changing-waveform

The purpose of this demo is to illustrate how observed gravitational-wave signals depend on binary masses and distances, and thus how these quantities can be measured from gravitational-wave data.

Reference waveform

The figure below (reference_waveform.jpg) shows an example gravitational waveform, corresponding to a $40+40\,M_\odot$ binary at a luminosity distance of $D_L = 1\,\mathrm{Gpc}$:

The overall amplitude of this signal is governed by both the luminosity distance and the redshifted chirp mass, given by the combination $M_c(1+z)$:

$$ \mathcal{A} = \frac{[M_c(1+z)]^{5/3}}{D_L} $$

The frequency derivative of the signal (aka the rate of the gravitational-wave "chirp") depends, in turn, on the redshift chirp mass.

$$ \dot f \propto [M_c(1+z)]^{5/3} $$

Changing the redshifted chirp mass

Dialing the redshifted chirp mass up or down changes both the predicted amplitude and frequency evolution of the signal (varying_mass.mp4):

Changing the luminosity distance

Shifting the luminosity distance, in turn, changes only the amplitude (varying_dist.mp4):

Qualitative "algorithm" for fitting a GW's mass and distance

The mass and distance of a GW source can thus be inferred as follows:

1. Given an observed gravitational-wave signal, vary the presumed redshifted chirp mass $M_c(1+z)$ until the predicted frequency evolution matches the observed evolution.
2. Then vary luminosity distance $D_L$ until the predicted amplitude matches as well
3. We're not quite finished. The final step is to assume a cosmological model, in order to predict the binary redshift $z$ as a function of luminosity distance. Then we can compute $(1+z)$ and finally obtain the binary mass $M_c$.