}-\frac{333\tilde{b}_{0}H^{2}q^{3}}{\tilde{t}_{0}-1}-\frac{4104\tilde{b}_{0}H^{2}q^{2}}{\tilde{t}_{0}-1}$
(39)
$\displaystyle-\frac{6588\tilde{b}_{0}H^{2}q}{\tilde{t}_{0}-1}-\frac{1584\tilde{b}_{0}H^{2}}{\tilde{t}_{0}-1}-4j-\frac{3}{2}(2q+5)q+12\Bigg{)}\div\Bigg{(}-\frac{18\tilde{b}_{0}H^{2}(q+7)}{\tilde{t}_{0}-1}-1\Bigg{)}$
$\displaystyle l$
$\displaystyle=\Bigg{(}\frac{1194\tilde{b}_{0}H^{2}j^{2}}{\tilde{t}_{0}-1}+\frac{3\tilde{b}_{0}H^{2}j(9q(79q+626)-26s+4644)}{\tilde{t}_{0}-1}+\frac{60\tilde{b}_{0}H^{2}m}{\tilde{t}_{0}-1}-\frac{6\tilde{b}_{0}H^{2}n}{\tilde{t}_{0}-1}$
(40)
$\displaystyle+\frac{18\tilde{b}_{0}H^{2}(-76qs+6q(q(86q+411)+444)-105s+598)}{\tilde{t}_{0}-1}+\frac{1}{2}j(20q+31)+6q(5q+1)-5s-66\Bigg{)}$
$\displaystyle\div\Bigg{(}-\frac{90\tilde{b}_{0}H^{2}}{\tilde{t}_{0}-1}-1\Bigg{)}$

In a general FLRW spacetime this can be described for small distances by the
luminosity distance which is expressed as

$\displaystyle d_{L}(z)$ $\displaystyle=$
$\displaystyle\frac{c}{H_{0}}[z+\frac{1}{2}(1-q_{0})z^{2}-\frac{1}{6}(1-q_{0}-3q_{0}^{2}+j_{0}+\frac{kc^{2}}{H_{0}^{2}a^{2}(t_{0})})z^{3}+$
(41)
$\displaystyle+\frac{1}{24}[2-2q_{0}-15q_{0}^{2}-15q_{0}^{3}+5j_{0}+10q_{0}j_{0}+s_{0}+\frac{2kc^{2}(1+3q_{0})}{H_{0}^{2}a^{2}(t_{0})}]z^{4}+\ldots]\,,$

where the cosmographic parameters are defined as

$\displaystyle H_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{a(t)}\frac{da(t)}{dt}|_{t=t_{0}}\equiv\frac{\dot{a}(t)}{a(t)}|_{t=t_{0}}~{},$
(42) $\displaystyle q_{0}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{H^{2}}\frac{1}{a(t)}\frac{d^{2}a(t)}{dt^{2}}|_{t=t_{0}}\equiv-\frac{1}{H^{2}}\frac{\ddot{a}(t)}{a(t)}|_{t=t_{0}}~{},$
(43) $\displaystyle j_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{H^{3}}\frac{1}{a(t)}\frac{d^{3}a(t)}{dt^{3}}|_{t=t_{0}}\equiv\frac{1}{H^{3}}\frac{a^{(3)}(t)}{a(t)}|_{t=t_{0}}~{},$
(44) $\displaystyle s_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{H^{4}}\frac{1}{a(t)}\frac{d^{4}a(t)}{dt^{4}}|_{t=t_{0}}\equiv\frac{1}{H^{4}}\frac{a^{(4)}(t)}{a(t)}|_{t=t_{0}}~{}.$
(45)

Notice that depending on the degree of Taylor series considered, the
cosmographic expressions derived could produce degenerate results. Moreover,
this may lead to a misunderstanding in the definition of the proper distance.
On the other hand, we can consider observations from recent surveys to
constrain the impact of this degeneracy through truncation of the Taylor
series. In thi way, we can use the luminosity distance relation to relate
cosmographic parameters with observational data. In this context, we consider
the luminosity distance as the most direct choice in the measurements of
distance for Type Ia Supernovae (SNeIa) and gamma-ray bursts (GRBs), along the
Hubble flow using cosmic chronometers (CCs).

To fit the luminosity distance redshift relation using this data and to solve
the convergence issue at high-$z$ via an adequate truncation of the series, it
is useful to rewrite $d_{L}$ as a function of the proposed variable
$y=z/(1+z)$. In this way, we can map $z\in(0,\infty)$ into $y\in(0,1)$, and
retrieve an optimal behaviour for the Taylor series at any redshift. This new
variable $y$ does not change the definition of the cosmographic parameters and
the luminosity distance at fourth order in such variable can be written as

$\displaystyle d_{L}(y)=$
$\displaystyle\frac{c}{H_{0}}\left\\{y-\frac{1}{2}(q_{0}-3)y^{2}+\frac{1}{6}\left[12-5q_{0}+3q^{2}_{0}-(j_{0}+\Omega_{0})\right]y^{3}+\frac{1}{24}\left[60-7j_{0}-\right.\right.$
(46)
$\displaystyle\left.\left.-10\Omega_{0}-32q_{0}+10q_{0}j_{0}+6q_{0}\Omega_{0}+21q^{2}_{0}-15q^{3}_{0}+s_{0}\right]y^{4}+\mathcal{O}(y^{5})\right\\}\,,$

where $\Omega_{0}=1+kc^{2}/H_{0}^{2}a^{2}(t_{0})$, is the total energy density
at current times. Therefore, we can write the luminosity distance logarithmic
Hubble relation as

$\displaystyle\ln{\left[\frac{d_{L}(y)}{y~{}}\right]}\rm Mpc^{-1}=$
$\displaystyle\frac{\ln{10}}{5}\left[\mu(y)-25\right]-\ln{y}=\ln{\left[\frac{c}{H_{0}}\right]}-\frac{1}{2}(q_{0}-3)y+\frac{1}{24}\left[21-4(j_{0}+\Omega_{0})+\right.$
(47)
$\displaystyle\left.+q_{0}(9q_{0}-2)\right]y^{2}+\frac{1}{24}\left[15+4\Omega_{0}(q_{0}-1)+j_{0}(8q_{0}-1)-5q_{0}+2q^{2}_{0}\right.$
$\displaystyle\left.-10q^{3}_{0}+s_{0}\right]y^{3}+\mathcal{O}(y^{4})\,,$

and the corresponding equation for the distance modulus is

$\displaystyle\mu(y)=$ $\displaystyle
25+\frac{5}{\ln{10}}\left\\{\ln{\left[\frac{c}{H_{0}}\right]}+\ln{y}-\frac{1}{2}(q_{0}-3)y+\frac{1}{24}\left[21-4(j_{0}+\Omega_{0})+q_{0}(9q_{0}-2)\right]y^{2}+\right.$
(48)
$\displaystyle+\left.\frac{1}{24}\left[15+4\Omega_{0}(q_{0}-1)+j_{0}(8q_{0}-1)-5q_{0}+2q^{2}_{0}-10q^{3}_{0}+s_{0}\right]y^{3}+\mathcal{O}(y^{4})\right\\}\,,$

which is important for SNeIa data. Notice that the higher order the term in
the expansion of $d_{L}$, the better the fit to the data sample we can obtain
since there will be more free parameters – but of course this comes at the
cost of greater degeneracy in these parameters. However, for a given data
there will be an upper bound on the order of the series which gives a
statistically significant fit to those data.

##  IV Observational cosmographic constraints

To perform the statistical analyses of the cosmography found and understand
current constraints, we need to focus on specific data sets and likelihood
functions. In this analysis we are going to consider three samples: a SNeIa
compilation, a newly calibrated GRB sample and the CCs to constrain the
cosmographic parameters from the standard $f(T,B)$ model.

  * •

Pantheon compilation. The recent SNeIa compilation known as the Pantheon
sample (SN) Scolnic:2017caz  consists of 1048 SNeIa compressed in 40 redshift
bins. As already analysed in the literature, SNeIa can provide estimates of
the distance modulus, $\mu$, the theoretically predicted value of which is
related to the luminosity distance $d_{L}$ we obtain in Eq.(46) and Eq.(48) as
follows

$\mu(z)=5\log{\left[\frac{d_{L}(z)}{1\text{Mpc}}\right]}+25,$ (49)

where the luminosity distance $d_{L}$ is given in Mpc. In this distance
modulus expression we should include the nuisance parameter, $\mathcal{M}$, as
an unknown offset of the supernovae absolute magnitude (and including other
possible systematics), which can also be degenerate with the value of $H_{0}$.
As is standard, we assume spatial flatness and suppose that $d_{L}$ can be
related to the comoving distance $D$ using
$d_{L}(z)=\frac{c}{H_{0}}(1+z)D(z),$ where $c$ is the speed of light. We
obtain $D(z)=\frac{H_{0}}{c}(1+z)^{-1}10^{\frac{\mu(z)}{5}-5}.$ The normalised
Hubble function $H(z)/H_{0}$ is derived from the inverse of the derivative of
$D(z)$ with respect to $z$ so that we can write down

$D(z)=\int^{z}_{0}\frac{H_{0}d\tilde{z}}{H(\tilde{z})},$ (50)

where $H_{0}$ is the value of the Hubble constant which we consider as a prior
to normalise $D(z)$. For our sample, we calibrated the data by using a value
obtained from late universe measurements corresponding to $H_{0}=73.8\pm
1.1\,\,\mbox{km s}^{-1}\mbox{Mpc}^{-1},$ from SH0ES + H0LiCOW, with the
corresponding value for the nuisance parameter $M=-19.24\pm 0.07$.

  * •

New GRB dataset. After the reconstruction/prior calibration of $d_{L}$ from
SN, we can use them to calibrate the luminosity correlations of the GRB data
set. The correlations obtained from this calibration can be expressed by
considering a generic exponential form $R=AQ^{b}$, where this assumption is
derived by consider the X-ray light curves of GRBs constructed from the
combination of Burst Alert Telescope and X- ray telescope data in the way
described in 2010GRBSHi  and fitted using one or two components from the
optional afterglow via the parametrised Amati relation and the Ghirlanda
relation Wang:2019fie , where this expression can be re-expressed in linear
form as $y=a+bx$, with $y\equiv\log R$, $x\equiv\log Q$ and $a=\log A$. Also,
the six luminosity correlations measured are reported in Escamilla-
Rivera:2021vyw .

To calibrate these six expressions with a SNeIa sample (in particular, for
Pantheon sample), we consider that GRBs radiate isotropically by computing
their bolometric peak flux, where uncertainty on $L$ propagates from the
uncertainties on bolometric peak flux $P_{\text{bolo}}=L/4\pi d^{2}_{L}$, and
$d_{L}$ the luminosity distance of the supernovae. With this new sample we
calibrated the luminosity correlations by maximizing the likelihood
DAgostini:2005mth

$\mathcal{L}(\sigma_{\text{int}},a,b)\propto\prod_{i}\frac{1}{\sqrt{\sigma^{2}_{\text{int}}+\sigma^{2}_{yi}+b^{2}\sigma_{xi}}}\times\text{exp}\left[-\frac{(y_{i}-a-bx_{i})^{2}}{2(\sigma^{2}_{\text{int}}+\sigma^{2}_{yi}+b^{2}\sigma^{2}_{xi})}\right].$
(51)

The best-fitting parameters and their
uncertainties,$(a,b,\sigma_{\text{int}})$, are reported in Tables 1 and 2 from
Escamilla-Rivera:2021vyw .

  * •

Cosmic Chronometers. We consider a sample of 31 model-independent measurements
which use the differential age method proposed by Jimenez:2001gg .

In order to determine viable cosmographic constraints that can reproduce the
observed cosmic acceleration for the standard $f(T,B)$ case, we consider the
three cases below which allow for progressively better cosmographic
constraints on the parameters.

  * •

Case (a). Fixed $q_{0}$. Using the expression in Eq. (35) and the values
$\tilde{t}_{0}=0$ and $\tilde{b}_{0}=1$ (which relates to a pure power-law in
the boundary term theory), we obtain $j_{0}=2.630^{+0.225}_{-0.229}$ and
$s_{0}=7.403^{+0.286}_{-0.283}$, using the full Pantheon + GRB + CC using a
Markov chain Monte Carlo (MCMC) analysis. For this case we obtain a Universe
that is in decelerating phase. This sets the first constraint on the form of
$f(T,B)$, where the $q_{0}$ cannot be fixed if we require to follow the
correct cosmic acceleration. See Fig. (1) for the posterior plot. Notice that
once we add GRB we obtain the confidence level (C.L) overlaps on the Pantheon
solely.

Figure 1: Case (a) using Pantheon compilation (green C.L.), Pantheon+GRB (blue
C.L.), Pantheon+CC (red C.L.) and Pantheon+GRB+CC (orange C.L.).

  * •

Case (b). Fixed $j_{0}$. Using the expression for $j$ (III.1) and the values
$\tilde{t}_{0}=0$ and $\tilde{b}_{0}=1$, we obtain
$q_{0}=-0.799^{+0.024}_{-0.025}$, $s_{0}=-3.409^{+0.287}_{-0.327}$, and
$l_{0}=-8.046^{+1.973}_{-1.346}$ using the full sample Pantheon + GRB+ CC with
an MCMC analysis. For this case we obtain a Universe that is in fact
accelerating as it is expected. See Fig. 2 for details of the posterior. Here,
we notice that the values of the higher derivative cosmographic parameters
starts to become reasonable.

Figure 2: Case (b) using Pantheon compilation (green C.L.), Pantheon+GRB (blue
C.L.), Pantheon+CC (red C.L.) and Pantheon+GRB+CC (orange C.L.).

  * •

Case (c). Fixed $s_{0}$. Using the expression for s (III.1) and the values
$\tilde{t}_{0}=0$ and $\tilde{b}_{0}=1$, we obtain $q_{0}=-0.630\pm 0.018$,
$j_{0}=1.828^{+0.115}_{-0.114}$, $l=-9.744^{+0.421}_{-0.193}$ and
$m_{0}=9.235^{+0.571}_{-1.246}$ using the full sample Pantheon + GRB+ CC.
Notice that here the cosmography can go beyond $m_{0}$. For this case we
obtain a Universe that is in fact accelerating as it is expected and the
addition of GRB explore the necessity of a high cosmography. See Fig. (3).

Figure 3: Case (c) using Pantheon compilation (green C.L.), Pantheon+GRB (blue
C.L.), Pantheon+CC (red C.L.) and Pantheon+GRB+CC (orange C.L.).

Table 1:  Best cosmographic fits for the standard case using a $f(T,B)$ power
law model. In first columns are denoted the three cases considered. We divided
the analysis using: Pantheon SN + CC (second block column) and Pantheon SN +
CC + GRB (third block column). The word fixed indicates a flat $\Lambda$CDM
prior imposed on each case. Results are discussed in the text.

Cases | Pantheon SN +CC | Pantheon SN + CC + GRB
---|---|---
$q_{0}$ | $j_{0}$ | $s_{0}$ | $l_{0}$ | $m_{0}$ | $q_{0}$ | $j_{0}$ | $s_{0}$ | $l_{0}$ | $m_{0}$
(a) | Fixed | $2.630^{+0.225}_{-0.229}$ | $7.403^{+0.286}_{-0.283}$ | - | - | Fixed | $2.710^{+0.205}_{-0.219}$ | $7.502^{+0.200}_{-0.189}$ | - | -
(b) | $-0.802^{+0.024}_{-0.225}$ | Fixed | $-3.411^{+0.273}_{-0.307}$ | $-8.582^{+1.754}_{-1.023}$ | - | $-0.799^{+0.224}_{-0.225}$ | Fixed | $-3.409^{+0.115}_{-0.114}$ | $-8.046^{+1.973}_{-1.346}$ | -
(c) | $-0.666^{0.040}_{-0.037}$ | $-0.836^{+0.419}_{-0.499}$ | Fixed | $-3.266^{+7.474}_{-4.878}$ | $1.209^{+6.170}_{-7.295}$ | $-0.630\pm 0.018$ | $1.828^{+0.225}_{-0.229}$ | Fixed | $-9.744^{+0.421}_{-0.193}$ | $9.235^{+0.571}_{-1.246}$

In Table 1, we show the results of the MCMC analyses using both SN+CC and
SN+CC+GRB data set compilations together with the three cases considered above
for the cosmographic parameters. What we find is that due to the inter-
relatedness of the cosmographic parameters, some high order derivative terms
cannot be constrained such as $l_{0}$ which id due to the high degree of
interdependence of the parameters. In this context, it follows suit that we
can only consider a certain amount of parameters depending on the amount of
data available. Another important point concerns which parameters can be left
fixed and which need to be constrained. In these analysis, we find that it is
best to fix the highest derivative term and constrain the lower parameters,
which leads to the best fit cosmographic parameters.

##  V Conclusions and discussion

Cosmography was used in this work to reconstruct elements of the $f(T,B)$
Lagrangian in the context of a power law model. The cosmographic parameters
(15,16) provide an interesting avenue to parameterizing the evolution of the
Universe using progressively higher order derivative terms in the scale
factor. Since they build on each other, these parameters can be related to
each other which causes complicated coupling problems in many cases. For this
reason, very high order derivative terms need enormous amounts of data to be
constrained to any degree using observational data. On the other hand, these
state parameters provide an efficient way to connect expansion terms with
derivatives in a cosmological model.

In the context of TG, the cosmographic parameters can readily be injected into
the scalar contributors of the theory (22). As expected progressively higher
time derivatives of these scalar are naturally related to higher time
derivatives of the Hubble parameter which is a crucial link with observations.
Thus, by using the Friedmann equations, the cosmographic parameters can then
be related to the form of $f(T,B)$ through the use of the Friedmann equations
which is the point at which this becomes dependent on the particular model
under consideration (28–32). The other key property to acknowledge at this
point is that it not possible to isolate any of these cosmographic parameters
for most model parameters values except for very specially chosen ones as
explored in Case A in Eq. (34). To this end, we use the luminosity distance
calculation to strengthen the link to observations using the SN data set since
it can be related to the distance modulus parameter, which is the true
parameter of the Pantheon data set.

In Sec. IV, we consider the CC, SN and the new GRB data sets. GRB data is very
interesting because it may shed new light on fundamental theory using new
physical phenomena. To probe these potential differences we consider three
settings of cosmographic parameters in Figs. (1,2,3) which respectively
constraint the cosmographic parameters for fixed $q_{0}$, $j_{0}$ and $l_{0}$.
While it is not possible in any of these analyses to completely constrain the
value of high derivative terms such as the lerk $l$. Saying that, as more
parameters are not fixed to a value, their values tend to the literature
values such as in the case of the deceleration parameter which readily turns
from a late time decelerating solution in Fig. (1) to an accelerating one in
Figs. (2,3). The complete set of results are collected in Table. 1 where the
impact of the GRB data on the various cosmographic parameter is shown more
clearly such as its preference for a weaker acceleration in the late Universe.

It would be interesting to consider other viable models in the $f(T,B)$
gravity extension to $\Lambda$CDM such as those models as proposed in Ref.
Escamilla-Rivera:2019ulu . However, it would make the analysis much more
involved. It would also be advantageous to consider other approaches to
constraining the evolution of the cosmographic parameters such as the non-
parametric ones proposed in Refs. Escamilla-Rivera:2021rbe ; Bernardo:2021mfs
; Briffa:2020qli .

###### Acknowledgements.

CE-R acknowledges the Royal Astronomical Society as FRAS 10147. CE-R and GRF
are supported by DGAPA-PAPIIT-UNAM Project IA100220. GRF acknowledges
financial support from CONACyT postgraduate grants program. The simulations
were performed in Centro de cómputo Tochtli-ICN-UNAM as part of CosmoNag. JLS
would also like to acknowledge funding support from Cosmology@MALTA which is
supported by the University of Malta. The authors would like to acknowledge
networking support by the COST Action CA18108.

##  Appendix A Derivations for the power law model cosmography

As we mentioned, consider a simple model as a power law model
$f(T,B)=\tilde{t}_{0}T^{m}+\tilde{b}_{0}B^{k}$, where mixed partial
derivatives vanish and it can be seen that its derivatives are given by the
following expressions:

$\displaystyle f_{T}$ $\displaystyle=m\tilde{t}_{0}T^{m-1}$ (52)
$\displaystyle f_{B}$ $\displaystyle=k\tilde{b}_{0}B^{k-1}$ (53)
$\displaystyle\dot{f}_{T}$ $\displaystyle=m(m-1)\tilde{t}_{0}\dot{T}T^{m-2}$
(54) $\displaystyle\dot{f}_{B}$
$\displaystyle=k(k-1)\tilde{b}_{0}\dot{B}B^{k-2}$ (55)
$\displaystyle\ddot{f}_{T}$
$\displaystyle=m(m-1)(m-2)\tilde{t}_{0}\dot{T}^{2}T^{m-3}+m(m-1)\tilde{t}_{0}\ddot{T}T^{m-2}$
(56) $\displaystyle\ddot{f}_{B}$
$\displaystyle=k(k-1)(k-2)\tilde{b}_{0}\dot{B}^{2}B^{k-3}+k(k-1)\tilde{b}_{0}\ddot{B}B^{k-2}$
(57) $\displaystyle\dddot{f}_{T}$
$\displaystyle=m(m-1)(m-2)(m-3)\tilde{t}_{0}\dot{T}^{3}T^{m-4}+3m(m-1)(m-2)\tilde{t}_{0}\dot{T}\ddot{T}T^{m-3}$
(58) $\displaystyle+m(m-1)\tilde{t}_{0}\dddot{T}T^{m-2}$
$\displaystyle\dddot{f}_{B}$
$\displaystyle=k(k-1)(k-2)(k-3)\tilde{b}_{0}\dot{B}^{3}B^{k-4}+3k(k-1)(k-2)\tilde{b}_{0}\dot{B}\ddot{B}B^{k-3}+k(k-1)\tilde{b}_{0}\dddot{B}B^{k-2}$
(59) $\displaystyle f^{\text{(iv)}}_{T}$
$\displaystyle=m(m-1)(m-2)(m-3)(m-4)\tilde{t}_{0}\dot{T}^{4}T^{m-5}+6m(m-1)(m-2)(m-3)\tilde{t}_{0}\dot{T}^{2}\ddot{T}T^{m-4}$
(60)
$\displaystyle+3m(m-1)(m-2)\tilde{t}_{0}\ddot{T}^{2}T^{m-3}+4m(m-1)(m-2)\tilde{t}_{0}\dot{T}\dddot{T}T^{m-3}+m(m-1)\tilde{t}_{0}T^{\text{(iv)}}T^{m-2}$
$\displaystyle f^{\text{(iv)}}_{B}$
$\displaystyle=k(k-1)(k-2)(k-3)(k-4)\tilde{b}_{0}\dot{B}^{4}B^{k-5}+6k(k-1)(k-2)(k-3)\tilde{b}_{0}\dot{B}^{2}\ddot{B}B^{k-4}$
(61)
$\displaystyle+3k(k-1)(k-2)\tilde{b}_{0}\ddot{B}^{2}B^{k-3}+4k(k-1)(k-2)\tilde{b}_{0}\dot{B}\dddot{B}B^{k-3}+k(k-1)\tilde{b}_{0}B^{\text{(iv)}}B^{k-2}$
$\displaystyle f^{\text{(v)}}_{B}$
$\displaystyle=k(k-1)(k-2)(k-3)(k-4)(k-5)\tilde{b}_{0}\dot{B}^{5}B^{k-6}+10k(k-1)(k-2)(k-3)(k-4)\tilde{b}_{0}\dot{B}^{3}\ddot{B}B^{k-5}$
(62)
$\displaystyle+15k(k-1)(k-2)(k-3)\tilde{b}_{0}\ddot{B}\dddot{B}B^{k-3}+5k(k-1)(k-2)\tilde{b}_{0}\dot{B}B^{\text{(iv)}}B^{k-3}+k(k-1)\tilde{b}_{0}B^{\text{(v)}}B^{k-2}.$

These derivatives can be written in terms of the cosmographic parameters
(15)-(16) using the expressions in (22) as:

$\displaystyle f_{T}$
$\displaystyle=6^{m-1}m\tilde{t}_{0}\left(H^{2}\right)^{m-1}$ (63)
$\displaystyle f_{B}$
$\displaystyle=\tilde{b}_{0}k\left(18H^{2}-6H^{2}(q+1)\right)^{k-1}$ (64)
$\displaystyle\dot{f}_{T}$
$\displaystyle=H^{3}\left(-2^{m}\right)3^{m-1}(m-1)m(q+1)\tilde{t}_{0}\left(H^{2}\right)^{m-2}$
(65) $\displaystyle\dot{f}_{B}$
$\displaystyle=\tilde{b}_{0}(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)\left(18H^{2}-6H^{2}(q+1)\right)^{k-2}$
(66) $\displaystyle\ddot{f}_{T}$
$\displaystyle=H^{6}2^{m+1}3^{m-1}(m-2)(m-1)m(q+1)^{2}\tilde{t}_{0}\left(H^{2}\right)^{m-3}$
(67)
$\displaystyle+H^{4}2^{m}3^{m-1}(m-1)m\tilde{t}_{0}\left(H^{2}\right)^{m-2}(j+q(q+5)+3)$
$\displaystyle\ddot{f}_{B}$
$\displaystyle=\tilde{b}_{0}(k-1)k\left(18H^{2}-6H^{2}(q+1)\right)^{k-2}\left(6H^{4}(-4j-3q(q+4)+s-6)+36H^{4}(j+q(q+5)+3)\right)$
(68)
$\displaystyle+\tilde{b}_{0}(k-2)(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)^{2}\left(18H^{2}-6H^{2}(q+1)\right)^{k-3}$
$\displaystyle\dddot{f}_{T}$
$\displaystyle=H^{9}\left(-2^{m+2}\right)3^{m-1}(m-3)(m-2)(m-1)m(q+1)^{3}\tilde{t}_{0}\left(H^{2}\right)^{m-4}$
(69)
$\displaystyle-H^{7}2^{m+1}3^{m}(m-2)(m-1)m(q+1)\tilde{t}_{0}\left(H^{2}\right)^{m-3}(j+q(q+5)+3)$
$\displaystyle+H^{5}2^{m}3^{m-1}(m-1)m\tilde{t}_{0}\left(H^{2}\right)^{m-2}(-j(3q+7)-3q(4q+9)+s-12)$
$\displaystyle\dddot{f}_{B}$
$\displaystyle=\tilde{b}_{0}(k-1)k\left(18H^{2}-6H^{2}(q+1)\right)^{k-2}\left(6H^{5}(10j(q+2)+l+30q(q+2)-5s+24)\right.$
(70)
$\displaystyle\left.+36H^{5}(-j(3q+7)-3q(4q+9)+s-12)\right)+\tilde{b}_{0}(k-3)(k-2)(k-1)k\left(6H^{3}(j+3q+2)\right.$
$\displaystyle\left.-36H^{3}(q+1)\right)^{3}\left(18H^{2}-6H^{2}(q+1)\right)^{k-4}+3\tilde{b}_{0}(k-2)(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)\left(18H^{2}\right.$
$\displaystyle\left.-6H^{2}(q+1)\right)^{k-3}\left(6H^{4}(-4j-3q(q+4)+s-6)+36H^{4}(j+q(q+5)+3)\right)$
$\displaystyle f^{\text{(iv)}}_{T}$
$\displaystyle=H^{12}2^{m+3}3^{m-1}(m-4)(m-3)(m-2)(m-1)m(q+1)^{4}\tilde{t}_{0}\left(H^{2}\right)^{m-5}$
(71)
$\displaystyle+H^{10}2^{m+3}3^{m}(m-3)(m-2)(m-1)m(q+1)^{2}\tilde{t}_{0}\left(H^{2}\right)^{m-4}(j+q(q+5)+3)$
$\displaystyle-H^{8}2^{m+3}3^{m-1}(m-2)(m-1)m(q+1)\tilde{t}_{0}\left(H^{2}\right)^{m-3}(-j(3q+7)-3q(4q+9)+s-12)$
$\displaystyle+H^{8}2^{m+1}3^{m}(m-2)(m-1)m\tilde{t}_{0}\left(H^{2}\right)^{m-3}(j+q(q+5)+3)^{2}$
$\displaystyle+H^{6}2^{m}3^{m-1}(m-1)m\tilde{t}_{0}\left(H^{2}\right)^{m-2}\left(j(3j+44q+48)+l+3q\left(4q^{2}+39q+56\right)-(4q+9)s+60\right)$
$\displaystyle f^{\text{(iv)}}_{B}$
$\displaystyle=\tilde{b}_{0}(k-1)k\left(18H^{2}-6H^{2}(q+1)\right)^{k-2}\left(6H^{6}\left(-10j^{2}-120j(q+1)\right.\right.$
(72)
$\displaystyle\left.\left.-3\left(2l+5\left(2q^{3}+18q^{2}-qs+24q-2s+8\right)\right)+m\right)+36H^{6}\left(j(3j+44q+48)+l\right.\right.$
$\displaystyle\left.\left.+3q\left(4q^{2}+39q+56\right)-(4q+9)s+60\right)\right)+3\tilde{b}_{0}(k-2)(k-1)k\left(18H^{2}-6H^{2}(q+1)\right)^{k-3}\left(6H^{4}(-4j\right.$
$\displaystyle\left.-3q(q+4)+s-6)+36H^{4}(j+q(q+5)+3)\right)^{2}+\tilde{b}_{0}(k-4)(k-3)(k-2)(k-1)k\left(6H^{3}(j+3q+2)\right.$
$\displaystyle\left.-36H^{3}(q+1)\right)^{4}\left(18H^{2}-6H^{2}(q+1)\right)^{k-5}+4\tilde{b}_{0}(k-2)(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)\left(18H^{2}\right.$
$\displaystyle\left.-6H^{2}(q+1)\right)^{k-3}\left(6H^{5}(10j(q+2)+l+30q(q+2)-5s+24)+36H^{5}(-j(3q+7)-3q(4q+9)+s-12)\right)$
$\displaystyle+6\tilde{b}_{0}(k-3)(k-2)(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)^{2}\left(18H^{2}-6H^{2}(q+1)\right)^{k-4}\left(6H^{4}(-4j\right.$
$\displaystyle\left.-3q(q+4)+s-6)+36H^{4}(j+q(q+5)+3)\right)$ $\displaystyle
f^{\text{(v)}}_{B}$
$\displaystyle=\tilde{b}_{0}(k-1)k\left(18H^{2}-6H^{2}(q+1)\right)^{k-2}\left(6H^{7}\left(140j^{2}+35j(6q(q+6)-s+24)+21l(q+2)\right.\right.$
(73)
$\displaystyle\left.\left.-7m+n-210(q+1)s+90\left(7q(q+2)^{2}+8\right)\right)-36H^{7}\left(50j^{2}+10j(q(8q+51)-s+36)+l(5q+11)-m\right.\right.$
$\displaystyle\left.\left.-5(14q+15)s+30q(q+2)(9q+20)+360\right)\right)+\tilde{b}_{0}(k-5)(k-4)(k-3)(k-2)(k-1)k\left(6H^{3}(j+3q+2)\right.$
$\displaystyle\left.-36H^{3}(q+1)\right)^{5}\left(18H^{2}-6H^{2}(q+1)\right)^{k-6}+5\tilde{b}_{0}(k-2)(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)\left(18H^{2}\right.$
$\displaystyle\left.-6H^{2}(q+1)\right)^{k-3}\left(6H^{6}\left(-10j^{2}-120j(q+1)-3\left(2l+5\left(2q^{3}+18q^{2}-qs+24q-2s+8\right)\right)+m\right)\right.$
$\displaystyle\left.+36H^{6}\left(j(3j+44q+48)+l+3q\left(4q^{2}+39q+56\right)-(4q+9)s+60\right)\right)+15\tilde{b}_{0}(k-3)(k-2)(k-1)k\left(18H^{2}\right.$
$\displaystyle\left.-6H^{2}(q+1)\right)^{k-3}\left(6H^{4}(-4j-3q(q+4)+s-6)+36H^{4}(j+q(q+5)+3)\right)\left(6H^{5}(10j(q+2)+l\right.$
$\displaystyle\left.+30q(q+2)-5s+24)+36H^{5}(-j(3q+7)-3q(4q+9)+s-12)\right)$
$\displaystyle+10\tilde{b}_{0}(k-4)(k-3)(k-2)(k-1)k\left(6H^{3}(j+3q+2)-36H^{3}(q+1)\right)^{3}\left(18H^{2}-6H^{2}(q+1)\right)^{k-5}$
$\displaystyle\left(6H^{4}(-4j-3q(q+4)+s-6)+36H^{4}(j+q(q+5)+3)\right).$

##  Appendix B Phenomenological Considerations

In this work we considered beforehand the model $f(T,B)=-T+\tilde{f}(T,B)$
with $\tilde{f}(T,B)=\tilde{t}_{0}T^{k}+\tilde{b}_{0}B^{m}$. This model is too
complex to be generally analysed in the context of cosmography, hence, we
decided to work with one of the simplest form of these kind of models, the
model where $\tilde{t}_{0}=0$, $\tilde{b}_{0}=1$ and $m=2$, which basically
makes the functional to be $f(T,B)=-T+B^{2}$. Although this model can be
somehow justified from a theoretical point of view (in the sense that it is a
simple extension of TEGR), it suffers from an ill phenomenology. Let us see
this in detail.

The first modified Friedman equation in $f(T,B)$ with the signature
$(-,+,+,+)$ is given by (see Eq. (19) in Escamilla-Rivera:2019ulu )

$\displaystyle-3H^{2}(3f_{B}+2f_{T})+3H\dot{f}_{B}-3\dot{H}f_{B}+\frac{1}{2}f=\kappa^{2}\rho_{m}.$
(74)

Since the boundary term and the torsion scalar are $B=6(3H^{2}+\dot{H})$ and
$T=6H^{2}$ respectively, the Eq. (74) can be rewritten as

$\displaystyle-\frac{1}{2}Bf_{B}-Tf_{T}+3H\dot{f}_{B}+\frac{1}{2}f=\kappa^{2}\rho_{m}.$
(75)

Consider the model $f(T,B)=-T+\alpha B^{2}$, which is exactly the same used
here if $\alpha=1$. With this model, the Eq. (75) becomes

$\displaystyle-\frac{1}{2}B(2\alpha
B)-T(-1)+3H(2\alpha\dot{B})+\frac{1}{2}(-T+\alpha B^{2})=\kappa^{2}\rho_{m}.$
(76)

Using the Eqs.(22), the form of $T$ and $B$, the Eq. (76) evaluated at $z=0$
is

$\displaystyle 6\alpha
H_{0}\left(6H_{0}^{3}(j_{0}+3q_{0}+2)-36H_{0}^{3}(q_{0}+1)\right)-\alpha\left(18H_{0}^{2}-6H_{0}^{2}(q_{0}+1)\right)^{2}+\frac{1}{2}\left(\alpha\left(18H_{0}^{2}-6H_{0}^{2}(q_{0}+1)\right)^{2}\right.$
(77) $\displaystyle\left.-6H_{0}^{2}\right)+6H_{0}^{2}=3H_{0}^{2}\Omega_{m0},$

which can be solved in terms of $\Omega_{m0}$ as

$\displaystyle\Omega_{m0}=-72\alpha H_{0}^{2}+12\alpha H_{0}^{2}j_{0}-6\alpha
H_{0}^{2}q_{0}^{2}-12\alpha H_{0}^{2}q_{0}+1,$ (78)

or from the amplitude coefficient equation $\alpha$ as

$\displaystyle\alpha=-\frac{1-\Omega_{m0}}{6H_{0}^{2}\left(2j_{0}-q_{0}^{2}-2q_{0}-12\right)}.$
(79)

If we substitute $\alpha=1$ which corresponds to our simplified model and the
values of the cosmographic parameters corresponding to $\Lambda$CDM
($q_{0}=-0.55$, $H_{0}=67.3$ and $j_{0}=1$), the density parameter is

$\displaystyle\Omega_{m0}\approx 2.5\times 10^{4}.$ (80)

On the other hand, if we substitute $q_{0}=-0.55$, $H_{0}=67.3$, $j_{0}=1$ and
$\Omega_{m0}=0.315$ into (79), the amplitude coefficient is

$\displaystyle\alpha\approx 2.7391\times 10^{-6}.$ (81)

Hence, even though the model can recover a reasonable set of values for the
cosmographic parameters, it is not completely phenomenological correct since
it leads to a wrong value of the critical matter density parameter. On the
other hand, this model does have some motivation from observational constraint
analysis Escamilla-Rivera:2019ulu .

## References

  * (1) S. Dodelson, Modern Cosmology.  Academic Press, 2003.  https://books.google.com.mt/books?id=3oPRxdXJexcC.
  * (2) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity.  John Wiley and Sons, New York, 1972.
  * (3) Supernova Search Team Collaboration, A. G. Riess et al., “Observational evidence from supernovae for an accelerating universe and a cosmological constant,” Astron.J. 116 (1998) 1009–1038, arXiv:astro-ph/9805201 [astro-ph].
  * (4) Supernova Cosmology Project Collaboration, S. Perlmutter et al., “Measurements of Omega and Lambda from 42 high redshift supernovae,” Astrophys.J. 517 (1999) 565–586, arXiv:astro-ph/9812133 [astro-ph].
  * (5) J. Binney, Galactic dynamics / James Binney and Scott Tremaine. Princeton series in astrophysics. Princeton University Press, Princeton, N.J, second edition. ed., 2008.
  * (6) J. L. Bernal, L. Verde, and A. G. Riess, “The trouble with $H_{0}$,” JCAP 10 (2016) 019, arXiv:1607.05617 [astro-ph.CO].
  * (7) E. Di Valentino et al., “Snowmass2021 - Letter of interest cosmology intertwined II: The hubble constant tension,” Astropart. Phys. 131 (2021) 102605, arXiv:2008.11284 [astro-ph.CO].
  * (8) E. Di Valentino, O. Mena, S. Pan, L. Visinelli, W. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, “In the realm of the Hubble tension—a review of solutions,” Class. Quant. Grav. 38 (2021) no. 15, 153001, arXiv:2103.01183 [astro-ph.CO].
  * (9) Planck Collaboration, N. Aghanim et al., “Planck 2018 results. VI. Cosmological parameters,” arXiv:1807.06209 [astro-ph.CO].
  * (10) E. Di Valentino et al., “Cosmology intertwined III: $f\sigma_{8}$ and $S_{8}$,” Astropart. Phys. 131 (2021) 102604, arXiv:2008.11285 [astro-ph.CO].
  * (11) S. Weinberg, “The Cosmological Constant Problem,” Rev. Mod. Phys. 61 (1989) 1–23.
  * (12) S. Appleby and E. V. Linder, “The Well-Tempered Cosmological Constant,” JCAP 07 (2018) 034, arXiv:1805.00470 [gr-qc].
  * (13) R. Gaitskell, “Direct detection of dark matter,” Ann. Rev. Nucl. Part. Sci. 54 (2004) 315–359.
  * (14) L. Baudis, “Dark matter detection,” J. Phys. G43 (2016) no. 4, 044001.
  * (15) G. Bertone, D. Hooper, and J. Silk, “Particle dark matter: Evidence, candidates and constraints,” Phys. Rept. 405 (2005) 279–390, arXiv:hep-ph/0404175 [hep-ph].
  * (16) E. M. Barboza, Jr. and J. S. Alcaniz, “A parametric model for dark energy,” Phys. Lett. B 666 (2008) 415–419, arXiv:0805.1713 [astro-ph].
  * (17) H. K. Jassal, J. S. Bagla, and T. Padmanabhan, “Observational constraints on low redshift evolution of dark energy: How consistent are different observations?,” Phys. Rev. D 72 (2005) 103503, arXiv:astro-ph/0506748.
  * (18) G. Efstathiou, “Constraining the equation of state of the universe from distant type Ia supernovae and cosmic microwave background anisotropies,” Mon. Not. Roy. Astron. Soc. 310 (1999) 842–850, arXiv:astro-ph/9904356.
  * (19) E. V. Linder, “Exploring the expansion history of the universe,” Phys. Rev. Lett. 90 (2003) 091301, arXiv:astro-ph/0208512.
  * (20) P. Astier, “Can luminosity distance measurements probe the equation of state of dark energy,” Phys. Lett. B 500 (2001) 8–15, arXiv:astro-ph/0008306.
  * (21) A. R. Cooray and D. Huterer, “Gravitational lensing as a probe of quintessence,” Astrophys. J. Lett. 513 (1999) L95–L98, arXiv:astro-ph/9901097.
  * (22) C. Escamilla-Rivera and A. Nájera, “Dynamical dark energy models in the light of Gravitational-Wave Transient Catalogues,” arXiv:2103.02097 [gr-qc].
  * (23) E. O. Colgáin, M. M. Sheikh-Jabbari, and L. Yin, “Can dark energy be dynamical?,” Phys. Rev. D 104 (2021) no. 2, 023510, arXiv:2104.01930 [astro-ph.CO].
  * (24) T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis, “Modified Gravity and Cosmology,” Phys. Rept. 513 (2012) 1–189, arXiv:1106.2476 [astro-ph.CO].
  * (25) S. Capozziello and M. De Laurentis, “Extended Theories of Gravity,” Phys. Rept. 509 (2011) 167–321, arXiv:1108.6266 [gr-qc].
  * (26) CANTATA Collaboration, E. N. Saridakis et al., “Modified Gravity and Cosmology: An Update by the CANTATA Network,” arXiv:2105.12582 [gr-qc].
  * (27) K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, “Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests,” Astrophys. Space Sci. 342 (2012) 155–228, arXiv:1205.3421 [gr-qc].
  * (28) T. Yang, A. Banerjee, and E. O. Colgáin, “Cosmography and flat $\Lambda$CDM tensions at high redshift,” Phys. Rev. D 102 (2020) no. 12, 123532, arXiv:1911.01681 [astro-ph.CO].
  * (29) S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. L. Said, J. Mifsud, and E. Di Valentino, “Teleparallel Gravity: From Theory to Cosmology,” arXiv:2106.13793 [gr-qc].
  * (30) R. Aldrovandi and J. G. Pereira, Teleparallel Gravity, vol. 173.  Springer, Dordrecht, 2013\.
  * (31) Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, “$f(T)$ teleparallel gravity and cosmology,” Rept. Prog. Phys. 79 (2016) no. 10, 106901, arXiv:1511.07586 [gr-qc].
  * (32) M. Krssak, R. van den Hoogen, J. Pereira, C. Böhmer, and A. Coley, “Teleparallel theories of gravity: illuminating a fully invariant approach,” Class. Quant. Grav. 36 (2019) no. 18, 183001, arXiv:1810.12932 [gr-qc].
  * (33) D. Lovelock, “The Einstein tensor and its generalizations,” J. Math. Phys. 12 (1971) 498–501.
  * (34) P. Gonzalez and Y. Vasquez, “Teleparallel Equivalent of Lovelock Gravity,” Phys. Rev. D 92 (2015) no. 12, 124023, arXiv:1508.01174 [hep-th].
  * (35) S. Bahamonde, K. F. Dialektopoulos, and J. Levi Said, “Can Horndeski Theory be recast using Teleparallel Gravity?,” Phys. Rev. D 100 (2019) no. 6, 064018, arXiv:1904.10791 [gr-qc].
  * (36) T. P. Sotiriou and V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys. 82 (2010) 451–497, arXiv:0805.1726 [gr-qc].
  * (37) V. Faraoni, “f(R) gravity: Successes and challenges,” in 18th SIGRAV Conference.  10, 2008.  arXiv:0810.2602 [gr-qc].
  * (38) R. Ferraro and F. Fiorini, “Modified teleparallel gravity: Inflation without inflaton,” Phys. Rev. D75 (2007) 084031, arXiv:gr-qc/0610067 [gr-qc].
  * (39) R. Ferraro and F. Fiorini, “On Born-Infeld Gravity in Weitzenbock spacetime,” Phys. Rev. D78 (2008) 124019, arXiv:0812.1981 [gr-qc].
  * (40) G. R. Bengochea and R. Ferraro, “Dark torsion as the cosmic speed-up,” Phys. Rev. D79 (2009) 124019, arXiv:0812.1205 [astro-ph].
  * (41) E. V. Linder, “Einstein’s Other Gravity and the Acceleration of the Universe,” Phys. Rev. D81 (2010) 127301, arXiv:1005.3039 [astro-ph.CO]. [Erratum: Phys. Rev.D82,109902(2010)].
  * (42) S.-H. Chen, J. B. Dent, S. Dutta, and E. N. Saridakis, “Cosmological perturbations in f(T) gravity,” Phys. Rev. D83 (2011) 023508, arXiv:1008.1250 [astro-ph.CO].
  * (43) S. Bahamonde, K. Flathmann, and C. Pfeifer, “Photon sphere and perihelion shift in weak $f(T)$ gravity,” Phys. Rev. D 100 (2019) no. 8, 084064, arXiv:1907.10858 [gr-qc].
  * (44) C. Escamilla-Rivera and J. Levi Said, “Cosmological viable models in $f(T,B)$ gravity as solutions to the $H_{0}$ tension,” Class. Quant. Grav. 37 (2020) no. 16, 165002, arXiv:1909.10328 [gr-qc].
  * (45) S. Bahamonde, C. G. Böhmer, and M. Wright, “Modified teleparallel theories of gravity,” Phys. Rev. D92 (2015) no. 10, 104042, arXiv:1508.05120 [gr-qc].
  * (46) S. Capozziello, M. Capriolo, and M. Transirico, “The gravitational energy-momentum pseudotensor: the cases of $f(R)$ and $f(T)$ gravity,” Int. J. Geom. Meth. Mod. Phys. 15 (2018) 1850164, arXiv:1804.08530 [gr-qc].
  * (47) S. Bahamonde and S. Capozziello, “Noether Symmetry Approach in $f(T,B)$ teleparallel cosmology,” Eur. Phys. J. C77 (2017) no. 2, 107, arXiv:1612.01299 [gr-qc].
  * (48) A. Paliathanasis, “de Sitter and Scaling solutions in a higher-order modified teleparallel theory,” JCAP 1708 (2017) no. 08, 027, arXiv:1706.02662 [gr-qc].
  * (49) G. Farrugia, J. Levi Said, V. Gakis, and E. N. Saridakis, “Gravitational Waves in Modified Teleparallel Theories,” Phys. Rev. D 97 (2018) no. 12, 124064, arXiv:1804.07365 [gr-qc].
  * (50) S. Bahamonde, M. Zubair, and G. Abbas, “Thermodynamics and cosmological reconstruction in $f(T,B)$ gravity,” Phys. Dark Univ. 19 (2018) 78–90, arXiv:1609.08373 [gr-qc].
  * (51) M. Wright, “Conformal transformations in modified teleparallel theories of gravity revisited,” Phys. Rev. D93 (2016) no. 10, 103002, arXiv:1602.05764 [gr-qc].
  * (52) G. Farrugia, J. Levi Said, and A. Finch, “Gravitoelectromagnetism, Solar System Test and Weak-Field Solutions in $f(T,B)$ Gravity with Observational Constraints,” Universe 6 (2020) no. 2, 34, arXiv:2002.08183 [gr-qc].
  * (53) S. Capozziello, M. Capriolo, and L. Caso, “Weak Field Limit and Gravitational Waves in $f(T,B)$ Teleparallel Gravity,” Eur. Phys. J. C 80 (2020) no. 2, 156, arXiv:1912.12469 [gr-qc].
  * (54) S. Bahamonde, J. Levi Said, and M. Zubair, “Solar System Tests in Modified Teleparallel Gravity,” arXiv:2006.06750 [gr-qc].
  * (55) F. W. Hehl, P. Von Der Heyde, G. D. Kerlick, and J. M. Nester, “General Relativity with Spin and Torsion: Foundations and Prospects,” Rev. Mod. Phys. 48 (1976) 393–416.
  * (56) M. Hohmann, “Variational Principles in Teleparallel Gravity Theories,” Universe 7 (2021) no. 5, 114, arXiv:2104.00536 [gr-qc].
  * (57) R. Weitzenböock, ‘Invariantentheorie’.  Noordhoff, Gronningen, 1923.
  * (58) T. Ortín, Gravity and Strings.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2004.  https://books.google.com.mt/books?id=sRlHoXdAVNwC.
  * (59) M. Zubair, L. R. Durrani, and S. Waheed, “Reconciling Tsallis holographic dark energy models in modified $f(T,B)$ gravitational framework,” Eur. Phys. J. Plus 136 (2021) no. 9, 943.
  * (60) M. Zubair and L. R. Durrani, “A study of the cosmologically reconstructed $f(T,B)$ gravity from the cosmological jerk parameter,” Eur. Phys. J. Plus 135 (2020) no. 8, 668.
  * (61) M. Zubair, S. Waheed, M. Atif Fayyaz, and I. Ahmad, “Energy constraints and the phenomenon of cosmic evolution in the $f(T,B)$ framework,” Eur. Phys. J. Plus 133 (2018) no. 11, 452, arXiv:1807.07399 [physics.gen-ph].
  * (62) M. Krššák and E. N. Saridakis, “The covariant formulation of $f(T)$ gravity,” Class. Quant. Grav. 33 (2016) no. 11, 115009, arXiv:1510.08432 [gr-qc].
  * (63) N. Tamanini and C. G. Boehmer, “Good and bad tetrads in $f(T)$ gravity,” Phys. Rev. D86 (2012) 044009, arXiv:1204.4593 [gr-qc].
  * (64) S. Capozziello, V. F. Cardone, H. Farajollahi, and A. Ravanpak, “Cosmography in f(T)-gravity,” Phys. Rev. D84 (2011) 043527, arXiv:1108.2789 [astro-ph.CO].
  * (65) A. Aviles, C. Gruber, O. Luongo, and H. Quevedo, “Cosmography and constraints on the equation of state of the Universe in various parametrizations,” Phys. Rev. D 86 (2012) 123516, arXiv:1204.2007 [astro-ph.CO].
  * (66) G. A. R. Franco, C. Escamilla-Rivera, and J. Levi Said, “Stability analysis for cosmological models in $f(T,B)$ gravity,” Eur. Phys. J. C 80 (2020) no. 7, 677, arXiv:2005.14191 [gr-qc].
  * (67) M. Caruana, G. Farrugia, and J. Levi Said, “Cosmological bouncing solutions in $f(T,B)$ gravity,” Eur. Phys. J. C 80 (2020) no. 7, 640, arXiv:2007.09925 [gr-qc].
  * (68) S. Bahamonde, A. Golovnev, M.-J. Guzmán, J. L. Said, and C. Pfeifer, “Black Holes in $f(T,B)$ Gravity: Exact and Perturbed Solutions,” arXiv:2110.04087 [gr-qc].
  * (69) D. M. Scolnic et al., “The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample,” Astrophys. J. 859 (2018) no. 2, 101, arXiv:1710.00845 [astro-ph.CO].
  * (70) S. Qi and T. Lu, “A new luminosity relation for gamma-ray bursts and its implication,”The Astrophysical Journal 717 (Jun, 2010) 1274–1278. http://dx.doi.org/10.1088/0004-637X/717/2/1274.
  * (71) Y.-Y. Wang and F. Y. Wang, “Calibration of Gamma-Ray Burst Luminosity Correlations Using Gravitational Waves as Standard Sirens,” Astrophys. J. 873 (2019) no. 1, 39, arXiv:1902.09677 [astro-ph.HE].
  * (72) C. Escamilla-Rivera, M. Carvajal, C. Zamora, and M. Hendry, “Neural networks and standard cosmography with newly calibrated high redshift GRB observations,” arXiv:2109.00636 [astro-ph.CO].
  * (73) G. D’Agostini, “Fits, and especially linear fits, with errors on both axes, extra variance of the data points and other complications,” arXiv:physics/0511182.
  * (74) R. Jimenez and A. Loeb, “Constraining cosmological parameters based on relative galaxy ages,” Astrophys. J. 573 (2002) 37–42, arXiv:astro-ph/0106145 [astro-ph].
  * (75) C. Escamilla-Rivera, J. Levi Said, and J. Mifsud, “Performance of Non-Parametric Reconstruction Techniques in the Late-Time Universe,” arXiv:2105.14332 [astro-ph.CO].
  * (76) R. C. Bernardo and J. Levi Said, “Towards a model-independent reconstruction approach for late-time Hubble data,” JCAP 08 (2021) 027, arXiv:2106.08688 [astro-ph.CO].
  * (77) R. Briffa, S. Capozziello, J. Levi Said, J. Mifsud, and E. N. Saridakis, “Constraining teleparallel gravity through Gaussian processes,” Class. Quant. Grav. 38 (2020) no. 5, 055007, arXiv:2009.14582 [gr-qc].