Gravitational-wave memory: Waveforms and phenomenology

The nonlinear gravitational-wave memory effect is a prediction of general relativity in which test masses are permanently displaced by gravitational radiation. We implement a method for calculating the expected memory waveform from an oscillatory gravitational-wave time series. We use this method to explore the phenomenology of gravitational-wave memory using a numerical relativity surrogate model. Previous methods of calculating the memory have considered only the dominant oscillatory ($\ell =2$, $m=|2|$) mode in the spherical harmonic decomposition or the post-Newtonian expansion. We explore the contribution of higher-order modes and reveal a richer phenomenology than is apparent with $\ell =|m|=2$ modes alone. We also consider the “memory of the memory” in which the memory is, itself, a source of memory, which leads to a small, $O({10}^{-4})$, correction to the memory waveform. The method is implemented in the python package GWMemory, which is made publicly available.

Figure 1

Including higher-order oscillatory modes significantly affects the predicted memory. Comparison of the $+$ (top panel) and $\times$ (bottom panel) polarizations of the memory time series when using only the $\ell =|m|=2$ oscillatory modes (dotted) and when using all modes with $\ell \le 4$ (solid). The colors are for binaries as follows: red is equal mass ($q=1$) and nonspinning ($S_1=S_2=\overrightarrow{0}$), green is equal mass with precessing spins ($S_{||}=0$, $S_{\perp }=0.8$), blue is unequal mass and nonspinning, black is unequal mass ($q\equiv m_1/m_2=2$) with precessing spins. In all cases, the late-time memory is different by $O(10\%)$ compared with the $\ell =|m|=2$ only case and is larger for large mass ratios and large, precessing, spins. For nonspinning binaries, this is due to the excitation of higher-order modes during merger and ringdown. Ignoring the higher-order modes completely removes the predicted $\times$ polarized memory. The systems shown are edge-on ($\iota =\pi /2$, $\varphi =0$) with total mass, $M=60M_{\odot }$, at a luminosity distance, $D_L=400\mathrm{Mpc}$.

Figure 2

The late time $+$ (solid) and $\times$ polarizations (dashed) of the memory amplitudes when including increasing numbers of modes in the oscillatory waveform (left to right). The horizontal axis $(\ell ,|m|{)}_{\mathrm{last}}$ indicates the last two oscillatory modes included in the calculation. The $(\ell =2,|m|=2{)}_{\mathrm{last}}$ modes make the dominant contribution to the $+$ polarization and have no $\times$ component for all spins and mass ratios. When the $(\ell =2,|m|=1{)}_{\mathrm{last}}$ modes are added we see that there is a nonzero $\times$ polarization for spinning systems. Including the $(\ell =3,|m|=2{)}_{\mathrm{last}}$ modes has the largest effect of all the higher-order modes on the $+$ contribution to the late-time memory. The colors are for binaries as follows: red is equal mass ($q=1$) and nonspinning ($S_1=S_2=\overrightarrow{0}$), green is equal mass with precessing spins ($S_{||}=0$, $S_{\perp }=0.8$), blue is unequal mass and nonspinning, black is unequal mass ($q\equiv m_1/m_2=2$) with precessing spins. The systems shown are edge-on ($\iota =\pi /2$, $\varphi =0$) with total mass, $M=60M_{\odot }$, at a luminosity distance, $D_L=400\mathrm{Mpc}$. We note that the memory is not necessarily maximized for edge-on systems when higher-order modes are included.

Figure 3

The spherical harmonic decomposition of the memory waveform for a range of mass ratios and spins. The absolute value of the late-time memory is shown as a function of the $(\ell ,m)$ spherical harmonic decomposition of the memory. The dominant term is the $\ell =2$, $m=0$ mode for equal mass nonprecessing binaries. For precessing binaries, the $\times$ terms in the memory integral lead to significant azimuthal dependence of the memory. This is seen in the $|m|=1$ modes. The colors are for binaries as follows: red is equal mass ($q=1$) and nonspinning ($S_1=S_2=\overrightarrow{0}$), green is equal mass with precessing spins ($S_{||}=0$, $S_{\perp }=0.8$), blue is unequal mass and nonspinning, black is unequal mass ($q\equiv m_1/m_2=2$) with precessing spins. The systems shown are edge-on ($\iota =\pi /2$, $\varphi =0$) with total mass, $M=60M_{\odot }$, at a luminosity distance, $D_L=400\mathrm{Mpc}$.

Figure 4

Angular dependence of the late-time $+$-(left) and $\times$-(right) polarized memory strain as a function of orientation angles $\iota$ (polar) and $\varphi$ (azimuth). The top panels show the late-time memory for a nonspinning equal-mass binary and the bottom panels the late-time memory for a precessing equal-mass binary. The top panel follows the analytic expression given an oscillatory waveform containing only the $\ell =|m|=2$ mode, $\delta h_{+}\propto {\sin }^2\iota (17+{\cos }^2\iota )$, $\delta h_{\times }=0$. The bottom panel demonstrates how precessing systems give rise to a more complex memory structure.

Figure 5

The maximum of the absolute value of the contribution to the memory entering at the $i$th iterative order for a range of mass ratios and spins. The peak of the oscillatory waveform corresponds to $i=0$. The first-order memory is $i=1$; we note that this is of the same order as the peak oscillatory strain. Each successive order of the memory is then on average 2 orders of magnitude smaller than the previous. The colors are for binaries as follows: red is equal mass ($q=1$) and nonspinning ($S_1=S_2=\overrightarrow{0}$), green is equal mass with precessing spins ($S_{||}=0$, $S_{\perp }=0.8$), blue is unequal mass and nonspinning, black is unequal mass ($q\equiv m_1/m_2=2$) with precessing spins. The systems shown are edge-on ($\iota =\pi /2$, $\varphi =0$) with total mass, $M=60M_{\odot }$, at a luminosity distance, $D_L=400\mathrm{Mpc}$.

Figure 6

The plus component of the predicted memory using waveforms generated using different models. We compare the numerical relativity surrogate (NRSur7dq2) used in the rest of the paper, an effective one-body model (SEOBNRv4) and a phenomenological model (IMRPhenomD). The predicted memory agrees for all models when only considering the $\ell =|m|=2$ oscillatory modes. As demonstrated above, including the higher-order oscillatory modes in the surrogate changes the predicted memory. The system shown is edge-on ($\iota =\pi /2$, $\varphi =0$), nonspinning, with total mass, $M=60M_{\odot }$, equal mass, and at a luminosity distance, $D_L=400\mathrm{Mpc}$.