National Aeronautics and Space Administration Jet Propulsion Laboratory California Institute of Technology Pasadena, California CC & M. Vallisneri, PRD 76, (2007); arXiv: LISA detections of Massive BH Binaries: parameter estimation errors from inaccurate templates

Vector space of all possible signals (to lowest order) natural inner product:

Vector space of all possible signals (to lowest order)

Remarks on Scalings so theoretical errors become relatively more important at higher SNR. One naturally thinks of LISA detections of MBH mergers, where SNR~1000. c.f. E Berti, Class. Quant. Grav. 23, 785 (2006)

LISA error boxes for MBHBs for pair of BHs merging at z =1, SNR~ 1000 and typical errors due to noise are: cf. Lang&Hughes, gr-qc/ Will need resolution to search for optical counterparts But how big are the theoretical errors? (neglecting lensing)

We want to evaluate: to lowest order to same order where is true GR waveform and is our best approximation (~3.5 PN). But we don’t know !

Since PN approx converges slowly, we adopt the substitute: Extra simplifying approximations for first-cut application: Spins parallel (so no spin-induced precession) Include spin-orbit term, but not spin-spin (,but not ) No higher harmonics (just m=2) Stationary phase approximation for Fourier transform Low-frequency approximation for LISA response

…so we evaluated using above substitutions and approximations. Check: is linear approx self-consistent? I.e., is ? No.

Back to the drawing board: Recall our goal was to find the best-fit params, i.e., the values that minimize the function There are many ways this minimization could be done, e.g., using the Amoeba or Simulated Annealing or Markov Chain Monte Carlo. But these are fairly computationally intensive, so we wanted a more efficient method.

ODE Method for minimizing Motivation: linearized approach would have been fine if only had been smaller. That would have happened if only the difference were smaller. This suggests finding the best fit by dividing the big jump into little steps: | | | | | | | | | | …….| | | | | | | | | |

ODE Method (cont’d) where and Integrate from to, with initial condition ; arrive at. Actually, this method is only guaranteed to arrive at a local best-fit, not the global best-fit, but in practice, for our problem, we think it does find the global best fit.

ODE Method (cont’d) Define the MATCH Between two waveforms by: Then we always find: despite the fact that “initial” match is always low:

One-step Method approx by value, which is use implies then approximate using ave. values

Comparison of our 2 quick estimates Original one-step formula: Improved one-step formula: The two versions agree in the limit of small errors, but for realistic errors the improved version is much more accurate (e.g., in much better agreement with ODE method). Improved version agrees with ODE error estimates to better than ~30%.

Why the improvement? A close analogy: say Two Taylor expansions: reliable << 1 cycle reliable as long as

Actually, considered 2 versions of plus hybrid version: Hybrid waveforms are basically waveforms that have been improved by also adding 3.5PN terms that are lowest order in the symmetric mass ratio. Motivation: lowest-order terms in can be obtained to almost arbitrary accuracy by solving case of tiny mass orbiting a BH, using BH perturbation theory. Such hybrid waveforms first discussed in Kidder, Will and Wiseman (1993).

Median results based on 600 random sky positions and orientations, for each of 8 representative mass combinations

(Crude) Summary of Results Mass errors: (noise errors scaled to SNR = 1000) Sky location errors:

Summary Introduced new, very efficient methods for estimating the size of parameter estimation errors due to inaccurate templates: -- ODE method -- one-step method (2nd, improved version) Applied methods to simplified version of MBHB mergers (no higher harmonics, no precession, no merger); found: -- for masses, theoretical errors are larger than random noise errors (for SNR = 1000), but still small for hybrid waveforms -- theoretical errors do not significantly degrade angular resolution, so should not hinder searches for EM counterparts

Future Work Improve model of MBH waveforms (include spin, etc.) Develop more sophisticated approach to dealing with theoretical uncertainties (Bayesian approach to models?) Apply new tools to many related problems, e.g.: --Accuracy requirements for numerical merger waveforms? --Accuracy requirements for EMRI waveforms? (2nd order perturbation theory necessary?) --Effect of long-wavelength approx on ground-based results? (i.e., the “Grishchuk effect”) --Quickly estimate param corrections for results obtained with “cheap” templates (e.g., for grid-based search using “easy-to-generate” waveforms, can quickly update best fit).