Sensitivity to periodic and quasi-periodic waves

6.2 Sensitivity to periodic and quasi-periodic waves

6.2.1 Sinusoidal waves and chirps

Sinusoidal sensitivity is traditionally stated as the amplitude h of a sinusoidal GW required to achieve a specified SNR, as a function of Fourier frequency [18 , 107 , 16 ]. Conventionally, the signal is averaged over the sky and over polarization state [49 , 120 , 16 ]. Figure 19 shows the all-sky sensitivity based on a smoothed version of the actual spectrum (black curve) for the Cassini 2001 – 2002 observation. Cassini achieves $∼$ 10^–16 all-sky sensitivity over a fairly broad Fourier band.

Figure 19:

Sensitivity of the Cassini 2001–2002 gravitational wave observations, expressed as the equivalent sinusoidal strain sensitivity required to produce SNR = 1 for a randomly polarized isotropic background as a function of Fourier frequency. This reflects both the levels, spectral shapes, and transfer functions of the instrumental noises (see Section 4) and the GW transfer function (see Section 3). Black curve: sensitivity computed using smoothed version of observed noise spectrum; blue curve: sensitivity computed from pre-observation predicted noise spectrum [105 ].

Early searches for periodic waves involved short duration observations (several hours to a few days; see, e.g., [4 , 18 ]) and thus ignorable modulation of an astronomical sinusoid due to changing geometry (non-time-shift-invariance of the GW transfer function; see Section 3). A true fixed-frequency signal would be reflected in the spectrum of the (noisy) Doppler times series as a “Rice-squared” random variable [91 ] at the signal frequency. Subsequent observations were over 10 – 40 days and the time dependence of the earth-spacecraft-source geometry became important: A sinusoidal excitation would be modulated into a non-sinusoidal Doppler response with power typically smeared over a few Fourier frequency resolution bins. Non-negligible modulation has both advantages and disadvantages. An advantage is that a real GW signal has a source-location-dependent signature in the data and this can be used to verify or refute an astronomical origin of a candidate. The disadvantage is that a simple spectral analysis is not sufficient for optimum detection (SNR losses of the simple spectral analysis technique are frequency and geometry dependent but can be $≃$ 3 dB or more in some observations) and the computational cost to search for even a simple astronomical-origin sinusoid becomes larger. Searches to date have addressed this with a hierarchical approach to the data analysis. First a suboptimal-but-simple spectral analysis is done. Candidates are then identified using an SNR threshold which is high enough to exclude Fourier components which, even with proper analysis, would be too weak to be classified as other-than-noise. The idea is to use a computationally inexpensive procedure (i.e. FFTs) to exclude candidates which could not, in principle, be raised to a reasonable threshold SNR even with accurate matched filtering. The frequencies of candidate signals passing the threshold are saved and matched filters are constructed using the known time-dependence of the earth-spacecraft geometry and for 20 points on the sky (the vertices of a dodecahedron projected onto the celestial sphere⁹ .)

Linear chirp processing adds an additional parameter (chirp rate) and requires detection thresholds to be set higher. In both the sinusoidal and chirp analyses, the pdfs of signal power have been used to assess candidates (see, e.g., [18 , 3 , 25 ]). Non-linear chirps, if source parameters are favorable, could be strong candidate signals. Analysis methods, anticipated sensitivity, and detection range for Cassini are discussed in [31 ].

6.2.2 Nonsinusoidal periodic waves

Calculations of Doppler response to GWs from a nonrelativistic binary system [120 ] show that the observed Doppler waveform can have a rich harmonic content. Monte Carlo calculations of GW strength from stars in highly elliptical orbits around the Galaxy’s central black hole have been given in [50 ]. For these stellar-mass secondaries generate wave amplitudes more than an order of magnitude weaker than Cassini-era Doppler sensitivity.