Isotropic compact stars in 4-dimensional Einstein-Gauss-Bonnet gravity
coupled with scalar field
– Reconstruction of model —

Although the fact that general relativity (GR) theory by Einstein is successful at present, that can forecast and elucidate the increase of observational data. Meanwhile, there are strong motivations to expect that it must be modified, due to its shortage in the quantization of gravity and explaining the recent observational; puzzles in modern cosmology yielding to the study of amended theories of gravity.

The Lovelock gravitational theories Lovelock (1971) are of special attraction since they are Lagrangian-based theories that could give conserved covariant field equations which do not include the derivatives higher than the second degree. In this regard, Lovelock’s theories are the physical extensions of GR. The Gauss-Bonnet (GB) theory is considered the first physical non-trivial expansion of Einstein’s GR. This theory is meaningful if its space-time is greater than 4-dimensional in which the GB invariant

$\displaystyle\mathcal{G}=R^{2}-4R_{\alpha\beta}R^{\alpha\beta}+R_{\alpha\beta\rho\sigma}R^{\alpha\beta\rho\sigma}\,,$

(1)

can create a rich phenomenology. Through the use of Chern’s theorem shen Chern (1945), it can be shown that in 4 dimensions, the GB expression is a non-dynamical term because the GB invariant becomes a total derivative. To make the GB expression a dynamical one in 4 dimensions, we must invoke a novel scalar field with a canonical kinetic term coupling to GB term Sotiriou and Zhou (2014a, b); Kanti et al. (1996); Kleihaus et al. (2011); Doneva and Yazadjiev (2018); Silva et al. (2018); Antoniou et al. (2018); Dima et al. (2020); Herdeiro et al. (2021); Berti et al. (2021) as stimulated, for example, by low-energy effective actions stem out of string theory, e.g., the Einstein-dilaton-GB models Zwiebach (1985); Kanti et al. (1996, 1998); Cunha et al. (2017). Actually, because of Lovelock’s theorem, all amended gravitational theories in 4 dimensions in principle will have extra degrees of freedom, which can be considered as new basic fields.

The exact solutions of the gravitational system supply scientific society with a simple test of space-time and evaluation of observable forecasts. Nevertheless, amended gravitational theories with new basic field(s) usually provide equations of motion with high intractability so that the evaluations become analytically out of the question. To face such an issue, one has to be coerced to apply either the perturbation theory, which is not well-qualified in the strong gravitational field or to defy numerical methods Sullivan et al. (2020). However, the field equations of GR coupled with a matter having conformal invariance since it possesses the constant Ricci scalar curvature on-shell, limiting the space-times and permitting analytic solutions to be easily derived. An example of such a theory that has conformal invariance and yields simple analytic solutions is the electro-vacuum, whose Reissner-Nordström (Kerr-Newman) solution was the first-ever discovered static (spinning) black hole (BH) with a matter source. Another model is the gravitational theory coupled with a conformally scalar field, in which the matter action obeys the conformal invariance and has the form,

$\displaystyle S_{\xi}=\int d^{4}x\sqrt{-g}\left(\frac{1}{6}R\xi^{2}+\left(\nabla\xi\right)^{2}\right)\,.$

(2)

where $R$ is the Ricci scalar and $\xi$ is the scalar field. The field equations of the above action give a solution with no-hair theorems (see e.g., Ref. Herdeiro and Radu (2015) for a review) and the static Bocharova-Bronnikov-Melnikov-Bekenstein BH Bocharova et al. (1970); Bekenstein (1975, 1974) has been much debated. Gravitational theory with a conformal scalar field and its solutions have been discussed throughout recent years because of its compelling properties (see e.g. Refs. Martinez et al. (2003, 2006); Anabalon and Maeda (2010); Padilla et al. (2014); Fernandes (2021); de Haro et al. (2007); Dotti et al. (2008); Gunzig et al. (2000); Oliva and Ray (2012); Cisterna et al. (2021); Caceres et al. (2020) and references therein).

As we discussed above, in 4 dimensions, the GB term is topological and does not yield any dynamical effect. Nevertheless, when the GB term is non-minimally coupled with any other field like a scalar field $\xi$, the output dynamics are non-trivial. Many cosmological proposals have been presented in recent literature Brax and van de Bruck (2003); Nojiri and Odintsov (2005); Nojiri et al. (2006); Cognola et al. (2006); Nojiri et al. (2010); Cognola et al. (2009); Capozziello et al. (2009); Maharaj et al. (2015); Bamba et al. (2008); Sadeghi et al. (2009); Guo and Schwarz (2010); Satoh (2010); Nozari and Rashidi (2013); Lahiri (2017, 2016); Nashed and Saridakis (2019); Mathew and Shankaranarayanan (2016); Nozari et al. (2015); Motaharfar and Sepangi (2016); Carter and Neupane (2006); De Laurentis et al. (2015); van de Bruck et al. (2016); Granda and Jimenez (2014); Granda and Loaiza (2012); Nojiri et al. (2005); Hikmawan et al. (2016); Kanti et al. (1999); Easther and Maeda (1996); Rizos and Tamvakis (1994); Starobinsky (1980); Brandenberger and Vafa (1989); Tseytlin and Vafa (1992); Nashed and El Hanafy (2017); Mukhanov and Brandenberger (1992); Brandenberger et al. (1993); Barrow (1993); Kobayashi (2005); Brassel et al. (2018); Damour and Polyakov (1994); Maeda (2006); Dehghani and Farhangkhah (2009); Nashed (2011); Angelantonj et al. (1995); Kaloper et al. (1995); Gasperini and Veneziano (1996); Rey (1996, 1997); Easther et al. (1996); Santillan (2017); Bose and Kar (1997); Kalyana Rama (1997a); Nashed and Capozziello (2021); Kalyana Rama (1997b, c); Brustein and Madden (1997, 1998) and references therein. In the frame of astrophysics, however, as far as we know, the GB theory with a non-minimal coupling of a scalar field via potential and coefficient function has not been tackled although there are some frontier works as in Silva et al. (2018). It is the aim of the present study to derive exact spherically symmetric interior solutions of this theory and discussed the obtained physical consequences. By using our formulation, we can construct a model which reproduces any given profile of the energy density $\rho$ for arbitrary EoS of matter. The mass $M$ and the radius $R_{s}$ of the compact star are determined by the profile of the energy density and therefore we can obtain an arbitrary relation between the mass $M$ and the radius $R_{s}$ of the compact star by adjusting the scalar potential and the coefficient function of the Gauss-Bonnet term in the action of EGBS, which could be a kind of degeneracy between the EoS and the functions characterizing the model Therefore we find that only the mass-radius relation is not sufficient to constrain the model.

The arrangement of the present study is as follows: In Section II, we give the cornerstone of the Einstein-Gauss-Bonnet gravity coupled with a scalar field (EGBS). In Section III, we apply the field equation of the EGBS theory to a spherically symmetric space-time and derive the full system of the differential equation. Here we show that we can construct a model which reproduces any given profile of the energy density $\rho$ for arbitrary EoS of matter. Also in Section III, we give the form of a polytropic equation of state (EoS) as an example and a form of one of the metric potentials as an input and then derived all the unknown functions including the profile of the scalar field, the coefficient function, the potential of the scalar field, and the form of another metric potential. Section IV states the physical conditions that must be satisfied for any real stellar configuration. In Section V, we discuss the physical properties analytically and graphically showing that the solutions have realistic physical properties. In Section VII, we discuss the issue of stability by using the adiabatic index and show that our model satisfies the adiabatic index, that is, the value of the index is greater than $4/3$, which is the condition of stability. The final Section is reserved for the conclusion and discussion of the present study.

Now we are going to consider the Einstein-Gauss-Bonnet gravity coupled with a scalar field (EGBS) in $N$ dimensions. This theory takes the following amended action,

$\displaystyle\mathcal{S}=\int d^{N}x\sqrt{-g}\left\{\frac{1}{2\kappa^{2}}R-\frac{1}{2}\partial_{\mu}\xi\partial^{\mu}\xi+V(\xi)+f(\xi)\mathcal{G}\right\}+S_{\mathrm{M}}\,,$

(3)

where $\xi$ is the scalar field and $V$ is the potential which is a function of $\xi$, $f(\xi)$ is an arbitrary function of the scalar field, and $S_{\mathrm{M}}$ is the matter action, where we assume that matter is to couple minimally to the metric, i.e., we are working in the so-called Jordan frame. In 4 dimensions, i.e., when $N=4$ the aforementioned action is physically non-trivial because the Gauss-Bonnet invariant term $\mathcal{G}$ is coupled with the real scalar field $\xi$ through the coupling $f(\xi)$. Because of this coupling, the Lagrangian is not a total derivative but contributes to the field equations of the system.

The variation of the action (3) w.r.t. the scalar field $\xi$ yields the following equation,

$\displaystyle\nabla^{2}\xi-V^{\prime}(\xi)+f^{\prime}(\xi)\mathcal{G}=0\,.$

(4)

The variation of the action (3) w.r.t. the metric $g_{\mu\nu}$ yields the following field equations,

	$\displaystyle T^{\mu\nu}=$	$\displaystyle\frac{1}{2\kappa^{2}}\left(-R^{\mu\nu}+\frac{1}{2}g^{\mu\nu}R\right)+\frac{1}{2}\partial^{\mu}\xi\partial^{\nu}\xi-\frac{1}{4}g^{\mu\nu}\partial_{\rho}\xi\partial^{\rho}\xi+\frac{1}{2}g^{\mu\nu}[f(\xi)G-V(\xi)]+2f(\xi)RR^{\mu\nu}$
		$\displaystyle+2\nabla^{\mu}\nabla^{\nu}\left(f(\xi)R\right)-2g^{\mu\nu}\nabla^{2}\left(f(\xi)R\right)+8f(\xi)R^{\mu}_{\ \rho}R^{\nu\rho}-4\nabla_{\rho}\nabla^{\mu}\left(f(\xi)R^{\nu\rho}\right)-4\nabla_{\rho}\nabla^{\nu}\left(f(\xi)R^{\mu\rho}\right)$
		$\displaystyle+4\nabla^{2}\left(f(\xi)R^{\mu\nu}\right)+4g^{\mu\nu}\nabla_{\rho}\nabla_{\sigma}\left(f(\xi)R^{\rho\sigma}\right)-2f(\xi)R^{\mu\rho\sigma\tau}R^{\nu}_{\ \rho\sigma\tau}+4\nabla_{\rho}\nabla_{\sigma}\left(f(\xi)R^{\mu\rho\sigma\nu}\right).$		(5)

Through the use of the below Bianchi identities,

	$\displaystyle\nabla^{\rho}R_{\rho\tau\mu\nu}=$	$\displaystyle\nabla_{\mu}R_{\nu\tau}-\nabla_{\nu}R_{\mu\tau}\,,$
	$\displaystyle\nabla^{\rho}R_{\rho\mu}=$	$\displaystyle\frac{1}{2}\nabla_{\mu}R\,,$
	$\displaystyle\nabla_{\rho}\nabla_{\sigma}R^{\mu\rho\nu\sigma}=$	$\displaystyle\nabla^{2}R^{\mu\nu}-{1\over 2}\nabla^{\mu}\nabla^{\nu}R+R^{\mu\rho\nu\sigma}R_{\rho\sigma}-R^{\mu}_{\ \rho}R^{\nu\rho}\,,$
	$\displaystyle\nabla_{\rho}\nabla^{\mu}R^{\rho\nu}+\nabla_{\rho}\nabla^{\nu}R^{\rho\mu}=$	$\displaystyle\frac{1}{2}\left(\nabla^{\mu}\nabla^{\nu}R+\nabla^{\nu}\nabla^{\mu}R\right)-2R^{\mu\rho\nu\sigma}R_{\rho\sigma}+2R^{\mu}_{\ \rho}R^{\nu\rho}\,,$
	$\displaystyle\nabla_{\rho}\nabla_{\sigma}R^{\rho\sigma}=$	$\displaystyle\frac{1}{2}\Box R\,,$		(6)

in Eq. (II), we obtain

	$\displaystyle T^{\mu\nu}=$	$\displaystyle\frac{1}{2\kappa^{2}}\left(-R^{\mu\nu}+\frac{1}{2}g^{\mu\nu}R\right)+\left(\frac{1}{2}\partial^{\mu}\xi\partial^{\nu}\xi-\frac{1}{4}g^{\mu\nu}\partial_{\rho}\xi\partial^{\rho}\xi\right)+\frac{1}{2}g^{\mu\nu}\left[f(\xi)G-V(\xi)\right]$
		$\displaystyle-2f(\xi)RR^{\mu\nu}+4f(\xi)R^{\mu}_{\ \rho}R^{\nu\rho}-2f(\xi)R^{\mu\rho\sigma\tau}R^{\nu}_{\ \rho\sigma\tau}-4f(\xi)R^{\mu\rho\sigma\nu}R_{\rho\sigma}$
		$\displaystyle+2\left(\nabla^{\mu}\nabla^{\nu}f(\xi)\right)R-2g^{\mu\nu}\left(\nabla^{2}f(\xi)\right)R-4\left(\nabla_{\rho}\nabla^{\mu}f(\xi)\right)R^{\nu\rho}-4\left(\nabla_{\rho}\nabla^{\nu}f(\xi)\right)R^{\mu\rho}$
		$\displaystyle+4\left(\nabla^{2}f(\xi)\right)R^{\mu\nu}+4g^{\mu\nu}\left(\nabla_{\rho}\nabla_{\sigma}f(\xi)\right)R^{\rho\sigma}-4\left(\nabla_{\rho}\nabla_{\sigma}f(\xi)\right)R^{\mu\rho\nu\sigma}.$		(7)

The field equations (4) and (II) are the full system of equations describing the theory under consideration. In the 4 dimensional case i.e., $N=4$, Eq. (II) yields to:

	$\displaystyle T^{\mu\nu}=$	$\displaystyle\frac{1}{2\kappa^{2}}\left(-R^{\mu\nu}+\frac{1}{2}g^{\mu\nu}R\right)+\left(\frac{1}{2}\partial^{\mu}\xi\partial^{\nu}\xi-\frac{1}{4}g^{\mu\nu}\partial_{\rho}\xi\partial^{\rho}\xi\right)-\frac{1}{2}g^{\mu\nu}V(\xi)$
		$\displaystyle+2\left(\nabla^{\mu}\nabla^{\nu}f(\xi)\right)R-2g^{\mu\nu}\left(\nabla^{2}f(\xi)\right)R-4\left(\nabla_{\rho}\nabla^{\mu}f(\xi)\right)R^{\nu\rho}-4\left(\nabla_{\rho}\nabla^{\nu}f(\xi)\right)R^{\mu\rho}$
		$\displaystyle+4\left(\nabla^{2}f(\xi)\right)R^{\mu\nu}+4g^{\mu\nu}\left(\nabla_{\rho}\nabla_{\sigma}f(\xi)\right)R^{\rho\sigma}-4\left(\nabla_{\rho}\nabla_{\sigma}f(\xi)\right)R^{\mu\rho\nu\sigma}.$		(8)

In the present study, we assume that the scalar field $\xi$ is a function of the radial coordinate $r$ and therefore the function $f(\xi)$ only depends on $r$, i.e., $f(r)\equiv f\left(\xi\left(r\right)\right)$, because we deal with static and spherically symmetric space-time,

$\displaystyle ds^{2}=-a(r)dt^{2}+\frac{dr^{2}}{a_{1}(r)}+r^{2}\left(d\theta^{2}+\sin^{2}\left(\theta\right)\right)d\phi^{2}\,.$

(9)

In the following section, we study the system of field equations (4) and (II) and try to find the analytic form of the unknown functions when $T^{\mu\nu}\neq 0$.

For the metric in (9), the $(t,t)$-component of the field equation Eq. (II) has the following form,

$\displaystyle-\rho=\frac{16a_{1}\left(1-a_{1}\right)f^{\prime\prime}+\left\{8\left(1-3a_{1}\right)f^{\prime}+2r\right\}a^{\prime}_{1}+2a_{1}+2Vr^{2}+r^{2}\xi^{\prime 2}a_{1}-2}{4r^{2}}\,,$

(10)

the $(r,r)$-component is given by

$\displaystyle p=\frac{2\left(4\left(1-3a_{1}\right)f^{\prime}+r\right)a_{1}a^{\prime}+a\left[2a_{1}-r^{2}a_{1}\xi^{\prime 2}-2+2Vr^{2}\right]}{4r^{2}a}\,,$

(11)

and the $(\theta,\theta)$ and $(\phi,\phi)$-components are,

$\displaystyle p=\frac{2a_{1}a\left(r-8f^{\prime}a_{1}\right)a^{\prime\prime}-16f^{\prime\prime}a_{1}^{2}aa^{\prime}+a_{1}\left(8f^{\prime}a_{1}-r\right)a^{\prime 2}+\left\{\left(r-24f^{\prime}a_{1}\right)a^{\prime}_{1}+2a_{1}\right\}aa^{\prime}+2a^{2}\left(a^{\prime}_{1}+r\left[\xi^{\prime 2}a_{1}+2V\right]\right)}{8a^{2}r}\,.$

(12)

The field equation of the scalar field (4) takes the following form

$\displaystyle 0=\frac{8a_{1}af^{\prime}\left(a_{1}-1\right)a^{\prime\prime}+2\xi^{\prime\prime}a_{1}\xi^{\prime}a^{2}r^{2}+4a^{\prime}f^{\prime}\left[a_{1}a^{\prime}\left(1-a_{1}\right)+a^{\prime}_{1}a\left(3a_{1}-1\right)\right]+ra\left(\left[\,a_{1}a^{\prime}r+a\left\{4a_{1}+a^{\prime}_{1}r\right\}\right]\xi^{\prime 2}-2arV^{\prime}\right)}{2r^{2}a^{2}\xi^{\prime}}\,.$

(13)

Here $\rho$ is the energy density and $p$ is the pressure of matter, which we assume to be a perfect fluid and satisfies an equation of state, $p=p\left(\rho\right)$. The energy density $\rho$ and the pressure $p$ satisfy the following conservation law,

$\displaystyle 0=\nabla^{\mu}T_{\mu r}=\frac{1}{2}\frac{a^{\prime}}{a}\left(\rho+p\right)+\frac{dp}{dr}\,.$

(14)

The conservation law is also derived from Eqs. (10), (11), (12), and (13). Here we have assumed $\rho$ and $p$ only depend on the radial coordinate $r$. Other components of the conservation law are trivially satisfied. If the equation of state $\rho=\rho(p)$ is given, Eq. (14) can be integrated as

$\displaystyle\frac{1}{2}\ln a=-\int^{r}dr\frac{\frac{dp}{dr}}{\rho+p}=-\int^{p(r)}\frac{dp}{\rho(p)+p}\,.$

(15)

Because Eq. (14) and therefore (15) can be obtained from Eqs. (10), (11), (12), and (13), as long as we use (15), we forget one equation in Eqs. (10), (11), (12), and (13). In the following, we do not use Eq. (13). Inside the compact star, we can use Eq. (15) but outside the star, we cannot use Eq. (15). Instead of using Eq. (15), we may assume the profile of $a=a(r)$ so that $a(r)$ and $a^{\prime}(r)$ are continuous at the surface of the compact star.

By Eq. (10) $+$ Eq. (11), we obtain

$\displaystyle V=-\rho+p+\frac{8\,a_{1}a\left(a_{1}-1\right)f^{\prime\prime}-4\left\{\left(1-3a_{1}\right)f^{\prime}+r\right\}\left(aa_{1}\right)^{\prime}+2a-2aa_{1}}{2ar^{2}}\,.$

(16)

On the other hand, Eq. (10) $-$ Eq. (11) gives,

$\displaystyle\xi^{\prime}=\pm\left\{\frac{2}{a_{1}}\left(\rho+p\right)+{\frac{8\,aa_{1}\left(a_{1}-1\right)f^{\prime\prime}-\left[4\left(1-3\,a_{1}\right)f^{\prime}+r\right]\left(aa^{\prime}_{1}-a_{1}a^{\prime}\right)}{a_{1}r^{2}a}}\right\}^{\frac{1}{2}}\,.$

(17)

Furthermore, Eq. (10) $-$ Eq. (12) gives,

	$\displaystyle 0=$	$\displaystyle 16\,\left[a_{1}a^{\prime}r-2\,a\left(a_{1}-1\right)\right]aa_{1}f^{\prime\prime}-2raa_{1}\left(r-8\,f^{\prime}a_{1}\right)a^{\prime\prime}+ra_{1}\left(r-8\,f^{\prime}a_{1}\right)a^{\prime 2}-\left[\left(r-24\,f^{\prime}a_{1}\right)a^{\prime}_{1}+2\,a_{1}\right]ara^{\prime}$
		$\displaystyle+2\,\left\{\left[8\left(1-3\,a_{1}\right)f^{\prime}+r\right]a^{\prime}_{1}-2+2\,a_{1}\right\}a^{2}+8a^{2}r^{2}\left(\rho+p\right)\,,$		(18)

which can be regarded with the differential equation for $f^{\prime}$ and therefore for $f$ if $a=a(r)$, $a_{1}=a_{1}(r)$, $\rho=\rho(r)$, and $p=p(r)$ are given and the solution is given by

	$\displaystyle f(r)=\,$	$\displaystyle-\int\left(\int{\frac{\left[a_{1}a^{\prime 2}r^{2}-2a_{1}a^{\prime\prime}ar^{2}-ra\left(a^{\prime}_{1}r+2\,a_{1}\right)a^{\prime}+{2\left\{a^{\prime}_{1}r-2+2a_{1}+4\left(\rho+p\right)r^{2}\right\}a^{2}}\right]}{2U(r)a_{1}a\left(a_{1}a^{\prime}r-2a\left(a_{1}-1\right)\right)}}{dr}-16c_{1}\right)Udr+c_{2}\,,$
	$\displaystyle U(r)\equiv\,$	$\displaystyle\mathrm{e}^{\int\frac{r{a_{1}}^{2}a^{\prime 2}-2ra{a_{1}}^{2}a^{\prime\prime}+\left(2a^{2}\left(3a_{1}-1\right)-3a_{1}a^{\prime}ar\right)a^{\prime}_{1}}{2a_{1}a\left\{a_{1}a^{\prime}r-2a\left(a_{1}-1\right)\right\}}{dr}}\,.$		(19)

Here $c_{1}$ and $c_{2}$ are constants of the integration.

Let us assume the $r$-dependencies of $\rho$ and $a_{1}$, $\rho=\rho(r)$ and $a_{1}=a_{1}(r)$. Then by using the EoS $p=p(\rho)$, we find the $r$-dependence of $p$, $p=p(r)=p\left(\rho\left(r\right)\right)$. Furthermore by using (15), we find the $r$-dependence of $a$, $a=a(r)$. However, Eq. (15) is not valid outside the compact star because $\rho$ and $p$, of course, vanish there. Then outside the compact star, we may properly assume the profile of $a(r)$ so that $a(r)$ and $a^{\prime}(r)$ are continuous at the surface, that is, the boundary of the compact star, and coincide with $a(r)$ and $a^{\prime}(r)$ obtained from (15). Therefore by using (III), we find the $r$-dependence of $f$, $f=f(r)$ and by using Eqs. (16) and (17), we find the $r$ dependencies of $V$ and $\xi$, $V=V(r)$ and $\xi=\xi(r)$. By solving $\xi=\xi(r)$ with respect to $r$, $r=r(\xi)$, we find $f$ and $V$ as functions of $\xi$, $f(\xi)=f\left(r\left(\xi\right)\right)$, $V(\xi)=V\left(r\left(\xi\right)\right)$ which realize the model which has a solution given by $\rho=\rho(r)$ and $a_{1}=a_{1}(r)$.

We should note, however, the expression of $\xi$ in (17) gives a constraint,

$\displaystyle\frac{2}{a_{1}}\left(\rho+p\right)+{\frac{8\,aa_{1}\left(a_{1}-1\right)f^{\prime\prime}-\left[4\left(1-3\,a_{1}\right)f^{\prime}+r\right]\left(aa^{\prime}_{1}-a_{1}a^{\prime}\right)}{a_{1}r^{2}a}}\geq 0\,,$

(20)

so that the ghost could be avoided. If Eq. (20) is not satisfied, the scalar field $\xi$ becomes pure imaginary. We may define a new real scalar field $\zeta$ by $\xi=i\zeta$ $\left(i^{2}=-1\right)$ but because the coefficient in front of the kinetic term of $\zeta$ becomes negative, $\zeta$ is a ghost, that is, a non-canonical scalar field. The existence of the ghost generates the negative norm states in the quantum theory and therefore the theory becomes inconsistent.

When we consider compact stars like neutron stars, we often consider the following equation of state,

1. 1.

   Energy-polytrope

   $\displaystyle p=K\rho^{1+\frac{1}{n}}\,,$

   (21)

   with constants $K$ and $n$. It is known that for the neutron stars, $n$ could take the value $0.5\leq n\leq 1$.

2. 2.

   Mass-polytrope

   $\displaystyle\rho=\rho_{m}+Np\,,\qquad\qquad p=K_{m}\rho_{m}^{1+\frac{1}{n_{m}}}\,,$

   (22)

   where $\rho_{m}$ is rest mass energy density and $K_{m}$, $N$ are constants,

Now let us study the case of the energy-polytrope (21), in detail, in which we can rewrite the EoS as follows,

$\displaystyle\rho=\tilde{K}p^{({1+\frac{1}{\tilde{n}}})}\,,\quad\tilde{K}\equiv K^{-\frac{1}{1+\frac{1}{n}}}\,,\quad\tilde{n}\equiv\frac{1}{\frac{1}{1+\frac{1}{n}}-1}=-1-n\,.$

(23)

For the energy-polytrope, Eq. (15) takes the following form,

$\displaystyle\frac{1}{2}\ln a=-\int^{p(r)}\frac{dp}{\tilde{K}p^{1+\frac{1}{\tilde{n}}}+p}=\frac{c}{2}+\tilde{n}\ln\left(1+{\tilde{K}}^{-1}p^{-\frac{1}{\tilde{n}}}\right)=\frac{c}{2}-\left(1+n\right)\ln\left(1+K\rho^{\frac{1}{n}}\right)\,.$

(24)

Here $c$ is a constant of the integration. Similarly, in case of mass-polytrope (22), we obtain

$\displaystyle\frac{1}{2}\ln a=\frac{\tilde{c}}{2}+\ln\left(1-K_{m}\rho^{\frac{1}{n_{m}}}\right)\,.$

(25)

Here $\tilde{c}$ is a constant of the integration, again.

Under one of the above equations of state, we may assume the following profile of $\rho=\rho(r)$ and $a_{1}=a_{1}(r)$, just for an example,

$\displaystyle\rho=\left\{\begin{array}[]{cc}\rho_{c}\left(1-\frac{r^{2}}{{R_{s}}^{2}}\right)&\ \mbox{when}\ r<R_{s}\\ 0&\ \ \mbox{when}\ r>R_{s}\end{array}\right.\,,\quad a_{1}=1-\frac{2Mr^{2}}{r^{3}+{r_{0}}^{3}}\,.$

(28)

Here $r_{0}$ is a constant, $\rho_{c}$ is a constant expressing the energy density at the center of the compact star, and $R_{s}$ is also a constant corresponding to the radius of the surface of the compact star, and $M$ is a constant corresponding to the mass of the compact star,

$\displaystyle M=4\pi\rho_{c}\int_{0}^{r}\psi^{2}\rho(\psi)d\psi=4\pi\rho_{c}\int_{0}^{r}d\psi\psi^{2}\left(1-\frac{\psi^{2}}{{R_{s}}^{2}}\right)=\frac{4\pi\rho_{c}r^{3}}{15}\left(5-\frac{3r^{2}}{{R_{s}}^{2}}\right)\,.$

(29)

When $r\to\infty$, $a_{1}$ behaves as $a_{1}(r)\sim 1-\frac{2M}{r}$ and therefore $M$ can be regarded as the mass of the compact star. Eq. (29) gives $M$-$r$ relation, that is, the relation between the mass and the radius of the compact star when $r=R_{s}$. We also note that we need to choose $r_{0}$ large enough so that $a_{1}$ is positive. In order that $a_{1}$ in (28) should be positive, we require

$\displaystyle\frac{2^{\frac{5}{3}}M}{3r_{0}}<1\,.$

(30)

We should also note that when $r\to 0$, $a_{1}$ behaves as $a_{1}(r)\sim 1-\frac{2Mr^{2}}{r_{0}^{3}}$¹¹1It is well known that the junction conditions for the matching of two spacetime manifolds have further restrictions in the EGBS gravity Davis (2003). With regards to a static configuration, this is not a real problem since the interior will match to vacuum, and so the pressure will still vanish at the surface $r=R_{s}$ as we will show below.. Therefore $a_{1}^{\prime}(r)$ vanishes at the center $r=0$, $a_{1}^{\prime}(r=0)=0$, and therefore, there is no conical singularity.

As an example, we use the energy-polytope as the equation of state by choosing $n=1$ just for simplicity. Then Eq. (24) gives,

$\displaystyle a=\frac{\mathrm{e}^{c}}{\left(1+K\rho_{c}\left(1-\frac{r^{2}}{{R_{s}}^{2}}\right)\right)^{4}}\,,$

(31)

which gives

$\displaystyle a^{\prime}=\frac{8\mathrm{e}^{c}K\rho_{c}r}{{R_{s}}^{2}\left(1+K\rho_{c}\left(1-\frac{r^{2}}{{R_{s}}^{2}}\right)\right)^{5}}\,,$

(32)

Outside the star, we assume $a(r)=a_{1}(r)$ in (28) and therefore

$\displaystyle a^{\prime}=\frac{2Mr\left(r^{3}-{r_{0}}^{3}\right)}{\left(r^{3}+{r_{0}}^{3}\right)^{2}}\,.$

(33)

Because $a(r)$ and $a^{\prime}(r)$ should be continuous at the surface $r=R_{s}$. we obtain

$\displaystyle\mathrm{e}^{c}=1-\frac{2M{R_{s}}^{2}}{{R_{s}}^{3}+{r_{0}}^{3}}\,,\quad\frac{8\mathrm{e}^{c}K\rho_{c}}{R_{s}}=\frac{2M{R_{s}}\left({R_{s}}^{3}-{r_{0}}^{3}\right)}{\left({R_{s}}^{3}+{r_{0}}^{3}\right)^{2}}\,.$

(34)

By deleting $\mathrm{e}^{c}$ in the two equations in (34), we obtain,

	$\displaystyle 0=$	$\displaystyle\left({r_{0}}^{3}\right)^{2}+\left(2{R_{s}}^{3}-2M{R_{s}}^{2}+\frac{M{R_{s}}^{2}}{2K\rho_{c}}\right){r_{0}}^{3}+{R_{s}}^{6}-2M{R_{s}}^{5}-\frac{M{R_{s}}^{5}}{4K\rho_{c}}$
	$\displaystyle=$	$\displaystyle\left({r_{0}}^{3}\right)^{2}+\left(2{R_{s}}^{3}-\frac{16\pi\rho_{c}R_{s}^{5}}{15}+\frac{2\pi R_{s}^{5}}{15K}\right){r_{0}}^{3}+{R_{s}}^{6}-\frac{16\pi\rho_{c}R_{s}^{8}}{15}-\frac{2\pi R_{s}^{8}}{15K}\,,$		(35)

where we have used Eq. (29) when $r=R_{s}$. Because $r_{0}$ should be positive, we find

		$\displaystyle{R_{s}}^{6}-\frac{16\pi\rho_{c}R_{s}^{8}}{15}-\frac{4\pi R_{s}^{8}}{15K}<0$
		or
		$\displaystyle 2{R_{s}}^{3}-\frac{16\pi\rho_{c}R_{s}^{5}}{15}+\frac{4\pi R_{s}^{5}}{15K}<0\quad\mbox{and}\quad{R_{s}}^{6}-\frac{16\pi\rho_{c}R_{s}^{8}}{15}-\frac{4\pi R_{s}^{8}}{15K}>0\,.$		(36)

Then by using (III), we find the $r$-dependence of $f$, $f=f(r)$ and by using Eqs. (16) and (17), the $r$ dependencies of $V$ and $\xi$, $V=V(r)$ and $\xi=\xi(r)$, are determined. If we can solve $\xi=\xi(r)$ with respect to $r$, $r=r(\xi)$, we find $f$ and $V$ as functions of $\xi$, $f(\xi)=f\left(r\left(\xi\right)\right)$, $V(\xi)=V\left(r\left(\xi\right)\right)$.

Just for further simplicity, we may choose

$\displaystyle 2r_{0}=R_{s}=4M=\frac{32\pi\rho_{c}R_{s}^{3}}{15}\,,\quad K\rho_{c}=\frac{7}{258}\,,\quad\mathrm{e}^{c}=\frac{5}{9}\,,$

(37)

which satisfy Eqs. (30), (34), and (III). For numerical calculation, we may further choose $R_{s}=1$.

Inside the compact star, by using Eqs. (28) and (31), we find the GB term $\mathcal{G}$ behaves as,

	$\displaystyle\mathcal{G}(r)=$	$\displaystyle\,-\frac{K\rho_{c}M}{64\left({R_{s}}^{2}+K\rho_{c}{R_{s}}^{2}-K\rho_{c}r^{2}\right)^{2}\left(r^{3}+{r_{0}}^{3}\right)^{3}}\left\{9r^{5}{r_{0}}^{3}K\rho_{c}+3{r_{0}}^{6}K\rho_{c}r^{2}-4{r_{0}}^{3}Mr^{4}K\rho_{c}+r^{5}MK\rho_{c}{R_{s}}^{2}\right.$
		$\displaystyle+3r^{3}{r_{0}}^{3}K\rho_{c}{R_{s}}^{2}+3{r_{0}}^{6}K\rho_{c}{R_{s}}^{2}-8{r_{0}}^{3}Mr^{2}K\rho_{c}{R_{s}}^{2}+r^{5}M{R_{s}}^{2}+6r^{8}K\rho_{c}-13r^{7}MK\rho_{c}$
		$\displaystyle\left.+3r^{3}{r_{0}}^{3}{R_{s}}^{2}+3{r_{0}}^{6}{R_{s}}^{2}-8{r_{0}}^{3}Mr^{2}{R_{s}}^{2}\right\}$
	$\displaystyle\approx$	$\displaystyle\,-\frac{64{R_{s}}^{2}{r_{0}}^{3}MK\rho_{c}}{{r_{0}}^{6}{R_{s}}^{4}\left(K\rho_{c}+1\right)}+\frac{64\left(8M{R_{s}}^{2}+8MK\rho_{c}{R_{s}}^{2}-9{r_{0}}^{3}K\rho_{c}\right)MK\rho_{c}r^{2}}{{r_{0}}^{6}{R_{s}}^{4}\left(K\rho_{c}+1\right)^{2}}+\frac{384MK\rho_{c}r^{3}}{{R_{s}}^{2}{r_{0}}^{6}\left(1+K\rho_{c}\right)}\,.$		(38)

Eq. (III) shows that the GB term does not vanish and it depends on the Mass of the star. Now we calculate the form of $f(r)$ by using the data given in Eqs. (31), (34), and (28). The explicit form of $f(r)$ is displayed in Appendix A. The form of $\xi(r)$, by using the data given in Eqs. (28), (31), and (34), is also displayed in Appendix A. Finally, we calculate the explicit form of $V(r)$ by using the data given in Eqs. (31), (34), and (28) and list the results in Appendix A.

To complete our study, we solve Eq. (51) asymptotically and obtain,

$\displaystyle\xi(r\rightarrow 0)\approx$

$\displaystyle C_{11}+C_{12}\sqrt{r}\ \Rightarrow\ r\approx C_{13}+C_{14}\xi+C_{15}\xi^{2}\,.$

(39)

The above equation is valid provided that the constant $C_{12}<0$. Now using Eq. (39) in (49), we obtain $f(\xi)$ as

$\displaystyle f(\xi)\approx C_{16}+C_{17}\xi+C_{18}\xi^{2}\,.$

(40)

Also using Eq. (39) in (53), we obtain $V(\xi)$ as:

$\displaystyle V(\xi)\approx C_{19}+C_{20}\xi+C_{21}\xi^{2}\,.$

(41)

A final remark that we should stress is the fact that the use of Eqs. (28), (31), (49), and the constraints (37) with $R_{s}=1$, one can show easily that the inequality (20) is hold.

We have four differential equations for seven unknown functions, as shown in Eqs. (10), (11), (12), and (13), that is $\rho$, $p$, $V$, $\xi$, $f$, $a$, and $a_{1}$. As a result, we need to require three additional conditions to close such a system. One of these extra conditions is the continuity equation given by Eq. (14). The second condition is the polytropic equation of state given by Eq. (21). The third one is the profile of the energy density of matter given by Eq. (28). When these additional conditions are combined with Eqs. (10), (11), (12), and (13), the system is in a closed form, allowing all seven unknown functions to be explicitly fixed.

For a physically reliable isotropic stellar model, the solution has to satisfy the below-listing conditions inside the stellar configurations,

• •

  The metric potentials $a(r)$ and $a_{1}(r)$, and the energy-momentum components $\rho$ and $p$ should be well defined at the center of the star and have a regular behavior and have no singularity in the interior of the star.

• •

  The density $\rho$ must be positive in the stellar interior i.e., $\rho\geq 0$. Moreover, its value at the center of the star must be finite, positive, and decreasing to the boundary of the stellar i.e., $\frac{d\rho}{dr}\leq 0$.

• •

  The pressure $p$ should have the positive value inside the fluid configuration i.e., $p\geq 0$. In addition, the derivative of the pressure should yield a negative value inside the stellar, i.e., $\frac{dp}{dr}<0$. At the surface of the stellar, $r=R_{s}$, the pressure $p$ should vanish.

• •

  For an isotropic fluid sphere, the inquiries of the energy conditions are given by the following inequalities in every point:

  1. 1.

     Null energy condition (NEC): $\rho>0$.

  2. 2.

     Weak energy condition (WEC): $p+\rho>0$.

  3. 3.

     Strong energy condition (SEC): $\rho+3p>0$.

• •

  The causality condition which should be satisfied to have a realistic model, i.e. the speed of sound should be less than $1$ (provided that the speed of light is $c=1$) in the interior of the star, i.e., $1\geq\frac{dp}{d\rho}\geq 0$.

• •

  To have a stable model, the adiabatic index must be more than $\frac{4}{3}$.

It is time to analyze the above conditions to see if we have a real isotropic star or not.

To test if our model given by Eqs. (22) and (24) agrees with a real stellar construction, we discuss the following issues: