Distortion of gravitational-wave packets due to their self-gravity

When a source emits a gravity-wave (GW) pulse over a short period of time, the leading edge of the GW signal is redshifted more than the inner boundary of the pulse. The GW pulse is distorted by the gravitational effect of the self-energy residing in between these shells. We illustrate this distortion for GW pulses from the final plunge of black hole binaries, leading to the evolution of the GW profile as a function of the radial distance from the source. The distortion depends on the total GW energy released $ϵ$ and the duration of the emission $\tau$, scaled by the total binary mass $M$. The effect should be relevant in finite box simulations where the waveforms are extracted within a radius of $\lesssim {10}^{2}M$. For characteristic emission parameters at the final plunge between binary black holes of arbitrary spins, this effect could distort the simulated GW templates for LIGO and LISA by a fraction of ${10}^{-3}$. Accounting for the wave distortion would significantly decrease the waveform extraction errors in numerical simulations.

Figure 1

A sketch of the effect under consideration. The coalescence of two BHs in a binary of total initial mass $M_0$ results in the emission of a burst of gravitational radiation which carries away a non-negligible $ϵ$ fraction of $M_0$. The remnant BH mass is $M_f$. The proper temporal width of the wave-packet for a hypothetical observer fixed at a radial distance $r$ is $\Delta \tau$. As the packet propagates outwards, it (1) expands due to gravitational redshift of the initial mass $M_0$ (solid lines), (2) contracts due to the mean self-gravity of the radiation (dotted line, $\Delta {\tau }^{\prime }$), and also (3) distorts its profile due to the self-gravity of the radiation (not shown). As a result the inner shells begin to catch up, and the proper time separation in excess of gravitational redshift from the front of the burst decreases with distance. Consequently, the net luminosity of the radiation burst changes with distance.

Figure 2

The residual self-gravitational distortion of shells after correcting for the bulk gravitational redshift. The thick curves show the change in the proper time duration of the signal at radial distance $r$ from the source between shells enclosing 7% (top) or 3% (bottom) of the total mass, dotted lines correspond to the leading order term given by Eq. (15). The vertical lines highlight the typical radii used in numerical simulations for GW extraction.

Figure 3

Our fits to the GW luminosity profile for binary BH inspiral merger simulations as a function of observer proper time $\tau$ and the evolution of the profile at various distances, $r$. The profile parameters are given in Sec. 2 and $ϵ=7\%$. Top: The absolute profile (in units of ${\mathrm{c}}^5/\mathrm{G}$) is shown for two extremes, a nearby distance ($r=12.5M_0$) and far away distance ($r={10}^3{M}_0$). Bottom: The difference between the GW luminosity profiles at infinity (i.e. $r={10}^3{M}_0$) and three cases of smaller $r$, in units of peak luminosity at infinity. The peaks of the profiles are set to $\tau =0$. The main effect responsible for the differences seen in this figure is the bulk gravitational redshift.

Figure 4

The residual self-gravitational distortion to the luminosity profiles after accounting for the average gravitational redshift, $z=M_0/r$ (top) or $(M_0-0.5\Delta m_{\mathrm{tot}})/r$ (bottom), respectively. Other parameters are the same as in Fig. 3.