Study of Compact Stars in $\mathcal{R}+\alpha\mathcal{A}$ Gravity

Investigation of relativistic stellar structures has now become an interesting research pursuit in astrophysics. The fundamental features of compact stars have encouraged several researchers not only in the prospect of usual general relativity ($\mathcal{GR}$) but also in broadly emerging gravitational alternative theories during the last few decades Skm ; Bhr ; Akp ; SMR ; Vum ; Bmk ; Hin ; Nag ; Hat ; Add .

Compact stars are originated as the result of gravitational collapse that happens due to the pressure of the relativistic objects. The gravitational collapse occurs at the center of the massive star which is highly exothermic and this happens when the internal pressure of the stars fails to maintain the pressure against the external force. As the result of gravitational collapse, either a compact star will be born, like neutron stars, white dwarfs, or it will be extended as a black hole. $\mathcal{GR}$ provides most suitable results in the study of stellar objects and develop basic understanding about gravitational theories. However, $\mathcal{GR}$ alone does not provide satisfactory outcomes in order to determine the mysterious nature of dark energy. Due to the limitation of $\mathcal{GR}$, the modified theories have gained the attention of researchers. As an alternative to the theory of $\mathcal{GR}$, modified theories of gravity have played an important role to somehow explain the accelerating expansion of the universe. Various modified gravitational theories are presented in literature, some of mostly debated are $f(\mathcal{R})$, $f(\mathcal{R},\mathcal{T})$, $f(\mathcal{G})$, $f(\mathcal{G},\mathcal{T})$, $f(\mathcal{R},\mathcal{G})$ Hbl ; MFS ; Stn ; Sma ; Ssz . A recently proposed theory namely $f(\mathcal{R},\mathcal{A})$ gravity has also gained some popularity due to the involvement of bivariate function of Ricci and anti-curvature scalars Luca .

We require a precise solution to the Einstein field equations ($\mathcal{EFE}$) to examine celestial compact structure models. Schwarzschild Sch was the first who discovered the $\mathcal{EFE}$ solution for the internal structure of compact stars in 1916. Tolman Tol and Oppenheimer Opp explored some feasible models of stellar objects to represent the relationship between internal pressure and the gravitational force. Bowers and Liang Bal introduced the idea of non-zero anisotropy in a stellar arrangement. Ruderman Rud was the first to discover that the nuclear density becomes anisotropic at the center of compact objects. Errehymy et al. Err investigated the solutions of dense stars in the domain of $\mathcal{GR}$. A new class of solution describing an anisotropic star which is satisfying Karmarkerfls condition was reported by Maurya et al. Sbs . The discussion of compact structures in modified gravity also seems interesting. Nojiri and Odintsov Nao discussed some viable and stable results of different models in the framework of $f(\mathcal{R})$ gravity. Starobinsky Star also presented the outcomes of various models of $f(\mathcal{R})$ gravity. Hu and Sawicki Has suggested several $f(\mathcal{R})$ gravity models. Cognola et al. Cog examined the physical behavior of the compact stars in the exponential type models of $f(\mathcal{R})$ gravity. Schlai Sli identified the embedding problem on geometrically significant spacetimes. Shamir et al. Mir explored the solutions of dense stars using Karmarkar condition in the background of $f(\mathcal{R})$ theory of gravity. Naz et al. Naz investigated the existence of solutions of anisotropic compact objects through embedding approach in the context of $f(\mathcal{R})$ gravity by applying Karmarkar condition.

Motivated from the interesting properties of stellar structures in modified theories of gravity, we aim to investigate anisotropic compact structures in the background of Ricci inverse ($\mathcal{RI}$) gravity. In particular, we provide a detailed investigation of stellar structure for twelve important compact stars from the recent literature namely $4U~{}1538-52$, $SAX~{}J1808.4-3658$, $Her~{}X-1$, $LMC~{}X-4$, $SMC~{}X-4$, $4U~{}1820-30$, $Cen~{}X-3$, $4U~{}1608-52$, $PSR~{}J1903+327$, $PSR~{}J1614-2230$, $Vela~{}X-1$, $EXO~{}1785-248$. Our work is managed as follows: Section II presents the fundamental concepts, structure, and the modified field equations of $\mathcal{RI}$ gravity. In Section III, the matching constraints for the chosen linear $\mathcal{RI}$ gravity model ($\mathcal{R}+\alpha\mathcal{A}$) are developed using Schwarzschild’s geometry. Section IV is devoted to calculate and analyze physical properties such as energy density, pressure profiles etc. The last section contains final remarks.

The inverse of the Ricci tensor known as the anticurvature tensor helps to form an alternative theory namely $f(\mathcal{R},\mathcal{A})$ gravity Luca , where $\mathcal{R}$ and $\mathcal{A}$ are the traces of curvature and anticurvature respectively. This theory can be explored to verify the widespread presence of spectral instabilities and to understand the effects of perturbations such as the propagation of gravitational waves, perturbation growth, or the Newtonian bound. Such theory is confirmed as a suitable choice to be regarded as a cosmological model after deriving the general field equations of the Lagrangian $f(\mathcal{R},\mathcal{A})$. According to a no-go theorem, every Lagrangian action $f(\mathcal{R},\mathcal{A})$ containing terms with any positive or negative exponent of anticurvature $\mathcal{A}$ can be used to justify dark energy Luca . Thus, it would be exceptionally intriguing to analyze this theory further. In this work, we focus ourselves to study compact structures in this theory. The starting point to derive the field equations is to define a tensor $\mathcal{A}^{\eta\xi}$ as inverse of $\mathcal{R}_{\eta\xi}$, such that

$$\mathcal{A}^{\eta\alpha}\mathcal{R}_{\alpha\xi}=\delta^{\eta}_{\xi}.$$

(1)

In order to develop the modified field equations, we need the following definition of the $f(\mathcal{R},\mathcal{A})$ gravitational action:

$$S=\int d^{4}x\sqrt{-g}[f(\mathcal{R},\mathcal{A})+\mathcal{L}_{m}],$$

(2)

where, the arbitrary function $f(\mathcal{R},\mathcal{A})$ depends on the Ricci scalar $\mathcal{R}$ and the anticurvature scalar $\mathcal{A}$, which are the traces of the Ricci tensor and the anticurvature tensor respectively. Here, $\mathcal{L}_{m}$ is matter Lagrangian and $g$ represents the determinant of the metric. We get the following $f(\mathcal{R},\mathcal{A})$ gravity field equation by varying the action mentioned in Eq. (2) with respect to the metric tensor

	$\displaystyle f_{\mathcal{R}}\mathcal{R}^{\eta\xi}-f_{\mathcal{A}}\mathcal{A}^{\eta\xi}-\frac{1}{2}fg^{\eta\xi}$	$\displaystyle+g^{\mu\eta}\nabla_{\beta}\nabla_{\mu}(\ f_{\mathcal{A}}\mathcal{A}_{\sigma}^{\beta}\mathcal{A}^{\xi\sigma})\ -\frac{1}{2}\nabla^{\kappa}\nabla_{\kappa}(\ f_{\mathcal{A}}\mathcal{A}_{\sigma}^{\eta}\mathcal{A}^{\xi\sigma})\$
		$\displaystyle-\frac{1}{2}g^{\eta\xi}\nabla_{\beta}\nabla_{\mu}(\ f_{\mathcal{A}}\mathcal{A}_{\sigma}^{\beta}A^{\mu\sigma})\ -\nabla^{\eta}\nabla^{\xi}f_{\mathcal{R}}+g^{\eta\xi}\nabla^{\kappa}\nabla_{\kappa}f_{\mathcal{R}}=\mathcal{T}^{\eta\xi},$		(3)

where, $f=f(\mathcal{R},\mathcal{A})$, $f_{\mathcal{R}}=\frac{\partial f}{\partial\mathcal{R}}$, $f_{\mathcal{A}}=\frac{\partial f}{\partial\mathcal{A}}$, $\mathcal{A}^{\eta\xi}$ is the anticurvature tensor which is the inverse of the Ricci tensor $\mathcal{R}_{\eta\xi}$. The stress-energy momentum tensor $\mathcal{T}_{\eta\xi}$ is chosen as

$$\mathcal{T}_{\eta\xi}=(\rho+p_{t})u_{\eta}u_{\xi}-p_{t}g_{\eta\xi}+(p_{r}-p_{t})\mathcal{\chi}_{\eta}\mathcal{\chi}_{\xi}.$$

(4)

Here, $\rho,$ $p_{r}$ and $p_{t}$ are energy density, radial and transverse pressure components respectively, while $u_{\eta}$ and $\mathcal{\chi}_{\eta}$ are four velocity vectors, which satisfy the relations $u^{\eta}u_{\eta}=-\mathcal{\chi}^{\eta}\mathcal{\chi}_{\eta}=1$. Furthermore, we consider a static spherically symmetric line element which is represented as

$$ds^{2}=e^{\nu}dt^{2}-e^{\lambda}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}),$$

(5)

where, $\nu$ and $\lambda$ are the functions of radial coordinate. Our stress-energy tensor is assumed to be anisotropic, with relativistic geometrized units $8\pi G=1$. Moreover, the embedded class-I family is represented by the metric tensor (5), if it fulfills the Karmarkar condition, i.e.

$$\mathcal{R}_{1414}\mathcal{R}_{2323}=\mathcal{R}_{1212}\mathcal{R}_{3434}+\mathcal{R}_{1224}\mathcal{R}_{1334},$$

(6)

with $\mathcal{R}_{2323}\neq 0$. The following differential equation for the given metric (5) is developed by using the well-known Karmarkar condition as

$$\lambda^{\prime}\nu^{\prime}-2\nu^{\prime\prime}-(\nu^{\prime})^{2}=\frac{\lambda^{\prime}\nu^{\prime}}{1-e^{\lambda}},$$

(7)

where $e^{\lambda}\neq 1$. Integrating Eq. (7), we obtain

$$e^{\nu}=\left[(A+B\int\sqrt{e^{\lambda}-1}dr)^{2}\right].$$

(8)

In this case, $A$ and $B$ are integration constants. Further, due to the highly non-linear nature of modified field equations, we have no other choice but to choose the component $g_{rr}=e^{\lambda(r)}$ in the following shape Bhr ; Spt

$$e^{\lambda}=1+a^{2}r^{2}(1+br^{2})^{x},$$

(9)

where one can choose any real value of $x$ other than zero. It is really important to mention here that in order to establish embedded class-I spacetime, the parameters $a$ and $b$ should not be zero. In the domain of $\mathcal{GR}$, Singh and Pant Spt investigated the properties of stellar structures by considering some positive values of $x$ for embedding class-I solutions of different models of compact stars. Later, Bhar et al. Bhr also used this metric potential by taking the value of $x=-4$, and the corresponding results are found stable in the context of $\mathcal{GR}$. Inspired by the work of Bhar et al. Bhr , we aim to expand the study in the framework of $\mathcal{RI}$ gravity by choosing $x=-4$. Interestingly, we tried some other possibilities for the choice of $x$ (both positive and negative), however, we could not obtain physically acceptable results in those cases. Manipulating Eqs. (8) and (9), the metric potential expression becomes

$$e^{\nu}=\left[\left(A-\frac{aB}{2b(1+br^{2})}\right)^{2}\right].$$

(10)

Further, to analyze the properties of compact stars, we probe the necessary conditions for the metric potentials of static spherically symmetric spacetime, namely $e^{\lambda(0)}=1$ and $((e^{\lambda(r)})^{\prime})_{r=0}=0$. The graphical response of $g_{rr}$ should not contain any singularities, and the curvature must be regular. From Fig. 1, it is evident that all above properties are satisfied and also the plot of $e^{\lambda}$ exhibits monotonically increasing behavior, reaching its maximum value near the boundary.

In this section, we provide some discussions about linear $\mathcal{R}+\alpha\mathcal{A}$ gravity model and the matching conditions. By substituting this model in Eq.(II), the field equations are simplified as

$$\mathcal{R}^{\eta\xi}-\frac{1}{2}\mathcal{R}g^{\eta\xi}-\alpha\mathcal{A}^{\eta\xi}-\frac{1}{2}\alpha\mathcal{A}g^{\eta\xi}+\frac{\alpha}{2}(2g^{\mu\eta}\nabla_{\beta}\nabla_{\mu}\mathcal{A}_{\sigma}^{\beta}\mathcal{A}^{\xi\sigma}-\nabla^{\kappa}\nabla_{\kappa}\mathcal{A}_{\sigma}^{\eta}\mathcal{A}^{\xi\sigma}-g^{\eta\xi}\nabla_{\beta}\nabla_{\mu}\mathcal{A}_{\sigma}^{\beta}\mathcal{A}^{\mu\sigma})=\mathcal{T}^{\eta\xi}$$

(11)

Using Eq. (11) and spacetime (5), modified field equations turn out to be

	$\displaystyle\rho$	$\displaystyle=e^{-\nu}\Gamma_{1}-\frac{1}{2}\Gamma_{2}-\alpha e^{\nu}\Gamma_{3}-\frac{1}{2}\alpha\Gamma_{4}+\frac{\alpha}{2}(e^{\nu-\lambda}\Gamma_{5}^{\prime\prime}+e^{\nu-\lambda}\nu^{\prime\prime}\Gamma_{5}+2e^{\nu-\lambda}\nu^{\prime}\Gamma_{5}^{\prime}+e^{\nu-\lambda}(\nu^{\prime})^{2}\Gamma_{5}-\frac{\lambda^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}^{\prime}$
		$\displaystyle-\frac{\nu^{\prime}\lambda^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}+\frac{2}{r}e^{\nu-\lambda}\Gamma_{5}^{\prime}+\frac{2\nu^{\prime}}{r}e^{\nu-\lambda}\Gamma_{5}-\Gamma_{6}^{\prime\prime}-\lambda^{\prime\prime}\Gamma_{6}-\frac{3}{2}\lambda^{\prime}\Gamma_{6}^{\prime}-\frac{(\lambda^{\prime})^{2}}{2}\Gamma_{6}-2re^{-\lambda}\Gamma_{7}^{\prime}-\frac{4}{r^{2}}\Gamma_{6}),$		(12)

	$\displaystyle p_{r}$	$\displaystyle=e^{-\lambda}\Gamma_{8}+\frac{1}{2}\Gamma_{2}-\alpha e^{\lambda}\Gamma_{9}+\frac{1}{2}\alpha\Gamma_{4}+\frac{\alpha}{2}(-\frac{(\nu^{\prime})^{2}}{2}e^{\nu-\lambda}\Gamma_{5}+\frac{\nu^{\prime}\lambda^{\prime}}{2}\Gamma_{6}+\frac{\nu^{\prime}}{2}\Gamma_{6}^{\prime}+4e^{-\lambda}\Gamma_{7}+\frac{2\lambda^{\prime}}{r}\Gamma_{6}+\frac{2}{r}\Gamma_{6}^{\prime}$
		$\displaystyle-\frac{\nu^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}^{\prime}+2re^{-\lambda}\Gamma_{7}^{\prime}),$		(13)

	$\displaystyle p_{t}$	$\displaystyle=\frac{1}{r^{2}}\Gamma_{10}+\frac{1}{2}\Gamma_{2}-\alpha r^{2}\Gamma_{11}+\frac{1}{2}\alpha\Gamma_{4}+\frac{\alpha}{2}(\frac{\nu^{\prime}}{2}r^{2}e^{-\lambda}\Gamma_{7}^{\prime}+re^{-\lambda}\nu^{\prime}\Gamma_{7}+r^{2}e^{-\lambda}\Gamma_{7}^{\prime\prime}+4e^{-\lambda}\Gamma_{7}+6re^{-\lambda}\Gamma_{7}^{\prime}$
		$\displaystyle-\frac{\lambda^{\prime}}{2}r^{2}e^{-\lambda}\Gamma_{7}^{\prime}-re^{-\lambda}\lambda^{\prime}\Gamma_{7}-\frac{\nu^{\prime}}{2}e^{\nu-\lambda}\Gamma_{5}^{\prime}+\frac{(\nu^{\prime})^{2}}{2}\Gamma_{6}+\Gamma_{6}^{\prime\prime}+\lambda^{\prime\prime}\Gamma_{6}+\frac{3}{2}\lambda^{\prime}\Gamma_{6}^{\prime}+\frac{(\lambda^{\prime})^{2}}{2}\Gamma_{6}+\frac{2}{r^{2}}\Gamma_{6}),$		(14)

where, the values of $\Gamma_{i}$’s $(i=1...11)$ are mentioned in Appendix and the prime represents the derivative with respect to $r$.

The star’s geometrical structure, whether viewed from the outside or the inside, has no impact on the interior boundary metric. The metric components must be continuous to the boundary regardless of the referential frame in this emerging situation. While evaluating several matching conditions and exploring compact objects in the context of $\mathcal{GR}$, the Schwarzschild solution seems to be the right approach. According to the Jebsen-Birkhoff theorem, given spherically symmetric spacetime, the solution of the Einstein field equations $\mathcal{(EFE)}$ must be asymptotically flat and static. In modified gravitational theories, the exterior geometry solution may vary from the Schwarzschild solution when we consider modified $\mathcal{(TOV)}$ equations Tol ; Opp with zero pressure and energy density. Furthermore, the Schwarzschild solution can be achieved in $\mathcal{RI}$ gravity by choosing a realistic and viable $\mathcal{R}+\alpha\mathcal{A}$ gravity model with non-zero density and pressure components. The Birkhoff theorem contradicts modified theories as a result Vfi . Numerous studies using Schwarzschild solution in matching constraints have produced impressive findings Ava ; Acn ; Agy ; Dmi . The radial pressure $p_{r}(r=R)=0$ is used to calculate the field equations solution under the specified boundary conditions at $r=R$. Thus, we compare the internal geometry solution to the Schwarzschild external geometry provided by

$$ds^{2}=\left(1-\frac{2M}{r}\right)dt^{2}-\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-r^{2}(d\theta^{2}+sin^{2}\theta d\phi^{2}),$$

(15)

where the entire mass contained within the star is represented by $M$. The following expressions are derived at $r=R$ by probing at the metric potentials as follows:

$\displaystyle g_{tt}^{+}=g_{tt}^{-},\qquad g_{rr}^{+}=g_{rr}^{-},\qquad\frac{\partial g_{tt}^{+}}{\partial r}=\frac{\partial g_{tt}^{-}}{\partial r},$

(16)

where, (+) refers to the external geometry and (-) represents the internal geometry. The values of the required unknown quantities can be obtained by comparing the inner and outer matrices as shown below:

	$\displaystyle a$	$\displaystyle=\frac{(1+bR^{2})^{2}}{R}\sqrt{\frac{\frac{2M}{R}}{1-\frac{2M}{R}}},$	$\displaystyle b$	$\displaystyle=\frac{4M-R}{R^{2}(9R-20M)},$		(17)
	$\displaystyle A$	$\displaystyle=\frac{aB}{2b(1+bR^{2})}+\sqrt{1-\frac{2M}{R}},$	$\displaystyle B$	$\displaystyle=\left(\frac{1}{2R}\right)\sqrt{\frac{2M}{R}}.$		(18)

We will use these matching conditions to analyze different compact star models in the background of $\mathcal{R}+\alpha\mathcal{A}$ gravity. It is worthwhile to mention here that we provide a detailed comparison by considering $12$ compact star models namely $4U~{}1538-52$, $SAX~{}J1808.4-3658$, $Her~{}X-1$, $LMC~{}X-4$, $SMC~{}X-4$, $4U~{}1820-30$, $Cen~{}X-3$, $4U~{}1608-52$, $PSR~{}J1903+327$, $PSR~{}J1614-2230$, $Vela~{}X-1$, $EXO~{}1785-248$. The values of the input parameters radius and mass of considered compact stars are given in Table I:

By using the values of the input parameters from Table I and manipulating Eqs. (17), the values of the output parameters $a$, $b$, $A$, and $B$ are shown in the following Table:

Further, the requirements for a well-behaved compact structure are specified as:

$\bullet$ The plots of $\rho$, $p_{r}$, and $p_{t}$ must always be positive, finite, and maximum at the center.
$\bullet$ The gradient of $\rho$, $p_{r}$, and $p_{t}$ should be negative.
$\bullet$ All of the energy requirements must be achieved.
$\bullet$ The equation of state ($EoS$) should satisfy the mandatory constraints, namely $0<\omega_{r}$, $\omega_{t}<1$.
$\bullet$ The equilibrium stability condition must be fulfilled by all of the forces.
$\bullet$ The parameters of speed of sound must be between $[0,1]$, i.e. $0<\upsilon_{r}^{2}$, and $\upsilon_{t}^{2}<1$.
$\bullet$ Adiabatic index must be greater than $\frac{4}{3}$ for an anisotropic fluid sphere.

In this section, we discuss some important physical properties of the observed compact stars in the context of considered $\mathcal{R}+\alpha\mathcal{A}$ gravity model. For this purpose, we provide the graphical representations of energy density, pressure components, energy conditions, mass function, adiabatic index, stability and equilibrium conditions. Moreover, for our given model, we take the parameter $\alpha=1.3\times 10^{-11}$ for various compact stars. The parameter $\alpha$ has a very limited range which provides us physically accepted results. We have checked two other values of parameter $\alpha=1.0\times 10^{-9}$, $\alpha=1.9\times 10^{-8}$ and found that there is a very slight difference in the behavior of the graphs towards the boundary. That difference is shown for one star $Vela~{}X-1$ as a magnified image in Fig. 2.

We assume a realistic $\mathcal{R}+\alpha\mathcal{A}$ gravity model by using the Karmarkar condition to investigate a novel family of embedded class-I solutions in the anisotropic framework. Since the Karmarkar condition provides a link between $g_{rr}$ and $g_{tt}$, it simplifies the solution generating process of $\mathcal{EFE}$ to a single metric potential. Assuming the metric potential $e^{\lambda}=1+\frac{a^{2}r^{2}}{(1+br^{2})^{4}}$ for this purpose, we use the Karmarkar condition to determine the second metric potential, $e^{\nu}=\left[\left(A-\frac{aB}{2b(1+br^{2})}\right)^{2}\right]$, where $a$, $b$, $A$, and $B$ are arbitrary constants. We also determine the unknown constants by applying matching constraints between the internal and Schwarzschild external geometries. We examine the physical behavior of twelve different compact stars to confirm that the discussed solutions are physically feasible. The primary objective of our research is to study stellar structures using a viable $\mathcal{R}+\alpha\mathcal{A}$ gravity model in the context of an anisotropic matter source. The notable results are listed below.

• •

  Metric potentials are frequently used to characterize the properties of spacetime. According to Fig. 1, both the metric potentials $g_{rr}=e^{\lambda}$ and $g_{tt}=e^{\nu}$ exhibit graphical behavior that is finite, positive, singularity-free, and satisfies the criteria, i.e., $e^{\lambda(r=0)}=1$ and $e^{\nu(r=0)}=\left[A-\frac{aB}{2b}\right]^{2}$. Since these graphs expand monotonically and reach their maximum values at the boundary, the progression of both potentials exhibits excellent outcomes, supporting the $\mathcal{R}+\alpha\mathcal{A}$ gravity model.

• •

  The trend of density and pressure components for $\mathcal{R}+\alpha\mathcal{A}$ gravity model is shown in Fig. 2. For this purpose, we consider the model parameter $\alpha=1.3\times 10^{-11}$. It is important to mention here that the parameter $\alpha$ has a very limited range which provides us physically accepted results. We have checked two other values of parameter $\alpha=1.0\times 10^{-9}$, $\alpha=1.9\times 10^{-8}$ and found that there is a very slight difference in the behavior of the graphs towards the boundary. That difference is shown for one star only $Vela~{}X-1$ as a magnified image in Fig. 2. The variations of $\rho$, $p_{r}$, and $p_{t}$ with respect to $r$ are finite and regular at the center. It is also evident that graphs of $\rho$, $p_{r}$, and $p_{t}$ reach their highest values at the center and show a diminishing response toward the border, supporting the stability of our model. The gradient of $\rho$, $p_{r}$, and $p_{t}$ has been depicted in Fig. 3. These graphs exhibit negative behavior, demonstrating the consistency of our suggested model.

• •

  It is clear from Fig. 4 that anisotropy is positive throughout the stellar objects. This behavior demonstrates that anisotropy has a repulsive nature, confirming its existence for compact objects.

• •

  The energy bounds for the suggested $\mathcal{R}+\alpha\mathcal{A}$ model are shown in Fig. 5. We observed that our given model fulfills all energy bounds.

• •

  The equilibrium conditions of $\mathcal{F}_{g}$, $\mathcal{F}_{h}$, and $\mathcal{F}_{a}$ of our physically acceptable model show balancing behavior in Fig. 6.

• •

  The graphical response of $EoS$ must fulfill the requirements $0<\omega_{r}$, $\omega_{t}<1$ for the consistency of the stellar objects. The corresponding graphical representation in Fig. 7 depicts that both $EoS$ ratios have a stable character.

• •

  The radial component sound velocity $\upsilon_{r}^{2}$ plot and the transverse component sound velocity $\upsilon_{t}^{2}$ plot should fall within the range [0, 1] for compact stars. According to Fig. 8, the stability constraints for our model are stable.

• •

  From Fig. 9, it is easy to notice that the graphs of the mass function and the compactness parameter demonstrate monotonically increasing responses. On the other hand, the graphical illustration of redshift analysis reveals monotonically decreasing behavior. The behavior shows that our given model achieves the stability criteria.

• •

  In Fig. 10, the adiabatic index of $\gamma_{r}$ and $\gamma_{t}$ for our given model is more than $\frac{4}{3}$, confirming the stability of our system.
  
  

As a final observation, we have used the Karmarkar condition to provide a stellar system that is both well-stable and singularity-free. We conclude that $\mathcal{R}+\alpha\mathcal{A}$ gravity model supports the existence of compact objects which follow observable patterns. Furthermore, it is worth mentioning that our suggested results are almost similar to the outcomes investigated by Naz et al. Naz in the context of $f(\mathcal{R})$ theory of gravity. It may be noted that due to the involvement of inverse curvature terms, we get more dense stars in our case.

The authors appreciate important comments from anonymous reviewer which improved the paper.

Following are the values of $\Gamma_{i}$’s,

$\displaystyle\Gamma_{1}=-\frac{a(1+br^{2})(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})B(-2b(1+br^{2})A+aB)}{2b(a^{2}r^{2}+(1+br^{2})^{4})^{2}}$

$\displaystyle\Gamma_{2}=\frac{(2a(2a^{3}br^{2}(1+br^{2})A-2ab(1+br^{2})^{4}(-3+5br^{2})A-a^{4}r^{2}B+a^{2}(-3+br^{2})(1+br^{2})^{3}B+2b(-3+br^{2})(1+br^{2})^{6}B))}{((a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB))}$

$\displaystyle\Gamma_{3}=-\frac{2b(a^{2}r^{2}+(1+br^{2})^{4})^{2}}{a(1+br^{2})(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})B(-2b(1+br^{2})A+aB)}$

	$\displaystyle\Gamma_{4}$	$\displaystyle=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(2b(1+br^{2})A-aB)}{2ab(1+br^{2})^{3}(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})B}$
		$\displaystyle+\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB)}{2a(1+br^{2})^{3}(-1+3br^{2})(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)}$
		$\displaystyle+\frac{2(a^{2}r^{2}+(1+br^{2})^{4})^{2}(2b(1+br^{2})A-aB)}{a(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)}$

$\displaystyle\Gamma_{5}=-\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{4}(2b(1+br^{2})A-aB)}{a^{2}(1+br^{2})^{4}(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})^{2}B^{2}(-2b(1+br^{2})A+aB)}$

$\displaystyle\Gamma_{5}^{\prime}=-\frac{8r(a^{2}r^{2}+(1+br^{2})^{4})^{3}(a^{4}r^{2}(-1+br^{2})+b(-5+br^{2})(1+br^{2})^{7}-a^{2}(1+br^{2})^{3}(2+br^{2}(-1+7br^{2})))}{a^{2}(1+br^{2})^{5}(2a^{2}r^{2}-(-3+br^{2})(1+br^{2})^{3})^{3}B^{2}}$

	$\displaystyle\Gamma_{5}^{\prime\prime}$	$\displaystyle=(8(a^{2}r^{2}+(1+br^{2})^{4})^{2}(2a^{8}r^{6}(3+br^{2}(-12+5br^{2}))+b(1+br^{2})^{14}(15+br^{2}(181+br^{2}(-67+7br^{2})))$
		$\displaystyle-a^{6}r^{4}(1+br^{2})^{3}(-25+br^{2}(93+br^{2}(-151+35br^{2})))-2a^{2}(1+br^{2})^{10}(-3+br^{2}(-37+br^{2}(-71+br^{2}$
		$\displaystyle(-211+34br^{2}))))+a^{4}r^{2}(1+br^{2})^{6}(31+3br^{2}(-23+br^{2}(125+br^{2}(-13+122br^{2}))))))/$
		$\displaystyle(a^{2}(1+br^{2})^{6}(-2a^{2}r^{2}+(-3+br^{2})(1+br^{2})^{3})^{4}B^{2})$

$\displaystyle\Gamma_{6}=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{3}(2b(1+br^{2})A-aB)(-2b(1+br^{2})A+aB)}{4a^{2}(1+br^{2})^{2}(-1+3br^{2})^{2}(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)^{2}}$

	$\displaystyle\Gamma_{6}^{\prime}$	$\displaystyle=(r(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-48b^{4}(-1+br^{2})(1+br^{2})^{9}A^{2}B-6a^{5}b(3+br^{2}+b^{2}r^{4}+3b^{3}r^{6})AB^{2}+a^{6}(3+br^{2}(-2+3br^{2}))B^{3}$
		$\displaystyle+a^{4}b(1+br^{2})^{2}B(12b(3+br^{2}(-2+3br^{2}))A^{2}-(-1+br^{2})(1+br^{2})(19+3br^{2})B^{2})+4ab^{3}(1+br^{2})^{7}A(16b(-2+3br^{2})A^{2}$
		$\displaystyle+(1+br^{2})(-11+9br^{2})B^{2})+4b^{2}(a+abr^{2})^{3}A(-2b(3+br^{2}(-2+3br^{2}))A^{2}+(1+br^{2})(-27+br^{2}(19+18br^{2}))B^{2})$
		$\displaystyle+2a^{2}b^{2}(1+br^{2})^{5}B(-6b(-17+br^{2}(10+19br^{2}))A^{2}-(-5+3br^{2})(B+br^{2}B)^{2})))/(2a^{2}(1+br^{2})^{3}(-1+3br^{2})^{3}$
		$\displaystyle(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)^{3})$

	$\displaystyle\Gamma_{6}^{\prime\prime}$	$\displaystyle=-(((a^{2}r^{2}+(1+br^{2})^{4})(48b^{5}(1+br^{2})^{16}(1+br^{2}(23+br^{2}(-37+21br^{2})))A^{2}B^{2}-8a^{9}br^{2}(1+br^{2})(15+$
		$\displaystyle br^{2}(-8+br^{2}(94+3br^{2}(-16+9br^{2}))))AB^{3}+a^{10}r^{2}(15+br^{2}(-8+br^{2}(94+3br^{2}(-16+9br^{2}))))B^{4}$
		$\displaystyle+4ab^{4}(1+br^{2})^{14}AB(-8b(7+25br^{2}(7+3br^{2}(-5+3br^{2})))A^{2}-(1+br^{2})(11+3br^{2}(79-93br^{2}+45b^{2}r^{4}))B^{2})$
		$\displaystyle+4a^{3}b^{3}(1+br^{2})^{10}AB(-4b(35+br^{2}(1100+3br^{2}(-618+br^{2}(-28+561br^{2}))))A^{2}-(1+br^{2})(43+br^{2}(1042+$
		$\displaystyle br^{2}(-700-546br^{2}+513b^{2}r^{4})))B^{2})+4a^{7}b(1+br^{2})^{3}AB(-8b^{2}r^{2}(15+br^{2}(-8+br^{2}(94+$
		$\displaystyle 3br^{2}(-16+9br^{2}))))A^{2}-(1+br^{2})(6+br^{2}(355+3br^{2}(4+br^{2}(98+39br^{2}(-2+3br^{2})))))B^{2})+4a^{5}b^{2}$
		$\displaystyle(1+br^{2})^{6}AB(-4b(6+br^{2}(325+br^{2}(106+3br^{2}(-56+br^{2}(-8+105br^{2})))))A^{2}-(1+br^{2})(41+br^{2}(1376+$
		$\displaystyle br^{2}(-871+9br^{2}(-31+15br^{2}(6+5br^{2})))))B^{2})+a^{8}(B+br^{2}B)^{2}(24b^{2}r^{2}(15+br^{2}(-8+br^{2}(94+3br^{2}$
		$\displaystyle(-16+9br^{2}))))A^{2}+(1+br^{2})(3+br^{2}(185+br^{2}(-24+br^{2}(280+3br^{2}(-49+45br^{2})))))B^{2})+2a^{2}b^{3}(1+br^{2})^{12}$
		$\displaystyle(64b^{2}(2+br^{2}(59+3br^{2}(-56+45br^{2})))A^{4}+2b(85+br^{2}(2152+3br^{2}(-650-480br^{2}+603b^{2}r^{4})))$
		$\displaystyle A^{2}B^{2}+(1+br^{2})^{2}(5+br^{2}(101+3br^{2}(-23+9br^{2})))B^{4})+a^{4}b^{2}(1+br^{2})^{8}(16b^{2}(1+br^{2})(3+br^{2}$
		$\displaystyle(149+3br^{2}(-77+45br^{2})))A^{4}+4b(114+br^{2}(3773+br^{2}(-2344+3br^{2}(-1310+9br^{2}(154+127br^{2})))))A^{2}B^{2}$
		$\displaystyle+(1+br^{2})^{2}(29+br^{2}(660+br^{2}(-242+3br^{2}(-100+63br^{2}))))B^{4})+a^{6}b(1+br^{2})^{4}(16b^{3}r^{2}(15+br^{2}$
		$\displaystyle(-8+br^{2}(94+3br^{2}(-16+9br^{2}))))A^{4}+8b(1+br^{2})(9+br^{2}(510+br^{2}(95+3br^{2}(25+br^{2}(-68+177br^{2})))))$
		$\displaystyle A^{2}B^{2}+(1+br^{2})^{2}(22+br^{2}(745+3br^{2}(-174+br^{2}(212-60br^{2}+81b^{2}r^{4}))))B^{4})))/(2a^{2}(1-3br^{2})^{4}$
		$\displaystyle(1+br^{2})^{4}(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)^{4}))$

$\displaystyle\Gamma_{7}=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{4}(2b(1+br^{2})A-aB)(-2b(1+br^{2})A+aB)}{a^{2}r^{2}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)^{2}}$

	$\displaystyle\Gamma_{7}^{\prime}$	$\displaystyle=(2(a^{2}r^{2}+(1+br^{2})^{4})^{3}(-8b^{3}(1+br^{2})^{12}(-1+3br^{2})A^{2}B+6a^{7}br^{4}(1+br^{2})AB^{2}-a^{8}r^{4}B^{3}+2a^{2}b(1+br^{2})^{8}B(12b$
		$\displaystyle(1+br^{2}(-12+7br^{2}))A^{2}-(-1+br^{2})(B+br^{2}B)^{2})+8ab^{2}(1+br^{2})^{9}A(-2b(1+br^{2}(-12+7br^{2}))A^{2}+(-1+2br^{2})(B+$
		$\displaystyle br^{2}B)^{2})+2(a+abr^{2})^{4}B(-2b^{2}r^{2}(9+br^{2}(8+63br^{2}))A^{2}+(-1+br^{2})(-1+11br^{2})(B+br^{2}B)^{2})+4a^{3}b(1+br^{2})^{5}A$
		$\displaystyle(2b^{2}r^{2}(3+br^{2}(2+23br^{2}))A^{2}-(3+br^{2}(-36+25br^{2}))(B+br^{2}B)^{2})-a^{6}(r+br^{3})^{2}B(3B^{2}+br^{2}(12bA^{2}+(4+13b$
		$\displaystyle r^{2})B^{2}))+2a^{5}br^{2}(1+br^{2})^{3}A(9B^{2}+br^{2}(4bA^{2}+(10+53br^{2})B^{2}))))/(a^{2}r^{3}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})$
		$\displaystyle(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)^{3})$

	$\displaystyle\Gamma_{7}^{\prime\prime}$	$\displaystyle=(2(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-48b^{4}(1+br^{2})^{22}(1+br^{2}(-2+5br^{2}))A^{2}B^{2}+8a^{13}br^{8}(1+br^{2})AB^{3}-a^{14}r^{8}B^{4}+2a^{12}r^{6}(1+br^{2})$
		$\displaystyle B^{2}(-12b^{2}r^{2}(1+br^{2})A^{2}-(3+br^{2}(7+5br^{2}(7+11br^{2})))B^{2})+8a^{9}b(r+br^{3})^{4}AB(4b^{2}r^{2}(6+br^{2}(11+3br^{2}(38+67br^{2})))$
		$\displaystyle A^{2}+(1+br^{2})(17+br^{2}(16+br^{2}(564+br^{2}(270+257br^{2}))))B^{2})+4a^{8}br^{4}(1+br^{2})^{5}(-8b^{3}r^{2}(3+br^{2}(5+br^{2}(61$
		$\displaystyle+107br^{2})))A^{4}-2b(1+br^{2})(51+br^{2}(36+br^{2}(1942+3br^{2}(348+533br^{2}))))A^{2}B^{2}+(1+br^{2})^{3}(-37+br^{2}(-48+br^{2}(-85$
		$\displaystyle+358br^{2})))B^{4})+a^{10}r^{4}(1+br^{2})^{3}(-16b^{4}r^{4}(1+br^{2})A^{4}-16b^{2}r^{2}(9+br^{2}(18+7br^{2}(22+39br^{2})))A^{2}B^{2}+(1$
		$\displaystyle+br^{2})(-17+br^{2}(-20+br^{2}(-462+br^{2}(-92+31br^{2}))))B^{4})+16ab^{3}(1+br^{2})^{19}AB(4b(3+br^{2}(-21+br^{2}(43-29br^{2})))$
		$\displaystyle A^{2}+3(1+br^{2}(-1+2br^{2}))(B+br^{2}B)^{2})+8b^{2}r^{4}(a+abr^{2})^{7}AB(4b(17+br^{2}(8+br^{2}(712+br^{2}(356+755br^{2}))))$
		$\displaystyle A^{2}-(-1+br^{2})(145+br^{2}(448+1367br^{2}))(B+br^{2}B)^{2})+16a^{3}b^{2}(1+br^{2})^{15}AB(12b(2+br^{2}(-23+br^{2}(186+br^{2}(-195$
		$\displaystyle+58br^{2}))))A^{2}+(9+br^{2}(-51+br^{2}(-3+br^{2})))(B+br^{2}B)^{2})+16a^{5}b(1+br^{2})^{11}AB(2b^{3}r^{4}(139+br^{2}(455+br^{2}(1381$
		$\displaystyle-983br^{2})))A^{2}+(6+br^{2}(-63+br^{2}(511+br^{2}(-853+231br^{2}))))(B+br^{2}B)^{2})-4a^{2}b^{2}(1+br^{2})^{16}(16b^{2}(3+br^{2}(-36$
		$\displaystyle+br^{2}(290+br^{2}(-300+91br^{2}))))A^{4}-24b(1+br^{2})^{2}(-3+br^{2}(19+br^{2}(-21+13br^{2})))A^{2}B^{2}+(3+b^{2}r^{4})(B$
		$\displaystyle+br^{2}B)^{4})+8a^{4}b(1+br^{2})^{12}(16b^{4}r^{4}(-17+br^{2}(-62+br^{2}(-173+112br^{2})))A^{4}-2b(1+br^{2})^{2}(18+br^{2}(-198+br^{2}$
		$\displaystyle(1605+br^{2}(-1988+543br^{2}))))A^{2}B^{2}-(3+br^{2}(-15+br^{2}(-23+19br^{2})))(B+br^{2}B)^{4})+4a^{6}(1+br^{2})^{8}(-4b^{4}r^{4}(17+$
		$\displaystyle br^{2}(4+br^{2}(758+324br^{2}+897b^{2}r^{4})))A^{4}+4b^{3}r^{4}(1+br^{2})^{2}(-213+br^{2}(-586+3br^{2}(-623+584br^{2})))A^{2}B^{2}-(3$
		$\displaystyle+br^{2}(-30+br^{2}(243+4br^{2}(-147+50br^{2}))))(B+br^{2}B)^{4})+8a^{11}br^{6}(1+br^{2})^{2}AB(6B^{2}+br^{2}(4b(1+br^{2})A^{2}+$
		$\displaystyle(13+br^{2}(88+153br^{2}))B^{2}))))/(a^{2}r^{4}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B$
		$\displaystyle+2b(1+br^{2})^{7}B)^{4})$

$\displaystyle\Gamma_{8}=\frac{2a(-1+3br^{2})(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)}{(1+br^{2})(a^{2}r^{2}+(1+br^{2})^{4})(2b(1+br^{2})A-aB)}$

$\displaystyle\Gamma_{9}=\frac{(1+br^{2})(a^{2}r^{2}+(1+br^{2})^{4})(2b(1+br^{2})A-aB)}{2a(-1+3br^{2})(-2ab(1+br^{2})A+a^{2}B+b(1+br^{2})^{3}B)}$

$\displaystyle\Gamma_{10}=\frac{ar^{2}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)}{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB)}$

$\displaystyle\Gamma_{11}=\frac{(a^{2}r^{2}+(1+br^{2})^{4})^{2}(-2b(1+br^{2})A+aB)}{ar^{2}(-2a^{3}br^{2}(1+br^{2})A+4ab(-1+br^{2})(1+br^{2})^{4}A+a^{4}r^{2}B+2a^{2}(1+br^{2})^{3}B+2b(1+br^{2})^{7}B)}$