Table Of ContentGeneral Relativity and Gravitation manuscript No.
(will be inserted by the editor)
Structure of neutron stars in R-squared gravity
Mariana Orellana 1,2,∗ · Federico Garc´ıa 1,2,∗∗ · Florencia A. Teppa
Pannia 2,∗∗ · Gustavo E. Romero 1,2,∗
3
1
0
2
n Received: date /Accepted: 31/12/2012
a
J
2 Abstract Theeffectsimpliedforthestructureofcom- Keywords modified gravity neutron stars gener-
2 · ·
pact objects by the modification of General Relativity alised gravity equation
produced by the generalization of the Lagrangianden-
]
O sity to the form f(R)=R+αR2, where R is the Ricci
C curvature scalar,have been recently explored. It seems 1 Introduction
. likely that this squared-gravity may allow heavier Neu-
h
p tron Stars (NSs) than GR. In addition, these objects Current cosmological observations interpreted in the
- can be useful to constrain free parameters of modified- standard cosmological model require the presence of a
o
gravity theories. The differences between alternative non-standardmattercontentinordertoexplaintheac-
r
t gravitytheories is enhanced in the stronggravitational celerated expansion of the Universe [1, 17, 18, 21, 23].
s
a regime.Inthisregime,becauseofthecomplexityofthe Along the last decade alternative cosmological models
[
field equations, perturbative methods become a good have been developed to reinterpret these data with-
1 choice to treat the problem. Following previous works outinvolvinganyunknown,doubtfulcomponentofthe
v inthefield,weperformedanumericalintegrationofthe energy-matter tensor (see, for instance, [4]). The ap-
9
structure equations that describe NSs in f(R)-gravity, pearance of Extended Theories of Gravity (ETGs) was
8
1 recovering their mass-radius relations, but focusing on stronglystimulatedbythe possibilitiestheymightpro-
5 particular features that arise fromthis approachin the vide in this context [5, 8, 27].
1. profiles of the NS interior. ETGs are based on corrections and generalizations
0 of Einstein’s General Relativity (GR) theory. We fo-
We show that these profiles run in correlation with
3
cus on a particular class, called f(R)-gravity theories,
1 the second-order derivative of the analytic approxima-
whicharebasedonamodificationoftheEinstein-Hilbert
: tion to the Equation of State (EoS), which leads to re-
v
action: the usual Lagrangian density is generalized re-
gionswheretheenclosedmassdecreaseswiththeradius
i
X placing the Ricci curvature scalar R by a function of
in a counter-intuitive way. We reproduce all computa-
r tions with a simple polytropic EoS to separate zeroth- itorhigh-orderinvariantsofthe curvaturetensor,such
a as R2, R Rµν, R Rµναβ, R(cid:3)R, R(cid:3)kR (see [5] for
order modified gravity effects. µν µναβ
a complete review).
Severalofthese models aresuccessfullyconstructed
tosatisfythecurrentSolarSystemandlaboratorytests
∗ Member of CONICET [15,16,19,28].Inparticular,thesimplestchoicef(R)=
∗∗ Fellowof CONICET R+αR2,alsocalled“R-squared”gravity,hasbeenfur-
1 Instituto Argentino de Radioastronom´ıa CCT La Plata ther studied as the basis for a viable alternative cos-
(CONICET), C.C.5, (1894) Villa Elisa, Buenos Aires, Ar- mological model, that can lead to the accelerated ex-
gentina
pansion of the Universe and is well consistent with the
2 Facultad de Ciencias Astron´omicas y Geof´ısicas, Universi-
temperatureanisotropiesobservedinCosmologicalMi-
dad Nacional de La Plata, Paseo del Bosque s/n, 1900 La
Plata, Argentina crowave Background [8]. But in contrast to gravity in
E-mail: [email protected] the weak-field regime, which has been subject to nu-
2 M. Orellana etal.
merous experimental tests, gravity in the strong-field equations, so-called TOV because of the pioneer work
regimeislargelyunconstrainedbyobservations[e.g.9]. of [29] and [20], can be expressed as:
However,otherauthors,makingamoredetailedmodel
dm(r)
ofthestructureofcompactstars,obtainedasetofmod- = 4πr2ρ(r) (2)
dr
ifiedTolman-Oppenheimer-Volkoff(TOV)equationsthat dp(r) c2ρ(r)+p(r) 4π m(r)
describeasphericallysymmetricmassdistribution,un- = G r2p(r)+ (3)
dr 2Gm(r) c2r (cid:18)c2 r (cid:19)
derhydrostaticequilibrium,insimplef(R)-gravity,and −
2Gm(r)
derived solutions that reproduce the correct behaviour e−2Λ(r) = 1 (4)
− c2r
attheweakgravitylimit[e.g.3,inthecontextofscalar-
dΦ(r) 1 dp
tensor theories of gravity]. Furthermore, working with = , (5)
dr −(c2ρ(r)+p(r))dr
a perturbative approach to solve the field equations,
Cooneyetal.[7]andSantos[25]havefoundmass-radius where m(r) is the total relativistic mass enclosed in
relationsforcompactstarsusingapolytropicEquation a sphere of radius r. The functions ρ(r) and p(r) are,
of State (EoS). More recently Arapog˘lu et al. [2] and respectively, the mass-energy density and the pressure
Deliduman et al. [10] applied the same approach using atthisradius.We explicitlykeepthe gravitationalcon-
a set of realistic EoS for Neutron Stars (NSs) in the stant,G,andthevelocityoflight,c,sincethequantities
R-squared and R Rµν gravities, respectively. areconsideredwiththeirfull-dimensionsforintegration
µν
Themass-radiusrelationsobtainedbyArapog˘luetal. purposes. Giving an explicit relation between ρ and p,
[2] and Deliduman et al. [10] indicate that such f(R) theso-calledEoS,theTOVequationscanbesolvedas-
modelscanaccommodateNSsuptomasseslargerthan suming a value for the central density, ρ(r = 0) = ρc,
thecurrentlyobservedones,whichareatmostM = and integrating the system to the pressure vanishes,
max
1.97 0.04M⊙forPSRJ1614-2230[11].TheR-squared p(r = R⋆) = 0. Here R⋆ is the radius of the star and
±
gravityintroducesanewparameterinthemodelthrough M⋆ =m(R⋆)thestellarmass.Everyρcgeneratesacou-
the value of α, the coefficient of R2. The freedom in pleofvaluesM⋆ andR⋆.Then,varyingthisparameter,
the choice of this parameter allows some EoS, which a mass-radius (M⋆ R⋆) relation is defined for every
−
are excluded within the framework of GR, to be rec- EoS.
onciled with the observations. Motivated by these re- The modified TOV equations can be obtained from
sults, we investigate in detail the structure of NSs un- the gravitationalfield equations. Adding the new term
der this model for two different EoS. One of them is a to the Hilbert-Einstein action, we have:
polytropic approximation that we use here to separate
c4
zeroth-ordermodified-gravitationaleffects,whereasthe S = d4x√ g(R+αR2)+Smatter, (6)
16πGZ −
otherprovidesarealisticrepresentationofnuclearmat-
ter at high densities. where g is the determinant of the metric gµν.
The paper is organized as follows. In Section 2, we Working in the metric formalism, the variation of
obtain the modified TOV equations following the per- theactionwithrespecttothemetricyieldsfourth-order
turbative approachto solve the field equations.In Sec- differential equations of gµν. This poses an enormous
tion 3, we present the EoS used to integrate the equa- obstacle to solve the problem thoroughly. For this rea-
tions and we briefly describe the numerical methods of son, we adopt the perturbative approach presented by
resolution. In Section 4, we present our results for the Cooney et al. [7] and Arapog˘lu et al. [2]. Rewriting
mass-radius relations,focusing on the behaviour of the f(R) = R +αR2 = R(1 +β), we consider the f(R)
profiles obtained for the NS interior.Final remarks are functionasaperturbationofaGRbackground.Hence,
shown in Section 5. the dimensionless quantity β αR comprises the de-
≡
viation from GR and the perturbative method can be
applied as long as β 1. Under this condition, we
| | ≪
2 Structure equations in R-squared gravity can work with equations of motion up to first order
in β without imposing any constrainat the levelof the
The equations that define the structure of a NS in GR actionandensuringthenatureofthevariationalprinci-
arededucedproposingthestatic,sphericallysymmetric ple[6].Neglectingtermswith (β2)orhigher,thefield
O
line element, ds, to be equations reduce, for this particular choice of f(R), to
ds2 = e2Φ(r)c2dt2+e2Λ(r)dr2+r2(dθ2+sin2θdϕ2). (1) G +α 2R R 1Rg +
− µν (cid:20) (cid:18) µν − 4 µν(cid:19)
(7)
Then, considering the Einstein equations for an ideal 8πG
+2(g (cid:3)R R) = T ,
energy-momentumtensorTνµ =diag{−ρc2,p,p,p},these µν −∇µ∇ν (cid:21) c4 µν
Structureof neutron stars inR-squared gravity 3
where G = R 1Rg is the Einstein tensor. In Mass-derivative terms are indicated, giving: dm/dr =
µν µν − 2 µν
the limit α 0, β 0, and the field equations of 4πr2ρ β A+B+C .Westudyeachtermcontribution
−→ −→ −
GR are recovered [see e.g. 5]. below. (cid:2) (cid:3)
We also assume a static and spherically symmetric Exact equations (10) and (11) are impractical be-
line element, given by equation (1). The perturbative cause β involves other derivatives through R, which
approachallows to expandthe functions presentin the makes β r-dependent in a complicated way1. Thus, we
metricintoaleadingterm(unperturbed),denotedwith integrate equations (10) and (11) assuming that β is
subscript 0, plus a corrective one, denoted with sub- well approximated by β βˆ αR , where R is the
0 0
≃ ≡
script 1, that is of first order in β: Λ = Λ +βΛ and RicciscalarlocallycalculatedinGRandαisaconstant
0 1
Φ=Φ +βΦ .Thehydrodynamicquantitiesarealsode- parameterwithsquareddistanceunits,compatiblewith
0 1
finedperturbatively:ρ=ρ +βρ andp=p +βp [2]. other authors approach [for instance 2]. Note that
0 1 0 1
Hence, new restrictions are imposed over the value of
8Gπ
β by the constraints βΦ1 ≪ Φ0, βΛ1 ≪Λ0, βρ1 ≪ ρ0, R0 = c4 (ρ0c2−3p0), (12)
andβp p .Fromnowon,weusetheprimeforradial
1 0
≪
derivatives. and then, contrary to the GR case, the derivatives of
Following [6] and [2], we define the mass assuming the EoS, dp/dρ and d2p/dρ2, also enter into (10) and
that the solution for the metric has the same form as (11), through R′ and R′′.
0 0
the exterior Schwarzschlid solution in GR, i.e. Inf(R)-gravitytheweightoftheperturbationisad-
justedbythevalueoftheαparameter.Inourwork,we
e−2Λ(r) =1 2GM∗, for r >R . (8) restrictourselvesto the constraintsreported by Santos
− c2r ⋆
[24], and references therein, which points to 108cm2 <
For the interior solution, α/3 < 1010cm2, based on astronomical observations
and nuclear experiments in terrestrial laboratories.
2Gm(r)
e−2Λ(r) =1 , for r <R , (9)
− c2r ⋆
where m also admits a perturbative expansion m = 3 Equations of state and numerical methods
m +βm , with m the zeroth-order mass that is ob-
0 1 0
tained integrating (2). Tosolvethesystemofequationsgivenby(10)and(11)
With this considerations, and taking into account is necessary to bring a relation between the pressure
that ρ andp satisfy Einstein’s equations,the derived p and the density ρ or the energy density ε, the so-
0 0
modified TOV equations are: called EoS. The EoS contains the information of the
behaviour of matter inside NSs through several orders
A/2 of magnitude in density. Because the properties of the
dm c2 matter at the highest densities in the central region of
=4πr2ρ 2β 4πr2ρ r2R +
dr − (cid:20)z 0−}|8G {0 NSsarenotwellunderstood,differentEoSareproposed
c2 3 R′ c2 R′′ and then constrained with observations of masses and
+ 2πρ r3 r+ m 0 r r m 0 . radii of actual NSs.
0 0 0
(cid:18) − G 2 (cid:19)R − (cid:18)2G − (cid:19) R (cid:21)
0 0 An analytical representation of the EoS is required
B/2 −C/2 for solving the structure of NSs in modified theories of
| {z } | {z }(10) gravity,where hydrostatic equilibrium equations are of
fourth-order.Insuchcasestheusualinterpolationtech-
nique fails to accurately represent high-order deriva-
dp c2ρ+p 4π m 4π tives[13].AnalyticalrepresentationsofseveralEoShave
dr = 2Gm c2r G(cid:26)(cid:18)c2r2p+ r (cid:19)−2β(cid:20)c2r2p0+ beenobtainedthroughaconsistentprocedurebyHaensel
− and Potekhin [14], who calculated the best-fit coeffi-
c2 2π c2 3 R′
+ r2R + p r3+ r m 0 . cients of a polynomial expansion both in the crust and
8G 0 (cid:18)c2 0 G − 2 0(cid:19)R (cid:21)(cid:27)
0 core density regimes. However, it must be emphasized
(11)
that these analytical EoS are approximations obtained
Note that 2β ... indicate the first order correction in 1 The Ricci scalar in terms of the functions of the metric
h i (1)is:
β into the gradients dm/dr and dp/dr, respectively. In
order to work up to first order in β, terms between 2e−2Λ(cid:2)r2{Φ′Λ′+(Φ′)2+Φ′′}+2r(Φ′−Λ′)−e2Λ+1(cid:3)
R=
the brackets have been evaluated at order zero. It is r2
important to note that dp/dr does not depend on R′′.
0
4 M. Orellana etal.
Fig. 1 Mass-radius (M⋆−R⋆) relations for the two selected EoS: SLY and POLY (left and right, respectively), considering
sevenvaluesfortheαparameter,whichareindicatedaboveinkm2units.Allthecurvescorrespondtovaluesofcentraldensity,
ρc,intherange1014.6−1015.9grcm−3.Thecrossesindicatethemaximummassforeachcurve,assuminganecessarycondition
for equilibrium:dM/dρc >0.
by fitting only the zeroth-order relation between ρ and ourresults. The secondEoSis a simpler polytropic ap-
p, because it is the relation needed to calculate the proximation given by
structure of NSs in GR. Thus, special care should be
ζ =2ξ+5.29355 . (16)
taken if high-order derivatives of these expressions are
usedduringthe calculation,asinthe caseweareinter- DespitethelatterisnotarealisticEoSthatthoroughly
ested in here (i.e. dp/dρ and d2p/dρ2). represents NSs, it is a toy model that allows to study
Taking this into account, and in order to compare zeroth-order modified gravity effects, separating them
our results with those already published in the litera- from more tricky effects arising in the case of a real-
ture,wecalculatemass-radius(M R )relationscon- istic EoS, with its complex analytical expression and
⋆ ⋆
−
sideringtwodifferentEoS:SLY[12,14]andPOLY[26]. from which the error propagating to the derivatives is
The first one is a realistic EoS that properly repre- out of our control. The precise value of the adiabatic
sents the behaviour of nuclear matter at high density. index, Γ =dlogp/dlogρ=dζ/dξ, is not relevantas Γ
Its analytic parametrizationis given by remains a derivable function. The reader is referred to
[22] for tighter constraints on Γ that point to a some-
a +a ξ+a ξ3
1 2 3 what larger value than the one adopted here.
ζ = f (a (ξ a ))
0 5 6
1+a ξ −
4
+(a +a ξ)f (a (a ξ))
7 8 0 9 10
− 3.1 Numerical Method
+(a +a ξ)f (a (a ξ))
11 12 0 13 14
−
+(a15+a16ξ)f0(a17(a18 ξ)) , (13) Solving the system of ordinary differential equations
−
formed by the equations (10) and (11) implies their
where
integration from the centre to the NS surface, using
the chosen EoS. Once the solution is found, a couple
ξ =log(ρ/g cm−3), ζ =log(P/dyncm−2), (14)
of values for the mass, M , and the radius, R , are es-
⋆ ⋆
tablished. In order to perform the integration, we use
anumericalcode basedona fourth-orderRunge-Kutta
1
f (x)= , (15) method on the radial coordinate. For this coordinate
0 ex+1
we implement a variable step which is systematically
and the coefficients a are tabulated [14]. This is the shortened close to the NS surface, to account for rapid
i
same expression used by [2], and we use it here to test variations of the physical parameters in this region.
Structureof neutron stars inR-squared gravity 5
Fig. 2 Profiles of the internal structure of NSs for two extreme cases of low and high central densities, ρc = 1014.6 and
1015.4 grcm−3,andforthreedifferentvaluesoftheαparameter (+0.2,0.0and–0.2km2),whereα=0.0correspondstoGR
case. On the left(right) panel profiles corresponding to the SLY(POLY) EoS are shown. Azoom-in of themass profile close
to the NS surface is shown in Figure 3. At low central density values the effect on the integrated mass can still represent a
deviation as much as ≤10% from theGRmass.
Fig. 3 Zoom-in of the the mass and density profiles close of the surface of the NS for the SLY EoS shown in Figure 2, for
ρc=1015.4and1014.6 grcm−3.Forthevalueoftheαparameter–0.2km2 (+0.2km2)themassincreases(decreases)roughly
10% respect to deGRcase(α=0), for both lowand high central densities.
DuringtheRunge-Kuttaloop,wealsosolvethe dif- NS crust and is the limit of validity for these kind of
ferentialequations correspondingto the metric compo- EoS,astheywereconceivedbeginningwithamodelfor
nents: g andg ,whichareinvolvedinthe criteriafor nuclear matter at high densities.
tt rr
the validity of the perturbative approach.In eachstep,
we first integrate the TOV equations in the frame of
GR to obtain zeroth-order values that then we use to 4 Results
calculate the first-order perturbative solution.
InFigure1wepresentthe mass-radius(M R )rela-
⋆ ⋆
−
We start the numerical integration from the centre tions obtained for SLY and POLY EoS (left and right
with a given centraldensity, ρ , and we finish the inte- panels, respectively), using seven values of the α pa-
c
gration at the surface, defining the NS radius, R , and rameter between –0.2 and +0.2 km2 and considering
⋆
mass, M , when the density reaches ρ = 106 gr cm−3. central densities, ρ , ranging from 1014.6 gr cm−3 to
⋆ c
This density corresponds to the outer boundary of the 1015.9 gr cm−3. Maximum masses achieved are indi-
6 M. Orellana etal.
cated by crosses in each curve, assuming a necessary
condition for equilibrium: dM/dρ >0. Our results for
c
SLY EoS are in accordance with those previously pre-
sentedby [2].Values ofα<0 km2 (α>0km2) canac-
commodate higher (lower) maximum masses than GR.
Inparticular,POLYconfigurationsare lesssensitive to
the value of α than those from SLY EoS.
InFigure2wepresenttheinternalprofilesfoundfor
thedensity,ρ(r),andmass,m(r),forourαextremeval-
ues and the GR case.The density andthe pressurefol-
low rather usual (resembling GR) profiles, where both
magnitudes monotonously decrease with radius. How-
ever,particulareffectsoff(R)arereflectedinthemass
profilesfortherealisticSLYEoS,andbecomemorepro-
nounced for the high-mass NSs (ρ =1015.4 gr cm−3).
c
These effects areevidentclosetothe NSsurface(at
r 10 km) where, in a narrow layer (∆r 0.2 km),
∼ ∼
an unexpected (counter-intuitive) decrease in m(r) ap-
pears before (α > 0 km2) or after (α < 0 km2) a dip
(peak) in the profile. In Figure 3 we present a zoom of
the mass and density profiles (left and right panels, re-
spectively) close to the NS surface, obtained using the
SLY EoS for logρ [gr cm−3] = 15.4 and 14.6. Besides
c
themodificationofM∗ ishigherforhighcentraldensity Fig. 4 Toppanel:Profileoftheofthefirst(dottedline)and
stars,therelativechangeinthetotalmasswithrespect second (continuous line) logarithmic derivatives of the SLY
totheGRcaseisroughly10%inbothhigh/lowcentral EoS in the NS interior. Middle and Bottom panels: Profiles
oftheratiobetweentheradialcomponentofthemetricator-
density cases.
derzero(GR),andatfirstorder(gr0r/grr)forα=+0.2and
In the frame of GR, a decreasing mass profile could −0.2km2,whichshouldbecloseto1.0asanecessarycondi-
only be accomplished by a fluid of negative density, tion of thepertubative method. Theperturbative deviations
because dm/dr = 4πr2ρ(r). However, in f(R)-gravity are closely related with the behaviour of the second-order
derivativeof theEoS.
these profiles can be explained as a consequence of the
modified geometry. In contrast, no such features are
presentintheprofilesobtainedwiththe POLYEoSfor
all the values considered for the parameters. it is the mass but not the pressure the one requiring
InFigure4wepresenttheprofileoftheratiogr0r/grr evaluations of R0′′, and thus depending on high-order
withthedensitythroughalltheNSinteriorfortheSLY derivatives of the EoS.
EoS,consideringthe same densities ofFigure 2.We re- To further analyse the originof the deviations from
callthat this ratioshouldbe closeto 1.0asa necessary the GR we explore the contribution to dm/dr of the
conditionforthevalidityofthepertubativemethod.On four terms involved in equation (10), which we call:
the upper panel of this figure we also plot the first and 4πr2ρ (GR) and A, B, C (perturbative terms). In Fig-
second logarithmic derivatives of the SLY EoS. From ure5wepresenteachtermcontributionwhenα=+0.2
the comparison of the trend, a strong coupling of the km2 forSLY(upper-leftpanel)andPOLY(upper-right
perturbative deviations of g with the second-order panel) EoSs, in the case of a high central density star
rr
derivativeoftheEoSisevident.Themodificationinthe (logρ [grcm−3]= 15.4).Forthe SLY EoSandforval-
c
metric radial component is mild, and only perceptible uesofradiir .9.5km,dm/drisdominatedbytheGR
when the second-order derivative becomes important, term.ClosertotheNSsurface,forr >9.5km,theterm
strongly oscillating, in the 1011 1014 gr cm−3 density C R′′/R becomesdominant,causingthefluctuation
− ∝ 0 0
range.SuchbehaviourisnotpresentforthePOLYEoS, inthemassprofile.Onthecontrary,forthePOLYEoS,
whichlogarithmicsecondderivativeisnull,maintaining which derivatives are strictly bounded smooth func-
g0 /g 1 allthroughthe NS interior.It is important tions, this counter-intuitive effect is not present and
rr rr ≃
to note that the function Φ(r) in the time component thetrendofdm/dr isdominatedbytheGRterm,with
of the metric is actually reflecting the behaviour of the verysmallmodificationsduetotheperturbativeterms.
pressure, whereas Λ(r) is governed by the mass, and In the lower panels of Figure 5 we zoom-in the upper
Structureof neutron stars inR-squared gravity 7
Fig. 5 Profileof themass gradient, dm/dr (thickline),closeto thesurfaceof theNSsforSLY(leftpanel)andPOLY(right
panel)EoSs.Deviationsof themassprofilefrom theGRcaseare muchmore importante fortherealisticSLYEoS.Note that
the dashed line indicates zeroth-order (GR) term and the continuous with plus signs line indicates the contribution of the
C ∝ R′0′/R0 term. Lower panels zoomed-in to show in detail the contribution of the minor perturbative terms, indicated in
thelegend.
Fig. 6 Same as Figure 5 for other three SLY cases: logρc [gr cm−3] = 15.4 (left) and 14.6 (center and right). The central
andrightplotscomparetheeffectsofchangingthesignof α,whichisindicatedin km2 units.Thesamevalues,i.e.α=−0.2
(left)and +0.2 (Fig 5, leftpannel)are shown fora large central density.
panelstoshowthebehaviouroftheminorperturbative a wider radial band 7–10.5 km, as the density changes
terms. in smoother way than in the high central density case.
In the latter the density drastically decays in a narrow
In Figure 6 we extend this analysis to compare the
and superficial range from 9.7 to 10.2 km.
behaviour of the mass derivative in four SLY cases:
logρ [gr cm−3] = 15.4 and 14.6 for α = +0.2 and
c
–0.2 km2. For low central density stars, the effect of
the second-order derivative (C term), is relatively less
important than in the high central density case. The
fluctuations in this term occur in the density range Inallcasesstudiedhere,the otherterms,namelyA
where the second order logarithmic derivatives of the and B, which correspond to R and R′ contribu-
∼ 0 ∼ 0
EoSbecomerelevant,whichinthis casecorrespondsto tions, respectively, are orders of magnitude lower.
8 M. Orellana etal.
5 Conclusions Acknowledgements We thank Dr. Arapog˘lu for explana-
tions. The authors appreciate helpful comments from Prof.
Santiago E. P´erez Bergliaffa. M.O. acknowledge support by
With the aim to investigate whether f(R)-theories are
theArgentineAgencyCONICETandANPCyTthroughgrants
viable to describe astrophysical scenarios like NSs, we
PICT2010-0213/PrestamoBIDandPICT-2007-00848.G.E.R.
have studied the particular R-squared case using both wassupportedbyPIP2010-0078 (CONICET)andtheSpan-
simplified and realistic EoS. We have followedthe gen- ish Ministerio de Innovacio´n y Tecnolog´ıa under grant AYA
2010-21782-C03-01.
eral steps presented by Cooney et al. [7] and Arapog˘lu
etal.[2]usingaperturbativeapproachappliedtosolve
the fourth order field equations. Concerning the mass-
radius (M R ) relations, we have obtained results
⋆ ⋆
− References
consistentwithformerstudies,findingthatforthehigh-
estabsolutevaluesadmittedfortheαparameter,f(R)-
1. Amanullah, R., Lidman, C., Rubin, D., Aldering,
theoriescanaccommodateheavierNSsthanGRforev-
G., Astier, P., Barbary,K., Burns,M. S., Conley, A.,
ery EoS. In this sense, it is important to remark that
Dawson, K. S., Deustua, S. E., Doi, M., Fabbro, S.,
there is no agreement on the maximum mass achiev-
Faccioli, L., Fakhouri, H. K., Folatelli, G., Fruchter,
able by NSs before they collapse to black holes, based
A. S., Furusawa, H., Garavini, G., Goldhaber, G.,
ontheuncertaintiespresentonthebehaviourofnuclear
Goobar, A., Groom, D. E., Hook, I., Howell, D. A.,
matteratthe highestdensitiesthroughtheirEoS.This
Kashikawa, N., Kim, A. G., Knop, R. A., Kowalski,
problem can not be split out from our lack of under-
M., Linder, E., Meyers,J., Morokuma,T., Nobili, S.,
standing of gravity [30].
Nordin,J.,Nugent,P.E.,O¨stman,L.,Pain,R.,Pana-
Notwithstanding,ourmostnotoriousresultconcerns
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