New gravitational self-force analytical results for eccentric orbits around a Schwarzschild black hole

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by  Donato Bini

New gravitational self-force analytical results for
eccentric orbits around a Schwarzschild black hole
Donato Bini1, Thibault Damour2, and Andrea Geralico1
1Istituto per le Applicazioni del Calcolo “M. Picone”, CNR, I-00185 Rome, Italy
2Institut des Hautes Etudes Scientifiques, 91440 Bures-sur-Yvette, France
(Dated: May 4, 2016)
We raise the analytical knowledge of the eccentricity-expansion of the Detweiler-Barack-Sago
redshift invariant in a Schwarzschild spacetime up to the 9.5th post-Newtonian order (included)
for thee2 and e4 contributions, and up tothe4th post-Newtonian order for thehighereccentricity
contributionsthroughe20. Weconvertthisinformationintoananalyticalknowledgeoftheeffective-
one-body radial potentials d¯(u), ρ(u) and q(u) through the 9.5th post-Newtonian order. We find
that our analytical results are compatible with current corresponding numerical self-force data.
I. INTRODUCTION
TABLEI.Presentanalytical knowledgeofδUen alongeccen-
tric orbits in a Schwarzschild spacetime.
6
Afruitfulsynergybetweenvariousmethodsforapprox-
1 imatingthegeneralrelativistictwo-bodyproblemhasde- n δUen Refs.
0
2 velopedoverthelastyears,withacceleratedprogressover 0 22.5PN Kavanagh et al.[25]
the last months. The concerned approximationmethods 2 9.5PN This paper
y
are: post-Newtonian (PN) theory, self-force (SF) the-
a 4 9.5PN This paper
M ory,andnumericalrelativity(NR).Thesynergybetween 6 4PN Hopperet al.[28]
these approximation methods was greatly facilitated by
3 theconstructionoftheoreticalbridgesconnectingthevar- 8 4PN Hopperet al.[28]
10 4PN Hopperet al.[28]
ious methods. Among these bridges, two have been par-

] ticularlyuseful: the effective-one-body(EOB)formalism 12 4PN This paper
c
[1–4],andthe firstlawofbinarymechanics[5–7]. Exam- 14 4PN This paper
q
- ples of synergiesbetween PN and SF facilitated by EOB 16 4PN This paper
gr and/or the first law are Refs. [8–30]. 18 4PN This paper
[ This paper is a follow-up of Ref. [27]. It concerns the 20 4PN This paper

first self-force (1SF) conservative dynamics of the eccen-
2
tric orbits of a small mass m around a (non-spinning)
v 1
8 largemassm2 (describedbyaSchwarzschildblackhole).
8 Our results complete the results of both Ref. [27] and of eccentricity powers of the alternative redshift function
9 the recent related Refs. [28, 29]. Before entering the de- δz1(p,e)= nδz1en(up)en, where z1 =1/U.]
2 tails ofournew resultswe summarizeinTable I howour To complete our results on the coefficients at or-
0 results go beyond present analytical knowledge in terms ders e2 anPd e4 of the redshift function δU(p,e) =
1. ofthedecompositionofthe(gauge-invariant)1SFcontri- δUen(u )en, we shall also transcribe below our
n p
0 butionδU(p,e)totheDetweiler-Barack-Sago[31,32]av- 9.5PN-accurate results in terms of the corresponding
6 erageredshiftU(p,e)inpowersofthe eccentricitye,i.e., EPOB potentials d¯(u) and q(u) q (u). [We also give
:1 δU(p,e)= nδUen(up)en. [Weuseinthepresentpaper the previously uncomputed 4PN≡valu4es of the higher-pr-
iv tdheenosatemsethneoPtiantvioenrseassienmRi-elaf.tu[2s7r].ecItnumparotfictuhlearc,ounpsi≡de1re/dp powersanalogsoftheO(p4r)EOBpotentialq(u)≡q4(u).]
X Finally, we shall also explicitly compute the 9.5PN-
eccentric orbit. In addition, we denote M m +m ,
1 2 accurate value of the gauge-invariant 1SF precession
ar µ m1m2/(m1+m2)andν µ/M =m1m2≡/(m1+m2)2 function ρ(u) defined in Ref. [8] and related there to
≡ ≡
in our EOB considerations.] the 1SF EOB potentials a(u) and d¯(u). The precession
Table I shows that our new results are of two differ- functionρ(u)isofparticularinterestbecauseitcanbedi-
ent types. On the one hand, we improve the PN knowl- rectly extracted from SF numerical computations of the
edge of the contributions to δU of order e2 and e4 to dynamics of slightly eccentric orbits [9] without making
the 9.5PN level (previous analytical knowledge was the use of the eccentric first law. Therefore a comparison
6.5PN level for δUe2 [27] and the 4PN one for δUe4 betweenour9.5PNanalyticalcomputationofthepreces-
[27, 33]). On the other hand, we combine the 4PN sion function ρ(u) (which combines SF theory with the
results of [33] with the eccentric first law [7] to com- eccentric first law [7]) and of a purely dynamical SF nu-
pute the 4PN-accurate values of δUen for the high val- mericalcomputation of the precessionof eccentric orbits
ues of n: n = 12,14,16,18,20 (previous similar 4PN- (asin[9])wouldbeausefulcheckoftheassumptionsun-
level knowledge concerned n = 6,8,10 [28]). [We also derlying the theoretical bridges (EOB and the first law)
givebelowthe 4PNknowledgeofthe correspondinghigh used in connecting SF and PN results.
2
II. NOVEL ANALYTICAL RESULTS FOR δUe2 the 6.5PNlevel obtained in our previous work [27] up to
AND δUe4 UP TO THE 9.5PN ORDER the 9.5PN level. Note that the conversion between PN
orderandmeaningful powersofu , or equivalently1 u or
p
Our new, 9.5PN-accurate, results for δUe2 and δUe4 x, depends onthe consideredSForEOBfunction. More
havebeenobtainedbyfollowingtheapproachofourpre- precisely, the nth PN order corresponds to: (1) a term
viouspapers[18,27]. Letusonlyrecallthatourapproach un+1 ina(u)(andδU(u,e));(2)aterm unind¯(u)or
combines standard Regge-Wheeler-Zerillifirst order per- ∝ρ(u); and(3) a term un−1 in q(u). Ther∝efore,our cur-
∝
turbationtheorywiththe Mano-Suzuki-Takasugi(MST) rent 9.5PN accuracy (obtained by using hypergeometric
[34, 35] hypergeometric-expansion technique (here used expansionsuptothemultipolarorderl=7)corresponds
up to the multipolar l = 7 solution included). The toerrorterms: Oln(u1p1)inδU(up,e);Oln(u10)ind¯(u)or
main steps of this, by now, well established procedure ρ(u); and O (u9) in q(u), where O (uq) denotes some
ln ln
are sketched in Appendix A. O(uq(lnu)p) with a non specified natural integer p 1.
≥
We have raised the analytical knowledge of δUe2 from Our result for δUe2 reads
5 41
δUe2(u )=u +4u2+7u3+ π2 u4
p p p p −3 − 32 p
(cid:18) (cid:19)
11141 29665 296 592 1458 3248
+ + π2 ln(u ) γ ln(3)+ ln(2) u5
− 45 3072 − 15 p − 15 − 5 15 p
(cid:18) (cid:19)
2238629 73145 8696 167696 17392 42282
+ π2+ ln(u ) ln(2)+ γ+ ln(3) u6
− 1575 − 1536 105 p − 105 105 35 p
(cid:18) (cid:19)
232618
πu13/2
− 1575 p
2750367763 9765625 41285072 5102288 673353 2551144
+ ln(5)+ ln(2)+ γ ln(3)+ ln(u )
p
198450 − 4536 2835 2835 − 280 2835
(cid:18)
9735101 13433142863
+ π4 π2 u7
262144 − 3538944 p
(cid:19)
2687231
+ πu15/2
4410 p
1040896 4163584 85422206699 936036 109568
+ ln(u )2+ γ + ln(3) ln(2) ln(u )
p p
1575 1575 − 5457375 175 − 1575
(cid:20) (cid:18) (cid:19)
471677766820151 171448137814 301990638447 1228515625 170844413398
+ ln(2) ln(3)+ ln(5) γ
1719073125 − 5457375 − 4312000 57024 − 5457375
1872072 219136 1872072 4163584 23854914937
+ γln(3) ln(2)γ+ ln(2)ln(3)+ γ2 π4
175 − 1575 175 1575 − 503316480
936036 77824 8655872 80420758955297
+ ln(3)2 ζ(3) ln(2)2 π2 u8
175 − 15 − 1575 − 2477260800 p
(cid:21)
66757650913
+ πu17/2
26195400 p
1 Thename we give tothe arguments inthe various EOB poten- but as the corresponding physical quantity u=GM/c2rEOB is
tials considered hereis arbitrary, because weare expanding the
corresponding functions (e.g. u → d¯(u)) in powers of their ar- numerically equal, moduloa O(ν) correction, both to up =1/p
and to the frequency parameter usually denoted x, one some-
gument. The traditional EOB notation for the argument is u,
timescallstheargumentup orx.
3
2994904 11979616 618506181077 5165694 55690528
+ ln(u )2+ γ+ ln(3)+ ln(2) ln(u )
p p
− 1225 − 1225 99324225 − 175 2205
(cid:20) (cid:18) (cid:19)
2205806334400049687 46585620571706 2452189382919 3191857421875
ln(2)+ ln(3) ln(5)
− 1720792198125 − 165540375 8008000 − 36324288
678223072849 1272610164394 10331388 111381056
ln(7)+ γ γln(3)+ ln(2)γ
− 46332000 99324225 − 175 2205
10331388 11979616 389897083139633
ln(2)ln(3) γ2+ π4
− 175 − 1225 16106127360
5165694 1020736 1391778208 79965804866374541
ln(3)2+ ζ(3)+ ln(2)2 π2 u9
− 175 105 11025 − 554906419200 p
(cid:21)
3936830890988503 100155852 2250424 120397684 665599064
+ π+ πln(3) π3+ πln(u )+ πln(2)
p
− 59935075200 6125 − 675 23625 165375
(cid:18)
240795368
+ πγ u19/2
23625 p
(cid:19)
91608512384 2694566979 105972007312260412 76708984375
+ ln(u )2+ ln(3)+ + ln(5)
p
− 9823275 53900 442489422375 1571724
(cid:20) (cid:18)
366434049536 2995825170944 86555681446617433123159 2694566979
γ ln(2) ln(u ) + ln(3)2
p
− 9823275 − 9823275 − 949139810994375 53900
(cid:19)
213354316911514424 1656928811171577752 995870224363383 38345561821484375
+ γ+ ln(2) ln(3) ln(5)
442489422375 442489422375 − 1079078000 − 56638646064
315073184 193778020814 113425393373 3608718872135173
+ ζ(3)+ ln(7) π6 π2
2835 868725 − 100663296 − 5651824640
16005605256259137079 76708984375 366434049536 5991650341888
+ π4+ ln(5)2 γ2 ln(2)γ
16492674416640 1571724 − 9823275 − 9823275
76708984375 239758989824 2694566979 76708984375
+ ln(2)ln(5) ln(2)2+ γln(3)+ γln(5)
785862 − 218295 26950 785862
2694566979
+ ln(2)ln(3) u10
26950 p
(cid:21)
28108289357 369663722 56216578714 15720247936467024947
+ πln(u )+ π3 πγ+ π
p
− 1157625 33075 − 1157625 114535928707200
(cid:18)
9003848366 839692089
+ πln(2) πln(3) u21/2+O (u11). (1)
231525 − 8575 p ln p
(cid:19)
The numerical values of the coefficients in the latter expansion read
δUe2(u )=u +4u2+7u3 14.31209731u4+( 345.3178497 19.73333333ln(u ))u5
p p p p− p − − p p
+( 1575.580014+82.81904762ln(u ))u6 463.9942859u13/2
− p p− p
+( 14960.48992+899.8744268ln(u ))u7+1914.327703u15/2
− p p p
+( 119420.1688 8298.710150ln(u )+660.8863492ln(u )2)u8+8006.189854u17/2
− − p p p p
+( 395945.586 14340.26852ln(u ) 2444.819592ln(u )2)u9
− − p − p p
+( 226044.9538+16010.17903ln(u ))u19/2
− p p
+(140039.6684ln(u ) 9325.658946ln(u )2 2966833.394)u10
p − p − p
+( 76281.00237ln(u )+436383.4353)u21/2+O (u11). (2)
− p p ln p
Similarly, we have extended the analyticalknowledge of δUe4 from 4PN (as obtainedin our previous work [27]) up
4
to 9.5PN, namely
1 705 123
δUe4(u )= 2u2+ u3+ π2 u4
p − p 4 p 8 − 256 p
(cid:18) (cid:19)
247931 89395 28431 292 64652 146
+ π2+ ln(3)+ γ ln(2)+ ln(u ) u5
360 − 6144 10 3 − 15 3 p p
(cid:18) (cid:19)
293423 25493859 601 1202 248378 9765625 275167
+ ln(3) ln(u ) γ+ ln(2) ln(5)+ π2 u6
4200 − 2240 − 5 p − 5 7 − 1344 1024 p
(cid:18) (cid:19)
430889
+ πu13/2
3150 p
4815135047 194385796 2260629 3470703125 794596 397298
+ ln(2) ln(3)+ ln(5) γ ln(u )
p
− 396900 − 945 − 320 36288 − 945 − 945
(cid:18)
58818333 16293066631
π4+ π2 u7
− 1048576 4718592 p
(cid:19)
13695499
+ πu15/2
47040 p
497764 1991056 66544956203 11934459 195652496
+ ln(u )2+ γ ln(3)+ ln(2) ln(u )
p p
1575 1575 − 3969000 − 175 1575
(cid:20) (cid:18) (cid:19)
2047686486671407 197388844553 359853720161877 4691575390625
+ ln(2)+ ln(3) ln(5)
13752585000 − 269500 275968000 − 8515584
678223072849 66544956203 23868918 391304992
ln(7) γ γln(3)+ ln(2)γ
− 6082560 − 1984500 − 175 1575
23868918 1991056 924796757543
ln(2)ln(3)+ γ2+ π4
− 175 1575 2013265920
11934459 37216 751271824 27703501682741
ln(3)2 ζ(3)+ ln(2)2+ π2 u8
− 175 − 15 1575 19818086400 p
(cid:21)
1023562537
+ πu17/2
1552320 p
10161819
+ ln(u )2
p
− 1225
(cid:20)
15523629993 46249898026747 14486589644 40647276 3173828125
+ ln(3)+ ln(2) γ+ ln(5) ln(u )
p
39200 305613000 − 11025 − 1225 14112
(cid:18) (cid:19)
46285104644347 2067345910491191 17923252135149887 148748447195686881
+ γ π4+ ln(2) ln(3)
152806500 − 85899345920 1986484500 − 25113088000
27786921439609375 421370306260043 1655592 4816187291152031551
ln(5)+ ln(7)+ ζ(3)+
− 16273281024 219648000 35 1529593065000
3173828125 6286441324 3173828125 40647276
+ ln(5)2 ln(2)2+ γln(5) γ2
14112 − 1225 7056 − 1225
15523629993 1712225112134041 15523629993 28973179288
+ ln(3)2 π2+ γln(3) ln(2)γ
39200 − 34681651200 19600 − 1025
3173828125 15523629993
+ ln(2)ln(5)+ ln(2)ln(3) u9
7056 19600 p
(cid:21)
137457732402576571 40118366 4292665162 2146332581 83149713482
+ π π3+ πγ+ πln(u )+ πln(2)
p
− 610248038400 − 4725 165375 165375 165375
(cid:18)
1427220891
πln(3) u19/2
− 6125 p
(cid:19)
5
2438262007 487959613018 4899895367447 114443682651 4876524014
+ ln(u )2+ ln(2)+ + ln(3) γ
p
− 198450 72765 4584195 431200 − 99225
(cid:20) (cid:18)
40750244140625 20377781024735400904328 9826562433694
ln(5) ln(u ) + γ
p
− 12573792 − 201332687180625 4584195
(cid:19)
176771772306908307 40750244140625 4876524014 114443682651
ln(3) ln(5)2 γ2+ ln(3)2
− 276243968000 − 12573792 − 99225 431200
36307823919194 536631960411 114443682651 19871612
+ ln(2)2+ ln(2)ln(3)+ γln(3)+ ζ(3)
1403325 215600 215600 63
131229423889414613 40750244140625 1167313947555 975919226036
ln(7) γln(5)+ π6+ ln(2)γ
− 8895744000 − 6286896 268435456 72765
628897515069490765625 707217483022033957 40750244140625
+ ln(5) π2 ln(2)ln(5)
14499493392384 − 1109812838400 − 6286896
10992948747002026551 476838331512979466
+ π4 ln(2) u10
10995116277760 − 9833098275 p
(cid:21)
16759623823 80294969715785936774437 6503164207213 100110344015981
+ π3+ π πln(u ) πln(2)
p
176400 45814371482880000 − 37044000 − 18522000
(cid:18)
2087606910177 6503164207213 206298828125
+ πln(3) πγ+ πln(5) u21/2+O (u11). (3)
1372000 − 18522000 296352 p ln p
(cid:19)
The numerical form of this expansion reads
δUe4(u )= 2.0u2+0.25u3+83.38296351u4+(737.1849552+48.66666667ln(u ))u5
p − p p p p p
+(2980.049710 120.2ln(u ))u6+429.7389577u13/2
− p p p
+(19588.97635 420.4211640ln(u ))u7+914.6615445u15/2
− p p p
+(62630.23815 4853.06274ln(u )+316.0406349ln(u )2)u8+2071.490767u17/2
− p p p p
+(18432.5611ln(u ) 8295.362449ln(u )2+837868.8305)u9
p − p p
+( 633183.2616+40773.40995ln(u ))u19/2
− p p
+(764293.4202ln(u ) 12286.53065ln(u )2+1154095.188)u10
p − p p
+(4816799.276 551514.2235ln(u ))u21/2+O (u11). (4)
− p p ln p
Using the relations explicitly written down in [7] we converted the new information on δUe2 and δUe4 into a
6
correspondingly improved knowledge of the EOB potentials d¯(x) and q(x), namely
1184 6496 2916 23761 533 592
d¯(x)=6x2+52x3+ γ ln(2)+ ln(3) π2 + ln(x) x4
15 − 15 5 − 1536 − 45 15
(cid:18) (cid:19)
2840 120648 19683 63707 294464 1420
+ γ+ ln(2) ln(3) π2+ ln(x) x5
− 7 35 − 7 − 512 175 − 7
(cid:18) (cid:19)
264932
+ πx11/2
1575
64096 6381680 1765881 9765625 135909 229504763
+ γ ln(2)+ ln(3)+ ln(5)+ π4+ π2
− 45 − 189 140 2268 262144 98304
(cid:18)
31721400523 32048
ln(x) x6
− 2116800 − 45
(cid:19)
21288791
πx13/2
− 17640
4187061434 876544 8108032 16216064 3744144
+ γ ln(x)γ+ ln(2)ln(x)+ ln(2)γ ln(2)ln(3)
99225 − 315 1575 1575 − 175
(cid:18)
3744144 18024943666 282753093897 16384 3091796875
γln(3)+ ln(2)+ ln(3)+ ζ(3) ln(5)
− 175 496125 2156000 3 − 66528
33089536 31596265477 3755930660113 876544 1872072
+ ln(2)2+ π4+ π2 γ2 ln(3)2
1575 251658240 247726080 − 315 − 175
629856 1340870864165051 219136 1872072 2093530717
+ ln(6) ln(x)2 ln(3)ln(x)+ ln(x) x7
55 − 5501034000 − 315 − 175 99225
(cid:19)
1173441809
πx15/2
− 3492720
281972594008247 232751488 31370368 62740736
+ γ+ ln(x)γ ln(2)ln(x) ln(2)γ
− 1986484500 11025 − 525 − 525
(cid:18)
174802536 174802536 107340333276983 25726492389393
+ ln(2)ln(3)+ γln(3)+ ln(2) ln(3)
1225 1225 283783500 − 49049000
1096192 1556814453125 678223072849 624682112 16273379175661
ζ(3)+ ln(5)+ ln(7) ln(2)2+ π4
− 35 6054048 23166000 − 2205 1073741824
2692389474594437 232751488 87401268 2751525936 58187872
+ π2+ γ2+ ln(3)2 ln(6)+ ln(x)2
92484403200 11025 1225 − 17875 11025
87401268 831440592970385544103 281464053976247
+ ln(3)ln(x) ln(x) x8
1225 − 440522802720000 − 3972969000
(cid:19)
144712674728544827 186756088 239421488 3490768 373512176
+ π πln(x)+ ln(2)π+ π3 πγ
1678182105600 − 33075 23625 945 − 33075
(cid:18)
200311704
πln(3) x17/2
− 6125
(cid:19)
145060456 4109882910365899 2205013489376 580241824 215213193
+ ln(x)2+ + ln(2) γ ln(3)
− 363825 − 19423404000 3274425 − 363825 − 770
(cid:20) (cid:18)
76708984375 8869707677468340294172589 4125670253137099
ln(5) ln(x)+ γ
− 785862 188984282366880000 − 9711702000
(cid:19)
23620001432239865033 18387195312716343 1763600530764453125 215213193
ln(2)+ ln(3)+ ln(5) γln(3)
− 2359943586000 4932928000 1812436674048 − 385
117281890332 7435264 43503165672743 13438960917574667 441262176956397691
+ ln(6) ζ(3) ln(7)+ π2 π4
125125 − 105 − 92664000 406931374080 − 1030792151040
215213193 580241824 5132203667744 76708984375 150232915593
ln(3)2 γ2+ ln(2)2 ln(5)2 π6
− 770 − 363825 1964655 − 785862 − 33554432
76708984375 215213193 4410026978752 76708984375
γln(5) ln(2)ln(3)+ ln(2)γ ln(2)ln(5) x9
− 392931 − 385 3274425 − 392931
(cid:21)
10310051408772977303753 1836704419 232145783843 83839907743
+ π π3+ πγ πln(2)
− 22907185741440000 − 66150 2315250 − 771750
(cid:18)
39949476291 232145783843
+ πln(3)+ πln(x) x19/2+O (x10) (5)
ln
171500 4630500
(cid:19)
7
and
496256 33048 5308
q(x)=8x2+ ln(2) ln(3) x3
45 − 5 − 15
(cid:18) (cid:19)
10856 40979464 14203593 9765625 93031 1295219 5428
+ γ ln(2)+ ln(3)+ ln(5) π2+ + ln(x) x4
105 − 315 280 504 − 1536 350 105
(cid:18) (cid:19)
88703
+ πx9/2
1890
617716 308858 65887036 36073593 8787109375 81030481
+ γ ln(x)+ ln(2) ln(3) ln(5)+ π2
− 315 − 315 63 − 112 − 27216 65536
(cid:18)
559872 7518451741
+ ln(6) x5
7 − 1270080
(cid:19)
714117331
πx11/2
− 846720
138169844888 69084922444 3250526464 13728528 13728528
+ γ+ ln(x) ln(2)γ+ ln(2)ln(3)+ γln(3)
1819125 1819125 − 4725 35 35
(cid:18)
527856862616 12960490645107 25344 27397616796875 678223072849
ln(2) ln(3)+ ζ(3)+ ln(5)+ ln(7)
− 16372125 − 6899200 5 9580032 2280960
2065918336 109837713789 1463044337673 451968 6864264 579887424
ln(2)2 π4+ π2 γ2+ ln(3)2 ln(6)
− 1575 − 83886080 91750400 − 175 35 − 385
451968 6864264 939101654498857 112992 1625263232
ln(x)γ+ ln(3)ln(x) ln(x)2 ln(2)ln(x) x6
− 175 35 − 3056130000 − 175 − 4725
(cid:19)
226615901761
+ πx13/2
167650560
29186389360543 3173828125 3173828125 9440966259
+ γ ln(2)ln(5) γln(5) ln(3)ln(x)
− 36786750 − 2646 − 2646 − 4900
(cid:18)
322866894016 9440966259 9440966259 97783791533166503
+ ln(2)γ ln(2)ln(3) γln(3) ln(2)
33075 − 2450 − 2450 − 2979726750
87139874452615209 2452928 2899973891640625 9257841833399257
+ ln(3) ζ(3) ln(5) ln(7)
6278272000 − 35 − 452035584 − 1482624000
210393017888 19047555410493 3975430726567129 168910688 9440966259
+ ln(2)2+ π4 π2+ γ2 ln(3)2
11025 10737418240 − 92484403200 3675 − 4900
2573147182608 3173828125 3173828125 161433447008
+ ln(6) ln(5)2 ln(5)ln(x)+ ln(2)ln(x)
175175 − 5292 − 5292 33075
690294961714478265797 29186389360543 42227672 168910688
+ ln(x)+ ln(x)2+ ln(x)γ x7
293681868480000 − 73573500 3675 3675
(cid:19)
8192870254937920639 15200768606 11631519958 4598822871 108705794
+ π ln(2)π πγ+ πln(3)+ π3
30981823488000 − 11025 − 496125 6125 14175
(cid:18)
5815759979
πln(x) x15/2
− 496125
(cid:19)
131228022231920707 23800697662770506993 4958146688407013943
+ γ+ ln(2) ln(3)
196661965500 65553988500 − 39463424000
(cid:20)
4492372832662738703125 2906254027437804 1519264 710656240002840019
ln(5) ln(6)+ ζ(3)+ ln(7)
− 43498480177152 − 67442375 315 10674892800
472332484052074531 9617337404302759049 105898193359375 226550022075
+ π2+ π4+ ln(5)2+ π6
678218956800 24739011624960 9430344 33554432
188687137328 7036471296 181114018983 4527732156900112
γ2 ln(6)2+ ln(3)2 ln(2)2
− 1091475 − 2695 15400 − 29469825
85388056818784 14072942592 105898193359375 14072942592
ln(2)γ ln(2)ln(6)+ γln(5) γln(6)
− 1091475 − 2695 4715172 − 2695
181114018983 105898193359375 181114018983 7190610346934768219939609
+ γln(3)+ ln(2)ln(5)+ ln(2)ln(3)
7700 4715172 7700 − 161986527743040000
7036471296 105898193359375 131815385968880707 188687137328 181114018983
+ ln(6)+ ln(5)+ γ+ ln(3)
− 2695 9430344 393323931000 − 1091475 15400
(cid:18)
42694028409392 47171784332
ln(2) ln(x) ln(x)2 x8
− 1091475 − 1091475
(cid:19) (cid:21)
8
309249455540719514934031 206298828125 431496991403 52009951116491
+ π πln(5) π3+ πγ
− 84580378122240000 − 111132 − 3175200 111132000
(cid:18)
88244053021571 21764539709991 52009951116491
+ πln(2) πln(3)+ πln(x) x17/2+O (x9). (6)
ln
4445280 − 2744000 222264000
(cid:19)
Asmentionedabove,anotherusefuldynamicalfunctionistheprecessionfunctionρ(u)introducedin[8]andrelated
there to the EOB potentials a(u) and d¯(u). Namely (denoting the argument of the function ρ as x)
ρ(x)=ρE(x)+ρa(x)+ρd¯(x), (7)
with
1 2x
ρ (x)=4x 1 − ,
E
− √1 3x
(cid:18) − (cid:19)
1
ρ (x)=a(x)+xa′(x)+ x(1 2x)a′′(x),
a
2 −
ρd¯(x)=(1−6x)d¯(x). (8)
We then find
397 123 5024 215729 2512 2916 1184 58265
ρ(x)=14x2+ π2 x3+ γ + ln(x)+ ln(3)+ ln(2)+ π2 x4
2 − 16 15 − 180 15 5 15 1536
(cid:18) (cid:19) (cid:18) (cid:19)
27824 6325051 1135765 202662 22672 11336
+ ln(2) + π2 ln(3) γ ln(x) x5
35 − 800 1024 − 35 − 7 − 7
(cid:18) (cid:19)
199876
+ πx11/2
315
4990303259 256727518799 435213 3606884 37648124 1803442
+ π2 + ln(3)+ γ ln(2)+ ln(x)
589824 − 6350400 20 945 − 945 945
(cid:18)
7335303 9765625
π4+ ln(5) x6
− 32768 2268
(cid:19)
1429274
πx13/2
− 225
3725312 419921875 3744144 3744144 253952
+ ln(2)2 ln(5) γln(3) ln(2)ln(3)+ ζ(3)
− 1575 − 6048 − 175 − 175 15
(cid:18)
13586432 230019793907682883 1872072 20598784 12659060941523
ln(x)γ ln(3)ln(x) ln(2)γ+ π2
− 1575 − 440082720000 − 175 − 1575 1238630400
681396625634 229716339147 3396608 1823766172754 10299392
+ γ+ ln(3) ln(x)2+ ln(2) ln(2)ln(x)
5457375 2156000 − 1575 5457375 − 1575
471044952937 340698312817 1872072 13586432
+ π4+ ln(x) ln(3)2 γ2 x7
251658240 5457375 − 175 − 1575
(cid:19)
18719967989
+ πx15/2
1455300
82814168955181 36686848 1920044921875 2269129471514627499419
+ ln(2)+ ln(x)γ+ ln(5)+
− 132432300 441 4036032 176209121088000
(cid:18)
148969692 36686848 103653376 9171712 297939384
+ ln(3)2+ γ2+ ln(2)γ+ ln(x)2+ γln(3)
1225 441 3675 441 1225
948480 297939384 678223072849 148969692
ζ(3)+ ln(2)ln(3)+ ln(7)+ ln(3)ln(x)
− 7 1225 23166000 1225
51826688 6517218707007553 1442495323220011
+ ln(2)ln(x) π2 ln(x)
3675 − 55490641920 − 3972969000
557542163367261 1444834607367211 2049476608 626168320805261
ln(3) γ ln(2)2 π4 x8
− 392392000 − 1986484500 − 11025 − 5368709120
(cid:19)
9
105699126344597143 4707645616 200311704 4004219056 2353822808
+ π πγ πln(3) ln(2)π πln(x)
524431908000 − 165375 − 6125 − 165375 − 165375
(cid:18)
43996688
+ π3 x17/2
4725
(cid:19)
42935456848 1995219783 20475902395612297 171741827392 2444046775616
+ ln(x)2+ ln(3) γ+ ln(2)
− 1091475 − 3850 − 262215954000 − 1091475 3274425
(cid:20) (cid:18)
76708984375 20657146063017097 10969454340865467 226626361596
ln(5) ln(x) γ+ ln(3)+ ln(6)
− 785862 − 131107977000 2466464000 125125
(cid:19)
76708984375 171741827392 1995219783 6754948737728
ln(5)2 γ2 ln(3)2+ ln(2)2
− 785862 − 1091475 − 3850 1964655
1995219783 76708984375 3956569170916362731724487183 8519104
γln(3) γln(5)+ + ζ(3)
− 1925 − 392931 18142491107220480000 315
29163592132507 76708984375 4888093551232 64674832921484375
ln(7) ln(2)ln(5)+ ln(2)γ ln(5)
− 46332000 − 392931 3274425 − 906218337024
3466357618648439 128148402261 1995219783 6269062781928031361
+ π2+ π6 ln(2)ln(3) π4
27128758272 16777216 − 1925 − 2748779069440
2702219779688690213
ln(2) x9
− 235994358600
(cid:21)
174754006268 2849519528 349508012536
+ πln(x) π3+ πγ
1157625 − 33075 1157625
(cid:18)
16864298172 2195209992943672765961 51802382504
+ πln(3) π+ πln(2) x19/2+O (x10). (9)
ln
42875 − 1431699108840000 385875
(cid:19)
III. ESTIMATING THE ORDER OF ation tells us that the large–N asymptotic values of the
MAGNITUDE OF THE COEFFICIENTS OF PN Taylor expansion coefficients f is of order
N
EXPANSIONS
f 3N. (10)
N
∼
Beforecomparingthenumericalvaluesofthese9.5PN-
One can, however, refine this exponential estimate by
accuratefunctions to correspondingpublished numerical
power-law corrections in N. Indeed, given a certain
SF estimates [9, 29], it is useful to have at hand a rough
function f(u) = f uN, its first derivative with re-
estimateofthetheoreticalerrorassociatedwithsuchPN- N N
spect to (wrt) u will be f′(u) = Nf uN−1, so that
expandedfunctions. Weshalldothisviatwocomplemen- (f′) = (N +1)fP . In other wNords,Neach derivative
tary approaches. Our first estimate will follow the spirit N N+1 P
adds an asymptotic factor N to the growth of the f ’s.
of Section IV in Ref. [21]. The idea there was to use N
Forinstance,theexistenceinEOBtheoryofthelink(7),
the existence of a power-law singularity at the lightring
(8) between the precession function ρ(u) and the first
[12] of the various SF or EOB potentials to estimate, for
two derivatives of the primary EOB radial (1SF) poten-
a given potential f(u) = f un+ǫN(u), both the
n<N n f tial a(u) suggests that, asymptotically,
order of magnitude of the PN expansion coefficients f ,
n
andthatoftheNthPNremPainderǫNf (u)=Of(uN),from ρN N2aN, (11)
the knowledge of its lightring singularity. The coarsest ∼
such estimate consists in saying that the radius of con- where a are the PN expansion coefficients of a(u) and
N
vergenceofa powerseries2, f uN, is determined by ρ those of ρ(u). [Here, we assume that the PN co-
N N N
the locationofthe singularityclosestto the originin the efficients d¯ of d¯(u) do not cancel the growth with N
N
P
complex u plane. Assuming that the closest singularity entailed by the two derivatives in the first equation (8).
is the lightring one at u = 1/3 determines the radius of Our numerical studies below will confirm this assump-
convergence as being u = 1. This simple consider- tion.]
| |conv 3
There is an alternative perspective on the additional
power-law growth (of the type of the factor N2 in (11)).
It consists in using more information about the singu-
2 Here,weformallyproceedasifthePNexpansioncontainedonly
larity structure of the considered function f(u) near its
integerpowers. Theexistenceoflogarithmiccorrections,starting
closestsingularity. Indeed, if we knew, for instance, that
at4PN[8,36],andofasub-series,startingat5.5PN[13,14,37],
containinghalf-integerpowers,indicatesthat,fromatheoretical f(u) had a power-law singularity near u= 1 of the type
3
point of view, a more subtle treatment should be applied. See
belowforthelogarithmiccorrections. fsing(u)=Kf(1 3u)−nf , (12)
−
10
(K denoting a constant), we would expect3 the expan- tigate the effect of the logarithms. It is technically con-
f
sioncoefficients f of f(u) to be asymptoticallyapprox- venient to work with the rescaled independent variable
N
imated by the expansion coefficients of its singular piece u 3u (with respect to which the singularity is located
3
≡
(12), namely at u =1), and expand a(u) a (u ) in powers of u
3 3 3 3
≡
fNsing =Kf −Nnf 3N ≈CfNnf−13N, (13) a(u)= N (aN+a′Nln(u3)+a′N′(ln(u3))2+···)uN3 . (14)
(cid:18) (cid:19) X
b b b
withCf =Kf/Γ(nf). Hereweseethatwhilethelocation Here,theaN arethesame(rescaled)coefficientsasabove
of the singularity determines the exponentially growing (obtained by replacing lnu by ln13). The higher loga-
wfaocutoldr 3bNe,dethteermsuibn-eldeabdyintghepopwoewre-lrawngroowftthhe∝siNngnufl−a1r criotlhummincscoobfeffiTacbielnetVsIai′Nn,Aa′Np′p,·e·n·diaxreBd.iWspelasyeeedthinatthteheotfihrestr
f
piece (12). Consistently with our r−emarks above, note logarithmic (rescalebd) cboefficients a′N are either compa-
thatactingonf(u)bykderivativeschangesnf intonf+ rabletothe aN,orslightlysmallerinabsolutevalue (the
k,andcorrespondinglyincreasesthepower-lawgrowthof signs, as well as the relative signs,bof aN and a′N fluctu-
the fN’s by +k. ate). The hbigher logarithmic terms appear only at u8,
Ref. [12]hasfoundthatthelightringsingularitystruc- and their coefficients a′N′,··· are founbd to bebgenerally
ture of the basic 1SF EOB potential a(u) was asing(u)= smaller. We shall neglect them in the following. Going
Kthae(n1e−xp3euc)t−analawrgiteh-NKbaeh≃av41ioarnad na =C N12.−1O/2n3eNwwouitlhd ba(auck)=to thNeaoNri(guin)uaNl P,bNwe-ecxapnantshieonnwcoreitffiectiheenitrscaoNm(bui)neodf
N a
C 1/(4Γ(1)) 0.14. We studied≃the evolution with N and u dependence as (neglecting higher logarithms)
a ≃ 2 ≃ P
N of the PN coefficients aN of a(u) by using the avail- a (u) (a +a′ ln(3u))3N; (15)
able high-PN results of Refs. [14, 25]. We confirmed the N ≃ N N
basic exponential growth aN ∼ 3N. Indeed, Table VI in with |aN| ∼ |a′N| ∼ 1.bThenb, in view of (8), we expect
Appendix B displays, in its first column, the values of acorrespondingapproximateasymptotic behaviorofthe
rthitehmresscraelpeldacPeNd bcoyelffin(ci1e)n;tsseaeNbe≡loawNf/o3rNth[weiltohgaarliltlhomgaic- PofNtheexbptyapnseionbcoefficientsofthe precessionfunctionρ(u)
3
dependence]. These rescaled coefficients are seen to re-
b
main (roughly) of order unity (in absolute magnitude), ρN(u)≃(ρN +ρ′Nln(3u))3N; (16)
even up to N = 23 for which 323 = 0.941432 1011.
× with ρ ρ′ N2.
More precisely, we have 0.1 . aN . 1, when N varies | N|∼| N|∼ b b
We tested this expectation on the 9.5PN-accurate ex-
between 3 and 20, while, for N = 21,22,23, we have
pansion of ρ(u) given above. An N2 scaling seems to be
a 2.254,1.459,3.313,respectively. [We do notknow b b
N b
| |≈ in reasonable agreement with the currently known PN
ifthefactthatthe lattervaluesareslightlylargerthan1
coefficients, and we found (using a (N 1)2 scaling and
signals the beginning of a growth for very large N’s.]
b −
relying on the 8PN and 9PN nonlogarithmic coefficients
We did not see any sign of the expected mild decay
a C N−1/2. This might be due to the more com- to fix the overall coefficient) for the coefficients of ρ (see
N a
≃ Table VII in Appendix B)
plicated singularity structure (beyond the leading-order
bpower-law) found in [12], or to the fact that the N−1/2 ρ′ . ρ 2.5(N 1)2. (17)
behavior sets in only for very large N’s. [Note also that, | N| | N|∼ −
after having factored the clear 3N growth, the rescaled A similar study of the PN expansion coefficients of the
coefficientsaN behaverathererratically,anddonotshow function d¯(u) (bsee Tabble VIII in Appendix B) leads to
any sign of converging towards a simple behavior.] If we a growth similar to the case of the function ρ(u), with
were only inbterested in estimating the PN error for val- simply a slightly smaller overall coefficient, i.e.
ues of u in the strong-field domain, and for values of
′
N around 10, we could simply use the simple estimate d¯ (u) (d¯ +d¯ ln(3u))3N; (18)
a C 3N with C 1. However, as we are also N ≃ N N
N a a
≃ | | ∼
interested in knowing what happens when u 0, we with b b
≪
should remember that the PN expansion coefficients run
′
logarithmically with u as u 0. We therefore kept the d¯ . d¯ 0.7(N 1)2. (19)
→ | N| | N|∼ −
full available high PN information [14, 25] to also inves-
Finally,thefactbthatthbeEOBpotentialq(u)isrelatedto
theredshiftcoefficientδUe0(u) a(u)byfourderivatives
∼
[7] suggests, in view of the argument above, that
3 Thisexpectation is basedon the usual integral Cauchy formula
giving the coefficients of the Laurent expansion of an analytic qN N4aN. (20)
∼
function. Bydeformingthecontourofintegrationsothatitgets
near the (closest) singularity one sees that the Cauchy integral We tested this expectation on the 9.5PN-accurate ex-
canbeapproximatedbyananalogousintegralinvolvingfsing(u). pansionofq(u)givenabove. AnN4 [or(N 1)4]scaling
−