Hierarchical Search for MBH mergers in Mock LISA data  Search for tracks in time-frequency plane; get first estimates of chirp mass, reduced mass, and coalescence time.  Set of (ever-more refined) grid-based searches in neighborhoods identified in step 1: better estimates of chirp mass, reduced mass, and coalescence time.  Final estimation for all params: Markov Chain Monte Carlo, using previous-stage results to reduce “burn-in” time. --D. Brown, J. Crowder, C. Cutler, S. Fairhurst, Ilya Mandel, Patrick Sutton, Michele Vallisneri 3-Stage Search:

Spectrogram of Training Set for Challenge w/ Matlab, after decimating by factor 16-- Stage 1: search for track in f-t plane

Iya Mandel wrote Matlab code to find smooth curve that best fits the f-t data, and extract chirp mass, mass ratio, and coalescence time.  First pass, uses equal time spacings  2nd pass, adjusts to get shorter as approach  Finds by fitting parabola to 3 freq bins near max  Does least squares fit to

Example: Challenge Training Set For Blind Challenge data, he got to ~0.1%, to ~10%, and to ~2K sec (~4 cycles). Definitely good enough for our purposes!

Stage 2: grid search using 2PN SPA templates (w/o LISA modulations)  Uses Duncan Brown’s C matched filtering code, swig-wrapped in Python to compute correlations with 2PN SPA templates.  Also uses Sathyaprakash’s LAL code for constructing template grid on (M1-M2) parameter space.  Basic idea is to calculate and look for maximum of |S|.

Stage 2: grid search (cont’d) Current strategy is quite ad hoc, but seems to work:  Start with a coarse grid around the params from the f-t stage  Find best fit, repeat with finer grid over smaller patch  Repeat 4 more times with ever finer grid on ever smaller patch  Do one final grid search with FF=0.995, using Challenge-1 templates constructed by Synthetic LISA (for arbitrarily chosen point on sky and other angles), instead of standard 2PN/SPA templates. These templates don’t get the LISA modulations right (since angles are wrong), but have “right” phase evolution and “right” taper at end. We find this iterative scheme gets chirp mass to 0.01%, reduced mass to few percent, and coalescence time to w/in ~ 100sec.

Stage 3: Markov Chain Monte Carlo (J.Crowder) Used Metropolis-Hastings algorithm to find best fit to two orthogonal channels constructed from the TDI variables: X and (X + 2Y)/. Started MCMC chains in right neighborhood, found by 2nd stage. We plan to optimize the F-statistic, which yields the best fit over automatically/analytically. But couldn’t quite implement that in time, so instead did brute force search over, and then inferred D from the SNR.

Stage 3: MCMC (cont’d)  Started multiple chains going on JPL’s supercomputer  Started ~100 chains at masses, coal.time from 2nd stage  Ran for ~10 hours, or 3500 steps in each chain  Used only small jumps (< 1%) for the masses, but a mixture of large and small jumps for the angles.  Most promising chains from 1st run were continued onto a 2nd run.

How well did we do on 1.2.1? (Our code wasn’t really ready, but we submitted our best fit anyway.) We found a secondary maximum with SNR= True signal SNR = Our fit was excellent near end, not very good at beginning. Overlap with true signal =

Our parameter estimation accuracy for 1st blind BBH challenge Re sky position, there’s a near degeneracy between opposite points on the sky, and we’re sitting on the wrong one (actually ~3 degrees off from the exact opposite point). Our extrinsic params were pretty far off, in a way that still gave a pretty good fit.

What we’ll improve very soon: We’ll calculate F-statistic for each. This automatically maximizes over extrinsic params. Also still have to finish some minor modifications to grid search to handle case where coalescence happens AFTER data ends.