Post-Newtonian corrections to the gravitational-wave memory for quasicircular, inspiralling compact binaries

TL;DR

The paper addresses the PN corrections to the nonlinear Christodoulou memory in gravitational waves from quasicircular inspiralling binaries. It employs the multipolar post-Minkowskian formalism to relate radiative, canonical, and source moments, and computes the memory contribution to radiative mass multipoles up to 3PN, including explicit expressions for the memory in the plus polarization and in the spin-weighted spherical-harmonic modes. The results show that 3PN corrections modestly reduce the memory amplitude while preserving angular structure, thereby completing the waveform to 3PN order when combined with prior oscillatory terms. The work also discusses nonhereditary DC terms, linear DC effects in bound binaries, challenges for NR extraction, and implications for memory detectability with GW detectors, highlighting the memory’s potential observability mainly for strong-signal, low-frequency sources like LISA-band binaries.

Abstract

The Christodoulou memory is a nonlinear contribution to the gravitational-wave field that is sourced by the gravitational-wave stress-energy tensor. For quasicircular, inspiralling binaries, the Christodoulou memory produces a growing, nonoscillatory change in the gravitational-wave "plus" polarization, resulting in the permanent displacement of a pair of freely-falling test masses after the wave has passed. In addition to its nonoscillatory behavior, the Christodoulou memory is interesting because even though it originates from 2.5 post-Newtonian (PN) order multipole interactions, it affects the waveform at leading (Newtonian/quadrupole) order. The memory is also potentially detectable in binary black-hole mergers. While the oscillatory pieces of the gravitational-wave polarizations for quasicircular, inspiralling compact binaries have been computed to 3PN order, the memory contribution to the polarizations has only been calculated to leading order (the next-to-leading order 0.5PN term has previously been shown to vanish). Here the calculation of the memory for quasicircular, inspiralling binaries is extended to 3PN order. While the angular dependence of the memory remains qualitatively unchanged, the PN correction terms tend to reduce the memory's magnitude. Explicit expressions are given for the memory contributions to the plus polarization and the spin-weighted spherical-harmonic modes of the metric and curvature perturbations. Combined with the recent results of Blanchet et al.(2008), this completes the waveform to 3PN order. This paper also discusses: (i) the difficulties in extracting the memory from numerical simulations, (ii) other nonoscillatory effects that enter the waveform at high PN orders, and (iii) issues concerning the observability of the memory.

Figures (2)

• Figure 1: (color online). Dependence of the post-Newtonian (PN) corrections to the Christodoulou memory on binary inclination and orbital separation. The plots in the top row show the memory contribution to $H_+$ [Eq. \ref{['eq:Hplus']}] at each cumulative PN order as a function of $\Theta$ (the polar angle to the observer; $\Theta=0$ points along the binary's orbital angular momentum) for $x=1/5$ and $\eta=0$ (left) and $\eta=0.25$ (right). The different curves represent terms up to the following cumulative PN orders: solid (black) 0PN; long-dashed (red) 1PN; short-dash-dotted (green) 2PN; long-dash-dotted (magenta) 2.5PN; short-dashed (blue) 3PN. The bottom row plots $xH_{+}$ [see Eqs. \ref{['eq:hplusfactor']} and \ref{['eq:Hplus']}] and shows the memory's dependence on the PN parameter $x$ (which equals $M/r$ at Newtonian order, where $r$ is the orbital separation in harmonic coordinates). The labeling scheme is the same as in the top row. The PN corrections do not qualitatively change the angular dependence, but tend to decrease the magnitude of the memory. Since the 2.5PN correction vanishes for $\eta=0.25$, the 2.5PN curve is identical to the 2PN one and is not displayed in the right column plots. For $\eta=0.25$ the 3PN curve is nearly coincident with the 2PN curve.
• Figure 2: (color online). Comparison of memory and nonmemory modes. The left plot shows the absolute value of some of the largest $h_{lm}$ modes---as well as the largest $m=0$ modes---as a function of the post-Newtonian (PN) parameter $x$. The $m\neq 0$ modes are the solid (red) curves. From top to bottom they are $h_{22}$, $h_{44}$, $h_{32}$, and $h_{42}$. These are computed from Eqs (9.3)-(9.4) of Ref. blanchet3pnwaveform. The remaining curves are the $m=0$ nonoscillatory modes. The long-dashed (blue) curves are the $h_{20}$ (top) and $h_{40}$ modes [Eqs. \ref{['eq:Hl0-mem']}] expanded to 3PN order. The short-dashed (navy) curve at the bottom is the $h_{30}$ mode [the nonlinear, nonhereditary DC term given by Eq. \ref{['eq:H30']}]. The right plot is similar except it shows the absolute value of the corresponding $\psi_{lm}$ modes (see equations in Appendix \ref{['app:psilm']}). Note how the relative values of the oscillatory ($m\neq0$) and the memory ($m=0$) modes change in the two plots. Although the memory is relatively large in the metric-perturbation modes, the memory modes are significantly suppressed relative to the other curvature-perturbation modes. Both plots are for equal-mass binaries ($\eta=1/4$) and span the range $x=1/30 - 1/5$. Recall that larger values of $x$ correspond to smaller orbital separations and later times. Note that the modes with odd $m$ vanish for equal-mass binaries. Several other numerically smaller modes would also appear on these plots, but are suppressed for clarity.