Impact of the second order self-forces on the dephasing of the gravitational waves from quasi-circular extreme mass-ratio inspirals

0.33 MB·English

by  Soichiro Isoyama

YITP-12-66
Impact ofthe second order self-forces onthe dephasing ofthe gravitationalwavesfrom
quasi-circular extreme mass-ratioinspirals
SoichiroIsoyama1,∗ RyuichiFujita2,† NorichikaSago3,‡ HideyukiTagoshi4,§ andTakahiroTanaka1¶
1 Yukawa Institute for Theoretical Physics, Kyoto university, Kyoto, 606-8502, Japan
2 Departament deF´ısica, Universitat delesIllesBalears, PalmadeMallorca, E-07122 Spain
3 Faculty of Arts and Science, Kyushu University, Fukuoka 819-0395, Japan
4 DepartmentofEarthandSpaceScience,GraduateSchoolofScience,Osakauniveristy,Osaka,560-0043, Japan
(Dated:October10,2012)
2 Theaccuratecalculationoflong-termphaseevolutionofgravitationalwave(GW)formsfromextreme(inter-
1 mediate)massratioinspirals(E(I)MRIs)isaninevitablesteptoextractinformationfromthissystem.Achieving
0 thisgoal,itisbelievedthatweneedtounderstandthegravitationalself-forces. However,itisnotquntatively
2 demonstratedthatthesecondorderself-forcesarenecessaryforthispurpose. Inthispaperwerevisittheprob-
lemtoestimatetheorderofmagnitudeofthedephasingcausedbythesecondorderself-forcesonasmallbody
t
c inaquasi-circularorbitaroundaKerrblackhole,basedontheknowledgeofthepost-Newtonian(PN)approx-
O
imationandinvokingtheenergybalanceargument. Inparticular,wefocusontheaverageddissipativepartof
theself-force, sinceitgivestheleadingorder contributionamong thevariouscomponents ofthem. Toavoid
9
thepossibility that theenergy fluxof GWsbecomes negative, wepropose anew simple resummation called

exponential resummation, whichassuresthepositivityoftheenergyflux. Inordertoestimatethemagnitude
]
c oftheyetunknownsecondorderself-forces,herewepointoutthescalingpropertyintheabsolutevalueofthe
q PNcoefficientsoftheenergyflux. Usingthesenewtools, weevaluatetheexpectedmagnitudeofdephasing.
- Ouranalysisindicatesthatthedephasingduetothesecondorderself-forcesforquasi-circularE(I)MRIsmay
r
g bewellcapturedbythe3PNenergyflux,onceweobtainallthespindependentterms,exceptforthecasewith
[ anextremelylargespinofthecentralKerrblack.

1
PACSnumbers:04.30Db,04.25Nx,04.30Tv,95.85.Sz
v
9
6
5
2
I. INTRODUCTION
.
0
1
Extrememassratioinspirals(EMRIs)andintermediatemassratioinspirals(IMRIs),inwhichastellarmassorseveraltens
2
solarmasscompactobjectinspiralsintoamoremassivecentralblackhole,haveattractedmuchinterestnotonlyasapromising
1
sourceofthegravitationalwaves(GWs)forfuturespacebornGWdetectors,butalsoasauniquecleanprobeofthespacetime
:
v region of strong gravity. To achieve the test of general relativity using GWs from E(I)MRIs, we need to predict sufficiently
i
X accuratewaveforms. ThisrequirementmotivatesustomodeltheE(I)MRIsasthemotionofasmallbodyinagivenbackgroud
spacetimewithgravitationalbackreaction. Thisbackreactionistreatedasgravitationalself-forces [1–6]anditshigherorder
r
a extensionwithrespecttothemassratiohasattractedmuchinterestinrecentyears [7–11]. Seefollowingreviewarticles, e.g.
,[12,13]andreferencesthereinformoredetails.
Infact,followingthescalingargument(Seee.g.[14,15]),thephaseofGWsfromaparticlewhoseorbitalfrequencysweeps
afewordersofmagnitudebeforetheplungecanbeexpandedas
M µ µ2
Φ= BH Φ(0)+ Φ(1)+O , (1)
µ M M2
(cid:20) BH (cid:18) BH(cid:19)(cid:21)
whereΦ(0) andΦ(1) areO(1)quantitiesindependentofµ,whichisthemassofasmallparticle,andM andaarethemass
BH
and the spin parameterof the Kerrblack hole, respectively. 1 On the one hand, Φ(0) in Eq. (1), we only needthe self-forces
up to the first order time-averageddissipative part. In addition, it has been long known that Φ(0) can be computedwith well
establishedbalanceargument,whichrelatesthefirstordertime-averageddissipativepartofself-forcestotheenergyandangular
momentumfluxesassociated to the globalkilling vectorson backgroundKerr black hole [19]. Thoughthe Carter constantis
∗Electronicaddress:isoyama˙at˙yukawa.kyoto-u.ac.jp
†Electronicaddress:ryuichi.fujita˙at˙uib.es
‡Electronicaddress:sago˙at˙artsci.kyushu-u.ac.jp
§Electronicaddress:tagoshi˙at˙vega.ess.sci.osaka-u.ac.jp
¶Electronicaddress:tanaka˙at˙yukawa.kyoto-u.ac.jp
1Throughoutthispaper,weassumethattheresonanceisabsentduringitsorbitalevolution.SeeRefs.[15–18]forfurtherdiscussionaboutresonantorbits.
2
notassociated to anyKillingvectorandsimplebalanceargumentisnotapplicable,the wayto computinglongtimeaveraged
evolutionofCarterconstanthasbeenalreadywellestablished[20–23]. Ontheotherhands, Φ(1), whichisreferredtoasthe
dephasing relative to Φ(0), depends on two differentcomponents of the self-forces: the first order conservative part, and the
averagedsecondorderdissipativepart. 2 Thefirstsub-leadingtermΦ(1) stillcanbeimportantsincepotentiallyitmaygivethe
correctionsignificantlygreaterthanunitytothephase [24–26].
Toobtainaccuratewaveform,therehavebeenmanyworksonself-forces. Asforthefirstorderconservativepart,thanksto
therecentmassivedevelopment,we arenowin partreadyforpracticalcomputationwithnumericalimplementation [27–31].
Particularly, in the case of Schwarzschild backgroundthe correction to the orbital frequencies [32] and Φ(1) [33] have been
already studied extensively. Even in the case of Kerr background the preliminary results of the self-forces in quasi-circular
orbitshavebeenreported[34].
By contrast, the averagedsecond orderdissipative partofself-forceshas beenso far studied onlyat the formallevelin the
contextofblackholeperturbation [7–11]. Itwillrequiremuchmoreefforttoestablishthemethodtocomputethesecondorder
dissipative part, especially in the case of Kerr background. Under such circumstances, a typical strategy to evaluate Φ(1) is
to makeuse of the standard post-Newtonian(PN) approximation,in which we assume slow motionof a satelite and its weak
gravitationalfield.BasedonthePNapproximation,HuertaandGair[35]evaluatedthesizeofdephasingcausedbythefirstorder
conservativeself-forcesandtheaveragedsecondorderself-forces,pickinguprepresentativeEMRIsinquasi-circularorbitson
aKerrblackhole.ThesamedephasingwasalsodiscussedbyYunesetal.[36]usingtheeffectiveonebodyformalismagainfor
representativeE(I)MRIs.
AnaiveexpectationisthatthePN approximationisnotsuitableformodellingthewaveformsofE(I)MRIs,especiallyfora
Kerr black hole with large spin. TypicalE(I)MRIsin circular orbitsspend the last few yearsof inspiralin the vicinity of the
innermost stable circularorbit(ISCO). Since the ISCO radiusreachesthe eventhorizonas the spin ofthe Kerrblack hole is
increasedtotheextremallimit,themotionofthebodybecomeshighlyrelativistic,exceedingthevalidityrangeofthestandard
PNapproximation[37]. However,thethingsarenotsotrivial. ThetimespentneartheISCObecomeslongerasweincreasethe
massratioM /µ,butinthatcasethemassofthelargecentralblackholealsobecomeslarger. Then,thetotalcyclesofGWs
BH
become smaller for a given observation time. As a result, the correction due to higher order self-forces mightbe suppressed
belowtheobservationalthreshold,despitethelossoftheaccuracyofthePN approximation. Therefore,itisnotsoobviousif
thereareE(I)MRIssuchthatreallyrequirethenotionofthesecondorderself-forces.
Thepreviousanalysesmentionedabove[35,36]arefocusingonthecorrectionscomingfromtheself-forcesatthecurrently
availablePNorderandarelimitedtorepresentativeE(I)MRIs.Togetaninsightintowhetherthesecondorderself-forcesbased
on the black hole perturbationare really necessaryto calculate the waveformsof quasi-circularE(I)MRIsor not, therefore, it
wouldbeusefultogiveanestimateofdephasingcomingfromtheaveragedsecondorderdissipativeself-forces,focusingonthe
yetunknownhigherPNtermsandsurveyingthewholeparameterregionofE(I)MRIs
Whatwediscussin thispaperisthe adiabaticevolutionofE(I)MRIsinquasi-circularorbitsona Kerrspacetime. Here the
adiabaticevolutionmeansanapproximationinwhichtheevolutionoftheorbitalfrequencyisdeterminedbytheenergybalance
argument,i.e.,therateofchangeofthetotalenergyofthebinaryisequatedtotheenergyfluxemittedtotheinfinity.
ToevaluatetheorderofmagnitudeoftheyetunknownhigherPNcorrections,weneedtorelyonsomeextrapolation.Forthis
purpose,wefirstintroduceasimplenewresummationoftheenergyflux,whichwecallthe“exponentialresummation”. When
thespinofaKerrblackholeislargeenough,thePNenergyfluxintheTaylorformcanbenegativeoutsidetheISCOradiusfor
somePN orders[38]. Ifthishappens,the estimatedtotalphasebeforetheplungedivergesandtheextrapolationto thehigher
PNorderwillnotmakesense. Oursimple“exponentialresummation”istheonethatensuresthepositivityoftheenergyflux.
AsaPNinputforthecorrectionsatthenextleadingorderinthemassratio,thebestoneavailablesofaristhe3.5PNenergy
fluxofGWs[39,40]withlinearspindependenttermsupto3PNorder,whichhasbeenrecentlyderivedbyBlanchetetal.[41].
Toestimate thepossiblemagnitudeoftheyetunknownhigherorderPN terms, wefocusona scalingpropertyamongthePN
coefficients in the energy flux. Using the 8PN energy flux in the test particle limit [42, 43], we will show that the absolute
valuesofthecoefficientsscaleroughlyasrequiredfromtheconvergenceofthePNseriesuptothelightringradius. (Thispoint
wasalsodiscussedbyNakanoetal. independently[44].) SincethisscalingbehaviorisrelatedtothePNconvergence,despite
thelackofthehigherPNtermsintheenergyflux,weconjecturethatthesamescalingpropertywillholdforthetermshigher
orderinthemassratio. Underthisassumption,weestimatetheorderofmagnitudeoftheunknownportionoftheenergyflux
comingfromhigherPNtermsatthenextleadingorderinthemassratiovia“extrapolation”. Gatheringthetoolsstatedabove,
we will investigate the impact on the dephasing from the averaged dissipative second-orderself-forces for various E(I)MRIs
systematically.Thisisthemaingoalofthispaper.
2Herethe“dissipative”partmeanstheself-forcesthatcausethetimevariationoftheconstantsofmotion,suchastheenergy,theangularmomentumaround
theaxisofsymmetryandtheCarterconstant.The“conservative”partmeanstheonegivingthecorrectiontotherelationbetweentheorbitalfrequenciesand
theconstantsofmotion[15,20]. Themeaningof“time-averaged”isaveragingoverasufficientlongperiodcomparedtothetimescalefortheevolutionof
thephasedifferencebetweentheoscillationsintheradialandthezenithangledirections.
3
Theremainderof thismanuscriptis organizedas follows. In Sec. IIwe brieflyreviewhowthe accumulatedphaseof GWs
fromadiabaticinspiraliscalculated,basedonthe balanceargument. InSec.III, tocurethenegativeenergyfluxofGWs that
appearersin the truncatedPN Taylorseries expansion,we proposethe exponentialresummationthat ensuresthe positivityof
theenergyflux. Sec.IVisdedicatedtothestudyofthescalingpropertyinthecoefficientsoftheenergyfluxinthetestparticle
limitthatbecomesmanifestowingtothebrandnew8PNenergyflux[43]. InSec.V,usingtheexponentialresummationandthe
scalingproperty,wewillestimatethedephasingcomingfromtheyetunknownpartofthesecondorderself-forcesforvarious
E(I)MRIs. We find that the unkown non-linearspin dependentterms at the lower PN order in the energyflux dominante the
unknowndephasing.WesummarizeourresultsandconcludeinSec.VI.
Inthismanuscript,weusegeometricalunitsG=c=1andthesignconventionofthemetricis(−,+,+,+).Thecoordinates
(t,r,θ,φ)denotetheBoyer-LindquistcoordinatesoftheKerrblackhole[45].Wefrequentlyusethedimensionlessspindefined
byq :=a/M ,andthesymmetricmassratiodefinedbyν :=µM /(M +µ)2.
BH BH BH
II. THEACCUMULATEDPHASEOFTHEGRAVITATIONALWAVEFROMANINSPIRALINGBINARY
We consider a binarycomposedof a small satellite bodywith the rest massµ in a quasi-circularorbitarounda Kerr black
holewiththemassM andthespinparametera. Weneglecttheeffectofthespinofthesatellite,whichisnegligibllysmall
BH
forE(I)MRIsthoughplaysnonegligibleroleinsomesituations [46,47]. Forabinaryinaquasi-circularorbit,theaccumulated
phaseofGWsiscalculatedas
x0 x3/2E′(x)
Φ := −2 dx , (2)
M E˙(x)
ZxISCO
whereE isthebindingenergyofthebinary,E˙ := dE/dtistheenergylossrate,andtheprimedenotesthedifferentiationwith
respectto x, which isa dimensionlessorbitalfrequencydefinedby x := (MΩ)2/3 with the orbitalfrequencyΩ and the total
massM := M +µ. HereE issupposedtodependonlyonx, neglectingtheeffectofvariationofmassduetothe energy
BH
absorbedbytheblackhole.Throughoutthepaperwetotallyneglecttheeffectofthistimedependentmassvariation.Wechoose
thelowerboundoftheintegralinEq.(2),x tothevalueofxattheinnermoststablecircularorbit(ISCO)andtheupper
ISCO
bound x to the one determined by the condition coming from the finite observation time. Throughoutthis paper, we adopt
0
t =1yearastheobservationtime.
obs
Ourprimaryinterestinthispaperistheorderofmagnitudeofthedephasingcomingfromthecorrectionsatthenextleading
order in the mass ratio. To evaluate it, we calculate the difference between Φ with and without the higher order corrections.
Checkingthedetectabilityofthisdifferencebyobservations,theinitialandthefinalfrequenciesshouldbekeptunchangedwhen
weevaluateit. Here,wefixbothx andx tothevaluesdeterminedbythetestparticlelimit. Namely, [48]
0 ISCO
−1/6
x := R3/2 +q R :=3+Z ∓ (3−Z )(3+Z +2Z ),
ISCO ISCO ISCO 2 1 1 2
Z := (cid:16)1+(1−q2)(cid:17)1/3[(1+q)1/3+(1−q)1/3], pZ := 3q2+Z2, (3)
1 2 1
q
wheretheupper(lower)signinthesecondequationischosenfortheco(counter)-rotatingcasewithq >(<)0andR:=r/M
BH
representsthedimensionlessradius.x isdeterminedby
0
x0 E′[0](x)
1(year)=− dx , (4)
E˙[0](x)
ZxISCO
wherethesuperscript[0]meanstheleadingordercontributioninthemassratio,i.e.,thetestparticlelimit,and
R3/2−2R1/2+q 1 1−qx3/2 2/3
E[0] =µ , Ω:= , R= . (5)
R3/4(R3/2−3R1/2+2q)1/2 M(R3/2+q) x3/2
(cid:18) (cid:19)
Asfortheenergy,wehereadoptedthebindingenergyofthebinaryinsteadoftheenergydirectlyrelatedtothefourmomentum
oftheparticle.Itwouldbemorenaturaltoconsiderthelatterinthecontextoftheself-forcecalculationbasedontheblackhole
perturbation,butitisnotgaugeinvariant.Hence,theformerismoresuitableforcomparisonwiththePNcalculation.Thesetwo
energiesaredifferentatthenextleadingorderin themassratioduetothe presenceofthe gravitationalfield energy,butthere
mustbeone-to-onecorrespondencebetweenthem,oncethegaugeiscompletelyfixed.
Theenergybalanceargumenttellsusthattheaveragedlossofthetotalenergyshouldbeequaltotheaveragedtotalenergy
fluxtothefuturenullinfinitybecauseoftheenergyconservation(seee.g.Ref.[19,49]).Hence,aslongaswedefinethebinding
energyappropriately,
dE
− =L, (6)
dt
4
holds,inthesenseaveragedoverasufficientlylongtimeandneglectingthehorizonabsorptionflux,whereListheenergyflux
emittedtothenullinfinity.Hence,weevaluatethephaseby
x0 x3/2 E′[0](x)
Φ[L]:=2 dx . (7)
M L(x)
ZxISCO
Intheaboveequation,asweareinterestedinthedissipativecorrections,wefixedE′ totheexpressioninthetestparticlelimit.
WeexpandLasL=L[0]+νL[1]+O(ν2).TocomputeL[0],wecanusetheTeukolskyformalism[50]andinvokethenumerical
codedevelopedbyFujitaandTagoshi[51,52].
The finite mass correctionsat the nextleadingorder, L[1], are in partprovidedby the standardPN calculations. In the PN
formalism, the energy flux of GWs emitted to the infinity from a quasi-circular binary is obtained up to 3.5PN order for the
spinindependentterms[39,40]andupto3PNorderforthetermslinearinspinwhenthespinvectorsareparalleltotheorbital
axis[41]. Inthepresentnotation,truncatedatO(ν),itisgivenby
32 1247 35 11 17 44711 9271
LT (x,q) = x5ν2 1+x − − ν +x3/2 4π− q+ qν +x2 − + ν
nPN 5 336 12 4 4 9072 504
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
8191 59 583 3749
+x5/2 − π− q+ν − π+ q
672 16 24 144
(cid:18) (cid:26) (cid:27)(cid:19)
6643739519 16 1712 856 65 134543 41 33
+x3 + π2− γ − log(16x)− πq+ − + π2+ πq ν
E
69854400 3 105 105 6 7776 48 2
(cid:18) (cid:26) (cid:27) (cid:19)
16285 214745
+x7/2 − + ν π+O(x4,ν2,q2,qx7/2) , (8)
504 1728
(cid:18) (cid:19) (cid:21)
whereγ =0.57721··· denotesEuler’sconstant.Wewillrefertotheaboveexpressionfortheenergyfluxtruncatedatthenth
E
PNorder3 as“thenPNTaylorflux”. WedenoteitbyLT ,andexpanditlikeLT :=LT[0] +νLT[1] +··· asbefore. We
nPN nPN nPN nPN
alsofrequentlyuse“thenormalizednPNflux”definedby
−1
32
LT (x,q):= ν2x5 LT (x,q). (9)
nPN 5 nPN
(cid:18) (cid:19)
Fromnowon,wedenotethePNfluxsolelycomposedoftheknownpartas“theknownnPNTaylorflux”anddistinguishingit
fromthefullPNfluxbyassociating“˜”likeL˜T . Wealsointroducethenotationtodenotetheresidualtermshigherorderin
nPN
PNexpansionandinspinby
LT[i] :=L[i] −LT[i] . (10)
>nPN full nPN
Then,L˜T[1] representstheyetunknownenergyfluxtobedeterminedfromthecomputationofthesecondorderdissipative
>3.5PN
self-forces. FurtherwedefineL˜T[0] andL˜T[0] bythetermsinL[0] thatcorrespondtoL˜T[1] andL˜T[1] ,respectively. Tobe
nPN >nPN nPN >nPN
precise,wedefineL˜T[0] bythesumofthespinindependenttermsupto3.5PNorderandthetermslinearintheblackholespin
nPN
upto3PNorder. L˜T[0] istheremainingtermsattheleadingorderinthemassratio,whichalsoincludesthenon-linearspin
>nPN
dependenttermsinallPNorders.4.
III. AMANIFESTLYPOSITIVEDEFINITEENERGYFLUX:EXPONENTIALRESUMMATION
Inthissectionwe proposeanewsimpleresummationscheme,whichcanbeeasilyappliedifwejustknowthenPNTaylor
flux.
A. LimitationoftheTaylorfluxrevisited
ItiseasytoimaginethatthePNTaylorfluxisnotsoaccuratewhentheorbitalradiusbecomessmall. Indeed,Zhangetal.[37]
pointedoutthatthenPNTaylorfluxinthetestparticlelimitrapidlylosestheaccuracyaroundtheISCOradius.However,itisa
3 WerefertoatermofO(xn)relativetotheleadingorderasthoseofnPNorder.
4Herewedonotdistinguishthelogarithmictermin3PNTaylorfluxinthetestparticlelimitfromtheother3PNterms. Weconfirmedthattheresultsinthe
presentpaperdonotchangemuchevenifweexcludethistermfromL˜T[0]
nPN
5
1
x 0
u
fl

r
o -1
yl
a
T
-2
N
P
n Exact
-3
d 2.5PN
e
z 4.0PN
ali -4 5.0PN
m
5.5PN
r
no -5 6.5PN q = 0.9
8.0PN
-6
0.55 0.6 0.65 0.7 0.75
x
FIG. 1: The normalized nPN Taylor flux in the test particle limit, LT[0](x,q), for q = 0.9 up to x (0.9) := 0.78014.... The
nPN ISCO
horizontalaxisisthedimensionlessfrequencyxofabody.Thelable“Exact”meanstheexactnumericalenergyflux,calculatedbyFujitaand
Tagoshi[51].
differentissuewhetherornottheTaylorfluxisaccurateenoughforourpresentpurposebecausewhatweareinterestedinhere
iswhetherornotthedephasinginGWwaveformsismeasurebale.
As we want to evaluate the correction to Φ in Eq. (7) caused by the yet unknown part of the flux L˜T[1] , we need some
>nPN
extrapolationmethod. Forthis purpose,laterit becomesnecessaryto evaluatesomethinglike the phaseforthe fluxtruncated
at nPN order,Φ[L˜T[0]]. At this pointwe find thatthe nPN Taylor flux is problematic. As mentionedearlier, the energyflux
nPN
truncatedatsomePNordersbecomenegativeoutsideR ifthespinofblackholeq issufficientlylarge,firstlypointedout
ISCO
byTagoshietal.[38].
Toseehowserioustheproblemis,werevisittheTaylorfluxinthetestparticlelimit.ThespinindependenttermsinLT[0] have
nPN
beenanalyticallycalculatedupto22PNorderandthespindependenttermsupto8PNorderbyFujita[42,43]. Theexpression
ofthenormalized8PNenergyfluxincludingcompletespindependenttermsisschematicallyexpandedas5
8 pmax
LT[0](x,q)= Lˆ(n,p)(q)xn(log(x))p , (11)
8PN
n=0 p=0
X X
wherep isthemaximumintegerthatdoesnotexceedn/3. InFIG.1,wedepictedLT[0](x,q),thenormalizednPNTaylor
max nPN
fluxinthetestparticlelimit(11)truncatedatvariousPNordersforq = 0.9. ThecurvesterminateattheISCOfrequency. We
alsoplottheexactnumericalenergyfluxinthetestparticlelimitasareference. Theexactnumericalenergyfluxismanifestly
positivedefiniteforx < x (0.9), whileallthenPN TaylorfluxesinFIG.1crosszeroatsomex = x (q) < x (0.9).
ISCO 0 ISCO
In additionto the alreadyknown2.5PN and 4PN cases 6 [38], we also find that 5PN, 5.5PN, 6.5PN and8PN fluxesbecome
negativebeforexreachesx (q). Evenforamoderatevaluesuchasq = 0.7,westillobservethefluxtocrosszero,sayat
ISCO
8PNorder.Oncethishappens,theintegrandof(7)diverges,andthenthephaseevaluatedbyusingthetruncatedfluxLT[0] does
nPN
notmakesense. Forareliableextrapolation,itisnecessarytousearesummedexpressionfortheenergyfluxthatgivesatleast
afiniteestimateofthephaseforthetruncationatanyPNorder.
5 TheexplicitexpressionforLT[0] willbemadeavailableelsewhereinanappropriateform[43].
8PN
6WhileTagoshietal.[38]reportedthat3PNTaylorfluxwithq = 0.9becamenegativeforsmallradius,wefindthatthe3PNTaylorfluxisalwayspositive
definiteforallradiusirrespectiveofthespinparameteroftheKerrblackhole.
6
B. Theexponentialresummationandanimprovedhybridenergyflux
To overcome the drawback of the nPN Taylor flux, a naive requirementwill be to guarantee the energy flux to be always
positive by resummation. Among various known resummation techniques [53–56], to the best of our knowledge, only the
factorizedresummationensuresthepositivityoftheenergyflux. ThisresummationwasproposedbyDamouretal.[57]forthe
equalmassnon-spinningbinariesin a circularorbit. Fujita and Iyer[58] andPan etal.[59] applieditto the testparticlein a
circularorbitaroundaSchwarzschildandaKerrblackhole,respectively.Here,weproposeanotherevensimplerresummation,
whichisdefinedby
32
Lexp (x,q):= ν2x5exp[Lexp (x,q)], (12)
nPN 5 nPN
with
LenxPpN(x,q):=log LTn′PN(x,q) , (13)
truncatedatnPNorder
(cid:12)
wheren′ ≥ n isunderstood. We refertothisresumm(cid:2)ationandthe(cid:3)(cid:12)energyflux (12) asthe exponentialresummationand“the
(cid:12)
nPNexponentialresummedflux”,respectively.Thisfluxismanifestlypositiveandcanincorporateanyhigherordercorrections
ifthecounterpartsinthenPNTaylorfluxisgiven. Furthermore,itsignificantlyimprovestheaccuracyofapproximation [43].
Fordefiniteness,weexplicitlywritethedefinitionofLexp[0] andLexp[1]as
nPN nPN
32 32
Lexp[0](x,q):= ν2x5exp[Lexp[0](x,q)], Lexp[1](x,q):= ν2x5exp[Lexp[0](x,q)] Lexp[1](x,q), (14)
nPN 5 nPN nPN 5 nPN nPN
(cid:18) (cid:19)
withLexp =Lexp[0]+νLexp[1]+···. WeshowtheexplicitformonlyforLexp[1] as
nPN nPN nPN nPN
35 17 30523 136229 101
Lexp[1](x,q) = − x+ qx3/2+ x2+ q− π x5/2
nPN 12 4 4032 4032 8
(cid:18) (cid:19)
43670915 41 π 70075
+ − + π2− q x3+ πx7/2+O(x4,qx7/2,q2). (15)
12192768 48 2 6048
(cid:18) (cid:19)
Combiningthe exactnumericalflux obtainedin the test particlelimit, L[0] , andthe PN flux, L , onecan obtain a better
full PN
estimatefortheenergyflux. InthecaseoftheTaylorflux,thesetwoarecombinedsimplybysummingL[0] andLT[1]. Inthe
full PN
caseoftheexponentialresummedenergyflux(12),weneedtodothissummationattheleveloftheexponent,Lexp. Namely,
expandingLexp inpowersofν,wereplaceLexp[0]withtheonecorrespondingtothenumericalflux. Then,weobtain
nPN nPN
Lhyb (x,q)=L[0] (x,q)exp νLexp[1](x,q)+··· , (16)
nPN full nPN
h i
whichwecall“thenPNhybridflux”.
Intheanalogouswayasbefore,weintroduce“theknownnPNexponentialresummedflux”and“theknownnPNhybridflux”,
anddistinguishthemfromtheirrespectivecounterpartsbyassociating“˜”. Herethetruncationfortheknownpartismadeat
thelevelofLexp[1]. Namely,L˜exp[1] istruncatedatthesamePNorderforeachorderofspinthatisincludedintheknownterms
nPN
L˜T[1]. For the exponentialresummedflux, we can also introduceL˜exp[0], the counterpartof L˜exp[1] in the test particlelimit,
nPN nPN nPN
asinthecaseofL˜T[0]. Namely,L˜exp[0] istruncatedatthesamePNorderforeachorderofspinthatisincludedintheknown
nPN nPN
termsL˜T[1]. Ontheotherhand,L˜hyb[0] doesnotmakesense. Inthefollowingweidentifythehybridfluxwiththeexponential
nPN nPN
resummedfluxfortheleadingorderinthemassratio. Wesummarizeournotationforvariousfluxesusedintherestofourpaper
inTableI,forreadability.
IV. THESCALINGLAWOFTHECOEFFICIENTSINTHEENERGYFLUX
Nowwetackleourmainissue: howtoevaluatethemagnitudeoftheyetunknownpartoftheenergyflux,L˜[1] . (When
>3.5PN
wediscussgeneralissuesindependentoftheformoftheenergyflux,wesimplysuppressthelabels“T”,“exp”and“hyb”.)Our
strategyhereistoestablishthescalingpropertyinthecoefficientsofthePNexpansionoftheenergyfluxinthetestparticlelimit
forfixedq. (SeealsoRef.[44].) TheargumentofthenPNexponentialresummedflux,Lexp[0],isexpandedas
nPN
n p
x x
Lexp[0](x,q):= C (q) log , (17)
nPN x (q) n,p x (q)
n (cid:18) lr (cid:19) p (cid:18) (cid:20) lr (cid:21)(cid:19)
X X
7
Symbol
ThefluxtruncatedatnPNorder L[i]
nPN
TheresidualpartofL[i] L[i]
nPN >nPN
TheknownpartoffluxtruncatedatnPNorder L˜[i]
nPN
TheresidualofL˜[i] L˜[i]
nPN >nPN
ThenormalizednPNTaylorflux LT[i]
nPN
TheexponentofnPNexponentialresummedflux Lexp[i]
nPN
TABLEI: Our notation for various fluxes in thispaper. Upper index [i] refers tothe order of truncation withrespect tothe mass ratio ν:
L=L[0]+νL[1]+O(ν2).Wealsoputlabel“T”,“exp”,“’hyb”and“full”whichrespectivelycorrespondtoTaylor,exponential,hybridand
numericalcompleteflux,todistinguishthefluxtype.
100
Exp resum
10
|0
n, 1
C
|
q = 0.998
0.1 q = 0.9
q = 0.5
q = 0.0
q = -0.5
q = -0.9
0.01
0 1 2 3 4 5 6 7 8 9
PN order: n
FIG.2: TheabsolutevalueofthecontributiontotheenergyfluxfromeachPNorder|Cn,0|atx=xlr(q),withthehorizontalaxisbeingthe
PNordern.Notethatthevaluesof|Cn,0|areaccidentalysmallwithsomefixedq.Wesafelyexludedthesevaluesfromtheplot.
where
2/3
1
x (q):= , (18)
lr R3/2+q!
lr
with
2
R (q):=2 1+cos arccosq , (19)
lr
3
(cid:20) (cid:18) (cid:19)(cid:21)
isthevalueofxcorrespondingtotheradiusofthecircularorbitonthelightring[48].
ItisknownthatforcircularorbitsthesourcetermoftheTeukolskyequationhasasimplepolewhentheorbitisonthelight
ringradius,i.e., atx = x (q) [60,61]. Thelightringradiusistheinnermostradiusforthepresenceofan(unstable)circular
lr
orbit,andtheretheparticleenergyperunitrestmassdiverges.Asaresult,theenergyfluxalsodivergesthere. Besidesthelight
ring singularity, another origin of singularity in solving the Teukolsky equation is the presence of quasi-normalmode poles,
whicharezerosoftheWronskian. However,theabsolutevalueofthefirstquasi-normalmodefrequency,whichisthesmallest,
isalmostidenticaltothefrequencyofGWsfromthesourceatthelightring[62]. Hence,weexpectthatx determines,atleast
lr
approximately, the convergenceradius when the energyflux is seen as an analytic function of x. If the PN expansionof the
energyfluxconvergesaslongasx<x (q),x=x (q)isthefirstplacewheretheseriesceasestoconverge.
lr lr
With this expectationin mind, we plotthe contributionfrom each PN orderwith x = x (q), |C |, in FIG. 2 for various
lr n,0
valuesofq. Hereq = 0.998,isaboundforpossiblemaximumspinofanastrophysicalblackholecalledThornelimit[63]. As
8
isexpected,|C |isroughlyindependentofthePNordern,astypicalattheboundaryofconvergencefor|q| < 0.9. Contrary
n,0
to ourexpectation,for|q| > 0.9, |C | continuestoincreaseupto 8PN,andtheseriesseemsto diverge. However,thisdoes
n,0
notmean that PN series doesnotconvergefor x ≈ x (q) with |q| ≈ 1. When we make a similar plotfor each partial wave
lr
contribution for a higher multipole component, the contribution to the energy flux peaks around a higher PN order, but the
seriesisconvergent,althoughwecannotcheckextremelyhighmultipolecomponents. Ontheotherhand,thesummationover
multipolecomponentsisknowntoconvergefromthenumericalcomputationoftheenergyfluxbyFujitaandTagoshi[51,52].
Thesefactsmayindicatethattheconvergencedependsontheordertotakesummation.ThepointofFIG.2isthat|C |shows
n,0
anicescalingpropertywithrespecttothePNorderforanyvalueofq. Evenforqclosetotheextremallimit,theplotbecomes
flatifwesubstitutealittlesmallervalueofxinsteadofx .
lr
Now we are in the position to discuss how to estimate the order of magnitude of the yet unknown part of the energyflux,
L˜[1] . The second order perturbedEinstein equation would schematically take the form (cid:3)h(2) = (T[z +δz]−T[z])+
>3.5PN
h(1)T[z]+(∇h(1))2,whereh(1)andh(2)are,respectively,thefirstandthesecondorderperturbationsinducedbyabody.Here
z isthebackgroundKerrgeodesicandδz istheO(ν)correctiontoit[65]. Weknowthatallthefirstorderperturbationshave
singularityonlyatx≈x (q). TheGreen’sfunction(cid:3)−1isbasicallythesameasinthelinearcase. Hence,evenforthesecond
lr
order perturbation, the convergenceradius in x will be the same as in the linear case, and thus a similar scaling for the PN
coefficientswillbeexpectedforthenextleadingorderinthemassratio,too.Then,wecanguesstheamplitudeofthehigherPN
coefficientsfromthefirstfewtermsinthePNexpansionthatareknownfromthestandardPNcalculation.
One mayworrythat (cid:3)−1 hassingularitiesat the quasi-normalmodefrequencies. When we considerthe secondorderper-
turbation,theremayarisehighfrequencycomponentswhichareabsentattheleveloflinearperturbation. However,thismight
nothappen. Thepointisthatthemetricperturbationcausedbya circulargeodesichasa helicalKillingvector7. Atthelevel
ofsecondordersourceterm,thishelicalsymmetryisbrokenduetothepresenceofthedeviationfromthegeodesic,δz. Apart
fromthiscontribution,however,thesourcetermkeepsthehelicalsymmetryandhencethefrequencythatappearsin a partial
wavelabelledby(ℓ,m)ismΩonly. Namely,higherfrequencymodesdonotarise. Evenifwetakeintoaccountδz,thetime
scaleassociatedwithδz isasslowasΩ−1×O(ν−1). Therefore,thepresenceofδz altersthefrequencymΩonlybyasmall
amountofO(νΩ). Thisessentiallydoesnotchangetheconvergenceradiusinx.
EvenifwegettheorderofmagnitudeofthePNcoefficients,thescalingpropertytellsusnothingabouttheactualsignofeach
term. Therefore,whenxiscloseto x (q), itbecomesdifficulttoguesstheorderofthemagnitudeoftheinfinitesummation.
lr
However,sincethehigherorderPNtermsaresuppressedbythepowerofx/x (q),thesummationisdominatedbyafewleading
lr
termswhenx/x (q)isreasonablysmall. Inthatcase,wedonothavetoworryabouttheinfinitesummation. Here,webravely
lr
step a little forwardbymakingthe followingproposalthatthe ratio betweenL˜[1] andL˜[1] willbe the same orderas the
nPN >nPN
ratiobetweenL˜[0] −L[0] andL˜[0] . Basedonthisassumption,weextrapolatetheknownresultstoestimatetheunknown
nPN 0PN >nPN
L˜[1] .
>nPN
V. THEDEPHASINGDUETOTHEAVERAGEDDISSIPATIVEPORTIONOFTHESECONDORDERSELF-FORCES
Themainfocusofthissectionistoevaluatethedephasingfromtheaverageddissipativepartofthesecondorderself-forces
inE(I)MRIs,basedontheideaoftheextrapolationproposedintheprecedingsection. Inthispaper,wemeasurethemagnitude
ofthedephasingduetohigherordercorrectionstotheenergyfluxδLby
x0 x3/2E′[0](x) 1 1
∆Φ[L,δL]:=−2 dx − . (20)
M L(x)+δL(x) L(x)
ZxISCO (cid:12) (cid:12)
(cid:12) (cid:12)
Heretheabsolutevalueofthedifferenceoffluxesistakenintheintegran(cid:12)(cid:12)d.Thefactor(L+δL)−1(cid:12)(cid:12)−L−1canchangeitssignature
inthedomainoftheintegralforsomeparameterregionofE(I)MRIs.Ifwedonottaketheabsolutevalue,therefore,theremight
beanaccidentalcancellationbetweenpositiveandnegativecontributions.Then,theorderofmagnitudeoftheevaluated∆Φcan
largelydeviatefromwhatwereallywanttomeasure. Noticethat,evenif∆Φdefinedwithouttakingtheabsolutevalueinthe
integrandstrictlyvanishes,thedeviationintheGWwaveformisstilldetectable.Therefore,toavoidthispossibleunderestimate
ofthedephasing,wetaketheabsolutevalueofthedifference.
Asageneralremark,wewouldliketomentionthemassdependencesofΦ[L[0]],thephaseforacertainreferencefluxL[0]
ref ref
whichincludestheleadingPNtermsinthetestparticlelimit, and∆Φ[L[0],δL[0]]and∆Φ[L[0],δL[1]],thedephasingsdueto
ref ref
7 ThehelicalKillingvectortα+Ωφα,thatremainstogenerateasymmetryevenfortheperturbedspacetime: wheretαandφαaretheasymptoticallytime
translationandrotationalKillingvectorsofthebackgroundKerrblackhole,respectively.Seee.g.Ref.[64]formoredetailes.
9
104 104
q = -0.9, spin full q = 0.9, spin full
102 Exp resum 102 Exp resum
ad] 100 ad] 100
Φ [r Φ [r
∆* 10-2 1PN ∆* 10-2 1PN
ν 1.5PN ν 1.5PN
2PN 2PN
10-4 2.5PN 10-4 2.5PN
3PN 3PN
3.5PN 3.5PN
10-6 10-6
10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2
(ν/M)*M (ν/M)*M
solar solar
FIG.3: ThedephasingduetothehigherPNcorrectionsinthetestparticlelimit,ν∆Φ[Lexp[0],Lexp[0]]asafunctionofν/M forq=±0.9.
nPN >nPN
δL[0] and δL[1], which are some portionsof the energyflux at the leading orderand the nextleading orderin the mass ratio,
respectively.FromtheintegralinEq.(4),whichdefinesx ,onecanfactoroutM /ν since
0 BH
E′[0] ∝M ν, L[0] ∝ν2, (21)
BH ref
for a given x. Hence, we find that x is a function of M /ν. Substituting δL[0] ∝ ν2 and δL[1] ∝ ν3 together with the
0 BH
relations(21)intothedefinitionsofν×Φ[L[0]],ν×∆Φ[L[0],δL[0]]and∆Φ[L[0],δL[1]],wefindthattheirmassdependences
ref ref ref
remainonlythroughx . Therefore,onecanconcludethatν×Φ[L[0]],ν×∆Φ[L[0],δL[0]]and∆Φ[L[0],L˜[1]]dependonM
0 ref ref ref BH
andν onlythroughthecombinationM /ν foralargemassratioν ≪1forafixedobservationperiodbeforetheplunge.
BH
A. Theerrorcausedbythepost-Newtoniantruncationinthetestparticlelimit
Beforeweevaluatethedephasingduetotheyetunknownpartoftheenergyfluxatthenextleadingorderinthemassratio,
we will assess the magnitude of the dephasing due to the PN truncation in the test particle limit ∆Φ[L[0] ,L[0] ] in this
nPN >nPN
subsectionandthatcomingfromtheknownPNterms,∆Φ[L[0] ,L˜[1] ],inthesucceedingsubsection.
full nPN
Here, we study the quantity ∆Φ[L[0] ,L[0] ] to examine the PN convergence in terms of the phase error for various
nPN >nPN
E(I)MRIs. We show the results only for the exponential resummed flux because the phase for the PN Taylor flux becomes
ill-definedforseveralPNorders. (Thehybridfluxisidenticaltotheexponentialresummedonebydefinitioninthetestparticle
limit.)
In FIG. 3 we plot ν∆Φ[Lexp[0],Lexp[0]] as a functionof ν/M for q = ±0.9. The main message of these plots is that the
nPN >nPN
PNconvergenceisalmostuniformexceptforthecaseoflargeblackholespinforlargeν/M. Whenq islarge,ISCObecomes
closetothelightringradiuswherePNconvergencebecomesveryslow. Eveninthatcase,forlowerPNorderstheconvergence
israthersmoothforlargeν/M. Thisisbecauseupto2PNorderthephasecorrectionisdominatedbythelowerboundofthe
x-integralin(20),iftheinitialseparationofthebinaryatthetime1yearbeforetheplungeissufficientlylarge.Thereasonwhy
wefocusonlargeν/M is,aswillbediscussedinAppend.V,thedephasingfromtheknownPNtermsisalreadywellsuppressed
below1radifν/M issmallenough.Thus,thePNconvergenceforsmallν/M doesnotaffectthefollowingdiscussion.
Noticealsothatx becomessmallerandsmallerforlargerν/M. The2.5PNcaseismarginalinthesensethatallrangeofx
0
almostequallycontributes.Bycontrast,thecontributionneartheISCOdominatesforthecorrectionsatthe3PNorderorhigher.
AlthoughthePNconvergenceisnotclearlyseenforq > 0.9,weshouldnotethatthephaseerrorisneverextremelyenhanced
forsomeparticularpost-Newtonianorder,largelyexceedingthevaluesfor3PNor3.5PNorder,whichwe mainlyfocusonin
thefollowingdiscussion.
B. Theexpecteddephasingfromtheunkownhigherorderpost-Newtonianterms
Now we moveon to our main issue: evaluatingthe dephasingdue to the yetunknownhigherorderPN terms, ∆Φ[L[0] +
full
L˜[1] ,L˜[1] ],basedontheideafortheextrapolationproposedinSec.IV.Thescalingargumentwilltellusthatthedephas-
3.5PN >3.5PN
10
102 102
101 101
100 100
ad] 10-1 ad] 10-1
[rss10-2 [rss10-2
e e
Φgu10-3 q=q0=.909.98 Φgu10-3 q=q0=.909.98
∆ ∆
10-4 q=0.5 10-4 q=0.5
spin linear q=0.0 spin linear q=0.0
10-5 3PN Hybrid q=-0.5 10-5 3.5PN Hybrid q=-0.5
q=-0.9 q=-0.9
10-6 10-6
10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2
(ν/M)*M (ν/M)*M
solar solar
FIG.4: Theexpectedresidualdephasing∆Φ [L[0] +L˜[1] ,L˜[1] ]causedbytheunknownpartoftheaverageddissipativesecond
guess full nPN >nPN
orderself-forces.Forn=3,the3.5PNknownspinindependentpartofthefluxatthenextleadingorderinthemassratioistreatedasunkown.
ingscomingfromtherespectivePNtermsattheleadingorderinthemassratiowillberoughlyproportionaltothecorresponding
nextleadingorderdephasings.Namely,
∆Φ[L[0] ,L˜[0] −L[0] ]:∆Φ[L˜[0] ,L˜[0] ]≈∆Φ[L˜[0] ,L˜[1] ]:∆Φ[L[0] +L˜[1] ,L˜[1] ], (22)
0PN nPN 0PN nPN >nPN full nPN full nPN >nPN
willhold. RecallthatL˜[0] consistsofthetermsattheleadingorderinmassratiouptothesamenPNorderthatisavailablein
nPN
L˜[1] .
nPN
Here, we have basically three options in the choice of the flux among the Taylor flux, the exponentialresummed flux and
thehybridflux. (Recallthatthehybridfluxisdefinedassuchthatisidenticaltotheexponentialresummedfluxattheleading
orderinthemassratio.)Ononehand,theTaylorfluxisproblematicsinceitcanbenegativebeforereachingISCOforsomePN
orders,say2.5PN,withmodelatevalueofthespinparameteraswementionedearlier. Insuchcasesthephasebeforetheplunge
isnotwelldefined. Ontheotherhand,thedifferencebetweentheexponentialresummedfluxandthehybridfluxisnegligible.
Therefore,wehereconsiderthehybridfluxonly.
Theresidualdephasingestimatedbyusing(22)
∆Φ[L˜[0] ,L˜[0] ]
∆Φ [L[0] +L˜[1] ,L˜[1] ]:= nPN >nPN ∆Φ[L˜[0] ,L˜[1] ], (23)
guess full nPN >nPN ∆Φ[L[0] ,L˜[0] −L[0] ] full nPN
0PN nPN 0PN
withn = 3andn = 3.5isdepictedinFIG.4. ThetrendoftheestimateddephasingisindependentofthetruncatedPNorder.
ThepointofFIG.4isthattheresidualdephasingisrathersupressedoverthewholerangeofthebainaryparameters.Indeed,the
residualdephasingisatmostabout10radinthecaseof3PNtruncation.Inthecaseof3.5PNtruncation,themaximumvalueof
theresidualdephasingisbiggerbyfactor2orso. Sincetheplotisgivenintheunitofradian,thevaluemustbedividedby2πto
translateitintothenumberofcycles. Fromtheaboveresults,onemaysaythattheresidualdephasingduetotheyetunknown
PNcorrectionsatthenextleadingorderinthemassratio,∆Φ [L[0] +L˜[1] ,L˜[1] ]isnotnegligible,adoptingonecycle
guess full nPN >nPN
asthecriteriaforthesignificantdephasing.
InFIG.4,weseeatinybumparoundνM /M ≈ 10−10−10−11 forsomevaluesofq. Thereasonwhythisbumpappears
⊙
at this position can be understood from the observation that L˜[0] − L[0] crosses zero within the domain of the integral,
nPN 0PN
x < x < x when the bump appearers. When L˜[0] −L[0] crosses zero, the factor in the denominator of Eq. (23)
0 ISCO nPN 0PN
∆Φ[L[0] ,L˜[0] − L[0] ] suffers from more or less accidental suppression even though we take the absolute value in the
0PN nPN 0PN
integrandof∆Φin(20). WedenotethevalueofxatwhichthefluxL˜[0] −L[0] crosseszerobyx . Then,ifx ≪x
nPN 0PN cross 0 cross
orx ≫ x , thissuppressiondoesnotproducemucheffectontheestimate of∆Φ[L[0] ,L˜[0] −L[0] ]. Therefore,the
0 cross 0PN nPN 0PN
suppression becomes significant only for x ≈ x . The value of x is rather close to the value at ISCO, x , but
0 cross cross ISCO
typicallynotextremelyclosetoit. Hence,x ≈ x occurswhenx isneitherextremelyclosetoISCOnorverysmalllike
0 cross 0
x ≪1,whichcorrespondstotheparameterregionνM /M ≈10−10−10−11. Sincenowwefindthatthistinybumpistobe
0 ⊙
attributedtoanaccidentalzeroinL˜[0] −L[0] ,thisbumpwouldberegardedasanartificialfeature.Ifitisfaretoremovethe
nPN 0PN
bumpsfromFIG.4,wewillfindthatthesignificantlylargedephasing(inthesenseofexceeding1cycle)willbeexpectedonly
forνM /M &10−9.
⊙