Energy–Momentum Squared Gravity Attached with Perfect Fluid Admitting Conformal Ricci Solitons

Abstract

In the present research note, we explore the nature of the conformal Ricci solitons on the energy–momentum squared gravity model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ that is a modification of general relativity. Furthermore, we deal with a subcase of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$-gravity model coupled with a perfect fluid, which admits conformal Ricci solitons with a time-like concircular vector field. Using the steady conformal Ricci soliton, we derive the equation of state for the perfect fluid in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model. In this series, we convey an indication of the pressure and density in the phantom barrier period and the stiff matter era, respectively. Finally, using a conformal Ricci soliton with a concircular vector field, we study the various energy constraints, black holes, and singularity circumstances for a perfect fluid coupled to $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity. Lastly, employing conformal Ricci solitons, we formulate the first law of thermodynamics, enthalpy, and the particle production rate in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity and orthodox gravity.

1. Introduction

In 1988, Hamilton [1] first presented the idea of the Ricci flow. A self-similar solution or Ricci soliton of its singularities are primarily explained by the Ricci flow, which was put forward in Riemannian Geometry [1]. The Ricci flow equation is [1]

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}{g}_t=-2\mathcal{S} \left(g_t\right),g_0=g\left(0\right),\end{array}$$

(1)

where $\mathcal{S}$ denotes the Ricci tensor of the Riemannian metric g.

Fischer [2] proposed the conformal Ricci flow, a new geometric flow that is a variation of the traditional Ricci flow. The unit volume restriction in the original equation is replaced in this version by a scalar curvature constraint. According to [2], the equation for the conformal Ricci flow is as follows:

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial g_t}{\partial t}}+2\left(\mathcal{S}\left(g_t\right)+{\displaystyle \frac{1}{n}}{g}_t\right)=-P{g}_t,\end{array}$$

(2)

$$R\left(g_t\right)=-1$$

for a dynamically evolving metric $g_t$, where $P=P\left(t\right)$, is a time-dependent non-dynamical scalar field, the Ricci tensor $\mathcal{S}\left(g_t\right)$ is a function of the metric tensor $g_t$, $R\left(g_t\right)=-1$ denotes the scalar curvature of the manifold $(M^n,g_t)$, which modifies the volume constraint $vol(M^n,g_t)=1$ of the evolving metric $g_t$ to a scalar curvature constraint $R\left(g_t\right)=-1$, and n represents the dimension of the manifold. Einstein metrics with the Einstein constant ${\displaystyle \frac{-1}{n}}$ are the equilibrium points of the conformal Ricci flow (2).

The conformal Ricci flow (2) equations are analogous to the Navier–Stokes equations of fluid mechanics

$${\displaystyle \frac{\partial u}{\partial t}}+{\nabla }_{u}u+\nu \Delta u=-gradP,$$

(3)

$$divu=0.$$

Due to this analogy, the time-dependent scalar field $P\left(t\right)$ is called a conformal pressure that maintains the scalar curvature constraint $R\left(g_t\right)=-1.$

The idea of a conformal Ricci soliton was first presented by Basu and Bhattacharyya in [3], and the associated equation is as follows:

$${\displaystyle \frac{1}{2}}{\mathfrak{L}}_{F}g+\mathcal{S}+\left[\mu -\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{n}}\right)\right]g=0.$$

(4)

where ${\mathfrak{L}}_F$ indicates the Lie derivative operator along the vector field F, and $\mu$ is a real constant. One can categorize the conformal Ricci soliton as increasing, stable, or shrinking if

•  $\mu >0$;
•  $\mu =0$;
•  $\mu <0$, respectively.

However, the traditional method for analyzing known cosmic dynamics is provided by Einstein’s formulation of the gravitational field equations [4,5]. The Einstein field equations provide the best estimate to the observable data. Accompanied by the inclusion of a hypothetical component of the cosmos known as Dark Matter $\left(DM\right)$ [6].

Furthermore, the Universe has an odd component called Dark Energy $\left(DE\right)$ that controls the matter–energy ratio, accelerates expansion, and is believed to be the primary component of the Universe. This led to the development of more complex theories of gravity by a number of mathematicians and physicists, such as $\mathcal{F}\left(\mathcal{R}\right)$-gravity [7]. These theories diverge from Einstein’s mainstream theory of gravity and may offer a trustworthy approximation to quantum gravity [8].

Advantage of Modified Gravity over the Standard Cosmological Models: The evolution of the universe is described by standard cosmological models, such as the $\Lambda$CDM model, which use General Relativity ($GR$) or Einstein’s theory of gravitation and elements like cold $DM$ and $DE$. In contrast, modified gravity frameworks suggest changes to $GR$ basic ideas in order to explain phenomena like the existence of dark energy and the universe’s late-time acceleration without requiring the addition of new elements. The key differences are in how they are mathematically expressed, for example, curvature-based $GR$ versus torsion-based teleparallel gravity $\mathcal{F}\left(\mathcal{T}\right)$ [9], and how the ensuing cosmic dynamics can vary at high redshifts and impact the creation of structures. Although $\Lambda$CDM is well established within $GR$, modified gravity theories are theoretical attempts to extend or modify $GR$ principles in order to build a more comprehensive and consistent theory of quantum gravity [10,11,12].

Moreover, $GR$ can be extended to the $\mathcal{F}\left(\mathcal{R}\right)$ gravity and transformed into a function $\mathcal{F}\left(\mathcal{R}\right)$ using the Einstein–Hilbert Lagrangian density, where $\mathcal{R}$ stands for the Ricci scalar [13]. Higher order curvature resolves the massive neutron stars in the $\mathcal{F}\left(\mathcal{R}\right)$-gravity, for example, stable star structure is one of the astronomical models that is unable to support $\mathcal{F}\left(\mathcal{R}\right)$-gravity (for more details, see [14,15]). Modified $GR$, such as the $\mathcal{F}\left(\mathcal{R}\right)$ theory of gravity, contains a higher order differential function of the Ricci scalar $\mathcal{R}$. The choice of the equation of state ($EoS$) for a massive star is crucial for the existence of solutions, and therefore, a polytropic equation of state can be used to overcome the possible problems related to the equation of state. Another approach to probe the viability of $\mathcal{F}\left(\mathcal{R}\right)$ theories in the strong gravity regime is to use a method called perturbative constraints or order reduction [16,17,18]. A better-suited theory that can correctly characterize huge mass neutron stars and estimate their maximum mass limit, and how this limit is affected by the sound speed for a broad range of $EoS$, would be modified gravity [15].

Harko et al. [19] presented a more extensive gravity notion, which they called $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity theory. They achieved this by assuming that $\mathcal{T}$ and $\mathcal{R}$ are arbitrary functions of the Lagrangian. $\mathcal{T}$ is the energy–momentum tensor trace in this instance. This idea well describes the late-time rapid expansion of the Universe.

Katirci and Kavuk [20] first introduced a covariant modification of $GR$, that allows for the possibility of a term proportional to ${\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$ in the action functional. Specific models of this gravity theory were studied in further research on this theory [21,22]. Roshan et al. [21] investigated the possibility of an early bounce in the energy–momentum squared gravity (EMSG) model, utilizing the specific functional supplied by

$$\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2,$$

where $\lambda$ is a constant.

In [21], the late time acceleration of the Universe has been explored within the context of energy–momentum squared gravity (EMSG) theories, taking into account the scenario of a pressureless fluid. In $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity theory, the gravitational Lagrangian depends on the contraction of the energy–momentum tensor with itself, i.e., ${\mathcal{T}}^2={\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$ [22,23], and the Ricci scalar $\mathcal{R}$. It should be mentioned that only in the presence of matter sources do the field equations deviate from $GR$. The term $\mathcal{F}\left({\mathcal{T}}_{ab}{\mathcal{T}}^{ab}\right)$ can be introduced in a variety of ways, leading to different iterations of the theory. For example, the variant with $\mathcal{F}\left({\mathcal{T}}_{ab}{\mathcal{T}}^{ab}\right)\propto {\left({\mathcal{T}}_{ab}{\mathcal{T}}^{ab}\right)}^{\lambda }$, where $\lambda$ is a constant, is energy–momentum squared gravity.

Much attention has been paid to this idea, which has been studied in a number of contexts. In [24], the use of energy–momentum powered gravity to analyze cosmic acceleration is presented. Cosmological theories in energy–momentum squared gravity, including brane-world cosmologies, loop quantum gravity, k-essence, and bulk viscous cosmology, have been discussed [22]. In energy–momentum log gravity, an extension of the traditional $\Lambda$CDM model has been studied [25]. Neutron star energy and momentum squared gravity limitations and their cosmological implications have been studied in [23]. Compact stars with spherically symmetric symmetry have been analyzed [26]. The structure of the phase space and its physical implications have been investigated for various gravity function types using dynamic system analysis $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ in [27].

On the hand, a spacetime in $GR$ and a cosmology model both exhibit a time-oriented 4-dimensional connected Lorentzian manifold. This displays a particular categorization of pseudo-Riemannian manifolds among the Lorentzian metric having signature $(-,+,+,+)$ that is crucial in $GR$. In the geometry of Lorentzian manifolds, we examine the characteristics of the vectors on the manifold. Thus, Lorentzian manifolds are emerging as the most practical framework for $GR$ analysis [28,29].

Definition 1

([29]). If the Ricci tensor $\mathcal{S}$ has the shape, then a quasi-Einstein–Lorentzian manifold is called a perfect fluid spacetime

$$\mathcal{S}=A_{1}g+A_2\eta \otimes \eta ,$$

(5)

wherein scalars $A_1$ and $A_2$ are present, and 1-form η.

Basically, the energy–momentum tensor [30] can be used to recast the equations of motion in $GR$. This energy–momentum tensor is represented as [28,29].

$${\mathcal{T}}_{ab}=(p+\sigma ){\eta }_a{\eta }_b-p{g}_{ab},$$

where $\sigma$ is the density, and p denotes the pressure of perfect fluid [31].

Specifically, in the $GR$, physical matter symmetry is associated with spacetime geometry. Actually, the symmetry of the metric tensor implies the categories of solutions to the field equations. The soliton is a notable symmetry related to the geometrical flow of spacetime.

Ahsan and Ali in [32] investigated spacetime in terms of the Ricci soliton. In [33], the authors also discussed Ricci solitons on perfect fluid spacetime. Using a range of methods, Siddiqi et al. (cf. [34,35,36]) used solitons to investigate perfect fluid spacetime.

Recently, Siddiqi, et al. [37,38] have introduced the characteristics of $\mathcal{F}\left(\mathcal{R}\right)$-gravity, $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity, and $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity [39] filled with perfect fluid matter admitting various solitons and gradient solitons. As a result, motivated by previous research, we are examining a subcase of $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity with perfect fluid admitting the conformal Ricci solitons in this paper.

The structure of the paper is as follows: In Section 2, we derive field equation for a subcase of $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$-gravity coupled with a perfect fluid and investigate solutions in a modified theory of gravity including a Lagrangian, in addition to deriving key quantities and equations, including the Ricci tensor, the scalar curvature, and a squared energy–momentum tensor term. In Section 3, we calculate the conformal Ricci soliton in the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ paired with a perfect fluid whose concircular vector field $\zeta$ is its timelike velocity vector field. In Section 4, we study the energy condition for the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ in terms of conformal Ricci solitons. In Section 5, we discuss some implications for the singularity theorems in the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ linked with perfect fluid and admit the conformal Ricci solitons. In Section 6, we explore some applications of the temperature evolution of gravitational systems in terms of the equation of state of $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity with perfect fluid admitting the conformal Ricci solitons.

2. Field Equation for $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$-Gravity Attached with Perfect Fluid

The EMSG model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ attached with perfect matter fluid is explored in this part of this paper by setting ${\mathcal{T}}^2={\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$.

We can derive a number of hypothetical models for different values of $\mathcal{R}$ and $\mathcal{T}$ [19] as this model depends on the physical parameters of the perfect fluid matter field. We are utilizing the following model as an example:

$$\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2,$$

(6)

where $\mathcal{F}\left(\mathcal{R}\right)$ and $\mathcal{F}\left(\mathcal{T}\right)$ represent the arbitrary functions of $\mathcal{R}$ and $\mathcal{T}$ and a scalar $\lambda$, respectively.

Therefore, the Einstein–Hilbert action term for the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity is

$${\mho }_E={\displaystyle \frac{1}{16\pi }}\int [2\Lambda +{\mathfrak{L}}_n+\mathcal{F}(\mathcal{R},{\mathcal{T}}_{ab}{\mathcal{T}}^{ab})]\sqrt{(-g)}{d}^{4}x,$$

(7)

where ${\mathfrak{L}}_n$ represents the Lagrangian, and $\Lambda$ represents the cosmological constant. The energy tensor of matter is supplied by

$${\mathcal{T}}_{ab}=g_{ab}{\mathcal{L}}_n-{\displaystyle \frac{2}{\sqrt{-g}}}{\displaystyle \frac{\delta \left(\sqrt{-g}{\mathcal{L}}_n\right)}{\delta g^{ab}}}.$$

(8)

Assume that ${\mathfrak{L}}_n$ depends only on $-g_{ab}$ and is independent of its derivatives.

Consequently, the variation of ${\mho }_E$ is

$$\delta {\mho }_E={\displaystyle \frac{1}{16\pi }}\int [{\mathcal{F}}_{\mathcal{R}}\delta \mathcal{R}+{\mathcal{F}}_{{\mathcal{T}}^2}\delta {\mathcal{T}}^2-{\displaystyle \frac{1}{2}}{g}_{ab}\mathcal{F}\delta g^{ab}-g_{ab}\Lambda \delta g^{ab}+{\displaystyle \frac{1}{\sqrt{-g}}}\delta \left(\sqrt{-g}\right){\mathfrak{L}}_n]\sqrt{-g}{d}^{4}x.$$

(9)

To ensure clarity, these have been described as

$$\mathcal{F}=\mathcal{F}(\mathcal{R},{\mathcal{T}}_{ab}{\mathcal{T}}^{ab}),$$

$${\mathcal{F}}_{\mathcal{R}}={\displaystyle \frac{\partial \mathcal{F}}{\partial \mathcal{R}}},$$

and

$${\mathcal{F}}_{{\mathcal{T}}^2}={\displaystyle \frac{\partial \mathcal{F}}{\partial {\mathcal{T}}^2}}.$$

The result is known to be produced by the Ricci scalar variation

$$\delta \mathcal{R}={\mathcal{S}}_{ab}\delta g^{ab}+g_{ab}\square \delta g^{ab}-{\nabla }_a{\nabla }_b\delta g^{ab}.$$

(10)

Using the variation of ${\mathcal{T}}^2$ with respect to the metric we turn up

$${\Theta }_{ab}\equiv {\displaystyle \frac{\delta \left({\mathcal{T}}_{ij}{\mathcal{T}}^{ij}\right)}{\delta g^{ab}}}={\mathcal{T}}_{ij}{\displaystyle \frac{\delta \left({\mathcal{T}}^{ij}\right)}{\delta g^{ab}}}+{\displaystyle \frac{\delta \left({\mathcal{T}}_{ij}\right)}{\delta g^{ab}}}{\mathcal{T}}^{ij}$$

(11)

$${\Theta }_{ab}={\displaystyle \frac{\delta \left({\mathcal{T}}_{ij}\right)}{\delta g^{ab}}}{\mathcal{T}}^{ij}+2{\mathcal{T}}_a^i{\mathcal{T}}_{bi}+{\displaystyle \frac{\delta \left({\mathcal{T}}_{ij}\right)}{\delta g^{ab}}}{\mathcal{T}}^{ij}.$$

(12)

The variation in the energy–momentum tensor given in (12) is expressed using (8) as

$${\displaystyle \frac{\delta \left({\mathcal{T}}_{ij}\right)}{\delta g^{ab}}}{\mathcal{T}}^{ij}=-{\mathfrak{L}}_n{\mathcal{T}}_{ab}+{\displaystyle \frac{1}{2}}{g}_{ab}{\mathfrak{L}}_n\mathcal{T}-{\displaystyle \frac{1}{2}}\mathcal{T}{\mathcal{T}}_{ab}-{\displaystyle \frac{{\partial }^2{\mathfrak{L}}_n}{\partial g^{ab}\partial g^{ab}}}{\mathcal{T}}^{ab}.$$

(13)

In view of (13), (12) turns into

$${\Theta }_{ab}=2{\mathcal{T}}_a^i{\mathcal{T}}_{bi}-\mathcal{T}{\mathcal{T}}_{ab}-2{\mathfrak{L}}_n({\mathcal{T}}_{ab}-{\displaystyle \frac{1}{2}}{g}_{ab}\mathcal{T})-4{\mathcal{T}}^{ij}{\displaystyle \frac{{\partial }^2{\mathfrak{L}}_n}{\partial g^{ab}\partial g^{ab}}}.$$

(14)

Now, in light of (9), the field equation of the energy–momentum squared gravity $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ is provided by

$${\mathcal{F}}_{\mathcal{R}}{\mathcal{S}}_{ab}=\Lambda g_{ab}+{\displaystyle \frac{1}{2}}\mathcal{F}{g}_{ab}+(g_{ab}{\nabla }_i{\nabla }^i-{\nabla }_a{\nabla }_b){\mathcal{F}}_{\mathcal{R}}+{\displaystyle \frac{1}{8\pi }}({\mathcal{T}}_{ab}-{\displaystyle \frac{1}{8\pi }}{\mathcal{F}}_{{\mathcal{T}}^2}{\Theta }_{ab}).$$

(15)

It is necessary to take into account the matter content in addition to the geometry aspects. In this instance, a perfect fluid is selected, and its energy–momentum tensor is defined as [29]

$${\mathcal{T}}_{ab}=(p+\sigma ){\eta }_a{\eta }_b-p{g}_{ab},$$

(16)

where $\eta$ is a unit time-like velocity four vector field, p is the pressure, and $\sigma$ is the energy density.

In the case of a perfect fluid described by the Lagrangian ${\mathfrak{L}}_n=-p$, ${\eta }^a{\eta }_b=1$, and the energy–momentum tensor specified in (16), it may be expressed as [26].

$${\Theta }_{ab}=-({\sigma }^2-p^2){\eta }_a{\eta }_b.$$

(17)

But its crucial to remember that the Ricci scalar has a constant value, meaning that the field Equation (15) reduces to

$${\mathcal{F}}_{\mathcal{R}}{\mathcal{S}}_{ab}=\Lambda g_{ab}+{\displaystyle \frac{1}{2}}\mathcal{F}{g}_{ab}+{\displaystyle \frac{1}{8\pi }}({\mathcal{T}}_{ab}-{\displaystyle \frac{1}{8\pi }}{\mathcal{F}}_{{\mathcal{T}}^2}{\Theta }_{ab}).$$

(18)

The energy–momentum squared gravity $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ model requires the development of such an analysis. Let us write the field Equation (15) considering the energy–momentum tensor for the perfect fluid given in (16).

For simplicity, it is chosen that $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$, where $\lambda$ is an integer. The Ricci tensor assumes the perfect fluid form, i.e.,

$${\mathcal{S}}_{ab}=\left\{\Lambda +{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2-8\pi p\right\}{g}_{ab}+\left\{{\mathcal{F}}_{{\mathcal{T}}^2}({\sigma }^2-p^2)+8\pi (\sigma +p)\right\}{\eta }_a{\eta }_b.$$

(19)

Hence, for a spacetime $(M^4,g)$ coupled with the perfect fluid in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity, the shape of the Ricci tensor is given as

$${\mathcal{S}}_{ab}=\alpha g_{ab}+\beta {\eta }_a{\eta }_b,$$

(20)

where

$$\alpha =\left\{\Lambda -8\pi p+{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2\right\},$$

(21)

$$\beta =\left\{{\mathcal{F}}_{{\mathcal{T}}^2}({\sigma }^2-p^2)+8\pi (\sigma +p)\right\}.$$

(22)

We make the assumption that $\alpha$ and $\beta$ are not both zero throughout this manuscript. For consistency, we also provide the proof here, even though the formulation of the Ricci tensor was discovered using a similar technique in [38]. Consequently, we obtain

Theorem 1.

A perfect fluid spacetime in the energy–momentum squared gravity model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$ has the following Ricci tensor

$${\mathcal{S}}_{ab}=\left\{\Lambda +{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2-8\pi p\right\}{g}_{ab}+\left\{{\mathcal{F}}_{{\mathcal{T}}^2}({\sigma }^2-p^2)+8\pi (\sigma +p)\right\}{\eta }_a{\eta }_b.$$

(23)

Corollary 1.

The scalar curvature of the energy–momentum squared gravity model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$ coupled with perfect fluid is given by

$$\mathcal{R}=8\pi (3p-\sigma )-{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)({\sigma }^2-p^2)-4\Lambda -2\lambda {\mathcal{T}}^2.$$

(24)

In index-free notation, Equation (19) can now be expressed as

$$\mathcal{S}=\alpha g+\beta \eta \otimes \eta .$$

(25)

3. Conformal Ricci Soliton on $\mathit{EMSG}$ Model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$ Coupled with Perfect Fluid

In this part of this paper, we estimate the conformal Ricci soliton $\left(CRS\right)$ in the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ coupled with perfect fluid whose time-like velocity vector field is $\zeta$, a concircular vector field.

Definition 2

([40]). A concircular vector field ζ $\left(CVF\right)$ on the Lorentzian spacetime manifold $(M^4,g)$ is characterized by

$${\nabla }_a\zeta =\Omega a$$

(26)

wherein Ω is a smooth function on $M^4$.

Next, adopting $\mathcal{V}=\zeta$, Equation (4) becomes

$${\displaystyle \frac{1}{2}}\left({\mathfrak{L}}_{\zeta }g\right)(a,b)+\mathcal{S}(a,b)+\left[\mu -\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)\right]g(a,b)=0,$$

(27)

for any time-like velocity four vector field $a,b\in \chi \left(M^4\right).$

$${\displaystyle \frac{1}{2}}[g({\nabla }_a\zeta ,b)+g(a,{\nabla }_b\zeta )]+\mathcal{S}(a,b)+\left[\mu -\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)\right]g(a,b)=0.$$

(28)

In light of (26) and (28), we gain

$$\mathcal{S}(a,b)+\left[\mu -\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right]g(a,b)=0.$$

(29)

By adding (25) to the equation above, we obtain

$$\left[\alpha +\mu -\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right]g(a,b)+\beta \eta \left(a\right)\eta \left(b\right)=0.$$

(30)

Entering $a=b=\zeta$ into (30) now, we discover

$$\mu =\beta -\left[\alpha +\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right].$$

(31)

Next, for the sake of convenience in this section, we use an abbreviation ${\Xi }^F$ to indicate “a spacetime $(M^4,g)$ in the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ attached with perfect fluid”. As such, we obtain the subsequent outcome.

Theorem 2.

If a ${\Xi }^F$ admits a $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ, then $CRS$ is increasing, stable, or decreasing, referring to the following:

1. 

$\left\{8\pi +(\sigma -p){\mathcal{F}}_{{\mathcal{T}}^2}\right\}>{\displaystyle \frac{1}{(p+\sigma )}}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda -8\pi p+\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}$;

2. 

$\left\{8\pi +(\sigma -p){\mathcal{F}}_{{\mathcal{T}}^2}\right\}={\displaystyle \frac{1}{(p+\sigma )}}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda -8\pi p+\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}$;

3. 

$\left\{8\pi +(\sigma -p){\mathcal{F}}_{{\mathcal{T}}^2}\right\}<{\displaystyle \frac{1}{(p+\sigma )}}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda -8\pi p+\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}$, respectively, provided $(p+\sigma )\ne 0.$

Remark 1.

An equation of state $\left(EoS\right)$ of form $\sigma +p=f\left(a\right)$ was examined by Srivastava in [41], where $f\left(a\right)$ is a function of scale factor $a\left(t\right)$ with the cosmic time $t.$ Srivastava explained that the $EoS$$\sigma =-p$, $\sigma =p$, $p={\displaystyle \frac{\sigma }{3}}$, and $p=0$ indicates the dark matter era, stiff matter era, radiation era, and dust matter era [41], respectively.

Now, part (2) of Theorem 2 entails the following results:

Theorem 3.

If a ${\Xi }^F$ admits the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ, then the $EoS$ is

$$p+\sigma ={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +{\mathcal{F}}_{{\mathcal{T}}^2}(\sigma -p)}}$$

(32)

Moreover, we assume that the source is of the stiff matter type, then $EoS$ is $p=\sigma$. This fact together with Equation (32) gives

$$p=\sigma ={\displaystyle \frac{1}{24\pi }}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda +\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}.$$

(33)

Corollary 2.

If the source of a ${\Xi }^F$ is a stiff matter type and admits the stable $CRS$, then the density σ and pressure p are determined by (33).

When a phantom barrier occurs, $\sigma =-p=-{\displaystyle \frac{1}{8\pi }}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda +\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}.$ Thus, we can articulate

Corollary 3.

If the source of a ${\Xi }^F$ is a phantom barrier type and admits the stable $CRS$, then the energy density and the pressure are evaluated as $\sigma =-p=-{\displaystyle \frac{1}{8\pi }}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda +\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}.$

4. Energy Conditions in $\mathit{EMSG}$ Model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$ Coupled with Perfect Fluid Admits Conformal Ricci Soliton

According to [42], we know the Ricci tensor $\mathcal{S}$ in the spacetime satisfies the condition

$$\mathcal{S}(\zeta ,\zeta )>0,$$

(34)

Equation (34) is known as the time-like convergence condition (TCC) for all time-like vector fields $\zeta \in \chi \left(M^4\right)$.

Equation (23) entails that

$$\mathcal{S}(\zeta ,\zeta )=\alpha +\beta .$$

The spacetime fulfills the $TCC$, that is, if $\mathcal{S}(\zeta ,\zeta )>0$.

$$\left\{8\pi +(\sigma -p){\mathcal{F}}_{{\mathcal{T}}^2}\right\}>{\displaystyle \frac{1}{(p+\sigma )}}\left\{{\displaystyle \frac{\mathcal{R}}{2}}+{\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda -8\pi p+\left({\displaystyle \frac{P}{2}}+{\displaystyle \frac{1}{4}}\right)+\Omega \right\}.$$

(35)

The spacetime obeys the cosmological strong energy constraint $\left(SEC\right)$ [43]. In light of the above fact and from (35), we can state that

Theorem 4.

If a ${\Xi }^F$ admits a $CRS$$(d,\zeta ,\mu )$ and satisfies $TCC$ with a $CVF$ ζ, provided $p+\sigma \ne 0,$ then $CRS$ is growing.

Remark 2.

Hawking and Ellis [44] demonstrated in 1973 that $TCC\Rightarrow SEC$, $TCC\Rightarrow NCC$ (Null Convergence Condition), and $SEC\Rightarrow NEC$ (Null Energy Condition), and as a result $TCC\Rightarrow NCC$.

The following outcomes are obtained by combining Remark 1 with Theorem 4:

Theorem 5.

If a ${\Xi }^F$ admits an increasing $CRS$$(d,\zeta ,\mu )$ with a $CVF$ ζ, and if (35) holds, then the ${\Xi }^F$ obeys $SEC$.

Corollary 4.

If a ${\Xi }^F$ admits an increasing $CRS$ with a $CVF$ ζ, and if (35) holds, then the ${\Xi }^F$ obeys $NCC$.

Corollary 5.

If a ${\Xi }^F$ admits a $CRS$ with a $CVF$ ζ and satisfies $SEC$, then the Ricci tensor $\mathcal{R}ic$ for the increasing $CRS$ is of the second Segre type [44].

5. Application of Singularity Theorem in $\mathit{EMSG}$ Model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$ Coupled with Perfect Fluid Admits a Conformal Ricci Soliton

Remark 3.

Vilenkin and Wall ([45]) proved that the spacetime $(M^4,g)$ obeys the $NCC$ based on Penrose’s singularity theorem. Then, $(M^4,g)$ has a trapped surface outside the black holes and some black holes.

Remark 4.

The Cauchy surface contains trapped (anti-trapped) surfaces, and the singularity theorem requires that some future and past-directed null geodesics must be incomplete. But once again a contradiction with the theorem is avoided if some black holes are formed in the future of the Cauchy surface and some time-reversed black holes (i.e., white holes), or “naked singularities”, are formed in the past of Cauchy surface. Moreover, the symmetry of gravity models causes the singularity. In view of Penrose’s singularity theorem [46] that follows, we will assume that the spacetime is globally hyperbolic, i.e., that it admits a Cauchy surface. We will also assume that there are no naked singularities which lie on the black hole horizon itself, and such singularities would still be naked, even though they are not ruled out by global hyperbolicity.

The spacetime obeys the NCC, has a non-compact connected Cauchy surface, and includes a trapped surface. Assuming in addition that the trapped surface lies (at least partly) outside the black 314 hole horizons and is not completely surrounded by horizons, Penrose proved in (Theorem 1, [46]) that the spacetime must contain naked singularities on the horizon.

We can express the following result in light of Theorem 4, Remark 2, Remark 3, and Corollary 4.

Theorem 6.

Let a ${\Xi }^F$ admit an increasing $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ, and if the spacetime $(M^4,g)$ holds $NCC$, then the ${\Xi }^F$ contains some black holes with a trapped surface which is outside the black holes.

The universe matter production rates and subsequent thermodynamical parameters, including the creation pressure, temperature evolution, and entropy evolution, were examined by Ricardo et al. [47] in 2024 within the framework of $EMSG$. In [48] the authors also discussed the cosmological models, thermodynamic consequences, theoretical formalism, and the characteristics of compact objects in $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2).$

In the present study, we determine how the evolution of the universe aligns with the first law of thermodynamics, and the particle production rate and enthalpy are produced using the equation of state of the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ admitting the conformal Ricci solitons with a conformal vector field.

6. Application of $\mathit{EoS}$ of $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$-Gravity in Temperature Evolution

In this section, we deduce the first law of thermodynamics in terms of the $EoS$ of the $EMSG$ model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ attached with perfect fluid admitting the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$$\zeta$.

Specifically, the heat $\left(dQ\right)$ is minimal for a homogeneous spacetime; hence, the first law can be represented as [49].

$${\displaystyle \frac{h}{n}}d\left(nV\right)=d\left(\rho V\right)+pdV,$$

(36)

where $n=N/V$ is the particle number density, p is the fluid’s ordinary thermodynamic pressure,

$$h=p+\sigma$$

is the enthalpy per unit volume, and $\sigma$ is the cosmic fluid’s energy density.

Equation (36) can be further developed by taking the time derivative and describing the space volume with the scale factor $\left(a\right(t\left)\right)$ as follows: $V\left(t\right)=a^3\left(t\right)$.

$$\dot{\rho }+3H(p+\sigma )={\displaystyle \frac{p+\sigma }{n}}(\dot{n}+3Hn),$$

(37)

where $H=\dot{a}/a$ is the Hubble function.

It is convenient to add two quantities to the above equation in order to develop it further. The number current is defined as

$$N^{\mu }=n{u}^{\mu }$$

(38)

as well as the rate of particle formation $(\Gamma )$. This amount indicates how many particles are being formed at any one time relative to the total number of particles, i.e.,

$$\Gamma ={\displaystyle \frac{1}{N}}{\displaystyle \frac{dN}{dt}}={\displaystyle \frac{1}{n}}{\displaystyle \frac{d}{dt}}\left(nV\right).$$

(39)

From Equations (38) and (39), the covariant derivative of the number current in the flat $FLRW$ geometry thus takes the following form:

$${\nabla }_{\mu }{N}^{\mu }=\dot{n}+3Hn=v\Gamma$$

(40)

Equation (40) is inserted into Equation (37) to obtain the particle production rate in terms of pressure, energy density, and the Hubble function.

$$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{\sigma +p}}.$$

(41)

It is possible to derive a cosmic temperature evolution once this framework has been constructed. In order to do this, we first assume that the temperature $\left(\mathbb{T}\right)$ and the particle number density $\left(n\right)$ are the functions of the energy density $\left(\sigma \right)$ and the pressure $\left(p\right)$, that is

$$p=p(n,\mathbb{T}),\sigma =\sigma (n,\mathbb{T}).$$

(42)

As a result, the first law of thermodynamics can be expressed as [48]

$${\left({\displaystyle \frac{\partial \sigma }{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial \sigma }{\partial \mathbb{T}}}\right)}_n\mathbb{T}+3(\sigma +p)H=(\sigma +p)\Gamma .$$

(43)

Moreover, the first law of thermodynamics in terms of pressure $p=p(n,\mathbb{T})$ is given as

$${\left({\displaystyle \frac{\partial p}{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial p}{\partial \mathbb{T}}}\right)}_n\mathbb{T}+3(\sigma +p)H=(\sigma +p)\Gamma .$$

(44)

The following equation can be obtained from the previous equation by applying some helpful thermodynamic law.

Next, in light of Equations (32), (41) and (43), one can articulate the following results:

Theorem 7.

If a ${\Xi }^F$ admits the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfies the $EoS$ (32), then the particle production rate is

$$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ](8\pi +{\mathcal{F}}_{{\mathcal{T}}^2}(\sigma -p))}}.$$

(45)

Theorem 8.

The first law of thermodynamics in a ${\Xi }^F$ admitting the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfying the $EoS$ (32) is

$${\left({\displaystyle \frac{\partial \sigma }{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial \sigma }{\partial \mathbb{T}}}\right)}_n\mathbb{T}={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +{\mathcal{F}}_{{\mathcal{T}}^2}(\sigma -p)}}(\Gamma -3H).$$

(46)

Now, adopting Remark 1 and (46) together, we turn up the next Theorem.

Theorem 9.

Let a ${\Xi }^F$ admit the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfy the $EoS$, then evolution of the universe given the first law of thermodynamics can be expressed as follows

${\Xi }^{\mathit{F}}$	Equation of State $\left(\mathit{E}\mathit{o}\mathit{S}\right)$	First Law of Thermodynamics
dark matter era	$p=-\sigma$	${\left({\displaystyle \frac{\partial \sigma }{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial \sigma }{\partial \mathbb{T}}}\right)}_n\mathbb{T}={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +2\sigma {\mathcal{F}}_{{\mathcal{T}}^2}}}(\Gamma -3H)$
Stiff fluid era	$p=\sigma$	${\left({\displaystyle \frac{\partial \sigma }{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial \sigma }{\partial \mathbb{T}}}\right)}_n\mathbb{T}={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8}}(\Gamma -3H)$
Radiation era	$p={\displaystyle \frac{\sigma }{3}}$	${\left({\displaystyle \frac{\partial \sigma }{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial \sigma }{\partial \mathbb{T}}}\right)}_n\mathbb{T}={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +2p{\mathcal{F}}_{{\mathcal{T}}^2}}}(\Gamma -3H)$
dust era	$p=0$	${\left({\displaystyle \frac{\partial \sigma }{\partial n}}\right)}_{\mathbb{T}}\dot{n}+{\left({\displaystyle \frac{\partial \sigma }{\partial \mathbb{T}}}\right)}_n\mathbb{T}={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +\sigma {\mathcal{F}}_{{\mathcal{T}}^2}}}(\Gamma -3H)$

In addition, again using Remark 1 and (45), we find the following:

Theorem 10.

If a ${\Xi }^F$ admits the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfies the $EoS$, then the evolution of the universe given the particle production rate is expressed as follows:

${\Xi }^{\mathit{F}}$	Equation of State $\left(\mathit{E}\mathit{o}\mathit{S}\right)$	Particle Production Rate Γ
dark matter era	$p=-\sigma$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ](8\pi +2\sigma {\mathcal{F}}_{{\mathcal{T}}^2})}}$
Stiff fluid era	$p=\sigma$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ]8\pi +2\sigma {\mathcal{F}}_{{\mathcal{T}}^2})}}$
Radiation era	$p={\displaystyle \frac{\sigma }{3}}$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ](8\pi +2p{\mathcal{F}}_{{\mathcal{T}}^2})}}$
dust era	$p=0$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ](8\pi +\sigma {\mathcal{F}}_{{\mathcal{T}}^2})}}$

Particularly, we consider that $\mathcal{F}\left(\mathcal{T}\right)=0$ or ${\mathcal{F}}_{{\mathcal{T}}^2}=0$, and then $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}$; that is, $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity reduces to the Einstein field equations $\left(EFE\right)$ of orthodox gravity or $GR.$ Now, we write the following:

Corollary 6.

If a perfect fluid spacetime obeying the $EFE$ admits the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfies the $EoS$, then the evolution of the universe given the particle production rate is expressed as follows

${\Xi }^{\mathit{F}}$	Equation of State $\left(\mathit{E}\mathit{o}\mathit{S}\right)$	Particle Production Rate Γ
dark matter era	$p=-\sigma$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ]8\pi }}$
Stiff fluid era	$p=\sigma$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ]8\pi }}$
Radiation era	$p={\displaystyle \frac{\sigma }{3}}$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ]8\pi }}$
dust era	$p=0$	$\Gamma =3H+{\displaystyle \frac{\dot{\rho }}{[\frac{\mathcal{R}}{2}+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega ]8\pi }}$

Remark 5.

In comparison with orthodox gravity, modified gravity theories provide methods for more effective gravitational particle generation because of oscillating scale factor $a\left(t\right)$ or gravitational backgrounds, which may have observable effects. These high-frequency oscillations in the corresponding gravitational fields are not naturally produced by standard $GR$ unless they are induced by particular matter content or early universe physics.

We have demonstrated that strong oscillations of curvature scalar $\mathcal{R}$ are induced in contracting astrophysical systems with increasing energy density. These oscillations begin as harmonic and progress to very anharmonic ones with a high frequency and a huge amplitude, which may be far more than the curvature value in normal $GR$. Such oscillations result in efficient particle production in a wide energy range, from a hundred $MeV$ up to the scalaron mass, m, which could be as large as 1010 $GeV$ (and maybe even larger). Such high frequency oscillations could be a source of ultra-high-energy cosmic rays (for more numerical results see [50]). The efficiency of particle production strongly depends upon the explicit form of the functions $\mathcal{F}\left(\mathcal{R}\right)$ and $\mathcal{F}(\mathcal{R},\mathcal{T}).$

Since enthalpy is a type of geometry transformation. Therefore, from (32), we have

$$h={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +{\mathcal{F}}_{{\mathcal{T}}^2}(\sigma -p)}}.$$

(47)

Consequently, we can state the following:

Theorem 11.

Let a ${\Xi }^F$ admit the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfy the $EoS$ (32), then the enthalpy in the ${\Xi }^F$ is given by (47).

Theorem 12.

Let a ${\Xi }^F$ admit the stable $CRS$$(g,\zeta ,\mu )$ with a $CVF$ ζ and satisfy the $EoS$ (32), then the evolution of the universe provides the enthalpy in the ${\Xi }^F$ as follows:

${\Xi }^{\mathit{F}}$	Equation of State $\left(\mathit{E}\mathit{o}\mathit{S}\right)$	Enthalpy h
dark matter era	$p=-\sigma$	$h={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +2\sigma {\mathcal{F}}_{{\mathcal{T}}^2}}}$
Stiff fluid era	$p=\sigma$	$h={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8}}$
Radiation era	$p={\displaystyle \frac{\sigma }{3}}$	$h={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +2p{\mathcal{F}}_{{\mathcal{T}}^2}}}$
dust era	$p=0$	$h={\displaystyle \frac{\frac{\mathcal{R}}{2}+\frac{1}{2}\lambda {\mathcal{T}}^2+\Lambda +\left(\frac{P}{2}+\frac{1}{4}\right)+\Omega }{8\pi +\sigma {\mathcal{F}}_{{\mathcal{T}}^2}}}$

7. Conclusions

A modified gravity theory with an Einstein–Hilbert Lagrangian density function that incorporates a squared energy–momentum tensor term and the Ricci scalar is examined in this study. In addition to introducing updated gravity theories, this paper develops important quantities and equations, such as the scalar curvature and Ricci tensor.

It is important to note that $\mathcal{F}\left(\mathcal{R}\right)$ and $\mathcal{F}(\mathcal{R},\mathcal{T})$ are two of several models of modified gravity theory. Additionally, the research presented in this work for $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity, is a generalization of the findings for the $\mathcal{F}\left(\mathcal{R}\right)$ and $\mathcal{F}(\mathcal{R},\mathcal{T})$ theories. It is also crucial to remember that our results restore the $\mathcal{F}\left(\mathcal{R}\right)$ gravity values when the matter term is turned off.

The function $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ has no universal form; instead, many forms are proposed by researchers depending on different cosmological circumstances. These include quadratic terms like $\mathcal{F}(\mathcal{R},\mathcal{T})=\mathcal{R}+\lambda {\mathcal{T}}^2$, additive functions like $\mathcal{F}(\mathcal{R},\mathcal{T})={\mathcal{F}}_1\left(\mathcal{R}\right)+{\mathcal{F}}_2\left(\mathcal{T}\right)$, and specific powers of ${\mathcal{T}}^2$. To match observational data, some studies even investigate more intricate combinations, such as logarithmic and power-law dependencies on $\mathcal{T}$ or

$$\mathcal{F}(\mathcal{R},\mathcal{T})=\mathcal{R}+{\lambda }_1{\mathcal{R}}^2+{\lambda }_2\mathcal{T}.$$

The present study holds for the specific case of $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ and not for all possible forms mentioned above. Particularly, in subclasses of high-energy structures, $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$-gravity provides a modified framework for understanding gravity and may be able to resolve singularities such as the early universe’s initial singularity. It also affects cosmological evolution, including the potential for a cosmic bounce and neutron stars and other compact objects.

The subcase of modified gravity that is the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$-gravity model filled with perfect fluid matter, which admits conformal Ricci solitons with a concircular vector field, is also addressed. The nature of conformal Ricci solitons on $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity occupied by a perfect fluid is investigated. The steady conformal Ricci soliton is used to derive the equation of state for the perfect fluid attached with the subcase of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model, providing information on pressure and density during the phantom barrier era.

We gain the strong energy, null energy, total convergence restrictions and black holes with a trapped surface, and singularity situations for a perfect fluid coupled with the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity utilizing a conformal Ricci soliton with a time-like concircular vector field. We derive a classification of the evolution of the universe for the particle production rate, enthalpy, and the first law of thermodynamics in the subcase of $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity in terms of conformal Ricci solitons with a time-like concircular vector field.

Moreover, the specific case of $EMSG$ introduces a non-minimal interaction between matter and geometry; the energy–momentum tensor is not conserved. Irreversible processes of matter formation could be described by a thermodynamic interpretation of this non-conservation.