National Aeronautics and Space Administration Jet Propulsion Laboratory California Institute of Technology Pasadena, California. LISA detections of Massive BH Binaries: parameter estimation errors from inaccurate templates. CC & M. Vallisneri, PRD 76, 104018 (2007); arXiv: 0707.2982.
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Jet Propulsion Laboratory
California Institute of Technology
LISA detections of Massive BH Binaries: parameter estimation errors from inaccurate templates
CC & M. Vallisneri, PRD 76, 104018 (2007); arXiv: 0707.2982
natural inner product:
Vector space of all possible signals
Vector space of all possible signals
so theoretical errors become relatively more important
at higher SNR.
One naturally thinks of LISA detections of MBH mergers,
c.f. E Berti, Class. Quant. Grav. 23, 785 (2006)
cf. Lang&Hughes, gr-qc/0608062
for pair of BHs merging at z =1,
SNR~ 1000 and typical errors due to noise are:
Will need resolution to search
for optical counterparts
But how big are the theoretical errors?
where is true GR waveform and is
our best approximation (~3.5 PN).
But we don’t know!
using above substitutions and approximations.
Check: is linear approx self-consistent? I.e., is
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
But these are fairly computationally intensive, so we
wanted a more efficient method.
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
| | | | | | | | | | …….| | | | | | | | | |
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.
Between two waveforms by:
despite the fact
that “initial” match
is always low:
approx by value,
using ave. values
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%.
Two Taylor expansions:
reliable << 1 cycle
reliable as long as
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
Sky location errors:
of parameter estimation errors due to inaccurate templates:
-- ODE method
-- one-step method (2nd, improved version)
(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
theoretical uncertainties (Bayesian approach to models?)
--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).