Matched filtering and waveform systematics

Write calibrated detector strain as signal plus noise. For an ideal stationary zero-mean Gaussian noise model, its one-sided power spectral density defines the frequency weighting. Adopt a Fourier transform with a negative two-pi-i-f-t exponent. Real time series then have redundant negative-frequency information, so an integral over positive frequencies uses the conventional factor four in the real noise-weighted inner product. A two-sided spectrum would require a corresponding change of normalization.

Strain is dimensionless, its Fourier transform has seconds as units, and a one-sided strain-noise PSD has inverse-hertz units. Dividing two Fourier amplitudes by the PSD and integrating over hertz therefore gives a dimensionless inner product. This unit check helps expose mistakes involving amplitude spectral density, which is the square root of the PSD and cannot simply replace it in the denominator.

Under the stated Gaussian model the log likelihood is minus one half the squared noise-weighted residual, up to a parameter-independent constant. Maximizing over an amplitude for a fixed template shape yields matched filtering. Searches also consider arrival time, phase and template parameters, and assess a statistic's background under the actual noise process. Nonstationary glitches or uncertain spectra can violate the simple assumptions, so an optimal formula is not itself a false-alarm probability.

Normalize waveforms by their noise-weighted norms to define an overlap, often maximized over relative time and phase. A best overlap below one reduces the signal-to-noise ratio recoverable with that template family. It measures similarity under the chosen PSD and band, not a universal pointwise waveform error. Optimizing more physical parameters can improve recovery while shifting their inferred values, so a high match does not guarantee negligible parameter bias.

Waveform errors can arise from truncated PN phasing, uncertain merger calibration, omitted precession or eccentricity, inaccurate higher modes and an inappropriate ringdown attachment. Detector calibration and PSD uncertainty affect the same comparison. A model should be validated against independent calculations over its intended parameter region. Agreement with its own fitted training cases does not establish accuracy everywhere, especially when extrapolating to mass ratios or spins not covered by those checks.

Chirp mass, mass ratio, spins, redshift, inclination, distance and antenna projection leave partially degenerate imprints. A stronger signal can narrow statistical errors while making a fixed modeling error more important. Real inference therefore compares calibrated detector data with propagated waveforms, priors and a noise model, and studies systematic robustness. The elementary filter geometry teaches how information is weighted; it cannot supply trustworthy source posteriors, discovery claims or a complete error budget from a toy signal alone.