1. Introduction

Researchers have been caught by the study of compact relativistic astrophysical objects. Compact stars are extremely dense natural objects with a mass that is concentrated in a tiny area, approaching the density of a nuclear nucleus [1–3]. Actually, in 1967, the first compact stars were discovered to be pulsars by controlling pulsating radio emissions [4].

According to the profound research on the cosmic microwave background (CMB), scientists have exposed some interesting secrets. Let us delve into this attractive realm. Dark energy makes up around 68.3% of the universe, according to the CMB analysis. Even though it is yet unknown, this enigmatic force is crucial in forming the universe. Alongside dark energy, we have dark matter, which accounts for 26.8% of the universe. Just like its counterpart, dark matter presents in mystery that has puzzled researchers for years [5, 6]. Several investigators have functioned on the nature of dark matter and dark energy, which remains mysterious such as Carroll et al. [7]; Nojiri and Odintsov [8]; Townsend and Wohlfarth [9]; Wohlfarth [10]; Roy [11]; Cline and Vinet [12].

The general theory of relativity (GR), which is considered a very useful and effective theory in the existing era, describes the gravitational interaction. The fields of astrophysics, cosmology, and gravitational-wave astronomy find the GR of interest, and it is potentially leading to major innovations since its introduction in 1915. The understanding of the physics of compact stellar objects has been impressively increased over the years due to the assumption of GR by the scientific community. The study of a compact star has expanded huge momentum. In current times, the substantial attention of researchers has been found in the modified theories of gravity. In the context of the modified theories of gravity, some interesting investigations can be found in [13–22]. Rastall gravity theory (RT) is one of the interesting theories to study the astrophysical objects. An interesting modification of the GR has been proposed in [23], by Rastall in 1972 (called RT), which has charmed the attentiveness of many researchers. Rastall realized that in a curved space-time, the conservation law of the energy momentum tensor (∇_vT^μν = 0) might not apply. He addressed this by introducing a novel modification to the framework where the covariant divergence of the energy momentum tensor is exactly proportional to derivative of the scalar curvature (∇_μT^μν ∝ R^ν). This adjustment allowed Rastall to establish a coupling parameter that, when set to its specific limiting value (known as zero coupling), revert back to the familiar form of GR. One of the remarkable aspects of RT is its relative simplicity compared to other modified theories. The field equations derived from Rastall’s framework are remarkably straightforward, making them accessible and less arduous to explore. This characteristic opens up the exciting possibilities for researchers to delve deeper into the intricacies of the modified gravitational framework proposed by Rastall.

The progressive development of our cosmos has been studied in the RT by numerous investigators. Moradpour studied the cosmological consequences and thermodynamical aspects of the RT and constraint the Rastall parameter by imposing the condition on the horizon entropy [24]. Al-Rawaf and Taha [25] studied the cosmological model in the RT and argued that these models in the RT are more consistent with the observational data. Moradpour et al. [26] studied a generalized version of the RT to study the current accelerating expansion of the universe caused by the dark energy considered in this investigation. The role of the Rastall model in the initial age of the universe (Cosmic inflation) has been investigated by Saleem and Hassan [27]. The interior arrangement of astrophysical objects should be expressed and the solar system test should be consistent with in order to verify a gravitational theory. In the Rastall paradigm, solar system tests have recently been studied [28]. Kiselev-like black hole (BH) solutions in RT have been studied in [29] with charged/uncharged perfect fluid. In addition, several BH results and their interesting consequences have been discovered in the RT. Spallucci and Smailagic [30] proposed the solution of the RT field equations by considering the Gaussian matter distribution. Sakti, Suroso, and Zen [31] investigated a new solution to the twisted and rotating electrically charged BH with quintessence matter field in the RT. Heydarzade, Moradpour, and Darabi [32] obtained the solution of the Reissner–Nordstrom BH in the background of the generic cosmological constant and also discussed the subclasses of the obtained results. Liang [33] investigated the quasinormal modes of the Schwarzschild BH surrounded by the quintessence field in the regime of the RT. The study of the thermodynamic analysis of BHs near the two types of perfect fluids (quintessence and dust) has been conducted in [34]. Batista et al. [35] considered two fluid model comprising baryons plus cold dark matter in the RT and observe the same results as in the Λ CDM model at the linear perturbative level whereas differs at the nonperturbative level. Ma and Zhao proposed the BH in the RT with noncommutative geometry [36]. Moradpour and Salako [37, 38] studied the limitations imposed on RT when applying the Newtonian limit.

Recently, Darabi et al. [39] conducted a comprehensive analysis comparing RT of gravity with the GR. Their findings challenged the Visser’s claim that the two theories are identical [40]. In fact, Darabi et al. argue that RT holds broader applicability than GR. These contrasting viewpoints suggest that RT remains an “open” theory, offering the potential to dare the challenges posed by astrophysical observations and, intriguingly, quantum gravity (for a more detailed conversation see, for reference to, [39]).

Moreover, Shahzad and Abbas investigated the RT to develop the numerous models of compact stars in this theory and compared the obtained results with the GR counterpart and observational data [41–43]. Shahzad and Abbas also studied a charged compact star model in RT [44]. Tahir, Abbas, and Shahzad [45] studied the instability of self-gravitating objects in the RT. Recently, the stability analysis in RT through the cracking technique has been investigated in [46]. Nashed and El Hanafy [47] propounded a new class of stellar structure in the RT consistent with the observational outcomes. Further, El Hanafy [48–50] investigated the implications of the RT on the mass and radius of Pulsars. Afshar, Moradpour, and Shabani [51] studied the primary inflationary and reheating eras in the RT concluding that the outcomes are well consistent with the observational data as compared to the GR.

Motivating from the recent interests of the researchers in this theory to investigate the astrophysical objects and to understand the more fascinating features of the RT, we develop a new model of compact stars in this theory by considering the Finch–Skea geometry in the static and spherically symmetric space-time. The sequence of our study is as follows: Section 2 deals with the formulation of field equations in the RT and acquires their solution by using a specific metric ansatz. In Section 3, we apply the junction conditions to formulate the unknown constants appeared due to the special metric potential. Section 4 describes the detailed physical analysis of the obtained results to verify the presented model. Section 5 devoted to the comparative study of the obtained model with the GR. The last section concludes our findings in detail in the present study.

2. Rastall Field Equations and Their Solution

Rastall’s hypothesis is built around the crucial point that  in general, in the curved space-time. Then, he revises the usual conservation law of the Energy-Momentum Tensor (EMT) as a nonconservation as follows

()

The above relation denotes Ricci scalar as R and Rastall parameter as β, which measures the deviation of the RT from GR. A nonminimal coupling of geometry and matter field is thereby affected, and the following field equations are yielded as

()

In this context, γ = βκ and κ denote the gravitational coupling constant of RT. It is then observed that equation (2) yields (−1 + βκ) = T, which implies that βκ = 1/4 is not permissible in the RT, since the trace of the EMT T is not necessary zero. The Newtonian limit is employed to define γ = βκ, which is termed as the Rastall dimensionless parameter; as a result, κ and λ are expressible in such a form [37]

()

()

Relation (3) reveals that the Einstein field equations (κ = 8π) can be regain in the β = 0 limit, equivalent to γ = 0. It is similarly apparent from relation (3) that κ undergoes divergence when γ = 1/6, necessitating the exclusion of γ = 1/6 case in the RT. The Rastall field equations are thereby expressible in the following form:

()

Equation (5) is found to yield the result R(6γ − 1) = 8πT, implying that the case γ = 1/6 is not allowable, in consistency with relation (3). In the Newtonian limit, both the cases γ = 1/6 and γ = 1/4 are necessitated to be not allowed in this modification.

For this study, we consider a static and spherically symmetric line element

()

The metric potentials  and e^ν(r) = 1 + Cr² of Finch–Skea ansatz [52], with A, B,  and C being constants. These metric potentials are non-singular and viable. These metric potentials are used to study the compact stellar structures with several matter distributions.

For this study, we take the anisotropic matter represented by

()

where density, radial, and transverse pressures are denoted by ρ, p_r, and p_t, respectively. The four-velocity ξ^σ and its unit normal vector η^σ satisfy the constriction ξ^σξ_σ = −1 and η^ση_σ = 1, respectively. The metric (6) and matter distribution (7) are considered, and the Rastall field equations (5) are expressible as

()

()

()

The system of equations (8)–(10) is solvable by the assumption of the Finch–Skea geometry. A system is formulated, comprising three independent equations and five unknown functions, which are ρ(r), p_r(r), p_t(r), λ(r) and ν(r). The Finch–Skea ansatz, that is,  and e^ν(r) = 1 + Cr², is utilized in the aforementioned system of equations, resulting in the following solution:

()

()

()

3. Matching Condition

In this segment, a match is made between the interior metric (6) and the suitable exterior space-time, which is the Schwarzschild metric [53], resulting in the relations for the unrevealed constants being expressed in tem of the mass and radius. The Schwarzschild metric is specified by the line element

()

where the sphere of radius r = R, and M is the gravitational mass. The metric coefficients g_ii, i = r, t, and the partial derivative ∂g_tt/∂_r are continuous at r = R (i.e., on the border) of the internal and external regions, resulting in the following relations:

()

()

()

The unknown constant is determined by relations (15)–(17), which yield the following values:

()

()

()

4. Main Features of the Model and Parameter Values