1 Introduction

The stars are among the most frequently observed celestial objects in the cosmos. These objects can be seen as light spots in the sky in a single view at night, but with to the advancement of sophisticated terrestrial and space telescopes, man is now able to study stars beyond our sun, ancient stars, and the edge of the universe. These telescopes have also enabled scientists to study the different stages of a star’s life cycle. Additionally, by analyzing the light emitted by stars, astronomers can gather valuable information about their composition, temperature, and distance from Earth. This information helps scientists understand the vastness and complexity of the universe as well as gain insights into the processes that govern the formation and evolution of stars. In a very special way, the remnants of the final phases of the evolution of these stellar objects are of great observational and theoretical interest given that they are natural laboratories of physical scenarios with very extreme conditions such as high densities, very intense magnetic fields, very strong gravitational fields, high temperatures, the presence of relativistic plasmas, etc. Between them are the white dwarfs, neutron stars, black holes, and even a hypothetical compact star composed of exotic matter [1,2,3].

In this sense, the need has arisen to create models that can explain the main characteristics of these complex stellar structures, such as their interior and the dynamics of the extreme conditions mentioned above, and even more since how the fundamental elements of the known universe are generated in them [4,5,6]. This study is highly multidisciplinary and involves several branches of physics such as nuclear physics, thermodynamics, statistical physics, particle physics, plasma physics, astrophysics, and general relativity in order to develop a comprehensive understanding of these intriguing astronomical entities [7, 8]. Now, for understanding the gravitational effects in these stellar objects, it is necessary to use general relativity (GR); in particular, the space-time and some macroscopic physical properties inside these objects can be modelled with great realism with the aid of EFE. The task of finding solutions that model the interior of a stellar compact objects is an arduous one given the non-linear nature of Einstein’s equations for self-gravitating spheres. However, already in 1916 Schwarzschild managed to find a first solution for the interior of a self-gravitating sphere of uniform density [9], and since then solutions have been built for the EFE that model the interior of these stellar objects.

Certainly, it is worth mentioning that these objects were initially modelled through isotropic fluid distributions to the first approximation, however it is currently known that this approximation is not the most appropriate, and in fact is known that there are several physical processes that take place in both high and low-density regimes that account for the deviation from local isotropy, between them we can mention phase transitions, solid cores, the presence of super-fluids, viscosity, rotation, intense magnetic fields, condensed phases states of pions and among others [10,11,12,13,14,15,16,17,18,19,20]. Even, it has been demonstrated more recently that pressure anisotropy is usually induced by the existence of shear forces, dissipation, and energy density inhomogeneities in any physical process of the kind envisaged in star evolution [21]. Consequently, models pertaining to the anisotropic pressure regime’s fluid distribution are more suitable for depicting a stellar compact object than are those using isotropic fluid. And more specifically, these solutions to be considered as models of physical interest must satisfy a set of physical acceptability conditions which guarantee that they can effectively model realistic stars. Particularly, these conditions have been studied in [22] for a stellar anisotropic fluid distribution.

There are several ways to obtain or construct stellar models in the regime of anisotropy pressure in the literature (see for example [23,24,25,26,27,28,29,30,31,32,33,34,35,36,37]). Recently, a novel and powerful framework generating new static and spherical symmetric solutions in the regime of anisotropic pressure has emerged called Gravitational Decoupling (GD) [38,39,40,41,42] in the context of the Randall–Sundrum brane-world [43, 44]. GD has been first formulated through the Minimal Geometrical Deformation (MGD) [42], which uses a known solution of EFE (commonly called “seed solution”) plus an additional unknown source to solve the entire system of EFE. Also, an extended version of MGD has been developed, called Extended Minimal Geometric Deformation (MGDe) [45]. In this context, a number of novel anisotropic stellar solutions have been built [46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99]. Even more recently, the use of cosmological solutions as “seed” into GD framework also has been explored; firstly in [100] the Einstein’s universe has been used as a seed solution in the GD framework by TGD (Temporal Geometric Deformation) in order to obtain a new stellar solution in the isotropic fluid regime, and also the same Einstein’s universe is extended to an isotropic interior solution through the MGDe in [101]. As well as, in [102] the known cosmological solution of Kohler–Chao–Tikekar [103, 104] is used in order to obtain a new anisotropic interior solution using the GD by MGD successfully. In certain measure, these works are unusual since the GD is commonly used with the intention of generating a new anisotropic solution from perfect fluid distributions, and not from cosmological solutions. However, the above mentioned works are pioneering in use these type of solutions as seed into the GD with the propose of obtaining bounded configuration related a stellar object.

Based on the aforementioned, the current study constructs both an anisotropic and an isotropic extension of the Kohler–Chao–Tikekar cosmological solution via GD via MGD. And thus, reinforce the idea that cosmological type solutions can also be used as seed solutions in the GD to obtain physically acceptable interior solutions.

2 Relativistic anisotropic fluid distributions

An anisotropic fluid distribution can arise in situations where there are asymmetries in the system, such as in extreme conditions found on a stellar compact object. So in such regard, to model a stellar compact object in the anisotropic regime of pressure, we need to use the EFE given by

$$\begin{aligned} G_{\mu \nu } = \kappa T_{\mu \nu }, \end{aligned}$$
(1)

where \(G_{\mu \nu }\) is the Einstein’s tensor and \(\kappa = \frac{8\pi G}{c^4}\). The application of these field equations is a necessary prerequisite since the gravitational effects of these stellar objects are significant and the classical Newtonian gravity is insufficient to adequately describe the behaviour of gravity [1].

Also, we need to consider the stellar compact object as a static, self-gravitating sphere whose interior space-time is given by the following metric:

$$\begin{aligned} ds^{2}=e^{\nu }dt^{2}-e^{\lambda }dr^{2} -r^{2}d\varOmega ^{2}, \end{aligned}$$
(2)

where \(d\varOmega ^{2}=d\theta ^{2}+\sin ^{2}\theta d\phi ^{2}\) stands for the differential solid angle on a unit 2-sphere, \(\nu \) and \(\lambda \) are functions only of radial coordinate r. While, the fluid matter distribution of this compact object is well described with an energy–momentum tensor \(T_{\mu \nu }\) given by

$$\begin{aligned} T_{\nu }^{\mu } = diag[\rho , -p_{r}, -p_{t},-p_{t}], \end{aligned}$$
(3)

where the density energy \(\rho \), the radial pressure \(p_{r}\), and the tangential pressure \(p_{t}\).

Thus, the following system of equations is formed if one applies this information about matter sector \(\lbrace \rho , p_r, p_t\rbrace \) and space-time (given by Eq. (2)) of the stellar compact object in the EFE (1):

$$\begin{aligned} \rho= & {} \frac{1}{\kappa }\left[ \frac{1}{r^2} + e^{-\lambda }\!\left( \frac{\lambda '}{r} - \frac{1}{r^2}\right) \right] \!, \end{aligned}$$
(4)
$$\begin{aligned} p_{r}= & {} \frac{1}{\kappa }\left[ -\frac{1}{r^2} + e^{-\lambda }\!\left( \frac{\nu '}{r} + \frac{1}{r^2}\right) \right] \!, \end{aligned}$$
(5)
$$\begin{aligned} p_{t}= & {} \frac{e^{-\lambda }}{4\kappa }\left[ 2\nu '' + {\nu '}^2 - \lambda '\nu ' + 2\frac{\nu '-\lambda '}{r} \right] .\! \end{aligned}$$
(6)

Here the primes denote differentiation with respect to the radial coordinate. It is worth noting that in order to close the system given by Eqs. (4)–(6), it is necessary to provide two extra conditions since we have a system of three differential equations with five unknown quantities: \(\lbrace \nu ,\lambda , \rho , p_r, p_t\rbrace \). Such conditions habitually are geometrical relations (for example, the Karmakar condition [32, 105], conformally symmetries [106]) or equations of state (EoS) that relate the physical quantities of the matter sector (for example, the polytropic equation of state [107,108,109,110], the Van der Waals equation of state [111, 112], the barotropic equation of state [113, 114], etc) and others.

3 Gravitational decoupling by MGD

The success of the GD framework by MGD is solve Eq. (1) with the particularity of

$$\begin{aligned} T_{\mu \nu }=T_{\mu \nu }^{s} + \vartheta _{\mu \nu } \end{aligned}$$
(7)

in an analytical way for the static and spherically symmetric sources. The resulting source \(T_{\mu \nu }\) of this problem should be then

$$\begin{aligned} \rho= & {} \rho ^{s}+\vartheta _{0}^{0}, \end{aligned}$$
(8)
$$\begin{aligned} p_{r}= & {} p_{r}^{s}-\vartheta _{1}^{1}, \end{aligned}$$
(9)
$$\begin{aligned} p_{t}= & {} p_{t}^{s}-\vartheta _{2}^{2}, \end{aligned}$$
(10)

where \(T_{\nu }^{\mu (s)} = diag[\rho ^{s},-p_{r}^{s},-p_{t}^{s},-p_{t}^{s}]\) usually is a known source (named as the “seed source”) and the other \(\vartheta _{\nu }^{\mu } = diag[\vartheta _{0}^{0},-\vartheta _{1}^{1},-\vartheta _{2}^{2},-\vartheta _{3}^{3}]\). It is possible if one consider that the space-time related with the seed source is deformed by the influence of the unknown source \(\vartheta _{\mu \nu }\) over \(T_{\mu \nu }^{s}\) in the following way

$$\begin{aligned} e^{-\mu } + h= & {} e^{-\lambda }, \end{aligned}$$
(11)
$$\begin{aligned} e^{\xi } + g= & {} e^{\nu }, \end{aligned}$$
(12)

where \(e^{\xi }\) and \(e^{\mu }\) are the temporal and radial component metrics of the seed space-time given by

$$\begin{aligned} ds^{2} = e^{\xi }dt^{2}-e^{\mu }dt^{2}-r^{2}d\varOmega ^{2}. \end{aligned}$$
(13)

Since in this work we use the MGD, we have to set \(g=0\), namely, the geometric deformation is only in the radial metric.

In order to guarantee the spherical symmetry of the effective solution, it is important to note that the functions \(\lbrace g,h\rbrace \) mentioned above are just radial functions.

Now, with this above idea one can decoupled the EFE given by Eq. (1) in two sets of differential equations: one describing a seed sector sourced by the conserved energy–momentum tensor, \(T_{\mu \nu }^{s}\)

$$\begin{aligned} \rho ^{s}= & {} \frac{1}{\kappa }\left[ \frac{1}{r^2} + e^{-\mu }\!\left( \frac{\mu '}{r} - \frac{1}{r^2}\right) \right] \!, \end{aligned}$$
(14)
$$\begin{aligned} p_r^{s}= & {} \frac{1}{\kappa }\left[ -\frac{1}{r^2} + e^{-\mu }\!\left( \frac{\nu '}{r} + \frac{1}{r^2}\right) \right] \!, \end{aligned}$$
(15)
$$\begin{aligned} p_{t}^{s}= & {} \frac{e^{-\mu }}{4\kappa }\!\! \left[ 2\nu '' + {\nu '}^2 - \mu ' \nu ' + 2\frac{\nu '-\mu '}{r} \right] \!, \end{aligned}$$
(16)

and the other set corresponds to quasi-Einstein field equations sourced by \(\vartheta _{\mu \nu }\)

$$\begin{aligned} \vartheta ^0_0= & {} \frac{1}{\kappa }\left[ - \frac{h}{r^2} -\frac{h'}{r}\right] , \end{aligned}$$
(17)
$$\begin{aligned} \vartheta ^1_1= & {} \frac{1}{\kappa }\left[ -h \left( \frac{\nu '}{r} + \frac{1}{r^2}\right) \right] \!, \end{aligned}$$
(18)
$$\begin{aligned} \vartheta ^2_2= & {} \frac{1}{\kappa }\left[ -\frac{h}{4} \left( 2\nu '' + {\nu '}^2 + 2\frac{\nu '}{r}\right) -\frac{h'}{4} \left( \nu ' + \frac{2}{r} \right) \right] \!. \end{aligned}$$
(19)

Precisely, the previous development of decoupling the original EFE into two subsets of systems of equations gives it the name of gravitational decoupling, but also its power when generating new solutions (for detailed information about the GD by MGD see Ref. [42]).

4 Solution I: the isotropic solution

In this section, we obtain a new interior solution in the regime of isotropic pressure of EFE using the framework of GD by MGD, in specific we use the Kohler–Chao–Tikekar cosmological solution [103, 104] as a seed solution, whose metrics components are given by

$$\begin{aligned} e^{\nu }= & {} A+B r^2, \end{aligned}$$
(20)
$$\begin{aligned} e^{-\mu }= & {} \frac{A+B r^2}{A+2 B r^2}, \end{aligned}$$
(21)

where A and B are constants. Now in order to close the EFE, we will use as a extra condition that

$$\begin{aligned} \varPi _{\vartheta } = \vartheta _{2}^{2}-\vartheta _{1}^{1}=0, \end{aligned}$$
(22)

namely, we consider that the cosmological fluid given by the Kohler–Chao–Tikekar interacts with an unknown \(\vartheta _{\mu \nu }\) isotropic source gravitationally. Thus, we obtain that

$$\begin{aligned} \left( \nu ' +\frac{2}{r}\right) h'+\left( 2\nu ''+\nu '^2-2\frac{\nu '}{r}-\frac{4}{r^2}\right) h = 0. \end{aligned}$$
(23)

Therefore, using the metric of the seed solution (20) and (21) in (23) we obtain

$$\begin{aligned} h(r) = \frac{\left( A+B r^2\right) \eta r^2}{A+2 B r^2}, \end{aligned}$$
(24)

where \(\eta \) is a integration constant. Thus, with this h deformation and (21) in Eq. (11) a new radial metric is obtained

$$\begin{aligned} e^{-\lambda } = \frac{\left( A+B r^2\right) \left( \eta r^2+1\right) }{A+2 B r^2}. \end{aligned}$$
(25)

Therefore, with this new metric given by (20) and (25) in EFE (4)–(6), we obtain a new isotropic matter sector given by

$$\begin{aligned} \rho= & {} \frac{B r^2 (2 B-7 A \eta )+3 A (B-A \eta )-6 B^2 \eta r^4}{\kappa \left( A+2 B r^2\right) ^2}, \end{aligned}$$
(26)
$$\begin{aligned} p= & {} \frac{A \eta +3 B \eta r^2+B}{\kappa \left( A +2 B r^2\right) }. \end{aligned}$$
(27)

Now, it is necessary to take into account that the space-time should be continuous even at the surface \(\varSigma \) of a stellar compact object. We have to match the interior metric with the outside, namely, the Schwarzschild exterior solution. Thus, it is required that

$$\begin{aligned} e^{\nu }\Big |_{\varSigma }= & {} e^{-\lambda }\Big |_{\varSigma } = 1 - \frac{2M}{R}, \end{aligned}$$
(28)
$$\begin{aligned} p_r(r) \Big |_{\varSigma ^{-}}= & {} p_r(r) \Big |_{\varSigma ^{+}} = 0, \end{aligned}$$
(29)

where M and R is the mass and radius of the stellar object, respectively.

So considering these matching conditions (28) and (29) for this model, we obtain the following relations:

$$\begin{aligned} \eta= & {} -\frac{M}{R^3}, \end{aligned}$$
(30)
$$\begin{aligned} A= & {} 1-\frac{3 M}{R}, \end{aligned}$$
(31)
$$\begin{aligned} B= & {} \frac{M}{R^3}. \end{aligned}$$
(32)

Indeed, in Figs. 1 and 2, the metrics profile for this solution is shown for several values of compactness factor \(u = \frac{M}{R}\). We can observe that \(e^{\nu }\) is a monotonously increasing function and \(e^{-\lambda }\) is monotonously decreasing of radial coordinate r; furthermore, we observe that \(e^{\nu (0)} = constant\) and \(e^{-\lambda (0)} = 1\). As so, the space-time in this model actually has the same behaviour as the space-time inside a compact stellar object.

Fig. 1
figure 1

\(e^{\nu }\) for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image
Fig. 2
figure 2

\(e^{-\lambda }\) for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image

The profile of \(\rho \) for different compactness factors is displayed in Fig. 3. It is observed that the energy density approaches the surface monotonically and is finite at the center. It is shown that the value of p diminishes monotonically with the radial coordinate and vanishes at the surface (Fig. 4). By other hand, the Dominant Energy Condition (DEC) \(\rho - p \ge 0\) is satisfied by these solution (see the Fig. 5). Therefore, once the previous conditions for the material sector have been satisfied, it can be said that this is indeed the material sector of a compact stellar object.

Fig. 3
figure 3

\(\rho \) for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image
Fig. 4
figure 4

p for Solution I for compactness factors of 0.20 (black line), 0.20 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image

On the other hand, we show the profile of \(z = g_{tt}^{-1/2}-1\) in Fig. 6. We can observe from this figure that the redshift is a monotonously decreasing function of the radial coordinate r, having its maximum value at the center of the stellar compact object. In addition, it is noticeable that the surface redshift value is below the universal bound of \(z_{bound} = 5.211\) [115].

Fig. 5
figure 5

\(\rho -p\) for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image
Fig. 6
figure 6

z for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image

In Fig. 7, we also display the sound radial velocity profile inside the stellar object. Notice that \(v =\sqrt{\frac{dp}{d\rho }}\) does not surpass the causal limit of light velocity (\(c=1\)). Note that the speed of sound is zero at the center of the stellar object, and it increases as one moves toward the surface. Therefore, we have checked that the obtained solution fulfils the fundamental physical conditions (see Ref. [22]) for the set of compactness factors of \(u = \frac{M}{R} \le 0.32\).

Fig. 7
figure 7

v for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image

On the other hand, we plot the profile of \(\rho ''\) in Fig. 8 to examine the stability of this solution versus the convective motion inside the stellar object. In order to garantize that any any fluid element displaced downward floats back to its initial position inside the stellar object is necessary that (See Ref. [116])

$$\begin{aligned} \rho '' \le 0. \end{aligned}$$
(33)

So in this case, we can observe that this condition is fulfilled by the inner regions of the stellar object, while the outer ones are not (Fig. 8). Note that these stable regions are wider, while the stellar compact object is less dense. Additionally, we examine how the adiabatic index \(\varGamma = \frac{\rho + p}{p}\frac{dp}{d\rho }\) behaves in the radial direction in order to analyse the stability of these isotropic solution against gravitational collapse. In such regard, it is necessary that the adiabatic index \(\varGamma \) must satisfy the following condition [117, 118]

$$\begin{aligned} \varGamma \ge \frac{4}{3} + \frac{19}{21}u \end{aligned}$$
(34)

in order to avoid a possible gravitational collapse (For more details about this criteria see Refs. [75, 119, 120]). We have checked that the condition given by Eq. (34) is satisfied for the compactness factors of \(0< u < 0.20\); namely, in the case of less compact objects, the model is stable against collapse (see Fig. 9).

As a result, a new physically feasible interior solution in the isotropic pressure regime has been developed. It exhibits inner stable zones against convection motion, but it is stable against gravitational collapse only when the compactness factor is set to \(0< u < 0.20\).

Fig. 8
figure 8

\(\rho ''\) for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.28 (orange line) and 0.30 (purple line)

Full size image
Fig. 9
figure 9

\(\varGamma \) for Solution I for compactness factors of 0.20 (black line), 0.25 (green line), 0.30 (orange line) and 0.32 (purple line)

Full size image

5 Solution II: the anisotropic solution

In this section, we use the isotropic solution found in Sect. 4 given by (20) and (25) as a seed solution in the framework of GD by MGD. In this case, we will use the energy density mimic constrain

$$\begin{aligned} \rho ^{s} = \vartheta _{0}^{0}, \end{aligned}$$
(35)

so in such a way we have that

$$\begin{aligned} h'+\frac{1}{r}h+\frac{1}{r}+e^{-\mu }\left( \mu '-\frac{1}{r}\right) = 0. \end{aligned}$$
(36)

So using (25) in Eq. (36), we obtain

$$\begin{aligned} h(r) = \frac{r^2 \left( A \eta +B \left( \eta r^2-1\right) \right) }{A+2 B r^2}+\frac{\sigma }{r}, \end{aligned}$$
(37)

where \(\sigma \) is a integration constant. For regularity reasons we set \(\sigma = 0\). Now using the geometric deformation h in the Eq. (11), we can obtain a new radial metric component

$$\begin{aligned} e^{-\lambda } = \frac{2 \eta r^2 \left( A+B r^2\right) +A}{A+2 B r^2}. \end{aligned}$$
(38)

After using the new space-time given by (20) and (38) in the EFE (4)–(6), we obtain the new matter sector

$$\begin{aligned} \rho= & {} -\frac{2 \left( 3 A^2 \eta +A B \left( 7 \eta r^2-3\right) +2 B^2 r^2 \left( 3 \eta r^2-1\right) \right) }{\kappa \left( A+2 B r^2\right) ^2}, \nonumber \\ \end{aligned}$$
(39)
$$\begin{aligned} p_{r}= & {} \frac{2 \eta \left( A+B r^2\right) \left( A+3 B r^2\right) -2 B^2 r^2}{\kappa \left( A+B r^2\right) \left( A+2 B r^2\right) }, \end{aligned}$$
(40)
$$\begin{aligned} p_{t}= & {} \frac{2 \eta \left( A+B r^2\right) ^2 \left( A+3 B r^2\right) -A B^2 r^2}{\kappa \left( A+B r^2\right) ^2 \left( A+2 B r^2\right) }. \end{aligned}$$
(41)

Additionally, based on the matching requirements (28) and (29), we may determine that

$$\begin{aligned} \eta= & {} \frac{M^2}{R^3 (R-2M)}, \end{aligned}$$
(42)
$$\begin{aligned} A= & {} 1-\frac{3 M}{R}, \end{aligned}$$
(43)
$$\begin{aligned} B= & {} \frac{M}{R^3}. \end{aligned}$$
(44)

In Figs. 10 and 11, we plot the metrics \(e^{\nu }\) and \(e^{-\lambda }\). Observe that \(e^{\nu }\) is a monotonously increasing function of radial coordinate with \(e^{\nu (0)} = constant\), and \(e^{-\lambda }\) is monotonously decreasing with \(e^{-\lambda (0)} = 1\). Also, in Figs. 12, 13, and 14, we plot the profile matter sector \(\lbrace \rho , p_{r}, p_{t}\rbrace \). These figures show that the three physical quantities are positive and regular. They reach their maximum values in the center of a compact star and then gradually decrease towards the surface. As can be seen, the radial pressure disappears at the surface, which is consistent with the matching condition provided by Eq. (29). Also, we plot the profile of the anisotropic function \(\varDelta = p_{t}-p_{r}\) in Fig. 15. In this figure, is visible that \(p_{t}-p_{r} > 0\) inside the stellar object; however, at the center, \(p_{r}(0)=p_{t}(0)\), which is necessary for the stability of the star.

Fig. 10
figure 10

\(e^{\nu }\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 11
figure 11

\(e^{-\lambda }\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 12
figure 12

\(\rho \) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 13
figure 13

\(p_{r}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 14
figure 14

\(p_{t}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 15
figure 15

\(p_{t}-p_{r}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image

Furthermore, we checked that Solution II fulfils the DEC conditions \(\rho - p_{r} \ge 0\) and \(\rho - p_{t} \ge 0\), namely, this solution is related to a gravitational source physically acceptable from the energetic point of view (see Figs. 16, 17).

Fig. 16
figure 16

\(\rho -p_{r}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 17
figure 17

\(\rho -p_{t}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image

In Fig. 18, the profile of the redshift function z is shown. We can observe that z is a monotonously decreasing function with its maximum value at the center of the star. It is also observed that its value at the surface does not exceed the \(z_{bound} = 5.211\) universal limit.

Fig. 18
figure 18

z for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image

We verified that the sound velocities in the radial and tangential directions inside the stellar object do not raise the causal limit of light velocity (\(c=1\)) in Fig. 19 and 20. Observe moreover that the two velocity profiles are functions of the r coordinate that decrease monotonically.

Therefore, with the aforementioned, we can infer that Solution II represents a stellar model related to an anisotropic fluid distribution, which is physically acceptable. Additionally, Solution II has certain stability in the inner shells of the stellar compact object; look up that these regions are wider than the external regions in this case (see Fig. 21). Also, note that these inner regions are wider for less compact objects. While the behaviour of the adiabatic index \(\varGamma \) is plotted in Fig. 22. We checked that this Solution II is stable against the gravitational collapse for all compactness factors taken into account, namely, for \(0 < u \le 0.15 \).

Fig. 19
figure 19

\(v_{r}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 20
figure 20

\(v_{t}\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 21
figure 21

\(\rho ''\) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image
Fig. 22
figure 22

\(\varGamma \) for Solution II for compactness factors of 0.10 (black line), 0.12 (green line), 0.14 (orange line) and 0.15 (purple line)

Full size image

6 Discussion

It is well known that the non-linear structure of the Einstein field equations (EFE) makes it difficult to find exact solutions for static and spherically symmetric space-times. This difficulty has motivated intense, active research in theoretical stellar astrophysics dedicated to developing new stellar models. In this regard, several methods have been developed in this area, and precisely the GD approach has been a ground-breaking result to address this problem and permit the determination of exact solutions for the interior configuration of stellar distribution.

In this work, we implemented the GD approach through MGD to obtain two new interior solutions, one in the isotropic regime of pressure and the other in the anisotropic regime. The isotropic interior solution was obtained using the Kohler–Chao–Tikekar cosmological solution as a seed solution with the imposition of the isotropy pressure of the extra source given by \(\varPi _\vartheta =0\), and with it, the system of EFE is completely closed. The resulting solution satisfies the fundamental physical acceptability conditions for any realistic stellar distribution. Furthermore, that this solution has certain stability against the convection motion for the inner regions of the stellar object which tending to growth when the stellar object is less compact, and that the stellar model is stable against gravitational collapse for the set compactness factor of \(u < 0.20\). It is worth mentioning, despite the fact that isotropic stellar solution approximations are not the best approximations to describe real stellar compact objects, this model can still be interesting because isotropic solutions can be used to expand the collection of known isotropic solutions to build realistic stellar models, or they can be used to perfectly describe phenomena like the initial states of a realistic star. They can even be used as models of cold fluid planets [121,122,123,124,125]. For example, also isotropic compact stars are useful for describing charged compact stars (see Refs. [126, 127]). Several investigations of charged isotropic spheres also can be review in [128, 129]. In this regard, this new solution adds to the limited number of physically feasible isotropic solutions already known.

Despite all of the above, we have to mentioned the importance of presence of anisotropy in interior solutions, this is to the fact that numerous cosmological studies over the last two decades led to the conclusion that our universe is not isotropic. For example, a slight deviation from isotropy was noted while examining inhomogeneous Supernova Ia [130,131,132]. There are several other inconsistencies (i.e., radio sources [133], infra-red galaxies [134] and gamma-ray bursts [135], etc.) that have been found through various observations. This observational data including several physical models lead to the indiscutible conclusion that the universe possesses anisotropy. This underscores the importance of creating anisotropic solutions, which would ultimately enable astronomers to gain a better understanding of the various developmental phases of the universe.

In this sense, the second solution obtained is in the anisotropic regime of pressure, which was obtained through the application of GD through MGD also; however, in this case, the seed solution used was the isotropic one obtained before, and for the extra condition, in order to close the EFE, we use the mimic constraint for the density energy, namely \(\rho ^{s} = \vartheta _{0}^{0}\). This solution, like the first one found, is a realistic solution. Thus, even its stability in terms of convective movement has a behaviour similar to the previous solution; that is, it presents regions of stability that are internal, while the most external ones are unstable, and also this regions are wider for the less compact objects. Furthermore, this solution is stable against the gravitational collapse for all compactness factors set of \(u \le 0.15\). In this point, it is worth mentioning that the Kohler–Chao–Tikekar cosmological solution has already been used to obtain an anisotropic interior solution in [102], which was obtained with the direct use of this cosmological solution as seed solution in GD (through MGD); however, in this work, we obtain a new isotropic solution (Solution I) using this cosmological solution as seed in GD, and with this last one, we obtain a new anisotropic solution (Solution II). That is to say, Solution II obtained here is a totally different solution from the one obtained in [102].

On the other hand, since the two solutions have internal layers with stability against convective movement and, for certain values of compactness, they are also not prone to gravitational collapse, this could be associated with the possibility of the existence of solid nuclei at the same time. This solid nuclei could potentially be made up of exotic forms of matter, such as strange quark matter or other hypothetical types of dense nuclear matter [136,137,138,139,140,141,142]. The presence of stable internal layers may indicate that these solid nuclei are able to withstand extreme pressures and the convective motion within the stellar compact object. Further research and observation will be necessary to confirm the existence of such solid nuclei and understand their implications for the properties of these fascinating stellar objects.

Finally, the present work represent a new study in which one can use a cosmological solution with the aim of generating new interior solutions through the use of GD. This goal has been addressed lastly in [100,101,102]. This MGD scheme was also followed by Cedeno and Contreras [143] with Kantowski-Sachs and Friedmann-Robertson-Walker cosmology metrics to find anisotropic extensions to this cosmological scenarios. As well as, the GD has been used in cosmology solutions in modified gravity solutions associated with a conformal gravitational sector, as in the case of braneworld and f(R) models by Sharif and his collaborators, where they have developed multiple anisotropic models from cosmological solutions and found them physically acceptable for certain parametric values [144, 145]. Therefore, the present work and those mentioned above can motivate us to explore new possible applications of GD using unusual solutions (such as cosmological ones) in order to show the great applicability of this method. By thinking outside the box and considering unconventional applications of GD, we may uncover new and exciting opportunities for scientific exploration in the area of theoretical astrophysics.