Exciting Modes due to the Aberration of Gravitational Waves: Measurability for Extreme-Mass-Ratio Inspirals

Gravitational waves from a source moving relative to us can suffer from special-relativistic effects such as aberration. The required velocities for these to be significant are on the order of $1000\mathrm{km}{\mathrm{s}}^{-1}$. This value corresponds to the velocity dispersion that one finds in clusters of galaxies. Hence, we expect a large number of gravitational-wave sources to have such effects imprinted in their signals. In particular, the signal from a moving source will have its higher modes excited, i.e., (3,3) and beyond. We derive expressions describing this effect and study its measurability for the specific case of a circular, nonspinning extreme-mass-ratio inspiral. We find that the excitation of higher modes by a peculiar velocity of $1000\mathrm{km}{\mathrm{s}}^{-1}$ is detectable for such inspirals with signal-to-noise ratios of $\gtrsim 20$. Using a Fisher matrix analysis, we show that the velocity of the source can be measured to a precision of just a few percent for a signal-to-noise ratio of 100. If the motion of the source is ignored, parameter estimates could be biased, e.g., the estimated masses of the components through a Doppler shift. Conversely, by including this effect in waveform models, we could measure the velocity dispersion of clusters of galaxies at distances inaccessible to light.

Figure 1

Velocity dispersion of galaxy clusters as a function of the redshift $z$ (bottom $x$ axis) and the distance $d$ (upper $x$ axis) from the data of Ref. [16] (blue squares) and Ref. [17] (red circles). We add the error bars for both the velocity dispersion and the distance, although the latter does not show up because it is smaller in size than the symbols.

Figure 2

Mismatch $1-M$ between the putative waveform of an EMRI at rest and the observed waveform of an EMRI moving with a typical c.m. velocity of $1000{\text{km s}}^{-1}$ (dotted blue line, left $y$ axis) and the SNR $\rho$ required to detect this mismatch (red continuous line, right $y$ axis). Both are shown for different viewing angles of the source $\theta$, where $\theta =0°$ corresponds to the source seen face on.

Figure 3

Confidence ellipses ($1\sigma$ level) for the measurement of the masses of the BHs, $m_{1,2}$, and the c.m. velocity of the source in the case of a source at rest (solid blue line) and a moving source (dotted red line). Note that for the moving source the masses measured represent the (Doppler-shifted) masses as seen in the observer frame.