1. Search for gravitational waves from binary black hole mergers:
parameter estimation and the Fisher matrix
Hee-Suk Cho (KISTI, KOREA)
2nd LeCosPA International Symposium
NTU

2. Outline
Gravitational waveforms from binary black holes (BBHs)
Search and parameter estimation (PE) overview
Fisher matrix for PE uncertainty prediction
Results and Summary

3. Inspiral-Merger-Ringdown stages

4. ==> need a Hybrid model
Constructing hybrid IMR waveform
Theoretical model: fast but inaccurate in MR region
NR simulation: accurate in IMR region but short Inspiral
due to a long computational time
==> need a Hybrid model
arXiv:
Hybrid = Inspiral:PN Merger - Ringdown:NR

5. Hybrid waveform samples
arXiv:

6. Defining IMR waveform model
Hybrid waveform
Fitting function
Fitting function to the hybrid waveforms
=> Analytic IMR waveform model

7. Analytic IMR models for BBHs
Binary System
Function Domain
PhenomA
NoSpin
Frequency
PhenomB
AlignedSpin
PhenomC
PhenomD
PhenomP
PrecessingSpin
Frequency-domain models are much faster than time- domain models in generation time
==> fast data analysis (search and parameter estimation)

8. Search method: Matched filtering
Matched filtering is the optimal filter for a signal of known shape in the stationary Gaussian noise
1) Known signal ? BBH waveforms are well modeled as IMR
2) Gaussian noise ? Average of real noises is roughly Gaussian But, it often gives false alarms, various additional filters applied to reduce false alarm rate

9. Detector noise sensitivity (Sn)
Matched filtering
Detector output (d)
Model waveform (T)
Detector noise sensitivity (Sn) or ⊗ /
template bank
matched filters are done at once
==> need computing resources
==> no output
==> strong output

10. Parameter estimation
Search pipeline just identifies the GW event.
We want to know the physical parameters of the source.
Once a detection is made, parameter estimation (PE) analysis is implemented.
PE pipeline explores the whole parameter space.
Number of parameters: 9 for NoSpinning, 12 for one spinning, 15 for two spinning.

11. ⊗ / Overlap ==> 9~15 parameters
Template bank-based analysis is not possible (search:2-3 parameters)
Continuous overlaps varying all parameters until the process converges
10^6 ~ 10^9 iterations, very long computational time

12. PDFs for a PrecessingSpin binary (a2=0)
PE result is given by probability density functions (PDFs)
==> Gaussian distribution

13. Parameter estimation limitation
PE aims to find the parameters of the GW source quickly and accurately when a real GW is detected.
PE performance test is done by using injection signals for various waveform models, various source parameters.
Only one PE test even needs a long computational time
Simple methods?

14. analytical derivatives
Fisher matrix
For known signal parameters with
high Signal-to-Noise Ratio (SNR) in
Gaussian noise,
the statistical errors can be calculated
semi-analytically using the Fisher matrix (FM).
PE performance test on errors can be
easily done for various waveform models,
various binary systems.
PDF
one numerical overlap
analytical derivatives

15. But semi-analytic, easily applicable to various
PE vs. FM
PE: no assumptions
Unknown signal, Real noises, explore whole
parameter space, long time
FM: with assumptions
Known signal parameters, high SNR, Gaussian noise,
only statistical errors
But semi-analytic, easily applicable to various
waveform models, detector models, binary models.

16. FM accuracy: NoSpin binary (10, 1.4) Msun
waveform1
waveform2 PE
noise1
noise2
noise3
noise4
90 % confidence region
SNR=20
FM result agrees well with PE errors !!

17. Result I: NoSpin BBHs - PhenomA model
Advanced LIGO detector sensitivity
SNR=20

18. Result II: AlignedSpin BBHs - PhenomC model
Strong correlation

19. Summary
Merging BBH systems suffer Inspiral-Merger-Ringdown phases, various IMR models have been developed.
In data analysis, search pipeline used matched filtering method, model waveforms are stored in the template bank.
Parameter estimation analysis continuously computes overlaps varying all parameters, so needs a very long computational time.
Fisher matrix is a simple semi-analytic method, used to predict the statistical error in parameter estimation for known source parameters of a song signal in Gaussian noise.
PE errors for NoSpin BBHs (<30Msun) ~0.2% for , ~2% for , errors for are much larger for AlignedSpin BBHs due to strong correlation between the mass ratio and spin

20. Thanks !!