Gravitational Wave Searches in PTA Data

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1 Gravitational Wave Searches in PTA Data Justin Ellis, Xavier Siemens and Jolien Creighton IAU XXVIII General Assembly IAUS 291 August 22,

2 Outline Introduce coherent and incoherent F-statistic search techniques for continuous gravitational waves. Outline detection and upper limit pipeline for CW searches using F-statistic. Report on preliminary CW upper limits for the Demorest et al. (2012) NANOGrav data set. Introduce new first-order likelihood method for stochastic gravitational wave background detection. Briefly outline future work. 2

3 The Current (?) State of Affairs Kocsis & Sesana, 2011 Roedig & Sesana, 2011 Overall we will be dominated by the stochastic GW background. However, single bright resolvable sources may stand out above the background. Expected that many sources will have nonnegligible eccentricity. (However algorithms that assume circular orbits should still be ok.) 3

4 Important Features of F-Statistics Fully implemented in time domain so no problems with irregular sampling Take timing model into account via projection matrices Can handle both uncorrelated and correlated colored noise Can confidently detect sources and estimate sky location and frequency 4

5 The Signal and Timing Model Define pre-fit residuals as: 4 Earth term Pulsar Term r α (t) = ai (ζ,ι,φ 0,ψ)A i α(θ, φ, ω 0,t) + p α (ζ,ι,φ 0,ψ,θ,φ,ω 0,L α,t) i=1 where each a i is a constant that depends on 4 extrinsic parameters and each A i α is a time dependent basis function that depends on 3 intrinsic parameters. Define a projection matrix R α for each pulsar contains the full linearized timing model and converts pre-fit residuals into post-fit residuals: R R R n r 1 r 2. r M = Rr = r 1 r 2. r M = r A tilde denotes fitting Residual [µs] Residual [µs] post-fit pre-fit Time [yr] 5

6 F e The and statistics F p Define two search statistics: F e -statistic: - Drop pulsar term from signal model and analytically maximize likelihood over 4 amplitudes that depend on extrinsic parameters. - Coherent statistic that depends on 3 unknown search parameters F p -statistic: - Put all pulsar dependence in 2M amplitudes and analytically maximize the likelihood over 2M amplitudes - Incoherent statistic that only depends on frequency (essentially a time domain analog to a weighted spectral summing statistic) Both statistics have well known statistical properties and false alarm rates can be determined through analytic calculations Ellis et al, 2012 arxiv: v1 (recently accepted to ApJ) 6

7 Outline of detection pipeline.tim file.par file tempo2 Residuals Design Matrix Noise Estimation P α F-statistic code Run Upper limit calculation no F 0 F 0 F detect yes Detection! Make Posterior probability plots and estimate parameters 7 inest

8 Example search OPTIMAL STRATEGIES FOR CONTINUOUS WAVE DETECTION 100 ns white noise 100 ns white noise + red noise (a) (b) (b). Figure Posterior 4. Posterior probability probability distribution 25 distribution functions pulsars functions for sky location for with sky location and orbital SNR=14 andfrequency orbital frequency for a network for injection a network SNR=14 SNR=14 injection with andwith without and without red noise. red we usedhave a PTA usedwith a PTA 25 pulsars. with 25 pulsars. The vertical Thelines vertical indicated lines indicated the injected the injected parameters parameters and the contours and the contours are the one, are the twoone, andtwo three and sigma threecontours. sigma conto (a): ns noise. white (b): noise. 100 (b): ns white 100 ns noise white andnoise uncorrelated and uncorrelated red noisered with noise amplitude with amplitude A =4.22 A = s ands 1.1 γ =4.1. and γ We =4.1. see that We see the that sky location the sky locatio and or frequency have all been have recovered all been recovered at the one-sigma at the one-sigma level in both levelcases. in both cases Implementation 4.3. Implementation of the detection of the detection statistics statistics tures of tures the of different the different detection detection statistics statistics and toand givetoorde 8 we Here willwe test will ourtest detection our detection statistics statistics on mock ondata mock sets data sets magnitude magnitude estimates estimates of expected of expected sensitivity sensitivity for real for data. real with injected SMBHB GW signals in the presence of white We have Weproduced have produced various various sensitivity sensitivity curves curves for both forthe b

9 Preliminary Upper Limits on NANOGrav Data Use loudest event statistic to produce upper limits in each frequency bin. For each frequency bin we inject GWs from SMBHBs with random binary orientation, sky location, and mass but fixed strain, h. Quadratic wall M sky location fitting Integration limit 10 9 M Perform 1000 realizations and determine the strain amplitude in which F inj > F 0 in 95% of realizations 3C66B About a factor of 7 more constraining than Yardley et al (2011) 9

10 Stochastic Background Detection I In general, the likelihood of our data given a stochastic GWB with amplitude p( r Ω) = 1 det 2π Σ exp 12 rt Σ 1 r Ω is: where r = r 1 r 2. and Σ = P 1 S S 1M S 21 P 2... S 2M very computationally intensive to invert (Order 10 mins) r M S M1 S M2... P M Must also search over intrinsic pulsar noise parameters simultaneously with the GWB parameters (amplitude and spectral index). This requires O(10^5) likelihood calls. So what can we do? 10

11 Stochastic Background Detection II Let Auto term Cross term Σ 1 =(P + S c ) 1 and expand to get: Σ 1 = P 1 S c + 2 P 1 S c P 1 S c + O( 3 ) P 1 And we also expand the determinant out in a similar way: ln det Σ =TrlnP 2 Tr(P 1 S c P 1 S c )+O( 4 ) We can now write a first-order log-likelihood: ln L = 1 2 Tr ln P + r T P 1 r r T P 1 S c P 1 r Now we only have to invert the individual auto-covariance matrices. This can result in speedups of up to a few orders of magnitude. 11

12 First Order Likelihood Tests MDC 1: 36 psr 100 ns white noise A=5e-14 MDC 2: 36 psr varying white noise A=5e-14 MDC 3: 36 psr varying white noise+red A=1e-14 Simulated data set: 15 psr, 100 ns white noise A=4e-15 12

13 First Order Likelihood Tests MDC 1: 36 psr 100 ns white noise A=5e-14 MDC 2: 36 psr varying white noise A=5e-14 MDC 3: 36 psr varying white noise+red A=1e-14 Simulated data set: 15 psr, 100 ns white noise A=4e-15 12

14 First Order Likelihood Tests MDC 1: 36 psr 100 ns white noise A=5e-14 MDC 2: 36 psr varying white noise A=5e-14 MDC 3: 36 psr varying white noise+red A=1e-14 Simulated data set: 15 psr, 100 ns white noise A=4e-15 12

15 Future Work Set upper limits on strain amplitude as function of frequency for continuous GWs using coherent F-statistic. Use F-statistic methods to place limits the coalescence rate of SMBHBs and possibly rule out areas of the mass-redshift parameter space. Add higher order terms to likelihood approximation and test on MDC datasets. Use First order likelihood to place upper limits on the amplitude or the stochastic GWB using NANOGrav data and compare to other methods. 13