Bayesian analysis for Pulsar Timing Arrays

1. Bayesian analysis for Pulsar Timing Arrays Rutger van Haasteren (Leiden) Yuri Levin (Leiden) Pat McDonald (CITA) Ting-Ting Lu (Toronto)

2. Pulsar Timing Array • GW timing residuals: Multidimentional Gaussian process; coherence matrix C (A, n)=G(t –t ) Q Jenet et al 04 Hill & Benders 1981 ai bj i j ab amplitude geometry GWB spectrum slope Phinney 01 Jaffe & Backer 03 Wyithe & Loeb 03

3. Pulsar Timing Array Real coherence matrix: C= C (A, n)+noise +noise +…+noise 1 20 ai bj 2 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 a a ai -1 P(A,n, N ,n | data)=exp[-X (2C) X - (1/2) log(det{C})] (Prior/Norm) x x a a X=(timing residuals) – (quadratic spidown) where

4. 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!

5. 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 0 0 i i j j i j

6. Results 10 pulsars 500 ns, 70 timings each over 9 yr

7. Results 10 pulsars 500 ns, 70 timings each over 9 yr

8. Results 10 pulsars 100 ns, 70 timings each over 9 yr

9. Results 10 pulsars 100 ns, 70 timings each over 9 yr

10. Results 10 pulsars, 50 ns timing error, 5 years, every 2.5 weeks A=10 E-15 n=-7/3

11. 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.

12. Example problem: finding the white noise amplitude b Pulsar observer: 2 2 2 b =(b + … +b )/N 1 N 0.5 Error = b/N

13. Example problem: finding the white noise amplitude b Bayesian Theorist: 2 2 2 P(data|b)=exp[(b + … +b )/2b -.5 log(b)] 1 N P(b|data)=(1/K) P(data|b) P (b) 0

14. Example problem: finding the white noise amplitude b Bayesian Theorist: 2 2 2 P(data|b)=exp[(b + … +b )/2b -.5 log(b)] 1 N P(b|data)=(1/K) P(data|b) P (b) 0 prior normalization

15. Example problem: finding the white noise amplitude b Bayesian Theorist: P b

16. Complication: white noise + jump a

17. Complication: white noise + jump a

18. Complication: white noise + jump a Pulsar observer: fit for a Lazy Bayesian Theorist: • Find P(a,b|data) • Integrate over a

19. Complication: white noise + jump a Pulsar observer: fit for a Lazy Bayesian Theorist: • Find P(a,b|data) • Integrate over a ANALYTICAL!

20. Complication: white noise + jump a Pulsar observer: fit for a Lazy Bayesian Theorist: • Find P(a,b|data) • Integrate over a 3. Get expression P(b|data), insensitive to jumps!

21. Jump removal:

22. Does not have to be jumps. ANYTHING of known functional 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.

23. Pulsar Timing Array C (A, n) ai bj Bayesian analysis: compute P(A,n| data), after “removing” unwanted components of known functional form easy

24. 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

25. 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

26. 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 concurrently with GWs. This is the real difficulty with the Bayesian Method.

27. Results

28. 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!

29. PhD position in Leiden • Supported by 5-yr VIDI grant • Collaboration with observers/other theorists essential • ….