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What a GW signal measures

This page gives the physical background that the merger rates and the Hubble constant rely on.

Amplitude and frequency

A compact binary emits a "chirp": a wave whose frequency and amplitude grow until the merger. To leading (Newtonian) order, its evolution is set by the chirp mass

\[ \mathcal{M} = \frac{(m_1 m_2)^{3/5}}{(m_1 + m_2)^{1/5}}, \qquad \dot f = \frac{96}{5}\,\pi^{8/3} \left(\frac{G\mathcal{M}}{c^3}\right)^{5/3} f^{11/3}, \]

and its strain amplitude, for a source at luminosity distance \(D_L\), is

\[ h \propto \frac{(G\mathcal{M})^{5/3} (\pi f)^{2/3}}{c^4\, D_L} \times (\text{orientation and antenna factors}). \]
  • The frequency evolution gives the masses: the chirp mass from the inspiral, the mass ratio and spins from higher-order effects and from the merger and ringdown.
  • The amplitude falls as \(1/D_L\). Once the masses are known from the phase evolution, the amplitude gives the distance directly, with no cosmological model and no distance ladder: GW sources are standard sirens (Schutz 1986 [17], Holz & Hughes 2005 [18]). The inclination of the orbit is partly degenerate with the distance, which is why distances are often uncertain by tens of percent.

Redshift and the detector frame

The expansion of the Universe stretches the signal on its way: every time scale is multiplied by \(1+z\), where \(z\) is the redshift, and every frequency divided by \(1+z\). A binary of masses \(m_\text{src}\) at redshift \(z\) produces exactly the signal of a binary of masses

\[ m_\text{det} = (1+z)\, m_\text{src} \]

at rest. The detector measures the detector-frame masses \(m_\text{det}\) (also called redshifted masses); the physical source-frame masses require the redshift.

The signal alone cannot give \(z\): a heavy nearby binary and a lighter distant one can produce the same signal. The distance is measured, but converting it to a redshift requires a cosmology.

Distance and cosmology

In a flat ΛCDM cosmology, the luminosity distance of a source at redshift \(z\) is

\[ D_L(z) = (1+z)\,\frac{c}{H_0} \int_0^z \frac{dz'}{\sqrt{\Omega_m (1+z')^3 + 1 - \Omega_m}} \;\approx\; \frac{c\,z}{H_0} \quad (z \ll 1), \]

where \(H_0\) is the Hubble constant and \(\Omega_m\) the matter density. So:

  • a measured \(D_L\) gives \(z\) only for an assumed \(H_0\) (and \(\Omega_m\));
  • source-frame masses, and every population property in the source frame, depend on the assumed cosmology. The catalogs quote them for the Planck 2015 cosmology.

Conversely, if the redshift of GW sources can be found by another route, the \(D_L\)–\(z\) relation measures \(H_0\):

Method Where the redshift comes from Example
Bright siren an electromagnetic counterpart and its host galaxy GW170817 (Abbott et al. 2017 [27])
Dark siren a statistical association with the galaxies of a catalog Gray et al. 2023 [26]
Spectral siren features in the source-frame mass distribution of the population Hubble constant

Comoving volume and the rate of mergers

The number of mergers observed per unit time from redshifts between \(z\) and \(z+dz\) is

\[ \frac{dN}{dt_\text{det}\,dz} = \frac{R(z)}{1+z}\,\frac{dV_c}{dz}, \]

where \(R(z)\) is the merger rate per unit comoving volume and per unit source-frame time, \(V_c\) the comoving volume, and the factor \(1/(1+z)\) the time dilation between the source and the detector. This is the redshift distribution used by the rates and spectral-siren analyses.