Influence of three parameters on decoupled charged compact stars generated by dark matter and their predicted radii in gravity theory

Article Content

Highlights

•
We have investigated new most general charged anisotropic solutions generated by dark matter.
•
The gravitational decoupling approach has been used to introduce dark matter.
•
The exact solution is derived using Durgapal–Fuloria metric along with Pseudo Isothermal dark matter density profile.
•
The obtained decoupled charged fluid solution is viable for modeling of compact objects.
•
The mass and radius have been fitted with known observed compact objects.

Abstract

In the context of

f (R, T)

gravity theory we have obtained an exact non-singular solution to the field equations for the anisotropic charged stellar system. The method of minimal geometrical deformation (MGD) with regard to the gravitational decoupling approach is employed to reduce the field equations into two sets of equations governed by the seed system and the new source system. An isotropic charged fluid configuration with modified Durgapal–Fuloria potential as a metric ansatz is considered for the seed system whereas density profile of Pseudo-Isothermal dark matter (PI-DM) is taken into account to mimic a component of the new source system. The anisotropy feature that essentially arises in the effective system is one of the important consequences of the gravitational decoupling approach. With this approach we get the solution to gravitationally decoupled field equations which describes the physical characteristics of an effective anisotropic and charged system. The effective system is found to be stable with regard to Herrera’s cracking concept, adiabatic condition, and Harrison–Zeldovich–Novikov criteria. The influence of the MGD parameter, coupling constant, and dark matter density parameter on the physical features and stability of the effective system have been analyzed and shown graphically. The mass–radius relation of the effective system is inspected in connection to the observational constraint of the massive stars such as pulsars and massive secondary companions involved in gravitational wave events. The maximum mass of the star is apparently constrained with the increasing values of the

f (R, T)

-gravity coupling constant, MGD parameter, DM density parameter, and charge. Enabling all the set of parameters the range of predicted radii of the observed massive stars was found to be 9.77 km–12.00 km.

Introduction

In recent times, the gravitational theory given by Einstein, i.e., general relativity (GR) seems to have confrontation [1], [2], [3] with the observational discoveries in connection to the anisotropies of cosmic microwave background radiation (CMBR) [4], study of dark energy models based on the high energy physics as well as the weak lensing data [5] and the Lyman-

α

forest power spectrum obtained from the Sloan Digital Sky Survey [6] which indicate the accelerated expanding of the universe. The observed accelerating Universe can be described with consideration of a positive cosmological constant in accordance to the observed acceleration in the Hubble fluid.

In the

Λ

CDM model [7], whereas a positive cosmological constant can account for about 70% to the whole energy of the Universe, the significant part about 25% of the rest of the energy is believed to be caused by the existence of dark matter that can be found in the large scale structure. On the other hand, the observed value of the dark energy being treated with the vacuum energy by Zel’dovich [8] in

Λ

CDM model seem to have major inconsistency by the

120

orders of magnitude in comparison with the predicted theoretical value of dark energy in quantum gravity [9]. This huge discrepancy is regarded as the ‘cosmological constant problem’ which is one of the key challenges in the field of modern cosmology. Therefore, it becomes an urgent matter of critical concern to the science community to come with more comprehensive explanations for the dark matter and dark energy.

Axions, fermions, and axion-like particles are treated to be as DM candidates to develop various models of DM which can describe physical features of compact star configurations consist of self-interacting DM particles. Some studies [10], [11] proposed that boson stars may have bosonic DM particles. Again, compact stars having fermionic DM particles are investigated in some research works [12], [13]. The physical features of astrophysical compact objects such as the relationship between mass and radius of neutron stars and quark stars [14] are expected to be effected significantly due to existence of self-interacting DM particles in the fluid-like structures of the compact objects. Conversely, it can be said that theoretical exploration as well as advanced astronomical measurements of physical properties of compact stars become essential in probing and constraining the attributes of DM [15], [16]. In this connection, there are some noteworthy research works [17], [18], [19], [20] which examined modifications in the stellar configurations due to exclusion or inclusion of interaction between the DM particles. As a consequence, compact objects of exotic nature like dark compact planets [21] are predicted theoretically to exist in universe. It is speculated that the sensors used in gravitational wave astronomy will be able to probe the possible existence of DM by enabling mass and interaction strength of DM particles in the neutron star mergers [22], [23].

The fact that masses of the observed massive pulsars [24], [25] and white dwarfs [26], [27], [28] exceed the Chandrasekhar mass limit, i.e.,

1.44 M_{⊙}

cannot be justified satisfactorily by GR. Since GR is unable to proper justification regarding the existence of dark matter and dark energy in the context of accelerating universe, it is needed to modify the Einstein’s GR to understand the universe at all scales. In the modified theories of gravity the field equations in GR are modified or extended to address various unsolved problems that cannot be explained in context of GR [29], [30], [31], [32], [33]. For instance, the exotic stellar configurations can be studied in the extended theory of gravity [34], [35] as framework of GR seems to be inadequate to explain this astrophysical exotic objects. Further, the application of modified theories of gravity in the field of astrophysics appears to be consistent with the observational data obtained from some local tests of gravity [36]. Subsequently, various modified gravitational theories have been recommended including

f (R)

gravity [29], [30], [37], [38] where

R

is the Ricci scalar, the

f (T)

gravity [39], [40], [41], [42] where

T

is the torsion scalar, the

f (G)

gravity [43], [44], [45] where

G

is the Gauss–Bonnet scalar, the

f (R, G)

gravity [46], [47], Brans–Dicke (BD) gravity [48], [49] etc.

Various research studies reveal inconsistent results obtained for the Solar system tests in the framework of

f (R)

gravity theory [50], [51], [52]. Despite the formulation of

f (R)

gravity theory with

f

-function [53], [54], it faced some difficulties to give proper explanation to CMBR tests, the strong lensing regime and the galactic scale [55], [56], [57], [58], [59]. With this regard, one can note that the

f (R)

gravity theory is found to be equivalent to the theories of scalar-tensor gravity which is actually the GR in terms of real scalar field resembling the

f

-function. In literature, supergravity [54], [60], [61] and braneworld [62] include the extensions of

f (R)

gravity theories.

As a consequence further modified gravity theories such as the

f (R, T)

theory of gravity [63] has been developed on the basis of the non minimally curvature and matter coupling by defining a function of the Ricci scalar

R

and the trace of the energy–momentum tensor

T

as the gravitational Lagrangian of the standard Einstein–Hilbert action. The influence of the imperfect fluid [63] or particle production [64] type quantum effects can be represented by including the reliance on

T

in the gravitational Lagrangian. It is remarkable to note that the covariant derivative of

T

do not vanish,i.e.,

\nabla_{μ} T^{μ ν} \neq 0

[63], [65]

f (R, T)

theory of gravity. Hence, the coupling between the geometry term (

R

) and matter (

T

) results an extra acceleration which compels the particles travel along a non-geodesic path in

f (R, T)

gravity. In cosmology, the possible occurrence of non-conservation of the

T

[66], [67] during the evolution of the universe is recommended in the process of the creation or destruction of matter. From the thermodynamical perspective non-conservation of the

T

have been investigated by [64]. The topic is further studied in the other research works [68], [69]. In this connection, one may note that the cosmic acceleration depends on the energy–momentum tensor [70], [71], [72].

In literature, one can find application of

f (R, T)

gravity theory in the field of cosmology [66], [73], [74], [75], [76] and astrophysics [77], [78], [79], [80], [81], [82], [83], [84], [85]. With a particular form of in the

f (R, T)

function presented as

f (R, T) = R + h (T)

a static analysis [86] revealed that matter-geometry coupling results the stellar configurations acting as a non-interacting two-fluid system. In addition, the analysis [86] explores the features of conservation of the effective

T

in connection to the geodesic path of particles in the background of

(R, T)

gravity theory. Again, a research study [76] have shown that the

f (R, T)

gravity theory is seen to be consistent with the tests based on Solar System. Moreover, the non-geodesic path [87], galactic effects due to the dark matter [88] and the tests regarding the gravitational lensing [89] are explained suitably by the

(R, T)

gravity theory. Various gravitational models [90], [91] based on the

(R, T)

gravity are in concurrence with the astronomical observations such as Hubble data set

H (z)

, the Baryon Acoustic Oscillation data BAO and the Planck 2015 observational data [92].

The instability problem of spherical gravitational body with anisotropy feature under the

f (R, T)

theory have been addressed by a perturbation scheme in research works [78], [79], [80]. The dynamics of gravitating bodies with axially symmetric spacetime under the

f (R, T)

gravity have been explored in another work [81]. There are other important research investigations [82], [83], [84] which explores various physical features of self-gravitating bodies under

f (R, T)

gravity. Further investigations studied the evolutionary behaviors of self-gravitating bodies and irregularity parameters arise due to existence of imperfect fluid in the self-gravitating bodies.

In recent years, exploring the nature of exact composition of compact stars (CSs) such as neutron stars (NSs), strange stars has evolved as a key area of active investigation in the field of relativistic astrophysics. The discovery of pulsars [93] supported observationally the existence of predicted CSs like NSs. In the past few decades Durgapal and Fuloria [94] had developed a new analytic stellar model for superdense stars in GR satisfying all the physically valid criteria and showing maximum allowable mass of neutron star as

4.17 M_{⊙}

. Recently, Gupta and Maurya [95] obtained a class of solution based on the charged analog of Durgapal–Fuloria model [94] for superdense star in the context of the Einstein–Maxwell spacetime. Thereafter another class of well behaved charged solution regarding to the Durgapal–Fuloria [94] model presented by Murad and Fatema [96] where the radius of the Crab pulsar found as 13.21 km. Later on anisotropic analog to the Durgapal–Fuloria [94] model have been investigated by a research study [97].

Ruderman [98] argued that the nuclear matter could possess anisotropic attributes due to very high density (

> 1 0^{15} {gm/cm}^{3}

). Consequently, the nuclear interaction with anisotropy characteristics at high densities can be considered in the stellar matter in compact stars under relativistic framework. In case of anisotropic stellar configuration radial pressure and tangential pressure are not equal inside the star. However, anisotropy vanishes at the center of the self-gravitating fluid sphere. The influence of anisotropic feature on maximum allowable mass and surface redshift have been studied by Bowers and Liang [99]. One can find various research works on anisotropic compact star models [100], [101], [102], [103] and anisotropic physical models in connection to Globular Clusters, Galactic Bulges and Dark Halos [104] in literature.

Stellar models of self-gravitating fluid systems have been studied in the background of GR and at present in modified theories of gravity to find physical viable explanations to the current unsolved problems in the field of astrophysics and cosmology. It is interesting to note that difficulties in obtaining analytical solution to the gravitational field equations may be eased by introducing physical complexity in the stellar systems. For instance, Ivanov [105] simplified the field equations for uncharged perfect fluid with inclusion of charge as a physical complexity to obtain an analytical solution. The simplification of the field equations can be enhanced if anisotropic feature is incorporated with charged stellar fluid. The assumption that the stellar fluid system can be coupled with electric field and anisotropic force seems reasonable at the CSs are extremely compact object with high densities. Further, gravitational decoupling approach leads the field equations of anisotropic charged fluid to the next level of simplification.

The technique of gravitational decoupling via minimally geometrical deformation (MGD) and extended geometrical deformation (EGD), as proposed by Ovalle [106], [107], is utilized to obtain novel solutions corresponding to the field equations in various gravitational theory in astrophysics. Eventually, MGD method generates two discrete systems of differential equations for the seed system and an extra source respectively from the field equations of any gravitational theory by modifying the radial component of the metric. Each system of differential equation can be solved separately and the solution of the effective system is obtained by employing superposition principle. It is to be noted that a solution obtained in a study [108] via MGD approach was investigated in the background of a braneworld scenario. Further, an anisotropic solution [109] of perfect fluid system have been proposed by introducing an extra gravitational source with help of MGD approach. Various significant attribute of an anisotropic analog of the Durgapal–Fuloria model via MGD have been explored in a study [110]. Moreover, anisotropic models with charged stellar configurations [111] have been studied using the MGD approach. One can find several works based on gravitational decoupling method in literature [112], [113], [114], [115], [116], [117], [118].

In addition to the investigation of exotic compact objects such as strange stars via the gravitational decoupling method, characteristics of black holes as well as wormholes have been explored in many studies [119], [120], [121]. In a recent work [122], features of geometrically deformed anisotropic charged stellar system have been explored by utilizing t gravitational decoupling method in the framework of

f (Q, T)

gravity theory where

Q

is non metricity scalar. With this regard, some important studies related to the influence of electric field in anisotropic stellar configurations can be found in literary works [123], [124], [125].

It is found that the choice of a Durgapal–Fuloria metric potential in association with Pseudo Isothermal dark matter as a new source in MGD approach is convenient to reduce the fluid equations to two systems of solvable equations. It is remarkable to note that the MGD approach reveals the influences of new source on the physical behavior of the anisotropic charges seed system. In the present article we want to study the anisotropic charged compact objects under MGD approach in the context of

f (R, T)

gravity [63] and to explore various physical properties of the object.

The present study is organized as follows: In Section 2 we have provided a basic formulation of

f (R, T

) gravity with gravitational decoupling for charged and anisotropic stellar system. Then the gravitationally decoupled field equations are presented in Section 3. The scheme of finding solution by MGD approach and the solution for the seed system as well as new source are presented in Section 4. Boundary condition is utilized to determine unknown constants in Section 5. Section 6 deals with the physical analysis of the solution obtained for the effective system. In the following Section 7 we investigates the stability condition for the effective stellar configuration. Additionally, we explore interesting features of the mass–radius relation in Section 8. At last we made some final remarks in Section 10.

Access through your organization

Check access to the full text by signing in through your organization.

Section snippets

Formulation $f (R, T$ ) gravity with gravitational decoupling

The gravitational Lagrangian density in the integral action for the gravitationally decoupled charged self-gravitating system in

f (R, T

) gravity consists of an arbitrary function

f (R, T)

L_{m}

corresponding to matter fields,

L_{X}

for a new source in accordance with GD and

L_{e}

for the electromagnetic field. Hence the integral action can be expressed as

S = \frac{1}{16 π} \int f (R, T) \sqrt{- g} d^{4} x + \int L_{m} \sqrt{- g} d^{4} x + β \int L_{X} \sqrt{- g} d^{4} x + \int L_{e} \sqrt{- g} d^{4} x,

where

c = G = 1

is assumed according to relativistic geometrized units and

β

is decoupling parameter.

It is

Main equations for charged anisotropic matter distributions

Let us consider the spacetime being static and spherically symmetric, which describes the interior of the object can be written in the following form

d s^{2} = e^{A (r)} d t^{2} - e^{B (r)} d r^{2} - r^{2} (d θ^{2} + {sin}^{2} θ d ϕ^{2}) .

The presence of the extra source

θ_{α γ}

in the effective energy tensor

T_{α γ}^{eff}

causes anisotropy within the effective matter distribution. We get the Einstein–Maxwell field equations for the charged anisotropic stellar configuration employing Eqs. (19), (24) together with spherically symmetric line element (25).

Solution to the gravitationally decoupled Einstein–Maxwell field equations

Finding analytic and exact solution to the system of Eqs. (26)–(28) is a difficult job. However, the MGD technique apparently makes the job of solving the field equations comparatively easier than the other direct methods. On the other hand, we are interested in finding the charged perfect fluid models in dark matter holes. Therefore, the MGD technique through the linear transformation over the metric function will be applied to investigate the charged perfect fluid models in dark matter holes.

Boundary conditions in $f (R, T)$ gravity

In this section we will match the interior solution with the exterior solution in a smooth way at the boundary of space–time geometry of compact configurations. For this the metric potentials should be continuous across the surface

Σ

as defined by

r = R

. In this connection, we have used notable Israel–Darmois junction conditions [128], [129].

{[d s^{2}]}_{Σ} = 0,

or equivalently

e^{A^{-} (r)} |_{r = R} = e^{A^{+} (r)} |_{r = R},

e^{B^{-} (r)} |_{r = R} = e^{B^{+} (r)} |_{r = R},

where foregoing expressions are of the first fundamental form.

A stellar model has

Physical attributes of the present model

We analyze the physical acceptance of the present charged and anisotropic stellar model developed with MGD technique and PI-DM source in

f (R, T)

gravity. The effective physical quantities such as

q

ρ^{eff}

P_{r}^{eff}

P_{t}^{eff}

and the anisotropy factor (

Δ

) be scrutinized in this section. The main salient features of the present model have been explored and discussed thoroughly to establish the physical validity of the model.

The graphical variation of the charge function with respect to radial distance

Stability for compact objects using cracking analysis

The effective system in the present study should fulfills the criteria of causal limit which states that sound speeds cannot be greater than the light speed inside the stellar configuration. We have presented the square of sound speed in radial and tangential direction as function of radial distance in the first six panels of Fig. 6 for different values of

χ

β

and

λ_{1}

. For each panel the magnitude of square of the sound speeds is less than unity which confirms the validity of the causal limit

Analysis of Mass–Radius relation for the effective system

In the present section, we investigate the mass–radius relation of the effective charged system with the presence of MGD effects and PI-DM source in the background of

f (R, T)

gravity. Regarding this we have presented the

M - R

curves for different values of

χ

β

λ_{1}

and

q

in panels of Fig. 10, Fig. 11. In the left panel of Fig. 10, the variation of

M - R

relation with regard to coupling constant

χ

has been shown where

χ = 0

is the GR case and non zero

χ

f (R, T)

case. The peak of the

M - R

curve

Impact of dark matter via decoupling parameter and negative values of $f (R, T)$ coupling parameter $χ$ on the stability of the model

In this section, we will focus on how dark matter affects the stability of the model through the decoupling parameter and negative values of the

f (R, T)

coupling parameter

χ

. This effect is shown in Fig. 12, which demonstrates four different cases: GR, GR+MGD,

f (R, T)

, and

f (R, T)

+MGD for two sets of parameters. In the left panel, we show the positive value of

χ

, while the right panel is for negative values of

χ

. As we notice from the left panel of Fig. 12, the stability of the model increases

Concluding remarks

In the present investigation we have studied the anisotropic charged compact objects under MGD approach in the context of

f (R, T)

gravity with linear form of

f (R, T) = R + 2 χ T

in order to explore various physical properties of the object. We have reduced the fluid equations to two systems of solvable equations with consideration of the choice of a Durgapal–Fuloria metric potential in association with Pseudo Isothermal dark matter as a new source. Notably, Durgapal–Fuloria metric potential has been

CRediT authorship contribution statement

S.K. Maurya: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Investigation, Formal analysis, Conceptualization. Fadhila Al Khayari: Writing – review & editing, Writing – original draft, Visualization, Validation, Methodology, Investigation, Formal analysis, Conceptualization. Asifa Ashraf: Writing – review & editing, Writing – original draft, Visualization, Validation, Investigation, Formal analysis, Conceptualization. M.K. Jasim: Writing

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The authors Fadhila Al Khayari and S. K. Maurya acknowledge that this research work is supported by the TRC Project, Oman (Grant No. BFP/GRG/CBS/23/072). The authors are also thankful for continuous support and encouragement from the administration of the University of Nizwa for this research work.

References (150)

T. Padmanabhan

Phys. Rep.

(2003)
T. Clifton et al.

Phys. Rep.

(2012)
S. Capozziello et al.

Phys. Rep.

(2011)
S. Nojiri et al.

Phys. Rep.

(2011)
S. Nojiri et al.

Phys. Lett. B

(2005)
T. Chiba

Phys. Lett. B

(2003)
E.H. Baffou et al.

Chin. J. Phys.

(2017)
P. de Bernardis

Nature

(2000)
D.N. Spergel

[Wmap]

Astrophys. J. Suppl.

(2003)
C. Schimd et al.

Astron. Astrophys.

(2007)

Related Articles

Contact us

Article Content

Highlights

Abstract

Introduction

Access through your organization

Section snippets

Formulation $f (R, T$ ) gravity with gravitational decoupling

Main equations for charged anisotropic matter distributions

Solution to the gravitationally decoupled Einstein–Maxwell field equations

Boundary conditions in $f (R, T)$ gravity

Physical attributes of the present model

Stability for compact objects using cracking analysis

Analysis of Mass–Radius relation for the effective system

Impact of dark matter via decoupling parameter and negative values of $f (R, T)$ coupling parameter $χ$ on the stability of the model

Concluding remarks

CRediT authorship contribution statement

Declaration of competing interest

Acknowledgments

References (150)

Phys. Rep.

Phys. Rep.

Phys. Rep.

Phys. Rep.

Phys. Lett. B

Phys. Lett. B

Chin. J. Phys.

Nature

[Wmap]

Astrophys. J. Suppl.

Astron. Astrophys.