Long Transient Gravitational Waves

• Long transient gravitational waves are signals lasting from tens of seconds to months, bridging the gap between short burst events and continuous emissions.
• Detection strategies utilize matched filtering, transient F-statistics, and machine learning to efficiently manage vast template banks and evolving signal morphologies.
• Astrophysical studies of these waves constrain neutron star interiors, glitch dynamics, and post-merger remnants, guiding future detector enhancements and multi-messenger approaches.

Long transient gravitational waves (GWs) are a class of gravitational-wave signals characterized by durations significantly longer than traditional burst events (≳1 s), but shorter than truly persistent continuous waves (CWs), typically ranging from tens of seconds to months or longer. These signals bridge the gap between burst searches (<10 s) and searches for steady (year-long) continuous emission, demanding specialized search methodologies and motivation from diverse astrophysical scenarios. Their detection and paper are critical for elucidating the dynamics of neutron-star interiors, the aftermath of catastrophic events such as binary mergers, supernovae, or pulsar glitches, and processes in accreting or highly magnetized compact stars.

1. Astrophysical Motivation and Source Classes

Long transient GWs arise in several astrophysical contexts distinguished by their emission durations, frequency content, and anticipated energy budgets:

• Neutron-Star Glitches: Sudden increases in spin frequency (Δf) observed in young pulsars (e.g. Vela, Crab) trigger post-glitch non-axisymmetric relaxation. This excites quadrupole deformations capable of radiating quasi-monochromatic GWs with durations from hours ($10^4$ s) to months ($10^7$ s). The recovery is governed by mutual coupling between the superfluid interior and the solid crust, and constrains physics such as the equation of state and superfluid angular momentum transfer (Collaboration et al., 2021, Modafferi et al., 2022).
• Newborn Neutron Stars and Magnetars: Core-collapse supernovae or binary neutron star mergers may leave behind highly deformed, rapidly rotating neutron stars—with large ellipticities or strong $r$-mode/Alfvén oscillations—which can emit long-lived GWs with rapidly evolving frequencies and amplitudes due to intense spin-down (characteristic timescales hours–weeks; braking index $n\sim3-7$) (Menon et al., 10 Dec 2025, Mérou et al., 10 Jul 2025, Attadio et al., 2 Jul 2024, Grace et al., 2023, Oliver et al., 2019, Miller et al., 2019, Quitzow-James et al., 2017).
• Fallback Accretion and Accretion Disk Instabilities (ADI): Black holes formed from massive star collapse can develop non-axisymmetric accretion disks, producing tens-to-hundreds of second downward-chirping GW signals. Neutron stars experiencing fallback accretion may undergo secular “spin-up,” emitting upward-chirping signals with durations $\sim10^2$–$10^4$ s (Collaboration et al., 2015, Macquet et al., 2021).
• Post-Merger Hyper-/Supramassive NS Remnants: Binary neutron star mergers (e.g., GW170817) can leave transient remnants with lifetimes up to hours or days, generating rapidly evolving narrowband GW emission as the remnant differentially spins down (Grace et al., 2023, Oliver et al., 2019).
• Magnetar Giant Flares: Soft gamma repeater flares can excite crustal or global Alfvén oscillations with QPOs, yielding GW emission persistent on $10^2$–$10^3$ s scales (Quitzow-James et al., 2017).
• Theoretical Long-Lived Mechanisms: Additional scenarios include free precession, $r$-mode instabilities, and exotic compact objects with secular or intermittent GW emission on days-to-months timescales (Thrane et al., 2015).

These source classes may exhibit nearly monochromatic emission (slow chirp), rapid frequency evolution, or broad stochastic spectral features, depending on underlying physics.

2. Signal Models and Parameterization

Long transient GW signals are best characterized as either quasi-monochromatic, power-law chirping, or amplitude-modulated oscillations:

• Quasi-Monochromatic/Transient-CW: The strain at the detector is

$$h(t) = h_0\big[F_+(t;\psi)A_+ + F_\times(t;\psi)A_\times\big]\cos[\Phi(t)+\phi_0]$$

with $F_+$, $F_\times$ the time-varying antenna pattern functions, $A_+ = (1+\cos^2\iota)/2$, $A_\times = \cos\iota$, and a phase model

$\Phi(t) = 2\pi\int_{t_0}^t f(t')\,dt'$

The instantaneous frequency may combine a Taylor expansion and a glitch relaxation term:

$$f(t) = f_0 + \dot{f}_0(t-t_0) + \Delta f\,e^{-(t-t_g)/\tau}H(t-t_g)$$

Signal envelopes are rectangular ($w_r$) or exponential ($w_e$) with duration $\tau$ (Collaboration et al., 2021, Modafferi et al., 2023, Moragues et al., 2022, Keitel et al., 2019).

• Power-Law Chirp/Spin-Down: For newborn neutron stars, the spin-down law

$\dot{f}_{\text{rot}}(t) = -k f_{\text{rot}}(t)^n$

yields a frequency evolution $f(t)$ that follows

$$f(t) = f_0\left(1 + \frac{t-t_0}{\tau}\right)^{-1/(n-1)}$$

The strain amplitude evolves as $h_0(t) \propto f(t)^2$ for quadrupole emission (Mérou et al., 10 Jul 2025, Miller et al., 2019, Attadio et al., 2 Jul 2024, Grace et al., 2023, Menon et al., 10 Dec 2025).

• Post-Merger Models: Piecewise polynomials or splines accurately accommodate rapidly evolving $f(t)$ following a BNS merger or core collapse (Grace et al., 2023).

In all cases, the phase and amplitude evolution, window function (rectangular, exponential), and relevant angular and polarization parameters define the template bank for matched-filter or semicoherent search methods.

3. Detection Methodologies and Statistical Frameworks

Long-transient GW searches employ both fully coherent matched filtering and robust unmodeled/statistical approaches, tuned to the expected signal morphology and computational constraints:

• Transient $\mathcal{F}$-Statistic: Extension of the standard continuous-wave $\mathcal{F}$-statistic, incorporating a transient window into the likelihood maximization over the four amplitude parameters. Time integration is restricted to $t_0 \leq t \leq t_0+T$ and matched to the transient duration. Exponential and rectangular windows are supported (Collaboration et al., 2021, Modafferi et al., 2023, Keitel et al., 2018, Keitel et al., 2019, Keitel, 2015).
• Template Grid Construction: The search grid is constructed in $(f, \dot{f})$ (and higher derivatives as needed), $t_0$, and $\tau$ using a mismatch metric that bounds worst-case fractional SNR loss (typically $\sim20\%$). For long durations, the number of templates scales as $T^{n}(f_{\max}-f_{\min})$ for $n$ phase parameters (Modafferi et al., 2022, Grace et al., 2023).
• Detection Thresholds and Significance: In Gaussian noise, $2\mathcal{F}$ follows a $\chi^2$ distribution with four degrees of freedom. Empirical background estimation using off-source segments or Monte Carlo time-slides yields extreme-value distributions (e.g., exponential-Gumbel fits) for total p-value calibration under multiple trials (Collaboration et al., 2021, Modafferi et al., 2022, Collaboration et al., 2015).
• Excess Power and Cross-Correlation Pipelines: For unmodeled or broadband signals, hierarchical pipelines such as PySTAMPAS perform data conditioning, construction of time-frequency maps, seed-based clustering, and cross-correlation of detector pairs. Detection statistics combine coherent SNR and incoherent residuals; performance is benchmarked by h$_{\text{rss}}^{50\%}$ (Macquet et al., 2021, Macquet et al., 2021, Collaboration et al., 2015, Thrane et al., 2010).
• Hough Transforms: For power-law frequency evolution (e.g., magnetar spindown), the Generalized Frequency Hough (GFH) maps time-frequency peaks to lines or curves in parameter space, populating a two-dimensional Hough map for candidate identification (Menon et al., 10 Dec 2025, Miller et al., 2019).
• Machine Learning Approaches: Convolutional neural networks (CNNs), optionally combined with denoiser autoencoders, are used either as fast triggers (classification of time-frequency maps, F-stat atoms, or reduced spectrogram blocks) or as hybrid detection statistics in concert with traditional matched-filtering (Attadio et al., 2 Jul 2024, Modafferi et al., 2023, Miller et al., 2019). CNNs, when carefully trained and validated, provide efficiencies at $\sim90\%$ and low false alarm probabilities ($\lesssim2\%$), with substantial computational speedup compared to full template-based searches.

4. Sensitivity, Upper Limits, and Computational Aspects

• Detection Sensitivity: For a monochromatic transient of amplitude $h_0$ and duration $\tau$,

$\rho^2 \approx \frac{h_0^2\,\tau}{S_n(f_0)}$

The minimal detectable $h_0$ at fixed SNR threshold scales as

$$h_0^{\min} \simeq \sqrt{ S_n(f_0)\, \rho_{\text{thr}}^2 / (2\tau) }$$

for rectangular envelopes. For broadband or rapidly evolving signals, h$_{\text{rss}}$ (root-sum-square strain) is used as a composite amplitude metric (Collaboration et al., 2021, Modafferi et al., 2022, Quitzow-James et al., 2017).

• Energy Constraints: The radiated GW energy over a transient is

$E_{\rm GW} = \frac{4\pi^2 c^3}{5G} d^2 f^2 h_0^2 T$

or, for a given energy budget, $$h_0(T) = \sqrt{ (5G E_{\rm GW}) / (4\pi^2 c^3 d^2 f^2 T) }$$ (Collaboration et al., 2021, Moragues et al., 2022, Keitel et al., 2019).

• Empirical Upper Limits: When no detection is made, frequentist upper limits $h_0^{95\%}(T)$ or $h_\text{rss}^{90\%}$ are set via simulated signal injection and recovery, as a function of transient duration, central frequency, and (for broadband signals) bandwidth (Collaboration et al., 2021, Quitzow-James et al., 2017, Grace et al., 2023).
• Computational Cost: Fully coherent transient $\mathcal{F}$-statistic searches over a broad parameter space are computationally intensive (up to $10^5$ CPU-hours per glitch event). GPU acceleration can yield $10$–$1000\times$ speedups, making O($10^8$)–O($10^{12}$) template searches feasible in days on modern hardware (Keitel et al., 2018, Mérou et al., 10 Jul 2025, Menon et al., 10 Dec 2025). CNN-based approaches, for detection, can run $10^6$–$10^7\times$ faster than classic Hough methods (Miller et al., 2019, Modafferi et al., 2023).
• Parameter-Space Coverage: Trade-offs between sensitivity and parameter space are managed by tuning template mismatch, coherence time, and the degeneracy utilization (e.g., $\xi$-space in inspiral searches). For inspiraling binaries, semi-coherent peakmap approaches with “degeneracy-aware” grid construction provide an optimal balance (Andrés-Carcasona et al., 7 Nov 2024).

5. Key Results and Astrophysical Implications

• Non-Detections and Implications: Recent high-sensitivity LIGO-Virgo (O3) searches set upper limits on long-transient GW emission in the range $h_0 \lesssim 10^{-25}$ for weeks-months durations. For some Vela-like glitches, these limits are comparable to or surpass the maximum GW emission allowed by the observed glitch energy, already constraining theoretical models of glitch relaxation, superfluid coupling, and NS dissipation (Collaboration et al., 2021, Moragues et al., 2022, Keitel et al., 2019, Modafferi et al., 2023).
• Model Constraints: In the context of the Yim & Jones model (post-glitch transient mountains, with a healing parameter $Q$), only large glitches with substantial $Q$ are within reach of current detectors. Third-generation detectors (Einstein Telescope, Cosmic Explorer) will access $35$–$40\%$ of known glitch events under optimistic energy constraints (Moragues et al., 2022).
• Exploration of Source Physics: Long-transient upper limits place lower bounds on neutron-star ellipticity, constrain $r$-mode amplitudes, and limit the efficiency of crustal energy transfer to GW emission in magnetar flares or post-merger remnants (Quitzow-James et al., 2017, Oliver et al., 2019, Grace et al., 2023).
• Algorithmic Advances: GFH-v2 and new GPU-accelerated pipelines now enable real-time, wide-sky follow-ups for post-merger or supernova triggers, expanding the practical horizon of long-transient GW searches to the Local Group ($\sim 4$ Mpc) with hours-to-days latency for order $10^8$ templates (Menon et al., 10 Dec 2025, Mérou et al., 10 Jul 2025).
• Sensitivity Horizons: With advanced detector noise, 90% efficiency horizon distances for newborn magnetars of ellipticity $3\times 10^{-3}$ are $\sim 3-4$ Mpc at $f_0 \sim 1$ kHz for direct searches triggered by SN 2023ixf, matching theoretical estimates (Menon et al., 10 Dec 2025).

6. Outlook and Future Directions

• Detector Enhancements: Ongoing upgrades (O4/O5) and 3G observatories will provide strain sensitivity at least $2\times$ better, enlarging both source reach and parameter space (Collaboration et al., 2021, Moragues et al., 2022, Menon et al., 10 Dec 2025).
• Algorithms and Pipelines: Future efforts aim to:
  • Combine machine-learning triggers with Bayesian posteriors or full matched filtering for parameter estimation (Attadio et al., 2 Jul 2024, Modafferi et al., 2023).
  • Optimize template bank placement in high-dimensional manifolds via analytic and empirical metrics (Andrés-Carcasona et al., 7 Nov 2024).
  • Incorporate astrophysical priors (e.g., braking index, ellipticity distributions), variable start times, and improved noise models (Menon et al., 10 Dec 2025).
  • Implement real-time, low-latency triggering for rapid electromagnetic or neutrino follow-up (Macquet et al., 2021, Mérou et al., 10 Jul 2025).
• Astrophysical Payoff: The identification of a long transient GW would provide unprecedented insights into neutron star internal physics, post-merger dynamics, superfluid processes, and the physics of compact binary evolution.
• Interdisciplinary Applications: Long-transient pipelines also inform searches for stochastic backgrounds exhibiting nonstationarity, constrain environmental noise artifacts, and enable multimessenger astrophysics with minimal electromagnetic guidance (Thrane et al., 2010, Thrane et al., 2015).

In summary, the field of long transient gravitational-wave search is at the confluence of advanced statistical signal processing, high-performance computing, and astrophysics. Systematic, physically-motivated searches leveraging both analytical techniques and machine learning now probe the regime where fundamental constraints can be placed on neutron-star interiors, glitch mechanisms, and post-merger evolution—ushering in a new era of GW transient astronomy (Collaboration et al., 2021, Modafferi et al., 2022, Macquet et al., 2021, Grace et al., 2023, Mérou et al., 10 Jul 2025, Menon et al., 10 Dec 2025).