Neutron Stars in Palatini R+αR^2 and R+αR^2+βQ Theories
EEur. Phys. J. C manuscript No. (will be inserted by the editor)
Neutron Stars in Palatini R + α R and R + α R + β Q Theories
Georg Herzog a,1 , Hèlios Sanchis-Alepuz b,2,3 Dipartimento di Fisica ’G. Occhialini’, Università degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy Institute of Physics, University of Graz, NAWI Graz, Universitätsplatz 5, 8010 Graz, Austria Silicon Austria Labs GmbH, Inffeldgasse 25F, 8010 Graz, AustriaReceived: date / Accepted: date
Abstract
We study solutions of the stellar structure equa-tions for spherically symmetric objects in Palatini f ( R ) = R + α R and f ( R , Q ) = R + α R + β Q in the mass-radiusregion associated to neutron stars. We illustrate the poten-tial impact of the R and Q terms by studying a range ofviable values of α and β . Similarly, we use different equa-tions of state (SLy, FPS, HS(DD2) and HS(TMA)) as a sim-ple way to account for the equation of state uncertainty. Ourresults show that for certain combinations of the α and β parameters and equation of state, the effect of modificationsof general relativity on the properties of stars is sizeable.Therefore, with increasing accuracy in the determination ofthe equation of state for neutron stars, astrophysical observa-tions may serve as discriminators of modifications of Gen-eral Relativity. General Relativity (GR) as the simplest realisation of a geo-metric description of gravity has so far shown, perhaps un-expectedly, total agreement with astrophysical and cosmo-logical observations [1, 2], as long as the general pictureis accepted that the dark sectors, as needed for the Λ CDMmodel, are related to the particle and energy content of theUniverse. Despite this tremendous success, there are reasonsto investigate modifications of GR. First, GR breaks down inthe high curvature regime. Second, it is only a classical fieldtheory without any quantum effects and which is not (per-turbatively) renormalisable as a quantum field theory unlessquadratic curvature corrections are added to the Lagrangian[3]. Moreover, it is conceivable that all or part of the effectscurrently attributed to dark sectors are in reality a manifes-tation of different gravitational dynamics [4–7]. a e-mail: [email protected] b e-mail: [email protected] Some of the (theoretical) problems of GR can be amelio-rated in modified gravitational theories, like f ( R ) and f ( R , Q ) theories in which the Einstein-Hilbert Lagrangian is mod-ified by adding polynomial terms in the Ricci scalar R orin the contraction of the Ricci curvature Q = R µν R µν (seee.g. [8–13] and references therein). From a different pointof view, if the different terms in a gravity action are consid-ered as quantum operators in the spirit of effective quantumfield theories, studying their renormalisation group evolu-tion indicates that f ( R ) and f ( R , Q ) terms render the theoryasymptotically safe (and hence viable as a quantum theory)in the Planckian and super-Planckian regime [14–18]. Fi-nally, classical theories of modified gravity can also serveas effective descriptions of their more fundamental quantumcounterparts (see e.g. [19, 20]).Studying the compatibility of modified theories of grav-ity with phenomenology is a highly non-trivial issue sincean interpretation of observations involves a combination ofmany different and uncertain physical mechanisms. As men-tioned above, cosmological observations compatible with GRand dark sectors can be also made compatible with certainmodifications of gravity and different contributions from darksectors [21]. Inflationary scenarios can also be expressed interms of modified theories of gravity (see e.g. [22–25]). Ithas been suggested that modifications of GR can be identi-fied via the detection of new modes in gravitational waves[26–33]. Another possibility to investigate deviations fromGR is with astrophysical observations. Different gravitationaldynamics reflects into different stellar equations of structure[2, 34–37] and hence, in particular, in mass to radius ratiosof neutron stars different to those expected from GR. Thisis particularly interesting with the growing volume of ob-servational data on neutron stars and the discovery of com-pact objects that are difficult to accomodate in a standard GRpicture (see e.g. [38–41]). These measurements are, unfor-tunately, riddled with enormous uncertainties coming from a r X i v : . [ g r- q c ] F e b the equation of state (EoS) describing the interior of neutronstars and can as well be interpreted as providing a handle onEoS assuming the theory of gravity is GR [42]. However, aswe will show in this paper, the changes in the mass-radiusrelation for neutron stars coming from the uncertainty in theEoS are sometimes comparable to the differences generatedby the modifications of the gravity theory, which implies thatif, in the future, reliable ab-initio calculations of the EoSwere available, astrophysical observations could be used asdiscriminators for gravity theories.In this paper we shall investigate the influence of twoparticular models of f ( R ) and f ( R , Q ) theories, namely f ( R ) = R + α R and f ( R , Q ) = R + α R + β Q on the mass-radius re-lation for neutron stars. We study those theories in the Pala-tini formalism in which the dynamical degrees of freedomare the metric and the affine connection [11]. We will per-form our calculations using a small number of representativeEoS with the goal of estimating the uncertainty stemmingfrom the EoS and from the theory of gravity (in our casesimply encoded in the values of α and β ).The paper is organised as follows. We sketch the deriva-tion of the equations of stellar structure for Palatini theoriesin Sec. 2 and discuss several aspects of the EoS chosen inSec. 3. Our results are shown in Sec. 4 and we conclude adiscussion of the implications of our results on the validityof theories studied herein and of possible future work. f ( R , Q ) PalatiniTheories
We discuss here the main aspects of the equations of stellarstructure solved in this work. We consider only the simpli-fied case of non-rotating and static stars, whose only observ-ables are their mass and radius. The equations for f ( R , Q ) theories were derived in [35] and we refer to that paperfor further details (see also [34, 43]). In GR, the equivalentequations are the well-known Tolman-Oppenheimer-Volkoff(TOV) equations.In Palatini f ( R , Q ) theories the action is given by S [ g , Γ , ψ m ] = κ (cid:90) d x √− g f ( R , Q ) + S m [ g , ψ m ] , (1)where R is the Ricci scalar, Q = R µν R µν the aforementionedcontraction of two Ricci tensors, with R µν = R ρµρν and theRiemann tensor given by R dabc ≡ ∂ b Γ dac − ∂ c Γ dab + Γ eac Γ deb − Γ eab Γ dec with Γ the connection coefficients, g µν is the metric, κ ≡ π G and S m [ g , ψ m ] is the matter action. In the Palatiniformalism the equations of motion are obtained after varyingthe action with respect to both, the metric and the connection f R R µν − f g µν + f Q R µα R α ν = κ T µν (2) ∇ β (cid:2) √− g ( f R g µν + f Q R µν ) (cid:3) = , (3) with f R ≡ ∂ f ∂ R and f Q ≡ ∂ f ∂ Q . One can introduce an auxil-iary metric h µν via the (matter-content dependent) mapping √− g ( f R g µν + f Q R µν ) ≡ √− hh µν such that the equationfor the connection becomes ∇ β (cid:2) √− hh µν (cid:3) = Γ can thus be written as the Levi-Civita connec-tion for h µν . Therefore, the Ricci tensor in Eqs. 2 and 3can be written as the standard metric Ricci tensor of themetric h which we denote as R µν ( h ) . Introducing Σ α ν : =( f R δ να + f Q B α ν ) , with B α ν given as B α ν = R αβ ( h ) g βν , therelation between g µν and h µν is [35] h µν = √ det Σ [ Σ − ] µ α g αν , h µν = g µα Σ α ν √ det Σ . (4)The Ricci tensor is given by R νµ ( h ) = R µα ( h ) h αν = √ det Σ (cid:18) f δ νµ + κ T νµ (cid:19) . (5)Note here that, when calculating R and Q from R ( h ) in f ( R , Q ) ,the Ricci tensor R µν ( h ) must be contracted with the physicalmetric g µν , that is, via the matrix B να above [35] R = B αα and Q = B να B αν . (6)In this paper we will only consider energy-momentumtensors of the form (7) T µν = ( ρ + P ) u µ u ν + Pg µν , and Lagrangians of the form f ( R ) = R + α R and f ( R , Q ) = R + α R + β Q . In these cases we can write the scalars R and Q in terms of ρ and P as R = − κ T , as in GR, and [35] (8) β Q = − (cid:18) ˜ f + ˜ f R f Q + κ P (cid:19) + f Q (cid:18) R + ˜ f R f Q (cid:19) ± (cid:115)(cid:18) R + ˜ f R f Q (cid:19) − κ ( ρ + P ) f Q , where ˜ f = f ( R ) = R + α R , ˜ f R = + α R and f Q = β . Thesign in front of the square root must be chosen so as to obtainthe right limit at low curvatures.The stellar structure equations are the dynamical equa-tions for the star’s interior metric, parametrised as g µν = diag (cid:18) − A ( r ) e ψ ( r ) , A ( r ) , r , r sin θ (cid:19) , (9)where A ( r ) will be written as A ( r ) = − M ( r ) r . After a te-dious but straightforward derivation (see [35] for details)one arrives at the stellar structure equations for Palatini inthe case of f ( R , Q ) = R + α R + β Q , which read (cid:18) Ω r Ω + r (cid:19) ψ r = A (cid:18) τ rr − Ω S τ tt (cid:19) − Ω r Ω (cid:18) Ω r Ω + S r S (cid:19) − r (cid:18) S r S − Ω r Ω (cid:19) + Ω rr Ω , (10) (11) (cid:18) Ω r Ω + r (cid:19) M r r = τ rr − Ω S τ tt + A (cid:18) Ω rr Ω + Ω r Ω (cid:18) r − Mr ( r − M ) − Ω r Ω (cid:19)(cid:19) , (12) P r = − P ( ) r [ − α ( r )] (cid:18) ± (cid:113) − β ( r ) P ( ) r (cid:19) , where, as before, a subscript denotes partial derivation, i.e. M r = ∂ M ∂ r and Ω rr = ∂ Ω ∂ r , and we introduced for compact-ness (13) α ( r ) = ( ρ + P ) (cid:18) Ω P Ω + S P S (cid:19) , β ( r ) = ( r ) Ω P Ω (cid:20) − ( ρ + P ) (cid:18) Ω P Ω − (cid:26) Ω P Ω − S P S (cid:27)(cid:19)(cid:21) , (14)(15) P ( ) r = ( ρ + P ) r ( r − M ) (cid:20) M − (cid:18) τ rr + Ω S τ tt (cid:19) r (cid:21) . Here, Ω and S are given by the relation between the auxiliarymetric h µν and the physical metric g µν h tt = σ √ σ σ g tt ≡ Sg tt and h i j = √ σ σ g i j ≡ Ω g i j , (16)where σ and σ appear upon rewriting the matrix Σ as Σ να = diag( σ , σ , σ , σ ) from Eq. 5 and are given by [35] σ = f R ± (cid:112) f Q (cid:113) λ − κ ( ρ + P ) , (17) σ = f R + (cid:112) f Q λ , (18)where (19) λ = (cid:112) f Q (cid:18) R + f R f Q (cid:19) ± (cid:115)(cid:18) R + f R f Q (cid:19) − κ ( ρ + P ) f Q . In order to recover the GR limit one has to use the positivesign in front of the square root in Eq. 12 and for σ but thenegative sign in λ [35]. Finally, τ refers to the right-handside of Eq. 5, namely R νµ ≡ τ νµ .Note that, in Eq. 11, we have Ω rr = Ω PP P r + Ω P P rr which involves the second derivative P rr , in contrast to thestandard TOV equations. As it turns out, it is possible to write P rr in terms of the first order derivative P r . The resultis [35] P rr P r = P ( ) rr P ( ) r + s β P ( ) r (cid:113) − β P ( ) r ( ± (cid:113) − β P ( ) r ) + α r − α + ± β r P ( ) r (cid:113) − β P ( ) r ( ± (cid:113) − β P ( ) r ) , (20)with α r = α P P r and β r = β P P r and (21) P ( ) rr P ( ) r = (cid:18) + ρ P ρ + P (cid:19) − Φ P r (cid:16) M − Φ r (cid:17) P r − (cid:32) ( r − M ) r ( r − M ) + Φ r M − Φ r (cid:33) + M r (cid:32) r − M + M − Φ r (cid:33) . where we introduced Φ ≡ ( τ rr + Ω S τ tt ) . In this way, Eqs. 10–12 are a closed (after fixing an equation of state) system ofequations expressed in terms of r , ρ ( P ) , P , M and their firstradial derivatives only. The essential ingredient to solve the equations of stellar struc-ture is an equation of state that relates the pressure to theenergy density inside the star.In oder to estimate the different effects on the mass andradius of neutron stars coming from the modifications of thegravity lagrangian and from the uncertainty in the EoS, weuse a number of different EoS in analytic as well as tabulatedform. Specifically, we use two different tabulated and twodifferent analytic EoS. The analytic equations of state weused are the so-called SLy and FPS [44] The SLy and FPSEoS are analytic parametrisations of the results of many-body calculations with unified effective nuclear Hamiltoni-ans [45, 46], describing all regions of the neutron star inte-rior from crust to core including its transitions. The analyticparamerisations take care that all thermodynamic conditionsinvolving derivatives of the EoS are fulfilled, an aspect thatwill become problematic when using tabular data directly,as we will see. The SLy and FPS EoS are both parametrisedby the following function ζ = a + a ξ + a ξ + a ξ f ( a ( ξ − a ))+( a + a ξ ) f ( a ( a − ξ ))+( a + a ξ ) f ( a ( a − ξ ))+( a + a ξ ) f ( a ( a − ξ )) , (22) Table 1
SLy and FPS parameters for Eq. (22), given in [44].i a i (FPS) a i (SLy) i a i (FPS) a i (SLy)1 6.22 6.22 10 11.8421 11.49502 6.121 6.121 11 − . − . − . − . where ξ = log ( ρ / g cm − ) and ζ = log ( P / dyn cm − ) and f ( x ) is given by [44] f ( x ) = e x + . (23)The fitted parameters a i in Eq. 22 are shown in Table 1. Fig. 1
The Sly and FPS equation of state and their first and secondderivatives.
Additionally to the two analytic EoS above, we also stud-ied neutron star solutions based on two other EoS in tabu-lated form. We choose EoS available in the CompOSE on-line service [47], namely the so called HS(DD2) [48, 49]and HS(TMA) [48, 50] EoS. These two EoS are both basedon the the statistical model presented in [48], which includesthe contribution of nuclei, nucleons, electrons, positrons andphotons (excluding neutrino contributions) and which re-quires as input the masses and binding energies of nucleiand an effective model for the nucleon-nucleon interaction,which is then treated in the relativistic mean-field (RMF) ap-proximation. The two EoS thus differ in the different modelused for that dynamical input. The HS(DD2) equation ofstate uses the density-dependent nuclear model called DD2 [49], with the nuclei properties given by the FRDM model[51]. The HS(TMA) equation of state uses instead the TMAmodel [50], with the masses of nuclei taken from [52].The drawback of using tabulated data to solve the equa-tions of stellar structure is that interpolation functions andnumerical derivatives thereof must be used. This procedureis ambiguous (as one can use many different forms for theinterpolating functions) and, as mentioned above, certain ther-modynamic relations must be preserved by the derivatives ofthe EoS (see e.g. [44]). This is a particularly difficult prob-lem to tackle in the case of modified Palatini theories, sincehigher derivatives of the EoS are needed as compared to GR.Indeed, in the stellar structure equations for Palatini f ( R ) and f ( R , Q ) the first and second derivative of the density ρ with respect to the pressure P appear. For the analyticallygiven EoS above this is relatively straightforward. Remem-bering that P ( ρ ) = ζ ( ξ ( ρ )) = ζ ( log ( ρ )) we have dP ( ρ ) d ρ = dPd ζ d ζ d ξ d ξ d ρ = P ρ d ζ d ξ , (24)and d Pd ρ = dd ρ (cid:18) dPd ρ (cid:19) = dPd ρ ρ d ζ d ξ − P ρ d ζ d ξ + P ρ d ζ d ξ , (25)in order to arrive at the first and second derivative of thedensity with respect to the pressure we now have to invertEqs. 24 and 25. One gets d ρ dP = dPd ρ , (26) d ρ dP = − d Pd ρ (cid:16) dPd ρ (cid:17) . (27)The result for the parametrization of the SLy and FPS EoStogether with the first and second derivative of the parametri-sations are shown in Fig. 1. A similar procedure must be ap-plied to the EoS given in form of tabulated data, but the dif-ferent ways in which the data can be interpolated and hencederivatives can be taken has an impact in the results, as wediscuss below. The mass and radius of a star are obtained by integratingfrom inside out the stellar structure equations after fixing acentral density and until the pressure reaches a pre-determinedthreshold. In our calculations, the threshold was set to P = P c · − where P c is the initial central pressure (determinedby the EoS from the central density) in case of the analyticEoS. For tabulated EoS we stopped the integration when reaching the last entry of the EoS tables. We considered cen-tral densities in the range ξ = [ . , ] (at r = . f ( R ) = R + α R and f ( R , Q ) = R + α R + β Q , the problem reduces to finding reasonablevalues for α and β , consistent with phenomenology. How-ever until now no solid experimental bounds on the valuesof α and β exist for theories in the Palatini formalism (see[53] for a study in Palatini). There exist, however, more re-strictive experimental bounds on α for theories in the metricformalism (where only the metric is considered a dynamicaldegree of freedom). The Gravity Probe B experiment con-strains α to α (cid:46) · cm , while the constraint comingthe Pulsar B in PSR J0737-3039 is four orders of magnitudehigher [54]. In [54] they derived an even more stringent con-straint on α from the Eöt-Wash experiment, constraining α to α (cid:46) − cm . However, this bound was derived in thelow curvature regime of a laboratory on earth. Since possi-ble modifications to GR will likely only be relevant in thehigh curvature regime, we do not take the bound from thisexperiment into consideration. Recently another bound for α has been derived by calculating the stability of stars inmetric f ( R ) = R + α R gravity [55]. Using polytropic EoSand looking for the maximum value of α which still fulfillsthe stability criteria they found α (cid:46) . · cm . To the bestof our knowledge, no bounds on β exist at the moment.Since the dynamics of metric and Palatini theories canbe very different, the bounds discussed above may not applyin our case. Lacking a thorough analysis along the lines of[54, 55] for Palatini theories, we used values for α and β for which an appreciable change in the mass-radius curvewas obtained and that, for the case of α , are still within thebounds obtained in the metric formalism and in [53].4.1 R + α R caseWe begin with the study of f ( R ) = R + α R theories andshow the mass-radius relation for a range of positive andnegative values of α . Note that an analogous study was per-formed in [43] for the SLy and FPS EoS. However, our re-suls are in contradiction with those presented in [43]. Whilein [43] the authors found only minor variations with respectto GR of the mass and radius of neutron stars, our calcula-tions show instead a significant deviation from GR in somecases. We show in Fig. 2 our results for the mass-radius relation using the SLy and FPS EoS, compared to the cor-responding GR result. As one can see, significant devia-tions from GR appear in the low-radius region even thoughsmaller deviations are also observed for less compact stars.Interestingly, for negative values of α , f ( R ) = R + α R the-ories generate heavier stars than GR and thus push the masslimit for the SLy and FPS EoS. Fig. 2
Mass-radius relation for GR and f ( R ) = R + α R , the SLy andFPS EoS and different values of the parameter α . The situation is less pronounced in the case of the tab-ulated HS(DD2) EoS, Fig. 3. In this case, the differencesin the mass-radius relation with respect to GR are still vis-ible but are significantly smaller than in the previous case.What remains true is that negative values of α allow for alarger maximum mass. In the case of the HS(TMA) EoS,Fig. 3, the deviations with respect to GR are dramatic for thelargest values of α we used. This is certainly an unexpectedbehaviour that we were able to trace back to a small differ-ence in the HS(TMA) EoS with respect to the other EoS we Fig. 3
Mass-radius relation for GR and f ( R ) = R + α R , the HS(DD2)and HS(TMA) EoS and different values of the parameter α . used , which induces a larger difference in the first and sec-ond derivative of the EoS. As can be seen in the first panelof Fig. 9 at around P = gcm s the HS(TMA) EoS differsslightly from all other EoS but the difference is amplified forthe first and second derivatives as can be seen in the secondand third panel of Fig. 9. Note finally, that the range of radiiobtained in the case of tabulated EoS is limited by the max-imum and minimum values of the data provided, unlike inthe case where we used analytic expressions.In addition to the mass-radius plots, it is instructive toanalyse the results as in Figs. 4–7, were we show the differ-ence in percentage between the f ( R ) and the GR solutions,for the mass and the radius of a star separately. In this waywe observe that the effect of the R term on the radius ofstars is generally smaller than on the mass, except for thelowest central-density regions. The shift in the mass is thus To confirm this hypothesis, we performed calculations using theHS(TM1) and SFHO EoS as well, which all show the same behaviourin their derivatives as HS(DD2) and do not show as large deviationsfrom GR as HS(TMA). the main responsible of the change of the mass-radius ratiosdiscussed above. The changes, moreover, are strongly de-pendent on the EoS; SLy, FPS and HS(TMA) exhibit largedeviations with respect to GR, but the latter shows a qualita-tively different behaviour as a function of the central density.For HS(DD2), the deviations with respect to GR are signif-icantly smaller than for the other EoS, for both mass andradius.
Fig. 4
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R and GR, for the SLY EoS. It is clear from our discussion so far, that as long as theambiguities in the EoS of neutron stars are not resolved, itis hardly possible to draw conclusions from mass and radiusobservations on the theory of gravity, as for a fixed f ( R ) (here, for a given α value) the mass-radius relation changesdramatically for different EoS. However, it is still interest-ing to speculate how relevant would astrophysical observa-tions be, if we knew the correct EoS of a neutron star. Forexample, the observation of the 2.2 M (cid:12) neutron star MSPJ0740+6620 [56] has been used as an argument to rule out Fig. 5
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R and GR, for the FPS EoS. certain EoS, including some used in this paper. This is canbe seen in Fig. 8, where we show the constraints put by MSPJ0740+6620 on EoS, assuming GR is the theory of gravityand also using f ( R ) = R + α R with α = − · cm (forwhich the increase of mass is the largest in our calculations).Clearly, some EoS can be rescued adding an R term to thetheory of gravity. Reversing the argument, wiht increasingcertainty in the EoS of neutron stars, observations could beused to constraint deviations of the theory of gravity fromGR.At this point, we would like to discuss in some detail thedifferences in the results induced by the different methodsused to calculate derivatives of the EoS in the case of tab-ulated data. To this end, we repeated the calculations withthe analytic EoS SLy, this time generating tabulated datafrom the analytic expressions as follows. In one case, thederivatives were calculated analytically as outlined in Sec. 3and separate data tables were generated from them; thesevalues then had to be interpolated (in the same way as the Fig. 6
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R and GR, for the HS(DD2) EoS. EoS data is interpolated). As a second method we calcu-lated the derivatives numerically with a finite difference ap-proach, using second order central differences. In Fig. 10the difference between the derivatives using those methodscan be seen. While the difference for the first derivative israther small and never exceeds 4%, the second derivativeshows very large differences, of about 200% which rises upto 400%, at some points. To exemplify the impact of suchdifferences in the mass-radius relations, we repeated the cal-culation for the SLy EoS with the different methods of cal-culating derivatives. The results are shown in Fig. 11. In theupper panel we show the calculation where the tabulateddata for exact derivatives is interpolated, which clearly leadsto the same results as when using the fully analytic version.However, in the lower panel, where the derivatives have beencalculated with a finite difference method, shows only a mi-nor difference between f ( R ) and GR, in fact in agreementwith the results published in [43]. This discussion points toa potential problem appearing when calculating with tabu- Fig. 7
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R and GR, for the HS(TDA) EoS. lated EoS and theories with high-order derivatives of matter;all calculations that rely on numerical differentiation mustbe taken with due caution.4.2 R + α R + β Q caseWe proceed now to discuss the results for the f ( R , Q ) = R + α R + β Q . As already indicated above, we consider val-ues of β for which an appreciable change in the mass-radiuscurve is obtained. This means that, even the small changesin the stars masses that we observe, as discussed next, maybe artificially large.We show in Figs. 12 and 13 the mass-radius relation fortwo values of β , with and without the R term (here the R term is excluded by setting α to a very small value, for com-putational convenience). The effect of the Q term appears tobe generally small, the deviation from GR mostly comingfrom the R term as we have seen in the previous section.The effect of the Q term seems to be more pronounced for Fig. 8
Maximum mass allowed by an f ( R ) = R + α R theory with α = − · cm for the different EoS used in this paper, comparedwith the observed mass of the MSP J0740+6620 neutron star. Fig. 9
The EoS used in this paper (first panel) together with their firstand second derivatives (second and third panel respectively).
Fig. 10
Difference between the exact and numeric derivative for thefirst (upper panel) and second derivative (lower panel) of the EoS.
Fig. 11
Mass-radius relation for fake tabulated SLy EoS using the ex-act values for the derivative (left panel) and the values derived by thefinite difference method (right panel).
Fig. 12
Mass-radius relation for f ( R , Q ) = R + α R + β Q for SLY(upper panel) and FPS (lower panel) EoS. analytic EoS than for the tabulated ones, which may pointto the problem with numerical derivatives that we discussedabove.As in the f ( R ) case, it is interesting to plot separately thedifferences between the f ( R , Q ) and the GR solutions, forthe mass and the radius of a star. We show them in Figs. 14–17. In contrast to the R term, for which the dependenceof the mass and radius as a function fo the central densityshowed a very different qualitative behaviour for differentEoS, in the case of the Q term these show a similar depen-dence on ξ c for all studied EoS. In all four cases the effectis somewhere below 5% on the mass, with the exception ofHS(DD2) which is up to 8% for low central densities. As forthe R case, it is interesting to note that the Q terms influ-ences the mass of the star much more than its radius, againwith the exception of lowest values of the central density. Inthat region, the radius and mass effects compensate in a waysuch that they are not visible in the mass-radius plots.To conclude, we wish to mention a peculiar feature ofthe solutions we obtained for f ( R ) gravities. In Fig. 18, we Fig. 13
Mass-radius relation for f ( R , Q ) = R + α R + β Q forHS(DD2) (upper panel) and HS(TDA) (lower panel) EoS. show a star profile (that is, the metric function M ( r ) as afunction of the radial coordinate r ) for the SLy EoS and ξ = .
1. The metric function M ( r ) shows an unexpectedcusp near the surface of the star. This feature, absent in GR,was already seen in [43] and predicted in [57], where it wasargued that it invalidates Palatini f ( R ) theories as physicallyviable. In [57] it was also speculated that a Q term may ame-liorate this odd behaviour of the metric inside the star. As wesee in Fig. 18, it is true that an f ( Q ) term (without R ) doesnot show a cusp . However, in an f ( R , Q ) model (i.e. withboth R and Q terms) the cusp remains; that is, the Q termis not able to compensate for the unexpected behaviour gen-erated by the R term. Note that, even though in Fig. 18 weshowed one particular solution only , in all cases we havestudied in this paper the qualitative behaviour of the metricfunction M ( r ) was analogous. We have studied changes of β by many orders of magnitude, and thecusp was absent in all cases. Fig. 14
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R + β Q and GR, for the SLy EoS. In this paper we studied neutron stars solutions in Palatini f ( R ) = R + α R and f ( R , Q ) = R + α R + β Q gravities. Weperformed calculations using four different EoS and differ-ent values for α and β .Our study shows that the differences in the masses andradius of stars induced by modifications of the gravity La-grangian can in fact be as large as the uncertainty inducedby the use of different EoS. Notably, it is the mass of thestar and not its radius that is mostly affected by modifica-tions of GR. We have also discussed separately the effect ofthe R and the Q terms. The deviations from GR induced bythe R term are qualitatively different for different EoS. The Q term instead shows a similar behaviour for all EoS, and ingeneral smaller than the effect from the R term.Even though it was not a goal of our investigations, wefound that some of the EoS which have previously beenruled out by neutron star observations, might become viable Fig. 15
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R + β Q and GR, for the FPS EoS. again if the α R correction is indeed part of the right theoryof gravity. For negative α the maximum mass allowed by agiven EoS tends to be increased. For some EoS, such as SLyor HS(TMA), it turns out that an α R term can produce solu-tions as heavy as the MSP J0740+6620, the heaviest neutronstar found so far [56].In this work we focused on the relatively simple problemof calculation the mass, radius and profiles of sphericallysymmetric stars. In the future, it is necessary to performmore detailed studies. For example, it is crucial to study thestability of a solution in order to understand whether it isviable as a physical solution crucial. For example, it couldbe that the solutions beyond the GR mass limit are unsta-ble and hence not realised in Nature. This is an even morepressing issue, considering the odd behaviour of the metricfunction M ( r ) for f ( R ) theories, as discussed in last section.Also, realistic neutron stars rotate and the system of equa-tions solved herein provides only an estimate of their proper-ties. Finally, on the technical side, it is necessary to establish Fig. 16
Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R + β Q and GR, for the HS(DD2) EoS. reliable and unambiguous methods to calculate high-orderderivatives of EoS, as we have seen.Clearly, no conclusive results can be obtained yet, atleast until the ongoing efforts to nail down the EoS for aneutron star from first principles are successful. However,our study indicates that, in the future, astrophysical observa-tions may serve as discriminators of modified gravity. Acknowledgements
We thank G. Olmo for a critical reading of themanuscript.
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Difference between the mass (upper panel) and the radius(lower panel) obtained, given a central density value ξ , with f ( R , Q ) = R + α R + β Q and GR, for the HS(TDA) EoS. Fig. 18
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