national aeronautics and space administration jet propulsion laboratory

National Aeronautics andSpace Administration

Jet Propulsion LaboratoryCalifornia Institute of TechnologyPasadena, California

CC & M. Vallisneri, PRD 76, 104018 (2007); arXiv: 0707.2982

LISA detections of Massive BH Binaries: parameter estimation errors from inaccurate templates

€

h(θ tr )€

h(θbf )

€

n€

s

Vector space of all possible signals

€

€

manifold h(θ i)€

•

€

•

(to lowest order)

€

(s1 | s2) ≡ 2˜ s 1

*( f )˜ s 2( f )df

Sh ( f )∫

natural inner product:

Vector space of all possible signals

€

hAP (θ tr )

€

hGR (θ tr )

€

hAP (θbf )

€

manifold hAP (θ i)€

manifold hGR (θ i)

€

•

(to lowest order)

€

Δ thθi

€

•

Remarks on Scalings

€

Δ nθi∝ SNR−1

€

Δ thθi is independent of SNR

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:

€

~ 106 Msun

€

Δ(ln M1) ~ Δ(ln M2) ~ 10−3

€

Δθsky ~ 0.1−1o

€

Δ(lnS1) ~ Δ(ln S2) ~ 10−3 −10−2

€

Δ(lnDL ) ~ 10−3 −10−2

cf. Lang&Hughes, gr-qc/0608062

Will need resolution to search for optical counterparts

€

~ 5o

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).

€

hGR (θbf )

€

hAP (θbf )

But we don’t know !

€

hGR (θbf )

Since PN approx converges slowly,we adopt the substitute:

€

hGR (θ) − hAP (θ)⇒ h3.5PN (θ) − h3PN (θ)

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

€

β

€

σ

…so we evaluated

€

Δθthi

using above substitutions and approximations.

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

€

hAP (θbf )− hAP (θ tr ) ≈ Δθ i∂i hAP ? No.

Back to the drawing board:

Recall our goal was to find the best-fit params, i.e., the values that minimize the function

€

θbfi

There are many ways this minimization could be done, e.g.,using the Amoeba or Simulated Annealing or Markov ChainMonte 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 ifonly had been smaller. That would have happenedif only the difference were smaller. Thissuggests finding the best fit by dividing the big jump intolittle steps:

€

Δθthi

€

hGR (θ) − hAP (θ)

€

hAP

| | | | | | | | | | …….| | | | | | | | | |

€

hGR

€

hλ →

€

θbf

€

θtr

€

θλ →

ODE Method (cont’d)

where

and

Integrate from to , with initial condition ;

arrive at .

€

λ =0

€

λ =1

€

θ |λ = 0= θbf

€

θ |λ =1= θ tr

Actually, this method is only guaranteed to arrive at a localbest-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:

€

< 0.50

One-step Method

approx by value,

which is

€

λ =0use

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, butfor 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:

€

f (x) ≡ e ig(x )say

Two Taylor expansions:

€

f (x) ≈ f (x0) + f '(x0) ⋅(x − x0)

€

f (x) ≈ exp[i{g(x0) + g'(x0) ⋅(x − x0)}]

reliable << 1 cycle

reliable as long as

€

g' '(x0) ⋅(x − x0)2 <<1

Actually, considered 2 versions of

plus hybrid version:

€

hGR (θ) − hAP (θ)⇒ h3.5PN (θ) − h3PN (θ)

€

hGR (θ) − hAP (θ)⇒ h3.5PN (θ) − h3PN* (θ)

Hybrid waveforms are basically waveforms that have been improved by also adding 3.5PN terms that are lowest order in the symmetric mass ratio .

€

h3PN*

€

h3PN

€

ηMotivation: lowest-order terms in can be obtained to almostarbitrary accuracy by solving case of tiny mass orbiting a BH,using BH perturbation theory. Such hybrid waveforms firstdiscussed in Kidder, Will and Wiseman (1993). €

η

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

(Crude) Summary of Results

Mass errors:

€

Δ n Mc

Mc

~ 10−5 −10−4

€

Δ th* Mc

Mc

~ 2 ×10−5 − 2 ×10−4

€

Δ th Mc

Mc

~ 10−3 −10−2

€

Δ nη

η~ 10−3 −10−2

€

Δ th* η

η~ 3×10−3 − 3×10−2

€

Δ thη

η~ 2 ×10−1 − 5 ×10−1

(noise errors scaled to SNR = 1000)

Sky location errors:

€

Δ nθsky ~ 1o

€

Δ th* θ sky ~ 0.1o

€

Δ thθ sky ~ 1o

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).