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Low-Frequency Gravitational Wave Searches Using Spacecraft Doppler Tracking

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## Low-Frequency Gravitational Wave Searches Using Spacecraft Doppler Tracking

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**Low-Frequency Gravitational Wave Searches Using Spacecraft**Doppler Tracking Cassini Radio Science GW Group* * J.W. Armstrong, R. Ambrosini, B. Bertotti, L. Iess, P. Tortora, H.D. Wahlquist Pulsar Timing Array Workshop, July 2005**Low-Frequency Gravitational Wave Searches Using Spacecraft**Doppler Tracking • The Doppler technique • Signal processing approaches + current sensitivity • Bursts • Periodic and quasi-periodic waves • Backgrounds • Data analysis ideas (which probably won’t work for ULF observations) • Data analysis ideas (which could well work for ULF observations) Pulsar Timing Array Workshop, July 2005**DSS25 and Cassini**Pulsar Timing Array Workshop, July 2005**Three-Pulse GW Response**Pulsar Timing Array Workshop, July 2005**Frequency/Timing Glitch**Pulsar Timing Array Workshop, July 2005**Antenna Mechanical Event**Pulsar Timing Array Workshop, July 2005**Plasma Events**Pulsar Timing Array Workshop, July 2005**Noises at = 1000 sec**Red: plasma at S, X, and Ka-band Blue: (hatched) uncalibrated troposphere at Goldstone Blue: (solid) after AMC/WVR calibration Green: antenna mechanical noise Asmar et al. Radio Science 40, RS2001 doi:10.1029/2004RS003101 (2005) Pulsar Timing Array Workshop, July 2005**Spectrum of Fractional Frequency Fluctuations**Armstrong et al. ApJ, 599, 806 (2003) Pulsar Timing Array Workshop, July 2005**Cartoon of Signal Phase-Space**Pulsar Timing Array Workshop, July 2005**Doppler Tracking and Pulsar Timing**s/c tracking pulsar timing Tracking mode: 2-way one-way GW coupling: 3-pulse 2-pulse Noise coupling: 1- and 2-pulse 1-pulse Characteristic time: T, TWLT T Noise sources: FTS FTS s/c buffetting PSR stability antenna mech station location plasma (solar wind) plasma (ISM) troposphere troposphere Pulsar Timing Array Workshop, July 2005**Signal Processing for Bursts**• If you know the waveform and the noise power spectrum, then matched filter • Subtlety: bogus tails of distribution of matched filter outputs caused by nonstationarity of the noise, even in absence of signal • Fix with local estimation of noise spectrum + histograms of SNR vs raw matched filter output • E.g. Iess & Armstrong in Gravitational Waves: Sources and Detectors, Ciufolini, ed., World Scientific, 1997; Armstrong (2002) http://cajagwr.caltech.edu/scripts/armstrong.ram • If you don’t know the waveform, try projecting data onto mathematical basis which has burst-like properties • “Burst-like”: localized in time; perhaps approx. localized in freq. • Wavelets (many flavors) • Empirical orthonormal functions? Pulsar Timing Array Workshop, July 2005**Signal Processing for Bursts (cont.)**• In Doppler tracking, you may not know the waveforms but you do know the signal and noise transfer functions • Use two-pulse noise transfer functions to characterize data intervals as “noise-like” (with a specific noise source) • Use three-pulse signal transfer functions to characterize data intervals as “candidate signal-like”, then follow up with detailed analysis • “Data sorting”, based only on noise and signal transfer functions, as a preprocessor for burst search • True GW burst must be internally consistent across multiple data sets (e.g., Cassini has multiple simultaneous data sets, but with different sensitivities) Pulsar Timing Array Workshop, July 2005**All-Sky Burst Sensitivity**Armstrong et al. ApJ, 599, 806 (2003) Pulsar Timing Array Workshop, July 2005**Directional Sensitivity for Mid-Band Burst**Pulsar Timing Array Workshop, July 2005**Signal Processing for Periodic and Quasi-Periodic Waves**• If sinusoid: • spectral analysis • E.g. Anderson et al. Nature 308, 158 (1984) Armstrong, Estabrook & Wahlquist ApJ 318, 536 (1987) Bertotti et al. A&A 296, 13 (1995) • If chirp: • dechirp with exp( i t2) followed by spectral analysis [arrow of time introduced] • E.g. Anderson et al. ApJ 408 287 (1993) Iess et al. in Gravitational Waves: Sources and Detectors, Ciufolini, ed., World Scientific, 323 (1997) Pulsar Timing Array Workshop, July 2005**Signal Processing for Periodic and Quasi-Periodic Waves**(cont.) • If periodic non-sinusoidal signal (e.g. nonrelativistic binary): • Harmonic summing/data folding • E.g. Groth ApJ Supp. Series 29, 285 (1975) • If binary system near coalescence: • Complicated time evolution of signal • May be helpful to do suboptimum pilot analysis by resampling based on assumed time-evolution of the phase • E.g. Bertotti, Vecchio, & Iess Phys. Rev. D. 59, 082001 (1999) Vecchio, Bertotti, & Iess gr-qc/9708033 Smith Phys. Rev. D36 2901 (1987) Pulsar Timing Array Workshop, July 2005**All-Sky Sinusoidal Sensitivity**Pulsar Timing Array Workshop, July 2005**Eccentric Nonrelativistic Binary Waveform**• • Waveforms can be complicated • • This example for Doppler tracking: • - Stellar mass object in orbit about BH at galactic center • - Cassini 2003 tracking geometry • E.g. Wahlquist GRG 19 1101 (1987) Freitag ApJ 583 L21 (2003) Pulsar Timing Array Workshop, July 2005**Signal Processing for Stochastic Background**• Isotropic BG limits can be derived from smoothed power spectrum of single s/c Doppler time series, since average transfer function to the Doppler is known • E.g. Estabrook & Wahlquist GRG 6, 439 (1975) Bertotti & Carr ApJ 236, 1000 (1980) Anderson & Mashoon ApJ 408, 287 (1984) Bertotti & Iess GRG 17, 1043 (1985) Giampieri & Vecchio CQG 27, 793 (1995) • Subtlety, related to estimation error statistics, the confidence with which the noise can be independently known, and use of the observed spectrum as an upper limit to the GW spectrum • E.g. Armstrong et al. ApJ 599, 806 (2003) Pulsar Timing Array Workshop, July 2005**Signal Processing for Stochastic Background (cont.)**• Using multiple spacecraft would be good, too • E.g. Estabrook & Wahlquist GRG 6, 439 (1975) Hellings Phys Rev. Lett. 43, 470 (1978) Bertotti & Carr ApJ 236, 1000 (1980) Bertotti & Iess GRG 17, 1043 (1985) • If BG not isotropic then correct, angle-dependent signal transfer function must be used Pulsar Timing Array Workshop, July 2005**Isotropic GW Background**Armstrong et al. ApJ, 599, 806 (2003) Pulsar Timing Array Workshop, July 2005**Signal Processing (good ideas which I suspect will notbe**useful for ULF GW processing) • Empirical orthonormal functions/Karhunen-Loeve expansion • Let the data themselves determine a mathematical basis for the data and hope that most of the variance projects onto a small number of basis vectors • Attractive as “template independent” search for signals • Probably useful for signal-dominated detector • In simulations with low SNR time series (unfortunately the practical s/c case) modes found were always the noise modes e.g., Helstrom Statistical Theory of Signal Detection (Pergamon: Oxford), 1968 Dixon and Klein “On the Detection of Unknown Signals” ASP Conf. Series, 129 (1993) Pulsar Timing Array Workshop, July 2005**Signal Processing (good ideas which I suspect will notbe**useful for ULF GW processing) • Bispectral analysis • Fourier decomposition of third moment: FT[<x(t) x(t+t1) x(t+t2)>] • Measures contribution to third moment from three Fourier components having frequencies adding to zero • Attractive theoretically as diagnostic of weak nonlinearities • Third moment may be intrinsically small • Convergence is slow e.g., Hasselmann, Munk, & MacDonald “Bispectra of Ocean Waves” in Time Series Analysis (Rosenblatt, ed.), (Weiley: New York) 1963 MacDonald Rev. Geophysics 27 449 (1989) Pulsar Timing Array Workshop, July 2005**Signal Processing (good ideas which I suspect willbe useful**for ULF GW processing) • Time-Frequency Analysis • Many ways to tile frequency-time (wavelets, chirplets, Gabor transforms); each can have special merit if you think your signal projects preferentially onto a specific mathematical basis • Template independent • Useful in Doppler tracking to characterize nonstationarities in the time series • Has been used in s/c tracking to “denoise” GLL time series by rejecting higher-frequency subbands Pulsar Timing Array Workshop, July 2005**Signal Processing (good ideas which I suspect willbe useful**for ULF GW processing) • Multi-taper spectral analysis • Very attractive theoretically: objective; synthesizes spectrum from average of spectra with the time series weighted by different windows • Achieves optimum resolution consistent with very low spectral leakage • Used successfully in geophysics on short, noisy, red time series • “Automatic” way to distinguish periodic signals in presence of steep continuum • Caveat: achieved some notoriety: outsiders found “too many signals” in space physics time series thought by insiders to be noise-only e.g. Percival and Walden Spectral Analysis for Physical Applications (Cambridge Univ. Press: Cambridge), 1993 Pulsar Timing Array Workshop, July 2005**Concluding Comments**• Low-frequency (i.e. ≈10-6-0.1 Hz) spacecraft observations are two-way and have well-defined transfer functions for f > 1/T2 • Noise analysis for s/c Doppler tracking in many ways similar to the ULF pulsar tracking problem: • Frequency standard noise • Plasma noise (ionosphere/solar wind for s/c; +ISM for pulsars) • “spacecraft buffeting” = intrinsic pulsar stability noise • Antenna mechanical noise (station location noise) • Tropospheric noise (wet + dry) • Signal processing and sensitivity analysis (noise/signal) similar Pulsar Timing Array Workshop, July 2005