National Aeronautics and

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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
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(to lowest order)
natural inner product
Vector space of all possible signals
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(to lowest order)
Vector space of all possible signals
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Remarks on Scalings

so theoretical errors become relatively more
important at higher SNR.
One naturally thinks of LISA detections of MBH
mergers, where SNR1000.
c.f. E Berti, Class. Quant. Grav. 23, 785 (2006)
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LISA error boxes for MBHBs
cf. LangHughes, gr-qc/0608062
for pair of BHs merging at z
1, SNR 1000 and typical errors due to noise are
(neglecting lensing)
Will need resolution to search for
optical counterparts
But how big are the theoretical errors?
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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 dont know !
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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 m2)
• Stationary phase approximation for Fourier
  transform
• Low-frequency approximation for LISA response

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

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ODE Method (contd)
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.
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ODE Method (contd)
Define the MATCH Between two waveforms by
Then we always find
despite the fact that initial match is always
low
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One-step Method
use
approx by value, which is
implies
then approximate using ave. values
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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.
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Why the improvement?A close analogy
say
Two Taylor expansions
reliable ltlt 1 cycle
reliable as long as
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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).
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Median results based on 600 random sky positions
and orientations, for each of 8 representative
mass combinations
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(Crude) Summary of Results
(noise errors scaled to SNR 1000)
Mass errors
Sky location errors
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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

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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).