Color–flavor locked strange stars in 4D Einstein–Gauss–Bonnet gravity

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Abstract

This work is devoted to the recently introduced Einstein–Gauss–Bonnet gravity in four dimensions. This theory can bypass Lovelock’s theorem and avoids Ostrogradsky instability. The integrated part of this theory is the GB term which gives rise to a non-trivial contribution to the gravitational dynamics in the limit D4. Our main interest is to explore a class of static, spherically symmetric compact objects made of strange matter. It is believed that at ultra-high densities, quark matter may exist in a variety of superconducting states, namely, the Color–Flavor-Locked (CFL) phase. With these equations of state (EoS) we explore the conditions under which compact stars are stable under extreme conditions in the context of 4D EGB gravity. In addition, the mass–radius relation is examined and the effects of the coupling constant ‘α’ on the stellar model, energy conditions, speed of sound and adiabatic stability are studied. With the aim to find observational constraints on the GB constant of 4D EGB gravity, the mass–radius curve is determined satisfying the maximum mass constraint of M2M for neutron stars. This feature may be relevant in view of recent observations claiming the existence of stable compact quark stars (hybrid neutron stars or strange stars).

Introduction

Low-energy effective string theories arising from various compactification schemes explore the fundamental nature and laws of our Universe, using the framework of String Theory. The main features of low-energy string theory which appears as an effective model of gravity in higher dimensions that involve higher order curvature terms in the action. In fact, this theory has been proposed with the hope that higher order corrections to Einstein’s GR might solve the singularity problem of black holes and the early universe. Indeed, there exists a lot of work on higher dimensional gravity theories ranging from condensed matter, string theory and gravitation, mainly. In this regard, Lovelock theory of gravity, as a natural generalization of Einstein’s general relativity, was proposed by Lovelock [1], [2]. In particular, the equations of motion of Lovelock gravity are of second order with respect to the metric, as the case of general relativity. Interestingly, Einstein–Gauss–Bonnet gravity [3] is considered as a special case of Lovelock’s theory of gravitation. The EGB gravity adds an extra term to the standard Einstein–Hilbert action, which is quadratic in the Riemann tensor. It appears naturally in the low energy effective action of heterotic string theory [4], [5], [6].

Recently, Glavan & Lin [7] have proposed a new covariant modified theory of gravity in 4-dimensional spacetime, namely, ‘Einstein–Gauss–Bonnet gravity’ (EGB). In D-dimensions, by rescaling the Gauss–Bonnet coupling α by a factor of α(D4), and taking the limit D4, the Gauss–Bonnet term gives rise to non-trivial dynamics. By this re-scaling one can bypass the conclusions of Lovelock’s theorem and avoid Ostrogradsky instability. Though the dimensional regularization of this was considered previously in [8]. The proposed new or modified version of the theory has attracted researchers for several novel predictions in cosmology and astrophysics, though the validity of this theory is at present under debate and doubts. For example, a static spherically symmetric vacuum black hole is obtained in [7], which differs from the standard vacuum-GR Schwarzschild BH. Related to this, other references about rotating and non-rotating black hole solutions and their physical properties have been discussed [9], [10], [11], [12], [13], [14], [15], [16], [17] and the references therein. Moreover, within the same context, the strong/ weak gravitational lensing by black hole [18], [19], [20], [21], spinning test particle [22], geodesics motion and shadow [23], thermodynamics of 4D EGB AdS black hole [24], Hawking radiation [25], [26], Quasinormal modes [27], [28], [29], wormhole solution [30], [31] have been addressed. Additionally, many aspects of this theory have indeed been explored [32], [33], [34], [35].

Hence, the 4D EGB gravity witnessed significant attention that includes finding astrophysical solutions and investigating their properties. In particular, the mass–radius relations are obtained for realistic hadronic and for strange quark star EoS [36]. Theoretically strange stars and their phenomenological properties have also been investigated with different EoS [37], [38]. Exact solutions have been found in [39]. Precisely speaking, we are interested in investigating the behavior of compact stars namely neutron/quark stars in regularized 4D EGB gravity. NSs are the remnants of supernovae, the most energetic events in the Universe since the Big Bang. The structure of NSs depend on the equation of state (EoS) under such extreme conditions is a demanding and still unsolved problem in which physicists around the world engage. However, measurements of the masses or radii of these objects can strongly constrain the NS matter EoS, and consequently the interior composition of NSs [40], [41]. Presently, the most stringent constraints for the EoS comes from the 2 M limit of the pulsars PSR J1614-2230 [42], PSR J0348+0432 [43], PSR J0740+6620 [44], and PSR J2215-5135 [45]. In summary, these celestial bodies provide us with a unique astrophysical laboratory for testing our understanding of fundamental aspects of physics.

In this article, we investigate the color–flavor-locked (CFL) matter [46] and its stability related to the strange quark star. It is well accepted that the Color–Flavor Locking (CFL) phase is the real ground state of Quantum Chromodynamics (QCD) at asymptotically large densities [47]. Since, quarks in the cores of neutron stars are likely to be in a paired phase [48], [49]. In addition, pairing affects the spectrum of quasi-particles and can change the transport properties qualitatively. At asymptotically high densities, one finds quark matter in the CFL phase [46]. Indeed, as a strongly interacting matter, the CFL matter is widely accepted to become ‘absolutely’ stable for sufficiently high densities [50].

Since, the theoretical foundation of all these models is QCD. However, at present it is not possible to obtain reliably the exact EoS of quark matter, and thus researchers impose severe constraints on the possible phases of dense QCD matter. In [51], authors have concluded that CFL is more stable than SQM as long as μ ms24Δ, with ms being the strange quark mass and Δ the pairing gap. If the baryon chemical potential is sufficiently large enough, the formation of Copper pairs is favored due to CFL producing color-superconducting phase, and is believed to be more stable than SQM [46]. Several more about CFL phase, it is electrically and color neutral in the absence of charged leptons when T=0 [52]. It has also been suggested that two-flavor color superconducting (2SC) phase is highly unlikely in the environment of compact objects because the free energy cost in 2SC phase is much higher than the CFL phase [53]. Furthermore, it was shown by various groups that the existence of the CFL phase can enhance the possibility of the existence of stable neutron stars or strange stars [54], [55], [56], [57].

Thus, self-bounded stars made up of CFL quark matter may discuss some recent astrophysical observational data that could shed new light on the possible existence of exotic and/or deconfined phases in some nearby neutron stars (NS). The plan of this paper is as follows: After the introduction in Section 1, we quickly review the field equations in the 4D EGB gravity and show that it makes a nontrivial contribution to gravitational dynamics in 4D in Section 2. In Section 3 we discuss the EoS for CFL strange matter. In Section 4, we discuss the numerical procedure used to solve the field equations. In Section 5, is devoted to report the general properties of the spheres in terms of the CFL strange quark matter. We analyzed the energy conditions as well as other properties of the spheres, such as sound velocity and adiabatic stability. Finally, in Section 6, we conclude.

Section snippets

Basic equations of EGB gravity

The field equation in D-dimensional EGB theory (we use geometrized units while deriving various equations, which is, c=G=1) is derived from the following action: IG=116πdDxgR+αD4LGB+Smatter,where R is the Ricci scalar which provides the general relativistic part of the action, and g denotes the determinant of the metric gμν. The parameter α is the Gauss–Bonnet coupling parameter of dimension [length]2. We consider α0 in this work. Here, LGB is the Einstein–Gauss–Bonnet Lagrangian given by L

Color–flavor locked strange matter

Here, we discuss the CFL quark matter in compact stars. If quark matter is in CFL phase, the thermodynamic potential for electric and color charge neutral CFL quark matter is given by [60] ΩCFL=6π20νp2(pμ)dp+3π20νp2p2+ms2μdp3Δ2μ2π2+B, of the order of Δ2. The symbols have their usual meanings. The first and second terms are contributions from massless u,d quarks and ms mass for s quark, while no interaction is considered. The next term is leading correction due to CFL of the power of Δμ

Numerical techniques and results

Here we will investigate the effects of (B,ms,Δ,α) on the physical properties of the compact stars. Employing CFL phase at sufficiently high densities and relatively low temperatures, we have to solve three equations with four unknown functions, which are m(r), Φ(r), p(r) and ρ(r). In addition to the CFL EoS given in (17), which explicitly depends on the bag constant B, strange quark mass ms and color superconducting gap Δ. As a first step, the TOV equation (11) and mass function (12) are

Energy conditions

In this section, we will briefly study the energy conditions, that are sets of inequalities depending on energy–momentum tensor. To be specific, we start by finding strange stars for weak energy condition (WEC), i.e. TμνUμUν, where Uμ is a timelike vector. For the given diagonal EM tensor, the WEC implies ρ(r)0andρ(r)+p(r)0,it follows that if WEC is satisfied then NEC also satisfied. The NEC is the assertion that for any null vector kμ, we should have Tμνkμkν0. The NEC is the simplest energy

Summary and discussion

In this work we have performed a detailed study on the properties of color–flavor locked quark matter in compact star interior. This paper presents a systematic study of static, spherically symmetric solutions in 4D Einstein–Gauss–Bonnet gravity, which bypasses the conclusions of Lovelock’s theorem and avoids Ostrogradsky instability. The hydrostatic equilibrium equations are obtained in order to test the new theory in strange stars whose mass–radius diagrams are obtained using CFL quark matter

CRediT authorship contribution statement

Ayan Banerjee: Ideas, Formulation, Writing - original draft. Ksh. Newton Singh: Application of mathematical formulation and computation.

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.

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