# Polytropic stars in Palatini gravity

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## Abstract

We have derived a modified Lane–Emden equation for the Starobinsky model in Palatini gravity which is numerically solvable. Comparing the results to the ones provided by General Relativity we observe a significant difference depending on the theory parameter for the \(M-R\) relations.

## 1 Introduction

The shortcomings [1, 2, 3, 4, 5] of General Relativity (GR) [6, 7, 8] make the search for some other proposals describing the gravitational phenomena necessary and appealing. The dark matter idea [9, 10], inflation [11, 12], the fact of the late-time cosmic acceleration [13, 14] with an explanation in the form of the exotic fluid called dark energy [1, 2, 3] are just the most widespread problems which we face. Looking for a generalization of Einstein’s theory is additionally supported by the fact that GR is non-renormalized while adding extra high curvature terms seems to improve the situation [15]. This is why Extended Theories of Gravity (ETG) [16, 17] have gained a lot of attention. However, many extensions introduce ghost-like instability. Nonetheless, attacking the Hilbert–Einstein action appears in many different ways: assumption on the “non-constancy” of the Nature constants [18, 19, 20], minimally or non-minimally coupled scalar fields added to the Lagrangian [21, 22], or more complicated functionals than the simple linear one used in GR, for example *f*(*R*) gravity [11, 23]. The extra geometric terms coming from the latter approach could explain not only dark matter issue [24, 25] but also the dark energy problem. Since the field equations also differ from the Einstein’s ones, they usually provide different behavior of the early Universe. One can formulate *f*(*R*) gravity in different ways: in the metric approach, [4, 5, 26, 27, 28] Palatini one [26, 29, 30, 31] as well as hybrid [32]. We will focus on the Palatini approach in this work.

The Palatini approach provides modified Friedmann equations [33, 34, 35] that can be compared with the observational data [36, 37, 38, 39, 40, 41]. It shows the potential of the Palatini formulation and it is still applied to gravitational problems [42, 43]. Moreover, there have appeared possibilities of observable effects in microscopic systems providing constraints on models parameters [44, 45]. There are also disadvantages reported: lack of perturbative approach [46], conflict with the Standard Model of particle physics [47, 48], the algebraic dependence of the post-Newtonian metric on the density [49, 50], and the complications with the initial values problem in the presence of matter [31, 51], although that issue was already solved in [52]. A similar discussion was performed in [53] where it was shown that the initial value problem is well-formulated in presence of the standard matter sources while the well-posedness of the Cauchy problem should be considered case by case: the Starobinsky one, which we are interested in, belongs to the well-posed class of models. There are also additional arguments showing that treating the extra terms in the fluid-like manner provides limitations [54, 55, 56]. However, as it was shown in [57], higher curvature corrections do not cause the above mentioned problems. Interestingly, the effective dynamics of Loop Quantum Gravity as well as brane-world cosmological background histories may be reproduced by the Palatini gravity giving the link to one of the approaches to Quantum Gravity [58, 59].

There are also astrophysical aspects of Palatini gravity, for instance black holes were considered in [60, 61, 62, 63, 64, 65], and also wormholes [66, 67, 68] and neutron stars [69, 70, 71, 72, 73]. Our concern is related to the last objects in Palatini gravity, especially that the recent neutron stars’ merger observation [74] will provide the possible confrontation of gravitational theories. Neutron stars seem to be perfect objects for testing theories at high density regimes: there are claims that using General Relativity in the case of strong gravitational fields and in the case of large spacetime curvature [75, 76, 77] is an extrapolation.

Stars in Palatini gravity were considered in [78, 79, 80, 81, 82] where it was claimed that there exist surface singularities of static spherically symmetric objects in the case of polytropic equation of state which can lead to infinite tidal forces on the star’s surface. An argument against that claim was introduced in [83]: the problem is caused by the particular equation of state which should not be used at the surface. Moreover, in [84] it was indicated that the polytropic equation of state is nothing fundamental but rather an approximation of the matter forming a star. The another important point was mentioned that Palatini gravity should be interpreted according to the Ehlers–Pirani–Schild (EPS) approach [85, 86, 87], which we are going to follow in this work. That means that a conformal metric is the one responsible for the free fall in comparison to metric which was used in [78]. As shown in [84], in this case the singularities are not generated and polytropic stars can be obtained in the Palatini framework.

Using this result, we are going to study non-relativistic stars with the polytropic equation of state in \(f(\hat{R})\) Palatini gravity. As a working example we will use the Starobinsky model, that is, \(f(\hat{R})=\hat{R}+\beta \hat{R}^2\) [11]. In order to do it for the wide class of the stellar objects, we will write down the modified Lane–Emden equation obtained from the generalized Tolman–Oppenheimer–Volkoff (TOV) equation which we studied in the context of star’s stability in [73]. That will allow to examine further different types of stars since the equations describing them can be solved numerically. There are already works considering modified TOV equations [88, 89, 90, 91, 92, 93, 94, 95, 96] as well as ones providing generalization of the Lane–Emden [97, 98, 99, 100, 101, 102].

We are using the Weinberg’s [103] signature convention, that is, \((-,+,+,+)\), with \(\kappa =-8\pi G/c^4\).

## 2 Stellar objects in Palatini gravity

### 2.1 Palatini formalism

*A*such that the above Eq. (3) is true for a metric

*g*, then there exists a 1-form \(\tilde{A}\) for any other metric \(\tilde{g}\) such that (3) selects the same connection \(\tilde{\varGamma }\). When we consider a triple which consists of the spacetime manifold

*M*and EPS-compatible structures on

*M*, then we deal with EPS geometry. We may put some extra conditions on it: when

*A*is fixed in the way that it depends on

*g*which had been chosen in the conformal structure than the geometry is called a Weyl geometry. It is integrable when there exists a connection \(\tilde{\varGamma }\) which is a Levi-Civita connection of \(\tilde{g}\). In that case there exists a relation between

*A*and the conformal factor \(\varOmega \): \(A_\mu =\partial _\mu \varOmega \) [105].

It should be noticed that GR is a special case of the EPS formalism where it is assumed that the connection \(\tilde{\varGamma }\) is a Levi-Civita connection of the metric *g* (the 1-form *A* is zero). Thus one treats the action of the theory as just metric-dependent. But we may consider the Einstein-Hilbert action which depends on two independent objects: the metric *g* and the connection \(\tilde{\varGamma }\). This approach is called Palatini formalism. Since one uses the simplest gravitational Lagrangian which is linear in scalar curvature *R*, Palatini approach turns out to provide that \(\tilde{\varGamma }\) is a Levi-Civita connection of the metric *g*. The difference is that this is the dynamical result, not a assumption as it happens in the previous case. However, the situation is different when we are interested in more complicated Lagrangians like the ones appearing in ETGs.

*g*and the connection \(\hat{\varGamma }\). From the field equations it turns out that the connection is a Levi-Civita connection of a metric conformally related to

*g*. Therefore, one should consider motion of a mass particle provided by the geodesic equation with the connection \(\hat{\varGamma }\). Clocks and distances in contrast are measured by the metric

*g*. Thus, we are supplied with the action

*T*is the trace of the energy-momentum tensor. If it is possible to solve (8) as \(\hat{R}=\hat{R}(T)\) we observe that \(f(\hat{R})\) is also a function of the trace of the energy momentum tensor, where \(T=g^{\mu \nu }T_{\mu \nu }\equiv 3p-c^2\rho \).

### 2.2 Generalized Tolman–Oppenheimer–Volkoff equation

*r*is the conformal coordinate which should be taken into account in the further analysis. The generalized energy density and pressure are

### 2.3 Modified Lane–Emden equation for the Starobinsky \(f(\hat{R})=\hat{R}+\beta \hat{R}^2\) Lagrangian

*K*and \(\gamma \) being the parameters of the polytropic EoS. The key observation is that for small values of

*p*the conformal transformation (7) preserves the polytropic equation of state [84]. Due to that fact, in the case of the Starobinsky model \(f(\hat{R})=\hat{R}+\beta \hat{R}^2\) we may write:

*r*give us

### 2.4 Solutions of the modified Lane–Emden equation

It would be also interesting to put some constraint on the upper bound. Since we are dealing with modofications to Newtonian gravity, we work with a weak gravitational field. Thus, the matter moves slowly and its velocity relative to Solar System center mass is \(v^2\le 10^{-7}\). An expression for the circular velocity of an object moving around a mass centre in the case of Starobinsky model in Palatini formalism under EPS interpretation was obtained in [41]. Using this expression one may try to bound the parameter \(\alpha \) and hence considering an object moving around the Sun on the roughly circular orbit with \(r=1\text {AU}\) one finds that the upper bound is around \(10^4\).

Ratios of numerical values of the radius and masses for the different types of stars (for example \(R_i,\;i=\{1,3\}\) denotes a radius ratio for \(n=1\), \(n=3\), respectively) for various values of \(\alpha \)

\(\alpha \) | \(R_1\) | \(M_1\) | \(R_3\) | \(M_3\) |
---|---|---|---|---|

1 | 4.05 | 2.01 | 1.94 | 4.79 |

0.5 | 1.75 | 1.52 | 1.2 | 2.02 |

0.3 | 1.36 | 1.37 | 1.06 | 1.5 |

0.1 | 1.1 | 1.21 | 0.99 | 1.14 |

0 | 1 | 1 | 1 | 1 |

\(-\)0.1 | 0.91 | 0.85 | 1.03 | 0.88 |

\(-\)0.3 | 0.78 | 0.56 | 1.23 | 0.73 |

\(-\)0.49 | 0.73 | 0.35 | 1.5 | 0.69 |

## 3 Conclusions

In contrast to the existing works on Palatini stars, we have used the EPS interpretation of the theory which provides different TOV equations. Together with the previous studies on neutron stars [73], galaxy rotation curves [41], and cosmology [86, 87, 116] the current proposal has added new arguments in favor of Palatini gravity under the EPS formulation.

We have derived the Lane–Emden equation coming from the Palatini modified equations describing the relativistic stellar object. Apart from the quadratic term in \(\theta ^n\) on the right-hand side it resembles the modified equations obtained already in the literature [97, 98, 99, 100, 101, 102]. The numerical solutions of the Eq. (35) pictured in the Figs. 1, 2 and 3 definitely shows that we deal with larger stars together with increasing the parameter \(\alpha \) for \(n=1\). The case \(n=3\) differs a lot: for the positive parameter \(\alpha \) the situation is similar like for \(n=1\) while for negative values (see the curves in the bottom of the Fig. 3) is opposite in the case of the radius: the star is larger with respect to decreasing \(\alpha \) while masses tend to decrease. Moreover, independently of the type of a star, because of the conformal transformation the case \(\alpha =-\frac{1}{2}\) is excluded while below \(\alpha =-\frac{1}{2}\) there are unphysical profiles.

The masses are significantly larger than in GR case, especially for bigger values of the parameter in both cases. Decreasing \(\alpha \) one obtains smaller masses where in the case of negative values of the parameter we deal with masses smaller than the ones predicted from GR. We have not considered the masses for the case \(n=1.5\) because the numerical solutions suffered by the extrapolation procedure and we do not treat the results reliable. We should also remember that in all cases we have used the simplified mass formula (39).

Although our studies should be viewed as a toy model, we consider it as a first step to the more accurate stellar description which we leave for the future projects. In this sense, the recent finding of a mapping between Palatini theories of gravity and GR [117, 118] may be helpful for this analysis. Work along these lines is currently underway.

## Notes

### Acknowledgements

The author would like to thank Gonzalo Olmo, Diego Rubiera-Garcia, Artur Sergyeyev, and Hermano Velten for their comments and helpful discussions. The work is supported by the NCN grant *DEC-2014/15/B/ST2/00089* (Poland) and FAPES (Brazil).

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