# Cosmic acceleration with bulk viscosity in an anisotropic $f(R,L_{m})$
background

Raja Solanki 0000-0001-8849-7688<EMAIL_ADDRESS>Department of
Mathematics, Birla Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. Bina Patel 0000-0002-3079-0061
<EMAIL_ADDRESS>Department of Mathematical Sciences, P.D. Patel
Institute of Applied Sciences, Charotar University of Science and Technology
(CHARUSAT), Changa, Anand 388421, Gujarat, India Lakhan V. Jaybhaye
0000-0003-1497-276X<EMAIL_ADDRESS>Department of Mathematics, Birla
Institute of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India. P.K. Sahoo 0000-0003-2130-8832
<EMAIL_ADDRESS>Department of Mathematics, Birla Institute
of Technology and Science-Pilani,
Hyderabad Campus, Hyderabad-500078, India.

###### Abstract

In this article, we investigate the observed cosmic acceleration in the
framework of a cosmological $f(R,L_{m})$ model dominated by bulk viscous
matter in an anisotropic background. We consider the LRS Bianchi type I metric
and derive the Friedmann equations that drive the gravitational interactions
in $f(R,L_{m})$ gravity. Further, we assume the functional form
$f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha}$, where $\alpha$ is a free model
parameter, and then find the exact solutions of fields equations corresponding
to our viscous matter dominated model. We incorporate the updated H(z) data
and the Pantheon data to acquire the best-fit values of parameters of our
model by utilizing the $\chi^{2}$ minimization technique along with the Markov
Chain Monte Carlo (MCMC) random sampling method. Further, we present the
behavior of physical parameters that describe the universe’s evolution phase,
such as density, effective pressure and EoS parameters, skewness parameter,
and the statefinder diagnostic parameters. We found that the energy density
indicates expected positive behavior, whereas the negative behavior of bulk
viscous pressure contributes to the universe’s expansion. The effective EoS
parameter favors the accelerating phase of the universe’s expansion. Moreover,
the skewness parameter shows the anisotropic nature of spacetime during the
entire evolution phase of the universe. Finally, from statefinder diagnostic
test, we found that our cosmological $f(R,L_{m})$ model lies in the
quintessence region, and it behaves like a de-Sitter universe in the far
future. We analyze different energy conditions in order to test the
consistency of the obtained solution. We found that all energy conditions
except SEC show positive behavior, while the violation of SEC favors the
recently observed acceleration with the transition from decelerated to an
accelerated epoch of the universe’s expansion in the recent past.

Keywords: $f(R,L_{m})$ gravity, LRS Bianchi type I metric, bulk viscosity,
skewness parameter, equation of state parameter, statefinder parameter, and
energy conditions.

##  I Introduction

The universe in the beginning and future may have anisotropies despite the
widely accepted assumption that the current universe is homogeneous and
isotropic. In 1992, Cosmic Background Explorer [1] revealed that the cosmic
microwave background possesses a small amount of anisotropy at cosmological
scales. The observational evidence, such as Wilkinson Microwave Anisotropy
Probe experiments [2] and Cosmic Background Imager [3], strongly support this
anisotropic nature of spacetime geometry. In 1998, the Supernovae Cosmology
Project led by Perlmutter and Riess confirmed the accelerating behavior of the
universe’s expansion [4, 5]. Moreover, recent advancements indicate the
variations in the strengths of microwaves received from different directions,
i.e., the universe’s expansion is anisotropic [6]. One can efficiently
describe the homogeneous and anisotropic nature of spacetime using Bianchi-
type cosmology. Several models based on Bianchi cosmology have appeared in the
literature [7, 8, 9, 10, 11, 12, 13]. In the present work, we incorporate
locally rotationally symmetric (LRS) Bianchi type-I metric that is near to the
FLRW metric and can be considered as a generalization of the isotropic one.

It is well known that the modified theories of gravity are capable of
bypassing the undetected dark energy issue. The cosmological models based on
the modifications of the Einstein action of general relativity (GR) that are
obtained by introducing the function $f(R)$ of the Ricci curvature $R$, first
appeared in [14, 15, 16]. This cosmological model could describe the observed
cosmic expansion without incorporating any dark energy component [17, 18].
Observational characteristics of $f(R)$ gravity models, along with the
constraints acquired by equivalence principle and solar system tests, are
investigated in the references [19, 20, 21, 22, 23]. The viable $f(R)$ gravity
models in the perspective of solar system tests do exist [24, 25, 26].
Odintsov et al. have studied energy conditions and the $H_{0}$ tension in the
context of $f(R)$ gravity models [27, 28]. One can follow references [29, 30,
31, 32, 33, 34, 35] to review some viable $f(R)$ gravity models.

Bertolami et al. [36] introduced a generalization of the $f(R)$ modified
gravity by assuming an explicit coupling between the generic $f(R)$ function
of the Ricci curvature scalar $R$ and the Lagrangian density of matter
$L_{m}$, in the Einstein-Hilbert action. T. Harko and F. S. N Lobo further
extended this model to the case of arbitrary geometry-matter couplings [37].
The cosmological models incorporating the non-minimal curvature and matter
couplings present some interesting applications in cosmology and astrophysics
[38, 39, 40]. Further, T. Harko and F. S. N Lobo proposed [41] $f(R,L_{m})$
gravity, which is the set of all curvature-matter coupling theories. In this
modified gravity, the energy-momentum tensor has the non-vanishing covariant
divergence, an extra force orthogonal to four velocities arises, and the
motion of the test particle is non-geodesic. Moreover, the $f(R,L_{m})$
modified theory of gravity does not satisfy the equivalence principle, and it
is constrained by the solar system tests [42, 43]. Nowadays, plenty of
interesting cosmological implications of $f(R,L_{m})$ gravity theory started
appearing in the literature; for instance, check references [44, 45, 46, 47,
48, 49, 50].

In this manuscript, we investigate the bulk viscous matter dominated
$f(R,L_{m})$ cosmological model with the anisotropic background. The
significance of viscosity coefficients in cosmological modeling has a long
history. From the hydrodynamics perspective, whenever a system loses its
thermal equilibrium state, an effective pressure is generated to retrieve its
thermal stability. Bulk viscosity in a matter content of the cosmos can be
viewed as a manifestation of such an effective pressure. We aim to investigate
the impact on the evolution phase of the universe when we consider the effect
of bulk viscosity coefficient $\zeta$ in the usual cosmic pressure. We assume
that the coefficient of viscosity $\zeta$ obeys a scaling law, and that
reduces the Einstein case to a form proportional to the Hubble parameter [51].
This scaling law considered in our study is quite useful. One can follow
references to review some attractive cosmological models incorporating the
viscosity in cosmic fluid [52, 53, 54, 55, 56, 57, 58].

The present work is organized as follows. In Sec II, we present the
fundamental formulations of $f(R,L_{m})$ gravity. In Sec III, we derive the
field equations corresponding to the LRS Bianchi type I metric. Further, we
consider a cosmological $f(R,L_{m})$ model and then obtain the Hubble
parameter value in terms of cosmic redshift. In Sec IV, we utilize the H(z)
and Pantheon observational data to estimate the model and bulk viscous
parameter values. In the next section Sec V, we present the physical behavior
of cosmological parameters and statefinder diagnostic corresponding to the
values of parameters obtained by the H(z), Pantheon, and H(z)+Pantheon data.
In addition, in Sec VI, we investigate the different energy conditions.
Finally, in Sec VII, we discuss and conclude our results.

##  II  $f(R,L_{m})$ Gravity Theory

The action ruling the gravitational interactions in $f(R,L_{m})$ gravity is
given as

$S=\int{f(R,L_{m})\sqrt{-g}d^{4}x}.$ (1)

Here $f(R,L_{m})$ is the generic function of the Ricci curvature $R$ and the
matter Lagrangian $L_{m}$.

The Ricci curvature term $R$ can be obtained by the following contraction,

$R=g^{\mu\nu}R_{\mu\nu},$ (2)

where $R_{\mu\nu}$ is the Ricci tensor and it is defined as

$R_{\mu\nu}=\partial_{\lambda}\Gamma^{\lambda}_{\mu\nu}-\partial_{\mu}\Gamma^{\lambda}_{\lambda\nu}+\Gamma^{\lambda}_{\mu\nu}\Gamma^{\sigma}_{\sigma\lambda}-\Gamma^{\lambda}_{\nu\sigma}\Gamma^{\sigma}_{\mu\lambda},$
(3)

with $\Gamma^{\alpha}_{\beta\gamma}$ denoting the components of Levi-Civita
connection.

One can obtained the following field equation of $f(R,L_{m})$ gravity by
varying the action (1) with respect to the metric $g_{\mu\nu}$,

$f_{R}R_{\mu\nu}+(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu})f_{R}-\frac{1}{2}(f-f_{L_{m}}L_{m})g_{\mu\nu}=\frac{1}{2}f_{L_{m}}T_{\mu\nu},$
(4)

where $f_{R}\equiv\frac{\partial f}{\partial R}$,
$f_{L_{m}}\equiv\frac{\partial f}{\partial L_{m}}$, and $T_{\mu\nu}$ is the
energy momentum tensor given by

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}.$ (5)

One can acquired the following relation by contracting the field equation (4)

$Rf_{R}+3\square f_{R}-2(f-f_{L_{m}}L_{m})=\frac{1}{2}f_{L_{m}}T.$ (6)

Here $R$, $T$, and $L_{m}$ being the Ricci curvature scalar, energy-momentum
scalar, and the matter Lagrangian term respectively. Furthermore, $\square
F=\frac{1}{\sqrt{-g}}\partial_{\alpha}(\sqrt{-g}g^{\alpha\beta}\partial_{\beta}F)$
for any scalar function $F$ .

Moreover, by employing the covariant derivative in equation (4) one can obtain
the following relation

$\nabla^{\mu}T_{\mu\nu}=2\nabla^{\mu}ln(f_{L_{m}})\frac{\partial
L_{m}}{\partial g^{\mu\nu}}.$ (7)

##  III The Cosmological $f(R,L_{m})$ Model with Anisotropic Background

Taking into account anomalies found in the observations of CMB, we consider
following the LRS Bianchi type-I metric to describe the spatially homogeneous
and anisotropic nature of the universe

$ds^{2}=-dt^{2}+A^{2}(t)dx^{2}+B^{2}(t)(dy^{2}+dz^{2}),$ (8)

where $A(t)$ and $B(t)$ are time dependent metric potentials. In particular,
one can recover the usual flat FLRW background for $A(t)=B(t)=a(t)$. Now, for
the line element (8), we obtained the Ricci scalar as

$R=2(\dot{H_{x}}+2\dot{H_{y}})+2({H_{x}}^{2}+3{H_{y}}^{2})+4{H_{x}}{H_{y}}.$
(9)

Here $H_{x}=\frac{\dot{A}}{A}$ and $H_{y}=H_{z}=\frac{\dot{B}}{B}$ represents
directional Hubble parameters.

The stress-energy tensor characterizing the universe filled with viscous
content reads as

$T_{\mu\nu}=(\rho+\bar{p})u_{\mu}u_{\nu}+\bar{p}g_{\mu\nu},$ (10)

where $\bar{p}=p-3\zeta H$ denote the effective pressure of the cosmic fluid
in the presence of viscosity coefficient $\zeta$ and $\rho$ is the usual
matter energy density with four velocities components $u^{\mu}=(1,0,0,0)$.

The assumption of viscous effects in the cosmic fluid content can be viewed as
an attempt to refine its description, relaxing its ideal properties and
contributing negatively to the total pressure that can drive the
aforementioned cosmic late-time acceleration. This can be checked in
References [59, 60, 61].

The field equations governing the dynamics of the viscous matter dominated
universe in $f(R,L_{m})$ with anisotropic background reads as

$(2{H_{x}}{H_{y}}+{H_{y}}^{2})f_{R}+\frac{1}{2}(f-Rf_{R}-L_{m}f_{L_{m}})+\dot{f_{R}}(H_{x}+2H_{y})=\frac{1}{2}f_{L_{m}}\rho,$
(11)
$(\dot{H_{x}}+{H_{x}}^{2}+2{H_{x}}{H_{y}}){f_{R}}-\ddot{f_{R}}-\dot{f_{R}}({H_{x}}+{2H_{y}})-\frac{1}{2}(f-{L_{m}}{f_{L_{m}}})=\frac{1}{2}f_{L_{m}}\bar{p},$
(12)
$(\dot{H_{y}}+2{H_{y}}^{2}+{H_{x}}{H_{y}}){f_{R}}-\ddot{f_{R}}-\dot{f_{R}}(H_{x}+{2H_{y}})-\frac{1}{2}(f-{L_{m}}{f_{L_{m}}})=\frac{1}{2}f_{L_{m}}(\bar{p}+\delta\rho).$
(13)

Here $\delta$ is a measure of deviation from isotropy along $y$ and $z$ axes,
called skewness parameter.

In this work, we study the following $f(R,L_{m})$ model in order to explore
the cosmological implications of the $f(R,L_{m})$ gravity [62, 63],

$f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha},$ (14)

where $\alpha$ is arbitrary parameter. The cosmological $f(R,L_{m})$ model
that we have considered is motivated by the generic function
$f(R,L_{m})=f_{1}(R)+f_{2}(R)G(L_{m})$ that represents arbitrary curvature-
matter coupling [63]. In particular, for $\alpha=1$ one can obtain usual
Friedmann equations of GR.

Now one can acquire the spatial volume element as

$V=AB^{2}.$ (15)

Then by using (15), the mean value of Hubble parameter can be calculated as

$H=\frac{1}{3}\frac{\dot{V}}{V}=\frac{1}{3}(H_{x}+2H_{y}).$ (16)

Further, we consider an extra ansatz to obtain the exact solution of the field
equations. We consider a physical relation between the expansion scalar
$\theta$ and shear scalar $\sigma$, specifically,
$\theta^{2}\propto\sigma^{2}$ . In particular, for
$\frac{\sigma}{\theta}=constant$, one can retrieve the isotropy of Hubble
expansion [64]. This gives rise to a following relation

$A=B^{n},$ (17)

and the corresponding directional Hubble parameter can be obtained as

$H_{x}=nH_{y},n\neq 0.$ (18)

For $n=1$, one can recover the usual flat FLRW cosmology.

Now, for the specific function (14) with $L_{m}=\rho$ [65], we obtained the
following Friedmann equations for non-relativistic viscous matter dominated
universe, by using equations (11)-(13)

$(2n+1){H_{y}}^{2}=(2\alpha-1){\rho}^{\alpha},$ (19)
$2\dot{H_{y}}+3{H_{y}}^{2}=(\alpha-1){\rho}^{\alpha}-{\alpha}{\rho^{\alpha-1}}\bar{p},$
(20)
$(n+1)\dot{H_{y}}+(n^{2}+n+1){H_{y}}^{2}=(\alpha-1){\rho}^{\alpha}-\alpha{\rho^{\alpha-1}}(\bar{p}+\delta\rho).$
(21)

By using equations (19) and (20), we obtain following first-order equation

$\dot{H_{y}}+\left[\frac{5\alpha-2+2n(1-\alpha)}{2(2\alpha-1)}\right]H_{y}^{2}={\frac{\alpha\zeta}{2}}(n+2)\bigg{(}\frac{2n+1}{2\alpha-1}\bigg{)}^{\frac{\alpha-1}{\alpha}}{H_{y}}^{\frac{3\alpha-2}{\alpha}}.$
(22)

On integrating the above equation, we have

$H_{y}=\biggl{\\{}H_{y_{0}}^{\frac{2-\alpha}{\alpha}}(1+z)^{\frac{(2-\alpha)[5\alpha-2+2n(1-\alpha)]}{2\alpha(2\alpha-1)}}+\frac{\alpha\xi(n+2)(2\alpha-1)^{\frac{1}{\alpha}}(2n+1)^{\frac{\alpha-1}{\alpha}}}{5\alpha-2+2n(1-\alpha)}\big{[}1-(1+z)^{{\frac{(2-\alpha)[5\alpha-2+2n(1-\alpha)]}{2\alpha(2\alpha-1)}}}\big{]}\biggr{\\}}^{\frac{\alpha}{2-\alpha}}.$
(23)

Here $H_{y}(0)=H_{y_{0}}$. Now by using equations (16) and (18), we obtained
the average value of the Hubble parameter as

$H=\big{(}\frac{n+2}{3}\big{)}H_{y}.$ (24)

By using equation (24) in (23), we obtained the expression for averaged Hubble
parameter in terms of redshift as

$H(z)=\big{(}\frac{n+2}{3}\big{)}\biggl{\\{}\big{(}\frac{3H_{0}}{n+2}\big{)}^{\frac{2-\alpha}{\alpha}}(1+z)^{\frac{(2-\alpha)[5\alpha-2+2n(1-\alpha)]}{2\alpha(2\alpha-1)}}+\frac{\alpha\xi(n+2)(2\alpha-1)^{\frac{1}{\alpha}}(2n+1)^{\frac{\alpha-1}{\alpha}}}{5\alpha-2+2n(1-\alpha)}\big{[}1-(1+z)^{{\frac{(2-\alpha)[5\alpha-2+2n(1-\alpha)]}{2\alpha(2\alpha-1)}}}\big{]}\biggr{\\}}^{\frac{\alpha}{2-\alpha}},$
(25)

where $H(0)=H_{0}$ is the present value of the Hubble parameter.

##  IV  Observational Constraints

In this section, we are going to investigate the viability of our theoretical
model by utilizing observational data sets. In this work, we incorporate
updated H(z) data sets and the Pantheon supernovae observations. In order to
perform statistical analysis, we use the $\chi^{2}$ minimization technique
along with the Markov Chain Monte Carlo (MCMC) random sampling method in the
emcee python library [66].

###  IV.0.1 H(z) datasets

It is well known that the expansion of the universe can be characterized by
the Hubble parameter. For early passively evolving galaxies, one can recover
the Hubble parameter values by measuring the $dt$ at a definite redshift
value. In this work, we use $57$ points of $H(z)$ data in the redshift range
$0.07\leq z\leq 2.41$ [67]. Moreover, there are two well-known approaches to
extracting the values of the Hubble parameter, namely the differential age
(DA) and line of sight BAO method. The set of $57$ H(z) data points can be
checked in the reference [68, 69]. Now, the chi-square function corresponding
to H(z) data points reads as

$\chi_{H}^{2}(H_{0},\alpha,\zeta,n)=\sum\limits_{k=1}^{57}\frac{[H_{th}(z_{k},H_{0},\alpha,\zeta,n)-H_{obs}(z_{k})]^{2}}{\sigma_{H(z_{k})}^{2}},$
(26)

where, the theoretically predicted value of the H(z) is denoted by $H_{th}$
whereas $H_{obs}$ represents the observed value of H(z) with standard error
$\sigma_{H(z_{k})}$. The contour plot for the model parameters $\alpha$,
$\zeta$, $n$, and $H_{0}$ corresponding to H(z) data sets at $1-\sigma$ and
$2-\sigma$ confidence interval is presented in Fig 1.

Figure 1: The $1-\sigma$ and $2-\sigma$ contour plot for the model parameters
using the H(z) data set.

The obtained best fit values are $\alpha=0.903^{+0.091}_{-0.10}$,
$\zeta=180.006^{+0.092}_{-0.10}$, $n=0.195\pm 0.096$, and $H_{0}=66.499\pm
0.097$.

###  IV.0.2 Pantheon datasets

In this work, we incorporate 1048 points of Pantheon supernovae samples with
corresponding distance moduli $\mu^{obs}$ in the redshift range
$z\in[0.01,2.3]$ [70]. The $\chi^{2}$ function corresponding to Pantheon data
sets is given as

$\chi^{2}_{SN}=\sum_{i,j=1}^{1048}\bigtriangledown\mu_{i}\left(C^{-1}_{SN}\right)_{ij}\bigtriangledown\mu_{j},$
(27)

Here $C_{SN}$ denotes the covariance matrix [70], and

$\displaystyle\quad\bigtriangledown\mu_{i}=\mu^{th}(z_{i},\theta)-\mu_{i}^{obs}.$

is the difference between the value of distance modulus obtained from the
cosmological observations and its theoretically predicted values estimated
from the given model with parameter space $\theta$. One can estimate the
distance modulus by using the relation $\mu=m_{B}-M_{B}$, where $m_{B}$ and
$M_{B}$ are the observed apparent magnitude and the absolute magnitude at a
given redshift (Retrieving the nuisance parameter following the recent
approach called BEAMS with Bias Correction (BBC) [71]). Moreover, its
theoretical value can be calculated as

$\mu(z)=5log_{10}\left[\frac{D_{L}(z)}{1Mpc}\right]+25,$ (28)

where

$D_{L}(z)=c(1+z)\int_{0}^{z}\frac{dx}{H(x,\theta)}.$ (29)

The contour plot for the model parameters $\alpha$, $\zeta$, $n$, and $H_{0}$
corresponding to Pantheon data sets at $1-\sigma$ and $2-\sigma$ confidence
interval is presented in Fig 2.

Figure 2: The $1-\sigma$ and $2-\sigma$ contour plot for the model parameters
using the Pantheon data set.

The obtained best fit values are $\alpha=0.899\pm 0.098$, $\zeta=179.99\pm
0.10$, $n=0.196\pm 0.098$, and $H_{0}=66.498\pm 0.097$.

###  IV.0.3 H(z)+Pantheon datasets

The $\chi^{2}$ function to obtain the mean value of the model parameters
corresponding to the combine H(z)+Pantheon data sets reads as

$\chi^{2}_{total}=\chi^{2}_{H}+\chi^{2}_{SN}.$ (30)

The contour plot for the model parameters $\alpha$, $\zeta$, $n$, and $H_{0}$
corresponding to H(z)+Pantheon data sets at $1-\sigma$ and $2-\sigma$
confidence interval is presented in Fig 3.

Figure 3: The $1-\sigma$ and $2-\sigma$ contour plot for the model parameters
using the H(z)+Pantheon data set.

The obtained best fit values are $\alpha=0.895\pm 0.098$, $\zeta=180.007\pm
0.099$, $n=0.195\pm 0.099$, and $H_{0}=66.50\pm 0.10$.

##  V Behavior of Physical Parameters

In this section, we are going to present the behavior of physical parameters
that describes the evolution phase of the universe, corresponding to the
extracted values of the model parameters from different observational data
sets.

Figure 4: Profile of the matter-energy density vs cosmic redshift $z$. Figure
5: Profile of the effective pressure vs cosmic redshift $z$. Figure 6: Profile
of the skewness parameter vs cosmic redshift $z$. Figure 7: Profile of the
effective EoS vs cosmic redshift $z$. Figure 8: Profile of evolutionary
trajectories of the given model in the $r-s$ plane.

The evolution profile of the matter-energy density and the effective pressure
is presented in Fig 4 and 5, respectively. We observe that the energy density
indicates expected positive behavior, whereas pressure with viscosity
coefficient shows negative behavior on the entire domain. Therefore the
presence of the viscosity coefficient in the cosmic content contributes to the
universe’s expansion. The profile of the skewness parameter presented in Fig 6
shows the anisotropic nature of the spacetime during the entire evolution
phase of the universe. Further, the effective EoS parameter presented in the
Fig 7 favors the accelerating phase of the universe’s expansion. It’s present
value corresponding to H(z), Pantheon, and combine H(z)+Pantheon data sets are
$\omega_{0}\approx-0.9$, $\omega_{0}\approx-0.85$, and $\omega_{0}\approx-0.8$
respectively.

The statefinder diagnostic proposed by V. Sahni [72] can geometrically
differentiate different models of dark energy through statefinder parameters
which are defined as

$r=\frac{\dddot{a}}{aH^{3}},$ (31)

and

$s=\frac{(r-1)}{3(q-\frac{1}{2})}.$ (32)

The value $(r>1,s<0)$ represents phantom scenario whereas $(r<1,s>0)$ denotes
the quintessence type dark energy, and the value $(r=1,s=0)$ mimics the
standard $\Lambda$CDM model. The behavior of our $f(R,L_{m})$ model in the
presence of viscous fluid in the $r-s$ plane is presented in Fig 8. The
present values of statefinder parameters corresponding to H(z), Pantheon, and
combine H(z)+Pantheon data sets are $(r,s)=(0.87,0.06)$, $(r,s)=(0.82,0.09)$,
and $(r,s)=(0.77,0.12)$ respectively [73]. We found that our cosmological
$f(R,L_{m})$ model lies in the quintessence region, and it behaves like the
de-Sitter universe in the far future.

##  VI Energy Conditions

In this section, we investigate different energy conditions in order to test
the consistency of the obtained solution of our model. The energy conditions
are relations imposed on the energy-momentum tensor in order to satisfy
positive energy. The energy conditions are obtained from the well-known
Raychaudhuri equation and are written as [74]

  * •

Null energy condition (NEC) : $\rho_{eff}+p_{eff}\geq 0$;

  * •

Weak energy condition (WEC) : $\rho_{eff}\geq 0$ and $\rho_{eff}+p_{eff}\geq
0$;

  * •

Dominant energy condition (DEC) : $\rho_{eff}\pm p_{eff}\geq 0$;

  * •

Strong energy condition (SEC) : $\rho_{eff}+3p_{eff}\geq 0$,

with $\rho_{eff}$ being the effective energy density.

Figure 9: Profile of the null energy condition vs cosmic redshift $z$. Figure
10: Profile of the dominant energy condition vs cosmic redshift $z$. Figure
11: Profile of the strong energy condition vs cosmic redshift $z$.

The profile of the NEC and the DEC with respect to cosmic redshift is
presented in Fig 9 and 10, respectively. Since WEC is the combination of NEC
and the positive energy density, we found that corresponding to the extracted
values of the model parameters from different observational data sets, NEC,
DEC, and WEC shows positive behavior, whereas SEC presented in Fig 11 favors a
transition from positive to negative behavior in the recent past. Thus, the
violation of SEC strongly supports the recently observed acceleration with the
transition from decelerated to the accelerated phase of the universe’s
expansion in the recent past.

##  VII Conclusion

From the hydrodynamics perspective, the assumption of viscous effects in the
cosmic fluid is quite natural since the perfect fluid is, after all, an
abstraction. In this work, we investigated the significance of viscosity
coefficients to describe the observed cosmic acceleration by taking into
account a cosmological $f(R,L_{m})$ model with an anisotropic background. We
considered $f(R,L_{m})=\frac{R}{2}+L_{m}^{\alpha}$, where $\alpha$ is an
arbitrary parameter, with the effective equation of state $\bar{p}=p-3\zeta H$
that is the Einstein case value with proportionality constant $\zeta$ used in
the Einstein theory [51]. In section III, we obtained the analytical
expression of the Hubble parameter H(z) by incorporating the assumed
$f(R,L_{m})$ viscous fluid model. Further, in section IV, we analyzed the
viability of the assumed $f(R,L_{m})$ model by incorporating the observational
data sets, specifically, H(z) data sets and the Pantheon supernovae data sets.
The obtained best fit values are $\alpha=0.903^{+0.091}_{-0.10}$,
$\zeta=180.006^{+0.092}_{-0.10}$, $n=0.195\pm 0.096$, and $H_{0}=66.499\pm
0.097$ for the H(z) datasets, $\alpha=0.899\pm 0.098$, $\zeta=179.99\pm 0.10$,
$n=0.196\pm 0.098$, and $H_{0}=66.498\pm 0.097$ for the Pantheon datasets, and
$\alpha=0.895\pm 0.098$, $\zeta=180.007\pm 0.099$, $n=0.195\pm 0.099$, and
$H_{0}=66.50\pm 0.10$ [75] for the combined H(z)+Pantheon datsets. In section
V, behavior of some cosmological parameters have been presented. We observed
that the energy density presented in Fig 4 decreases with the expansion of the
universe, whereas the bulk viscous pressure presented in Fig 5 indicates the
negative behavior. The skewness parameter presented in Fig 6 favors the
anisotropic type evolution of the universe during the entire time regime.
Further, the effective EoS parameter in Fig 7 shows the accelerating nature of
the cosmic expansion with the present values $\omega_{0}\approx-0.9$,
$\omega_{0}\approx-0.85$, and $\omega_{0}\approx-0.8$ corresponding to H(z),
Pantheon, and the combined H(z)+Pantheon data sets respectively. Finally, from
Fig 8 we found that our cosmological $f(R,L_{m})$ model exhibits quintessence
behavior, and it favors the de-Sitter type expansion in the far future. In
section VI, energy conditions have been analyzed to interpret the viability of
the obtained solution. All energy conditions except SEC exhibit positive
behavior (see Fig 9 and 10), whereas the violation of SEC presented in Fig 11
strongly supports the accelerating nature of cosmic expansion with the
transition from decelerated to accelerated era.

## Data Availability Statement

There are no new data associated with this article.

## Acknowledgments

R.S. acknowledges UGC, New Delhi, India for providing Senior Research
Fellowship (UGC-Ref. No.: 191620096030). L.V.J. acknowledges UGC, Govt. of
India, New Delhi, for awarding JRF (NTA Ref. No.: 191620024300). PKS
acknowledges Science and Engineering Research Board, Department of Science and
Technology, Government of India for financial support to carry out Research
project No.: CRG/2022/001847 and IUCAA, Pune, India for providing support
through the visiting Associateship program. We are very much grateful to the
honourable referees and the editor for the illuminating suggestions that have
significantly improved our research quality and presentation.

## References

  * [1] C.L. Bennett, et al., Four-year $COBE^{1}$ DMR cosmic microwave background observations: maps and basic results, ApJ 464, L1, (1996).
  * [2] G. Hinshaw, et al., Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological parameter results, ApJS 208 , 19, (2003).
  * [3] B.S. Mason, et al, The anisotropy of the microwave background to l= 3500: deep field observations with the cosmic background imager, ApJ 591, 540, (2003).
  * [4] A.G. Riess, et al., Observational evidence from Supernovae for an accelerating universe and a cosmological constant, Astron. J. 116, 1009 (1998).
  * [5] S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 High-Redshift Supernovae, Astrophys. J. 517, 565 (1999).
  * [6] K. Migkas, et al, Probing cosmic isotropy with a new X-ray galaxy cluster sample through the $L_{X}$-T scaling relation, A & A 636, A15, (2020).
  * [7] H. Amirhashchi, Viscous dark energy in Bianchi type V spacetime, Phy Rev. D 96, 123507, (2017).
  * [8] M.F. Shamir, Locally rotationally symmetric Bianchi type I cosmology in $f(R,T)$ gravity, Eur. Phys. J. C 75, 354 (2015).
  * [9] S. Capozziello, Curvature quintessence, Int. J. Mod. Phys. D 11, 483, (2002).
  * [10] M.E. Rodrigues, et al., Anisotropic universe models in $f(T)$ gravity, Phys. Rev. D 86, 104059, (2012).
  * [11] V.K. Bhardwaj, et al., Cosmographic analysis of a closed bouncing universe with the varying cosmological constant in $f(R,T)$ gravity, Can. J. Phys. 100, 11 , (2022).
  * [12] A. Najafi and H. Hossienkhani, Using Pantheon and Hubble parameter data to constrain the Ricci dark energy in a Bianchi I Universe, Commun. Theor. Phys. 74, 065401, (2022).
  * [13] C. Tadmon and G.D. Seugmo, New exact solutions to the Eddington-inspired-Born-Infeld equations for a relativistic perfect fluid in a Bianchi type I spacetime, Commun. Theor. Phys. 74, 065403, (2022).
  * [14] H.A. Buchdahl, Non-linear Lagrangians and cosmological theory, Mon. Not. R. Astron. Soc. 150, 1 (1970).
  * [15] R. Kerner, Cosmology without singularity and nonlinear gravitational Lagrangians, Gen. Relativ. Gravit. 14, 453 (1982).
  * [16] H. Kleinert and H.J. Schmidt, Cosmology with curvature-saturated gravitational lagrangian $R$, Gen. Relativ. Gravit. 34, 1295 (2002).
  * [17] S.M. Carroll, et al., Is cosmic speed-up due to new gravitational physics?, Phys. Rev. D 70, 043528 (2004).
  * [18] S. Capozziello et al., Cosmological viability of $f(R)$-gravity as an ideal fluid and its compatibility with a matter dominated phase, Phys. Lett. B 639, 135 (2006).
  * [19] S. Tsujikawa, Observational signatures of $f(R)$ dark energy models that satisfy cosmological and local gravity constraints, Phys. Rev. D 77, 023507 (2008).
  * [20] S. Capozziello and S. Tsujikawa, Solar system and equivalence principle constraints on $f(R)$ gravity by the chameleon approach, Phys. Rev. D 77, 107501 (2008).
  * [21] A.A. Starobinsky, Disappearing cosmological constant in $f(R)$ gravity, JETP Lett. 86, 157-163 (2007).
  * [22] A. Dixit, et al., Observational constraint in Kantowski-Sachs $f(R)$ gravity model with strange quark matter, Indian J. Phys. (2023).
  * [23] J. Wang, et al., Effect of arbitrary matter-geometry coupling on thermodynamics in f(R) theories of gravity, Commun. Theor. Phys. 74, 085401, (2022).
  * [24] S. Nojiri and S.D. Odintsov, Modified gravity with negative and positive powers of curvature: unification of inflation and cosmic acceleration, Phys. Rev. D 68, 123512 (2003).
  * [25] V. Faraoni, Solar system experiments do not yet veto modified gravity models, Phys. Rev. D 74, 023529 (2006).
  * [26] L. Amendola and S. Tsujikawa, Phantom crossing, equation-of-state singularities, and local gravity constraints in $f(R)$ models, Phys. Lett. B 660, 125 (2008).
  * [27] S.D. Odintsov, D.S.C. Gomez, and G.S. Sharov, Analyzing the $H_{0}$ tension in $f(R)$ gravity models, Nucl. Phys. B 966, 115377 (2021).
  * [28] S. Capozziello, S. Nojiri and S.D. Odintsov, The role of energy conditions in $f(R)$ cosmology, Phys. Lett. B 781, 99-106 (2018).
  * [29] S. Nojiri and S.D. Odintsov, Unifying inflation with $\Lambda$CDM epoch in modified $f(R)$ gravity consistent with solar system tests, Phys. Lett.B 657, 238 (2007).
  * [30] S. Nojiri and S.D. Odintsov, Modified $f(R)$ gravity unifying $R^{m}$ inflation with the $\Lambda$CDM epoch, Phys. Rev. D 77, 026007 (2008).
  * [31] J. Santos et al., Energy conditions in $f(R)$ gravity, Phys. Rev. D 76, 083513 (2007).
  * [32] S. Nojiri, S.D. Odintsov, and V.K. Oikonomou, Unifying inflation with early and late-time dark energy in $f(R)$ gravity, Phys. Dark Univ. 29, 100602 (2020).
  * [33] S. D. Odintsov and V.K. Oikonomou, Inflationary attractors in $f(R)$ gravity, Phys. Lett. B 807, 135576 (2020).
  * [34] S. D. Odintsov and V.K. Oikonomou, Dark energy oscillations in mimetic $f(R)$ gravity, Phys. Rev. D 94, 044012 (2016).
  * [35] A. Paliathanasis, Similarity solutions for the Wheeler-DeWitt equation in $f(R)$-cosmology, Eur. Phys. J. C 79, 1031 (2019).
  * [36] O. Bertolami et al., Extra force in $f(R)$ modified theories of gravity, Phys. Rev. D 75, 104016 (2007).
  * [37] T. Harko, Modified gravity with arbitrary coupling between matter and geometry, Phys. Lett. B 669, 376 (2008).
  * [38] T. Harko, Galactic rotation curves in modified gravity with nonminimal coupling between matter and geometry, Phys. Rev. D 81, 084050 (2010).
  * [39] T. Harko, The matter Lagrangian and the energy-momentum tensor in modified gravity with nonminimal coupling between matter and geometry, Phys. Rev. D 81, 044021 (2010).
  * [40] V. Faraoni, Viability criterion for modified gravity with an extra force, Phys. Rev. D 76, 127501 (2007).
  * [41] T. Harko and F.S.N. Lobo, $f(R,L_{m})$ gravity, Eur. Phys. J. C 70, 373-379 (2010).
  * [42] V. Faraoni, Cosmology in Scalar-Tensor Gravity, Kluwer Academic, Dordrecht (2004).
  * [43] O. Bertolami, R.O. Ramos, and L.L. Graef, Gravitational particle production and the validity of effective descriptions in loop quantum cosmology, arXiv, arXiv:gr-qc/0602016.
  * [44] B.S. Goncalves and P.H.R.S. Moraes, Cosmology from non-minimal geometry-matter coupling, arXiv, arXiv:2101.05918.
  * [45] G.A. Carvalho, et al., Quark stars with 2.6 $M_{O}$ in a non-minimal geometry-matter coupling theory of gravity, Eur. Phys. J. C 82, 1096 (2022).
  * [46] R.V. Labato, G.A. Carvalho, and C.A. Bertulani, Neutron stars in $f(R,L_{m})$ gravity with realistic equations of state: joint-constrains with GW170817, massive pulsars, and the PSR J0030+0451 mass-radius from NICER data, Eur. Phys. J. C 81, 1013 (2021).
  * [47] L.V. Jaybhaye, S. Mandal, and P.K. Sahoo, Constraints on energy conditions in $f(R,L_{m})$ gravity, Int. J. Geom. Methods Mod. 19, 2250050 (2022).
  * [48] T. Harko and S. Shahidi, Palatini formulation of the conformally invariant $f(R,L_{m})$ gravity theory, Eur. Phys. J. C 82, 1003 (2022).
  * [49] T. Harko and M.J. Lake, Cosmic strings in $f(R,L_{m})$ gravity, Eur. Phys. J. C 75, 60 (2015).
  * [50] N.S. Kavya et al., Constraining anisotropic cosmological model in $f(R,L_{m})$ gravity Phys. Dark Univ. 38, 101126 (2022).
  * [51] I. Brevik, Viscosity in modified gravity, Entropy 2012(14), 2302-2310 (2012).
  * [52] I. Brevik and O. Gron, Universe models with negative bulk viscosity, Astrophys. Space Sci. 347, 399 (2013).
  * [53] I. Brevik et al., Viscous cosmology for early-and late-time universe, Int. J. Mod. Phys. D 26, 1730024 (2017).
  * [54] 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).
  * [55] A.V. Astashenok, S.D. Odintsov, and A.S. Tepliakov, The unified history of the viscous accelerating universe and phase transitions, Nucl. Phys. B 974, 115646 (2022).
  * [56] A. Sasidharan and T. K. Mathew, Bulk viscous matter and recent acceleration of the universe, Eur. Phys. J. C 75, 348 (2015).
  * [57] A. Pradhan, A. Dixit, and D.C. Maurya, Quintessence behavior of an anisotropic bulk viscous cosmological model in modified $f(Q)$ gravity, Symmetry 14(12), 2630 (2022).
  * [58] S. Kaur and C.P. Singh, Viscous cosmology in holographic dark energy with Granda-Oliveros cut-off, Commun. Theor. Phys. 75, 025401, (2023).
  * [59] S.D. Odintsov et al., Testing the equation of state for viscous dark energy,  Phys. Rev. D 101, 044010(2020).
  * [60] J.C. Fabris et al., Bulk viscosity driving the acceleration of the Universe,  Gen. Rel. Grav. 38, 495(2006).
  * [61] X.H. Meng and X. Dou, Singularity and entropy of the viscosity dark energy model,  Comm. Theor. Phys. 52, 377(2009).
  * [62] L. V. Jaybhaye et al., Cosmology in $f(R,L_{m})$ gravity, Phys. Lett. B 831, 137148 (2022).
  * [63] T. Harko and F. S. N. Lobo, Generalized curvature-matter couplings in modified gravity, Galaxies 2014(2), 410-465 (2014).
  * [64] A. De, et al., Isotropization of locally rotationally symmetric Bianchi-I universe in $f(Q)$-gravity, Eur. Phys. J. C 82, 72 (2022).
  * [65] T. Harko et al., Gravitational induced particle production through a nonminimal curvature-matter coupling, Eur. Phys. J. C 75, 386 (2015).
  * [66] D. F. Mackey et al., emcee: the MCMC hammer, Publ. Astron. Soc. Pac. 125, 306 (2013).
  * [67] G. S. Sharov and V. O. Vasiliev, How predictions of cosmological models depend on Hubble parameter data sets, Math. Model. Geom. 6, 1 (2018).
  * [68] R. Solanki et al., Cosmic acceleration with bulk viscosity in modified f (Q) gravity, Phys. Dark Univ. 32, 100820 (2021).
  * [69] A. Dixit, et al., Observational constraints for an axially symmetric transitioning model with bulk viscosity parameterization, arXiv, arXiv:2212.10413, (2022).
  * [70] 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, 101(2018).
  * [71] R. Kessler and D. Scolnic, Correcting type Ia supernova distances for selection biases and contamination in photometrically identified samples Astrophys. J. 836,56 (2017).
  * [72] V. Sahni et al., Statefinder-a new geometrical diagnostic of dark energy, JETP Lett. 77, 201 (2003).
  * [73] G. Gadbail, S. Arora, P. K. Sahoo, Viscous cosmology in the Weyl-type $f(Q,T)$ gravity, Eur. Phys. J. C 81, 1088 (2021).
  * [74] A. Raychaudhuri, Relativistic cosmology. I,  Phys. Rev. 98, 1123 (1955).
  * [75] A. Pradhan, et al, Modeling transit dark energy in $f(R,Lm)$ gravity, Int. J. Geom. Methods Mod. 20, 2350105, (2023).