A free parametrized TOV: Modified Gravity from Newtonian to Relativistic Stars

0.14 MB·

by  Hermano Velten

A free parametrized TOV: Modified Gravity from
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Newtonian to Relativistic Stars
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n
a
J
2 HermanoVelten∗†
1 CPT,AixMarseilleUniversité,UMR7332,13288Marseille,France

UniversidadeFederaldoEspíritoSanto(UFES),Vitória,Brazil
]
O E-mail: [email protected]
C
AdrianoM.Oliveira
.
h InstitutoFederaldoEspíritoSanto(IFES),Guarapari,Brazil
p
UniversidadeFederaldoEspíritoSanto(UFES),Vitória,Brazil
-
o E-mail: [email protected]
r
t
s AnetaWojnar
a
[ IFT,UniversityofWroclaw,pl. M.Borna9,50-204,Wroclaw,Poland,

INFNSez. diNapoli,Univ. diMonteS.Angelo,Ed. G,ViaCinthia,I-80126Napoli,Italy.
1
E-mail: [email protected]
v
0
0
WetestafreeadhocparametrizationoftheTolman-Oppenheimer-Volkoff(TOV)equation. We
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3 do nothave in mind anyspecific extendedtheoryof gravity (ETG)but each new parameterin-
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troducedhasaphysicalinterpretation. Ouraimisfullypedagogicalratherthanaproposalfora
.
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newETG.Givenarealisticneutronstarequationofstatewemapthecontributionsofeachnew
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6 parameterintoashiftintrajectoriesofthemass-radiusdiagram. Thisexerciseallowsustomake
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thecorrespondencebetweeneachTOVsectorwithpossiblemodificationsofgravityandclarifies
:
v
howneutronstarobservationsarehelpfulfordistinguishingtheories.
i
X
r
a
TheModernPhysicsofCompactStars2015
30September2015-3October2015
Yerevan,Armenia
Speaker.
∗
†ThiscontributionwassupportedbyA*MIDEXproject(nANR-11-IDEX-0001-02)fundedbythe“Investissements
d’avenir" FrenchGovernment program, managed bytheFrench National ResearchAgency (ANR).HV andAO also
thankCNPqandFAPES.AWacknowledges support fromINFNSez. diNapoli(IniziativeSpecificheTEONGRAV).
Moreover,HVandAWwouldliketothanktheOrganizersforwarmhospitalityduringtheconference. Wealsothank
DavidEdwinAlvarezCastilloforacriticalreadingofthemanuscript.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
AfreeparametrizedTOV HermanoVelten
1. Introduction
One of the main challenges in modern astrophysics concerns the equation of state (EoS) of
neutron stars (NSs). Recent observations point outtoM 2M [1,2]objects preferring therefore
∼ ⊙
stiffnuclear EoS.However,itisworthnoting thatthislatterconclusion isvalidonlyinthegeneral
relativity (GR)domain.
Another important property of NSs is the stellar radius R. The mass-radius (M R)diagram
−
has become a comprehensive tool for matching theoretical predictions to observations. Modifica-
tions of GR can produce shifts in the trajectories along the M R diagram but sometimes higher
−
masses are achieved at the price of having undesirable larger radii configurations. Thecurrent ra-
dius determination isnotsoprecise asinthemasscasebutfuture satellites willplace veryprecise
constraints on both M and R for the same object. As specific examples one can cite The Neutron
starInterior CompositionExplorer(NICER)missionandtheSKAradioproject.
Although GR is a very well tested theory and its predictions have been confirmed with solar
system and binary pulsars observations the cosmological arena still challenges it. Actually, by
adoptingGRweareleftwiththestrongindicationthat 95%oftheenergybudgetoftheuniverse
∼
is unknown. According to the most recent observations, this fraction is divided into the probably
fraction of 25%fordarkmatterand 70%fordarkenergy.
∼ ∼
One possible solution relies in modifying GR on large (cosmological) scales. By replacing
eitherthedarkmatterorthedarkenergycomponentbysomeextendedtheoryofgravity(ETG)the
alreadyconfirmedpredictionofGRshouldbereconquered. Thishappensusuallyviatheinclusion
of screening mechanisms around anastrophysical environment where theETGpredictions should
suffer a metamorphopsia back to the GR ones. If this is actually the case, it is therefore expected
that the NS interior and its habitat remains described by GR. However, the use of NSs for con-
straining ETGshasbecome afruitful routeofinvestigation. See[3]forafewrecent worksonthis
topic.
The first usual procedure is to compute the analogue of the Tolman-Oppenheimer-Volkoff
(TOV) equation [4, 5]—the GR structure for static and spherical objects—for your favorite ETG.
In general, departures from the standard TOV equation are non-trivial. However, it can be useful
to understand how masses and radii of NSs are shifted in the M R plane according to specific
−
changes in the TOV structure. In this sense, a recent and interesting work [6] proposed a “Post-
TOV” formalism based on the parametrized post-Newtonian theory in which non-GR effects can
be separately studied inrelativistic stars. Wealso point out the analysis done inRef. [7]in which
thespecificpressurecontribution totheactivegravitational massinNSsisstudied indetail.
Our aim in this contribution is to propose a free parametrization for the TOV equation. The
curiousreadercangodirectlytoEqs. (3.1)and(3.2). Insomesense,thisisanextensionof[7]. Our
approachshouldbeseenasapedagogicalguidetounderstand theroleplayedbydifferentphysical
contributions totheTOVequation ratherthannewaETG.
Inthe next section wereview aquick derivation of the TOVequation. Section 3presents our
free parametrization for stellar equilibrium equations. Wewill adopt one realistic EoS for the NS
interiorandshowinsection4resultsontheM Rplanewhenvaryingsuchnewparameters. Inthe
−
finalsectionwediscussinmoredetailtheroleplayedbyeachparameter. Weusec=ℏ=1units.
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AfreeparametrizedTOV HermanoVelten
2. The Tolman-Oppenheimer-Volkoff equation
In this section we review the derivation of the TOV equation which can be found in most
textbooks aboutGRorrelativistic astrophysics.
The first and simplest step in studying stellar objects is to consider that matter is spherically
symmetricdistributed inastaticgeometryi.e.,timereversalaboutanyoriginoftime,
ds2= B(r)dt2+A(r)dr2+r2(dq 2+sin2q df 2) . (2.1)
−
wherehereafter B(r) B;A(r) A. Generalrelativity isbasedonEinstein’s equations
≡ ≡
1
Rmn gmn R=8p GTmn , (2.2)
−2
whereRmn istheRiccitensorandRisthecurvaturescalar(traceoftensorRmn ). Themattercontent,
inthiscase,theinternaldescription ofthestar,isencapsulated intheenergy-momentum tensor
Tmn = (r +p)um un +pgmn , (2.3)
where respectively r and p are the density and pressure of the fluid which depend on the radial
m
coordinate only. Also,um isthe4-velocity (withum u = 1). Noticedthatthefluidisatrest,then
−
ur =uq =uf =0andut = ( gtt)−1/2= √B.
− − − mn
InordertoderivetheTOVequation weneedtoconsider theconservation T ;n =0,whichis
concomitant withBianchi’sidentities. Thisreads
1dB 2 dp
= . (2.4)
B dr −r +p dr
Theinformation fromthecomponents Gtt,Grr andGqq provides theequation
d r
=1 8p Gr r2 . (2.5)
dr(cid:16)A(cid:17) −
Itssolution, demandingthatA(0)isfinite,is
r
M(r) 4p r 2r (r)dr . (2.6)
′ ′ ′
≡Z0
This solution issimilar to the Newtonian equation and therefore one calls M(r)the massfunction
foragivenradiusr. Adoptingthisassociation withtheclassical framework,thegravitational mass
insidethestarwillbecalculated usingM=M(R)whereRisthefinitestellarradius.
Finally,itisalsopossible tocombinealltheseequations intothefinalform
dp GM(r)r 1+rp 1+4Mp (rr3)p
= (cid:16) (cid:17)(cid:16) (cid:17) , (2.7)
dr − r2 1 2GM(r)
− r
whichisknowastheTOVequation [4,5].
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AfreeparametrizedTOV HermanoVelten
3. Parametrized TOV
The TOV equation (2.7) represents the full general relativistic equilibrium configuration for
stars. The compactness h =2GM/R of a star measures the relevance of GR effects. The New-
tonian counterpart is enough for low compactness h 1 stars. For instance, in white dwarfs
≪
h 10 6 whereas in main sequence stars h 10 4. Typical NS compactness values are in
WD − MS −
∼ ∼
therange0.2.h .0.4. Ingeneral,relativisticeffectscanbecapturedbypressurecontributions
NS
and curvature effects. Also, having in mind that typical predictions of ETG replace the gravita-
tional coupling by some effective quantity we therefore propose for the equilibrium equation the
followingparametrization
dp G(1+a )M(r)r 1+brp 1+cM4p(rr3)p
= (cid:16) (cid:17)(cid:16) (cid:17) . (3.1)
dr − r2 1 g2GM(r)
− r
Wealsogeneralize themassfunction M(r)in(3.1)bywriting
dM(r)
=4p r2(r +s p) . (3.2)
dr
Note that this is an effective mass which is used in the integration of (3.1). For s =0 this defi-
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nition should be different from the conventional mass as calculated in (2.6). However, the actual
gravitational mass remains being M M(R). Since this definition can also be written in terms of
≡
l
themetriccomponents(asusuallydefinedviaA(r)=e )thereexistsaalternativevisualizationfor
thatwhichreads
M(r)=r(1 e l (r)) . (3.3)
−
−
Now,thereare5newparameters, namelya ,b ,g ,c ands .
a parametrizes the effective gravitational coupling, i.e., G =G(1+a ). In particular, in
eff
•
f(R)theories onehasa =1/3[8]. InGRa =0.
b couplestotheinertialpressure. Theterm(r +p)appearsfromthehydrostaticequilibrium
• Tmn;n =r =0,whereristheradialcoordinate. InGRb =1.
g isanintrinsic curvature contribution whichisabsent inthe Newtonian physics, i.e., inthe
•
classical caseg =0. InGRg =1.
c measures theactivegravitational effects ofpressure which isaremarkable feature ofGR.
•
Itseffecthasalready investigated inRef. [7]. InGRc =1.
s changes thewaythemassfunction iscomputed taking intoaccount possible gravitational
•
effectsofpressure. InGRs =0.
ThestrategywhichwillbeadoptedhereafteristoassumeonerealisticEoSandthenvarysuch
parameters. This somehow follows the reasoning discussed in Ref. [9]. In this reference, it is
argued thattheoretical EoSusedforNSsareonlyanorder ofmagnitude larger thantypical values
innucleonscatteringexperiments. Ontheotherhand,curvatureeffectswithinNSsaremanyorders
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AfreeparametrizedTOV HermanoVelten
of magnitude above the domain where GR is confined. Therefore, it is safer to claim we have a
betterknowledgment abouttheNSEoSthanthegravitational theoryinsidetheseobjects.
Someparticular configurations oftheseparametershavethefollowinginterpretation:
a =b =g =c =s =0: Thiscorresponds totheNewtonianhydrostatic equilibrium which
•
givesrisetotheLane-Endemequation.
a =0;b =1;g =c =0;s =3: Theneo-Newtonianhydrodynamic—aproposalwhichtries
•
to include relativistic inspired pressure effects at Newtonian level—has been applied to the
stellarequilibriumprobleminRef. [10]. ContrarytotheNewtoniancase,theneo-Newtonian
case leads to the existence of maximum masses for NS as well as the GR prediction. How-
ever, the maximum masses found in the neo-Newtonian formalism are slightly higher than
theGRones.
a =0;b =0;g =c =s =0: Thiswouldconsist inthesimplest manifestation ofmodified
• 6
gravity in Newtonian stars. In general, some f(R) theories gives rise to the modification
a =1/3togetherwiths =0.
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a =b =g =0;c =0;s =0: Configurationtestedin[7]. Theparameter c wouldberelated
• 6
tothegeneration ongravitational fieldbypressure whichisabsentintheclassical context.
In(3.2)wehaveassumedaquitesimplemodificationofthestandardGRmassequationwhich
disappears when p= p(R)=0, where R is the star’s radius. In general, instead of s p one deals
with a much complicated function that may depend even on higher derivatives of pressure. That
dependence isvisibleifoneconsiders themodifiedEinstein’sequations as
Gmn =Tmneff , Tmneff =s¯Tmnmat+Wmn , (3.4)
whereTmatisthestandardperfectfluidenergy-momentumtensor,s¯ isacouplingindicatingoneof
mn
the ETG’s andWmn originates from geometric corrections. Following that approach one gets that
thehydrostaticequilibriumequationsdiffersfrom(2.4)(seeforexample[11]). TheTOVequations
arisingfromsuchequationshavemuchmorecomplicatedforms. SomeofsuchTOV-likeequations
canbewrittenasaparametrized TOVequations considered inthatworkbutthere isstillroom for
amoredetailed discussion inthis direction (in progress). Here, wewould like tostress how small
modificationsofGRstellarequationsaffecttheM Rdiagram. Itisimportanttopointoutdifficul-
−
ties appearing in integrating the mass function coming from the new possible modifications. The
mass of the NS is calculated as M = R4p r2r (r)dr independent of the functional form involved
0
for the effective mass M in these equRations. In GR we have always s =0 and therefore there is
nodifferencebetweentheeffectivemassandthephysicalmeaningfulgravitational massM. More-
over,thereimmediatelygivesriseaquestiononthestabilityoftheconsideredsystem(seeTheorem
2 in page 306 of Ref. [12]) which should be also investigated in the terms of ETG’s. Here, as in
GR, we have adopted the condition dM/dr >0 for determining the maximum mass. However,
c
thestabilityproblemforstaticequilibrium configurations—stars—in ETGisstillanopenproblem
inthefield.
Toclosethesystemofequations anEoSofthetype
p p(r ) , (3.5)
≡
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AfreeparametrizedTOV HermanoVelten
should be specified. Equilibirum configurations are found by numerically solving the coupled
system (3.1), (3.2) and (3.5) with the condition on the central pressure p(0)= p (corresponding
0
toacentraldensityr )anddemandingtha p(R)=r (R)=0.
0
4. Numerical results ontheMass-Radius diagram
Inorder tonumerically solve the setof equations (3.1), (3.2)and (3.5) wedetermine the EoS
ofthe stellar interior. Theusual technique converts dp/dr into dp/dr dr /dr in(3.5) insuch way
that the central density (r (r =0)=r ) becomes a free parameter. A given r value determines
0 0
onesinglepointintheM Rdiagram. Byvaryingr someordersofmagnitudearoundthenuclear
0
−
saturation densityoneobtainsacurveinthisplaneM R.
−
AmongavastnumberofpossibleEoSfoundintheliteraturetheBSkfamilyprovideanunified
description of the stellar interior treating in a consistent way transitions between outer and inner
crust (and core). The BSk structure has 23 free coefficients which have to be fitted numerically.
Each BSk equation of state correspond to one specific numerical fit. For example, the unified
BSk19, BSk20, and BSk21 EoSs approximate, respectively, the EoSs FPS [13, 14] (soft), APR
[15](moderate), andV18[16](stiff)—see also[17]fordiscussion andreferences.
In general, predictions for the maximum masses in BSk models slightly differ. BSk19 EoS
does not allow masses larger than 2M and therefore observations of very massive neutron stars
⊙
withradius 13KmwouldfavortheBSk20andBSk21fits. Anuptodatecompilationofobserved
∼
neutron starsviadifferent observational methodscanbefoundin[2].
We will use in this contribution the BSk20 only. Our strategy is to fix the GR configuration
and letone ortwoparameters change simultaneously. ThiswillbeshowninFig. 1. Ineach panel
ofthisfigurethesolidblacklinerepresents theGRconfiguration.
5. Discussion
Inthiscontribution wehaveproposedafree“ad-hoc”parametrizationoftherelativisticequa-
tionforequilibrium. Ouraimisfullypedagogical ratherthanaproposalforanewextendedtheory
forgravity. BykeepingtheBSk20EoSfortheneutronstarinterior—whichisrealisticinthesense
thatequilibriumconfigurationsachieve2M usingGR—weseektheimpactofthenewparameters
a ,b ,g ,c ands . Theycanbevisualized in⊙Eq. (3.1).
The simplest expected manifestation of modified gravity theories occurs via a redefinition of
theeffectivegravitationalcouplingG G(1+a ). Theeffectsofa ontheM Rplaneareseen
eff
→ −
inthepanelsbelonging torowA(seethecaptionofFig. 1). Thelargerthea value,thesmallerthe
typical radiusoftheequilibrium configurations. Themaximummassisalsoreduced.
The parameter b is clearly related to the maximum mass. The larger the inertial effects of
pressure (b ),thesmallerthemaximummass.
Concerning the parameter c we have confirmed the findings of [7], i.e., the self-gravity of
pressure contributes toreducing themaximum allowedNSwithalmostnoimpactontheradiusof
thestar.
Theimpactofparameterg isrelevanttopanels AIII,BI andCI. IntheNewtonianhydrostatic
equilibrium g =0. However,byre-performing theNewtonianclassical casewithamodifiedNew-
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AfreeparametrizedTOV HermanoVelten
3.0 3.0 3.0 3.0
Α=0,Β=0 Α=0,Γ=0 Α=0,Χ=0 Α=0,Σ=3
2.5 Α=0,Β=1 2.5 Α=0,Γ=1 2.5 Α=0,Χ=1 2.5 Α=0,Σ=0
Α=1(cid:144)3,Β=0 Α=1(cid:144)3,Γ=0 Α=1(cid:144)3,Χ=0 Α=1(cid:144)3,Σ=3
2.0 Α=1(cid:144)3,Β=1 2.0 Α=1(cid:144)3,Γ=1 2.0 Α=1(cid:144)3,Χ=1 2.0 Α=1(cid:144)3,Σ=0
Ÿ Ÿ Ÿ Ÿ
MM(cid:144) 1.5 ΑΑ==--11(cid:144)(cid:144)33,,ΒΒ==01 MM(cid:144) 1.5 ΑΑ==--11(cid:144)(cid:144)33,,ΓΓ==01 MM(cid:144) 1.5 ΑΑ==--11(cid:144)(cid:144)33,,ΧΧ==01 MM(cid:144) 1.5 ΑΑ==--11(cid:144)(cid:144)33,,ΣΣ==30
1.0 1.0 1.0 1.0
0.5 0.5 0.5 0.5
0.0 0.0 0.0 0.0
8 10 12 14 16 18 20 22 8 10 12 14 16 18 20 22 8 10 12 14 16 18 20 22 8 10 12 14 16 18 20 22
RHkmL RHkmL RHkmL RHkmL
3.0 3.0 3.0
Β=1,Γ=0 Β=1,Χ=0 Β=1,Σ=3
2.5 Β=1,Γ=1 2.5 Β=1,Χ=1 2.5 Β=1,Σ=0
Β=0,Γ=0 Β=0,Χ=0 Β=0,Σ=3
2.0 Β=0,Γ=1 2.0 Β=0,Χ=1 2.0 Β=0,Σ=0
Ÿ Ÿ Ÿ
MM(cid:144) 1.5 MM(cid:144) 1.5 MM(cid:144) 1.5
1.0 1.0 1.0
0.5 0.5 0.5
0.0 0.0 0.0
8 10 12 14 16 18 20 22 8 10 12 14 16 18 20 22 8 10 12 14 16 18 20 22
RHkmL RHkmL RHkmL
3.0 3.0
Χ=1,Γ=0 Γ=1,Σ=3
2.5 Χ=1,Γ=1 2.5 Γ=1,Σ=0
Χ=0,Γ=0 Γ=0,Σ=3
2.0 Χ=0,Γ=1 2.0 Γ=0,Σ=0
Ÿ Ÿ
MM(cid:144) 1.5 MM(cid:144) 1.5
1.0 1.0
0.5 0.5
0.0 0.0
8 10 12 14 16 18 20 22 8 10 12 14 16 18 20 22
RHkmL RHkmL
3.0
Χ=1,Σ=3
2.5 Χ=1,Σ=0
Χ=0,Σ=3
2.0 Χ=0,Σ=0
Ÿ
MM(cid:144) 1.5
1.0
0.5
0.0
8 10 12 14 16 18 20 22
RHkmL
Figure 1: Mass-radius diagram for various choices of the parameters a ;b ;g ;c ;s . In order to mention
a specific panelin the textwe will introducethe followingnotation: Thepanelin the first row (A) and in
thefirstcolumn(I),top-left,iscalledAI.Itthispaneltheeffectsofvaryinga andb canbeseenwhilethe
otherparametersarefixedtotheGRconfiguration(g =1;c =1;s =0). Thesamestrategyisadoptedin
theremainingpanels. IneachpanelthestandardGRconfigurationisdenotedbythesolid-blackline. The
massiscalculatedviaM= R4p r2r drevenifs =0wasadoptedwhennumericallysolvingthesystemof
0 6
equationsi.e.,wealwaysploRtthegravitationalmassMratherthantheeffectivemassM. Effectsofvarying
s canbeseeninpanelsAIV,BIII,CIIandDI.
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AfreeparametrizedTOV HermanoVelten
tonian potential of the typeV(r)= GM, where A is a constant, it is possible to obtain a similar
−r A
contribution with a singularity in the d−enominator of the TOV equation like the g =1 case. The
existence ofamaximum mass(absent intheNewtonian case) isapure relativistic prediction. But
itisworthnotingthatthecurveswithg =0produceindeedamaximummass,butwithaverylarge
valuewhichcannotbeseenintheseplots.
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