23 July 2002 LISA Sympsium, Penn State

Templates for MTemplates for M** BHs spiraling into a M BHs spiraling into a MSMSM BHBH

B S Sathyaprakash and B F SchutzCardiff University and AEI

independently by

C Cutler and L BarakAEI

23 July 2002 2LISA Sympsium, Penn State

Issues concerning templatesIssues concerning templates

Waveform families Approximations Approximations

Test mass Vs comparable Test mass Vs comparable mass modelsmass models

Analytical Vs numerical Analytical Vs numerical schemesschemes

Post-Newtonian ordersPost-Newtonian orders From source calculations to

detector response Extraction of waveform in Extraction of waveform in

different directionsdifferent directions Doppler modulation due to Doppler modulation due to

motion of the detector motion of the detector relative to the sourcerelative to the source

Number of independent templates Crude estimates using Crude estimates using

counting argumentscounting arguments Rough estimates from Rough estimates from

covariance matrix (principal covariance matrix (principal component analysis to component analysis to determine the number of determine the number of independent parameters)independent parameters)

Accurate estimates using Accurate estimates using geometric approach and/or geometric approach and/or Monte-Carlo simulations Monte-Carlo simulations

Template placement algorithms

23 July 2002 3LISA Sympsium, Penn State

Massive BH with a stellar mass BH companion 10107 7 M > M > M11 > 10 > 104 4 M, M, M22 < 100 < 100 M

eccentric orbit both bodies spinning random orientation unknown direction arbitrary initial phase

SourceSource

23 July 2002 4LISA Sympsium, Penn State

Waveform FamiliesWaveform Families

For computing number of templates it suffices to employ the simplest possible model that includes all the parameters of the source and all the dominant modulation effects: By second post-Newtonian By second post-Newtonian

order the model has all the order the model has all the parameters of the source parameters of the source

distance direction (2) masses (2) initial angular momentum (3) initial spins (6) initial eccentricity initial phase instant of merger

Use high-post-Newtonian order analytical results in the test mass approximation to explore the parameter space and identify important parameters and non-linear effects

For instance, is test mass For instance, is test mass approximation good approximation good enough? enough?

Is the effect of spins at Is the effect of spins at lowest orders sufficient? lowest orders sufficient?

What is the effect of higher What is the effect of higher order post-Newtonian order post-Newtonian effects, etc.effects, etc.

23 July 2002 5LISA Sympsium, Penn State

?? Typical ?? Waveforms?? Typical ?? Waveforms

23 July 2002 6LISA Sympsium, Penn State

Number of templates - Simple Number of templates - Simple Counting ArgumentsCounting Arguments

There are totally 17 parameters No templates are needed to search for instant of

coalescence and distance to the source A rough estimate for the number of templates in

each parameter direction is: Band width Band width xx Duration = Duration = 1010-3-3 Hz Hz xx 11yr = yr = 3 x 103 x 1044

If each parameter direction requires 3 x 104 templates 15 source parameters require 1067

templates Phinney and Thorne distinguish between amplitude-

type (9) and phase-type (6) parameters and get (10(105 5 Cyc)Cyc)6 6 x (10x (102 2 Cyc)Cyc)9 = 9 = 101048 48 TemplatesTemplates

23 July 2002 7LISA Sympsium, Penn State

Principal Component AnalysisPrincipal Component Analysisfamiliar to many in GW communityfamiliar to many in GW community

given a signal h(t,p) compute the information matrix

gkm = (hk , hm)

where (a, b) denotes the inner product of vectors a and b defined by matched filtering and a subscript denotes derivative of the signal w.r.t. parameter p k

inverse of the information matrix is the covariance matrix

km = [g -1 ]km

define variance-covariance matrix by

C kk= kk , if k = mC km= km

/ (kk mm) 1/2, if k != m

non-diagonal elements lie in the range [-1,1]

if |C km | ~ 1 means that the parameters are correlated:

diagonalize, principal components are the largest eigenvalues

number of nearly equal large components gives the effective dimensionality of the parameter space

Applying this to non-spinning BH binaries automatically shows that there is only 1 ind. param as opp. 4

23 July 2002 8LISA Sympsium, Penn State

Number of templates - Principal Number of templates - Principal Component AnalysisComponent Analysis

Note that diagonlization takes us to a new set of parameters (in the geometrical language a new coordinate system) that is related to the set of physical parameters via a linear transformation; but the system is not ingtegrable

Principal component analysis (preliminary and limited study) shows that

three or four parameters are most important and others

are probably not of significance

Conjecture: number of templates required isN = (3 x 104 )p x 10q x 2n-p-q

where p is the number of principal components, q is the number of subsidiary components and n-p-q is the number of least important components

For p = 4, q = 4, n =15, N = 1024

The actual number is likely to be much smaller

23 July 2002 9LISA Sympsium, Penn State

Hierarchical Search StrategiesHierarchical Search Strategies

Two-step hierarchical search Number of templates along each principal parameter goes Number of templates along each principal parameter goes

does down by a factor of 5does down by a factor of 5 Interpolation on top of hierarchical search (theory of

quasi-band limited signals, Pinto et al 00, 01, 02) A gain by a factor of 2 for each principal componentA gain by a factor of 2 for each principal component

These and other hierarchical searches should bring down the number of templates at least by a factor of 10

N = 1020

23 July 2002 10LISA Sympsium, Penn State

Signal Model: (Kidder, Apostolatos et al)h(t) h(t) == -A(t) -A(t) cos [cos [22t t )) + + t t )) + + ((t t )])]

A(t, m1, m2, A(t, m1, m2, N, L, S1, S2N, L, S1, S2) ) == Amplitude modulation Amplitude modulation t , m1, m2, tc, t , m1, m2, tc, cc) = ) = Inspiral phase carrier signalInspiral phase carrier signal t, m1, m2, t, m1, m2, N, L, S1, S2N, L, S1, S2)) == Phase modulation Phase modulation ((t, m1, m2, t, m1, m2, N, L, S1, S2N, L, S1, S2) = ) = Thomas precessionThomas precession

Use Numerical derivatives to compute the covariance matrix Sample at points where precessional effects could be greatestSample at points where precessional effects could be greatest

The metric on the signal manifold is defined byC C == 1 - 1 - gkm dp dp kk dp dp mm, , where g gkmkm == (h (hk k , h, hmm))

Choosing a minimal match of Choosing a minimal match of MMMM the number of templates the number of templates NN

g g 1/2 1/2 dpdp n n/ (1 - MM) / (1 - MM) n/2 n/2

Covariance Matrix And the Covariance Matrix And the MetricMetric

Owen 96; Owen and Sathyaprakash 98Owen 96; Owen and Sathyaprakash 98

23 July 2002 11LISA Sympsium, Penn State

Why are there only three/four Why are there only three/four independent parameters?independent parameters?

Of the two masses only chirp mass or chirp time is relevant

eccentricity is a major player Of the two direction-cosines

only co-latitude is dominant

Component of small body’s spin in the orbital plane and perpendicular to the plane containing the primary’s spin and AM are important - but secondary effects?

Component of large body’s spin in the orbital plane is important

binary’s angular momentum alone cannot cause precession

rest of the parameters are possibly not very relevant

23 July 2002 12LISA Sympsium, Penn State

Ongoing workOngoing work the analysis is highly time consuming since the

parameter space is huge; an MPI code capable of running on a cluster is being written

however, analytic solution is desirable even if some simplification is necessary

when the masses are equal or if the spin of stellar mass component is zero analytic solution to the waveform evolution exists; this is the so-called simple precession (Apostolatos, Cutler, Sussman and Thorne)

this can be used to derive the information matrix analytically which should then give a good handle on the metric and the number of templates.