Parameter Estimation for Gravitational Waves from Compact Binaries

ChungleeKim

(WestVirginiaUniversity)

APCTP International School on Numerical Relativity and Gravitational WavesAugust 2, 2011, Pohang, Korea

Outline

Lecture 2.

Examples of parameter estimation (LIGO/Virgo mock data challenge)

Lecture 1.

IntroductionCompact Binary CoalescencesBayesian inferenceParameter estimation (Markov Chain Monte Carlo method)

Introduction

theoretical modeling : constructing mathematical expressions to describe phenomena based on our understanding of the nature

experiment/observation: (re)producing/measuring/detecting phenomena (fully or partially) in a controlled environment

In this lecture series, we will discuss

“How to interprete an observation (data)?”

“How to measure physical quantities relevant to compact binary inspirals from the LIGO/Virgo data?”

Modeling, observation, and data analysis

theoretical modeling : constructing mathematical expressions to describe phenomena based on our understanding of the nature

experiment/observation: (re)producing/measuring/detecting phenomena (fully or partially) in a controlled environment

In this lecture series, we will discuss

“How to interprete an observation (data)?”

“How to measure physical quantities relevant to compact binary inspirals from the LIGO/Virgo data?”

Modeling, observation, and data analysis

GW waveform, source characteristics

GW signals hidden in detector noise

Goal:Calculate “posteria” probability density functions (PDFs) of physical quantities relevant to compact binary inspiralsthat best match with a “detected” GW signal in terrestrial interferometers

Inference (n.) a conclusion reached on the basis of evidence and reasoning.

parameter (n.) a numerical or other measurable factor forming one of a set that defines a system or sets the conditions of its operation.

Definitions from the Oxford dictionaries

Stretagy: Bayesian inference. “parameter estimation”

Parameter estimation can provide useful information for

1. source identification

2. feedback to astrophysical models of GW sources

3. EM follow-up observations

parameter estimation is most useful when GWs are actually detected

see, last week’s lectures for theoretical aspectsAndo, Chen, Landry’s lectures for detectors

Brown, Koranda, Marion’s lectures for detection pipelines

Target to search: “GWs from inspiral-merge-ringdown phases of compact binaries consisting of a NS or stellar-mass BH”

Compact Binary Coalescences (CBCs)

fgw,CBC ~ typically a few 100 Hz

NS-NS, NS-BH, BH-BH

simple, well-defined waveforms (Post-Newtonian approximation is available)

advanced LIGO/Virgo & LCGT will be able to detect GWs from these sources

non-spinning, standard “chirp” waveform

effects of spin

Fairhurst et al. (2009)

GW detectors’ sensitivity in 2006-2007

Compact Binary Coalescences (CBCs)

Types of CBCs

1. NS-NSdominated by inspiral, PN waveforms, negligible spin, narrow mass domain (~1-3 solarmass), simplest to analyze, existence is confirmed by radio pulsar observations

2. NS-BHNS mass/structure, moderate/low BH spins (PN still works), contribution of merging+ringdown

3. BH-BH

BH spins important, great testbeds for full GR, large h (GW amplitude) due to heavior binary components (“brightest sources among CBCs”), optimal detection distance (LIGO + Virgo) DBH-HB > 2 Gpc

(1 parsec = 1pc=3 1013 km)

Rdetection=0.4-400 per yr

Rdetection=0.4-300 per yr

Rdetection=0.4-1000 per yr

Rdetection= 1 event per yr up to a few events per day

Bayesian inference

Bayesian Inference

Richard Price published Bayes’ note, “An essay towards solving a problem in the doctrine of chances” in 1763

rev. Thomas Bayes (1701(?)–1761)

“In the Bayesian framework, probability is a measure of the degree of belief about a proposition”

applications of Bayesian framework in astrophysics and cosmology:

cosmic microwave background, mass of an extra–solar planet, abundance of dark matter in the Universe, and GW source characteristics

Trotta (2009)

Bayesian vs Frequentist

p(Rupper) = p(Rlower)

! Rupper

Rlower

p(R)dR

where

credible range (%) =

In Bayesian inference, a credible range (in %) is the range of parameters corresponding to a given probability.

Under the Frequentist point of view, a confidence interval indicates that one would observe a value within this interval at a given freqeuncy (or probability).

Rtrue

Rlower Rupper

If a confidence interval for the length of a desk is given at 95% probability, one should expect that 95 out of 100 measurements will have values within the given confidence interval. A credible region, however, represents a range of parameters within which the true value is included at the given probability.

Bayes’ theorem

H: hypothesis (“There is a GW signal embbeded in the LIGO/Virgo data”)d: dataI : information we already know (priors, often expressed as or )!" !"

Bayes’ theorem

p(H|d, I) : posterior distribution. This is what we want to calculate. It provides a “probability” of the existence of a GW signal that is matched with our template in a credible interval (cf) confidence interval

p(d|H,I) : likelihood that describes observations/data (ex: Poisson, Gaussian). conditional probability to obtain a data d given H and I

p(H|I) : assumed prior distribution of model parameter(s)

p(d|I) : normalization constant. “Bayesian evidence”

H: hypothesis (“There is a GW signal embbeded in the LIGO/Virgo data”)d: dataI : information we already know (priors, often expressed as or )!" !"

posterior ! prior" likelihood

Bayesian evidence and model selectionBayes’ theorem

posterior =prior! likelihood

normalization

How to compare different model waveforms A and B?(ex) spining vs non-spinning waveforms for a CBC

Bayesian evidence ranks a “quality” of a model (waveforms), so that one can select which one best describes the data (a GW signal candidate)

Parameter estimation for GWs from CBCs

- Markov Chain Monte Carlo algorithm

A detection can be determined by a sinal-to-noise ratio (SNR)

SNR (inspiral only)

GW amplitude(frequency domain)

h(f) = h(f | D, M1, M2)~ ~

CBC parameter space is complex and multi-dimensional

9 (non-spinning, circular), 12 (spinning, circular), and 17 parameters (a spinning black-hole binary in an eccentric orbit)marginalized posterior pdf

detector noise

{!"|i = 0, 1, ...n}

Bayes’ theorem

p(H|d, I) : posterior distribution. This is what we want to calculate. It provides a “probability” of the existence of a GW signal that is matched with our template in a credible interval (cf) confidence interval

p(d|H,I) : likelihood that describes observations/data (ex: Poisson, Gaussian). conditional probability to obtain a data d given H and I

p(H|I) : assumed prior distribution of model parameter(s)

p(d|I) : normalization constant. “Bayesian evidence”

posterior =prior! likelihood

normalization

H: hypothesis (“There is a GW signal embbeded in the LIGO/Virgo data”)d: dataI : information we already know (priors, often expressed as or )!" !"

Markov chain Monte Carlo sampling algorithm

Markov chain: discrete, random, stochastic threads of process that only depends on the previous process (examle: Brownian motion, druken person’s walk)

Monte Carlo simulation : independent sampling based on a distribution with many iterations (N > 1e6)

Markov chain Monte Carlo (MCMC) sampling algorithm

“to construct a sequence of points in parameter space (called “a chain”), whose density is proportional to the posterior pdf” Trotta (2006)

parameter spacechain 1

chain 2

Initial values are randomly chosen.Each chain (a sequence of updated likelihoods, given a parameter set) becomes independent of the choice of the starting point.All chains should converge after many iterations (N > 1e6), and the results must be consistent (all chains should end up with the same mode that is most likely)

starting point 1

starting point 2

L1

L2

Feroz and Hobson (2008)

p(!i!1, !i)p(!i) = p(!i, !i!1), p(!i!1) detailed balance

(p(!i), p(!i!1)) = (p(!i!1), p(!i)) convergence condition

detailed balance

chain calculation

p : pdf implied by a jump probability~

“reversable”

Example of MCMC: Marc van der Sluys’ homepage: http://www.astro.ru.nl/~sluys/index.php?title=SPINspiral

Challenges in parameter estimation

1) efficiency (mapping, local maxima, tempering, jump proposals, burnin)

“How to smartly map out the CBC parameter space in a reasonable computing time in order to find a global maximum in the likelihood surface?”

2) robust (accuracy, consistency)

“Is the result (posterior) consistent between different waveforms/priors?”

Parameter estimation for GWs from CBCs is typically done by numerical methods that are often computationally intensive, requiring high-performance computers

Summary of lecture 1

1. Parameter esimation in Bayesian framework allows one to perform a quantitative analysis of GW signals and to calculate statistical significance of the physical parameters that best match a detected signal.

2. In order to deal with complex, multi-dimensional parameter space of GW signals, an efficient, robust sampling, computing technique (e.g., Markov chain Monte Carlo) is adapted in the GW parameter estimation pipeline.

3. This requires computationally intensive numerical simulations.

4. The efficiency of the parameter estimation technique depends on both sampling methods and computing power.

5. Realistic GW waveforms and priors that can describe different CBCs are prerequisite to improve the accuracy of parameter estimation (e.g., higher-order PN corrections, merger+ringdown contributions) Astrophysics and GR contribution