Neutron star models in frames of f (R) gravity

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1 Neutron star models in frames of f R) gravity Artyom V. Astashenok Citation: AIP Conference Proceedings 166, 99 14); View online: View Table of Contents: Published by the American Institute of Physics

2 Neutron star models in frames of f R) gravity Artyom V. Astashenok I. Kant Baltic Federal University, Institute of Physics and Technology, 3641, 14, Nevsky st., Kaliningrad, Russia Abstract. Neutron star models in perturbative f R) gravity are considered with realistic equations of state. In particular, we consider the FPS and SLy equations of state. The mass-radius relations for f R) =R + βre R/R 1) model and for R models with cubic corrections are obtained. In the case of R gravity with cubic corrections, we obtain that at high central densities ρ > 1ρ ns, where ρ ns = g/cm 3 is the nuclear saturation density), stable star configurations exist. The minimal radius of such stars is close to 9 km with maximal mass 1.9M SLy equation) or to 8.5 km with mass 1.7M FPS equation). This effect can give rise to more compact stars than in. If observationally identified, such objects could constitute a formidable signature for modified gravity at astrophysical level. Keywords: modified gravity, neutron stars PACS: 97.6.Jd, 4.5.Kd INTRODUCTION The current accelerated expansion of the universe has been confirmed by several independent observations. Standard candles and distance indicators point out an accelerated expansion which cannot be obtained by ordinary perfect fluid matter as source for the cosmological Friedmann equations [1, ]. This evidence gives rise to difficulties in order to explain the evolution of large scale structures. In particular, the discrepancy between the amount of luminous matter revealed from observations and the critical density needed to obtain a spatially flat universe could be solved if one assumes the existence of a non-standard cosmic fluid with negative pressure, which is not clustered in large scale structure. In the simplest scenario, this dark energy, can be addressed as the Einstein Cosmological Constant and would contribute about 7% to the global energy budget of the universe. A very straightforward approach is to look for explanations for dark energy within the realm of known physics. Assuming this point of view, one can propose alternative theories of gravity extending the Einstein theory in this sense one deals with modified gravity), keeping its positive results, without requiring dark components, up to now not detected at experimental level. In this perspective, it can be shown that the accelerated expansion can be obtained without using new fundamental ingredients but enlarging the gravitational sector see for example [3]). However, this approach needs new signatures in order to be accepted or refuted. In particular, exotic astrophysical structures, which cannot be addressed by standard gravity, could constitute a powerful tool to address this problem. The study of relativistic stars in modified gravity could have very interesting consequences to address this issue. In fact, new theoretical stellar structures emerge and they could have very important observational consequences constituting the signature for the Extended Gravity. In particular, considering the simplest extension of General Relativity, namely the f R) gravity, some models can be rejected because do not allow the existence of stable star configurations [4, 5, 6, 7, 8]. On the other hand, stability can be achieved in certain cases due to the so called Chameleon Mechanism [9, 1]. In this framework, we investigate two f R) models, namely f R) =R + βrexp R/R ) 1 and R model with cubic [ f R)=R+αR 1+γR)] corrections. In particular, we consider the FPS and SLy equations of state. One of the results is that, if cubic term, at some densities, is comparable with the quadratic one, stable star configurations exist at high central densities. Clearly, such objects cannot be achieved in the context of General Relativity [11] so their possible observational evidences could constitute a powerful probe for modified gravity [1, 13, 14]. II Russian-Spanish Congress on Particle and Nuclear Physics at all Scales, Astroparticle Physics and Cosmology AIP Conf. Proc. 166, ); doi: 1.163/ AIP Publishing LLC /$3. 99

3 Let us start from the action for f R) gravity MODIFIED TOV EQUATIONS IN f R) AVITY S = c4 d 4 x gfr)+s matter, 1) 16πG where g is determinant of the metric g μν and S matter is the action of the standard perfect fluid matter. The field equations for metric g μν can be obtained by varying with respect to g μν. It is convenient to write function f R) as f R)=R + αhr), ) where hr) is, for now, an arbitrary function. In this notation, the field equations are 1 + αh R )G μν 1 αh h RR)g μν α μ ν g μν )h R = 8πGT μν /c 4. 3) Here G μν = R μν 1 Rg μν is the Einstein tensor and h R = dr dh is the derivative of hr) with respect to the scalar curvature. We are searching for the solutions of these equations assuming a spherically symmetric metric with two independent functions of radial coordinate, that is: ds = e φ c dt + e λ dr + r dθ + sin θdφ ). 4) The energy momentum tensor in the r.h.s. of Eq. 3) is that of a perfect fluid. The components of the field equations can be written as 8πGρ/c = r + e λ 1 rλ )r + αh R r + e λ 1 rλ )r ) 1 αh h RR)+e λ α[h Rr 1 rλ )+h R], 5) 8πGP/c 4 = r + e λ 1 + rφ )r + αh R r + e λ 1 + rφ )r ) 1 αh h RR)+e λ αh Rr 1 + rφ ), 6) where prime denotes derivative with respect to radial distance, r. For the exterior solution, we assume a Schwarzschild solution. For this reason, it is convenient to define the change of variable [15, 16] e λ = 1 GM c r. 7) The value of parameter M on the surface of a neutron stars can be considered as a gravitational star mass. The following relation GdM c dr = 1 ) 1 e λ 1 rλ ), 8) is useful for the derivative dm/dr. The hydrostatic condition equilibrium can be obtained from the Bianchi identities which give conservation equation of the energy-momentum tensor, μ T μν =, that, for a perfect fluid, is dp dr = ρ + P/c ) dφ,. 9) dr The second TOV equation can be obtained by substitution of the derivative dφ/dr from 9) in Eq.6). Then we use the dimensionless variables defined according to the substitutions M = mm, r r g r, ρ ρm /r 3 g, P pm c /r 3 g, R R/r g. Here M is the Sun mass and r g = GM /c = km. In terms of these variables, Eqs. 5), 6) can be rewritten, after some manipulations, as 1 + αrgh R + 1 ) dm αr gh Rr dr = 4πρr 1 4 αr rg h h R R 1 m ) )) h R + h R, 1) r r 1

4 8π p = 1 + αrgh ) m R r 3 1 m r 1 αr g where = d/dr. The trace of field Eqs. 3) gives the third equation: h h R R 4 ) ) r 1 + αr gh R )+αrgh R 1 m ) ) h R, r r 1 dp ρ + p) dr 11) 3α h R + αh R R αh R = 8πG c 4 3P + ρc ). 1) In dimensionless variables, we have 3αr g r 3m r dm rdr 1 m r ) ) dp d ρ + p)dr dr + 1 m r ) d ) h R +αr gh R R αr gh R = 8πρ 3p). dr 13) One has to note that the combination αrghr) is a dimensionless function. We need to add the EoS for matter inside star to the Eqs. 1), 11), 13). NEUTRON STAR MODELS IN f R) AVITY The solution of Eqs. 1)-13) can be achieved by using a perturbative approach see [17] for details). For a perturbative solution the density, pressure, mass and curvature can be expanded as p = p ) + α p 1) +..., ρ = ρ ) + αρ 1) +..., 14) m = m ) + αm 1) +..., R = R ) + αr 1) +..., where functions ρ ), p ), m ) and R ) satisfy to standard TOV equations assumed at zeroth order. Terms containing h R are assumed to be of first order in the small parameter α, so all such terms should be evaluated at Oα) order. We have, for the m = m ) + αm 1), the following equation dm dr = 4πρr αr 4πρ ) h R + 1 ) 4 h h RR) + 1 α r 3m ) 4πρ ) r 3) ) d dr + rr m) ) d dr h R 15) and for pressure p = p ) + α p 1) r m dp ρ + p dr = 4πr p + m r αr 4π p ) h R + 1 ) 4 h h RR) α r 3m ) + π p ) r 3) dh R dr. 16) The Ricci curvature scalar, in terms containing h R and h, has to be evaluated at O1) order, i.e. R R ) = 8πρ ) 3p ) ). 17) We consider two EoS for the description of the behavior of nuclear matter at high densities: the SLy [19] and FPS [] equations. Model 1. Let s consider the simple exponential model f R)=R + βrexp R/R ) 1), 18) where R is a constant. Similar models are considered in cosmology, see for example [18]. We can assume, for example, R = rg.forr << R this model coincides with quadratic model of f R) gravity. The neutron stars models in frames of quadratic gravity is investigated in detail in [17]. It is interesting to consider the model 18) for the investigations of higher order effects. For the FPS and SLy EoS interesting effects can occur. For β.15 at high central densities ρ c g/cm 3 ), we have the dependence of the neutron star mass from central density see Figs. 1-). Although 11

5 β = β =.3 M/M 1 M/M 1 β = β = R, km logρ/g cm 3 ) FIGURE 1. The mass-radius diagram upper panel) and dependence of mass from central density down panel) for neutron stars in f R) model 18) in comparison with General Relativity by using a FPS equation of state. The constraints derived from observations of three neutron stars from [1] is depicted by the dotted contour hereinafter). β = β =.3 M/M 1 M/M 1 β = β = R, km logρ/g cm 3 ) FIGURE. The mass-radius diagram left panel) and dependence of mass from central density right panel) for neutron stars in f R) model 18) in comparison with General Relativity for a SLy equation of state. the applicability of perturbative approach at high densities is doubtful, it indicates the possibility of a stabilization mechanism in f R) gravity. This mechanism, as can be seen from a rapid inspection of Figs. 1-, leads to the existence of stable neutron stars which are more compact objects than in General Relativity. In principle, the observation of such objects could be an experimental probe for f R) gravity. Model. It is interesting to investigate also the R model with a cubic correction: f R)=R + αr 1 + γr). 19) The case where γr O1) for large R is more interesting. In this case the cubic term comparable with quadratic term. Of course we consider the case when αr 1+γR) << R. In this case the perturbative approach is valid although the cubic term can exceed the value of quadratic term. For small masses, the results coincides with R model. For narrow region of high densities, we have the following situation: the mass of neutron star is close to the analogue mass in General Relativity with dm/dρ c >. This means that this configuration is stable. For α = cm, γ = 1 in units r g) the maximal mass of neutron star at high densities ρ > g/cm 3 is nearly 1.88M and radius is about 9 km SLy equation, see Fig. 3). For γ = the maximal mass is 1.94M and radius is about 9. km. In the General Relativity, for SLy equation, the minimal radius of neutron stars is nearly 1 km. Therefore such a model of f R) gravity can give rise to neutron stars with smaller radii than in General Relativity. For FPS equation of state, we have the similar situation. Therefore such theory can describe assuming only the SLy equation), the existence of peculiar neutron stars with mass M the measured mass of PSR J [1]) and compact stars R 9 km) with masses M M see [, 3, 4]). 1

6 α 9 =5 α 9 =7.5 M/M R, km FIGURE 3. The mass-radius diagram for neutron stars in f R) model with cubic corrections 19)α 9 = α/1 9 cm ) for γ = 1 in comparison with General Relativity assuming a SLy equation of state. The parameters of stable configurations on the second branch of stability are 1.8M < M < 1.89M,9.4 < R < 9.36 km and < ρ c < g/cm 3. The parameter γ is measured in units rg. For comparison, the M R relation in General Relativity is also drawn for dm/dρ c <. One has to note that at given values of γ, the cubic term is greater than quadratic only at high densities. For γ = 1 at ρ > g/cm 3 that corresponds to central regions of star) and its maximum value γr 3 max R. The perturbative approach is valid at these values of parameters. CONCLUSION We have considered the mass-radius relations for neutron stars in a gravity models of the form f R)=R+Re R/R 1) and for the R models with cubic corrections. We also investigated the dependence of the maximal mass from the central density of the structure. In the case of quadratic gravity with cubic corrections, we found that, for high central densities ρ > 1ρ ns, where ρ ns = g/cm 3 is the nuclear saturation density) stable star configurations exist. In other words, we have a second branch of stability with respect to the one existing in General Relativity. The minimal radius of such stars is close to 9 km with maximal mass 1.9M SLy equation) or to 8.5 km with maximal mass 1.7M FPS equation). This effect gives rise to more compact stars than in General Relativity and could be extremely relevant from an observational point of view. In fact, it is interesting to note that using an equation of state in the framework of f R) gravity with cubic term gives rise to two important features: the existence of an upper limit on neutron star mass M ) and the existence of neutron stars with radii R km and masses 1.7M. REFERENCES 1. S. Perlmutter et al. [Supernova Cosmology Project Collaboration], Astrophys. J. 517, ) [arxiv:astro-ph/981133].. A. G. Riess et al. [Supernova Search Team Collaboration], Astron. J. 116, ) [arxiv:astro-ph/9851]. 3. S. Capozziello and V. Faraoni, Beyond Einstein Gravity Springer) New York 1). 4. F. Briscese, E. Elizalde, S. Nojiri and S. D. Odintsov, Phys. Lett. B 646, 15 7) [arxiv:hep-th/61]. 5. M. C. B. Abdalla, S. Nojiri and S. D. Odintsov, Class. Quant. Grav., L35 5) [arxiv:hep-th/49177]. 6. K. Bamba, S. Nojiri and S. D. Odintsov, JCAP 81, 45 8) [arxiv: [hep-th]]. 7. T. Kobayashi and K. i. Maeda, Phys. Rev. D 78, ) [arxiv:87.53 [astro-ph]]. 8. S. Nojiri and S. D. Odintsov, Phys. Lett. B 676, 94 9) [arxiv: [hep-th]]. 9. S. Tsujikawa, T. Tamaki and R. Tavakol, JCAP 95, 9) [arxiv:91.36 [gr-qc]]. 1. A. Upadhye and W. Hu, Phys. Rev. D 8, 64 9) [arxiv: [astro-ph.co]]. 11. W. Becker Ed.) Neutron Stars and Pulsars, Vol. 357, Astrophysics and Space Science Library, Springer, New York 9). 1. M. De Laurentis and I. De Martino, [arxiv:13. [gr-qc]]. 13. J. Antoniadis et al., Science 34, ) [arxiv: [astro-ph.he]]. 14. P. C. Freire P.C. et al., MNRAS 43, 338 1). 15. H. Stephani, General Relativity, Cambridge Univ. Press, Cambridge 199). 16. A. Cooney, S. De Deo and D. Psaltis, Phys. Rev. D 8, ) [arxiv:9148 [astro-ph.he]]. 17. S. Arapoglu, C. Deliduman, K. Yavuz Eksi, JCAP 117, 11) [arxiv: v3[gr-qc]]. 18. E. Elizalde et al., Phys. Rev. D 83 11) 866 [arxiv:11.8 [hep-th]]. 19. F. Douchin, P. Haensel, Phys. Lett. B 485, 17 ).. V.R. Pandharipande, D.G. Ravenhall, Hot nuclear matter in Nuclear Matter and Heavy Ion Collisions, Reidel, Dordrecht 1989) pp

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