The African Review of Physics (2020) 15: 0012 Compact Star Structure with a Cosmological Constant Tapaswini Mahala1, Sidhartha Biswal2 and Dipanjali Behera3 Department of Physics, Indira Gandhi Institute of Technology, Sarang, Dhenkanal, Odisha-759146, India 1 E-mail:[email protected] 2 E-mail:[email protected] 3 E-mail:[email protected] (Corresponding author) Compact star structures are investigated in the framework of General Relativity using a modified TOV equation. The modified TOV equation is derived by incorporating the cosmological constant in the field equations. A quark equation of state signified by a bag constant is used to integrate the TOV equations. Depending on the choice of the cosmological constant and bag constant, stellar properties such as mass and radius of the compact star are found to change effectively. Also, we have obtained the mass-radius and mass-density relationship of compact star. 1. Introduction Long after Hubble’s discovery regarding the cosmic as the simplest candidate for DE which provides an expansion, the acceleration of cosmic expansion has anti-gravity effect that drives the cosmic been confirmed from observations of type Ia acceleration. The incorporation of cosmological supernova, large scale structure and Baryon constant in the Einstein field equations is not new Acoustic Oscillations [1, 2]. This idea has led to and has a checked history of its inclusion or drop novel ideas in cosmology and astrophysics and to from the field equations. the development of the idea of dark energy (DE). DE is usually described by a cosmological fluid of In astrophysics, the study of dynamics of almost constant density with negative pressure. compact stars such as Neutron stars, Quark stars General Relativity (GR) has been a successful and strange stars are considered to be all time theory of gravity in addressing many issues in important problems. A Neutron star is formed when cosmology and astrophysics. All along its journey a massive star collapses under the inward for hundred years, it has passed so many tests. gravitational pull of its own mass. When a dying Recently, the detection of gravitational waves and star collapses, the core becomes denser and the the first ever snap shot of the outer part of a black elementary particles in the core like proton and hole have supported greatly to the robustness of the electron merge together and form neutrons and theory. However, the late time cosmic acceleration neutrinos. Neutrinos fly off leaving behind a phenomenon cannot be explained in the framework mixture of muons, electrons, protons and neutrons. of GR. In order to address this cosmic speed up In general, the radius of Neutron stars is about 10 issue, many authors have incorporated some km and a mass between 1.4 and 2.16 solar masses dynamical scalar fields in the matter side of the (푀푠푢푛). At the time of the birth, Neutron stars are Einstein Field equations. These scalar fields, hot and have surface temperature of around 6 × 5 usually quintessence field, represent as the 10 K. candidate for DE. Cosmological constant is viewed 110 The African Review of Physics (2020) 15: 0012 Although, white dwarfs, neutron stars, and black Section-2, the modified TOV equations have holes are the three known stellar remnants of our been derived that contain a cosmological universe, other theoretical compact objects such as constant. The stellar structure has been discussed Quark stars and strange stars might also exist [3, in Section-3 where we have used the relativistic 4, 5]. A Quark star is a hypothetical star that is Fermi gas equation of state of compact star in composed of quark∼ matter. Quark star is formed − MIT bag model. Section-4 contains the summary when neutrons having up and down quarks break and conclusions of the work. apart because of high gravitational pressure of Neutron star. After the proposal of Quark star by 2. TOV equations in presence of a finite Ivanenko and Kurdgelaidze [3, 4], there have been Cosmological constant a lot of research interest in investigating the structure of Quark star. Quark stars are comprised We consider an uncharged and spherically of de-confined up, down and strange quarks and symmetric metric ought to be color superconducting. The maximum 푑푠2 = 푒휈푑푡2 − 푒휆푑푟2 − 푟2푑푟2 − 푟2 푠푖푛2휃 푑휑2, mass of Quark star is ~ 2 푀푠푢푛 and has a radius around 10 − 12 KM with density 1017g/cm3. (1) Quark stars are smaller in size compared to Where, the metric potentials ν and λ are arbitrary neutron stars and possess high rotational functions of radial distance. frequency. They have a tendency to cool faster. The non-vanishing components of the Ricci Due to a balance between the inward 훼 훼 훽 훼 tensor 푅휇휈 = 휕휈Γ휇훼 − 휕훼Γ휇휈 + Γ휇훼 Γ훽휇 − gravitational pull and the outward pressure, an 훽 훼 equilibrium state of the star is reached. A suitable Γ휇휈 Γ훽훼 for the above metric are given by hydrostatic equilibrium equation is required for the understanding of the stellar structural −휈′2 휈′휆′ 휈′′ 휈′ 푅 = [ + − − ] 푒휈−휆, (2) properties. The first such hydrostatic equilibrium 푡푡 4 4 2 푟 equation using GR has been obtained by Tolman, Oppenheimer and Volkoff [6–8]. These equations 휈′′ 휈′2 휆′ 휈′휆′ 푅 = + − − , (3) are known as the TOV equations which are 푟푟 2 4 푟 4 integrated for a given nuclear equation of state to obtain stellar properties. Recently there has been 휆′ 휈′ 푅 = −1 + 푒−휆 − 푟푒−휆 + 푟푒−휆, (4) an increased interest in obtaining modified TOV 휃휃 2 2 equations in modified gravity theories [9, 10]. Using a Quark-Meson- Coupling model, Nayak et 1 1 푅 = − 푟 푠푖푛2휃 푒−휆휆′ + 푟 푠푖푛2휃 푒−휆휈′ al. have investigated the effect of cosmological 휑휑 2 2 constant on compact star [11]. Burikham et al. − 푠푖푛2휃 + 푒−휆푠푖푛2휃. (5) have considered a finite cosmological constant in D- dimension to obtain the minimum mass of spherically symmetric object [12]. Stellar Here the primes denote derivatives with respect structures have also been studied in literature with to the radial distance, 푟. The Ricci scalar 푅 = 휇휈 a finite cosmological constant in TOV equations 푔 푅휇휈 can be obtained as [13–15]. −휈′2 휈′휆′ 2휈′ 2휆′ 2 2 푅 = 푒−휆 [ − 휈′′ + − + − ] + In the present paper, the stellar structure of 2 2 푟 푟 푟2 푟2 compact star has been investigated in the . (6) framework of modified TOV equations. The modified TOV equations are obtained by The Einstein field equations with a cosmological incorporating a cosmological constant in the field constant Λ are given by equations. The paper is organized as follows: In 111 The African Review of Physics (2020) 15: 0012 So that 1 퐺 ≡ 푅 − 푔 푅 + Λ푔 = −8휋푇 , (7) 휇휈 휇휈 2 휇휈 휇휈 휇휈 2 푑푃 휈′ = − ( ) . (15) 휀+푃 푑푟 Where, 퐺 is the Einstein tensor and 푇 is the 휇휈 휇휈 energy momentum tensor. For a perfect fluid The gravitational mass of a spherical shell of distribution of the universe, 푇 is given by 휇휈 radius r can be obtained from 푟 푚(푟) = 4휋 ∫ 휀(푟)푟2 푑푟 (16) 푇 = (휀 + 푃)푢 푢 − 푃푔 , (8) 0 휇휈 휇 휈 휇휈 Which, can be manipulated algebraically to Where, P and ε are respectively the pressure and yield energy density. The time like 4-velocity vector uμ μ 푑푚 푑 Λ푟3 satisfies uμu = 1. From this condition one can 2 = − [푒−휆푟 − 푟 + ]. (17) 푑푟 푑푟 3 have t −ν/2 u = e . From the above equation (17), it is straight forward to obtain Now the 푡푡 and 푟푟 components of 퐺휇휈 are obtained as 2푚(푟) Λ푟3 푒−휆 = 1 − − . (18) 푟 3 −휆′ 1 1 퐺 = 푒휈 [푒−휆 { + } − + Λ], (9) 푡푡 푟 푟2 푟2 Using equations (13), (15) and (18), we obtain the expression for the hydrodynamic −휈′ 1 1 equilibrium as 퐺 = − + 푒휆 [ + Λ]. (10) 푟푟 푟 푟2 푟2 푑푃 4휋푃푟3 Λ푟3 = −(휀 + 푃)푚 (1 + − ) 푑푟 푚 3푚 The Einstein field equations for the 3 −1 2 2푚 Λ푟 × [푟 (1 − − )] (19) spherically symmetric object can now be 푟 3 explicitly written as This is the modified TOV equation 휆′ 1 1 containing a finite cosmological constant. If 8휋휀 = 푒−휆 ( − ) + − Λ, (11) 푟 푟2 푟2 we put Λ = 0 in the above equation, it reduces to the usual TOV equation in General 휈′ 1 1 Relativity. 8휋푃 = 푒−휆 ( + ) − + Λ. (12) 푟 푟2 푟2 Equations (16) and (19) are numerically Additions of the above equations yields solved simultaneously which requires some boundary conditions. At the center of the star with, 푟 = 0, we consider 푒−휆 8휋(휀 + 푃) = (휆′ + 휈′). (13) 푟 푚(0) = 0, 휀(0) = 휀푐, 푃(0) = 푃푐 . (20) From the energy momentum conservation equation, 푇 = 0, we have 휇휈;휈 The TOV equations are integrated for a given central density 휀푐 and equation of state = 푑푃 휈′ 푃(휀) , from the center of the star (푟 = 0) to its = −(휀 + 푃) (14) 푑푟 2 surface when the pressure vanishes i.e. 112 The African Review of Physics (2020) 15: 0012 푃(푅) = 0 . The corresponding 푅 and 푀 = where 푛 is a free integral parameter. For the 푚(푅) provide the radius and mass of the sake of investigation of the effects of the star. cosmological constant on the stellar structure, we have considered three different values of n namely 푛 = 0, 5, 10. The mass-radius relation 3. Stellar Structure of Compact Star and mass-density relation of compact star are calculated for the equation of state in (21) and A reliable equation of state is required as some representative values of 퐵 .