Imprints of dark matter on the structural properties of minimally deformed compact stars

Understanding the nature of the dark content of our universe remains largely unexplained, posing a significant challenge for both theoretical and observational cosmology. The $\Lambda$-cold dark matter (DM) model suggests that DM constitutes about $27\%$ of the total cosmic mass density [1, 2], yet it remains elusive to current astronomical instrumentation. The initial evidence regarding the existence of DM emerges from discrepancies between predicted and observed rotation curves of spiral galaxies [3, 4]. Neutralino has appeared as a primary candidate among the potential explanations for the principal constituent and origin of DM. It is classified within the group of the lightest supersymmetric particles [5, 6]. The investigation by Hadjimichef et al. [7] proposed a novel DM-based model for modeling ultra-dense stellar configurations, including white dwarfs, neutron stars (NS), and black holes. This model considers DM as a standard model fermion gauge singlets against a dark energy background within the context of pseudo-complex general relativity (GR). On the other hand, the mass-radius relationship can be impacted by the presence of DM on the periphery of NS, as discussed in [8]. Rezaei et al. [9, 10] used a polytropic EoS to study how spin-polarized self-interacting DM affects NS structure. Building on earlier research, scientists investigated the effects of concentrated DM on the structures of self-gravitational entities as detailed in [11, 12]. Furthermore, Ciarcelluti et al. [13] explored the potential presence of a DM core within NS.

Analytical closed-form solutions of the field equations of general relativity (GR) are essential resources for comprehending the strong gravity regime in ultra-dense stars. Studies show that GR provides a well-established toolbox of solutions for isotropic objects, relying on simplified assumptions of spherical symmetry and a staticity. However, a significant portion of these solutions lack physical relevance and often fail to pass basic tests based on astrophysical observations [14]. On the other hand, incorporating anisotropic matter content as an approximation to study stellar systems has consistently garnered attention and remains a dynamically evolving area of study in astrophysical scenarios. Herrera et al. [15] discussed various theoretical lines of evidence suggesting that diverse and intriguing physical phenomena can induce a significant prevalence of local anisotropies across low and high-density regimes. Supplementing the analysis of high-density regimes, Ruderman’s investigations into the formation of more realistic stellar objects offer insights into the potential anisotropy of nuclear matter [16]. According to these studies, nuclear matter may exhibit anisotropic behavior, especially at high-density levels $\thicksim 3\times 10^{17}\text{kg/m}^{3}$. In these regimes, a relativistic treatment of nuclear interactions becomes significant. The origin of anisotropic behavior is inherently linked to the deviation from the standard pressure term within the stress-energy tensor (SET). In an isotropic self-gravitational distribution, the radial pressure ($P_{r}$) and tangential pressure ($P_{t}$) are equal: $P_{r}=P_{t}$. However, stellar fluids featuring anisotropic matter content are characterized by a SET where $P_{r}\neq P_{t}$. To enhance the understanding of gravitational interactions and the formation of stellar structures for a broader context, it is valuable to acknowledge the studies presented in [17, 18, 19, 20, 21, 22, 23, 24, 25, 26]

Building upon the seminal research conducted by Bowers and Liang [27], a significant amount of attempts have been dedicated to further elucidating the behavior and characteristics of astrophysical entities filled with anisotropic matter content. Additionally, the incorporation of matter configurations featuring anisotropic content has been extensively probed in the study of stellar structures [28, 29, 30, 31, 32, 33, 34, 35, 36]. In this direction, the research conducted by Lake and Delgaty [14] represented a significant advancement in our understanding of stellar interiors. Through the utilization of the spherically symmetric metric ansatz with isotropic fluid, their investigation revealed that only a small fraction (9 out of 127) of known isotropic solutions were physically relevant. This significant constraint triggered a surge in research, focusing on unraveling the fresh insights brought about by local anisotropies within compact configurations. Anisotropic behavior in self-gravitational stellar fluids can be attributed to several physical mechanisms, including viscosity, rotation, the presence of superfluid, magnetic field, or the existence of a solid core (see [15] and references therein). Anisotropies can also be induced by certain phenomena like phase transitions or pion condensation, among other contributing factors [37, 38]. Understanding the behavior of astrophysical dark configurations under an anisotropic framework is a well-studied topic in cosmology and astrophysics. This pressure anisotropy holds particular significance in the analysis of ultra-compact entities such as black holes, NS, wormholes, and other unique stellar formations [39, 40, 41]. Researchers have investigated these effects using both standard four-dimensional metric ansatzes [42] and within the framework of higher-dimensional braneworld scenarios [43, 44, 45].

The inclusion of anisotropy as a fluid approximation in the modeling of collapsing stellar structures such as neutron stars or quark stars give rise to several interesting and significant features: i) it increases the stability of the stellar fluid; ii) it enables the construction of highly compact stellar objects; iii) the gravitational surface redshift achieves higher magnitudes compared to its isotropic equivalent; iv) the inclusion of an additional term increases the hydrostatic equilibrium, which acts to counterbalance gravitational collapse into a point singularity provided that the anisotropic scalar $\Pi$, remains positive throughout. Despite the increasing interest in local anisotropies, there are still several significant challenges in effectively including them into distributions of stellar matter. Here are some essential questions that require attention:

1. 1.

   What are the efficient approaches for including the effects of localized pressure variations (anisotropies) for describing the interior of compact configurations?

2. 2.

   Does the feasibility of the resultant solution rely on the particular method employed to incorporate anisotropy?

3. 3.

   Do the numerical results obtained from anisotropic models align with experimental data from observations of real stellar structures?

An EMT with varying pressure in the principal directions of the stellar fluid or the inclusion of an electromagnetic field are the most straightforward approaches to generate anisotropies into the self-gravitational objects. Interestingly, the approach of decoupling the spherical stellar distribution using the minimal geometric deformation (MGD), introduced within the framework of GR emerges as a natural means to introduce anisotropies in compact stars [46, 47]. Originating in Randall-Sundrum braneworld scenarios [48, 49, 50, 51, 52, 53, 54, 55, 56], the notion of MGD decoupling has a long and well-established history. This scheme has also found applications in constructing black hole solutions by using the minimal deformation (radial metric deformation) in the classical Schwarzschild metric. Following its success in different frameworks, the MGD scheme was successfully adapted for use within the arena of GR and enables the researchers to anisotropize the isotropic stellar models [46, 47]. As we will explore in detail in the next section, the MGD approach relies on two fundamental components: i) the system includes two independent gravitational-field sources terms represented by $T_{\mu\nu}$ and $\Theta_{\mu\nu}$; ii) The MGD approach specifically deforms the radial metric potential through a geometric deformation. This deformation enables the decoupling of the anisotropic gravitational source into two sets of differential equations, each corresponding to a distinct field source. Importantly, the structure of gravitational decoupling usually considers $\widetilde{T}_{\mu\nu}$ as a known isotropic matter content. The function of additional-field source $\Theta_{\mu\nu}$ is therefore to insert localized anisotropies within the considered system. Consequently, this framework can be regarded as a means of extending isotropic solutions to anisotropic domains. Here, both the particular geometric deformation and the constituents of the $\Theta_{\mu\nu}$-field source are initially unknown. The primary objective becomes establishing additional constraints to determine them. Numerous strategies have been suggested to tackle this issue, such as mimic constraint (density-like and pressure-like constraints), anisotropic mechanism, or imposing a well-behaved decoupling function.

The MGD-based decoupling approach has shown remarkable versatility, with a significant increase in its applications across various scenarios [57, 58, 59, 60, 61, 62, 63]. Significantly, progress has been achieved in both the inverse problem and its extended counterpart [64, 65]. In light of the outstanding adaptability that the MGD decoupling offers, the onset of the $\Theta_{\mu\nu}$-field source remains an open question. In this direction, the authors of [66, 67] analyzed the intriguing possibility of interpreting $\Theta_{\mu\nu}$ as DM, while its contribution within the cosmological scenarios, such as cold DM and $\Lambda$ cold DM model has been presented in [68]. Motivated by the previous research, this study explores the possible existence of DM stellar structures within the arena of GR by employing MGD decoupling. Choosing suitable parameters that align with observational data ensure that the resulting DM solutions are physically realistic and mathematically sound. Our approach considers that the temporal component of the $\Theta$-field source mimics the Einasto DM profile. Consequently, the deformation function, $f(r)$ incorporates this DM contribution to the minimally deformed Adler-Finch-Skea metric and the remaining components of the SET. The mimicking of the Einasto DM profile with the temporal component of $\Theta$-sector makes this work different from the previous studies [67, 69, 70] that consider the isothermal DM profile for modeling stellar structures. The Einasto density model has been proven significant in modeling the compact configurations emersed DM haloes [71].

In this work, we have examined the influence of DM on the structural features of a spherically symmetric, anisotropic stellar configuration by employing the MGD version of gravitational decoupling. The Einasto profile, which is a popular framework for modeling DM halos in galaxies, is employed in our model to describe the DM component. Although the existence of DM may raise questions about the validity of the vacuum exterior, we want to emphasize that our DM profile is limited to the stellar interior and does not extend significantly beyond the stellar radius. The DM distribution is significantly compressed by the strong gravitational potential of the star core, even though this profile typically extends to large radii. The dense surroundings inside the star serve as a gravitational trap, confining the DM particles and preventing them from leaving. This means that most of the mass of DM is contained within the star radius, and only small amounts are found outside of it. Therefore, the impact of DM on the exterior metric is negligible, validating the use of the vacuum Schwarzschild metric. Several observational studies have investigated the distribution of dark matter (DM) within stellar interiors in recent years. The authors of [72] explore the impact of DM on the mass-radius relationship of neutron stars (NS). They show that the observable characteristics of NS can be affected by the presence of DM, providing indirect evidence for the confinement of DM within the stellar interior. Furthermore, the studies discussed in [73] examine the implications of DM annihilation on the stellar interiors and indicate how DM particles are confined within the core of stellar distributions.

Lavallaz and Fairbairn [74] studied the accretion of DM onto the internal features of NS. This accretion mechanism supports the confinement of DM within stars. Freese et al. [75] probed potential effects of DM on the evolution of self-gravitational compact stars. They investigated how DM particles confined within stellar configurations can affect their physical characteristics, suggesting the presence of DM in stellar interiors. Furthermore, the increased mass concentration has been already examined in [76] by taking into account that the Kalb-Ramond field influences the matter field. A primary difference between this scheme and the present investigation is the inclusion of the Kalb-Ramond field, which influences the exterior metric. This contrasts with the procedure of MGD-decoupling, where the contributions of extra-field source $\Theta_{\mu\nu}$ are neglected [47]. Thus, the outer manifold remains characterized by a vacuum spacetime. In this respect, Tello-Ortiz [67] constructed minimally deformed DM stellar configurations based on the GD scheme by matching the interior metric with the vacuum Schwarzschild metric. On the other hand, Maurya et al. [70] presented some viable models describing the spherically symmetric MGD-based, DM stelar structures by using the pseudo-isothermal density (PID) profile. Furthermore, the evolution of relativistic distributions with DM induced anisotropy based on the PID profile has been discussed in [69]. These studies were based on the matching of the interior peacetime with the outer Schwarzschild metric. Moreover, several other investigations justify the matching of the interior metric with exterior vacuum geometry by suppressing the contributions of the $\Theta_{\mu\nu}$ sector [77, 78, 79, 80].

The remainder of this paper is structured as follows: The relevance of the Einasto density model in modeling the DM haloes within the galactic structures along with the key concepts are presented in Sec. II. In Sec. III, we provide a concise overview of the Class I approach and the notion of gravitational decoupling utilizing minimal geometric deformation. Section IV covers the investigation of anisotropic stellar solutions endowed with spherical symmetry by employing the familiar Adler-Finch–Skea metric and Einasto density profiles through the MGD approach. Section V captures the influence of the decoupling parameter $\chi$ on the thermodynamical features of the minimally deformed DM Adler-Finch-Skea metric. Finally, the last section describes the main findings of this study.

A metric is classified as Class I, indicating it can be embedded in a 5-dimensional pseudo-Euclidean space, when there exists a symmetric tensor $K_{\mu\nu}$ serving as the second fundamental form, which satisfies the Gauss–Codazzi equations

		$\displaystyle R_{\mu\nu\sigma\epsilon}=\varepsilon\left(K_{\mu\sigma}K_{\nu\epsilon}-K_{\mu\epsilon}K_{\nu\sigma}\right),$		(6)
		$\displaystyle\nabla_{\sigma}K_{\mu\nu}=\nabla_{\nu}K_{\mu\sigma},$		(7)

where $\varepsilon$ can be positive or negative depending on whether the normal vector to the manifold is timelike (-) or spacelike (+). Furthermore, $R_{\mu\nu\sigma\epsilon}$ and $\nabla_{\mu}$ denote the Riemann tensor and covariant derivative, respectively. The line element corresponding to the spherically symmetric fluid sphere in the Schwarzschild-like coordinates, $x^{\mu}\equiv x^{0,1,2,3}=(t,r,\theta,\phi)$, reads

$\displaystyle ds^{2}=e^{\alpha}dt^{2}-e^{\beta}dr^{2}-r^{2}d\Omega^{2},$

(8)

where $d\Omega^{2}=dr^{2}-\sin^{2}\theta d\theta^{2}$. To maintain the static nature of the aforementioned metric, both $\alpha$ and $\beta$ must be functions of the radial variable $r$ only. Additionally, when taking these factors into account, the timelike four-velocity adopts the following form

$\displaystyle\xi^{\mu}=(e^{-\alpha/2},0,0,0)\quad\textmd{such that}\quad\xi^{t}\xi_{t}=1.$

(9)

Due to spherical symmetric, the non-null components associated with the tensor $K_{\mu\nu}$ are: $K_{tt}$, $K_{rr}$, and $K_{\theta\theta}=\sin^{2}\theta K_{\phi\phi}$. Substituting these components in Eq. (6), we obtain

$\displaystyle R_{rtrt}=\frac{R_{r\theta r\theta}R_{\phi t\phi t}+R_{r\theta\theta t}R_{r\phi r\phi}}{R_{\theta\phi\theta\phi}},$

(10)

which gives rise to the following differential equation

$\displaystyle 2\frac{\alpha^{\prime\prime}}{\alpha^{\prime}}+\alpha^{\prime}=\frac{\beta^{\prime}e^{\beta}}{e^{\beta}-1},\quad\textmd{such that}\quad e^{\beta}\neq 1,$

(11)

and

$\displaystyle e^{\beta}=1+Fe^{\alpha}\alpha^{\prime 2}.$

(12)

Here, $F$ is an integration constant. After rearranging Eq. (11), we have

$\displaystyle e^{\alpha}=\left[B+C\sqrt{e^{\beta}-1}\right]^{2},$

(13)

which describes a relationship between the geometric variables $\alpha(r)$ and $\beta(r)$.

Now, we revisit the basic structure of the gravitational decoupling scheme employing the MGD-decoupling. In the previous section, we discuss the effectiveness of gravitational decoupling as a technique to split the seed source into two independent gravitational sectors. This separation is achieved through a mechanism that maintains the essential geometric properties of the underlying problem. Consequently, after decoupling, the respective conservation relations remain valid for each gravitational sector. The covariant conservation of both the gravitational-field sources implies that their interaction is gravitational. However, there stems a question regarding the separation of the gravitational sector fulfilling the original fluid features. For a more profound understanding of this phenomenon, we consider a matter distribution characterized as an isotropic/perfect fluid. This distribution is described by the following SET

$\displaystyle\widetilde{T}_{\mu\nu}=(\widetilde{\rho}+\widetilde{P})\xi_{\mu}\xi_{\nu}-\widetilde{P}g_{\mu\nu},$

(14)

where $\widetilde{\rho}$ is the energy density and $\widetilde{P}$ is the isotropic pressure. We now extend the gravitational field source $T_{\mu\nu}$ by including the additional field source $\Theta_{\mu\nu}$, decoupled through a dimensionless constant $\chi$ as

$\displaystyle\widetilde{T}_{\mu\nu}\mapsto T_{\mu\nu}=\widetilde{T}_{\mu\nu}+\chi\Theta_{\mu\nu}.$

(15)

The introduction of this additional field term, $\Theta_{\mu\nu}$, unveils novel physical phenomena that extend beyond the confines of the established theoretical framework of GR. Despite the remarkable success of GR in understanding a vast array of gravitational interactions, including cosmological matters, the solar system, black holes, neutron stars, and even hypothetical structures like wormholes, it is widely acknowledged that modifications or extensions may be necessary to fully incorporate certain gravitational aspects. We now consider a self-gravitating compact object, potentially representing an NS or a quark star, described by the source terms introduced in Eqs. (8) and (15). Then, owing to the spherically symmetric metric ansatz and the isotropic nature of the matter distribution, we have $\widetilde{T}^{r}_{r}=\widetilde{T}^{\theta}_{\theta}=\widetilde{T}^{\phi}_{\phi}=\widetilde{P}$. However, if we consider the system (15) instead of (14), then the introduction of the new sector may lead to a distinct behavior, wherein $T^{r}_{r}\neq T^{\theta}_{\theta}=T^{\phi}_{\phi}$. Consequently, the relativistic configuration exhibits an anisotropic behavior. The manifestation of this induced anisotropy in the matter content is critically dependent on the particular features of the $\Theta_{\mu\nu}$ field. This field can take the form of a scalar, vector, or tensor field. Furthermore, it could be interpreted as a signature of a dark sector component, such as DM or dark energy [66, 68, 64]. In order to separate the field sources $\widetilde{T}_{\mu\nu}$ and $\Theta_{\mu\nu}$, it is advantageous to understand how these sources relate to the geometry of spacetime. This correspondence is developed by varying the action

$\displaystyle\mathcal{A}^{\circ}=\mathcal{A}_{\mathrm{EH}}+\mathcal{A}_{\textmd{matter}},$

(16)

with respect to $g^{\mu\nu}$. In the above expression, $\mathcal{A}_{\mathrm{EH}}$ denotes the usual Einstein-Hilbert action, defined as

$\displaystyle\mathcal{A}_{\mathrm{EH}}=\frac{1}{16\pi}\int d^{4}x\sqrt{|g|}\mathrm{R}.$

(17)

Here, $\mathrm{R}$ denotes the contraction of the Ricci tensor $R_{\mu\nu}$, commonly referred to as the Ricci scalar, while $g$ represents the trace of $g_{\mu\nu}$. On the other hand, the matter content is characterized by the following action

$\displaystyle\mathcal{A}_{\textmd{matter}}=\int d^{4}x\sqrt{|g|}\mathscr{L}_{\textmd{matter}},$

(18)

where $\mathscr{L}_{\textmd{matter}}$ is the matter Lagrangian. Generally, this Lagrangian density could encompass a broader range of fields representing diverse forms of matter distributions, which are not described by GR.

$\displaystyle\mathscr{L}_{\textmd{matter}}=\mathscr{L}_{\widetilde{\textmd{M}}}+\chi\mathscr{L}_{\mathbb{X}},$

(19)

where $\mathscr{L}_{\widetilde{\textmd{M}}}$ captures the contributions of the seed source, while $\mathscr{L}_{\mathbb{X}}$ encodes the characteristics of the new matter fields. These additional contributions may be regarded as corrections to GR [64]. The gravitational-field equations describing the gravity-matter correspondence are obtained from the variational principle, which reads

$\displaystyle\frac{\delta\mathcal{A}^{\circ}}{\delta g^{\mu\nu}}=0\Rightarrow G_{\mu\nu}\equiv R_{\mu\nu}-\frac{1}{2}\mathrm{R}g_{\mu\nu}=-8\pi T_{\mu\nu}.$

(20)

Here, $G_{\mu\nu}$ denotes the usual Einstein tensor and

$\displaystyle T_{\mu\nu}=-2\frac{\delta\mathscr{L}_{\widetilde{\textmd{M}}}}{\delta g^{\mu\nu}}+g_{\mu\nu}\mathscr{L}_{\widetilde{\textmd{M}}}+\chi\left(-2\frac{\delta\mathscr{L}_{\mathbb{X}}}{\delta g^{\mu\nu}}+g_{\mu\nu}\mathscr{L}_{\mathbb{X}}\right),$

(21)

where

$\displaystyle\widetilde{T}_{\mu\nu}=-2\frac{\delta\mathscr{L}_{\widetilde{\textmd{M}}}}{\delta g^{\mu\nu}}+g_{\mu\nu}\mathscr{L}_{\widetilde{\textmd{M}}},$

(22)

and

$\displaystyle\Theta_{\mu\nu}=-2\frac{\delta\mathscr{L}_{\widetilde{\textmd{M}}}}{\delta g^{\mu\nu}}+g_{\mu\nu}\mathscr{L}_{\widetilde{\textmd{M}}}.$

(23)

The divergence-free nature of $G_{\mu\nu}$ provides the following result

$\displaystyle\nabla_{\mu}G^{\mu\nu}=0\Rightarrow\nabla_{\mu}T^{\mu\nu}=0,$

(24)

which ensures the covariant conservation of the SET, $T_{\mu\nu}$.

$\displaystyle\nabla_{\mu}T^{\mu\nu}=0\Rightarrow\nabla_{\mu}\widetilde{T}^{\mu\nu}+\chi\nabla_{\mu}\Theta^{\mu\nu}=0.$

(25)

The solution of the above equation can be described in two ways:

(i)

$\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}\Theta^{\mu\nu}=0$,

(ii)

$\nabla_{\mu}T^{\mu\nu}=-\nabla_{\mu}\Theta^{\mu\nu}$.

The first option suggests that each source is covariantly conserved, implying a purely gravitational interaction between them. On the other hand, the second option indicates an energy exchange between both the fields [89, 90]. In this investigation, we are mainly concerned with the second case. Consequently, the gravitational system $\{ds^{2},T_{\mu\nu}\}$ produces the following set of differential equations

	$\displaystyle G^{t}_{t}$	$\displaystyle=-8\pi T^{t}_{t}:\frac{1}{r^{2}}-\left(\frac{1}{r^{2}}-\frac{\beta^{\prime}}{r}\right)e^{\beta}=-8\pi(\widetilde{\rho}+\chi\Theta^{t}_{t}),$		(26)
	$\displaystyle G^{r}_{r}$	$\displaystyle=-8\pi T^{r}_{r}:-\frac{1}{r^{2}}+\left(\frac{1}{r^{2}}-\frac{\alpha^{\prime}}{r}\right)e^{\beta}=-8\pi(-\widetilde{P}+\chi\Theta^{r}_{r}),$		(27)

$\displaystyle G^{\theta}_{\theta}$

$\displaystyle=-8\pi T^{\theta}_{\theta}:\frac{e^{-\beta}}{4}\left(2\frac{\beta^{\prime}}{r}-2\frac{\alpha^{\prime}}{r}+\beta^{\prime}\alpha^{\prime}-\alpha^{\prime 2}-2\alpha^{\prime\prime}\right)=-8\pi(-\widetilde{P}+\chi\Theta^{\theta}_{\theta}),$

(28)

where $G^{\theta}_{\theta}=G^{\phi}_{\phi}$ due to spherical symmetry. The mass function for the self-gravitational, spherically symmetric configuration of matter can be defined by employing the Misner and Sharp scheme as

$\displaystyle R^{3}_{232}=1-e^{-\beta}=\frac{2m}{r},$

(29)

or, alternatively

$\displaystyle m(r)=4\pi\int^{r}_{0}\left[\rho(x)\right]x^{2}dx,$

(30)

which can be represented in a particular form accounting for the effects of density homogeneity plus the change induced by density inhomogeneity as

$\displaystyle m=\frac{4\pi}{3}r^{3}\rho-\frac{4\pi}{3}\int^{r}_{0}\left[\rho^{\prime}(x)\right]x^{3}dx.$

(31)

Then, by using Eqs. (26)-(29), we obtain

$\displaystyle\alpha^{\prime}=\frac{2m+8\pi r^{3}P_{r}}{(r-2m)}.$

(32)

The covariant conservation of the gravitational-field source $T^{\mu\nu}$ produces the following result

$\displaystyle 0\equiv T^{\mu\nu}_{~{}~{}~{};\nu}=\frac{dT^{\mu\nu}}{dx^{\nu}}+\Gamma^{\mu}_{\nu\epsilon}T^{\epsilon\nu}+\Gamma^{\nu}_{\nu\epsilon}T^{\mu\epsilon},$

(33)

which gives the conservation equation for the spherically symmetric stellar distribution for $\mu=1$, given by

$\displaystyle\frac{dT^{r}_{r}}{dr}$

$\displaystyle=-\frac{1}{2}g^{tt}\frac{dg_{tt}}{dr}\left(T^{t}_{t}-T^{r}_{r}\right)-g^{\theta\theta}\frac{dg_{\theta\theta}}{dr}\left(T^{r}_{r}-T^{\theta}_{\theta}\right),$

(34)

from where do we get

$\displaystyle\widetilde{P}^{\prime}$

$\displaystyle=-\frac{\alpha^{\prime}}{2}(\widetilde{\rho}+\widetilde{P})+\chi\left[\left(\Theta^{r}_{r}\right)^{\prime}-\frac{\alpha^{\prime}}{2}\left(\Theta^{t}_{t}-\Theta^{r}_{r}\right)-\frac{2}{r}\left(\Theta^{\theta}_{\theta}-\Theta^{r}_{r}\right)\right].$

(35)

Then, by simple observation, we can write

		$\displaystyle T^{t}_{t}=\widetilde{\rho}+\chi\Theta^{t}_{t}\equiv\rho,$		(36)
		$\displaystyle T^{r}_{r}=-\widetilde{P}-\chi\Theta^{r}_{r}\equiv P_{r},$		(37)
		$\displaystyle T^{\theta}_{\theta}=-\widetilde{P}-\chi\Theta^{\theta}_{\theta}\equiv P_{t},$		(38)

where clearly we have

		$\displaystyle T^{\mu}_{\nu}=\textmd{Diag}\Big{[}\rho,-P_{r},-P_{\theta},-P_{\phi}\Big{]},$		(39)
		$\displaystyle\Theta^{\mu}_{\nu}=\textmd{Diag}\left[\Theta^{t}_{t},-\Theta^{r}_{r},-\Theta^{\theta}_{\theta},-\Theta^{\phi}_{\phi}\right].$		(40)

We now employ the GD approach by considering a solution for the seed gravitational-field source $\widetilde{T}_{\mu\nu}$, i.e.,

$\displaystyle T_{\mu\nu}=\widetilde{T}_{\mu\nu}+\chi\cancelto{{0}}{\Theta_{\mu\nu}},$

(41)

which can be formally written as

$\displaystyle ds^{2}=e^{u(r)}dt^{2}-\frac{dr^{2}}{y(r)}-r^{2}d\Omega^{2}.$

(42)

The effects emerging from the generic gravitational-field source $\Theta_{\mu\nu}$ can be observed through the geometric deformation of the line element (42), as follows:

		$\displaystyle\alpha(r)\mapsto\alpha(r)=\eta(r)+\chi h(r),$		(43)
		$\displaystyle e^{-\beta(r)}\mapsto e^{-\beta(r)}=y(r)+\chi f(r),$		(44)

where $h$ and $f$ denote the deformation functions. Henceforth, we exclusively examine the MGD decoupling, characterized by minimal deformation where $h=0$. Consequently, only $e^{-\beta(r)}$ undergoes modification, with $\alpha=\eta$. Therefore, we summarize the primary characteristics of the MGD-decoupling scheme as follows:

• •

  According to (26), one can conclude that the energy density relies solely on $e^{\beta(r)}$ only. Thus, one can conclude that the linear mapping (44) appears as the only possibility to decouple the gravitational-field sources $\widetilde{T}_{\mu\nu}$ and $\Theta_{\mu\nu}$. In this context, the linear mapping that introduces deformations only in the radial metric potential is not realizable without considering the extended scenario [64, 91].

• •

  The radial metric deformation $f(r)$ should be purely radial for the preservation of spherical symmetry.

• •

  The geometric variable $y(r)$ acts as the primary influence on the behavior of $f(r)$. As $y(r)$ is a strictly increasing function, the sign or monotonicity (increasing/decreasing) of $f(r)$ is irrelevant. The crucial aspect is that $y(r)$ must dominate the behavior of $f(r)$ to ensure a physically viable solution. Additionally, $f(r)\rightarrow 0$ at the center of the stellar distribution.

• •

  Since the metric variable $y(r)$ is closely linked with Misner-Sharp mass $m(r)$, the linear mapping (44) extends the classical description of the mass of the self-gravitational compact configuration by introducing an additional component. Thus, the minimally deformed form of $m(r)$ reads

$\displaystyle m(r)=\frac{r}{2}\left[1-y(r)-\chi f(r)\right].$

(45)

Next, employing the linear mapping (44) into the gravitational system (26)-(28) provides the following differential equations for the seed gravitational-field source

		$\displaystyle 8\pi\widetilde{\rho}=\frac{1-y}{r^{2}}-\frac{y^{\prime}}{r},$		(46)
		$\displaystyle 8\pi\widetilde{P}=-\frac{1}{r^{2}}+y\left(\frac{1}{r^{2}}+\frac{\alpha^{\prime}}{r}\right),$		(47)
		$\displaystyle 8\pi\widetilde{P}=\frac{y}{4}\left(2\frac{\alpha^{\prime}}{r}+2\alpha^{\prime\prime}+\alpha^{\prime 2}\right)+\frac{y^{\prime}}{r}\left(\alpha^{\prime}+\frac{2}{r}\right),$		(48)

and the corresponding conservation equation reads

$\displaystyle\widetilde{P}^{\prime}+\frac{\alpha^{\prime}}{2}(\widetilde{\rho}+\widetilde{P})=0.$

(49)

Now, the stellar system describing the imprints of additional field source $\Theta_{\mu\nu}$ can be written as

		$\displaystyle 8\pi\Theta^{t}_{t}=-\left(\frac{f}{r^{2}}+\frac{f^{\prime}}{r}\right),$		(50)
		$\displaystyle 8\pi\Theta^{r}_{r}=-f\left(\frac{1}{r^{2}}+\frac{\alpha^{\prime}}{r}\right),$		(51)
		$\displaystyle 8\pi\Theta^{\theta}_{\theta}=-\frac{f}{4}\left(2\frac{\alpha^{\prime}}{r}+2\alpha^{\prime\prime}+\alpha^{\prime 2}\right)-\frac{f^{\prime}}{4}\left(\alpha^{\prime}+\frac{2}{r}\right),$		(52)

whose conservation equation likewise reads

$\displaystyle\left(\Theta^{r}_{r}\right)^{\prime}-\frac{\alpha^{\prime}}{2}\left(\Theta^{t}_{t}-\Theta^{r}_{r}\right)-\frac{2}{r}\left(\Theta^{\theta}_{\theta}-\Theta^{r}_{r}\right)=0.$

(53)

Here are some key observations regarding the systems representing the seed and $\Theta$ gravitational-field sources:

1. 1.

   The gravitational system encoding the imprints of decoupling fluid $\Theta_{\mu\nu}$ correlates with the quasi-Einstein filed equations; however, they deviate by a factor of $1/r^{2}$.

2. 2.

   The independent conservation of both field sectors implies a purely gravitational interaction between them.

3. 3.

   The self-gravitational system filled isotropic matter content transitions to an anisotropic state if and only if $\Theta^{r}_{r}\neq\Theta^{\theta}_{\theta}$.

Therefore, the value of the anisotropic scalar for the considered system turns out to be

$\displaystyle\Pi(r,\chi)\equiv P_{r}-P_{t}=\chi(\Theta^{\theta}_{\theta}-\Theta^{r}_{r}),$

(54)

which clearly describes the role of the $\Theta$ field source in generating anisotropy within the isotropic matter configuration. Importantly, for a physically viable self-gravitational model, one must have $\Pi>0$ throughout the range, $0\leq r\leq R$.

The seed system is characterized by Eqs. (46)-(48) is already established by the seed metric. However, the $\Theta$-system is not closed as described by the set of Eqs. (50)-(52). The inclusion of $\Theta$-field source introduces four unknowns: $\{\Theta^{t}_{t},\Theta^{r}_{r},\Theta^{\theta}_{\theta},f(r)\}$. However, only three equations govern their behavior. Consequently, it becomes necessary to impose an additional constraint to establish a well-defined system. When it comes to the closure of the $\ Theta$ system, various approaches have been implemented, leading demonstrably to well-behaved interior solutions in most cases. Among different gravitational techniques, the typical ones include: i) the mimic constraint methodology [47, 92, 93, 94]; ii) enforcing a suitable deformation function $f(r)$ [95, 96]; and iii) employing a regularity requirement on the altered segment driven by the renowned anisotropy condition as elaborated in [97].

However, we consider an alternative scheme to close the $\Theta$-gravitational field source. As highlighted in [64] and further investigated in the context of cosmology [68], the enigmatic birth of the $\Theta$-gravitational sector may be theoretically grounded by postulating that this field constitutes one of the mysterious, unseen constituents of our universe. By drawing a parallel to DM, we consider a hypothesis suggesting a potential connection between the $\Theta$-field and a DM constituent. Building on this conjecture, we associate the temporal aspect of the $\Theta$-field with a DM density distribution. In this direction, we have explicitly assumed $\Theta^{t}_{t}=\rho$, as elaborated upon in the subsequent section.

Now, the definition of mass function for the seed gravitational sector $\widetilde{T}_{\mu\nu}$ reads

$\displaystyle m_{s}(r)=\frac{r}{2}(1-y)=4\pi\int^{r}_{0}\left[\widetilde{\rho}(x)\right]x^{2}dx,$

(55)

which implies

$\displaystyle m=m_{s}-\chi\frac{r}{2}f(r)\equiv m_{s}+\chi m_{\Theta},$

(56)

where the mass functions $m_{s}$ and $m_{\Theta}$ can be expressed as

		$\displaystyle m_{s}=\frac{4\pi}{3}\widetilde{\rho}r^{3}-\frac{4\pi}{3}\int^{r}_{0}[\widetilde{\rho}(x)]x^{3}dx,$		(57)
		$\displaystyle m_{\Theta}=\frac{4\pi}{3}\Theta^{t}_{t}r^{3}-\frac{4\pi}{3}\int^{r}_{0}[\Theta^{t}_{t}(x)]x^{3}dx.$		(58)

The generic physical and mathematical features associated with the minimally deformed Finch-Skea metric will be analyzed in the following sections.

Let us consider the matching constraints that guarantee the smooth matching between the interior $(0\leq r\leq R)$ of a stellar configuration and the outer geometry $(r>R)$ at the boundary surface $\Sigma:r=R$. We describe the interior geometry using the generic metric (8). This metric can be expressed in terms of the MGD transformation (44) as follows

$\displaystyle ds^{2}=e^{\alpha^{-}}dt^{2}-\left(1-\frac{2m}{r}\right)^{-1}dr^{2}-r^{2}d\Omega^{2},$

(59)

with $m=m_{s}-\frac{r}{2}\chi f$. However, it is important to note that the exterior metric may deviate from a strict vacuum state. In general, the inclusion of the additional field source, $\Theta_{\mu\nu}$, may produces new fields that do not exist in a standard vacuum. Therefore, the generic form of the outer metric can be defined as

$\displaystyle ds^{2}=e^{\alpha^{+}}dt^{2}-e^{\beta^{+}}dr^{2}-r^{2}d\Omega^{2},$

(60)

where metric functions $e^{\alpha^{+}}$ and $e^{\beta^{+}}$ can be determined by solving the effective four-dimensional exterior Einstein stellar structure equations

$\displaystyle R_{\mu\nu}-\frac{1}{2}R^{\mu}_{\mu}\textsl{g}_{\mu\nu}=\chi\Theta_{\mu\nu}.$

(61)

Applying the MGD decoupling technique reduces the aforementioned stellar structure equations to the system of Eqs. (50)–(52), where the metric variable $\alpha$ is determined by the Schwarzschild solution. Now, the continuity of the first fundamental form at the stellar surface reads

$\displaystyle\left[ds^{2}\right]_{\Sigma}=0,$

(62)

where $\left[\mathcal{G}\right]\equiv\mathcal{G}(r\rightarrow R^{+})-\mathcal{G}(r\rightarrow R^{-})\equiv\mathcal{G}_{R}^{+}-\mathcal{G}_{R}^{-}$, for any function $\mathcal{G}=\mathcal{G}(r)$, which gives

$\displaystyle\alpha^{-}(R)=\alpha^{+}(R),$

(63)

and

$\displaystyle 1-\frac{2M_{\textmd{Sch}}}{R}+\chi f_{R}=e^{-\beta^{+}(R)},$

(64)

where $M_{\textmd{Sch}}=m_{s}(R)$ and $f_{R}$ represents the MGD deformation at the stellar surface. Similarly, the continuity of the second fundamental form is expressed by

$\displaystyle\left[G_{\mu\nu}r^{\nu}\right]_{\Sigma}=0,$

(65)

where $r_{\mu}$ denotes the unit-radial radial. Then, using the above expression in gravitational equations of motion (20), we have

$\displaystyle\left[T_{\mu\nu}r^{\nu}\right]_{\Sigma}=0,$

(66)

which implies

$\displaystyle\left[\widetilde{P}-\chi\Theta^{r}_{r}\right]_{\Sigma}=0,$

(67)

$\displaystyle\widetilde{P}_{R}-\chi\left(\Theta^{r}_{r}\right)_{R}^{-}=-\chi\left(\Theta^{r}_{r}\right)_{R}^{+}.$

(68)

This expression signifies an important result describing the second fundamental form corresponding to the Einstein stellar structure Eqs. (20) and the Schwarzschild vacuum filled with the decoupling fluid $\Theta^{+}_{\mu\nu}$, namely, $\{\Theta^{+}_{\mu\nu}\neq 0,T^{+}_{\mu\nu}=0\}$. Then, substituting Eq. (85) for the interior geometry in Eq. (68), we get

$\displaystyle\widetilde{P}_{R}+\frac{\chi f_{R}}{8\pi}\left(\frac{1}{R^{2}}+\frac{\alpha^{\prime}_{R}}{R}\right)=-\chi\left(\Theta^{r}_{r}\right)_{R}^{+},$

(69)

where $\alpha^{\prime}_{R}=\left(\frac{\partial\alpha^{-}}{\partial r}\right)_{r=R}$. Now, substituting Eq. (85) for the exterior geometry in Eq. (69), we have

$\displaystyle\widetilde{P}_{R}+\frac{\chi f_{R}}{8\pi}\left(\frac{1}{R^{2}}+\frac{\alpha^{\prime}_{R}}{R}\right)=\frac{\chi s^{\ast}_{R}}{8\pi}\left[\frac{1}{R^{2}}+\frac{2M^{\ast}}{R^{3}}\left(1-\frac{2M^{\ast}}{R}\right)^{-1}\right].$

(70)

Here, the function $s^{\ast}_{R}$ captures the geometric deformation associated with the outer Schwarzschild metric due to the addition of the extra field source $\Theta_{\mu\nu}$, that is,

$\displaystyle ds^{2}=\left(1-\frac{2M^{\ast}}{R}\right)dt^{2}-\left(1-\frac{2M^{\ast}}{R}+\chi s^{\ast}\right)^{-1}dr^{2}-r^{2}d\Omega^{2}.$

(71)

Finally, Eqs. (63), (64) and (70) describe the necessary and sufficient conditions that must be satisfied to ensure a smooth matching between the minimally deformed inner geometry (59) and the outer geometry describing a spherically symmetric vacuum field, with the minimally deformed Schwarzschild metric (71). It is crucial to understand that the exterior metric may not be entirely free of matter or energy. The source term, $\Theta_{\mu\nu}$, permits the presence of various fields within this region. The junction constraint (71) leads to an important result: if the outer geometry is described by the exact Schwarzschild metric, then $s^{\ast}(r)=0$, which in turn provides

$\displaystyle P_{R}=\widetilde{P}_{R}+\frac{\chi f_{R}}{8\pi}\left(\frac{1}{R^{2}}+\frac{\alpha^{\prime}_{R}}{R}\right)=0.$

(72)

Thus, the essential condition for a stellar configuration to be in equilibrium within a true Schwarzschild vacuum is that the effective pressure at its surface must be zero. An interesting result emerges when the interior geometric deformation ($f_{R}<R$) weakens the gravitational field and is positive. In this scenario, a standard outer Schwarzschild vacuum can only be compatible with a non-zero inner source term $\Theta_{\mu\nu}$ if the pressure of the isotropic stellar matter at the star’s surface is negative ($\widetilde{P}_{R}<0$). This implies the presence of regular matter with a solid crust, as discussed in reference [45]. Alternatively, we can achieve the pressure condition $\widetilde{P}_{R}=0$ through a different approach. Instead of directly solving for it, we can impose a constraint on Eq. (72), that is, $\beta\left(\Theta^{r}_{r}\right)^{-}_{R}\thicksim\widetilde{P}_{R}$ in Eq. (68). Consequently, the interior deformation, $f_{R}$, vanishes.

This section captures the formulation of a minimally deformed DM model for self-gravitational distributions incorporating Einasto density parametrization. We begin by considering the seed metric employed throughout this work. Subsequently, we construct the self-gravitational structures characterizing the seed and $\Theta$-field sources with the help of the Einasto DM profile. Building upon the groundwork established previously, we extend our understanding of time-independent, isotropic stellar spheres, which satisfy GR-field equations, towards anisotropic domains. This extension requires incorporating elements beyond GR. For this purpose, we have considered the Adler-Finch-Skea ansatz as a seed metric. This model has been employed to represent various self-gravitational compact configurations, including neutron stars and quark stars, by incorporating an isotropic matter content. In this direction, Tello-Ortiz [98] et al. have constructed relativistic anisotropic stellar systems by employing the combination of MGD decoupling with the Class I approach. This is achieved by considering the Adler-Finch-Skea solution as a seed metric. Furthermore, Maurya et al. [99] explored realistic anisotropic fluid sphere-like stellar structures within the context of the extended MGD approach by considering the Class I method. To assess the feasibility of this approach, they used the hybrid Adler–Finch–Skea solution as the seed space-time solution. Then, we have the following Class I seed metric

$\displaystyle\alpha(r)=\ln\left[\left(A+Br^{2}\right)^{2}\right],$

(73)

$\displaystyle y(r)=\frac{1}{1+Cr^{2}}.$

(74)

Then, the fluid configuration filling the stellar interior, characterized by the above-mentioned geometry, reads

		$\displaystyle 8\pi\widetilde{\rho}(r)=\frac{C(3+Cr^{2})}{(1+Cr^{2})^{2}},$		(75)
		$\displaystyle 8\pi\widetilde{P}(r)=\frac{4B-C(A+Br^{2})}{(A+Br^{2})(1+Cr^{2})}.$		(76)

Notably, this isotropic stellar model corresponds to the GR-relativistic equations of motion in the absence of the decoupling fluid $\Theta_{\mu\nu}$. We determine the constants $A$, $B$, and $C$ appearing in Eqs. (73)–(76) by applying the junction constraints defined in Eqs. (62) and (65), between the inner and outer geometry. According to these constraints, the values of $A$, $B$, and $C$ termed as

	$\displaystyle\frac{A}{BR^{2}}=$	$\displaystyle\frac{2R-3M_{\textmd{Sch}}}{M_{\textmd{Sch}}},$		(77)
	$\displaystyle B^{2}R^{4}=$	$\displaystyle\frac{M^{2}_{\textmd{Sch}}}{4R(R-2M_{\textmd{Sch}})},$		(78)
	$\displaystyle\frac{2(1+CR)}{CR}=$	$\displaystyle\frac{R}{M_{\textmd{Sch}}}.$		(79)

Notably, Eq. (79) guarantees a smooth geometric transition at $r=R$. However, this expression will change when we include the additional source term, $\Theta_{\mu\nu}$. Let us now turn on the deformation parameter $\chi$ within the interior stellar configuration. The radial and metric potentials (defined in Eqs. (73) and (44), respectively), whereas the additional field source $\Theta_{\mu\nu}$ and the deformation parameter $f(r)$ are interconnected through the stellar structure Eqs. (50)–(52). Therefore, additional information must be provided to close the system of Eqs. (50)–(52). We have multiple strategies to fully solve this system. One possibility is to use an EoS to define the relationship between density and pressure within the source term $\Theta_{\mu\nu}$. An alternative method is to impose a restriction based on physical principles on the interior deformation $f(r)$. Importantly, we must ensure that any solution we find remains physically realistic. This is a significant challenge, and the following section addresses it by presenting three new, exact solutions that are demonstrably compatible with physical principles.