IL NUOVO CIMENTO VOL. 111 B, N. 12 Dicembre 1996

Gravitational-wave chirps: accumulating phase errors due to residual orbital eccentricity

V. PIERRO and I. M. PINTO

Dipartimento di Ingegneria dell'InJbrmazione ed Ingeg~Je,~a Elettrica Universit~t di Salerno - Salerno, Italy

(ricevuto il 15 Maggio 1996; approvato il 9 Luglio 1996)

Summary. -- We compute to lowest order the phase error due to the circular- orbit assumption in modelling gravitational-wave chirps from coalescing binaries. The error increases steeply as one proceeds toward coalescence, and can be comparable to ~ even for very small (residual) initial orbital eccentricities, thus deteriorating severely the detector's performance.

PACS 04.20 - Classical general relativity.

Gravitational-wave detection and (source) parameter estimation stems from the principle of successful overlapping of the function representing the detector output f, with another function representing the sought signal ~ , called template. The degree of overlapping can be conveniently measured by the (squared) functional scalar product

(1) 0 : <(f, fs )2 >

averaged over the (unknown) signal initial phase (arrival time), and successful overlapping here means that the above scalar product (occasionally called optimum non-coherent correlator, or matched filter) exceeds some suitably chosen threshold. The above criterion is optimum in that for any given signal-to-noise ratio SNR, it pro- rides the largest detection probability PD for a fLxed false-alarm probability P7 [1].

It is well known that moderate amplitude errors in the template do not deteriorate the detector's performance in a significant fashion. On the other hand, even modest phase errors can seriously affect it [1].

Recently, attention has been called on the possibility that oversimplified gravitational-wave chirp modelling may result into accumulating phase errors [2]. As a special result, the detector's performance degradation due to the neglect of the relativistic periastron advance has been computed (to lowest PN order) in[3], under the assumption of a circular orbit. In this paper we focus on orbital eccentricity as a further potential source of accumulating phase errors, in the frame of the

1517

1518 v. PIERRO and I. M. PINTO

Peters-Mathews (henceforth PM) model, augmented to include (to lowest PN order) the relativistic periastron advance (henceforth RPA).

In this connection the following remarks are in order. According to folklore scientific wisdom, one could ignore residual eccentricity as a potential source of orbital phase errors, since in the late evolutionary stages a binary orbit becomes nearly circular, by the very emission of gravitational radiation. Presumed quantitative support to this widespread opinion traces back to a frequently (mis)quoted remark by Krolak, whereby -the phase error due to residual orbital eccentricity computed in the Newtonian approximation is two orders of magnitude smaller than the first post-Newtonian correction computed for a circular orbit,, [4]. While this statement is indeed correct, it does not imply by any means that the phase error due to residual orbital eccentricity should be consistently neglected. In fact, the relative magnitude of the various phase errors is devoid of significance, as far as the operation of a matched filter is concerned, and only the total error mod 2Jr is relevant. For instance, between a phase error 2' 10"z and an additional phase error 1.5:r, it is the second one, which is smaller by n orders of magnitude, which would almost destroy the performance of a matched filter, while the frrst one is harmless.

Assuming the obselwation to start at t = 0, the instantaneous phase q)(t) to be used in computing the gravitational waveforms is given, but for an (unknown) additive initial phase, by

t (2) q~(t) = f d~ 2:r + q)a(~,'

o T(~) '

where T is the time of return to periastron and q~ad,- is the relativistic angle of periastron advance.

In the following we rely on the exact solution of the Peters-Mathews orbital damping equations obtained in[5] and summarized in appendix A, and use the following (dimensionless) orbital parameters:

cTo I M1 - Mz [ (3) Xo - , A -

~rg M1 + M2

where To is the orbital period at t = 0, M1, Me are the companion masses, rg = = 2G(M1 + M2)/c e is the source gravitational radius, and c is the velocity of light in vacuo. To lowest PN order, the angle of periastron advance per orbit is given by [6]

T)-2/a (4) Cadv = 6:r~'O 2/a T0 (1 - e2) -1 ,

where by T and e are the instantaneus orbital period and eccentricity, respect- ively.

Letting 0 = t /Te, T e being the binary lifetime given by eq. (A.4), the instantaneous phase can be conveniently written as follows:

(5) ~b(O) = ~b PM (0) -}- ~b RPA (0) ,

GRAVITATIONAL-WA~E CHIRPS: ETC. 1519

where

(6) OP~ (0) = 2 s ~ o ~ de/dO de

and

(7)

e(-O )

(PRpA(O) = 6:~fg 2/a - - (1 -- e2) - ' 1 _ de. de/d 0

Here Tr is the lifetime of a circular binary having the same Z0 and A parameters, viz

5 ZSo/3 (8) Tc = 128zC 1 U ~1 e To.

Using eqs. (A.1), (A.2) and (A.4) of appendix A, based on the Peters-Mathews model for orbital damping under gravitational-wave emission, the above integrals can be computed exactly in terms of known functions (see, e.g. [7], sect. 15.3.1). Hence

_ [ 121 2~-2175/e299 (9) q~PM 8 Tr - e~) 5/2 1 +

2z 5 [ 304 e~ } "

�9 , ~ " - - ; e - - - ~ F 1 . . . . , ~ ; - - ; e 2

~F~ 2299 19' 19 304 e0 ) 2299 19 19 304

and

- 8 Z 0 -2~ Tc(1 - 1 + - - e02 3O4

�9 2 2 9 9 - ' - " ' ' 2299 19' 19 304 '

valid for any eo, wherein 2F1 is Gauss hypergeometric function (the implicit dependence of the orbital phase on 0 through e can be made explicit by resorting to eq. (A.7) of appendix A). The first 30 coefficients of the series expansion of the hypergeometric functions appearing in (9), (10) in powers of e z are collected under table I and II, respectively, and are sufficient to obtain 16 figure accuracy(1), for 0 ~< e ~<e~<~0.9.

In the limit e0-~ 0, one recovers from eq. (6) the well-known result

(11)

(1) The power series (9) and (10) are alternating, and hence their truncation error is of the same size as the first term omitted.

1520 v . P I E R R O and I. M. P I N T O

TABLE I. - Coefficie,~ts of series exp(~t~sion qf hypergeomet~ic functions in eq. (9) in powers Of e 2.

0 1 .000000000000000"100 16

1 - 9 . 4 7 1 2 4 0 0 1 9 5 5 3 5 3 0 . 1 0 -a 17

2 1 . 2 7 4 4 0 1 0 5 9 8 3 5 8 5 1 . 1 0 -a 18

3 - 2 . 556389591355666"10 a 19

4 6 . 1 4 6 5 2 9 0 9 6 8 3 7 7 3 7 " 1 0 ~ 20

5 - 1 . 6 4 0 9 5 7 6 4 8 4 5 4 4 2 6 . 1 0 -~ 21

6 4 .691272540283231"10 ~ 22 7 - 1 . 4 0 7 5 7 0 8 5 7 5 1 2 5 8 0 . 1 0 ~ 23 8 4 .377930431013402"10 T 24

9 - t . 4 0 0 0 7 0 3 5 8 3 6 5 8 0 5 - 1 0 7 25

10 4 . 5 7 7 8 1 4 3 3 5 8 0 9 6 3 1 . 1 0 s 26

11 - 1 . 5 2 4 1 2 0 3 3 2 3 9 6 6 4 2 ' 1 0 s 27

12 5 . 1 5 1 2 0 1 1 3 0 2 6 9 3 5 2 ' 1 0 u 28

13 - 1 . 7 6 3 2 3 7 5 3 8 2 0 7 1 7 5 . 1 0 -~ 29 14 6 . 1 0 1 4 2 1 6 8 3 5 8 1 3 4 0 - 1 0 ~o ~0

15 - 2 . 1 3 1 2 5 1 1 0 1 1 7 0 3 6 7 . 1 0 1o

7 . 5 0 5 9 7 4 3 5 1 2 6 4 3 2 7 ' 1 0 11

- 2 . 6 6 2 7 1 9 7 8 2 3 2 4 5 4 5 . 1 0 - n

9 .506876933608390"10 12

- 3 . 4 1 3 8 8 0 6 9 2 4 6 1 3 2 5 . 1 0 -12

1 , 2 3 2 2 6 9 1 9 8 6 8 5 7 2 1 . 1 0 - t~ - 4 . 4 6 8 8 3 6 5 9 6 7 2 0 3 0 6 ' 1 0 la

1 .627531242775566"10 la

- 5 . 9 5 0 4 3 7 0 6 3 4 4 9 5 4 0 . 1 0 14 2 .183293044870461"10 24

- 8 . 0 3 7 0 1 5 3 2 5 1 4 6 9 1 0 . 1 0 - ~

2 . 9 6 7 4 7 7 9 8 1 0 3 4 5 0 0 . 1 0 -15

- 1 . 0 9 8 7 3 4 4 3 9 8 8 7 0 7 8 . 1 0 -15 4 . 0 7 8 7 1 6 8 3 5 5 1 2 2 3 9 . 1 0 -16

- 1 . 5 1 7 7 5 6 3 1 6 6 4 3 5 1 1 . 1 0 -16 5 . 6 6 0 5 4 ~ 4 5 8 5 7 2 5 6 6 . 1 0 -17

TABLE II . - CoejJicie~ts qf series cxpa~t~sion qf hypergeometric .functions in eq. (10) in powers o f e

0 1 . 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 . 1 0 o 16

1 - 5 . 5 3 1 5 0 9 6 9 5 2 9 0 8 5 9 . 1 0 2 17

2 9 .393737106305590"10 a 18 3 - 2 . 1 5 8 7 9 3 2 3 1 1 5 8 2 4 2 ' 1 0 a 19

4 5 . 7 2 5 0 7 1 6 3 2 8 5 5 0 1 6 . 1 0 4 20 5 - 1 . 6 5 0 9 8 6 4 4 0 8 4 1 8 7 0 1 0 4 21

6 5 .030613651775844"10 5 22 7 - 1 , 5 9 3 7 5 6 4 2 8 1 0 1 6 3 0 . 1 0 ~ 23

8 5 . 1 9 7 9 6 1 5 8 5 9 9 9 0 6 8 . 1 0 -~ 24 9 - 1 .733823763326178"10 G 25

10 5 .887848431380474"10 v 26 11 - 2 . 0 2 8 8 7 3 4 0 8 9 6 8 7 2 4 . 1 0 ~ 27 12 7 . 0 7 6 6 8 0 7 2 5 3 5 4 4 1 8 . 1 0 - s 28 13 - 2 . 4 9 3 7 9 1 0 4 4 1 6 5 6 2 5 - 1 0 - s 29

14 8 . 8 6 5 5 0 4 7 6 4 9 6 0 6 7 0 . 1 0 ~ 30 15 - 3 . 1 7 5 7 5 3 3 8 0 4 7 0 4 2 8 . 1 0 9

1 . 1 4 5 1 8 0 5 5 4 9 6 9 3 4 9 . 1 0 -9 - 4 . 1 5 3 7 7 6 0 2 4 9 7 8 2 4 6 . 1 0 -1o

1 . 5 1 4 5 0 0 9 7 5 7 0 0 5 4 6 . 1 0 - l ~

- 5 . 5 4 7 7 1 4 7 7 1 2 1 8 5 4 5 . 1 0 - n 2 . 0 4 0 6 7 3 9 8 4 7 0 8 2 3 7 . 1 0 -11

- 7 . 5 3 4 8 4 1 6 6 0 5 8 4 3 4 8 . 1 0 -12 2 . 7 9 1 6 7 9 5 0 5 1 9 4 9 5 3 . 1 0 -12

- 1 . 0 3 7 5 7 0 2 9 0 9 5 4 8 8 3 . 1 0 -12

3 . 8 6 7 3 7 2 8 5 7 6 2 8 2 7 7 . 1 0 -13

- 1 . 4 4 5 3 0 9 5 3 1 5 6 1 7 1 8 . 1 0 -13 5 . 4 1 4 5 6 2 5 7 2 9 4 6 4 7 6 - 1 0 -24

- 2 . 0 3 3 0 3 4 9 9 0 6 2 3 2 0 9 . 1 0 _24

7 . 6 4 9 5 3 5 2 1 2 3 3 5 8 1 0 . 1 0 -15

- 2 . 8 8 3 8 3 9 0 1 7 1 8 1 7 0 4 . 1 0 -~5 1 . 0 8 9 1 7 1 1 3 5 7 0 4 7 2 9 . 1 0 -1~

w h i c h h a s b e e n u s e d t h r o u g h o u t t h e t e c h n i c a l l i t e r a t u r e to m o d e l t h e c h i r p s i g n a l

p h a s e f o r a c i r c u l a r - o r b i t c o a l e s c i n g b i n a r y , a n d f r o m (7)

(12) q~RPA TTe = 16'r)~~ 1 - 1 - ~ / j : = ~ ' R P A ,

w h i c h h a s b e e n i n t r o d u c e d i n [3] .

W e n o w d e f i n e a n orbital phase er'ror d u e t o t h e c i r c u l a r - o r b i t a s s u m p t i o n , viz

,4, (0) ) (13) q0 e~~' ((p p• - q) ~e~ ) + (cp aeA - ~ aeA -

G R A V I T A T I O N A L - W A V E C H I R P S : E T C . 1521

The steep increase of the (sealed) orbital phase errors Aq)=q~p~_.~(o)v, pM and 6q~ = 0RPA- q~V)A as t---) To is displayed in fig. 1 and 2, for initial eccentricities in the range 0 ~< eo ~ 0.3. The final accumulated phase errors Aq~ ace= Aq~(O = 1) and

0.10

0 .00 i ,

0.00 I I I l l l l I I I I I l l I I I I I I I I l l [ l l l l l I l l l I l l I l l ' I I l l l I

0.20 0.40 0.60 0.80 1.00 tlT~

0.40

0.30

7 i ~ o 0.20-

Fig. 1. - Scaled orbital phase error Aq~ vs. scaled time, for eo = 0.05 (0.05) 0.3

0.40

0.30,

u 0.20- ~ .

co

0.10"

0 . 0 0 , ,

0 . 0 0

%=0.3....

I l l l l l l l l l l l l l l l l l l l l l l l l l l i l l l l l l i i l l l l l l | l l l l

0.20 0.40 0.60 0.80 1.00 t/To

Fig. 2. - Scaled orbital phase error 6q~ vs. scaled time, for eo = 0.05 (0.05) 0.3.

1522 v. PIERRO and [. M. PINTO

6q$ ~< = 6q$(0 = 1) are given by

AO ace 8 To{ ( 121 ) -2175/2299 (14) 2:r - 5 To ( 1 - e 2)51~ 1 + --304 e~

and

(15)

( )[ ( /5J811 124 15. 34 121eo 2 _ 1 - 1 - '2F1 2 - ~ ' ~ ' ~ ' ~ To] JJ

- 8 Z o 2j3 ( 1 - e ~ ) 3/2 1 + - - C o 2 304

( ) [ ( /3J811 994 9 28 121e~ - 1 - 1 - �9 ~F~ 2-~9'-~; ~' ~ To/ JJ

valid for a n y Co. The accumulated phase errors are shown in fig. 3 and 4, for 0 ; eo ~< 0.3. The principal parts of the accumulated phase errors as eo ~ 0 are easily obtained using eq. (8) and (A.4), by expanding all hypergeometric functions to lowest order in Co. The result is

(16) 2n 16n 1 - A z [ ~ - e314 '

•q• f i C C

(17) 2n

5 Xo ( 1 5 7 ) 3/s

where 157/43 is the coefficient of e~ in the Taylor series of (A.4).

0.40

0.3(

l ~ 0.20

0.10

0.00 0.00

I

I

t t , i i , i i i ' , , , i t i t i i i i

0.10 0.20 0.30 e o

Fig. 3. - Accumulated scaled phase error Aq~(O = 1) vs. e o c (0, 0.3).

4.00

3.oo..

I ~ 2.002 E �9 c o

1 .oo

o.oo ~ o.o0

| , J . . . . . , i | i ! i t | i i i i , , i , i J i i i 1 , i i i

0.10 0.20 0.30

GRAVITATIONAL-WAVE CHIRPS: ETC. 1523

e 0

Fig. 4. - Accumulated scaled phase error 5q~(0 -- 1) vs. eo ~ (0, 0.3).

In order to illustrate the possible relevance of these results, we consider two simple prototype cases. For a twin-pulsar coalescing binary with M1 = M2 = 1.4Mo, whose observation starts when To = 0.01 s with a residual eccentricity as small as e0 = 0.01, one has from eqs. (3) and (8) 40 = 114.48, Te = 33.56, and hence from eqs. (14) and (15) A~ ae~ -- 0.76~ and 5q~ a~ ~ 1.17~. For a massive coalescing binary with M1 = M2 -- = 10M o whose observation star ts when T o - - 1 s with a residual eccentricity as small as eo--5"10 -4, one has ;~o = 1602.68, Tc = 2729.12, and hence Aq~ acc~ 1.82z~ and 5q~ acc~ 1.32~. For these prototypical cases, the total accumulated phase e r ror due to residual orbital eccentrici ty is indeed non-negligible, as far as the detector 's performance is concerned.

The above findings as well as those discussed in [2] and [3] suggest that even a tiny residual orbital eccentricity should not be neglected, in principle, in building reliable PN templates for matched filter detection of gravitational-wave chirps from coalescing binaries.

APPENDIX A

PM equations for orbital damping

The Peters-Mathews equations describing the binary orbital damping due to gravitational radiation have been completely solved in [5]. We summarize here the main results relevant to this paper.

1524 v. PIERRO and ~. M. PINTO

According to the PM model, the orbital period evolves according to [8]

(A.1) eo ~ - e'~ ] [ 1 + (121/3o4)eg '

where To is the initial period, and eo the initial eccentricity. On the other hand, the evolution of orbital eccentricity is ruled by the following equation [5]:

(A.2) dde0 _ 4819F(e~)e-29/19(1 - ee)a/2 ( 1+ --e2304121) 1181/2299 '

where

(A.3) F(x) x24/1~~Fl(243 1 1 8 1 4 3 1 2 1 ) = - - - - - , - - ; x, - - - x . 19 ' 2 ' 2299 19 304

F~ is the Appel hypergeometric function, 0 = t /Te, and T~ is the binary lifetime, which turns out to be [5]

(A.4)

t 121 :3480/2299 T~ = T~(1- e~ 1+ 304 e~) ,

( 121 2t F~ 24 3 1181 --,43"e~, - eo , 19' 2 ' 2299' 19 3 ~ ]

TABLE III. - Pad~ coeScients c~,1, fij> relevant to eq. (A.7).

q aq

1.000000000000000 3.921538106840176 5.924580997538873 4.316016227481374 1.553597118197367 2.580370538488808.10 Ol 1.785020146795427.10 oz

P /%

4.674674581748377 8.898707726220437 8.879493674733567 4.983029168755820 1.567813009804013 2.582367452151288.10 ol 1.781322070711866.10 -o2 1.985994588046824.10 -~

GRAVITATIONAL-WAVE CHIRPS: ETC. 1525

Tc being defined by eq. (8). Equation (A~3) admits the following exact solution[5]:

t F(e 2 ) (A.5) Te - 1 r(e~ 2) ,

which can be formally inverted in a neighbourhood of t = T~ as follows:

(A.6) e2 ~-e m=ldm Te "

The partial sums of (A.6), however, become rapidly inaccurate as the initial eccentricity is increased, i.e. for t ~ 0. A substantial improvement in accuracy is obtained by recasting (A.6) into a Pad6 approximant [9]. The handy approximant [7, 8]

7 [ [ t \ ](19/24)q )

( t )~. . . . . . . . . q~=laq[F(e2)(1-- ~)J 11/2 s [ ~ / t ~](19/24)p '

(A.7) e -~e l +p~lflp[F(e~ 1 - -~e)]

whose coefficients have been collected in table III , provides at least 8 accurate decimal figures, throughout the whole range e ~< eo ~< 0.9, by comparison with eq. (A.6)[5].

R E F E R E N C E S

[1] LEVINE B., Radiotechnique statistique (MIR) 1973. [2] CUTLER C. et al., Phys. Rev. Lett., 70 (1993) 2984. [3] PIERRO V. and PINTO I. M., Phys. Left. A, 173 (1993) 121. [4] KROLAK A., in Gravitational Wave Data Analysis, edited by B. SCHUTZ (Pergamon Press)

1986. [5] PIERRO V. and PINTO I. M., Nuovo Cimento, B, 111 (1996) 331. [6] MISNER C. W., THORNE K. and WHEELER J. A., Gravitation (Freeman) 1973. [7] ABRAMOWITZ M. and STEGUN I. A., Tables of Functions (Dover) 1965. [8] PETERS P. C., Phys. Rev. B, 136 (1964) 1224. [9] CUYT A., Padg Approximants (Springer-Verlag) 1980.