Templates for M* BHs spiraling into a MSM BH B S Sathyaprakash and B F Schutz Cardiff University and AEI independently by C Cutler and L Barak AEI 23 July 2002 LISA Sympsium, Penn State

Issues concerning templates Waveform families Approximations Test mass Vs comparable mass models Analytical Vs numerical schemes Post-Newtonian orders From source calculations to detector response Extraction of waveform in different directions Doppler modulation due to motion of the detector relative to the source Number of independent templates Crude estimates using counting arguments Rough estimates from covariance matrix (principal component analysis to determine the number of independent parameters) Accurate estimates using geometric approach and/or Monte-Carlo simulations Template placement algorithms LISA Sympsium, Penn State 23 July 2002

LISA Sympsium, Penn State Source Massive BH with a stellar mass BH companion 107 M8 > M1 > 104 M8, M2 < 100 M8 eccentric orbit both bodies spinning random orientation unknown direction arbitrary initial phase LISA Sympsium, Penn State 23 July 2002

LISA Sympsium, Penn State Waveform 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 order the model has all the 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 approximation good enough? Is the effect of spins at lowest orders sufficient? What is the effect of higher order post-Newtonian effects, etc. LISA Sympsium, Penn State 23 July 2002

LISA Sympsium, Penn State ?? Typical ?? Waveforms LISA Sympsium, Penn State 23 July 2002

Number of templates - Simple Counting 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 x Duration = 10-3 Hz x 1yr = 3 x 104 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 (105 Cyc)6 x (102 Cyc)9 = 1048 Templates LISA Sympsium, Penn State 23 July 2002

Principal Component Analysis familiar 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 G km = [g -1 ]km define variance-covariance matrix by C kk= G kk , if k = m C km= G km / (G kk G 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 LISA Sympsium, Penn State 23 July 2002

Number of templates - Principal Component 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 is N = (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 LISA Sympsium, Penn State 23 July 2002

Hierarchical Search Strategies Two-step hierarchical search Number of templates along each principal parameter goes does 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 component These and other hierarchical searches should bring down the number of templates at least by a factor of 10 N = 1020 LISA Sympsium, Penn State 23 July 2002

Covariance Matrix And the Metric Owen 96; Owen and Sathyaprakash 98 Signal Model: (Kidder, Apostolatos et al) h(t) = -A(t) cos [2F(t ) + f(t ) + df(t )] A(t, m1, m2, N, L, S1, S2) = Amplitude modulation F(t , m1, m2, tc, fc) = Inspiral phase carrier signal f(t, m1, m2, N, L, S1, S2) = Phase modulation df(t, m1, m2, N, L, S1, S2) = Thomas precession Use Numerical derivatives to compute the covariance matrix Sample at points where precessional effects could be greatest The metric on the signal manifold is defined by C = 1 - gkm dp k dp m, where gkm = (hk , hm) Choosing a minimal match of MM the number of templates N g 1/2 dp n/ (1 - MM) n/2 LISA Sympsium, Penn State 23 July 2002

Why are there only three/four 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 binary’s angular momentum alone cannot cause precession Component of large body’s spin in the orbital plane is important 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? rest of the parameters are possibly not very relevant LISA Sympsium, Penn State 23 July 2002

LISA Sympsium, Penn State Ongoing 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. LISA Sympsium, Penn State 23 July 2002