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 ai bj Jenet et al 04 Hill & Benders 1981 amplitude slope Phinney 01 Jaffe & Backer 03 Wyithe & Loeb 03 j i ab GWB spectrum geometry

3. Pulsar Timing Array C= C (A, n)+noise +noise +…+noise ai 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 where X=(timing residuals) – (quadratic spidown) a a a a ai

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 0i i i j jj 00

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: b =(b + … +b )/N 2 2 2 1 N Error = b/N 0.5

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

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

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

16. Complication: white noise + jump a

18. Pulsar observer: fit for a Lazy Bayesian Theorist: 1.Find P(a,b|data) 2.Integrate over a

19. Complication: white noise + jump a Pulsar observer: fit for a Lazy Bayesian Theorist: 1.Find P(a,b|data) 2.Integrate over a ANALYTICAL!

20. 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!

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 ….