Search for gravitational waves from binary black hole mergers: parameter estimation and the Fisher matrix Hee-Suk Cho (KISTI, KOREA) 2nd LeCosPA International Symposium 2015.12.14-NTU

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

Inspiral-Merger-Ringdown stages

==> 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:0704.3764 Hybrid = Inspiral:PN + Merger - Ringdown:NR

Hybrid waveform samples arXiv:0710.2335

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

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)

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

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

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.

⊗ / 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

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

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?

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

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.

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 !!

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

Result II: AlignedSpin BBHs - PhenomC model Strong correlation

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

Thanks !!