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 symmetries of asymptotically flat spacetimes and fluxes of conserved charges conjugate to these symmetries. While the phenomenology of GW memory effects is well understood in general relativity (GR), it is less well understood in the many modifications to GR. We recently computed asymptotically flat solutions, symmetries, conserved quantities, and GW memory effects in one such modified theory: Brans-Dicke theory. In this paper, we apply our results from this earlier work to compute the GW memories from compact binaries in the post-Newtonian (PN) approximation. In addition to taking the PN 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) GWs. We focus on the tensor (as opposed to scalar) GW memory effect, which we compute through Newtonian order, and the small differences induced by scalar radiation 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 -1 PN order-effect, then in this approximation, the displacement memory has a logarithmic dependence on the PN parameter and the spin memory has a relative -1 PN-order correction; these corrections 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 GR.