#
Observational constraints on non-minimally coupled curvature-matter models of
gravity from the analysis of Pantheon data

Biswajit Jana<EMAIL_ADDRESS>Ramakrishna Mission Vivekananda
Educational and Research Institute, Belur Math, Howrah 711202, India  Anirban
Chatterjee Corresponding Author<EMAIL_ADDRESS>&<EMAIL_ADDRESS>Indian Institute of Technology Kanpur, Kanpur 208016, India  Kumar Ravi
<EMAIL_ADDRESS>Ramakrishna Mission Vivekananda Educational and Research
Institute, Belur Math, Howrah 711202, India  Abhijit Bandyopadhyay
<EMAIL_ADDRESS>Ramakrishna Mission Vivekananda Educational and
Research Institute, Belur Math, Howrah 711202, India

(August 29, 2024)

###### Abstract

We considered non-minimally coupled curvature-matter models of gravity in a
FRW universe filled with perfect fluid and investigated its cosmological
implications in the light of Pantheon compilation of 1048 Supernova Ia data
points along with 54 data points from Observed Hubble Data. The non-minimal
curvature-matter coupling has been introduced by adding a term
$\int\left[\lambda R^{n}\mathcal{L}_{m}\right]\sqrt{-g}d^{4}x$ to the usual
action for Einstein gravity involving the Einstein Hilbert action and
minimally coupled matter action. We investigate the observational constraints
on the non-minimal models by choosing two different kinds of parametrization
of fluid-pressure profiles using a dimensionless parameter $k$. The interplay
of the three parameters $\lambda,n,k$ plays a pivotal role in testing the
consistency of non-minimally coupled fluid-curvature scenarios with the
observed data. We found there exist large domains in the
($\lambda,n,k$)-parameter space for which models with non-minimal curvature-
matter mixing stand as viable cosmological models reproducing the observed
features of late-time cosmic evolution. We also commented on the possibility
of ‘gravitationally induced particle creation’ in the context of SNe Ia data.

##  1 Introduction

Observation of type Ia Supernovae (SNe Ia) events and their luminosity
distance and redshift measurements by Riess et.al. [1] and Perlmutter et.al.
[2] have independently confirmed that present universe is in an accelerated
state of expansion and a transition from a decelerated to the current
accelerated phase happened during it’s late time phase of evolution. The
reason for this late-time cosmic acceleration is attributed to a hypothetical
unclustered form of energy, called ‘Dark Energy’ (DE), which has become a
general label for the late time cosmic acceleration. On the other hand,
observations of rotation curves of spiral galaxies [3], gravitational lensing
[4], Bullet Cluster [5] and other colliding clusters provide evidence in
favour of existence of non-luminous matter called ‘Dark Matter’ (DM) in the
universe. Such matter indirectly manifest their existence only through
gravitational interactions. Experiments like WMAP [6] and Planck [7] have
revealed that dark energy (DE) and dark matter (DM) together contribute $\sim
96\%$ of total energy density of the present-day universe, with $\sim$69$\%$
and $\sim$27$\%$ as their respective shares. The rest of the 4$\%$
contribution comes from radiation and baryonic matter.

To support his theory of the static universe, Einstein initially considered a
cosmological constant term $\Lambda g_{\mu\nu}$ in the geometric part of his
field equations of general relativity, but later discarded it, in favour of
Hubble’s observation of expanding universe. However, with the discovery of
late-time cosmic acceleration in the later part of last century, the
cosmological constant term again came under the spotlight because of its
potential to provide a straight-way solution to Einstein’s field equation
resulting in an accelerated expansion. The corresponding phenomenological
model that fits the observed features of cosmic acceleration is called
$\Lambda$-CDM model, where CDM refers to cold dark matter content of the
present universe. Unfortunately, the $\Lambda$-CDM model is plagued with the
coincidence problem [8] problem and the fine-tuning problem [9]. These
limitations motivate investigation of other models to account for dark energy.
One class of such models are field theoretic models of dark energy involving
modification of energy-momentum tensor in the Einstein’s field equations due
to presence of a field as one content of the universe, other than matter and
radiation. Such class of models include both quintessence [10, 11, 12, 13, 14,
15, 16, 17, 18], and $k$-essence [19, 20, 21, 22, 23, 24, 25, 26] models of
scalar fields. Another class of models involve modification of the geometric
part of Einstein’s equations, i.e. the Einstein-Hilbert action in order to
address issue of late time cosmic acceleration. Such models include $f(R)$
gravity models, scalar-tensor theories, Gauss-Bonnet gravity, and braneworld
models of dark energy [27]. In most of these scenarios, the universe is
considered to be homogeneous and isotropic at large scales, with its metric
described by the Friedmann-Robertson-Walker (FRW) metric specified by a time-
dependent scale factor $a(t)$ and a curvature constant $K$. There also exist
approaches, where an inhomogeneous universe, described by a perturbed FRW
metric, has been considered as the spacetime background.

In this article, we examine a model of the universe with its content as an
ideal hypothetical fluid, characterised by its energy-density $(\rho)$ and
pressure $(p)$, which is non-minimally coupled to the spacetime curvature. The
non-minimal curvature-matter coupling has been introduced by adding a term
$\int d^{4}x\sqrt{-g}\lambda R^{n}{\cal L}_{m}$ to the usual action for
Einstein gravity involving the Einstein Hilbert action and minimally coupled
matter action. $R$ is the Ricci Scalar and ${\cal L}_{m}$ (referred to as
matter Lagrangian) is the Lagrangian of the hypothetical fluid. The coupling
constant ($\lambda$) and the power ($n$) of $R$ in the non-minimal coupling
term, are considered as the parameters of the model. We explored the
constraints on these parameters from the combined analysis of Pantheon
compilation of 1048 SNe Ia data points [28] and 54 data points from Observed
Hubble data [29, 30, 31, 32, 33].

In the general framework of non-minimal models, the geometric part of
Einstein’s equation is altered by modifying the Einstein-Hilbert action to
$\int d^{4}x\sqrt{-g}f(R)$, replacing $R$ by a general function $f(R)$ [34,
35, 36]. In the context of modified $f(R)$ theories of gravity, it has been
shown in [37] that the covariant derivative of energy-momentum tensor is non-
vanishing ($\nabla_{\mu}T^{\mu\nu}\neq 0$), when a $f(R)$-${\cal L}_{m}$
(curvature-matter) coupling is present in the theory which may cause a
departure from geodesic motion manifesting existence of a new force.
Consequences of such models in stellar equilibrium have been investigated in
[38, 39]. An equivalence between a suitable scalar theory and the generic
model with non-minimal coupling between curvature and matter have been
established in [40, 41, 42, 43, 44, 45, 46]. A viability criterion to avoid
instabilities in such models has been derived in [47]. The effect of
curvature-matter couplings on the dynamics of particles and fields have been
analysed in [48]. In [38, 39], it has been argued that the choice ${\cal
L}_{m}=p$, ($p$ being the pressure of the matter fluid) is a ‘natural’ one for
the matter of Lagrangian density, as it reproduces the equations of
hydrodynamics of perfect fluid and in the context of curvature-matter coupling
such choices result in vanishing of the extra force. However, there may be
some other choices resulting in vanishing the extra force (see [36, 49] for
references). In this work we have used ${\cal L}_{m}=p$ and in the framework
of our considerations the field equations following from the total action
involving $S_{\rm non-minimal}$ establishes connection between
$a(t),\rho(t),p(t)$ through their time derivatives and various other functions
like the Hubble function $H(t)$ and the Ricci scalar $R(t)$. We exploit these
equations for investigating constraints on non-minimal models from the
analysis of observed data.

Using a model-independent construction of Hubble’s function $H$ in terms of
redshift $z$ without invoking any specific cosmological models, we obtained
the dependence of Hubble’s parameter $H$ on $z$ from the analysis of the SNe
Ia data (Pantheon + OHD). We used the FRW scale factor $a$ to be normalised to
unity at present epoch ($z=0$). Using the relation $1/a=1+z$, the temporal
behaviour of cosmological quantities expressed in terms of $z$ may be
equivalently expressed in terms of $a$ or another dimensionless time parameter
chosen as $\tau\equiv\ln a$, which we used in our analysis. Exploiting the
profile of the Hubble’s parameter for the redshift domain accessible in the
SNe Ia observations, obtained from the analysis of cosmological data set, we
obtain the time profile of FRW scale factor $a$ and it’s higher-order
derivatives $\dot{a}$, $\ddot{a}$. This also enables us to express temporal
behaviour of the Ricci scalar $R$ and it’s time derivatives which are involved
in our analysis.

To investigate the observational constraints on models with non-minimal
curvature-matter couplings (specified in terms of parameters $\lambda$ and $n$
in this paper), we considered two different types of fluid pressure models
with temporal behaviour of pressure $p$ modelled as $p\sim e^{ak}$
(exponential model) and $p\sim a^{k}$ (power law model) where $k$ is a
dimensionless parameter. Consequently the three parameters ($\lambda,n,k$)
appear in the evolution equations and control the model-based computations of
$\rho$ and $p$. Using the observational inputs from SNe Ia data, we obtained
the regions in the model parameter space ($\lambda,n,k$), for which the
computed energy density ($\rho$) of the fluid remains positive for all time
during the late time phase of evolution. This is an essential requirement for
viability of cosmological models. We have also seen that there exist a small
range of parameter values around $(\lambda=-0.1,n=0.2,k=1)$ for which the
computed temporal profiles of $\rho$ and $p$ mimic the corresponding profiles
obtained from the analysis of the data using $\Lambda$-CDM model in the
context of usual minimal coupling scenario.

The non-vanishing covariant derivative of the energy-momentum tensor in the
context of non-minimally coupled curvature-matter scenarios implies an
exchange of energy between curvature and matter sectors. We computed the rate
of the energy exchange for different benchmark values of the model parameters
($\lambda,n,k$) and found a monotonous decrease in the absolute value of the
energy exchange rate as time approaches towards the present epoch. This
implies that the rate of exchange of energy between the two sectors is more
significant during relatively early phases of late-time cosmic evolution. The
cosmological implications of curvature-matter coupling scenarios from the
viewpoint of thermodynamics have been investigated in [50, 51, 52]. There it
has been shown that curvature-matter coupling may be responsible for
generation of a large amount of comoving entropy during late-time evolutionary
phase of the universe leading to a possible interpretation of the exchange of
energy between curvature and matter sector in terms of gravitationally induced
particle creation in FRW universe. We have shown from the analysis of SNe Ia
data that for some values of the model parameters ($\lambda,n,k$) possibility
of such an interpretation is always allowed.

Rest of the paper is arranged in the following manner. In Sec. 2 we discussed
the framework of non-minimally coupled $f(R)$ model in the context of a flat
FRW universe and obtained the corresponding generic modified evolution
equations of the universe in presence of non-minimal curvature-matter
coupling. We choose two different types of fluid-pressure models, and obtained
the equations corresponding to each of the models. In Sec. 3 we discussed the
methodology of analysis of Pantheon + OHD data sets for obtaining temporal
behaviour of different relevant cosmological quantities during the late-time
phase of cosmic evolution. In Sec. 4 we presented the constraints on model
parameters $\lambda$, $n$ and $k$ obtained by using the results extracted from
the analysis of the observed data and discussed the results. We summarize the
conclusions of the paper in Sec. 5.

##  2 Theoretical framework of curvature-matter coupling scenario

In the framework of modified theories of gravity with non-minimal curvature-
matter coupling, the action may be written in the form [34]

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int\left[{1\over
2}f_{1}(R)+\left[1+\lambda f_{2}(R)\right]{\cal
L}_{m}\right]\sqrt{-g}\;d^{4}x\,,$ (1)

where $f_{1}(R)$ and $f_{2}(R)$, in general, are two arbitrary functions of
the Ricci scalar $R$ and ${\cal L}_{m}$ is the Lagrangian density for matter.
$g$ is the determinant of the spacetime metric tensor $g_{\mu\nu}$ and
$\lambda$ is a constant representing the strength of the non-minimal coupling
between curvature term $f_{2}(R)$ and matter lagrangian ${\cal L}_{m}$.
Following the approach of metric formalism in the context of the modified
theories of gravity, the variation of the action with respect to the field
$g_{\mu\nu}$ yields the modified field equations as

$\displaystyle F_{1}R_{\mu\nu}-{1\over
2}f_{1}g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}F_{1}+g_{\mu\nu}\square F_{1}$
$\displaystyle=$ $\displaystyle(1+\lambda f_{2})T_{\mu\nu}-2\lambda F_{2}{\cal
L}_{m}R_{\mu\nu}$ (2)
$\displaystyle+2\lambda(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square){\cal
L}_{m}F_{2}\,,$

where we put $8\pi G=1$ ($G$ = Gravitational Constant), $F_{i}=df_{i}/dR$
($i=1,2$) and $T_{\mu\nu}$ is the matter energy-momentum tensor given by

$\displaystyle T_{\mu\nu}=-{2\over\sqrt{-g}}{\delta(\sqrt{-g}{\cal
L}_{m})\over\delta(g^{\mu\nu})}\,.$ (3)

In this article, we investigated the scenario of non-minimal coupling between
curvature and matter with geometric part of field equation driven by pure
Einstein gravity i.e. with $f_{1}(R)=R$ and considered the matter part to be
non-minimally coupled (with coupling strength $\lambda$) to curvature through
the function $f_{2}(R)=R^{n}$ (where $n$ is some constant). In this model, the
matter part of the universe is described by a perfect fluid characterised by
energy density $\rho$ and pressure $p$. Following the comprehensive discussion
in [35, 36], we assume $\mathcal{L}_{m}=p$, which is a ‘natural choice’ for
Lagrangian density for perfect fluids, which correctly reproduces the
hydrodynamical equations for a perfect fluid. This particular choice has
interesting consequences in the analysis of curvature-matter coupling
scenarios implying vanishing of extra force owing to departure from motion
along the geodesic which may arise due to non-vanishing covariant derivative
of $T_{\mu\nu}$ in the context of curvature-matter coupling [37]. There are,
however, other choices of $\mathcal{L}_{m}$ investigated in [34, 36].

We conducted an exploration of the intricacies associated with non-minimal
coupling between the curvature and matter by employing a power law form of the
function $f_{2}(R)=R^{n}$, where $n$ is a constant exponent. Apart from the
strength of the coupling $\lambda$, such consideration introduces a single
additional parameter $n$ into the analytical framework. These two parameters
$(\lambda,n)$, along with a third parameter $k$, used to express temporal
variation of the fluid pressure, constitute a 3-dimensional parameter space
$(\lambda,n,k)$ facilitating the exploration of non-minimally coupled theories
with fluids having a wide range of pressure variation modes, with an optimum
number of parameters. It’s important to recognize that the power law
representation $f_{2}(R)=R^{n}$ for any real $n$ may be represented through
the expansion of $f_{2}(R)$ around a non-zero reference point $R=R_{0}$ in the
form of infinite series: $f_{2}(R)=\sum_{m}c_{m}(R-R_{0})^{m}$, where $m$ is
an integer, encompassing negative values as well, when $n$ is negative.
Nevertheless, the scenario of non-minimal couplings may theoretically be
explored using particular $f_{2}(R)$ expressions consisting of a finite number
of terms incorporating diverse integral powers (both positive and negative) of
$R$. Consideration of such forms would then introduce more parameters (the
coefficients of the different integral powers of $R$) into the analysis scheme
compared to the power-law version of $f_{2}(R)$. Under such considerations,
obtaining observational constraints on non-minimally coupled scenarios
necessitate exploring parameter spaces of significantly higher dimensions.
This also requires assessment of distinct parameter combinations for each
specific expression of $f_{2}(R)$ to encompass the wide spectrum of $f_{2}(R)$
patterns. In contrast, opting for the power law form $f_{2}(R)=R^{n}$ (as
adopted for this study), covers an alternative array of varied $f(R)$ profiles
(demonstrating either monotonic increase or decrease) achieved solely by
modifying the exponent $n$. This choice renders the investigation more
manageable in terms of the triad $(\lambda,n,k)$. We have also come to
recognize that the adoption of the power law form $f_{2}(R)=R^{n}$ has proven
advantageous in terms of computational time and efficiency, in pinpointing the
precise value of $n$ for which the evaluated temporal profile of energy
density and fluid pressure mimic the corresponding patterns seen in the
$\Lambda$-CDM model. To summarise our considerations for the present analysis,
we have chosen $f_{1}(R)=R,f_{2}(R)=R^{n},F_{2}(R)=nR^{n-1},{\cal L}_{m}=p$.
For such choices Eq. (2) can be expressed as

$\displaystyle R_{\mu\nu}-{1\over 2}Rg_{\mu\nu}$ $\displaystyle=$
$\displaystyle(1+\lambda
R^{n})T_{\mu\nu}-2\lambda(nR^{n-1})pR_{\mu\nu}+2\lambda(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square)p(nR^{n-1})\,.$
(4)

The energy density $\rho$ and pressure $p$ of the fluid are respectively
obtained from ‘$00$’ and ‘$ii$’ components of the energy-momentum tensor
$T_{\mu\nu}$. The ‘$00$’ component of Eq. (4) is

$\displaystyle R_{00}-{1\over 2}Rg_{00}$ $\displaystyle=$
$\displaystyle(1+\lambda R^{n})T_{00}-2\lambda p(nR^{n-1})R_{00}-6\lambda
H(\dot{p}nR^{n-1}+pn(n-1)R^{n-2}\dot{R})\,.$ (5)

For a FRW spacetime background the above equation takes a form which after
some rearrangements may be written as

$\displaystyle\rho$ $\displaystyle=$ $\displaystyle\frac{1}{1+\lambda
R^{n}}\left[3H^{2}+6n\lambda\\{(n-1)HR^{n-2}\dot{R}-(H^{2}+\dot{H})R^{n-1}\\}p+6n\lambda
HR^{n-1}\dot{p}\right]\,.$ (6)

On the other hand, the ‘$ii$’-component of Eq. (4) is

$\displaystyle R_{11}-{1\over 2}Rg_{11}$ $\displaystyle=$
$\displaystyle(1+\lambda R^{n})T_{11}-2\lambda
pnR^{n-1}R_{11}+2\lambda(\nabla_{1}\nabla_{1}-g_{11}\square)(pnR^{n-1})\,,$
(7)

which in the FRW spacetime background takes the form

$\displaystyle-(2\dot{H}+3H^{2})$ $\displaystyle=$
$\displaystyle\Big{[}(1+\lambda R^{n}-2\lambda
nR^{n-1}(\dot{H}+3H^{2})+4n(n-1)\lambda HR^{n-2}\dot{R}$ (8)
$\displaystyle+2\lambda\\{n(n-1)(n-2)R^{n-3}\dot{R}^{2}+n(n-1)R^{n-2}\ddot{R}\\}\Big{]}p$
$\displaystyle+4n\lambda\Big{\\{}HR^{n-1}+(n-1)R^{n-2}\dot{R}\Big{\\}}\dot{p}+2n\lambda
R^{n-1}\ddot{p}\,.$

In the framework of curvature-matter coupling in FRW spacetime background Eqs.
(6) and (8) are the master equations. Note that Eqs. (6) and (8) involve time
derivatives of the pressure of the fluid which is a consequence of the non-
minimal curvature-matter coupling. Setting the non-minimal coupling constant
$\lambda=0$ in the master equation, we retrieve the usual Friedmann equations
which do not contain any derivative term of pressure.

To investigate the observational constraints on such curvature-matter coupling
models, we considered two types of fluid pressure models with specific
temporal profile of the fluid pressure. In FRW background, we choose to
express the modelled temporal profiles of the fluid pressure in terms of the
FRW scale factor. We discuss below the two models referred to as exponential
and power-law models as per the nature of dependences of the pressure on the
scale factor in the corresponding models.

Exponential model : In this model we take the fluid pressure $p$ as
$p=p_{0}\exp(ak)$, where $k$ is a dimensionless parameter and $p_{0}$ is a
constant having the dimension of pressure. $p_{0}e^{k}$ gives the pressure of
the fluid at present epoch $a=1$, in this model. We then have
$\dot{p}=kp\dot{a}$ and $\ddot{p}=k^{2}p\dot{a}^{2}+kp\ddot{a}$. Using
$\dot{p}=kp\dot{a}$ in Eq. (6), we may express the energy density for a given
value of $\lambda,n,k$ and pressure $p$ corresponding to this model as

$\displaystyle\rho(t;\lambda,n,k)=\frac{1}{1+\lambda
R^{n}}\left[3H^{2}+6n\lambda\Big{\\{}(n-1)HR^{n-2}\dot{R}-(H^{2}+\dot{H})R^{n-1}+kHR^{n-1}\dot{a}\Big{\\}}p\right]\,.$
(9)

Note that the above equation also implies dependence of the
$\rho(t;\lambda,n,k)$ on the parameter $p_{0}$ due to occurrence of the
pressure term $p=p_{0}\exp(ak)$ in the right hand side of Eq. (9). But note
that, we may use Eq. (8) with $\dot{p}=kp\dot{a}$ and
$\ddot{p}=k^{2}p\dot{a}^{2}+kp\ddot{a}$ to express the pressure
$p=p_{0}\exp(ak)$ as

$\displaystyle p(t;\lambda,n,k)$ $\displaystyle=$
$\displaystyle-(2\dot{H}+3H^{2})\,\Big{[}(1+\lambda R^{n}-2\lambda
nR^{n-1}(\dot{H}+3H^{2})+4n(n-1)\lambda HR^{n-2}\dot{R}$ (10)
$\displaystyle+2\lambda\Big{\\{}n(n-1)(n-2)R^{n-3}\dot{R}^{2}+n(n-1)R^{n-2}\ddot{R}\Big{\\}}$
$\displaystyle+4n\lambda
k\Big{\\{}HR^{n-1}+(n-1)R^{n-2}\dot{R}\Big{\\}}\dot{a}+2n\lambda
kR^{n-1}(k\dot{a}^{2}+\ddot{a})\Big{]}^{-1}\,.$

We may now use this expression of Eq.(10) for $p$ in Eq. (9) to get rid of its
explicit $p_{0}$ dependence. So that $\rho(t;\lambda,n,k)$ in Eq. (9) is
computable at any $t$ for any given choice of parameter set $(\lambda,n,k)$.

Power-law model : In this model we take $p=p_{0}a^{k}$, where $k$ is again a
dimensionless constant and $p_{0}$ is a constant, which in this model gives
the value of pressure at present epoch ($a=1$). Here we thus have
$\dot{p}=kpH$ and $\ddot{p}=p(k^{2}H^{2}+k\dot{H})$. Using $\dot{p}=kpH$ in
Eq. (6) we may similarly express the energy density for a given set of values
of $\lambda,n,k$ and pressure $p$ as

$\displaystyle\rho(t;\lambda,n,k)=\frac{1}{1+\lambda
R^{n}}\left[3H^{2}+6n\lambda\Big{\\{}(n-1)HR^{n-2}\dot{R}-(H^{2}+\dot{H})R^{n-1}+kH^{2}R^{n-1}\Big{\\}}p\right]\,.$
(11)

Here also, to get rid of the parameter $p_{0}$ which appears in the pressure
term $p=p_{0}a^{k}$ in the right hand side, we exploit the Eq. (8) with
$\dot{p}=kp\dot{a}$ and $\ddot{p}=k^{2}p\dot{a}^{2}+kp\ddot{a}$ to express the
instantaneous values of pressure $p=p_{0}a^{k}$ in terms of parameters
($\lambda,n,k$) as

$\displaystyle p(t;\lambda,n,k)$ $\displaystyle=$
$\displaystyle-(2\dot{H}+3H^{2})\,\Big{[}(1+\lambda R^{n}-2\lambda
nR^{n-1}(\dot{H}+3H^{2})+4n(n-1)\lambda HR^{n-2}\dot{R}$ (12)
$\displaystyle+2\lambda\Big{\\{}n(n-1)(n-2)R^{n-3}\dot{R}^{2}+n(n-1)R^{n-2}\ddot{R}\Big{\\}}$
$\displaystyle+4n\lambda\Big{\\{}HR^{n-1}+(n-1)R^{n-2}\dot{R}\Big{\\}}kH+2n\lambda
R^{n-1}(k^{2}H^{2}+k\dot{H})\Big{]}^{-1}$

and use this expression for pressure $p$ in Eq. (11) to evaluate instantaneous
value of energy density for any given set of parameters $(\lambda,n,k)$.

Apart from the model parameters $(\lambda,n,k)$, the expressions for $\rho$
and $p$ (in both exponential and power law models) involve other cosmological
parameters - the FRW scale factor $a$, its time derivatives, the Hubble
parameter $H$ and its time derivatives, the Ricci scalar
$R(=12H^{2}+6\dot{H}$) and its time derivatives. We have obtained the temporal
behaviour of all these cosmological quantities during the late time phase of
the cosmic evolution from a combined analysis of recently released Pantheon
(SNe Ia) data. The accessible range of red-shift in the date sets
corresponding to Pantheon data set is $0\leq z\leq 2.3$ corresponds to a
cosmic time domain $0.23\leq t\leq 1$, where $t$ is normalised to unity at the
present epoch. Using the time dependences of the said cosmological quantities
as extracted from the observed data, we obtained the temporal behaviour of the
energy density and pressure of the fluid for both exponential and power law
models using Eqs. (9), (10), (11) and (12). We obtained range of values of
parameters $(\lambda,n,k)$ for which the evaluated values of energy density
function $\rho(t;\lambda,n,k)$ remains positive at all time. The reach of the
recent SNe Ia data accommodates within itself the scenario of matter-curvature
non-minimal coupling for the constrained values of the model parameters.

##  3 Obtaining constraints on the cosmological quantities from combined
analysis of Pantheon and Observed Hubble data

The cosmological data from Supernovae Ia observation provides information
about late time cosmic evolution of the universe. In this section we briefly
outline the technical details of the analysis of the observational data which
involves Pantheon (SNe Ia) data involving 1048 data points and Observed Hubble
Data (OHD) involving 54 data points [29, 30, 31, 32, 33] and discussed how we
finally extracted the temporal behaviour of the FRW scale factor $a$ and its
derivatives which are essential ingredients for constraining the non-minimally
coupled matter-curvature models as discussed in Sec. 2.

The SNe Ia data provides an estimation of luminosity distances ($d_{L}$) at
the redshift values ($z$) corresponding to different SNe Ia events. The
‘Pantheon Sample’ [28] is a compilation of the subset of 279 PS1 SNe Ia data
points (over a redshift range $0.03<z<0.68$) along with useful compilations of
SNe Ia data from SDSS [53], SNLS [54], various low-z [55], and HST samples
[56] to form the largest combined sample of SNe Ia consisting of a total of
1048 SNe Ia over the redshift range $0.01<z<2.3$. We have used the above
mentioned data samples to construct the Hubble function $H(z)$ without
invoking any specific cosmological models. For such a model-independent
construction, we use Padé approximant of order (2,1) [57] for luminosity
distance function, which is a closed-form parametrization of the luminosity
distance in terms of the redshift expressed as

$\displaystyle d_{L}(z,\alpha,\beta)=\frac{c}{H_{0}}\left(\frac{z(1+\alpha
z)}{1+\beta z}\right)\,,$ (13)

where $c$ is the speed of light and $H_{0}$ is the value of the Hubble
parameter at the present epoch which may be defined through a dimensionless
quantity $h$ by $H_{0}=100h$ km s-1 Mpc-1. $\alpha$ and $\beta$ are the
parameters to be determined from cosmological data. The Pantheon sample of
1048 data points provide the redshift, apparent magnitude ($m$) at maximum
brightness of the SNe Ia events, and also the covariance and correlations
among the different data points. The observed data at any redshift $z_{i}$ is
expressed through the distance modulus $\mu_{\rm obs}(z_{i})$ which is given
in terms of the absolute magnitude $M$ and observed apparent magnitudes
$m_{\rm obs}(z_{i})$ by the relation

$\displaystyle\mu_{\rm obs}(z_{i})$ $\displaystyle=$ $\displaystyle m_{\rm
obs}(z_{i})-M\,.$ (14)

The form of the distance modulus $\mu_{\rm th}(z,\alpha,\beta)$ corresponding
to the assumed parametric form of $d_{L}(z,\alpha,\beta)$ given in Eq. (13)
may be obtained by using the expression

$\displaystyle\mu_{\rm th}(z,\alpha,\beta)$ $\displaystyle=$ $\displaystyle
5\log_{10}\left[\frac{H_{0}d_{L}(z,\alpha,\beta)}{c}\right]+\mu_{0}=5\log_{10}\left[\frac{z(1+\alpha
z)}{1+\beta z}\right]+\mu_{0}\,,$ (15)

where $\mu_{0}=42.38-5\log_{10}h$. Computing $\mu_{\rm
th}(z_{i},\alpha,\beta)-\mu_{\rm obs}(z_{i})$ for the data points $i$, we may
find the $\chi^{2}$-function for the SN-data date defined by the relation

$\displaystyle\chi^{2}_{0}$ $\displaystyle=$
$\displaystyle\sum_{i,j=1}^{N}\Big{[}\mu_{\rm th}(z_{i},\alpha,\beta)-\mu_{\rm
obs}(z_{i})\Big{]}C^{-1}_{ij}\Big{[}\mu_{\rm th}(z_{j},\alpha,\beta)-\mu_{\rm
obs}(z_{j})\Big{]}\,,$ (16)

where $N=1048$ is the total number of data points considered and $C$ is the
covariance matrix of the data as released in [28], which includes both
statistical and systematic uncertainties. Using Eqs. (14) and (15), the
quantities $\mu_{\rm th}-\mu_{\rm obs}$ appearing in Eq. (16) may be expressed
as

$\displaystyle\Big{[}\mu_{\rm th}(z_{i},\alpha,\beta)-\mu_{\rm
obs}(z_{i})\Big{]}$ $\displaystyle=$ $\displaystyle
5\log_{10}\left[\frac{z_{i}(1+\alpha z_{i})}{1+\beta z_{i}}\right]-m_{\rm
obs}(z_{i})+M^{\prime}\,,$ (17)

where $M^{\prime}\equiv\mu_{0}+M=42.38-5\log_{10}h+M$. Since the Pantheon data
set comprises of apparent magnitudes only so this data-set alone cannot
constrain $H_{0}$. The parameter $H_{0}(=100h$ km s-1 Mpc${}^{-1})$ enters in
the $\chi^{2}$ function through the parameter $M^{\prime}$. Marginalising over
the parameter $M^{\prime}$, called the nuisance parameter, we define an
appropriate $\chi^{2}$ function for this data set [58] as

$\displaystyle\chi^{2}_{\rm SN}$ $\displaystyle=$
$\displaystyle-2\ln\int_{-\infty}^{\infty}\exp\left[-\frac{1}{2}\chi^{2}_{0}(\alpha,\beta,M^{\prime})\right]dM^{\prime}\,,$
(18)

which may be put in the following form

$\displaystyle\chi^{2}_{\rm SN}$ $\displaystyle=$ $\displaystyle
P-\frac{Q^{2}}{R}+\ln(R/2\pi)\,,$ (19)

where $P=(m_{\rm obs}-m_{\rm th})^{T}C^{-1}(m_{\rm obs}-m_{\rm th})$,
$Q=(m_{\rm obs}-m_{\rm th})^{T}C^{-1}\mathbf{1}$ and
$R=\mathbf{1}^{T}C^{-1}\mathbf{1}$. Here $(m_{\rm obs}-m_{\rm th})$ represents
a column matrix of the residuals with $i^{\rm th}$ entry as $\Big{[}m_{\rm
th}(z_{i},\alpha,\beta)-m_{\rm obs}(z_{i})\Big{]}$ and $(m_{\rm obs}-m_{\rm
th})^{T}$ denotes the corresponding transposed matrix. $m_{\rm obs}(z_{i})$ is
the observed value of apparent magnitude at redshift $z_{i}$ and corresponding
theoretical value $m_{\rm th}(z)$ is given by the formula $m_{\rm
th}(z)=5\log_{10}\left[\frac{z(1+\alpha z)}{1+\beta z}\right]+25$. The symbol
$\mathbf{1}$ represents a column array of ones of same length as $m_{\rm
obs}$.

In our analysis we have also used 31 redshift vs Hubble parameter data points
from chronometer observations [59] and 23 data points from the line of sight
Baryonic Acoustic Oscillations(BAO) data [29, 30, 31, 32, 33]. For a
cosmological model-independent construction of the Hubble parameter, we use
the expression

$\displaystyle
H(z,\alpha,\beta)=\left[\frac{1}{c}\frac{d}{dz}\left\\{\frac{d_{L}(z,\alpha,\beta)}{(1+z)}\right\\}\right]^{-1}\,,$
(20)

where $d_{L}(z,\alpha,\beta)$ is as given in Eq. (13). The residual for the
Observed Hubble data involving total 54 data points fitting is given by

$\displaystyle\chi^{2}_{\rm OHD}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{54}\left[\frac{H(z_{i},\alpha,\beta)-H_{\rm
obs}(z_{i})}{\sigma_{i}}\right]^{2}\,,$ (21)

where $H_{\rm obs}(z_{i})$ and $\sigma_{i}$ give the observed value of Hubble
parameter and its corresponding uncertainty respectively at redshift $z_{i}$
corresponding to the $i^{\rm th}$ data point of OHD data set.

To get estimates of the parameters $H_{0},\,\alpha,\,\beta$ along with their
uncertainties, we perform a Markov chain Monte Carlo (MCMC) Bayesian parameter
estimation using a uniform prior to all the parameters. We used the python
packages, available in the public domain, emcee [60] and GetDist [61] for
generating generate MCMC samples and for plotting posterior distributions of
$H_{0}$, $\alpha$, and $\beta$ respectively. After performing various tests
for the independence, convergence of the MCMC samples and thinning of the
samples accordingly, we get following estimates from the combined analysis of
Pantheon data and OHD. In Fig. 1 we presented the posterior distribution plot
of $H_{0}$, $\alpha$, and $\beta$ which depicts the corresponding 1$\sigma$
and 2$\sigma$ uncertainties for these parameters.

$\displaystyle\alpha=1.23^{+0.06}_{-0.05},\quad\beta=0.45^{+0.03}_{-0.03},\quad{\rm
and}\quad H_{0}=68.31^{+1.03}_{-1.03}\mbox{ km s${}^{-1}$ Mpc${}^{-1}$}\,.$
(22)

Figure 1: The posterior distribution plot of $H_{0}$, $\alpha$, and $\beta$

The obtained best-fit values of the parameters $\alpha$, $\beta$ and $H_{0}$
along with their respective uncertainties are then exploited to compute the
redshift dependences (temporal profiles) of the Hubble parameter $H$ and the
normalised Ricci Scalar $R/H_{0}^{2}$ making use of Eq. (20) and
$R=12H^{2}+6\dot{H}^{2}$. We have shown the plots of $H$ and $R/H_{0}^{2}$ vs
$z$ respectively in the left and right panels of Fig. 2. The solid lines in
Fig. 2 correspond to the best-fit curve of each figure. The $1\sigma$ and
$3\sigma$ uncertainties in the obtained dependences are also shown by dashed
lines.

Figure 2: Left Panel: Best-fit curve with 1$\sigma$ and 3$\sigma$
uncertainties obtained for $H(z)$ vs $z$ from analysis of Pantheon + OHD data.
Observed values of $H$ (with error bars) corresponding to the 54 data points
from OHD are also shown. Right Panel: Best-fit curve with 1$\sigma$ and
3$\sigma$ uncertainties obtained for $R(z)/H_{0}^{2}$ vs $z$ from the analysis

The profile of the $H(z)$ function, thus extracted from the analysis of the
observational data, may further be used to compute the temporal behaviour of
the FRW scale factor $a(t)$ and it’s time derivatives $\dot{a}$, $\ddot{a}$.
The steps for this computation are briefly outlined below. The FRW scale $a$
is normalised to unity at the present epoch ($z=0$) and is related to redshift
by the relation $1/a=1+z$. Since $H=\dot{a}/a$, this corresponds to
$dt=-\frac{dz}{(1+z)H(z)}$ which on integration gives

$\displaystyle\frac{t(z)}{t_{0}}$ $\displaystyle=$ $\displaystyle
1-\frac{1}{t_{0}}\int_{z}^{0}\frac{dz^{\prime}}{(1+z^{\prime})H(z^{\prime})}\,,$
(23)

where $t_{0}$ denotes the present epoch which is also normalised to unity at
present epoch ($t_{0}=1$). Using the values of $H(z)$ corresponding to the
$H(z)$ profile as depicted in left panel of Fig. 2, we numerically compute the
integral in the right hand side of Eq. (23). Using Eq. (23) and the equation
$1/a=1+z$, we may obtain numerically compute simultaneous values of $a$ and
$t$ at any given redshift $z$ which amounts to obtaining values of
$a(t)$-vs-$t$ eliminating $z$ from the two equations. This leads to extraction
of the temporal behaviour of the scale factor $a(t)$ from the observational
data for the late time domain of the cosmic evolution accessible through
Supernova Ia observations.

To perform this, we vary $z$ from zero (present epoch) to $\sim 2.4$ (i.e.
within the domain of $z$ relevant for Pantheon data set) in small steps
($\Delta z=0.01$). We numerically compute value of the integral in Eq. (23)
and also the value of $a(z)=1/(1+z)$ at each $z$-step, to obtain the sets of
values ($t(z),a(z)$) at each step. Using the normalisation of scale factor
$a(t)$ as $a=1$ at present epoch ($z=0$ or $t=1$), we found that the range
$0<z<2.3$ corresponds to $t$-range: $1>t(z)>0.23$. Values of ($t(z),a(z)$)
over the domain $0<z<2.3$ provides temporal behaviour of scale factor over the
time range $0.23<t<1$. Using the obtained profile of $a(t)$, we also obtain a
profile of $\dot{a}(t)$ and $\ddot{a}(t)$ using direct numerical
differentiation techniques. We have shown the plots of the profiles $a(t)$,
$\dot{a}(t)$ and $\ddot{a}(t)$ respectively in the left, middle and right
panels of Fig. 3. The appearance of a minima at $t\sim 0.53$ in the time-
profile of $\dot{a}$ (middle panel) or equivalently,the corresponding change
of sign of $\ddot{a}$ in the plots of right panel figure clearly signifies the
transition from decelerated to an accelerated phase of expansion during the
late time of cosmic evolution.

Figure 3: Temporal behaviour of scale factor at the best-fit of Pantheon data
sets

##  4 Results and Discussion

In this section we present the results regarding the observational constraints
on non-minimally coupled curvature-matter models for two types of fluid
pressure scenarios - the exponential and power law profiles as discussed in
Sec. 2. The energy density of the fluid in the context of the two scenarios
are given by Eqs. (9) and Eq. (11) respectively. The temporal evolution of the
energy density $\rho(t;\lambda,n,k)$ has been expressed in terms of
cosmological quantities like $a,\dot{a},H,R$ etc. The expression also involves
the dimensionless parameters: $\lambda,n,k$. Temporal profile of the
cosmological quantities during the late time phase of the cosmic evolution
have been extracted from observational data and presented in Sec. 3. The
constant $\lambda$ gives the strength of the non-minimal coupling between
matter and curvature sector. The parameter $k$ is involved in the modelling of
fluid pressure and $n$ denotes the power of the gravity term
($f_{2}(R)=R^{n}$) that is coupled to the matter lagrangian. The energy
density of a fluid always being a positive quantity, the constraints on
parameters $(\lambda,n,k)$ for such models come from imposition of the
constraint $\rho(t;\lambda,n,k)>0$ for all $t$, where $\rho$ at any $t$ is
evaluated with the values of cosmological quantities ($a,\dot{a},H,R$) at that
$t$ as obtained from the analysis of observational data.

We presented the observational constraints on the parameters $(\lambda,n,k)$
by depicting the allowed area of the $k$-$n$ parameter space for certain
chosen values of the coupling parameter $\lambda$. Obtaining these allowed
regions required a thorough scanning of the parameter space and the pragmatic
approach involves predefining a range for the parameters $n$ and $k$ that
requires scanning, alongside selecting a scope for the parameter $\lambda$. In
this study, we purposefully choose the relevant range of the coupling strength
$|\lambda|$ spanning from 0.1 to 10. This choice is motivated by our intention
to comprehensively investigate the cumulative impact of all terms involved in
the primary action. When the coupling strength takes on higher values
($|\lambda|>10$), the significance of $f_{1}(R)=R$ diminishes compared to the
prevalence of other terms. Conversely, for lower coupling values
($|\lambda|<0.1$), the impact of non-minimal coupling becomes overshadowed by
the prominence of other terms within the action. We, thus, purposefully choose
to focus on benchmark values of $\lambda$ that fall within the above mentioned
range in order to illustrate our findings. When determining the range of
parameter space to be scanned for obtaining observationally allowed ranges of
$n$ and $k$, we systematically investigated various distinct regions within
the parameter space. In the parameter space domain corresponding to
exceedingly high or low values of $n$ and $k$ (beyond the limits of
$-10\leqslant n\leqslant 10$ and $-20\leqslant k\leqslant 20$), we did not
observe any distinctive, disconnected and closed domains which are
observationally allowed or disallowed, that could provide novel and
significant insights into those specific portions of the parameter space.
Hence, in presenting the outcomes in Fig. 4, we have depicted a confined
region within the parameter space, defined by $-10\leqslant n\leqslant 10$ and
$-20\leqslant k\leqslant 20$. Within this defined realm, conspicuous patterns
of various disallowed domains have come to light, providing a scope to
comprehend the interrelations between the parameters $n$ and $k$ in congruence
with observations.

We presented the results for both the fluid pressure models (exponential and
power law) for four benchmark values: $\lambda=-0.1,0.1,1,10$. The allowed
regions in the parameter space, which correspond to $\rho>0$, are shown by
shaded regions in Fig. 4. The left and right panels of the figure correspond
to the parameter space constraints obtained for exponential and power-law
fluid-pressure models respectively. We observe from Fig. 4 that, for both
classes of fluid pressure models, a wide region of the explored portion of the
parameter space is allowed which in turn implies that the Supernova Ia data
(Pantheon) and Observed Hubble data robustly allows non-minimally coupled
matter-curvature scenarios. However for negative values of $\lambda$, the
positive values of the $n$ (which is the power of $R$ in the matter-curvature
coupling term) gets severely constrained.

Figure 4: Left Panel: Allowed region of $k$-$n$ parameter space for
$\lambda=-0.1,0.1,1,10$ for exponential model (discussed in text). Right
Panel: Allowed region of $k$-$n$ parameter space for $\lambda=-0.1,0.1,1,10$
for power law model (discussed in text)

In Eqs. (10) and (12) we have also expressed the temporal profile of the fluid
pressure $p(t;\lambda,n,k)$ for exponential and power law models respectively.
To see the time evolution of energy density and pressure in the context of
non-minimally coupled matter-curvature scenarios we have shown the plots of
the temporal profile of energy density and pressure for both fluid models in
Fig. 5 for certain benchmark values of the parameter set $(\lambda,n,k)$
chosen from the allowed domains corresponding to both fluid models.

Figure 5: Left Panel: Temporal behaviour of energy density (in the left) and
pressure (in the right) for different benchmark values of the parameters
$(\lambda,n,k)$ for exponential model (discussed in text). Right Panel:
Temporal behaviour of energy density (in the left) and pressure (in the right)
for different benchmark values of the parameters $(\lambda,n,k)$ for power law
model (discussed in text)

For convenience, we use a dimensionless time parameter $\tau$ defined as
$\tau=\ln a$ to show the time profile of the energy density and pressure in
Fig. 5. The late-time domain of cosmic evolution is accessible in SNe Ia
observations of Pantheon sample which corresponds to the range
$-1.18\leq\tau\leq 0$ with $\tau=0$ corresponding to the present epoch
$(a=1)$. We also normalise the energy density $\rho(\tau)$ and pressure
$p(\tau)$ by the energy density value at present epoch $\rho_{0}$ to get rid
of dimensions and plotted $\rho(\tau)/\rho_{0}$ and $p(\tau)/\rho_{0}$ as a
function of $\tau$ for different choices of the parameters $k,n,\lambda$. In
the left (right) panel of Fig. 5 we showed these plots for
$\rho(\tau)/\rho_{0}$ and $p(\tau)/\rho_{0}$ and for the exponential (power
law) model. All the curves are shown in Fig. 5 correspond to $k=1$. The
4-curves in each of plots correspond to 4 different choices of the set
$(\lambda,n)$ viz. (0.1, 1), (1,1), (10,1), (-0.1,0.2), all of which are well
within the observationally allowed region of parameter values for each of the
models as depicted in Fig. 4. For comparison, in these plots, we also showed
the best-fit (solid curve) and $3\sigma$ uncertainties (shaded region) of the
energy density and pressure profiles that result from usual analysis of
observational data corresponding to the $\Lambda-$CDM model.

We see from Fig. 5 that, for exponential and power-models the energy density
and pressure profile corresponding to parameter values
($\lambda=-0.1,n=0.2,k=1$) remain mostly well within the $3\sigma$ region of
the corresponding quantities as extracted from the observation. For other 3
sets of parameter values $(\lambda,n,k)$ considered here, viz. (0.1, 1,1),
(1,1,1), (10,1,1), the energy density curves are outside the corresponding
3$\sigma$ range obtained from observation. A close look at the plots of
pressure profile for exponential and logarithmic models in Fig. 5 reveals that
the fluid pressure profile for exponential models also lies almost entirely
within its the 3$\sigma$ observed limits, for the full range of time ($\tau$)
accessible in SNe Ia observations, while for power law model, the
corresponding curve is outside the 3$\sigma$ observed limits for a small
temporal regime during a relatively earlier part of late-time cosmic
evolution. Therefore, non-minimally coupled matter-curvature scenarios with
fluid pressure modelled as $p\leavevmode\nobreak\ \sim e^{ak}$ mimics the
outcome of the $\Lambda-$CDM model in terms of energy density and pressure
profiles within their 3$\sigma$ limits, for the model parameters taking values
in the close proximity of the values $\lambda=-0.1,n=0.2,k=1$. We may mention
here, that the $\Lambda-$CDM model which, though fits the data well, is
plagued with the coincidence and fine-tuning problems, whereas the models with
non-minimally coupled matter-curvature scenarios are free from such problems.

In the light of the above results we also investigated the issue of energy
transfer between curvature and fluid sectors owing to the considered non-
minimal coupling between them. For the non-minimally coupled curvature-fluid
models considered in this article, the energy-balance equation takes the form
[50],

$\displaystyle\nabla^{\mu}T_{\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{\lambda F_{2}(R)}{1+\lambda
f_{2}(R)}\Big{(}g_{\mu\nu}\mathcal{L}_{m}-T_{\mu\nu}\Big{)}\nabla^{\mu}R\,.$
(24)

For an ideal fluid with energy density $\rho$ and pressure $p$ in FRW
spacetime background and with $\mathcal{L}_{m}=p$ as considered in the work,
the $\nu=0$ component of Eq. (24) takes the form

$\displaystyle\dot{\rho}+3H(\rho+p)$ $\displaystyle=$
$\displaystyle-\frac{\lambda F_{2}(R)}{1+\lambda
f_{2}(R)}\Big{(}p+\rho\Big{)}\dot{R}\,\equiv Q\,.$ (25)

Note that, in absence of any non-minimal coupling $(\lambda=0)$, the above
energy balance equation reduces to usual continuity equation
$\dot{\rho}+3H(\rho+p)=0$ of FRW universe. The term $Q$ on right hand side of
Eq. (25) is a measure of rate of energy exchange between matter and curvature
sectors. In the context of Eq. (25) we may mention that, thermodynamical
implications of curvature-matter coupling scenarios at cosmological scales
have been investigated in [50, 51, 52], where from the perspective of
thermodynamics of open systems, the energy balance equation (25) has been
interpreted to describe particle creation in FRW universe. Such an
interpretation requires referring to the term $-Q/3H$ as the creation pressure
$p_{c}$ associated with the particle creation, so as to cast the energy
balance equation in the form $\dot{\rho}+3H(\rho+p+p_{c})=0$, with creation
pressure subject being to the constraint that $p_{c}\equiv-Q/3H<0$, implying
the requirement that $Q$ always remains positive.

In the context of non-minimally coupled curvature-fluid models considered here
with $f_{2}(R)=R^{n}$, we may write down the expression for the time profile
of the $Q$-function from Eq. (25) corresponding to the parameter set
$(\lambda,n,k)$ as

$\displaystyle Q(t;\lambda,n,k)$ $\displaystyle=$ $\displaystyle-\frac{\lambda
nR^{n-1}}{1+\lambda
R^{n}}\Big{[}p(t;\lambda,n,k)+\rho(t;\lambda,n,k)\Big{]}\dot{R}$ (26)

where the energy density $\rho(t;\lambda,n,k)$ and fluid pressure
$p(t;\lambda,n,k)$ profiles corresponding to both exponential and power law
models are given by Eqs. (9) - (12). We use Eq. (26) to compute the function
$Q(t;\lambda,n,k)$ for given choices of $\lambda,n,k$. Time dependences of
cosmological parameters viz. $a$, $H$, $R$ etc., as extracted from observed
data are instrumental in the determination of the temporal profile of $Q$. In
Fig. 6 we have shown the time dependence of $Q$ for both exponential (left
panel) and power law (right panel) models by plotting $Q(\tau)/\rho_{0}$ vs
$\tau$ for the same 4 benchmark values of parameters ($\lambda,n,k$) as taken
earlier.

Figure 6: Temporal behaviour the function $Q(t;\lambda,n,k)$ (discussed in
text) for different sets of benchmark values of parameters $Q(t;\lambda,n,k)$
for exponential (left panel) and power-law (right panel) models

We observe from Fig. 6 that, corresponding to the benchmark cases with $n=1$
(i.e. $f_{2}(R)=R$) and $k=1$ (which correspond to $p\sim e^{a}$ for
exponential model and $p\sim a$ for power law model), $Q(\tau)$ remains always
positive. So, certain domains of $(\lambda,n,k)$ parameter space of non-
minimally coupled matter-curvature models are allowed from the combined
analysis of Pantheon data and OHD, for which the possibility of interpreting
the energy balance equation in terms of particle creation [50] remains open.
On the other hand, the estimation of $Q(\tau)$ for parameter values
($\lambda=-0.1,n=0.2,k=1$), (which mimics energy density and pressure profile
obtained from the data analysis with $\Lambda$-CDM model as explained earlier)
gives negative values of $Q$ for the entire time range probed in SNe Ia
observations. We also observe from Fig. 6 that, the absolute value of the rate
of energy transfer between curvature and matter sectors and hence the effect
of the considered non-minimal coupling, monotonically decreases as time
approaches towards the present epoch.

##  5 Conclusion

In this work we considered non-minimally coupled curvature-matter models of
gravity and investigated its cosmological implications in the light of
luminosity distance and redshift measurements of Supernova Ia events. The non-
minimal curvature-matter coupling has been introduced by adding a term
$\int\left[\lambda R^{n}\mathcal{L}_{m}\right]\sqrt{-g}d^{4}x$ to the usual
action for Einstein’s gravity involving the Einstein Hilbert action and
minimally coupled matter action. The parameters $\lambda$ and $n$ fix the
strength and nature of the non-minimal coupling. To explore consequences of
such non-minimal couplings in relation to evolutionary aspects of the universe
at large scales, we considered homogeneous and isotropic spacetime geometry of
the expanding universe described by a flat FRW metric involving the time
dependent scale factor $a(t)$. The matter content of the universe is modelled
as a perfect fluid characterised by energy density $\rho(t)$ and pressure
$p(t)$ and chosen the form of matter Lagrangian as $\mathcal{L}_{m}=p$. Such a
choice for Lagrangian density correctly reproduces the hydrodynamical
equations for the perfect fluid and also may cause to vanish the extra force
owing to departure from motion along the geodesic arising from a non-vanishing
covariant derivative of energy-momentum tensor in coupled scenarios. For such
considerations, the field equations corresponding to the modified action
containing non-minimal term take forms exhibiting connection between the
$a(t),\rho(t),p(t)$ through their time derivatives and various other functions
like the Hubble function $H(t)$ and the Ricci scalar $R(t)$. The functions
$H(t)$, $R(t)$ are however also directly related to the scale factor $a(t)$
and its derivatives. The evolution equations also involve the parameters
$\lambda$ and $n$ related to the curvature-matter coupling and the minimal
coupling scenario corresponds to $\lambda=0$, for which the equations reduce
to the Friedman equations. Unlike Friedman’s equations, their corresponding
analogues which govern the evolutionary dynamics in presence of non-minimal
curvature-matter coupling, involves time derivates of fluid pressure $p$. We
investigate the observational constraints on the non-minimal models choosing
two different ansatzes for $p(t)$ referred to in the paper as the ‘exponential
model’ and ‘power law model’, where the temporal profile of the fluid pressure
has been parametrized in terms of a dimensionless parameter $k$ as $p\sim
e^{ka}$ and $p\sim a^{k}$ respectively. Consequently, in the context of this
work, the interplay of the three parameters $(\lambda,n,k)$ plays a pivotal
role in testing the consistency of non-minimally coupled fluid-curvature
scenarios with the observed data.

From a comprehensive analysis of Pantheon compilation of 1048 SNe Ia data
points and 54 data points on measurement of Hubble parameter from OHD, we
obtained time dependences of the relevant cosmological parameters
$a(t),H(t),R(t)$ and their time derivatives over the late time phase of cosmic
evolution. Taking into account all these information and using the evolution
equations for non-minimally coupled scenarios, we numerically computed the
energy density $\rho(t,\lambda,n,k)$ and pressure $p(t,\lambda,n,k)$ profiles
for different values of $(\lambda,n,k)$ thoroughly scanning a portion of the
parameter space: [$-0.1\leqslant\lambda\leqslant 10;-10\leqslant n\leqslant
10;-20\leqslant k\leqslant 10$]. We also obtained and presented in Fig. 4, the
regions in the parameter space corresponding to parameter values giving
$\rho(t,\lambda,n,k)>0$ for all $t$ which is a trivial but essential
requirement for viability of cosmological models. We found there exist large
domains in the $(\lambda,n,k)$-parameter space for which models with non-
minimal curvature-matter mixing stand as viable cosmological models
reproducing the observed features of late time cosmic evolution. We have also
seen that there exist a small range of parameter values around
($\lambda=-0.1,n=0.2,k=1$) for which the computed temporal profiles of $\rho$
and $p$ mimic the corresponding profiles obtained from the analysis of the
data using $\Lambda-$CDM model in the context of usual minimal coupling
scenario.

The energy-momentum tensor has a non-vanishing covariant derivative
($\nabla^{\mu}T_{\mu\nu}\neq 0$) in the realm of non-minimally coupled
curvature-matter scenarios and this implies exchange of energy between
curvature and matter sectors. We found that, the absolute value of the rate of
energy transfer between the two sectors monotonically decreases as time
approaches towards the present epoch. This dynamics of energy exchange may be
expressed by an equation portraying the energy balance between the two sectors
in FRW background as $\dot{\rho}+3H(\rho+p+p_{c})=0$, with the function
$p_{c}(t)$ (multiplied by $3H$) providing a measure of the rate of energy
transfer owing to the non-minimal coupling between curvature and matter. From
the perspective of thermodynamics of open systems, as extensively discussed in
[50, 51, 52], curvature-matter coupling allows production of a substantial
amount of comoving entropy during late time evolutionary phase of the
universe. This opens up the possibility to interpret the energy-balance
equation describing particle creation in FRW universe, with $p_{c}$ realised
as the (creation) pressure related to the particle creation. Such an
interpretation however requires $p_{c}$ to be negative. Our investigation
shows, there exist values of model parameters $(\lambda,n,k)$, for which the
requirement $p_{c}<0$ (for all time) is met. Thus the ‘particle creation’
interpretation of energy balance equation in non-minimally coupled curvature-
matter scenarios remains an open possibility in the context of SNe Ia data.

### Acknowledgement

We express our gratitude to the referees for providing valuable suggestions.
A.C. would like to thank Indian Institute of Technology, Kanpur for supporting
this work by means of Institute Post-Doctoral Fellowship
(Ref.No.DF/PDF197/2020-IITK/970).

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