Slowly rotating neutron and strange stars in R 2 gravity

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    Slowly rotating neutron and strange stars in R2 gravity

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  • JCAP10(2014)006

    ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

    Slowly rotating neutron and strange

    stars in R2 gravity

    Kalin V. Staykov,a,b Daniela D. Doneva,b,c Stoytcho S. Yazadjieva,b

    and Kostas D. Kokkotasb,d

    aDepartment of Theoretical Physics, Faculty of Physics, Sofia University,Sofia 1164, Bulgaria

    bTheoretical Astrophysics, Eberhard Karls University of Tubingen,Tubingen 72076, Germany

    cINRNE Bulgarian Academy of Sciences,1784 Sofia, Bulgaria

    dDepartment of Physics, Aristotle University of Thessaloniki,Thessaloniki 54124, Greece


    Received July 14, 2014Revised August 20, 2014Accepted September 3, 2014Published October 2, 2014

    Abstract. In the present paper we investigate self-consistently slowly rotating neutron andstrange stars in R-squared gravity with Lagrangian f(R) = R+aR2, where a is a parameter.For this purpose we first derive the equations describing the structure of the slowly rotatingcompact stars in f(R)-gravity and then simultaneously solve numerically the exterior andthe interior problem. The structure of the slowly rotating neutron stars is studied for twodifferent hadronic equations of state and a strange matter equation of state. The moment ofinertia and its dependence on the stellar mass and the R-squared gravity parameter a is alsoexamined in details. The numerical results show that the neutron star moment of inertiacan be up to 30% larger compared to the corresponding general relativistic models. This ismuch higher than the change in the maximum mass induced by R-squared gravity and isbeyond the EOS uncertainty. In this way the future observations of the moment of inertia ofcompact stars could allow us to distinguish between general relativity and f(R) gravity, andmore generally to test the strong field regime of gravity.

    Keywords: modified gravity, neutron stars

    ArXiv ePrint: 1407.2180

    c 2014 IOP Publishing Ltd and Sissa Medialab srl doi:10.1088/1475-7516/2014/10/006


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    1 Introduction 1

    2 Basic equations 2

    3 Results 4

    4 Conclusions 8

    1 Introduction

    The f(R) theories are natural generalizations of Einsteins theory and they are widely ex-plored alternative theories of gravity trying to explain the accelerated expansion of the uni-verse. Their essence is that the standard Einstein-Hilbert Lagrangian is replaced by a func-tion of the Ricci scalar curvature R. Many different classes of f(R) theories were constructedand examined (for a review see [13]). In the current paper we will be concentrated on theso-called R2-gravity, where the standard Einstein-Hilbert Lagrangian is replaced by R+aR2.

    Even though the f(R) theories are normally employed to explain cosmological observa-tions, the various astrophysical phenomena can also be used to impose constraints on thesetheories. Some of the most suitable objects in this direction are the neutron stars because oftheir high compactness and the rich spectrum of observations. Moreover the neutron starscan serve as a tool to test the strong field regime of the alternative theories of gravity. Asthe investigations show, nonlinear effects can appear in alternative theories of gravity whenstrong fields are considered for both neutron stars and black holes [47], which are not presentfor weak fields.

    Static neutron stars in f(R) theories of gravity were examined up to now in manypapers [818]. From the results in [18] one can conclude that the mass and the radii of neutronstars can increase considerably for certain values of the parameters, but these changes arestill comparable to the changes induced by different nuclear equations of state (EOSs). Sothe observations of the neutron star masses and radii alone can not be used to put constraintson the free parameters of the theory.

    Our purpose in the current paper is to take the first steps toward thoroughly studying ro-tating compact stars in f(R) theories of gravity and their astrophysical manifestations. Simi-lar studies were already performed in other generalized theories of gravity (see e.g. [7, 1925]).A very first step in this direction and an important extension of the results in [18] is to con-sider the slow rotation approximation (in linear order of the angular velocity ). Even thoughthis approximation is expected to be valid only for rotational frequencies below a few hundredHz, it turns out that most of the observed neutron stars fall into this category. A drawbackis that it can not account for the changes in the mass and radius of the neutron stars dueto rotation, because these effects are of the order 2. But the slow rotation approximationcan give us information about the frame dragging around neutron stars and the momentof inertial. This can be already observationally relevant, because it is expected that in thenear future the observations of double neuron stars would allow us to measure the momentof inertia with a good accuracy [26, 27]. Moreover the slow rotation approximation can beused to study the r-modes for rotating neutron stars and the associated gravitational waveemission [2830].


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    In the present paper we study models with both nuclear matter EOS and strange starEOS. The solutions are obtained numerically using a shooting method and the limiting cases(the general relativistic one and the case when a ) are checked also with the codedeveloped in [7]. Up to our knowledge this is the first self-consistent study of strange stars inf(R) theories of gravity. It is important because strange stars have very distinct propertiescompared to the standard neutron stars and our investigations can help us to determine upto what extend the results and conclusions in the present paper are EOS independent.

    Studying rapidly rotating neutron stars is also important, because as the results in thecase of scalar-tensor theories of gravity [7] show, rapid rotation can magnify significantly thedeviations from general relativity (GR). It is expected that the same will be true also forrapidly rotating neutron stars in f(R) theories, but this study is very complex and involvedand we leave it for a future publication.

    2 Basic equations

    The f(R) theories are described by the following action

    S =1


    d4xgf(R) + Smatter(g , ) , (2.1)

    with R being the scalar curvature with respect to the spacetime metric g . Smatter is theaction of the matter fields collectively denoted by . In order for the f(R) theories to be freeof tachyonic instabilities and the appearance of ghosts, the following inequalities have to besatisfied [13]


    dR2 0, df

    dR> 0 , (2.2)

    respectively. In the special case of R2 gravity the above inequalities give a0 and 1+2aR 0.It is well-known that the f(R) theories are mathematically equivalent to the Brans-Dicke

    theory (with BD = 0) given by the action

    S =1


    d4xg [RU()] + Smatter(g , ) , (2.3)

    where the gravitational scalar and the potential U() are defined by = df(R)dR


    U() = R dfdR

    f(R), respectively. In the case of R2 gravity we have = 1 + 2aR and theBrans-Dicke potential is U() = 14a( 1)


    In many cases it proved useful to study the scalar-tensor theories in the so-called Einsteinframe with metric g defined by the conformal transformation g

    = g . The Einstein

    frame action can be written in the form

    S =1



    g [R 2g V ()] + Smatter(

    e 2

    3g ,


    , (2.4)

    where R is the Ricci scalar curvature with respect to the Einstein frame metric g and

    the new scalar field is defined by =32 ln. The Einstein frame potential V () is

    correspondingly V () = A4()U(()) with A() defined as A2() = 1() = e 2

    3. In

    the case of R2 gravity one can also show that V () = 14a


    1 e2





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    In accordance with the main purpose of the present paper we consider stationary andaxisymmetric spacetimes as well as stationary and axisymmetric fluid and scalar field con-figurations. Keeping only first-order terms in the angular velocity = u/ut, the spacetimemetric can be written in the form [31]

    ds2 = e2(r)dt2 + e2(r)dr2 + r2(

    d2 + sin2 d2)

    2(r, )r2sin2ddt . (2.5)The last term in the expression for the metric reflects the influence of the rotation in

    linear order in , i.e. O(). The effect of rotation on the other metric functions and thescalar field is of order O(2). The influence of the rotation on the fluid energy density andpressure is also of order O(2). For the fluid four-velocity u, up to linear terms in , onefinds u = ut(1, 0, 0,), where ut = e(r).

    Taking into account all the symmetries and conditions imposed above, the dimensionallyreduced Einstein frame field equations containing at most terms linear in , are the following






    1 e2)]

    = 8GA4()+ e2(





    2V (), (2.6)




    dr 1


    1 e2)

    = 8GA4()p+ e2(




    12V (), (2.7)





    dr d






    dr= 4G()A4()( 3p)e2 + 1


    dV ()

    de2, (2.8)


    dr= (+ p)



    dr+ ()




    , (2.9)






    r2 sin3


    sin3 ]

    = 16GA4()(+ p) , (2.10)

    where we have defined

    () =d lnA()

    d= 1

    3and = . (2.11)

    Here p and are the pressure and energy density in the Einstein frame and they are connectedto the Jordan frame quantities p and via = A4() and p = A4()p respectively.

    The above system of equations, supplemented with the equation of state for the starmatter and appropriate boundary conditions, describes the interior and the exterior of theneutron star. Evidently in the exterior of the neutron star we have to set = p = 0.

    What is important for this system of equations is the fact that the equation for isseparated from the other equations which form an independent subsystem.1 This subsystemis just the system of reduced field equations for the static and spherically symmetric case. Thenatural boundary conditions at the center of the star are (0) = c,(0) = 0, while at infinitywe have limr (r) = 0, limr (r) = 0 as required by the asymptotic flatness [18]. Thecoordinate radius rS of the star is determined by the condition p(rS) = 0 while the physicalradius of the star as measured in the physical (Jordan) frame is given by RS = A[(rS)]rS .

    The equation for is in fact an elliptical partial differential equation on a sphericallysymmetric background. This fact and the asymptotic behaviour of at spacial infinity allowus to considerably simplify this equation. Expanding in the form [31]








    , (2.12)

    1The equation for the metric function is separated from the rest of the equations for any scalar-tensortheory in the slow rotation approximation. However, this is no longer true for rapidly rotating stars as onecan see from the equations given in [7].


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    where Pl are Legendre polynomials and substituting into the equation for we find








    l(l + 1) 2r2

    l(r) = 16GA4()(+ p)l(r) . (2.13)

    In asymptotically flat spacetimes, the asymptotic of the exterior solution of (2.13) isl const1 rl2 + const2 rl1. Taking into account that 2J/r3 (or equivalently 2J/r3) for r with J being the angular momentum of the star and comparingit with the above asymptotic for , we conclude that l = 1, i.e. l = 0 for l 2. In otherwords, is a function of r only. Therefore the equation for the is








    = 16GA4()(+ p)(r) . (2.14)

    The natural boundary conditions for this equation are


    dr(0) = 0 and lim

    r = . (2.15)

    The first condition ensures the regularity of at the center of the star.One of the most important quantities we consider in the present paper is the inertial

    moment I of the compact star. It is defined in the standard way

    I =J

    . (2.16)

    Using equation (2.14) for and the asymptotic form of one can find another expressionfor the inertial moment, namely

    I =8G



    0A4()(+ p)er4



    dr . (2.17)

    In the next section where we present our numerical results we shall use the dimensionlessparameter a a/R20 and the dimensionless inertial moment I I/MR20 where M is thesolar mass and R0 is one half of the solar gravitational radius R0 = 1.47664 km (i.e. the solarmass in geometrical units).

    3 Results

    The last supplement in solving the field equations (2.6)(2.9), (2.14) is to specify the EOS.We will be working with two classes of EOSs realistic hadronic EOS and quark matterEOS. Let us note that these are the first self-consistent and non-perturbative with respectto a results of strange stars in f(R) theories of gravity, even in the static case. The hadronicEOSs are Sly4 [32] and APR4 [33], which both reach the two solar mass barrier and are inagreement with the current estimates of the neutron star radii [3438]. The strange matterEOS is taken to be of the form

    p = b( 0) , (3.1)where b and 0 are parameters obtained from fitting different quark star EOS. In this paperwe will work with b = 1/3 and 0 = 4.2785 1014g/cm3, which corresponds to the SQSB60equation of state given in [39]. This strange star EOS is just a little bit below the two solarmass barrier, but we consider it as a representative example and we expect that other strangematter EOS will lead to qualitatively similar results.


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    9 10 11 120.0






    9 10 11 12 7 8 9 10 11 12

    GR a=0.3 a=1 a=10 a=102





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