Projected constraints on the dispersion of gravitational waves using advanced ground- and space-based interferometers

Certain alternative theories of gravity predict that gravitational waves will disperse as they travel from the source to the observer. The recent binary black hole observations by Advanced-LIGO have set limits on a modified dispersion relation from the constraints on their effects on gravitational-wave propagation. Using an identical modified dispersion, of the form $E^2=p^2{c}^2+\mathbb{A}{p}^{\alpha }{c}^{\alpha }$, where $\mathbb{A}$ denotes the magnitude of dispersion and $E$ and $p$ are the energy and momentum of the gravitational wave, we estimate the projected constraints on the modified dispersion from observations of compact binary mergers by third-generation ground-based detectors such as the Einstein Telescope and Cosmic Explorer as well as the space-based detector Laser Interferometer Space Antenna. We find that third-generation detectors would bound dispersion of gravitational waves much better than their second-generation counterparts. The Laser Interferometer Space Antenna, with its extremely good low-frequency sensitivity, would place stronger constraints than the ground-based detectors for $\alpha \le 1$, whereas for $\alpha >1$, the bounds are weaker. We also study the effect of the spins of the compact binary constituents on the bounds.

Figure 1

Strain sensitivities of ET-D (black curve), CE-wb (red curve), aLIGOZeroDetHighPower (blue curve), and LISA (green curve) overlaid. The corresponding signal-to-noise ratios as a function of the total mass of the compact binary are on the right panel. The sources considered consist of nonspinning (solid curves), with $\chi =0.4$ (dashed curves) and with $\chi =-0.4$ (dash-dotted curves). The sources targeted by CE, aLIGO, and ET are at a redshift of 0.2, and the LISA sources are at a redshift of 0.5.

Figure 2

Upper bounds on $\mathbb{A}$ obtained with equal-mass nonspinning sources for future ground-based detectors and the planned space-based detector LISA. For ground-based detectors, we use the future design of aLIGO (in blue), the 3G detectors ET (in black), and CE (in red). The total masses vary from $20–1000{\mathrm{M}}_{\odot }$ for aLIGO and CE and lie between $20–3000{\mathrm{M}}_{\odot }$ for ET. The total masses vary between $10^5–{10}^7{\mathrm{M}}_{\odot }$ for sources targeted by LISA (in green). All bounds are obtained for $\alpha =0$ (dashed lines), $\alpha =1$ (dash-dotted lines), $\alpha =2.5$ (solid lines), and $\alpha =3$ (dotted lines). The sources targeted by the ground-based detectors are at a redshift of 0.2, and the LISA sources occur at a redshift of 0.5.

Figure 3

Upper bounds on $\mathbb{A}$ varying with total mass for spinning sources overlaid with nonspinning sources for CE and aLIGO (left) and LISA (right). The bounds are for $\alpha =0$ (top panel) and $\alpha =1$ (bottom panel). The plots are made for $\chi =0.4$ (dashed lines), $\chi =-0.4$ (dash-dotted lines), and nonspinning sources. In general, spins deteriorate the bounds, though $\chi =-0.4$ is found to perform almost comparably with the nonspinning counterparts for higher mass sources. It is discussed in Sec. 4b.

Figure 4

Upper bounds on $\mathbb{A}$ varying with total mass for spinning sources overlaid with nonspinning sources for CE and aLIGO (left) and LISA (right). The bounds are for $\alpha =3$ (top panel) and $\alpha =2.5$ (bottom panel). The plots are made for $\chi =0.4$ (dashed lines), $\chi =-0.4$ (dash-dotted lines), and nonspinning sources. A more detailed discussion occurs in Sec. 4b.