Trans-dimensional signal modeling in PTA data

Transcription

1 Trans-dimensional signal modeling in PTA data Justin Ellis 1* 1 JPL/Caltech * Einstein Fellow! Einstein Fellows Symposium October 18, California Institute of Technology. Government sponsorship acknowledged. 1

2 Pulsar Timing Preliminaries Highly magnetized rotating neutron star Extremely stable clocks time-of-arrival (TOA) Pulsar Timing Residual t = t measured t model Measured Pulse time-of-arrival (TOA) Averaged Residual [µs] ] -1 Model that accounts for many delay factors but not GWs

3 polation (top panel), respectively. The bottom panel is the recovered spectrum from the powerlaw analysis (density) and the free spectral coefficients (error bars). In both density plots, the dashed line is the injected PSD and the solid black lines represent the median and 90% confidence region. In the bottom panel, the gray error bars represent the median and 90% confidence interval on each spectral coefficient.. Motivation Intermediate Case Spectral modeling of red noise. Power-law = parameters. Free spectrum > 30 parameters. Which is better? Would like a way to use free parameters where they are needed. The NANOGrav Coll 0. Timing summary for PSR J Colors are blue: 1. GHz, purple:.3 GHz, green: 80 MHz, orange: 30 MHz, red: 37 MH Some pulsars show transientfigure individual points are semi-transparent; darker regions arise from the overlap of many points. B noise events. How do we model them? 0 Would like a way to model them without a-priori choosing number of basis functions. Whitened ] Residual [µs] 31 Figure 3. Timing summary for PSR J Colors are: Blue: 1. GHz, Purple:.3 GHz, Green: 80 MHz, Orange: 30 MHz, Red: 37 MHz. In the top panel, individual points are semi-transparent; darker regions arise from the overlap of many points. - B Averaged s] Residual [µs] Residual Power [s ] For the intermediate case we have injected a realizananog RAV N INE - YEAR DATA S ET 1 tion of a gaussian process that follows a broken power-law 13 November 1 distribution. In this case we see in Figure 5 that the spec- 813:5 (31pp), The Astrophysical Journal, 015 trum is well recovered with the adaptive technique and 1 that the data strongly prefer an additional control point 15 at the turnover frequency in the spectrum. Furthermore, we see from Figure 5 that, overall, the data prefers to 1 describe the spectrum by three parameters. This is ex 17 actly the behavior that we wish to see in that the most parsimonious model to describe a broken power-law con A RZOUMANIAN ET AL. Z. 7 tains three parameters. Meanwhile, the bottom panels of Figure 5 shows that the power-law noise model exfrequency [Hz] tremely overestimates the low frequency noise and the free spectral model loosely models the power around the FIG. 5: Same as Fig. 3 but for the intermediate case. turnover in the spectrum. This case really demonstrates the power of the adaptive technique in which it correctly Figure 7. Timing summary for PSR B Colors are: Blue: 1. GHz, Purple:.3 GHz, Green: 80 MHz, Orange: 30 MHz, Red: 37 MHz. In the top panel, individual points are semi-transparent; darker regions arise from the overlap of many points. 3

4 Model everything and let the data sort it out Trans-dimensional Reversible Jump! Markov Chain Monte Carlo Similar to standard MCMC but now the model is another parameter Let the data pick the best model. Marginalize over models and their parameters.! BayesWave: Cornish & Littenberg, CQG 3, (015) BayesLine: Littenberg and Cornish, PRD 91, 0803 (015) PTA Noise: Ellis & Cornish, PRD 93, 0808 (01) PTA GW Burst: Ellis & Cornish in prep

5 Modeling Model Transient signals with sum of Morlet-Gabor Wavelets NX wave (t; A, f 0,Q,t 0, 0) =Ae (t t 0) / cos( f 0 (t t 0 )+ 0 ) s + (t) =F + (, ', ) N + X i=1 i(t) s (t) =F (, ', ) Model red spectrum with free control points and linear interpolation in log-space. N X i=1 i(t) 5

6 Transient Noise Event Simulation. B TOAs with simulated white noise burst. Test model with uniform SNR prior and log-uniform amplitude prior Signal recovered well in both cases. Model much simpler with uniform SNR prior. Utility Residuals [µs] Utility Residuals [µs] Uniform SNR prior PDF Number of Wavelets Normalized Residuals log-uniform prior PDF Number of Wavelets 3 0 Normalized Residuals

7 Adaptive PSD estimation FIG. 7: Same content as Fig. 3 for the extreme case. 7 parameters while still incorporating the uncertainty asso13 lines thecomponents, median andall90% confidence withrepresent more or less while producingregion. In the ciated bottomreconstruction panel, the gray error bars represent the median and an 1 accurate of the power spectral density. 90% confidence interval on each spectral coefficient Intermediate Case the rigidness of the power-law model, number it overestimates thecomponents from the FIG. 8: Favored of spectral power at low frequencies and underestimates the power BayesSpecPTA analysis of the extreme case. In this case, at the high frequency we peaks. see that the peak. injectedfurthermore, spectrum has two The data can either be 0.3 bydoes even the free spectral method not significantly described a power law or a moreconcomplicated spectra that 0. requires strain the PSD at either peak. control points. Utility FIG. : Favored number of spectral components from the BayesSpecPTA analysis of the intermediate case. In this case, 0.0 where the injected spectrum has a singe turnover the spec0.3 is best described by three parameters as one would extrum 0. and two parameters (powerlaw) is strongly disfavored pect, 0.1 more than 3 components is allowed. while Extreme Case For the so-called extreme case we have injected a realization 1 of a gaussian process that follows a distribution with two distinct peaks. In this case we see in Figure [1] Z. Arzoumanian, A. Brazier, S. Burke-Spolaor, S. J. 18 Chamberlin, S. Chatterjee, J. M. Cordes, P. B. DemorFor the intermediate case we have injected a realiza1 est, X. Deng, T. Dolch, J. A. Ellis, et al., submitted to 1 tion of a gaussian process that follows a broken power-law ApJ (015). 13 []13L. Lentati, P. Alexander, M. P. from Hobson, distribution. In this case we see in Figure 5 that thefig. spec: Favorednumber of spectral components thes. Taylor, J. Gair, S. T. Balan, and R. van Haasteren, 1 trum is well recovered with the adaptive technique BayesSpecPTA analysis of the null case. In this case, wherephys. Rev. and 1 D 87, 01 (013), injected spectrum is a simple powerlaw, the spectrum is that the data strongly prefer an additional controlthe point [3] R. van Haasteren and M. Vallisneri, Phys. Rev. D 90, 15 15parameters as one would expect. best described bytwo at the turnover frequency in the spectrum. Furthermore, 01 (01), we see from Figure 5 that, overall, the data prefers to [] R. van Haasteren and M. Vallisneri, MNRAS, (015), describe the spectrum by three parameters. This is ex[5]17m. A. Woodbury, Memorandum report, (1950). actly the behavior that we wish to see in that theidentifies most the most []parsimonious C. M. F. Mingarelli and containing T. Sidery, Phys. Rev. D 90, model three realiza- wer-law parsimonious 8model to describe a broken7 power-law constill he specparameters while incorporating 8the uncertainty asso- 7 tains three parameters. Meanwhile, the bottom ciated panelswith more or less components, que and all while producing Frequency [Hz]power-law noise model exol point of Figure 5 shows that the Frequency an accurate reconstruction of the power spectral[hz] density. ermore, tremely overestimates the low frequency noise and the efers FIG. to free spectral model loosely the power 3: Results of BayesSpecPTA and themodels two standard anal-around the FIG. 5: Same as Fig. 3 but for the intermediate case. s is exyses (powerlaw and free spectral coefficients) on the really null case. turnover in the spectrum. This case demonstrates he most The top two recovered spectrum thepanels powershow of thethe adaptive technique in (middle which it correctly aw con- 7 C. VII Combination of Both Residual Power [s ] 0.0 polation (top panel), respectively. The bottom panel is the 0.03 recovered spectrum from the powerlaw analysis (density) and freemost spectral coefficients (error bars).three In both density parsimonious model containing 1 the the identifies plots, the dashed line is the injected PSD and the solid black Utility FIG. 3: Results of of BayesSpecPTA and the two FIG. : Favored number spectral components from thestandard analyses (powerlaw free spectral on the null case. BayesSpecPTA analysis and of the null case. Incoefficients) this case, where 0.0 the injected spectrum is a simple powerlaw, the spectrum is The top two panels show the recovered spectrum (middle 0.05described best by two as one would expect. panel) and theparameters favored control points for the spectral inter- 0 an accurate reconstruction of the power spectral density. 7 from the bottom panel of Figure 7 we see that, because of Residual Power [s ] Number of Spectral Components 13 DISCUSSION 1 AND CONCLUSIONS (01), [7] N. J. Cornish and T. B. Littenberg, Classical and Quantum Gravity 3, (015), [8] T. B. Littenberg and N. J. Cornish, Phys. Rev. D 91, 0803 (015), [9] W. Vousden, W. M. Farr, and I. Mandel, ArXiv e-prints 1 (015), [] Z. Arzoumanian, A. S. Burke-Spolaor, S. Cham15 Brazier, berlin, S. Chatterjee, B. Christy, J. M. Cordes, N. Cornish, K. Crowter, P. B.1Demorest, et al., ArXiv e-prints (015), Residual Power [s ] Residual Power [s ] rd analull case. (middle al interel is the ity) and density id black. In the ian and 7 that the spectrum is well recovered with the adaptive 0.0 technique and that the0.18 high frequency peak is clearly dis0.35 tinguishable while the0.1 low frequency peak is not as conas can be seen in both Figures 7 FIG. :0.30 Favored number of spectral components from the strained. In this case,0.1 and 8, the model complexity is in strong competition with BayesSpecPTA analysis of the null case. In this case, where the goodness-of-fit in that data nearly equally prefer eithe injected spectrum is a simple powerlaw, the spectrum is 0. ther a shallow power-law ( parameters) that models the 0.0 best described by two parameters as one would expect power at the peaks of the PSD but also the frequencies complex spectrum that models in between, or the more 0. both the peaks and valleys 0.0 of the PSD. These features show the power of this0.0 kind of analysis in that one does 0.05 identifies the most parsimonious model containing three not have to a-priori choose a model for the spectrum and 0.00 while still incorporating the uncertainty asso- try to find the best model 0.00 from the data but instead we parameters ciated with more or less components, all while producing allow the data to constrain the model while marginaliznumber of Spectral Components Number ofcomparison, Spectral Components ing over our uncertainty in that model. For Residual Power [s ] Residual Power [s ] Utility Frequency [Hz] FIG. 7: Same content as Fig. 3 for the extreme case.

8 Improvement of GW limits If transient noise event is present and not modeled, GW limits could suffer severely Wavelet model does not absorb GW power in the absence of a transient signal Plan to use this method for upcoming GW limits Spectral Index Wavelet Model Standard Model log A gw.5 3 Utility Number of Wavelets log A gw 3.5 Spectral Index

9 Analysis on B Bayes factor for > 0 wavelets is ~exp(13) Residuals [µs] Utility PDF Number of Wavelets Normalized Residuals 9 Real B data shows a very significant transient noise event Corresponds to observation run in which half of observing bandwidth data was corrupted Plan to carry out this analysis on all NANOGrav pulsars in the future

10 Simulated data: SNR=, 1 pulsars, 5 years Simulated stochastic burst with duration 30 days and bandwidth 3.e-8 Hz

11 Recovered signals: SNR=, 1 pulsars, 5 years Simulated stochastic burst with duration 30 days and bandwidth 3.e-8 Hz

12 Simulated data: SNR=, 1 pulsars, 5 years Simulated stochastic burst with duration 00 days and bandwidth 1.3e-7 Hz

13 Recovered signals: SNR=, 1 pulsars, 5 years Duration 30 days. Bandwidth 3.e-8 Hz Duration 30 days. Bandwidth 1.3e-7 Hz Duration 00 days. Bandwidth 3.e-8 Hz Duration 00 days. Bandwidth 1.3e-7 Hz Performance degrades for signals with larger TF volume (fixed SNR)

14 Distinguishing between GW bursts from glitches On average, for every Earth term burst signal, there will be an equal number of uncorrelated pulsar term burst from different sources. These are effectively glitches. Probability of two stochastic glitches matching in two pulsars is tiny Probability of a match in N>>1 pulsars is essentially zero

15 Future Directions Apply to Dispersion Measure as well as TOAs ummary for PSR J Colors are blue: 1. GHz, purple:.3 GHz, green: 80 MHz, orange: 30 MHz, red: 37 MHz. In the top panel, semi-transparent; darker regions arise from the overlap of many points. Use wavelets per observing backend to model systematics Quite a difficult sampling problem ummary for PSR J Colors are blue: 1. GHz, purple:.3 GHz, green: 80 MHz, orange: 30 MHz, red: 37 MHz. In the top panel, semi-transparent; darker regions arise from the overlap of many points. NANOGrav generic burst search 15