Strange Quark Stars in 4D Einstein–Gauss–Bonnet Gravity

In modern gravity theories, higher derivative gravity (HDG) theories have attracted considerable attention, as alternative theories beyond general relativity (GR). Among many impressive outcomes, HDG shows quite different aspects from that in four dimensions, and Einstein–Gauss–Bonnet (EGB) theory (Lanczos 1938) is one of them. EGB theory is a natural extension of GR to higher dimensions, which emerges as a low-energy effective action of heterotic string theory (Zwiebach 1985; see the extended discussion in Wiltshire 1986; Boulware & Deser 1985; Wheeler 1986 for more information) as string theory yields additional higher-order curvature correction terms to the Einstein action (Callan et al. 1985). Interestingly, the EGB Lagrangian is a linear combination of Euler densities continued from lower dimensions, and has been widely studied in astrophysics and cosmology. EGB theory, which contains quadratic powers of the curvature, is a special case of Lovelock's theory of gravitation (Lovelock 1971, 1972) and is free of ghosts. In 4D spacetime EGB and GR are equivalent, as the Gauss–Bonnet (GB) term does not make any contribution to the dynamical equations.

Following recent theoretical developments, Glavan and Lin (2020) proposed a four-dimensional (4D) EGB gravity theory by rescaling the coupling constant α → α/(D − 4) and then, taking the limit D → 4, a nontrivial black hole solution was found. Guo & Li (2020) have studied geodesic motions of timelike and null particles in the spacetime of a spherically symmetric 4D EGB black hole. It was suggested that one can bypass Lovelock's theorem, and the GB term gives rise to a nontrivial contribution to the gravitational dynamics. Moreover, this theory is free from the Ostrogradsky instability with the following rescaling. However, it seems that the regularization procedure was originally traced back to Tomozawa (1986) with finite one-loop quantum corrections to Einstein gravity. One can say that this interesting proposal has opened up a new window for several novel predictions, though the validity of this theory is at present under debate. The spherically symmetric black hole solutions and their physical properties have been discussed (Glavan & Lin 2020) and are claimed to differ from the standard vacuum-GR Schwarzschild black hole. In the same framework, a static and spherically symmetric GB black hole was used to reveal many interesting features (for a review, see, for instance, Ghosh & Kumar 2020; Konoplya & Zhidenko 2020; Kumar & Ghosh 2020; Kumar & Kumar 2020; Liu et al. 2021; Zhang et al. 2020b; Wei & Liu 2020). In Kumar & Ghosh (2020) and Naveena et al. (2020) a rotating black hole solution was found using the Newman–Janis algorithm. They showed that the rotating black hole has an additional GB parameter α over the Kerr black hole, and it produces a deviation from Kerr geometry. However, it is well known (Hansen & Yunes 2013) that the Newmann–Janis trick is not generally applicable in higher-curvature theories. Thus, a rotating solution still remains to be found in 4D EGB gravity. Alongside this, geodesic motion and shadow (Zeng et al. 2020), the strong/ weak gravitational lensing by black hole (Heydari-Fard et al. 2020; Islam et al. 2020; Jin et al. 2020; Kumar et al. 2020), spinning test particles (Zhang et al. 2020c), thermodynamic anti-de Sitter (AdS) black holes (Ali & Mansoori 2021), Hawking radiation (Konoplya & Zinhailo 2020; Zhang et al. 2020a), quasinormal modes (Aragon et al. 2020; Churilova 2021; Mishra 2020), and wormhole solutions (Jusufi et al. 2020; Liu et al. 2020), have been extensively analyzed and attracted a great deal of recent attention (see Jusufi 2020; Ma & Lu 2020; Yang et al. 2020a, 2020b for more details). More recently, the study of the possible existence of a thermal phase transition between AdS to dS asymptotic geometries in vacuum in the context of novel 4D EGB gravity has been proposed in Samart & Channuie (2020).

However, there are several works (Ali 2020; Arrechea et al. 2021; Gurses et al. 2020; Mahapatra 2020) debating that the procedure of taking the D → 4 limit in Glavan & Lin (2020) may not be consistent. Let us mention a few examples. It was shown in Gurses et al. (2020) that there exist no 4D equations of motion constructed from the metric alone that could serve as the equations of motion for such a theory. The 4D theory must introduce additional degrees of freedom. In any case, it cannot be a pure metric theory of gravity. Moreover, from the perspective of scattering amplitudes, it was demonstrated that the limit leads to an additional scalar degree of freedom, confirming the previous analysis. Taking these issues together, the approach proposed by Glavan and Lin may be incomplete. In other words, a description of the extra degree of freedom is required. Since then, several regularization schemes, e.g., see Lu & Pang (2020), Kobayashi (2020), and Fernandes et al. (2020), have been proposed to overcome these shortcomings. In fact, Lu & Pang (2020) have shown that the Kaluza–Klein approach of the D → 4 limit leads to a class of scalar–tensor theory that belongs to the Horndeski class. Thus, it is important to check the equivalence of the actions in the regularized and novel 4D EGB theory for a particular kind of static spherically symmetric spacetime.

Nevertheless, 4D EGB gravity has received significant attention, including finding astrophysical solutions and investigating their properties. In particular, the mass–radius relations have been obtained for realistic hadronic and for strange quark star equations of state (EoSs; Doneva & Yazadjiev 2020). Specifically, we are interested investigating the behavior of compact stars, namely strange quark stars in regularized 4D EGB gravity. Matter at densities exceeding that of nuclear matter will have to be discussed in terms of quarks. As mentioned in Haensel et al. (1986), for quark matter models massive neutron stars may exist in the form of strange quark stars. Usually the quark matter phase is modeled in the context of the MIT bag model as a Fermi gas of u, d, and s quarks. At finite densities and zero or low temperature, quark matter can exhibit substantial rich phase structures resulting from different pairing mechanisms due to the coupling of color, flavor, and spin degrees of freedom (see e.g., Rajagopal & Wilczek 2001; Alford & Rajagopal 2002; Alford et al. 2007). In addition, a variety of different condensates underlying fundamental descriptions may be plausible.

An expectation is that quark matter might play an important role in cosmology and astrophysics (see, e.g., Alford 2004; Farhi & Jaffe 1984). In cosmology, it may provide an explanation of a source of density fluctuation and, as a consequence, of how galaxies form generated by the quark–hadron transition. In astrophysics, quark matter is an interplay between GR effects and the EoS of nuclear particle physics. These objects are present in the form of the stellar equilibrium including neutron stars with a quark core, supermassive stars, white dwarfs, and even strange quark stars. Nevertheless, in all possible applications of quark matter from cosmology and astrophysics, our lack of knowledge of the exact equation poses the main source of uncertainty in describing stars. In order to study the stable/unstable configurations and even other physical properties of stars, realistic EoSs have to be proposed. The color–flavor-locked (CFL) phase appearing in three-flavor (up, down, strange) matter suggests the importance of condensates (Alford et al. 2001; Rajagopal & Wilczek 2001; Lugones & Horvath 2002; Steiner et al. 2002) and is shown to be the asymptotic ground state of quark matter at low temperature (Schäfer & Wilczek 1999). For instant, Banerjee & Singh (2020) studied a class of static and spherically symmetric compact objects made of strange matter in the CFL phase in 4D EGB gravity.

The structure of the present work is as follows: we briefly review how to derive the field equations in the context of 4D EGB gravity in Section 2 and show that they make a nontrivial contribution to gravitational dynamics in 4D in Section 3. In Section 4 we discuss a class of static and spherically symmetric compact objects invoking the EoS parameters in quark matter phases using massless quark and cold star approximations. In Section 5 we discuss the numerical procedure used to solve the field equations. In the same section, we report the general properties of the spheres in terms of the massless quark and cold star approximations. We analyze the energy conditions as well as other properties of the spheres, such as sound velocity and adiabatic stability. We conclude our findings in Section 6.

In this section, we will give a short recap of the regularization technique developed in Fernandes et al. (2020), and apply it to the novel 4D EGB theory to find the regularized action. This regularization method leads to a well defined action which is free from divergences, and produces well behaved second-order field equations. The general action in the framework of GB theory in D-dimensional spacetime is given by

$$\begin{eqnarray}&&S=\displaystyle \frac{{c}^{4}}{16\pi {G}_{D}}{\int }_{{ \mathcal M }}{d}^{D}x\sqrt{-g}\left(R+\hat{\alpha }{{ \mathcal L }}_{\mathrm{GB}}\right)+{{ \mathcal S }}_{\mathrm{matter}},\end{eqnarray} \tag{ 1 }$$

where g denotes the determinant of the metric g_{μ ν} and $\hat{\alpha }$ is the GB coupling constant. ${{ \mathcal S }}_{\mathrm{matter}}$ is the action of the standard perfect fluid matter and the GB term is

$$\begin{eqnarray}&&{{ \mathcal L }}_{\mathrm{GB}}={R}^{2}-4{R}_{\mu \nu }{R}^{\mu \nu }+{R}_{\mu \nu \alpha \beta }{R}^{\mu \nu \alpha \beta }\,.\end{eqnarray} \tag{ 2 }$$

Varying the action (9) results in the following equations of motion:

$$\begin{eqnarray}&&\begin{array}{l}{G}_{\mu \nu }+\hat{\alpha }{H}_{\mu \nu }=\displaystyle \frac{8\pi G}{{c}^{4}}{T}_{\mu \nu }\\ \quad \mathrm{where}\,{T}_{\mu \nu }=-\displaystyle \frac{2}{\sqrt{-g}}\displaystyle \frac{\delta \left(\sqrt{-g}{{ \mathcal S }}_{m}\right)}{\delta {g}^{\mu \nu }},\end{array}\end{eqnarray} \tag{ 3 }$$

where G_{μ ν} is the Einstein tensor and H_{μ ν} is a tensor carrying the contributions from the GB term, which yield

$$\begin{eqnarray}\begin{array}{rcl}{G}_{\mu \nu } & = & {R}_{\mu \nu }-\displaystyle \frac{1}{2}R\,{g}_{\mu \nu },\\ {H}_{\mu \nu } & = & 2\left({{RR}}_{\mu \nu }-2{R}_{\mu \sigma }R{{}^{\sigma }}_{\nu }\right.\\ & & \left.-\,2{R}_{\mu \sigma \nu \rho }{R}^{\sigma \rho }+{R}_{\mu \sigma \rho \delta }{R}_{\nu }^{\sigma \rho \delta }\right)-\displaystyle \frac{1}{2}\,{g}_{\mu \nu }\,{{ \mathcal L }}_{\mathrm{GB}},\end{array}\end{eqnarray} \tag{ 4 }$$

where R_{μ ν} is the Ricci tensor, R_{μ σ ν ρ} is the Riemann tensor, and R is the Ricci scalar. Note that for D > 4, the equation of motion (3) is the well-known EGB theory. However, in D = 4 the GB term vanishes identically and hence the field Equations (3) reduce to the Einstein theory. However, if $\hat{\alpha }$ is rescaled as $\hat{\alpha }\to \tfrac{\alpha }{D-4}$ and taking the limit D → 4, the GB gives nontrivial contributions to a well defined action principle in four dimensions. In what follows, we will show how this method works for static and spherically symmetric spacetime and then investigate contributions of the GB term to a compact stellar object.

For the stellar configurations, we assume that the energy–momentum tensor T_{μ ν} is a perfect fluid matter source:

$$\begin{eqnarray}&&{T}_{\mu \nu }=(\epsilon +P){u}_{\nu }{u}_{\nu }+{{Pg}}_{\nu \nu },\end{eqnarray} \tag{ 5 }$$

where P = P(r) is the pressure,  ≡ (r) is the energy density of matter, and u_ν is a D-velocity. Here, we consider the D-dimensional spherically symmetric metric anstaz describing the interior of the star:

$$\begin{eqnarray}&&{{ds}}_{D}^{2}=-W(r){{dt}}^{2}+H(r){{dr}}^{2}+{r}^{2}d{{\rm{\Omega }}}_{D-2}^{2},\end{eqnarray} \tag{ 6 }$$

where $d{{\rm{\Omega }}}_{D-2}^{2}$ is the metric on the unit (D − 2)-dimensional sphere and W = W(r) and H = H(r) are functions of radial coordinate r. In the limit D → 4, considering the metric (6) and Equation (5) for the perfect fluid we obtain the components of (t,t) and (r,r) in the following forms:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{8\pi G}{{c}^{4}}\epsilon & = & \displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{1}{{rH}}\left(\displaystyle \frac{H^{\prime} }{H}-\displaystyle \frac{1}{r}\right)\\ & & -\,\displaystyle \frac{\alpha (H-1)}{{r}^{4}{H}^{3}}\left({H}^{2}-H-2H^{\prime} r\right),\end{array}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{8\pi G}{{c}^{4}}P & = & -\displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{1}{{rH}}\left(\displaystyle \frac{1}{r}+\displaystyle \frac{W^{\prime} }{W}\right)\\ & & +\,\displaystyle \frac{\alpha (H-1)}{{r}^{4}{H}^{2}W}\left(W(H-1)+2W^{\prime} r\right),\end{array}\end{eqnarray} \tag{ 8 }$$

where primes denote derivative with respect to r. This new theory has stimulated a series of research works concerning cosmological as well as astrophysical solutions, even though there are several criticisms against the model, including the fact that the proposed rescaling substitutes a vanishing factor with an undetermined one (see, for example, Ali 2020; Fernandes et al. 2020; Gurses et al. 2020; Hennigar et al. 2020; Kobayashi 2020; Lu & Pang 2020; Shu 2020). In addition, several alternative regularization methods have been proposed including the Kaluza–Klein reduction procedure (Lu & Pang 2020; Kobayashi 2020), the conformal subtraction procedure (Fernandes et al. 2020; Hennigar et al. 2020), and Arnowitt–Deser–Misner decomposition analysis (Aoki et al. 2020). Thus, the regularization scheme is not unique. We follow the approach as proposed in Fernandes et al. (2020), Lu & Pang (2020), and Yang et al. (2020a) using the form

$$\begin{eqnarray}&&\begin{array}{l}S=\displaystyle \frac{{c}^{4}}{16\pi G}{\displaystyle \int }_{{ \mathcal M }}{{\rm{d}}}^{4}x\sqrt{-g}\left[R+\alpha \left(4{G}^{\mu \nu }{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}_{\nu }\phi \right.\right.\\ \quad \left.-\,\phi {{ \mathcal L }}_{\mathrm{GB}}+4\square \phi {\left({\rm{\nabla }}\phi \right)}^{2}+2{\left({\rm{\nabla }}\phi \right)}^{4}\right]+{{ \mathcal S }}_{\mathrm{matter}},\end{array}\end{eqnarray} \tag{ 9 }$$

which can be seen to be free of divergences, and ϕ is a scalar function of the spacetime coordinates. Interestingly, the scalar acts as a Lagrange multiplier in the action, allowing the GB term itself to appear in the 4D field equations. Hence, we clearly see that the GB term does not vanish in the limit D = 4, and it has an effect on gravitational dynamics in 4D. Moreover, the action (9) belongs to a subclass of Horndeski gravity (Horndeski 1974; Kobayashi 2019) with G₂ = 8α X² − 2Λ₀, G₃ = 8α X, G₄ = 1 + 4α X and ${G}_{5}=4\alpha \mathrm{ln}X$ (where $X=-\tfrac{1}{2}{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}^{\mu }\phi$). Namely, the idea is to consider a Kaluza–Klein ansatz (Lu & Pang 2020; Kobayashi 2020)

$$\begin{eqnarray}&&{{ds}}_{D}^{2}={{ds}}_{4}^{2}+{e}^{2\phi }d{{\rm{\Omega }}}_{D-4}^{2},\end{eqnarray} \tag{ 10 }$$

or conformal subtraction (Hennigar et al. 2020; Fernandes et al. 2020), where the subtraction background is defined under a conformal transformation ${\tilde{g}}_{{ab}}={e}^{2\phi }{g}_{{ab}}$ and a counterterm, i.e., $-\alpha {\int }_{{ \mathcal M }}{{\rm{d}}}^{4}\sqrt{-\tilde{g}}\,\tilde{{ \mathcal G }}$ is added to the original action (Fernandes et al. 2020). In the above expression $d{{\rm{\Omega }}}_{D-4}^{2}$ is the line element on the internal maximally symmetric space and ϕ an additional metric function that depends only on the external D-dimensional coordinates. Now, varying the action (9), one can obtain the gravitational field equations of the regularized 4D EGB theory as (Fernandes et al. 2020)

$$\begin{eqnarray}&&{G}_{\mu \nu }=\alpha {\hat{{ \mathcal H }}}_{\mu \nu }+\displaystyle \frac{8\pi G}{{c}^{4}}{T}_{\mu \nu },\end{eqnarray} \tag{ 11 }$$

where ${\hat{{ \mathcal H }}}_{\mu \nu }$ is defined by

$$\begin{eqnarray}\begin{array}{rcl}{\hat{{ \mathcal H }}}_{\mu \nu } & = & 2R\left({{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }\phi -{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}_{\nu }\phi \right)\\ & & +\,2{G}_{\mu \nu }\left({\left({\rm{\nabla }}\phi \right)}^{2}-2\square \phi \right)\\ & & +\,4{G}_{\nu \alpha }\left({{\rm{\nabla }}}^{\alpha }{{\rm{\nabla }}}_{\mu }\phi -{{\rm{\nabla }}}^{\alpha }\phi {{\rm{\nabla }}}_{\mu }\phi \right)\\ & & +\,4{G}_{\mu \alpha }\left({{\rm{\nabla }}}^{\alpha }{{\rm{\nabla }}}_{\nu }\phi -{{\rm{\nabla }}}^{\alpha }\phi {{\rm{\nabla }}}_{\nu }\phi \right)\\ & & +\,4{R}_{\mu \alpha \nu \beta }\left({{\rm{\nabla }}}^{\beta }{{\rm{\nabla }}}^{\alpha }\phi -{{\rm{\nabla }}}^{\alpha }\phi {{\rm{\nabla }}}^{\beta }\phi \right)\\ & & +\,4{{\rm{\nabla }}}_{\alpha }{{\rm{\nabla }}}_{\nu }\phi \left({{\rm{\nabla }}}^{\alpha }\phi {{\rm{\nabla }}}_{\mu }\phi \right.\\ & & \left.-\,{{\rm{\nabla }}}^{\alpha }{{\rm{\nabla }}}_{\mu }\phi \right)+4{{\rm{\nabla }}}_{\alpha }{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}^{\alpha }\phi {{\rm{\nabla }}}_{\nu }\phi \\ & & -\,4{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}_{\nu }\phi \left({\left({\rm{\nabla }}\phi \right)}^{2}+\square \phi \right)\\ & & +4\square \phi {{\rm{\nabla }}}_{\nu }{{\rm{\nabla }}}_{\mu }\phi -{g}_{\mu \nu }\left[2R\left(\square \phi -{\left({\rm{\nabla }}\phi \right)}^{2}\right)\right.\\ & & +\,4{G}^{\alpha \beta }\left({{\rm{\nabla }}}_{\beta }{{\rm{\nabla }}}_{\alpha }\phi -{{\rm{\nabla }}}_{\alpha }\phi {{\rm{\nabla }}}_{\beta }\phi \right)+2{\left(\square \phi \right)}^{2}\\ & & -\left.\,{\left({\rm{\nabla }}\phi \right)}^{4}+2{{\rm{\nabla }}}_{\beta }{{\rm{\nabla }}}_{\alpha }\phi \left(2{{\rm{\nabla }}}^{\alpha }\phi {{\rm{\nabla }}}^{\beta }\phi -{{\rm{\nabla }}}^{\beta }{{\rm{\nabla }}}^{\alpha }\phi \right)\right].\end{array}\end{eqnarray} \tag{ 12 }$$

and by varying with respect to the scalar field, we get

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{8}{{ \mathcal L }}_{\mathrm{GB}} & = & {R}^{\mu \nu }{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}_{\nu }\phi -{G}^{\mu \nu }{{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }\phi \\ & & -\,\square \phi {\left({\rm{\nabla }}\phi \right)}^{2}+{\left({{\rm{\nabla }}}_{\mu }{{\rm{\nabla }}}_{\nu }\phi \right)}^{2}\\ & & -\,{\left(\square \phi \right)}^{2}-2{{\rm{\nabla }}}_{\mu }\phi {{\rm{\nabla }}}_{\nu }\phi {{\rm{\nabla }}}^{\mu }{{\rm{\nabla }}}^{\nu }\phi .\end{array}\end{eqnarray} \tag{ 13 }$$

The trace of the field Equation (11) is found to satisfy

$$\begin{eqnarray}&&R+\displaystyle \frac{\alpha }{2}{{ \mathcal L }}_{\mathrm{GB}}=-\displaystyle \frac{8\pi G}{{c}^{4}}T.\end{eqnarray} \tag{ 14 }$$

Fernandes et al. (2020) argued that the trace of the field equation is in exactly same form as in the original 4D EGB theory. In continuation with the above it is also argued that there may be a hidden scalar degree of freedom in the original theory. Note that when ${{ \mathcal L }}_{\mathrm{GB}}=0$, the scalar field, Equation (13), can be seen to be exactly equivalent, which means that the counterterm added to the action must vanish on-shell. In other words, an on-shell action is identical to the action of the original theory, and thus classical evolution of the gravity–matter system is independent of the hidden scalar field (Fernandes et al. 2020). This claim needs to be scrutinized carefully, as we do here.

With the spherical coordinates (6), the nonvanishing components of the gravitational field equations (Fernandes et al. 2020) are written in terms of metric components:

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{8\pi G}{{c}^{4}}\epsilon & = & \displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{1}{{rH}}\left(\displaystyle \frac{H^{\prime} }{H}-\displaystyle \frac{1}{r}\right)\\ & & +\,\alpha \left(\displaystyle \frac{6H^{\prime} \phi ^{\prime} }{{r}^{2}{H}^{3}}-\displaystyle \frac{2H^{\prime} \phi ^{\prime} }{{r}^{2}{H}^{2}}+\displaystyle \frac{H^{\prime} W^{\prime} \phi {{\prime} }^{2}}{{H}^{3}W}\right.\\ & & +\,\displaystyle \frac{2H^{\prime} \phi {{\prime} }^{3}}{{H}^{3}}+\displaystyle \frac{6H^{\prime} \phi {{\prime} }^{2}}{{{rH}}^{3}}+\displaystyle \frac{4\phi ^{\prime\prime} }{{r}^{2}H}-\displaystyle \frac{4\phi ^{\prime\prime} }{{r}^{2}{H}^{2}}\\ & & -\,\displaystyle \frac{2\phi {{\prime} }^{2}}{{r}^{2}H}-\displaystyle \frac{2\phi {{\prime} }^{2}}{{r}^{2}{H}^{2}}-\displaystyle \frac{4W^{\prime} \phi {{\prime} }^{2}}{{{rH}}^{2}W}-\displaystyle \frac{W{{\prime} }^{2}\phi {{\prime} }^{2}}{{H}^{2}{W}^{2}}\\ & & \left.-\,\displaystyle \frac{2W^{\prime} \phi ^{\prime} \phi ^{\prime\prime} }{{H}^{2}W}+\displaystyle \frac{\phi {{\prime} }^{4}}{{H}^{2}}-\displaystyle \frac{4\phi {{\prime} }^{2}\phi ^{\prime\prime} }{{H}^{2}}-\displaystyle \frac{8\phi ^{\prime} \phi ^{\prime\prime} }{{{rH}}^{2}}\right),\end{array}\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{8\pi G}{{c}^{4}}P & = & -\displaystyle \frac{1}{{r}^{2}}+\displaystyle \frac{1}{{rH}}\left(\displaystyle \frac{1}{r}+\displaystyle \frac{W^{\prime} }{W}\right)\\ & & +\,\alpha \left(\displaystyle \frac{2H^{\prime} \phi {{\prime} }^{3}}{{H}^{3}}-\displaystyle \frac{H^{\prime} W^{\prime} \phi {{\prime} }^{2}}{{H}^{3}W}+\displaystyle \frac{H{{\prime} }^{2}\phi {{\prime} }^{2}}{{H}^{4}}-\displaystyle \frac{4H^{\prime} \phi {{\prime} }^{2}}{{{rH}}^{3}}\right.\\ & & -\,\displaystyle \frac{4H^{\prime} \phi ^{\prime} \phi ^{\prime\prime} }{{H}^{3}}+\displaystyle \frac{6W^{\prime} \phi ^{\prime} }{{r}^{2}{H}^{2}W}-\displaystyle \frac{2W^{\prime} \phi ^{\prime} }{{r}^{2}{HW}}\\ & & -\,\displaystyle \frac{2\phi {{\prime} }^{2}}{{r}^{2}H}+\displaystyle \frac{6\phi {{\prime} }^{2}}{{r}^{2}{H}^{2}}+\displaystyle \frac{6W^{\prime} \phi {{\prime} }^{2}}{{{rH}}^{2}W}+\displaystyle \frac{2W^{\prime} \phi ^{\prime} \phi ^{\prime\prime} }{{H}^{2}W}\\ & & \left.+\,\displaystyle \frac{4{\phi }^{{\prime\prime} 2}}{{H}^{2}}+\displaystyle \frac{3{\phi }^{{\prime} 4}}{{H}^{2}}-\displaystyle \frac{4{\phi }^{{\prime} 2}\phi ^{\prime\prime} }{{H}^{2}}+\displaystyle \frac{8\phi ^{\prime} \phi ^{\prime\prime} }{{{rH}}^{2}}\right).\end{array}\end{eqnarray} \tag{ 16 }$$

To prove an equivalence between the two theories, one must quantify the solution of ϕ, making both theories have the same solution. Thus it is convenient to eliminate the matter part by subtracting Equations (7) and (15) and Equations (8) and (16). The resultant field equations read

$$\begin{eqnarray}\begin{array}{rcl}0 & = & \displaystyle \frac{2H^{\prime} }{{r}^{3}{H}^{3}}-\displaystyle \frac{2H^{\prime} }{{r}^{3}{H}^{2}}+\varphi \left(\displaystyle \frac{6H^{\prime} }{{r}^{2}{H}^{3}}-\displaystyle \frac{2H^{\prime} }{{r}^{2}{H}^{2}}\right)\\ & & +\,{\varphi }^{2}\left(\displaystyle \frac{H^{\prime} W^{\prime} }{{H}^{3}W}+\displaystyle \frac{6H^{\prime} }{{{rH}}^{3}}-\displaystyle \frac{2}{{r}^{2}H}\right.\\ & & \left.-\,\displaystyle \frac{2}{{r}^{2}{H}^{2}}-\displaystyle \frac{W{{\prime} }^{2}}{{H}^{2}{W}^{2}}-\displaystyle \frac{4W^{\prime} }{{{rH}}^{2}W}\right)\\ & & +\,\displaystyle \frac{2{\varphi }^{3}H^{\prime} }{{H}^{3}}-\displaystyle \frac{2}{{r}^{4}H}+\displaystyle \frac{1}{{r}^{4}{H}^{2}}+\varphi ^{\prime} \left(\displaystyle \frac{4}{{r}^{2}H}-\displaystyle \frac{4}{{r}^{2}{H}^{2}}\right.\\ & & \left.-\,\displaystyle \frac{4{\varphi }^{2}}{{H}^{2}}-\varphi \left[\displaystyle \frac{2W^{\prime} }{{H}^{2}W}+\displaystyle \frac{8}{{{rH}}^{2}}\right]\right)+\displaystyle \frac{{\varphi }^{4}}{{H}^{2}}+\displaystyle \frac{1}{{r}^{4}},\end{array}\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}\begin{array}{rcl}0 & = & {\varphi }^{2}\left(\displaystyle \frac{H{{\prime} }^{2}}{{H}^{4}}-\displaystyle \frac{H^{\prime} W^{\prime} }{{H}^{3}W}-\displaystyle \frac{4H^{\prime} }{{{rH}}^{3}}-\displaystyle \frac{2}{{r}^{2}H}+\displaystyle \frac{6}{{r}^{2}{H}^{2}}+\displaystyle \frac{6W^{\prime} }{{{rH}}^{2}W}\right)\\ & & +\,\displaystyle \frac{2{\varphi }^{3}H^{\prime} }{{H}^{3}}+\varphi ^{\prime} \left(\varphi \left[\displaystyle \frac{2W^{\prime} }{{H}^{2}W}-\displaystyle \frac{4H^{\prime} }{{H}^{3}}+\displaystyle \frac{8}{{{rH}}^{2}}\right]-\displaystyle \frac{4{\varphi }^{2}}{{H}^{2}}\right)\\ & & +\,\displaystyle \frac{2}{{r}^{4}H}-\displaystyle \frac{1}{{r}^{4}{H}^{2}}+\displaystyle \frac{2W^{\prime} }{{r}^{3}{H}^{2}W}-\displaystyle \frac{2W^{\prime} }{{r}^{3}{HW}}\\ & & +\,\varphi \left(\displaystyle \frac{6W^{\prime} }{{r}^{2}{H}^{2}W}-\displaystyle \frac{2W^{\prime} }{{r}^{2}{HW}}\right)+\displaystyle \frac{4\varphi {{\prime} }^{2}}{{H}^{2}}+\displaystyle \frac{3{\varphi }^{4}}{{H}^{2}}-\displaystyle \frac{1}{{r}^{4}},\end{array}\end{eqnarray} \tag{ 18 }$$

where the new variable, φ, is defined as $\varphi =\phi ^{\prime}$ and $\varphi ^{\prime} =\phi ^{\prime\prime}$. To eliminate $\varphi ^{\prime}$ from Equations (17) and (18), we rewrite the above expressions as

$$\begin{eqnarray}\begin{array}{rcl}0 & = & \left(\sqrt{H}-r\varphi -1\right)\left({r}^{3}{HW}{{\prime} }^{2}\varphi -{rW}^{\prime} W\right.\\ & & \times \,\left[H^{\prime} {r}^{2}\varphi +2{H}^{3/2}r\varphi +2{H}^{2}-2H(3r\varphi +1)\right]\\ & & +\,\left(\sqrt{H}+r\varphi +1\right){W}^{2}\\ & & \times \,\left[{H}^{2}-8{\sqrt{H}}^{3}+H\left(2r\varphi -{r}^{2}{\varphi }^{2}+7\right)\right.\\ & & \left.\left.+\,2H^{\prime} \sqrt{H}r-2H^{\prime} r(r\varphi +2)\right]\right),\end{array}\end{eqnarray} \tag{ 19 }$$

$$\begin{eqnarray}\begin{array}{rcl}0 & = & \displaystyle \frac{H-{\left(r\varphi +1\right)}^{2}}{{\left(W^{\prime} {r}^{2}\varphi +W\left[2{\left(r\varphi +1\right)}^{2}-2H\right]\right)}^{2}}\\ & & \times \,\left({W}^{3}\left[H-{\left(r\varphi +1\right)}^{2}\right]\left[{H}^{2}\left(4H^{\prime} r-9{r}^{4}{\varphi }^{4}\right.\right.\right.\\ & & -\left.\,12{r}^{3}{\varphi }^{3}-10{r}^{2}{\varphi }^{2}-4r\varphi +3\right)-4H{{\prime} }^{2}{r}^{2}\\ & & +\,4H^{\prime} {Hr}\left({r}^{2}{\varphi }^{2}-2r\varphi -1\right)\\ & & \left.+3{H}^{4}+{H}^{3}\left(6{r}^{2}{\varphi }^{2}+4r\varphi -6\right)\right]\\ & & +\,2{r}^{3}{HW}{{\prime} }^{2}W\varphi \left[+\left(3{r}^{3}{\varphi }^{3}+6{r}^{2}{\varphi }^{2}\right.\right.\\ & & +\,\left.\left.11r\varphi +4\right)-H^{\prime} {r}^{2}\varphi -{H}^{2}(3r\varphi +4)\right]\\ & & +\,2{rHW}^{\prime} {W}^{2}\left[H\left(6{r}^{5}{\varphi }^{5}+20{r}^{4}{\varphi }^{4}+33{r}^{3}{\varphi }^{3}\right.\right.\\ & & -\,\left.2{r}^{2}\varphi (H^{\prime} -19\varphi )+19r\varphi +4\right)+2H^{\prime} {r}^{2}\varphi +{H}^{3}(3r\varphi +4)\\ & & -\left.\left.\,{H}^{2}\left(9{r}^{3}{\varphi }^{3}+22{r}^{2}{\varphi }^{2}+22r\varphi +8\right)\right]+2{r}^{5}{H}^{2}W{{\prime} }^{3}{\varphi }^{2}\right).\end{array}\end{eqnarray} \tag{ 20 }$$

In doing so so, one may see that the above field Equations (19) and (20) are identities when $\left(\sqrt{H}-r\varphi -1\right)$ equals zero. Obviously, the above scalar field equation is an identity when the scalar field φ satisfies the relation

$$\begin{eqnarray}&&\varphi =\displaystyle \frac{1}{r}\left(\sqrt{H}-1\right).\end{eqnarray} \tag{ 21 }$$

Thus, the moral of the story is that the two 4D EGB theories are equivalent by substituting the solution of φ(r) in (21) in the field equation in (11) in the static spherically symmetric spacetime (6).

In the previous section we gave a recap of regularized 4D EGB gravity, and showed that regularized 4D EGB theory is equivalent to the original one in a spherically symmetric spacetime. Thus we anticipate the use of novel 4D EGB gravity to avoid an impasse in this work. Here, we begin by assuming the general action of EGB gravity in D dimensions and deriving the equations of motion for the underlying theory. For the moment we take the action as

$$\begin{eqnarray}&&{{ \mathcal I }}_{G}=\displaystyle \frac{{c}^{4}}{16\pi G}\int {d}^{D}x\sqrt{-g}\left[R+\alpha {{ \mathcal L }}_{\mathrm{GB}}\right]+{{ \mathcal S }}_{\mathrm{matter}},\end{eqnarray} \tag{ 22 }$$

where all the notations and symbols have their usual meanings, and with the EGB Lagrangian defined as ${{ \mathcal L }}_{\mathrm{GB}}$ in Equation (11). The corresponding field equations can be derived by varying the action with respect to the metric tensor g_{μ ν} , which is exactly same as Equation (3). It is interesting to note that the trace of Equation (3) is

$$\begin{eqnarray}\begin{array}{rcl}{g}^{\mu \nu }\left({G}_{\mu \nu }+\alpha {H}_{\mu \nu }\right) & = & -R+\alpha \left(2D-8\right){R}_{\mu \nu }{R}^{\mu \nu }\\ & & +\,\alpha \left(2-D/2\right){R}^{2}+\alpha \left(2-D/2\right){R}_{\mu \nu \sigma \rho }{R}^{\mu \nu \sigma \rho }\\ & = & -R-\displaystyle \frac{\alpha (D-4)}{2}{{ \mathcal L }}_{\mathrm{GB}}.\end{array}\end{eqnarray} \tag{ 23 }$$

It has thus been argued that, for D = 4, the GB term has no effect on the gravitational dynamics. However, rescaling the GB dimensional coupling constant α according to α → α/(D − 4), the trace of the field Equation (3) yields

$$\begin{eqnarray}&&R+\displaystyle \frac{\alpha }{2}{{ \mathcal L }}_{\mathrm{GB}}=-\displaystyle \frac{8\pi G}{{c}^{4}}T,\end{eqnarray} \tag{ 24 }$$

which is exactly the same form as the trace of the field equations obtained from regularizing the 4D EGB theory. In this way, the GB term can yield a nontrivial contribution to the gravitational dynamics even in 4D. Thus, we can conclude that the two 4D EGB theories are equivalent in a static spherically symmetric spacetime. In the same way, Lin et al. (2020) have demonstrated the equivalence of these two theories in a cylindrically symmetric spacetime. For a complete description of the compact star, we use the regularization process (see Cognola et al. 2013; Glavan & Lin 2020) in which the spherically symmetric solutions are exactly same as those of other regularized theories (Casalino et al. 2021; Hennigar et al. 2020; Lu & Pang 2020; Ma & Lu 2020).

The line element of the static and spherically symmetric metric describing a stellar structure in 4D EGB theory has the following form:

$$\begin{eqnarray}&&{{ds}}^{2}=-{e}^{2{\rm{\Phi }}(r)}{c}^{2}{{dt}}^{2}+{e}^{2{\rm{\Lambda }}(r)}{{dr}}^{2}+{r}^{2}d{{\rm{\Omega }}}^{2}.\end{eqnarray} \tag{ 25 }$$

The above line element is equivalent to that in (6) for e^2Φ(r) c² = W(r) and e^2Λ(r) = H(r). Finally, the Tolman–Oppenheimer–Volkoff (TOV) equations for this theory of gravity are nothing other than (tt), (rr), and the hydrostatic continuity Equation (3) yields

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{2}{r}\displaystyle \frac{d{\rm{\Lambda }}}{{dr}} & = & {e}^{2{\rm{\Lambda }}}\,\left[\displaystyle \frac{8\pi G}{{c}^{4}}\epsilon -\displaystyle \frac{1-{e}^{-2{\rm{\Lambda }}}}{{r}^{2}}\right.\\ & & \times \,\left.\left(1-\displaystyle \frac{\alpha (1-{e}^{-2{\rm{\Lambda }}})}{{r}^{2}}\right)\right]{\left[1+\displaystyle \frac{2\alpha (1-{e}^{-2{\rm{\Lambda }}})}{{r}^{2}}\right]}^{-1},\end{array}\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{2}{r}\displaystyle \frac{d{\rm{\Phi }}}{{dr}} & = & {e}^{2{\rm{\Lambda }}}\,\left[\displaystyle \frac{8\pi G}{{c}^{4}}P+\displaystyle \frac{1-{e}^{-2{\rm{\Lambda }}}}{{r}^{2}}\right.\\ & & \left.\times \,\left(1-\displaystyle \frac{\alpha (1-{e}^{-2{\rm{\Lambda }}})}{{r}^{2}}\right)\right]{\left[1+\displaystyle \frac{2\alpha (1-{e}^{-2{\rm{\Lambda }}})}{{r}^{2}}\right]}^{-1},\end{array}\end{eqnarray} \tag{ 27 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{dP}}{{dr}}=-(\epsilon +P)\displaystyle \frac{d{\rm{\Phi }}}{{dr}}.\end{eqnarray} \tag{ 28 }$$

As usual, the asymptotic flatness imposes Φ( ∞ ) = Λ(∞) = 0 while the regularity at the center requires Λ(0) = 0.

It is advantageous to define the gravitational mass within a sphere of radius r, such that ${e}^{-2{\rm{\Lambda }}}=1-\tfrac{2{Gm}(r)}{{c}^{2}r}$. Now, we are ready to write the TOV equations in a form we wish to use. So, using (27) and (28), we obtain the modified TOV equations as

$$\begin{eqnarray}&&\displaystyle \frac{{dP}}{{dr}}=-\displaystyle \frac{G\epsilon (r)m(r)}{{c}^{2}{r}^{2}}\displaystyle \frac{\left[1+\tfrac{P(r)}{\epsilon (r)}\right]\left[1+\tfrac{4\pi {r}^{3}P(r)}{{c}^{2}m(r)}-\tfrac{2G\alpha m(r)}{{c}^{2}{r}^{3}}\right]}{\left[1+\tfrac{4G\alpha m(r)}{{c}^{2}{r}^{3}}\right]\left[1-\tfrac{2{Gm}(r)}{{c}^{2}r}\right]}.\end{eqnarray} \tag{ 29 }$$

If we take the limit α → 0, the above equation reduces to the standard TOV equation of GR. Replacing the last equality in Equation (26), we obtain the gravitational mass:

$$\begin{eqnarray}&&m^{\prime} (r)=\displaystyle \frac{6\alpha {Gm}{\left(r\right)}^{2}+4\pi {r}^{6}\epsilon (r)}{4\alpha {Grm}(r)+{c}^{2}{r}^{4}},\end{eqnarray} \tag{ 30 }$$

using the initial condition m(0) = 0. Then we use the dimensionless variables $P(r)={\epsilon }_{0}\bar{P}(r)$ and $\epsilon (r)={\epsilon }_{0}\bar{\epsilon }(r)$ and $m(r)={M}_{\odot }\bar{M}(r)$, with ₀ = 1 MeV fm⁻³. As a result, the above two equations become

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{d\bar{P}(r)}{{dr}} & = & -\displaystyle \frac{G\bar{\epsilon }(r){M}_{\odot }\bar{M}(r)}{{c}^{2}{r}^{2}}\\ & & \times \,\displaystyle \frac{\left[1+\tfrac{\bar{P}(r)}{\bar{\epsilon }(r)}\right]\left[1+\tfrac{4\pi {r}^{3}{\epsilon }_{0}\bar{P}(r)}{{c}^{2}{M}_{\odot }\bar{M}(r)}-\tfrac{2G\alpha {M}_{\odot }\bar{M}(r)}{{c}^{2}{r}^{3}}\right]}{\left[1+\tfrac{4G\alpha {M}_{\odot }\bar{M}(r)}{{c}^{2}{r}^{3}}\right]\left[1-\tfrac{2{{GM}}_{\odot }\bar{M}(r)}{{c}^{2}r}\right]}\\ & = & -\displaystyle \frac{{c}_{1}\bar{\epsilon }(r)\bar{M}(r)}{{r}^{2}}\displaystyle \frac{\left[1+\tfrac{\bar{P}(r)}{\bar{\epsilon }(r)}\right]\left[1+\tfrac{{c}_{2}{r}^{3}\bar{P}(r)}{\bar{M}(r)}-\tfrac{2{c}_{1}\alpha \bar{M}(r)}{{r}^{3}}\right]}{\left[1+\tfrac{4{c}_{1}\alpha \bar{M}(r)}{{r}^{3}}\right]\left[1-\tfrac{2{c}_{1}\bar{M}(r)}{r}\right]},\end{array}\end{eqnarray} \tag{ 31 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{M}_{\odot }\displaystyle \frac{d\bar{M}(r)}{{dr}} & = & \displaystyle \frac{6\alpha {{GM}}_{\odot }^{2}\bar{M}{\left(r\right)}^{2}+4\pi {r}^{6}{\epsilon }_{0}\bar{\epsilon }(r)}{4\alpha {{GrM}}_{\odot }\bar{M}(r)+{c}^{2}{r}^{4}}\\ \displaystyle \frac{d\bar{M}(r)}{{dr}} & = & \displaystyle \frac{6{c}_{1}\alpha \bar{M}{\left(r\right)}^{2}+{c}_{2}{r}^{6}\bar{\epsilon }(r)}{4{c}_{1}\alpha r\bar{M}(r)+{r}^{4}},\end{array}\end{eqnarray} \tag{ 32 }$$

where ${c}_{1}\equiv \tfrac{{{GM}}_{\odot }}{{c}^{2}}=1.474\ \mathrm{km}$ and ${c}_{2}\equiv \tfrac{4\pi {\epsilon }_{0}}{{M}_{\odot }{c}^{2}}=1.125\times {10}^{-5}\ {\mathrm{km}}^{-3}$. The relationship between mass M and radius R can be straightforwardly illuminated using Equation (32) with a given EoS. Therefore, the final two Equations (31) and (32) can be numerically solved for a given EoS, P = P(). In the next section, we will discuss the strange matter hypothesis.

To understand what kind of matter compact stars may be built up from, we assume an EoS is the most important step, which encompasses all the information regarding the stellar inner structure. Here, we solve the hydrostatic equilibrium Equations (31) and (32) numerically for a specific EoS,  = f(P), where is the energy density and P is the pressure. For each possible EoS, there is a unique family of stars, parametrized by, say, the central density and the central pressure. The standard procedure is to derive the expressions P = f(ρ) and  = g(ρ), with ρ being the baryon density, and then obtain an  − P pair for every value of ρ. Fitting a curve to these data results in the EoS.

4.1. Massless Quark Approximation

Here, we begin with the discussion outlined above. One assumes that the asymptotically free quarks are confined in a finite region of space called a bag. The bag constant B is basically considered as the inward pressure required to confine quarks inside the bag. It is usually given in units of energy per unit volume. In this particular model, we assume that the quark matter distribution is governed by the MIT bag EoS. For simplicity, it is assumed that u, d, and s quarks are noninteracting and massless. Thus, according to the MIT bag model, the quark pressure P is defined as

$$\begin{eqnarray}&&P=\displaystyle \sum _{f=u,d,s}{P}^{f}-B,\end{eqnarray} \tag{ 33 }$$

where P_f is the pressure due to each flavor. The energy density of each flavor _f is related to the corresponding pressure P_f by the relation ${P}_{f}=\tfrac{1}{3}{\epsilon }_{f}$. The energy density due to the quark matter distribution in the MIT bag model is governed by

$$\begin{eqnarray}&&\epsilon =\displaystyle \sum _{f=u,d,s}{\epsilon }_{f}+B\,.\end{eqnarray} \tag{ 34 }$$

Using Equations (33) and (34) with the relation between _f and P_f , we end up with the well-known simplified MIT bag model and the EoS takes the following simple form:

$$\begin{eqnarray}&&\epsilon =3P+4B.\end{eqnarray} \tag{ 35 }$$

What we have to do next is solve three equations with four unknown functions, which are m(r), Φ(r), P(r), and (r). Notice that the EoS for the massless quark approximations explicitly depends on the bag constant B and the pressure P(r). Due to the long-range effects of confinement of quarks, the stability of a strange quark star is essentially determined by the value of B, which can be seen from Figure 1 in the left panel. We then consider the customized TOV equations, Equation (31), and mass function, Equation (32). Mass is measured in solar mass units (M_⊙), radius in km, and energy density and pressure are in MeV fm⁻³. In the present analysis, we treat the values of B and α as free constant parameters, since the parameter B can vary from 57 to 94 MeV fm⁻³ (Witten 1984). For the study of quark matter with massless strange quarks, we consider B = 70 MeV fm⁻³.

Zoom In Zoom Out Reset image size

Given the set of differential Equations (31) and (32) together with the EoS (35), we adopt a numerical approach for integrating and calculating the maximum mass and other properties of the strange quark matter star. To do so, one can consider the boundary conditions P(r₀) = P_c and M(R) = M, and integrate Equation (31) outward to a radius r = R in which fluid pressure P vanishes for P(R) = 0. This leads to the strange star radius R and mass M = m(R). The initial radius r₀ = 10⁻⁵ and mass m(r₀) = 10⁻³⁰ are set to very small numbers rather than zero to avoid discontinuities, as they appear in denominators within the equations.

We start from the center of the star for a certain value of central pressure, P(r₀) = 800 MeV fm⁻³ and the radius of the star is identified when the pressure vanishes or drops to a very small value. For such a choice, we plot pressure and density versus distance from the center of the strange star (see Figure 2). At that point we record the mass–radius (M−R) relation of the star in Figure 3. As one can see, (M − R) depends on the choice of the value of the coupling constant α. For α > 0 the mass of the star for a given radius increases with fixed value of B. In all the presented cases, one can note that there are significant differences for positive and negative values of α, but the case α = 0 is equivalent to pure GR. Moreover, as seen from Table 1 and comparing the results to GR, one may obtain a maximum mass for strange stars with positive α. Therefore, we argue a confirmed determination of a compact star with 2M_⊙, which is actually very close to those of realistic neutron star models (Haensel et al. 1986).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 1. Parameters of Strange Quark Stars Using Values of the 4D EGB Coupling Constant, α

Massless Quark				Cold Star
α	${M}_{\max }$	R	c	${M}_{\max }$	R	c
km2	(M⊙)	(km)	(MeV fm−3)	(M⊙)	(km)	(MeV fm−3)
−5.0	1.61	9.68	1.16 × 103	1.27	8.02	1.55 × 103
−2.5	1.72	9.82	1.25 × 103	1.39	8.19	1.72 × 103
0	1.82	9.93	1.35 × 103	1.52	8.33	1.91 × 103
2.5	1.93	10.03	1.45 × 103	1.65	8.44	2.13 × 103
5.0	2.04	10.12	1.56 × 103	1.78	8.53	2.35 × 103

Note. Using Figure 10, we recorded the maximum mass of the stars M in units of the solar mass M_⊙ with their radius R in km and the central energy density _c for both the massless quark model and the cold star approximation.

Download table as:  ASCIITypeset image

4.2. Cold Star Approximation

This section contains a discussion of zero temperature (T = 0) and m ≠ 0. Detailed calculations of the pressure, energy density, and baryon number density can be found, for example, in Glendenning (2000b). In this scenario, we add the electrons to the system with their statistical weights (=2) due to the spin. Performing the standard calculations, we obtain (Glendenning 2000a)

$$\begin{eqnarray}\begin{array}{rcl}P & = & -B+\displaystyle \sum _{f}\left[\displaystyle \frac{1}{4{\pi }^{2}}\left({\mu }_{f}{k}_{f}\left({\mu }_{f}^{2}-\displaystyle \frac{5}{2}{m}_{f}^{2}\right)+\displaystyle \frac{3}{2}{m}_{f}^{4}\mathrm{ln}\left(\displaystyle \frac{{\mu }_{f}+{k}_{f}}{{m}_{f}}\right)\right)\right]\\ & & +\,\displaystyle \frac{1}{12{\pi }^{2}}\left[{\mu }_{e}{k}_{e}\left({\mu }_{e}^{2}-\displaystyle \frac{5}{2}{m}_{e}^{2}\right)+\displaystyle \frac{3}{2}{m}_{e}^{4}\mathrm{ln}\left(\displaystyle \frac{{\mu }_{e}+{k}_{e}}{{m}_{e}}\right)\right],\end{array}\end{eqnarray} \tag{ 36 }$$

$$\begin{eqnarray}\begin{array}{rcl}\epsilon & = & B+\displaystyle \sum _{f}\left[\displaystyle \frac{3}{4{\pi }^{2}}\left({\mu }_{f}{k}_{f}\left({\mu }_{f}^{2}-\displaystyle \frac{1}{2}{m}_{f}^{2}\right)+\displaystyle \frac{1}{2}{m}_{f}^{4}\mathrm{ln}\left(\displaystyle \frac{{\mu }_{f}+{k}_{f}}{{m}_{f}}\right)\right)\right]\\ & & +\,\displaystyle \frac{1}{4{\pi }^{2}}\left[{\mu }_{e}{k}_{e}\left({\mu }_{e}^{2}-\displaystyle \frac{1}{2}{m}_{e}^{2}\right)+\displaystyle \frac{1}{2}{m}_{e}^{4}\mathrm{ln}\left(\displaystyle \frac{{\mu }_{e}+{k}_{e}}{{m}_{e}}\right)\right],\end{array}\end{eqnarray} \tag{ 37 }$$

where k_f is the Fermi momentum for flavor f with ${k}_{f}={\left({\mu }_{f}^{2}-{m}_{f}^{2}\right)}^{1/2}$ and ${k}_{e}={\left({\mu }_{e}^{2}-{m}_{e}^{2}\right)}^{1/2}$. Notice that there are four independent variables appearing in the above equations, i.e., μ_u , μ_d , μ_s , and μ_e . Strange stars are composed of uds quarks. Hence, we constrain the chemical potentials of the quarks to a single independent variable μ such that μ_d  = μ_s  = μ and μ_u  + μ_e  = μ. Thus, for the two independent variables μ and μ_e , two equations are necessary to produce a set of chemical potentials and solve the system for a pair of values for and P.

Knowing the quark chemical potentials, the relation between the pressure and energy density of the quarks can be verified. The effects of the finite strange quark mass on the energy density () and the pressure (P) for neutral quark matter including electrons shows that there is a sizable difference in the energy density and the pressure between zero strange quark mass and nonvanishing strange quark mass. However, the EoS in this case basically exhibits a nonlinear behavior between and P, and this nonlinearity is very hard to solve. This is because the quark chemical potentials increase when we increase the baryon number density, while the electron chemical potential is negligible. We have thoroughly quantified the behaviors of the chemical potentials versus baryon number density using quark masses as given in Glendenning (2000b).

Fortunately, it has been also noticed from Haensel et al. (2007) that the resulting EoS  = (P) can be approximated by a nonideal bag model which is written in the following form:

$$\begin{eqnarray}&&\epsilon =a\,B+b\,P,\end{eqnarray} \tag{ 38 }$$

with a and b being arbitrary constants. In our case, we find that  = 3.05 P + 368, taking the value B = 70 MeV fm⁻³. Using the numerical calculations, the chemical potentials and the number density ρ are simultaneously obtained. After substituting the results into Equation (36) and Equation (37), we finally end up with the EoS displaying a relationship between the energy density and pressure. However, the linear behavior is maintained by the relation given in Equation (38), as illustrated in Figure 1 (right panel).

The input data for the numerical calculation are as mentioned above. The pressure and density versus radial distance from the center of the cold star, i.e., quark matter at zero temperature, are presented in Figure 4. In all the curves in Figure 4, note that the pressure and density are maximum at the center and decrease monotonically toward the boundary. In turn, to study the mass–radius relation and the mass versus central density for cold quark matter, EoSs are given for five representative values of α in Figure 5. For a given central density, the star mass grows with increasing α. The maximum mass increases with increasing α and we find that, for α = 5, the maximum mass becomes M_max = 1.78 M_⊙. At that point we record the mass of the star, 1.52M_⊙ when α = 0 in GR. For more clarity, the properties of stars with maximal mass are reported in Table 1 and are compared to GR (α = 0). Finally, in Figure 6, the mass–radius diagram is represented for two models (massless quark and cold star approximation) for different values of α. Recent discoveries of millisecond pulsars have shown that the neutron star mass distribution is much wider, extending firmly up to  ∼ 2M_⊙, and has already ruled out many soft EOSs. Figure 6 clearly shows that massless quark stars can achieve much higher masses and radii than cold stars, which are actually very close to realistic neutron star models with  ∼ 2M_⊙. Doneva & Yazadjiev (2020) have obtained compact stars considering the hadronic and strange quark star EoSs. For the strange star EoS, the M − R dependence is almost indistinguishable from GR for small masses and larger deviations exist only close to the maximum mass, which is similar to our solution. We discuss here the case corresponding to α = 5 and the value of B = 70 MeV fm^–3, as an example, showing in Figure 6 that if we set a limit on the maximum mass of a compact star, the corresponding maximum mass in the GR case cannot be achieved from the same values. Interestingly, all values of ${M}_{{\rm{\max }}}$ for massless quarks are higher than Chandrasekhar limit, which is about 1.4M_⊙.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

A profile of solutions that covers the full range of values for α and the bag constant B is presented in Figures 7 and 8. Our choice of parameter α has a significant influence on the maximum masses and radius relation. In this case the maximum masses of quark stars monotonically increase with increasing α values. Observational constraints on the GB constant were explored in Clifton et al. (2020), which is 0 ≲ α ≲ 10² km² based on observations of binary black holes. By analyzing Figure 7 for the massless quark star, it can be understood that one may achieve the maximum mass above ${M}_{\max }\sim 2{M}_{\odot }$ for 2.0 < α < 3.0 km² and B < 60 MeV fm^–3. Using Figure 8 for the cold star, we can obtain the maximum mass above ${M}_{\max }\sim 2{M}_{\odot }$ for α > 5.0 km² and B < 60 MeV fm^–3. The theory and findings suggest that our proposed model is in agreement with the current results for positive values of α.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

For completeness, we would also like to show that the equations of stellar structure admit stable solutions, and explore the physical properties in the interior of the fluid sphere. In the following, we discuss the compactness and the stability of stars.

5.1. Compactness

The qualitative effect of the compactness 2MG/Rc² for each EoS is illustrated in Figure 9 for particular values of the bag constant B and coupling constant α. Regarding the stars with the same M and α, it is noted that the compactness decreases when the bag constant increases. Additionally, at the same values of M, B, and α, the compactness in the case of the cold star EoS is higher than that of the massless quark EoS. Let us next define the Schwarzschild radius r_g  = 2GM/c². Interestingly, as mentioned in Haensel et al. (2007), the compactness parameter r_g /R characterizes the importance of relativistic effects for a star of mass M and radius R. Figure 9 shows that the trend of stellar compactness lies in the range 0.5 < r_g /R < 0.6 for both stars corresponding their respective EoS.

Zoom In Zoom Out Reset image size

5.2. Stability Test

Our particular interest is to study the stability of the compact strange star. A necessary (but insufficient) condition for stability of a compact star is that the total mass be an increasing function of the central density, dM/d _c  > 0 (Arbanil & Malheiro 2016; Glendenning & Kettner 2000). In Figure 10, we plot the dependence of masses of compact stars in solar units on their central density for the massless quark (left panel) and cold star case (right panel). Here, we vary the central density between the range 100 and 3500 MeV fm^–3. The dotted section of each curve corresponds to the unstable configuration where dM/d _c  < 0. The maxima of the mass–central density relations are easily determined and summarized in Table 1 for the EoSs investigated in this work.

Zoom In Zoom Out Reset image size

Apart from the above discussion, here we need to further study the stability by examining the adiabatic index (γ) based on our EoSs concerning quark matter models. It is noted that the adiabatic index is a basic ingredient of the instability criterion, and is related to thermodynamical quantities. The dynamical theory of infinitesimal, adiabatic, and radial oscillations of relativistic stars was first investigated more than 56 years ago by Chandrasekhar (1964). The main conclusion regarding this study is that the critical adiabatic index γ_cr , for the onset of instability, increases due to relativistic effects from the Newtonian value γ = 4/3. For an adiabatic perturbation, the adiabatic index, which for adiabatic oscillations is related to the sound speed, is defined by (Chandrasekhar 1964; Merafina & Ruffini 1989)

$$\begin{eqnarray}&&\gamma \equiv \left(1+\displaystyle \frac{\epsilon }{P}\right){\left(\displaystyle \frac{{dP}}{d\epsilon }\right)}_{S},\end{eqnarray} \tag{ 39 }$$

where dP/d is the speed of sound in units of the speed of light and the subscript S indicates constant specific entropy. Equation (39) is the adiabatic index associated with perturbation of the balance of forces that act on it; in other words, a perturbation of hydrostatic equilibrium.

Thus, the "effective" γ must be greater than γ_cr i.e., γ > γ_cr for the configuration to be stable against radial perturbations. Moustakidis (2020) has shown that these conditions are also applicable to describe compact objects including white dwarfs, neutron stars and supermassive stars. In Haensel et al. (2007), the value of γ lies between 2 and 4 for the EoS related to neutron star matter. Glass & Harpaz (1983) estimate a value of γ > 4/3 for relativistic polytropes depending on the ratio /P at the center of the star. A more fruitful discussion was found in Chavanis (2002) for the stability of an extended cluster with ρ_e /ρ₀ ≪ 1 in Newtonian gravity with γ > 4/3.

Plots of γ against values of EoS parameters are shown in Figure 11. It can be seen that our model is stable against radial adiabatic infinitesimal perturbations and increasing values of γ mean a growth in pressure for a given increase in energy density.

Zoom In Zoom Out Reset image size

In this paper, we investigate the features of 4D EGB gravity in extreme circumstances such as those arising within highly compact static spherically symmetric bodies. We consider the self-bound strange matter hypothesis. The interesting part of this theory is that the resulting regularized 4D EGB gravity has nontrivial dynamics and is free from the Ostrogradsky instability.

There exists considerable evidence that compact stars are partially or totally made up of quark matter. But the existence of quark stars is still controversial and their EoS is also uncertain. Here, we first considered the static spherically symmetric D-dimensional metric and derived corresponding field equations taking a limit of D → 4 at the level of field equations. We then numerically solved the field equations for the strange matter hypothesis. To clarify the astrophysical implications of our work, we discussed two important scenarios: quark matter phases consisting of massless quarks, and quark matter at zero temperature.

To gain a better understanding of the physical properties, we quantified the maximal mass from the central density and mass–radius relation of the stellar structure. The mass–radius results are graphically shown, and strictly depend on the values of the coupling constant and the chosen EoS. Then, we showed that for the limit α → 0, the obtained TOV in 4D EGB gravity reduces to standard Einstein theory, and the solutions are compared in Table 1. From Figure 6, we found that massless quark stars can achieve much higher masses and radii than cold stars within the constraint of  ∼ 2M_⊙, since negative α reduces the maximum mass of a compact star for a given EoS. Furthermore, we obtain interesting results on their physical properties such as compactness and the corresponding effective adiabatic index, γ, which appears in the stability formula introduced by Chandrasekhar. We found a value of γ > γ_cr for the critical adiabatic index, for both equations of state considered here. In other words, the stability in all cases is ensured for quark matter EoSs.

Finally, it is notable that studies of other compact objects such as neutron stars and white dwarfs using the same context and its modified TOV equation will be interesting. However, we will leave these topics for future work.

The authors are grateful to the referee for a careful reading of the paper and valuable suggestions and comments. T.T. would like to thank the financial support from the Science Achievement Scholarship of Thailand (SAST). P.C. acknowledges the Mid-Career Research Grant 2020 from National Research Council of Thailand under contract No. NFS6400117.