Bayesian analysis for Pulsar Timing Arrays

Rutger van Haasteren (Leiden)Yuri Levin (Leiden)Pat McDonald (CITA)Ting-Ting Lu (Toronto)

Pulsar Timing Array

• GW timing residuals:

Multidimentional Gaussian process;

coherence matrix

C (A, n)=G(t –t ) Qai bj

Jenet et al 04Hill & Benders 1981

amplitudeslope

Phinney 01Jaffe & Backer 03Wyithe & Loeb 03

j i ab

GWBspectrum

geometry

Pulsar Timing Array

C= C (A, n)+noise +noise +…+noiseai bj

Real coherence matrix:

1 2 20

Bayesian solution:

• parametrize each pulsar noise reasonably: N exp[-n f]+σ• construct multidimantional probability distribution

• marginalize over quadradic spindowns – analytical• marginalize over pulsar noises – numerical

P(A,n, N ,n | data)=exp[-X (2C) X - (1/2) log(det{C})] (Prior/Norm) x x-1

where X=(timing residuals) – (quadratic spidown)

a

a

a

a

ai

Markov Chain Monte-Carlo

• Direct integration unrealistic• Markoff chain cleverly explores parameter

space, dwelling in high-probability regions• Typically need few 10000 points for reliable

convergence• Can do white pulsar noises-last year’s talk

• However, problems with chain convergence when one allows for colored

pulsar noises. Need something different!

Maximum-likelihood method

• Find global maximum of log[P(A, n, N )]• Run a chain in the neighbourhood until enough points to fit a quadratic form:

log(P)=log(P ) – (p –p ) Q (p – p ) • Approximate P as a Gaussian and

marginalize over pulsar noises

a

0 i i i j j j

0 0

Results

10 pulsars 500 ns, 70 timings each over 9 yr

Results

10 pulsars 500 ns, 70 timings each over 9 yr

Results

10 pulsars 100 ns, 70 timings each over 9 yr

Results

10 pulsars 100 ns, 70 timings each over 9 yr

Results

10 pulsars, 50 ns timing error, 5 years, every 2.5 weeksA=10 E-15 n=-7/3

our algorithm:

• Does not rely on estimators – explores the

full multi-dimensional likelihood function• Measures simultaneously amplitude AND

slope of the gravitational-wave background• Deals easily with unevenly sampled data,

variable number of tracked pulsars, etc.• Deals easily with systematics-quadratic spindowns,

zero resets, pointing errors, and human errors of known functional form.

Example problem: finding the white noise amplitude b

Pulsar observer:

b =(b + … +b )/N2 2 2

1 N

Error = b/N0.5

Example problem: finding the white noise amplitude b

Bayesian Theorist:

P(data|b)=exp[(b + … +b )/2b -.5 log(b)]2 21 N

2

P(b|data)=(1/K) P(data|b) P (b)0

Example problem: finding the white noise amplitude b

Bayesian Theorist:

P(data|b)=exp[(b + … +b )/2b -.5 log(b)]2 21 N

2

P(b|data)=(1/K) P(data|b) P (b)0

normalizationprior

Example problem: finding the white noise amplitude b

Bayesian Theorist:

b

P

Complication: white noise + jump a

Complication: white noise + jump a

Complication: white noise + jump a

Pulsar observer: fit for a

Lazy Bayesian Theorist:

1. Find P(a,b|data)

2. Integrate over a

Complication: white noise + jump a

Pulsar observer: fit for a

Lazy Bayesian Theorist:

1. Find P(a,b|data)

2. Integrate over a

ANALYTICAL!

Complication: white noise + jump a

Pulsar observer: fit for a

Lazy Bayesian Theorist:

1. Find P(a,b|data)

2. Integrate over a

3. Get expression P(b|data), insensitive to jumps!

Jump removal:

Does not have to be jumps. ANYTHING of knownfunctional form, i.e.:

•Quadratic/cubic pulsar spindowns•Annual variations•Periodicity due to Jupiter•Zero resets•ISM variations, if measured independently

can be removed analytically when writing down P(b).

Don’t care if pre-fit by observers or not.

Pulsar Timing Array

C (A, n)ai bj

Bayesian analysis:

compute P(A,n| data), after

“removing” unwanted components of known functional form

easy

Pulsar Timing Array

C (A, n)ai bj

Bayesian analysis:

compute P(A.n| data), after

“removing” unwanted components of known functional form

Complication 1: low-frequency cut-off

Pulsar Timing Array

C (A, n)ai bj

Bayesian analysis:

compute P(A.n| data), after

“removing” unwanted components of known functional form

Complication 1: low-frequency cut-off

Pulsar Timing Array

C (A, n)ai bj

Bayesian analysis:

compute P(A.n| data), after

“removing” unwanted components of known functional form

Complication 2: pulsar noises, measured concurrentlywith GWs. This is the real difficulty with the BayesianMethod.

Results

Strengths of B. approach• Philosophy

• No loss of info, no need to choose optimal estimator

• No noise whitening, etc. Irregular time intervals, etc.

• Easy removal of unwanted functions

Weaknesses:

• Computational cost

Need better algorithms!

PhD position in Leiden

• Supported by 5-yr VIDI grant

• Collaboration with observers/other theorists essential

• ….