The Palatini star: exact solutions of the modified Lane-Emden equation

TL;DR

This paper derives exact solutions for specific cases of the Palatini-modified Lane-Emden equation and compares the properties of Palatini-Newtonian neutron stars with classical Newtonian models, revealing that negative parameters lead to heavier and larger stars.

Contribution

It provides the first exact solutions for the Palatini-modified Lane-Emden equation for n=0 and n=1, and applies these to model neutron stars within this modified gravity framework.

Findings

Palatini-Newtonian neutron stars are heavier and larger for negative Starobinsky parameters.

Exact solutions for n=0 and n=1 cases of the modified Lane-Emden equation.

Comparison shows significant differences from classical Newtonian models.

Abstract

Two exact solutions for $n=0$ and $n=1$ of the Palatini-modified Lane-Emden equation are found. We have employed these solutions to describe a Palatini-Newtonian neutron star and compared the result with the pure Newtonian counterpart. It turned out that for the negative parameter of the Starobinsky model the star is heavier and larger.

Figures

Equations

$$S=S_{\text{g}}+S_{\text{m}}=\frac{1}{2\kappa }\int \sqrt{-g}f(\hat{R})d^{4}x+S_{\text{m}}(g_{\mu \nu },{\psi }_m),$$

$$f^{\prime }(\hat{R}){\hat{R}}_{\mu \nu }-\frac{1}{2}f(\hat{R})g_{\mu \nu }=$$

$${\hat{\nabla }}_{\beta }(\sqrt{-g}{f}^{\prime }(\hat{R})g^{\mu \nu })=$$

$$h_{\mu \nu }=f^{\prime }(\hat{R})g_{\mu \nu }.$$

$${\bar{R}}_{\mu \nu }-\frac{1}{2}{h}_{\mu \nu }\bar{R}$$

$$\Phi \bar{R}$$

$${\left(\frac{\Pi }{\Phi (r{)}^2}\right)}^{\prime }$$

$$\mathcal{M}(r)$$

$$\bar{Q}=\bar{\rho }+\frac{1}{2}\bar{U}=\frac{\rho }{{\Phi }^2}+\frac{U}{2{\Phi }^2}=\frac{Q}{{\Phi }^2},$$

$$\bar{\Pi }=\bar{p}-\frac{1}{2}\bar{U}=\frac{p}{{\Phi }^2}-\frac{U}{2{\Phi }^2}=\frac{\Pi }{{\Phi }^2}.$$

$$\frac{1}{{\xi }^2}\frac{d}{d\xi }\left({\xi }^2\frac{d\theta }{d\xi }\right)+{\theta }^n=0;$$

$$r$$

$$r_c^2$$

$$\frac{1}{\bar{\xi }}\frac{d^2}{d{\bar{\xi }}^2}\left(\left[1+\frac{2\alpha }{n+1}{\theta }^n\right]\bar{\xi }\theta \right)=-{\theta }^n,$$

$${\xi }^2{\theta }^n\Phi +\frac{{\Phi }^{-1/2}}{1+\frac{1}{2}\xi {\Phi }_{\xi }/\Phi }\frac{d}{d\xi }\left(\frac{{\xi }^2{\Phi }^{3/2}}{1+\frac{1}{2}\xi {\Phi }_{\xi }/\Phi }\frac{d\theta }{d\xi }\right)=0.$$

$$\frac{1}{\bar{\xi }}\frac{d^2}{d{\bar{\xi }}^2}\left(\left[1+2\alpha \right]\bar{\xi }\theta \right)=-1,$$

$$\theta =-\frac{{\bar{\xi }}^2}{6(1+2\alpha )}+C_0+\frac{C_1}{\bar{\xi }},$$

$$\theta =-\frac{{\xi }^2}{6}+C_0+\frac{C_1}{\xi },$$

$$\theta =-\frac{{\xi }^2}{6}+1.$$

$$\theta =\frac{15-{\xi }^2}{2\alpha (10+{\xi }^2)},$$

$$\beta \sim -5.02978\cdot 10^6{\text{m}}^2.$$

$${\omega }_n$$

$${\gamma }_n$$

$${\delta }_n$$

$$M$$

$$R$$

$${\rho }_c$$

$$T$$

$${\omega }_n$$

$${\delta }_n$$

$${\omega }_n$$

$${\delta }_n$$

$$\frac{M}{M_N}=\frac{{\omega }_1}{{\omega }_N}\sim 1.23,\frac{R}{R_N}\sim 1.23,\frac{\rho }{{\rho }_N}\sim 0.99$$

Full text

Alternative theories of gravity; neutron stars; exact solutions; Lane-Emden equation

I Introduction

Although there is no doubt of the beauty and the overall correctness of the General Relativity ein1 ; ein2 , the theory suffers from a number of shortcomings Copeland:2006wr ; Nojiri:2006ri ; Capozziello:2007ec ; Carroll:2004de ; Sotiriou:2008ve which could be addressed by extensions, provided by the Extended Theories of Gravity (ETGs), of the Einstein’s proposal. Apart from the cosmological arguments on the need of looking for modified theories of gravity Cap_Laur ; Cap_Far ; Starobinsky:1980te ; Guth:1980zm ; Huterer:1998qv ; Sami , there are also indications coming from astrophysics strengthening reasons for searching for the alternatives. In the light of the very recent observation of the pulsar PSR J2215+5135 with the mass $M_{1}=2.27^{+0.17}_{-0.15}M_{\odot}$ for the neutron star lina putting in trouble exotic forms of matter (hyperons or deconfined quarks) and the previous discoveries of the massive neutron stars as $2.01\pm 0.04M_{\odot}$ (PSR J0348+0432) as , and $1.97\pm 0.04M_{\odot}$ (PSR J1614-2230) craw , the ETGs Cap_beyond ; cap_invar allow to exceed the maximal mass of $2M_{\odot}$ demonstrated by GR without introducing exotic components.

Roughly speaking, in the ETGs one changes the geometric part of the action which can be achieved by, for example, generalizing the Einstein-Hilbert action Lagrangian density to an arbitrary function of the scalar curvature ($f(R)$ gravity Starobinsky:1980te ; Sotiriou:2008ve ; buchdahl ; cap_jcap ; cap_not ; DeFelice:2010aj ; Will:1993te ; Palatini:1919di ; Capozziello:2011et ; Ferraris:1992dx ), by adding minimally or non-minimally coupled scalar field brans ; Bergmann , or by assuming that physical constants are not actually constants Dabrowski:2012eb ; Leszczynska:2014xba ; Salzano:2016pny .

In the present paper we are interested in the $f(\hat{R})$ Palatini gravity under the Ehlers-Pirani-Schild (EPS) interpretation, which in more detail was discussed in eps ; mauro ; fatibene . The Palatini formulation of $f(\hat{R})$ gravity has advantages with respect to the metric one since the field equation are second-order differential equations for the metric $g$ while in the metric formulation we deal with the fourth-order equations. Since we are interested in stars, it should be mentioned that astrophysical objects have been already widely discussed in the context of Palatini gravity: studies on black holes can be found in diego1 ; diego2 ; diego3 ; diego4 ; diego5 ; diego6 ; man , on wormholes in worm ; worm2 ; diego7 ; diego8 , and on neutron stars in kain ; reij ; pano ; anab ; anet ; Barausse:2007ys ; barau ; barau2 ; pani ; sham ; olmo_ns ; mana .

Let us just briefly recall the Palatini formalism which assumes that the spacetime geometry is described by two independent structures: a class of Lorentzian metrics which are conformally related to each other, and a connection which later on, as a dynamical feature of the action, will turn out to be a Levi-Civita connection with respect to a metric conformally related to the metric $g$. Thus, we consider a standard action of the $f(\hat{R})$ gravity of the form

[TABLE]

where $\hat{R}=\hat{R}^{\mu\nu}g_{\mu\nu}$ is the Palatini-Ricci scalar constructed using two independent objects, the connection $\tilde{\Gamma}$ and the metric $g$. The variation with respect to the metric $g$ and the independent connection $\tilde{\Gamma}$ is given, respectively, by the formulas

[TABLE]

where $T_{\mu\nu}$ is the energy momentum tensor and $f^{\prime}(\hat{R})=df/d\hat{R}$.

Equations (3) imply that the connection $\hat{\nabla}_{\beta}$ is the Levi-Civita connection of the conformal metric $h$:

[TABLE]

Let us observe that Palatini gravity is equivalent to GR when one considers the linear function $f$ of the form $f(\hat{R})=\hat{R}-2\Lambda$, and the independent connection turns out to be Levi-Civita connection of the metric $g$.

We are going to use here the following interesting fact: the equations in question can be rewritten DeFelice:2010aj in the terms of the conformal metric $h_{\mu\nu}$ BSS ; SSB and the scalar field defined as $\Phi=f^{\prime}(\hat{R})$:

[TABLE]

where the bar quantities are defined as follows: $\bar{U}(\Phi)=(\hat{R}\Phi-f(\hat{R}))/\Phi^{2}$, and the appropriate energy momentum tensor reads $\bar{T}_{\mu\nu}=\Phi^{-1}T_{\mu\nu}$. One should also bear in mind that $\hat{R}_{\mu\nu}=\bar{R}_{\mu\nu},\bar{R}=h^{\mu\nu}\bar{R}_{\mu\nu}=\Phi^{-1}\hat{R}$ and $h_{\mu\nu}\bar{R}=\ g_{\mu\nu}\hat{R}$. The above equations are field equations of the scalar-tensor action for the metric $h_{\mu\nu}$ and (non-dynamical) scalar field $\Phi$.

Using these properties and the interpretation of the Palatini gravity it was shown aneta ; anet that the generalized Tolman-Oppenheimer-Volkoff (TOV) equations read

[TABLE]

where the generalized energy density $Q$ and the pressure $\Pi$ are defined as

[TABLE]

Recall that $\bar{U}$ and $\Phi$ depend on the choice of the model we are interested in.

In what follows we will focus on the Starobinsky model $f(\hat{R})=\hat{R}+\beta\hat{R}^{2}$ Starobinsky:1980te considered in aneta2 in the context of the Lane-Emden equation. The generalized versions of Lane-Emden equation were already studied in aneta2 ; riazi ; capH ; saito ; andre ; sak ; koy , and some of them were obtained from the generalized TOV equations aneta ; AltNS1 ; AltNS2 ; AltNS3 ; AltNS4 ; AltNS5 ; AltNS6 ; AltNS7 ; aw1 .

We follow the Weinberg convention weinberg , and hence the signature of the metric is $(-+++)$ while $\kappa=-8\pi G/c^{4}$.

II Exact solutions of the Lane-Emden equation

The standard Lane-Emden equation coming from the Newtonian limit of the GR equilibrium equations (TOV equations) and polytropic equation of state $p=K\rho^{\gamma}$, where $K$ and $\gamma$ are polytropic parameters, has the following form:

[TABLE]

here $n=\frac{1}{\gamma-1}$.

Recall the relation among dimensionless quantities appearing in (9) and the physical ones:

[TABLE]

where $p_{c}$ and $\rho_{c}$ are central pressure and energy density, respectively.

Equation (9) possesses three exact solutions for $n=\{0,1,5\}$ and, generally speaking, it proved pretty much impossible to find exact solutions of the generalized Lane-Emden equations considered in riazi ; capH ; saito ; andre ; sak ; koy . Thus, for other values of $n$ or in the case of generalized Lane-Emden equations, the approximate and/or numerical methods were applied in order to examine the properties of stars.

As shown in aneta2 , the generalized Lane-Emden equation for the Starobinsky Lagrangian $f(\hat{R})=\hat{R}+\beta\hat{R}^{2}$ in the Einstein frame is given by

[TABLE]

where $\alpha=\kappa c^{2}\beta\rho_{c}$, with $\rho_{c}$ being a central density. The extra term including $\alpha$ occurring in (12) comes from the potential $U$ appearing in (5a) and definitions (8) whose form depends on the Lagrangian. The details of the derivation can be found in aneta2 .

Under the conformal transformation $\bar{\xi}^{2}=\Phi\xi^{2}$, where $\Phi=1+2\alpha\theta^{n}$, the above equation takes the form

[TABLE]

An approximation of (13) was analyzed from the perspective of finding the numerical solutions in aneta2 . Notice that for $\alpha=0$ we have $\Phi=1$ and (13) boils down to (9).

Now note that for $n=0$ equation (12) becomes linear,

[TABLE]

and its general solution is readily found to read

[TABLE]

where $C_{i}$ are arbitrary constants.

The transformation $\bar{\xi}^{2}=\Phi\xi^{2}$ in this case amounts to a rescaling $\bar{\xi}=(1+2\alpha)^{1/2}\xi$, so the general solution of (13) for $n=0$ reads

[TABLE]

where we have rescaled $C_{1}$ for the sake of convenience.

If $C_{1}\neq 0$ then this solution has singularity at $\xi=0$, which is unphysical, so we should assume that $C_{1}=0$. Upon further imposing the boundary conditions in the Jordan frame, $\theta(0)=1$, $\theta^{\prime}(0)=0$, we find that (16) becomes

[TABLE]

Notice that it has the same form as the solution of the standard Lane-Emden equation (9) for $n=0$ describing incompressible stars, and for this reason it has been discussed in many textbooks (see e.g. weinberg ).

Our key observation now is that for $n=1$ the generalized Lane-Emden equation (13) has an exact solution

[TABLE]

which appears to be new and can be employed to describe a Newtonian neutron star.

III Newtonian neutron star in Palatini gravity

As we have already mentioned, the solutions of the Lane-Emden equation have to satisfy the boundary conditions, namely $\theta(0)=1$ and $\theta^{\prime}(0)=0$. The solution (18) for $n=1$ turns out to satisfy them only if $\alpha=\frac{3}{4}$.

This value lies in the allowed range $\alpha>-1/2$. Indeed, because of the boundary condition $\theta(0)=1$ the conformal factor $\Phi$ for $\xi=0$ takes the value $1+2\alpha$. Now, for $\alpha=-1/2$ this value becomes zero which is not allowed since the conformal factor may not change sign, cf. aneta2 for details. Moreover, the numerical analysis in aneta2 also indicates that the values of $\alpha$ smaller than $-1/2$ are unphysical.

This means that when we assume that the value of the central density of an average neutron star is $\rho_{c}\sim 8\cdot 10^{17}\frac{\text{kg}}{\text{m}^{3}}$, the Starobinsky parameter is found to be

[TABLE]

Now we are able to find the star’s mass, radius, and central density. Recall that by introducing the dimensionless quantities edd ; weinberg :

[TABLE]

we can write down the star’s mass, the mass-radius relation, central density, and temperature, in the form

[TABLE]

where the last equation for the temperature profile is given by the assumption that the gas is ideal with the equation of state $T=\frac{k_{B}\rho T}{\mu}$, with $k_{B}$ and $\mu$ being the Boltzmann constant and mean molecular weight, respectively.

However, let us notice that the above quantities were defined for the General Relativity setting. Since we wish to apply them to the Palatini gravity, we must remember that the above expressions are the ones in Einstein frame. Thus, the quantities $\omega_{n}$ and $\delta_{n}$ should be rewritten; notice that the mass (7) is still written in the Einstein frame because the $r$-coordinate comes from the conformal metric $h$. This can be directly verified by applying the generalized Lane-Emden equation (13) to (7) while taking into account that the conformal transformation in the case of Starobinsky model preserves the polytropic equation of state for small values of $p$ mana . Thus, in the case of Palatini gravity we should have written

[TABLE]

Then, applying the conformal transformation relation $\bar{\xi}^{2}=\Phi\xi^{2}$, we find that

[TABLE]

Now we are able to calculate the physical values of the star’s mass. Moreover, since the solution for $n=1$ in the case of the standard Lane-Emden equation is known, $\theta_{N}=\frac{\sin{\xi}}{\xi}$, we can compare Newtonian neutron stars in the two models. Here and below we will denote the solution and values obtained from the standard Lane-Emden equation by the subscript $N$.

We plot the solutions in Figure 1. Notice that Figure 1 also represents the dimensionless temperature profile as $T\sim\theta$.

The neutron star’s radius $\xi_{R}$ is defined by the first zero of the solution, $\theta(\xi_{R})=0$, and hence we find that $\xi_{R}=\sqrt{15}$ while $\omega_{1}\sim 3.87$, and $\delta_{1}=5$. Therefore, we immediately can compare the masses and radii of the stars

[TABLE]

where we have used $(\xi_{R})_{N}=\pi$ and $\omega_{N}=\pi$ weinberg .

IV Conclusions

We have found two exact solutions for $n=0$ and $n=1$ of the modified Line-Emden equation (13) using the Einstein-Jordan frame correspondence Afonso:2018bpv ; Afonso:2018mxn ; beltran ; olek , keeping in mind that physical properties of the star such as mass and radius are the ones appearing in Jordan frame, which we demonstrated clearly in the discussion preceding the formulas (27) and (28). We have not examined the result for $n=0$ since it would describe an incompressible star. We should notice that the latter solution has exactly the same form as the solution of the standard Lane-Emden equation. However, the case of the Newtonian neutron star (the solution for $n=1$), although being just a toy model, show that one deals with the stellar objects bigger than the stars of the pure GR models. Let us also stress that in this work we have used the exact formula for obtaining the star’s mass, that is, the equations (23) and (29) in contrast to the previous work aneta2 where the approximated formula for mass was used.

We expect that the examination of the full relativistic theory, described by the modified TOV equations (6) and (7), will also provide a similar conclusion, that is, that the Palatini star is larger and heavier for the negative Starobinsky parameter (positive $\alpha$ parameter), without a need of introducing exotic matter in their internal structure. It should be clarified here that the result was obtained for the negative Starobinsky parameter $\beta$: the parameter $\alpha$ appearing in the modified Lane-Emden equation consists of the constant $\kappa$, which had been defined to be negative.

Moreover, requiring that the solution (18) has to satisfy the boundary conditions accompanying the Lane-Emden equation, we obtained the exact value of the parameter $\alpha=\frac{3}{4}$ which led to the fact that the Starobinsky parameter is negative and of the order $10^{6}\text{m}^{2}$ under the assumption of the central energy density $\rho_{c}\sim 8\cdot 10^{17}\frac{\text{kg}}{\text{m}^{3}}$.

Let us also mention a discussion in beltran that in the Palatini theories (for example, $f(\hat{R})$ gravity and Born-Infeld inspired gravity, cf. e.g. b1 ; b2 ; b3 ; b4 ) the energy density must be continuous and differentiable function in order to avoid divergences in the field equations. This comes from the fact that one obtains $\hat{R}=\hat{R}(\rho)$ which appears in the conformal function $\Phi$ taking part in the conformal transformation which includes the derivatives of $\rho$.

Therefore, we would like to conclude that the Palatini gravity is a viable alternative to General Relativity since it passes Solar System tests (vacuum solutions are equivalent to General Relativity with cosmological constant) barraco ; al1 ; al2 , introduces inflation preserving late time accelerated expansion BSSW1 ; BSSW2 ; BSS ; SSB , gives clues on the Dark Matter problem wojn_galax , and provides conditions for existence of stable relativistic stars anet that are similar to General Relativity.

Acknowledgements.

AW acknowledges financial support from FAPES (Brazil). The research of AS was supported in part by the Grant Agency of the Czech Republic (GA ČR) under grant P201/12/G028 and under RVO funding for IČ47813059. AS is pleased to thank the Institute for Theoretical Physics of the Wrocław University and, in particular, Andrzej Borowiec, for the warm hospitality extended to him in the course of his visits to Wrocław where the present research project was initiated. Furthermore, the authors warmly thank Andrzej Borowiec, Gonzalo Olmo, Diego Rubiera-Garcia, and Hermano Velten for stimulating discussions and comments.