Search for nonlinear memory from subsolar mass compact binary mergers

Abstract: We present the first results of the search for nonlinear memory from subsolar mass binary black hole (BBH) mergers during the second observing run (O2) of the LIGO and Virgo detectors. The oscillatory chirp signal from the inspiral and merger of low mass BBHs ($M_\mathrm{Tot} \leq 0.4 M_\odot$) are at very high frequencies and fall outside the sensitivity band of the current ground-based detectors. However, the non-oscillatory memory signal during the merger saturates towards the lower frequencies and can be detected for those proposed BBHs. We show in this work that the morphology of the memory signal depends minimally upon the source parameters of the binary, thus only the overall amplitude of the signal changes and hence the result can be interpolated for extremely low mass BBH mergers. We did not find any signal which can be interpreted as a memory signal and we place upper limits on the rate of BBH mergers with $M_\mathrm{Tot} \leq 0.4 M_\odot$ for the first time.

• The paper presents the first targeted search for nonlinear gravitational wave memory from subsolar mass binary mergers using Advanced LIGO and Virgo O2 data.
• It demonstrates that the bandpassed memory burst shows a parameter-independent morphology with linear amplitude scaling, enabling extrapolation to lower mass systems.
• The analysis sets upper limits on the merger rate of subsolar mass binaries, emphasizing the potential to probe exotic astrophysical populations.

This paper (2005.03306) presents the first search specifically targeting the nonlinear gravitational wave (GW) memory effect from subsolar mass binary black hole (BBH) mergers using data from the second observing run (O2) of the Advanced LIGO and Advanced Virgo detectors. Subsolar mass compact objects (below 1 solar mass, $M_\odot$) are not expected to form through conventional stellar evolution, making their detection a potential signature of exotic physics, such as primordial black holes or dark matter interactions. While the standard oscillatory GW signal from low-mass BBHs is emitted at high frequencies, potentially falling outside the sensitivity band of current ground-based detectors, the nonlinear memory effect, which is a non-oscillatory, persistent change in spacetime geometry, manifests as a burst-like signal during the merger and saturates towards lower frequencies. This low-frequency characteristic makes it potentially detectable even if the corresponding high-frequency oscillatory signal is not, a scenario referred to as "orphan memory."

The nonlinear memory (also known as Christodoulou memory) arises from the gravitational waves themselves sourcing additional gravitational waves, reflecting the nonlinearity of General Relativity. The memory waveform typically shows a gradual increase during the inspiral, a sharp jump during the merger, and then saturates at a final value. This merger jump, particularly for short-duration events like those from low-mass binaries, appears as a burst signal. In the frequency domain, a short burst leads to power extending to low frequencies, with the amplitude spectral density proportional to $1/f$. This property is crucial because ground-based detectors like LIGO and Virgo are most sensitive to GWs in the frequency range of tens to hundreds of Hertz, while their sensitivity decreases significantly at higher frequencies where the oscillatory signals from subsolar binaries would be.

To implement a search for this memory signal, the authors first needed to generate theoretical waveforms. Since standard numerical relativity (NR) simulations often struggle to accurately extract the non-oscillatory memory component, the memory contribution was computed separately from existing NR surrogate models of the oscillatory waveforms. The oscillatory strain $h_+ - i h_\times$ is decomposed into spin-weighted spherical harmonics $h^{\ell m}$. The memory contribution to a specific mode $h^{\ell m}_{mem}$ is calculated via an integral over time of the product of derivatives of pairs of oscillatory modes $\dot{h}^{\ell'm'}$ and $\dot{\bar h}^{\ell''m''}$ (Eq. \ref{eq:hlmmem}). This involves angular integrals represented by $G^{\ell \ell' \ell''}_{m m'm''}$ (Eq. \ref{eq:G}) which involve Wigner 3-$j$ symbols. The dominant memory contribution comes from the $h^{20}_{mem}$ mode, primarily sourcing the $h_+$ polarization, especially for equal-mass, non-precessing binaries. The calculation uses the NRSur7dq4 (Varma et al., 2018) surrogate model, which includes modes up to $\ell \leq 4$ and covers spinning and unequal-mass binaries.

Due to the low-frequency cutoff of terrestrial detectors (around 10 Hz), the theoretical memory waveform, which is a step function or saturating signal, is effectively high-pass filtered. This filtering transforms the merger jump into a short-duration burst signal with oscillations around the central peak (Figure \ref{fig:memwaveform}). The morphology of this bandpassed memory burst was found to be minimally dependent on the specific source parameters (mass ratio, spins) for low-mass systems. Crucially, the peak amplitude of the bandpassed memory scales approximately linearly with the total mass for subsolar systems (Figure \ref{fig:massamp}). Furthermore, the spectral content of the memory signal for various subsolar mass systems appears similar, with the central frequency of the burst remaining around the detector's most sensitive region (~100 Hz), unlike the oscillatory signal which shifts to higher frequencies for lower masses (Figure \ref{fig:memfreq}). This parameter-independent morphology and linear amplitude scaling allows the sensitivity results obtained from injecting a few specific waveforms to be extrapolated to a wider range of subsolar masses and parameters.

The search for these memory signals was performed on approximately 118 days of coincident data from the LIGO Hanford and Livingston detectors during O2. The analysis employed the coherent WaveBursts (cWB) algorithm [klimenko-2008, klimenko-2016], which is an unmodeled search method designed to detect generic GW transients by identifying excess power in the time-frequency domain across a network of detectors. The search configuration used was based on the all-sky search for short-duration transients in O2 [all-skyo2], focusing on the low-frequency range of 32-1024 Hz.

A key implementation consideration for this search was the presence of various noise artifacts (glitches) in the detector data. The cWB analysis categorizes low-frequency triggers into two bins: LF1, which contains non-stationary lines and short-duration "blip" glitches, and LF2 for others. Unfortunately, the morphology of the subsolar mass memory signal causes it to fall predominantly into the LF1 bin. The high rate of blip glitches in this bin significantly affects the background noise level and reduces the search sensitivity for memory signals, as there was no specific veto applied to discriminate memory from blips in this analysis. The background was estimated by time-shifting the data streams of the two detectors to break correlations from real astrophysical signals. The data was processed in approximately 5-day chunks to account for the non-stationary nature of the detector noise.

No signals consistent with the memory from a subsolar mass BBH merger were detected in the O2 data. This null result was used to place upper limits on the merger rate of subsolar mass BBHs. The sensitivity of the search was characterized by computing the detection "range" (the distance within which a signal could be detected with an inverse false alarm rate of at least 1 year) and the sensitive volume $\langle V \rangle$ (Eq. \ref{eq:V}). The range was estimated by injecting simulated memory waveforms corresponding to equal-mass binaries with total masses of 0.02, 0.2, and 2 $M_\odot$, with and without aligned spins, into the O2 data and determining the detection efficiency as a function of distance. The linear scaling of the memory amplitude with mass for subsolar systems (Figure \ref{fig:massrange}) allowed for extrapolation of the sensitive range to even lower masses. The upper limit on the merger rate density $R_i$ for a given population $i$ was then calculated using the sensitive volume and the total coincident observation time $T$ (114.78 days) as $R_i = \frac{2.3}{\langle V T \rangle_i}$ at the 90% confidence level (Eq. \ref{eq:rate}).

The resulting rate upper limits (Figure \ref{fig:rate}) are several orders of magnitude less stringent than those obtained by previous searches for the oscillatory part of subsolar mass binaries [LIGO_subsolar-2018, LIGO_subsolar-2019] in the mass range where the oscillatory signal is still detectable (e.g., above 0.4 $M_\odot$). This is expected because the memory signal is inherently weaker than the oscillatory signal (roughly an order of magnitude difference in amplitude) and its detection efficiency depends on the inclination angle differently, generally reducing its detectability compared to the optimal orientation for the oscillatory signal. However, the crucial advantage of the memory search is its sensitivity to very low mass systems (down to $\approx 0.01 M_\odot$) and potentially highly spinning systems, where the oscillatory component is pushed to very high frequencies, far outside the detector band, making template-based searches computationally infeasible or impossible.

In conclusion, this work demonstrates the feasibility of searching for gravitational wave memory from subsolar mass compact binary mergers and provides the first upper limits on their merger rate based solely on the memory signature. While less sensitive than oscillatory searches where applicable, the memory channel offers a unique probe of parameter space inaccessible to standard methods, potentially revealing exotic astrophysical populations. Future improvements could include tuning the search algorithm specifically for memory signals and developing better methods to discriminate memory signals from instrumental glitches, particularly "blips," which currently limit sensitivity in the low-frequency band where memory signals reside. Detecting a memory-only signal without a corresponding oscillatory counterpart would strongly indicate an astrophysical source in a previously unexplored regime.