Pulsar Timing Array Workshop, July 2005

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

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

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

binary near coalescence

sinusoid

chirp

time

freq

uen

cy

stochastic

burst

Pulsar Timing Array Workshop, July 2005

Doppler Tracking and Pulsar Timing

s/c tracking pulsar timing

Tracking mode: 2-way one-wayGW coupling: 3-pulse 2-pulseNoise coupling: 1- and 2-pulse 1-pulseCharacteristic time: T, TWLT TNoise sources: FTS FTS

s/c buffetting PSR stabilityantenna mech station

locationplasma (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 not

be 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 not

be useful for ULF GW processing)

• Bispectral analysis– Fourier decomposition of third moment: FT[<x(t) x(t+1)

x(t+2)>]

– 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 will be 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

Pulsar Timing Array Workshop, July 2005

Signal Processing (good ideas which I suspect will be 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