# Non-adiabatic particle production scenario in algebraically coupled
quintessence field with dark matter fluid

Saddam Hussain<EMAIL_ADDRESS>Department of Physics, Indian Institute of
Technology Kanpur, UP 208016, India

###### Abstract

We investigate the dynamics of an algebraically coupled quintessence field
with a dark matter fluid, considering a scenario involving non-adiabatic
particle production, through the action principle by modifying the interaction
Lagrangian. The interaction parameter serves as the source of dark matter
particle and entropy production. As particle creation occurs due to the
interaction between the field and fluid sectors, the system manifests an
additional pressure. Our analysis includes studying the system’s dynamics by
considering an exponential type of interaction corresponding to the field’s
exponential potential. We find that the system exhibits phantom behavior at
the current epoch before stabilizing in the accelerating future epoch of the
universe.

##  I Introduction

Over the past two decades, observations have shed light on the dynamics of the
cosmos at its largest scale, revealing evidence that the expansion of the
universe is accelerating [1, 2, 3, 4, 5, 6, 7, 8, 9]. One of the most accepted
explanations for this late-time cosmic acceleration is the existence of an
exotic component known as the cosmological constant ($\Lambda$), which exerts
negative pressure. The largest portion of the universe’s energy budget
consists of the cosmological constant, accounting for about $70\%$.
Additionally, approximately $25\%$ of the energy budget is dominated by a non-
relativistic, pressureless fluid commonly referred to as dark matter, with the
remaining portion composed of baryonic matter.

Although the $\Lambda$CDM model, where CDM represents cold dark matter, is
favored due to its ability to describe most observational evidence, it faces
several theoretical shortcomings, including the cosmological constant problem,
fine-tuning issues, and the cosmological coincidence problem [10, 11, 12, 13,
14, 15, 16]. Numerous alternatives have been proposed to address these issues,
either by modifying the gravitational sector [17, 18, 19, 20] or by modifying
the matter sector [21, 22, 23, 24, 25, 26, 27]. In many instances, scalar
fields serve as viable candidates for dark energy (DE) and are often minimally
coupled with pressureless dark matter (DM) fluid. However, beyond their
gravitational signatures, these enigmatic forms of matter pose puzzles to the
scientific community. Consequently, numerous possible scenarios have been
intensely investigated, including (i) non-gravitational interactions between
dark matter and dark energy [28, 29, 30, 31, 32, 33, 34] and (ii) non-minimal
coupling between matter fields and curvature [35, 36, 37, 38, 39, 40, 41, 42,
43, 44, 45, 46, 47].

A recent approach has emerged to investigate the non-gravitational interaction
between dark sectors through the variational principle [40, 41, 48]. In these
studies, dark energy is governed by a scalar field, while the action for the
dark matter fluid is modeled using the relativistic fluid action proposed by
Brown [49]. This action encompasses the energy density of the fluid $\rho$,
the particle flux number $J^{\mu}$, and several Lagrange multipliers.
Additionally, a non-gravitational interaction term is introduced at the action
level, consisting of fluid and field variables, denoted by $\mathcal{L}_{\rm
int}=-\sqrt{-g}f(n,s,\phi)$, where $f$ is an arbitrary interaction function
depending on the fluid number density $n$, entropy per particle $s$, and
scalar field $\phi$. One immediate constraint studied under these models is
the conservation of number density $\nabla_{\mu}(nu^{\mu})=0$, where $u^{\mu}$
represents the fluid’s four-velocity, and the conservation of entropy
$\nabla_{\mu}(snu^{\mu})=0$. Consequently, the normal component of the fluid’s
covariant derivative of the energy-momentum tensor vanishes, i.e.,
$u_{\mu}\nabla_{\nu}T^{\mu\nu}=0$, implying $\rho\propto a^{-3}$, where $a$ is
the scale factor. However, the inclusion of $\phi$ in the interacting
Lagrangian modifies the field evolution, resulting in dynamics different from
the minimally coupled scenario. In light of the current cosmological crisis,
where the discrepancy between the measured values of the Hubble parameter
$H_{0}$ and the amplitude of matter density $S_{8}$ between high and low-
redshift data exceeds the $4.4\sigma$ level [50, 51, 52], exploring scenarios
where both the field and fluid sectors are simultaneously affected becomes
crucial.

This paper explores a scenario where the dynamics between the quintessence
field and dark matter fluid are investigated by modifying the interaction
Lagrangian $f(n,s,\phi)\rightarrow f(n,s,\phi,\varphi)$, where $\varphi$ is a
fluid Lagrange multiplier. This modification results in alterations to the
thermodynamic constraints such that $\nabla_{\mu}(nu^{\mu})\neq 0$ and
$\nabla_{\mu}(snu^{\mu})\neq 0$. With the number density of the dark matter
fluid no longer conserved, energy flow from the quintessence field to the dark
matter can lead to the creation of matter particles, consequently inducing an
additional pressure known as creation pressure $P_{c}$ [53, 54, 55, 56, 57,
58]. By analyzing this scenario from the action principle, the interaction
function serves as a source of particle and entropy production. We obtained
the fluid’s equations of motion and temperature evolution from thermodynamic
principles. Additionally, we provide a background dynamics by considering an
exponential type of interaction
$f\propto\rho^{\beta}e^{\delta\kappa\phi+\gamma\varphi}$, corresponding to the
field’s exponential potential. Stability analysis of the interacting system is
conducted using the standard linearization technique [59, 60, 61, 62], with
proper constraints on thermodynamic quantities to ensure positivity of entropy
and number density throughout the evolution. Our findings indicate that the
interacting system exhibits a stable accelerating critical point in the future
epoch of the universe. However, at the present epoch $a=1$, the effective
equation of state (EoS) crosses the phantom barrier, i.e., $\omega_{\rm
eff}\leq-1$. Furthermore, we utilize $43$ Hubble data and $1701$ Pantheon+
data to numerically simulate the current model against the $\Lambda$CDM model,
revealing compatibility with the data for $f\propto\rho^{3}$.

We organized the paper as follows: In sec. II, we set up the action for the
algebraically coupled field-fluid system and obtained the governing background
equations. In sec. III, we present brief thermodynamic relations corresponding
to the fluid component. A detailed picture of the conservation of energy-
momentum tensor is presented in sec. IV. The dynamical system’s stability is
discussed in sec. V. Finally, a brief conclusion is outlined in sec. VI.

##  II Action for the algebraic interaction

The action describing the algebraically (non-minimally) coupled field-fluid
scenario is given by [40]:

$S=\int_{\Omega}d^{4}x\sqrt{-g}\frac{R}{2\kappa^{2}}-\sqrt{-g}\rho(n,s)+J^{\mu}(\varphi_{,\mu}+s\theta_{,\mu}+\alpha_{A}{\beta}_{,\mu}^{A})-\sqrt{-g}\mathcal{L}_{\phi}(\phi,\partial{}_{\mu}\phi)-\alpha_{1}\sqrt{-g}f(n,s,\phi,\varphi),$
(1)

where,

$J^{\mu}=\sqrt{-g}nu^{\mu},\quad|J|=\sqrt{-g_{\mu\nu}J^{\mu}J^{\nu}},\quad
n=\frac{|J|}{\sqrt{-g}},\quad u^{\mu}u_{\mu}=-1\,.$ (2)

In this action, the first term corresponds to the Einstein-Hilbert action,
where $g$ denotes the determinant of the metric tensor $g^{{\mu\nu}}$, $R$
represents the Ricci scalar, and $\kappa^{2}=8\pi G$. The second and third
terms together represent the action for a relativistic fluid, where the energy
density of the dark matter fluid is denoted as $\rho$, depending on the fluid
number density $n$ and the entropy density per particle $s$. The relativistic
fluid Lagrangian contains the fluid flux density $J^{\mu}$ and Lagrange
multipliers $\varphi$, $\theta$, $\alpha^{A}$, $\beta_{A}$. Note that Greek
indices range from 0 to 3, and $A$ runs from 1 to 3. It’s important to
distinguish between $\alpha_{1}$ and $\alpha^{A}$ as they are distinct
quantities unless specified. The commas followed by Greek indices indicate
covariant derivatives. The field action is given by:

$\mathcal{L}_{\phi}=\frac{1}{2}\partial{}_{\mu}\phi\partial{}^{\mu}\phi+V(\phi).$
(3)

Here, the Lagrangian represents a standard quintessence field with the
potential $V(\phi)$. The remaining term in the action corresponds to the
interaction parameter $f$, which depends on fluid and field parameters, while
$\alpha_{1}$ is a constant parameter. Taking the variation of the action Eq.
(1) with respect to the metric $g^{{\mu\nu}}$ yields the Einstein field
equation:

$R_{{\mu\nu}}-\dfrac{1}{2}Rg_{{\mu\nu}}=\kappa^{2}\left(T_{{\mu\nu}}^{M}+T_{{\mu\nu}}^{\phi}+T_{{\mu\nu}}^{\rm
int}\right).$ (4)

Here, the stress tensor of the matter components is defined as
$T_{{\mu\nu}}=\frac{-2}{\sqrt{-g}}\frac{\delta S}{\delta g^{{\mu\nu}}}$. The
energy-momentum tensor corresponding to the relativistic fluid is:

$T^{{\mu\nu}}_{M}=\rho
u^{\mu}u^{\nu}+\left(n\frac{\partial{}\rho}{\partial{}n}-\rho\right)(u^{\mu}u^{\nu}+g^{{\mu\nu}}).$
(5)

Similarly, corresponding to the interaction part:

$T^{{\mu\nu}}_{\rm
int}=\alpha_{1}fu^{\mu}u^{\nu}+\alpha_{1}\left(n\frac{\partial{}f}{\partial{}n}-f\right)(u^{\mu}u^{\nu}+g^{{\mu\nu}}).$
(6)

The field’s stress tensor becomes:

$T_{\phi}^{{\mu\nu}}=-g^{{\mu\nu}}\left(\frac{1}{2}\partial{}_{\alpha}\phi\partial{}^{\alpha}\phi+V(\phi)\right)+\partial{}^{\mu}\phi\partial{}^{\nu}\phi.$
(7)

To evaluate the energy density and corresponding pressure of these matter
components, we compare with the stress tensor for the perfect fluid
$T^{{\mu\nu}}=\rho u^{\mu}u^{\nu}+P(g^{{\mu\nu}}+u^{\mu}u^{\nu})$. Hence, the
energy density and pressure of the fluid become:

$\rho_{M}=\rho,\quad
P_{M}=\left(n\frac{\partial{}\rho}{\partial{}n}-\rho\right).$ (8)

The energy density and pressure corresponding to the algebraic interaction
become:

$\rho_{\rm int}=\alpha_{1}f,\quad P_{\rm
int}=\alpha_{1}\left(n\frac{\partial{}f}{\partial{}n}-f\right).$ (9)

The field energy density and pressure become:

$\rho_{\phi}=\frac{1}{2}\dot{\phi}^{2}+V(\phi),\quad
P_{\phi}=\frac{1}{2}\dot{\phi}^{2}-V(\phi).$ (10)

In the flat FLRW metric $ds^{2}=-dt^{2}+a(t)^{2}d\vec{x}^{2}$, the Friedmann
equations become:

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle\kappa^{2}(\rho_{M}+\rho_{\phi}+\rho_{\rm int}),$ (11)
$\displaystyle 2\dot{H}+3H^{2}$ $\displaystyle=$
$\displaystyle-\kappa^{2}(P_{M}+P_{\phi}+P_{\rm int}).$ (12)

Upon taking the variation of the action with respect to $\phi$, the field
equation becomes:

$\ddot{\phi}+3H\dot{\phi}+\dfrac{dV(\phi)}{d\phi}+\alpha_{1}\frac{\partial{}f}{\partial{}\phi}=0.$
(13)

##  III Thermodynamic relations

In this section, we will examine the thermodynamic aspect of the fluid in
light of interaction. Upon varying the action with respect to the fluid
variables, the corresponding equations of motion become:

$\displaystyle J^{\mu}:$ $\displaystyle\qquad
u_{\mu}\left(\frac{\partial{}\rho}{\partial{}n}+\alpha_{1}\frac{\partial{}f}{\partial{}n}\right)+\varphi_{,\mu}+s\theta_{,\mu}+\beta_{A}\alpha^{A}_{,\mu}=0,$
(14) $\displaystyle s:$
$\displaystyle\qquad-\left(\frac{\partial\rho}{\partial
s}+\alpha_{1}\frac{\partial{}f}{\partial{}s}\right)+nu^{\mu}\theta_{,\mu}=0\,,$
(15) $\displaystyle\varphi:$
$\displaystyle\qquad\nabla_{\mu}(nu^{\mu})+\alpha_{1}\dfrac{\partial{}f}{\partial{}\varphi}=0\,,$
(16) $\displaystyle\theta:$ $\displaystyle\qquad(sJ^{\mu})_{,\mu}=0\,,$ (17)
$\displaystyle\beta_{A}:$ $\displaystyle\qquad J^{\mu}\alpha^{A}_{,\mu}=0\,,$
(18) $\displaystyle\alpha^{A}:$
$\displaystyle\qquad(\beta_{A}J^{\mu})_{,\mu}=0.$ (19)

Due to the modifications in the interaction parameter, which now includes the
Lagrange parameter $\varphi$, the number density of the fluid from Eq. (16) is
no longer conserved, and the interaction parameter $f$ acts as a source of
fluid particle number density. In the flat FLRW background metric, this
equation reads as:

$\dot{n}+3nH=-\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}.$ (20)

As the non-minimal coupling acts as a source of the fluid’s particle density,
the corresponding contribution in entropy from Eq. (17) becomes:

$\dot{s}=\frac{s}{n}\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}$ (21)

Therefore, due to the dependence of $\varphi$ in the interaction function, the
fluid particle density may increase or decrease, resulting in a change in
entropy. Consequently, the evolution of the fluid sector becomes non-
adiabatic. As the interaction parameter induces changes in number density and
entropy, the corresponding change in energy density of the fluid sector can be
explored using the thermodynamic relation:

$d(\rho V)=dQ-PdV.$ (22)

Here, $dQ$ denotes the heat received by the fluid sector during time $dt$, and
$V=a^{3}$ signifies the comoving volume. For the closed system, the above
relation can be re-expressed as:

$d(\rho/n)=Tds-Pd(1/n).$ (23)

Here, $dq=dQ/N$, $n=N/V$, $s=S/N$, and
$T=\frac{1}{n}\frac{\partial\rho}{\partial s}\big{|}_{n}$ is the temperature
of the fluid [54]. As the particle production takes place, the system can
absorb heat. The above relation can then be extended to the open system as:

$d(\rho V)=TV(d(sn)-sdn)-PdV+\frac{\rho+P}{n}d(nV).$ (24)

On differentiating both sides with respect to time $t$, we obtain:

$\dot{\rho}+3H(\rho+P)=Ts\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}-\alpha_{1}\frac{\rho+P}{n}\frac{\partial{}f}{\partial{}\varphi}.$
(25)

From this, we can express the creation pressure or non-minimally induced
pressure $P_{c}$ as:

$P_{c}=\frac{1}{3H}\left(\frac{\rho+P}{n}-\frac{s}{n}\frac{\partial{}\rho}{\partial{}s}\right)\alpha_{1}\frac{\partial{}f}{\partial{}\varphi}.$
(26)

Therefore, as the fluid’s particle number density changes, the system can
exhibit an additional pressure known as creation pressure. The density
evolution can be rewritten as:

$\dot{\rho}+3H(\rho+P+P_{c})=0.$ (27)

From Eq. (20) and Eq. (21), it is clear that if $\alpha_{1}\frac{\partial
f}{\partial\varphi}<0$, the interaction contributes to an increment in the
number density of the fluid particles. As a consequence, the change in entropy
corresponding to the fluid sector decreases. Although the entropy of the
matter sector decreases, it’s important to note that the entropy of the system
as a whole may not decrease. Hence, the second law of thermodynamics for the
entire system remains intact.

Using the above equation, the evolution of the temperature $T$ of the fluid
can be determined. We utilize the total derivative of energy density as:

$d\rho(n,T)=\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}dn+\left(\frac{\partial{}\rho}{\partial{}T}\right)_{n}dT,$
(28)

After taking the time derivative and using equations Eq. (27) and Eq. (20), we
obtain:

$\dot{T}=\frac{1}{\left(\partial{}\rho/\partial{}T\right)_{n}}\left(-3H(\rho+P+P_{c})-\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}\dot{n}\right).$
(29)

To evaluate $\left(\frac{\partial\rho}{\partial n}\right)_{T}$, we use the
thermodynamic relation:

$ds=\frac{1}{nT}d\rho-\frac{\rho+P}{n^{2}T}dn,$ (30)

and using the partial derivative property, we obtain:

$\left(\frac{\partial{}s}{\partial{}n}\right)_{T}=\frac{1}{nT}\left(\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}-\frac{\rho+P}{n}\right),\quad\left(\frac{\partial{}s}{\partial{}T}\right)_{n}=\frac{1}{nT}\left(\frac{\partial{}\rho}{\partial{}T}\right)_{n}.$
(31)

Further using the property
$\frac{\partial{}^{2}s}{\partial{}T\partial{}n}=\frac{\partial{}^{2}s}{\partial{}n\partial{}T}$,
we get:

$h=\rho+P=n\left(\frac{\partial{}\rho}{\partial{}n}\right)_{T}+T\left(\frac{\partial{}P}{\partial{}T}\right)_{n}\,.$
(32)

Here, $h$ is the enthalpy per unit volume. Inserting this relation into
$\dot{T}$ and using
$\frac{(\partial{}P/\partial{}T)_{n}}{(\partial{}\rho/\partial{}T)_{n}}\equiv(\partial{}P/\partial{}\rho)_{n}$,
we finally obtain:

$\dot{T}/T=\frac{s\alpha_{1}\partial{}f/\partial{}\varphi}{(\partial{}\rho/\partial{}T)_{n}}+\frac{\dot{n}}{n}\left(\frac{\partial{}P}{\partial{}\rho}\right)_{n}.$
(33)

Using Eq. (21), this relation is similar to the one that has been obtained in
ref. [63]. The first term on the right-hand side shows the contribution from
the non-conserved entropy of the fluid. To find the temperature evolution, we
choose the model corresponding to the pressureless non-relativistic fluid. The
energy density of the non-relativistic ideal gas $(k_{\rm B}=1)$ is:

$\rho=Mn+\frac{3}{2}nT,$ (34)

where $M$ is the mass of gas particles. Corresponding to this ideal gas model,
the pressure, $P=n(\partial\rho/\partial n)-\rho=0$, vanishes. Using this
relation, the temperature relation simplifies to:

$\dfrac{\dot{T}}{T}=\dfrac{2\dot{s}}{3}.$ (35)

This gives the variation of temperature with entropy irrespective of any form
of interaction parameter $f$. Hence the entropy becomes:

$s=s_{0}+\frac{3}{2}\ln(T/T_{0}).$ (36)

Here, the parameters with subscript $0$ are integration constants representing
the present value of the respective parameters.

##  IV Conservation of energy momentum tensor

In the preceding section, we explored the thermodynamic behavior of the fluid
sector, employing thermodynamic relations to determine the corresponding
evolution of fluid density. In this section, we delve into evaluating the
covariant derivative of the stress tensor of the entire system to investigate
the transportation of energy flow between the fluid and field sectors. To do
so, we first redefine the matter stress tensor as:

$T^{{\mu\nu}}_{A}=(\rho+\alpha_{1}f)u^{\mu}u^{\nu}+\left(n\frac{\partial{}\rho}{\partial{}n}-\rho+\alpha_{1}n\frac{\partial{}f}{\partial{}n}-\alpha_{1}f\right)(u^{\mu}u^{\nu}+g^{{\mu\nu}}).$
(37)

With this redefinition, the Einstein tensor becomes
$G_{{\mu\nu}}=\kappa^{2}(T_{{\mu\nu}}^{A}+T_{{\mu\nu}}^{\phi})$ 111Here $A$
denotes a label.. Upon taking the covariant derivative of the stress tensor,
we obtain:

$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+\alpha_{1}\dot{f}+3H(\rho+\alpha_{1}f+P_{M}+P_{\rm
int})=Q^{0}.$ (38)

Utilizing Eq. (20) and Eq. (21), we can express:

$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M})-\alpha_{1}^{2}\frac{\partial{}f}{\partial{}\varphi}\frac{\partial{}f}{\partial{}n}+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}+\alpha_{1}^{2}\frac{\partial{}f}{\partial{}s}\left(\frac{s}{n}\frac{\partial{}f}{\partial{}\varphi}\right)+\alpha_{1}\dot{\varphi}\frac{\partial{}f}{\partial{}\varphi}=Q^{0}.$
(39)

To eliminate $\dot{\varphi}$ from the above equation, we contract Eq. (14)
with $J^{\mu}$ and then use Eq. (18), we get:

$\frac{\partial{}\rho}{\partial{}n}-s\dot{\theta}+\alpha_{1}\frac{\partial{}f}{\partial{}n}=\dot{\varphi},$
(40)

which, when inserted into the above equation, yields:

$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M})+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}+\alpha_{1}\frac{\partial{}\rho}{\partial{}n}\frac{\partial{}f}{\partial{}\varphi}+\alpha_{1}^{2}\frac{\partial{}f}{\partial{}s}\left(\frac{s}{n}\frac{\partial{}f}{\partial{}\varphi}\right)-s\dot{\theta}\frac{\partial{}f}{\partial{}\varphi},$
(41)

The time derivative of the temperature gradient $\dot{\theta}$ can be
eliminated by using Eq. (15), resulting in:

$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M})+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}+\alpha_{1}\frac{\partial{}\rho}{\partial{}n}\frac{\partial{}f}{\partial{}\varphi}-\alpha_{1}\frac{s}{n}\frac{\partial{}f}{\partial{}\varphi}\frac{\partial{}\rho}{\partial{}s}.$
(42)

Using the definition of the pressure of the fluid
$P_{M}+\rho=n\frac{\partial\rho}{\partial n}$ and comparing this with Eq.
(25), we obtain:

$\nabla_{\mu}T^{\mu
0}_{A}=\dot{\rho}+3H(\rho+P_{M}+P_{c})+\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}=Q^{0},$
(43)

where $P_{c}=\dfrac{1}{3H}\left(\dfrac{\rho+P_{M}}{n}\alpha_{1}\dfrac{\partial
f}{\partial\varphi}-\alpha_{1}\dfrac{s}{n}\dfrac{\partial
f}{\partial\varphi}\dfrac{\partial\rho}{\partial s}\right)$. Considering Eq.
(25), the covariant derivative of the fluid sector yields:

$\nabla_{\mu}T^{\mu
0}_{A}=\alpha_{1}\dot{\phi}\frac{\partial{}f}{\partial{}\phi}=Q^{0}.$ (44)

The covariant derivative of the field sector can be determined using Eq. (13)
as:

$\nabla_{\mu}T^{\mu
0}_{\phi}=-\alpha_{1}\frac{\partial{}f}{\partial{}\phi}\dot{\phi}=-Q^{0}.$
(45)

This exercise demonstrates that the total energy density of the system remains
conserved, and both sectors exchange energy through the interaction term:

$Q^{0}=\alpha_{1}\frac{\partial{}f}{\partial{}\phi}\dot{\phi}.$ (46)

##  V Background dynamics

In this section, we will conduct a stability analysis of the system using the
standardized linearization technique. We observed that the interaction
function alters not only the field’s equation of motion but also the fluid’s
equation of motion. As the non-minimal coupling parameter $f$ acts as a source
of particle creation and entropy generation, the system transitions to a non-
adiabatic state. To elucidate the background evolution of the system, we
define a set of dimensionless dynamical variables as:

$x=\frac{\kappa\dot{\phi}}{\sqrt{6}H},\ y=\frac{\kappa\sqrt{V}}{\sqrt{3}H},\
z=\frac{\kappa^{2}f}{3H^{2}},\
\Omega_{\phi}=\frac{\kappa^{2}\rho_{\phi}}{3H^{2}},\
\Omega_{M}=\frac{\kappa^{2}\rho}{3H^{2}}.$ (47)

Expressing the first Friedmann equation in dynamical variables puts constraint
on the fluid fractional energy density as:

$\Omega_{M}=1-x^{2}-y^{2}-\alpha_{1}z,$ (48)

and the effective equation of state (EoS) corresponding to the entire system
becomes:

$\omega_{\rm eff}=\frac{P_{\rm tot}}{\rho_{\rm
tot}}=x^{2}-y^{2}+\alpha_{1}(\beta-1)z.$ (49)

Based on the definition of dynamical variables, the variables range as
follows:

$-\infty<x<+\infty,\quad 0\leq
y<\infty,\quad-\infty<z<+\infty,\quad\Omega_{M}\geq 0,\quad\Omega_{\phi}\geq
0.$ (50)

To construct the autonomous equations corresponding to the primary variables
of the system $x,y,z$, one needs to choose the form of the interaction
function,

$f=M^{4-4\beta}\rho^{\beta}e^{(\delta\kappa\phi+\gamma\varphi)},$ (51)

and the potential of the quintessence field is:

$V(\phi)=V_{0}e^{\lambda\kappa\phi}.$ (52)

Before constructing the autonomous equations, we evaluate $\dot{z}$:

$\dot{z}=\frac{\kappa^{2}}{3H^{2}}\left(-3H\beta f+\delta\kappa
f\dot{\phi}+\gamma f(\rho/n-sT)\right)-z\frac{\dot{H}}{H}.$ (53)

To evaluate this, we have used $T=\frac{1}{n}\frac{\partial\rho}{\partial s}$,
Eq. (40), and $\dot{\theta}=T+\alpha_{1}\frac{1}{n}\frac{\partial f}{\partial
s}$. However, to close the system, we still need to define additional
dynamical variables corresponding to $n,s,T$, as these are time-dependent
quantities. Hence, the additional dynamical variables are:

$\chi=\frac{\kappa^{2}n}{3H},\quad\xi=T/H,\quad s=s.$ (54)

We choose the variables corresponding to the number density $n$, temperature
$T$ of the fluid, and entropy $s$. With these definitions, the thermodynamic
variables are constrained within the following ranges:

$\chi>0,\quad\xi>0,\quad s\geq 0.$ (55)

In terms of these defined variables, the aforementioned dynamical equation in
$z^{\prime}$ becomes:

$z^{\prime}=\frac{\dot{z}}{H}=-3\beta z+\delta zx\sqrt{6}+\gamma
z(\Omega_{M}/\chi-s\ \xi)+\frac{3}{2}z(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}).$
(56)

The autonomous system of equations that describes the dynamics of the entire
system are:

$\displaystyle x^{\prime}$ $\displaystyle=$ $\displaystyle-3x-\frac{3\lambda
y^{2}}{\sqrt{6}}-\frac{\delta\alpha_{1}3z}{\sqrt{6}}+\frac{3}{2}x(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$
(57) $\displaystyle y^{\prime}$ $\displaystyle=$
$\displaystyle\dfrac{\sqrt{6}yx\lambda}{2}+\frac{3}{2}y(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$
(58) $\displaystyle z^{\prime}$ $\displaystyle=$ $\displaystyle-3\beta
z+\delta zx\sqrt{6}+\gamma z(\Omega_{M}/\chi-s\
\xi)+\frac{3}{2}z(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$ (59)
$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle-\alpha_{1}\gamma
z-3\chi+\frac{3}{2}\chi(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$ (60)
$\displaystyle\xi^{\prime}$ $\displaystyle=$ $\displaystyle\frac{2\xi
s\alpha_{1}z\gamma}{3\chi}+\frac{3}{2}\xi(1+x^{2}-y^{2}+(\beta-1)z\alpha_{1}),$
(61) $\displaystyle s^{\prime}$ $\displaystyle=$
$\displaystyle\frac{s\alpha_{1}z\gamma}{\chi}.$ (62)

Here, we obtained the dynamics corresponding to the temperature parameter
using Eq. (33), where we evaluated $(\partial\rho/\partial T)_{n}$ by
considering the ideal gas model given in Eq. (34). Due to the non-adiabatic
particle production mechanism, the degrees of freedom to describe the dynamics
of the system increase to 6 dimensions. It’s important to note that although
we have assumed dark matter to behave as an ideal gas, with its energy density
given by Eq. (34), this alone cannot provide a constraint on either $\chi$ or
$\xi$. An additional variable is needed to capture the dynamics of $H$ to
close the autonomous system of equations. As a result, the dimension of the
phase space remains 6D. Hence, keeping $\chi$ and $\xi$ as dynamical variables
reduces the complexity of obtaining the critical points as no logarithmic
function will appear. The autonomous equations in
$y^{\prime},z^{\prime},\xi^{\prime},s^{\prime}$ are invariant under the
transformations $(y\mapsto-y,z\mapsto-z,\xi\mapsto-\xi,s\mapsto-s)$,
presenting an invariant sub-manifold at $y=z=\xi=s=0$. This implies that no
trajectory originating in $y,z,\xi,s\geq 0$ can cross the $y=z=\xi=s=0$ plane.
To constrain the model parameters and obtain the stability of the system, we
determine the critical points of the system by equating the right-hand side of
the autonomous equations to zero. The differential equation corresponding to
$s^{\prime}$ vanishes when either $s=0$ or $z=0$. As the coordinate $z=0$ does
not serve the non-minimal coupling scenario of the system, hence we will not
consider those fixed points where $z$ vanishes. On the other hand, the co-
ordinate $s=0,z\neq 0$ makes the entire system adiabatic. Therefore, we
initially study the nature of critical points corresponding to the system in a
restricted 5D phase space by fixing $s$ to a particular value, say $s\neq 0$.
The critical points corresponding to the autonomous system
$(x^{\prime}-\xi^{\prime})$ are:

  * •

Point $P_{1}$: The co-ordinates of the point are
$\bigg{(}x=\frac{3\sqrt{6}(\beta-1)\gamma-\sqrt{2}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{2\gamma\delta(2s+3)},y=0,z=\frac{3\left(-9\beta\gamma+9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}\right)}{\alpha_{1}\gamma\delta^{2}(2s+3)^{2}},\chi=-\frac{-9\beta\gamma+9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{3\delta^{2}(2s+3)},\xi=\frac{\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}-6\beta\gamma
s+6\gamma s}{6\left(3\beta^{2}-6\beta-2\delta^{2}+3\right)}\bigg{)}$. As the
potential variable $y$ vanishes at this point, the point becomes dependent on
the field’s kinetic component. The effective equation of state at this point
is $\omega_{\rm eff}=\frac{2s-3}{2s+3}$, depending solely on entropy. For
$s=3/2$, the point corresponds to a non-accelerating matter solution, whereas
it acts as an accelerating point, i.e., $-1\leq\omega_{\rm eff}<-1/2$, for
$0\leq s<\frac{1}{2}$. The point can only be considered a viable fixed point
when the thermodynamical quantities $\xi,\chi$ yield positive values. For
$s\geq 3/2$, these conditions can’t be satisfied regardless of the choices of
other model parameters. This point exists for $\gamma>0,s=0.1$,
$\beta>0.82\sqrt{\delta^{2}}+1$, and for $\gamma<0,s=1/10$,
$\beta<1-0.82\sqrt{\delta^{2}}$. Although the existence of the point imposes
tight constraints on the model parameters, the constraint on $\lambda$ can be
imposed by determining the stability of the point. After linearizing the
autonomous systems Eq. (57)–Eq. (61) at the critical points up to first order,
we construct the Jacobian matrix $J_{ij}=\frac{dx_{i}}{dx_{j}}$. If the real
part of the eigenvalues are all positive (negative), the point becomes an
unstable (stable) point. For the alternate sign of the real part, the point
becomes a saddle point. Since we consider the autonomous equations Eq.
(57)–Eq. (61), the Jacobian becomes a $5\times 5$ dimensional matrix yielding
five eigenvalues. The real part of the eigenvalues corresponding to this point
has been plotted in Fig.[1], considering $\delta=2$ and $\gamma=1$. The real
part of the eigenvalues takes alternate signs, indicating that the point
becomes a saddle point. Apart from these model parameters, the point remains a
saddle point for any choice of $\delta$.

Figure 1: The stability of the point $P_{1}$ in the parameter space of
$(\beta,\lambda)$ for $(\alpha_{1}=1,s=0.1,\delta=2,\gamma=1)$.

  * •

Point $P_{2}$: The co-ordinates of the point are
$\bigg{(}x=\frac{3\sqrt{3}(\beta-1)\gamma+\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{\sqrt{2}\gamma\delta(2s+3)},y=0,z=-\frac{3\left(9\beta\gamma-9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}\right)}{\alpha_{1}\gamma\delta^{2}(2s+3)^{2}},\chi=\frac{9\beta\gamma-9\gamma+\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}}{3\delta^{2}(2s+3)},\xi=-\frac{\sqrt{3}\sqrt{\gamma^{2}\left(27\beta^{2}-54\beta+2\delta^{2}\left(4s^{2}-9\right)+27\right)}+6(\beta-1)\gamma
s}{6\left(3\beta^{2}-6\beta-2\delta^{2}+3\right)}\bigg{)}$. Similar to the
aforementioned point, the potential variable $y$ vanishes at this point and
the effective EoS takes the same form $\omega_{\rm eff}=\frac{-3+2s}{3+2s}$.
Following the same argument as above i.e., ensuring $\xi$ and $\chi$ are
positive, the point becomes viable under the following conditions. Assuming
the entropy $s=0.1$, the conditions are satisfied for $\gamma>0$ and
$0.815\sqrt{\delta^{2}}+1\leq\beta<0.816\sqrt{\delta^{2}}+1$. This imposes a
stringent constraint on $\beta$ for positive $\gamma$. Similarly, for
$\gamma<0$, $1-0.816\sqrt{\delta^{2}}<\beta\leq 1-0.815\sqrt{\delta^{2}}$. For
$\delta\neq 0$ and $\lambda\neq 0$, the point always becomes a saddle point.

  * •

Point $P_{3}$: The co-ordinates of the point are
$\bigg{(}x=-\frac{2\sqrt{6}s}{3\lambda+2\lambda
s},y=\frac{\sqrt{9\delta\lambda^{2}-4\delta\left(\lambda^{2}-6\right)s^{2}+36(\beta-1)\lambda
s}}{\sqrt{\lambda^{2}(2s+3)^{2}((\beta-1)\lambda+\delta)}},z=\frac{(2s-3)\left(3\lambda^{2}+2\left(\lambda^{2}-6\right)s\right)}{\alpha_{1}\lambda(2s+3)^{2}((\beta-1)\lambda+\delta)},\chi=-\frac{\gamma(2s-3)\left(3\lambda^{2}+2\left(\lambda^{2}-6\right)s\right)}{9\lambda(2s+3)((\beta-1)\lambda+\delta)},\xi=\frac{\gamma\lambda\left(9-4s^{2}\right)}{6(3(\beta-1)\lambda+2(\beta-1)\lambda
s+4\delta s)}\bigg{)}$. At this point, both the kinetic and potential
variables of the field are finite, and the effective equation of state becomes
$\omega_{\rm eff}=\frac{2s-3}{2s+3}$. To discuss the existence of the point,
we initially fix some values of the model parameters, considering $\beta=1$.
The point exists if
$\lambda>2\sqrt{3}\sqrt{\frac{s}{2s+3}},\delta>0,\gamma>0$, and the
corresponding stability has been determined for $s=0.01$ in
Fig.[LABEL:fig:stability_p3_beta1]. In the parameter space of
$(\delta,\gamma)$, the point becomes stable for any choice of $\gamma\neq 0$.
We also examine the existence and stability for other values of $\beta$,
particularly $s=0.1,\beta=2$. We find that the thermodynamic constraints Eq.
(55) are satisfied when $\gamma>0$, $\delta\geq
4.98,\lambda>0.61,\lambda<-\delta$, or for $\gamma>0$, $\delta\leq
4.98,\lambda<-0.61,\lambda>-\delta$. If $\gamma<0$, then $\delta>4.98$,
$-0.13\delta<\lambda<-0.61$, or $\delta<-4.98,0.61<\lambda<-0.13\delta$.
Hence, with these inputs, if we select the model parameters
$\gamma>0,\beta=2,s=0.1$, we determine that the critical point can only be a
stable fixed point when $\delta\geq 5,\lambda>2\delta+3$. It’s worth noting
that the value of $\alpha_{1}$ does not affect the dynamics in any way, since
it can always be absorbed in $z$. Similarly, for $\beta=3,s=0.01$, the
thermodynamic parameters become positive if
$\gamma>0,\lambda>0.19,\delta>-2\lambda$. The stability corresponding to this
range of parameters is shown in Fig.[LABEL:fig:stability_p3_beta3].

(a)

(b)

Figure 2: The stability of the point $P_{3}$ (a) for
$\beta=1,s=0.01,\alpha_{1}=1,\gamma=1$, (b) for
$\beta=3,s=0.01,\alpha_{1}=1,\gamma=1$.

As we have seen, the system yields some non-trivial critical points for $s\neq
0$, and among them, one critical point is both accelerating and stable,
depending on the choice of model parameters. Due to this characteristic, the
point signifies the late-time phase of the universe. Having determined the
ranges of model parameters that produce viable stable critical points, we will
now evaluate the dynamics of the entire system considering the autonomous
equations Eq. (57)–Eq. (62), which will depict the different phases of the
universe. As discussed earlier, taking the equation in $s^{\prime}$ yields
critical points when either $s=0$ or $z=0$. As the system is non-minimally
coupled with the fluid, one expects that during the matter and future epoch,
the variable $z$ must be non-vanishing. Hence, $s=0$ serves as the viable co-
ordinate of the critical points. The critical points corresponding to $s=0$
and the corresponding nature of the fixed points are tabulated in Tabs.[1, 2].
The critical points are given similar nomenclature as the aforementioned fixed
points.

s=0
---
Points | $x$ | $y$ |  $z$ |  $\chi$ |  $\xi$
$P_{1\pm}$ |  $\frac{\pm p+\sqrt{6}\beta-\sqrt{6}}{2\delta}$ 222$p=\sqrt{6\beta^{2}-12\beta-4\delta^{2}+6}$ | 0 |  $\frac{\mp r-3\beta+3}{\text{$\alpha$1}\delta^{2}}$ 333 $r=\sqrt{9\beta^{2}-18\beta-6\delta^{2}+9}$ |  $\pm\frac{\gamma\left(r+3\beta-3\right)}{3\delta^{2}}$ |  0
$P_{2\mp}$ | $\frac{\mp r+3\beta-3}{\sqrt{6}\delta}$ | 0 |  $\frac{\pm r-3\beta+3}{\alpha_{1}\delta^{2}}$ |  $\mp\frac{\gamma\left(r-3\beta+3\right)}{3\delta^{2}}$ |  Any
$P_{3}$ | 0 | $\frac{\sqrt{\delta}}{\sqrt{(\beta-1)\lambda+\delta}}$ |  $-\frac{\lambda}{\alpha_{1}((\beta-1)\lambda+\delta)}$ |  $\frac{\gamma\lambda}{3((\beta-1)\lambda+\delta)}$ |  Any
$P_{4\mp}$ | $\mp 1$ | 0 |  0 |  0 |  0
$P_{5}$ | $\frac{\sqrt{\frac{3}{2}}(\beta-1)}{\delta}$ | $\frac{\sqrt{\frac{\left(-3\beta^{2}+6\beta+2\delta^{2}-3\right)((\beta-1)\lambda+\delta)}{\delta}}}{\sqrt{2}\sqrt{\delta((\beta-1)\lambda+\delta)}}$ |  $\frac{-3\beta-\delta\lambda+3}{\text{$\alpha$1}\delta^{2}}$ |  $\frac{2\gamma(3\beta+\delta\lambda-3)}{3\delta((\beta-1)\lambda+2\delta)}$ |  0
Table 1: The critical points corresponding to autonomous system of equations
Eq. (57)–Eq. (62). s=0
---
Points |  $\omega_{\rm eff}$ |  $(\chi,\xi>0)$ | Stability
$P_{1\pm}$ |  $-1$ |  $\gamma>0,\beta\geq\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}+1$,
$\gamma<0,\beta\leq 1-\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}$ | Indeterminate
$P_{2\mp}$ |  $-1$ |  $\gamma>0,\beta\geq\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}+1$,
$\gamma<0,\beta\leq 1-\sqrt{\frac{2}{3}}\sqrt{\delta^{2}}$ | Saddle
$P_{3}$ |  $-1$ |  $\gamma>0,\beta\lambda+\delta>\lambda,\lambda>0,\delta>0$ | see Figs.[LABEL:fig:stability_p3_beta1, LABEL:fig:stability_p3_beta3]
$P_{4\mp}$ |  $\mp 1$ |  Any | Unstable
$P_{5}$ |  $-\frac{\beta\lambda+\delta-\lambda}{\delta}$ |  $\beta\geq 1,\lambda>0,\delta>0,\gamma>0$ | Indeterminate
Table 2: The condition of existence of the fixed points and their nature.

The critical points $P_{1\pm}$ produce an accelerating solution; however, when
$\xi$ becomes zero, it leads to the temperature of the fluid reaching absolute
zero, which is not physically viable. Moreover, the indeterminacy of the
Jacobian further complicates the assessment of stability for this point. As
observed in the case when $s\neq 0$, the point cannot exhibit a stable
solution. Therefore, this point cannot be considered to describe the late-time
era of the universe.

The critical points $P_{2\mp}$ also exhibit accelerating solutions, and $\xi$
can take any positive value. However, considering the existence range given in
Tab.[2], for any values of $\beta$, the point becomes a saddle point. Thus,
these points are also not considered viable descriptions for the late-time
epoch of the universe.

The critical point $P_{3}$ can exhibit an accelerating solution where the
field’s kinetic part vanishes, and the corresponding stability for
$s\rightarrow 0$ shows similar behavior as discussed previously in
Figs.[LABEL:fig:stability_p3_beta1, LABEL:fig:stability_p3_beta3]. We have
also noted that for $s=0$, one of the eigenvalues becomes zero, rendering the
standard linearization technique inapplicable for determining the stability of
the point. However, we can employ a simple numerical trick to determine the
stability even without resorting to the center manifold theorem. Since the
system is non-adiabatically coupled, at any instant of time, the entropy of
the dark matter fluid must be positive or zero. Hence, we numerically evolved
the system by choosing initial conditions near zero, i.e., $s>0$, and observed
that the system stabilizes at $P_{3}$ with $s=0$ in the far future, as
demonstrated in Fig.[LABEL:fig:coord_p3_beta3]. Consequently, this point is a
physically viable fixed point for describing the late-time phase of the
universe.

The other critical points $P_{4\mp}$ rely solely on the field’s kinetic
component, while the co-ordinate $z$ vanishes. This leads to the autonomous
equation in $s^{\prime}$ becoming zero, i.e., $s^{\prime}=0$, without
requiring $s=0$. Consequently, the variable $s$ can take any value. The
effective equation of state at this point is $1$, indicating a stiff matter
solution, which reflects the minimally coupled behavior of quintessence in the
far past epoch. Since $s$ can take any value, this suggests that during the
past epoch, the system was non-adiabatically coupled.

At critical point $P_{5}$, all the field components are finite, but the
parameter corresponding to the temperature vanishes. For $\beta=1$, the
effective equation of state becomes $-1$. If $\beta>1$, then
$0<\lambda<-\frac{\delta}{2\beta-2}$. However, since $\xi$ vanishes for any
choice of model parameters and the Jacobian becomes indeterminate, as
discussed earlier, this point cannot be considered physically relevant.

(a)

(b)

Figure 3: The numerical evolution of cosmological parameters for
$(\gamma=1,\beta=3,\alpha_{1}=1,\delta=2,\lambda=0.5)$ with the initial
conditions $x(0)=0.02,y(0)=0.7,z(0)=-0.3,\chi(0)=0.03,\xi(0)=0.009,s(0)=0.15$.

After carefully studying the nature of critical points for $s\neq 0$ and
$s=0$, we can conclude that the physically viable critical point that
describes the behavior of the late-time epoch of the universe is $P_{3}$.
Therefore, to analyze the system’s overall dynamics, we numerically simulate
the autonomous equations Eq. (57)–Eq. (62), by selecting the benchmark points
$\beta=3$, $\delta=2$, $\lambda=0.5$, $\alpha_{1}=1$. Corresponding to these
benchmark points, the coordinates of the point $P_{3}$ become:
$(x=0,y=0.82,z=-0.17,\chi=0.06,\xi=\xi)$, and the corresponding field and
fluid fractional densities become $\Omega_{\phi}=0.67$, $\Omega_{M}=0.5$. The
numerical evolution of the cosmological parameters has been demonstrated in
Fig.[LABEL:fig:evo_p3_beta3] against $N=\ln a$, ranging from the past epoch to
the future epoch of the universe.

At the very past epoch of the universe, the field’s kinetic component
dominates, showing the behavior described by points $P_{4\mp}$. At this epoch,
the effective equation of state and the field’s equation of state
$\omega_{\phi}\equiv\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ both become $1$. As the
universe enters the matter-dominated phase, the effective equation of state
becomes zero, and it continuously increases towards negative values. However,
a steeper fall can be observed in the behavior of $\omega_{\phi}$. Unlike the
standard model of cosmology ($\Lambda$CDM) or other versions of minimally
coupled quintessence field with dark matter fluid, as the system enters the
current epoch ($a=1$ or $N=0$), the dark energy dominates over the fluid
component and the effective equation of state becomes $-1$. However, in the
current scenario, the fluid energy density dominates over the field density,
forcing the effective equation of state to cross the phantom barrier ($-1$)
for a brief period of time. This is the consequence of non-adiabatic energy
transfer occurring between the field and the fluid.

As the system further evolves into the future epoch, the model stabilizes to
$\omega_{\rm eff}=-1$ with a dominating field energy density. However, because
of the interaction, the fluid energy density does not get diluted. At any
instant of time, the fluid and field energy densities can take values greater
than $1$, however, the total energy density of the system $\Omega_{\rm
tot}=\Omega_{M}+\Omega_{\phi}+\alpha_{1}z$ always remains $1$ as shown by the
gray dashed line parallel to the $x$-axis. One way to interpret this behavior
is as discussed in ref.[64], where it is noted that it’s nearly impossible to
distinguish between dark sectors and gravity; therefore, they are degenerate.
Alongside the cosmological parameters, we have also shown the evolution of
dynamical variables in Fig.[LABEL:fig:coord_p3_beta3], and found that the
thermodynamic variables throughout the evolution follows the constraint Eq.
(55).

In addition to stability, we have also assessed the viability of the model by
comparing it with the $\Lambda$CDM model using available observational data
from Hubble $(43)$ and Pantheon+ $(1701)$ (url) datasets [65, 66]. Utilizing
the relation between the scale factor $a$ and redshift $\tilde{z}$,
$a=1/(1+\tilde{z})$, we can derive the evolution of the Hubble parameter as
follows:

$\frac{dH}{d\tilde{z}}=\frac{3}{2}\frac{1}{(1+\tilde{z})}H(\tilde{z})(1+\omega_{\rm
eff}).$ (63)

The Hubble parameter has been fitted with observational Hubble dataset (OHD)
alongside the $\Lambda$CDM model, using dark matter density $\Omega_{m}=0.25$,
cosmological constant fractional density $\Omega_{\Lambda}=0.75$, and
$H_{0}=67$ (in units of km/s/Mpc), as shown in Fig.[LABEL:fig:hubble_beta3].
The current model has been simulated using the same benchmark point and
initial conditions as shown in Fig.[LABEL:fig:evo_p3_beta3], with $H_{0}=66$,
to better match observations. The non-linear evolution of $H(\tilde{z})$ near
$\tilde{z}\rightarrow 0,1.5$ shows a significant deviation from the standard
model. By testing the model with larger datasets using sampling techniques
like MCMC, it may contribute to lowering the discrepancy in the Hubble value.

To fit the model with the Pantheon+ data, we utilize the distance modulus
$\mu$ relation with redshift,

$\mu=5\log_{10}(d_{L})+25,$ (64)

where the luminosity distance is given by,

$d_{L}=2.99\times
10^{5}(1+\tilde{z})\int_{0}^{\tilde{z}}\frac{1}{H(\tilde{z})}d\tilde{z}\ ,$
(65)

The evolution of $\mu$ for the $\Lambda$CDM model overlaps with the current
model, as shown in Fig.[LABEL:fig:supernova_dist_1]. It’s noteworthy that we
have selected the model parameter $\beta=3$ because the evolution of the
Hubble parameter closely resembles that of $\Lambda$CDM for this case.

(a)

(b)

Figure 4: (a) The Hubble evolution and (b) the distance modulus corresponding
to $\Lambda$CDM is evaluated with
$\Omega_{M}=0.25,\Omega_{\Lambda}=0.75,H(0)=67.0$. The current model has been
fitted for $(\gamma=1,\beta=3,\alpha_{1}=1,\delta=2,\lambda=0.5)$ with the
initial conditions
$x(0)=0.02,y(0)=0.7,z(0)=-0.3,\chi(0)=0.03,\xi(0)=0.009,s(0)=0.15,H(0)=66.0$.

##  VI Conclusion

The study delves into a non-minimal coupling scenario between the quintessence
field and a pressureless fluid, examined thoroughly through the variational
principle. By incorporating one of the Lagrange parameters $\varphi$ into the
interacting Lagrangian, we altered the conservation laws governing the fluid’s
number density and entropy density. Consequently, the fluid sector introduces
an additional pressure termed creation pressure. Upon evaluating the covariant
derivative of the energy-momentum tensor, it becomes apparent that individual
components are no longer conserved, facilitating the flow of energy from the
field to the fluid through the field derivative of the interaction function
$f_{,\phi}\dot{\phi}$.

We assessed the background stability of the model by considering an
exponential type of interaction. The system yielded a stable critical point
capable of producing an accelerating solution in the far future epoch of the
universe, dominated by the scalar field energy density with a finite fluid
energy density. Furthermore, numerical simulations corresponding to the
interaction $f\propto\rho^{3}$ revealed that during the current epoch, the
fluid density predominates, resulting in a phantom equation of state for a
brief period.

To evaluate the viability of the model, we conducted simulations with the
Hubble and Pantheon+ data sets alongside the $\Lambda$CDM model. The model
demonstrates the ability to mimic $\Lambda$CDM behavior while simultaneously
exhibiting deviations. In future work, we intend to conduct a comprehensive
statistical analysis to further assess the model’s viability with various data
sets.

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