Towards Claiming a Detection of Gravitational Memory

Author(s)

Zosso, Jann, Magaña Zertuche, Lorena, Gasparotto, Silvia, Cogez, Adrien, Inchauspé, Henri, Jacobs, Milo

Abstract

Gravitational memory is a zero-frequency effect associated with a permanent change in the asymptotic spacetime metric induced by radiation. While its universal manifestation is a net change of proper distances, gravitational-wave detectors are intrinsically insensitive to the final offset and can only probe the associated transition. A central challenge for any claim of detection therefore lies in defining a physically meaningful and operationally robust model of this time-dependent signal, which is uniquely attributable to gravitational memory and distinguishable from purely oscillatory radiation. In this work, we propose a general solution to this challenge. Building on a self-contained review of the theory of gravitational memory, we discuss a theoretical framework for defining and modeling a gravitational memory rise, in particular applicable to compact binary coalescences. Specializing to space-based detectors, we analyze the response of LISA to gravitational radiation including a memory contribution, with particular emphasis on mergers of supermassive black hole binaries, which offer the most promising prospects for a first single-event detection. The framework developed here provides the theoretical foundation for statistically well-defined hypothesis testing between memory-free and memory-full radiation and quantitative assessments of detection prospects. As such, these results establish a principled pathway toward a future observational claim of gravitational memory.

Figures

Caption

Penrose diagram of a conformally compactified asymptotically flat spacetime in asymptotic light-cone coordinates $\{u,r,\theta,\phi\}$, with $u=t-r$ and where time $t$ flows vertically. A localized source emits null radiation (yellow) toward future null infinity $\scri^+$, defined as the $r\to\infty$ limit at fixed retarded time $u$. The angular coordinates are not shown, but the asymptotic two-spheres at retarded times $u_0$ and $u$ are depicted schematically as blue circles. The BMS balance laws state that the supermomentum flux reaching $\scri^+$ between $S^2_{u_0}$ and $S^2_u$ is exactly balanced by the change in supermomentum charge between these two times. [Figure adapted from~\cite{DAmbrosio:2022clk,Zosso:2024xgy}.]

Caption

Illustration of type $(a)$ and type $(b)$ waveform models. The type $a$ waveform (orange) includes no memory and is based off the extrapolation method (EXT) while the CCE waveform (blue) is type $b$ and includes memory.

Caption

\textit{Top panel:} We show the full CCE waveform strain (blue, waveform type $b$), the $(2,0)$ mode of the extrapolated surrogate model (green, waveform type $a$), and the memory content (red) of the $(2,0)$ mode as calculated from the extrapolated waveform through Eq.~\eqref{eq:memorymodes20}. \textit{Bottom panel:} A close-up around the merger (orange dashed line) of the $(2,0)$ mode used to compare the CCE (purple, waveform type $b$), EXT (green, waveform type $a$), and memory calculation (dashed red). \textit{Parameters:} $Q=1.5$, $\chi = 0.6$.

Caption

\textit{Top:} Dominant oscillatory waveform and memory for an equal-mass, nonspinning binary (edge-on). The green band marks the interval $t \in [-30M,\,30M]$, during which approximately $66\%$ of the total radiated energy is emitted. \textit{Bottom:} Instantaneous gravitational-wave frequency $f_{\rm GW}$ of the dominant $(2,2)$ mode, together with the characteristic memory-growth timescale $\dot{f}_{\rm H}/f_{\rm H}$, and the corresponding energy flux $dE^{\rm GW}/dt$.

Caption

Characteristic strain in frequency space of the memory (solid) and of the dominant GW signal (dashed) the system in Fig.~\ref{fig:memwith freq} with zero spin (blue) and with spin $\chi=0.8$ (grey). The signal of the oscillatory GWs follows the typical shape of an IMR event, with a powerlaw increase in frequency during inspiral of $|\tilde h|\sim f^{-7/6}$, followed by a merger feature that ends in a sharp, damped ringdown at the highest frequencies. The memory signal on the other hand approaches a constant value $\Delta h_{\rm mem}/(2\pi)$ (dot-dashed) at low frequencies and decays at frequencies higher than $\sim f_L^{\rm M}$, as explained in the main text.

Caption

Spectrogram of a radiation signal composed of the dominant $(2,2)$ oscillatory mode and the $(2,0)$ nonlinear memory contribution, generated by an aligned-spin MBHB system. Two distinct power blobs are distinguishable: one exhibiting the typical chirp-like behavior of the oscillatory $(2,2)$ mode, spread in time but narrow in frequency; the second, the memory piece, well localized at merger time, but spread in frequency with the expected $1/f$ behavior and high-frequency damping. This plot was made using the waveform model $(a)$ with Eq.~\eqref{eq:memorymodes20} for its memory. \textit{Parameters:} $M=5\times10^5 M_\odot$, $Q=1$, $\chi = 0.95$, $\theta=\tfrac{\pi}{2}$. [The spectrogram is computed from \texttt{scipy.signal.spectrogram}, based on a short-time Fourier transform, and was optimized for visualization of the scale separations (e.g. time-frequency settings, colorbar saturation). It is using $N=5$ Hanning windows with $95 \%$ overlap, hence tolerating high correlations between pixels and tending to broaden the time localization of the signals power; see also~\cite{Inchauspe:2024ibs}]

Caption

As in Fig.~\ref{fig:memwith freq}, but for a system with aligned spins $\chi=0.8$. For reference, we show the instantaneous frequency evolution for the zero spin case in a dashed line.

Caption

Time domain TDI-A channel obtained, after the response of the links to the radiation signals in Fig.~\ref{fig:WaveformWithMem}. The total waveform is shown in blue together with the $(2,0)$ SWSH mode split into the pure memory component $h^L_{20}$ in red and the purely oscillatory part $h^\text{EXT}_{20}$ in green. The TDI-A time domain channel shows that the memory component is suppressed by the LISA response and TDI post-treatment compared to the oscillatory signals. The parameters are the same as in Fig.~\ref{fig:WaveformWithMem} $Q=1.5$, $\chi=0.6$, with additional physical parameters adapted to the LISA scale $M_z = 10^6 M_\odot$, $d_l = 10^4 \text{Mpc}$, $\theta = \pi/2$, $\psi=0$, and sky coordinates $\alpha = 0.74$ (right ascension), $\delta = 0.29$ (declination) as defined for instance in~\cite{Cogez:2025memoryLISA,LISA_RosettaStone}.

Caption

Frequency domain TDI-A channel obtained, after the response of the links, from Fig.~\ref{fig:WaveformWithMem}. The (2,2)+mem waveform is in blue and the memory component alone in red. The analytical PSD of the Science Requirements Document (SciRD)~\cite{LISA_SciRD} noise model was added in green, as a reference. This highlight the possible visibility of the memory component, here in the mHz region. The parameters are the same as in Fig.~\ref{fig:Comparing20Components}.

Caption

\small Memory SNR waterfall of the total binary redshifted mass against the mass ratio for three different ``memory models'', showcasing the importance of a well-defined signal of the memory rise. \textit{Left:} Memory defined as in Eq.~\eqref{eq:memorymodes}, where the spacetime averaging selects out the $(2,0)$ mode. \textit{Middle:} Memory defined as the full $(2,0)$ mode within the waveform model $(b)$, including the oscillatory feature associated to the ringdown. \textit{Right:} Memory defined as Eq.~\eqref{eq:memorymodes}, including $m\neq0$ modes, without any averaging over high-frequency scales [Fig.~\ref{fig:MultiModeMemory}], which corresponds to a memory model defined as the null part of the BMS balance laws. \textit{Parameters:} $\chi=0.4$, $\iota = \pi/3$, $d_l = 10^4 \textrm{Mpc}$, $\varphi_{ref} = 1$, $\psi = 0$, $\alpha = 0.74$, $\delta = 0.29$.

Caption

Example of the LISA frequency domain response for an out-of-band merger source. The inspiral at $t\simeq 1.10\,\mathrm{h}$ before merger (blue) and the memory signal (red) are shown. The SNR is $197$ in total, and $31$ for the memory alone. \textit{Parameters:} $\chi=0.7$, $\theta=\pi/2$, $d_l=10^2\,\mathrm{Mpc}$, $\varphi_{\rm ref}=0$, $\psi=0$, $\alpha=0.74$, $\delta=0.29$ and $M_z=10^4 M_\odot$.

Caption

Mean and dispersion values of $\Delta \log_{10}\mathcal{L} \approx \log_{10}\mathcal{B}$ for various parameters sets. The black dotted line shows the fitted power law linking $\SNRmem$ and $\log_{10}\mathcal{B}$, and the red dashed line correspond to the $\log_{10}\mathcal{B} = 2$ threshold. Different total redshifted mass $M_z$ parameters are distinguished by different colors (blue for $M_z=10^5 M_\odot$, orange for $M_z=10^6 M_\odot$, and green for $M_z=10^7 M_\odot$), both for computed points and the estimated dispersion, represented as colored areas. The dot points are computed using waveform model $(b)$ and are used to perform the fit. The red star points correspond to additional runs computed using {\tt SEOBNRv5HM}~\cite{pySEOBNR, SEOBNRv5HM} waveform, which includes higher modes, and are used to test the model.

Caption

Memory waterfall plot with stars corresponding to the computed $\log_{10}$Bayes factor. The color of the stars corresponds to the Jeffreys scale~\cite{Jeffreys_1998}, indicated by the colorbar under the figure. The light gray dashed line represents the ISO-SNR contour $\SNRmem = 3$. This plot used the waveform model $(b)$, including only the (2,2)-mode. \textit{Parameters:} $Q=1$, $\chi=0.4$, $\theta = \pi/3$, $\psi = 0$, $\alpha = 0.74$, $\delta = 0.29$.

Caption

Conversion of the Fig.\ref{fig:MemoryWaterfallPlotWithBF} SNR waterfall plot into a detectability plot. The main colorbar (on the right) provides information on how likely we are to detect memory for a given set of parameters. The black line shows the $\SNRmem = 3$ threshold. We kept stars from the previous Bayes factor computations to compare with the prediction, using the same colorbar as in Fig.~\ref{fig:MemoryWaterfallPlotWithBF}.

Caption

Probability of having an iteration with $\SNRmem$ greater than a given value (x-axis). Each color correspond to a population model in Ref.~\cite{Barausse_2020, Barausse_Lapi_2021}. Solid lines corresponds to 4-years iterations and dotted lines to 10-years. The red area covers the region where we are under the threshold $\SNRmem^{\textrm{thresh}} = 3$. The gray dashed line shows the value $\SNRmem = 5$ over which memory should be always detected. The $\SNRmem$ computed here include higher mode contribution.

Caption

Output of SWSH modes $h_{\ell m}$ of the memory formula Eq.~\eqref{eq:memorymodes} (or equivalently Eq.~\eqref{eq:memory-pre-3j}) without averaging over high-frequency scales for a particular BBH merger. \textit{Parameters: $Q=6$, $\chi = 0.4$}