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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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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


(to lowest order)

natural inner product:

Vector space of all possible signals


(to lowest order)

Vector space of all possible signals


Remarks on scalings
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

LISA error boxes for MBHBs

cf. Lang&Hughes, 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?


We want to evaluate

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

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

…so we evaluated

using above substitutions and approximations.

Check: is linear approx self-consistent? I.e., is

No.

?


Back to the drawing board

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

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

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 d1

Define the MATCH

Between two waveforms by:

ODE Method (cont’d)

Then we

always find:

despite the fact

that “initial” match

is always low:


One step method

One-step Method

use

approx by value,

which is

implies

then approximate

using ave. values


Comparison of our 2 quick estimates

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

Why the improvement?A close analogy:

say

Two Taylor expansions:

reliable << 1 cycle

reliable as long as


Actually considered 2 versions of

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

(noise errors scaled to SNR = 1000)

(Crude) Summary of Results

Mass errors:

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

Summary


Future work

  • Improve model of MBH waveforms size (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).

Future Work


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