The Effect of Gravitational Decoupling on Constraining the Mass and Radius for the Secondary Component of GW190814 and Other Self-bound Strange Stars in f(Q) Gravity Theory

Over the years, compact objects such as neutron stars (NSs), pulsars, and strange stars have served as cosmic laboratories for determining the nature of matter at ultrahigh densities. While Einstein's classical theory of general relativity (GR) can account for many observed features such as compactness, mass–radius relations, and surface redshifts of these objects, it has fallen short in accounting for peculiar observations of NSs with masses exceeding M = 2 M_⊙. The LIGO Scientific and Virgo Collaboration observations of gravitational waves such as the GW190814 and GW170817 events have also cast light on the shortcomings of classical GR in accounting for supermassive black holes (BHs; Abbott et al. 2016, 2017, 2020). In particular, the gravitational-wave event GW190814 suggests that the source of the signals originated in a compact binary coalescence of a 22.2–24.3 M_⊙ BH and a compact object having a mass in the range of 2.50–2.67 M_⊙. The GW170817 event of 2017 August 17 is thought to be the merger of two NSs with masses in the range 0.86–2.26 M_⊙.

There have been various proponents put forward to account for the observed signals of gravitational events, including the nature of matter (equation of state; EOS) of stars making up the binary duo and modified gravity theories. The GW170817 event and its electromagnetic counterparts provided researchers with a new tool to propose more exotic EOSs for the NSs involved in this merger. The electromagnetic signal emanating from GW170817 was composed of two parts: a short gamma-ray burst GRB170817A with a delay of approximately 2 s with respect to the GW signal, and a kilonova, AT2017gfo, peaking in luminosity a few days after the merger. These observed delays have led researchers to speculate on the nature of the binary components. For example, the delay in the gamma-ray burst has prompted some to believe that the remnant arising from the merger was most likely a hypermassive star that collapsed to a BH within a few milliseconds (Ruiz et al. 2018). Similarly, the kilonova signal points to a not-too-soft EOS (Radice et al. 2018). In order to account for the possible ranges for the tidal deformability, 400 ≤ Λ ≤ 800, and the radius of the 1.5 M_⊙ component that lies within 11.8 km ≤ R_1.5 ≤ 13.1, km, the GW170817 event has been modeled as the merger of a hadronic star with a strange star (Burgio et al. 2018). Furthermore, to account for small radii and not too small Λ, it has been proposed that the stellar matter undergoes strong phase transitions at supranuclear densities, giving rise to quark matter.

The observed signal associated with GW190814 has led to speculation of the nature of the secondary component owing to several factors, including the mass of ${2.59}_{-0.09}^{+0.08}{M}_{\odot }$, a lack of significant tidal deformations, and no accompanying electromagnetic signals. This points to the presence of either an NS or a BH. We are thus faced with a conundrum where the secondary component is either the heaviest NS or the lightest BH ever observed in a double compact object system. On the other hand, the GW190814 event has been modeled as a second-generation merger, i.e., a triple hierarchical system giving rise to a remnant from a primary binary NS that was then captured by the 23 M_⊙ BH (Lu et al. 2021). Alternatively, the double merger scenario can be the result of a tight NS–NS scattering off a massive BH. These scenarios all call on the very nature of matter (EOS) of the components making up the binary or triple hierarchical system. Using covariant density functional theory, Fattoyev et al. (2020) explored the possible 2.6 M_⊙ stable bounded configuration while ensuring that existing constraints on the composition of NSs and the ground-state properties of finite nuclei are preserved. The energy density functionals used in their investigation predicted high pressures prevalent in stellar matter. A softening of the EOS accomplished by the addition of more interactions at high densities does not eradicate the problem with the internal pressure of the compact star. Their findings point to the secondary component of the GW190814 event as being most likely a BH. In a later study, Tews et al. (2021) employed the NMMA framework to ascertain whether GW190814 was the result of an NS–BH or BH–BH merger. Their starting point was to use a set of 5000 EOSs that extended beyond 1.5 times the nuclear saturation density. Their conclusion was that GW190814 arose from a binary BH merger with a probability of >99.9%. In an attempt to provide a theoretical basis for the existence of a 2.6 M_⊙ NS, Godzieba et al. (2021) employed a Markov Chain Monte Carlo approach to generate about two million phenomenological EOSs with and without first-order phase transitions for the interior matter distribution. The only imposition in their study was the requirements of GR and causality. They observed that if the secondary component of GW190814 was indeed a nonrotating NS, then this constrains the EOS to densities of the order of 3ρ_nuc. It is still possible to have an NS with a mass greater than 2.5 M_⊙ and an R_1.4 of 11.75 km. In a more recent study, Mohanty et al. (2023) employed a total of 60 EOSs to investigate the impact of anisotropy in NSs. Using a numerical approach, they showed that it is possible to generate NS masses greater than 2.0 M_⊙ within the GR framework by varying the degree of anisotropy within the stellar core.

Researchers have also adopted alternative explorations within modified theories of gravity to account for the peculiar masses observed in gravitational events. Using a set of astronomical data and associated constraints on mass–radius relations, Tangphati et al. (2022) modeled the secondary component of the GW190814 event as a quark star (QS) obeying a color-flavored locked-in (CFL) EOS within the framework of f(R, T) gravity. They showed that the curvature coupling constant arising in f(R, T) theory impacts the maximum allowable masses and radii for stable compact objects to exist. By varying the bag constant and the color superconducting gap energy, masses up to 2.86 M_⊙ with a radius R = 12.43 km were possible. Using Einstein–Gauss–Bonnet (EGB) gravity, Tangphati et al. modeled anisotropic QSs satisfying a CFL EOS (Tangphati et al. 2021). They obtained masses up to 2.83 M_⊙, which exceeded the observed masses of pulsars. Such unconventional masses were possible through the variation of the EGB coupling constant. Further work in R² gravity, coupled with a CFL EOS, has led to predicted masses of the order of 1.97 M_⊙, which falls in the observed range of the static NS PSR J1614–2230. In their attempt to model the mass of the secondary component inferred from the GW190814 event, Astashenok et al. imposed a stiff EOS on the stellar fluid making up the compact object within the f(R) framework (Astashenok et al. 2021). They concluded that the secondary component could be an NS, BH, or rapidly rotating NS and ruled out the possibility of it being a strange star. In a separate work, Astashenok et al. claimed that it was possible to have supermassive NSs with masses in the region of 3 M_⊙ provided that their spin was nonzero (Astashenok et al. 2020). These stars were modeled within the context of f(R) gravity, where f(R) = R + α R² and the quadratic contribution arises in the strong gravitation regime.

In recent work on the modeling of compact objects the so-called $f({ \mathcal Q })$ gravity was employed with very interesting results, especially with regard to the upper bound on the mass limit of these bounded configurations. The study of anisotropic compact objects in which nonmetricity ${ \mathcal Q }$ drives gravitational interactions in the presence of a quintessence field was carried out by Mandal et al. (2022). By choosing an exponential form for $f({ \mathcal Q })$, they demonstrated that their models described physically realizable compact stellar objects such as Her X-1, SAX J1808.4–3658, and 4U 1820–30. In more recent work, Maurya et al. assumed that the interior matter distribution of a compact object obeyed an MIT bag EOS with the stellar fluid being composed of an anisotropic fluid originating from the superposition of two solutions via the complete geometric deformation (CGD; Maurya et al. 2022a). They showed that contributions from the nonmetricity and the decoupling parameter tend to stabilize mass configurations beyond 2.5 M_⊙.

On the observational front, LIGO, together with the advanced VIRGO observations of gravitational-wave events, has given us a glimpse into the possible nature of the sources of these signals. With the Laser Interferometer Space Antenna (LISA) and the Einstein Telescope (ET), researchers hope to probe deeper for new physics during the merger of binary NSs. The increased resolution of these probes will enable researchers to gain a better understanding of the physics at play within the core of these massive stars. While the first observations from LISA are expected to be around 2030 and those from ET a little later, researchers are currently generating simulated catalogs of standard siren events for LIGO–Virgo, LISA, and ET. Utilizing these mock catalogs and a simple parameterization for the nonmetricity function, $f({ \mathcal Q })$, which replicates a ΛCDM cosmological background, Ferreira et al. (2022) were able to calculate redshifts and Ω_m and compare their results to Type Ia supernova data. The idea is to find signatures or telltale signs of nonmetricity in observations of gravitational events.

Motivated by the work of Godzieba et al. (2021) and Tews et al. (2021), among others, in which they utilized a wide spectrum of EOSs to simulate the matter configurations for the secondary components of GW190814 and GW170817, we employ a quadratic EOS within the $f({ \mathcal Q })$ gravity and CGD framework to model compact objects. We pay particular attention to the contributions from the quadratic term on observed mass–radius limits of NSs and pulsars, which help constrain the free parameters in our model.

The paper is organized as follows: In Section 2, model equations for $f({ \mathcal Q })$ gravity with an extra source have been provided. In Section 3, the extended gravitationally decoupled solution in $f({ \mathcal Q })$ gravity is studied, along with mimicking of the density constraint (i.e., $\rho ={\theta }_{0}^{0}$ in Section 3.1) and mimicking of the pressure constraint (i.e., ${p}_{r}={\theta }_{1}^{1}$ in Section 3.2). In Section 4, the matching conditions for the astrophysical system in connection to the exterior spacetime have been elaborated. The physical analysis of completely deformed strange star models and astrophysical implications of the problem are give in Section 5, where the regular behavior of strange star models is discussed under the cases (i) for solution $\rho ={\theta }_{0}^{0}$ (in Section 5.1.1) and (ii) for solution ${p}_{r}={\theta }_{1}^{1}$ (in Section 5.1.2), whereas we have provided the constraining upper limit of maximum mass for strange stars via M–R diagrams under the cases (i) a linear EOS with constancy in bag constant, fixed decoupling parameter, and varying EOS parameter (in Section 5.2.1); (ii) a linear EOS with constancy in bag constant, fixed EOS parameter, and varying decoupling constant (in Section 5.2.2), (iii) a quadratic EOS with fixed bag constant, fixed decoupling constant, and varying quadratic EOS parameter (in Section 5.2.3); and (iv) a quadratic EOS with fixed bag constant, fixed quadratic EOS parameter, and constant versus varying decoupling constant for mimicking of radial pressure, ${p}_{r}={\theta }_{1}^{1}$ (in Section 5.2.4). We have discussed the energy exchange between the fluid distributions for ${\hat{T}}_{{ij}}$ and θ_ij under the cases (i) for solution $\rho ={\theta }_{0}^{0}$ (in Section 6.1) and (ii) for solution ${p}_{r}={\theta }_{1}^{1}$ (in Section 6.2). In Section 7 the comparative study of the model's arising in GR, GR+CGD, $f({ \mathcal Q })$, and $f({ \mathcal Q })$+CGD gravity has been presented, and a few concluding remarks are given in Section 8.

Let us now have an exposure to the $f({ \mathcal Q })$ gravity and hence gravitationally decoupled systems in terms of symmetric teleparallel paradigm. The first essential point is that in $f({ \mathcal Q })$ theory the gravitational interaction is triggered by the nonmetric scalar ${ \mathcal Q }$. Therefore, it is always possible that, using a second Lagrangian ${{ \mathcal L }}_{\theta }$ for a different source θ_ij , one may express the modified action of $f({ \mathcal Q })$ gravity for gravitationally decoupled systems in the following way:

$$\begin{eqnarray}\begin{array}{rcl}{ \mathcal S } & = & \displaystyle \int \Space{0ex}{0.25ex}{0ex}(\displaystyle \frac{1}{2}\,f({ \mathcal Q })+{\lambda }_{k}^{\varrho {ij}}{R}_{\varrho {ij}}^{k}+{\lambda }_{k}^{{ij}}{T}_{{ij}}^{k}+{{ \mathcal L }}_{m}\Space{0ex}{0.25ex}{0ex})\sqrt{-g}\,{d}^{4}x\\ & & +\,\alpha \displaystyle \int {{ \mathcal L }}_{\theta }\,\sqrt{-g}\,{d}^{4}x.\end{array}\end{eqnarray} \tag{ 1 }$$

In the above Equation (1), ${{ \mathcal L }}_{m}$ is the matter Lagrangian density, g is a determinant of the metric tensor (i.e., g = ∣g_ij ∣), α represents a decoupling constant, and ${\lambda }_{k}^{\varrho {ij}}$ defines the Lagrange multipliers. However, within the context of the $f({ \mathcal Q })$ gravity, the metric tensor g_ij and the connection ${{\rm{\Gamma }}}_{\,\,{ij}}^{k}$ are considered individually and the necessary connection for the nonmetricity term can be expressed as follows:

$$\begin{eqnarray}&&{Q}_{{kij}}={{\rm{\nabla }}}_{{}_{k}}\,{g}_{{ij}}={\partial }_{i}\,{g}_{{jk}}-{{\rm{\Gamma }}}_{\,\,{ij}}^{l}\,{g}_{{lk}}-{{\rm{\Gamma }}}_{\,\,{ik}}^{l}\,{g}_{{jl}},\end{eqnarray} \tag{ 2 }$$

where ∇_k defines the covariant derivative and ${{\rm{\Gamma }}}_{\,\,{ij}}^{k}$ refers to an affine connection. The final affine connection configuration is defined as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{{ij}}^{k}={\{}_{\,i\,j}^{\,\,k}\}+{K}_{\,\,{ij}}^{k}+{L}_{\,{ij}}^{k},\end{eqnarray} \tag{ 3 }$$

where ${\{}_{\,i\,j}^{\,\,k}\}$, ${K}_{\,\,{ij}}^{k}$, and ${L}_{\,{ij}}^{k}$ define the Levi–Civita connection, contortion tensor, and disformation tensor, respectively, which are described as

$$\begin{eqnarray}\begin{array}{rcl} & & {\{}_{\,i\,j}^{\,\,k}\}=\displaystyle \frac{1}{2}{g}^{{kl}}({\partial }_{i}{g}_{{lj}}+{\partial }_{j}{g}_{{li}}-{\partial }_{l}{g}_{{ij}}),\\ & & {L}_{\,\,{ij}}^{k}=\displaystyle \frac{1}{2}{Q}_{\,\,{ij}}^{k}-{Q}_{(i\,\,\,j)}^{\,\,\,k},\,{K}_{\,\,{ij}}^{k}=\displaystyle \frac{1}{2}{T}_{\,\,{ij}}^{k}+{T}_{(i\,\,\,j)}^{\,\,\,k},\end{array}\end{eqnarray} \tag{ 4 }$$

where ${T}_{\,i\,j}^{k}$ defines the antisymmetric part of the affine connection, i.e., ${T}_{\,\,i\,j}^{k}=2{{\rm{\Gamma }}}_{\,\,[i\,j]}^{l}$.

In terms of the nonmetricity tensor, the superpotential is defined as

$$\begin{eqnarray}&&{P}_{\,\,{ij}}^{k}=\displaystyle \frac{1}{4}\left[-{Q}_{\,\,{ij}}^{k}+2{Q}_{(i\,j)}^{k}+{Q}^{k}{g}_{{ij}}-{\tilde{Q}}^{k}{g}_{{ij}}-{\delta }_{(i}^{k}{Q}_{j)}\right].\,\end{eqnarray} \tag{ 5 }$$

In particular, there are only two unique traces of the nonmetricity tensor Q_kij because of the symmetry of the metric tensor g_ij , which can be written as

$$\begin{eqnarray}&&{Q}_{k}\equiv {Q}_{k\,\,i}^{\,\,i},\,\,\,\,\ {\tilde{Q}}^{k}\equiv {Q}_{i}^{\,\,{ki}}.\end{eqnarray} \tag{ 6 }$$

Let us define the nonmetricity scalar in the background of connection, as it will be useful in the current analysis and the calculated form of the nonmetricity scalar can be given as

$$\begin{eqnarray}&&{ \mathcal Q }=-\displaystyle \frac{1}{4}{Q}_{{ijk}}{Q}^{{ijk}}+\displaystyle \frac{1}{2}{Q}_{{ijk}}{Q}^{{kji}}+\displaystyle \frac{1}{4}{Q}_{i}{Q}^{i}-\displaystyle \frac{1}{2}{Q}_{i}{\tilde{Q}}^{i}.\end{eqnarray} \tag{ 7 }$$

Furthermore, we define T_ij and θ_ij for the current analysis as

$$\begin{eqnarray}{\hat{T}}_{{ij}}=\displaystyle \frac{2}{\sqrt{-g}}\displaystyle \frac{\delta (\sqrt{-g}\,{{ \mathcal L }}_{m})}{\delta {g}^{{ij}}}\,\,\,\&\,{\theta }_{{ij}}=\displaystyle \frac{2}{\sqrt{-g}}\displaystyle \frac{\delta (\sqrt{-g}\,{{ \mathcal L }}_{\theta })}{\delta {g}^{{ij}}}.\,\end{eqnarray} \tag{ 8 }$$

The modified Einstein–Hilbert action within the purview of symmetric teleparallel gravity (i.e., Equation (1)) can produce the following gravitational field equations by variation of action with respect to the metric tensor g^ij :

$$\begin{eqnarray}&&\begin{array}{l}\displaystyle \frac{2}{\sqrt{-g}}{{\rm{\nabla }}}_{k}(\sqrt{-g}\,{f}_{{ \mathcal Q }}\,{P}_{\,\,{ij}}^{k})+\displaystyle \frac{1}{2}{g}_{{ij}}f+{f}_{{ \mathcal Q }}({P}_{i\,{kl}}\,{Q}_{j}^{\,\,{kl}}\\ -2\,{Q}_{{kli}}\,{P}_{\,\,\,j}^{{kl}})={T}_{{ij}},\,\mathrm{where}\,{T}_{{ij}}=({\hat{T}}_{{ij}}+\alpha \,{\theta }_{{ij}}),\,\,\end{array}\end{eqnarray} \tag{ 9 }$$

where ${f}_{{ \mathcal Q }}=\tfrac{\partial f}{\partial { \mathcal Q }}$, T_ij is the stress–energy–momentum tensor, and θ_ij represents the involvement of an extra source term.

On the other hand, by varying the action with respect to the connection, one may obtain the following relation:

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{n}\,{\lambda }_{k}^{{jin}}+{\lambda }_{k}^{{ij}}=\sqrt{-g}\,{f}_{{ \mathcal Q }}{P}_{{ij}}^{k}+{H}_{k}{}^{{ij}},\end{eqnarray} \tag{ 10 }$$

where H_k ^ij represents the density related to the hypermomentum tensor and can be defined as

$$\begin{eqnarray}&&{H}_{k}{}^{{ij}}=-\displaystyle \frac{1}{2}\displaystyle \frac{\delta {L}_{m}}{\delta {{\rm{\Gamma }}}^{k}{}_{{ij}}}.\end{eqnarray} \tag{ 11 }$$

Now, by using the antisymmetric property of i and j in the Lagrangian multiplier coefficients, Equation (10) gives the following relation:

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{i}{{\rm{\nabla }}}_{j}(\sqrt{-g}\,{f}_{{ \mathcal Q }}\,{P}_{\,\,{ij}}^{k}+{H}_{\,\,{ij}}^{k})=0.\,\end{eqnarray} \tag{ 12 }$$

Additionally, we can retrieve the constraint over the connection, ${ \triangledown }_{i}{ \triangledown }_{j}({H}_{\,\,{ij}}^{k})=0$, according to Equation (12) as follows:

$$\begin{eqnarray}&&{{\rm{\nabla }}}_{i}{{\rm{\nabla }}}_{j}(\sqrt{-g}\,{f}_{{ \mathcal Q }}\,{P}_{\,\,{ij}}^{k})=0.\end{eqnarray} \tag{ 13 }$$

Due to a lack of curvature and torsion, it can be more explicitly parameterized by a number of functions, and finally, the affine connection gets the following form:

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\,\,{ij}}^{k}=\left(\displaystyle \frac{\partial {x}^{k}}{\partial {\xi }^{l}}\right){\partial }_{i}{\partial }_{j}{\xi }^{l}.\end{eqnarray} \tag{ 14 }$$

Under the current scenario, an invertible relation is ξ^k = ξ^k (xⁱ ). As a consequence, there always does exist a possibility of discovering a coordinate system that eliminates the ${{\rm{\Gamma }}}_{\,\,{ij}}^{k}$ connection, i.e., ${{\rm{\Gamma }}}_{\,\,{ij}}^{k}=0$. It is worth noting that the coincident gauge is the covariant derivative ∇_i reduced to the partial one ∂_i . In any other coordinate system, where this affine relationship does not vanish, the metric development would be altered, resulting in a completely new theory (Dimakis et al. 2021). Therefore, we find a coincident gauge coordinate and nonmetricity by Equation (2), which can be simplified as

$$\begin{eqnarray}&&{Q}_{{kij}}={\partial }_{k}\,{g}_{{ij}}.\end{eqnarray} \tag{ 15 }$$

One consequence of the operations stated above is that the metric makes the computation easier, except for the standard GR, where for the action, diffeomorphism invariance no longer exists. In principle, a covariant formulation of $f({ \mathcal Q })$ gravity can be used by determining the affine connection in the absence of gravity before choosing the affine connection in Equation (14). In this paper, we look into the gravitationally decoupled solutions for compact objects under $f({ \mathcal Q })$ gravity.

For the current analysis, we consider the generic spherically symmetric metric, which is given as

$$\begin{eqnarray}&&{{ds}}^{2}={e}^{{\rm{\Phi }}(r)}{{dt}}^{2}-{e}^{\mu (r)}{{dr}}^{2}-{r}^{2}(d{\theta }^{2}+{\sin }^{2}\theta \,d{\phi }^{2}).\end{eqnarray} \tag{ 16 }$$

In the present study, we consider that the spacetime is filled by the anisotropic matter distribution, and then the total energy–momentum tensor (T_ij ) can be described as

$$\begin{eqnarray}&&{T}_{{ij}}=\epsilon \,{u}^{i}\,{u}_{j}-{ \mathcal P }\,{K}_{j}^{i}+{{\rm{\Pi }}}_{j}^{i},\end{eqnarray} \tag{ 17 }$$

where

$$\begin{eqnarray}\begin{array}{rcl} & & { \mathcal P }=\displaystyle \frac{{P}_{r}+2{P}_{\perp }}{3};\,{{\rm{\Pi }}}_{j}^{i}={\rm{\Pi }}\left({\xi }^{i}{\xi }_{j}+\displaystyle \frac{1}{3}{K}_{j}^{i}\right);\\ & & \mathrm{with}\,{\rm{\Pi }}={P}_{r}-{P}_{\perp };\,{K}_{j}^{i}={\delta }_{j}^{i}-{u}^{i}{u}_{j},\end{array}\end{eqnarray} \tag{ 18 }$$

and the fluid's four-velocity vector uⁱ and unit-space-like vector ξⁱ are given by {i = 0, 1, 2, 3},

$$\begin{eqnarray}&&{u}^{i}=({e}^{-{\rm{\Phi }}/2},0,0,0)\,\mathrm{and}\,{\xi }^{i}=(0,{e}^{-\mu /2},0,0),\,\end{eqnarray} \tag{ 19 }$$

such that ξⁱ u_i = 0 and ξⁱ ξ_i = −1. Moreover, denotes the total energy density, while P_r and P_⊥ represent the total radial and tangential pressure, respectively, for the gravitationally decoupled system. In this regard, the components of the energy–momentum tensor for the gravitationally decoupled system under the spherically symmetric line element (16) are

$$\begin{eqnarray}&&{T}_{0}^{0}=\epsilon ,\,{T}_{1}^{1}=-{P}_{r},\,\,{T}_{2}^{2}={T}_{3}^{3}=-{P}_{\perp },\end{eqnarray} \tag{ 20 }$$

For the metric Equation (16), we can calculate the nonmetricity scalar as follows:

$$\begin{eqnarray}&&{ \mathcal Q }=-\displaystyle \frac{2{e}^{-\mu (r)}[1+r\,{{\rm{\Phi }}}^{{\prime} }(r)]}{{r}^{2}},\end{eqnarray} \tag{ 21 }$$

where the prime denotes the derivative over the radial coordinate r only.

In the above expression, ${ \mathcal Q }$ is based on the zero affine connections, which can be absorbed via the equations of motion (9) for the anisotropic fluid (18) as follows:

$$\begin{eqnarray}&&\epsilon =-\displaystyle \frac{f({ \mathcal Q })}{2}+{f}_{{ \mathcal Q }}({ \mathcal Q })\left[{ \mathcal Q }+\displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{{e}^{-\mu }}{r}({{\rm{\Phi }}}^{{\prime} }+{\mu }^{{\prime} })\right],\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{P}_{r}=\displaystyle \frac{f({ \mathcal Q })}{2}-{f}_{{ \mathcal Q }}({ \mathcal Q })\left[{ \mathcal Q }+\displaystyle \frac{1}{{r}^{2}}\right],\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\perp } & = & \displaystyle \frac{f({ \mathcal Q })}{2}-{f}_{{ \mathcal Q }}({ \mathcal Q })\left[\displaystyle \frac{{ \mathcal Q }}{2}-{e}^{-\mu }\left\{\displaystyle \frac{{\rm{\Phi }}^{\prime\prime} }{2}+\left(\displaystyle \frac{{{\rm{\Phi }}}^{{\prime} }}{4}+\displaystyle \frac{1}{2r}\right)\right.\right.\\ & & \left.\left.\times ({{\rm{\Phi }}}^{{\prime} }-{\mu }^{{\prime} })\right\}\right],\end{array}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&0=\displaystyle \frac{\cot \,\theta }{2}\,{{ \mathcal Q }}^{{\prime} }\,{f}_{{ \mathcal Q }{ \mathcal Q }}({ \mathcal Q }),\end{eqnarray} \tag{ 25 }$$

where ${f}_{{ \mathcal Q }}({ \mathcal Q })=\tfrac{\partial f({ \mathcal Q })}{\partial { \mathcal Q }}$ and ${f}_{{ \mathcal Q }{ \mathcal Q }}(Q)=\tfrac{{\partial }^{2}f({ \mathcal Q })}{\partial {{ \mathcal Q }}^{2}}$.

It should be noted that the nonzero off-diagonal metric components derived from the specific gauge choice for the field equations in the context of f(T) theory put some constraints on the functional form of f(T) (Ferraro & Fiorini 2011). As a consequence, this imposes restrictions on the functional form of $f({ \mathcal Q })$ theory. In this connection, Wang et al. (2022) derived the possible functional forms for $f({ \mathcal Q })$ gravity in the framework of the static and spherically symmetric spacetime by taking an anisotropic matter distribution. More specifically, they have shown that the exact Schwarzschild solution can exist only when ${f}_{{ \mathcal Q }{ \mathcal Q }}({ \mathcal Q })=0$, while the solution obtained by taking nonmetricity scalar ${ \mathcal Q }^{\prime} =0$ or ${ \mathcal Q }={{ \mathcal Q }}_{0}$, where Q₀ is constant, shows the deviation from the exact Schwarzschild solution (detailed analysis of the above derivation is given in Section 4). Therefore, in order to solve the system of field equations in $f({ \mathcal Q })$ gravity theory for obtaining self-gravitating compact objects, we derive the functional form of $f({ \mathcal Q })$ by taking only ${f}_{{ \mathcal Q }{ \mathcal Q }}$ to be zero as

$$\begin{eqnarray}&&{f}_{{ \mathcal Q }{ \mathcal Q }}({ \mathcal Q })=0\,\Rightarrow \,{f}_{{ \mathcal Q }}({ \mathcal Q })={\beta }_{1}\,\Rightarrow \,f({ \mathcal Q })={\beta }_{1}\,{ \mathcal Q }+{\beta }_{2},\,\,\end{eqnarray} \tag{ 26 }$$

where β₁ and β₂ are constants.

At this point, we would like to mention that for the compatibility of static spherically symmetric spacetime with the coincident gauge, if one assumes the affine connection to be zero and $f({ \mathcal Q })$ gravity theory has vacuum solutions (i.e., T_ij = 0), then the off-diagonal component can be given by

$$\begin{eqnarray}&&\displaystyle \frac{\cot \,\theta }{2}\,{{ \mathcal Q }}^{{\prime} }\,{f}_{{ \mathcal Q }{ \mathcal Q }}=0,\end{eqnarray} \tag{ 27 }$$

where ${ \mathcal Q }$ has been provided by Equation (21).

By virtue of Equation (27), we require ${f}_{{ \mathcal Q }{ \mathcal Q }}=0$, which implies that $f({ \mathcal Q })$ is a linear function (Zhao 2022). This aspect is very essential, as the nonlinear function of ${ \mathcal Q }$ would give rise to inconsistent equations of motion. This implies that we need a more generalized form of the spherically symmetric metric related to a fixed coincident gauge (Zhao 2022). Therefore, in the present study, to make the spherically symmetric coordinate system compatible with the affine connection ${{\rm{\Gamma }}}_{{ij}}^{k}=0$, we have opted for the linear functional form by considering ${f}_{{ \mathcal Q }{ \mathcal Q }}=0$ to derive the equations of motion.

Now, inserting Equations (21) and (26) into Equations (22)–(24), the equations of motion can be obtained as follows:

$$\begin{eqnarray}&&\epsilon =\displaystyle \frac{1}{2\,{r}^{2}}[2\,{\beta }_{1}+2\,{e}^{-\mu }\,{\beta }_{1}(r\,{\mu }^{{\prime} }-1)-{r}^{2}\,{\beta }_{2}],\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}&&{P}_{r}=\displaystyle \frac{1}{2\,{r}^{2}}[-2\,{\beta }_{1}+2\,{e}^{-\mu }\,{\beta }_{1}(r\,{{\rm{\Phi }}}^{{\prime} }+1)+{r}^{2}\,{\beta }_{2}],\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}\begin{array}{rcl}{P}_{\perp } & = & \frac{{e}^{-\mu }}{4\,r}[2\,{e}^{\mu }\,r\,{\beta }_{2}+{\beta }_{1}(2+r{{\rm{\Phi }}}^{{\prime} }\,)({{\rm{\Phi }}}^{{\prime} }-{\mu }^{{\prime} })\\ & & +2\,r\,{\beta }_{1}\,{\rm{\Phi }}^{\prime\prime} ].\,\,\end{array}\end{eqnarray} \tag{ 30 }$$

The vanishing of the covariant derivative of the effective energy–momentum tensor is ${ \triangledown }_{i}{T}_{j}^{i}=0$, which yields

$$\begin{eqnarray}&&-\displaystyle \frac{{H}^{{\prime} }}{2}(\epsilon +{P}_{r})-\left({P}_{r}\right)^{\prime} +\displaystyle \frac{2}{r}({P}_{\perp }-{P}_{r})=0.\,\end{eqnarray} \tag{ 31 }$$

It should be noted that the above Equation (31) is nothing but the usual Tolman–Oppenheimer–Volkoff (TOV) equation (Oppenheimer & Volkoff 1939; Tolman 1939), with $f({ \mathcal Q })={\beta }_{1}{ \mathcal Q }+{\beta }_{2}$. Therefore, in connection with the proposed compact stellar model, we would like to employ gravitational decoupling under the CGD approach to get a solution to the system of Equations (28)–(30). For this specific purpose, the gravitational potentials Φ(r) and μ(r) are essential to modify as follows:

$$\begin{eqnarray}&&{\rm{\Phi }}(r)\longrightarrow H(r)+\alpha \,\eta (r)\,\end{eqnarray} \tag{ 32 }$$

$$\begin{eqnarray}&&{e}^{-\mu (r)}\longrightarrow W(r)+\alpha \,{\rm{\Psi }}(r),\,\,\end{eqnarray} \tag{ 33 }$$

where Φ(r) and μ(r) are two arbitrary deformation functions via the decoupling constant β, whereas in the resultant parts η(r) and Ψ(r) are the geometric deformation functions for the temporal and radial metric components, respectively.

In the above Equations (32) and (33), for α = 0, the standard $f({ \mathcal Q })$ gravity theory can be easily recovered. Essentially, to continue in the present work, we must consider the nonzero value for both the deformation functions, i.e., η(r) ≠ 0 and Ψ(r) ≠ 0. The above transformations (32) and (33) can easily divide the decoupled system, namely Equations (28)–(30), into two subsystems: (i) the first system reflects the field equation in $f({ \mathcal Q })$ gravity under T_{i j} , and (ii) the second system represents the additional source θ_ij . To include all these systems, we need to specify the energy–momentum tensor T_{i j} in the following form:

$$\begin{eqnarray}&&{\hat{T}}_{j}^{i}=\rho \,{\chi }^{i}\,{\chi }_{j}-{{ \mathcal P }}_{{ \mathcal Q }}\,{h}_{j}^{i}+{\hat{{\rm{\Pi }}}}_{j}^{i},\end{eqnarray} \tag{ 34 }$$

where

$$\begin{eqnarray}\begin{array}{rcl} & & {{ \mathcal P }}_{{ \mathcal Q }}=\displaystyle \frac{{p}_{r}+2{p}_{\perp }}{3};\,{\hat{{\rm{\Pi }}}}_{j}^{i}={{\rm{\Pi }}}_{{ \mathcal Q }}({\zeta }^{i}{\zeta }_{j}+\displaystyle \frac{1}{3}{h}_{j}^{i});\\ & & \,\mathrm{with}\,\,{{\rm{\Pi }}}_{{ \mathcal Q }}={p}_{r}-{p}_{t};\,{h}_{j}^{i}={\delta }_{j}^{i}-{\chi }^{i}{\chi }_{j},\end{array}\end{eqnarray} \tag{ 35 }$$

and χⁱ (fluid's four-velocity vector) and ζⁱ (unit-space-like vector) are given by

$$\begin{eqnarray}&&{\chi }^{i}=({e}^{-H/2},0,0,0)\,\mathrm{and}\,{\zeta }^{i}=(0,\sqrt{W},0,0),\end{eqnarray} \tag{ 36 }$$

such that ξⁱ ζ_i = 0 and ζⁱ ζ_i = −1. The , P_r , and P_⊥ can be written as

$$\begin{eqnarray}&&\epsilon =\rho +\alpha \,{\theta }_{0}^{0},\,{P}_{r}={p}_{{}_{r}}-\alpha \,{\theta }_{1}^{1},\,{P}_{\perp }={p}_{{}_{t}}-\alpha \,{\theta }_{2}^{2},\,\,\end{eqnarray} \tag{ 37 }$$

and the corresponding total anisotropy,

$$\begin{eqnarray}&&{\rm{\Delta }}={P}_{\perp }-{P}_{r}={{\rm{\Delta }}}_{{ \mathcal Q }}+{{\rm{\Delta }}}_{\theta },\end{eqnarray} \tag{ 38 }$$

where $\,{{\rm{\Delta }}}_{{ \mathcal Q }}={p}_{{}_{t}}-{p}_{{}_{r}}\,\,\mathrm{and}\,\,{{\rm{\Delta }}}_{\theta }=\alpha ({\theta }_{1}^{1}-{\theta }_{2}^{2}).$

One may note that in the present anisotropic compact stellar system there are two types of anisotropies, namely T_{i j} and θ_{i j} . Another anisotropy, Δ_θ , comes into the picture owing to gravitational decoupling, which has a definite role in the transformation processes.

By putting Equations (32) and (33) in the system of Equations (28)–(30), the set of equations of motion dependent on the gravitational potentials (H(r) and W(r), or when α = 0), are produced:

$$\begin{eqnarray}&&\rho =\displaystyle \frac{{\beta }_{1}(1-W)}{{r}^{2}}-\displaystyle \frac{W^{\prime} {\beta }_{1}}{r}-\displaystyle \frac{{\beta }_{2}}{2},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{p}_{{}_{r}}=\displaystyle \frac{{\beta }_{1}(W-1)}{{r}^{2}}+\displaystyle \frac{H^{\prime} W{\beta }_{1}}{r}+\displaystyle \frac{{\beta }_{2}}{2},\end{eqnarray} \tag{ 40 }$$

$$\begin{eqnarray}\begin{array}{rcl}{p}_{{}_{t}} & = & \displaystyle \frac{{\beta }_{1}(W^{\prime} H^{\prime} +2H^{\prime\prime} W+{H}^{{\prime} 2}W)}{4}+\displaystyle \frac{{\beta }_{1}(W^{\prime} +H^{\prime} W)}{2r}\\ & & +\displaystyle \frac{{\beta }_{2}}{2},\,\end{array}\end{eqnarray} \tag{ 41 }$$

and according to the TOV Equation (31),

$$\begin{eqnarray}&&-\displaystyle \frac{{H}^{{\prime} }}{2}(\rho +{p}_{{}_{r}})-\left({p}_{{}_{r}}\right)^{\prime} +\displaystyle \frac{2}{r}({p}_{{}_{t}}-{p}_{{}_{r}})=0.\,\end{eqnarray} \tag{ 42 }$$

Consequently, the spacetime that follows can provide the corresponding solution:

$$\begin{eqnarray}&&{{ds}}^{2}=-{e}^{H(r)}{{dt}}^{2}+\displaystyle \frac{{{dr}}^{2}}{W(r)}+{r}^{2}d{\theta }^{2}+{r}^{2}{\sin }^{2}\theta d{\phi }^{2}.\end{eqnarray} \tag{ 43 }$$

Moreover, the system of field equations for the θ-sector is derived by turning on α as

$$\begin{eqnarray}&&{\theta }_{0}^{0}=-{\beta }_{1}\Space{0ex}{0.25ex}{0ex}(\displaystyle \frac{{\rm{\Psi }}}{{r}^{2}}+\displaystyle \frac{{{\rm{\Psi }}}^{{\prime} }}{r}\Space{0ex}{0.25ex}{0ex}),\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray}&&{\theta }_{1}^{1}=-{\beta }_{1}\Space{0ex}{0.25ex}{0ex}[\displaystyle \frac{{\rm{\Psi }}}{{r}^{2}}+\displaystyle \frac{({\rm{\Phi }}^{\prime} {\rm{\Psi }}+W\,\eta ^{\prime} )}{r}\Space{0ex}{0.25ex}{0ex}],\end{eqnarray} \tag{ 45 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\theta }_{2}^{2} & = & -{\beta }_{1}\left[\displaystyle \frac{({{\rm{\Psi }}}^{{\prime} }{\rm{\Phi }}^{\prime} +2{\rm{\Phi }}^{\prime\prime} {\rm{\Psi }}+{{\rm{\Phi }}}^{{\prime} 2}{\rm{\Psi }}+W^{\prime} \,\eta ^{\prime} )}{4}+\displaystyle \frac{({{\rm{\Psi }}}^{{\prime} }+{\rm{\Phi }}^{\prime} {\rm{\Psi }})}{2r}\right]\\ & & -{\beta }_{1}\left[\displaystyle \frac{W}{4}(2\,{\eta }^{{\prime} ^{\prime} }+\alpha \,{\eta }^{{\prime} \,2}+\displaystyle \frac{2\,\eta ^{\prime} }{r}+2\,H^{\prime} \,\eta ^{\prime} )\right],\end{array}\end{eqnarray} \tag{ 46 }$$

and the associated conservation is

$$\begin{eqnarray}&&-\displaystyle \frac{{\rm{\Phi }}^{\prime} }{2}({\theta }_{0}^{0}-{\theta }_{1}^{1})+{\left({\theta }_{1}^{1}\right)}^{{\prime} }+\displaystyle \frac{2}{r}({\theta }_{1}^{1}-{\theta }_{2}^{2})=\displaystyle \frac{\eta ^{\prime} }{2}({p}_{r}+\rho ).\end{eqnarray} \tag{ 47 }$$

However, the mass function for both systems is given by

$$\begin{eqnarray}&&{m}_{{ \mathcal Q }}=\displaystyle \frac{1}{2}{\int }_{0}^{r}\rho (x){x}^{2}{dx}\,\mathrm{and}\,{m}_{\theta }=\displaystyle \frac{1}{2}\,{\int }_{0}^{r}{\theta }_{0}^{0}(x){x}^{2}{dx},\,\end{eqnarray} \tag{ 48 }$$

where the relevant mass functions for the sources T_ij and θ_ij are ${m}_{{ \mathcal Q }}(r)$ and m_θ (r), respectively. Then, in the context of $f({ \mathcal Q })$ gravity, the interior mass function of the minimally deformed spacetime (16) may be expressed as

$$\begin{eqnarray}&&\tilde{m}(r)={m}_{{ \mathcal Q }}(r)-\displaystyle \frac{{\beta }_{1}\,\alpha }{2}\,r\,{\rm{\Psi }}(r).\end{eqnarray} \tag{ 49 }$$

Therefore, from all the previous steps toward the generation of the mass functions, the need for, as well as advantage of, CGD decoupling becomes very straightforward; as such, it can be pointed out that one can extend any known solutions associated with the action ${{ \mathcal S }}_{{ \mathcal Q }}$ with solution {T_ij , W, H} of the system (e.g., Equations (39)–(41)) into the domain beyond $f({ \mathcal Q })$ gravity theory associated with the action ${ \mathcal S }$. It should be noted that in the process, equations of motion are displayed in Equations (28)–(30) and the unconventional gravitational system of equations is given in Equations (44)–(48) to determine {θ_ij , Ψ, η}.

Let us now generate the θ version of any {T_ij , W, H} solution as

$$\begin{eqnarray}&&\{{T}_{{ij}},H(r),W(r)\}\Longrightarrow \{{T}_{{ij}}^{\mathrm{tot}},{\rm{\Phi }}(r),\mu (r)\},\end{eqnarray} \tag{ 50 }$$

which describes a definite way to investigate the results that is beyond the symmetric teleparallel gravity.

In this section, we will solve both systems of Equations (36)–(39) and (41)–(44) related to the sources T_{μ ν} and θ_{μ ν} . It is mentioned that the energy–momentum tensor T_ij describes an anisotropic fluid matter distribution; therefore, θ_ij may enhance the total anisotropy of the system, which helps in preventing the gravitational collapse of the system. Furthermore, if we look at the second system, it shows clearly that the solution of the second system depends on the first system. Then, it is mandatory to solve the first system initially. For solving the first system of Equations (36)–(39) in $f({ \mathcal Q })$ gravity, we use a generalized polytropic EOS of the form

$$\begin{eqnarray}&&{p}_{r}=a\,{\rho }^{1+1/n}+b\rho +c,\end{eqnarray} \tag{ 51 }$$

where a, b, and c are constant parameters with proper dimensions and n denotes a polytropic index.

Let us trace back the generic history of the polytropic EOS ${p}_{r}=a{\rho }^{1+\tfrac{1}{n}}$, which has been extensively used to analyze the physical attributes of the compact stellar objects in the context of various requirements (Bekenstein 1960; Cosenza et al. 1981; Abramowicz 1983; Herrera & Barreto 2004; Herrera et al. 2004; Herrera & Barreto 2013; Takisa & Maharaj 2013; Herrera et al. 2014, 2016; Azam & Mardan 2017b). In connection with the cosmological scenario, Chavanis (2014a) first tried to generalize the polytropic EOS in the form ${p}_{r}=\gamma {\rho }^{1+\tfrac{1}{n}}+\beta \rho$. In the context of the late universe Chavanis (2014b) further studied this EOS by considering negative indices, which were applied to the study of quantum fluctuations and constant-density cosmology (Freitas & Goncalves 2014). However, this EOS is mostly applicable for a stellar system with vanishing pressure when the density goes to zero. Therefore, for the self-bound compact objects, a further modified EOS ${p}_{r}=\gamma {\rho }^{1+\tfrac{1}{n}}+\beta \rho +\chi$ was extensively considered by several investigators (Azam et al. 2015, 2016; Azam & Mardan 2017a; Naeem et al. 2021).

It is interesting to note that the polytropic EOS (51) may represent an MIT bag EOS for the specifications $a\,=0,\,b=\tfrac{1}{4}$, and $c=-\tfrac{4}{3}{{ \mathcal B }}_{g}$, where ${{ \mathcal B }}_{g}$ is a bag constant. Due to the high nonlinearity of the field equations, as a simple case we assume the value for the polytropic index to be n = 1. In this context one may note that the contribution for the quadratic term (i.e., a ρ²) appears in the EOS to express the neutron liquid in Bose–Einstein condensate form, whereas the linear terms (i.e., b ρ + c) come from the free quarks model of the MIT bag model, with b = 1/3 and $c=-4{{ \mathcal B }}_{g}/3$.

In addition to the above studies, there are various works employing more realistic EOSs. Analytical representations of more realistic EOSs based on parameters arising in nuclear physics, such as the FPS EOS (Pandharipande & Ravenhall 1989), SLy EOS (Douchin & Haensel 2001), and the unified EOSs BSK19, BSk20, and BSk21 (Potekhin et al. 2013), among others, have been previously employed using numerical techniques to generate NS models. The quadratic EOS can be viewed as a truncation of these more realistic EOSs. The models derived using the quadratic EOS can be used as first approximation to test more complicated numerical codes. Furthermore, Haensel and Potekhin (Haensel & Potekhin 2004) pointed out that unified EOSs (presented in the form of tables) introduce ambiguities in the calculations of various parameters in NS modeling. These pathologies arise in the interpolation between the tabulated points, as well as the calculation of derivatives regarding thermodynamical quantities. They further highlight the point that analytical EOSs ensure that shortcomings are circumnavigated and allow for high-precision NS modeling.

Now, by using Equations (39) and (40) in Equation (51), one can find the differential equation

$$\begin{eqnarray}\begin{array}{rcl} & & 4{\beta }_{1}{r}^{2}(a{\beta }_{2}-a{\beta }_{1}{W}^{{\prime} 2}-a{\beta }_{2}W+{bW}-b+W-1)+4{\beta }_{1}{r}^{3}\\ & & \times (-a{\beta }_{2}W^{\prime} +{bW}^{\prime} +H^{\prime} W)+{\beta }_{2}{r}^{4}(-a{\beta }_{2}+2b+2)-8a{\beta }_{1}^{2}\\ & & \times W^{\prime} (W-1)r-4a{\beta }_{1}^{2}{\left(W-1\right)}^{2}-4{{cr}}^{4}=0.\end{array}\end{eqnarray} \tag{ 52 }$$

Then, Equation (52) leads directly to

$$\begin{eqnarray}&&H(r)=\int G(W,W^{\prime} ,\bar{r})d\bar{r},\end{eqnarray} \tag{ 53 }$$

where

$$\begin{eqnarray*}\begin{array}{rcl}G(W,W^{\prime} ,r) & = & \displaystyle \frac{1}{4{\beta }_{1}{r}^{3}W}[-4{\beta }_{1}{r}^{2}(a{\beta }_{2}-a{\beta }_{1}{W}^{{\prime} 2}-a{\beta }_{2}W\\ & & +{bW}-b+W-1)-4{\beta }_{1}{r}^{3}(-a{\beta }_{2}W^{\prime} +{bW}^{\prime} )\\ & & -{\beta }_{2}{r}^{4}(-a{\beta }_{2}+2b+2)+8a{\beta }_{1}^{2}W^{\prime} (W-1)r\\ & & -4a{\beta }_{1}^{2}{\left(W-1\right)}^{2}+4{{cr}}^{4}].\end{array}\end{eqnarray*}$$

The above differential equation (53) depends on one undermined unknown W. Therefore, we assume a well-known ansatz for W for the Buchdahl model to solve the above differential equation,

$$\begin{eqnarray}&&W(r)=\displaystyle \frac{1+{{Lr}}^{2}}{1+{{Nr}}^{2}},\end{eqnarray} \tag{ 54 }$$

where L and N are constants with dimensions length⁻² and length⁻⁴, respectively. Now, plugging Equation (54) into Equation (53) and integrating, we obtain the solution for H(r) of the form

$$\begin{eqnarray}\begin{array}{rcl} & & H(r)=\displaystyle \frac{1}{8}[\displaystyle \frac{1}{{\beta }_{1}{L}^{2}(L-N)}(\mathrm{log}({{Lr}}^{2}+1)\{4{\beta }_{1}{L}^{3}(3a{\beta }_{2}-6a{\beta }_{1}\\ & & \times N-3b-1)+{L}^{2}[{\beta }_{2}(a{\beta }_{2}-2b-2)+8{\beta }_{1}N(1-2a{\beta }_{2}+2b)\\ & & +4a{\beta }_{1}^{2}{N}^{2}]-2{LN}({\beta }_{2}(a{\beta }_{2}-2b-2)+2{\beta }_{1}N(-a{\beta }_{2}+b+1))\\ & & +{\beta }_{2}{N}^{2}(a{\beta }_{2}-2b-2)+36a{\beta }_{1}^{2}{L}^{4}+4c{\left(L-N\right)}^{2}\})+\displaystyle \frac{1}{{\beta }_{1}L}{\beta }_{1}\\ & & \times [{{Nr}}^{2}({\beta }_{2}(a{\beta }_{2}-2b-2)+4c)]+\displaystyle \frac{{\beta }_{1}}{N-L}[4\mathrm{log}({{Nr}}^{2}+1){\beta }_{1}\\ & & \times \{a(9{\beta }_{1}{L}^{2}+2{\beta }_{2}L-6{\beta }_{1}{LN}+{\beta }_{1}{N}^{2}-2{\beta }_{2}N)\\ & & -2b(L-N)\}]+\displaystyle \frac{16a{\beta }_{1}(2L-N)}{{{Nr}}^{2}+1}+\displaystyle \frac{8a{\beta }_{1}(L-N)}{{\left({{Nr}}^{2}+1\right)}^{2}}]+F,\end{array}\end{eqnarray} \tag{ 55 }$$

where F is an arbitrary constant of integration. Using the expression for W(r) and H(r), we find the expressions for ρ, p_r , and p_t as

$$\begin{eqnarray}\begin{array}{rcl} & & \rho (r)=\displaystyle \frac{1}{2{\left({{Nr}}^{2}+1\right)}^{2}}[{N}^{2}(2{\beta }_{1}{r}^{2}-{\beta }_{2}{r}^{4})-{\beta }_{2}-2{\beta }_{1}L\\ & & \times ({{Nr}}^{2}+3)+N(6{\beta }_{1}-2{\beta }_{2}{r}^{2})],\end{array}\end{eqnarray} \tag{ 56 }$$

$$\begin{eqnarray}\begin{array}{rcl} & & {p}_{r}(r)=\displaystyle \frac{1}{4{\left({{Nr}}^{2}+1\right)}^{4}}[({\beta }_{2}+2{\beta }_{1}L({{Nr}}^{2}+3)+{N}^{2}\\ & & \times ({\beta }_{2}{r}^{4}-2{\beta }_{1}{r}^{2})+N(2{\beta }_{2}{r}^{2}-6{\beta }_{1}))\{a({\beta }_{2}+2{\beta }_{1}L\\ & & \times ({{Nr}}^{2}+3)+{\beta }_{2}{N}^{2}{r}^{4}-2{\beta }_{1}{N}^{2}{r}^{2}-6{\beta }_{1}N\\ & & +2{\beta }_{2}{{Nr}}^{2})-2b{\left({{Nr}}^{2}+1\right)}^{2}\}]+c,\end{array}\end{eqnarray} \tag{ 57 }$$

$$\begin{eqnarray}\begin{array}{rcl} & & {p}_{{}_{t}}(r)=\displaystyle \frac{1}{64}\left[\displaystyle \frac{{f}_{1}^{2}(r)({{Lr}}^{2}+1){\beta }_{1}{r}^{2}}{{{Nr}}^{2}+1}+\displaystyle \frac{8{f}_{1}(r)({{Lr}}^{2}+1){\beta }_{1}}{{{Nr}}^{2}+1}\right.\\ & & +32{\beta }_{2}+\displaystyle \frac{1}{({{Lr}}^{2}+1){\left({{Nr}}^{2}+1\right)}^{5}}[8(L-N)(4{{cr}}^{2}\\ & & \times {\left({{Nr}}^{2}+1\right)}^{4}+{r}^{2}{\beta }_{2}(-2b+a{\beta }_{2}-2){\left({{Nr}}^{2}+1\right)}^{4}\\ & & +4{\beta }_{1}({N}^{2}(b-a{\beta }_{2}+1){r}^{4}+3N(b-a{\beta }_{2}+1){r}^{2}+L(N\\ & & \times (a{\beta }_{2}+1){r}^{2}-b({{Nr}}^{2}+3)+3a{\beta }_{2}+1){r}^{2}+2)\\ & & \times {\left({{Nr}}^{2}+1\right)}^{2}+4a{\left(L-N\right)}^{2}{r}^{2}{\left({{Nr}}^{2}+3\right)}^{2}{\beta }_{1}^{2})]\\ & & +\displaystyle \frac{1}{{{Nr}}^{2}+1}\left(\displaystyle \frac{8{f}_{2}(r)({{Lr}}^{2}+1){\beta }_{1}}{{\left({{Lr}}^{2}+1\right)}^{2}{\left({{Nr}}^{2}+1\right)}^{4}{\beta }_{1}}\right.\\ & & +\left.\left.\displaystyle \frac{8{f}_{3}(r)({{Lr}}^{2}+1){\beta }_{1}}{{\left({{Nr}}^{2}+1\right)}^{3}(L{\beta }_{1}{r}^{2}+{\beta }_{1})}\right)\right],\end{array}\end{eqnarray} \tag{ 58 }$$

where the coefficients used in the above expressions are mentioned in the Appendix.

In the above set of Equations (56)–(58), we have the complete spacetime geometry for the seed solution. However, corresponding to the θ-sector, we need to find the solution of the second system of Equations (44)–(46). However, note that there are three independent equations with five unknowns. This situation, therefore, demands two additional pieces of information to close the θ system, e.g., Ψ and η(r). It is known that physical viability Φ(r) should have a monotonic increasing feature toward the boundary, and so H(r) + α η(r) must be an increasing function of r. Hence, for simplicity, we assume η(r) = H(r), which provides eventually Φ(r) = (1 + α)H(r).

Related to the system of Equations (44) and (45) for the source θ_ij , after imposing the constraint β₂ ≠ 0, we opt for the following preferences:

$$\begin{eqnarray}&&{\rm{\Psi }}(r)=W(r)-1+\displaystyle \frac{{\beta }_{2}{r}^{2}}{6{\beta }_{1}}\,\end{eqnarray} \tag{ 59 }$$

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Psi }}(r) & = & \displaystyle \frac{1}{1+r\,{\rm{\Phi }}^{\prime} (r)}-\displaystyle \frac{W(r)[1+r\{H^{\prime} (r)+\eta ^{\prime} (r)\}]}{1+r\,{\rm{\Phi }}^{\prime} (r)}\\ & & -\displaystyle \frac{{\beta }_{2}{r}^{2}}{2{\beta }_{1}}.\end{array}\end{eqnarray} \tag{ 60 }$$

One can note that Ψ(r) is free from any singularity and also Ψ(0) = 0. As a result, these allow us to mimic (i) ${\theta }_{0}^{0}$ with the energy density ρ, i.e., $\rho ={\theta }_{0}^{0}$ from Equation (59), and (ii) ${\theta }_{1}^{1}$ with the radial pressure p_r , i.e., ${p}_{r}={\theta }_{1}^{1}$ from Equation (60). Several authors (Sharif & Majid 2020a; Sharif & Saba 2020b; Maurya et al. 2020, 2021; Sharif & Aslam 2021; Maurya et al. 2022b, 2023) have successfully applied this technique in modeling compact objects in GR, as well as modified gravity theories, and its gravitational cracking concept under gravitational decoupling (Contreras & Fuenmayor 2021). Motivated by these works, we use the mimic approach in the present study for the system of Equations (39)–(41). Therefore, we adopt the following approaches: (i) mimicking of the density constraint (i.e., $\rho ={\theta }_{0}^{0}$), and (ii) mimicking of the radial pressure constraint (i.e., ${p}_{r}={\theta }_{1}^{1}$) (for details see Ovalle 2017).

3.1. Mimicking of the Density Constraint (i.e., $\rho ={\theta }_{0}^{0}$)

To solve the θ-sector, here we mimic the seed energy density ρ to ${\theta }_{0}^{0}$ from Equations (39) and (44), and we find the first-order linear differential equation in Ψ(r) to be

$$\begin{eqnarray}&&{\rm{\Psi }}^{\prime} +\displaystyle \frac{{\rm{\Psi }}}{r}+\displaystyle \frac{1}{2{\beta }_{1}r}[2{\beta }_{1}(1-W-r\,W^{\prime} )-{r}^{2}\,{\beta }_{2}]=0.\end{eqnarray} \tag{ 61 }$$

Now we obtain the expression of deformation function Ψ(r) after integrating the above equation by using the known potential H(r) as

$$\begin{eqnarray}&&{\rm{\Psi }}(r)=\displaystyle \frac{{r}^{2}({\beta }_{2}+6{\beta }_{1}L-6{\beta }_{1}N+{\beta }_{2}{{Nr}}^{2})}{6{\beta }_{1}(1+{{Nr}}^{2})}.\end{eqnarray} \tag{ 62 }$$

The arbitrary constant of integration has been taken to be zero to ensure the nonsingular nature of Ψ(r) at the center. Furthermore, we take the deformation function η(r) = H(r) as mentioned above in order to find the expression for the θ-sector. Hence, the θ-sector components are obtained as

$$\begin{eqnarray}\begin{array}{rcl}{\theta }_{0}^{0}(r) & = & \displaystyle \frac{1}{2{\left({{Nr}}^{2}+1\right)}^{2}}[{N}^{2}(2{\beta }_{1}{r}^{2}-{\beta }_{2}{r}^{4})-{\beta }_{2}-2{\beta }_{1}L\\ & & \times ({{Nr}}^{2}+3)+N(6{\beta }_{1}-2{\beta }_{2}{r}^{2})],\end{array}\end{eqnarray} \tag{ 63 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\theta }_{1}^{1}(r)=\displaystyle \frac{-1}{6(1+{{Nr}}^{2})}[{\beta }_{2}+6{\beta }_{1}{\theta }_{11}(r)((\alpha +2){{Lr}}^{2}-(\alpha +1)\\ \,\times {{Nr}}^{2}+1)+{\beta }_{2}(\alpha +1){\theta }_{11}(r){r}^{2}({{Nr}}^{2}+1)+6{\beta }_{1}L\\ \,-6{\beta }_{1}N+{\beta }_{2}{{Nr}}^{2}],\,\end{array}\end{eqnarray} \tag{ 64 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\theta }_{2}^{2}(r)=\displaystyle \frac{-1}{24{\left({{Nr}}^{2}+1\right)}^{2}}[2{\theta }_{22}(r)({{Nr}}^{2}+1)[6{\beta }_{1}(1+(\alpha +2){{Lr}}^{2}\\ \,-(\alpha +1){{Nr}}^{2})+(\alpha +1){\beta }_{2}{r}^{2}({{Nr}}^{2}+1)]+{\theta }_{23}(r)],\end{array}\end{eqnarray} \tag{ 65 }$$

where the explicit expressions for θ₂₂(r) and θ₂₃(r) are given in the Appendix.

3.2. Mimicking of the Pressure Constraint (i.e., ${p}_{r}={\theta }_{1}^{1}$)

In this mimic constraints approach, we are mimicking the seed radial pressure p_r with the ${\theta }_{1}^{1}(r)$ from Equations (40) and (45), and we obtain the expression for deformation function Ψ(r) as

$$\begin{eqnarray}\begin{array}{rcl} & & {\rm{\Psi }}(r)=-\displaystyle \frac{1}{({{Nr}}^{2}+1){{\rm{\Psi }}}_{21}(r)}[2{r}^{2}({{Lr}}^{2}+1)\{a{\beta }_{2}^{2}-2{\beta }_{1}L[-6a{\beta }_{2}\\ & & \,+{N}^{3}({r}^{6}(1-2a{\beta }_{2})+4a{\beta }_{1}{r}^{4})+{N}^{2}({r}^{4}(3-10a{\beta }_{2})+24a\\ & & \,\times {\beta }_{1}{r}^{2})+{{\rm{\Psi }}}_{22}(r)\}],\,\,\end{array}\end{eqnarray} \tag{ 66 }$$

Finally, using the same deformation function η(r) = H(r), we find the θ-components for this solution as

$$\begin{eqnarray}&&{\theta }_{0}^{0}(r)=\displaystyle \frac{2{\beta }_{1}[4{{Nr}}^{2}({{Lr}}^{2}+1)({{Nr}}^{2}+1){{\rm{\Psi }}}_{00}(r)+{f}_{4}(r)]}{3{f}_{3}(r){\left({{Nr}}^{2}+1\right)}^{2}},\,\,\end{eqnarray} \tag{ 67 }$$

$$\begin{eqnarray}&&\begin{array}{l}{\theta }_{1}^{1}(r)=\displaystyle \frac{1}{4{\left({{Nr}}^{2}+1\right)}^{4}}[({\beta }_{2}+2{\beta }_{1}L({{Nr}}^{2}+3)+{N}^{2}({\beta }_{2}{r}^{4}\\ -2{\beta }_{1}{r}^{2})+N(2{\beta }_{2}{r}^{2}-6{\beta }_{1}))\{a({\beta }_{2}+2{\beta }_{1}L({{Nr}}^{2}+3)\\ +{\beta }_{2}{N}^{2}{r}^{4}-2{\beta }_{1}{N}^{2}{r}^{2}-6{\beta }_{1}N+2{\beta }_{2}{{Nr}}^{2})\\ -2b{\left({{Nr}}^{2}+1\right)}^{2}\}]+c,\end{array}\end{eqnarray} \tag{ 68 }$$

where the explicit expressions for Ψ₂₁(r) and Ψ₂₂(r), and Ψ₀₀(r) are given in the Appendix.

At this juncture, let us apply the boundary condition to obtain the expressions for the constants and the physical parameters to study the features of the compact star. For this purpose, we need to match the interior spacetime smoothly with a suitable exterior vacuum solution at the pressureless bounding surface (i.e., at r = R) for the functional form of $f({ \mathcal Q })={\beta }_{1}{ \mathcal Q }+{\beta }_{2}$.

Following the analysis provided by Wang et al. (2022) for the off-diagonal component as it appeared in Equation (20), the solutions of f(Q) gravity are restricted to the following two cases:

$$\begin{eqnarray}&&{f}_{{ \mathcal Q }{ \mathcal Q }}=0\,\,\Rightarrow f({ \mathcal Q })={\beta }_{1}\,{ \mathcal Q }+{\beta }_{2},\end{eqnarray} \tag{ 69 }$$

$$\begin{eqnarray}&&{ \mathcal Q }^{\prime} =0\,\Rightarrow \,{ \mathcal Q }={{ \mathcal Q }}_{0}.\end{eqnarray} \tag{ 70 }$$

where β₁, β₂, and ${{ \mathcal Q }}_{0}$ constants.

One can note that this result is similar to the one in f(T) gravity theory related work in Böhmer et al. (2011), where they have considered the constraint case f_TT = 0 or $T^{\prime} =0$ for the spherically symmetric static distributions with diagonal tetrad as background formalism. One can also note that for the cosmological constant assumed to be β₂/β₁, the first solution Equation (69) is equivalent to GR as it reduces to symmetric teleparallel general relativity.  We shall return to this point without getting into the technical details.

We have the physical = P_r = P_⊥ = 0 in the case of vacuum. Therefore, the equations of motion (21)–(24), due to Equation (69), take the forms

$$\begin{eqnarray}&&{\rm{\Phi }}^{\prime} +\mu ^{\prime} =0,\end{eqnarray} \tag{ 71 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\beta }_{2}}{{\beta }_{1}}-\displaystyle \frac{2}{{r}^{2}}={ \mathcal Q },\end{eqnarray} \tag{ 72 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{\beta }_{2}}{2}+{\beta }_{1}{e}^{-\mu }\left[\displaystyle \frac{{\rm{\Phi }}^{\prime\prime} }{2}+\left(\displaystyle \frac{{\rm{\Phi }}^{\prime} }{4}+\displaystyle \frac{1}{2r}\right)\left({\rm{\Phi }}^{\prime} -\mu ^{\prime} \right)\right]=0.\end{eqnarray} \tag{ 73 }$$

Now, from Equation (71), one can get in a straightforward way the following one:

$$\begin{eqnarray}&&{\rm{\Phi }}(r)=-\mu (r)+{\mu }_{0},\end{eqnarray} \tag{ 74 }$$

where μ₀ is a constant of integration.

It should be noted that, for the sake of convenience, μ₀ can be absorbed via the rescaling of the time coordinate t to ${e}^{-{\mu }_{0}/2}$. As a result, the rr component in Equation (14) becomes the inverse of the tt component, so that one may obtain

$$\begin{eqnarray}&&{\rm{\Phi }}(r)=-\mu (r).\end{eqnarray} \tag{ 75 }$$

In Equation (72), we consider the term β₂/β₁ as the cosmological constant of Einstein Λ with a reversed sign due to the convention of nonmetricity of Equation (7).

Now, from the relation (16), along with Equations (71) and (72), we get the expression for μ as

$$\begin{eqnarray}&&{e}^{\mu }={\left(1+\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{\beta }_{2}}{6{\beta }_{1}}{r}^{2}\right)}^{-1},\end{eqnarray} \tag{ 76 }$$

where μ₁ is an integration constant.

In a similar way Φ(r) can be found from Equations (75) and (76) as follows:

$$\begin{eqnarray}&&{e}^{{\rm{\Phi }}}=\left(1+\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{\beta }_{2}}{6{\beta }_{1}}{r}^{2}\right).\end{eqnarray} \tag{ 77 }$$

Hence, the explicit line element can be provided as

$$\begin{eqnarray}\begin{array}{rcl} & & {{ds}}^{2}=-\left(1+\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{\beta }_{2}}{6{\beta }_{1}}{r}^{2}\right){{dt}}^{2}+{\left(1+\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{\beta }_{2}}{6{\beta }_{1}}{r}^{2}\right)}^{-1}\\ & & {\,\times \,{dr}}^{2}+{r}^{2}(d{\theta }^{2}+{\sin }^{2}\theta \,d{\phi }^{2}).\end{array}\end{eqnarray} \tag{ 78 }$$

A few interesting observations from the above metric (78) are as follows: straightforwardly it represents the Schwarzschild (anti–)de Sitter solution with (i) the cosmological constant Λ = β₂/β₁ and (ii) the mass of the stellar object ${\mu }_{1}\,=\,2{ \mathcal M }$. This comparison and resemblance indicate that the Schwarzschild (anti–)de Sitter solution exists only for the linear $f({ \mathcal Q })$ gravity, which is equivalent to GR.

Let us now treat the following situations: (i) solution for ${ \mathcal Q }={{ \mathcal Q }}_{0}$ as obtained in Equation (70), which acts as a constraint on the functional form of $f({ \mathcal Q })$, and (ii) the nonmetricity scalar constant ${{ \mathcal Q }}_{0}$ which can be considered as equivalent to the cosmological constant Λ, as shown in Equation (72). Hence, in the present case the nonmetricity scalar ${ \mathcal Q }$ can be expressed as

$$\begin{eqnarray}&&{{ \mathcal Q }}_{0}=-\displaystyle \frac{2{e}^{-\mu }}{r}\left({\rm{\Phi }}^{\prime} +\displaystyle \frac{1}{r}\right).\end{eqnarray} \tag{ 79 }$$

Now, under the constant nonmetricity scalar (i.e., ${ \mathcal Q }={{ \mathcal Q }}_{0}$) and vacuum case (i.e., = P_r = P_⊥ = 0), the equations of motion (22)–(24) become

$$\begin{eqnarray}&&{f}_{{ \mathcal Q }}({{ \mathcal Q }}_{0})\displaystyle \frac{{e}^{-\mu }}{r}({\rm{\Phi }}^{\prime} +{\mu }^{{\prime} })=0,\end{eqnarray} \tag{ 80 }$$

$$\begin{eqnarray}&&-\displaystyle \frac{{f}_{{ \mathcal Q }}({{ \mathcal Q }}_{0})}{2}+{f}_{{ \mathcal Q }}({{ \mathcal Q }}_{0})\Space{0ex}{0.25ex}{0ex}({{ \mathcal Q }}_{0}+\displaystyle \frac{1}{{r}^{2}}\Space{0ex}{0.25ex}{0ex})=0,\end{eqnarray} \tag{ 81 }$$

$$\begin{eqnarray}&&\begin{array}{l}{f}_{{ \mathcal Q }}({{ \mathcal Q }}_{0})\left[\displaystyle \frac{{{ \mathcal Q }}_{0}}{2}+\displaystyle \frac{1}{{r}^{2}}+{e}^{-\mu }\left\{\displaystyle \frac{{\rm{\Phi }}^{\prime\prime} }{2}+\left(\displaystyle \frac{{\rm{\Phi }}^{\prime} }{4}+\displaystyle \frac{1}{2r}\right)\right.\right.\\ \left.\left.\,\times \left({\rm{\Phi }}^{\prime} -\mu ^{\prime} \right)\right\}\right]=0.\,\end{array}\end{eqnarray} \tag{ 82 }$$

The above set of Equations (80) and (81) readily provide

$$\begin{eqnarray}&&f({{ \mathcal Q }}_{0})=0,\,\mathrm{or}\,\,{f}_{{ \mathcal Q }}({{ \mathcal Q }}_{0})=0.\end{eqnarray} \tag{ 83 }$$

These two restrictions immediately give a clue that the functional form of $f({ \mathcal Q })$ can be expressed in terms of power series expansion of $f({ \mathcal Q })$ around ${ \mathcal Q }={{ \mathcal Q }}_{0}$ which may be given as

$$\begin{eqnarray}\begin{array}{rcl} & & f({ \mathcal Q })={{ \mathcal Q }}_{1}{\left({ \mathcal Q }-{{ \mathcal Q }}_{0}\right)}^{1}+{{ \mathcal Q }}_{2}{\left({ \mathcal Q }-{{ \mathcal Q }}_{0}\right)}^{2}+{{ \mathcal Q }}_{3}{\left({ \mathcal Q }-{{ \mathcal Q }}_{0}\right)}^{3}\\ & & \,+{{ \mathcal Q }}_{4}{\left({ \mathcal Q }-{{ \mathcal Q }}_{0}\right)}^{4}+..........\,\,\end{array}\end{eqnarray} \tag{ 84 }$$

and which can be written in the general form as

$$\begin{eqnarray}&&f({ \mathcal Q })=\displaystyle \sum _{n=1}{{ \mathcal Q }}_{n}{\left({ \mathcal Q }-{{ \mathcal Q }}_{0}\right)}^{n},\end{eqnarray} \tag{ 85 }$$

${{ \mathcal Q }}_{1}$, ${{ \mathcal Q }}_{2}$, ${{ \mathcal Q }}_{3}$, ..... being the constant coefficients. In the present situation of an $f({ \mathcal Q })$ gravity-related nontrivial solution one should keep in mind that Equation (83) should be satisfied by the functional form of $f({ \mathcal Q })$. This is essential to obtain new solutions that are distinctly different from GR.

Now, from Equation (80), one may get another situation, which is

$$\begin{eqnarray}&&{{\rm{\Phi }}}^{{\prime} }+{\mu }^{{\prime} }=0.\end{eqnarray} \tag{ 86 }$$

Then, substituting Equation (86) in Equation (79), we get

$$\begin{eqnarray}&&{e}^{{\rm{\Phi }}(r)}=\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{Q}_{0}}{6}{r}^{2}\,\mathrm{and}\,{e}^{-\mu (r)}=\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{Q}_{0}}{6}{r}^{2}.\end{eqnarray} \tag{ 87 }$$

Therefore, the metric (14) eventually takes the form as follows:

$$\begin{eqnarray}\begin{array}{rcl}{{ds}}^{2} & = & -\left(\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{Q}_{0}}{6}{r}^{2}\right){{dt}}^{2}+{\left(\displaystyle \frac{{\mu }_{1}}{r}-\displaystyle \frac{{Q}_{0}}{6}{r}^{2}\right)}^{-1}{{dr}}^{2}\\ & & +{r}^{2}d{\theta }^{2}+{r}^{2}{\sin }^{2}\theta \,d{\phi }^{2}.\end{array}\end{eqnarray} \tag{ 88 }$$

It is noticeable that the above line element is not the same as the Schwarzschild solution which implies that the exact Schwarzschild solution does not exist for the nontrivial functional form of $f({ \mathcal Q })$.

By taking into account the above discussion, we opt the Schwarzschild anti–de Sitter spacetime in $f({ \mathcal Q })$ gravity under the functional form (69), which can be provided as

$$\begin{eqnarray}\begin{array}{rcl}{{ds}}_{+}^{2} & = & -\left(1-\displaystyle \frac{2{ \mathcal M }}{r}-\displaystyle \frac{{\rm{\Lambda }}}{3}\,{r}^{2}\right){{dt}}^{2}+\displaystyle \frac{{{dr}}^{2}}{\left(1-\tfrac{2{ \mathcal M }}{r}-\tfrac{{\rm{\Lambda }}}{3}\,{r}^{2}\right)}\\ & & +{r}^{2}(d{\theta }^{2}+{\sin }^{2}\theta \,d{\phi }^{2}),\end{array}\end{eqnarray} \tag{ 89 }$$

where ${ \mathcal M }={\hat{M}}_{{ \mathcal Q }}/{\beta }_{1}$ and Λ = β₂/2β₁, where ${\hat{m}}_{{ \mathcal Q }}(R)={\hat{M}}_{{ \mathcal Q }}$. Therefore, it is clearly observed that when β₁ = 1 and β₂ = 0, the Schwarzschild anti–de Sitter spacetime (89) reduces to the Schwarzschild exterior solution.

On the other hand, the minimally deformed interior spacetime for the region (0 ≤ r ≤ R) is given by

$$\begin{eqnarray}\begin{array}{rcl}{{ds}}_{-}^{2} & = & -{e}^{H(r)+\alpha \,\eta (r)}\,{{dt}}^{2}+{[W(r)+\alpha \,{\rm{\Psi }}(r)]}^{-1}{{dr}}^{2}\\ & & +{r}^{2}(d{\theta }^{2}+{\sin }^{2}\theta \,d{\phi }^{2}).\end{array}\end{eqnarray} \tag{ 90 }$$

As usual, here we employ the Israel–Darmois matching conditions (Darmois 1927; Israel 1966), i.e., to satisfy the first and second forms at the interface and mathematically, this can be given as

$$\begin{eqnarray}&&{e}^{{{\rm{\Phi }}}^{-}(r)}{| }_{r=R}={e}^{{{\rm{\Phi }}}^{+}(r)}{| }_{r=R}\,{\rm{and}}\,{e}^{{\lambda }^{-}(r)}{| }_{r=R}={e}^{{\lambda }^{+}(r)}(r){| }_{r=R},\,\,\end{eqnarray} \tag{ 91 }$$

$$\begin{eqnarray}&&\begin{array}{l}{[{G}_{i\,\varepsilon }\,{r}^{\varepsilon }]}_{{\rm{\Sigma }}}\equiv \mathop{\mathrm{lim}}\limits_{r\to {R}^{+}}({G}_{\epsilon \,\varepsilon })-\mathop{\mathrm{lim}}\limits_{r\to {R}^{-}}({G}_{\epsilon \,\varepsilon })=0\,\\ \qquad \Longrightarrow \,{[{T}_{\epsilon \,\varepsilon }^{\mathrm{eff}}\,{r}^{\varepsilon }]}_{{\rm{\Sigma }}}={[({T}_{\epsilon \,\varepsilon }+\alpha \,{\theta }_{\epsilon \,\varepsilon }){r}^{\varepsilon }]}_{{\rm{\Sigma }}}=0,\,\end{array}\end{eqnarray} \tag{ 92 }$$

The conditions (91) and (92) yield

$$\begin{eqnarray}&&\begin{array}{l}{e}^{H(R)+\alpha \eta (R)}=\left(1-\displaystyle \frac{2{ \mathcal M }}{R}-\displaystyle \frac{{\rm{\Lambda }}}{3}\,{R}^{2}\right)\,\\ \mathrm{and}\,W(R)+\alpha \,\psi (R)=\left(1-\displaystyle \frac{2{ \mathcal M }}{R}-\displaystyle \frac{{\rm{\Lambda }}}{3}\,{R}^{2}\right),\end{array}\end{eqnarray} \tag{ 93 }$$

$$\begin{eqnarray}&&{P}_{r}(R)={p}_{r}(R)+\alpha \,{\beta }_{1}\,\left[{\psi }_{{}_{{\rm{\Sigma }}}}\left(\displaystyle \frac{1}{{R}^{2}}+\displaystyle \frac{H{{\prime} }_{{}_{{\rm{\Sigma }}}}}{R}\right)+\displaystyle \frac{{W}_{{}_{{\rm{\Sigma }}}}\,\eta {{\prime} }_{{}_{{\rm{\Sigma }}}}}{R}\right]=0.\,\,\end{eqnarray} \tag{ 94 }$$

In order to calculate the numerical values, we employ Equations (93) and (94) and thus determine the unknown parameters such as the constant (F), mass (${ \mathcal M }$), and arbitrary constant (c).

5.1. Regular Behavior of Strange Star (SS) Models

5.1.1. For Solution 3.1 ($\rho ={\theta }_{0}^{0}$)

We now turn our attention to the physical analysis of the models obtained for the $\rho ={\theta }_{0}^{0}$ sector. The energy density is plotted in Figure 1. Keep in mind that for α = 0 we recover the standard $f({ \mathcal Q })$ gravity. Starting from the left of Figure 1, the first and second panels reveal the behavior of the energy density as a function of the radial coordinate for a fluid obeying the MIT bag model EOS. We observe that the energy density is regular at all interior points of the fluid configuration, attaining a maximum at the center. It is clear that contributions from the decoupling constant α lead to higher core densities in the linear regime as compared to the standard $f({ \mathcal Q })$ gravity models. The third and fourth panels reveal the trend in the energy density for a fluid obeying a quadratic EOS. We note that for the quadratic EOS, contributions from α lead to a significant increase in the energy density, particularly in the central regions of the star. Moreover, the closeness of the contours reveals that these higher densities lead to more compact configurations. If we compare the linear EOS to the quadratic EOS for the nonzero decoupling constant, we observe that stars with quadratic EOS have higher core densities and are more compact than their linear EOS counterparts. The first and third panels reveal that in the absence of the decoupling constant (α = 0) the energy density in the linear regime dominates its quadratic counterpart. The linear model predicts high core densities, but as one moves to the surface layers of the star, the densities are of similar magnitude. A comparison of the second and fourth panels (MIT vs. quadratic EOS for nonvanishing decoupling parameter) reveals that the density for the quadratic model is higher than the linear model at each interior point of the bounded configuration.

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We now consider the trends in the behavior of the radial pressure (plotted in Figure 2) throughout the interior of the compact object for the $\rho ={\theta }_{0}^{0}$ sector. A comparison of the first and third panels shows that the pressure in the quadratic models dominates over the linear models when the decoupling parameter vanishes. This increased pressure in the quadratic models leads to greater stability against the inwardly driven gravitational force, with this effect being enhanced in the central regions of the star. The surface pressure in the quadratic models dominates their linear counterparts, thus leading to stable surface layers. It is clear from the second and fourth panels that the pressure is enhanced in the presence of the decoupling parameter. In addition, the radial pressure in the quadratic models dominates their linear counterparts at each interior point from the center through to the stellar surface, with the highest pressures achieved in models with quadratic EOS and nonzero α.

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Before we embark on a discussion of the trends in anisotropy in our models, we highlight some work that motivates the inclusion of pressure anisotropy in stellar configurations. In our models the radial and transverse stresses at each interior point of the compact object are unequal. Obviously, equality of the radial and transverse components of pressure does ensure the spherical symmetry of the model (Dev & Gleiser 2002). However, there are various reasons for consideration of the origin of anisotropy inside the compact stars (Ruderman 1972; Sawyer 1972; Jones 1975; Sokolov 1980; Kippenhahn & Weigert 1990; Weber 1999; Liebling & Palenzuela 2012), and review work on the local anisotropy by Herrera & Santos (1997) may be informative in this regard. Very recently, Herrera (2020) has investigated the conditions for the (in)stability of the isotropic pressure condition in collapsing spherically symmetric, dissipative fluid distributions. Herrera demonstrated that an initially isotropic configuration upon leaving hydrostatic equilibrium evolves into an anisotropic regime.

The anisotropy parameter is displayed in Figure 3. We observe that the anisotropy changes sign within the compact object. We recall that a positive anisotropy factor signifies a repulsive anisotropic force that is directed outward (Hansraj et al. 2022). This helps stabilize the stellar configuration against gravity. In Figure 3, the leftmost panel shows that the anisotropy parameter is minimum at the center of the fluid and is negative up to a finite radius, r₀. This negative anisotropic factor is accompanied by an inwardly driven force that sums with the gravitational force, leading to an unstable interior. As one moves beyond r = r₀, the anisotropy factor becomes positive. The repulsive force associated with Δ > 0 stabilizes the surface layers of the star. In direct comparison, the third panel reveals a peculiar behavior of the anisotropy factor. It starts off negative in the central regions and remains negative up to a large radius r₁ where r₁ > r₀. It appears that the quadratic EOS model is more unstable than its linear counterpart in the central regions of the compact object. As one moves farther out, the anisotropy becomes positive. The first and third panels indicate that the quadratic EOS model has a narrower band of stable surface layers as compared to the linear EOS model. Its quadratic counterpart with a vanishing decoupling parameter (third panel) shows a marked difference between the two models. In the quadratic model, we see an interesting variation in the anisotropy parameter, i.e., starting from the center of the star, the anisotropy is negative for some central region, 0 < r < r₀, and then becomes more negative as one moves outward toward the boundary. The anisotropy remains negative for r₀ < r < r₁ where r₁ > r₀. Beyond r₁, the anisotropy becomes positive, rendering the surface layers stable owing to the repulsive anisotropic force. The change in sign of Δ within the quadratic EOS model can be attributed to phase transitions in different regions within the stellar fluid. In the second and fourth panels, we observe the effect of a nonvanishing decoupling constant in both the linear and quadratic EOS models, respectively. In the second panel, we observe that Δ < 0 from the center to some finite radius. Thereafter, Δ remains positive up to the boundary of the star. The fourth panel shows that anisotropy is negative within the central regions of the configuration. The anisotropy factor starts off with a finite negative value from r = 0 up to some radius r = r₀. Thereafter Δ becomes more negative as one moves outward. After some radius r₁, the anisotropy factor changes sign and becomes positive. A comparison of the second and fourth panels clearly shows that the surface layers of the linear model are more stable than their quadratic counterpart. It is interesting to observe the behavior of the anisotropy in the quadratic models, particularly the change of sign of Δ in different regions within the stellar fluid. The change in the nature of the anisotropic force (from attraction to repulsion) leads to an unstable core but stable surface layers. This interesting behavior in anisotropy was also observed (Maurya et al. 2023). In this work, they modeled compact objects in $f({ \mathcal Q })$ gravity in which the stellar fluid obeyed the MIT bag model EOS.

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5.1.2. For Solution 3.2 (${p}_{r}={\theta }_{1}^{1}$)

In this subsection, we move our attention to the physical analysis of the models discovered for the solution ${p}_{r}={\theta }_{1}^{1}$. Figure 4 exhibits the energy density plot. As usual, α = 0 leads to the standard $f({ \mathcal Q })$ gravity. Initially, let us look at the left two panels of Figure 4; the first and second plots show the behavior of the energy density against the radial coordinate r for a fluid distribution following the MIT bag model EOS. It is evident that the energy density obtained here is regular at all internal points of the stellar structure and attaining a maximum at the core of the object. From the first and second panels, it is clear that the contribution of the decoupling constant (α) allows higher core densities in the MIT bag model EOS as compared to the pure $f({ \mathcal Q })$ gravity stellar models. Furthermore, the third and fourth panels indicate the trend in the energy density for the fluid distribution in the quadratic EOS. We notice that the contributions from the decoupling parameter α in the context of the quadratic EOS give rise to a substantial growth in the energy density, particularly in the central regions of the star, which lead to more compact configurations. On the other hand, we also detect that stars obeying the quadratic EOS have higher central densities and are more compact than their linear EOS counterparts as happened in the first solution.

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Now we move to Figure 5 to check the behavior of the radial pressure within the stellar object for the ${p}_{r}={\theta }_{1}^{1}$ sector. Comparing the first and third panels in the absence of gravitational decoupling, we observe that the radial pressure of models under the quadratic EOS dominates over models in a linear regime. Furthermore, this increment in the pressure for quadratic models leads to greater stability in the central regions of the star against the inwardly directed gravitational force. From the second and fourth panels, it is clear that the pressure decreases in the presence of the gravitational decoupling parameter. Apart from this, the radial pressure of the model in the context of the quadratic EOS dominates their linear counterparts everywhere inside the star from the center to the surface, and the highest pressure is achieved in models obeying the quadratic EOS under vanishing α.

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A scrutiny of Figure 6 reveals that the anisotropy is regular at all interior points of each of the models displayed in the four panels. In addition, the anisotropy is positive and increases from the center of the star toward the boundary. This is in direct contrast to our model's $\rho ={\theta }_{0}^{0}$ sector. A comparison of the first and third panels shows the trend in anisotropy in the absence of the decoupling constant for the linear and quadratic EOS, respectively. The central anisotropy is lower in the linear model compared to its quadratic counterpart. As one approaches the surface layers, we observe that anisotropy increases, with the increase being more significant over a larger portion of the surface layers in the linear model. Since Δ > 0, the repulsive anisotropy renders the surface layers more stable than the central core regions. Comparatively, the relative magnitudes of the anisotropy show that the linear model is more stable than the quadratic model for the vanishing decoupling parameter. We now turn our attention to the second and third panels of Figure 6. In these models, the anisotropy increases from the center of the configuration toward the boundary. The increase in Δ is more profound at each interior point in the linear model. This indicates that the contribution from the repulsive nature of the anisotropic force renders the linear model more stable. A peculiar observation in the behavior of the anisotropy is observed in the linear model. While the contributions from anisotropy increase steadily from the center r₀ to some finite radius r₁, we observe a decrease in Δ for r₁ < r ≤ b. This trend is not observed in the quadratic model. If we now compare the first and second panels, i.e., the linear models for α = 0 and nonvanishing α, respectively, we observe that the decoupling parameter stabilizes the central region by enhancing the anisotropy. A comparison of the third and fourth panels clearly shows the effect of the decoupling parameter on Δ in the quadratic models. We note that the contributions from α lead to enhanced anisotropy throughout the stellar configuration, thus leading to greater stability of concentric matter shells centered about the origin.

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5.2. Constraining Upper Limit of Maximum Mass for SS via M–R Diagrams

In this section we investigate the effect of the decoupling parameter (α), bag constant (${{ \mathcal B }}_{g}$), and EOS parameters on the mass–radius relations of various compact objects and compare our theoretical values to observational constraints. Based on the EOS

$$\begin{eqnarray}&&{p}_{r}=a\,{\rho }^{1+1/n}+b\rho +c,\end{eqnarray} \tag{ 95 }$$

we classify the stellar fluid as

• (1)  
  a ≠ 0, b ≠ 0 → quadratic EOS
• (2)  
  a ≠ 0, b = 0 → pure quadratic EOS
• (3)  
  a = 0, b ≠ 0 → linear EOS (MIT bag model)

5.2.1. Linear EOS with Constancy in Bag Constant, Fixed Decoupling Parameter, and Varying EOS Parameter

We have up to this point tested the physical viability of our solutions in describing self-gravitating stellar objects. We have shown, through tests based on regularity, causality, and stability, that these solutions describe physically realizable stellar configurations. In this section, we constrain the free parameters in our solutions by using observational data of compact stars LMC X-4, Cen X-3, PSR J1614–2230, PSR J0740+6620, and the supposedly secondary component of GW190814. The robustness of our solutions allows us to predict the observed mass–radius relations of these stars. Starting off with solution A emanating from the $\rho ={\theta }_{0}^{0}$ sector, we refer to Table 1 and Figure 7 (left panel). Here we focus on the MIT bag models where we have fixed the bag constant and decoupling parameter while varying the linear EOS parameter, b. At the outset we must point out that a fixed linear EOS parameter b leads to the prediction of a category of stars with similar trends in mass and radii and not individual stars. From Table 1, we observe that for small values of b, linear contributions from the energy density can account for low-mass stars such as LMC X-4 and Cen X-3. Our model shows that by fixing the parameter b, we can obtain masses of stars beyond 2 M_⊙ . For example, for b = 0.38, our model predicts the existence of a class of stars befitting of the secondary component of the GW190814 event with a mass range of 2.5–2.67 M_⊙ and a radius of ${15.74}_{-0.21}^{+0.40}$. In the right panel, we see the trend in the M–R curves for the ${p}_{r}={\theta }_{1}^{1}$ sector in the linear regime. A similar trend in the increase in the linear EOS parameter on the radii of compact stars is observed in this case. However, there is one notable difference: while the predicted radii increase as b increases, the ${p}_{r}={\theta }_{1}^{1}$ sector predicts smaller radii, i.e., more compact stellar models. For example, in the case of the GW190814 event, the secondary component has a radius of ${14.02}_{-0.08}^{+0.09}$ km for the upper value of b, which is 10% less than its $\rho ={\theta }_{0}^{0}$ counterpart.

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Table 1. The Predicted Radii of Compact Stars LMC X-4, Cen X-3, PSR J1614–2230, PSR J0740+6620, and GW190814 for the MIT Bag Model (See Figure 7)

Objects	$\tfrac{M}{{M}_{\odot }}$	Predicted R km					Predicted R km
		For Solution $\rho ={\theta }_{0}^{0}$					For Solution ${p}_{r}={\theta }_{1}^{1}$
		b = 0.18	b = 0.23	b = 0.28	b = 0.33	b = 0.38	b = 0.13	b = 0.18	b = 0.23	b = 0.28	b = 0.33
LMC X-4 (Rawls et al. 2011)	1.29 ± 0.05	${9.39}_{-0.48}^{+0.24}$	${11.65}_{-0.01}^{+0.02}$	${13.01}_{-0.08}^{+0.06}$	${14.05}_{-0.1}^{+0.1}$	${14.94}_{-0.10}^{+0.12}$	${8.23}_{-0.04}^{+0.06}$	${9.45}_{-0.11}^{+0.05}$	${10.32}_{-0.09}^{+0.11}$	${11.05}_{-0.11}^{+0.10}$	${11.67}_{-0.13}^{+0.13}$
Cen X-3 (Rawls et al. 2011)	1.49 ± 0.08	⋯	${11.63}_{-0.06}^{+0.13}$	${13.25}_{-0.08}^{+0.05}$	${14.44}_{-0.14}^{+0.12}$	${15.43}_{-0.16}^{+0.14}$	${8.45}_{-0.08}^{+0.02}$	${9.79}_{-0.06}^{+0.09}$	${10.72}_{-0.14}^{+0.13}$	${11.55}_{-0.19}^{+0.16}$	${12.21}_{-0.18}^{+0.18}$
PSR J1614–2230 (Demorest et al. 2010)	1.97 ± 0.04	⋯	⋯	${12.89}_{-0.11}^{+0.17}$	${14.81}_{-0.01}^{+0.01}$	${16.11}_{-0.02}^{+0.01}$	⋯	${10.21}_{-0.01}^{+0.02}$	${11.44}_{-0.05}^{+0.05}$	${12.39}_{-0.04}^{+0.07}$	${13.21}_{-0.06}^{+0.05}$
PSR J0740+6620 (Cromartie et al. 2020)	${2.14}_{-0.17}^{+0.2}$	⋯	⋯	⋯	${14.68}_{-0.57}^{+0.13}$	${16.18}_{-0.10}^{+0.02}$	⋯	${10.19}_{-0.53}^{+0.01}$	${11.63}_{-0.21}^{+0.08}$	${12.63}_{-0.27}^{+0.20}$	${13.47}_{-0.36}^{+0.27}$
GW190814 (Lu et al. 2021)	2.5–2.67	⋯	⋯	⋯	⋯	${15.74}_{-0.21}^{+0.40}$	⋯	⋯	${11.64}_{-0.07}^{+0.13}$	${13.03}_{-0.06}^{+0.02}$	${14.02}_{-0.08}^{+0.09}$

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5.2.2. Linear EOS with Constancy in Bag Constant, Fixed EOS Parameter, and Varying Decoupling Constant

We now look at Table 2 in conjunction with Figure 8, in which we have fixed the bag constant and the EOS parameter b while varying α. From Table 2 and the left panel of Figure 8, we observe that an increase in the decoupling parameter results in a decrease in radii, leading to more compact configurations. A similar trend is observed in the ${p}_{r}={\theta }_{1}^{1}$ sector. We also observe that for the chosen values of the bag constant and EOS parameter, our solutions are able to predict a wider range of radii for all compact objects under investigation. A comparison of the left and right panels reveals that the allowable radii for the compact objects under investigation are more tightly bounded for the $\rho ={\theta }_{0}^{0}$ sector. The left panel shows that low-mass stars such as LMC X-4 (1.29 ± 0.05 M_⊙) and Cen X-3 (1.49 ± 0.08 M_⊙) exist for radii greater 10 km. For the ${p}_{r}={\theta }_{1}^{1}$, there is a wider range of radii for low-mass stars closer to 9 km. More noticeable is the prediction of the existence of higher-mass compact objects beyond 2 M_⊙. For small α, there exist stars with masses closer to 3 M_⊙. For the vanishing of the decoupling parameter (α = 0), stellar masses exceed 3 M_⊙. For the right panel, we note that the maximum mass is on the order of 3.5 M_⊙. In their study, Burgio et al. (2018) questioned the existence of compact stars with small radii giving rise to the GW170817/AT2017gfo signals. In one of their proposals, the so-called "twin-star" scenario, they concluded that the merger involved the coalescence of a hadronic star and a quark matter star with radii in the range 10.7 km < R_1.5 < 12km, with R_1.5 (14.28–15.41 km for ${\theta }_{0}^{0}=\rho$ and 9.70–14.18 km for ${\theta }_{1}^{1}={p}_{r}$) being the radius of a 1.5 M_⊙ compact star. Figure 8 accounts for the existence of a myriad of observed compact objects, including those falling into the "twin-star" category scenario, with the latter being more suitable for the ${p}_{r}={\theta }_{1}^{1}$ sector.

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Table 2. The Predicted Radii of Compact Stars LMC X-4, Cen X-3, PSR J1614–2230, PSR J0740+6620, and GW190814 for the MIT Bag Model (See Figure 8)

Objects	$\tfrac{M}{{M}_{\odot }}$	Predicted R km					Predicted R km
		For Solution $\rho ={\theta }_{0}^{0}$					For Solution ${p}_{r}={\theta }_{1}^{1}$
		α = 0.0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.0	α = 0.1	α = 0.2	α = 0.3	α = 0.4	α = 0.5
LMC X-4 (Rawls et al. 2011)	1.29 ± 0.05	${14.86}_{-0.10}^{+0.09}$	${14.58}_{-0.08}^{+0.12}$	${14.36}_{-0.12}^{+0.10}$	${14.14}_{-0.07}^{+0.07}$	${13.93}_{-0.08}^{+0.06}$	${13.64}_{-0.15}^{+0.12}$	${12.44}_{-0.14}^{+0.13}$	${11.71}_{-0.18}^{+0.17}$	${10.88}_{-0.14}^{+0.12}$	${10.02}_{-0.13}^{+0.10}$	${9.17}_{-0.02}^{+0.14}$
Cen X-3 (Rawls et al. 2011)	1.49 ± 0.08	${15.14}_{-0.14}^{+0.11}$	${15.10}_{-0.16}^{+0.09}$	${14.82}_{-0.11}^{+0.10}$	${14.54}_{-0.09}^{+0.08}$	${14.26}_{-0.09}^{+0.08}$	${14.18}_{-0.20}^{+0.16}$	${13.19}_{-0.18}^{+0.18}$	${12.28}_{-0.21}^{+0.15}$	${11.38}_{-0.10}^{+0.18}$	${10.00}_{-0.19}^{+0.18}$	${9.67}_{-0.19}^{+0.15}$
PSR J1614–2230 (Demorest et al. 2010)	1.97 ± 0.04	${16.42}_{-0.02}^{+0.03}$	${15.90}_{-0.01}^{+0.01}$	${15.39}_{-0.01}^{+0.01}$	${14.92}_{-0.01}^{+0.01}$	${14.40}_{-0.02}^{+0.04}$	${15.11}_{-0.07}^{+0.05}$	${14.16}_{-0.04}^{+0.07}$	${13.23}_{-0.05}^{+0.07}$	${12.36}_{-0.05}^{+0.05}$	${11.46}_{-0.03}^{+0.05}$	${10.58}_{-0.05}^{+0.04}$
PSR J0740+6620 (Cromartie et al. 2020)	${2.14}_{-0.17}^{+0.2}$	${16.79}_{-0.14}^{+0.05}$	${16.11}_{-0.04}^{+0.01}$	${15.48}_{-0.22}^{+0.07}$	${14.82}_{-0.50}^{+0.17}$	${13.96}_{-1.31}^{+0.81}$	${15.37}_{-0.29}^{+0.23}$	${14.43}_{-0.34}^{+0.27}$	${13.51}_{-0.34}^{+0.28}$	${13.60}_{-0.35}^{+0.26}$	${11.74}_{-0.33}^{+0.27}$	${10.83}_{-0.31}^{+0.24}$
GW190814 (Lu et al. 2021)	2.5–2.67	${17.42}_{-0.02}^{+0.02}$	${16.32}_{-0.16}^{+0.16}$	${14.86}_{-0}^{+0.09}$	⋯	⋯	${15.84}_{-0.07}^{+0.05}$	${14.96}_{-0.08}^{+0.06}$	${14.09}_{-0.10}^{+0.07}$	${13.19}_{-0.08}^{+0.07}$	${12.28}_{-0.08}^{+0.07}$	${11.31}_{-0.08}^{+0.03}$

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5.2.3. Quadratic EOS with Fixed Bag Constant, Fixed Decoupling Constant, and Varying Quadratic EOS Parameter

Here we focus on Table 3 and Figure 9. It is clear in this scenario that neither of our solutions predicts very well the existence of compact objects beyond 2 M_⊙. The lower-mass stars have larger radii, thus indicating that these candidates have low densities compared to their linear EOS counterparts as displayed in Figure 8. In a study by Astashenok et al. (2020), they showed that the observations of GW190814 can be accounted for within the framework of f(R) gravity and the inclusion of rotation. They utilized three frameworks, i.e., classical GR; f(R) gravity with f(R) = R + α R², where α is quadratic curvature correction; and finally f(R) = R¹⁺ , where is a measure of a small deviation from GR. Utilizing different EOSs, they showed that in the absence of rotation the models of compact objects for both GR and f(R) gravity with varying α lead to masses less than 2.5 M_⊙ and radii between 10 and 14 km. With the inclusion of rotation, the f(R) models can easily account for masses greater than 2.5 M_⊙, with this accounting for the observed mass of the secondary component of GW190814. For observed rotational frequencies of NSs, f(R) gravity predicts the existence of a 2.0 M_⊙ compact object. In the case of the f(R) = R¹⁺ framework, they obtained masses in the range of 2.5–2.67 M_⊙ for in the range of 0.005–0.008. Interestingly, the radii of these compact objects in f(R) gravity with rotation can vary from 12 to 18 km. We have also obtained models of compact objects with large radii in this category of stars.

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Table 3. The Predicted Radii of Compact Stars LMC X-4, Cen X-3, PSR J1614–2230, PSR J0740+6620, and GW190814 for Figure 9

Objects	$\tfrac{M}{{M}_{\odot }}$	Predicted R km					Predicted R km
		For Solution $\rho ={\theta }_{0}^{0}$					For Solution ${p}_{r}={\theta }_{1}^{1}$
		a = 3	a = 6	a = 9	a = 12	a = 15	a = 25	a = 30	a = 35	a = 40	a = 45
LMC X-4 (Rawls et al. 2011)	1.29 ± 0.05	${13.71}_{-0.06}^{+0.04}$	${14.44}_{-0.05}^{+0.07}$	${15.13}_{-0.07}^{+0.07}$	${15.78}_{-0.07}^{+0.08}$	${17.0}_{-0.10}^{+0.08}$	${13.11}_{-0.06}^{+0.05}$	${13.74}_{-0.08}^{+0.09}$	${14.31}_{-0.11}^{+0.11}$	${14.81}_{-0.12}^{+0.10}$	${15.25}_{-0.15}^{+0.13}$
Cen X-3 (Rawls et al. 2011)	1.49 ± 0.08	${13.84}_{-0.03}^{+0.01}$	${14.63}_{-0.04}^{+0.02}$	${15.36}_{-0.06}^{+0.03}$	${16.05}_{-0.09}^{+0.05}$	${16.71}_{-0.12}^{+0.06}$	${13.15}_{-0.15}^{+0.05}$	${13.95}_{-0.03}^{+0.04}$	${14.62}_{-0.10}^{+0.05}$	${15.21}_{-0.14}^{+0.10}$	${15.72}_{-0.15}^{+0.14}$
PSR J1614–2230 (Demorest et al. 2010)	1.97 ± 0.04	⋯	${13.72}_{-0.27}^{+0.34}$	${14.88}_{-0.12}^{+0.17}$	${15.80}_{-0.11}^{+0.09}$	${16.62}_{-0.06}^{+0.06}$	⋯	⋯	⋯	${14.19}_{-0}^{+0.5}$	${15.75}_{-0.20}^{+0.13}$
PSR J0740+6620 (Cromartie et al. 2020)	${2.14}_{-0.17}^{+0.2}$	⋯	⋯	⋯	${14.99}_{-0.81}^{+0.91}$	${16.11}_{-1.46}^{+0.58}$	⋯	⋯	⋯	⋯	${15.78}_{-0.45}^{+0.87}$
GW190814 (Lu et al. 2021)	2.5–2.67	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯

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5.2.4. Quadratic EOS with Fixed Bag Constant, Fixed Quadratic EOS Parameter, and Constant versus Varying Decoupling Constant for Mimicking of Radial Pressure, ${p}_{r}={\theta }_{1}^{1}$

In this subsection, we pay particular attention to the M–R curves arising in the composite linear-quadratic EOS and the effect of the decoupling constant. Table 4 shows that these models predict the existence of low-mass stars with an increase in the decoupling constant, leading to higher-mass stars with larger radii, very similar to the trends obtained by Astashenok et al. (2021) in describing rotating NSs in f(R) = R + α R² gravity. It is clear from Table 4 and Figure 10 that the mixed EOS model fails to predict masses that can account for the LIGO observation of approximately 2.6 M_⊙ of the secondary component of the binary coalescence GW190814. It appears that when the linear and quadratic EOS parameters are switched on simultaneously, the decoupling constant quenches any increase in the mass of the compact object. While these models fail to predict masses above 2.6 M_⊙, the quadratic EOS predicts the existence of well-known pulsars and NSs to a very good approximation.

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Table 4. The Predicted Radii of Compact Stars LMC X-4, Cen X-3, PSR J1614–2230, PSR J0740+6620, and GW190814 for Solution ${p}_{r}={\theta }_{1}^{1}$ Corresponding to Figure 10

Objects	$\tfrac{M}{{M}_{\odot }}$	Predicted R km					Predicted R km
		Quadratic Model+MIT Bag Model with Different b					Quadratic Model+MIT Bag Model with Different α
		b = 0.23	b = 0.33	b = 0.43	b = 0.53	b = 0.63	α = 0	α = 0.05	α = 0.1	α = 0.15	α = 0.20
LMC X-4 (Rawls et al. 2011)	1.29 ± 0.05	${10.67}_{-0.68}^{+0.28}$	${12.01}_{-0.02}^{+0.06}$	${12.92}_{-0.01}^{+0.01}$	${13.65}_{-0.06}^{+0.05}$	${14.26}_{-0.09}^{+0.08}$	${17.19}_{-0.04}^{+0.01}$	${16.66}_{-0.06}^{+0.06}$	${16.07}_{-0.09}^{+0.08}$	${15.43}_{-0.09}^{+0.12}$	${14.81}_{-0.14}^{+0.12}$
Cen X-3 (Rawls et al. 2011)	1.49 ± 0.08	⋯	⋯	${12.56}_{-0.87}^{+0.26}$	${13.65}_{-0.18}^{+0.06}$	${14.45}_{-0.03}^{+0.01}$	${17.05}_{-0.17}^{+0.12}$	${16.74}_{-0.07}^{+0.02}$	${16.29}_{-0.05}^{+0.02}$	${15.77}_{-0.10}^{+0.05}$	${15.21}_{-0.14}^{+0.10}$
PSR J1614–2230 (Demorest et al. 2010)	1.97 ± 0.04	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	${14.12}_{-0}^{+0.65}$
PSR J0740+6620 (Cromartie et al. 2020)	${2.14}_{-0.17}^{+0.2}$	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯
GW190814 (Lu et al. 2021)	2.5–2.67	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯	⋯

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In comparing the models derived from the $\rho ={\theta }_{0}^{0}$ and ${p}_{r}={\theta }_{1}^{1}$ sectors, we compared and contrasted their predicting powers when it comes to low-mass stars, as well as extreme masses bordering on the mass of the lightest BH. We have shown that the predicted stellar masses are sensitive to the decoupling constant α, the linear EOS parameter b, the quadratic EOS parameter a, and the bag constant. By varying these constants in particular sets, our models describe a family of compact objects with varying EOSs ranging from the simplest linear EOS (MIT bag model) to the pure quadratic EOS and the more complex quadratic EOS. The tabulated data and plots reveal that the quadratic EOS accounts for a wide spectrum of observed compact objects in the low-mass limit and stars that qualify as the secondary component of the GW190814 event.

Let us now discuss the energy exchange requirement in connection to the extended gravitational decoupling. Ovalle (2019) showed in his work that both the sources ${\hat{T}}_{{ij}}$ and θ_ij can be decoupled in a successful way as long as the exchange of energy is there in between them. Denoting ${{ \mathcal G }}_{{ij}}$ as the equations of motion of the line element in f(Q) gravity, Equation (9) can be considered as follows:

$$\begin{eqnarray}&&\begin{array}{l}{{ \mathcal G }}_{{ij}}=\displaystyle \frac{2}{\sqrt{-g}}{{\rm{\nabla }}}_{k}(\sqrt{-g}\,{f}_{Q}\,{P}_{\,\,{ij}}^{k})+\displaystyle \frac{1}{2}{g}_{{ij}}f+{f}_{Q}({P}_{i\,{kl}}\,{Q}_{j}^{\,\,{kl}}\\ \,-2\,{Q}_{{kli}}\,{P}_{\,\,\,j}^{{kl}})={T}_{{ij}}={\hat{T}}_{{ij}}+\alpha \,{\theta }_{{ij}}.\end{array}\end{eqnarray} \tag{ 96 }$$

The conservation equation can be found by Bianchi identity ${ \triangledown }^{i}{{ \mathcal G }}_{{ij}}=0$, given by

$$\begin{eqnarray}&&\displaystyle \frac{H^{\prime} }{2}({T}_{1}^{1}-{T}_{0}^{0})+\left({T}_{1}^{1}\right)^{\prime} -\displaystyle \frac{2}{r}({T}_{2}^{2}-{T}_{1}^{1})=\displaystyle \frac{\alpha \eta ^{\prime} }{2}({T}_{0}^{0}-{T}_{1}^{1}).\,\,\,\end{eqnarray} \tag{ 97 }$$

The above Equation (97) can be provided in a more suitable form, which is

$$\begin{eqnarray}\begin{array}{rcl} & & -\displaystyle \frac{H^{\prime} }{2}({\hat{T}}_{0}^{0}-{\hat{T}}_{1}^{1})+\left({\hat{T}}_{1}^{1}\right)^{\prime} -\displaystyle \frac{2}{r}({\hat{T}}_{2}^{2}-{\hat{T}}_{1}^{1})-\displaystyle \frac{\alpha \eta ^{\prime} }{2}({\hat{T}}_{0}^{0}-{\hat{T}}_{1}^{1})\\ & & \Space{0ex}{3.04em}{0ex}\displaystyle \frac{\alpha {\rm{\Phi }}^{\prime} }{2}([{T}^{\theta }{]}_{1}^{1}-[{T}^{\theta }{]}_{0}^{0})+\alpha \left([{T}^{\theta }{]}_{1}^{1}\right)^{\prime} =\displaystyle \frac{2\alpha }{r}([{T}^{\theta }{]}_{2}^{2}-[{T}^{\theta }{]}_{1}^{1}).\end{array}\end{eqnarray} \tag{ 98 }$$

Now there is a notable point in the context of the TOV equation for $f({ \mathcal Q })$ that under the linear functional form it corresponds to the static and spherically symmetric line element of GR. This ensures that ${{ \mathcal G }}_{{ij}}^{\{H,W\}}$ for the metric (43) should satisfy its corresponding Bianchi identity. Again, this also suggests that the energy–momentum tensor ${\hat{T}}_{{ij}}$ should be conserved with the spacetime geometry {H, W} of Equation (42). Hence, one can provide

$$\begin{eqnarray}&&{ \triangledown }_{i}^{\{H,W\}}\,{\hat{T}}_{j}^{i}=0.\end{eqnarray} \tag{ 99 }$$

It is also instructive, in connection to Equation (13), that

$$\begin{eqnarray}&&{ \triangledown }_{i}\,{\hat{T}}_{j}^{i}={ \triangledown }_{i}^{\{H,W\}}\,{\hat{T}}_{j}^{i}-\displaystyle \frac{\alpha \,\eta ^{\prime} }{2}({\hat{T}}_{0}^{0}-{T}_{1}^{1}){\delta }_{\nu }^{1}.\end{eqnarray} \tag{ 100 }$$

As a linear combination of the Einstein field Equations (40)–(90), we obtain from Equation (99) the following explicit form:

$$\begin{eqnarray}\begin{array}{rcl} & & -\displaystyle \frac{H^{\prime} }{2}({\hat{T}}_{0}^{0}-{\hat{T}}_{1}^{1})+\left({\hat{T}}_{1}^{1}\right)^{\prime} -\displaystyle \frac{2}{r}({\hat{T}}_{2}^{2}-{\hat{T}}_{1}^{1})=0.\,\,\end{array}\end{eqnarray} \tag{ 101 }$$

This at once indicates that the source ${\hat{T}}_{{ij}}$ can be decoupled from the system of Equations (39)–(41) in a well-defined manner, and eventually, based on the condition (98), one may get from Equation (99) the following forms:

$$\begin{eqnarray}&&{ \triangledown }_{\epsilon }\,{\hat{T}}_{j}^{i}=-\frac{\alpha \,\eta ^{\prime} }{2}({\hat{T}}_{0}^{0}-{\hat{T}}_{1}^{1}){\delta }_{j}^{1}\end{eqnarray} \tag{ 102 }$$

and

$$\begin{eqnarray}&&{ \triangledown }_{i}\,{\theta }_{j}^{i}=\displaystyle \frac{\alpha \,\eta ^{\prime} }{2}({\hat{T}}_{0}^{0}-{\hat{T}}_{1}^{1}){\delta }_{j}^{1}.\end{eqnarray} \tag{ 103 }$$

At this juncture, it should be noted that (i) in Equations (39)–(41) the divergence has been calculated in connection to the deformed spacetime (16); (ii) Equation (103) is nothing but a linear combination of "quasi-Einstein" field equations, i.e., (45)–(47) under the platform $f({ \mathcal Q })$ gravity; and (iii) as long as there is an exchange of energy between the sources ${\hat{T}}_{{ij}}$ and θ_ij , decoupling can be successfully performed. Following Contreras & Stuchlik (2022) and Ovalle et al. (2022), the energy exchange between the sources can be expressed as follows:

$$\begin{eqnarray}&&{\rm{\Delta }}E=\displaystyle \frac{{\eta }^{{\prime} }}{2}({p}_{r}+\rho ).\end{eqnarray} \tag{ 104 }$$

Therefore, with p_r and ρ being two positive physical quantities, the above Equation (104) can help explore the following situations: (i) if ${\eta }^{{\prime} }\gt 0$, then ΔE > 0, which implies ${ \triangledown }_{i}\,{\theta }_{j}^{i}\gt 0$, i.e., the new source θ_ij supplies energy to the environment; and (ii) if ${\eta }^{{\prime} }\lt 0$, then ΔE < 0, which implies ${ \triangledown }_{i}\,{\theta }_{j}^{i}\lt 0$, i.e., the new source θ_ij extracts energy from the environment.

It is noted that temporal deformation is the same for both solutions; therefore, the expressions of energy exchange will be the same for both cases, but the amount of energy exchange will be different for both cases. Now inserting the expressions for seed pressure and density along with the temporal deformation function η, we find

$$\begin{eqnarray}\begin{array}{ccl}{\rm{\Delta }}E & = & \frac{r}{8}\Space{0ex}{2.5ex}{0ex}[\frac{1}{4{\left({{Nr}}^{2}+1\right)}^{4}}\{(2{\beta }_{1}(L-N)({{Nr}}^{2}+3)+{\beta }_{2}\\ & & \times {\left({{Nr}}^{2}+1\right)}^{2})(2a{\beta }_{1}(L-N)({{Nr}}^{2}+3)+(a{\beta }_{2}-2b)\\ & & \times {\left({{Nr}}^{2}+1\right)}^{2})\}+c-\frac{{\beta }_{2}}{2}-\frac{{\beta }_{1}(L-N)({{Nr}}^{2}+3)}{{\left({{Nr}}^{2}+1\right)}^{2}}\Space{0ex}{2.5ex}{0ex}]\\ & & \times [\{4{\beta }_{1}{L}^{3}(3a{\beta }_{2}-6a{\beta }_{1}N-3b-1)+{L}^{2}\{{\beta }_{2}(a{\beta }_{2}-2b-2)\\ & & +8{\beta }_{1}N(1-2a{\beta }_{2}+2b)+4a{\beta }_{1}^{2}{N}^{2}\}-2{LN}[{\beta }_{2}(a{\beta }_{2}-2b-2)\\ & & +2{\beta }_{1}N(1-a{\beta }_{2}+b)]+{\beta }_{2}{N}^{2}(a{\beta }_{2}-2b-2)+36a{\beta }_{1}^{2}{L}^{4}\\ & & +4{a}_{3}{\left(L-N\right)}^{2}\}\frac{1}{{\beta }_{1}L(L-N)({{Lr}}^{2}+1)}\\ & & +\frac{N({\beta }_{2}(a{\beta }_{2}-2b-2)+4c)}{{\beta }_{1}L}+\frac{1}{(N-L)({{Nr}}^{2}+1)}\\ & & \times [4N(a\{9{\beta }_{1}{L}^{2}+2{\beta }_{2}L-6{\beta }_{1}{LN}+{\beta }_{1}{N}^{2}-2{\beta }_{2}N\}\\ & & -2b(L-N))]-\frac{16a{\beta }_{1}N(2L-N)}{{\left({{Nr}}^{2}+1\right)}^{2}}-\frac{16a{\beta }_{1}N(L-N)}{{\left({{Nr}}^{2}+1\right)}^{3}}].\end{array}\end{eqnarray} \tag{ 105 }$$

6.1. For Solution 3.1 ($\rho ={\theta }_{0}^{0}$)

In this subsection we discuss the amount of energy exchange (ΔE) between the generic fluid θ_ij and anisotropic fluid ${\hat{T}}_{{ij}}$ for solution 3.1. To see this distribution, we plotted Figure 11 to show the energy exchange between the relativistic fluids via the density plots. The first panel is plotted in the context of the pure MIT bag model by taking bag constant value ${{ \mathcal B }}_{g}=60\,\mathrm{MeV}\ {\mathrm{fm}}^{-3}$ and decoupling constant α = 0.1. We observe that the ΔE is positive and minimum at the center and starts increasing toward the boundary and attaining its maximum value within the star rather than the boundary. The maximum value of ΔE is 0.00054 km⁻³. Now we move to the second panel, which is plotted for the pure MIT bag model by taking ${{ \mathcal B }}_{g}=60\,\mathrm{MeV}\ {\mathrm{fm}}^{-3}$ with decoupling constant α = 0.2. We observe that the same situation occurs as happened before, but the amount of energy exchange increases, ${\rm{\Delta }}{E}_{\max }\approx 0.00058\,{\mathrm{km}}^{-3}$.

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The third and fourth panels show the distribution of energy for the quadratic model with decoupling constant values α = 0.1 and α = 0.2, respectively, for the same value of bag constant ${{ \mathcal B }}_{g}=60\,\mathrm{MeV}\ {\mathrm{fm}}^{-3}$. The pattern of energy change within the star for the quadratic model is similar to the MIT bag model, but ∣ΔE∣_qua > ∣ΔE∣_MIT. The maximum value of ΔE is 0.00063 and 0.00067 km⁻³ at α = 0.1 and α = 0.2, respectively.

Finally, we conclude that the interaction between both fluids increases significantly when moving toward the boundary and reaches its maximum value within the star, not at the boundary in all cases for solution 3.1 ($\rho ={\theta }_{0}^{0}$). In addition, ΔE is positive throughout in the radial direction, and its magnitude of maximum value increases when the decoupling constant α increases. Furthermore, the generic source θ_ij gives more energy to the environment in the presence of the quadratic EOS as compared to the MIT bag model EOS.

6.2. For Solution 3.2 (${p}_{r}={\theta }_{1}^{1}$)

This section contains the analysis of energy exchange (ΔE) distributions between the generic fluid θ_ij and anisotropic fluid ${\hat{T}}_{{ij}}$ for solution 3.2. For this purpose, the density plot for Figure 12 reveals the flow of energy between the relativistic fluids. The first two panels are plotted in the context of the pure MIT bag model for decoupling constant α = 0.1 and α = 0.2, respectively, using the bag constant value ${{ \mathcal B }}_{g}=60\,\mathrm{MeV}\ {\mathrm{fm}}^{-3}$. We observe that the ΔE is positive and minimum at the center and starts increasing when moving toward the boundary, but the maximum value of ΔE is achieved within the star, not at the boundary. On the other hand, we observe one interesting point here: no impact of the gravitational decoupling on the energy exchange fluid distributions is noticed under the mimic-to-pressures constraint approach. Therefore, the same maximum value of ΔE ≈ 0.0005 km⁻³ is observed for both values of α = 0.1 and 0.2. From the third and fourth panels, we show the distribution of energy for the quadratic model with decoupling constant values α = 0.1 and 0.2, respectively, for the same value of bag constant ${{ \mathcal B }}_{g}=60\,\mathrm{MeV}\ {\mathrm{fm}}^{-3}$. We find that the behavior of energy exchange within the star for the quadratic model is similar to the MIT bag model but ∣ΔE∣_qua > ∣ΔE∣_MIT as happened in solution 3.1. The maximum value of ΔE is 0.00054 km⁻³ at both α = 0.1 and α = 0.2.

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For the above solution, we finally can conclude that the interaction between both fluids also increases significantly when moving toward the boundary and reaches its maximum value within the star, not at the boundary, in all scenarios for solution 3.2 (${p}_{r}={\theta }_{1}^{1}$), and ΔE is positive throughout in the radial direction. Furthermore, the magnitude of the maximum value for ΔE is independent of decoupling constant α, i.e., ΔE remains the same for each α. In fact, here also the generic source θ_ij gives more energy to the environment in the presence of the quadratic EOS as compared to the MIT bag model EOS.

In this section we highlight the key differences arising in GR and $f({ \mathcal Q })$ gravity with and without CGD via our modeling framework. To this end, we draw the reader's attention to Figure 13. For the first case ($\rho ={\theta }_{0}^{0}$), we have plotted the M–R curves for pure GR (β₁ = 1, α = 0) and pure $f({ \mathcal Q })$ (β₁ = 1.1, α = 0) in Figure 13 (top). We note that for the same EOS the $f({ \mathcal Q })$ gravity can generate higher M_max than GR iff β₁ > 1; otherwise, the reverse will happen. However, when CGD turns on, i.e., α ≠ 0, the corresponding EOS gets softer, resulting in lower M_max for both GR and $f({ \mathcal Q })$ gravity. Therefore, when the tt component of the CGD-induced stress tensor ${\theta }_{i}^{j}$ mimics the density, the M_max is lowered owing to softening EOS. This can be ascribed to the effective density, i.e., = (1 + α)ρ increasing with the CGD strength, leading to a denser interior, which will eventually trigger many exotic processes such as hyperon and kaon productions. On the other hand, the second solution, where ${\theta }_{1}^{1}$ mimics pressure, leads to the lowering of the effective pressure P_r = (1 − α)p_r . In view of the EOS within the core, lower radial pressure supports lower interior density, which may suppress any exotic phase transitions. This makes the EOS stiffer when the CGD strength increases, leading to higher M_max (see Figure 13, bottom panel) and compactness parameter. Furthermore, it can also be observed that both $f({ \mathcal Q })$ models yield roughly the same masses and radii when CGD is turned off. However, when CGD turns on, the first case immediately decreases the M_max and corresponding radius, while the second solution yields a slightly higher maximum mass with smaller radius. In the GR limit, both solutions yield the same M–R curves, as the seed solution is unaffected by β.

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In this section, we provide succinct conclusions about our findings in a point-wise sequence as follows:

• (i)  
  Inspired by the recent gravitational event, i.e., GW190814, which brings to light the coalescence of a 23 M_⊙ BH with a yet-to-be-determined secondary component, we try to model compact objects under the framework of $f({ \mathcal Q })$ gravity theory along with the method of gravitational decoupling.
• (ii)  
  We assume a suitable quadratic EOS for the interior matter distribution of a compact star, especially an NS/QS, which in the appropriate limit reduces to the MIT bag model.
• (iii)  
   The field equations arising in the $f({ \mathcal Q })$ gravity framework are subject to gravitational decoupling and bifurcate into the $\rho ={\theta }_{0}^{0}$ and ${p}_{r}={\theta }_{1}^{1}$ sectors leading two distinct classes of solutions.
• (iv)  
  Both families of solutions are subjected to rigorous tests, qualifying them to describe a class of compact stellar objects that include NSs, strange stars, and the possible progenitor of the secondary component of GW190814.
• (v)  
  Using observational data of mass–radius relations for compact objects, e.g., LMC X-4, Cen X-3, PSR J1614–2230, and PSR J0740+6620, we show that it is possible to generate stellar masses and radii beyond 2.5 M_⊙.
• (vi)  
  The outcomes of the present work reveal that the quadratic EOS is versatile enough to account for a range of low-mass stars, as well as typical stellar candidates describing the secondary component of GW190814.

Based on the abovementioned steps, the obtained results have already been classified with detailed discussions in several tables and figures. Let us now put some salient features of the outcomes as follows:

• 1.  
  Graphical plots for the basic physical parameters. Distributions of the energy density (Figures 1 and 4), the radial pressure (Figures 2 and 5), and the anisotropy (Figures 3 and 6) are exhibited for different values of the model parameters. Discussions at length have been given already in Section 5.1.1 in connection to solution 3.1 ($\rho ={\theta }_{0}^{0}$) and in Section 5.1.2 in connection to solution 3.2 (${p}_{r}={\theta }_{1}^{1}$). We note that all the features are satisfactory as far as physical attributes are concerned.
• 2.  
  Constraining upper limit of maximum mass via the graphical plots for M–R diagrams. For different choices of the linear EOS with constancy in bag constant, fixed decoupling parameter, and varying EOS parameter we have shown results in Tables 1–4 and Figures 7–10, which are as follows.In Table 1 and Figure 7, we observe linear contributions from the energy density that do account for low-mass stars such as LMC X-4 and Cen X-3.From Table 2 and Figure 8 (left panel), we observe that an increase in the decoupling parameter results in a decrease in radii, leading to more compact configurations. The left panel also shows that low-mass stars such as LMC X-4 (1.29 ± 0.05 M_⊙) and Cen X-3 (1.49 ± 0.08 M_⊙) do exist for radii greater than 10 km.Figure 8 predicts a wide range of masses and radii applicable to observed compact objects, including the "twin-star" scenario, the latter of which is favored in the ${p}_{r}={\theta }_{1}^{1}$ sector. Interestingly, the radii of these compact objects in f(R) gravity with rotation can vary between 12 and 18 km.We ascertain from Table 4 and Figure 10 that the quadratic EOS model fails to predict masses that can account for the LIGO observation of approximately 2.6 M_⊙ of the secondary component of the binary coalescence GW190814. It appears that when the linear and quadratic EOS parameters are switched on simultaneously, the decoupling constant quenches any increase in the mass of the compact object. While these models fail to predict masses above 2.6 M_⊙, the quadratic EOS predicts the existence of well-known pulsars and NSs to a very good approximation. The tabulated data and plots reveal that the quadratic EOS that has the linear limit successfully predicts the existence of low-mass stars, as well as NSs with masses beyond 2 M_⊙.
• 3.  
  Graphical plots for the energy exchange between the fluid distributions. We now discuss the most important aspect of our findings, i.e., the energy exchange arising from the extended gravitational decoupling. It is noted that temporal deformation is the same for both the solutions, and hence the expressions of energy exchange will be the same for both cases, but the amount of energy exchange will be different for both cases.

  • (i)  
    For solution 3.1 ($\rho ={\theta }_{0}^{0}$): In Figure 11 we have exhibited the energy exchange between the relativistic fluids via the density plots. The first panel is plotted in the context of the pure MIT bag model, where we observe that the ΔE is positive and minimum at the center and starts increasing toward the boundary and attaining its maximum value within the star rather than the boundary. Now we move to the second panel, which is plotted for the pure MIT bag model; one can note that the same situation occurs as happened before, but the amount of energy exchange increases to ${\rm{\Delta }}{E}_{\max }\approx 0.00058\,{\mathrm{km}}^{-3}$.The third and fourth panels show the distribution of energy for the quadratic model, where the pattern of energy change within the star for the quadratic model is similar to the MIT bag model but ∣ΔE∣_qua > ∣ΔE∣_MIT. The maximum value of ΔE is 0.00063 and 0.00067 km⁻³ at α = 0.1 and 0.2, respectively.Based on the above observations, one may conclude that the interaction between both fluids increases significantly when moving toward the boundary and reaches its maximum value within the star, not at the boundary in all cases for solution 3.1 ($\rho ={\theta }_{0}^{0}$). In addition, ΔE is positive throughout in the radial direction, and its magnitude of maximum value increases when the decoupling constant α increases. Furthermore, the generic source θ_ij gives more energy to the environment in the presence of the quadratic EOS as compared to the MIT bag model EOS.
  • (ii)  
    For solution 3.2 (${p}_{r}={\theta }_{1}^{1}$), the density plot for Figure 12 has been shown to observe the flow of energy exchange between the relativistic fluids. The first two panels are plotted in the context of the pure MIT bag model, where we observe that ΔE is positive and minimum at the center and starts increasing when moving toward the boundary, but the maximum value of ΔE is achieved within the star, not at the boundary. On the other hand, we observe one interesting point here, that no impact of the gravitational decoupling on the energy exchange fluid distributions is noticed under the mimic-to-pressures constraint approach. From the third and fourth panels, we show the distribution of energy for the quadratic model and find that the behavior of energy exchange within the star for the quadratic model is similar to the MIT bag model, but ∣ΔE∣_qua > ∣ΔE∣_MIT as happened in solution 3.1.

Therefore, we can conclude that the interaction between both fluids also increases significantly when moving toward the boundary and reaches its maximum value within the star, not at the boundary in all the situations for solution 3.2 (${p}_{r}={\theta }_{1}^{1}$), and ΔE is positive throughout in the radial direction. Furthermore, the magnitude of the maximum value for ΔE is independent of decoupling constant α, i.e., ΔE remains the same for each α. In fact, here also the generic source θ_ij gives more energy to the environment in the presence of the quadratic EOS as compared to the MIT bag model EOS.

The overall findings of the models presented reveal that it is possible to put suitable constraints on the upper limit of the mass–radius relation of the secondary component of GW190814 and other self-bound strange star configurations under gravitational decoupling in $f({ \mathcal Q })$ gravity theory, which may provide the observational signature of the objects in a significant way.

S.K.M. is grateful for continuous support and encouragement from the administration of the University of Nizwa to carry out the research work. G.M. is very grateful to Prof. Gao Xianlong from the Department of Physics, Zhejiang Normal University, for his kind support and help during this research. Further, G.M. acknowledges grant No. ZC304022919 to support his Postdoctoral Fellowship at Zhejiang Normal University. K.N.S. and S.R. are also grateful to the authorities of the Inter-University Centre for Astronomy and Astrophysics, Pune, India, for providing the research facilities, where S.R. specifically expresses thanks to ICARD of IUCAA at GLA University. The authors are also grateful to Prof. Jorge Ovalle for his help in the production of the contour diagrams.

No new data were generated or analyzed in support of this research.

The appendix gives the explicit expressions for θ₂₂(r), θ₂₃(r), Ψ₂₁(r), Ψ₂₂(r), Ψ₀₀(r), and the coefficients used in Equations (56)–(58).

$$\begin{eqnarray*}\begin{array}{rcl} & & {\theta }_{11}(r)=\displaystyle \frac{1}{4}\Space{0ex}{2.5ex}{0ex}[\displaystyle \frac{1}{{\beta }_{1}L(L-N)({{Lr}}^{2}+1)}(-4{\beta }_{1}{L}^{3}(1-3a{\beta }_{2}\\ & & +6a{\beta }_{1}N+3b)+{L}^{2}({\beta }_{2}(a{\beta }_{2}-2b-2)-8{\beta }_{1}N(2a{\beta }_{2}\\ & & -2b-1)+4a{\beta }_{1}^{2}{N}^{2})-2{LN}({\beta }_{2}(a{\beta }_{2}-2b-2)+2{\beta }_{1}\\ & & \times N(-a{\beta }_{2}+b+1))+{\beta }_{2}{N}^{2}(a{\beta }_{2}-2b-2)+36a{\beta }_{1}^{2}\\ & & \times {L}^{4}+4c{\left(L-N\right)}^{2})+\displaystyle \frac{N({\beta }_{2}(a{\beta }_{2}-2b-2)+4c)}{{\beta }_{1}L}\\ & & +\displaystyle \frac{4N(a(9{\beta }_{1}{L}^{2}+2{\beta }_{2}L-6{\beta }_{1}{LN}+{\beta }_{1}{N}^{2}-2{\beta }_{2}N))}{(N-L)({{Nr}}^{2}+1)}\\ & & -\displaystyle \frac{16a{\beta }_{1}N(2L-N)-2b({{Nr}}^{2}+1)}{{\left({{Nr}}^{2}+1\right)}^{2}}-\displaystyle \frac{16a{\beta }_{1}N(L-N)}{{\left({{Nr}}^{2}+1\right)}^{3}}\Space{0ex}{2.5ex}{0ex}],\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {\theta }_{22}(r)=-\displaystyle \frac{1}{4{\beta }_{1}{\left({{Lr}}^{2}+1\right)}^{2}{\left({{Nr}}^{2}+1\right)}^{4}}[-a{\beta }_{2}^{2}-4{\beta }_{1}{L}^{2}\\ & & \times (9a{\beta }_{1}+{N}^{4}{r}^{10}(1-a{\beta }_{2}+b)+2{N}^{3}{r}^{8}(-5a{\beta }_{2}\\ & & +3a{\beta }_{1}N+5b+2)+{N}^{2}{r}^{6}(-20a{\beta }_{2}+53a{\beta }_{1}N+20b\\ & & +6)+{{Nr}}^{4}(-14a{\beta }_{2}+101a{\beta }_{1}N+14b+4)+{r}^{2}(-3a\\ & & \times ({\beta }_{2}+3{\beta }_{1}N)+3b+1))+L\{4{\beta }_{1}{\left({{Nr}}^{2}+1\right)}^{2}[-3a{\beta }_{2}\\ & & +{N}^{3}{r}^{6}(1-a{\beta }_{2})+3{N}^{2}{r}^{4}(1-3a{\beta }_{2})-3{{Nr}}^{2}(a{\beta }_{2}-1)\\ & & +b({N}^{3}{r}^{6}+9{N}^{2}{r}^{4}+3{{Nr}}^{2}+3)+1]+{\beta }_{2}{r}^{2}({{Nr}}^{2}-1)\\ & & \times {\left({{Nr}}^{2}+1\right)}^{4}(-a{\beta }_{2}+2b+2)+4a{\beta }_{1}^{2}N(3{N}^{4}{r}^{8}+25{N}^{3}\\ & & \times {r}^{6}+31{N}^{2}{r}^{4}-45{{Nr}}^{2}+18)\}+12a{\beta }_{1}^{2}{L}^{3}{r}^{2}({N}^{3}{r}^{6}\\ & & +9{N}^{2}{r}^{4}+19{{Nr}}^{2}+3)-3a{\beta }_{2}^{2}{N}^{5}{r}^{10}+4a{\beta }_{1}{\beta }_{2}{N}^{5}{r}^{8}\\ & & +4a{\beta }_{1}^{2}{N}^{5}{r}^{6}-13a{\beta }_{2}^{2}{N}^{4}{r}^{8}+8a{\beta }_{1}{\beta }_{2}{N}^{4}{r}^{6}+52a{\beta }_{1}^{2}\\ & & \times {N}^{4}{r}^{4}-22a{\beta }_{2}^{2}{N}^{3}{r}^{6}+16a{\beta }_{1}{\beta }_{2}{N}^{3}{r}^{4}+108a{\beta }_{1}^{2}{N}^{3}{r}^{2}\\ & & -36a{\beta }_{1}^{2}{N}^{2}-18a{\beta }_{2}^{2}{N}^{2}{r}^{4}+24a{\beta }_{1}{\beta }_{2}{N}^{2}{r}^{2}+12a{\beta }_{1}{\beta }_{2}N\\ & & -7a{\beta }_{2}^{2}{{Nr}}^{2}+2b{\beta }_{2}+6b{\beta }_{2}{N}^{5}{r}^{10}-4{\beta }_{1}{{bN}}^{5}{r}^{8}+26b{\beta }_{2}{N}^{4}\\ & & \times {r}^{8}-8{\beta }_{1}{{bN}}^{4}{r}^{6}+44b{\beta }_{2}{N}^{3}{r}^{6}-16{\beta }_{1}{{bN}}^{3}{r}^{4}+36b{\beta }_{2}\\ & & \times {N}^{2}{r}^{4}-24{\beta }_{1}{{bN}}^{2}{r}^{2}-12{\beta }_{1}{bN}+14b{\beta }_{2}{{Nr}}^{2}+2{\beta }_{2}-4c\\ & & \times {\left({{Nr}}^{2}+1\right)}^{4}({{Lr}}^{2}({{Nr}}^{2}-1)+3{{Nr}}^{2}+1)+6{\beta }_{2}{N}^{5}{r}^{10}\\ & & -4{\beta }_{1}{N}^{5}{r}^{8}+26{\beta }_{2}{N}^{4}{r}^{8}-16{\beta }_{1}{N}^{4}{r}^{6}+44{\beta }_{2}{N}^{3}{r}^{6}-24{\beta }_{1}\\ & & \times {N}^{3}{r}^{4}+36{\beta }_{2}{N}^{2}{r}^{4}-16{\beta }_{1}{N}^{2}{r}^{2}-4{\beta }_{1}N+14{\beta }_{2}{{Nr}}^{2}].\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {\theta }_{23}(r)={\theta }_{11}^{2}(r){r}^{2}({{Nr}}^{2}+1)[6{\beta }_{1}\{{{kLr}}^{2}+k+({\alpha }^{2}+2\alpha +3)\\ & & \times {{Lr}}^{2}-{\alpha }^{2}{{Nr}}^{2}-2\alpha {{Nr}}^{2}-{{Nr}}^{2}+2\}+{\left(\alpha +1\right)}^{2}{\beta }_{2}{r}^{2}\\ & & \times ({{Nr}}^{2}+1)]+4{\theta }_{11}(r)(3{\beta }_{1}((\alpha +2){{Lr}}^{2}({{Nr}}^{2}+2)\\ & & -(\alpha +1)-{N}^{2}{r}^{4}-2(\alpha +1){{Nr}}^{2}+1)+(\alpha +1){\beta }_{2}\\ & & \times {\left({{Nr}}^{3}+r\right)}^{2})+4({\beta }_{2}+6{\beta }_{1}L+{\beta }_{2}{N}^{2}{r}^{4}-6{\beta }_{1}N\\ & & +2{\beta }_{2}{{Nr}}^{2}),\\ & & {{\rm{\Psi }}}_{21}(r)=4{\beta }_{1}{\left({{Nr}}^{2}+1\right)}^{2}[{{Nr}}^{2}(\alpha -3\alpha a{\beta }_{2}+\alpha {{Lr}}^{2}(a{\beta }_{2}-1)\\ & & +a{\beta }_{2}({{Lr}}^{2}-3)+2)+3\alpha a{\beta }_{2}{{Lr}}^{2}+3a{\beta }_{2}{{Lr}}^{2}-(\alpha +1){N}^{2}{r}^{4}\\ & & \times (a{\beta }_{2}-1)-(\alpha +1){{br}}^{2}(L-N)({{Nr}}^{2}+3)-\alpha {{Lr}}^{2}+1]\\ & & +(\alpha +1){\beta }_{2}{r}^{2}{\left({{Nr}}^{2}+1\right)}^{4}(a{\beta }_{2}-2b-2)+4(\alpha +1)a{\beta }_{1}^{2}{r}^{2}\\ & & \times {\left(L-N\right)}^{2}{\left({{Nr}}^{2}+3\right)}^{2}+4(\alpha +1){{cr}}^{2}{\left({{Nr}}^{2}+1\right)}^{4},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {{\rm{\Psi }}}_{22}(r)=+N(36a{\beta }_{1}+{r}^{2}(3-14a{\beta }_{2}))+2b{\left({{Nr}}^{2}+1\right)}^{2}({{Nr}}^{2}\\ & & +3)+1]+4a{\beta }_{1}^{2}{L}^{2}{\left({{Nr}}^{2}+3\right)}^{2}+a{\beta }_{2}^{2}{N}^{4}{r}^{8}-4a{\beta }_{1}{\beta }_{2}{N}^{4}{r}^{6}\\ & & +4a{\beta }_{1}^{2}{N}^{4}{r}^{4}\ +4a{\beta }_{2}^{2}{N}^{3}{r}^{6}-20a{\beta }_{1}{\beta }_{2}{N}^{3}{r}^{4}+24a{\beta }_{1}^{2}{N}^{3}{r}^{2}\\ & & +36a{\beta }_{1}^{2}{N}^{2}+6a{\beta }_{2}^{2}{N}^{2}{r}^{4}-28a{\beta }_{1}{\beta }_{2}{N}^{2}{r}^{2}-12a{\beta }_{1}{\beta }_{2}N\\ & & +4a{\beta }_{2}^{2}{{Nr}}^{2}-2b{\beta }_{2}-2b{\beta }_{2}{N}^{4}{r}^{8}+4{\beta }_{1}{{bN}}^{4}{r}^{6}-8b{\beta }_{2}{N}^{3}{r}^{6}\\ & & +20{\beta }_{1}{{bN}}^{3}{r}^{4}-12b{\beta }_{2}{N}^{2}{r}^{4}+28{\beta }_{1}{{bN}}^{2}{r}^{2}+12{\beta }_{1}{bN}-8b{\beta }_{2}\\ & & \times {{Nr}}^{2}-{\beta }_{2}+4c{\left({{Nr}}^{2}+1\right)}^{4}-{\beta }_{2}{N}^{3}{r}^{6}(1+4{{Nr}}^{2})+2{\beta }_{1}{N}^{4}{r}^{6}\\ & & +6{\beta }_{1}{N}^{3}{r}^{4}-6{\beta }_{2}{N}^{2}{r}^{4}+6{\beta }_{1}{N}^{2}{r}^{2}+2{\beta }_{1}N-4{\beta }_{2}{{Nr}}^{2},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {{\rm{\Psi }}}_{00}(r)=[2a{\beta }_{2}^{2}-{\beta }_{1}L({N}^{2}({r}^{4}(3-6a{\beta }_{2})+8a{\beta }_{1}{r}^{2})-14a{\beta }_{2}\\ & & +N(24a{\beta }_{1}+{r}^{2}(6-20a{\beta }_{2}))+2b(3{N}^{2}{r}^{4}+10{{Nr}}^{2}+7)\\ & & +3)+4a{\beta }_{1}^{2}{L}^{2}({{Nr}}^{2}+3)+2{{aN}}^{3}({\beta }_{2}^{2}{r}^{6}-3{\beta }_{1}{\beta }_{2}{r}^{4}+2{\beta }_{1}^{2}{r}^{2})\\ & & +12a{\beta }_{1}^{2}{N}^{2}+6a{\beta }_{2}^{2}{N}^{2}{r}^{4}-20a{\beta }_{1}{\beta }_{2}{N}^{2}{r}^{2}-14a{\beta }_{1}{\beta }_{2}N\\ & & +6a{\beta }_{2}^{2}{{Nr}}^{2}-4b{\beta }_{2}-4b{\beta }_{2}{N}^{3}{r}^{6}+6{\beta }_{1}{{bN}}^{3}{r}^{4}-12b{\beta }_{2}{N}^{2}{r}^{4}\\ & & +20{\beta }_{1}{{bN}}^{2}{r}^{2}+14{\beta }_{1}{bN}-12b{\beta }_{2}{{Nr}}^{2}-2{\beta }_{2}+8c{\left({{Nr}}^{2}+1\right)}^{3}\\ & & -2{\beta }_{2}{N}^{3}{r}^{6}+3{\beta }_{1}{N}^{3}{r}^{4}-6{\beta }_{2}{N}^{2}{r}^{4}+6{\beta }_{1}{N}^{2}{r}^{2}+3{\beta }_{1}N-6{\beta }_{2}\\ & & \times {{Nr}}^{2}][4{\beta }_{1}{\left({{Nr}}^{2}+1\right)}^{2}({{Nr}}^{2}(\alpha -3\alpha a{\beta }_{2}+\alpha {{Lr}}^{2}(a{\beta }_{2}-1)\\ & & +a{\beta }_{2}({{Lr}}^{2}-3)+2)+3\alpha a{\beta }_{2}{{Lr}}^{2}+3a{\beta }_{2}{{Lr}}^{2}-(\alpha +1){{Nr}}^{4}\\ & & \times (a{\beta }_{2}-1)-(\alpha +1){{br}}^{2}(L-N)({{Nr}}^{2}+3)-\alpha {{Lr}}^{2}+1)\\ & & +(\alpha +1){\beta }_{2}{r}^{2}{\left({{Nr}}^{2}+1\right)}^{4}(a{\beta }_{2}-2b-2)+4(\alpha +1)a{\beta }_{1}^{2}{r}^{2}\\ & & \times {\left(L-N\right)}^{2}{\left({{Nr}}^{2}+3\right)}^{2}+4(\alpha +1){{cr}}^{2}{\left({{Nr}}^{2}+1\right)}^{4}]\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{1}(r)=\displaystyle \frac{1}{{\left({{Nr}}^{2}+1\right)}^{3}({\beta }_{1}+{\beta }_{1}{{Lr}}^{2})}[a{\beta }_{2}^{2}-4{\beta }_{1}L(-3a{\beta }_{2}+{N}^{3}\\ & & \times ({r}^{6}(1-a{\beta }_{2})+2a{\beta }_{1}{r}^{4})+{N}^{2}({r}^{4}(3-5a{\beta }_{2})+12a{\beta }_{1}{r}^{2})\\ & & +N(18a{\beta }_{1}+{r}^{2}(3-7a{\beta }_{2}))+b{\left({{Nr}}^{2}+1\right)}^{2}({{Nr}}^{2}+3)+1)\\ & & +4a{\beta }_{1}^{2}{L}^{2}{\left({{Nr}}^{2}+3\right)}^{2}+a{\beta }_{2}^{2}{N}^{4}{r}^{8}-4a{\beta }_{1}{\beta }_{2}{N}^{4}{r}^{6}+4a{\beta }_{1}^{2}{N}^{4}{r}^{4}\\ & & +4a{\beta }_{2}^{2}{N}^{3}{r}^{6}-20a{\beta }_{1}{\beta }_{2}{N}^{3}{r}^{4}+24a{\beta }_{1}^{2}{N}^{3}{r}^{2}+36a{\beta }_{1}^{2}{N}^{2}\\ & & +6a{\beta }_{2}^{2}{N}^{2}{r}^{4}-28a{\beta }_{1}{\beta }_{2}{N}^{2}{r}^{2}-12a{\beta }_{1}{\beta }_{2}N+4a{\beta }_{2}^{2}{{Nr}}^{2}-2b{\beta }_{2}\\ & & -2b{\beta }_{2}{N}^{4}{r}^{8}+4{\beta }_{1}{{bN}}^{4}{r}^{6}-8b{\beta }_{2}{N}^{3}{r}^{6}+20{\beta }_{1}{{bN}}^{3}{r}^{4}-12b{\beta }_{2}\\ & & \times {N}^{2}{r}^{4}+28{\beta }_{1}{{bN}}^{2}{r}^{2}+12{\beta }_{1}{bN}-8b{\beta }_{2}{{Nr}}^{2}-2{\beta }_{2}+4b\\ & & \times {\left({{Nr}}^{2}+1\right)}^{4}-2{\beta }_{2}{N}^{4}{r}^{8}+4{\beta }_{1}{N}^{4}{r}^{6}-8{\beta }_{2}{N}^{3}{r}^{6}+12{\beta }_{1}{N}^{3}{r}^{4}\\ & & -12{\beta }_{2}{N}^{2}{r}^{4}+12{\beta }_{1}{N}^{2}{r}^{2}\\ & & +4{\beta }_{1}N-8{\beta }_{2}{{Nr}}^{2}],\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{2}(r)=2[\{(N-L)({{aN}}^{4}{\beta }_{2}^{2}{r}^{8}-2{{bN}}^{4}{\beta }_{2}{r}^{8}-2{N}^{4}{\beta }_{2}{r}^{8}+4{{aN}}^{3}\\ & & \times {\beta }_{2}^{2}{r}^{6}-8{{bN}}^{3}{\beta }_{2}{r}^{6}-8{N}^{3}{\beta }_{2}{r}^{6}-4{{aN}}^{4}{\beta }_{1}^{2}{r}^{4}+6{{aN}}^{2}{\beta }_{2}^{2}{r}^{4}\\ & & -8{{bN}}^{3}{\beta }_{1}{r}^{4}-12{{bN}}^{2}{\beta }_{2}{r}^{4}-12{N}^{2}{\beta }_{2}{r}^{4}+8{{aN}}^{3}{\beta }_{1}{\beta }_{2}{r}^{4}-40a\\ & & \times {N}^{3}{\beta }_{1}^{2}{r}^{2}+4{aN}{\beta }_{2}^{2}{r}^{2}-16{{bN}}^{2}{\beta }_{1}{r}^{2}-8{bN}{\beta }_{2}{r}^{2}-8N{\beta }_{2}{r}^{2}\\ & & +16{{aN}}^{2}{\beta }_{1}{\beta }_{2}{r}^{2}+4b{\left({{Nr}}^{2}+1\right)}^{4}-84{{aN}}^{2}{\beta }_{1}^{2}+4{{aL}}^{2}(2{N}^{3}{r}^{6}\\ & & +17{N}^{2}{r}^{4}+36{{Nr}}^{2}+9){\beta }_{1}^{2}+a{\beta }_{2}^{2}-8{bN}{\beta }_{1}-2b{\beta }_{2}+8{aN}{\beta }_{1}\\ & & \times {\beta }_{2}-2{\beta }_{2}-4L{\beta }_{1}[((1-a{\beta }_{2}){r}^{8}+2a{\beta }_{1}{r}^{6}){N}^{4}+4{r}^{4}\\ & & \times ((1-2a{\beta }_{2}){r}^{2}+4a{\beta }_{1}){N}^{3}+((6-16a{\beta }_{2}){r}^{4}+26a{\beta }_{1}{r}^{2}){N}^{2}\\ & & -4((3a{\beta }_{2}-1){r}^{2}+3a{\beta }_{1})N+b{\left({{Nr}}^{2}+1\right)}^{2}({N}^{2}{r}^{4}+6{{Nr}}^{2}\\ & & +3)-3a{\beta }_{2}+1])\}]{r}^{2},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{3}(r)={{aN}}^{4}{\beta }_{2}^{2}{r}^{8}-2{{bN}}^{4}{\beta }_{2}{r}^{8}-2{N}^{4}{\beta }_{2}{r}^{8}+4{{aN}}^{3}{\beta }_{2}^{2}{r}^{6}\\ & & +4{{bN}}^{4}{\beta }_{1}{r}^{6}+4{N}^{4}{\beta }_{1}{r}^{6}-8{{bN}}^{3}{\beta }_{2}{r}^{6}-8{N}^{3}{\beta }_{2}{r}^{6}-4{{aN}}^{4}{\beta }_{1}\\ & & \times {\beta }_{2}{r}^{6}+4{{aN}}^{4}{\beta }_{1}^{2}{r}^{4}+6{{aN}}^{2}{\beta }_{2}^{2}{r}^{4}+20{{bN}}^{3}{\beta }_{1}{r}^{4}+12{N}^{3}{\beta }_{1}{r}^{4}\\ & & -12{{bN}}^{2}{\beta }_{2}{r}^{4}-12{N}^{2}{\beta }_{2}{r}^{4}-20{{aN}}^{3}{\beta }_{1}{\beta }_{2}{r}^{4}+24{{aN}}^{3}{\beta }_{1}^{2}{r}^{2}\\ & & +4{aN}{\beta }_{2}^{2}{r}^{2}+28{{bN}}^{2}{\beta }_{1}{r}^{2}+12{N}^{2}{\beta }_{1}{r}^{2}-8{bN}{\beta }_{2}{r}^{2}-8N{\beta }_{2}{r}^{2}\\ & & -28{{aN}}^{2}{\beta }_{1}{\beta }_{2}{r}^{2}+4b{\left({{Nr}}^{2}+1\right)}^{4}+36{{aN}}^{2}{\beta }_{1}^{2}+4{{aL}}^{2}\\ & & \times {\left({{Nr}}^{2}+3\right)}^{2}{\beta }_{1}^{2}+a{\beta }_{2}^{2}+12{bN}{\beta }_{1}+4N{\beta }_{1}-2b{\beta }_{2}-12{aN}{\beta }_{1}{\beta }_{2}\\ & & -2{\beta }_{2}-4L{\beta }_{1}[\{(1-a{\beta }_{2}){r}^{6}+2a{\beta }_{1}{r}^{4}\}{N}^{3}+[(3-5a{\beta }_{2}){r}^{4}\\ & & +12a{\beta }_{1}{r}^{2}]{N}^{2}+((3-7a{\beta }_{2}){r}^{2}+18a{\beta }_{1})N+b{\left({{Nr}}^{2}+1\right)}^{2}\\ & & \times ({{Nr}}^{2}+3)-3a{\beta }_{2}+1],\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{4}(r)=-2N({{Lr}}^{2}+1)[4{{cr}}^{2}(\alpha +1){\left({{Nr}}^{2}+1\right)}^{4}+{r}^{2}(\alpha +1){\beta }_{2}\\ & & \times (a{\beta }_{2}-2b-2){\left({{Nr}}^{2}+1\right)}^{4}+4{\beta }_{1}({N}^{2}(\alpha +1)(1-a{\beta }_{2}-1){r}^{4}\\ & & -L\alpha {r}^{2}-b(L-N)({{Nr}}^{2}+3)(\alpha +1){r}^{2}+3{aL}{\beta }_{2}{r}^{2}+3{aL}\alpha {\beta }_{2}{r}^{2}\\ & & +N[L\alpha (a{\beta }_{2}-1){r}^{2}+\alpha +a({{Lr}}^{2}-3){\beta }_{2}-3a\alpha {\beta }_{2}+2]{r}^{2}+1)\\ & & \times {\left({{Nr}}^{2}+1\right)}^{2}+4a{\left(L-N\right)}^{2}{r}^{2}{\left({{Nr}}^{2}+3\right)}^{2}(\alpha +1){\beta }_{1}^{2}]{f}_{5}(r)\\ & & +2L({{Nr}}^{2}+1)[4{{cr}}^{2}(\alpha +1){\left({{Nr}}^{2}+1\right)}^{4}+{r}^{2}(\alpha +1){\beta }_{2}(-2b\\ & & +a{\beta }_{2}-2){\left({{Nr}}^{2}+1\right)}^{4}+4{\beta }_{1}(-{N}^{2}(\alpha +1)(a{\beta }_{2}-1){r}^{4}\\ & & -L\alpha {r}^{2}-b(L-N)({{Nr}}^{2}+3)(\alpha +1){r}^{2}+3{aL}{\beta }_{2}{r}^{2}+3{aL}\alpha \\ & & \times {\beta }_{2}{r}^{2}+N[L\alpha (a{\beta }_{2}-1){r}^{2}+\alpha +a({{Lr}}^{2}-3){\beta }_{2}-3a\alpha {\beta }_{2}+2]\\ & & \times {r}^{2}+1){\left({{Nr}}^{2}+1\right)}^{2}+4a{\left(L-N\right)}^{2}{r}^{2}{\left({{Nr}}^{2}+3\right)}^{2}(\alpha +1){\beta }_{1}^{2}]\\ & & \times {f}_{6}(r)+{f}_{44}(r),\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{4}(r)=-2({{Lr}}^{2}+1)({{Nr}}^{2}+1)[{{aN}}^{4}{\beta }_{2}^{2}{r}^{8}-2{{bN}}^{4}{\beta }_{2}{r}^{8}-{N}^{4}\\ & & \times {\beta }_{2}{r}^{8}+4{{aN}}^{3}{\beta }_{2}^{2}{r}^{6}+4{{bN}}^{4}{\beta }_{1}{r}^{6}+2{N}^{4}{\beta }_{1}{r}^{6}-8{{bN}}^{3}{\beta }_{2}{r}^{6}\\ & & -4{N}^{3}{\beta }_{2}{r}^{6}-4{{aN}}^{4}{\beta }_{1}{\beta }_{2}{r}^{6}+4{{aN}}^{4}{\beta }_{1}^{2}{r}^{4}+6{{aN}}^{2}{\beta }_{2}^{2}{r}^{4}+20b\\ & & \times {N}^{3}{\beta }_{1}{r}^{4}+6{N}^{3}{\beta }_{1}{r}^{4}-12{{bN}}^{2}{\beta }_{2}{r}^{4}-6{N}^{2}{\beta }_{2}{r}^{4}-20{{aN}}^{3}\\ & & \times {\beta }_{1}{\beta }_{2}{r}^{4}+24{{aN}}^{3}{\beta }_{1}^{2}{r}^{2}+4{aN}{\beta }_{2}^{2}{r}^{2}+28{{bN}}^{2}{\beta }_{1}{r}^{2}+6{N}^{2}{\beta }_{1}{r}^{2}\\ & & -8{bN}{\beta }_{2}{r}^{2}-4N{\beta }_{2}{r}^{2}-28{{aN}}^{2}{\beta }_{1}{\beta }_{2}{r}^{2}+4c{\left({{Nr}}^{2}+1\right)}^{4}+36a\\ & & \times {N}^{2}{\beta }_{1}^{2}+4{{aL}}^{2}{\left({{Nr}}^{2}+3\right)}^{2}{\beta }_{1}^{2}+a{\beta }_{2}^{2}+12{bN}{\beta }_{1}+2N{\beta }_{1}-2b{\beta }_{2}\\ & & -12{aN}{\beta }_{1}{\beta }_{2}-{\beta }_{2}-2L{\beta }_{1}(\{(1-2a{\beta }_{2}){r}^{6}+4a{\beta }_{1}{r}^{4}\}{N}^{3}\\ & & +[(3-10a{\beta }_{2}){r}^{4}+24a{\beta }_{1}{r}^{2}]{N}^{2}+((3-14a{\beta }_{2}){r}^{2}+36a{\beta }_{1})N\\ & & +2b{\left({{Nr}}^{2}+1\right)}^{2}({{Nr}}^{2}+3)-6a{\beta }_{2}+1)]{f}_{7}(r)+3({{Lr}}^{2}+1)\\ & & \times ({{Nr}}^{2}+1)[4{{cr}}^{2}(\alpha +1){\left({{Nr}}^{2}+1\right)}^{4}+{r}^{2}(\alpha +1){\beta }_{2}(-2b\\ & & +a{\beta }_{2}-2){\left({{Nr}}^{2}+1\right)}^{4}+4{\beta }_{1}({N}^{2}(\alpha +1)(1-a{\beta }_{2}){r}^{4}-L\alpha {r}^{2}\\ & & -b(L-N)({{Nr}}^{2}+3)(\alpha +1){r}^{2}+3{aL}{\beta }_{2}{r}^{2}+3{aL}\alpha {\beta }_{2}{r}^{2}+N\\ & & \times (L\alpha (a{\beta }_{2}-1){r}^{2}+\alpha +a({{Lr}}^{2}-3){\beta }_{2}-3a\alpha {\beta }_{2}+2){r}^{2}+1)\\ & & \times {\left({{Nr}}^{2}+1\right)}^{2}+4a{\left(L-N\right)}^{2}{r}^{2}{\left({{Nr}}^{2}+3\right)}^{2}(\alpha +1){\beta }_{1}^{2}]{f}_{8}(r)\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{5}(r)=[{{aN}}^{4}{\beta }_{2}^{2}{r}^{8}-2{{bN}}^{4}{\beta }_{2}{r}^{8}-{N}^{4}{\beta }_{2}{r}^{8}+4{{aN}}^{3}{\beta }_{2}^{2}{r}^{6}\\ & & +4{{bN}}^{4}{\beta }_{1}{r}^{6}+2{N}^{4}{\beta }_{1}{r}^{6}-8{{bN}}^{3}{\beta }_{2}{r}^{6}-4{N}^{3}{\beta }_{2}{r}^{6}\\ & & -4{{aN}}^{4}{\beta }_{1}{\beta }_{2}{r}^{6}+4{{aN}}^{4}{\beta }_{1}^{2}{r}^{4}+6{{aN}}^{2}{\beta }_{2}^{2}{r}^{4}+20{{bN}}^{3}{\beta }_{1}{r}^{4}\\ & & +6{N}^{3}{\beta }_{1}{r}^{4}-12{{bN}}^{2}{\beta }_{2}{r}^{4}-6{N}^{2}{\beta }_{2}{r}^{4}-20{{aN}}^{3}{\beta }_{1}{\beta }_{2}{r}^{4}\\ & & +24{{aN}}^{3}{\beta }_{1}^{2}{r}^{2}+4{aN}{\beta }_{2}^{2}{r}^{2}+28{{bN}}^{2}{\beta }_{1}{r}^{2}+6{N}^{2}{\beta }_{1}{r}^{2}-8{bN}\\ & & \times {\beta }_{2}{r}^{2}-4N{\beta }_{2}{r}^{2}-28{{aN}}^{2}{\beta }_{1}{\beta }_{2}{r}^{2}+4c{\left({{Nr}}^{2}+1\right)}^{4}+36{{aN}}^{2}\\ & & \times {\beta }_{1}^{2}+4{{aL}}^{2}{\left({{Nr}}^{2}+3\right)}^{2}{\beta }_{1}^{2}+a{\beta }_{2}^{2}+12{bN}{\beta }_{1}+2N{\beta }_{1}-2b{\beta }_{2}\\ & & -12{aN}{\beta }_{1}{\beta }_{2}-{\beta }_{2}-2L{\beta }_{1}(((1-2a{\beta }_{2}){r}^{6}+4a{\beta }_{1}{r}^{4}){N}^{3}\\ & & +((3-10a{\beta }_{2}){r}^{4}+24a{\beta }_{1}{r}^{2}){N}^{2}+((3-14a{\beta }_{2}){r}^{2}+36a{\beta }_{1})\\ & & \times N+2b{\left({{Nr}}^{2}+1\right)}^{2}({{Nr}}^{2}+3)-6a{\beta }_{2}+1)]{r}^{2},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{6}(r)=[{{aN}}^{4}{\beta }_{2}^{2}{r}^{8}-2{{bN}}^{4}{\beta }_{2}{r}^{8}-{N}^{4}{\beta }_{2}{r}^{8}+4{{aN}}^{3}{\beta }_{2}^{2}{r}^{6}+4{{bN}}^{4}\\ & & \times {\beta }_{1}{r}^{6}+2{N}^{4}{\beta }_{1}{r}^{6}-8{{bN}}^{3}{\beta }_{2}{r}^{6}-4{N}^{3}{\beta }_{2}{r}^{6}-4{{aN}}^{4}{\beta }_{1}{\beta }_{2}{r}^{6}\\ & & +4{{aN}}^{4}{\beta }_{1}^{2}{r}^{4}+6{{aN}}^{2}{\beta }_{2}^{2}{r}^{4}+20{{bN}}^{3}{\beta }_{1}{r}^{4}+6{N}^{3}{\beta }_{1}{r}^{4}-12{{bN}}^{2}\\ & & \times {\beta }_{2}{r}^{4}-6{N}^{2}{\beta }_{2}{r}^{4}-20{{aN}}^{3}{\beta }_{1}{\beta }_{2}{r}^{4}+24{{aN}}^{3}{\beta }_{1}^{2}{r}^{2}+4{aN}{\beta }_{2}^{2}{r}^{2}\\ & & +28{{bN}}^{2}{\beta }_{1}{r}^{2}+6{N}^{2}{\beta }_{1}{r}^{2}-8{bN}{\beta }_{2}{r}^{2}-4N{\beta }_{2}{r}^{2}-28{{aN}}^{2}{\beta }_{1}\\ & & \times {\beta }_{2}{r}^{2}+4c{\left({{Nr}}^{2}+1\right)}^{4}+36{{aN}}^{2}{\beta }_{1}^{2}+4{{aL}}^{2}{\left({{Nr}}^{2}+3\right)}^{2}{\beta }_{1}^{2}\\ & & +a{\beta }_{2}^{2}+12{bN}{\beta }_{1}+2N{\beta }_{1}-2b{\beta }_{2}-12{aN}{\beta }_{1}{\beta }_{2}-{\beta }_{2}-2L{\beta }_{1}\\ & & \times (((1-2a{\beta }_{2}){r}^{6}+4a{\beta }_{1}{r}^{4}){N}^{3}+((3-10a{\beta }_{2}){r}^{4}+24a{\beta }_{1}{r}^{2})\\ & & \times {N}^{2}+((3-14a{\beta }_{2}){r}^{2}+36a{\beta }_{1})N+2b{\left({{Nr}}^{2}+1\right)}^{2}({{Nr}}^{2}+3)\\ & & -6a{\beta }_{2}+1)]{r}^{2},\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{7}(r)=[4{cr}(\alpha +1){\left({{Nr}}^{2}+1\right)}^{4}+r(\alpha +1){\beta }_{2}(-2b+a{\beta }_{2}-2)\\ & & \times {\left({{Nr}}^{2}+1\right)}^{4}+4r{\beta }_{1}(-2{N}^{2}(\alpha +1)(a{\beta }_{2}-1){r}^{2}-b(L-N)\\ & & \times (2{{Nr}}^{2}+3)(\alpha +1)+N(2L\alpha (a{\beta }_{2}-1){r}^{2}+\alpha +a(2{{Lr}}^{2}-3)\\ & & \times {\beta }_{2}-3a\alpha {\beta }_{2}+2)+L(3a{\beta }_{2}+\alpha (3a{\beta }_{2}-1))){\left({{Nr}}^{2}+1\right)}^{2}\\ & & +8{Nr}{\beta }_{1}(-{N}^{2}(\alpha +1)(a{\beta }_{2}-1){r}^{4}-L\alpha {r}^{2}-b(L-N)\\ & & \times ({{Nr}}^{2}+3)(\alpha +1){r}^{2}+3{aL}{\beta }_{2}{r}^{2}+3{aL}\alpha {\beta }_{2}{r}^{2}+N(L\alpha (a{\beta }_{2}\\ & & -1){r}^{2}+\alpha +a({{Lr}}^{2}-3){\beta }_{2}-3a\alpha {\beta }_{2}+2){r}^{2}+1)({{Nr}}^{2}+1)\\ & & +4a{\left(L-N\right)}^{2}r{\left({{Nr}}^{2}+3\right)}^{2}(\alpha +1){\beta }_{1}^{2}+8a{\left(L-N\right)}^{2}{{Nr}}^{3}\\ & & \times ({{Nr}}^{2}+3)(\alpha +1){\beta }_{1}^{2}+16{cN}{\left({{Nr}}^{3}+r\right)}^{3}(\alpha +1)+4N\\ & & \times {\left({{Nr}}^{3}+r\right)}^{3}(\alpha +1){\beta }_{2}(-2b+a{\beta }_{2}-2)]r,\end{array}\end{eqnarray*}$$

$$\begin{eqnarray*}\begin{array}{rcl} & & {f}_{8}(r)=[{{aN}}^{4}{\beta }_{2}^{2}{r}^{8}-2{{bN}}^{4}{\beta }_{2}{r}^{8}-{N}^{4}{\beta }_{2}{r}^{8}+4{{aN}}^{3}{\beta }_{2}^{2}{r}^{6}+4{{bN}}^{4}\\ & & \times {\beta }_{1}{r}^{6}+2{N}^{4}{\beta }_{1}{r}^{6}-8{{bN}}^{3}{\beta }_{2}{r}^{6}-4{N}^{3}{\beta }_{2}{r}^{6}-4{{aN}}^{4}{\beta }_{1}{\beta }_{2}{r}^{6}\\ & & +4{{aN}}^{4}{\beta }_{1}^{2}{r}^{4}+6{{aN}}^{2}{\beta }_{2}^{2}{r}^{4}+20{{bN}}^{3}{\beta }_{1}{r}^{4}+6{N}^{3}{\beta }_{1}{r}^{4}-12{{bN}}^{2}\\ & & \times {\beta }_{2}{r}^{4}-6{N}^{2}{\beta }_{2}{r}^{4}-20{{aN}}^{3}{\beta }_{1}{\beta }_{2}{r}^{4}+24{{aN}}^{3}{\beta }_{1}^{2}{r}^{2}+4{aN}{\beta }_{2}^{2}{r}^{2}\\ & & +28{{bN}}^{2}{\beta }_{1}{r}^{2}+6{N}^{2}{\beta }_{1}{r}^{2}-8{bN}{\beta }_{2}{r}^{2}-4N{\beta }_{2}{r}^{2}-28{{aN}}^{2}{\beta }_{1}{\beta }_{2}{r}^{2}\\ & & +4c{\left({{Nr}}^{2}+1\right)}^{4}+36{{aN}}^{2}{\beta }_{1}^{2}+4{{aL}}^{2}{\left({{Nr}}^{2}+3\right)}^{2}{\beta }_{1}^{2}+a{\beta }_{2}^{2}\\ & & +12{bN}{\beta }_{1}+2N{\beta }_{1}-2b{\beta }_{2}-12{aN}{\beta }_{1}{\beta }_{2}-{\beta }_{2}-2L{\beta }_{1}\\ & & \times \{((1-2a{\beta }_{2}){r}^{6}+4a{\beta }_{1}{r}^{4}){N}^{3}+((3-10a{\beta }_{2})\times {r}^{4}+24a\\ & & \times {\beta }_{1}{r}^{2}){N}^{2}+((3-14a{\beta }_{2}){r}^{2}+36a{\beta }_{1})N+2b{\left({{Nr}}^{2}+1\right)}^{2}\\ & & \times ({{Nr}}^{2}+3)-6a{\beta }_{2}+1\}].\end{array}\end{eqnarray*}$$