#  $f(\bar{R},L(X))$-gravity in the context of dark energy with power law
expansion and energy conditions

Arijit Pandaa<EMAIL_ADDRESS>Goutam Mannab
<EMAIL_ADDRESS>
b Corresponding author  Aninda Karmakar<EMAIL_ADDRESS>Department of
Physics, Prabhat Kumar College, Contai, Purba Medinipur-721404, India  Saibal
Ray<EMAIL_ADDRESS>Department of Physics, Government College of
Engineering & Ceramic Technology, 73, Abinash Chandra Banerjee Lane,
Kolkata-700 010, India  Md. Rabiul Islam<EMAIL_ADDRESS>a Department
of Physics, Raiganj University, Raiganj, Uttar Dinajpur-733 134, West Bengal,
India.

###### Abstract

The motto of this work is to generate a general formalism of
$f(\bar{R},L(X))-$gravity in the context of dark energy under the framework of
the K-essence emergent geometry with the Dirac-Born-Infeld (DBI) variety of
action, where $\bar{R}$ is the familiar Ricci scalar, $L(X)$ is the DBI type
non-canonical Lagrangian with $X={1\over
2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ and $\phi$ is the scalar field
of the K-essence geometry. The emergent gravity metric $\bar{G}_{\mu\nu}$ and
the well known gravitational metric $g_{\mu\nu}$ are not conformally
equivalent. We have constructed a modified field equation using the metric
formalism in $f(\bar{R},L(X))$-gravity incorporating the corresponding
Friedmann equations in the framework of the background gravitational metric
which is of Friedmann-Lemaître-Robertson-Walker (FLRW) type. The solution of
modified Friedmann equations have been deduced for the specific choice of
$f(\bar{R},L(X))$, which is of Starobinsky type, using power law expansion
method. The consistency of the model with the accelerating phase of the
Universe has been shown, when we restrict ourselves to consider the value of
the dark energy density, as $\dot{\phi}^{2}=\frac{8}{9}=0.888<1$, which
indicates that the present Universe is dark energy dominated. Graphical plots
for the energy density ($\rho$), pressure ($p$) and equation of state
parameter ($\omega$) w.r.t. time ($t$) based on parametric values are
interestingly consistent with the dark energy domination and hence
accelerating features. We also put some light on the corresponding energy
conditions and constraints of the $f(\bar{R},L(X))$ theory with one basic
example.

##  I Introduction

Einstein’s gravitational theory, General relativity (GR), was reshaped by Weyl
in 1919 [1] and Eddington in 1923 [2] through incorporating higher-order
invariants in its action. We are bound to quantize GR by the non-
renormalizability of the theory in a conventional way. Utiyama and De Witt [3]
in 1962 showed that, renormalization at one loop is possible if, the Einstein-
Hilbert action is supplemented by higher-order curvature terms and made
renormalizable. In addition, when quantum corrections or string theory entered
in the scenario, the effective low energy gravitational action admits higher-
order curvature invariants [4]. Being inspired by this argument, scientists
have been tried to modify the Einstein-Hilbert action in applied higher order
theories of gravity.

Therefore, it was evident that higher-order curvature invariants, concerning
Ricci scalar, should be taken into consideration. We should keep it in mind
that, the corrected theories of GR are considered to be important only at
scales which are close to the Planck scale, i.e., in the early universe or
near the black hole singularities, such as curvature-driven inflation scenario
[5]. Sotiriou and Felice [6, 7] made a significant contribution by introducing
the well-known “$f(R)$ theory of gravity”. This $f(R)$ gravity was
reconstructed by Dunsby et al. [8] from the background Friedman-Lemaitre-
Robertson-Walker (FLRW) expansion history. To be particular, they have found
that the only real-valued Lagrangian for which $f(R)$ can generate an exact
$\Lambda CDM$ expansion history for a universe which is filled with dust-like
matter and is equivalent to the Einstein-Hilbert Lagrangian with positive
cosmological constant. A general formalism was proposed by Mukherjee and
Banerjee [9] to observe the late time dynamics of the universe for any
analytic $f(R)$ gravity model, along with a cold dark matter. The fruits of
energy conditions in $f(R)$ gravity in various scenarios of cosmology has also
been investigated in [10, 11, 12, 13, 14]. The energy conditions using the
Raychaudhuri equations in expanding universe was also studied in [15, 16, 17].

Harko et al. [18] generalized the $f(R)$ gravity taking the gravitational
Lagrangian as an arbitrary function of the Ricci scalar ($R$) along with the
matter Lagrangian ($L_{m}$). They have obtained the gravitational field
equations of $f(R,L_{m})$ gravity, using the metric formalism, as well as the
equations of motion for test particles. Also, later on, the general energy
conditions of the $f(R,L_{m})$ gravity was developed by Wang et al. [19]. In
[20, 21, 22], the authors have described the power-law cosmic expansion in
higher derivative gravity.

On the basis of the Dirac-Born-Infeld (DBI) model [23, 24, 25], Manna et al.
[26, 27, 28] have developed the simplest form of the K-essence emergent
gravity metric $\bar{G}_{\mu\nu}$ which is obviously different from the usual
gravitational metric $g_{\mu\nu}$. The K-essence model [29, 30, 31, 32, 33,
34, 35] is a scalar field model where the kinetic energy of the field
dominates over the potential energy of the field. The K-essence field
Lagrangian is non-canonical in nature and it confirms the non dependency of
the Lagrangian on the field explicitly. The K-essence theory with non-
canonical kinetic terms and the relativistic field theories with canonical
kinetic terms are different in the non-trivial dynamical solutions of the
K-essence equation of motion. Besides spontaneously breaking Lorentz
invariance it also changes the metric for the perturbations around these
solutions. Thus, the perturbations propagate in the so-called emergent or
analogue curved space-time with the metric. The general form of the Lagrangian
for K-essence model is $L=-V(\phi)F(X)$ where $\phi$ is the K-essence scalar
field and $X=\frac{1}{2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$.

A modified version of $f(R)$ gravity was also studied by Nojiri et al. [36],
in which they consider the higher order kinetic terms of a scalar field, added
in the action of vacuum $f(R)$ gravity. Actually, in the form of action, they
added a general class of the K-essence Lagrangian $G(X)$ with the vacuum
$f(R)$ gravity Lagrangian. In the context of slow-roll approximation they have
investigated the inflationary aspects of their theory. Odintsov et al. [37]
have studied the consequences of K-essence terms in the late-time
phenomenology of $f(R)$ gravity in the presence of cold dark matter and
radiation. Perfect fluids with the similar model was considered in [36].
Several cosmological quantities of cosmological interest, such as the dark
energy equation of state parameter, the dark energy density parameter, and
some state finder quantities were also some of their findings. Recently,
Oikonomou et al. [38] have discussed the phase space of a simple K-essence
$f(R)$ gravity theory. They have worked with both canonical and phantom scalar
fields and found that the canonical scalar K-essence theory is structurally
more fascinating in comparison with the phantom theory.

At this juncture, let us discuss the importance of the K-essence theory with
specific choice of DBI type Lagrangian: It is now unavoidable to admit the
accelerating universe after analysing the observations of Large-Scale
Structure, searches of type Ia Supernovae and measurements of Cosmic Microwave
Background anisotropy [39]. It has also been acknowledged that our universe is
now dominated by a component named dark energy which poses negative pressure.
Scientists proposed cosmological constant or vacuum density to be the
candidate for such an exotic component of the universe. But the barrier of
cosmological coincidence problem created hurdles for us and raised the
question that why does the strange dark energy component possesses a tiny
energy density $(O(meV^{4}))$ compared to the simple expectation based on
quantum field theory. Additionally, the occurrence of the acceleration at such
a late stage of evolution continuously poked cosmologists for better theories.
The problem with most of the dark energy models (take cosmological constant as
example) are that they require extraordinary fine tuning of the initial energy
density which is of the orders of $100$ or more smaller than the initial
matter-energy density.

Here comes the new class of scalar field model with amazing dynamic properties
that avoids the fine-tuning problem, known as K-essence theory [40]. The most
promising feature of this model is that it brings the negative pressure from
its nonlinear kinetic energy of the scalar field. It has already been
investigated that for a wide class of theories, there exist attractor solution
[41, 42] in which the scalar field changes its speed of evolution to give the
required equation of state of the K-essence theory of different epoch on the
basis of the changing equation of state of the background. While the universe
was going through the radiation dominated epoch, K-essence theory was driven
to be sub-dominant and imitated the equation of state (EOS) of the radiation
so that the ratio of K-essence theory to radiation density remains constant.
When the universe stepped into the dust dominated epoch K-essence theory
failed to mimic the dust-like EOS for dynamical reasons but it deceased its
energy value quickly by several orders of magnitude and settled upon at a
constant value. Afterwards the field outgrew the matter density and took the
universe into cosmic acceleration at a time, roughly corresponding to the
current age of the universe. At last, the EOS of K-essence theory slowly
reclines to an value between the range $0$ and $-1$.

The well-known Quintessence Trackers models [43, 44, 45, 46, 47, 48, 49]
almost give the same result as given by the K-essence theory but with one
problem. It is good that it can mimic the equation of state of matter and
radiation of background EOS, but it requires an adjustable parameter which
need to be finely tuned to achieve the energy density for which the pressure
becomes negative at present age.

The unique characteristics of K-essence theory is that it tracks the
background energy density only in the radiation epoch. The sharp transition of
positive pressure to negative pressure at the matter-radiation equality occurs
automatically with the help of its dynamics only. K-essence theory was unable
to dominate before matter-radiation equality as it was busy to track the
radiation background. Since its energy density inevitably drops several orders
of magnitude at the transition to dust domination, it was also impossible to
dominate immediately just after the dust domination. On the other hand, as the
matter density decreases more rapidly than the energy density with expanding
universe, K-essence theory took the dominating role at the age roughly around
the current epoch. Thus, the whole problem of Cosmic Coincidence Problem
(i.e., why we live in the particular era during which the dark matter and dark
energy densities are roughly equal) vanishes to the fact that we came to
observe the universe at the right time after matter-radiation equality.

There is another interesting characteristic of K-essence model that it can
produce a dark energy component where the sound speed ($c_{s}$) does not
exceed the speed of light. These models are observationally distinguished from
standard scalar field quintessence models with a canonical kinetic term (for
which $c_{s}=1$), and may provide a mechanism to reduce cosmic microwave
background (CMB) fluctuations on large angular scales [50, 51, 52]. Some
cosmological behaviours and the stability of the K-essence model in FLRW
spacetime has been studied by Yang et al. [53]

Historically, a theory with a non-canonical kinetic term was first proposed by
Born and Infeld [23] in order to get rid of the infinite self-energy of the
electron. Similar type of theories was also studied in the literature [24,
25]. The literature [54, 55, 56, 57, 58, 59, 60, 61, 62] of string theory,
brane cosmology, D-branes, etc. have also used the DBI type non-canonical
Lagrangian.

Motivated by the above said importance of the K-essence theory which is a
theoretical model to investigate the effects of the presence of dark energy in
cosmological scenarios, in this paper we study the $f(R)-$gravity in the
context of K-essence emergent gravity, i.e., dark energy in a general manner.
We have made the generalization of the $f(\bar{R},L(X))$ theory by using the
metric formalism where $\bar{R}$ is the Ricci scalar of the K-essence geometry
and $L(X)$ is the DBI type non-canonical Lagrangian. Also, we have calculated
the energy conditions and modified Friedman equations in $f(\bar{R},L(X))$
gravity, where we have considered the flat FLRW type metric as the background
gravitational metric. The modified field equation, Friedmann equations and
energy conditions for the new $f(\bar{R},L(X))$ gravity theory are different
from the usual $f(R,L_{m})$ [18] and $f(R)$ [6] gravity theories. Also, we
have solved the modified Friedmann equations using power law cosmic expansion
method.

The paper is organized as follows: In Sec. II, we have briefly discussed about
the K-essence emergent geometry with the help of the following works [26, 27,
28, 29, 30, 31, 32, 33]. In Sec. III, we have formulated the $f(\bar{R},L(X))$
gravity in the context of the K-essence emergent geometry. We also have
derived the modified field equations and the condition of the requirement of
the conservation of the energy-momentum tensor in the $f(\bar{R},L(X))$
gravity. The modified Friedmann equations were brought into limelight in Sec.
IV, considering the background gravitational metric as flat FLRW and the
K-essence scalar field as a function of time only. The solution of Friedmann
equations have been solved for specific choice of $f(\bar{R},L(X))$ using
power law method in Sec. V, whereas in Sec. VI we develop the energy
conditions and constraints of the $f(\bar{R},L(X))$ gravity with example. The
last section Sec. VI contains some general discussion and crucial conclusion
of our work. Also, we have briefly discussed the $f(R)$-gravity and
$f(R,L_{m})-$gravity and corresponding energy conditions [10, 11, 12, 13, 19,
14, 73] in Sec. VIII (Appendix).

##  II K-essence theory: a glimpse of background and development

In this section, we will discuss about the development of the effective metric
for the emergent spacetime corresponding to a general background geometry and
a very general K-essence scalar field vector. The K-essence scalar field
$\phi$, minimally coupled with the background space-time metric $g_{\mu\nu}$
has action [29, 30, 31, 32, 33]

$\displaystyle S_{k}[\phi,g_{\mu\nu}]=\int d^{4}x{\sqrt{-}g}L(X,\phi),$ (1)

where $X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$ is the
canonical kinetic term.

The energy-momentum tensor is

$\displaystyle T_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle{-2\over\sqrt{-g}}{\delta S_{k}\over\delta
g^{\mu\nu}}=-2\frac{\partial L}{\partial
g^{\mu\nu}}+g_{\mu\nu}L=-L_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi+g_{\mu\nu}L,$
(2)

with $L_{\mathrm{X}}={dL\over dX},~{}L_{\mathrm{X}X}={d^{2}L\over
dX^{2}},~{}L_{\mathrm{\phi}}={dL\over d\phi}$ and $\nabla_{\mu}$ is the
covariant derivative defined with respect to the gravitational metric
$g_{\mu\nu}$.

The equation of motion of the scalar field is

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=G^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi+2XL_{X\phi}-L_{\phi}=0,$
(3)

where

$\displaystyle
G^{\mu\nu}\equiv\frac{c_{s}}{L_{X}^{2}}[L_{X}g^{\mu\nu}+L_{XX}\nabla^{\mu}\phi\nabla^{\nu}\phi],$
(4)

with $1+{2XL_{XX}\over L_{X}}>0$ and
$c_{s}^{2}(X,\phi)\equiv{(1+2X{L_{XX}\over L_{X}})^{-1}}$.

The inverse metric of $G^{\mu\nu}$ takes the form

$\displaystyle G_{\mu\nu}={L_{X}\over
c_{s}}[g_{\mu\nu}-{c_{s}^{2}}{L_{XX}\over
L_{X}}\nabla_{\mu}\phi\nabla_{\nu}\phi].$ (5)

A further conformal transformation [26, 27] $\bar{G}_{\mu\nu}\equiv{c_{s}\over
L_{X}}G_{\mu\nu}$ gives

$\displaystyle\bar{G}_{\mu\nu}={g_{\mu\nu}-{{L_{XX}}\over{L_{X}+2XL_{XX}}}\nabla_{\mu}\phi\nabla_{\nu}\phi}.$
(6)

Using Eq. (2), the effective emergent metrics (6) can be written as [31, 32]

$\displaystyle\bar{G}_{\mu\nu}=\left(1-\frac{LL_{XX}}{L_{X}(L_{X}+2XL_{XX})}\right)g_{\mu\nu}+\frac{L_{XX}}{L_{X}(L_{X}+2XL_{XX})}T_{\mu\nu}.$
(7)

Here one must always have $L_{X}\neq 0$ for $c_{s}^{2}$ to be positive
definite and only then Eqs. (1) – (4) will be physically meaningful.

It is clear that for non-trivial space-time configurations of $\phi$, the
emergent metric $\bar{G}_{\mu\nu}$ is in general, not conformally equivalent
to $g_{\mu\nu}$. So $\phi$ has different properties as of the canonical scalar
fields with the local causal structure. It is distinct from those defined with
$g_{\mu\nu}$. Further, if $L$ is not an explicit function of $\phi$, the
equation of motion (3) reduces to

$\displaystyle-{1\over\sqrt{-g}}{\delta
S_{k}\over\delta\phi}=\bar{G}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}\phi=0.$ (8)

We shall take the Dirac-Born-Infeld (DBI) type non-canonical Lagrangian [23,
24, 25, 32, 33] as

$\displaystyle L(X,\phi)=1-\sqrt{1-2X}.$ (9)

It should be noted that here we ignore any type of potential part of the
Lagrangian, since in the K-essence theory, the kinetic energy dominates over
the potential energy of the K-essence scalar field. Then
$c_{s}^{2}(X,\phi)=1-2X$ and thus the effective emergent metric (6) becomes

$\displaystyle\bar{G}_{\mu\nu}=g_{\mu\nu}-\nabla_{\mu}\phi\nabla_{\nu}\phi\equiv
g_{\mu\nu}-\partial_{\mu}\phi\partial_{\nu}\phi,$ (10)

since $\phi$ is a scalar.

Using Eq. (2), this can also be written in terms of $T_{\mu\nu}$ and
$\nabla_{\mu}\phi$, (10) as

$\displaystyle\bar{G}_{\mu\nu}L=L_{X}\nabla_{\mu}\phi\nabla_{\nu}\phi+T_{\mu\nu}-L\nabla_{\mu}\phi\nabla_{\nu}\phi.$
(11)

Following [63, 26] the new Christoffel symbols can be related to the old ones
by

$\displaystyle\bar{\Gamma}^{\alpha}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\Gamma^{\alpha}_{\mu\nu}+(1-2X)^{-1/2}\bar{G}^{\alpha\gamma}[\bar{G}_{\mu\gamma}\partial_{\nu}(1-2X)^{1/2}$
(12) $\displaystyle+$
$\displaystyle\bar{G}_{\nu\gamma}\partial_{\mu}(1-2X)^{1/2}-\bar{G}_{\mu\nu}\partial_{\gamma}(1-2X)^{1/2}]$
$\displaystyle=$
$\displaystyle\Gamma^{\alpha}_{\mu\nu}-\frac{1}{2(1-2X)}[\delta^{\alpha}_{\mu}\partial_{\nu}X+\delta^{\alpha}_{\mu}\partial_{\nu}X].$

The geodesic equation for the K-essence theory in terms of the new Christoffel
connections $\bar{\Gamma}$ now becomes

$\displaystyle\frac{d^{2}x^{\alpha}}{d\lambda^{2}}+\bar{\Gamma}^{\alpha}_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}=0,$
(13)

where $\lambda$ is an affine parameter.

Now we introduce the covariant derivative $D_{\mu}$ [31, 32], associated with
the emergent metric $\bar{G}_{\mu\nu}$ $(D_{\alpha}\bar{G}^{\alpha\beta}=0)$
as

$\displaystyle
D_{\mu}A_{\nu}=\partial_{\mu}A_{\nu}-\bar{\Gamma}^{\lambda}_{\mu\nu}A_{\lambda},$
(14)

and the inverse emergent metric is $\bar{G}^{\mu\nu}$ such that
$\bar{G}_{\mu\lambda}\bar{G}^{\lambda\nu}=\delta^{\nu}_{\mu}$.

Therefore, the “emergent” Einstein’s equation is

$\displaystyle\bar{E}_{\mu\nu}=\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R}=\kappa
T_{\mu\nu},$ (15)

where $\kappa=8\pi G$ is constant, $\bar{R}_{\mu\nu}$ is Ricci tensor and
$\bar{R}~{}(=\bar{R}_{\mu\nu}\bar{G}^{\mu\nu})$ is the Ricci scalar of the
emergent space-time.

##  III $f(\bar{R},L(X))$ Gravity in the context of K-essence emergent space-
time

We are now considering the action of the modified gravity in the context of
the K-essence emergent space-time, which takes the following form ($\kappa=1$)

$\displaystyle S=\int d^{4}x\sqrt{-\bar{G}}~{}f(\bar{R},L(X)),$ (16)

where $f(\bar{R},L(X))$ is an arbitrary function of the Ricci scalar $\bar{R}$
and the non-canonical Lagrangian density $L(X)$ corresponding to the K-essence
theory, and

$\sqrt{-\bar{G}}=\sqrt{-det({\bar{G}_{\mu\nu}})}.$

Based on [18], varying the action $S$ with respect to the K-essence emergent
gravity metric $\bar{G}^{\mu\nu}$ we obtain

$\displaystyle\delta
S=\int\Big{[}f_{\bar{R}}(\bar{R},L)\delta\bar{R}+f_{L}(\bar{R},L)\frac{\delta
L}{\delta\bar{G}^{\mu\nu}}\delta\bar{G}^{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}f(\bar{R},L)\delta\bar{G}^{\mu\nu}\Big{]}\times\sqrt{-\bar{G}}~{}d^{4}x,$
(17)

where we have denoted
$f_{\bar{R}}(\bar{R},L)=\frac{\partial{f(\bar{R},L)}}{\partial\bar{R}}$ and
$f_{L}(\bar{R},L)=\frac{\partial{f(\bar{R},L)}}{\partial L}$.

Now we obtain the variation of the Ricci scalar for the K-essence emergent
gravity metric

$\displaystyle\delta\bar{R}=\delta(\bar{R}_{\mu\nu}\bar{G}^{\mu\nu})=\delta\bar{R}_{\mu\nu}\bar{G}^{\mu\nu}+\bar{R}_{\mu\nu}\delta\bar{G}^{\mu\nu}=\bar{R}_{\mu\nu}\delta\bar{G}^{\mu\nu}+\bar{G}^{\mu\nu}(D_{\lambda}\delta\bar{\Gamma}^{\lambda}_{\mu\nu}-D_{\nu}\delta\bar{\Gamma}^{\lambda}_{\mu\lambda}),$
(18)

where

$\displaystyle\bar{R}_{\mu\nu}=\partial_{\mu}\bar{\Gamma}^{\alpha}_{\alpha\nu}-\partial_{\alpha}\bar{\Gamma}^{\alpha}_{\mu\nu}+\bar{\Gamma}^{\alpha}_{\beta\mu}\bar{\Gamma}^{\beta}_{\alpha\nu}-\bar{\Gamma}^{\alpha}_{\alpha\beta}\bar{\Gamma}^{\beta}_{\mu\nu},$
(19)
$\displaystyle\bar{\Gamma}^{\alpha}_{\mu\nu}=\frac{1}{2}\bar{G}^{\alpha\beta}[\partial_{\mu}\bar{G}_{\beta\nu}+\partial_{\nu}\bar{G}_{\mu\beta}-\partial_{\beta}\bar{G}_{\mu\nu}],$
(20)

and the variation of $\delta\bar{\Gamma}^{\lambda}_{\mu\nu}$ is

$\displaystyle\delta\bar{\Gamma}^{\lambda}_{\mu\nu}=\frac{1}{2}\bar{G}^{\lambda\alpha}[D_{\mu}\delta\bar{G}_{\nu\alpha}+D_{\nu}\delta\bar{G}_{\mu\alpha}-D_{\alpha}\delta\bar{G}_{\mu\nu}].$
(21)

Thus, the expression for the variation of Ricci scalar $\delta\bar{R}$ is

$\displaystyle\delta\bar{R}=\bar{R}_{\mu\nu}\delta\bar{G}^{\mu\nu}+\bar{G}_{\mu\nu}D_{\alpha}D^{\alpha}\delta\bar{G}^{\mu\nu}-D_{\mu}D_{\nu}\delta\bar{G}^{\mu\nu}.$
(22)

Therefore, variation of the action (17) is

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\Big{[}f_{\bar{R}}(\bar{R},L)\bar{R}_{\mu\nu}\delta\bar{G}^{\mu\nu}+f_{\bar{R}}(\bar{R},L)\bar{G}_{\mu\nu}D_{\alpha}D^{\alpha}\delta\bar{G}^{\mu\nu}-f_{\bar{R}}(\bar{R},L)D_{\mu}D_{\nu}\delta\bar{G}^{\mu\nu}$
(23) $\displaystyle+$ $\displaystyle f_{L}(\bar{R},L)\frac{\delta
L}{\delta\bar{G}^{\mu\nu}}\delta\bar{G}^{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}f(\bar{R},L)\delta\bar{G}^{\mu\nu}\Big{]}\sqrt{-\bar{G}}d^{4}x.~{}~{}~{}~{}~{}$

After partially integrating second and third terms of the above Eq. (23), we
get

$\displaystyle\delta S$ $\displaystyle=$
$\displaystyle\int\Big{[}f_{\bar{R}}(\bar{R},L)\bar{R}_{\mu\nu}+\bar{G}_{\mu\nu}D_{\mu}D^{\mu}f_{\bar{R}}(\bar{R},L)-D_{\mu}D_{\nu}f_{\bar{R}}(\bar{R},L)$
(24) $\displaystyle+f_{L}(\bar{R},L)\frac{\delta
L}{\delta\bar{G}^{\mu\nu}}-\frac{1}{2}\bar{G}_{\mu\nu}f(\bar{R},L)\Big{]}\delta\bar{G}^{\mu\nu}\sqrt{-\bar{G}}d^{4}x.$

Therefore, using principle of least action, i.e. $\delta S=0$, we have the
modified field equation for $f(\bar{R},L(X))$ theory

$\displaystyle
f_{\bar{R}}(\bar{R},L)\bar{R}_{\mu\nu}+\bar{G}_{\mu\nu}D_{\alpha}D^{\alpha}f_{\bar{R}}(\bar{R},L)-D_{\mu}D_{\nu}f_{\bar{R}}(\bar{R},L)-\frac{1}{2}\bar{G}_{\mu\nu}f(\bar{R},L)+f_{L}(\bar{R},L)\frac{\delta
L}{\delta\bar{G}^{\mu\nu}}=0.$ (25)

Now, we evaluate the term $\frac{\delta L}{\delta\bar{G}^{\mu\nu}}$ as

$\displaystyle\frac{\delta L}{\delta\bar{G}^{\mu\nu}}=\frac{\delta L}{\delta
X}\frac{\delta X}{\delta g^{\mu\nu}}\frac{\delta
g^{\mu\nu}}{\delta\bar{G}^{\mu\nu}}=\frac{1}{2}L_{X}D_{\mu}\phi
D_{\nu}\phi(1+D_{\alpha}\phi D^{\alpha}\phi),$ (26)

since for the scalar field $\nabla_{\mu}\phi\equiv\partial_{\mu}\phi\equiv
D_{\mu}\phi$.

Using Eqs. (11), (25) and (26), we obtain the expression for modified field
equation for the $f(\bar{R},L(X))$ theory in terms of $T_{\mu\nu}$ as

$\displaystyle
f_{\bar{R}}(\bar{R},L)\bar{R}_{\mu\nu}+\left(\bar{G}_{\mu\nu}\bar{\square}-D_{\mu}D_{\nu}\right)f_{\bar{R}}(\bar{R},L)-\frac{1}{2}\left[f(\bar{R},L)-Lf_{L}(\bar{R},L)(1+D_{\alpha}\phi
D^{\alpha}\phi)\right]\bar{G}_{\mu\nu}$ $\displaystyle+$
$\displaystyle\frac{1}{2}Lf_{L}(\bar{R},L)D_{\mu}\phi
D_{\nu}\phi\left[1+D_{\alpha}\phi
D^{\alpha}\phi\right]=\frac{1}{2}f_{L}(\bar{R},L)T_{\mu\nu}\left[1+D_{\alpha}\phi
D^{\alpha}\phi\right]=\frac{1}{2}f_{L}(\bar{R},L)\bar{T}_{\mu\nu},$

where $\bar{\square}=D_{\mu}D^{\mu}$ and
$\bar{T}_{\mu\nu}=T_{\mu\nu}[1+D_{\alpha}\phi D^{\alpha}\phi]$.

The above Eq. (LABEL:40) is different from the usual Eq. (105) of $f(R,L_{m})$
theory in the presence of the K-essence scalar field (vide Appendix). If we
consider the emergent gravity metric $\bar{G}_{\mu\nu}$ is conformally
equivalent to the gravitational metric $g_{\mu\nu}$ and $L$ can be matter
Lagrangian then we get back to the usual $f(R,L_{m})$ theory in the absence of
the K-essence scalar field. Also, if we consider $f(\bar{R},L(X))\equiv
f(R,L_{m})\equiv\frac{1}{2}R+L_{m}$, i.e., the Hilbert-Einstein Lagrangian
form, then from (LABEL:40), we lead to the standard Einstein field equation
$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=T_{\mu\nu}$.

Contracting the above field equation (LABEL:40) with $\bar{G}^{\mu\nu}$, we
have the modified trace equation for the $f(\bar{R},L(X))$ theory is

$\displaystyle
f_{\bar{R}}(\bar{R},L)\bar{R}+3\bar{\square}f_{\bar{R}}(\bar{R},L)-2[f(\bar{R},L)-f_{L}(\bar{R},L)L(1+D_{\alpha}\phi
D^{\alpha}\phi)]$ (28) $\displaystyle+$
$\displaystyle\frac{1}{2}Lf_{L}(\bar{R},L)D_{\mu}\phi
D^{\mu}\phi[1+D_{\alpha}\phi
D^{\alpha}\phi]=\frac{1}{2}f_{L}(\bar{R},L)\bar{T},$

where $\bar{T}=\bar{T}^{\mu}_{\mu}$ is trace of the energy-momentum tensor.

On subtracting Eq. $(\ref{41})\times\bar{G}_{\mu\nu}$ from Eq.
$(\ref{40})\times 3$, we get

$\displaystyle
f_{\bar{R}}(\bar{R},L)(\bar{R}_{\mu\nu}-\frac{1}{3}\bar{G}_{\mu\nu}\bar{R})+\frac{1}{6}\bar{G}_{\mu\nu}[f(\bar{R},L)-Lf_{L}(\bar{R},L)(1+D_{\alpha}\phi
D^{\alpha}\phi)]$ $\displaystyle+$
$\displaystyle\frac{1}{3}Lf_{L}(\bar{R},L)D_{\mu}\phi
D_{\nu}\phi[1+D_{\alpha}\phi
D^{\alpha}\phi]=\frac{1}{2}f_{L}(\bar{R},L)(\bar{T}_{\mu\nu}-\frac{1}{3}\bar{G}_{\mu\nu}\bar{T})+D_{\mu}D_{\nu}f_{\bar{R}}(\bar{R},L),$

which is an another form of the modified field equation in the presence of
K-essence scalar field $\phi$.

By taking covariant divergence with respect to $D^{\mu}$ of Eq. (LABEL:40), we
have

$\displaystyle
D^{\mu}[f_{\bar{R}}(\bar{R},L)]\bar{R}_{\mu\nu}-(\bar{\square}D_{\nu}-D_{\nu}\bar{\square})f_{\bar{R}}(\bar{R},L)+f_{\bar{R}}(\bar{R},L)D^{\mu}(\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R})$
$\displaystyle+\frac{1}{2}\bar{G}_{\mu\nu}f_{\bar{R}}(\bar{R},L)D^{\mu}(\bar{R})-\frac{1}{2}D^{\mu}[f(\bar{R},L)]\bar{G}_{\mu\nu}+\frac{1}{2}D^{\mu}[Lf_{L}(\bar{R},L)(1+D_{\alpha}\phi
D^{\alpha}\phi)]\bar{G}_{\mu\nu}$
$\displaystyle+\frac{1}{2}D^{\mu}[Lf_{L}(\bar{R},L)D_{\mu}\phi
D_{\nu}\phi(1+D_{\alpha}\phi
D^{\alpha}\phi)]=\frac{1}{2}D^{\mu}[f_{L}(\bar{R},L)\bar{T}_{\mu\nu}].$ (30)

Using the identities on purely geometrical grounds [64, 65, 66]
$D^{\mu}(\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R})=D^{\mu}\bar{E}_{\mu\nu}=0$,
$D^{\mu}[f_{\bar{R}}(\bar{R},L)]\bar{R}_{\mu\nu}=(\bar{\square}D_{\nu}-D_{\nu}\bar{\square})f_{\bar{R}}(\bar{R},L)$
and also using Eqs. (11) and (26), the above Eq. (30) becomes

$\displaystyle
D^{\mu}[f_{L}(\bar{R},L)\bar{T}_{\mu\nu}]=-f_{L}(\bar{R},L)D^{\mu}[\bar{G}_{\mu\nu}L]+D^{\mu}\left[Lf_{L}(\bar{R},L)(1+D_{\alpha}\phi
D^{\alpha}\phi)(\bar{G}_{\mu\nu}+D_{\mu}\phi D_{\nu}\phi)\right]$
$\displaystyle\Rightarrow
D^{\mu}\bar{T}_{\mu\nu}=D^{\mu}~{}ln[f_{L}(\bar{R},L)]\left[L_{X}D_{\mu}\phi
D_{\nu}\phi(1+D_{\alpha}\phi D^{\alpha}\phi)\right]$
$\displaystyle+D^{\mu}\left[LD_{\mu}\phi D_{\nu}\phi(1+D_{\alpha}\phi
D^{\alpha}\phi)+L\bar{G}_{\mu\nu}D_{\alpha}\phi D^{\alpha}\phi\right]$
$\displaystyle\Rightarrow
D^{\mu}\bar{T}_{\mu\nu}=2D^{\mu}~{}ln[f_{L}(\bar{R},L)]\frac{\delta
L}{\delta\bar{G}^{\mu\nu}}+D^{\mu}\left[LD_{\mu}\phi
D_{\nu}\phi(1+D_{\alpha}\phi D^{\alpha}\phi)+L\bar{G}_{\mu\nu}D_{\alpha}\phi
D^{\alpha}\phi\right].$ (31)

Thus, the requirement of the conservation of the energy-momentum tensor
$(D^{\mu}\bar{T}_{\mu\nu}=0)$ for the K-essence Lagrangian, gives an effective
functional relation as

$\displaystyle 2D^{\mu}~{}ln[f_{L}(\bar{R},L)]\frac{\delta
L}{\delta\bar{G}^{\mu\nu}}+D^{\mu}[LD_{\mu}\phi D_{\nu}\phi(1+D_{\alpha}\phi
D^{\alpha}\phi)+L\bar{G}_{\mu\nu}D_{\alpha}\phi D^{\alpha}\phi]=0.$ (32)

##  IV Modification of the Friedmann equations

We consider the gravitational metric $g_{\mu\nu}$ to be a flat Friedmann-
Lemaître-Robertson-Walker (FLRW) metric and the line element for this is,

$\displaystyle ds^{2}=dt^{2}-a^{2}(t)\sum_{i=1}^{3}(dx^{i})^{2},$ (33)

with $a(t)$ being the scale factor as usual.

From Eq. (10), we have the components of the emergent gravity metric as

$\displaystyle\bar{G}_{00}=(1-\dot{\phi}^{2})~{};~{}\bar{G}_{ii}=-(a^{2}(t)+(\phi^{{}^{\prime}})^{2})~{};~{}\bar{G}_{0i}=-\dot{\phi}\phi^{{}^{\prime}}=\bar{G}_{i0},$
(34)

where we consider $\phi\equiv\phi(t,x^{i})$,
$\dot{\phi}=\frac{\partial\phi}{\partial t}$ and
$\phi^{{}^{\prime}}=\frac{\partial\phi}{\partial x^{i}}$.

So the line element of the FLRW emergent gravity metric is

$\displaystyle
dS^{2}=(1-\dot{\phi}^{2})dt^{2}-[a^{2}(t)+(\phi^{{}^{\prime}})^{2}]\sum_{i=1}^{3}(dx^{i})^{2}-2\dot{\phi}\phi^{{}^{\prime}}dtdx^{i}.$
(35)

Now from the emergent gravity equation of motion (8) we have

$\displaystyle\bar{G}^{00}(\partial_{0}\partial_{0}\phi-\Gamma^{0}_{00}\partial_{0}\phi-\Gamma^{i}_{00}\partial_{i}\phi)+\bar{G}^{ii}(\partial_{i}\partial_{i}\phi-\Gamma^{0}_{ii}\partial_{0}\phi-\Gamma^{i}_{ii}\partial_{i}\phi)$
(36) $\displaystyle+$
$\displaystyle\bar{G}^{0i}(\partial_{0}\partial_{i}\phi-\Gamma^{0}_{0i}\partial_{0}\phi-\Gamma^{i}_{oi}\partial_{i}\phi)+\bar{G}^{i0}(\partial_{i}\partial_{0}\phi-\Gamma^{0}_{i0}\partial_{0}\phi-\Gamma^{i}_{i0}\partial_{i}\phi)=0.$

For simplification, we consider the homogeneous K-essence scalar field $\phi$,
i.e., $\phi(t,x^{i})\equiv\phi(t)$ then $\bar{G}_{00}=(1-\dot{\phi}^{2})$,
$\bar{G}_{0i}=\bar{G}_{i0}=0=\partial_{i}\phi$, $\bar{G}_{ii}=-a^{2}(t)$ and
$X={1\over
2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi=\frac{1}{2}\dot{\phi}^{2}$. This
consideration is possible in this case since, the dynamical solutions of the
K-essence scalar fields spontaneously break Lorentz symmetry. Therefore, the
flat FLRW emergent gravity line element (35) and the equation of motion (36)
become

$\displaystyle
dS^{2}=(1-\dot{\phi}^{2})dt^{2}-a^{2}(t)\sum_{i=1}^{3}(dx^{i})^{2},$ (37)

and

$\displaystyle\frac{\dot{a}}{a}=H(t)=-\frac{\ddot{\phi}}{\dot{\phi}(1-\dot{\phi}^{2})},$
(38)

where $H(t)=\frac{\dot{a}}{a}$ is the usual Hubble parameter (always
$\dot{a}\neq 0$). The Eq. (38) gives the relation between Hubble parameter and
the time derivatives of the K-essence scalar field. Note that, in the above
space-time (37) always $\dot{\phi}^{2}<1$. If $\dot{\phi}^{2}>1$, the
signature of this space-time will be ill-defined. Also $\dot{\phi}^{2}\neq 0$
condition holds good instead of $\dot{\phi}^{2}=0$, which leads to non-
applicability of the K-essence theory. Also, $\dot{\phi}^{2}\neq 1$ because
$\Omega_{matter}+\Omega_{radiation}+\Omega_{darkenergy}=1$ and we can measured
$\dot{\phi}^{2}$ as dark energy density in units of the critical density,
i.e., it is nothing but $\Omega_{darkenergy}$ [26, 27, 28, 67]. Therefore
$\dot{\phi}^{2}$ takes the value in between $0$ and $1$.

The Ricci tensors and Ricci scalar of the emergent gravity space-time are

$\displaystyle\bar{R}_{ii}$ $\displaystyle=$
$\displaystyle-\frac{a^{2}}{1-\dot{\phi}^{2}}\left[\frac{\ddot{a}}{a}+2\left(\frac{\dot{a}}{a}\right)^{2}+\frac{\dot{a}}{a}\frac{\dot{\phi}\ddot{\phi}}{1-\dot{\phi}^{2}}\right]=-\frac{a^{2}}{1-\dot{\phi}^{2}}\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}(2-\dot{\phi}^{2})\right]$
(39) $\displaystyle=$
$\displaystyle-\frac{a^{2}}{1-\dot{\phi}^{2}}\left[\dot{H}+H^{2}(3-\dot{\phi}^{2})\right],$

$\displaystyle\bar{R}_{00}=3\frac{\ddot{a}}{a}+3\frac{\dot{a}}{a}\frac{\dot{\phi}\ddot{\phi}}{1-\dot{\phi}^{2}}=3\frac{\ddot{a}}{a}+3\left(\frac{\dot{a}}{a}\right)^{2}\dot{\phi}^{2}=3\left[\dot{H}+H^{2}(1-\dot{\phi}^{2})\right],$
(40)

and

$\displaystyle\bar{R}$ $\displaystyle=$
$\displaystyle\frac{6}{1-\dot{\phi}^{2}}\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}+\frac{\dot{a}}{a}\frac{\dot{\phi}\ddot{\phi}}{1-\dot{\phi}^{2}}\right]=\frac{6}{1-\dot{\phi}^{2}}\left[\frac{\ddot{a}}{a}+\left(\frac{\dot{a}}{a}\right)^{2}(1-\dot{\phi}^{2})\right]$
(41) $\displaystyle=$
$\displaystyle\frac{6}{1-\dot{\phi}^{2}}\left[\dot{H}+H^{2}(2-\dot{\phi}^{2})\right],$

where we have used the relation (38) and $\dot{H}\equiv\frac{\partial
H}{\partial t}=\frac{a\ddot{a}-\dot{a}^{2}}{a^{2}}$.

Combining Eqs. (39) and (40) with (41) we get

$\displaystyle\bar{R}_{00}=\frac{1}{2}(1-\dot{\phi}^{2})\bar{R}-3H^{2},$ (42)

$\displaystyle\bar{R}_{ii}=-\frac{a^{2}}{(1-\dot{\phi}^{2})}\left[\frac{1}{6}\bar{R}(1-\dot{\phi}^{2})+H^{2}\right].$
(43)

We assume that the energy-momentum tensor is an ideal fluid type which is

$\displaystyle T_{\mu}^{\nu}$ $\displaystyle=$ $\displaystyle
diag(\rho,-p,-p,-p)=(\rho+p)u_{\mu}u^{\nu}-\delta_{\mu}^{\nu}p$ $\displaystyle
T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\bar{G}_{\mu\alpha}T^{\alpha}_{\nu},$ (44)

where $p$ is pressure and $\rho$ is the matter density of the cosmic fluid. In
the co-moving frame we have $u^{0}=1$ and $u^{\alpha}=0$ ; $\alpha=1,2,3$ in
the K-essence emergent gravity space-time.

Now the question is whether this type of energy-momentum tensor is valid or
not in case of a perfect fluid model when the kinetic energy
($\dot{\phi}^{2}$) of the K-essence scalar filed is present. The answer is yes
since our Lagrangian is $L(X)=1-\sqrt{1-2X}$. This class of models is
equivalent to perfect fluid models with zero vorticity and the pressure
(Lagrangian) can be expressed through the energy density only [32, 67].

Now we evaluate the $ii$ and $00$ components of the modified field equation
(LABEL:40) using (44) and considering $\phi\equiv\phi(t)$ only:

$\displaystyle
F\bar{R}_{ii}+(\bar{G}_{ii}\bar{\square}-D_{i}D_{i})F-\frac{1}{2}[f-Lf_{L}(1+\dot{\phi}^{2})]\bar{G}_{ii},=\frac{1}{2}f_{L}a^{2}(t)\bar{p}~{}~{}~{}~{}$
(45)

and

$\displaystyle
F\bar{R}_{00}+(\bar{G}_{00}\bar{\square}-D_{0}D_{0})F-\frac{1}{2}[f-Lf_{L}(1+\dot{\phi}^{2})]\bar{G}_{00}+\frac{1}{2}Lf_{L}\dot{\phi}^{2}(1+\dot{\phi}^{2})=\frac{1}{2}f_{L}(1-\dot{\phi}^{2})\bar{\rho},$
(46)

with $F=f_{\bar{R}}(\bar{R},L)\equiv\frac{\partial
f(\bar{R},L)}{\partial\bar{R}}$, $\bar{p}=p(1+\dot{\phi}^{2})$ and
$\bar{\rho}=\rho(1+\dot{\phi}^{2})$.

Now, we calculate the terms $\bar{G}_{00}\bar{\square}F$ and
$\bar{G}_{ii}\bar{\square}F$, using the determinant of the flat FLRW emergent
gravity metric $\sqrt{-\bar{G}}=a^{3}\sqrt{1-\dot{\phi}^{2}}$ and the relation
(38):

$\displaystyle\bar{G}_{00}\bar{\square}F=\ddot{F}+3\frac{\dot{a}}{a}\dot{F}+\dot{F}\frac{\dot{\phi}\ddot{\phi}}{(1-\dot{\phi}^{2})}=\ddot{F}+H\dot{F}(3-\dot{\phi}^{2}),$
(47)

and

$\displaystyle\bar{G}_{ii}\bar{\square}F=D_{i}D_{i}F-\frac{a^{2}}{(1-\dot{\phi}^{2})}\left[\ddot{F}+2H\dot{F}(1-\dot{\phi}^{2})\right],$
(48)

where we have used, $(\partial_{i}t)^{2}=\frac{a^{2}}{1-\dot{\phi}^{2}}$ for
flat FLRW emergent gravity metric.

Now we substitute Eqs. (42) and (47) in the above Eq. (46), so that we have
the first modified Friedmann equation as

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{F}[-\frac{1}{2}\bar{\rho}f_{L}(1-\dot{\phi}^{2})+3H\dot{F}+(1-\dot{\phi}^{2})\frac{1}{2}(F\bar{R}-f)+\frac{1}{2}Lf_{L}(1+\dot{\phi}^{2})]$
$\displaystyle=$
$\displaystyle\frac{1}{F}[-\frac{1}{2}\bar{\rho}f_{L}(1-\dot{\phi}^{2})+3H\dot{\bar{R}}F_{\bar{R}}+3HF_{L}L_{X}\dot{X}+(1-\dot{\phi}^{2})\frac{1}{2}(F\bar{R}-f)+\frac{1}{2}Lf_{L}(1+\dot{\phi}^{2})].$

We also substitute Eqs. (39), (41) and (48) in the $ii$-components of Eq. (45)
and thereafter rearranging, we get the second modified Friedmann equation for
the flat FLRW K-essence emergent gravity space-time under $f(\bar{R},L(X))$
theory. Hence

$\displaystyle
2\dot{H}+H^{2}(3-2\dot{\phi}^{2})=\frac{1}{F}[\frac{1}{2}\bar{p}f_{L}(1-\dot{\phi}^{2})+\ddot{F}+2H\dot{F}(1-\dot{\phi}^{2})-\frac{1}{2}(1-\dot{\phi}^{2})(f-\bar{R}F)$
$\displaystyle+\frac{1}{2}Lf_{L}(1-\dot{\phi}^{2})(1+\dot{\phi}^{2})]$
$\displaystyle=$
$\displaystyle\frac{1}{F}\Big{[}\frac{1}{2}\bar{p}f_{L}(1-\dot{\phi}^{2})+\ddot{\bar{R}}F_{\bar{R}}+(\dot{\bar{R}})^{2}F_{\bar{R}\bar{R}}+2H\dot{\bar{R}}F_{\bar{R}}(1-\dot{\phi}^{2})-\frac{1}{2}(1-\dot{\phi}^{2})(f-\bar{R}F)$
$\displaystyle+\frac{1}{2}Lf_{L}(1-\dot{\phi}^{2})(1+\dot{\phi}^{2})\Big{]}+\frac{1}{F}\Big{[}2H(1-\dot{\phi}^{2})F_{L}L_{X}\dot{X}+F_{LL}(L_{X}\dot{X})^{2}+F_{L}L_{XX}(\dot{X})^{2}+F_{L}L_{X}\ddot{X}\Big{]}.$

Since the Lagrangian ($L$) of the K-essence theory is function of $X(={1\over
2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi)$, therefore we can write

$\displaystyle\dot{F}=F_{\bar{R}}\dot{\bar{R}}+F_{L}L_{X}\dot{X}~{}and~{}\ddot{F}=F_{\bar{R}}\ddot{\bar{R}}+(\dot{\bar{R}})^{2}F_{\bar{R}\bar{R}}+F_{LL}(L_{X}\dot{X})^{2}+F_{L}L_{XX}(\dot{X})^{2}+F_{L}L_{X}\ddot{X}.$
(51)

The above Friedmann equations in the presence of the kinetic energy of the
K-essence scalar field are different from the usual $f(R)$ gravity model. Note
that, if we consider $f(\bar{R},L(X))\equiv f(R)$ and $\bar{G}_{\mu\nu}\equiv
g_{\mu\nu}$ then, the above modified Friedmann equations (LABEL:62) and
(LABEL:63) reduces to the usual Friedmann equations of $f(R)$ gravity, with
$\kappa=1$ and $T_{\mu\nu}$ replaced by $\frac{1}{2}T_{\mu\nu}$ [6, 7] as

$\displaystyle
3H^{2}=\frac{1}{F}[\frac{-\rho}{2}+\frac{RF-f}{2}-3H\dot{R}F_{R}],$ (52)

and

$\displaystyle
2\dot{H}+3H^{2}=\frac{1}{F}[\frac{p}{2}+(\dot{R})^{2}F_{RR}+2H\dot{R}F_{R}+\ddot{R}F_{R}-\frac{f-RF}{2}].$
(53)

##  V Solution to the Friedmann equation using power law

We choose Starobinsky type model [5, 6] to investigate our theory and evaluate
some cosmological values of the universe. This model has achieved popularity
as, the inflationary predictions produced by this theory seems very much
consistent with the observational data. The coefficient of $R^{2}$ curvature
term single-handedly obtains the slow-roll inflation with tremendous success,
without the introduction of outside inflation field by hand. Other reasons for
Starobinksy model to be treated as an important model has been discussed in
[81]. Now we write $f(\bar{R},L)$ as

$\displaystyle f(\bar{R},L)=\bar{R}+\alpha\bar{R}^{2}+L,$ (54)

and $L$ can be written for homogeneous K-essence scalar field as

$\displaystyle L=1-\sqrt{1-2X}=1-\sqrt{1-\dot{\phi}^{2}}.$ (55)

Therefore, we get

$\displaystyle f_{L}=\frac{\partial f}{\partial L}=1,\quad{F}=\frac{\partial
f}{\partial\bar{R}}=1+2\alpha\bar{R},\quad{F}_{\bar{R}}=2\alpha,\quad{F}_{L}=0.$
(56)

Using these values and after some algebraic calculations we can write
Friedmann equation (LABEL:62) as

$\displaystyle(1+2\alpha\bar{R})3H^{2}=-\frac{1}{2}\bar{\rho}(1-\dot{\phi}^{2})+\frac{1}{2}(1-\dot{\phi}^{2})\alpha\bar{R}^{2}+6\alpha
H\dot{\bar{R}}+\dot{\phi}^{2}(1-\sqrt{1-\dot{\phi}^{2}}).$ (57)

Analogous to [20, 21, 22], let us now assume, there exists an exact power–law
solution to the field equations, i.e., the scale factor behaves as

$\displaystyle a(t)=a_{0}t^{m},$ (58)

where $m(>0)$ is a fixed real number.

The definition of $H$ gives us

$\displaystyle
H=\frac{\dot{a}}{a}=\frac{m}{t},\quad\dot{H}=-\frac{m}{t^{2}},\ddot{H}=\frac{2m}{t^{3}},\quad\dot{\ddot{H}}=-\frac{6m}{t^{4}}.$
(59)

Now, taking Eq. (41) into consideration we can evaluate the value of Ricci
scalar as

$\displaystyle\bar{R}=\frac{6}{(1-\dot{\phi}^{2})t^{2}}\Big{[}-m+m^{2}(2-\dot{\phi}^{2})\Big{]},$
(60)

and then using Eq. (38) we get

$\displaystyle\dot{\bar{R}}$ $\displaystyle=$
$\displaystyle\frac{6}{1-\dot{\phi}^{2}}\Big{[}\ddot{H}+4H\dot{H}(1-\dot{\phi}^{2})-2H^{3}\dot{\phi}^{2}\Big{]}=\frac{12}{t^{3}(1-\dot{\phi}^{2})}\Big{[}m-m^{3}\dot{\phi}^{2}-2m^{2}(1-\dot{\phi}^{2})\Big{]}.~{}~{}~{}~{}$
(61)

Now putting the values of Eqs. (58) – (61) in (57), we simply get

$\displaystyle\frac{1}{2}\bar{\rho}(1-\dot{\phi}^{2})$
$\displaystyle=\frac{90\alpha m^{2}}{t^{4}(1-\dot{\phi}^{2})}-\frac{180\alpha
m^{3}}{t^{4}(1-\dot{\phi}^{2})}+\frac{180\alpha
m^{3}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})}$ (62)
$\displaystyle-\frac{108\alpha
m^{4}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})}+\frac{18\alpha
m^{4}\dot{\phi}^{4}}{t^{4}(1-\dot{\phi}^{2})}-\frac{3m^{2}}{t^{2}}+\dot{\phi}^{2}(1-\sqrt{1-\dot{\phi}^{2}}).$

Now from the second Friedmann equation (LABEL:63), we have

$\displaystyle(1+2\alpha\bar{R})\Big{[}2\dot{H}+H^{2}(3-2\dot{\phi}^{2})\Big{]}=\frac{1}{2}\bar{p}(1-\dot{\phi}^{2})+2\alpha\ddot{\bar{R}}$
$\displaystyle+4\alpha
H\dot{\bar{R}}(1-\dot{\phi}^{2})+\frac{1}{2}\alpha\bar{R}^{2}(1-\dot{\phi}^{2})+\dot{\phi}^{2}(1-\sqrt{1-\dot{\phi}^{2}})$
(63)

or

$\displaystyle\frac{1}{2}\bar{p}(1-\dot{\phi}^{2})$
$\displaystyle=-\frac{2m}{t^{2}}+\frac{3m^{2}}{t^{2}}+\frac{2m\dot{\phi}^{2}}{t^{2}}-\frac{5m^{2}\dot{\phi}^{2}}{t^{2}}-\frac{186\alpha
m^{2}}{t^{4}(1-\dot{\phi}^{2})}+\frac{240\alpha
m^{2}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})}$ (64)
$\displaystyle+\frac{84\alpha m^{3}}{t^{4}(1-\dot{\phi}^{2})}-\frac{252\alpha
m^{3}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})}+\frac{12\alpha
m^{4}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})}+\frac{72\alpha
m}{t^{4}(1-\dot{\phi}^{2})}-\frac{42\alpha
m^{4}\dot{\phi}^{4}}{t^{4}(1-\dot{\phi}^{4})}$ $\displaystyle+\frac{96\alpha
m^{3}\dot{\phi}^{4}}{t^{4}(1-\dot{\phi}^{2})}-\frac{1}{2}\dot{\phi}^{2}(1-\sqrt{1-\dot{\phi}^{2}})+\frac{1}{2}\dot{\phi}^{4}(1-\sqrt{1-\dot{\phi}^{2}}).~{}~{}~{}$

For our case, the energy-momentum conservation relation is

$\displaystyle D^{\mu}\bar{T}_{\mu\nu}=0,$ (65)

with $\bar{T}_{\mu\nu}=T_{\mu\nu}[1+D_{\alpha}\phi D^{\alpha}\phi]$.

Now, using Eqs. (44) and (65), we have the conserving equation as

$\displaystyle\dot{\bar{\rho}}=3\frac{\dot{a}}{a}(\bar{\rho}+\bar{p}),$ (66)

where $\bar{\rho}$ and $\bar{p}$ already have been defined. It is essential to
mention here that, $\bar{\rho}$ and $\bar{p}$ are not same as the normal
$\rho$ and $p$.

Now considering the power law, we get from Eq. (66)

$\displaystyle\bar{\rho}=\rho(1+\dot{\phi}^{2})=\rho_{0}t^{-3m(1+\omega)},$
(67)

where $\omega=\frac{\bar{p}}{\bar{\rho}}=\frac{p}{\rho}$.

Now, putting the value of $\bar{\rho}$ in Friedmann equation (62) we have

$\displaystyle\frac{1}{2}\rho_{0}t^{-3m(1+\omega)}$
$\displaystyle=\frac{90\alpha
m^{2}}{t^{4}(1-\dot{\phi}^{2})^{2}}-\frac{180\alpha
m^{3}}{t^{4}(1-\dot{\phi}^{2})^{2}}+\frac{180\alpha
m^{3}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})^{2}}-\frac{108\alpha
m^{4}\dot{\phi}^{2}}{t^{4}(1-\dot{\phi}^{2})^{2}}$ (68)
$\displaystyle+\frac{18\alpha
m^{4}\dot{\phi}^{4}}{t^{4}(1-\dot{\phi}^{2})^{2}}-\frac{3m^{2}}{t^{2}(1-\dot{\phi}^{2})}+\frac{\dot{\phi}^{2}}{(1-\dot{\phi}^{2})}(1-\sqrt{1-\dot{\phi}^{2}}).~{}~{}~{}$

On the other hand, to maintain the energy-momentum conservation, the relation
(32) must be satisfied. So, the effective functional relation (32) for
homogeneous K-essence scalar field reduces to

$\displaystyle 3\dot{\phi}^{2}-2=2\sqrt{1-\dot{\phi}^{2}},$ (69)

where we have used Eqs. (55) and (56).

Solving the above Eq. (69) we have either, $\dot{\phi}^{2}=0$, which is not
acceptable for our case or

$\displaystyle\dot{\phi}^{2}=\frac{8}{9}=0.888=\text{constant}.$ (70)

It should be noted that, the exact solution of field equations (Eq. no. 2 in
[22]) are already obtained by the assumption of power law form of the scale
factor using Starobinsky Model in [22]. The results of that case is

$\displaystyle\rho_{\phi}=\frac{3n^{2}}{t^{2}}-\frac{\rho_{0}}{t^{3n(1+\omega)}}+\frac{54\alpha
n^{2}(2n-1)}{t^{4}},$ (71)

and

$\displaystyle p_{\phi}=\frac{n(2-3n)}{t^{2}}+\frac{18\alpha
n(2n-1)(4-3n)}{t^{4}}-\frac{\omega\rho_{m0}}{t^{3n(1+\omega)}},$ (72)

where $\rho_{\phi}$ and $p_{\phi}$ is the energy density and pressure of the
scalar field and $n$ is synonymous to $m$ for our case.

Rearranging Eq. (71) we get

$\displaystyle\rho_{0}t^{-3n(1+\omega)}=\frac{3n^{2}}{t^{2}}+\frac{108\alpha
n^{3}}{t^{4}}-\frac{54\alpha n^{2}}{t^{4}}-\frac{1}{2}\dot{\phi}^{2}-V(\phi),$
(73)

where they have defined, $\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi)$,
$p_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$, $V(\phi)$ is the scalar
potential.

Singh et al. [22] in their paper have used canonical Lagrangian and usual
field equations of $f(R)$-gravity but, in our case we have used non-canonical
Lagrangian and corresponding field equations (LABEL:40). This is the basic
difference between these two articles. Also, it should be clearly mentioned
that, the scalar field of them is not identical with the K-essence scalar
field.

Now let us concentrate upon the deceleration parameter using the expression

$\displaystyle q=-\frac{1}{H^{2}}\frac{\ddot{a}}{a}=\frac{1}{m}-1,$ (74)

where we have used Eq. (58). From the above expression, it is obvious that for
our present epoch the deceleration parameter should have a negative value to
support the acceleration of the universe. Therefore, we can conclude from Eq.
(74) that the $m$ takes the value greater than $1$. Negative value of $m$ can
not be considered, since observations show the universe is expanding.

As we know that, the value of $\dot{\phi}^{2}$ is less than $1$, so neglecting
the higher order terms $O(\dot{\phi}^{4}$) in Eq. (62) and (64), and using Eq.
(70), we get the equation of state (EOS) parameter of this scenario as:

$\displaystyle\omega=\frac{-(2+13m)t^{2}+\alpha(648+246m-1260m^{2}-96m^{3})}{-3mt^{2}+\alpha
m(810-180m-864m^{2})}.$ (75)

Figure 1: (Color online) Variation of $\rho$ and $p$ with $t$ for different
values of $m$ ($=1.5,2$) and $\alpha$$(=1,3,5,7,9)$ Figure 2: (Color online)
Variation of $\omega$ with $t$ for different values of $m$ ($=1.5,2$) and
$\alpha$$(=1,3,5,7,9)$

The variation of $\rho$ and $p$ (using Eqs. (62) and (64) and omitting
$O(\dot{\phi}^{4}$) terms) with time ($t$) has been plotted in Fig. 1 for
different choices of the positive power law parameter ($m=1.5,2$) and the
positive coefficient of $R^{2}$ in Starobinsky model ($\alpha=1,3,5,7,9$).
Fig. 2 shows the variation of EOS parameter $\omega$ with $t$ for the
aforementioned values of $m$ and $\alpha$. As we know that, the values of
$\rho$ and $p$ should differ in signature for dark energy dominated era and
simultaneously the value of EOS parameter ($\omega$) should approach a value
close to $-1$, therefore, the above two figures concludes that the choice of
positive $m$ and positive $\alpha$ is ruled out for our model to produce dark
energy conditions.

The negative values of $\alpha$ has already been considered in [83] for
$f(R,T)$ gravity. So, let’s check the results of our model for positive value
of $m(=2,3)$ and negative values of $\alpha(=-0.9,-0.7,-0.5,-0.3,-0.1)$. Fig.
3 depicts the variation of $\rho$ and $p$ with time ($t$) for above values of
the parameters. Fig. 4 shows the variation of EOS parameter $(\omega)$ with
time $(t)$. From Fig. 3, it is evident that the value of $\rho$ and $p$ have
the expected nature at a particular region of time. Simultaneously, Fig. 4
produces the anticipated value of $\omega$ which is $-1$ for dark energy
epoch. We discuss the results more elaborately obtained in Figs. 3 and 4 in
the following subsection.

Figure 3: (Color online) Variation of $\rho$ and $p$ with $t$ for different
values of $m$ ($=3,2$) and $\alpha(=-0.9,-0.7,-0.5,-0.3,-0.1)$ Figure 4:
(Color online) Variation of $\omega$ with $t$ for different values of $m$
($=2,3$) and $\alpha$$(=-0.1,-0.3,-0.5,-0.7,-0.9)$

###  V.1 Observational Verification of the Model

Before entering into this section we would like to discuss two significant
works, one has been done by Tripathi et al. [82] and the other one is done by
Moraes et al. [83]. In [82] they have constrained the dark energy models for
low redshifts and compared the data with the observations of Supernova Type Ia
data, Baryon Acoustic Oscillation data and Hubble parameter measurements. On
the other hand, in [83] the authors have studied various cosmological aspects
with the help of Starobinsky model in the framework of $f(R,T)$ gravity. They
have found the nature of material content of the universe, i.e. $\rho$ and $p$
in both decelerated and accelerated regime of the universe.

$m$ | $\alpha$ | $t$ |  $\omega$ ($3\sigma$ confidence) [82] | Observation [82]
---|---|---|---|---
2 | $-0.9$ | $14.96-15.5$ | $-0.95\geq\omega\geq-1.13$ | SNIa+ BAO+ H(z)
3 | $22.8-23.87$
2 | $-0.7$ | $13.2-13.67$
3 | $20.1-21.05$
2 | $-0.5$ | $11.16-11.55$
3 | $17.06-17.8$
2 | $-0.3$ | $8.64-8.95$
3 | $13.22-13.78$
2 | $-0.1$ | $4.98-5.16$
3 | $7.63-7.97$
Table 1: Table for Observational Verification of the Model.

The variation of $\rho$ and $p$ with time obtained in Fig. 3 is quite similar
with the variation obtained in [83], though our models differ from each other.
Fig. 3 shows that at early time ($t\rightarrow 0$) the pressure was positive.
But after a certain value of time it takes negative value which may be
correlated with the effect of the negative pressure fluid, responsible for the
accelerating universe. We have shown a table which depicts that for different
choices of the positive $m$ and the negative $\alpha$, we get a range of $t$
(from Fig. 4) where the value of $\omega$ satisfies with the observational
data of Supernova Type Ia data, Baryon Acoustic Oscillation data and Hubble
parameter measurements (Observational data are taken from [82]). Furthermore,
if we concentrate on the range of $t$ that has been shown in Table 1 and match
those values with Fig. 3, then can be observed that at those particular region
of time, the value of $p$ takes the negative sign whereas the $\rho$ is
positive.

##  VI Energy conditions in $f(\bar{R},L(X))$-Gravity

With the help of the modified field equation (LABEL:40) for $f(\bar{R},L(X))$
theory, the emergent Einstein’s equation (15) can be written as ($\kappa=1$)

$\displaystyle\bar{R}_{\mu\nu}-\frac{1}{2}\bar{G}_{\mu\nu}\bar{R}=T_{\mu\nu}^{eff},$
(76)

where

$\displaystyle T_{\mu\nu}^{eff}$ $\displaystyle=$
$\displaystyle\frac{1}{F}[\frac{1}{2}f_{L}\bar{T}_{\mu\nu}-\frac{1}{2}\bar{R}F\bar{G}_{\mu\nu}-(\bar{G}_{\mu\nu}\bar{\square}-D_{\mu}D_{\nu})F+\frac{1}{2}\bar{G}_{\mu\nu}(f-Lf_{L}(1+D_{\alpha}\phi
D^{\alpha}\phi))$ (77) $\displaystyle-$
$\displaystyle\frac{1}{2}Lf_{L}D_{\mu}\phi D_{\nu}\phi(1+D_{\alpha}\phi
D^{\alpha}\phi)]$ $\displaystyle=$
$\displaystyle\frac{1}{F}[\frac{1}{2}f_{L}\bar{T}_{\mu\nu}+\frac{1}{2}\bar{G}_{\mu\nu}(f-F\bar{R})-(\bar{G}_{\mu\nu}\bar{\square}-D_{\mu}D_{\nu})F-\frac{1}{2}Lf_{L}\bar{G}_{\mu\nu}(1+D_{\mu}\phi
D^{\mu}\phi)^{2}],$

with $F=f_{\bar{R}}=\frac{\partial f(\bar{R},L(X))}{\partial\bar{R}}$.

The trace of the effective energy momentum tensor (77) is

$\displaystyle
T^{eff}=\frac{1}{F}[\frac{1}{2}f_{L}\bar{T}+2(f-F\bar{R})-3\bar{\square}F-2Lf_{L}(1+D_{\mu}\phi
D^{\mu}\phi)^{2}].$ (78)

Now from Eq. (76), we have the emergent Ricci tensor in terms of the effective
energy momentum tensor as

$\displaystyle\bar{R}_{\mu\nu}=T_{\mu\nu}^{eff}-\frac{1}{2}\bar{G}_{\mu\nu}T^{eff}.$
(79)

Let $\bar{u}^{\mu}$ be the tangent vector field to a congruence of time-like
geodesics in the K-essence emergent space-time manifold endowed with the
metric $\bar{G}_{\mu\nu}$ ($\bar{G}_{\mu\nu}\bar{u}^{\mu}\bar{u}^{\nu}=1$),
then the strong energy condition (SEC) (108) in $f(\bar{R},L(X))$ modified
gravity can be expressed as

$\displaystyle\bar{R}_{\mu\nu}\bar{u}^{\mu}\bar{u}^{\nu}=(T_{\mu\nu}^{eff}\bar{u}^{\mu}\bar{u}^{\nu}-\frac{1}{2}T^{eff})\geq
0.$ (80)

On the other hand, if we consider $\bar{k}^{\mu}$ be the tangent vector along
the null geodesic congruence ($\bar{G}_{\mu\nu}\bar{k}^{\mu}\bar{k}^{\nu}=0$),
then the null energy condition (NEC) (110) in $f(\bar{R},L(X))$ gravity is

$\displaystyle\bar{R}_{\mu\nu}\bar{k}^{\mu}\bar{k}^{\nu}=T_{\mu\nu}^{eff}\bar{k}^{\mu}\bar{k}^{\nu}\geq
0.$ (81)

So, considering an additional condition [19]
$\frac{f_{L}(\bar{R},L)}{f_{\bar{R}}(\bar{R},L)}>0$, and the K-essence scalar
field to be homogeneous, i.e., $\phi(x^{i},t)\equiv\phi(t)$ and using the
perfect fluid energy momentum tensor (44), we have the SEC and NEC in the
$f(\bar{R},L(X))$ gravity are

$\displaystyle SEC$ $\displaystyle:$
$\displaystyle~{}~{}\bar{\rho}+3\bar{p}-\frac{2}{f_{L}}(f-F\bar{R})+2L(1+\dot{\phi}^{2})^{2}$
(82) $\displaystyle+$
$\displaystyle\frac{6}{f_{L}(1-\dot{\phi}^{2})}[\ddot{F}(1-\frac{2}{3}\dot{\phi}^{2})+H\dot{F}(1-\dot{\phi}^{2}+\frac{2}{3}\dot{\phi}^{4})]\geq
0,$ $\displaystyle NEC$ $\displaystyle:$
$\displaystyle~{}~{}\bar{\rho}+\bar{p}+\frac{2}{f_{L}}(\ddot{F}-H\dot{F}\dot{\phi}^{2})\geq
0,$ (83)

where $\bar{\rho}=\rho(1+\dot{\phi}^{2})$ and $\bar{p}=p(1+\dot{\phi}^{2})$.

To evaluate the effective density $\bar{\rho}^{eff}$ and effective pressure
$\bar{p}^{eff}$ in the K-essence emergent $f(\bar{R},L(X))$ gravity, we
consider two following equations

$\displaystyle
T_{\mu\nu}^{eff}\bar{u}^{\mu}\bar{u}^{\nu}-\frac{1}{2}\bar{G}_{\mu\nu}T^{eff}\bar{u}^{\mu}\bar{u}^{\nu}=\bar{\rho}^{eff}+3\bar{p}^{eff}$
(84)

and

$\displaystyle
T_{\mu\nu}^{eff}\bar{k}^{\mu}\bar{k}^{\nu}=\bar{\rho}^{eff}+\bar{p}^{eff}.$
(85)

Solving these Eqs. (LABEL:62) and (LABEL:63), we get

$\displaystyle\bar{\rho}^{eff}$ $\displaystyle=$
$\displaystyle\bar{\rho}+\frac{1}{f_{L}}(f-F\bar{R})-\frac{6}{f_{L}(1-\dot{\phi}^{2})}[\frac{1}{3}\ddot{F}\dot{\phi}^{2}+H\dot{F}(1-\frac{1}{3}\dot{\phi}^{4})]-L(1+\dot{\phi}^{2})^{2},$
$\displaystyle\bar{p}^{eff}$ $\displaystyle=$
$\displaystyle\bar{p}-\frac{1}{f_{L}}(f-F\bar{R})+\frac{3}{f_{L}(1-\dot{\phi}^{2})}[\frac{1}{3}\ddot{F}(2-\dot{\phi}^{2})$
(87) $\displaystyle+$ $\displaystyle
H\dot{F}(1-\frac{2}{3}\dot{\phi}^{2}+\frac{1}{3}\dot{\phi}^{4})]+L(1+\dot{\phi}^{2})^{2}.$

From the above Eqs. (LABEL:99) and (87), we have WEC and DEC respectively for
the K-essence emergent $f(\bar{R},L(X))$ gravity as

$\displaystyle WEC$ $\displaystyle:$
$\displaystyle~{}~{}\bar{\rho}+\frac{1}{f_{L}}(f-F\bar{R})-\frac{6}{f_{L}(1-\dot{\phi}^{2})}[\frac{1}{3}\ddot{F}\dot{\phi}^{2}+H\dot{F}(1-\frac{1}{3}\dot{\phi}^{4})]-L(1+\dot{\phi}^{2})^{2}\geq
0,$ $\displaystyle DEC~{}$ $\displaystyle:$
$\displaystyle\bar{\rho}-\bar{p}+\frac{2}{f_{L}}(f-F\bar{R})-\frac{2}{f_{L}(1-\dot{\phi}^{2})}[\ddot{F}(1+\frac{1}{2}\dot{\phi}^{2})$
$\displaystyle+$ $\displaystyle
3H\dot{F}(\frac{3}{2}-\frac{1}{3}\dot{\phi}^{2}-\frac{1}{6}\dot{\phi}^{4})]-2L(1+\dot{\phi}^{2})^{2}\geq
0.$

These energy conditions (82), (83), (LABEL:101) and (LABEL:102) of the
K-essence emergent $f(\bar{R},L(X))$ gravity are different from the usual
$f(R,L_{m})$-gravity (112) and $f(R)$-gravity (111) in the presence of the
K-essence scalar field $\phi$. Also note that if we consider
$f(\bar{R},L(X))\equiv R$ and $\bar{G}_{\mu\nu}\equiv g_{\mu\nu}$, then we can
get back to the usual energy conditions of GR, i.e., $SEC:~{}\rho+3p\geq 0$;
$NEC:~{}\rho+p\geq 0$; $WEC:~{}\rho\geq 0$ and $DEC:~{}\rho\geq|p|$.

One may notice that we briefly have discussed the energy conditions of $f(R)$
and $f(R,L_{m})$ gravity in Appendix.

###  VI.1 Constraints on K-essence emergent $f(\bar{R},L(X))$-gravity

The inequalities of the energy conditions (82), (83), (LABEL:101) and
(LABEL:102) can also be expressed in terms of the deceleration ($q$), jerk
($j$), and snap ($s$) parameters such that the Ricci scalar and its
derivatives for a spatially flat K-essence emergent FLRW geometry (37) are

$\displaystyle\bar{R}=\frac{6}{1-\dot{\phi}^{2}}\left[\dot{H}+H^{2}(2-\dot{\phi}^{2})\right]=\frac{6H^{2}}{1-\dot{\phi}^{2}}\left[1-q-\dot{\phi}^{2}\right],$
(90)
$\displaystyle\dot{\bar{R}}=\frac{6}{1-\dot{\phi}^{2}}\left[\ddot{H}+4H\dot{H}(1-\dot{\phi}^{2})-2H^{3}\dot{\phi}^{2}\right]=\frac{6H^{3}}{1-\dot{\phi}^{2}}\left[(j-q-2)+2\dot{\phi}^{2}(1-2q)\right],$
(91)
$\displaystyle\ddot{\bar{R}}=\frac{6}{1-\dot{\phi}^{2}}[(\dot{\ddot{H}}+4\dot{H}^{2}+4H\ddot{H})-2\dot{\phi}^{2}(2\dot{H}^{2}+3H\ddot{H}-2H^{4}+3H^{2}\dot{H}]$
$\displaystyle=\frac{6H^{4}}{1-\dot{\phi}^{2}}[(s+q^{2}+8q+6)-2\dot{\phi}^{2}(3+3j+10q+2q^{2})],$
(92)

where [19, 11, 74, 75, 76, 77]

$\displaystyle
q=-\frac{1}{H^{2}}\frac{\ddot{a}}{a}~{};~{}j=\frac{1}{H^{3}}\frac{\dot{\ddot{a}}}{a}~{};~{}s=\frac{1}{H^{4}}\frac{\ddot{\ddot{a}}}{a}.$
(93)

Now from Eq. (51), we evaluate the values of $\dot{F}$ and $\ddot{F}$ (using
Eq. (38)) in terms of $q,~{}j,~{}s$ as

$\displaystyle\dot{F}$ $\displaystyle=$
$\displaystyle\frac{6H^{3}}{1-\dot{\phi}^{2}}F_{\bar{R}}[(j-q-2)+2\dot{\phi}^{2}(1-2q)]-HF_{L}L_{XX}\dot{\phi}^{2}(1-\dot{\phi}^{2}),$
(94) $\displaystyle\ddot{F}$ $\displaystyle=$
$\displaystyle\frac{6H^{4}}{1-\dot{\phi}^{2}}F_{\bar{R}}[(s+q^{2}+8q+6)-2\dot{\phi}^{2}(3+3j+10q+2q^{2})]$
(95) $\displaystyle+$
$\displaystyle\frac{36H^{6}}{(1-\dot{\phi}^{2})^{2}}F_{\bar{R}\bar{R}}[(j-q-2)+2\dot{\phi}^{2}(1-2q)]^{2}+H\dot{\phi}^{4}(1-\dot{\phi}^{2})^{2}\left(F_{LL}L_{X}^{2}+F_{L}L_{XX}\right)$
$\displaystyle-
F_{L}L_{X}\dot{\phi}^{2}(1-\dot{\phi}^{2})[\dot{H}-2H^{2}(1-2\dot{\phi}^{2})].~{}~{}$

Therefore, putting these values of $\dot{F}$ and $\ddot{F}$ in the energy
conditions (82), (83), (LABEL:101) and (LABEL:102) we have the energy
conditions in terms of $q$, $j$ and $s$. We can easily check that these energy
conditions in terms of $q$, $j$ and $s$ are also different from the
$f(R,L_{m})$-gravity [19] in the presence of the K-essence scalar field
$\phi$.

###  VI.2 An example of the energy conditions

Considering the Starobinsky Model, i.e., Eq. (54) and hence we obtain the
following results as $f_{L}=1$, $F=1+2\alpha\bar{R}$,
$\dot{F}=2\alpha\dot{\bar{R}}$ and $\ddot{F}=2\alpha\ddot{\bar{R}}$.

Using these results we get the energy conditions from (82), (83), (LABEL:101)
and (LABEL:102) as follows:

$\displaystyle SEC$ $\displaystyle:$
$\displaystyle\bar{\rho}+3\bar{p}+2\left[\alpha\bar{R}^{2}+L\dot{\phi}^{2}(2+\dot{\phi}^{2})\right]+\frac{12\alpha\ddot{\bar{R}}}{1-\dot{\phi}^{2}}\left(1-\frac{2}{3}\dot{\phi}^{2}\right)+\frac{12\alpha
H\dot{\bar{R}}}{1-\dot{\phi}^{2}}\left(1-\dot{\phi}^{2}+\frac{2}{3}\dot{\phi}^{4}\right)\geq
0$ $\displaystyle NEC$ $\displaystyle:$
$\displaystyle\bar{\rho}+\bar{p}+4\alpha\left(\ddot{\bar{R}}-H\dot{\bar{R}}\dot{\phi^{2}}\right)\geq
0$ (97) $\displaystyle WEC$ $\displaystyle:$
$\displaystyle\bar{\rho}-\alpha\bar{R}^{2}-\frac{4\alpha}{1-\dot{\phi}^{2}}\left[\ddot{\bar{R}}\dot{\phi}^{2}-3\dot{\bar{R}}(1-\frac{1}{3}\dot{\phi}^{4})\right]-L\dot{\phi}^{2}(2-\dot{\phi}^{2})\geq
0$ (98) $\displaystyle DEC$ $\displaystyle:$
$\displaystyle\bar{\rho}-\bar{p}-2\alpha\bar{R}^{2}-\frac{4\alpha}{1-\dot{\phi}^{2}}\Big{[}\ddot{\bar{R}}(1-\frac{1}{2}\dot{\phi}^{2})-3H\dot{\bar{R}}(\frac{3}{2}-\frac{1}{3}\dot{\phi}^{2}-\frac{1}{6}\dot{\phi}^{4})\Big{]}-2L\dot{\phi}^{2}(2+\dot{\phi}^{2})\geq
0$

Again, if we put the values of $\bar{R}$, $\dot{\bar{R}}$ and $\ddot{\bar{R}}$
from (90), (91) and (92) in the above equations (LABEL:110), (97), (98) and
(LABEL:113), we easily reconstruct the energy conditions in terms of the
deceleration ($q$), jerk ($j$), and snap ($s$) parameters.

##  VII Discussion and conclusion

In this work, we present a new type of modified theory, viz.
$f(\bar{R},L(X))$-gravity, with a general formalism in the context of the dark
energy (using K-essence emergent geometry) where $\bar{R}$ is the Ricci scalar
of this geometry, $L(X)$ is the DBI type non-canonical Lagrangian with
$X={1\over 2}g^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$, $\phi$ is the
K-essence scalar field. The K-essence emergent metric $\bar{G}_{\mu\nu}$ is
not conformally equivalent to the gravitational metric $g_{\mu\nu}$. This new
type of modified theory is a general mixing between $f(R)$ gravity and
K-essence emergent gravity based on the DBI model.

Let us discuss some salient features of the present study which are as
follows:
(1) It is to be noted that the modified field equation (LABEL:40) is different
from the usual $f(R)$ and $f(R,L_{m})$ gravities. If we consider
$f(\bar{R},L(X))\equiv R$ and $\bar{G}_{\mu\nu}\equiv g_{\mu\nu}$ then we can
easily get back to the standard Einstein field equation. The effective
functional relation for the requirement of the conservation of the energy-
momentum tensor is also different from the $f(R,L_{m})$-gravity. We derive the
modified Friedmann equations for the $f(\bar{R},L(X))$-gravity considering the
background gravitational metric as flat FLRW and the K-essence scalar field
$\phi$ being simply a function of time only, which are quite different from
the Friedmann equations of the standard $f(R)$ gravity.

(2) For the particular choice (viz. Starobinksy type), Eq. (54) of
$f(\bar{R},L)$ and from the requirement of the energy-momentum conservation
(32), the kinetic energy of the K-essence scalar field is a constant. This
value of $\dot{\phi}^{2}$($=0.888$) is less than unity which is comparable
with the range of $\dot{\phi}^{2}$. It is also to be noted that the K-essence
theory can be used to investigate the effects of the presence of dark energy
on cosmological scenarios. In this context, if we consider $\dot{\phi}^{2}$ be
dark energy density in unit of critical density as [26, 27, 28], then the
value of dark energy density, i.e., $\dot{\phi}^{2}=\frac{8}{9}=0.888$
indicates that the present universe is dark energy dominated.

It is well known that the present observational value [79, 80] of dark energy
density is approximately $0.75$. Therefore, we note that in the context of
dark energy regime our result is in good agreement with the observational
data. Now-a-days people believe that the dark energy is one of the reasons for
the accelerating Universe. So, our value of dark energy density may indicate
that the Universe is more accelerating. From Figs. 3 and 4, we also observe
that our model is observationally verified for certain values of parameters.
According to Fig. 3, the negativity of pressure can be achieved after a
certain value of time which may be responsible for the accelerating universe.
The variation of EOS ($\omega$) with time ($t$) in Fig. 4 shows that $\omega$
approaches the negative values which corresponds to observational result [82]
of the present universe.

Also, we would like to put here the following two special aspects which emerge
from of the present investigation:
(i) This model can open up an alternative window to explore the current cosmic
acceleration without a stringent condition of invoking an exotic component as
the dark energy. In other words, this theory seems interesting from a purely
gravitational theory standpoint, rather than the cosmological context of dark
energy whose very existence is still an issue of doubt [78] in the context of
the latest analysis of data from the Planck consortium [79, 80]. However, the
arbitrariness in the choice of different functional forms of $f(\bar{R},L(X))$
based on DBI Lagrangian gives rise to the problem of how to constrain the many
possible $f(\bar{R},L(X))$ gravity theories on physical grounds. In this
context, we have shed some light on this issue by discussing some constraints
on general $f(\bar{R},L(X))$ gravity from the so-called energy conditions.

(ii) Also, we have derived the null, strong, weak and dominant energy
conditions in the framework of $f(\bar{R},L(X))$ gravity from the Raychaudhuri
equations. These energy conditions are different from the usual $f(R,L_{m})$
and $f(R)$ theories. With the help of the specific form of $f(\bar{R},L(X))$,
we also have derived these energy conditions in the $f(\bar{R},L(X))$-gravity.

However, we intend to report on this interesting theory in the near future
encompassing the multifarious aspects of Cosmology.

##  Appendix A Appendix:

###  A.1 : The modified field equation in $f(R,L_{m})$-Gravity

Let us consider here the well known $f(R)$ and $f(R,L_{m})$ gravity, where $R$
is the Ricci scalar with respect to the gravitational metric $g_{\mu\nu}$ and
$L_{m}$ is the matter Lagrangian. The total action for the $f(R)$ gravity is
[6, 7]

$\displaystyle S=\frac{1}{2\kappa}\int
d^{4}x\sqrt{-g}f(R)+S_{M}(g_{\mu\nu},\psi),$ (100)

where $S_{M}$ is matter term, $\psi$ denotes the matter fields, $\kappa=8\pi
G$, $G$ is the gravitational constant, $g$ is the determinant of the
gravitational metric and $R~{}(=g^{\mu\nu}R_{\mu\nu})$ is the Ricci scalar.

Varying which with respect to the gravitational metric we achieve the modified
field equation as

$\displaystyle
f^{\prime}(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-[\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\Box]f^{\prime}(R)=\kappa
T_{\mu\nu},$ (101)

with

$\displaystyle T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta S_{M}}{\delta
g^{\mu\nu}},$ (102)

where $f^{\prime}(R)=\frac{\partial f(R)}{\partial R}$, $\nabla_{\mu}$ is
covariant derivative with respect to the gravitational metric and
$\square\equiv\nabla^{\mu}\nabla_{\mu}$.

On the other hand, the action for the $f(R,L_{m})$ gravity is [18, 19]

$\displaystyle S=\frac{1}{2\kappa}\int d^{4}x\sqrt{-g}f(R,L_{m}),$ (103)

where $f(R,L_{m})$ is an arbitrary function of the Ricci scalar $R$, and the
Lagrangian density corresponding to matter $L_{m}$. The energy-momentum tensor
is

$\displaystyle
T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}L_{m})}{\delta
g^{\mu\nu}}=-2\frac{\partial L_{m}}{\partial
g^{\mu\nu}}+g_{\mu\nu}L_{m},~{}~{}$ (104)

where the Lagrangian density $L_{m}$ is only matter dependent on the metric
tensor components $g_{\mu\nu}$.

The modified field equations of the $f(R,L_{m})$-gravity model is

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

where $f_{R}(R,L_{m})=\partial f(R,L_{m})/\partial R$ and
$f_{L_{m}}(R,L_{m})=\partial f(R,L_{m})/\partial L_{m}$. However, if
$f(R,L_{m})=R/2+L_{m}$, then the above Eq. (105) reduces to the usual field
equation $R_{\mu\nu}-(1/2)g_{\mu\nu}R=\kappa T_{\mu\nu}$.

###  A.2 : Brief review of energy conditions in General Relativity

Following most of the techniques of [10, 11, 12, 13, 19, 14, 73], we will
derive the energy conditions for modified ($f(R)$, $f(R,L_{m})$, etc.)
gravities. From these theories we can approach to the Null Energy Condition
(NEC) and Strong Energy Condition (SEC) in the context of GR. The origin of
these energy conditions comes from the Raychaudhuri equations. Let $u^{\mu}$
be the tangent vector field to a congruence of time-like geodesics in a space-
time manifold endowed with a metric $g_{\mu\nu}$. Therefore, the Raychaudhuri
equation [68, 69, 70, 71, 72] is

$\displaystyle\frac{d\theta}{d\tau}=-\frac{1}{3}\theta^{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}+\omega_{\mu\nu}\omega^{\mu\nu}-R_{\mu\nu}u^{\mu}u^{\nu},$
(106)

where $R_{\mu\nu}$ is the Ricci tensor corresponding to the metric
$g_{\mu\nu}$, and $\theta$, $\sigma_{\mu\nu}$, and $\omega_{\mu\nu}$ are the
expansion, shear, and rotation associated with the congruence, respectively.
While in the case of a congruence of null geodesics defined by the vector
field $k^{\mu}$, the Raychaudhuri equation [71] is given by

$\displaystyle\frac{d\theta}{d\tau}=-\frac{1}{2}\theta^{2}-\sigma_{\mu\nu}\sigma^{\mu\nu}+\omega_{\mu\nu}\omega^{\mu\nu}-R_{\mu\nu}k^{\mu}k^{\nu}.$
(107)

These equations are purely based on geometric statements, and as such it makes
no reference to any gravitational field equations. In other words, the
Raychaudhuri equation can be thought of as geometrical identities which do not
depend on any gravitational theory. These equations are provide the evolution
of the expansion of a geodesic congruence. However, since the GR field
equations relate $R_{\mu\nu}$ to the energy-momentum tensor $T_{\mu\nu}$, the
combination of Einstein and Raychaudhuri equations can be used to restrict
energy-momentum tensors on physical ground. Indeed, the shear is a “spatial”
tensor, given by $\sigma^{2}\equiv\sigma_{\mu\nu}\sigma^{\mu\nu}\geq 0$.

Thus it is clear from Raychaudhuri equation that for any hypersurface
orthogonal congruences ($\omega_{\mu\nu}\equiv 0$) the condition for
attractive gravity (convergence of timelike geodesics or geodesic focusing)
reduces to ($R_{\mu\nu}u^{\mu}u^{\nu}\geq 0$), which by virtue of Einstein’s
equation implies

$\displaystyle
R_{\mu\nu}u^{\mu}u^{\nu}=(T_{\mu\nu}-\frac{T}{2}g_{\mu\nu})u^{\mu}u^{\nu}\geq
0,$ (108)

where $T$ is trace of the energy momentum tensor $T_{\mu\nu}$ ($\kappa=1$).
Here Eq. (108) is nothing but the SEC stated in a coordinate-invariant way in
terms of $T_{\mu\nu}$ and vector fields of fixed (time-like) character. Thus,
in the context of GR, the SEC ensures the fact that the gravity is attractive.
In particular, for a perfect fluid of density $\rho$ and pressure $p$

$\displaystyle T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}-pg_{\mu\nu}$ (109)

and the restriction given by Eq. (108) takes the familiar form for the SEC,
i.e., $\rho+3p\geq 0$.

On the other hand, the condition for the convergence (geodesic focusing) of
hypersurface orthogonal ($\omega_{\mu\nu}\equiv 0$) congruences of null
geodesics along with Einstein’s equation implies

$\displaystyle R_{\mu\nu}k^{\mu}k^{\nu}=T_{\mu\nu}k^{\mu}k^{\nu}\geq 0$ (110)

which is the condition for NEC written in a coordinate-invariant way.

Thus, in GR the NEC ultimately encodes the null geodesic focusing due to the
gravitational attraction. For the energy-momentum tensor of a perfect fluid
(109) the above condition (110) reduces to the well-known form of the NEC,
i.e., $\rho+p\geq 0$.

The Weak Energy Condition (WEC) states that $T_{\mu\nu}u^{\mu}u^{\nu}\geq 0$
for all time-like vectors $u^{\mu}$, or equivalently for perfect fluid it is
$\rho>0$ and $\rho+p>0$. The Dominant Energy Condition (DEC) includes the WEC
as well as the additional requirement that $T_{\mu\nu}u^{\mu}$ is a non space-
like vector i.e., $T_{\mu\nu}T^{\nu}_{\lambda}u^{\mu}u^{\lambda}\leq 0$. For a
perfect fluid, these conditions together are equivalent to the simple
requirement that $\rho\geq|p|$, the energy density must be nonnegative, and
greater than or equal to the magnitude of the pressure.

In $f(R)$-gravity [10, 11], the energy conditions for perfect fluid are given
by

$\displaystyle
SEC:~{}~{}\rho+3p-f+Rf^{\prime}+3(\ddot{R}+\dot{R}H)f^{\prime\prime}+3\dot{R}^{2}f^{\prime\prime\prime}\geq
0,$ $\displaystyle
NEC:~{}~{}\rho+p+(\ddot{R}-\dot{R}H)f^{\prime\prime}+\dot{R}^{2}f^{\prime\prime\prime}\geq
0,$ $\displaystyle
WEC:~{}~{}\rho+\frac{1}{2}(f-Rf^{\prime})-3\dot{R}Hf^{\prime\prime}\geq 0,$
$\displaystyle
DEC:~{}\rho-p+f-Rf^{\prime}-(\ddot{R}+5\dot{R}H)f^{\prime\prime}-\dot{R}^{2}f^{\prime\prime\prime}\geq
0,$ (111)

where $f^{\prime}=\frac{\partial f(R)}{\partial R}$.

In $f(R,L_{m})$-gravity [19], the energy conditions are

$\displaystyle SEC$ $\displaystyle:$
$\displaystyle~{}~{}\rho+3p-\frac{2}{f_{L_{m}}}[f-Rf^{\prime}]+\frac{6}{f_{L_{m}}}[\dot{R}^{2}f^{\prime\prime\prime}+\ddot{R}f^{\prime\prime}+H\dot{R}f^{\prime\prime}]-2L_{m}\geq
0.$ $\displaystyle NEC$ $\displaystyle:$
$\displaystyle~{}~{}\rho+p+\frac{2}{f_{L_{m}}}[\dot{R}^{2}f^{\prime\prime\prime}+\ddot{R}f^{\prime\prime}]\geq
0,$ $\displaystyle WEC$ $\displaystyle:$
$\displaystyle~{}~{}\rho+\frac{1}{f_{L_{m}}}[f-Rf^{\prime}]-\frac{6}{f_{L_{m}}}H\dot{R}f^{\prime\prime}+L_{m}\geq
0,$ $\displaystyle DEC$ $\displaystyle:$
$\displaystyle~{}~{}\rho-p+\frac{2}{f_{L_{m}}}[f-Rf^{\prime}]-\frac{2}{f_{L_{m}}}[\dot{R}^{2}f^{\prime\prime\prime}+\ddot{R}f^{\prime\prime}+6H\dot{R}f^{\prime\prime}]+2L_{m}\geq
0,$ (112)

where $f^{\prime}=\frac{\partial f(R,L_{m})}{\partial R}$.

## Acknowledgement

A.P. and G.M. acknowledge the DSTB, Government of West Bengal, India for
financial support through the Grants No.: 322(Sanc.)/ST/P/S&T/16G-3/2018 dated
06.03.2019. The authors also thank to Dr. Pradipta Panchadhyayee, Associate
Professor, Department of Physics, Prabhat Kumar College, Contai for his
valuable suggestions.

## References

  * [1] H. Weyl, A new extension of relativity theory, Ann. Phys. 59, 101 (1919).
  * [2] A.S. Eddington, The mathematical theory of relativity, Cambridge University Press, Cambridge (1923).
  * [3] R. Utiyama, B.S. DeWitt, Renormalization of a classical gravitational field interacting with quantized matter fields, J. Math. Phys. 3, 608 (1962).
  * [4] N.D. Birrell, P.C.W. Davies, Quantum fields in curved spacetime, Cambridge University Press, Cambridge (1982).
  * [5] A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91, 9 (1980).
  * [6] T.P. Sotiriou, V. Faraoni, $f(R)$ theories of gravity, Rev. of Mod. Phys. 82, 451 (2010).
  * [7] A. De Felice, S. Tsujikawa, $f(R)$ theories, Living Rev. Relativity 13, 3 (2010).
  * [8] P.K.S. Dunsby et al., $\lambda$CDM universe in $f(R)$ gravity, Phys. Rev. D 82, 023519 (2010).
  * [9] A. Mukherjee, N. Banerjee, Acceleration of the universe in f(R) gravity models, Astrophys. Space Sci. 352, 893 (2014).
  * [10] K. Atazadeh et al., Energy conditions in $f(R)$ gravity and Brans-Dicke theories, Int. J. Mod. Phys. D 18, 1101 (2009).
  * [11] J. Santos et al., Energy conditions in $f(R)$ gravity, Phys. Rev. D 76, 083513 (2007).
  * [12] S. Capozziello et al., The role of energy conditions in $f(R)$ cosmology, Phys. Lett. B 781, 99 (2018).
  * [13] J. Wang et al., Energy conditions and stability in generalized $f(R)$ gravity with arbitrary coupling between matter and geometry, Phys. Lett. B 689, 133 (2010).
  * [14] S.E. Perez Bergliaffa, Phantom and Quintessence fields coupled to scalar curvature in general $f(R)$ gravity theory, Phys. Lett. B 642, 311 (2006).
  * [15] F.D. Albareti et al., The Raychaudhuri equation in homogeneous cosmologies, JCAP 03, 012 (2014).
  * [16] F.D. Albareti et al., On the non-attractive character of gravity in f(R) theories, JCAP 07, 009 (2013).
  * [17] K.D. Krori et al., Raychaudhuri equation, big bang and accelerating universe, Ind. J. Phys. 82(5), 531 (2008).
  * [18] T. Harko, F.S.N. Lobo, $f(R,L_{m})$ gravity, Eur. Phys. J. C 70, 373, (2010).
  * [19] J. Wang, K. Liao, Energy conditions in $f(R,L_{m})$ gravity, Class. Quantum Gravit., 29, 215016 (2012).
  * [20] N. Goheer, R. Goswami P. Dunsby, K. Ananda, Coexistence of matter dominated and accelerating solutions in $f(G)$ gravity, Phys. Rev. D 79, 121301(R) (2009).
  * [21] N. Goheer, J. Larena, P.K.S. Dunsby, Power-law cosmic expansion in $f(R)$ gravity models, Phys. Rev. D 80, 061301(R) (2009).
  * [22] C.P. Singh, V. Singh, Power-law expansion and scalar field cosmology in higher derivative theory, Int. J. Theor. Phys. 51, 1889 (2012).
  * [23] M. Born, L. Infeld, Foundations of the new field theory, Proc. Roy. Soc. Lond A 144, 425 (1934).
  * [24] W. Heisenberg, On the theory of explosive showers in cosmic rays, Zeit. Phys. 113 61 (1939).
  * [25] P.A.M. Dirac, An extensible model of the electron, Proc. R. Soc. Lond. A 268, 57 (1962).
  * [26] D. Gangopadhyay, G. Manna, The Hawking temperature in the context of dark energy, Eur. Phys. Lett. 100 49001 (2012).
  * [27] G. Manna, D. Gangopadhyay, The Hawking temperature in the context of dark energy for Reissner–Nordström and Kerr background, Eur. Phys. J. C 74 2811 (2014).
  * [28] G. Manna, B. Majumder, The Hawking temperature in the context of dark energy for Kerr–Newman and Kerr–Newman–AdS backgrounds, Eur. Phys. J. C 79, 553 (2019).
  * [29] M. Visser, C. Barcelo, S. Liberati, Analogue models of and for gravity, Gen. Relativ. Gravit. 34 1719 (2002).
  * [30] E. Babichev, V. Mukhanov, A. Vikman, Escaping from the black hole?, JHEP 0609, 061 (2006).
  * [31] E. Babichev, V. Mukhanov, A. Vikman, K- essence, superluminal propagation, causality and emergent geometry, JHEP 0802 101 (2008).
  * [32] A. Vikman, K-essence: Cosmology, causality and Emergent Geometry, Dissertation an der Fakultat fur Physik, Arnold Sommerfeld Center for Theoretical Physics, der Ludwig-Maximilians-Universitat Munchen, Munchen, den 29.08.2007.
  * [33] E. Babichev, V. Mukhanov, A. Vikman, Looking beyond the Horizon, WSPC-Proceedings (October 23, 2008).
  * [34] R.J. Scherrer, Purely kinetic K-essence as unified dark matter, Phys. Rev. Lett. 93 011301 (2004).
  * [35] L.P. Chimento, Extended tachyon field, Chaplygin gas, and solvable K-essence cosmologies, Phys. Rev. D 69 123517 (2004).
  * [36] S. Nojiri et. al., K-essence $f(R)$ gravity inflation, Nuc. Phys. B 941, 11 (2019).
  * [37] S.D. Odintsov et. al., $f(R)$ gravity K-essence late-time phenomenology, Phys. Dark Univ., 29,100563, (2020)
  * [38] V.K. Oikonomou, N. Chatzarakis, The phase space of K-essence $f(R)$ gravity theory, Nuc. Phys. B 956, 115023 (2020).
  * [39] N. Bahcall et al., The cosmic triangle: Revealing the state of the universe, Science 284, 1481 (1999) [and references therein].
  * [40] C. Armendariz-Picon et al., Essentials of K-essence, Phys. Rev. D 63, 103510, (2001)
  * [41] C. Armendariz-Picon et al., Dynamical solution to the problem of a small cosmological constant and late-time cosmic acceleration, Phys. Rev. Lett. 85, 4438 (2000)
  * [42] Jin U Kang et al., Attractor scenarios and superluminal signals in K-essence cosmology, Phys. Rev. D.76, 083511 (2007)
  * [43] R.R. Caldwell et al., Cosmological imprint of an energy component with general equation of state, Phys. Rev. Lett. 80, 1582 (1998).
  * [44] J. Frieman et al., Cosmology with ultralight pseudo Nambu-Goldstone bosons, Phys. Rev. Lett. 75, 2077, (1995).
  * [45] P.J.E. Peebles, B. Ratra, Cosmology with a time-variable cosmological constant, Astrophys. J. Lett. 325, L17 (1988).
  * [46] B. Ratra, P.J.E. Peebles, Cosmological consequences of a rolling homogeneous scalar field, Phys. Rev. D 37, 3406 (1988).
  * [47] I. Zlatev, P.J. Steinhardt, A tracker solution to the cold dark matter cosmic coincidence problem, Phys. Lett. B 459, 570 (1999).
  * [48] P. J. Steinhardt, L. Wang, I. Zlatev, Cosmological tracking solutions, Phys. Rev. D59, 123504 (1999).
  * [49] I. Zlatev, Limin Wang, P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant, Phys. Rev. Lett. 82, 896 (1999)
  * [50] J.K. Erickson et al., Measuring the speed of sound of quintessence’, Phys. Rev. Lett. 88, 121301, (2002).
  * [51] S. DeDeo, R.R. Caldwell, P.J. Steinhardt, Effects of the sound speed of quintessence on the microwave background and large scale structure, Phys. Rev. D 67, 103509 (2003).
  * [52] R. Bean, O. Dore, Probing dark energy perturbations: the dark energy equation of state and speed of sound as measured by WMAP, Phys. Rev. D 69, 083503 (2004).
  * [53] R. Yang et al., The evolution of the power law K-essence cosmology, Astrophys. Space Sci. 356, 399 (2014).
  * [54] A.D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108, 389 (1982).
  * [55] A. Albrecht, P.J. Steinhardt, Cosmology for grand unified theories with radiatively induced symmetry breaking, Phys. Rev. Lett. 48, 1220 (1982).
  * [56] G. Dvali, S.-H. Henry Tye, Brane inflation, Phys. Lett. B 450, 72 (1999).
  * [57] S. Kachru et al., Towards inflation in string theory, JCAP 0310, 013 (2003).
  * [58] M. Alishahiha, E. Silverstein, D. Tong, DBI in the sky: non-Gaussianity from inflation with a speed limit, Phys. Rev. D 70, 123505 (2004).
  * [59] E. Silverstein, D. Tong, Scalar speed limits and cosmology: Acceleration from Deceleration, Phys. Rev. D 70, 103505 (2004).
  * [60] X. Chen, Multithroat brane inflation, Phys. Rev. D 71, 063506 (2005).
  * [61] S. Weinberg, Effective field theory for inflation, Phys. Rev. D 77, 123541 (2008).
  * [62] X. Chen et. al., Observational signatures and non-Gaussianities of general single-field inflation, JCAP 0701, 002 (2007)
  * [63] R.M. Wald, General Relativity, Univ. Chicago Press (1984).
  * [64] C. Misner, K.S. Thorne, J. Wheeler, Gravitation, W.H. Freeman and Company (1970).
  * [65] T. Koivisto, A note on covariant conservation of energy–momentum in modified gravities, Class. Quantum Gravit. 23 4289 (2006).
  * [66] T. Harko, Modified gravity with arbitrary coupling between matter and geometry, Phys. Lett. B 669, 376 (2008).
  * [67] D. Gangopadhyay, G. Manna, Cosmology in presence of dark energy in an emergent gravity scenario, arXiv:gr-qc/1502.06255 (2015).
  * [68] A. Raychaudhuri, Relativistic cosmology. I, Phys. Rev. 98, 1123 (1955).
  * [69] A. Raychaudhuri, Relativistic and Newtonian cosmology, Z. Astrophysik, 43, 161 (1957).
  * [70] A. Raychaudhuri, Singular state in relativistic cosmology, Phys. Rev. 106, 172 (1957).
  * [71] M. Blau, Lecture Notes on General Relativity, http://www.blau.itp.unibe.ch/GRLecturenotes.html (2020).
  * [72] I. Bhattacharyya, S. Ray, A generalized form of the Raychaudhuri equation, Int. J. Mod. Phys. D, 30, 2150092 (2021)
  * [73] S. Carroll, Spacetime and Geometry: An Introduction to General Relativity, Addison Wesley, New York (2004).
  * [74] E.R. Harrison, Observational tests in cosmology, Nature (London) 260, 591 (1976).
  * [75] P. Landsberg, Q in cosmology, Nature (London) 263, 217 (1976).
  * [76] M. Visser, Jerk, snap and the cosmological equation of state, Class. Quantum Gravit. 21, 2603 (2004).
  * [77] M. Visser, Cosmography: Cosmology without the Einstein equations, Gen. Relativ. Gravit. 37, 1541 (2005).
  * [78] J.T. Nielsen, A Guffanti, S. Sarkar, Marginal evidence for cosmic acceleration from Type Ia supernovae, Sci. Rep. 6 3559 (2016).
  * [79] P.R. Ade et al., Planck 2015 results. XIII. Cosmological parameters, Astron. Astrophys. 594, A13 (2016).
  * [80] N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Planck Collaboration, Astron. Astrophys. 641, A6 (2020).
  * [81] A. Kehagias et al., Remarks on the Starobinsky model of inflation and its descendants, Phys. Rev. D 89 043527 (2013).
  * [82] A. Tripathi et. al., Dark energy equation of state parameter and its evolution at low redshift, JCAP 06, 12 (2017)
  * [83] P.H.R.S. Moraes et. al. A cosmological scenario from the Starobinsky model within the $f(R,T)$ Formalism, Adv. Astron. 8574798 (2019).