Modified $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-Gravity Coupled with Perfect Fluid Admitting Hyperbolic Ricci Soliton Type Symmetry

Abstract

In the present research note, we discuss the energy–momentum squared gravity model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ coupled with perfect fluid. We obtain the equation of state for the perfect fluid in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model. Furthermore, we deal with the energy–momentum squared gravity model $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ coupled with perfect fluid, which admits the hyperbolic Ricci solitons with a conformal vector field. We provide a clue in this series to determine the density and pressure in the radiation and phantom barrier periods, respectively. Also, we investigate the rate of change in hyperbolic Ricci solitons within the same vector field. In addition, we determine the different energy conditions, black holes and singularity conditions for perfect fluid attached to $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity in terms of hyperbolic Ricci solitons. Lastly, we deduce the Schrödinger equation for the potential ${\mathcal{U}}_n$ with hyperbolic Ricci solitons in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid and a phantom barrier.

1. Introduction

When specific conditions are met, nonlinear systems supporting solitons can regularly produce self-similarity and fractals on increasingly smaller scales. These forms of fractals are seen in the majority of soliton-supporting systems in nature. Fractals are real physical phenomena rather than just a mathematical illustration. It appears that all systems in nature that contain solitons share this universal capacity of soliton-supporting systems to form fractals, regardless of the type of waves (density, electromagnetic, etc.), the medium in which they travel, or the degree of nonlinearity. Additionally, the distribution of matter in the universe is considered a fractal medium model.

Einstein’s version of the gravitational field equations provides the standard way to analyze known cosmic dynamics [1,2]. The best approximation to the observable data is provided by Einstein’s field equation.

The dynamical evolution of the Universe is not well described by Standard $GR$. Numerous modified gravity models have been presented in this context, all attempting to extend and generalize $GR$ [3,4,5,6]. Traditionally, extrapolating the geometric component of the Einstein–Hilbert action was the first step in surpassing the conventional gravitational models. $\mathcal{F}\left(\mathcal{R}\right)$-gravity is among the first models of this kind [7].

The two main issues are that (i) $GR$ is a classical theory, meaning that, unlike the other fundamental forces of nature, it does not exist in a quantum form; and (ii) the universe is expanding more quickly than predicted by numerous observational tests [8]. Since $GR$ is unable to explain these issues, different hypotheses are developed and looked at.

Due to the above situations, a number of mathematicians and physicists developed more sophisticated theories, including the $\mathcal{F}\left(\mathcal{R}\right)$-gravity theory [7], the $\mathcal{F}\left(\mathcal{G}\right)$-gravity [9], and the $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity theory [10], among others. These theories resulted from the Einstein–Hilbert action. Extended theories deviate from the standard gravity theory proposed by Einstein and could potentially provide a reliable approximation to quantum gravity [11].

By using the Einstein–Hilbert Lagrangian density, $GR$ can be extended to the $\mathcal{F}\left(\mathcal{R}\right)$ gravity and become a function $\mathcal{F}\left(\mathcal{R}\right)$, wherein $\mathcal{R}$ represents the Ricci scalar. The enormous neutron stars in the $\mathcal{F}\left(\mathcal{R}\right)$-gravity are resolved by higher-order curvature; see, for instance, Refs. [3,4,5]. However, there are issues over the applicability of the $\mathcal{F}\left(\mathcal{R}\right)$ gravity due to its limitations regarding equilibrium in relation to the solar system and its inability to back the inclusion of a variety of astronomical models, including steady star structure (see [12,13] for more information).

A more comprehensive gravity model was introduced by Harko et al. [14], who named it $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity theory. They did this by assuming that the Lagrangian is an arbitrary function of $\mathcal{T}$ and $\mathcal{R}$. In this case, $\mathcal{T}$ represents the energy–momentum tensor trace ($EMT$). The late-time swift growth of the cosmos was described with effectiveness using this concept.

A covariant modification of $GR$ that permits the possibility of a term proportional to ${\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$ in the action functional was initially presented by Katirci and Kavuk [15] in 2014.

Additional research on this theory was performed in [16,17], wherein particular models of this gravity theory were examined. The energy–momentum squared gravity ($EMSG$) model’s potential for an early bounce was examined by Roshan et al. [16], using the particular function provided by

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

where $\lambda$ is a constant.

In [16], the late-time accelerated Universe has been studied within the context of $EMSG$ theories, taking into account the scenario of a pressureless fluid. The gravitational Lagrangian in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$-gravity theory is dependent upon the Ricci scalar $\mathcal{R}$ and the contraction of the $EMT$ with itself, i.e., ${\mathcal{T}}^2={\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$ [17,18]. There are various ways to introduce the expression $\mathcal{F}\left({\mathcal{T}}_{ab}{\mathcal{T}}^{ab}\right)$, which results in various versions of the theory. For instance, $EMSG$ is the version 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. This concept has drawn a lot of interest and has been investigated in several settings. The study of cosmic acceleration using $EMSG$ is presented in [19]. There has been discussion of cosmological models in energy–momentum squared gravity, such as bulk viscous fluid cosmology, loop quantum gravity, k-essence, and brane-world cosmologies [17]. Research has been conducted on an expansion of the conventional $\Lambda$CDM model in energy momentum log gravity [20]. Limitations on the squared gravity of energy and momentum from neutron stars and their consequences for cosmology have been examined [18]. An analysis has been performed on compact stars with spherical symmetry [21]. Using a dynamic system analysis, the structure of spacetime and its physical consequences have been examined for different versions of gravity functions $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ [22].

A time-constrained four-dimensional Lorentzian-connected manifold $(M^4,g)$ with a Lorentzian metric g is represented both in terms of cosmology and the spacetime of the $GR$. For $GR$ analysis, connected Lorentzian manifolds are thus turning out to be the most useful framework [23,24].

Perfect fluid spacetime $\left(PFS\right)$ is the name given to quasi-Einstein Lorentzian manifolds if the Ricci tensor has the desired shape [24]

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

(1)

where the one-form $\eta$ is metrically equal to a unit time-like vector field, and there are scalars $A_1$ and $A_2$.

It is possible to recast the $GR$ formally using the momentum energy tensor [25]. Next, this momentum tensor is expressed as density, anisotropic pressure, isotropic pressure, and energy flow, when a competent timelike vector field is present [23,24].

In particular, spacetime geometry is related to physical matter symmetry in the $GR$. More precisely, the metric of symmetry usually simplifies the categories of solutions to field equations. The soliton, linked to spacetime’s geometric movement, is a prominent example of symmetry.

After Dai et al. [26] introduced the notion of hyperbolic geometric flow in 2010, Faraji et al. [27,28] introduced the theories of hyperbolic Ricci solitons and gradient hyperbolic Ricci solitons. Recently, Blaga and Ozgur have explored the concept of hyperbolic Ricci solitons in several ways (for further details, see [29,30,31]).

2. Hyperbolic Ricci Solitons

In 1988, Hamilton [32] first presented the ideas behind Ricci flow. In addition, geometric flow theory, and especially the Ricci flow, has attracted the attention of numerous mathematicians throughout the past 20 years.

The Ricci flow [32] is formed by the family of metrics $g\left(t\right)$ on a Riemannian manifold M, if

$${\displaystyle \frac{\partial }{\partial t}}{g}_{ij}=-2\mathcal{R}ic{g}_{ij},g\left(0\right)=g_0.$$

(2)

In 2010, Kong and Liu, however, investigated the hyperbolic Ricci flow [26].

A system of second-order nonlinear evolution of partial differential equations makes up this flow. Hyperbolic Ricci flow describes the wave properties of metrics and manifold curvatures. In addition, waves can be formed by gravity. Spacetime rippling caused by gravity waves propagate throughout the universe. Basically, a gravitational wave is an oscillation of spacetime curvature moving away from the Earth.

The ensuing evolution equation characterizes the hyperbolic Ricci flow [26], which is hence motivated by the Ricci flow.

$${\displaystyle \frac{{\partial }^2}{\partial t^2}}{g}_{ij}=-2\mathcal{R}ic{g}_{ij},$$

(3)

$${\displaystyle \frac{\partial }{\partial t}}{g}_{ij}=h_{ij},g\left(0\right)=g_0,$$

where $h_{ij}$ is a symmetric two-tensor field. It follows that a self-similar solution of hyperbolic Ricci flow, called a hyperbolic Ricci soliton $\left(HRS\right)$, is explained as in [27].

Definition 1 

([27]). A Riemannian manifold $(M^n,g)$ is called a hyperbolic Ricci soliton if there exists a vector field ζ on M and real scalars μ and λ such that

$$\mathcal{R}ic+{\displaystyle \frac{1}{2}}{L}_{\zeta }{L}_{\zeta }g+\theta L_{\zeta }g=\mu g.$$

(4)

In (4), θ shows the types of solitons of the underlying type, respectively. Moreover, μ represents the rate of change in the solutions. Depending on the constant μ, the $HRS$’s rate of change can be either shrinking, expanding, or approximately stable, regardless of whether $\mu <0$, $\mu >0$, or $\mu =0$.

An $HRS$$(g,\theta ,\zeta ,\mu )$ is known as a gradient hyperbolic Ricci soliton $\left(GHRS\right)$ [27] if there exists a potential function f such that $\zeta =\nabla f$. This makes it possible to rewrite (4) as

$$L_{\nabla f}\left(Hessf\right)+2\theta Hessf+\mathcal{R}ic=\mu g,$$

(5)

where the Lie derivative of the Ricci tensor and a real number are denoted by ${\mathfrak{L}}_{\mathcal{V}}$, $\mathcal{R}ic$, and $\theta$, respectively. A hyperbolic Ricci soliton ($RS$) is said to be expanding, stable, or declining, with reference to (4), depending on whether $\theta >0$, $\theta =0$, or $\theta <0$, respectively.

$PFS$ was examined by researchers in [33] using the Ricci soliton in the $GR$ model. In [34], Venkatesha and Aruna also discussed Ricci solitons on $PFS$ in $GR$ with a torse-forming vector field. Siddiqi et al. (cf. [35,36,37]) analyzed $PFS$ using solitons in $GR$ with a variety of techniques.

Recently, in 2022, Siddiqi et al., [38,39] explored the characteristics of spacetime with the $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity model filled with perfect fluid matter admitting the Ricci soliton, gradient Ricci soliton, Yamabe soliton, and gradient Yamabe soliton. Since the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model is an extension of $\mathcal{F}\left(\mathcal{R}\right)$-gravity and $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity model, this generalized version of modified gravity leads to a different version of the theory. Thus, inspired by other studies, in this paper, we study a $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model with perfect fluid matter admitting a hyperbolic Ricci soliton with a time-like conformal-type vector field. The study developed in this paper is a generalization of the results obtained for $\mathcal{F}\left(\mathcal{R}\right)$ and $\mathcal{F}(\mathcal{R},\mathcal{T})$ theories [38,39].

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

The $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model with perfect matter fluid is discussed in this section with ${\mathcal{T}}^2={\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$.

Since this model relies on the perfect fluid matter field’s physical properties, we can derive a variety of hypothetical models for various values of $\mathcal{R}$ and $\mathcal{T}$ [14]. As an example, we are using the model that follows.

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

(6)

where the arbitrary functions of $\mathcal{R}$ and $\mathcal{T}$, and a scalar $\lambda$, respectively, are denoted by $\mathcal{F}\left(\mathcal{R}\right)$ and $\mathcal{F}\left(\mathcal{T}\right)$. The modified Einstein–Hilbert action term is taken to be

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

(7)

where the Lagrangian is denoted by ${\mathbf{L}}_m$, $\Lambda$ is the cosmological constant. The matter’s energy tensor is provided by

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

(8)

Suppose that ${\mathbf{L}}_m$ is independent of its derivatives and solely dependent on $-g_{ab}$.

Now, the variation in ${\Pi }_E$ is

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

(9)

For the sake of clarity, these have been defined $\mathcal{F}=\mathcal{F}(\mathcal{R},{\mathcal{T}}_{ab}{\mathcal{T}}^{ab})$, ${\mathcal{F}}_{\mathcal{R}}={\displaystyle \frac{\partial \mathcal{F}}{\partial \mathcal{R}}}$ and ${\mathcal{F}}_{{\mathcal{T}}^2}={\displaystyle \frac{\partial \mathcal{F}}{\partial {\mathcal{T}}^2}}$. It is known that the Ricci scalar variation yields the outcome

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

(10)

Adopting the variation in ${\mathcal{T}}^2$ with respect to the metric, we obtain

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

(11)

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

(12)

Then, (8) is used to express the alteration in the energy–momentum tensor given in (12) as

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

(13)

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

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

(14)

Now, in light of (9), the field equations of the $EMSG$$\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ model are given by

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

(15)

The matter content needs to be taken into account in addition to the geometric components. In this case, the energy–momentum tensor of a $PFS$ is defined as

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

(16)

where $\sigma$ is the energy density, p is the pressure and $\eta$ is a unit time-like vector field. Taking the Lagrangian ${\mathbf{L}}_m=-p$, ${\eta }^a{\eta }_b=1$, which describes a perfect fluid, the $EMT$ defined in (16) can be written as

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

(17)

The field Equation (15) appears to decrease to a constant value, though, as the Ricci scalar is a constant and the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model admits a stable static universe. Therefore, the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ models presented here can provide a setting in which a static universe is connected to an asymptotic emergent universe scenario.

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

(18)

In order to develop such an analysis for the $EMSG$$\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ model.

Let us compose the field Equation (15) examining the perfect fluid’s energy–momentum tensor provided in (16).

For ease of understanding, $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)=\mathcal{R}+\lambda {\mathcal{T}}^2$ is chosen, where $\lambda$ is an integer. Assuming it has a perfect fluid form, the Ricci tensor is

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

(19)

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

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

(20)

where

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

(21)

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

(22)

During the entire study, we assume that $\alpha$ and $\beta$ are not both zero. The formulation of the Ricci tensor was found using a similar method in [39]; however, for consistency, we also present the proof here. As such, we acquire

Theorem 1. 

The $PFS$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity has the following Ricci tensor

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

(23)

Corollary 1. 

The scalar curvature in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity coupled with $PFS$ is given by

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

(24)

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

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

(25)

Referring to Equations (2) and (20), we gain the next result.

Theorem 2. 

A spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ model coupled with perfect fluid is a $PFS$.

4. Hyperbolic Ricci Soliton on $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$-Gravity Model Coupled with Perfect Fluid

In this section, we examine the hyperbolic Ricci soliton $\left(HRS\right)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with $PFS$ whose conformal vector field $\zeta$ is the timelike velocity vector field.

According to Kuhnel and Rademacher [40], a conformal vector field $\mathcal{V}$$\left(CVF\right)$ on Lorentzian spacetime $(M^4,g)$ is defined by

$$L_{\mathcal{V}}g=2\Omega g$$

(26)

wherein $\Omega$ is a smooth function on $M^4$. $\mathcal{V}$ is killing if $\Omega =0$ (KVF), and homothetic when $\Omega$ is constant. Next, adopting $\mathcal{V}=\zeta$, Equation (4) becomes

$${\displaystyle \frac{1}{2}}{L}_{\zeta }\left(L_{\zeta }g\right)(a,b)+\theta \left(L_{\zeta }g\right)(a,b)+\mathcal{R}ic(a,b)=\mu g(a,b).$$

(27)

In light of (26), we gain

$$\mathcal{R}ic(a,b)=(\mu -2{\Omega }^2-2\theta \Omega )g(a,b).$$

(28)

Inserting (25) in the above equation, we obtain

$$(\alpha +\theta +\Omega )g(a,b)+\beta \eta \left(a\right)\eta \left(b\right)=0.$$

(29)

Now plugging $a=b=\zeta$ in (29), we find

$$\theta ={\displaystyle \frac{\mu }{2\Omega }}-\left\{{\displaystyle \frac{(\alpha +\beta )}{2\Omega }}+\Omega \right\}.$$

(30)

Now, employing Equations (43) and (22), we derive,

$$p+\sigma ={\displaystyle \frac{[\mu +8\pi \sigma +\lambda {\mathcal{T}}^2+\Lambda ]-[4\pi (3p-\sigma )+2{\Omega }^2+2\theta \Omega ]}{(\sigma -p)[{\mathcal{F}}_{{\mathcal{T}}^2}-\frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}]}}.$$

(31)

Remark 1. 

Referring to [41], $p=-\sigma +\mathcal{F}\left(r\right)$ is the equation of state for dark energy, with t being the cosmic time and $\mathcal{F}\left(r\right)$ being a function of the scale factor $"r\left(t\right)"$. Moreover, Cosmic time t is the standard time coordinate (time-like velocity vector field) for specifying the field equations of $GR.$ In order to ensure that the universe has the same density everywhere at all times, a homogenous, expanding universe may develop such a time coordinate t.

The author of [41] also demonstrated that the case $\omega =-1$ yields $\omega ={\displaystyle \frac{p}{\sigma }}=-1$ and produces a phantom barrier, whereas $\omega >-1$ and $\omega <-1$ reflect a transition from non-phantom to phantom. In addition, the limit $-1$ would correspond to the Λ or dark-energy era with a negative pressure that causes the expansion to an accelerating universe.

Therefore, in view of (31) and Remark 1, we can articulate the following results:

Theorem 3. 

If the matter of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with $PFS$ admits a HRS $(g,\theta ,\mu )$ with CVF ζ, then $\mathcal{E}o\mathcal{S}$ is given by (31).

In the case of the phantom barrier, $\sigma =-p={\displaystyle \frac{\mu +\lambda {\mathcal{T}}^2+\Lambda -2\Omega (\Omega +\theta )}{24\pi }}$. As a result, we may infer

Corollary 2. 

If the source of matter in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with $PFS$ admits an HRS $(g,\theta ,\mu )$, with CVF ζ as a phantom barrier type, then the pressure and energy density are measured as

$$\sigma =-p={\displaystyle \frac{\mu +\lambda {\mathcal{T}}^2+\Lambda -2\Omega (\Omega +\theta )}{24\pi }}.$$

(32)

As such, in light of (30), we gain the subsequent outcome of HRS with CVF in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with $PFS$.

Theorem 4. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity coupled, with $PFS$ admitting an $HRS$$(g,\theta ,\mu )$ with a $CVF$ζ, then $HRS$ is growing, stable or decreasing, referred to as

1. 

${\displaystyle \frac{1}{2\Omega }}[\mu +({\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda )]>\left\{{\displaystyle \frac{({\sigma }^2-p^2)}{2\Omega }}[{\mathcal{F}}_{{\mathcal{T}}^2}-{\displaystyle \frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}}]+{\displaystyle \frac{2\pi }{\Omega }}(3p+\sigma )+\Omega \right\}$,

2. 

${\displaystyle \frac{1}{2\Omega }}[\mu +({\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda )]=\left\{{\displaystyle \frac{({\sigma }^2-p^2)}{2\Omega }}[{\mathcal{F}}_{{\mathcal{T}}^2}-{\displaystyle \frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}}]+{\displaystyle \frac{2\pi }{\Omega }}(3p+\sigma )+\Omega \right\}$, and

3. 

${\displaystyle \frac{1}{2\Omega }}[\mu +({\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda )]<\left\{{\displaystyle \frac{({\sigma }^2-p^2)}{2\Omega }}[{\mathcal{F}}_{{\mathcal{T}}^2}-{\displaystyle \frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}}]+{\displaystyle \frac{2\pi }{\Omega }}(3p+\sigma )+\Omega \right\}$, respectively, provided $\Omega \ne 0.$

The following results are now implied by Equation (32) and Corollary 2.

Corollary 3. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity coupled with $PFS$ represents a dark matter era and admits an HRS $(g,\theta ,\mu )$ with CVF ζ, then $HRS$ is growing, stable or decreasing, referred to as

1. 

${\displaystyle \frac{1}{2\Omega }}[\mu +\lambda {\mathcal{T}}^2+\Lambda )]>\{{\displaystyle \frac{1}{\Omega }}(1+2p\pi )$,

3. 

${\displaystyle \frac{1}{2\Omega }}[\mu +\lambda {\mathcal{T}}^2+\Lambda )]=\{{\displaystyle \frac{1}{\Omega }}(1+2p\pi )$, and

3. 

${\displaystyle \frac{1}{2\Omega }}[\mu +\lambda {\mathcal{T}}^2+\Lambda )]<\{{\displaystyle \frac{1}{\Omega }}(1+2p\pi )$, respectively, provided $\Omega \ne 0.$

Moreover, from (30) and (32), we obtain the rate of change in HRS in terms of the following relations:

$$\mu =2\theta \Omega +\left\{(\alpha +\beta )+2{\Omega }^2\right\}.$$

(33)

$$\mu =24p\pi +2\Omega (\Omega +\theta )-(\lambda {\mathcal{T}}^2+\Lambda ).$$

(34)

Consequently, we can state the following theorems:

Theorem 5. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity coupled with $PFS$ admits an $HRS$$(g,\theta ,\mu )$ with a $CVF$ζ, then the rate of change in $HRS$ is growing.

Theorem 6. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity coupled with $PFS$ represents a dark matter era and admits an HRS $(g,\theta ,\mu )$ with CVF ζ, then the rate of change in $HRS$ is growing, stable or decreasing, referred to as

1. 

$2\Omega (\Omega +\theta )+24p\pi >(\Lambda +\lambda {\mathcal{T}}^2)$,

2. 

$2\Omega (\Omega +\theta )+24p\pi =(\Lambda +\lambda {\mathcal{T}}^2)$, and

3. 

$2\Omega (\Omega +\theta )+24p\pi <(\Lambda +\lambda {\mathcal{T}}^2)$, respectively.

5. Energy Conditions in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$-Gravity Model Coupled with Perfect Fluid Admits Hyperbolic Ricci Soliton

In this section, we assume that the Ricci tensor $\mathcal{R}ic$ in the spacetime satisfies the requirement (see [42]).

$$\mathcal{R}ic(\zeta ,\zeta )>0,$$

(35)

where the time-like vector field $\zeta \in \chi \left(M^4\right)$. Consequently, the time-like convergence condition (TCC) is defined as Equation (35).

From (28) and (23), it gives

$$\mathcal{R}ic(\zeta ,\zeta )=\mu -2\Omega (\Omega +\theta ).$$

The spacetime admits an $HRS$$(d,\zeta ,\theta ,\mu )$ with a $CVF$$\zeta$ obey the $TCC$, if $\mathcal{R}ic(\zeta ,\zeta )>0$. Therefore, from above equation, we obtain

$$\mu >2\Omega (\Omega +\theta ).$$

(36)

$${\displaystyle \frac{1}{2\Omega }}[\mu +({\displaystyle \frac{1}{2}}\lambda {\mathcal{T}}^2+\Lambda )]>\left\{{\displaystyle \frac{({\sigma }^2-p^2)}{2\Omega }}[{\mathcal{F}}_{{\mathcal{T}}^2}-{\displaystyle \frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}}]+{\displaystyle \frac{2\pi }{\Omega }}(3p+\sigma )+\Omega \right\}$$

(37)

The spacetime is compliant with the cosmological strong energy condition $\left(SEC\right)$ [43]. In light of the above-provided data and from (36), we can state that

Theorem 7. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with $PFS$ admits an $HRS$$(g,\zeta ,\theta ,\mu )$ with a $CVF$ζ and satisfies $TCC$, then $HRS$ is growing.

Remark 2. 

In 1973, Hawking and Ellis [44] showed that

• $TCC\Rightarrow SEC$, $TCC\Rightarrow NCC$ (null convergence condition),
• $SEC\Rightarrow NEC$ (null energy condition).
• Therefore, $TCC\Rightarrow NCC$ as well.

Combining Remark 1 with Theorem 7 yields the following results:

Theorem 8. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid admits a growing $HRS$$(g,\zeta ,\theta ,\mu )$ with a $CVF$ζ, if (36) holds, then the $PFS$$(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity satisfies $SEC$.

Corollary 4. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid admits a growing $HRS$ with a conformal vector field ζ, if (36) holds, then the $PFS$$(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity satisfies $NCC$.

Corollary 5. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid admits an $HRS$ with a $CVF$ζ and satisfies $SEC$, then the Ricci tensor $\mathcal{R}ic$ in growing $RS$ is of the second Segre type [44].

6. Application of Singularity Theorem in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$-Gravity Model Coupled with Perfect Fluid Admits a Hyperbolic Ricci Soliton

Remark 3. 

According to Penrose’s singularity theorem, Vilenkin and Wall ([45]) demonstrated that the spacetime M obeys the $NCC$. Then, M contains some black holes as well as a trapped surface outside the black holes.

From Theorem 7, Remark 1, Remark 2 and Corollary 4, follows.

Theorem 9. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid admits a growing $HRS$$(g,\zeta ,\theta ,\mu )$ with a $CVF$ζ and if the spacetime $(M^4,g)$ holds $NCC$, then the perfect fluid spacetime $(M^4,g)$ contains some black holes with a trapped surface, which is outside the black holes in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity.

Corollary 6. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity coupled with perfect fluid admits a growing $HRS$$(g,\zeta ,\theta ,\mu )$ with a $KVF$ζ and if the spacetime $(M^4,g)$ holds $NCC$, then the perfect fluid spacetime $(M^4,g)$ contains some black holes with a trapped surface, which is outside the black holes in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity.

7. Schrödinger Equation for $\mathcal{F}(\mathcal{R},{\mathcal{T}}^{\mathbf{2}})$-Gravity Model Coupled with Perfect Fluid

In [46], Yurov and Astashenok proved a Schrödinger equation with the exact solutions ${\Psi }_n$ in cosmology for Einstein’s gravity such that

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}={\mathcal{U}}_n,$$

(38)

where the ”potential” is ${\mathcal{U}}_n=n^2\sigma -{\displaystyle \frac{3n}{2}}(\sigma +p),$ and n is the principal quantum number. For $n=1,2,3,\cdots$, we may write the energy of each quantum state. Thus, we have

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=n^2\sigma -{\displaystyle \frac{3n}{2}}(\sigma +p).$$

(39)

Now, in light of (31) and (39), the Schrödinger equation for the potential of the ${\mathcal{U}}_n$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid that admits $HRS$ with a $CVF$$\zeta$ is

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=n^2\sigma -{\displaystyle \frac{3n}{2}}{\displaystyle \frac{[\mu +8\pi \sigma +\lambda {\mathcal{T}}^2+\Lambda ]-[4\pi (3p-\sigma )+2{\Omega }^2+2\theta \Omega ]}{(\sigma -p)[{\mathcal{F}}_{{\mathcal{T}}^2}-\frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}]}},$$

(40)

where the potential is ${\mathcal{U}}_n=n^2\sigma -{\displaystyle \frac{3n}{2}}{\displaystyle \frac{[\mu +8\pi \sigma +\lambda {\mathcal{T}}^2+\Lambda ]-[4\pi (3p-\sigma )+2{\Omega }^2+2\theta \Omega ]}{(\sigma -p)[{\mathcal{F}}_{{\mathcal{T}}^2}-\frac{{\mathcal{F}}^{\prime }\left({\mathcal{T}}^2\right)}{2}]}}.$

Therefore, we entail the following

Theorem 10. 

If a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid admits $HRS$ with a $CVF$ζ, then the Schrödinger equation for the potential ${\mathcal{U}}_n$ in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity is given by Equation (41).

Corollary 7. 

If the source of matter is of phantom barrier type in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled with perfect fluid and admits a HRS $(g,\theta ,\mu )$ with CVF ζ, then the Schrödinger equation for the potential ${\mathcal{U}}_n$ is

$${\displaystyle \frac{{\ddot{\Psi }}_n}{{\Psi }_n}}=n^2\sigma -{\displaystyle \frac{3n}{2}}\left[{\displaystyle \frac{\mu +\lambda {\mathcal{T}}^2+\Lambda -2\Omega (\Omega +\theta )}{24\pi }}\right],$$

(41)

where the potential is ${\mathcal{U}}_n=n^2\sigma -{\displaystyle \frac{3n}{2}}\left[{\displaystyle \frac{\mu +\lambda {\mathcal{T}}^2+\Lambda -2\Omega (\Omega +\theta )}{24\pi }}\right].$

Remark 4. 

The rapid acceleration of the universe may be explained by modified gravity theories, which avoid using the enigmatic idea of dark energy. The recent cosmological dynamics of the Universe may be explained by matter–geometry coupling, and matter itself may have a more fundamental role in the description of the gravitational processes that typically assumed quantum cosmology of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ gravity theory, in which the effective Lagrangian of the gravitational field is given by an arbitrary function of the Ricci scalar $\mathcal{R}$ and the trace of the matter energy–momentum tensor $\mathcal{T}.$

Additionally, these models, like $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$, demonstrate that matter itself might be more fundamental than previously thought in explaining the gravitational dynamics that are typically assumed [47]. They can also serve as a link between the classical and quantum realms. For instance, the gravitational action’s dependency on the $EMT$$\mathcal{T}$ trace could be caused by imperfect quantum fluids or quantum effects (conformal anomaly) (see [47] for additional details).

The Friedmann–Robertson–Walker form is adopted for the classical background metric in order to construct the quantum cosmology of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ gravity. It is assumed that the matter content of the very early Universe consists of a perfect fluid, described by only two thermodynamic parameters, the thermodynamic pressure and the energy density, respectively.

We begin by obtaining the generic form of the gravitational Hamiltonian, the quantum potential, and the canonical momenta in order to introduce the canonical quantization method for the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ gravity theory.

The Wheeler-de Witt equation [47] of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ modified gravity theory is written down once these quantities are explicitly determined. This equation describes the quantum properties of the very early Universe when quantum effects dominated the dynamic evolution of the system. The quantum cosmological characteristics of a specific model of the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ theory may be presented and thoroughly examined [47].

As an example, we look at the field equations in the Friedmann metric [46]

$${\displaystyle \frac{{\dot{a}}^2}{a^2}}=\sigma -{\displaystyle \frac{k}{a^2}}.$$

(42)

$${\displaystyle \frac{\ddot{a}}{a}}=-{\displaystyle \frac{1}{2}}(\sigma +3p).$$

(43)

Let $a=a\left(t\right)$, $p=p\left(t\right)$, $\sigma =\sigma \left(t\right)$ be a solution of (42) and (43) for $k=0$. Then, the function ${\Psi }_n=a^n$ is a solution of the Schrödinger equation

$${\ddot{\Psi }}_n=U_n\left(t\right){\Psi }_n,$$

(44)

wherein the ”potential” is

$$U_n\left(t\right)=n^2\sigma -{\displaystyle \frac{3n}{2}}(\sigma +p).$$

(45)

8. Conclusions

Modified $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity is a class of theories that generalize $GR$, $\mathcal{F}\left(\mathcal{R}\right)$, and $\mathcal{F}(\mathcal{R},\mathcal{T})$-gravity models by adding higher-order terms of the form ${\mathcal{T}}_{ab}{\mathcal{T}}^{ab}$ to the matter Lagrangian. For a fluid as perfect as the content of matter, it has been shown that a spacetime $(M^4,g)$ in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model is a perfect fluid spacetime. Moreover, field equations aim to reduce to a constant value, yet, since the Ricci scalar is a constant in $\mathcal{F}(\mathcal{R},the{\mathcal{T}}^2)$-gravity model admits a stable static universe. Specifically, we discussed a hyperbolic Ricci soliton on spacetime in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model attached to the perfect fluid, where the time-like velocity vector field is a conformal vector field. We gain an equation of state for the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model attached to the perfect fluid that admits a hyperbolic Ricci soliton with a time-like conformal vector field. In addition, we characterized the nature of the hyperbolic Ricci soliton and the rate of change in the hyperbolic Ricci soliton in the same framework. We have explored that the spacetime in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model attached to the perfect fluid satisfied the strong energy condition and the null convergence condition in terms of a hyperbolic Ricci soliton. Lastly, this study demonstrated the implications of the singularity theorem and the occurrence of a black hole of spacetime, admitting the hyperbolic Ricci soliton in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model coupled to the perfect fluid.

9. Follow up Plans

The spacetime in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model with imperfect fluid matter attached is a possible area of study. Furthermore, Yamabe, conformal Ricci, and Ricci–Yamabe solitons can be applied to spacetime in the $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$-gravity model, which is connected to a perfect fluid with a concircular vector field and a torse-forming vector field. Additionally, a study of the gradient hyperbolic Ricci soliton on spacetime in $\mathcal{F}(\mathcal{R},{\mathcal{T}}^2)$ filled with perfect and imperfect fluid matter might be anticipated.