Compressibility of the Quark Stars in Einstein-Gauss-Bonnet Gravity

Abstract

To study the behavior of some of the quark stars (QSs) in the Einstein-Gauss-Bonnet gravity, we develop analytically the equation of state (EoS). We show that the Gauss-Bonnet coupling affects the star compressibility. This last relates the radial pressure with the thermodynamic volume and analyzes their behavior through the quark stars. We have found that the TOV equation is well behaved as the van der Waals equation. We discuss the characteristics of the radial and tangential compressibility in the surface and inside the star. We have determined the thickness of the region that contains the quarks in the QS. We show that the quark fluid exists in the region between the Schwarzschild radius and the radius of stars. We study the stability under radial and tangential perturbations.

1. Introduction

A new class of compact stars called strange stars can exist if the strange matter hypothesis is true Refs. [1, 2]. There are three different kinds of compact stars [2]: white dwarfs, neutron stars (NSs), and black holes (BHs). Even more intriguing is the existence of quark matter in an NS, which is the possible existence of a new, less conventional class of compact stars, the so-called strange stars or quark stars (QSs) [3]. In the original model, the compact stars consist of three fundamental quarks: up (u), down (d), and strange (s). The pulsar masses predicted through observations impose a strong constraint on the internal composition of the QS. The maximum mass-radius relation of the stars allows us to understand the nature of the matter inside these stars and to determine the limit conditions for the formation of the black hole. The creation of a quark star starts from an explosive transition called the quark-nova. At the highest densities, the ground state is a symmetric state in the superfluid of the color-flavor locked (CFL) phase [4]. The relationship between the pressure and density within the star can be found by a specific equation of state (EoS). Observations of NSs allow for a more fundamental understanding of nuclear physics. In order to consider the PSR J0348+0432 and MSP J0740 + 6620 as quark stars, the importance of isospin and quark mass effects at finite temperature were explored in [5]. The tidal deformability increases with the temperature, and we can describe the most massive compact star MSR J0740 + 6620 as quark star [6]. Some recent works have been done to study the anisotropic matter as spherical symmetry interior fluid, such as neutron stars [7] and boson stars [8]. Tolman-Oppenheimer-Volkoff (TOV) equation [9] plays a very important role in the description of the anisotropic matter in relativistic stars [10]. Moreover, the TOV formalism has been explored in Gauss-Bonnet gravity [11].

Recently, Glavan and Lin [12] introduced a general covariant modified theory of gravity in 4-spacetime dimensions, which propagates only the massless graviton and also bypasses Lovelock’s theorem [13]. Their intriguing idea was to multiply the Gauss-Bonnet (GB) gravity term by the factor $1/\left(D-4\right)$ and take the limit $D\to 4$, which offers a new four-dimensional gravitational theory with only two dynamical degrees of freedom. The four-dimensional symmetrical static and spherical black hole solution in Einstein-Gauss-Bonnet (EGB) gravity were obtained [14], as well as solutions of a static and spherically symmetric compact stars [15, 16, 17]. Many researchers have studied the mass-radius profile and the maximum mass in EGB gravity [18, 19]. Consequently, it is possible to describe the matter inside compact objects and the dynamical evolution of the matter at high density and the behavior of violent events. The aim of this chapter is to explore a new analytical solution to the TOV equation describing a spherically symmetric static distribution of anisotropic fluid in the stars. We show that TOV is acting like the van der Waals equation of state. In this case, we determine the star compressibility and link it with all the parameters of this model. We apply our result to the compact stars PSR J1614-2230 and SAX J1808.4-3658 by considering the spherically symmetric line element.

The chapter is planned as follows: In Section 2, we briefly review the Gauss-Bonnet gravity in four dimensions in coupling with a scalar field. In Section 3, we study the spherically symmetric metric ansatz describing the interior of the star. In Section 4, we explore a new analytical solution to the TOV and discuss the relationship between compressibility factors and the EoS. In Section 5, we discuss the mass-radius profiles for the quark matter. In the same section, we discuss the relationship between the surface redshift and compressibility. In Section 6, we analyze the stability of the QS under radial and tangential perturbations. We conclude our findings in Section 7.

Mass is measured in solar mass units ($M_{\odot }$) with $M_{\odot }=\mathrm{1,989}\times {10}^{30}\mathit{kg}$, radius in $\mathit{km}$, and energy density and pressure are in ${\mathit{\text{MeVfm}}}^{-3}$ with $c=3\times {10}^5\mathit{km}/s$, $G={\mathrm{6,6742.10}}^{-17}N$. ${\mathit{km}}^2$. ${\mathit{kg}}^{-2}$, $k_B=1.38\times {10}^{-23}{m}^2{\mathit{kgs}}^{-2}{K}^{-1},$$\mathit{MeV}/{\mathit{fm}}^3=1.7827\times {10}^{15}\mathit{kg}/m^3$.

2. Einstein-scalar-Gauss-Bonnet gravity

In this section, we explain in detail how to construct the equation of motion of the Einstein-Gauss-Bonnet gravity. We begin by reviewing the ansatz for the metric, matter, and scalar field. Consider now the scalar-Gauss-Bonnet (sGB) gravity in four dimensions [20]:

$$\mathcal{S}=\frac{1}{2{\kappa }^2}\int d^{4}x\sqrt{-g}\left(R+f\left(\varphi \right)\mathcal{G}-\frac{1}{2}{g}^{\mathit{\mu \nu }}{\partial }_{\mu }\varphi {\partial }_{\nu }\varphi -V\left(\varphi \right)\right)+{\mathcal{S}}_m,$$E1

where $M_p=1/\kappa =1.221\times {10}^{-19}\mathit{GeV}$ is the reduced Planck mass, $R$ is the Ricci scalar, $S_m$ is the matter action, and $f\left(\varphi \right)$ is a functional coupling of the scalar field $\varphi$. In the above equation $\left(\mu ,\nu \right)=\left(\mathrm{0,1,2,3}\right)$. We define the GB term as

$$\mathcal{G}\equiv R^2-4R_{\mathit{\mu \nu }}{R}^{\mathit{\mu \nu }}+R_{\mathit{\text{μνρσ}}}{R}^{\mathit{\text{μνρσ}}}.$$E2

The variation with respect to the field $\varphi$ gives us the equation of motion for the scalaron field

$$\square \varphi ={\partial }_{\varphi }{V}_{\mathit{eff}}\left(\varphi \right),$$E3

where $\square \equiv {\nabla }_{\mu }{\nabla }^{\mu }$ and the effective potential is

$$V_{\mathit{eff}}\left(\varphi \right)=V\left(\varphi \right)-f\left(\varphi \right)\mathcal{G}.$$E4

Varying the action (1) over the metric $g_{\mathit{\mu \nu }}$, we obtain the following equations of motion:

$$G^{\mathit{\mu \nu }}+{\mathcal{K}}^{\mathit{\mu \nu }}+f\left(\varphi \right){\mathcal{H}}^{\mathit{\mu \nu }}=\frac{1}{2}\left[{\kappa }^2{T}^{\mathit{\mu \nu }}-{\mathcal{T}}_{\varphi }^{\mathit{\mu \nu }}+g^{\mathit{\mu \nu }}{V}_{\mathit{eff}}\left(\varphi \right)\right],$$E5

where the Einstein tensor is $G^{\mathit{\mu \nu }}=R^{\mathit{\mu \nu }}-\frac{1}{2}{g}^{\mathit{\mu \nu }}R$, the matter stress tensor is $T^{\mathit{\mu \nu }}=-\frac{2}{\sqrt{-g}}\frac{\delta {\mathcal{S}}_m}{\delta g_{\mathit{\mu \nu }}}$. On the other hand, the ${\mathcal{K}}^{\mathit{\mu \nu }}$ and ${\mathcal{H}}^{\mathit{\mu \nu }}$ are given by

$${\mathcal{K}}^{\mathit{\mu \nu }}=4\left(G^{\mathit{\mu \nu }}\square +\frac{1}{2}R{\nabla }^{\mu }{\nabla }^{\nu }+\left(g^{\mathit{\mu \nu }}{R}^{\mathit{\rho \sigma }}-R^{\mathit{\text{μρνσ}}}\right){\nabla }_{\rho }{\nabla }_{\sigma }-\left(R^{\mathit{\nu \rho }}{\nabla }_{\rho }{\nabla }^{\mu }+R^{\mathit{\mu \rho }}{\nabla }_{\rho }{\nabla }^{\nu }\right)\right)f\left(\varphi \right),$$E6

$${\mathcal{H}}^{\mathit{\mu \nu }}=2R^{\mathit{\text{μρστ}}}{R}_{\mathit{\rho \sigma \tau }}^{\nu }-{\mathit{RR}}^{\mathit{\mu \nu }}+\frac{1}{2}{R}_{\rho }^{\mu }{R}^{\mathit{\nu \rho }}-R^{\mathit{\text{μρστ}}}{R}_{\mathit{\rho \sigma \tau }}^{\nu }.$$E7

The tensor ${\mathcal{K}}^{\mathit{\mu \nu }}$ represents an operator which acts on $f\left(\varphi \right)$. The energy-momentum tensor for the scalar field is

$${\mathcal{T}}_{\varphi }^{\mathit{\mu \nu }}={\nabla }^{\mu }\varphi {\nabla }^{\nu }\varphi -\frac{1}{2}{g}^{\mathit{\mu \nu }}{\nabla }_{\rho }\varphi {\nabla }^{\rho }\varphi .$$E8

To find Einstein’s limit, we can take ${\mathcal{K}}^{\mathit{\mu \nu }}+f\left(\varphi \right){\mathcal{H}}^{\mathit{\mu \nu }}-\frac{1}{2}{g}^{\mathit{\mu \nu }}{V}_{\mathit{eff}}\left(\varphi \right)=\frac{1}{2}{\mathcal{T}}_{\varphi }^{\mathit{\mu \nu }},$ this last equation is similar to the Einstein’s equation. The stress tensor for anisotropic compact star is given as

$$T^{\mathit{\mu \nu }}=\left(\rho +P_t\right)u^{\mu }{u}^{\nu }+P_t{g}^{\mathit{\mu \nu }}+\left(P-P_t\right){\chi }^{\mu }{\chi }^{\nu },$$E9

with energy density $\rho \left(r\right)\equiv c^2\rho \left(r\right)$, transverse pressure $P_t\left(r\right)$, and radial pressure $P\left(r\right)$ of the homogeneously distributed quark matter, where $u^{\mu }$ is the four-velocity of the fluid, and ${\chi }^{\mu }$ is the unit space-like vector in the radial direction. The form the functional $f\left(\varphi \right)$ can take [21]

$$f\left(\varphi \right)=e^{-\mathit{\gamma \varphi }},$$E10

where $\gamma$ is a constant, which corresponds to EGB gravity coupled with dilaton that arises as a low-energy limit of the string theory [22]. We suppose that in the exterior region of the QS, there is a presence of scalar fields $\varphi$. On the other hand, inside the QS is replaced by the Gauss-Bonnet (GB) coupling $\alpha$. The GB coupling is measured in ${\mathit{km}}^2$:

$$\left\{\begin{array}{l}f\left(\varphi \right)=\frac{\alpha }{D-4},D\to 4\text{stars}\text{interior}\\ f\left(\varphi \right)=e^{-\mathit{\gamma \varphi }},D=4\text{stars}\text{exterior}\end{array}\right..$$E11

In the star interior, we have rescaled the coupling constant $\alpha \to \alpha /\left(D-4\right)$, and in the QS surface, we choose $f\left(\varphi \right)=\frac{\alpha }{D-4}$. The negative (positive) $\alpha$ leads to a decrease (increase) of the quark star radius and the maximum mass [23]. If $\alpha <0$_, the solution is still the anti de Sitter (AdS) space; if $\alpha >0$_, the solution is the de Sitter (dS) space [24]. We investigate in detail the impact of the Gauss-Bonnet coupling on properties of anisotropic quark stars, such as mass, radius, and the factor of compactness. Considering the limit $D\to 4$, it has an effect on gravitational dynamics in 4D. Additionally, the GB coupling must be continuous at the boundary $\left(r=R\right)$ of the star: $\alpha \equiv e^{-\mathit{\gamma \varphi }}$. Next, by studying the star surface, we show that the coupling $\alpha$ described the interior structure of the star. On the other hand, the function $f\left(\varphi \right)$ describes the star exterior region.

3. Interior of the quark stars

We consider non-rotating quark stars assuming an interacting EoS for strange matter phase and allowing anisotropy in the pressure. Here, we start by the metric of the four-dimensional spherically symmetric metric ansatz describing the interior of the star [3] by:

$${\mathit{ds}}_D^2=-e^{2\tau \left(r\right)}{\mathit{dt}}^2+e^{2\sigma \left(r\right)}{\mathit{dr}}^2+r^{2}d{\mathrm{\Omega }}^2,$$E12

where $d{\mathrm{\Omega }}^2=d{\theta }^2+{sin}^2\mathit{\theta d}{\phi }^2$ is the metric on the unit two-dimensional sphere, $\tau \left(r\right)$ and $\sigma \left(r\right)$ are functions of radial coordinate $r$. Next, we choose $f\left(\varphi \right)=\alpha$ and $V\left(\varphi \right)=2\mathrm{\Lambda }\approx 0$, where $\mathrm{\Lambda }$ is the cosmological constant, we find

$$G^{\mathit{\mu \nu }}+\alpha \left({\mathcal{H}}^{\mathit{\mu \nu }}+\frac{1}{2}{g}^{\mathit{\mu \nu }}\mathcal{G}\right)=\frac{{\kappa }^2}{2}{T}^{\mathit{\mu \nu }}.$$E13

The non-vanishing components for a static, spherically symmetric perfect-fluid star in EGB theory read:

$$\left(1+\frac{2\alpha \left(1-e^{-2\sigma }\right)}{r^2}\right)\frac{2}{r}{e}^{-2\sigma }\frac{\mathit{d\sigma }}{\mathit{dr}}=\frac{{\kappa }^2}{2}\rho -\frac{1-e^{-2\sigma }}{r^2}\left(1-\frac{\alpha \left(1-e^{-2\sigma }\right)}{r^2}\right),$$E14

$$\left(1+\frac{2\alpha \left(1-e^{-2\sigma }\right)}{r^2}\right)\frac{2}{r}{e}^{-2\sigma }\frac{\mathit{d\tau }}{\mathit{dr}}=\frac{{\kappa }^2}{2}P+\frac{1-e^{-2\sigma }}{r^2}\left(1-\frac{\alpha \left(1-e^{-2\sigma }\right)}{r^2}\right),$$E15

$$\frac{{\mathit{dP}}_r}{\mathit{dr}}=\left(\rho +P\right)\frac{\mathit{d\tau }}{\mathit{dr}}+\frac{2}{r}\left(P_t-P\right).$$E16

To study the general structure of the solution, we substitute by

$$e^{-2\sigma }=1-\frac{2\mathit{Gm}\left(r\right)}{r}\text{or}\sigma =-\frac{1}{2}ln\left(1-\frac{2\mathit{Gm}\left(r\right)}{r}\right),$$E17

where $m\left(r\right)$ is the mass contained in a sphere of radius $r$. Here, $M=m\left(R\right)={\int }_0^{R}4\pi r^3\mathit{\rho dr}$ is the total mass of the quark stars of the boundary $r=R$. We get the Tolman-Oppenheimer-Volkoff (TOV) equation:

$$\frac{\mathit{dP}}{\mathit{dr}}=-\frac{2\mathit{Gm\rho }}{r^2{c}^2}\left(1+\frac{P}{\rho }\right)\left(1+\frac{4\pi r^{3}P}{3{\mathit{mc}}^2}-\frac{2\mathit{G\alpha m}}{r^3{c}^2}\right){\left(1+\frac{4\mathit{G\alpha m}}{3r^3{c}^2}\right)}^{-1}{\left(1-\frac{2\mathit{Gm}}{{\mathit{rc}}^2}\right)}^{-1}+\frac{2}{r}\left(P_t-P\right).$$E18

The thermal emission from the surface of the quark star is of key importance to determine its mass and its radius. Next, we show how quark matter appears in compact stars by studying $M-R$ relationship. The global behavior of $M\left(R\right)$ only exists for models located well beyond the maximum of the mass, which are considered unstable. Next, we review the structure equations describing the QS interior solutions by studying the stellar mass $M$ against the energy density

$$\frac{2G\left(\rho +P\right)}{r^2{c}^2}\left(\frac{4\pi r^{3}P}{3c^2}+m-\frac{2\mathit{G\alpha }{m}^2}{r^3{c}^2}\right){\left(1+\frac{4\mathit{G\alpha m}}{3r^3{c}^2}\right)}^{-1}{\left(1-\frac{2\mathit{Gm}}{{\mathit{rc}}^2}\right)}^{-1}=\frac{2}{r}\left(P_t-P\right)-P^{\prime },$$E19

where $M$ denotes the mass of the star, and (${}^{\text{'}}$) represents $d/\mathit{dr}$. In the limit of vanishing GB coupling constant $\left(\alpha \approx 0\right)$_, we can only recover the GR branch.

$$m_{\mathit{GR}}\left(r\right)=\frac{\frac{r^2{c}^2}{2G}\left(\frac{2}{r}\left(P_t-P\right)-P^{\prime }\right)-\left(\rho +P\right)\frac{4\pi r^{3}P}{3c^2}}{\rho -P+2P_t-r{P}^{\prime }}.$$E20

The mass $m_{\mathit{GR}}$ vanishes for $r=0$. The mass function $m\left(r\right)$ of the stars is increasing with $r$. The internal structure of the QS depends on the EoS used to describe its composition. We set $\upsilon =\frac{4\pi r^3}{3}$, and we propose that $16\mathit{\text{πGαm}}\ll 9\upsilon c^2$, Eq. (19) takes the following form

$$\left(P+\frac{{\mathit{mc}}^2}{\upsilon }-\alpha \frac{8\pi {\mathit{Gm}}^2}{3{\upsilon }^2}\right)\left(\upsilon -\alpha \frac{16\mathit{\pi Gm}}{9c^2}\right)=\frac{r^2{c}^4}{2G\left(\rho +P\right)}\left[\frac{2}{r}\left(P_t-P\right)-P^{\prime }\right]\left(1-\frac{2\mathit{Gm}}{{\mathit{rc}}^2}\right).$$E21

The left side of this equation is similar to that of the van der Waals equation. In the next section, we will study this last equation in the surface of the QS.

4. Fluid in the stars surface

The TOV equation can be solved analytically for a given EoS $P=P\left(\mathcal{V}\right)$ that relates the radial pressure with the thermodynamic volume $\mathcal{V}$. We assume that the pressure is constant at the boundary $r=R$ of the quark stars with a thickness $\mathit{\delta R}$. The new equation is obtained from Eq. (21) and the requirement that $\mathit{dP}/\mathit{dR}=0$:

$$\left(P+\frac{{\mathit{Mc}}^2}{\mathcal{V}}-\frac{8\mathit{\pi G\alpha }{M}^2}{3{\mathcal{V}}^2}\right)\frac{\mathcal{V}}{\left(1+\frac{16\mathit{\text{πGαM}}}{9\mathcal{V}{c}^2}\right)}=\frac{{\mathit{Rc}}^4\left(P_t-P\right)}{G\left(\rho +P\right)}\left(1-\frac{2\mathit{GM}}{{\mathit{Rc}}^2}\right),$$E22

where $M$ denotes the mass of the star. We take $\mathcal{V}=4\pi R^3/3$. We propose that $16\mathit{\text{πGαM}}\ll 9\mathcal{V}{c}^2$, we get

$$\left(P+\frac{{\mathit{Mc}}^2}{\mathcal{V}}-\alpha \frac{8\pi {\mathit{GM}}^2}{3{\mathcal{V}}^2}\right)\left(\mathcal{V}-\frac{16\mathit{\text{πGαM}}}{9c^2}\right)=\frac{{\mathit{Rc}}^4\left(P_t-P\right)}{G\left(\rho +P\right)}\left(1-\frac{2\mathit{GM}}{{\mathit{Rc}}^2}\right).$$E23

This equation is analogous to the van der Waals equation of state, which is defined as a function of $P$. According to this equation, the GB coupling represents the average attraction between particles in the van der Waals fluid on the QS surface. The term ${\mathit{Mc}}^2$ represents the internal energy of QS. Signature of the van der Waals like small-large charged AdS black hole phase transition in quasinormal modes [25]. This van der Waals equation provided for intermolecular interaction by adding to the observed pressure $P$ in the equation of state the term ${\mathit{Mc}}^2/\mathcal{V}-\alpha 8\pi {\mathit{GM}}^2/3{\mathcal{V}}^2$. We set $\tilde{P}=P+\frac{{\mathit{Mc}}^2}{\mathcal{V}}-\frac{8\mathit{\pi G\alpha }{M}^2}{3{\mathcal{V}}^2}$, $\tilde{\mathcal{V}}=\mathcal{V}-\frac{16\mathit{\text{πGαM}}}{9c^2}$ and the temperature

$$T=\frac{c^4\left(P_t-P\right)}{k_{B}G\left(\rho +P\right)}\left(R-R_S\right),$$E24

where $R_S=2\mathit{GM}/c^2$ denotes the Schwarzschild radius of a compact object set by gravitational collapse to a black hole. We get the equation of state for an ideal gas $\tilde{P}\tilde{\mathcal{V}}=T$. In Eq. (24) the temperature is not expressed in terms of GB coupling. In a stable fluid sphere, the equation of state parameters $\omega \equiv P/\rho$ should be positive. The pressure $P$ and density $\rho$ should be positive inside the stars and should satisfy the energy conditions $\rho +P\ge 0$ [26]. For this purpose, we write $\left(P_t-P\right)\left(R-R_S\right)\ge 0$. In other words, we have two cases: (i) As a result, the condition concerning the quark stars is when $R>R_S$ and $P_t>P$_, (ii) When $R=R_S$ or $P_t=P$, the temperature of QS becomes zero. At this point, there is a transformation of QS toward the black hole, and (iii) The condition when $R<R_S$ and $P_t<P$, concerning the QS inside of the black hole. The radial and transverse compressibility factors are respectively:

$$Z=\frac{P\mathcal{V}}{k_{B}T}=1-\frac{{\mathit{Mc}}^2}{k_{B}T}+\frac{8\mathit{\pi \alpha }}{9\mathcal{V}}\left(\frac{3{\mathit{GM}}^2}{k_{B}T}-R_S\right),$$E25

and

$$Z_t=\frac{P_t\mathcal{V}}{k_{B}T}=\frac{G\mathcal{V}\left(\rho +P\right)}{c^4\left(R-R_S\right)}+Z.$$E26

For an ideal fluid, the compressibility factor becomes $Z=1$ (or $Z_t=1$). The case of $Z>1$ describes the repulsion between the fluid particles and generates force in an outward direction (repulsive). In the case of $Z<1$, there is a strong attraction between these particles. For a given star temperature, the radial compressibility grows with increasing $\alpha$. We discuss the behaved nature of the radial and transverse pressures $\left(P,P_t\right)$ then compressibility factors $\left(Z,Z_t\right)$ of the stars: (i) SAX J1808.4-3658 with mass $M=1.2M_{\odot }$ and radius $R=7.5\mathit{km}$. (ii) PSR J1614-2230 with mass $M=1.97M_{\odot }$ and radius $R=9.69\mathit{km}$ [26]. The pressure is to be computed out to that radial distance where $P\left(r=R\right)=\mathit{cte}$ [2]. Next using $T=160\mathit{MeV}$ and ${10}^3\mathit{MeV}/{\mathit{fm}}^3\le \left(\rho +P\right)\le {10}^{10}\mathit{MeV}/{\mathit{fm}}^3$ for Eqs. (24) and (25).

• For SAX J1808.4-3658 with mass $M=1.2M_{\odot }$ and radius $R=7.5\mathit{km}$ we have $Z=\left(8\times \alpha -9.54\times {10}^8\right){10}^{49}$ and $Z_t-Z=3.28\times {10}^{-29}\left(\rho +P\right)$. Taking the value $\alpha \sim 1.192\times {10}^8{\mathit{km}}^2$, the compressibility will be zero. For $\alpha \gg {10}^8{\mathit{km}}^2$, we get $Z>1$. From [3, 23] we have $-5{\mathit{km}}^2\le \alpha \le 5{\mathit{km}}^2$, the GB coupling will not have a great effect on radial compressibility. We can obtain $Z\approx -9.54\times {10}^{57}<1$, which describes a strong interaction between the particles. This case corresponds exactly with the strong interaction between the quarks. The quarks in SAX J1808.4-3658 exist in the region $R-R_S=3.98\mathit{km}$.

• For PSR J1614-2230 with mass $M=1.97M_{\odot }$ and radius $R=9.69\mathit{km}$, we have $Z=\left(\alpha -1.56\times {10}^8\right){10}^{50}$ and $Z_t-Z=7.25\times {10}^{-29}\times \left(\rho +P\right)$. If $-5{\mathit{km}}^2\le \alpha \le 5{\mathit{km}}^2$, the GB coupling will not have a great effect on the radial compressibility. The fluid of quarks in PSR J1614-2230 exists in the region $R-R_S=3.89\mathit{km}$. From Eqs. (24) and (26) we get

$$\left(P_t-P\right)\mathcal{V}=\left(Z_t-Z\right)k_{B}T.$$E27

The above equation is similar to that of the ideal gas. In the limits $P_t\to 0$ and $P\to 0$, considering Eq. (27) we obtain respectively the components of radial and tangential pressure in the following forms: $P\mathcal{V}={\mathit{Zk}}_{B}T$ and $P_t\mathcal{V}=Z_t{k}_{B}T$. This result shows that the case $\left(P_t-P\right)>0$ indicates the inhomogeneity in pressures, which generates a repulsive force. The case $\left(P_t-P\right)<0$ indicates the native anisotropy and an attractive force. Eq. (25) takes the following simple form

$$Z=1-\frac{R_c}{R_{\mathit{ps}}}+\frac{8\mathit{\pi \alpha }}{9\mathcal{V}}\left(R_c-R_S\right),$$E28

where $R_c=\frac{3{\mathit{GM}}^2}{k_{B}T}$ and $R_{\mathit{ps}}=\frac{3\mathit{GM}}{c^2}$ are the radii of the photon sphere for a Schwarzschild black hole. On the other hand, The tangential compressibility factor $Z_t$ does not depend on $\alpha$, which shows that the GB term does not influence the tangential structure of QS. Hence, we see that the factor $Z_t$ does not vanish in the limit $\alpha \to 0$, and it affects the gravitational dynamics of the QS. Studying the impact of the GB coupling according to the QS compressibility. From Eq. (28) yields

$$\left\{\begin{array}{l}{Z}_S=\frac{8\mathit{\pi \alpha }}{9\mathcal{V}}\left(R_c-R_S\right),\text{for}{R}_c=R_{\mathit{ps}}\\ Z_{\mathit{ps}}=1-\frac{R_c}{R_{\mathit{ps}}},\text{for}{R}_c=R_S\end{array}\right.,$$E29

where $Z_S$ and $Z_{\mathit{ps}}$ are, respectively, the compressibility inside and outside of the star. Furthermore, one can also see that: (i) The radius $R_c$ is the critical radius that determines the nature of the compressibility. (ii) The interior compressibility factor $Z_S$ depends on the coupling $\alpha$, which shows that $\alpha$ describes the interior structure of the star. (iii) The exterior compressibility $Z_{\mathit{ps}}$ does not depend on the coupling $\alpha$, which corresponds to the proposition Eq. (11). The compressibility $Z_S$ vanishes for $M=5.1\times {10}^{-41}T$. For $M=1.5\times {10}^{-40}T$, there remains only the $Z_S$, that is, the compressibility corresponding to the GB branch. Eq. (29) leads to the important consequence: The two branches of compressibility are necessary to produce the photon sphere and the Schwarzschild black hole. Thus, the compressibilities $Z_S$ and $Z_{\mathit{ps}}$ are crucial in our effort to set the limit from where there is the formation of a black hole and to understand the true nature of matter inside a star.

5. Mass-radius profiles

To understand what kind of matter compact in the interior of stars, we assume that the all information regarding the stellar inner structure is given by the EoS. Next, we show the mass of the stars versus the radius in the quark stars surface. Eq. (27) takes the following form:

$$P=P_t-\left(Z_t-Z\right)\frac{k_{B}T}{\mathcal{V}}.$$E30

In the context of the MIT bag model [27], the quark matter phase is modeled as a Fermi gas of u, d, and s quarks. According to the MIT bag model, the pressure $P$ is defined as

$$P=\sum _{f=u,d\text{and}s}{P}_f-B,$$E31

with $B$ is the bag constant. For the massless quark case, the equation of state is $P=\left(\rho -4B\right)/3$. We propose that

$$\sum _{f=u,d\text{and}s}{P}_f\equiv P_t.$$E32

According to this relation, the pressure $P_t$ is generated by the quark fluid, we will later justify this correspondence. Also we have $\mathrm{\Delta }Z=Z_t-Z\equiv \pm \frac{B}{k_{B}T}$. The compressibility of QS increases when the temperature decreases. In this case, when $Z_t=Z$, we obtain $T=\infty$. From Eq. (26) we obtain

$$M\left(R\right)=\frac{c^2}{2G}R\mp \frac{2\pi k_B\left(\rho +P\right)T}{3c^{2}B}{R}^3.$$E33

In the present analysis, we treat the values of $M$ concerning $R$. The bag parameter $B$ is included between $57\mathit{MeV}/{\mathit{fm}}^3$ and $94\mathit{MeV}/{\mathit{fm}}^3$ [28]. For the lowest value of the bag parameter $B=57\mathit{MeV}/{\mathit{fm}}^3$, we find the maximum masses of compact stars.

$$M_{\max }\left(M_{\odot }\right)=3.388\times {10}^{-4}R-4.51\times {10}^{-10}\left(\rho +P\right)R^3.$$E34

As one can see, the $\left(M-R\right)$ profile depends on the choice of the value of the constant $B$.

We introduce the factor of compactness: $\mathcal{C}=2\mathit{GM}/{\mathit{Rc}}^2$ and the surface redshift: $z_s=-1+{\left(1-\mathcal{C}\right)}^{-1/2}$.

$$\mathcal{C}=1-\frac{\pi R^2\left(\rho +P\right)}{3{\mathit{Gc}}^4\left(Z_t-Z\right)},$$E35

$$1+z_s=\sqrt{\frac{3{\mathit{Gc}}^4\left(Z_t-Z\right)}{\pi R^2\left(\rho +P\right)}}.$$E36

In the above expression, for $Z_t=Z$, we get $\mathcal{C}=-\infty$ and $z_s=-1$. The red shift depends on the pressure, density, and the GB coupling constant $\alpha$. $z_s$ is related to the emission produced by photons from the surface of the quark stars, which is of great importance to astronomers. The compactness and red shift grow with increasing and decreasing $\alpha$, respectively.

6. Stability analysis

To study the stability of the QS under radial and tangential perturbations, we employ the causality condition checked $0\le v_{r\text{or}t}^2\le c^2$. In this section, we discuss the effective sound speeds, which are related to the energy density, transverse pressure, and radial pressure and defined by

$$v_r^2=\frac{\mathit{\delta P}}{\mathit{\delta \rho }},v_t^2=\frac{\delta P_t}{\mathit{\delta \rho }}.$$E37

In stable anisotropic models, the sound speeds $v_r^2$ and $v_t^2$ within the fluid spheres should be positive. From Eq. (38) we get

$$v_t^2=\frac{k_B\mathit{GT}}{c^2\left(R-R_S\right)}.$$E38

The temperature $T$ is weak if the distance between $R_S$ and $R$ is very small. In this case, the quark fluid exists in the region $R-R_S$. Eq. (39) is valid for a stable system if $R>R_S$, which shows that there is a tangential propagation of the particles. This justifies why the pressure due to each flavor is equal to the transverse pressure Eq. (32). The transverse speed $v_t$ depends on the thermal variation of QS. In this case, $v_t$ is defined as the thermal velocity of the thermal motion of particles that make up the quark fluid

$$v_r^2=-\frac{c^2}{1+\frac{c^4\left(R-R_S\right)}{k_B\mathit{GT}}}=-\frac{v_t^2{c}^2}{c^2+v_t^2}.$$E39

For the QS ($R>R_S$), we have $v_r^2<0$, which shows that the system is unstable in the radial direction. Moreover, the interval $R<R_S$ has also been taken into consideration and represents a black hole. Usually, the QS is always unstable in the radial direction.

In addition, the QS is stable during the propagating of the transverse sounds. After the QS-BH transition, there is the propagation of the sounds $v_r$ because of $R<R_S$. For the stellar configurations, we assume that the quark fluid exists between $R_S$ and $R$. The stability of the QS is essentially determined by the value of $B$ as $v_t^2{c}^2\left(R-R_S\right)\left(Z_t-Z\right)\mathbf{=}\mathit{GB}$, which shows that the quarks always move in the tangential direction of QS. If $Z_t=Z$ or $R=R_S$, we get $B=0$, which shows that there is the disappearance of the asymptotically free quarks of a finite region of space (bag) in these both cases.

7. Conclusion

In the present work, we have studied the compact stars in 4D Einstein-Gauss-Bonnet (EGB) gravity. The Gauss-Bonnet (GB) coupling describes the interior structure of the star. On the other hand, the coupling function $f\left(\varphi \right)$ describes the star exterior region. The aim of this chapter is to explore a new analytical solution to the Tolman-Oppenheimer-Volkoff (TOV) equation describing a spherically symmetric static distribution of anisotropic fluid in the stars. It is solved in two regions: the interior and the surface of the star. To study the behavior of the quark stars (QSs) in the EGB gravity, we develop the equation of state (EoS). That relates the radial pressure with the thermodynamic volume and analyzes their behavior through the QS. We have found that the TOV equation can be expressed as the van der Waals equation. For such a choice of the radial and transverse pressure, we got the ideal gas equation. We have determined the thickness of the region that contains the quarks in the QS. We have employed the values of the pressure and the density to study the mass-radius $\left(M-R\right)$ profile. For a fixed value of bag constant, when the tangential compressibility $Z_t$ is greater than the radial compressibility $Z$, $M$ is linear as a function of $R$ for $M_{\max }\approx 1.2M_{\odot }$. After this value, the mass increases if the star radius decreases. Furthermore, we have investigated the properties of the quark matter phase we have used the expression of the pressure in the context of the MIT bag model. We have also studied the effects of the parameters on star compressibility. We have found the relation between the compressibility factors of the star interior and the Gauss-Bonnet coupling. The compressibility of the star exterior is expressed as a function of the photon sphere radius. We have studied the stability of the QS under radial and tangential perturbations. These compressibilities have a noticeable effect on the surface red shift.