Stellar Structure Model in Hydrostatic Equilibrium in the Context of $ F

Stellar structure model in hydrostatic equilibrium in the context of f(R)-gravity Ra´ıla André∗ and Gilberto M. Kremery Departamento de F´ısica, Universidade Federal do Paraná, 81531-980 Curitiba, Brazil. (Dated: August 29, 2017) In this work we present a stellar structure model from f(R)-gravity point of view capable to describe some classes of stars (White Dwarfs, Brown Dwarfs, Neutron stars, Red Giants and the Sun). 2 This model was based on f(R)-gravity field equations for f(R) = R+f2R , hydrostatic equilibrium equation and a polytropic equation of state. We compared the results obtained with those found by the Newtonian theory. It has been observed that in these systems, where high curvature regimes emerge, stellar structure equations undergo modifications. Despite the simplicity of this model, the results were satisfactory. The estimated values of pressure, density and temperature of the stars are within those determined by observations. This f(R)-gravity model has been proved to be necessary to describe stars with strong fields such as White Dwarfs, Neutron stars and Brown Dwarfs, while stars with weaker fields, such as Red Giants and the Sun, were best described by the Newtonian theory. I. INTRODUCTION developed. In [19], using the method of perturbative con- straints and corrections of the form Rn+1, they showed The f(R)-gravity is a class of theories that represent that the predicted mass-radius relation for neutron stars an approach to gravitational interaction. In this con- differs from that calculated in the General Relativity. text, General Relativity (GR) has to be extended in or- In [20], the mass-radius diagram for static neutron star der to solve several issues. This approach considers mod- were obtained in f(R) gravity for two functions: f(R) = 2 1+ ifications of the Einstein-Hilbert action in order to in- R + αR (1 + γR) and f(R) = R . New terms related clude higher-order curvature invariants with respect to to curvature corrections emerged and modified the evolu- the Ricci scalar [1, 2]. From the astrophysical and cos- tion of the mass-radius relation. In [21], realistic models 2 mological point of view, the objective is to explain phe- of relativistic stars in f(R) = R+αR gravity have been nomena such as dark energy and dark matter under a explored. In this context, the authors presented a study geometric pattern [3{11] with the possibility that gravi- on the existence of neutron and quark stars for various tational interaction depends on the scales. In this sense, α with no intermediate approximation in the system of in principle, these theories do not require the introduc- equations. On the other hand, in [22], quark star models tion of new particles and preserve all sucessful results of with realistic equation of state in nonperturbative f(R) Einstein's theory, based on the same fundamental phys- gravity have been considered. The authors showed that it ical principles. It is also known that some f(R) gravity is possible discriminate modified theories of gravity from models can pass in test performed in the weak-field solar General Relativity due to the gravitational redshift of the system [10]. In addition, a considerable number of viable termal spectrum emerging from the surface of the star. In 2 f(R) models are known, among which we highlight [12{ [23], a stellar structure model in f(R) = R+αR theory 16]. Futhermore, no extended gravity model, until this was considered, using the method of matched asymptotic moment, can handle all phenomenology ranging from the expansion to handle the higher order derivatives in field quantum scale to the cosmological scale [17]. Another equations. Solutions were found for uniform density stars problem is that the description of the f(R) theory is sub- matching to the Schwarzschild solution outside the star. The mass-radius relations were obtained, observing the arXiv:1707.07675v2 [gr-qc] 26 Aug 2017 stantially equivalent to the description associated with the hypothesis of dark components. This interpretation dependence of maximium mass on α. In [24], neutron arises from the fact that the degrees of freedom present in star models in perturbative f(R) have been considered with realistic equations of state. The mass-radius rela- the f(R) theory can be expressed by an effective energy- h i momentum tensor able of giving rise to the dark matter tions for f(R) = R + βR exp( −R ) − 1 and R2 models R0 effects [2, 18]. From this picture emerges the necessity with logarithmic and cubic corrections were obtained. In of observations capable of preserving or excluding one of the case of cubic corrections, stable star configurations at these theories. high central density were obtained. Such an effect could In this framework, several works related to relativis- give rise to more compact stars than in General Rela- tic stars considering extended gravity theories have been tivity. In [25], considering a logarithmic f(R) theory, the authors showed that the model exhibts a chameleon effect which completely eliminates the effect of the mod- ification on scale exceeding a few radii, but close to the ∗ [email protected] surface of the neutron star, the deviation from General y kremer@fisica.ufpr.br Relativity can significantly affect the surface redshift. In 2 [26], the authors showed that for a simple hyperon equa- of the star and R as the Ricci scalar. tion of state it is possible to obtain the maximal neutron star mass (which satisfies the recent observational data) in higher-derivative models with power-law terms II. THE LANE-EMDEN EQUATION FOR A 2 3 as f(R) = R+γR +βR . And in [27], the authors stud- MODEL DESCRIBED BY THE f(R) THEORY ied neutron stars with strong magnetic fields. They took into account models derived from f(R) and f(G) the- Considering a spherically symmetric self-gravitating ories where function of the Ricci curvature invariant R system in equilibrium we adopt the hydrostatic equilib- and the Gauss-Bonnet invariant G are respectively con- rium equation represented below sidered. In this model, the maximal mass of neutron star had a considerable increasing in f(R) with cubic correc- dφ 1 dp = ; (1) tions. dr ρ dr In the model discussed in this work we propose second- order corrections in the Ricci scalar adopting f(R) = where ρ(r) is the matter density, p is the pressure and φ is 2 R + f2R . Such orders in the correction are an exten- the gravitational potential. The equation (1) is a Newto- sion of General Relativity and are particularly interest- nian limit of the equation resulting from the conservation ing in cosmology, since they allow the construction of a of the stress-energy tensor for a perfect fluid in hydro- self-consistent inflationary model [28]. We present the static equilibrium. It can be also achieved through the results obtained for a stellar model in hydrostatic equi- Newtonian limite of the Tolman-Oppeheimmer-Volkoff librium according to the f(R)-gravity theory. The goal equation (equation employed to describe a spherically is to control precisely how the results deviate from those symmetric astrophysical system in equilibrium in GR obtained through General Relativity in order to see how [31, 32]). strong gravity regimes affect the pressure, temperature We assume a polytropic equation of state and density. We also compare these results with those related to the Newtonian theory, already known in the p = kργ ; (2) literature [29]. In our model we consider a polytropic equation of where k is the polytropic constant and γ is the polytropic state, since this equation plays an importante role in stel- exponent. Then we insert equation (2) into equation (1), lar structure models. This equation correctly represents obtaining the stellar gas behavior and, consequently, solves the fun- dφ dρ damental problem of these structures together with the = γkργ−2 : (3) hydrostatic equilibrium equation. The motivation to em- dr dr ploy the polytropic equation in the study of stellar struc- For γ 6= 1, the integration of the above equation results ture is the simple nature of the polytropic structure and in its correspondence with known classes of stars. Such sim- n plicity provides a basis for the incorporation of additional 1 " # γ − 1 γ−1 1 φ effects (such as rotation), and thus an insight into the na- ρ = φ γ−1 = ; (4) γk (n + 1)k ture of the effects on true stars [30]. The interest of this model, in the present context, is where the chosen integration constant was φ = 0 on the manifested in the fact that, due to the expressive grav- surface (ρ = 0). The constant n is known as the poly- itational field, the interior of the stars can be seen as tropic index and is defined as n = 1 . Through the appropriate places to test alternative theories of grav- γ−1 equations (2) and (4), we obtain the following expression ity. In these regions, high curvature regimes can emerge for the pressure and modify the stellar structure. In this way, we aim to show that the pressure, temperature and density can be ρφ consistently reached by the extended theories of gravity, p = : (5) 2 (n + 1) such as f(R)-gravity for f(R) = R + f2R and how the expected changes occur in the values of these quantities. To describe a stellar structure model by f(R)-gravity This paper is organized as follows: In section II we [33, 34] we adopt the action represented below derive the modified Poisson and Lane-Emden equations Z through the Newtonian limit of f(R)-gravity.

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