Gravitational-wave memory effects in Brans-Dicke theory: Waveforms and effects in the post-Newtonian approximation

Gravitational-wave (GW) memory effects produce permanent shifts in the GW strain and its time integrals after the passage of a burst of GWs. Their presence is closely tied to the symmetries of asymptotically flat spacetimes and the corresponding fluxes of conserved charges conjugate to these symmetries. While the phenomenology of GW memory effects (particularly for compact-binary mergers) is now well understood in general relativity, it is less well understood in the many modifications to general relativity. We recently, however, computed asymptotically flat solutions, symmetries, conserved quantities, and GW memory effects in one such modified theory with an additional scalar degree of freedom, Brans-Dicke theory. In this paper, we apply our results from this earlier work to compute the GW memory effects from compact binaries in the post-Newtonian approximation. In addition to taking the post-Newtonian limit of these effects, we work in the approximation that the energy and angular momentum losses through scalar radiation are small compared to the energy and angular momentum losses through (tensor) gravitational radiation. We focus on the tensor (as opposed to scalar) GW memory effect, which we compute through Newtonian order, and the small differences induced by the radiation of scalar waves at this order. Specifically, we compute the nonlinear parts of the tensor displacement and spin GW memory effects produced during the inspiral of quasicircular, nonprecessing binaries in Brans-Dicke theory. Because the energy radiated through the scalar dipole moment appears as a negative post-Newtonian order-effect, then in this approximation, the displacement memory has a logarithmic dependence on the post-Newtonian parameter and the spin memory has a relative minus-onepost-Newtonian-order correction; these corrections, however, are ultimately small because they are related to the total energy and angular momentum radiated in the scalar field, respectively. At Newtonian order, the scalar radiation also gives rise to a sky pattern of the memory effect around an isolated source that differs from that of the memory effect in general relativity.