The Delay Time Distribution of Quasi-Periodic Eruptions

Quasi-periodic eruptions (QPEs) are quasi-periodic X-ray bursts observed in the nucleus of a galaxy. Multiple pieces of observational evidence link QPEs to tidal disruption events (TDEs), which occur when stars are disrupted after approaching a supermassive black hole too closely. Post-starburst galaxies are overrepresented among the host galaxies of both TDEs and QPEs, though the mechanism causing this overrepresentation is unknown. While their physical origin is unclear, the delay time distribution (DTD) of QPEs, or rate of QPEs as a function of time since a burst of star formation, can constrain what mechanisms influence the QPE rate and possible QPE formation channels. We compile a catalog of 10 QPE host galaxies with optical spectra, model the stellar populations with Bagpipes, and retrieve the age of the most recent burst of star formation to construct the DTD of QPEs. We find that the QPE rate increases with post-burst age to reach a peak at ∼1 Gyr relative to a control sample, similar to the observational TDE DTD, though we cannot rule out a flat distribution of burst ages relative to a control sample. However, the fraction of QPE host galaxies with high (>1%) burst mass fractions is larger than the fraction of galaxies with high burst mass fractions in either a sample of TDE host galaxies or a sample of control galaxies. If the preferred QPE formation channel requires extreme mass ratio inspirals (EMRIs), then such EMRIs may be more readily produced by large, ∼1-Gyr-old bursts of star formation.

The main conclusions are:

• The observational QPE DTD, constructed from the subset of QPE host galaxies that have high burst mass fractions and a sample of control galaxies, increases with burst age until it peaks at ∼1 Gyr, then decreases. However, our sample size is not sufficient to distinguish any trends from a DTD that is flat over burst age.

• QPEs are overrepresented among host galaxies that have recently had a strong burst of star formation. The fraction of the QPE host galaxy sample with high burst mass fractions is significantly higher than the fraction of the TDE host galaxy sample with high burst mass fractions (70% vs. 35.7%).

• PSB and QBS galaxies are both overrepresented in our sample, albeit with large uncertainties. PSB (QBS) galaxies are overrepresented among QPE host galaxies by a factor of 49.0±46.5 (8.6±5.4) when compared to galaxies in SDSS DR8. PSB (QBS) galaxies are overrepresented among QPE host galaxies by a factor of 27.4±26.2 (6.7±4.3) when compared to the stellar mass-matched and redshift-matched expanded control sample.

TDE and QPE host galaxies are found to share some similarities, though the degree of overlap will have to be studied further with an expanded sample. The observations of stronger burst mass fractions in the QPE hosts relative to TDE hosts and control galaxies implies that, if EMRIs are necessary for QPEs to be observed, the EMRI rate may also depend on the host galaxy SFH. Further study of QPEs can lead to estimates of the EMRI rate and their contribution to the stochastic gravitational wave background.

Improvements to AI Systems:

1. Temporal Population-Synthesis Modeling

• Improve AI systems that model stellar populations and galaxy evolution by incorporating burst-age-dependent event rates (e.g., QPE/TDE likelihood as a function of post-starburst age).

• The improved system can predict the probability of transient events (QPEs, TDEs) in a galaxy based on its star-formation history, enabling more accurate transient surveys and follow-up targeting.

2. Bayesian Hierarchical Inference for Small Samples

• Enhance AI frameworks for constructing delay time distributions (DTDs) from sparse catalogs (e.g., 10 host galaxies) by using hierarchical Bayesian models that explicitly account for selection effects, mass/redshift matching, and control-sample normalization.

• The improved system can produce robust DTD constraints with credible intervals, even when sample sizes are small, and automatically flag when flat vs. peaked distributions are statistically indistinguishable.

3. Galaxy–Transient Co-Evolution Classifier

• Develop an AI classifier that integrates optical spectra (e.g., from Bagpipes stellar population fits) with transient occurrence data to identify which galaxy properties (burst mass fraction, post-burst age, spectral type like PSB/QBS) most strongly predict QPE or TDE activity.

• The improved system can rank candidate host galaxies for future X-ray/optical monitoring, prioritizing those with high burst mass fractions and 1 Gyr post-burst ages.

4. Simulation-to-Observation Inference for EMRI Rates

• Improve AI systems that simulate extreme mass ratio inspiral (EMRI) formation by linking them to observed host-galaxy star-formation histories.

• The improved system can generate synthetic galaxy populations with varying burst mass fractions and ages, then predict the resulting EMRI rate and its contribution to the stochastic gravitational wave background (detectable by LISA). This enables direct comparison with observed QPE rates.

5. Anomaly Detection for QPE-like Transients

• Enhance AI-based transient detection pipelines to use host-galaxy context (e.g., post-burst age, burst mass fraction) as priors, not just light-curve features.

• The improved system can more efficiently identify candidate QPEs in large X-ray surveys (e.g., eROSITA, Athena) by flagging galaxies with recent strong starbursts, reducing false positives and enabling faster discovery of rare events.

6. Causal Inference for Formation Channels

• Improve AI systems that infer causal pathways between galaxy evolution and nuclear transients by using counterfactual reasoning (e.g., What would the QPE rate be if the burst mass fraction were lower?) based on the observed DTD and control samples.

• The improved system can disentangle whether the QPE rate is driven by burst age, burst mass, or both, and test whether EMRI formation is the dominant channel vs. other mechanisms (e.g., disk instability).

Quasi-periodic eruptions (QPEs) are quasi-periodic X-ray bursts observed in the nucleus of a galaxy. Multiple pieces of observational evidence link QPEs to tidal disruption events (TDEs), which occur when stars are disrupted after approaching a supermassive black hole too closely. Post-starburst galaxies are overrepresented among the host galaxies of both TDEs and QPEs, though the mechanism causing this overrepresentation is unknown. While their physical origin is unclear, the delay time distribution (DTD) of QPEs, or rate of QPEs as a function of time since a burst of star formation, can constrain what mechanisms influence the QPE rate and possible QPE formation channels. We compile a catalog of 10 QPE host galaxies with optical spectra, model the stellar populations with Bagpipes, and retrieve the age of the most recent burst of star formation to construct the DTD of QPEs. We find that the QPE rate increases with post-burst age to reach a peak at 1 Gyr relative to a control sample, similar to the observational TDE DTD, though we cannot rule out a flat distribution of burst ages relative to a control sample. However, the fraction of QPE host galaxies with high (>1%) burst mass fractions is larger than the fraction of galaxies with high burst mass fractions in either a sample of TDE host galaxies or a sample of control galaxies. If the preferred QPE formation channel requires extreme mass ratio inspirals (EMRIs), then such EMRIs may be more readily produced by large, 1-Gyr-old bursts of star formation.