Compact Stars Admitting Noether Symmetries in Energy-Momentum Squared Gravity

+is paper investigates the geometry of compact stellar objects through the Noether symmetry approach in the energy-momentum squared gravity. +is newly developed theory overcomes the problems of big bang singularity and provides the viable cosmological consequences in the early time universe. Moreover, its implications occur in high curvature regime where the deviations of energy-momentum squared gravity from general relativity is confirmed.We consider theminimal couplingmodel of this modified theory and formulate symmetry generators as well as corresponding conserved quantities. We use conservation relation and apply some suitable initial conditions to evaluate the metric potentials. Finally, we explore some interesting features of the compact objects for appropriate values of the model parameters through numeric analysis. It is found that compact stellar objects in this particular framework depend on the model parameters as well as conserved quantities. We conclude that Noether symmetries generate solutions that are consistent with the astrophysical observational data and hence confirm the viability of this procedure.


Introduction
Noether symmetry approach is recognized as the most efficient method to investigate the analytic solutions that help to find the conserved parameters of the field equations corresponding to symmetry generators. e main motivation comes from various conservation laws (energy, momentum, angular momentum, etc.) which are outcomes of some kind of symmetry being present in a system. e conservation laws are the key factors in the study of various physical processes and familiar Noether theorem, which implies that every differentiable symmetry of the action leads to the law of conservation. is theorem is significant because it provides a correlation between conserved quantities and symmetries of a physical system [1]. A lot of fascinating work has been done in this background [2][3][4][5].
Modified gravitational theories are considered as the most favorable and propitious techniques to uncover the cosmic mysteries. Such theories can be formulated by adding the functions of curvature invariants in the geometric part of the Einstein-Hilbert action. e natural modification is obtained by replacing the Ricci scalar (R) with its generic function in the Einstein-Hilbert action so-called f(R) theory of gravity. ere has been a crucial literature [6][7][8] available to understand the viable characteristics of this gravity. is theory has further been generalized by introducing some couplings between curvature invariant and the matter part. ese couplings describe different cosmic eras and the rotation curves of galaxies. Such interactions also yield nonconserved stress-energy tensor indicating the existence of an additional force. ese coupling models are the key aspects to understand the cosmic accelerated expansion and dark matter/dark energy interactions [9]. e nonminimal coupling between the curvature invariant and matter Lagrangian (L m ) has been established in [10] known as f(R, L m ) gravity. Harko et al. [11] formulated such coupling in f(R) theory referred to as f(R, T) gravity (T represents the trace of stress-energy tensor). A more generic theory in which matter is nonminimally coupled to geometry was proposed in [12], referred to as f(R, T, R μ] T μ] ) theory, where R μ] is the Ricci tensor and T μ] demonstrates the stress-energy tensor. One such modifications gave rise to f(R, T ϕ ) theory, where ϕ defines the scalar field [13]. e presence of singularities can be considered a major problem in general relativity (GR) because of its prediction at high energy level, where GR is no longer valid due to the expected quantum impacts. However, there is no specific formalism for quantum gravity. In this regard, energymomentum squared gravity (EMSG) is considered as the most favorable and prosperous technique which resolves the singularity of the big bang in nonquantum description. is modification of GR is formulated by adding the analytic function T μ] T μ] in the generic action which is also referred as f(R, T 2 ) gravity where T μ] T μ] is denoted by T 2 [14]. It provides the contribution of squared terms (ρ 2 , p 2 , and ρp where ρ and p are the matter variables) in the field equations that are used to explore the various fascinating cosmological consequences. is theory has a regular bounce with minimum scale factor (a min ) and finite maximum energy density (ρ max ) at early times. Consequently, it can solve the singularity of the big bang with a classical prescription. e cosmological constant does not play a crucial role in the background of the standard cosmological model. e repulsive nature of the cosmological constant supports to resolve singularity only after the matter-dominated era in the EMSG. It is worthwhile to mention here that this theory overcomes the spacetime singularity but does not change the cosmological evolution.
Several researchers have carried out further studies on this theory. Board and Barrow [15] investigated the range of exact solutions for isotropic spacetime, presence of singularities, cosmic accelerated expansion, and evolution with a particular model of this theory. Nari and Roshan [16] studied physical viability and stability of compact stars in this framework. Morares and Sahoo [17] studied nonexotic matter wormholes while Akarsu [18] explored possible constraints from neutron stars in the same frame. Bahamonde et al. [19] investigated the minimal and nonminimal coupling models of EMSG and observed that these models describe the current cosmic accelerated expansion. ere has been a recent literature [20][21][22] that indicates various cosmological applications of this modified theory. It is evident from the above set of literature that EMSG requires more focus, and hence motivations to analyze such theory are very high. ere are many open issues that can be studied and this would add as well to improve our current knowledge about different modified theories of gravity.
Noether symmetries have various important applications in modified gravitational theories. Capozziello et al. [23] found static and nonstatic spherical solutions through Noether symmetry technique in f(R) theory. Roshan and Shojai [24] investigated cosmological models of f(R, G) theory (G is the Gauss-Bonnet invariant) by adopting Noether symmetry technique. Hussain et al. [25] studied the Noether gauge symmetry approach in f(R) theory. Kucukakca [26] analyzed this approach in scalar-tensor teleparallel theory to explore some physically viable cosmological models. Bahamonde [27] investigated the wormhole geometry through the Noether symmetry approach in the same gravity. Sharif and Fatima [28] studied Noether symmetries of FRW spacetime for both dust as well as vacuum cases in f(G) theory. ey also formulated exact cosmological models of this gravity and investigated the current cosmic accelerated expansion in terms of scale factor. Shamir and Ahmad [29,30] used this technique to explore different cosmological models with isotropic and anisotropic matter configurations in the background of f(G, T) theory. Sharif and his collaborators [31][32][33][34][35][36][37] analyzed accelerated expansion and evolution of the universe by using this approach.
ere has been a recent literature [38][39][40][41][42] that indicates various cosmological applications of this approach in various modified theories of gravity. e attributes and outcomes of self-gravitating objects have great interest for the researchers because of their fascinating features and relativistic geometries in astrophysics as well as cosmology.
e final outcome of this phenomenon is the gravitational collapse which is responsible for the formation of new celestial objects named as compact stars. Such compact objects are assumed to be very dense due to extensive masses and short radii. ese dense objects may be well described by GR and modified theories [43][44][45]. Abbas [46] examined the equilibrium state of compact objects and also analyzed their physical attributes in modified Gauss-Bonnet gravity. Zubair and Abbas [47] analyzed the geometry of compact stars with anisotropic matter configuration in f(R) theory. Recently, Shamir and Naz [39] studied the geometry of compact objects through the Noether symmetry approach in f(G) gravity. e impact of modified theories is well-known to analyze the geometry of compact stars and matter configuration at large densities [48][49][50][51][52].
Since EMSG is established to overcome the singularities, it is significant to analyze the inner region of massive objects where energy factor is quite strong to see the deviations of EMSG from GR. In this paper, we formulate Noether symmetry generators and corresponding conserved quantities for a minimal coupling model of EMSG, i.e., f(R, T 2 ) � αR n + β(T 2 ) m , n, m ≠ 0, 1, and α, β ≠ 0 [19]. We then discuss some salient features of compact objects such as effective matter variables, energy conditions, compactness parameter, gravitational redshift, stability against equilibrium forces, and sound speed for particular values of the model parameters. e paper is organized as follows: in Section 2, we formulate the field equations of static spherical system in the background of EMSG. Section 3 gives a brief description of Noether symmetry technique. In Section 4, we find the expression of metric potentials by using conserved quantities with suitable initial conditions. Section 5 is devoted to analyze some physical characteristics of compact star to explore the viability of the model through graphs. We summarize and discuss the results in the last section.

Basic Formalism of Energy-Momentum Squared Gravity
In this section, we formulate the field equations for EMSG in the presence of perfect fluid. e action for this theory is determined as follows [14]: where κ 2 and g represent the coupling constant and determinant of the line element, respectively. We consider coupling constant as a unity for the sake of simplicity. e action indicates that this theory has extra degrees of freedom. erefore, due to additional force and matter-dominated era, it is expected that some useful consequences would be obtained to study the current cosmic issues in this gravity. e variation of the action corresponding to the metric tensor leads to the following field equations: where It is noted that for f(R, T 2 ) � f(R), the field equations of this gravity reduce to f(R) theory while GR is recovered We assume matter distribution as a perfect fluid: where ρ m , p m , and U μ demonstrate the energy density, pressure, and four-velocity, respectively. e Lagrangian corresponding to matter distribution (4) is defined as L m � p m , and manipulating equation (3), we obtain Rearranging equation (2), we have where is the additional effects of EMSG named as correction terms, and T eff μ] determines the effective stress-energy tensor expressed as In order to study the characteristics of compact stars, we consider static spherical spacetime as follows [53]: e respective field equations turn out to be ese equations are highly nonlinear as well as complicated due to the presence of multivariate function and its derivatives. We consider the Noether symmetry approach to obtain the analytic solutions of f(R, T 2 ) field equations. e conservation law does not hold in this theory, but we obtain conserved quantities in the background of the Noether symmetry approach. ese are helpful to obtain physically viable solutions and hence analyze the geometry of compact objects.

Pointlike Lagrangian and Noether Symmetry
Noether symmetry provides a fascinating procedure to develop new cosmological models and related structures in modified gravitational theories. Here, we formulate the pointlike Lagrangian for static spherical spacetime in the background of EMSG. We determine the corresponding equations by using Noether symmetry technique.
is method provides a unique nature of the vector field within the tangent space associated with it. Hence, the vector field behaves as a symmetry generator and gives conserved quantities which are then useful to examine exact solutions of the modified field equations. e canonical form of action (1) gives Using Lagrange multiplier approach, we have

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We see that if R − R � 0 and T 2 − T 2 � 0, then the above action reduces to action (1). Substituting the values from equation (12) in (11) and eliminating the boundary terms with the help of integration by parts, we have e Euler-Lagrange equations are given as follows: where q i represents the generalized coordinates of n-dimensional space. By using Lagrangian (13), equation (14) turns out to be e corresponding Hamiltonian, H � q i′ (zL/zq i′ ) − L, turns out to be e generators of Lagrangian (13) are considered as follows: where ϱ ≡ ϱ(λ, ϑ, R, T 2 ) and ζ i ≡ ζ i (λ, ϑ, R, T 2 )(i � 1, 2, 3, 4) are unknown coefficients of the vector field Y. e Lagrangian must fulfill the condition of invariance for the vector field over the tangent space to assure the existence of 4 Advances in Astronomy Noether symmetries. In this regard, Y acts as a symmetry generator that constructs the conserved quantities. e condition of invariance can be expressed as follows: where ψ represents the boundary term, Y [1] is the first order prolongation, and D demonstrates the total rate of change. is can also be expressed as follows: where ζ i′ � Dζ i′ − q i′ Dψ. e first integral of motion corresponds to Noether symmetry generator Y which is determined as follows: is is the most significant part of Noether symmetries which is also known as a conserved quantity. It is interesting to mention here that the first integral plays a remarkable role to obtain physically viable solutions. e first integrals of motion are the main factors to determine the characteristics of massive objects in modified gravitational theories. By considering equation (18) and comparing the coefficients, we have a set of partial differential equations named as determining equations. In this case, we obtain the following system of equations:

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Noether symmetry method minimizes the complexity of the system and helps to determine the exact solutions. However, it is complicated to derive a nontrivial solution without taking any specific EMSG model. e analysis of compact stars through Noether symmetry technique in the curvature-matter coupling model would provide interesting consequences. We investigate the presence of symmetry generators with corresponding conserved quantities and investigate structure of compact objects for f(R, T 2 ) gravity model. e minimal model is defined as follows: is model determines a higher complexity of the phase space characteristics with three main eras of the universe (radiation-dominated, matter-dominated, and de Sitter) and solutions exhibit accelerated expansions. We consider m � 2 � n for the sake of convenience.
It is worthwhile to explore perfect fluid as it explains exact matter of various celestial objects like stars, galaxies, etc. e cosmic matter configuration can also be examined by dust fluid only when negligible amount of radiations is present.
e interaction of radiations with dust particles supports to develop compact objects. In the following equation, we examine features of compact stars and derive exact solutions of the EMSG model for dust matter distribution, i.e., T m μ] � ρ m U μ U ] . e simultaneous solutions of equations (21)-(30) yield where ci represents the arbitrary constants. e generators of the Noether symmetry and corresponding conserved quantities become

Metric Potentials and Boundary Conditions
e conserved quantities obtained through the Noether symmetry approach play a significant role to investigate various physical characteristics of compact objects. is approach has successfully been executed in axial and spherically symmetric spacetimes [54,55]. e solutions at the boundary of compact objects are determined by smooth matching of internal and external geometries. e metric potentials of both interior and exterior spacetimes are joined at the surface boundary through the relationship We use the above mentioned conserved quantities to obtain the metric potentials that are helpful to analyze the realistic features of compact objects. Using equation (35) in (33) and (34), it follows ese equations cannot be solved analytically due to their complicated and highly nonlinear nature. Hence, we adopt a numerical method with suitable initial conditions to examine the viable behavior of the metric elements.
In order to check the existence of singularities, we examine the viable behavior of line elements. For a physically realistic and stable cosmological model, the metric potentials must be nonsingular, positive as well as regular inside the geometry of compact objects. e graphical behavior of the metric elements obtained by first, second, and third conserved quantities (I 1 , I 2 , I 3 ) are presented in Figures 1-3, respectively, which shows that the metric elements fulfill all the required conditions. We would like to mention here that for physically viable compact stars, the following conditions must be satisfied: (i) e effective energy density should be positive inside the stellar object as well as at the surface boundary (ρ eff > 0, 0 ≤ r ≤ R). (ii) e effective pressure should be positive inside the stellar object (p eff > 0) and should be zero (p eff � 0) at the surface boundary (r � R). (iii) e gradient of effective matter variables should be negative for 0 ≤ r ≤ R, i.e., (dρ eff /dr) r�0 � 0, (d 2 ρ eff /dr 2 ) r�0 < 0, (dp eff /dr) r�0 � 0, and (d 2 p eff /dr 2 ) r�0 < 0}. is condition shows that the effective matter variables must be decreasing at the boundary of the surface. (iv) e sound speed must be less than the speed of the light.
ese physical characteristics are important to determine the geometry of compact objects. 6 Advances in Astronomy   Figure 4: Plots of effective matter variables corresponding to r. 8 Advances in Astronomy In the following, we discuss physical attributes of compact objects for the metric potentials obtained only by the first conserved quantity (I 1 ) because the effective matter variables become undefined for the metric potentials obtained by the second conserved quantity (I 2 ) while the third conserved quantity (I 3 ) is quite complicated as we could not find the appropriate value of the metric elements.

Physical Characteristics of Compact Objects
Here, we study physical features of the compact stars through graphical analysis of the effective matter variables, energy bounds, compactness parameters, gravitational redshifts, and stability analysis against equilibrium forces and speed of sound.

Evolution of the Effective Matter Variables.
e effective matter variables inside the compact object should be maximum at the center. For this purpose, we analyze the behavior of compact objects for the range 0 ≤ r ≤ 10. We plot the graphs for small radii to have the smooth nature of compact objects. Figure 4 shows that the behavior of effective matter variables is positive and represents the decreasing nature at the boundary of the stellar structure.
is assures high compactness of the compact star at the center. In fact, we observe that (dρ eff /dr) � 0, (d 2 ρ eff /dr 2 ) < 0, and (dp eff /dr) � 0, (d 2 p eff /dr 2 ) < 0 at r � 0 which determines the compact behavior of the star. Here, we note that stellar geometries depend upon the first integrals of motion. e energy conditions play a crucial role to investigate the physical existence of cosmological geometries as well as viable matter distribution. For a physically realistic geometry of compact objects, these conditions must be satisfied. ese conditions are considered as quite helpful to examine the nature of matter (normal/exotic) inside the geometry of compact stars. ese bounds can be categorized into null (NEC), weak (WEC), strong (SEC), and dominant (DEC) energy conditions. In curvature-matter coupled gravity, these bounds are expressed as follows [56]: acceleration term that exists due to the additional impacts of curvature-matter coupled gravity. For the physically viable geometry, the energy density must be finite as well as positive 10 Advances in Astronomy everywhere and also have maximum value at the core of the star. Figure 5 indicates that all required energy bounds fulfill at every point inside the stellar structure and hence, it confirms the viability as well as consistency of our chosen EMSG model.

Compactness and Surface
Redshift. e ratio between mass and radius of a stellar object is known as compactness factor. It can be seen clearly from the profile of the mass function given in Figure 6 that the mass of the star is directly proportional to the radius, and M(r) ⟶ 0 as r ⟶ 0, which shows that the mass function is regular at the center of the star. e compactness factor u is defined as follows: e gravitational redshift (Z s ) acts as a crucial parameter to interpret the smooth relationship between particles in the celestial object. In the framework of compactness parameter, the gravitational redshift is expressed in the following form: e graphical evolution of the compactness factor and surface redshift is given in Figure 7. ese plots manifest that the behavior of u(r) and (Z s ) is increasing as required.

e Modified TOV Equation.
e conservation equation is determined as follows: We investigate the equilibrium state of the compact stars through the modified TOV equation with dust matter configuration as follows: is equation determines the combination of two forces, i.e., hydrostatic force F F and gravitational force F g that define the equilibrium state of the stellar structure. In the light of equation (41), these forces can be divided as F g � (λ ′ /2)(ρ eff + p eff ) and F F � (dp eff /dr). e null impact of these forces (F F + F g � 0) ensures the presence of physically realistic geometry of compact objects [57,58]. e graphical interpretation of F F and F g for distinct values of α and β is given in Figure 8, which shows that these forces   Advances in Astronomy counter-balance each other's effect and confirm the equilibrium state of our stellar system.

Stability Analysis.
e stability of compact objects has great importance for a physically viable and consistent model. In order to check the stability of our considered model, we take into account Herrera's cracking method [59]. According to this technique, the square of sound speed (v 2 s ) must satisfy the condition (0 ≤ v 2 s ≤ 1). e sound speed is determined as follows: v 2 s � dp eff dρ eff .
(42) Figure 9 shows the graphical behavior of sound speed indicating that v 2 s satisfies the required condition. is indicates the stability of our solution in this background.

Concluding Remarks
Noether symmetries are much helpful to find solutions of the dynamical system. ese can also provide some viable conditions so that cosmological models can be selected according to current observations [60].
e Lagrange multipliers are used to minimize the dynamical system that ultimately helps to evaluate analytical solutions. In this paper, we have investigated the physical attributes of compact objects via Noether symmetry technique. For this purpose, we have taken static spherical spacetime with perfect fluid configuration in the context of EMSG. We have formulated the Lagrangian of this gravity and evaluated the symmetry generators with corresponding conserved quantities to analyze the solutions of modified equations of motion. e analytic solutions of Noether equations have been studied for the minimal coupling model of this theory by assuming dust fluid just for the sake of simplicity. e presence of conserved quantities is the key aspect in discussing the geometry of compact objects. e main findings of this analysis can be summarized as follows: (i) For a physically realistic and stable model, e λ and e ϑ must be positive, finite, and nonsingular everywhere inside the stellar structure. e graphical representation (Figures 1-3) of both the metric potentials shows the viability and stability of these quantities.
(ii) e effective matter variables must be maximum at the center of compact objects. We are unable to achieve the clear graphs for the complete range (0 ≤ r ≤ 10) of radii. erefore, we have plotted the graphs for the small radius to present the smooth behavior of compact objects. Figure 4 indicates that the effective matter variables have maximum value at the core of compact object and then decreases towards the surface boundary which shows physically viable behavior. (iii) We have shown ( Figure 5) that all energy conditions are well satisfied for our considered model exhibiting the physically viable matter.
(iv) We have found ( Figure 6) the direct proportionality of the mass function to the radius, M(r) ⟶ 0 as r ⟶ 0, suggesting that the mass function is regular at the center of the star. (v) e graphical analysis of the compactness parameter and gravitational redshift function is found to be increasing as required (Figure 7). (vi) It is found through TOV equation that gravitational F H and hydrostatic forces F G are in equilibrium for our proposed model (Figure 8) is ensures the stability of our system. (vii) Finally, we have examined the causality condition through the speed of sound for the compact star.
We have obtained (Figure 9) that our gravity model is consistent with this condition.
We have found that compact stars in this modified gravity via Noether symmetry technique depend on the first integrals of motion as well as model parameters α and β. We have shown that all physical characteristics of compact objects obey the physically viable pattern with small range of radii. We can conclude that Noether symmetry technique in the framework of energy-momentum squared gravity provides a physically realistic and stable model. It is worthwhile to mention here that this is the first investigation of compact objects through Noether symmetry technique in the energymomentum squared gravity [61].

Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.