#  Viscous cosmology in $f(T)$ gravity

Jing Yang  Rui-Hui Lin<EMAIL_ADDRESS>Xiang-Hua Zhai<EMAIL_ADDRESS>Division of Mathematics and Theoretical Physics, Shanghai Normal University,
100 Guilin Road, Shanghai 200234, China

###### Abstract

We propose a new model for the viscosity of cosmic matters, which can be
applied to different epochs of the universe. Using this model, we include the
bulk viscosities as practical corrections to the perfect fluid models of the
baryonic and dark matters since the material fluids in the real world may have
viscosities due to thermodynamics. Such inclusion is put to the test within
the framework of $f(T)$ gravity that is proved to be successful in describing
the cosmic acceleration, where $T$ denotes the torsion scalar. We perform an
observational fit to our model and constrain the cosmological and model
parameters by using various latest cosmological datasets. Based on the fitting
result, we discuss several cosmological implications including the dissipation
of matters, the evolutionary history of the universe, $f(T)$ modification as
an effective dark energy, and the Hubble tension problem. The corresponding
findings are (i) The late time dissipation will make the density parameters of
the matters vanish in the finite future. Moreover, the density ratio between
the baryonic and dark matters will change over time. (ii) The radiation
dominating era, matter dominating era and the accelerating era can be
recovered and the model can successfully describe the known history of the
universe. (iii) The $f(T)$ modification is the main drive of the acceleration
expansion and currently mimics a phantom-like dark energy. But the universe
will eventually enter a de Sitter expansion phase. (iv) The Hubble tension
between local and global observations can be significantly alleviated in our
model.

##  I Introduction

Cosmological and astrophysical observations including the Type-Ia supernovae
(SNIa), cosmic microwave background radiation (CMB), baryon acoustic
oscillations (BAOs) and large-scale structure (LSS) suggest that our universe
is now in an acceleration phase of expansion. One of the straightforward ways
to explain this acceleration is to introduce the cosmological constant
$\Lambda$ back to play the role of the so-called dark energy and construct the
$\Lambda$ cold dark matter ($\Lambda$CDM) model. This model proves to be quite
accurate and successful. However, from the theoretical point of view, the
cosmological constant $\Lambda$ lacks a physical origin and its observed value
is too small to be explained by any fundamental theories. Moreover, this
simple model also suffers from the cosmic coincident problem, i.e., the
densities of matter and dark energy of the universe are of the same order. To
alleviate these problems, various dark energy models have been proposed and
studied (for comprehensive reviews, see, e.g., Refs. [1, 2, 3, 4]). Another
angle to address these problems is to consider modifications of gravity and
see General Relativity (GR) as an approximation to the real theory of gravity
yet to be discovered. One of the simplest schemes to modify Einstein’s gravity
is the $f(R)$ gravity, where the curvature scalar $R$ in the gravitational
Lagrangian is replaced by a function $f$ of $R$ (see, e.g., Refs. [5, 6, 7, 8]
for extensive reviews). On the other hand, based on an equivalent description
of Einstein’s gravity, known as the Teleparallel Equivalent of General
Relativity (TEGR) [9, 10] where the torsion scalar $T$ takes the place of the
curvature scalar $R$, the $f(T)$ modification of gravity is proposed and
widely studied (see, e.g., Refs. [11, 12, 13, 14]).

In most studies of cosmology, may that be within the framework of GR or
modified gravities, the universe media are usually simplified as perfect
fluids. From the hydrodynamic point of view, a less ideal fluid may involve
the dissipative effects that lead to a deviation from the thermal equilibrium.
Such dissipative processes in the cosmic expansion include heat conduction,
bulk viscosity, and shear viscosity, which have been carefully studied (see,
e.g., Refs. [15, 16, 17]). Due to the homogeneous and isotropic assumption of
the cosmic background, considering the bulk viscosity as the only relevant
dissipative phenomenon in cosmology is appropriate. Pioneer studies to include
bulk viscosities in cosmology follow conceptually different trains of thought,
partly such as

  * $\bullet$

Viscosity as a sole drive of acceleration: Either some part or the unity of
the universe medium has viscosity which is the sole mechanism to drive the
cosmic acceleration [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. This idea
utilizes the fact that the bulk viscosity of the fluid may act as an effective
pressure, possibly negative, to recover the local thermodynamic equilibrium
[29], and explores the possibility that this effective pressure may be the
culprit of the accelerated expansion of the universe. Such a consideration can
be seen as one of the approaches to a unifying model of (dark) matter and dark
energy.

  * $\bullet$

Dust matter and viscous dark sector: The known contents of the universe are
still without viscosity, while the dark sector that has already the mechanism
to drive the acceleration may be viscous [30, 31, 32, 33, 34, 35, 36, 37].
Since the properties of dark energy or dark matter or the unification of both
are still not very conclusive, this idea puts more physical and realistic
considerations into the model of dark sector.

  * $\bullet$

Viscous matter and effective dark energy: The material contents may have
viscosities, while the accelerated expansion is of geometric or gravitational
origin, e.g., cosmological constant or modified gravities [38, 39, 40, 41, 36,
42, 43, 44, 45, 46, 47, 48]. This idea sees the viscosities to be the physical
properties that may be possessed by any material fluid in the real world and
considers the viscosity as one of the realistic corrections of imperfectness
to the perfect fluid model. The inclusion of viscosities in the fluid models
is put to the test within the various frameworks of successful cosmological
models.

In this work, we follow the last route and intend to include the viscosities
in the cosmological context of the $f(T)$ gravity. In Refs. [44, 45, 46], the
authors have analyzed the cosmology in $f(T)$ gravity where the universe is
filled with a single viscous fluid by considering specific evolutionary
functions given by $H(t)\propto 1/t^{\lambda}$, where $H$ is the Hubble
parameter, $t$ is the cosmic time and $\lambda$ is a constant parameter.
Moreover, $f(T)$ gravity has also been used to reconstruct the generalized
Chaplygin gas model with viscosity [49]. In the current work, to accommodate
some of the existing late time viscous models, we propose a simple unifying
bulk viscous model of the matters that can be applied to different epochs of
the universe. Using this new model, we intend to study the evolutionary
history of the universe with bulk viscous matter in $f(T)$ gravity, where
different cosmic contents, i.e., radiation and viscous baryonic and dark
matters, are included. The motivation of this consideration is twofold.
Firstly, we can model the whole evolutionary history of the universe including
the radiation dominating era, matter dominating era and the acceleration era.
As such, we can include the rather stringent datasets of CMB and BAO in
addition to the late time SNIa datasets and various estimations of $H$
dependence on the redshift $z$ (denoted as Hz in the following) in the
parameter fitting. Secondly, the physics before the last scattering can be
different from the standard model not only for the modification of $f(T)$
gravity but also for the bulk viscosity. This could be significant. For
example, the recently revealed tension between the Hubble constant $H_{0}$
measured locally and that fitted by the global observation [50] may, among
various possibilities, be a result of modified prerecombination physics [51,
52, 53].

The paper is arranged as follows. Section II contains brief descriptions of
the bulk viscous model of matter contents and the $f(T)$ gravity. In Sect.
III, we use the observational datasets of CMB, BAO, SNIa, and Hz to fit and
evaluate the model, presenting constraints and best-fit values of the
cosmological and model parameters. Some cosmological implications including
the dissipation of the matters, the evolutionary history of the universe,
$f(T)$ modification as an effective dark energy, and the Hubble tension
problems are presented in Sect. IV. We conclude our study in the last section.

Throughout the paper, we use the natural unit with $c=8\pi G=1$, where $c$ is
the speed of light and $G$ is the gravitational constant.

##  II Overview of the $f(T)$ gravity and the viscous cosmology

###  II.1 Cosmological model with bulk viscous matter

We assume a Friedmann-Lemaître-Robertson-Walker (FLRW) metric for the spatial
flat universe, which is given by

$\mathrm{d}s^{2}=\mathrm{d}t^{2}-a(t)^{2}\delta_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j},$
(1)

where $a(t)$ is the cosmic scale factor.

For the matter source we consider an imperfect fluid with viscosity, where the
dissipative effects resulted from microscopic interactions are described by
transport coefficients. Since the universe is treated to be isotropic and
expanding, only bulk viscosity should be relevant and it can be expressed as a
first order deviation from the thermodynamic equilibrium. The energy-momentum
tensor for such a fluid is then given by [15, 16, 17, 29, 36],

$\bar{\mathcal{T}}_{\mu}^{\;\nu}=\left(\rho+\bar{p}\right)u_{\mu}u^{\nu}-\bar{p}\delta_{\mu}^{\nu},$
(2)

with

$\bar{p}=p-\zeta\nabla_{\mu}u^{\mu},$ (3)

where $p$ and $\rho$ are the pressure and energy density of the fluid,
respectively, $u^{\mu}$ is the 4-velocity, and $\zeta$ is the bulk viscosity
arising from deviation from the local thermodynamic equilibrium. The viscosity
is then encoded in a viscous pressure term $-\zeta\nabla_{\mu}u^{\mu}$ in
addition to the usual material pressure. In the cosmological scenario, this
term can be written as $-3\zeta H$, where $H=\dot{a}/a$ is the Hubble
parameter.

The continuous equation $\nabla_{\nu}\bar{\mathcal{T}}_{\mu}^{\;\nu}=0$ is
equivalent to

$\nabla_{\nu}\mathcal{T}_{\mu}^{\;\nu}=\nabla_{\nu}\left[\left(\delta_{\mu}^{\nu}-u_{\mu}u^{\nu}\right)\left(\bar{p}-p\right)\right],$
(4)

where $\mathcal{T}_{\mu}^{\;\nu}$ is the energy-momentum tensor of a usual
perfect fluid. For cosmological metric and the assumption $p=0$,

$\nabla_{\nu}\mathcal{T}_{0}^{\;\nu}=-3H\bar{p},$ (5)

where the right-hand side can be either seen as a sink of energy if $\bar{p}$
is positive, or a source if $\bar{p}$ is negative. It reflects that there is
energy transference between the kinetic energy of cosmological expansion and
the material fluid due to the bulk viscosity.

The transport coefficient $\zeta$ should depend on the dynamics of the
material fluid. A general model for viscous effects that emerge at late time
is proposed and can be written as [54, 25, 40, 36, 55, 43]

$\zeta=\zeta_{0}+\zeta_{1}H,$ (6)

where both $\zeta_{0}$ and $\zeta_{1}$ are constants. This can be viewed as an
agreement with a generalized equation of state (EOS) of a dark fluid [56, 57]
since the pressure now can be written as

$\bar{p}=p(\rho)+w_{0}H+w_{1}H^{2},$ (7)

where $w_{0}=-3\zeta_{0}$ and $w_{1}=-3\zeta_{1}$ are new constant
coefficients of EOS. Another widely considered model of material bulk
viscosity is of the form [58, 59, 60, 36, 61, 55]

$\zeta\sim\rho^{s}.$ (8)

For late time universe when matter is dominating and $H^{2}\sim\rho_{m}$,
these two forms (6) and (8) can reach some common ground for $s=1/2$.

However, for early times of the universe, both forms will lead to an unbounded
viscosity or a growing Hubble parameter, which seems to be contrary to our
conventional understanding of the evolutionary history of the universe.
Therefore, to cover the early times, we consider the merits of both forms of
the viscosity and propose a simple new form

$\zeta=\zeta_{0}\sqrt{\Omega}+\zeta_{1}\Omega H,$ (9)

where $\Omega=\rho/(3H^{2})$ is the density parameter of the corresponding
content with bulk viscosity. While Eq. (9) preserves the property (8), it will
reduce to the form (6) at the era when the universe is dominated by the
corresponding viscous content with $\Omega\sim 1$, and will be suppressed when
the viscous content is not dominating.

In the current work, for simplicity, we only consider the imperfectness and
bulk viscosity of the material contents including both the baryonic and dark
matters, leaving the relativistic contents to be described by the EOS
$w=p/\rho=\frac{1}{3}$. In addition, we assume that the viscosities of the
baryonic and dark matters have the same coefficients. That is,

$\begin{split}\overline{p_{b}}=&p_{b}-3H(\zeta_{0}\sqrt{\Omega_{b}}+\zeta_{1}\Omega_{b}H),\\\
\overline{p_{m}}=&p_{m}-3H(\zeta_{0}\sqrt{\Omega_{m}}+\zeta_{1}\Omega_{m}H),\end{split}$
(10)

where the subscripts $b$ and $m$ indicate the baryonic and dark matters,
respectively. One can easily check that

$\overline{p_{i}}=p_{i}-\zeta_{0}\sqrt{3\rho_{i}}-\zeta_{1}\rho_{i},$ (11)

where $i=b,m$. So, the $\zeta_{1}$ term is in effect a correction to the
pressureless EOS $w_{i}=0$, and one can assume the material pressure $p_{i}=0$
without loss of generality.

It is noted here that the viscous effects of the baryonic and dark matters
must be considered separately even it is simply assumed that they have the
same viscous coefficients. Otherwise the viscosity will become another
interaction between the baryonic and dark matters other than gravitation. At
late time, this does not seem very important in that the same viscous
coefficients and EOSs make sure both the baryonic and dark matters evolve
synchronously. However, before the last scattering, the propagation of the
sound wave of the photon-baryon fluid depends closely on the EOS of the baryon
matter, but is not so sensitive to that of the dark matter. Specifically,
around some primordial and initial clumps of matter, the pressure of the
photon-baryon fluid resists the gravitational clustering. The resulting sound
waves propagate at the sound speed $c_{s}$ given by

$c_{s}^{2}=\frac{\partial
p}{\partial\rho}=\frac{\overline{p_{b}}^{\prime}+p_{\gamma}^{\prime}}{\rho_{b}^{\prime}+\rho_{\gamma}^{\prime}},$
(12)

where the subscript $\gamma$ indicates the photon and the prime denotes
derivative with respect to some common variable, e.g., the scale factor $a$.
The propagations continued until the universe became transparent for photons,
leaving dramatic oscillations CMB anisotropic data we see today, as well as
imprints in the galaxy distributions. This is the key to our model in the era
before the last scattering and include the CMB and BAO datasets in the
following section to perform the observational fit.

###  II.2 Viscous cosmology in $f(T)$ gravity

Although the bulk viscosity may provide a possibly negative pressure, as seen
in Eq. (10), it can hardly be the main cause of the late time acceleration of
the universe. As we will see in the next section, the densities and
viscosities of the matters are quite stringently constrained by the
observation. The effective pressures from viscosities do not seem to be of the
same order of the energy densities and hence cannot account for the missing
dark energy that are around 70 percent of the cosmic contents. Therefore, the
drive of the acceleration is still needed elsewhere. In this work, we consider
the cosmic evolution under the framework of $f(T)$ gravity, attributing the
late time acceleration to the modification of gravitational theory in the
teleparallel description.

On the parallelizable spacetime manifold $\mathcal{M}$ with a metric $g$, one
can generally find a set of tetrad $\\{e_{A}^{\;\mu}\\}$ and its dual
$\\{e^{A}_{\;\mu}\\}$, such that

$e^{A}_{\;\mu}e_{B}^{\;\mu}=\delta_{B}^{A},\quad
e_{A}^{\;\mu}e^{A}_{\;\nu}=\delta_{\nu}^{\mu}.$ (13)

They are related to the metric tensor via

$g_{\mu\nu}=\eta_{AB}e^{A}_{\;\mu}e^{B}_{\;\nu},\quad\eta_{AB}=g_{\mu\nu}e_{A}^{\;\mu}e_{B}^{\;\nu}.$
(14)

Then, the torsion scalar $T$ is defined as

$T\equiv T^{\alpha}_{\;\mu\nu}S_{\alpha}^{\;\mu\nu},$ (15)

where the torsion tenser is

$T^{\alpha}_{\hskip 3.5pt\mu\nu}=e_{A}^{\hskip
3.5pt\alpha}\left(\partial_{\mu}e^{A}_{\hskip
3.5pt\nu}-\partial_{\nu}e^{A}_{\hskip 3.5pt\mu}\right)$ (16)

and the superpotential is

$S_{\alpha}^{\>\beta\gamma}=\frac{1}{4}\left(T_{\alpha}^{\>\beta\gamma}+T^{\gamma\beta}_{\>\;\>\;\alpha}-T^{\beta\gamma}_{\>\;\>\;\alpha}\right)+\frac{1}{2}\left(\delta^{\beta}_{\alpha}T^{\lambda\gamma}_{\>\;\>\;\lambda}-\delta^{\gamma}_{\alpha}T^{\lambda\beta}_{\>\;\>\;\lambda}\right).$
(17)

TEGR uses this scalar $T$ as its gravitational Lagrangian, and the $f(T)$
gravity considers an arbitrary function of $T$ instead, i.e.,

$\mathcal{S}=-\frac{1}{2}\int|e|f(T)\mathrm{d}^{4}x+\int|e|\mathcal{L}_{M}\mathrm{d}^{4}x,$
(18)

where $|e|=\det(e^{A}_{\alpha})=\sqrt{-g}$ is the determinant of the tetrad
$e^{A}_{\alpha}$, and $\mathcal{L}_{M}$ is the Lagrangian of matter.

Varying the above action (18) with respect to the tetrad $e^{A}_{\alpha}$, one
can obtain the field equation

$\frac{2}{|e|}\partial_{\beta}\left(|e|S_{\sigma}^{\>\alpha\beta}e_{A}^{\>\sigma}f_{T}\right)+\frac{f}{2}e_{A}^{\>\alpha}=\mathcal{T}_{\beta}^{\alpha}e_{A}^{\>\beta},$
(19)

where $f_{T}$ denotes $\mathrm{d}f/\mathrm{d}T$, and the energy-momentum
tensor $\mathcal{T}_{\beta}^{\alpha}$ of matter is given by

$\frac{\delta(|e|\mathcal{L}_{M})}{\delta
e^{A}_{\>\alpha}}=|e|\mathcal{T}_{\beta}^{\alpha}e_{A}^{\>\beta}.$ (20)

For the cosmological metric given in Eq. (1), one can easily find that
$T=-6H^{2}$. Then, since the Hubble parameter $H$ is generally a decreasing
function of cosmic time, one would expect that a $f(T)$ model with a
modification term more significant at smaller $|T|$ is appropriate in that it
could give explanation to the late time (smaller $H$ and $|T|$) acceleration
and would not mess up the early time evolution. We therefore choose a concrete
$f(T)$ model in the form

$f(T)=T+\frac{\alpha}{T},$ (21)

where $\alpha$ is the model parameter. Studies show that the modifications
with inverse powers of $T$ may indeed provide the drive of the accelerated
expansion [62, 63, 64].

Taking the model (21) and the energy-momentum (2) into the field equation
(19), one can obtain the modified Friedmann equations

$\begin{split}3H^{2}-\frac{\alpha}{4H^{2}}&=\rho_{r}+\rho_{b}+\rho_{m},\\\
2\dot{H}\left(1+\frac{\alpha}{12H^{4}}\right)&=-p_{r}-\overline{p_{b}}-\overline{p_{m}}-\rho_{r}-\rho_{b}-\rho_{m},\end{split}$
(22)

where the subscript $r$ denotes the radiation including both the photons and
relativistic neutrinos.

##  III Observational fit

In this section, we perform an observational fit to the cosmological model we
introduce in the previous section. In most studies of viscous cosmology, the
authors usually focus on a certain epoch of the evolutionary history of the
universe. Since the newly proposed model of viscosity in the current work
depends on $\Omega$ instead of $H$ or $\rho$, we can consider the cosmic
evolution as a whole and, more importantly, utilize the observational datasets
of both early and late time eras in a joint fitting procedure.

For the late time SNIa data, we consider the latest Pantheon compilation [65]
consisting of a total of $1048$ SNIa with the cosmic redshift $z$ ranging from
$0.01$ to $2.3$. We also utilize $57$ data points of Hz [66]. These two sets
of data represent the observations of the acceleration phase of the universe
and provide the late time constraints of the model.

For earlier era, we consider the CMB and BAO datasets. The CMB temperature
power spectrum is sensitive to the matter densities, and it also measures
precisely the sound horizon, which is given by

$r_{s}^{*}=\int_{z^{*}}^{\infty}\frac{c_{s}}{H(z)}\mathrm{d}z,$ (23)

where $z^{*}\sim 1000$ is the cosmic redshift of the last-scattering surface
and $c_{s}$ is the sound speed of the photon-baryon fluid before the last
scattering. In viscous cosmology, such consideration is only possible when the
viscosities of baryonic and dark matters are separated in that $c_{s}$ is
closely related to the EOS of baryonic matter but not sensitive to that of
dark matter. By using the viscosity model (10) proposed in the previous
section, we can substitute $c_{s}$ with Eq. (12). We will be using the CMB
data inferred from Refs. [67, 68]. Moreover, the BAO measurement provides a
standard ruler to probe the angular diameter distance versus the redshift by
performing a spherical average of their scale measurement. We will use the BAO
data given in Refs. [69, 70, 71].

We perform a cosmological fit for the viscous model in $f(T)$ gravity based on
a Markov chain Monte Carlo method using the datasets mentioned above. As
references, we also perform the same fitting procedure for the concord
cosmological model $\Lambda$CDM, and a viscous model that is described by Eq.
(10) but still leaves the dark energy to the mysterious cosmological constant
(denoted as v$\Lambda$CDM, for detailed analysis of similar models, see Ref.
[41, 36, 42]). The fitting results and best-fitted parameters are listed in
Table 1, where $\Omega_{b0}$ and $\Omega_{m0}$ are the values of $\Omega_{b}$
and $\Omega_{m}$ at present time, respectively, and
$h=H_{0}/(100\>\mathrm{km/s/Mpc})$. In the fitting procedure, we consider the
radiation density parameter at present $\Omega_{r0}=\Omega_{\gamma
0}(1+0.2271N_{\text{eff}})$, where the photon density parameter at present is
fixed $\Omega_{\gamma 0}=2.469\times 10^{-5}h^{-2}$ in this work and
$N_{\text{eff}}$ is the effective number of neutrino species with
$N_{\text{eff}}=3.046$ for the current standard value. Figure 1 illustrates
the $1\sigma(68.3\%)$ and $2\sigma(95.4\%)$ confidence levels of the
parameters of our model.

Parameters |  $\Lambda$CDM | v$\Lambda$CDM | our model
---|---|---|---
$\Omega_{b0}h^{2}$ | $0.0222^{+0.0001}_{-0.0001}$ | $0.0222^{+0.0003}_{-0.0003}$ | $0.0220^{+0.0003}_{-0.0003}$
$\Omega_{m0}$ | $0.2529^{+0.0019}_{-0.0027}$ | $0.2519^{+0.0079}_{-0.0080}$ | $0.2268^{+0.0073}_{-0.0071}$
$H_{0}$ (km/s/Mpc) | $68.39^{+0.20}_{-0.22}$ | $68.29^{+0.72}_{-0.71}$ | $73.15^{+0.74}_{-0.75}$
$\zeta_{0}/H_{0}$ | - | $-0.0014^{+0.0013}_{-0.0038}$ | $-0.0195^{+0.0077}_{-0.0076}$
$\zeta_{1}$ | - | $0.0001^{+0.0008}_{-0.0003}$ | $0.0029^{+0.0010}_{-0.0010}$
Table 1: Best-fitted parameters for $\Lambda$CDM, v$\Lambda$CDM, and our
viscous model in $f(T)$ gravity. Figure 1: Constraints on the parameters from
$1\sigma$ to $2\sigma$ confidence level obtained by the fitting procedure.

##  IV Cosmological discussions

###  IV.1 Matter density parameters

As shown in Table 1, the two viscous coefficients in our model, $\zeta_{0}$
and $\zeta_{1}$, have different signs, which, according to Eq. (10), means the
effective pressures $\bar{p}$ may have different signs depending on the
densities. And Eq. (5) shows that the signs of $\bar{p}$ indicate the
direction of energy transference. So, the matters may acquire or lose energy
due to the universe expansion through viscosity at different cosmic time.
Moreover, the viscous model (10) suggests that the baryonic and dark matters
will acquire or lose energy at different rates once they have different
densities. Figures 2 and 3 illustrate the evolutions of the effective
pressures and density parameters of the baryonic and dark matters, where in
the latter the standard $\Lambda$CDM case has also been plotted for reference.

Figure 2: Evolutions of the effective pressures of the total matter, baryonic
matter and dark matter, plotted in unit of $H_{0}^{2}$. Figure 3: Evolutions
of density parameters $\Omega_{b}$ and $\Omega_{m}$.

At early time (large redshift $z$), the effective pressures are negative and
the matters acquire energy from the expansion, while at later time (smaller
redshift $z$), the pressures are positive and the matters lose energy to the
cosmic expansion. The changes of signs of the effective pressures happen at
$z=5.54462,\>8.87689$, and $4.77607$ for total matter, baryonic matter, and
dark matter, respectively. In the future ($-1<z<0$), the effective pressures
will return to zero because the matters dissipate and the densities vanish
successively, which will happen at $z=-0.803174$, and $-0.884895$ for the
baryonic and dark matters, respectively.

As mentioned before, the different paces of evolutions of the baryonic and
dark matters will lead to an evolving ratio $K$ between the baryonic and dark
matters, which is plotted in Fig. 4.

Figure 4: Evolution of the ratio between the baryonic and dark matter density
parameters.

In the future, when $\Omega_{b}$ reaches zero at $z=-0.803174$, $K$ vanishes.
In the remote past, $K=\Omega_{b}/\Omega_{m}=0.2009$, which is in fact
inferred from the power spectrum of CMB. On the other hand, while $K$ is
difficult to observe, the baryon-to-photon ratio $\eta$ plays a significant
role in the light element abundances. And since the photon number in the
universe can be directly observed, the visible difference in the evolutions of
$\Omega_{b}$ between our model and the standard $\Lambda$CDM model may then be
subjected to the test of observations. There are several epochs of the cosmic
evolutionary history where $\eta$ can be determined independently. For the Big
Bang nucleosynthesis epoch ($z\sim 10^{9}$), $\eta$ can be derived from the
abundances of primordial elements (see, e.g., [72]). For the recombination
epoch ($z\sim 1100$), inference of $\eta$ from CMB in fact has already been
taken into account in this work when we include the CMB and BAO datasets. At
later time ($2\lesssim z\lesssim 6$), the abundances of light elements, and
hence $\eta$, can be determined from the so-called Lyman $\alpha$ forest (see,
e.g., Refs. [73, 74, 75]), where for the standard model, the cosmic gas doing
the absorption is considered nearly primordial in that it has not been
processed during many generations of stars. In our model, the energy
transference between the baryonic matter and the cosmic expansion leads to the
deviation of $\Omega_{b}$ from the standard model, which may result in
possibly observable difference of such absorption.

Most of these results rely on the modeling of the evolutionary history as a
whole so that one can see the effects of the unified form of viscosities in
different eras. Moreover, the different evolutionary paces of the baryonic and
dark matters essentially come from the separation of viscosities of the two
matters in our model, which makes it possible to discern or even observe the
effects of viscosities. However, the time-dependent ratio between the baryonic
and dark matters is not a special result of simply setting the same form of
bulk viscosities for them. In fact, as long as the baryonic and dark matters
do not interact through viscosity (so that dark matter is still dark and
prerecombination BAO can happen), they most likely feel the viscosity
differently, and acquire/lose energy at different paces. The model with
synchronized evolution and constant ratio is actually the special one and
possibly needs some fine tuning.

###  IV.2 Evolutionary history

Due to the rather stringent constraints from the CMB dataset [67, 68], the
best-fitted parameters for $\Lambda$CDM is quite close to the Planck CMB
results [68] even though we have included late time datasets of SNIa and Hz.
To check the evolutionary history of the universe, one can, as usually done in
the cosmological study of modified gravities, consider the extra geometric
term from the $f(T)$ modification as an effective fluid. Then, this $f(T)$
fluid has its own density, namely,

$\rho_{f(T)}\equiv\frac{\alpha}{4H^{2}}.$ (24)

Using the best-fitted parameters in Table 1, we illustrate the evolutions of
the density parameters $\Omega=\rho/3H^{2}$ in Fig. 5. The radiation
dominating era, matter dominating era, and late time accelerating era are
obviously recovered. Note that in the future, the density parameters of the
baryonic and dark matters will vanish in finite time due to the existence of
viscosities, as discussed in the previous subsection.

Figure 5: The evolutions of the density parameters. $\Omega_{f(T)}$ denotes
the effective density parameter of the $f(T)$ modification.

The EOS of the effective $f(T)$ fluid is given by

$w_{f(T)}=-1-\frac{1}{3}\frac{\mathrm{d}\ln\rho_{f(T)}}{\mathrm{d}\ln a},$
(25)

the evolution of which is illustrated in Fig. 6. At present time with $z=0$
and $\ln a=0$, $w_{f(T)}^{(0)}=-1.081$, whereas the EOS of matter
$w_{M}^{(0)}=0.140$. Or, equivalently, the density parameter of the $f(T)$
fluid $\Omega_{f(T)}=0.732$, while the contribution from the effective
pressures of the viscosities $\bar{p}_{M}/3H_{0}^{2}=0.012$. This suggests
that the main contribution to the accelerating expansion at present time comes
from the $f(T)$ modification. In fact, the viscosity model (10) we have
introduced in this work allows for consideration of the physics of the
prerecombination era, for which the observational datasets give rather
stringent constraints and suppress the possible magnitude of the viscosities.

At future infinity with $z\rightarrow-1$ and $\ln a\rightarrow+\infty$,
$w_{f(T)}$ approach to $-1$, indicating a de Sitter fate of the universe.
Before that, the matter densities will reach zero in the finite future.

Figure 6: The evolution of the EOS of the geometric term from $f(T)$
modification. The best-fitted parameters given in Table 1 have be used.

Therefore, we can conclude that the inclusion of bulk viscosities of the
matters and considering the $f(T)$ modification of gravity as the main drive
of the present acceleration will not violate the known history of the
universe. The density parameters of the baryonic and dark matters will vanish
in the finite future due to the viscosities. The universe will continue to
evolve without them, and will finally enter a de Sitter expansion phase under
the effect of $f(T)$ modification.

###  IV.3 $f(T)$ modification as an effective dark energy

Although the $f(T)$ model that drives the accelerated expansion is not
considered as a dynamical field in this work, it effectively plays the role of
dark energy and does evolve over the cosmic time. To discriminate different
time-varying models of dark energy or effective dark energy, the statefinder
parameters $(r,s)$ have been introduced [76]. This pair $(r,s)$ is defined as

$r=\frac{\dddot{a}}{aH^{3}},\quad s=\frac{r-1}{3\left(q-\frac{1}{2}\right)},$
(26)

with $q=-{a\ddot{a}}/{\dot{a}^{2}}$ being the deceleration parameter. Since
$H$ and $q$ are composed from the scale factor $a$ and its first and second
order of derivatives, the geometric parameter $r$ is then naturally related to
the next order of derivative. And $s$ is a combination of $r$ and $q$. Based
on the values of the pair $(r,s)$, the (effective) dark energy models can be
put into three categories:

  * $\bullet$

$r>1,\>s<0$, Chaplygin gas model;

  * $\bullet$

$r<1,\>s>0$, quintessence models;

  * $\bullet$

$r=1,\>s=0$, the standard $\Lambda$CDM model.

Using the best-fitted parameters in Table 1, we plot the $r$-$s$ diagram of
our model in Fig. 7, where we also plot the specific quintessence and
Chaplygin gas models described in Ref. [76] for comparison. One can see that
our model mimics a kind of Chaplygin gas model, and eventually it will arrive
at the $\Lambda$CDM point $(1,0)$. Therefore, the late time behavior described
in the previous subsection, i.e., that the universe is currently experiencing
a phantom-like dark energy but eventually will enter a de Sitter phase, is
originated from the feature of the $f(T)$ model.

Figure 7: The statefinder pair $(r,s)$ for our model, as well as a specific
quintessence model and a Chaplygin gas model described in [76]. The solid
rounded points on the lines indicate the present states. The solid rounded
point at $(0,1)$ denotes the $\Lambda$CDM model.

Another useful tool to tell apart time-dependent dark energy models is the
$Om$ diagnostic [77]. $Om$ is defined as

$Om\left(z\right)=\frac{H^{2}\left(z\right)/H_{0}^{2}-1}{\left(1+z\right)^{3}-1},$
(27)

which provides a null test of the (effective) dark energy being a cosmological
constant $\Lambda$. The sloping of the trajectory on the $Om$-$z$ diagram is
used to categorize the different types of (effective) dark energy:

  * $\bullet$

a negative slope indicates a quintessence type of dark energy;

  * $\bullet$

a positive slope indicates a phantom-like dark energy;

  * $\bullet$

$\Lambda$CDM model gives a horizontal line.

In Fig. 8 we plot the $Om$-$z$ diagram of our model, as well as those of a
specific quintessence model and a Chaplygin gas model described in Ref. [76]
for comparison. Obviously, the viscous $f(T)$ model under consideration
provides a phantom-like effective dark energy like the Chaplygin gas model,
which is in accordance with the effective EOS $w_{f(T)}$ shown in Fig. 6.

Figure 8: $Om$-$z$ diagram for our model, as well as a specific quintessence
model and a Chaplygin gas model described in [76] for contrast.

###  IV.4 Hubble tension

Recently, more precise observations seem to reveal that there are
discrepancies among the values of the cosmological parameters fitted by the
different datasets (see, e.g., Ref. [50] for a historical review). One of the
most shocking tensions is between the inferences of $H_{0}$ from the Planck
CMB data, reporting $H_{0}=67.4\pm 0.5\>\mathrm{km/(s\>Mpc)}$ [68], and that
from local distance ladder measurements by Hubble Space Telescope (HST) which
reports $H_{0}=74.03\pm 1.42\>\mathrm{km/(s\>Mpc)}$ [78]. The two different
inferred values of $H_{0}$ are at a $4.4\sigma$ difference. From Table 1,
another immediate observation is that in our model $H_{0}$ is raised to a
value close to the astrophysical measurements by HST. Comparing with the
v$\Lambda$CDM column, one can see that adding bulk viscosity corrections to
the material contents in the $\Lambda$CDM does not help with this tension very
much. A semi-quantitative explanation is that an extra geometric term in the
form $1/T\sim 1/H^{2}$ makes the sloping of $H$ over the cosmic time more
gently than a dark content with a constant energy density. This changes both
the sound horizon $r_{s}^{*}$ given in (26) and the angular diameter distance
to the last-scattering surface

$d_{A}^{*}=\int_{0}^{z^{*}}\frac{\mathrm{d}z}{H(z)},$ (28)

since both of which are integrands of $1/H$ over the cosmic time (represented
by the redshift $z$). While the peak spacing of the CMB spectrum determines
quite precisely the ratio $\theta^{*}=r_{s}^{*}/d_{A}^{*}$, a more gently
sloping $H$ may then lead to a larger $H_{0}$. In this sense, the alleviation
of the Hubble tension in our model in fact comes from the $1/T$ term of
modification. In the current work, the new model of viscosity allows one to
describe the early and late time evolution as a whole, which makes it possible
to compare the local and global measurements of $H_{0}$ and hence to discuss
Hubble tension problem. However, the viscosity terms that are not negligible
in this model also change the physics before the last scattering a little bit,
giving a slightly smaller sound speed of the photon-baryon fluid and hence a
smaller sound horizon $r_{s}^{*}$, which in turn slightly lower the $H_{0}$
value. Nonetheless, the fitting result shows that the inclusion of viscosities
does not diminish the model’s successful alleviation of the Hubble tension.

##  V Concluding remarks

In this work, we have studied the bulk viscous cosmology under the framework
of $f(T)$ gravity. In order to consider the whole evolutionary history
including both the early time prerecombination physics and late time
acceleration, we have proposed a new simple viscous model for the baryonic and
dark matters that absorbs the merits of the viscous models in the literature.
Due to the stringent constraints from the observations, viscosities are not
likely significant enough to account for the accelerating expansion of the
universe, we therefore treat the viscosities as the corrections of
imperfectness to the fluid models of the baryonic and dark matters, and leave
the major drive of the acceleration to the $f(T)$ modification.

Using the CMB, BAO, Hz, and the Pantheon compilation of SNIa data, we have
performed an observational fit for the cosmological and model parameters. The
fitting result shows that the two viscous coefficients in our model have
different signs, which means both the baryonic and dark matters will lose or
acquire energy due to the viscosities at different times. Since the viscosity
model considered in this work is related to the material density, the baryonic
and dark matters will have different rates of energy transference, and the
ratio between the densities of them will not be a constant. Both the baryonic
and dark matters acquire energy to the cosmic expansion at early time, while
they lose energy at late time. The changes of directions of energy
transference happen at redshifts $z=8.87689$, and $4.77607$ for the baryonic
and dark matters, respectively. In the future, they will vanish in finite time
due to dissipation, which will happen at $z=-0.803174$ and $-0.884895$ for the
baryonic and dark matters, respectively.

With the fitting result, it is shown that our model can recover the known
history of the universe including the radiation dominating era, matter
dominating era, and the acceleration era. The main contribution to the
accelerating expansion comes from the $f(T)$ fluid of the modification. The
diagnostic analyses show that our model mimics a phantom-like dark energy.
Nonetheless, at the future infinity, the universe will eventually enter a de
Sitter expansion phase.

The Hubble constant $H_{0}$ is fitted to be $73.15$ ( km/s/Mpc ),
significantly alleviating the Hubble tension. This is mainly because the
$f(T)$ modification makes the sloping of $H$ over the cosmic time more gently,
which raises the value of $H_{0}$ while fitting the CMB peak spacings. The
viscosity in fact acts in an opposite direction in this problem, namely, it
makes the sound horizon $r_{s}^{*}$ slightly smaller and hence, lowers $H_{0}$
a little bit.

Conceptually, viscosity arises from the interaction between material
particles. However, a viscosity model that contains only the geometric part,
e.g., Eq. (6), will lead to a corresponding term in the field equation that is
purely geometrical, which can in effect be interpreted as some modification to
the spacetime geometry. That is, starting from some modified gravity model
without viscosity, one may arrive in the same field equation. Or, one can
understand this the other way around. The viscosity may be yet another
physical interpretation of modified gravity. Similarly, a viscous model
involves both the energy of the matter and the spacetime geometry can also be
interpreted as some modified gravity that considers nonminimal interactions
between matter and gravity. For example, a nonminimal coupling $f(T)$ model
without viscosity (see, e.g., Refs. [63, 64]) may have the same behavior as
the viscous model considered in this work. This is also referred to as the
degeneracy of the viscous cosmologies [41].

Nonetheless, viscosity is one of the realistic ways to consider material
interactions of the cosmic contents. The viscous model we employ in this work,
although based on the models in the literature, is chosen for simplicity. And
we have also assumed that it has the same form and has the same coefficients
for both the baryonic and dark matters. A more sophistic model may start from
the thermodynamic theory that takes the dark matter model into account, and
should leave the viscous coefficients all to the observational fit. Moreover,
beyond this study of background evolution of the cosmological model with
viscosities in $f(T)$ gravity, the growth of perturbation should also be
interesting once viscosity is not negligible [27]. Future works can be done
considering these issues.

## Acknowledgement

This work is supported by the National Science Foundation of China under Grant
No. 12105179.

## References

  * Copeland  _et al._ [2006] E. J. Copeland, M. Sami, and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006), arXiv:hep-th/0603057 .
  * Bahamonde  _et al._ [2018] S. Bahamonde, C. G. Böhmer, S. Carloni, E. J. Copeland, W. Fang, and N. Tamanini, Dynamical systems applied to cosmology: dark energy and modified gravity, Phys. Rept. 775-777, 1 (2018), arXiv:1712.03107 [gr-qc] .
  * Frusciante and Perenon [2020] N. Frusciante and L. Perenon, Effective field theory of dark energy: A review, Phys. Rept. 857, 1 (2020), arXiv:1907.03150 [astro-ph.CO] .
  * Ishak [2019] M. Ishak, Testing General Relativity in Cosmology, Living Rev. Rel. 22, 1 (2019), arXiv:1806.10122 [astro-ph.CO] .
  * De Felice and Tsujikawa [2010] A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13, 3 (2010), arXiv:1002.4928 [gr-qc] .
  * Sotiriou and Faraoni [2010] T. P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82, 451 (2010), arXiv:0805.1726 [gr-qc] .
  * Capozziello and De Laurentis [2011] S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept. 509, 167 (2011), arXiv:1108.6266 [gr-qc] .
  * Nojiri and Odintsov [2011] S. Nojiri and S. D. Odintsov, Unified cosmic history in modified gravity: from F(R) theory to Lorentz non-invariant models, Phys. Rept. 505, 59 (2011), arXiv:1011.0544 [gr-qc] .
  * Aldrovandi and Pereira [2013] R. Aldrovandi and J. G. Pereira,  _Teleparallel Gravity: An Introduction_, Vol. 173 (Springer, 2013).
  * Maluf [2013] J. W. Maluf, The teleparallel equivalent of general relativity, Annalen der Physik 525, 339 (2013), https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.201200272 .
  * Bengochea and Ferraro [2009] G. R. Bengochea and R. Ferraro, Dark torsion as the cosmic speed-up, Phys. Rev. D 79, 124019 (2009).
  * Linder [2010] E. V. Linder, Einstein’s other gravity and the acceleration of the universe, Phys. Rev. D 81, 127301 (2010).
  * Cai  _et al._ [2016] Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. 79, 106901 (2016), arXiv:1511.07586 [gr-qc] .
  * Nojiri  _et al._ [2017] S. Nojiri, S. Odintsov, and V. Oikonomou, Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution, Phys. Rept. 692, 1 (2017), arXiv:1705.11098 [gr-qc] .
  * Eckart [1940] C. Eckart, The Thermodynamics of irreversible processes. 3.. Relativistic theory of the simple fluid, Phys. Rev. 58, 919 (1940).
  * Israel [1976] W. Israel, Nonstationary irreversible thermodynamics: A Causal relativistic theory, Annals Phys. 100, 310 (1976).
  * Hiscock and Salmonson [1991] W. A. Hiscock and J. Salmonson, Dissipative Boltzmann-Robertson-Walker cosmologies, Phys. Rev. D 43, 3249 (1991).
  * Ren and Meng [2006a] J. Ren and X.-H. Meng, Modified equation of state, scalar field and bulk viscosity in friedmann universe, Phys. Lett. B 636, 5 (2006a), arXiv:astro-ph/0602462 .
  * Meng  _et al._ [2007] X.-H. Meng, J. Ren, and M.-G. Hu, Friedmann cosmology with a generalized equation of state and bulk viscosity, Commun. Theor. Phys. 47, 379 (2007), arXiv:astro-ph/0509250 .
  * Cataldo  _et al._ [2005] M. Cataldo, N. Cruz, and S. Lepe, Viscous dark energy and phantom evolution, Phys. Lett. B 619, 5 (2005), arXiv:hep-th/0506153 .
  * Brevik and Gorbunova [2005] I. H. Brevik and O. Gorbunova, Dark energy and viscous cosmology, Gen. Rel. Grav. 37, 2039 (2005), arXiv:gr-qc/0504001 .
  * Wilson  _et al._ [2007] J. R. Wilson, G. J. Mathews, and G. M. Fuller, Bulk viscosity, decaying dark matter, and the cosmic acceleration, Phys. Rev. D 75, 043521 (2007).
  * Colistete  _et al._ [2007] R. Colistete, J. C. Fabris, J. Tossa, and W. Zimdahl, Bulk viscous cosmology, Phys. Rev. D 76, 103516 (2007).
  * Li and Barrow [2009] B. Li and J. D. Barrow, Does bulk viscosity create a viable unified dark matter model?, Phys. Rev. D 79, 103521 (2009).
  * Avelino and Nucamendi [2010] A. Avelino and U. Nucamendi, Exploring a matter-dominated model with bulk viscosity to drive the accelerated expansion of the Universe, JCAP 08, 009, arXiv:1002.3605 [gr-qc] .
  * Brevik  _et al._ [2011] I. Brevik, E. Elizalde, S. Nojiri, and S. D. Odintsov, Viscous little rip cosmology, Phys. Rev. D 84, 103508 (2011).
  * Gagnon and Lesgourgues [2011] J.-S. Gagnon and J. Lesgourgues, Dark goo: Bulk viscosity as an alternative to dark energy, JCAP 09, 026, arXiv:1107.1503 [astro-ph.CO] .
  * Fabris  _et al._ [2011] J. C. Fabris, P. L. C. de Oliveira, and H. Velten, Contraints on unified models for dark matter and dark energy using H(z), Eur. Phys. J. C 71, 1773 (2011), arXiv:1106.0645 [astro-ph.CO] .
  * Okumura and Yonezawa [2003] H. Okumura and F. Yonezawa, New expression of the bulk viscosity, Physica A: Statistical Mechanics and its Applications 321, 207 (2003), statphys-Taiwan-2002: Lattice Models and Complex Systems.
  * Hu and Meng [2006] M.-G. Hu and X.-H. Meng, Bulk viscous cosmology: statefinder and entropy, Phys. Lett. B 635, 186 (2006), arXiv:astro-ph/0511615 .
  * Feng and Li [2009] C.-J. Feng and X.-Z. Li, Viscous Ricci Dark Energy, Phys. Lett. B 680, 355 (2009), arXiv:0905.0527 [astro-ph.CO] .
  * Setare and Sheykhi [2010] M. R. Setare and A. Sheykhi, Viscous dark energy and generalized second law of thermodynamics, Int. J. Mod. Phys. D 19, 1205 (2010), arXiv:1103.1067 [physics.gen-ph] .
  * Hipólito-Ricaldi  _et al._ [2010] W. S. Hipólito-Ricaldi, H. E. S. Velten, and W. Zimdahl, Viscous dark fluid universe, Phys. Rev. D 82, 063507 (2010).
  * Montiel and Breton [2011] A. Montiel and N. Breton, Probing bulk viscous matter-dominated models with Gamma-ray bursts, JCAP 08, 023, arXiv:1107.0271 [astro-ph.CO] .
  * Wang  _et al._ [2017] D. Wang, Y.-J. Yan, and X.-H. Meng, Constraining viscous dark energy models with the latest cosmological data, Eur. Phys. J. C 77, 660 (2017), arXiv:2103.14788 [astro-ph.CO] .
  * Brevik  _et al._ [2017] I. Brevik, O. Grøn, J. de Haro, S. D. Odintsov, and E. N. Saridakis, Viscous Cosmology for Early- and Late-Time Universe, Int. J. Mod. Phys. D 26, 1730024 (2017), arXiv:1706.02543 [gr-qc] .
  * da Silva and Silva [2021] W. J. C. da Silva and R. Silva, Growth of matter perturbations in the extended viscous dark energy models, Eur. Phys. J. C 81, 403 (2021), arXiv:2011.09516 [astro-ph.CO] .
  * Velten and Schwarz [2012] H. Velten and D. J. Schwarz, Dissipation of dark matter, Phys. Rev. D 86, 083501 (2012).
  * Brevik [2012] I. Brevik, Viscosity in modified gravity, Entropy 14, 2302 (2012), arXiv:1211.2562 [gr-qc] .
  * Singh and Kumar [2014] C. P. Singh and P. Kumar, Friedmann model with viscous cosmology in modified $f(R,T)$ gravity theory, Eur. Phys. J. C 74, 3070 (2014), arXiv:1406.4258 [gr-qc] .
  * Velten  _et al._ [2013] H. Velten, J. Wang, and X. Meng, Phantom dark energy as an effect of bulk viscosity, Phys. Rev. D 88, 123504 (2013), arXiv:1307.4262 [astro-ph.CO] .
  * Hu and Hu [2020] J. Hu and H. Hu, Viscous universe with cosmological constant, Eur. Phys. J. Plus 135, 718 (2020).
  * Arora  _et al._ [2020] S. Arora, X.-h. Meng, S. K. J. Pacif, and P. K. Sahoo, Effective equation of state in modified gravity and observational constraints, Class. Quant. Grav. 37, 205022 (2020), arXiv:2007.07717 [gr-qc] .
  * Sharif and Rani [2013] M. Sharif and S. Rani, Viscous Dark Energy in $f(T)$ Gravity, Mod. Phys. Lett. A 28, 1350118 (2013), arXiv:1405.5232 [gr-qc] .
  * Jawad  _et al._ [2016] A. Jawad, S. Chattopadhyay, and S. Rani, Viscous pilgrim $f(T)$ gravity models, Astrophys. Space Sci. 361, 231 (2016).
  * Davood Sadatian [2019] S. Davood Sadatian, Effects of viscous content on the modified cosmological F(T) model, EPL 126, 30004 (2019).
  * Gadbail  _et al._ [2021] G. N. Gadbail, S. Arora, and P. K. Sahoo, Viscous cosmology in the Weyl-type f(Q, T) gravity, Eur. Phys. J. C 81, 1088 (2021), arXiv:2110.02726 [gr-qc] .
  * Arora  _et al._ [2022] S. Arora, S. K. J. Pacif, A. Parida, and P. K. Sahoo, Bulk viscous matter and the cosmic acceleration of the universe in f(Q,T) gravity, JHEAp 33, 129 (2022), arXiv:2106.00491 [gr-qc] .
  * Saha and Chattopadhyay [2022] S. Saha and S. Chattopadhyay, Viscous generalised Chaplygin gas under the purview of f(T) gravity and the model assessment through probabilistic information theory, Phys. Scripta 97, 045006 (2022).
  * Freedman [2017] W. L. Freedman, Cosmology at a Crossroads, Nature Astron. 1, 0121 (2017), arXiv:1706.02739 [astro-ph.CO] .
  * Evslin  _et al._ [2018] J. Evslin, A. A. Sen, and Ruchika, Price of shifting the hubble constant, Phys. Rev. D 97, 103511 (2018).
  * Aylor  _et al._ [2019] K. Aylor, M. Joy, L. Knox, M. Millea, S. Raghunathan, and W. L. K. Wu, Sounds Discordant: Classical Distance Ladder & $\Lambda$CDM -based Determinations of the Cosmological Sound Horizon, Astrophys. J. 874, 4 (2019), arXiv:1811.00537 [astro-ph.CO] .
  * Verde  _et al._ [2019] L. Verde, T. Treu, and A. G. Riess, Tensions between the Early and the Late Universe, Nature Astron. 3, 891 (2019), arXiv:1907.10625 [astro-ph.CO] .
  * Ren and Meng [2006b] J. Ren and X.-H. Meng, Cosmological model with viscosity media (dark fluid) described by an effective equation of state, Phys. Lett. B 633, 1 (2006b), arXiv:astro-ph/0511163 .
  * Sasidharan  _et al._ [2018] A. Sasidharan, N. D. J. Mohan, M. V. John, and T. K. Mathew, Bayesian analysis of bulk viscous matter dominated universe, Eur. Phys. J. C 78, 628 (2018), arXiv:1803.08235 [gr-qc] .
  * Capozziello  _et al._ [2006] S. Capozziello, V. F. Cardone, E. Elizalde, S. Nojiri, and S. D. Odintsov, Observational constraints on dark energy with generalized equations of state, Phys. Rev. D 73, 043512 (2006).
  * Odintsov  _et al._ [2020] S. D. Odintsov, D. S.-C. Gómez, and G. S. Sharov, Testing the equation of state for viscous dark energy, Phys. Rev. D 101, 044010 (2020).
  * Padmanabhan and Chitre [1987] T. Padmanabhan and S. M. Chitre, Viscous universes, Phys. Lett. A 120, 433 (1987).
  * Szydlowski and Hrycyna [2007] M. Szydlowski and O. Hrycyna, Dissipative or conservative cosmology with dark energy?, Annals Phys. 322, 2745 (2007), arXiv:astro-ph/0602118 .
  * Hipolito-Ricaldi  _et al._ [2009] W. S. Hipolito-Ricaldi, H. E. S. Velten, and W. Zimdahl, Non-adiabatic dark fluid cosmology, JCAP 06, 016, arXiv:0902.4710 [astro-ph.CO] .
  * Mohan  _et al._ [2017] N. D. J. Mohan, A. Sasidharan, and T. K. Mathew, Bulk viscous matter and recent acceleration of the universe based on causal viscous theory, Eur. Phys. J. C 77, 849 (2017), arXiv:1708.02437 [gr-qc] .
  * Feng  _et al._ [2015] C.-j. Feng, F.-f. Ge, X.-z. Li, R.-h. Lin, and X.-h. Zhai, Towards realistic $f(T)$ models with nonminimal torsion-matter coupling extension, Phys. Rev. D 92, 104038 (2015), arXiv:1511.07935 [gr-qc] .
  * Lin  _et al._ [2017] R.-H. Lin, X.-H. Zhai, and X.-Z. Li, Solar system tests for realistic $f(T)$ models with non-minimal torsion–matter coupling, Eur. Phys. J. C 77, 504 (2017), arXiv:1610.04956 [gr-qc] .
  * Lin  _et al._ [2018] R.-H. Lin, Q. Wen, X.-H. Zhai, and X.-Z. Li, Diagnostics for generalized power-law torsion–matter coupling $f(T)$ model, Int. J. Mod. Phys. D 28, 1950031 (2018), arXiv:1803.04134 [gr-qc] .
  * Scolnic  _et al._ [2018] D. M. Scolnic _et al._ (Pan-STARRS1), The Complete Light-curve Sample of Spectroscopically Confirmed SNe Ia from Pan-STARRS1 and Cosmological Constraints from the Combined Pantheon Sample, Astrophys. J. 859, 101 (2018), arXiv:1710.00845 [astro-ph.CO] .
  * Sharov and Vasiliev [2018] G. S. Sharov and V. O. Vasiliev, How predictions of cosmological models depend on Hubble parameter data sets, Math. Model. Geom. 6, 1 (2018), arXiv:1807.07323 [gr-qc] .
  * Betoule  _et al._ [2014] M. Betoule _et al._ (SDSS), Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples, Astron. Astrophys. 568, A22 (2014), arXiv:1401.4064 [astro-ph.CO] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Beutler  _et al._ [2011] F. Beutler, C. Blake, M. Colless, D. H. Jones, L. Staveley-Smith, L. Campbell, Q. Parker, W. Saunders, and F. Watson, The 6dF Galaxy Survey: Baryon Acoustic Oscillations and the Local Hubble Constant, Mon. Not. Roy. Astron. Soc. 416, 3017 (2011), arXiv:1106.3366 [astro-ph.CO] .
  * Padmanabhan  _et al._ [2012] N. Padmanabhan, X. Xu, D. J. Eisenstein, R. Scalzo, A. J. Cuesta, K. T. Mehta, and E. Kazin, A 2 per cent distance to $z$=0.35 by reconstructing baryon acoustic oscillations - I. Methods and application to the Sloan Digital Sky Survey, Mon. Not. Roy. Astron. Soc. 427, 2132 (2012), arXiv:1202.0090 [astro-ph.CO] .
  * Anderson  _et al._ [2013] L. Anderson _et al._ , The clustering of galaxies in the SDSS-III Baryon Oscillation Spectroscopic Survey: Baryon Acoustic Oscillations in the Data Release 9 Spectroscopic Galaxy Sample, Mon. Not. Roy. Astron. Soc. 427, 3435 (2013), arXiv:1203.6594 [astro-ph.CO] .
  * Sarkar [2004] S. Sarkar, Measuring the baryon content of the universe: BBN vs CMB, in  _13th Rencontres de Blois on Frontiers of the Universe_ (2004) pp. 53–63, arXiv:astro-ph/0205116 .
  * Tytler  _et al._ [2000] D. Tytler, J. M. O’Meara, N. Suzuki, and D. Lubin, Deuterium and the baryonic density of the universe, Physics Reports 333-334, 409 (2000).
  * Thuan and Izotov [2002] T. X. Thuan and Y. I. Izotov, The primordial helium-4 abundance determination: systematic effects, Space Sci. Rev. 100, 263 (2002), arXiv:astro-ph/0112348 .
  * Coc  _et al._ [2002] A. Coc, E. Vangioni-Flam, M. Casse, and M. Rabiet, Constraints on Omega b from nucleosynthesis of Li-7 in the standard big bang model, Phys. Rev. D 65, 043510 (2002), arXiv:astro-ph/0111077 .
  * Sahni  _et al._ [2003] V. Sahni, T. D. Saini, A. A. Starobinsky, and U. Alam, Statefinder: A New geometrical diagnostic of dark energy, JETP Lett. 77, 201 (2003), arXiv:astro-ph/0201498 .
  * Sahni  _et al._ [2008] V. Sahni, A. Shafieloo, and A. A. Starobinsky, Two new diagnostics of dark energy, Phys. Rev. D 78, 103502 (2008), arXiv:0807.3548 [astro-ph] .
  * Riess  _et al._ [2019] A. G. Riess, S. Casertano, W. Yuan, L. M. Macri, and D. Scolnic, Large Magellanic Cloud Cepheid Standards Provide a 1% Foundation for the Determination of the Hubble Constant and Stronger Evidence for Physics beyond $\Lambda$CDM, Astrophys. J. 876, 85 (2019), arXiv:1903.07603 [astro-ph.CO] .