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

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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

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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 −1 2푚 Λ푟3 × [푟2 (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.

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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 . In the numerical procedure, we have defined an an input in the TOV equation. We consider allowed range of central density for the here the relativistic Fermi gas quark equation compact star. In fact, the range of central of state to model the compact stars. In MIT density considered for the present work is bag model, the compact star equation of state is given by [16] 4.1 퐵 − 20 퐵.

1 푃 = (휀 − 4퐵), (21) 3 3.1 Mass-Radius Relation where 퐵 is the bag constant which is typically about 56 푀푒푉 푓푚−3 . However, the fixing of the MIT bag constant value as adopted in In Figures 1 to 3, we have shown the Quantum Meson Coupling model is calculated mass-radius relations for compact questionable and may depend on the local star for different values of the bag constant and cosmological constant. In Fig 1, the density [17]. From hadron phenomenology at mass and radius of compact star is shown for 푇 = 0 and at normal nuclear matter density a bag constant 퐵 = 50 푀푒푉 푓푚−3 and the the value of 퐵 in the range 50 − 100 three representative values of cosmological 푀푒푉 푓푚−3 is acceptable. In view of this, constant described through the free we have considered three different values of 훬 parameter, 푛 = 0, 5, 10 . The curves the bag constant namely 퐵 = 50, 60 and obtained are smooth and continuous and 70 푀푒푉 푓푚−3 for the compact star structure show respective peaks. The curve for the calculation. The cosmological constant may 푛 = 0 case provides the mass-radius be considered either positive or negative and relationship in the absence of a cosmological stellar structure may be obtained for different constant ( ). For this case ( ), density profile. Christian Bohmer [18] has 훬 = 0 훬 = 0 obtained stellar solutions in the framework of the maximum stable star mass ( 푀푚푎푥 ) is Einstein’s gravity with a cosmological obtained as 1.9558 푀푠푢푛 at a corresponding constant considering both positive and radius of11.37 퐾푀 . Here 푀푠푢푛 is the mass negative values of Λ and constant stellar of the Sun. Similarly, for the case 훬 = 5 휀0, density. However, the announcement of late the maximum mass of the stable star time cosmic acceleration phenomenon two becomes 2.0498 푀푠푢푛 and the corresponding decades ago from different observations has radius is 11.77 퐾푀 . We get a maximum led to the prediction of a small but positive mass of 2.1779 푀푠푢푛 and radius 12.30 퐾푀 value of 훬 . In view of the observational for 훬 = 10휀0. It is interesting to note from requirement for the cosmological constant, the figures that, for a given bag constant, we consider here some representative positive 푀푚푎푥 of the stable compact star values of 훬 in the units of normal nuclear configuration and the corresponding radius −3 increases with an increase in the value of the matter energy density 휀0 = 140 푀푒푉 푓푚 : cosmological constant. In Fig. 2, the mass-radius relation of stable Λ = 푛휀0, (22) compact star is shown for three

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representative values of 훬 for a bag constant, configuration. The decrement of maximum 퐵 = 60 푀푒푉 푓푚−3 . As in the previous star mass with bag constant is clearly visible figure, here also, the mass-radius from the figure. relationship curves are continuous and smooth showing respective peaks. In this case we obtained a maximum mass of

1.7859 푀푠푢푛 with a radius of 10.38 퐾푀 for 훬 = 0 . For, 훬 = 5 휀0 , the maximum mass and radius are obtained as

1.8555 푀푠푢푛 and 10.67 퐾푀 . Similarly for 훬 = 10휀0, we have the maximum mass as 1.9445 푀푠푢푛 and the corresponding radius as 11.05 퐾푀 . In this case also, we observed

that, 푀푚푎푥 of a stable compact star increases with an increase in 훬.

The mass-radius relation for a bag constant B = 70MeV fm−3 is shown for the different

values of Λ in Fig. 3. Similar trend to the earlier cases have also been observed in this Fig.1: Mass-Radius relation of compact star case. For, Λ = 0 , the maximum mass is for different values of Λ for a Bag constant, 퐵 = 50 푀푒푉 푓푚−3. obtained as 1.6536 Msun with a radius of 9.61 KM. When, Λ = 5 ε0 , the maximum mass and radius of respectively become

1.7081 Msun and 9.83 KM . We have the maximum stable mass as 1.7743 Msun and radius as 10.09 KM for a cosmological

constant of훬 = 10휀0. The maximum masses with their corresponding radii for stable compact star configuration as obtained in this work are summarized in Table 1. One can note from the table that, for a given value of bag constant, there is a substantial increase in the maximum mass of the stable star with an increase in 훬. Similarly, for a given value of cosmological constant, the maximum star mass decreases with an increase in the bag constant. We have plotted the maximum mass as a function of the bag constant in Figure 4, to show the effect of the bag constant on the maximum mass of a stable stellar

Fig.2: Mass-Radius relation of compact star for different values of Λ for a Bag constant, 퐵 = 60 푀푒푉 푓푚−3.

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Fig.3: Mass-Radius relation of compact star Fig.4: Maximum Mass of a stable star as a for different values of Λ for a Bag constant, function of Bag constant. 퐵 = 70 푀푒푉 푓푚−3.

3.2 Mass-Central density Relation Table-1: Mass and Radius data of Compact stars as calculated using the Modified TOV The effect of cosmological constant and the equations incorporating a cosmological bag constant on the mass-central density

constant. relation (푀 − 휀푐 ) have been explored for the compact star configuration in Figures 5-7.

The curves in the figures represent the 푀 − 휀푐 Bag 풏 Maximum Radius relation for a given bag constant and three Constant Mass different values of cosmological constant. [KM] (푩) The curves are smooth and continuous. In all [푴풔풖풏] the figures the behavior of the central density [푴풆푽 풇풎−ퟑ] are the same in the sense that, central density slowly increases initially with an increase in 0 1.9558 11.37 the mass of the compact star and at certain 50 5 2.0498 11.77 mass it suddenly rises to large values. The mass, 푀푡 at which this sudden rise occurs, 10 2.1779 12.30 depends on the choice of the bag constant and the cosmological constant. For a given value 0 1.7859 10.38 of bag constant, this mass 푀푡 increases with 60 5 1.8555 10.67 an increase in 훬 . Typically this mass is beyond1.5 푀푠푢푛 . 10 1.9445 11.05

0 1.6536 9.61 4. Conclusions 70 5 1.7081 9.83 Observations show the existence of stable Neutron stars and Quark stars with mass 10 1.7743 10.09 greater than 2 Solar-mass. To our apathy, most known nuclear equations of state (baring some stiff equations of state) cannot

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Fig.5: Central density- Mass relation of Fig.7: Central density- Mass relation of compact star for different values of Λ for a compact star for different values of Λ for a −3 Bag constant, 퐵 = 50 푀푒푉 푓푚 . Bag constant, 퐵 = 70 푀푒푉 푓푚−3.

a compact star. In principle, the cosmological constant can either be positive or negative. However, the late time cosmic speed up issue as has been raised by recent observations from Supernova speculated a small but positive cosmological constant. In view of this, we have considered three different positive values of the cosmological constant in the TOV equation. Also, we have assumed three possible values of the bag constant in the equation of state. The mass-radius relation of the compact star has been obtained by numerically integrating the modified TOV equation. We observed that, an increase in Fig.6: Central density- Mass relation of cosmological constant leads to an increase in compact star for different values of Λ for a the maximum mass of the stable compact Bag constant, 퐵 = 60 푀푒푉 푓푚−3. star. On the other hand, for a given cosmological constant, an increase in the bag constant decreases the maximum mass of the compact star. The mass-central density be able to predict neutron stars of such relationship curves are obtained to be smooth masses. In order to obtain Neutron star of and continuous. For a given value of bag mass higher than 2 Solar mass from constant, a higher value of cosmological theoretical calculations, it is necessary to constant favors a lower central density of the change either the equation of state or the stable star. Further for a given cosmological background theory leading to the TOV constant, the equation of state with higher equation. In the present work, a modified values of bag constant favor a higher central TOV equation has been derived for a finite density. cosmological constant. Using a relativistic Fermi gas equation of state for quark matter, we have studied the stellar structure of

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