Cosmic hide and seek: the volumetric rate of X-ray quasi-periodic eruptions

Multi-wavelength extragalactic nuclear transients, particularly those detectable as multi-messengers, are among the primary drivers for the next-generation observatories. X-ray quasi-periodic eruptions (QPEs) are the most recent and perhaps most peculiar addition to this group. Here, we report a first estimate of the volumetric rate of QPEs based on the first four discoveries with the eROSITA X-ray telescope onboard the Spectrum Roentgen Gamma observatory. Under the assumption, supported by a suite of simulated light curves, that these four sources sample the intrinsic population somewhat homogeneously, we correct for their detection efficiency and compute a QPE abundance of $\mathscr{R}_{{\rm vol}} = 0.60_{-0.43}^{+4.73} \times 10^{-6}\,$Mpc$^{-3}$ above an intrinsic average $\log L_{\rm 0.5-2.0\,keV}^{\rm peak} > 41.7$. Since the exact lifetime of QPEs ($\tau_{\rm life}$) is currently not better defined than between a few years or few decades, we convert this to a formation rate of $\mathscr{R}_{\rm vol}/\tau_{\rm life}\approx 0.6 \times 10^{-7} (\tau_{\rm life}/10\,\mathrm{y})^{-1}\,$Mpc$^{-3}\,$year$^{-1}$. As a comparison, this value is a factor $\sim10\,\tau_{\rm life}$ times smaller than the formation rate of tidal disruption events. The origin of QPEs is still debated, although lately most models suggest that they are the electromagnetic counterpart of extreme mass ratio inspirals (EMRIs). In this scenario, the QPE rate would thus be the first-ever constraint (i.e. a lower limit) to the EMRI rate from observations alone. Future discoveries of QPEs and advances in their theoretical modeling will consolidate or rule out their use for constraining the number of EMRIs detectable by the LISA mission.