Universal I-Q relations for rapidly rotating neutron and strange starsin scalar-tensor theories

Daniela D. Doneva,1,2,* Stoytcho S. Yazadjiev,3,1,† Kalin V. Staykov,3,1,‡ and Kostas D. Kokkotas1,4,§1Theoretical Astrophysics, Eberhard Karls University of Tübingen, Tübingen 72076, Germany

2INRNE—Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria3Department of Theoretical Physics, Faculty of Physics, Sofia University, Sofia 1164, Bulgaria

4Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece(Received 9 August 2014; published 18 November 2014)

We study how rapid rotation influences the relation between the normalized moment of inertia I andquadrupole moment Q for scalarized neutron stars. The questions one has to answer are whether theequation of state universality is preserved in this regime and what are the deviations from general relativity.Our results show that the I − Q relation is nearly equation of state independent for scalarized rapidlyrotating stars, but the differences with pure Einstein’s theory increase compared to the slowly rotating case.In general, smaller negative values of the scalar field coupling parameters β lead to larger deviations, butthese deviations are below the expected accuracy of the future astrophysical observations if one considersvalues of β in agreement with the current observational constraints. An important remark is that althoughthe normalized I − Q relation is quite similar for scalar-tensor theories and general relativity, theunnormalized moment of inertia and quadrupole moment can be very different in the two theories. Thisdemonstrates that although the I − Q relations are potentially very useful for some purposes, they might notserve us well when trying to distinguish between different theories of gravity.

DOI: 10.1103/PhysRevD.90.104021 PACS numbers: 04.50.Kd, 97.60.Jd

I. INTRODUCTION

Neutron stars are among the best candidates to testthe strong field regime of general relativity (GR) due to thehigh compactness of the matter in their core and to thevariety of observed phenomena. One of the most significantobstacle in this direction is the uncertainty in the nuclearmatter equation of state (EOS). In many cases theseuncertainties are comparable or even bigger than the effectsinduced by different modifications of GR. Strong con-straints on the EOS can be set [1–5], but still the accuracy isbelow the desired one for testing different generalizationsof Einstein’s theory. A way to circumvent these uncertain-ties is to search for equation of state independent character-istics. An important step in this direction was made recentlyin [6,7] (see also [8–10]), where it was found that relationsbetween the normalized moment of inertia (I), quadrupolemoment (Q) and the tidal Love number (λ) exist (the so-called I-Love-Q relations) which are practically indepen-dent of the equation of state. In a subsequent series ofpapers these results were generalized to large tidal defor-mations [11] and fast rotation [12–14]. Magnetized neutronstars on the other hand can break the universality for strongmagnetic fields and low rotational rates. In [13] it wasshown that similar EOS independent relations can be

formulated for the higher multipole moments and theresults were later extended in [9]. Other universal relationswere studied in [15–20].Different astrophysical implications of the I-Love-Q

relations were proposed. One of the most important isbreaking the degeneracy between the spins and the quadru-pole moments of neutron star inspirals. A development inthis direction is important due to the expected detection ofgravitational waves in the very near future. Another possibleimplication of the I-Love-Q relations is to use them as aprobe for modified theories of gravity. For example if aspecific alternative theory gives significantly differentI-Love-Q relations compared to GR, then a measurementof two of these quantities would allow us to determine thepossible deviations from general relativity and set constraintson the alternative theory of gravity. This is for example thecase with the Dynamical Chern-Simons gravity where theI-Love-Q relations are still pretty much EOS independentbut they are significantly different from the GR case [6,7].On the other hand the corresponding relations in Eddington-inspired Born-Infeld gravity, scalar-tensor theories of gravityand Einstein-Gauss-Bonnet-dilaton theory [21] are almostindistinguishable from Einstein’s theory [22–24]. Thereforethe I-Love-Q relations cannot serve as a test for thesetheories. But from another point of view these results showthat the relations are universal not only with respect to theEOS but also to certain beyond-GR corrections. Thus if oneof the quantities in the I-Love-Q trio is measured, one caninfer the other two using formulas which are universal for awhole plethora of gravitational theories.

*daniela.doneva@uni‑tuebingen.de†yazad@phys.uni‑sofia.bg‡kalin.v.staikov@gmail.com§kostas.kokkotas@uni‑tuebingen.de

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Except the case of Einstein-Gauss-Bonnet-dilaton grav-ity [24], all of the studies of I-Love-Q relations inalternative theories of gravity up to now are limited tothe slow rotation regime. One of the reasons is thatconstructing rapidly rotating neutron star solutions inalternative theories of gravity is very involved. Also theslow rotation approximation is sufficient in practice formany cases, for example the inspiraling neutron stars aresupposed to be rotating with relatively low rotationalfrequencies. But the rotational rates of neutron stars insome cases, such as the millisecond pulsars and newbornneutron stars, can reach very high values and the extensionto rapid rotation is important. Moreover the studies ofneutron stars in alternative theories of gravity show thatrotation can magnify the deviation from GR significantlyand potentially lead to observational consequences [25].In the present paper we concentrate on the I-Q relation

for rapidly rotating neutron stars in scalar-tensor theory ofgravity (STT). This is one of the most natural and widelyexplored extensions of general relativity where in additionto the spacetime metric, a scalar field appears which is alsoa mediator of the gravitational interaction. Different classesof scalar-tensor theories were considered in the literaturewith probably the most famous one being the Brans-Dicketheory. But in the past two decades special attention wasgiven to a specific class of scalar-tensor theories which isindistinguishable from GR in the weak field regime butinteresting nonlinear effects can develop for strong fields.Such a nonlinear phenomenon is the scalarization ofneutron stars discovered in [26]. These results weregeneralized later to slow [23,27,28] and rapid rotation[25]. The general idea behind the scalarization is that forcertain ranges of the parameters new solutions with non-trivial scalar field can exist in addition to the pure GRsolutions, and the scalarized neutron stars are energeticallymore favorable compared to their GR counterpart. Thecurrent observations of binary pulsars set tight constraintson the coupling parameter in this scalar-tensor theory andfor the nonrotating case the scalarized solutions differ onlyslightly from the general relativistic ones [4,29,30]. But asthe results in [25] show, the rotation can enhance the effectof the scalar field considerably and leads to much moresignificant deviations from GR.A natural question that arises is whether the I-Love-Q

relations are different for scalarized neutron stars. As wehave already mentioned, the studies in slow rotationapproximation [23] show that these relations are practicallythe same as in GR if one considers values of the parametersin agreement with the current observational constrains.Our goal in the current paper is to extend these results to

the case of rapid rotation and to check if the universality ofthe relations is preserved. For this purpose we have tomake also a proper derivation of the quadrupole momentof rotating scalarized neutron stars in quasi-isotropiccoordinates.

II. BASIC EQUATIONS

In this section we very briefly present the necessarybackground for the scalar-tensor theories and the rapidlyrotating stars in their framework without going into toomuch detail. For a more detailed discussion we refer thereader to [25].The scalar-tensor field equations in the Einstein frame

are

Rμν −1

2gμνR ¼ 8πG�Tμν þ 2∂μφ∂νφ

− gμνgαβ∂αφ∂βφ − 2VðφÞgμν;

∇μ∇μφ ¼ −4πG�kðφÞT þ dVðφÞdφ

; ð1Þ

where kðφÞ ¼ d lnðAðφÞÞ=dφ with AðφÞ being the con-formal function relating the Jordan and Einstein framemetrics. In what follows we consider only scalar-tensortheories with VðφÞ ¼ 0. Tμν is the Einstein frame energy-momentum tensor Tμν. In modeling the rotating stars weuse the energy-momentum tensor of a perfect fluid Tμν ¼ðϵþ pÞuμuν þ pgμν with energy density ϵ, pressure p and4-velocity uμ.The spacetime metric for rotating stars can be presented

in the form

ds2¼−e2νdt2þρ2B2e−2νðdϕ−ωdtÞ2þe2ζ−2νðdρ2þdz2Þ;ð2Þ

where all the metric functions depend on the coordinates ρand z. We will consider the natural from a physical pointof view case when the geometry and the scalar fieldare invariant under the reflection symmetry through theequatorial plane, i.e. when the metric functions and thescalar field are invariant under the map θ → π − θ.In comparison with [25], in the present paper we use a

slightly different but equivalent representation of the metricand the dimensionally reduced field equations.The dimensionally reduced Einstein frame scalar-tensor

field equations describing the structure of the rotating stars(with constant angular velocity) are given by

B−1DiðBDiνÞ ¼ 1

2ρ2B2e−4νDiωDiωþ 4πe2ζ−2ν

�ðϵþ pÞ 1þ v2

1 − v2þ 2p

�; ð3Þ

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ρ−1DiðρDiBÞ ¼ 16πBe2ζ−2νp; ð4Þ

DiDiω ¼ ½4Di − 2ρ−1Diρ − 3B−1DiB�Diω − 16πρ−1B−1e2ζ−2νðϵþ pÞ v1 − v2

; ð5Þ

ρ−1∂zζþB−1ð∂ρB∂zζþ ∂zB∂ρζÞ−1

2ρ−2B−1∂ρðρ2∂zBÞ−

1

2B−1∂ρ∂zB¼ 2∂ρν∂zν−

1

2ρ2B2e−4ν∂ρω∂zωþ 2∂ρφ∂zφ; ð6Þ

ρ−1∂ρζ þ B−1ð∂ρB∂ρζ − ∂zB∂zζÞ −1

2ρ−2B−1∂ρðρ2∂ρBÞ þ

1

2B−1∂2

zB

¼ ð∂ρνÞ2 − ð∂zνÞ2 −1

4ρ2B2e−4ν½ð∂ρωÞ2 − ð∂zωÞ2� þ ð∂ρφÞ2 − ð∂zφÞ2; ð7Þ

B−1DiðBDiφÞ ¼ 4πkðφÞðϵ − 3pÞe2ζ−2ν; ð8Þ

Dip ¼ −ðϵþ pÞ�Diν −

v1 − v2

Div

�− kðφÞðϵ − 3pÞDiφ; ð9Þ

where Di is the covariant derivative with respect to the3-dimensional flat metric dl2 ¼ dρ2 þ ρ2dϕ2 þ dz2 andv ¼ ðΩ − ωÞBρe−2ν is the proper velocity of the stellarfluid with Ω being the angular velocity of the star.In order to derive the formula for the quadrupole moment

we will need the system of differential equations thatdescribes the exterior of the star, i.e. the stationary andaxisymmetric, vacuum scalar-tensor equations. This systemis formally obtained from Eqs. (3)–(9) by settingϵ ¼ p ¼ 0. Using these stationary and axisymmetric vac-uum equations one can find the asymptotic behavior of themetric functions and the scalar field. From a physical pointof view it is more convenient and clear to present theasymptotic behavior in the quasi-isotropic coordinates rand θ defined by

ρ ¼ r sin θ; z ¼ r cos θ: ð10Þ

In these coordinates, keeping only terms up to order ofr−3 we have

ν ≈ −Mrþ�b3þ ν2M3

P2ðcos θÞ��

Mr

�3

; ð11Þ

B ≈ 1þ b

�Mr

�2

; ð12Þ

ω ≈2Jr3

; ð13Þ

ζ ≈ −�1

4

�1þ D2

M2

�þ 1

3

�bþ 1

4

�1þ D2

M2

��½1 − 4P2ðcos θÞ�

��Mr

�2

; ð14Þ

φ ≈ −Drþ�1

3

�DM

�bþ φ2

M3P2ðcos θÞ

��Mr

�3

; ð15Þ

whereM and J are the mass and the angular momentum,Dis the scalar charge defined by

D ¼ 1

4π

IS2∞

DiφdSi; ð16Þ

b, ν2 and φ2 are constants and P2ðcos θÞ is the secondLegendre polynomial. In the present paper as in [25] we

consider the case lim∞φ ¼ φ∞ ¼ 0. Proceeding further, wecan, just as in general relativity, derive the formula for thequadrupole moment from the asymptotic expansion of themetric functions. After some algebra we find the followingformula for the quadrupole moment in our case

Q ¼ −ν2 −4

3

�bþ 1

4

�1þ D2

M2

��M3: ð17Þ

As can be seen the quadrupole moment formula explic-itly involves the scalar charge and reduces to that in generalrelativity for D ¼ 0.

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The other quantity we will need in the present paper isthe moment of inertia I which is defined as usual, namely

I ¼ JΩ

ð18Þ

with Ω being the angular velocity of the star.The quadrupole moment and the moment of inertia have

been defined in the Einstein frame. However, we need thesequantities in the physical Jordan frame. For the moment ofinertia one can show that it is the same in both frames [25].The relation between the quadrupole moment in theEinstein and the Jordan frame depends in general on theparticular scalar-tensor theory. In the present paper weconsider a class of scalar-tensor theories defined byA2ðφÞ ¼ eβφ

2

[and VðφÞ ¼ 0] with β being a negativeparameter β < 0. For this particular class of scalar-tensortheories, taking into account the asymptotic expansions ofthe Einstein frame quantities given in Eqs. (11)-(15) andespecially that φ∞ ¼ 0, one can easily find the asymptoticexpansion of the Jordan frame metric ~gμν ¼ A2ðφÞgμνwhich shows that the quandrupole moment in the Jordanframe ~Q is the samewith that in the Einstein frame, ~Q ¼ Q.

III. NUMERICAL RESULTS

We calculate the rotating neutron star solutions in scalar-tensor theories of gravity with a modified version of theRNS code [31] developed in [25]. The normalized momentof inertia I ¼ I

M3 and the normalized quadrupole moment

Q ¼ − QM3χ2

are examined, where χ ¼ JM2. More specifically

our goal is to check the universality of the I − Q relation forrapidly rotating scalarized neutron stars. As we havealready commented, the I − Q relation is practically indis-tinguishable from GR for slow rotation [23], if oneconsiders values of β which are in agreement with thecurrent constrains coming from the binary pulsar experi-ments β > −4.5 [4,29,30]. In our calculations we will usemainly β ¼ −4.5 which represents the lower limit on β. Inorder to demonstrate the sensitivity of our results andconclusions to the value of the coupling parameter β, wealso show some calculations for smaller β, more preciselyβ ¼ −5 and β ¼ −6 which are already ruled out by theobservations.In order to quantify the deviation from slow rotation we

will build sequences of models with constant values of thenormalized parameter α ¼ fM, where f ¼ Ω=2π is therotational frequency of the star, and in our system of units fis given in kHz andM is in solar masses. As it was shown in[14], this choice of α leads to EOS independent relations inthe GR case for each value of α.1 The last bit of informationwe have to fix is the set of equations of state. We employsix hadronic EOS which span a very wide range of

stiffness. These are WFF2 [32], APR [33], GCP [34],HLPS [35], FPS [36] and the zero temperature limit ofShen EOS [37,38]. For the sake of completeness weconsider also two strange star EOS—the SQSB40 andSQSB60 given in [39].In Fig. 1 we present the I − Q relation and the associated

deviations from universality in the case of scalar-tensortheories with β ¼ −4.5 and for the pure general relativisticcase. The results for all hadronic EOSs are shown. For eachvalue of α we make a fourth order polynomial fit to theinput data of the form

ln I ¼ a0 þ a1 ln Qþ a2ðln QÞ2 þ a3ðln QÞ3 þ a4ðln QÞ4:ð19Þ

FIG. 1 (color online). I − Q relations for GR (continuous blacklines) and for STT (red lines) for all hadronic EOSs and forseveral values of the parameter α. The case of α ¼ 0.32 is given asdashed black line as representative for slow rotation (GR and STTsolutions in this case are almost indistinguishable). In the middlepanel the relative deviation of the STT solutions from the GRpolynomial fits is presented. In the bottom panel the relativedeviations of STT solutions from the fits to the STT data areplotted.

1Other normalized rotational parameters also lead to EOSindependent relations [13,14].

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The calculated models span the range from slow rotation tothe Kepler (mass shedding) limit. We present the results forα up to 2.23. The slow rotation limit on the other hand,corresponding to α ¼ 0.32 in the figure, is shown as adashed line for both GR and STT. The reason is that for slowrotation the two cases are practically indistinguishable whenβ ¼ −4.5 and the scalarization can hardly be noticed in thegraph. In Fig. 1 only the case of hadronic EOS is shown, butour results show that strange stars lead to practically thesame I − Q relation for both the scalarized and nonscalar-ized cases. The only notable difference is that quark starscan reach higher values of Q and I for rapid rotation, due tothe higher oblateness close to their Kepler limit.

In the two lower panels of Fig. 1 the relative devia-tions of the solutions from the polynomial fits are shown,defined as

ΔI ¼ jI − IfitjIfit

: ð20Þ

We plot two deviations–ΔISTT (bottom panel) which rep-resents the deviation of the data in STT from the calculatedSTT fits, i.e. the deviation from EOS universality, andΔISTT=GR (middle panel) which is the deviation of thescalarized neutron stars from the general relativistic fits.Several conclusions can be made using these results.

First it is important to note that for all values of α thedeviation from EOS universality for scalarized neutronstars is below approximately 1%, similar to the pure GRcase. This means that for a fixed value of α the I − Qrelation is indeed nearly EOS independent for scalarizedneutron stars. The difference between the I − Q relationsfor neutron stars in GR and STTon the other hand increaseswith rotation, which is an expected result as rapid rotationcan significantly magnify the deviations from GR in theneutron star equilibrium properties [25]. But still thedeviations are below roughly 5% even for the most rapidlyrotating models shown on the graph. This difference isabove the deviation from EOS universality and it issufficient to make a clear distinction between the I − Qrelations for scalarized and nonscalarized solutions (at leastfor rapid rotation). But the deviation is below the expectedobservational accuracy and therefore it would be difficult toset further constrains on STT using the I − Q relations.Let us now turn to the question of the sensitivity of our

results to the value of the scalar field coupling parameter β.In Fig. 2 we examined the behavior of the scalarized

FIG. 2 (color online). Comparison between I − Q relations forGR and for STT, for several values of β ¼ −4.5;−5;−6. Thepresented data are for EOS APR.

FIG. 3 (color online). The unnormalized moment of inertia (left panel) and quadrupole moment (right panel) as functions of theneutron star mass for several sequences with fixed rotational frequency. The presented results are for EOS APR.

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solutions for different values of β in the case of EOS APR.We should note that the cases with β ¼ −5 and β ¼ −6 arealready ruled out by the binary pulsar experiments and theyare used just to examine the qualitative behavior. As onecan see, when the value of β is decreasing, the I − Qrelation can change considerably. This is indeed anexpected result because in general smaller negative valuesof β lead to larger deviations from GR, and for examplescalarized neutron stars with β ¼ −6 would alreadyhave very different equilibrium properties even in thenonrotating limit.It is important to note that even though the normalized

I − Q relations are very close to GR for the case ofβ ¼ −4.5, the unnormalized moment of inertia and quadru-pole moment can deviate strongly from pure Einstein’stheory. In Fig. 3 the unnormalized I and Q are shown asfunctions of mass for several fixed values of the rotationalfrequency. It is evident that as the frequency increases thedeviations from GR are magnified and the difference canreach much larger values compared to the normalized I − Qrelations. In the graph we have shown models with rota-tional periods up to approximately 1 ms, but if oneconsiders even faster rotation the deviations increasefurther, reaching close to the Kepler limit above 50%for the moment of inertia and 100% for the quadrupolemoment. This can potentially lead to observationalmanifestations.

IV. CONCLUSIONS

In the present paper we investigated the normalizedI − Q relations for rapidly rotating neutron stars in scalartensor theories of gravity. Our goal was to check theuniversality, both with respect to EOS and to the gravita-tional theory, that was observed for slow rotation [23]. Thework is motivated by the fact that rapid rotation cansignificantly influence the scalarization and increase thedifferences from GR [25]. Our results show the EOSuniversality is well preserved but the rotation can magnify

the deviation from GR. Still the differences are belowroughly 5% (for the current lower limit of the scalar-fieldcoupling parameter β ¼ −4.5) which is supposed to besmaller than the expected observational uncertainties.We studied also the dependence of the I − Q relations for

rapidly rotating neutron stars on the coupling parameter βand showed that as β is decreased, the differences from GRincrease. This is an expected effect as smaller negative βlead to larger deviations from GR in the neutron starequilibrium properties. Still, even for values of β that arealready ruled out by the observations, the changes in thenormalized I − Q relations are not so strong.An important observation is that although the normalized

moment of inertia and quadrupole moment are very close tothe pure general relativistic case, the unnormalized I and Qcan change significantly due to the neutron star scalariza-tion. This can also lead to distinct observational signatures.Another important conclusion that can be drawn from ourresults is that although the normalized I − Q relations havethe very nice property of being independent from the EOS,they can suppress the deviations between general relativityand some alternative theories. That is why they alone are oflimited use for testing Einstein’s theory.

ACKNOWLEDGMENTS

We would like to thank George Pappas for discussionsand Emanuele Berti for a critical reading of the manuscriptand constructive suggestions. D. D. would like to thankthe Alexander von Humboldt Foundation for a stipend.K. K., S. Y. and K. S. would like to thank the ResearchGroup Linkage Programme of the Alexander vonHumboldt Foundation for the support. The support bythe Bulgarian National Science Fund under GrantNo. DMU-03/6, by the Sofia University Research Fundunder Grant No. 63/2014 and by the German ScienceFoundation (DFG) via Grant No. SFB/TR7 is gratefullyacknowledged. Partial support comes from “New-CompStar,” COST Action MP1304.

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