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Self-force gravitational waveforms for extreme-andintermediate-mass-ratio inspirals

Kristen A. Lackeos, Lior M. Burko

To cite this version:Kristen A. Lackeos, Lior M. Burko. Self-force gravitational waveforms for extreme-and intermediate-mass-ratio inspirals. Physical Review D, American Physical Society, 2012, 86, 084055 (10 p.).10.1103/PhysRevD.86.084055. insu-01254887

Self-force gravitational waveforms for extreme- and intermediate-mass-ratio inspirals

Kristen A. Lackeos1 and Lior M. Burko1,2,3

1Department of Physics, University of Alabama in Huntsville, Huntsville, Alabama 35899, USA2Universite d’Orleans, Observatoire des Sciences de l’Univers en Region Centre,

LPC2E Campus CNRS, 45071 Orleans, France3Department of Physics, Chemistry, and Mathematics, Alabama A&M University, Normal, Alabama 35762, USA

(Received 7 June 2012; revised manuscript received 8 September 2012; published 31 October 2012)

We present the first orbit-integrated self force effects on the gravitational waveform for an intermediate

(extreme) mass ratio inspiral source. We consider the quasi-circular motion of a particle in the spacetime

of a Schwarzschild black hole and study the dependence of the dephasing of the corresponding

gravitational waveforms due to ignoring the conservative piece of the self force. We calculate the

cumulative dephasing of the waveforms and their overlap integral, and discuss the importance of the

conservative piece of the self force in detection and parameter estimation. For long templates the inclusion

of the conservative piece is crucial for gravitational-wave astronomy, yet may be ignored for short

templates with little effect on detection rate. We then discuss the effect of the mass ratio and the start point

of the motion on the dephasing.

DOI: 10.1103/PhysRevD.86.084055 PACS numbers: 04.25.dg, 04.25.Nx, 04.70.Bw

I. INTRODUCTION AND SUMMARY

The detection of gravitational waves (GW) and the onsetof the new field of gravitational-wave astronomy is one ofthe most exciting challenges for science in the XXI cen-tury, completing what is sometimes alluded to as Einstein’sUnfinished Symphony. The detection of GW will open anew window onto the universe, that in addition to revealingexciting information on exotic systems such as blackholes or cosmic strings is expected to also unravel as yetunexpected sources.

One of the interesting sources for low-frequency GWarethe so called I(E)MRI sources, or intermediate (extreme)mass ratio inspirals. Those are the GWemitted by a systemincluding a smaller compact object whose orbit decays intoa much larger massive black hole (MBH). Typical sourcesare stellar mass black holes inspiraling into a supermassiveblack hole, like those residing at the center of galaxies, andalso intermediate mass black holes (IMBHs) inspiralinginto MBHs. The importance of such sources is that becauseof the extreme mass ratio the smaller compact object canbe viewed as a test particle, thus probing the spacetime ofthe larger black hole and its surroundings. Inter alia, suchsources will allow us to test directly the Kerr hypothesis,and allow us to map the spacetime surrounding such exoticobjects. Moreover, the detection of I(E)MRIs will allow usto determine the mechanisms that shape stellar dynamics ingalactic nuclei with unprecedented precision [1].

The orbits of I(E)MRIs are typically highly relativistic,and exhibit exciting phenomena, e.g., extreme periastronand orbital plane precessions. Because the orbital evolutiontime scale (‘‘radiation reaction time scale’’) is much longerthan the orbital period(s), over short time scales the orbit isapproximately geodesic, yet on long time scales it deviatesstrongly from geodesic motion of the background. Instead,the smaller objects moves along a geodesic of a perturbed

spacetime. Alternatively, one may construe the orbit as anaccelerated, nongeodesic motion in the spacetime of theunperturbed central object, where the acceleration iscaused by the self force (SF) of the smaller object [2].Detection and parameter estimation of GW from I(E)

MRIs relies on the construction of theoretical templates.A number of approximation schemes for such templatesare available. First, the energy balance approach (‘‘theradiative approximation’’) uses balance arguments forotherwise conserved quantities, and relates the flux in thesequantities to infinity and down the event horizon of theblack hole with the particle’s orbit, so that the latter can beadjusted to agree with the fluxes [3]. As the orbital evolu-tion time scale is typically much longer than the orbitalperiod(s), the radiative approximation is very satisfactoryduring the adiabatic phase of the motion. As the particle’sorbit is affected by the fluxes away from it, when the orbitis not stationary one encounters complex retardationeffects. Most currently available EMRI waveforms havebeen obtained by such an approach. This approach, how-ever, ignores conservative effects that do not register in theconstants of motion.These retardation effects are completely avoided when

one considers a local approach to orbital evolution in termsof the SF. (One should bear in mind, however, that the SFitself is a nonlocal quantity, with contributions arising fromthe quasilocal neighborhood of the particle and possiblybeyond [4].) In addition, the local approach to the calcu-lation of orbital evolution via the SF is not restricted to theadiabatic regime, it avoids the complications associatedwith the rate of change of the Carter constant, and, mostimportantly, it includes also conservative effects that arediscarded when one uses balance arguments. Over the lastdecade much progress has been made in the computationand understanding of the SF in General Relativity

PHYSICAL REVIEW D 86, 084055 (2012)

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(for recent reviews of the SF in General Relativitysee Refs. [2,5]).

The computation of the fully relativistic SF [6] allowsone to include conservative effects in the waveformtemplates, and study the importance of the conservativeeffects. True self consistent orbit and waveforms includethe instantaneous solution of the coupled SF integratedequations of motion and the perturbation equations, orequivalently the interaction of the particle with its ownfield over its half-infinite past world line [7]. Very recently,for the scalar field toy model, such self consistentSchwarzschild orbits and waveforms were presented [8].Here, we are making the simplifying assumption that theeffects of the difference between the SF that is calculatedfor the actual orbit (the self consistent approach [7,9]) andthat which is calculated for a geodesic of the same instan-taneous orbital parameters, is smaller than the effects of thelatter and hence negligible at first order. This approxima-tion is valid for as long as the orbital evolution is adiabatic,that is as long as the orbital evolution time scale is muchlonger than the orbital period(s). In a Schwarzschild back-ground of massM, the adiabatic approximation holds when

the mass ratio :¼ =M is such that " 1=2, where "measures the distance to the innermost stable orbit, spe-cifically " ¼ p 6 2 where p is the semilatus rectumand is the orbital eccentricity [10]. In practice, ourapproximation is to a leading order in beyond geodesicmotion. We neglect terms that are linear in second-orderSFs, although our method is amenable to their inclusionwhen they become available. This approximation is validfor at least a part of the relevant parameter space [11], butas their inclusion would contribute linearly to the dephas-ing, the contribution of the conservative piece of the SF(hereafter CSF) may be isolated as is done here. Using trueself consistent waveforms will both produce more accuratewaveforms, and allow us to test the accuracy of thisapproximation. Most importantly, our approach allows usto see for the first time the effect of the CSF on GWemittedfrom IMRI sources.

We present here the first waveforms obtained withinclusion of the CSF, and study its effect within the simpleclass of quasi-circular orbits around a Schwarzschild blackhole. Specifically, we study the effect of the system’s massratio on the dephasing that occurs when one neglects theCSF. We find weak dependence of the dephasing on themass ratio, in accord with expectations based on the scalingof the number of orbits with the inverse of the mass ratio,and the scaling of the dephasing effect of the CSF per orbitwith themass ratio.We also find that the dephasing dependsquadratically on the initial point of the motion for the rangeof parameters we tested. We reiterate that second-orderdissipative effects are ignored in this Paper. Their inclusionwill guarantee the full consistency of the model, and will becomparable to the self-consistent approach. Recently, sig-nificant advance in the understanding of the second-order

self force has been achieved [12,13], although the second-order terms have not been calculated in practice for anyconfiguration yet. The inclusion of the second-order dissi-pative effects awaits further development to both theory andcomputational techniques.After the completion of this work we became aware of

Warburton et al. [14]. The approach of Ref. [14] is similarto our osculating method, except that [14] estimates thedephasing by the difference in the azimuthal angle ’ of theorbit, whereas we actually compute the waveforms and findtheir dephasing. The generalization of our quasi-circularorbit to bound orbits of varying eccentricity—as is done inRef. [14]—is straightforward, as the osculating orbitsequations of motion already include the eccentricityparameter. Lastly, we compute the waveforms using twoindependent computational methods, specifically the oscu-lating method and the direct method. The direct methoddoes not appear to us to be convenient for generalization togeneric Kerr orbits, and the osculating method has a clearadvantage over it for such orbits.The organization of this paper is as follows: In Sec. II

we discuss the computational and numerical methods thatwe use. In Sec. III we discuss our results for the orbits(III A), the waveforms (III B) and the dependence of thedephasing on the mass ratio and the initial point of themotion (III C).

II. METHOD

We use the fully relativistic SF obtained by Barack andSago [6] for circular Schwarzschild geodesics to drive theorbital evolution (our computation allows for an easyreplacement with a different force expression, say onethat includes second order dissipative effects, or spin-orbitcoupling effects when becoming available), and comparethe resulting waveforms with those obtained from theenergy balance approach and those obtained when onlydissipation is left in the SF, setting by hand the CSF tovanish. In practice, we consider a point source in a quasi-circular orbit around a Schwarzschild black hole M with amass ratio :¼ =M, and the motion starts at a value ofthe semilatus rectum p0 until it gets close to the InnermostStable Circular Orbit (ISCO) at p ¼ 6. Such sources arerelevant to the NGO capabilities: NGO will have thecapability to detect GW emitted by an IMBH in the massrange 102–4M spiraling into a MBH in the mass range3 105–107M such that the mass ratio is 103–102 outto cosmological redshift z 2–4. In addition, advancedLIGO could detect compact stellar sources spiralinginto an IMBH in the same mass ratio range [15]. Wespecialize below to this mass ratio range, 2½103–102. Although the linearized approach used hereis intended to be used only when 1 and one may notsimply extend the range of to high values and still expectaccurate results, the error involved from neglecting Oð2Þterms in the self force (specifically its dissipative piece) is

KRISTEN A. LACKEOS AND LIOR M. BURKO PHYSICAL REVIEW D 86, 084055 (2012)

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comparable to the accuracy of our computation. In thissense, to within the accuracy of our numerics, we arejustified in studying the IMRI case as long as we do notraise beyond 102. Throughout we present waveformsfor the fundamental mode, m ¼ 2.

We integrate the equations of motion using the Barack-Sago SF which was calculated for momentary circulargeodesics. As the Barack-Sago SF is tabulated for a selectchoice of orbital radii, we interpolate to intermediatevalues such that the original accuracy is maintained.Specifically, we match two asymptotic expansions: at largedistances (which we take in practice to be r > 8M) we takethe standard post-Newtonian expansion for the luminosityin gravitational waves, and construct from it the temporalcomponent of the SF. To 5.5 PN order the PN expressiondoes not provide us with sufficient accuracy to reproduceall the r 8M data points of the Barack-Sago data. Wetherefore add an effective remainder term that appears likea 6PN term, and fit its two free parameters to agree with allthe large distance tabulated data to all significant figures.The radial component, or CSF, is modeled by a PN-likeexpansion with four free parameters. At short distanceswe expand the SF such that convergence is fast and onlyfour free parameters are needed for either component.Specifically, we expand the radial component about theISCO at p ¼ 6. The expansion functions are as follows:

ftr8M¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 3Mr

q

ð1 2Mr Þ

M

r

5

M

2

a0 þa1M

rþa2

M

r

2þa3

M

r

3þ . . .

(1)

ftr8M ¼ 32

5

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 3Mr

q

ð1 2Mr Þ

M

r

5

M

2

PN5:5 þ

aþ6 þ aþ6L lnM

r

M

r

6 þ . . .

(2)

frr8M¼

12M

r

M

r

2

M

2

b0 þb1

16M

r

þb2

16M

r

2þb3

16M

r

3þ

(3)

frr8M ¼

M

r

2

M

2

bþ0 þ bþ1M

rþ bþ2

M

r

2

þ bþ3

M

r

3 þ

; (4)

where PN5:5 stands for the standard 112—post-Newtonian

expression (converting Eq. (3.1) in Ref. [16] from lumi-nosity to ft—see Appendix ). Fitting the free parameters,we find the values appearing in Table I.The simplicityof ourmodel allows us to easily separate the

CSF effects. Specifically, for quasi-circular Schwarzschildorbits we may write fSF ¼ fSFt t

þ fSF’ ’ þ fSFr t

r where

the last term on the right hand side (rhs) is purely conserva-tive, and the first two are purely dissipative. We may there-fore study the conservative effects by turning off by hand thelast term on the rhs.We integrate the SF driven orbit using two independent

methods: one method is the osculating orbit approach [17][specifically Eqs. (43)–(47) therein], with special caregiven to the requirement that the orbit is quasi-circular.Specifically, free evolution may take the orbit away fromquasi-circularity because integration using the osculatinggeodesics method cannot keep the value of the eccentricityas precisely zero. As both variables and (see Ref. [17]for definitions) are dynamical, the eccentricity mustevolve along the orbit too. This behavior is shown inFig. 1. Interestingly, the inclusion of the conservative pieceof the SF amplifies the resulting eccentricity.The second method is the direct integration of the orbit.

The direct integration method takes the local equation ofmotion to be uru

¼ 1fSF (Newton’s second law,

with covariant differentiation compatible with the back-ground metric) and integrates its solution. Both codes arenumerically stable and convergent. Specifically, the oscu-lating code converges with 5th order, and the direct codeconverges with 4th order (Fig. 2).Comparing the two independent methods for finding the

SF driven waveforms is not trivial because of a difficulty infinding identical initial conditions. Specifically, the oscu-lating method requires as initial data only the specificationof the initial position vector, which in our case is taken tobe a circular geodesic at some initial p0. The directmethod, however, requires both the position and the veloc-ity vectors to be specified, such that the constraint equationuu

¼ 1 is also satisfied. The main difficulty is that in

the initial data for the osculating method the initial ur0 ¼ 0,

TABLE I. The fit parameters for the SF. These parameters reproduce the accuracy of Ref. [6]to all significant figures for all data points. Our results for aþ6;6L are very inaccurate predictions

for the corresponding PN parameters, as our fit ignores all higher-order terms.

a0 4.57583 aþ6 331.525 b0 1.32120 bþ0 1.999991

a1 31.8117 aþ6L 2081:57 b1 1.2391 bþ1 6:9969

a2 267:250 b2 1:297 bþ2 6.29

a3 1049.27 b3 1.07 bþ3 24:6

SELF-FORCE GRAVITATIONAL WAVEFORMS FOR . . . PHYSICAL REVIEW D 86, 084055 (2012)

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such that it corresponds to an incorrect initial radial veloc-ity for the direct method. In practice, we find the initial datafor the direct code by generating the orbital parameters atp0 by running the osculating code from some p p0

down to p0, and then take the position and velocity vectorsat p0 as the initial data for the direct method. The residualdisagreement in the initial data can be controlled to becompatible with our numerical error tolerance. We then usethe obtained orbits to generate the waveforms using a code

for the sourced Teukolsky equation with hyperboloidalslicing, a code which converges at 2nd order [18].As noted above, our approximation holds only for as

long as " 1=2. We therefore do not integrate theequations of motion in practice all the way down to theISCO at p ¼ 6, and stop the integration at a finite distancefrom the ISCO. In practice, we stop the integration atpfinal ¼ 6:15 0:10. Stopping at pfinal is enough to esti-mate the dephasing at the ISCO: the dephasing is asmooth function of the time t along the particle’s worldline. In practice, we extrapolate r as a function of t to theISCO to determine the time at which the particle arrives atthe ISCO, and then we extrapolate the phase of the wave-form to the same value of the time to estimate the phase ofthe waveform when the particle arrives at the ISCO. Wemay then find the difference of the total phases betweentwo waveforms to find the dephasing .

III. RESULTS

A. The orbit

We next choose ¼ 102 and p0 ¼ 10. There is ofcourse nothing special about this choice of the parameters,except that we couldn’t make a much higher choice for and justify it with the linearized approximation used.Below we study the dependence of the effect on theparameters , p0. The orbit is displayed in Figs. 3–6 forthe three codes. Notably, the two independent self forcecodes reproduce the orbit to high level of agreement, with adifference much smaller than the difference between eitherand the orbit generated in the energy balance approach.This difference is attributed to the effect of the conserva-tive piece of the self force.The orbit, of course, is a gauge dependent quantity.

Indeed, the position vector changes trivially under gauge

0 100 200 300 4004

4.5

5

5.5

6

φ

Con

verg

ence

Ord

er

r tτ

0 100 200 300 4004

4.5

5

5.5

6

φ

Con

verg

ence

Ord

er

ur

ut

uφ

2000 3000 4000 5000 60003.9

4

4.1

4.2

4.3

4.4

t / M

Con

verg

ence

Ord

er r tφ

2000 3000 4000 5000 60003.9

3.95

4

4.05

4.1

t / M

Con

verg

ence

Ord

er

ur

ut

uφ

(A)

(C)

(B)

(D)

FIG. 2 (color online). Convergence tests for the two codes. Toppanels: the 4-position (A) and 4-velocity (B) for the osculatingcode as functions of the azimuthal angle ’. Lower panels: the4-position (C) and 4-velocity (D) for the direct code anda function of time t. In all cases shown here ¼ 102 andr0 ¼ 10M, and the CSF is included.

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

2000

4000

6000

8000

t / M

τ / M

8000 8500 90006500

7000

7500

0 1000 2000 3000 4000 5000 6000 7000 8000 90006

7

8

9

10

t / M

r / M

Energy Balance Direct Osculating

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

100

200

300

t / M

φ

8000 8500 9000300

350

FIG. 3. The Orbit. The 4-position for three orbital evolutioncodes: energy balance (dotted), direct evolution (dashed), andosculating code (solid).

0 1000 2000 3000 4000 5000 6000 7000 8000 9000−0.04

−0.02

0

0.02

0.04

t / M

α

0 1000 2000 3000 4000 5000 6000 7000 8000 9000−0.04

−0.02

0

0.02

0.04

t / M

β

0 1000 2000 3000 4000 5000 6000 7000 8000 90000

0.02

0.04

t / M

ε

Osculating w/o conservative effects Osculating

FIG. 1. The osculating-code variables and as functions ofthe time t (upper two panels), and the effective eccentricity ofthe orbit (lower panel) as a function of t in the osculating case.We show the results from the osculating orbits method with theinclusion of the CSF (solid curves) and with turning off the CSF(dotted curves). In all cases shown here ¼ 102 and p0 ¼ 10.

KRISTEN A. LACKEOS AND LIOR M. BURKO PHYSICAL REVIEW D 86, 084055 (2012)

084055-4

transformations, x ! x þ . We can, however, creategauge invariant quantities in a specific gauge choice (in ourcase, the Lorenz gauge), and then those quantities areguaranteed to remain unchanged in any other gauge. Twoindependent gauge invariant quantities are ut (‘‘gravita-tional redshift,’’ ‘‘helical Killing vector of the perturbedspacetime’’) and the angular frequency [19]. In Fig. 7we plot ut as a function of with and without the con-servative piece of the self force. Notably, to the accuracyof our numerical computation the two curves overlap. Thatis, we find—as expected—that ut as a function of is

insensitive to the conservative piece of the self force [5].This conclusion implies that when an actual data stream isused and this gauge invariant figure is plotted, one may usea simplified radiation-reaction scheme, that does notinclude the conservative effects in its analysis.

−6.5 −6.4 −6.3 −6.2 −6.1 −6 −5.9 −5.8 −5.7 −5.6 −5.57

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8

x / M

y / M

FIG. 6 (color online). The Orbit. The shape of the orbit for thethree orbital evolution codes: energy balance (dotted), directevolution (dashed), and osculating code (solid). A small portionof the orbit shown in Fig. 5 is magnified to show detail.

−10 −8 −6 −4 −2 0 2 4 6 8 10

−10

−8

−6

−4

−2

0

2

4

6

8

10

x / M

y / M

FIG. 5 (color online). The Orbit. The shape of the orbit for thethree orbital evolution codes: energy balance (dotted), directevolution (dashed), and osculating code (solid).

0 1000 2000 3000 4000 5000 6000 7000 8000 90001.2

1.25

1.3

1.35

1.4

t / M

ut

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

−6

−4

−2

0x 10−3

t / M

ur

Energy Balance Direct Osculating

8000 8200 8400 8600 8800 90001.3

1.35

1.4

0 1000 2000 3000 4000 5000 6000 7000 8000 90000.040.050.060.070.08

t / M

uφ

8000 8200 8400 8600 8800 90000.06

0.07

0.08

FIG. 4. The Orbit. The 4-velocity for three orbital evolutioncodes: energy balance (dotted), direct evolution (dashed), andosculating code (solid).

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.0651.18

1.2

1.22

1.24

1.26

1.28

1.3

1.32

1.34

1.36

1.38

Ω M

ut

Osculating w/ conservative effects Osculating w/o conservative effects

FIG. 7. The gauge invariant figure of ut as a function of theangular frequency. The curve without (dashed, ) and with theconservative effects (solid, h) are shown, together with equal-tincrement marks starting at t ¼ 1; 500M (at the bottom left) inincrements of 1; 000M. The two curves are indistinguishable tothe numerical accuracy of our computation (using the osculatingorbits code in both cases).

SELF-FORCE GRAVITATIONAL WAVEFORMS FOR . . . PHYSICAL REVIEW D 86, 084055 (2012)

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There is, however, an aspect of the gauge invariantfigure 7 that is sensitive to the conservative effects, spe-cifically the speed with which the data point moves alongthe curve. The way the conservative effects are manifestedin the gauge invariant plot is not is the shape of the curve,but in the time it takes the signal to move along it. One maytherefore observe the CSF effect by monitoring the motionof the data point representing the system along its curve onthe ut plane. We note that one additional effect of theCSF is the shift in the ISCO [20], which we do not considerhere.

B. The waveforms

We show the waveform for the case that the CSF isturned off in Fig. 8 for the same parameters discussedabove, specifically p0 ¼ 10 and ¼ 102. We find thatthe waveforms obtained with the SF keeping only its dis-sipative pieces (and turning off its CSF) overlap with theenergy balance waveform. Notice that the two waveformsin the figure are indistinguishable. The calculation methodis very different in these two cases: In the SF case theorbital evolution is local; the orbit evolves because of alocal force acting on the particle; in the energy balancecase the fluxes to infinity and down the event horizon arecalculated, and then the energy escaping over a period ofthe orbit is removed from the particle, and a new orbit withthe new values of the constants of motion is found. Theagreement of the waveforms is therefore a nontrivial test ofthe correctness of the calculation. The two waveformsoverlap nearly exactly, with total cumulative dephasing atthe order of 103 radians.

We next reintroduce the CSF. We integrate the SF drivenorbit using two independent methods: the osculating orbit

approach and direct integration of the orbit. Figure 9 showsthe waveform for the energy balance approach (same as inFig. 8) and for the two independent methods of calculatingthe SF driven orbit (including the CSF). The latter twowaveforms are in agreement with each other with smalldephasing (see below) between them, and a much largerdephasing of either with the energy balance waveform.The waveform dephasing of Fig. 9 is shown explicitly in

Fig. 10. We find that the total cumulative dephasing of thewaveforms at the endpoint of the evolution at pfinal is ¼ 10:3 0:4 radians. The error estimate comes fromthe numerical errors in each calculation method and fromresidual incompatibility of initial data in the two SF drivencases, which we estimate by comparing the waveformsobtained from the osculating orbit approach and the directintegration approach.The dependence of on the position p is a simple

monotonic unction of the time (Fig. 10) which we canextrapolate from pfinal down to the ISCO at p ¼ 6. Wefind that at the ISCO the dephasing is ¼ 14 1 radi-ans. Dephasing of 14 1 radians corresponds to about 2.2cycles over the entire motion of the particle over 107.8cycles. We next consider the following simulation of adetection event. Say the actual data stream is modeled bythe waveforms obtained with the full SF expression, i.e.,including the CSF, and that the theoretical template isobtained by turning off the CSF, or equivalently by usingthe energy balance waveform. By how much is the overlapintegral of the waveforms reduced because of our igno-rance of the CSF?

0 1000 2000 3000 4000 5000 6000 7000 8000 9000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10−3

t / M

Re

( ψ4 )

8300 8400 8500 8600 8700 8800 8900 9000−1

0

1x 10−3

Osculating Energy balance

Osculating Energy balance

FIG. 8 (color online). The real part of the waveform c 4 whenthe CSF is turned off (solid), and using the energy balanceapproach (dashed, ). The SF waveform was calculated withthe orbit evolving using the osculating method [17].

1000 2000 3000 4000 5000 6000 7000 8000 9000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10−3

t / M

Re

( ψ4 )

Osculating Direct Energy Balance

400 600 800 1000−2−1

012

x 10−4

8200 8400 8600 8800−1

−0.5

0

0.5

1x 10

−3

FIG. 9. The real part of the waveform c 4 when the CSF isincluded (solid for the osculating orbit method, and dashed forthe direct integration method), and using the energy balanceapproach (dotted). The three waveforms are in phase at thebeginning of the waveforms, but the former two dephase withtime from the latter (and to a much smaller extent from eachother).

KRISTEN A. LACKEOS AND LIOR M. BURKO PHYSICAL REVIEW D 86, 084055 (2012)

084055-6

Figure 11 shows the overlap integral as a function of thetime, when we take a window of length L of the energybalance waveforms (specifically from their late chirp part),and shift it along the waveform of the full SF. At eachpoint we calculate the overlap integral of the window witha local piece of the second waveform, and plot the localoverlap integral as a function of the start time of thewindow. Because the window was taken from the latepart of the waveform, we find that the overlap integral isvery small at first, and becomes large only at the latepart of the other waveform. We then take the maximumof the overlap integral to be the one corresponding tothe chosen window L. More precisely, we calculate

Cmax ¼ maxhc SFðtÞjc EBðtÞi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

hc SFðtÞjc SFðtÞihc EBðtÞjc EBðtÞip . Larger values of

L would reduce the overlap integral even further. InFig. 12 we show Cmax as a function of the window sizeL. As L increases, at some value Cmax would drop below apredetermined value that marks our tolerance for detectionor parameter estimation. Many times this threshold is takento be C ¼ 0:96, because then detection rate would drop by10%. Here, this threshold is obtained when Lthreshold ¼816:6M, which corresponds to just over 14 wavelengthsof the emitted GW. If L * Lthreshold, the exclusion of theCSF effects would cause a significant drop in the Cmax that

would reduce the detection rate by 10% or more. In such acase ignoring the CSF would have an important effect ondetection or parameter estimation of the GW. However,short waveforms (i.e., L < Lthreshold) do not require theCSF effects to be included if reduction of the detection

0 1000 2000 3000 4000 5000 6000 7000 8000 9000−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

t / M

C (

t)

L = 820 M L = 350 M

FIG. 11 (color online). The local overlap integral for a windowof length L ¼ 350M (dash-dotted) and of length L ¼ 820M(solid) taken from the energy balance waveform and a piece ofequal length of the full SF (including conservative effects)waveform obtained from the osculating orbit method, as afunction of the starting point of the latter. The window is takenhere from the end of the waveform (the late chirp part).The global maximum of the overlap integral is 0.9900 (forL ¼ 350M) and 0.9594 (for L ¼ 820M).

300 400 500 600 700 800 9000.95

0.955

0.96

0.965

0.97

0.975

0.98

0.985

0.99

0.995

1

Window Size [t / M]

Cm

ax

data pointsquadratic fit

FIG. 12 (color online). The maximal overlap integral forwindows of varying length L taken from the energy balancewaveform and a piece of equal length of the full SF (includingconservative effects) obtained from the osculating orbit method,as a function of the starting point of the latter. The circles aredata points, and the curve is the quadratic fit Cmax ¼ 1:00101:7988 105L 3:9154 108L2, with fit parameterR2 ¼ 0:9983.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000−2

0

2

4

6

8

10

12

t / M

deph

asin

g

8200 8400 8600 88006

8

10

12

14

AVE OS − EB DIR − EB DIR − OS

FIG. 10 (color online). The dephasing of the waveforms for thesame data shown in Fig. 9. We show the dephasing from theenergy balance waveform of the waveforms obtained from bothmethods to calculate the orbital evolution when the conservativeeffects are included, and the dephasing of the two latter methodsfrom each other. Dashed curve: dephasing of the waveformsfrom the osculating orbit method from the energy balancemethod; Dotted curve: dephasing of the waveforms from thedirect integration method from the energy balance method;Thick solid curve: the average dephasing of the waveform.Dash-dotted curve: The difference between the dotted anddashed curves. Insert: same as in the main figure, with theextrapolated dephasing down to the ISCO shown in a thin solidcurve.

SELF-FORCE GRAVITATIONAL WAVEFORMS FOR . . . PHYSICAL REVIEW D 86, 084055 (2012)

084055-7

rate by less than this tolerance is acceptable. The overlapintegral increases rapidly as the template window is takenfrom earlier times. This result suggests that when otherparts than the very end of the waveform is of interest, thefull SF is even less significant than we have found.

C. Varying the values of parameters

Our model of quasi-circular Schwarzschild orbitsdepends on two variables: the mass ratio and the startpoint p0. Here we vary each parameter independently andfind the dependence of the dephasing [between theosculating orbits case (that includes the CSF) and theenergy balance case (that neglects the CSF and considersonly dissipative effects)] on either parameter.

1. Varying the mass ratio

First we study the variation of the dephasing with chang-ing the parameter , the mass ratio. The greatest problemwith varying is its effect on the computation time. On theone hand we cannot justifiably increase beyond 102,because then the linearization approximation breaks down.On the other hand, lowering to very small values, whilesatisfying the linearization requirement more confidently,results in longer physical evolution times and correspond-ingly also longer computation times.

In practice we reduce the value of by a full order ofmagnitude, and sample values in the range [103–102],and fix p0 ¼ 8. (We decrease the value of p0 from itsprevious value of 10 to save on computation time for thelower values of .)

Figure 13 shows a family of curves displaying the de-phasing between the cases of the osculating orbits methodand the energy balance method. Each curve in Fig. 13 endsat the point we stop the integration when the orbits gets tooclose to the ISCO for the adiabatic approximation to stillhold. These curves are very smooth and simple functions,which we extrapolate to the times at which the particlearrives at the ISCO. The extrapolated value of the dephas-ing at the ISCO is also shown in Fig. 13. We find only littlevariation in the dephasing as changes over a full order ofmagnitude, consistent with the expectation that the dephas-ing has only a weak dependence on . Indeed, one couldexpect from scaling arguments that the dephasing per orbitscales with while the number of orbits scales with 1

(both scalings are indeed found in our simulations—seeFig. 14), so that the total dephasing is at the leading order atOð0Þ. Here we show that not only is the dephasing atOð1Þ, but in the range tested and with our numericalresolution is indistinguishable from a constant value, orat the most is a very weak function of .

In Fig. 14 we show five dimensionless quantities con-structed from the evolution time T from p0 down to theISCO, the total phase of the waveform in the energybalance and osculating methods case (including the CSFterm), and the differences in arrival time and the dephasing

between the latter two. We find that T Oð1Þ, thatOð0Þ, T=MOð0Þ and that in both the energybalance and osculating orbits cases the total phase is alinear function of with a very small slope, that is at the

0 2000 4000 6000 8000 10000 12000 14000 16000 180000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

t / M

deph

asin

g [

rad

]

FIG. 13 (color online). The dephasing as function of timefor a family of mass ratios for the values (from right to left): ¼ 0:001, 0.002, 0.003, 0.005, 0.006, 0.007, and 0.010 (solidcurves). The circles display the extrapolated values whenthe particle arrives at the ISCO, and the dashed line showstheir average at 1:726 0:021. In all cases the motion startsat p0 ¼ 8.

1 2 3 4 5 6 7 8 9 10x 10−3

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

η

N

(1/10) η (T/M)∆Φ

(∆ T) / Mη Φ [EB]η Φ [OS]

FIG. 14 (color online). Five dimensionless quantities (denotedcollectively by N) as functions of the mass ratio : T=M2

(divided by 10 to keep the scale similar with the other fourquantities) ( ), the dephasing (*), the difference in arrivaltime T=M, and for the energy balance case (h) and theosculating orbits case (). Here, and T=M are between theosculating orbits and the energy balance cases, and T is the totaltime of motion from p0 down to the ISCO. In all cases themotion starts at p0 ¼ 8 and the values shown are extrapolationsto the ISCO.

KRISTEN A. LACKEOS AND LIOR M. BURKO PHYSICAL REVIEW D 86, 084055 (2012)

084055-8

magnitude of our computational error. We therefore cannotrule out that we see in addition to the leading Oð1Þ termalso a higher-order Oð0Þ term. Notice that the lastfour quantities are comparable to each other, and that

the variation in all five (at the most 10%) is very smallcompared with the full order of magnitude variationin .

2. Varying the initial position p0

Next we fix the mass ratio and vary the starting pointof the motion p0. In practice we choose the value of ¼ 102. Figure 15 shows the dephasing between theosculating orbits case (that includes the CSF) and theenergy balance case (that neglects the CSF) for the rangep0 2 ½8; 10. Naturally, the dephasing grows with p0. Thedata presented are consistent a quadratic dependenceof on p0.

ACKNOWLEDGMENTS

The authors wish to thank Gaurav Khanna for discus-sions and for use of his numerical code. This work hasbeen supported by a NASA EPSCoR RID Grant and byNSF Grants No. PHY-0757344, No. PHY-1249302, andNo. DUE-0941327. L. M. B. is grateful to AlessandroSpallicci for hospitality.

APPENDIX: THE 5.5 POST-NEWTONIAN TERM

The PN5:5 term used in Eq. (2) is given by ([16]):

PN5:5¼11247

336

M

r

þ4

M

r

3244711

9072

M

r

28191

672

M

r

52þ

6643739519

698544001712

105þ16

32

3424

105ln2856

105lnM

r

M

r

316285

504

M

r

7=2þ

323105549467

3178375200þ232597

44101369

1262þ39931

294ln2

47385

1568ln3þ232597

8820lnM

r

M

r

4þ

265978667519

7451136006848

10513696

105ln23424

105lnM

r

M

r

9=2

þ

2500861660823683

2831932303200þ916628467

7858620424223

6804283217611

1122660ln2þ47385

196ln3þ916628467

15717240lnM

r

M

r

5

þ

8399309750401

101708006400þ177293

1176þ8521283

17640ln2142155

784ln3þ177293

2352lnM

r

M

r

11=2:

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2

4

6

8

10

12

14

p0

deph

asin

g [

rad

]

FIG. 15 (color online). The dephasing as function of theinitial position p0. The circles display the extrapolated valueswhen the particle arrives at the ISCO, and the curve shows aquadratic fit. In all cases the mass ratio ¼ 102.

SELF-FORCE GRAVITATIONAL WAVEFORMS FOR . . . PHYSICAL REVIEW D 86, 084055 (2012)

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084055-10