Emerging Anisotropic Compact Stars in Gravity

Eur. Phys. J. C (2017) 77:674 DOI 10.1140/epjc/s10052-017-5239-1

Regular Article - Theoretical Physics

Emerging anisotropic compact stars in f (G, T) gravity

M. Farasat Shamira, Mushtaq Ahmadb National University of Computer and Emerging Sciences, Lahore Campus, Lahore, Pakistan

Received: 6 July 2017 / Accepted: 13 September 2017 / Published online: 12 October 2017 © The Author(s) 2017. This article is an open access publication

Abstract The possible emergence of compact stars has d p (mc2 + 4pπr 3)(ρc2 + p) dm =−G and = 4ρπr 2, been investigated in the recently introduced modified GaussÐ dr r 2c4 − 2Gmc2r dr Bonnet f (G, T ) gravity, where G is the GaussÐBonnet term (1) and T is the trace of the energy-momentum tensor (Sharif and Ikram, Eur Phys J C 76:640, 2016). Specifically, for this where p, ρ, and m is the pressure, density, and mass of the modified f (G, T ) theory, the analytic solutions of Krori and star, respectively, varying with radial coordinate r.Atr = R Barua have been applied to an anisotropic matter distribution. (the coordinate radius of the compact star), the total mass of To determine the unknown constants appearing in the Krori the compact star is determined as and Barua metric, the well-known three models of the compact stars, namely 4U1820-30, Her X-I, and SAX J 1808.4- R ( ) = π 2ρ . 3658 have been used. The analysis of the physical behaviour M R 4 r dr (2) 0 of the compact stars has been presented and the physical features like energy density and pressure, energy conditions, The combination of these OppenheimerÐVolkoff equations p static equilibrium, stability, measure of anisotropy, and reg- with the equation of the sate (EoS) parameter ω = ρ when ularity of the compact stars, have been discussed. solved numerically results in different configurations of the compact stars with divisions of low and high densities [3,4]. Schwarzschild [5] was the pioneer to present the spher- 1 Introduction ically symmetric exact solutions of the Einstein field equations. The outcome of the first solution was the exploration of In astrophysics, compact stars are generally referred to as the space-time singularity which gave the idea of a black hole. white dwarfs, neutron stars (including the hybrid and quark The second non-trivial solution predicted the bounded com- stars), and black holes. White dwarfs and neutron stars orig- μ( ) = 2M(R) < 8 pactness parameter i.e., R R 9 in hydrostatic inate due to the degeneracy pressure produced by the funda- equilibrium for some static and spherically symmetric con- mental particles responsible for their formation. These stars figured structure [6]. The investigation of compact stars (neu- are massive but volumetrically smaller objects, and therefore tron stars, dark stars, quarks, gravastars, and black holes) has they come with high densities. Usually, the exact nature of become now an interesting research pursuit in astrophysics these compact stars is not known to us, but they are believed to despite not being a new one. Baade and Zwicky [7] stud- be the massive objects with a small radius. Excluding black ied the compact stellar objects and argued that the super- holes, all the other types of compact stars are sometimes nova may turn into a smaller dense compact object, which also known as degenerate stars. In general relativity (GR), came true later on after the discovery of pulsars, which are an analysis of configured equilibrium is essential due to the highly magnetized rotating neutrons [8,9]. Ruderman [10] reason of compact stars having huge mass and density. This was first to explore that, at the core of the compact stars, can be started in a general relativistic way by considering the nuclear density turns anisotropic. A number of investiga- the OppenheimerÐVolkoff equations [2] for static spherically tions have been made to find the solutions of the field equa- symmetric and hydrostatic equilibrium, given as tions for spherically symmetric anisotropic configurations in different contexts [11Ð13]. The pressure of the fluid sphere a e-mail: [email protected] splits into tangential and radial pressures in anisotropic con- b e-mail: [email protected] figurations. Different investigations reveal that the repulsive 123 674 Page 2 of 12 Eur. Phys. J. C (2017) 77 :674 forces, which are determining for compact stars, are pro- the presence of an extra force. It is expected that the the- duced due to anisotropy. Kalam et al. [14] showed that the ory may describe the late-time cosmic acceleration for some Krori and Barua metric [15] establishes the necessary condi- special choices of f (G, T ) gravity models. We discussed the tions for the advocacy of an effective and stable approach in Noether symmetry approach to find the exact solutions of modelling the compact objects. From an integrated TolmanÐ the field equations in an f (G, T ) theory of gravity [49]. In OppenheimerÐVolkoff (TOV) equation, the numerical simu- Ref. [50], we investigated the same modified GaussÐBonnet lations may be used to study the nature of the compact stars, f (G, T ) gravity and used the Noether symmetry method- using the EoS parameter. Rahaman et al. [16,17]usedthe ology to discuss some cosmologically important f (G, T ) EoS of the Chaplygin gas to explore their physical charac- gravity models with anisotropic background reported for a teristics by extending the Krori and Barua models. Mak and locally rotationally symmetric Bianchi type I universe. We Harko [18] used some standard models for spherically sym- used two models to explore the exact solutions and found that metric compact objects and explored exact solutions to find the specific models of modified GaussÐBonnet gravity may the physical parameters such as the energy density, radial be used to reconstruct CDM cosmology without involv- and tangential pressures, thereby concluding that inside these ing any cosmological constant. Thus it seems interesting to stars, the parameters would remain positive and finite. Hos- further explore the universe in this theory. sein et al. [19] studied the effects on anisotropic stars due This paper aims to investigate the possible emergence of to the cosmological constant. Various physical properties, compact stars by constructing some viable stellar solutions such as mass, radius, and moment of inertia, of neutron stars in a f (G, T ) theory of gravity by choosing some specific have been investigated and a comprehensive comparison has models. The plan of our present study is as follows: In Sect. 2, been established with GR and modified theories of gravity we give the fundamental formalism of f (G, T ) gravity with [20]. Some interesting investigations related to the structure an anisotropic matter distribution. Section 3 is dedicated to of slowly rotating neutron stars in R2 gravity have been the matching of the metric conditions. Some physical features accomplished by making use of two distinct hadronic and of the present study in the context of the f (G, T ) gravity a strange matter EOS parameters [21]. For a comprehensive model under consideration are given in Sect. 4.Lastly,we study, some fascinating results can be found in [22Ð28]. present some conclusive discussions. As an alternative to the theory of GR, modified theories of gravity have played an important and pivotal role in revealing the hidden facts as regards the accelerating expansion of the 2 Anisotropic matter distribution in f (G, T) gravity universe. After being motivated by the original theory and using the complex Lagrangian, modified theories of gravity The general action for the modified f (G, T ) model is [1] ( ) ( , ) (G) ( , G) like f R , f R T , f , and f R have been struc- 1 √ √ G A = d4x −g[R + f (G, T)]+ d4x −gL , (3) tured, where R is the Ricci scalar, is the GaussÐBonnet 2κ2 M invariant term, and T is the trace of the energy-momentum tensor. Some reviews and important discussions relating to where the function f (G, T ) contains the GaussÐBonnet term different modified theories of gravity have been published G and the trace of the energy-momentum tensor T , κ denotes by different researchers [29Ð42]. Das et al. [43] presented the coupling constant, g is for the determinant of the metric exact conformal solutions to describe the interior of a star in tensor, R is the Ricci scalar, and LM represents the matter modified teleparallel gravity. In another work, Das et al. [44] part of the Lagrangian. The GaussÐBonnet term G is defined explored several physical features of the model admitting as conformal motion to describe the behaviour of the compact 2 ζη ζημν G = R − 4RζηR + RζημνR , (4) stars using modified f (R, T ) gravity. Sharif and Yousuf [45] investigated the stability conditions of collapsing object by where Rζημν, and Rζη is for the Riemann and Ricci tensors, considering the non-static and spherically symmetric space- respectively. The variation of Eq. (3) with respect to gζη, and time. The field equations of f (R, T ) modified theory have by setting κ = 1, gives the following fourth order non-linear been explored by implementing the perturbation approach field equations: [46]. The possible formation of compacts stars in a modi- 2 μν fied theory of gravity by using the Krori and Barua metric Gζη =[2Rgζη∇ + 2R∇ζ ∇η + 4gζηR ∇μ∇ν 2 μ μ for spherically symmetric anisotropic compact stars has been + 4Rζη∇ − 4Rζ ∇η∇μ − 4Rη ∇ζ ∇μ discussed [47,48]. In a recently published paper [1], Sharif μ ν 1 and Ikram presented a new modified f (G, T ) theory of grav- − 4Rζμην∇ ∇ ] fG + gζη f −[Tζη + ζη] 2 ity and studied various energy conditions for the FriedmannÐ μ μν × f −[2RRζη − 4R Rμη − 4RζμηνR RobertsonÐWalker (FRW) universe. They found that the mas- T ζ μνδ 2 sive test particles follow non-geodesic geometry lines due to + 2Rζ Rημνδ] fG + κ Tζη, (5) 123 Eur. Phys. J. C (2017) 77 :674 Page 3 of 12 674

ζ where =∇2 =∇ζ ∇ is the d’Alembertian opera- the neighbourhood of r. To investigate the existence of the = − 1 = (G, ) = αGn + λ tor, Gζη Rζη 2 gζηR is the Einstein tensor, ζη compact stars for the model f T T,wehave μν δTμν ∂ f (G,T) ∂ f (G,T) α = λ = = ≡ (G, ) G ≡ ≡ considered 1, 2, and n 2. For these paramet- g δgζη , f f T , f ∂G , and fT ∂T . The Einstein equations can be recovered by putting simply ric values, the energy density and all the energy conditions f (G, T) = 0, whereas the field equations for f (G) are repro- remain positive for the model under investigation. It is worth duced by replacing f (G, T) with f (G) in Eq. (5). The energy- mentioning here that one may opt for some other choices of momentum tensor, denoted by Tζη, can be defined as these values for further analysis. The explicit expressions for the energy density ρ, the radial pressure p , and the tangential √ r pressure p are obtained: 2 δ( −gLM) t Tζη =−√ . (6) −g δgζη e−2b ρ = 8eb(−1 + eb)r 2(1 + 2λ) Moreover, the metric dependent energy-momentum tensor 8r 4(1 + λ)(1 + 2λ) may have the form 2b 4 b 2 − 4e r (1 + λ) f1 + fG − 16(−1 + e ) λ ∂LM Tζη = gζηL − . 2 2 4 2 3 M 2∂ ζη (7) + r r (1 + 2λ)a − 2r (1 + 2λ)a b g − 4a b 2(−3 + eb)(1 + λ) + r 2(1 + 2λ)a The usual anisotropic energy-momentum tensor Tζη is given as + a 2 8(λ + eb(1 + λ)) 2 2 2 Tζη = (ρ + pt)Vζ Vη − ptgζη + (pr − pt)ξζ ξη, (8) + r (1 + 2λ)(b + 4a ) + 4 − 2λb + { (− + b)( + λ)} where pt and pr represent the tangential and radial pressures, a 4 1 e 1 ρ respectively, while denotes the energy density. The four + r 2(1 + 2λ)a + 2r 4 f {−8(2 + 5λ) velocity is denoted by Vζ and the radial four vector by ξα; G they satisfy + r{b (10 + 27λ − 2rλb )} − { + λ + ( + λ) } } α −a α α α −b α α r 8 18 r 2 3 b a V = e 2 δ , V Vα = 1,ξ = e 2 δ ,ξξα =−1. (9) 0 1 − 8r(2 + 5λ)(1 − 2rb + r 2a ) f G 2 2 b In this paper, we have chosen specifically the following + r a e rλ − 2(8 + 18λ + r(2 + 3λ)b ) fG (G, ) f T model [49]: − 4r(2 + 5λ) fG (G, ) = (G) + ( ), f T f1 f2 T (10) b + 2e 16(2 + 5λ) fG + 2rb {r + 2rλ (G) where f1 is an analytic function comprised of a GaussÐ + (2 + 3λ) f }+r 3λa + 4r(2 + 5λ) f n G G Bonnet term. In particular, we consider f1(G) = αG ,a power law model of f (G) gravity proposed by Cognola et al. + ra 2(−32 − 74λ + rb (2λ + r(2 + 3λ)b )) fG [51] with α being an arbitrary real constant, and n a positive b real number. Here we take f (T ) = λT, with λ being some − e λ r(−4 + rb ) + 4 fG 2 positive real number. Further, for the investigations of com- + 4r − 2(4 + 9λ) + r(2 + 5λ)b f , (12) pact stars we take the static, spherically symmetric space- G time as e−2b p = − 4e2br 4(1 + λ) f r 8r 4(1 + λ)(1 + 2λ) 1 2 = a(r) 2 − b(r) 2 − 2 θ 2 − 2 2θ φ2. ds e dt e dr r d r sin d (11) b 2 + fG − 16(−1 + e ) λ ( ) = 2 + We parameterize metric (11) by taking a r Br C and + r 2 r 2(1 + 2λ)a 4 − 2r 2(1 + 2λ)a 3b b(r) = Ar 2, given by Krori and Barua [15] and with the help , − { (− + b)( + λ) of some physical assumptions, the arbitrary constants A B 4a b 2 3 e 1 and C will be calculated. The above set of functions are estab- + r 2(1 + 2λ)a }+a 2 8(−1 + eb)(1 + λ) lished to reach a singularity free structure for compact stars. Therefore, our main concern is to present these functions to + r 2(1 + 2λ)(b 2 + 4a ) the metric in a way so as to achieve the structure for the com- b pact star in this extended model, free of the singularities in + 4 2(1 + λ)b 2 + a {4(−1 + e )(1 + λ) 123 674 Page 4 of 12 Eur. Phys. J. C (2017) 77 :674 + r 2(1 + 2λ)a } zero pressure and energy density, the solution outside the star can differ from Schwarzschild’s solution. However, it + { b(− + b)} ( + λ) 2r 4e 1 e r 1 2 is expected that the solutions of the modified TOV equa- + 4 fG − 8λ + r − b (4 + λ + 2rλb ) tions with energy density and pressure (maybe non-zero) may accommodate Schwarzschild’s solution with some spe- + λ(− + ) + 2λ 2{ (− + ) r 2 rb a r a 2 2 rb fG cific choice of the f (G, T ) gravity model. Perhaps this is + r(eb − 4 f )}−8rλ(1 − 2rb the reason that Birkhoff’s theorem may not hold in modified G gravity. A detailed investigation of the issue in the context of + r 2a ) f + ra − 2{12 + 34λ G f (G, T ) gravity can be an interesting task. Many authors have + rb (8 + 14λ + rλb )} f considered the Schwarzschild solution for this purpose, giv- G b ing some interesting results [52Ð55]. Now to solve the field + e − r(4 + 4λ + rλb ) + 4(2 + 3λ) fG = equations under the restricted boundary conditions at r R, the pressure pr = 0, the interior metric (11) requires these + 4rλ(−2 + rb ) fG matching conditions. This can be done by taking a smooth + 2eb[2{8λ + r(4 + 7λ)b } f match at r = R to Schwarzschild’s exterior metric, given by G + rλ(r 2a + 4 f )] , (13) G −1 2 2M 2 2M 2 2 2 2 2 − ds = 1− dt − 1− dr −r (dθ +sin θdφ ), e 2b r r p = − 2e2br 4(1 + λ) f t 4( + λ)( + λ) 1 (15) 4r 1 1 2 b 2 + 2 fG 4(−1 + e ) (1 + λ) yielding + r 2 {−1 + 2eb(1 + λ)}a 2 ∂ − ∂ + − = +, − = + , gα = gα , − (− + b)( + λ) + 2 + (− + b)( gtt gα grr grr (16) 2 3 e 1 a b b 4 1 e 1 ∂r ∂r + λ)a − r f [ λ 4 G 8 where (+) corresponds to the exterior solution and (−) to + r{b (−7λ − 2r(1 + λ)b ) + r(2 + 6λ the interior solution. Now from a comparison of exterior and + r(2 + 3λ)b )a }] + 8rλ(1 − 2rb + r 2a ) f interior metrics, the constants A, B, and C are obtained: G b −1 2M + 2e {−16λfG − rb (r + 2rλ − 2λfG) A = ln 1 − , (17) R2 R +r 3(1 + λ)a − 4rλf }+r 2a 2 ebr(1 + λ) M 2M G B = − , 3 1 (18) R R + 2 2 + 6λ + r(2 + 3λ)b fG + 4rλfG = − 2M − M − 2M . C ln 1 1 (19) + ra − 2[−10λ + rb {2 + 6λ R R R + r(2 + 3λ)b }] f + eb[−r{−2 + r(1 + λ)b } The approximated values of the mass M and radius R of the G compact stars Her X-1, SAX J1808.4-3658 and 4U1820-30 + 4λfG]+4r(2 + 6λ − rλb ) fG , (14) are used to calculate the values of constants A and B [56,57]. μ = M(R) The result R defines the compactness of the star and where a prime denotes the radial derivative. −1/2 the expression Zs = (1−2μ) −1 determines the surface redshift Zs. For the compact stars under consideration, the values of Zs are given in Table 1. 3 Matching with Schwarzschild’s exterior metric Now making use of the Krori and Barua metric, Eqs. (12)Ð (14) take the form Whatever the geometry of the star is, either derived inter- −2Ar2 e 2 nally or externally, the intrinsic boundary metric remains ρ = 4 − λ − e2Ar λ the same. Thus, one confirms that the components of the 2r 4(1 + λ)(1 + 2λ) 2 metric tensor irrespective of the coordinate system across + 2eAr {λ + Br2(1 + (−A + B)r 2)(1 + λ) the surface of the boundary will remain continuous. No + r 2{−2A2r 2λ + 2B(−1 + 3Ar 2)(1 + λ) doubt, in GR, the Schwarzschild solutions have been the + 4 6( + λ) − 3 4(− + 2)( + λ) lead in guiding us to choose from the diverse possibili- B r 1 2 2B r 1 Ar 1 2 2 2 2 2 ties of the matching conditions while investigating the stel- + B r (1 + 4λ + Ar (−2 + Ar )(1 + 2λ))}} fG lar compact objects. Now when we come to the case of modified theories of gravity, modified TOV equations with + r − 4(2 + 5λ) + r 2(10A − 24B 123 Eur. Phys. J. C (2017) 77 :674 Page 5 of 12 674

Table 1 The approximate Compact stars MR(km) μ = M A (km−2) B (km−2) Z values of the masses M,radiiR, R s compactness μ,andthe Her X-1 0.88M 7.7 0.168 0.006906276428 0.004267364618 0.23 constants A,andB for the compact stars Her X-1, SAXJ SAXJ1808.4-3658 1.435M 7.07 0.299 0.01823156974 0.01488011569 0.57 1808.4-3658, and 4U 1820-30 4U1820-30 2.25M 10.0 0.332 0.01090644119 0.009880952381 0.73

− 4B(A + 2B)r 2 + 4A(A − B)Br4 +{27A + Br2 + B(−A + B)r 4) 2 2 2 Ar2 3 n Ar2 − 55B − 2(2A + 2AB + 9B )r × (1 + λ)) fG + r(e r (−8 e 4 + 6A(A − B)Br }λ) 1 − 2 × Be 2Ar (1 + (−3A + B)r 2 2 2n + Ar [ + λ + 2(− λ + ( + λ))] r e 8 20 r B A 2 3 fG n Ar2 2 2 + e (−1 + (A − B)r )) α(1 + λ) + r(eAr (−2 + 4Ar 2 − 4λ + 2r 2(4A + 3B 2 − ( + λ + 2( + λ)) + + B(−A + B)r 2)λ + eAr (2 + 4λ − 8nr 2 2B 2 Br 1 2A 2 Ar2 1 − 2 × (1 + 2λ + Br (1 + λ))) + 4(4(−1 + e )λ × Be 2Ar (1 + (−3A + B)r 2 2n 2 r + r 2(A(−(− + eAr )λ n 7 Ar2 2 + e (−1 + (A − B)r )) + 4Ar 2(1 + λ)) − 2B2r 2(1 + 3λ 2 2 Ar2 × α(1 + λ))) + 4(−2 − 5λ + eAr (2 + 5λ) + Ar (2 + 3λ)) + B(−2 − (11 + e ))λ + 2( ( + 2)( + λ) + 2Ar 2(−1 + Ar 2(2 + 3λ))))) f 2r A 2 Br 2 5 G − ( + λ + 2( + λ)))) ) , − (λ − Ar2 λ + 2( − λ B 6 14 Br 2 5 fG (20) 4r e 2r B 2A 2 + ( + (− + ) 2)λ)) ) , e−2Ar B 4 A B r fG (22) pr = 4 − (−1 2r 4(1 + λ)(1 + 2λ) where 2 + eAr )2λ + 2A2r 4(1 + λ) 2 2 2 2 Ar2 2 e−2Ar (−16ABr2 + 2(1 − eAr )(4B − 4ABr2 + 4B2r 2)) n − 2Br (1 − 3Ar + e (−1 + Ar ))(1 + λ) f = α , 1 2 + 4 8( + λ) − 3 6(− + 2)( + λ) r B r 1 2 2B r 1 Ar 1 2 (23) 2 + B2r 4(−1 + 2eAr (1 + λ) = 1 −2+3n −2Ar2 (− 2 2 fG 2 Be 1 + Ar (−2 + Ar )(1 + 2λ)) fG r 3 + ) ( −2Ar2 ( + (− + ) 2 + ( Ar2 ( Ar2 ( + λ − n 2 n n Be 1 3A B r r e r e 2 4 8 r − + + Ar2 (− + ( − ) 2))) 2 2 n(− 1 − 2 e 1 A B r r 1 × Be 2Ar (1 + (−3A + B)r 2 2n + 2 4 − ( 2 + 4) r 6A r 2A r Br 2 n 2 + eAr (−1 + (A − B)r 2)) + eAr (1 + Ar 2 + A(−A + B)r 4))α , (24)

× α(1 + λ)) − 2(1 + 2λ −1+2n 3Ar2 4 e (−1 + n)nα + 2( + ( + ( − ) 2)λ))) fG = Br 2 1 A B r B(−1 + (3A − B)r 2 + eAr2 (1 + (−A + B)r 2))3 2 2 + 4(−4λ − r (4A + 6B + 8ABr 2 2 1 − 3Ar 2 Ar × − Be 2 Ar cosh + (A + 19B + 2(2A2 + 6AB + B2)r 2 r 2n 2 2 + ( − ) 4)λ) + Ar ( λ + 2( ( 2 n 2A A B Br e 4 r B 2 + ( + ( + ) 2) Ar 1 2A B r sinh + 3λ) + A(4 + 7λ)))) fG 2 2 Ar2 2 2 × (2 − 4n − 3(2B + A(−9 + 4n))r + 4r(−1 + e + 2(A − B)r (2 + Br ))λfG ) , + 2A(9B − 6Bn + A(−21 + 10n))r 4 (21) + ( 2 − + 2(− + )) 6 2 A 3B 16ABn A 15 32n r e−2Ar p = 4(1 + A2r 4 − 2A2(3A2 − 4AB + B2) t 2r 4(1 + λ)(1 + 2λ) 8 2 ×(−3 + 4n)r + (−2 + 4n + 3(2B − 2 4 + λ + 2Ar ( + λ) B r e 1 2 2 + A(−9 + 4n))r + 2A(16A − 9B + 2Br2(−1 + 3Ar 2)(1 + λ) + 2eAr (−1

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− 9An + 6Bn)r 4 + A(−3B2 + A2(33 − 52n) the same row provide us with a similar sort of conclusions. + 2AB(−3 + 10n))r 6 Furthermore, from the other plots of the anisotropic radial pressure p (Fig. 2), it is evident that the radius of the star for + 2A2(37A2 − 26AB + 5B2)(−1 r this model is R = 10 km, which further gives us a density of + ) 8) ( 2) + 2( − − n r cosh Ar Ar 11 4n 1.5828 × 1015 gcm 3, a high value with a small radius of + 2(B(3 − 2n) + A(−12 + 7n))r 2 + (B2 10 km, showing that our f (G, T ) model is compatible with + 2AB(1 − 6n) + A2(−27 the structure of the ultra-compact star [10,59]. A comparison of this with the already existing data labels this compact star + 44n))r 4 − 2A(7A − 3B)(5A − B)(−1 as quark/strange star [60]. All the three plots in Fig. 3 indicate + n)r 6)sinh(Ar 2)). (25) that the tangential pressure pt remains positive and finite and show their decreasing behaviour, which is required for the Now we investigate the nature and some interesting features viability of the compact star model. of the compact star, specifically for the assumed model of The variations of the radial derivatives of the density and f (G, T ) gravity. radial pressure are shown in Figs. 4 and 5, respectively. One can see the decreasing evolution of the first r-derivatives, i.e. ρ d < 0 and d pr < 0. It may be noted here that at r = 0, these 4 Physical aspects of f (G, T) gravity model dr dr derivatives just disappear except for the radial pressure for the SAX J 1808.4-3658 candidate. The mathematical calcu- In this section, some interesting physical aspects of compact lations for the second derivative test both for ρ and pr tell us stars, such as the energy density and pressure evolutions, 2ρ 2 that d < 0 and d pr < 0, indicating the maximum values energy conditions, equilibrium conditions, stability and adia- dr2 dr2 batic index analysis, compactness, and redshift analysis, shall of the density and radial pressure at the centre. This further be discussed. suggests the compact nature of the star.

4.1 Energy density and pressure evolutions 4.2 Energy conditions

The plot of the energy density for the strange star candidate For the viability of the model, the energy bounds must be Her X-1 (Fig. 1) shows that as r → 0, ρ goes to maximum, satisﬁed due to their signiﬁcant importance in analysing and this in fact indicates the high compactness of the core the theoretical data. NEC (null energy conditions), WEC of the star, validating that our model under investigation is (weak energy conditions), SEC (strong energy conditions), viable for the outer region of the core. The other two graphs in and DEC (dominant energy conditions) are given as

Fig. 1 Plot of the density evolution of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

Fig. 2 Plots of the radial pressure evolution of the strange star candidates Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

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Fig. 3 Plots of the transverse pressure evolution of the strange star candidates Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

dρ Fig. 4 Plots of dr with increasing radius of the strange star candidates Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

d pr Fig. 5 Variations of dr with increasing radius of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

NEC : ρ + pr ≥ 0,ρ+ pt ≥ 0, where

WEC : ρ ≥ 0,ρ+ p ≥ 0,ρ+ p ≥ 0, − r t 1 2 a b MG(r) = r e 2 a (27) SEC : ρ + pr ≥ 0,ρ+ pt ≥ 0,ρ+ pr + 2pt ≥ 0, 2 : ρ>| |,ρ>| |. DEC pr pt is the gravitational mass of a sphere of radius r. Now putting Eq. (27) into Eq. (26), it follows The evolution of all these energy conditions have been well a d pr 2 satisﬁed as represented graphically for the strange star can- − (ρ + pr) − + (pt − pr) = 0. (28) didate Her X-1 in Fig. 6. Hence our solutions are physically 2 dr r viable. Equation (28) gives the information as regards the stellar conﬁguration equilibrium under the combined effect of different forces, like the anisotropic force Fa, the hydrostatic 4.3 TolmanÐOppenheimerÐVolkoff (TOV) equation force Fh, and the gravitational force Fg. Their summation to zero eventually describes the equilibrium condition of the In anisotropic case, the generalized TOV equation is given form as Fg + Fh + Fa = 0, (29) MG(r)(ρ + pr) b−a d pr 2 e 2 + − (p − p ) = 0, (26) r 2 dr r t r where 123 674 Page 8 of 12 Eur. Phys. J. C (2017) 77 :674

Fig. 6 Plots of energy conditions with respect to radius r (km) of the strange star candidate Her X-1

d pt = v2 dρ st. It can be seen from Figs. 8 and 9 that the evolution of the radial and transversal sound speeds for the strange star candidate Her X-1 are within the bounds of stability as discussed, but in the case of SAX J 1808.4-3658, and 4U 1820- 30 strange star candidates, the radial sound speeds evolution expressed in the radial coordinater temporarily violates these stability conditions. However, for the same candidates the requirements for the transversal sound speeds are satisﬁed. Within the matter distribution, the estimation of the poten- tially stable and unstable eras can be obtained from the difference of the sound propagation speeds, found by the expres- v2 − v2 < |v2 − v2 | < sion st sr, satisfying the inequality 0 st sr 1. ( ) ( ) Fig. 7 The plot of gravitational force Fg , hydrostatic force Fh and This can be seen clearly from the plots of Fig. 10. Thus, ( ) anisotropic force Fa for the strange star candidate Her X-1 with respect overall the stability may be attained for compact stars in the to the radial coordinate r (km) f (G, T ) gravity model, particularly for the strange star candidate Her X-1. a F =− (ρ + p ), g 2 r d p F =− r , 4.5 Adiabatic index analysis h dr 2 Fa = (pt − pr). (30) For the case of an anisotropic ﬂuid spherical star, as proposed r in [62,63], the stability depends on the adiabatic index γ , and From Fig. 7, it can be noticed that under the mutual effect of for the radial and tangential cases, we, respectively, have the three forces Fg, Fh and Fa, the static equilibrium might be = achieved. It is mentioned here that at some point if pr pt ρ + ρ + = pr d pr pt d pt then Fa 0, which suggests that the equilibrium becomes γr = and γt = . (31) pr dρ pt dρ independent of the anisotropic force Fa.

4.4 Stability analysis The stability of a Newtonian sphere should be satisﬁed if γ>4 γ = 4 3 , and 3 is the condition for the occurrence of For the stability, the radial and transversal sound speeds neutral equilibrium [64]. Due to the presence of the effective v2 v2 denoted by sr and st, respectively, should satisfy the bounds, pressure, the anisotropic relativistic sphere obeys the more ≤ v2 ≤ ≤ v2 ≤ d pr = v2 0 sr 1 and 0 st 1[61], where dρ sr and complicated stability condition given by 123 Eur. Phys. J. C (2017) 77 :674 Page 9 of 12 674

v2 Fig. 8 Variations of sr with respect radius r (km) of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

v2 Fig. 9 Variations of st with respect radius r (km) of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

v2 −v2 Fig. 10 Variations of st sr with respect to radius r (km) of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

4 4 p − p 8π ρ p r γ> + t0 r0 + 0 r0 , It can be seen clearly from the profile of the mass function (32) |p |r |p | max 3 3 r0 3 r0 given in Fig. 11 that the mass of the star is directly propor- tional to the radius, and M(r) → 0asr → 0, which shows that the mass function is regular at the centre of the star. where ρ , p , and p are the initial energy density, tangen- 0 t0 r0 Moreover, for the spherically symmetric anisotropic perfect tial pressure, and radial pressure in static equilibrium, respec- fluid case, the ratio of the mass to the radius, according to tively, satisfying Eq. (26). In our case, the adiabatic index has Buchdahl [6], should be bounded like 2M ≤ 8 . In our case, been calculated analytically for the strange star candidate Her r 9 the situation is very good as we get 2M = 0.4987 and the X-1, giving us γr = 1.3757 and γt = 1.0597, which shows r the complete stability in the radial case but some deviation condition is clearly satisfied. in the tangential case. 4.7 Compactness and redshift analysis

4.6 MassÐradius relationship Compactness μ(r) of the star is expressed as 4 r ´ μ(r) = πr 2ρdr´. (34) The mass of the compact star as a function of radius r is given r as 0 Therefore, the redshift Z is determined as S r ( ) = π ´2ρ ´. −1 M r 4 r dr (33) Z + 1 =[−2μ(r) + 1] 2 . (35) 0 S 123 674 Page 10 of 12 Eur. Phys. J. C (2017) 77 :674

4.8 EoS parameter and the measurement of anisotropy

Now for the anisotropic case, the radial and transversal forms of the EoS parameter can be written as

ω = pr ω = pt . r ρ and t ρ (36)

The evolution of these EoS parameters with the increasing radius is shown in Figs. 13 and 14, which clearly demon- strates that all the six plots satisfy the inequalities 0 <ωr < 1 and 0 <ωt < 1. This further advocates the effectiveness of the considered model. Fig. 11 Evolution of the mass function M(r) for the strange star can- The measurement of the anisotropy denoted by is given didate Her X-1 with respect to the radial coordinate r (km) by 2 = (p − p ), (37) The graphical evolution of the surface redshift is given in r t r Fig. 12. The value of the function Z S for the case of the strange star candidate Her X-1 is calculated as ZS ≈ 0.22, which gives the information as regards the anisotropic which is within the desired bound of ZS ≤ 2. behaviour of the model. remains positive if pt > pr,

Fig. 12 Plots of the compactness μ(r) and surface redshift (Zs) of the strange star candidate Her X-1 with respect to the radial coordinate r (km)

Fig. 13 Variations of the radial EoS parameter with respect to the radial coordinate r(km) of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

Fig. 14 Variations of the tangential EoS parameter with respect to the radial coordinate r (km) of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the ﬁrst, second, and third graphs, respectively

123 Eur. Phys. J. C (2017) 77 :674 Page 11 of 12 674

Fig. 15 Variations of anisotropic measure with respect to the radial coordinate r (km) of the strange star candidate Her X-1, SAX J 1808.4-3658, and 4U 1820-30; represented by (left to right) the first, second, and third graphs, respectively suggesting the anisotropy being drawn outward, and for the mating the radius of the star (R ≈ 10) from the evolution of reverted situation, i.e., pt < pr, the anisotropy turns neg- the radial pressure. The evolution of the EoS parameters with ative, which corresponds to being directed inward. For our the increasing radius satisfies the inequalities 0 <ωr < 1 situation, the variations of the anisotropic measurement and 0 <ωt < 1 for the radial and tangential EoS parameters, with respect to the radial coordinate r show the decreasing respectively, which favours the acceptance of the model of negative behaviour for the strange star candidate Her X-1 our study. We have also shown through the graphical repre- and SAX J 1808.4-3658 suggesting that pt < pr.Forthe4U sentation that all the energy conditions, namely NEC, WEC, 1820-30 candidate, it remains positive for a fraction of the SEC, and DEC, are satisfied for the proposed f (G, T ) gravity radius r where some repulsive anisotropic force followed by model in the case of Her X-1, favouring the physical viability the massive matter distribution appears, and very soon it gets of the model. negative after r = 0.25 (Fig. 15). The static equilibrium, to some extent, has been established by plotting the three forces Fg, Fh, and Fa comprised in the TOV equation. The evolution of the radial and transver- v2 v2 5 Concluding discussion sal sound speeds, denoted by sr and st, respectively, for the strange star candidate Her X-1 are within the bounds of In this paper, we have presented some useful discussions stability [61], but in the case of the strange star candidates related to the emergence of compact stars in the newly intro- SAX J 1808.4-3658, and 4U 1820-30, the radial sound speed v2 duced f (G, T ) theory of gravity by considering the model sr evolutions temporarily violate these stability conditions. f (G, T ) = αGn + λT . We have tested this model for However, for the same candidates the requirements for the v2 the strange star candidates Her X-1, SAX J 1808.4-3658, transversal sound speeds st are satisfied. Within the matter and 4U 1820-30 for an anisotropic case by using the Krori distribution, the estimation of the strongly stable and unsta- and Barua approach of the metric function [15], that is, ble eras from the difference of the propagations of the sound 2 2 < |v2 − v2 | < a = Br +C, b = Ar . The arbitrary constants A, B, and C speeds satisfies the inequality 0 st sr 1 for all the are calculated by smoothly matching the interior metric con- candidates as shown in Fig. 11. Thus, overall the stability is ditions with Schwarzschild’s exterior metric conditions. This attained for the compact star f (G, T ) model, particularly for phenomenon makes us understand the nature of the compact the strange star candidate Her X-1. stars by expressing their masses and radii in terms of the We have also investigated the dynamical stability by ana- arbitrary constants. lytically calculating the adiabatic index γ of the model both By using these constants in our investigation for the for the radial and tangential pressures for the strange star strange star candidates Her X-1, SAX J 1808.4-3658, and candidate Her X-1, giving us γr = 1.3757 > 4/3 and 4U 1820-30 the energy density, and radial and tangential γt = 1.0597, which shows the complete stability in radial pressures have been plotted with respect to the radial coor- case but a slight deviation in the tangential case. We have dinate r indicating that when r approaches zero, the density found the direct proportionality of the mass function to the goes to its maximum for all the three strange star candidates. radius, and M(r) → 0asr → 0, suggesting that the mass The same is the situation for the tangential pressure but dif- function is regular at the centre of the star. Moreover, for the ferent behaviour in the case of the radial pressure for the 4U spherically symmetric anisotropic fluid case, the ratio of the 2M = . 1820-30 candidate. Mainly, this situation admits the theory mass to the radius has been calculated as r 0 4987, sat- 2M ≤ 8 that the core of compact stars under consideration is intensely isfying r 9 as proposed by Buchdahl [6]. The evolution compact, particularly in the case of the strange star candidate of the compactness of the star for Her X-1 favours the model. Her X-1. We have succeeded to determine the density of the The values of the surface redshift function ZS are within the 15 −3 emerging compact star (≈ 1.5828 × 10 gcm ) after esti- bound of ZS ≤ 2 and, for the case of the strange star candi- 123 674 Page 12 of 12 Eur. Phys. J. C (2017) 77 :674 date Her X-1, the redshift value is calculated as ZS ≈ 0.22, 25. S. Capozziello, M.D. Laurentis, S.D. Odintsov, Eur. Phys. J. C 72, 2068 (2012) which satisfies the upper bound ZS ≤ 2. This further indicates the stability of the model of our study. Conclusively, for 26. S. Capozziello, M.D. Laurentis, S.D. Odintsov, A. Stabile, Phys. Rev. D 83, 064004 (2011) the case of the strange star candidate Her X-1, all the physical 27. A.V. Astashenok, S. Capozziello, M.D. Laurentis, S.D. Odintsov, parameters have been more consistent to favour the f (G, T ) JCAP 1501, 001 (2015) gravity model of our study as compared to the other two can- 28. S. Capozziello, M.D. Laurentis, R. Farinelli, S.D. Odintsov, Phys. didates SAX J 1808.4-3658 and 4U 1820-30. The overall Rev. D 93, 023501 (2016) 29. A.D. Felice, S. Tsujikaswa, Living Rev. Relativ. 13, 3 (2010) consistency for the model may be improved by considering 30. K. Bamba, S. Capozziella, S. Nojiri, S.D. Odintsov, Astrophys. some more suitable choices of the physical parameters. Space Sci. 342, 155 (2012) 31. A.V. Astashenok, S. Capozziello, S.D. Odintsov, Phys. Rev. D 89, Acknowledgements Many thanks to the anonymous reviewer for valu- 103509 (2014) able comments and suggestions to improve the paper. This work was 32. A.V. Astashenok, S. Capozziello, S.D. Odintsov, JCAP 01, 001 supported by National University of Computer and Emerging Sciences (2015) (NUCES). 33. G. Cognola, E. Elizalde, S. Nojiri, S.D. Odintsov, S. Zerbini, Phys. Rev. D 73, 084007 (2006) Open Access This article is distributed under the terms of the Creative 34. M.F. Shamir, Astrophys. 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