Coherent Bayesian inference on compact binary inspirals using a - PowerPoint PPT Presentation

Coherent Bayesian inference on compact binary inspirals using a network of interferometric gravitational wave detectors over 1 , Renate Meyer 1 , and Nelson Christensen 2 Christian R¨ 1 The University of Auckland Auckland, New Zealand 2 Carleton College Northfield, MN, U.S.A.

Overview: 1. gravitational waves 2. measuring gravitational waves 3. the binary inspiral signal 4. prior & model 5. MCMC details 6. example application C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 1

Gravitational waves • general relativity: space-time curved by masses • implication: existence of gravitational waves (pointed out in 1916) • existence proven in 1979 • measurement attempted since 1960s • no direct measurement yet C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 2

Gravitational waves • very weak effect • emitted by rapid ly moving, heavy objects • event candidates: – supernovae – big bang – binary star systems – . . . C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 3

C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 4

C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 5

time C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 6

‘‘plus’’ ( + ) time ‘‘cross’’ ( × ) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 7

time C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 8

C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 9

C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 10

C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 11

Hanford, WA Pisa, Italy Livingston, LA Hannover, Germany C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 12

Measuring gravitational waves • laser interferometry • output: a time series • problems: signal detection, parameter estimation , . . . C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 13

Binary inspiral events • binary star system, orbiting around their barycentre • energy is radiated in the form of gravitational waves • orbits shrink, rotation accelerates • → “chirping” GW signal (increasing frequency and amplitude) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 14

The “chirp” signal (3.5PN phase / 2.5PN amplitude approximation) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 15

The 9 signal parameters • masses : m 1 , m 2 • luminosity distance : d L • sky location : declination δ , right ascension α • orientation : inclination ι , polarisation ψ , coalescence phase φ 0 • coalescence time : t c C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 16

Prior information • different locations / orientations equally likely • masses: uniform across [ 1 M ⊙ , 10 M ⊙ ] • events spread uniformly across space: P( d L ≤ x ) ∝ x 3 • but: certain SNR required for detection • cheap SNR substitute : signal amplitude A • primarily dependent on masses , distance , inclination : A ( m 1 , m 2 , d L , ι ) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 17

• introduce sigmoid function linking amplitude to detection probability : 100% 90% detection probability 10% 0% A(2,2,60,0) A(2,2,50,0) (log−) amplitude C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 18

Resulting (marginal) prior density 200 luminosity distance (d L ) 150 100 50 0 5 10 15 20 total mass (m t = m 1 + m 2 ) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 19

Marginal prior density 200 luminosity distance (d L ) 150 100 50 0 π 2 0 π inclination angle ( ι ) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 20

Marginal prior densities individual masses (m 1 , m 2 ) inclination angle ( ι ) 2 4 6 8 10 π 2 0 π (sun masses) (radian) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 21

Prior • prior ‘considers’ Malmquist effect (selection effect) • more realistic settings once detection pipeline is set up (“selection” of signals done by the signal detection algorithm) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 22

Model • data from several interferometers • noise assumed gaussian , coloured ; interferometer-specific spectrum • noise independent between interferometers ⇒ coherent network likelihood is product of individual ones • likelihood computation based on Fourier transforms of data and signal C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 23

MCMC details • Metropolis -algorithm • Reparametrisation , most importantly: chirp mass m c , mass ratio η • Parallel Tempering 1 several tempered MCMC chains running in parallel 1 sampling from p ( θ ) p ( θ | y ) for ‘temperatures’ 1 = T 1 ≤ T 2 ≤ . . . Ti 1 W.R. Gilks et al.: Markov chain Monte Carlo in practice (Chapman & Hall / CRC, 1996). C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 24

Example application • simulated data : 2 M ⊙ - 5 M ⊙ inspiral at 30 Mpc distance measurements from 3 interferometers: SNR LHO (Hanford) 8.4 LLO (Livingston) 10.9 Virgo (Pisa) 6.4 network 15.2 • data : 10 seconds (LHO/LLO), 20 seconds (Virgo) before coalescence, noise as expected at design sensitivities • computation speed : 1–2 likelihoods / second C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 25

Hanford Livingston Pisa 0.00 = t c −0.15 −0.10 −0.05 (seconds) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 26

declination ( δ ) right ascension ( α ) −0.55 −0.50 −0.45 −0.40 4.60 4.65 4.70 4.75 4.80 (radian) (radian) coalescence time (t c ) luminosity distance (d L ) 9012.340 9012.344 9012.348 10 20 30 40 50 60 (seconds) (Mpc) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 27

chirp mass (m c ) mass ratio ( η ) 2.685 2.695 2.705 2.715 0.18 0.19 0.20 0.21 0.22 0.23 0.24 (sun masses) individual masses (m 1 , m 2 ) 2 3 4 5 (sun masses) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 28

0.23 0.22 mass ratio ( η ) 0.21 0.20 0.19 2.685 2.690 2.695 2.700 2.705 2.710 2.715 chirp mass (m c ) C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 29

−24 ° −26 ° −28 ° declination δ −30 ° −32 ° −34 ° 18.2 h 18 h 17.8 h 17.6 h 17.4 h right ascension α C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 30

Additional examples • lower (total) signal-to-noise ratio (SNR) • ‘unbalanced’ SNR: SNR LHO (Hanford) 9.6 LLO (Livingston) 13.9 Virgo (Pisa) 0.2 network 16.9 C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 31

Low total SNR 40 − 10 ° distance d L (Mpc) declination δ − 20 ° 30 − 30 ° 20 − 40 ° 10 18 h 17 h 16 h π 2 0 π inclination ι (rad) right ascension α C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 32

Low SNR at one interferometer data included data excluded 40 ° 40 ° declination δ declination δ 30 ° 30 ° 20 ° 20 ° 10 ° 10 ° 19 h 18 h 17 h 19 h 18 h 17 h right ascension α right ascension α C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 33

Parallel tempering MCMC • several parallel MCMC chains • tempering : sampling from tempered distributions chain temperature sampling from 1 T 1 = 1 p ( θ ) p ( y | θ ) 1 2 T 2 = 2 p ( θ ) p ( y | θ ) 2 1 3 T 3 = 4 p ( θ ) p ( y | θ ) 4 . . . . . . . . . p ( θ ) • additional swap proposals between chains C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 34

MCMC chain 1 — temperature = 1 − 20 ° − 25 ° declination δ − 30 ° − 35 ° − 40 ° 18 h 30 ′ 18 h 00 ′ 17 h 30 ′ 17 h 00 ′ right ascension α C. R¨ over, R. Meyer and N. Christensen: Coherent Bayesian inference on compact binary inspirals... 35