2404/2404.00941.md

Title: Repeating Nuclear Transients as Candidate Electromagnetic Counterparts of LISA Extreme Mass Ratio Inspirals

URL Source: https://arxiv.org/html/2404.00941

Markdown Content: Back to arXiv

This is experimental HTML to improve accessibility. We invite you to report rendering errors. Use Alt+Y to toggle on accessible reporting links and Alt+Shift+Y to toggle off. Learn more about this project and help improve conversions.

Why HTML? Report Issue Back to Abstract Download PDF Abstract 1Introduction 2QPP Frequencies of LISA-relevant EMRIs Today 3Modeling RE J1034+396 as an EMRI 4Discussion 5Conclusion References License: CC BY 4.0 arXiv:2404.00941v2 [astro-ph.HE] 01 Jul 2024 Repeating Nuclear Transients as Candidate Electromagnetic Counterparts of LISA Extreme Mass Ratio Inspirals Shubham Kejriwal,1 Vojtěch Witzany,2 Michal Zajaček,3 Dheeraj R. Pasham,4 and Alvin J. K. Chua1,5 1Department of Physics, National University of Singapore, Singapore 117551 2Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University, CZ-180 00 Prague, Czech Republic 3Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic 4Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA, USA 5Department of Mathematics, National University of Singapore, Singapore 119076 shubhamkejriwal@u.nus.edu (SK)vojtech.witzany@matfyz.cuni.cz (VW)zajacek@physics.muni.cz (MZ)dheeraj@space.mit.edu (DP)alvincjk@nus.edu.sg (AC) Abstract

Extreme-mass-ratio inspirals (EMRIs) are one of the primary targets for the recently adopted millihertz gravitational-wave (GW) observatory LISA. Some previous studies have argued that a fraction of all EMRIs form in matter-rich environments, and can potentially explain the dozens of soft X-ray band ( ∼ 10 − 1 ⁢ keV ), low-frequency ( ∼ 0.1 mHz) periodic phenomena known as quasi-periodic eruptions (QPEs) and quasi-periodic oscillations (QPOs). Here, using a representative EMRI population retrofitted with cutoffs on LISA-band SNRs and luminosity distances to account for the sensitivity of current instruments, we estimate the mean frequency band in which QPEs and QPOs originating from detectable LISA EMRIs may be emitting an X-ray signal “today” (i.e., in 2024) to be 0.46 ± 0.22 mHz. We also model the well-known QPO source, RE J1034+396, which falls in this frequency band, as an EMRI assuming its primary black hole mass to be 10 6 − 10 7 ⁢ 𝑀 ⊙ . Through a prior-predictive analysis, we estimate the orbiting compact object’s mass to be 46 − 40 + 10 ⁢ 𝑀 ⊙ and the source’s LISA-band SNR as ≈ 14 , highlighting it as a candidate multi-messenger EMRI target. We also highlight the role of current and near-future X-ray and UV observatories in enabling multi-messenger observations of EMRIs in conjunction with LISA, and conclude with a discussion of caveats of the current analysis, such as the exclusion of eccentricity and inclination from the model, and the measurability of sub-solar mass compact object EMRIs.

keywords: black hole physics – accretion, accretion discs – gravitational waves – transients: black hole mergers – transients: black hole - neutron star mergers – galaxies: nuclei †pubyear: 2024 †pagerange: Repeating Nuclear Transients as Candidate Electromagnetic Counterparts of LISA Extreme Mass Ratio Inspirals–A 1Introduction

Extreme-mass-ratio inspirals (EMRIs) are black hole binary systems in which a stellar-mass compact object (CO) of mass 𝜇 ∼ 1 − 10 2 ⁢ 𝑀 ⊙ completes ∼ 10 5 orbits around a supermassive black hole (MBH) of mass 𝑀 ∼ 10 5 − 10 7 ⁢ 𝑀 ⊙ in its strong-gravity regime ( 𝑟 ∼ 10 ⁢ 𝐺 ⁢ 𝑀 / 𝑐 2 ) over a period of ∼ 1 − 10 years. They emit millihertz gravitational waves signal (GW) in the process (Amaro-Seoane et al., 2015; Barack & Cutler, 2004), and are one of the main targets of the recently adopted European Space Agency (ESA)-led space-based GW observatory, Laser Interferometer Space Antenna (LISA) (eLISA Consortium et al., 2013; Amaro-Seoane et al., 2017; Colpi et al., 2024). EMRIs have the potential to be unique scientific laboratories for conducting tests of general relativity (GR), constraining the astrophysical MBH mass-function, and inferring cosmological parameters (Amaro-Seoane et al., 2007; Amaro-Seoane, 2018; Gair et al., 2013; Babak et al., 2017; Berry et al., 2019; Laghi et al., 2021).

Over the years, various EMRI formation channels have been proposed (Amaro-Seoane, 2018): (i) Direct capture, where the many-body scattering in the nuclear star cluster around an MBH pushes a CO into the gravitational-wave loss cone (Hils & Bender, 1995; Sigurdsson & Rees, 1997); (ii) The tidal disruption of a CO binary by the MBH (Hills mechanism; Hills, 1988) where one component is ejected as a hypervelocity star and the other component becomes more tightly bound to the MBH (Miller et al., 2005); (iii) The capture of COs and their stellar precursors by the accretion disk in an active galactic nucleus (AGN) and their subsequent migration to tighter orbits around the MBH due to disk-CO interactions (Paczynski, 1978; Syer et al., 1991; Rauch, 1995; Karas & Šubr, 2001; Pan & Yang, 2021); and (iv) The formation of stellar objects later evolving into COs in situ within the accretion disk and their continual migration towards the MBH while embedded within the disk (Paczynski, 1978; Collin & Zahn, 1999; Levin & Beloborodov, 2003; Levin, 2007; Sigl et al., 2007). While the disk-assisted formation channels (iii)-(iv) are only relevant for the ∼ 10 − 2 − 10 − 1 fraction of galactic nuclei that are (or recently have been) AGN, in some population models such as in that of Pan & Yang (2021) they have significantly higher formation rates than the “dry” channels (i)-(ii) and, consequently, produce a major fraction of the predicted LISA EMRIs.

In addition to GW radiation, EMRI sources formed in channels (iii)-(iv) can trigger high energy electromagnetic (EM)-band emissions owing to CO interactions with the matter in the MBH accretion disk, making them viable candidates for multi-messenger detections (Lehto & Valtonen, 1996; Semerák et al., 1999; Ivanov et al., 1998; Pihajoki, 2016; Suková et al., 2021). Channels (i)-(ii) can also lead to CO – hot flow interactions in low-luminosity galactic nuclei. However, because of lower gas densities, the emerging weak EM signal limits such GW sources to the local Universe and rather fine-tuned setups with a low likelihood, such as a pulsar on a tight orbit around Sgr A* (Eckart et al., 2017).

Identification of individual multi-messenger sources would be invaluable for simplifying data analysis schemes such as the LISA global-fit (Vallisneri, 2009; Cornish & Crowder, 2005; Littenberg & Cornish, 2023). This can be done by setting tight constraints on the extrinsic parameter set, while lifting degeneracies arising from the GW-only inference of beyond vacuum-GR EMRI parameters (Speri et al., 2023; Kejriwal et al., 2023). Such observations will also enable independent constraints on the redshift of the host galaxy and the properties of the galactic environment, without invoking statistical methods like the dark siren method (Schutz, 1986a; Del Pozzo, 2012; see also Gray et al., 2023 for a review). Multi-messenger detections of EMRIs would thus help to decouple the MBH parameters from different galaxy components (Kormendy & Ho, 2013; Karas et al., 2021) and significantly improve the constraining power of LISA on the luminosity distance-redshift relation in cosmology (Schutz, 1986b; Holz & Hughes, 2005). The precise knowledge of the parameters of an accreting MBH would additionally allow the resolution of long-standing conundrums such as the radio loud/quiet dichotomy of AGN (Sikora et al., 2007) and their conjectured relation to MBH spin (see, e.g., Tchekhovskoy et al., 2010).

Independent of the specific model, one can expect the CO-disk interactions to manifest as regular time variability in the optical to X-ray bands, corresponding to the temperatures of the accretion disk at various distances from the MBH (Shakura & Sunyaev, 1973). Another possible signature could come in the X-ray/radio band in the case of launched outflows (Suková et al., 2021; Li et al., 2023; Pasham et al., 2024a). Variability in the soft X-ray band (around ∼ 10 − 1 ⁢ keV or ∼ 10 6 ⁢ K ) corresponds to regions in the disk within a few gravitational radii near the MBH. At such distances, GW radiation is also non-negligible. The characteristic period of this variability should then correspond to the orbital period of the CO inferred from the GW signal of the source, which will be ∼ hours to minutes for EMRIs relevant to LISA (see Section 2).

In line with this description, the recently discovered quasi-periodic eruptions (QPEs), which are signals of semi-regular bursts in soft X-ray energy bands around ∼ 10 − 1 ⁢ keV , and which can be traced back to centers of known galaxies, have been proposed to be EM counterparts of environment-rich EMRIs (Suková et al., 2021; Chen et al., 2022; Metzger et al., 2022; Pasham et al., 2024a; Linial & Metzger, 2023; Franchini et al., 2023; Zhou et al., 2024). Notably, QPEs may also be explained by accretion disk instabilities around the MBH (Miniutti et al., 2019) or gravitational lensing during the inspiral of massive black hole binaries (MBHBs) (Ingram et al., 2021), but such models only explain a subset of the observed signals. Coincidentally, some QPEs have been found in galaxies hosting MBHs of masses ∼ 10 6 ⁢ 𝑀 ⊙ (Wevers et al., 2022; King, 2023b), which is consistent with the mass range of expected LISA-relevant EMRIs ( 10 5 − 10 7 ⁢ 𝑀 ⊙ ) further motivating a multi-messenger search.

Another type of possibly related phenomenon are the less abrupt quasi-periodic oscillations (QPOs) in the X-ray fluxes of AGNs, such as the ones reported in RE J1034+396 (Gierliński et al., 2008), 2XMM J123103.2+110648 (Lin et al., 2013), Sw J1644+57 (Wang et al., 2014), MS 2254.9-3712 (Alston et al., 2015), 1H 0707-495 (Pan et al., 2016), Mrk 766 (Boller, Th. et al., 2001; Zhang et al., 2017), ASASSN-14li (Pasham et al., 2019) or MCG-06-30-15 (Gupta et al., 2018), all with the estimated periods of oscillations falling within few hours. While many studies have tried to make a connection between AGN QPOs and their well-known kHz analogues in X-ray binaries (van der Klis, 2000; Remillard & McClintock, 2006; Ingram & Motta, 2019), their mechanisms may be completely unrelated. Indeed, it has been suggested in the literature that AGN QPEs and QPOs may originate from the same source under different configurations (King, 2023a; Linial & Metzger, 2023).

Hence, in this paper, we proceed with the assumption that AGN QPEs and QPOs are both caused by EMRIs, jointly referring to them as quasi-periodic phenomena (QPPs). The relative differences in the intensity of peaks and duty cycles between QPEs and QPOs will then be assumed to be caused by variations in the particular parameters of the sources such as the orbital inclination of the EMRIs or the accretion states of the disks (see Section 4.2.4 for a detailed discussion).

1.1Executive Summary of Results

An effective multi-messenger study of LISA-relevant EMRIs would require us to first determine the QPP frequency band in which their EM counterparts will most likely be observable. We find in Section 2 that EMRIs in the LISA observation window (assuming an operation time from the year 2037 to 2041) should emit QPP signals “today”, i.e. in 2024, in a wide frequency range with a mean of ≈ 0.46 mHz and a standard-deviation of 𝜎 ≈ 0.22 mHz. As a demonstrative case study, in Section 3 we analyze the well-known QPO source RE J1034+396, which has a suitable oscillatory signature with the period of ∼ 1 hr in the relevant EM energy band of ∼ 0.2 − 10 ⁢ keV (Gierliński et al., 2008; Jin et al., 2020). Assuming an MBH mass of 10 6 − 10 7 ⁢ 𝑀 ⊙ , we model it as an EMRI to obtain prior-predictive distributions on the CO’s mass of 46 − 40 + 10 ⁢ 𝑀 ⊙ , the source’s LISA-band SNR with a mean of ≈ 14 , and LISA-band detection probability of ≈ 0.25 with an SNR cutoff of 15 , highlighting it as a potential multi-messenger EMRI candidate. Given the QPP frequency band, in Section 4.1 we then highlight some current and near-future X-ray observatories that are sufficiently sensitive to signals in this band, and can thus be employed to obtain a catalog of EM counterparts of LISA-relevant EMRIs. We discuss some caveats of our analysis in Section 4.2, and conclude in Section 5. We assume the standard flat Λ CDM model with 𝐻 0

70 ⁢ km ⁢ s − 1 ⁢ Mpc − 1 and Ω m

0.3 throughout the text.

2QPP Frequencies of LISA-relevant EMRIs Today 2.1EMRI catalog

We begin by assuming that the astrophysical EMRI population is described by the M1 population catalog of Babak et al. (2017)1. However, the source parameters in the M1 catalog are given at plunge, i.e. when the CO is near the MBH’s innermost stable orbit. Pozzoli et al. (2023) back-evolved these plunge-time parameters to a random time 𝑡 ∈ [ 0 , 𝑇 back ] years before plunge (where 𝑇 back

20 ⁢ ( 𝑀 / 10 4 ⁢ 𝑀 ⊙ ) years) using the leading-order Peters & Mathews waveform model (Peters & Mathews, 1963), given by:

d ⁢ 𝑓 orb d ⁢ 𝑡

96 ⁢ 𝐺 5 / 3 5 ⁢ 𝑐 5 ⁢ ( 2 ⁢ \uppi ) 8 / 3 ⁢ ℳ 5 / 3 ⁢ 𝑓 orb 11 / 3 ⁢ ℱ ⁢ ( 𝑒 ) ,

(1)

d ⁢ 𝑒 d ⁢ 𝑡

− 𝐺 5 / 3 15 ⁢ 𝑐 5 ⁢ ( 2 ⁢ \uppi ) 8 / 3 ⁢ ℳ 5 / 3 ⁢ 𝑓 orb 8 / 3 ⁢ 𝒢 ⁢ ( 𝑒 ) ,

(2)

where 𝑓 orb is the orbital frequency of the EMRI, 𝑒 is its eccentricity, and

ℳ

( 𝑀 ⁢ 𝜇 ) 3 / 5 ( 𝑀 + 𝜇 ) 1 / 5 ,

(3)

ℱ ⁢ ( 𝑒 )

1 + ( 73 / 24 ) ⁢ 𝑒 2 + ( 37 / 96 ) ⁢ 𝑒 4 ( 1 − 𝑒 2 ) 7 / 2 ,

(4)

𝒢 ⁢ ( 𝑒 )

304 ⁢ 𝑒 + 121 ⁢ 𝑒 3 ( 1 − 𝑒 2 ) 5 / 2 .

(5)

Pozzoli et al. (2023) thus describes a more realistic EMRI catalog such that only a fraction of EMRIs will plunge in the LISA observation window. We employ this modified catalog in our study. Additionally, note that the purpose of the M1 catalog is to synthesize a generous set of EMRIs that is significantly larger than what will be detected by LISA, which is then used to explore its detection capabilities. As such, 𝑇 back for 𝑀 ∼ 10 6 ⁢ 𝑀 ⊙ can be as large as ∼ 1000 years. Sources this far away from plunge will produce negligible ( ≪ 1 ) SNRs in the LISA band in general. For simplicity, we thus only consider sources that plunge within 𝑡 ≤ 100 years after the start of LISA observation window. We additionally require the sources to have eccentricities, 𝑒 ≤ 0.8 , mass-ratios, 𝜇 / 𝑀 ≤ 10 − 4 , and inclination angles between the orbital plane and accretion disk, 𝜄 ≥ 0.01 . We assume the plane of the accretion disk is perpendicular to the spin-axis of the MBH, but briefly discuss the effect of misaligned planes in Section 4.2.4. These preliminary cutoffs are based on conservative estimates on the distribution of environment-rich EMRI parameters and validity of the GW waveform model (see also discussion in Sections 3 and 4.2.3).

Equations  (1)-(5) do not account for contributions from the CO-disk interactions, due to the unavailability of such models for generic orbit (eccentric and inclined) Kerr MBH EMRIs. While their inclusion may introduce significant dephasing in the long-term inspiral evolution, they are only expected to contribute to the inspiral at the next-to-leading order, as shown by previous studies (see e.g., Kocsis et al., 2011; Barausse et al., 2014; Speri et al., 2023, and also subsection 4.2.3). However, we stress that these effects may significantly influence the long-term evolution of the EMRI and should be explored in future studies. A more realistic analysis should also use higher-order inspiral evolution models with a larger parameter space (including, for example, the evolution of inclination, MBH spin, etc.), in contrast to the leading post-Newtonian (PN) order Peters & Mathews Schwarzschild EMRI model employed here for its computational feasibility.

2.2Back-evolving to current orbital frequencies

We randomly distribute the M1 catalog parameters in the expected LISA operation window (Gregorian calendar year 2037 to 2041)2, and back-evolve the EMRI’s orbital frequency and eccentricity parameters to the year 2024, i.e. “today”, using Eqs. (1) and (2) (see Fig.1). From the EMRI orbital frequencies “today”, 𝑓 orb , init , we can then calculate the distribution of QPP frequencies for the population’s EM counterparts, 𝑓 QPP , as

𝑓 QPP ≡ 𝑛 col × 𝑓 orb , init

(6)

where 𝑛 col is the number of CO-disk collisions per orbit, assumed to be 2 throughout our analysis and supported by previous estimates (Linial & Metzger 2023, 2024, see also Suková et al. 2021, Xian et al. 2021, and Franchini et al. 2023). Note that this relation is only valid for EMRIs evolving in circular orbits, and while environment-rich EMRIs with COs evolving in the disk are expected to rapidly circularize under the effect of CO-disk interactions (Tanaka & Ward, 2004; see also Kley & Nelson, 2012), such interactions will be small in EMRIs with misaligned accretion disk and CO orbital planes. While we use the approximate relation in Eq. (6) in our study as a leading-order estimate, we emphasize that such a relation should be generalized to include eccentric EMRIs in future work. We also discuss the potential impact of introducing non-negligible eccentricity and inclination in the model in Section 4.2.4.

In Fig. 1, we visualize the back-evolved detector-frame orbital frequency and eccentricity trajectories for the considered EMRI population in the LISA band till 2001.We also estimate their LISA-band SNRs, 𝜌 , using the solar system barycenter frame point-detector approximation of Bonetti & Sesana (2020) (see Eq. (17), (18), and (23) in their paper). The cross-section of these trajectories evaluated for the year 2024 thus gives the distribution of detector-frame orbital frequencies 𝑓 orb , init

𝑓 orb , source / ( 1 + 𝑧 ) and eccentricities 𝑒 init “today”, and are summarized in Fig. 2.

Figure 1:Back evolution of the trajectories of detector-frame orbital frequency, 𝑓 orb , (upper panel) and eccentricity, 𝑒 , (lower panel) for EMRIs that are uniformly distributed in LISA’s observation window (orange band), i.e. years 2037-2041 to 2001. Time “today”, 2024, is demarcated by the black dashed line. The boldness of each trajectory is proportional to the EMRI SNR in the LISA observation time calculated using the approximate model described in the text. 2.3Simultaneous EM and GW observability Name Type ref. 𝑧

𝐿 𝑥

𝑓 MBH mass range (erg/s) ( × 10 − 4 Hz) ( × 10 6 ⁢ 𝑀 ⊙ ) GSN 069 QPE Miniutti et al. (2019) 0.0181 1.1 × 10 43 0.3 0.4 - 2 eRO-QPE1 QPE Arcodia et al. (2021) 0.0505 3.3 × 10 42 0.2 0.1 - 1 eRO-QPE2 QPE Arcodia et al. (2021) 0.0175 1.0 × 10 42 1.2 0.1 - 1 eRO-QPE3 QPE Arcodia et al. (2024) 0.024 4.2 × 10 41 0.1 0.9 - 5.3 eRO-QPE4 QPE Arcodia et al. (2024) 0.044 1.27 × 10 43 0.2 17 - 68 RX J1301.9+2747 QPE Giustini et al. (2020) 0.0237 1.6 × 10 42 0.6 0.8 - 2.8 RE J1034+396 QPO Gierliński et al. (2008) 0.042 4.0 × 10 43 2.8 1 - 10 2XMMJ123103,2+110648 QPO Lin et al. (2013) 0.13 2.5 × 10 42 0.7 0.09 - 1.1 ASASSN-14li QPO Pasham et al. (2019) 0.0206 2.0 × 10 43 76.3 0.6 - 12.5 ASASSN-20qc QPOut∗ Pasham et al. (2024a) 0.0136 ( 2 − 4 ) × 10 44 0.019 16-160 Swift J0230+28 QPE + QPOut(?)∗ Guolo et al. (2023) 0.0053 ( 3 − 6 ) × 10 42 0.019 1.6-10 Table 1:Summarized properties of QPEs and QPOs discovered so far in the frequency band of ∼ 0.1 mHz. From left to right, we specify the source’s name, type (QPO/QPE/QPOut), redshift ( 𝑧 ), peak soft X-ray (0.2-2 keV or 0.2-10 keV) luminosity, the QPP frequency ( 𝑓 ), and current lower and upper bounds on the source’s MBH mass. Asterisk (∗): QPOuts stands for the newly-discovered phenomenon of quasi-periodic outflows where the X-ray modulation is driven by a narrow bandpass dominated by an outflow. Recurrent spectral features reminiscent of outflows were reported from Swift J0230+28, but their statistical robustness is yet to be verified.

The M1 catalog of Babak et al. (2017) has EMRIs at redshifts up to 𝑧

4.5 or luminosity distance 𝑑 𝐿 ∼ 40 Gpc, which is much larger than the distances covered by current state-of-the-art X-ray band instruments. The observable volume is further restricted by the requirement of small integration times necessary for capturing the ∼ hours variability of relevant QPP sources. Similarly, for LISA-band EMRI observations, detectability of sources will be constrained by their GW SNRs. A large fraction of EMRIs in the M1 catalog have SNRs ≤ 1 even after setting the cutoffs mentioned in Section 2.1, making them undetectable. For a simultaneous observation in EM and GW bands, we thus must set realistic cutoffs on the observable cosmological volume (or equivalently, the maximum luminosity distance, 𝑑 𝐿 ) and the lowest detectable LISA-band SNR, 𝜌 .

The luminosity distance cutoff can be set by considering both the sensitivity threshold of current X-ray instruments and the required exposure time to significantly detect the QPP EM emission. For the mean X-ray luminosity of QPE sources, 𝐿 ¯ X

2 × 10 42 ⁢ erg ⁢ s − 1 (see Table 1), a luminosity distance of 𝑑 L ≲ 1 ⁢ Gpc results in the X-ray flux density threshold of 𝑓 X , sens ≡ 𝐿 X / 4 ⁢ 𝜋 ⁢ 𝑑 L 2 ≳ 1.7 × 10 − 14 ⁢ erg ⁢ s − 1 ⁢ cm − 2 . Since current X-ray detectors (Swift, XMM-Newton, and eROSITA), meet the sensitivity threshold 𝑓 X , sens for suitable integration times of a few tens of minutes, we treat the luminosity distance 𝑑 𝐿

1 Gpc as the cutoff (see also Section 4.1 for a more detailed discussion of the capabilities of current and future X-ray detectors).

For the LISA-band SNR cutoff, 𝜌 , we choose a value of 15, given the unavailability of a universally accepted threshold (which should scale with the information volume of the signal space (Moore et al., 2019)). Our specific choice is motivated by the estimates produced as part of the mock LISA data challenges (MLDC), which hinted at an EMRI measurability threshold of 𝜌 ≈ 20 for a two-year observation window (Babak et al., 2010), and accounting for the accumulation of SNR over the full LISA observation window, assumed here to be four years (Amaro Seoane et al., 2021). The SNR cutoff for an EMRI’s detectability would also depend on the number of parameters under simultaneous inference. If, for example, an EM counterpart detection tightly constrains some extrinsic EMRI parameters, the SNR cutoff can be expected to further drop, justifying our choice of 𝜌 cutoff

15 . However, this cutoff ignores the effect of data gaps and glitches in the LISA data stream which complicates inference and can thus potentially increase the SNR requisites. In general, their contributions are expected to be negligible in the inference of LISA-relevant sources (Dey et al., 2021; Baghi et al., 2019) and are thus ignored in our analysis.

Fig. 3 describes the distribution of the detector-frame orbital frequencies and eccentricities “today” (in 2024) after passing the EMRI population through a cutoff of 𝑑 L ≤ 1 Gpc and 𝜌 ≥ 15 .3 The corresponding 1-D sample distribution of 𝑓 orb , init has mean ⟨ 𝑓 orb , init ⟩ ≈ 2.3 × 10 − 4 Hz and 1 𝜎 standard-deviation of 1.1 × 10 − 4 Hz. Using Eq. 6 with 𝑛 col

2 , a rough estimate on the frequency of LISA-relevant EMRI QPPs is obtained as ⟨ 𝑓 QPP ⟩ ≈ 4.6 ± 2.2 × 10 − 4 Hz. This estimate represents a generous spread in the ∼ 0.1 mHz frequency band and numerous candidate and confirmed QPEs and QPOs in this frequency band have been reported in the literature (Miniutti et al., 2019; Arcodia et al., 2021, 2024; Giustini et al., 2020; Chakraborty et al., 2021; Gierliński et al., 2008; Jin et al., 2020; Alston et al., 2015; Lin et al., 2013; Boller, Th. et al., 2001; Pan et al., 2016; Pasham et al., 2019; Wang et al., 2014; Gupta et al., 2018). However, follow-up studies have speculated upon the statistical robustness of some of these observations (see for example Vaughan (2005) and Zhang et al. (2023)).

A list of QPPs that fall near the proposed frequency band, and with statistically significant observations, is presented in Table 1. For completeness, we additionally list a new class of nuclear transients called QPOuts (quasi-periodic outflows) discovered by Pasham et al. (2024b) which are also proposed to be (early) electromagnetic counterparts of large mass-ratio inspirals. In Pasham et al. (2024b), the authors observed quasiperiodic dips in the X-ray band caused by the enhanced absorption due to an intermittent fast outflow. The QPOut was interpreted as possibly caused by an intermediate-mass black hole orbiting a supermassive black hole as supported by the simulations of Suková et al. (2021). However, only one such confirmed transient has been observed so far (but see also Guolo et al. 2023) and is most likely far from merger. We thus leave its analysis for future studies.

As we discuss in Section 4.1, some current and near-future EM observatories should be able to detect QPP signals in the estimated frequency band, enabling potential multimessenger observations of LISA-EMRIs.

Figure 2:The joint distribution (top panel) and 1-D marginals (bottom panel) of the detector-frame orbital frequency, 𝑓 orb , init and eccentricity 𝑒 init of LISA-band EMRIs “today”, i.e. in 2024. Each marker in the top panel is darkened in proportion to the source’s SNR. In the bottom panel, the distributions of sources satisfying the LISA-band detectability cutoff based on these SNRs are also depicted. Figure 3:Distribution of EMRI sources with respect to detector-frame 𝑓 orb , init and their luminosity distance from the detector, 𝑑 𝐿 , “today”, i.e. in 2024. Sources in the grey band have 𝑑 𝐿 ≥ 1 Gpc. The stars mark all sources that simultaneously satisfy 𝑑 𝐿 ≤ 1 Gpc and the cutoff on the LISA-band SNR, 𝜌 ≥ 15 . 1D marginal distribution of the detector frame 𝑓 orb , init of sources that satisfy the cutoffs is depicted along the horizontal axis, which thus describes the distribution of orbital frequencies of candidate multi-messenger EMRIs. 3Modeling RE J1034+396 as an EMRI

RE J1034+396, an AGN located at 𝑧

0.042 , is a confirmed QPO with a frequency of several times 10 − 4 Hz (Alston et al., 2014; Jin et al., 2020). Several estimates on the mass of its MBH are available, (Bian & Huang, 2010; Chaudhury et al., 2018; see also Czerny et al., 2016 for the comparison of the inferred MBH mass using different methods), but it is generally agreed to be in the range of 10 6 − 10 7 ⁢ 𝑀 ⊙ . It has been observed multiple times over the span of about a decade (2007-2018) — In 2007, the QPO’s time period was observed to be 𝑇 2007 ∼ 3730 ± 80 ⁢ 𝑠 (Gierliński et al., 2008) which in 2018 reduced to 𝑇 2018 ∼ 3550 ± 60 ⁢ 𝑠 (Jin et al., 2020). If RE J1034+396 is modeled as an EMRI, the reduction in periodicity can be associated with GW-induced radiation reaction of the system, and/or loss of energy via CO-disk interactions. However, according to Table 1 in Jin et al. (2020) (see also Table 2 in Alston et al., 2014), the QPO frequency fluctuated between 0.25 − 0.27 mHz during 2007 - 2011 without any secularly evolving features. This can still be explained under the environment-rich EMRI model through eccentric orbit CO-disk interactions, in which the torque forces generated by the disk’s density wave perturbations both inside and outside of the CO orbit compete with each other, sometimes canceling each other out to create “migration traps”, or even exerting a net-outwards force on the CO (Bellovary et al., 2016; Secunda et al., 2020). A detailed analysis of this effect, however, is beyond the scope of our study, and we assume a secular drift between the reported QPO time-periods in 2007 ( 3730 ± 80 s) and 2018 ( 3550 ± 60 s) in our model, which mark the earliest and latest reported values of the QPO in RE J1034+396.

3.1CO mass estimate for RE J1034+396

Assuming two collisions per orbit, from Eq. (6), the corresponding orbital frequencies of the EMRI source in 2007 and 2018 are

𝑓 orb , 𝑖

1 2 ⁢ 𝑇 𝑖

{ 1.34 × 10 − 4 ⁢ Hz
𝑖

2007

1.41 × 10 − 4 ⁢ Hz
𝑖

2018

(7)

such that the observed rate of change of the orbital frequency, 𝑓 ˙ orb , obs , can be approximated to linear-order as

𝑓 ˙ orb , obs ≈ 𝑓 orb , 2018 − 𝑓 orb , 2007 2018 − 2007 ≈ 6.18 × 10 − 7 Hz yr − 1 .

(8)

We can now use the distribution of 𝑓 ˙ orb , obs to optimize the corresponding theoretical prediction,

𝑓 ˙ orb , th ≡ 𝑓 ˙ orb , th ⁢ ( 𝑀 , 𝜇 , 𝑎 , 𝑒 0 , 𝜄 , 𝑓 orb , 2007 )

(9)

by minimizing the following 𝜒 2 -error function,

𝜒 2 ≡ ( 𝑓 ˙ orb , obs − 𝑓 ˙ orb , th ) 2 .

(10)

Here 𝜇 is the unknown mass of the CO, 𝑎 is the dimensionless spin of the MBH, 𝑒 0 is the initial eccentricity of the CO’s orbit, and 𝜄 is the initial inclination of the CO’s orbital plane (which does not usually evolve significantly in the parameter regime we are considering). The inclusion of { 𝑎 , 𝑒 0 , 𝜄 } in the parameter set allows us to gauge their correlations with the calculated CO mass. A discussion on the choice of prior on { 𝑎 , 𝑒 0 , 𝜄 } and their effect on our analysis is presented in Appendix A. We adopt 𝑓 ˙ orb , th from the waveform model of Fujita & Shibata (2020), as implemented in the 5PNAAK waveform model of the FastEMRIWaveform (FEW) toolkit (Chua et al., 2021; Katz et al., 2021). Due to the unavailability of CO-disk interaction models for inclined and eccentric orbit EMRIs, we exclude their effects from our analysis. While these effects are subdominant in comparison to contributions from eccentricity and inclination, we stress that generic orbit environment-rich EMRI models will be necessary before drawing strong conclusions.

The first three panels in Fig. 4 describe the distribution on the parameter set { 𝑀 , 𝑓 orb , 2007 , 𝑓 ˙ orb , obs } , from which we infer the best-fit CO mass of

𝜇 ≈ 46 − 40 + 10 ⁢ 𝑀 ⊙ ,

as depicted in the figure’s fourth panel. This estimate comfortably falls in the range of CO masses expected for LISA-relevant EMRIs ( ≈ 1 − 100 ⁢ 𝑀 ⊙ ). We also refer to Fig. 11 in the Appendix which explicitly shows the distribution with respect to all other considered parameters, i.e. { 𝑎 , 𝑒 0 , 𝜄 } . Note, however, that the correlation coefficient, 𝒞 𝑖 ⁢ 𝑗 ≡ 𝜎 𝑖 ⁢ 𝑗 / 𝜎 𝑖 2 ⁢ 𝜎 𝑗 2 , between 𝜇 and { 𝑎 , 𝑒 0 , 𝜄 } is relatively small with 𝒞 𝜇 ⁢ 𝑎 ∼ 0.01 , 𝒞 𝜇 ⁢ 𝑒 0 ∼ − 0.005 , and 𝒞 𝜇 ⁢ 𝜄 ∼ 0.02 which implies that the prior choice of { 𝑎 , 𝑒 0 , 𝜄 } does not significantly affect the distribution of 𝜇 in this model. 𝜇 ’s correlation with 𝑓 ˙ orb , obs is highest at 𝒞 𝜇 ⁢ 𝑓 ˙ orb , obs ∼ 0.9 , while with the initial orbital frequency is also high at 𝒞 𝜇 ⁢ 𝑓 orb , 2007 ∼ − 0.76 . Comparatively, 𝜇 is moderately correlated with 𝑀 , 𝒞 𝜇 ⁢ 𝑀 ∼ − 0.5 .

Figure 4:Marginalized distributions over { 𝑀 , 𝑓 orb , 2007 , 𝑓 ˙ orb , obs , 𝜇 , SNR } of RE J1034+396 modeled as an extreme-mass-ratio inspiral. { 𝑀 , 𝑓 orb , 2007 , 𝑓 ˙ orb , obs } are estimated from current observational constraints on RE J1034+396. The thus obtained prior-predictive distribution on 𝜇 is described in the fourth panel, and the corresponding LISA-band SNR in the final panel. The full 8-dimensional distribution (including { 𝑎 , 𝑒 0 , 𝜄 } ) is available in Appendix A as Fig. 11. Figure 5:Prior-predictive distribution of LISA-band SNRs of QPO RE J1034+396 modeled as an EMRI, with an indicative exponential fit. We assume the MBH mass to be in range 10 6 − 10 7 ⁢ 𝑀 ⊙ , mass of the orbiting body 𝜇 ≈ 46 − 40 + 10 ⁢ 𝑀 ⊙ , and conservative prior distributions on { 𝑎 , 𝑒 , 𝜄 } as described in the text. 3.2LISA band detectability of RE J1034+396

The availability of the full EMRI parameter set, ( 𝑀 , 𝜇 , 𝑎 , 𝑒 0 , 𝜄 , 𝑓 orb , 2007 ) , now allows us to evolve RE J1034+396 to LISA’s operation time, 2037, and calculate the SNR, 𝜌 , of the GW signal, given as (Barack & Cutler, 2004)

𝜌 ≡ ⟨ ℎ | ℎ ⟩

(11)

where ℎ ≡ ℎ ⁢ ( 𝑀 , 𝜇 , 𝑎 , 𝑒 0 , 𝜄 , 𝑓 orb , 2007 , 𝝍 ext ) is the waveform template corresponding to the signal, and 𝝍 ext

{ 𝑑 𝐿

0.18 , 𝜃 𝑆

2.77 , 𝜙 𝑆

0.69 } is the extrinsic parameter set defined by the luminosity distance 𝑑 𝐿 and sky localisation parameters 𝜃 𝑆 , 𝜙 𝑆 of the source. The one-sided detector noise weighted inner-product ⟨ ⋅ | ⋅ ⟩ between two discrete time-domain signals 𝛼 ⁢ ( 𝑡 ) and 𝛽 ⁢ ( 𝑡 ) is given as (Chua & Cutler, 2022):

⟨ 𝛼 | 𝛽 ⟩ ≡ 4 ⁢ Re ⁢ ∑ 𝜒 ∑ 𝑓

0 𝑓 𝑁 𝛿 ⁢ 𝑓 ⁢ 𝛼 ~ 𝜒 ∗ ⁢ ( 𝑓 ) ⁢ 𝛽 ~ 𝜒 ⁢ ( 𝑓 ) + 𝛼 ~ 𝜒 ⁢ ( 𝑓 ) ⁢ 𝛽 ~ 𝜒 ∗ ⁢ ( 𝑓 ) 𝑆 n , 𝜒 ⁢ ( 𝑓 ) .

(12)

Here 𝛼 ~ ⁢ ( 𝑓 ) , 𝛽 ~ ⁢ ( 𝑓 ) are the frequency domain Fourier transformations of 𝛼 ⁢ ( 𝑡 ) , 𝛽 ⁢ ( 𝑡 ) , 𝑓 𝑁 is the Nyquist sampling frequency, 𝜒 ≡ I , II are the two independent data-channels of LISA in its long-wavelength approximation (LWA) configuration, and 𝑆 n ⁢ ( 𝑓 ) is the power spectral density of LISA’s noise channel 𝑛 𝜒 evaluated at detector frequency 𝑓 (Robson et al., 2019).

The prior-predictive distribution of the source’s LISA-band SNR is approximately exponential, with a sample mean of 𝜌 ≈ 14 (see Fig. 5). From this distribution, we can estimate the probability of detecting RE J1034+396 in the LISA band as p detection ⁢ ( 𝜌 ≥ 15 ) ≈ 0.25 for a LISA-band SNR cutoff of 15. Note that, in our model, the CO is on average separated from the MBH by about 18.8 ⁢ 𝐺 ⁢ 𝑀 / 𝑐 2 when entering the LISA band and evolves to an average of 18.4 ⁢ 𝐺 ⁢ 𝑀 / 𝑐 2 at the end of LISA observation period. If instead a model accounting for CO-disk interactions in generic orbits is used, the system will lose energy more rapidly and thus attain smaller separations when entering the LISA band, boosting the strength of its GW emission and hence its LISA-band SNR. Additionally, recall that the SNR cutoff of 15 was set based on conservative estimates for a GW-only inference. A multiband observation may enable the detection of sources with smaller SNRs. Illustratively, for smaller LISA-band SNR cutoffs, we have p detection ⁢ ( 𝜌 ≥ 12 )

0.33 and p detection ⁢ ( 𝜌 ≥ 10 )

0.41 . These estimates suggest that RE J1034+396 may be a potential target for multimessenger EMRI observations.

4Discussion 4.1Targeted QPP searches using current & planned observatories 4.1.1Detectability threshold in soft X-rays

For the electromagnetic detection of repeating X-ray transients, generally two aspects should be met: (i) sensitivity close to the eruption peak flux density and (ii) a sufficient EM-band SNR reached for the integration time that is shorter than the periodicity of a given source.

Conditions (i) and (ii) can be combined to estimate the minimal count rate required to detect the flare robustly to constrain its spectral energy distribution (SED), in particular its temperature. We consider the 5 𝜎 threshold to be 25 counts. Since the QPE flares last for about an hour and repeat every few hours, the observation should last for at least a couple of hours, which results in the count rate of 25 / 7200 ∼ 0.003 counts/s. For currently operational X-ray telescopes (except for Chandra which has a deteriorating soft X-ray sensitivity), this count rate translates into the limiting flux density threshold of 𝑓 𝑋 ≈ 6.3 × 10 − 14 ⁢ erg ⁢ cm − 2 ⁢ s − 1 for Swift XRT (Burrows et al., 2005), and 𝑓 𝑋 ≈ 3.2 × 10 − 15 ⁢ erg ⁢ cm − 2 ⁢ s − 1 for XMM/PN (Hasinger et al., 2001; Brunner et al., 2008) 4. In both cases, we considered the energy range of 0.3 − 2.0 keV and the thermal spectrum with the temperature of 100 eV. For the NICER telescope, we adopt the minimal count rate of 0.51 counts/s in the 0.3-2.0 keV band, which corresponds to 𝑓 𝑋 ≈ 4.3 × 10 − 13 ⁢ erg ⁢ cm − 2 ⁢ s − 1 (Remillard et al., 2022). Assuming the characteristic X-ray luminosity of the eruptions to be 𝐿 𝑋

5 × 10 42 ⁢ erg ⁢ s − 1 , we estimate the maximum luminosity distance (cosmological volume) which the current detectors can probe using the simple relation, 𝑑 L

[ 𝐿 𝑋 / ( 4 ⁢ 𝜋 ⁢ 𝑓 𝑋 ) ] 1 / 2 . Thus, we get 𝑑 L ∼ 814 Mpc and ∼ 3613 Mpc for the Swift and the XMM-Newton telescopes, respectively. For the NICER X-ray timing telescope, the limiting luminosity distance is ∼ 312 Mpc. Overall, this large scatter in the cosmological volume justifies the distance cut-off of 1 Gpc. The caveat of the simple estimates above is the assumption that the EM counterparts of gas-rich EMRIs are all similar to the currently detected soft X-ray QPEs. In principle, some of them can be more or less luminous and could thus manifest in different wavebands, in particular harder X-ray or softer bands such as UV, which directly affects the number of detected sources. Below we discuss some of those aspects considering the current and future survey telescopes, including radio and UV surveys in addition to X-ray telescopes.

Figure 6:The flux density or flux sensitivity (in erg ⁢ s − 1 ⁢ cm − 2 ) as a function of the cumulative exposure time (in hours). Left: Sensitivity levels as a function of exposure time (in hours) for currently used instruments capable to detect soft X-ray emission (NICER, Einstein Probe, Swift XRT, XMM EPIC, eROSITA). The stars represent already detected repeating X-ray transients with a peak flux density and periodicity (QPPs; see Table 1). Right: Same as in the left panel but for some of the planned X-ray missions (Athena, Lynx, AXIS, STROBE-X). These levels should be considered as estimates since the actual sensitivity depends on the spectrum of the source, cosmic and galactic X-ray background, and the real-time performance of the detector in space. 4.1.2Detectability by specific X-ray missions

Currently detected soft X-ray repeating sources (QPPs) are plotted in Fig. 6 along with the flux sensitivity limits of current (left panel) and future-planned detectors in the soft X-ray band (right panel). From this, we infer that the NICER telescope could detect most currently observed sources since they are above the flux sensitivity limit (blue dotted line in Fig. 6; Remillard et al., 2022), while the SRG/eROSITA survey in soft X-rays (0.2-2.3 keV) (green solid line; Predehl et al., 2021) and Swift XRT (0.2-10 keV) (maroon dash-dotted line; Burrows et al., 2005) should be able to detect all of these sources. XMM-Newton EPIC detector (0.5-2 keV) (cyan dotted line; Hasinger et al., 2001; Brunner et al., 2008) is the most sensitive in the X-ray domain from the list of current instruments. The Einstein probe launched on January 9, 2024 5, will have a wide field of view ( ∼ 3600 ⁢ deg 2 ) using the Lobster-eye technology (the Wide-field X-ray Telescope, WXT) and will be able to detect about half of the currently detected transients (dashed orange line; Yuan et al., 2022) at 0.5-4 keV. The future Athena mission (Nandra et al., 2013), currently under preparation, with a field of view of 0.4 ⁢ deg 2 (Wide Field Imager) and angular resolution of 5-10 arcseconds, will be able to detect even fainter sources in the soft X-ray band with the flux density at least two orders of magnitude lower (see magenta and orange lines in the right panel of Fig. 6). The concept of the Lynx mission (Gaskin et al., 2019) with a smaller field of view of 0.1 deg 2 and an angular resolution of 0.5” should reach even lower flux densities (dashed green line). 6 The concept of Advanced X-ray Imaging Satellite (AXIS) mission (0.5-10 keV; Mushotzky et al., 2019; AXIS Time-Domain et al., 2023) with the exquisite spatial resolution ( < 1.5 ′′ ) across 24’ field of view will reach the flux sensitivity intermediate between the planned Athena and the current XMM EPIC detector. The planned STROBE-X Wide Field Monitor as well as its X-ray Concentrator Array (2-50 keV and 0.2-12 keV, respectively, Ray et al., 2019) will reach the sensitivity to detect just the brightest QPP sources (dashed and dotted lime lines in Fig. 6). The detection and classification of new X-ray QPP sources will benefit from the combination of current and future survey telescopes, such as eROSITA and Einstein probe, with the dedicated monitoring of the detected QPPs with more sensitive detectors, e.g. on-board of XMM-Newton and Athena. We note that Athena and Lynx missions are currently planned to be launched in the mid-2030s, hence they are not relevant for the detection of EMRI EM counterparts “today”, though they will be crucial for such detections and monitoring in the era of future space-borne GW detectors, possibly including LISA.

4.1.3Dependence on the accretion state and prospects for detectability in the UV domain

The prospect of detecting EMRIs in the electromagnetic spectrum can also be a function of the relative accretion rate, as suggested by Linial & Metzger (2024). Within their model where the QPEs are generated by the ejection of shocked plasmoids as the CO/star punches through the accretion disk, the flares can effectively be detected when their radiation is spectrally harder and hence brighter in the soft X-ray band than that of the underlying accretion flow. This is not met for accretion rates close to the Eddington limit, i.e. when the relative accretion rate (or equivalently the Eddington ratio) is 𝑚 ˙ ≡ 𝑀 ˙ / 𝑀 ˙ Edd ∼ 1 (where 𝑀 ˙ Edd is the Eddington accretion rate as defined in Linial & Metzger, 2024). As the relative accretion rate 𝑚 ˙ drops to 𝑚 ˙ ∼ 0.1 , e.g. following a tidal disruption event with a decreasing accretion rate, QPE emission in the soft X-ray band is above the quiescent disc emission due to the higher temperature associated with QPEs. Linial & Metzger (2024) discuss that for a further drop in the accretion rate to 𝑚 ˙ ∼ 0.01 , the QPEs become too dim in the X-ray domain due to softer radiation. At this stage, the eruptions could rather be traceable in the UV domain. An example of such sources is provided by the AGN QPE source, GSN069, with its generally decreasing accretion rate following two TDE-like outbursts. Eruptions in the source were not detected for accretion rates 𝑚 ˙ ≥ 0.5 , but (re)appeared for lower accretion rates (Miniutti et al., 2019, 2023a, 2023b). Future monitoring of this source by X-ray and UV instruments will reveal how the eruption SED changes with the drop in the accretion rate. A drop in eruption amplitudes that can be potentially linked to a decrease in accretion rate is also traced for eRO-QPE1 (Pasham et al., 2024b; Chakraborty et al., 2024). Note that a UV-band luminosity of ∼ 10 41 ⁢ ergs − 1 , typical for such sources, corresponds to the apparent AB magnitude of ∼ 21.8 mag for nearby sources ( 𝑧 ∼ 0.025 ). Such sensitivity will be reached by the future-planned wide field-of-view NUV satellite ULTRASAT (Ben-Ami et al., 2022; Shvartzvald et al., 2023), which will perform continuous monitoring of the transient UV sky (230-290 nm) thanks to its wide field-of-view of 204 ⁢ deg 2 . It will also provide target-of-opportunity triggers for the planned two-band (FUV, 140-190 nm, and NUV, 260-360 nm) UV photometry mission QUVIK (Werner et al., 2024; Krtička et al., 2024; Zajaček et al., 2023), which will be able to perform a high-cadence dedicated monitoring of UV QPP candidates down to ∼ 22 and ∼ 20 AB magnitudes in the NUV and FUV bands (detector SNR of 5 in 1000 seconds) with the point spread function of ≲ 2.5 ′′ . In addition, the UVEX mission (Kulkarni et al., 2021) with the planned launch in the early 2030s will perform medium-resolution spectroscopy in the range 115.0–265.0 nm in addition to deep two-band monitoring of the UV sky with its sensitivity reaching ∼ 24.5 AB magnitude (detector SNR of 5 in 900 seconds). Sensitive UV observations may also reveal long-period UV QPEs tracing EMRIs crossing the disc on wider orbits, i.e. at lower relative velocities producing softer thermalized shocks. Alternatively, short-period QPEs around low-mass central black holes of ≲ 10 5.5 ⁢ 𝑀 ⊙ are also expected to be detected in the UV domain (Linial & Metzger, 2024).

4.1.4Prospects for coincident detection in the radio band

The detection of gas-rich EMRI in the radio domain is highly uncertain, depending mainly on the presence of the magnetized outflow or jet in EMRI hosts, i.e. whether the source is radio-loud or quiet. The viewing angle also affects the jet contribution via the Doppler-boosting factor. One of the most promising sources for hosting a binary MBH – OJ287 – is a blazar that exhibits a quasiperiodic precession/nutation pattern in the temporal evolution of the jet position as well as its radio brightness on the timescale of ∼ 20 years (Britzen et al., 2018, 2023), which is about twice longer than the typical optical variability timescale in this source (Valtonen et al., 2016). However, most of the QPP hosts in Fig. 1 are radio quiet and hence the prospects for tracing the EMRI presence in the radio variability is at best at the limit of the sensitivity of current radio telescopes. From the repeating soft X-ray transients (QPE-like) only Swift J0230+28 exhibited one radio flare coincident with the start of the X-ray flare (Guolo et al., 2023). The radio flare detected by the Very Large Array (VLA; D configuration) was unresolved and coincident with the point source in the nucleus. The flux density at 10 GHz was 𝐹 𝜈

93 ± 7 ⁢ 𝜇 ⁢ Jy (13 𝜎 detection). Given the distance of the host galaxy ( ∼ 165 ⁢ Mpc ), the radio luminosity at 10 GHz is 𝐿 10 ⁢ G ⁢ H ⁢ z ∼ 3 × 10 37 ⁢ erg ⁢ s − 1 and the ratio of the radio luminosity to the corresponding soft X-ray luminosity of the flare is 𝐿 10 ⁢ G ⁢ H ⁢ z / 𝐿 𝑋

3 × 10 37 / ( 10 42 ) ∼ 3 × 10 − 5 , hence the energy going into the radio emission is significantly smaller than the detected X-ray luminosity. The radio flare could be related to the synchrotron emission due to star-disc interaction shocks or the magnetic reconnection event in the magnetized plasma of the hot accretion flow. For the known QPPs, one can plan observational campaigns of simultaneous X-ray/radio monitoring with the VLA, whose sensitivity threshold is as low as 10 ⁢ 𝜇 ⁢ Jy (RMS noise) at 10 GHz (bandwidth 500 MHz, field of view of 4.2 arcmin) for the on-source time of 24.9 minutes 7. In the second half of 2020s, the Square Kilometer Array (SKA-1, mid-frequency) should reach the RMS noise level of 1.2 ⁢ 𝜇 ⁢ Jy for 1 hour-long observation at 12.5 GHz (bandwidth 3.75 GHz), which should reveal radio variability of even fainter QPPs. With the field of view of 6.7 arcmin (SKA1-mid, 12.5 GHz) and a high sensitivity, the SKA will survey the sky faster than previous radio arrays, making it possible to visit the sky more frequently (Braun et al., 2019). For detecting radio flares from previously unidentified QPPs, hence candidates for EMRIs, a large sky coverage in the radio domain is desirable. The concept of the Canadian Hydrogen Intensity Mapping (CHIME; CHIME Collaboration et al., 2022) experiment with the total sky coverage of 31 000 deg 2 for one-year integration is in principle beneficial for detecting new radio transients such as fast radio bursts. However, the confusion noise level of ∼ 0.1 Jy currently limits the instrument to much brighter transients than those expected from CO-accretion flow interactions in nominally non-jetted hosts (at least thousand times brighter than the Swift J0230-like radio eruption).

4.2QPP luminosity and dependence on binary parameters

The dynamics of the putative binaries behind QPPs will depend on a number of their properties, such as the mass of the perturbing COs, the properties of the accretion disk, or the orbital parameters. Here we discuss how these will affect the character and observability of the QPPs.

4.2.1Radius of disk-perturber interaction and QPE luminosity

Using the results on the orbital frequency of EMRI detectable EM candidates (see Fig. 3) we set the fiducial orbital period to 𝑃 orb

1 hour (QPE-like eruption occurs every ∼ 0.5 hours for the CO-disc interaction). The primary MBH mass is set to 𝑀

10 6 ⁢ 𝑀 ⊙ unless stated otherwise. Then the radius of the circular orbit of the CO is

𝑟

( 𝑃 ⁢ 𝑐 3 𝐺 ⁢ 𝑀 ) 2 / 3 ⁢ 𝑟 g ∼ 24 ⁢ ( 𝑀 10 6 ⁢ 𝑀 ⊙ ) − 2 / 3 ⁢ ( 𝑃 1 ⁢ hour ) 1 / 3 ⁢ 𝑟 g ,

(13)

where 𝑟 g ≡ 𝐺 ⁢ 𝑀 / 𝑐 2 ∼ 1.5 ⁢ km ⁢ ( 𝑀 / 𝑀 ⊙ ) is the gravitational radius of the MBH.

The influence radius 𝑅 inf , within which the compact object or the perturber with the mass 𝜇 affects the surrounding gas gravitationally, depends on the relative velocity with respect to the accretion disc material 𝑣 rel and the disc sound speed 𝑐 s at the distance 𝑟 , (e.g. Suková et al., 2021)

𝑅 inf

2 ⁢ 𝐺 ⁢ 𝜇 𝑣 rel 2 + 𝑐 s 2 .

(14)

Approximating both the disc velocity and the perturber velocity with Keplerian radial profiles, and adopting the inclination 𝜄 between the perturber and the accretion disc, we may estimate the relative velocity as follows,

𝑣 rel

[ 2 ⁢ 𝐺 ⁢ 𝑀 𝑟 ⁢ ( 1 − cos ⁡ 𝜄 ) ] 1 / 2 .

(15)

The sound speed is calculated as 𝑐 s

( 𝑘 B ⁢ 𝑇 disc / 𝑚 p ) 1 / 2 , where 𝑘 B is the Boltzmann constant, 𝑇 disc is the temperature of the assumed standard disk, and 𝑚 p is the proton mass.

Specifically, we can expect the accretion disk sonic speed to be 𝑐 s ≲ 𝑣 k ∼ 𝐺 ⁢ 𝑀 / 𝑟 , where 𝑣 k is the Keplerian speed at radius 𝑟 . For a radiatively efficient accretion disk we will have 𝑐 s / 𝑣 k ≪ 1 and the height of the disk 𝐻 / 𝑟 ∼ 𝑐 s / 𝑣 k ≪ 1 (Shakura & Sunyaev, 1973), whereas for hot radiatively inefficient flows we will have 𝐻 / 𝑟 ∼ 𝑐 s / 𝑣 k of order one (Yuan & Narayan, 2014). We thus obtain 𝑅 inf as

𝑅 inf

𝑟 ⁢ 𝜇 𝑀 ⁢ ( 1 − cos ⁡ 𝜄 + 1 2 ⁢ ( 𝐻 𝑟 ) 2 ) − 1

(16)

This estimate for 𝑅 inf ignores the gravitational influence of the primary MBH, which will dominate the gas outside the distance given by the Hill (tidal) radius 𝑅 Hill from the CO, where

𝑅 Hill ∼ 𝑟 ⁢ ( 𝜇 3 ⁢ 𝑀 ) 1 / 3 .

(17)

It is easy to see that for moderate inclinations and small mass ratios 𝜇 / 𝑀 , the Hill radius will always be larger than 𝑅 inf , so eq. (16) provides a good estimate for the interaction cross-section. For low inclinations sin ⁡ 𝜄 ≲ ( 𝜇 / 𝑀 ) 1 / 3 and ( 𝐻 / 𝑟 ) ≲ ( 𝜇 / 𝑀 ) 1 / 3 , however, the Hill radius acts as a cap for the radius affected by the perturber.

Additionally, we conjecture that a QPE will only be created by an orbiter that is both not embedded in the disk, and punches through the disk at a relative speed that is supersonic, both of which lead to the same condition sin ⁡ 𝜄 ≳ 𝐻 / 𝑟 .8 For sin ⁡ 𝜄 ≲ 𝐻 / 𝑟 the orbiter is embedded in the disk and we surmise that this instead leads to a QPO.

For example, at the distance of the CO corresponding to one hour orbital period, the sound speed of the accretion-disk gas is ∼ 35 ⁢ km ⁢ s − 1 for a fiducial relative accretion rate of 𝑚 ˙

0.1 and an innermost stable circular orbit set to 6 ⁢ 𝑟 g , while the Keplerian velocity is ∼ 61 000 ⁢ km ⁢ s − 1 . As a result, the orbit will be non-embedded in the disk and cross it supersonically for 𝜄 ≳ 0.03 ∘ . Similarly, the orbit will not reach the cap on the influence radius set by the Hill radius for 𝜄 ≳ 0.5 ∘ ⁢ ( 𝜇 / 𝑀 ⊙ ) 1 / 3 ⁢ ( 𝑀 / 10 6 ⁢ 𝑀 ⊙ ) − 1 / 3 .

To estimate the X-ray luminosity of QPE-like eruptions, we then adopt a toy model of CO-disc interactions, which lead to shocks and ejection of shocked spherical, optically thick clouds above the disc plane (see e.g. Franchini et al., 2023). For the initial cloud radius, we adopt 𝑅 inf and for its temperature, we set 𝑇 cloud ∼ 10 6 ⁢ K , which was inferred for QPEs (Miniutti et al., 2019; Arcodia et al., 2021, 2024). Then the soft X-ray luminosity (in the 0.2-2 keV band) can be estimated as follows,

𝐿 X

4 ⁢ 𝜋 ⁢ 𝑅 inf 2 ⁢ ∫ 0.2 ⁢ keV 2.0 ⁢ keV 2 ⁢ ℎ ⁢ 𝜈 3 𝑐 2 ⁢ d ⁢ 𝜈 exp ⁡ ( ℎ ⁢ 𝜈 / 𝑘 B ⁢ 𝑇 cloud ) − 1 ,

(18)

where 𝜈 is the frequency corresponding to the photon energy 𝐸 𝜈

ℎ ⁢ 𝜈 and ℎ is the Planck constant. We plot examples of the spectral energy densities corresponding to this model in Figure 7.

Linial & Metzger (2023) present a similar CO-disk interaction model, taking into account inefficient photon production during the optically thick phase of an expanding shocked cloud. To reach 𝐿 𝑋 ∼ 10 42 ⁢ erg ⁢ s − 1 , the CO mass should fall into the IMBH range with 𝜇 ∼ 10 4 ⁢ 𝑀 ⊙ ,

𝐿 X

8.9 × 10 41 ⁢ ( 𝑚 ˙ 0.1 ) 1 / 3 ⁢ ( 𝑀 10 6 ⁢ 𝑀 ⊙ ) 5 / 9

× ( 𝜇 10 4 ⁢ 𝑀 ⊙ ) 2 / 3 ⁢ ( 𝑃 QPP 0.5 ⁢ hours ) − 2 / 9 ⁢ erg ⁢ s − 1 .

(19)

We indicate the luminosity range according to Eq. (19) in Figure 7 using the cyan rectangle, which corresponds to the CO masses of 𝜇

10 − 10 4 ⁢ 𝑀 ⊙ .

Figure 7:Exemplary spectral energy distributions (SEDs) corresponding to shocked clouds of material ejected during the passages of a lighter compact object (CO) around the MBH. We assumed the orbital period of 1 hour and the MBH mass of 10 6 ⁢ 𝑀 ⊙ . To reach typical QPE luminosities of 10 42 ⁢ erg ⁢ s − 1 (red dotted line) in the 0.2-2 keV band (blue shaded region), the initial cloud size should be ∼ 10 11 ⁢ cm , which corresponds to ∼ 0.7 gravitational radii of the MBH. The compact objects that are able to eject such clouds fall into the range of ∼ 100 − 1000 ⁢ 𝑀 ⊙ , with the inclination of 5 − 15 degrees with respect to the accretion disc, respectively. Solar and subsolar COs typically generate X-ray flares of ≲ 10 39 ⁢ erg ⁢ s − 1 (grey shaded region), hence at least three orders of magnitude below the typical QPE luminosities. The dashed black vertical line represents the peak of the SEDs. The white star stands for the mean QPE luminosity inferred from seven detected QPE sources ( log ⁡ ( 𝐿 x ⁢ [ erg ⁢ s − 1 ] )

42.3 ± 0.6 ; see Table 1). The cyan rectangle represents the X-ray luminosity range for COs of 𝜇

10 − 10 4 ⁢ 𝑀 ⊙ according to the CO-disk interaction model by Linial & Metzger (2023). 4.2.2Theoretical uncertainties of disk-perturber interaction

The Shakura & Sunyaev (1973) 𝛼 -disk used for various estimates above corresponds only to the effective structure of an accretion disk averaged over the orbital period, disk height, and azimuthal angle. As such, it cannot be used for the dynamical analysis of its interactions with bodies embedded in the disk (i.e., smaller than the disk height) and/or moving on the orbital time-scale, since on those scales the structure is dominated by magneto-rotational-instability-driven turbulence (see Balbus & Hawley (1991) and especially the discussion in Balbus & Hawley (1998)).

Another difficult aspect of the interaction are the dynamics of the ejected plasma leading to the observed lightcurve of the QPE. In some models the ejected material is assumed to expand in the form of a “bubble” for some time before becoming optically thin and releasing the radiation (Lehto & Valtonen, 1996; Ivanov et al., 1998; Pihajoki, 2016; Linial & Metzger, 2023). However, the expansion implies significant adiabatic cooling of the plasma, which requires involving other subtle mechanisms to explain the relatively high 10 6 ⁢ K effective temperatures observed in QPEs (Linial & Metzger, 2023). Additionally, the CO will eject the material in the form of a trailing wake as seen in the Hoyle-Lyttleton scenario, which is likely incompatible with the expanding bubble model. Furthermore, for some configurations, such as lower-inclination and retrograde orbits, the break-out emission of a bow shock associated with the supersonic CO can dominate the EM signal (Tagawa & Haiman, 2023). As such, both the action of the perturber on the disk, and the possible back-reaction of the disk on the perturber is currently uncertain and will likely be only resolved by first-principle simulations of the interaction of the plasma of the accretion disk with the orbiting body.

Furthermore, the model of star-disk interactions described above and the estimated X-ray luminosity rely on the standard geometrically thin and optically thick accretion disc. However, most of the EMRI hosts in the local Universe are expected to be low-luminosity sources. Such sources host hot accretion flows that are advection-dominated and radiatively inefficient (Yuan & Narayan, 2014). In this case, plasma is more diluted and warmer, hence the shocks due to interactions with the CO are also weaker because of lower Mach numbers. The ejected mass and the associated electromagnetic energy is also lower due to a smaller surface density (Linial & Metzger, 2023). For the hot flow, the variability could rather be driven by the accretion-rate variations due to propagating density waves induced by the CO bow shock (see e.g. Suková et al., 2021).

4.2.3Mass of compact object and detectability

The mass range of orbiting COs in EMRIs/IMRIs is rather broad – from a few Solar masses (neutron stars, white dwarfs, stellar black holes; Shapiro & Teukolsky, 1983) to several thousand or even ten thousand in the case of intermediate-mass black holes (IMBHs; Greene et al., 2020; Pasham et al., 2024a). The range can be extended to lower, sub-solar masses if we admit the occurrence of brown dwarfs in galactic nuclei (see, e.g., Amaro-Seoane, 2019; Gourgoulhon et al., 2019; Zajaček, 2021) or hypothetical light primordial black holes (see, e.g., Abbott et al., 2022; Bird et al., 2023).

In Fig. 7, we plotted spectral energy distributions (SEDs) of emitting shocked clouds ejected from the accretion disc due to the passages of lighter compact objects. To reach typical QPE X-ray luminosities of 𝐿 𝑋 ∼ 10 42 ⁢ erg ⁢ s − 1 (0.2-2.0 keV; red dotted line), the initial cloud size should be 𝑅 in ∼ 10 11 ⁢ cm . This corresponds to the influence radius of a CO with mass 𝜇 ∼ 10 2 − 10 3 ⁢ 𝑀 ⊙ orbiting around the MBH at the inclination of 5 − 15 degrees with respect to the accretion disc (prograde sense), respectively. This range corresponds to rather low inclinations. If the mutual inclination is increased so that the CO and the disc are approximately perpendicular to each other, then the required CO mass is in the IMBH range, 𝜇 ∼ 29 000 ⁢ 𝑀 ⊙ , since the relative velocity increases to 𝑣 rel ∼ 86 400 ⁢ km ⁢ s − 1 . If we keep the same inclination of 15 degrees and decrease the perturber mass by one order of magnitude, the influence radius decreases by one order and the X-ray luminosity by two orders of magnitude, see Eqs. (14), (18), and Fig. 7.

For Solar-mass COs, the X-ray luminosity is typically 𝐿 𝑋 ≲ 10 39 ⁢ erg ⁢ s − 1 (solid orange line in Fig. 7) for the MBH mass of ≳ 10 5 ⁢ 𝑀 ⊙ and inclination of ≳ 5 degrees, which corresponds to the relative velocity of ≳ 2500 ⁢ km ⁢ s − 1 (the orbital distance is 111 ⁢ 𝑟 g for 𝑀

10 5 ⁢ 𝑀 ⊙ , one hour orbital period). For the minimal inclination of 𝜄 ∼ 0.5 ∘ around 𝑀

10 6 ⁢ 𝑀 ⊙ , the X-ray luminosity could reach 𝐿 𝑋 ∼ 10 42 ⁢ erg ⁢ s − 1 assuming 𝑇 cloud ∼ 10 6 ⁢ K , which is, however, a special “grazing” setup. Using the CO-disk interaction model of Linial & Metzger (2023), the luminosity estimate is 𝐿 𝑋 ∼ 2 × 10 39 ⁢ erg ⁢ s − 1 for 𝑀

10 6 ⁢ 𝑀 ⊙ and 𝑚 ˙

0.1 . For the interactions of a Solar-type star with the disk, the same model predicts 𝐿 ⋆ ∼ 1.4 × 10 42 ⁢ erg ⁢ s − 1 . However, Solar-type stars are tidally disrupted at 𝑟 ∼ 47 ⁢ 𝑟 g , hence they are not relevant for QPPs with 𝑓 QPP ∼ 0.5 mHz. On the other hand, brown dwarfs with the mass of ∼ 0.08 ⁢ 𝑀 ⊙ and the radius of ∼ 0.1 ⁢ 𝑅 ⊙ would disrupt at ∼ 𝑟 t / 𝑟 g ∼ 11 ⁢ ( 𝑅 ⋆ / 0.1 ⁢ 𝑅 ⊙ ) ⁢ ( 𝑀 / 10 6 ⁢ 𝑀 ⊙ ) − 2 / 3 ⁢ ( 𝜇 / 0.08 ⁢ 𝑀 ⊙ ) − 1 / 3 , hence the brown dwarf-disk interactions with 𝑓 QPP ∼ 0.5 mHz are plausible. The X-ray flares would have luminosities one order of magnitude below the QPE luminosities (Linial & Metzger, 2023),

𝐿 dwarf

∼ 3 × 10 41 ⁢ ( 𝑅 ⋆ 0.1 ⁢ 𝑅 ⊙ ) 2 / 3 ⁢ ( 𝑀 10 6 ⁢ 𝑀 ⊙ )

× ( 𝑚 ˙ 0.1 ) 1 / 3 ⁢ ( 𝑃 QPP 0.5 ⁢ hours ) − 2 / 3 ⁢ erg ⁢ s − 1 .

(20)

Hence, the prospects for detecting X-ray flares from orbiting and inclined solar-mass and subsolar-mass COs are currently weak, but they can improve with the new generation of sensitive X-ray imaging instruments, such as Athena and Lynx. For instance, for the source at the redshift of 𝑧

0.05 , the eruption X-ray flux density for the Solar-mass CO is predicted to be 𝐹 X ∼ 10 39 ⁢ erg ⁢ s − 1 / [ 4 ⁢ 𝜋 ⁢ ( 222.3 ⁢ Mpc ) 2 ] ∼ 1.7 × 10 − 16 ⁢ erg ⁢ s − 1 ⁢ cm − 2 , which is at the sensitivity limit of Athena and Lynx for one hour exposure time, see Fig. 6. Solar and subsolar-mass COs have characteristic influence radii smaller than one gravitational radius for typical parameters ( 𝑀

10 6 ⁢ 𝑀 ⊙ ). Using Eq. (14), we obtain 𝑅 inf ∼ 7 × 10 − 4 ⁢ 𝑟 g for 𝜄

15 ∘ , and using Eq. (17), we get 𝑅 Hill ∼ 0.2 ⁢ 𝑟 g . Hence, this limits the overall strength of the accretion-disc perturbation and the resulting emission modulation.

We can also make rough estimates for CO masses causing significant perturbations of geometrically thick accretion disks based on the simulations of Suková et al. (2021). In that paper it was found that in order to detect significant quasiperiodic variability in the inflow (accretion) rate, the size of the influence radius of the CO should be of the order of 𝑅 inf ∼ 0.1 − 1.0 ⁢ 𝑟 g for the CO distance of ∼ 10 ⁢ 𝑟 g . For 𝑀

10 6 ⁢ 𝑀 ⊙ , one gravitational radius we obtain the required CO mass in the range 𝜇 ∼ 160 − 42 000 ⁢ 𝑀 ⊙ (inclination between 5 to 90 degrees with respect to the disc midplane) to induce significant quasiperiodic variability. We can also compare this with the estimate using the Hill radius in eq. (17); the mass of the CO with the Hill radius equal to 1 ⁢ 𝑟 g is

𝜇

𝑀 3 𝑃 orb 2 ⁢ 12 ⁢ 𝜋 2 ⁢ 𝐺 2 𝑐 6 ,

∼ 221 ⁢ ( 𝑀 10 6 ⁢ 𝑀 ⊙ ) 3 ⁢ ( 𝑃 orb 1 ⁢ hour ) − 2 ⁢ 𝑀 ⊙ ,

(21)

which is within the same order of magnitude as the estimate using 𝑅 inf .

Figure 8:Comparison of different dynamical timescales for COs of different masses. Left: Timescales expressed in years as a function of the distance from the MBH of 10 6 ⁢ 𝑀 ⊙ (in gravitational radii). Different dynamical processes (merger, grinding, hydrodynamical drag) are listed in the legend. Solid lines correspond to the CO mass of 100 ⁢ 𝑀 ⊙ , while dashed lines correspond to the CO mass of 1 ⁢ 𝑀 ⊙ . The vertical solid line at the distance of ∼ 24 ⁢ 𝑟 g corresponds to the orbital period of one hour. The hydrodynamical and griding timescales are estimated for the standard disk with the relative accretion rate of 𝑚 ˙

0.1 and the initial inclination of the CO set to 15 ∘ . The stars represent the positions of detected QPP sources (see Table 1) assuming that the CO orbital period is twice as long as the measured QPP period. The distance of the CO is estimated assuming a circular orbit around a 10 6 ⁢ 𝑀 ⊙ MBH. Right: Timescales (in years) as a function of the CO mass (in Solar masses). The distance from the MBH of 10 6 ⁢ 𝑀 ⊙ is fixed to ∼ 24 ⁢ 𝑟 g , which corresponds to the orbital period of one hour. In this panel, we also depict separately the hydrodynamical-drag and grinding timescales for standard and hot accretion-flow discs adopting the relative accretion rates of 0.1 and 0.01, respectively.

Another issue with solar-mass and subsolar-mass compact objects is that they will not merge with the MBH within ∼ 10 years, even for distances of the order of 10 gravitational radii. In Fig. 8 (left panel), we compare the merger timescale using the evolution model of Peters & Mathews (1963) for a CO of 100 ⁢ 𝑀 ⊙ (solid black line) with the one corresponding to 1 ⁢ 𝑀 ⊙ CO (dashed black line). For the CO of 100 ⁢ 𝑀 ⊙ , the merger timescale is ∼ 10 years for the mean distance of ∼ 24 ⁢ 𝑟 g . We compare the merger timescales with the hydrodynamical grinding (or alignment) timescales (Syer et al., 1991, assuming the standard disk with 𝑚 ˙

0.1 and the initial inclination of 15 degrees) as well as with the hydrodynamical drag timescale related to dynamical friction due to the accretion-disc material (Ostriker, 1999; Narayan, 2000), assuming the limiting case when the CO is embedded within the disk. We can notice that for heavier COs hydrodynamical timescales (alignment and drag) are shorter by a factor 100. By the same factor, the merger timescale is also shorter. The interaction of the embedded CO with the realistic accretion disk is expected to be more complex than the inspiral motion given by the classical hydrodynamic drag. Density waves can lead to both inward and outward type I migration and eventually to the formation of “migration” traps (Paardekooper et al., 2010; Bellovary et al., 2016; Metzger et al., 2022). However, these dynamical effects are beyond the scope of the comparison of order-of-magnitude estimates provided here.

The inspiral due to gravitational-wave emission dominates inside ∼ 35 ⁢ 𝑟 g (merger timescales is shorter than the hydrodrag or dynamical friction timescale), while the merger timescale is shorter than the characteristic grinding timescale inside ∼ 350 ⁢ 𝑟 g . This indicates that inside ∼ 35 ⁢ 𝑟 g , where the EM counterparts of LISA EMRI sources are expected to be located, the gravitational-wave emission should be the main driver of the inspiral, with the dynamical friction force being progressively negligible (the associated timescale is longer than the merger timescale). At larger radii, the dynamical friction force due to the accretion-disc material as well as the hydrodynamical drag causing the alignment are expected to play dominant roles in bringing lighter perturbing objects into the closer vicinity of the MBH and aligning them with the accretion-flow plane.

In the right panel of Fig. 8, we fix the distance of the CO to ∼ 24 ⁢ 𝑟 g , which corresponds to one-hour orbital period for 10 6 ⁢ 𝑀 ⊙ MBH, and we show the dependency between the characteristic timescales and the CO mass. We see that the merger timescale is ∼ 10 years for ∼ 100 ⁢ 𝑀 ⊙ , while heavier COs would merge too soon (before the LISA launch), while lighter, solar-mass COs would merge on the timescale of ∼ 1000 years. Similarly, it should be noted that the peak gravitational-wave strain of EMRIs scales linearly with 𝜇 , so the GW detection distance of light COs will be smaller as compared to heavy COs even if they were close to merger within the LISA window. Combining the favourable merger timescale for the LISA observing window with the strong enough electromagnetic signal (see Subsec. 4.2.1) gives preference for the CO mass of the order of ∼ 100 ⁢ 𝑀 ⊙ , hence at the border between the stellar-mass and the intermediate-mass black holes.

The origin and the dynamics of IMBHs in galactic nuclei are quite different from stellar-mass COs and the analysis of these differences is beyond the scope of the current manuscript. There may in principle also be new telltale features in the EM signature for higher-mass secondaries. For example, in the case of an IMBH orbiting in the plane of the accretion disk, one generically expects the perturbation to open a gap in the disk (Syer & Clarke, 1995; Štolc et al., 2023). This gap later closes due to increasing radiation pressure at lower radii when the system is approaching the LISA frequency window, and the closure process leads to increased EM activity preceding the inspiral detection (Kocsis et al., 2011). However, Kocsis et al. (2011) estimate that for a standard 𝛼 -disk the gap is open only above the radius

𝑟 gap 𝑟 g

max ⁢ { 20 ⁢ ( 𝛼 0.01 ) 1 / 5 ⁢ 𝑚 ˙ 0.1 ⁢ ( 𝑀 10 5 ⁢ 𝑀 ⊙ ) 2 / 5 ⁢ ( 𝜇 100 ⁢ 𝑀 ⊙ ) − 2 / 5

30 ⁢ ( 𝛼 0.01 ) 1 / 2 ⁢ 𝑚 ˙ 0.1 ⁢ ( 𝑀 10 5 ⁢ 𝑀 ⊙ ) 1 / 2 ⁢ ( 𝜇 100 ⁢ 𝑀 ⊙ ) − 1 / 2 } .

(24)

At higher secondary masses, this transition occurs closer to the primary BH and thus makes it more relevant for our discussion. For values 𝜇 ∼ 100 ⁢ 𝑀 ⊙ and 𝑀 ∼ 10 5 ⁢ 𝑀 ⊙ ,  𝛼 ∼ 0.01 and accretion at 10 % of the Eddington rate the gap would close below 𝑟 gap ∼ 30 ⁢ 𝑟 g or for orbital periods above ∼ 500 seconds. This implies that the closure of the gap occurs very close to the periods of hours to minutes of the QPPs we consider here. The associated increase of EM activity thus represents a new possible avenue for detecting and characterizing these systems.

4.2.4Eccentricity and disk precession

The M1 EMRI catalogue from Babak et al. (2017) used in Figs 1, 2 and 3 corresponds to the MBH mass function of Barausse (2012), with the assumption that typical MBHs are rapidly spinning, 𝑎 MBH ∼ 0.98 ⁢ 𝑀 , typical mass of the secondary CO ∼ 10 ⁢ 𝑀 ⊙ , and specific assumptions about the cusp erosion, regrowth, and structure of the star cluster in the MBH environments (see Babak et al., 2017, for more details). In particular, the M1 model assumes that the dominant formation channel for EMRIs is the already mentioned dynamical capture in a “loss-cone” scenario (Hils & Bender, 1995; Sigurdsson & Rees, 1997). This leads to inspirals that have eccentricities 𝑒 ≳ 0.1 during the entire course of their inspiral as can be seen from Figs. 1 and 2, and only weak correlation of the orbital plane with the direction of the central MBH rotation. In other words, the EMRIs in question have generic inclinations and eccentricities as illustrated in Fig. 9. On the other hand, the other already mentioned channels including the capture and separation of a binary by the Hills mechanism (Miller et al., 2005), and disk-assisted migration (Pan & Yang, 2021) lead to low-eccentricity EMRIs with low inclinations in the latter cases and generic inclinations in the former. Hence, it is important to discuss what these distributions of orbital parameter imply for EM observations. Here we show in particular that generic orbits will lead to subtle timing features that will make the matching of QPPs to EMRIs nontrivial.

If the perturbing CO evolves along an eccentric trajectory, this introduces a more complex two-periodic structure of the time-variability. With the leading time-scale being the azimuthal orbital period Ω 𝜙 ∼ 2 ⁢ 𝜋 / 𝑃 orb , the second time-scale is the pericentre precession frequency, which reads to leading post-Newtonian and mass-ratio order (Robertson, 1938)

Ω p

3 ⁢ ( 2 ⁢ 𝐺 ⁢ 𝑀 ⁢ 𝜋 ) 2 / 3 𝑃 orb 5 / 3 ⁢ 𝑐 2 ⁢ ( 1 − 𝑒 2 ) ∼ 0.13 hour ⁢ 1 1 − 𝑒 2 ⁢ ( 𝑀 10 6 ⁢ 𝑀 ⊙ ) 2 / 3 ⁢ ( 𝑃 orb 1 ⁢ hour ) − 5 / 3 ,

(25)

where 𝑒 is the eccentricity of the orbit. Specifically, if one assumes that the variability is caused by the CO punching through an accretion disk in the equatorial plane of the primary, these intersections will occur at different times and radii for an inclined eccentric trajectory with “beat” patterns in the time domain. For example, taking a moderate eccentricity 𝑒 ∼ 0.3 and the orbital period and the MBH mass as 1 hour and 10 6 ⁢ 𝑀 ⊙ , respectively, the pericenter precession will complete a cycle in roughly 44 hours, so the beat pattern will span ∼ 88 eruptions. An example of such an orbit and its intersection times and radii are plotted in Figs 9 and 10.

Figure 9:The intersections (red) of an eccentric and inclined free test particle orbit (blue) through an accretion disk (orange) lying in the equatorial plane of a Kerr black hole with spin 𝑎

0.98 (black). The orbit is prograde with the orbital parameters semi-latus rectum 𝑝

24 ⁢ 𝑟 g , eccentricity 𝑒

0.3 and inclination 𝑖

60 ∘ (defined as in Schmidt, 2002). The interplay of eccentricity, pericenter precession, Lense-Thirring precession, and other sub-leading relativistic effects lead to a complex pattern. Figure 10:30 intersection times and radii of the orbit from Fig. 9 where the times correspond to the fiducial MBH mass 𝑀

10 6 ⁢ 𝑀 ⊙ .

Nevertheless, the EM flares are not necessarily produced immediately after the intersection, so identifying this secondary frequency may be subtle, as already discussed in Section 4.2.2. In particular, since the eccentric orbits intersect the disk at different radii and thus interact with a plasma slab with different temperatures, densities, and geometrical thickness, the delays between impact and radiation release will vary among impacts. This introduces additional timing features into the model that depend on the disk properties, as exemplified in the case of the fit of the parameters of the putative eccentric MBH binary in OJ 287 (Dey et al., 2018; Zwick & Mayer, 2023).

Another non-trivial timing feature may emerge from the fact that the orbital plane of the perturber will generally precess due to the spin-orbital coupling. This is to leading post-Newtonian order and mass ratio given by the Lense-Thirring frequency (Damour & Schäfer, 1988)

Ω LT

8 ⁢ 𝜋 2 ⁢ 𝐺 ⁢ 𝑀 ⁢ 𝑎 𝑃 orb 2 ⁢ 𝑐 3 ⁢ ( 1 − 𝑒 2 ) 3 / 2

∼ 0.11 hour ⁢ 𝑎 ( 1 − 𝑒 2 ) 3 / 2 ⁢ ( 𝑀 10 6 ⁢ 𝑀 ⊙ ) ⁢ ( 𝑃 orb 1 ⁢ hour ) − 2 ,

(26)

where 𝑎 ∈ ( − 1 , 1 ) is the dimensionless spin of the MBH. For an accretion disk aligned with the MBH spin this does not lead to significant effects in the temporal spacing of the flares. However, if the accretion disk is not aligned with the equatorial plane, the interplay of the differential precession of the accretion disk (Bardeen & Petterson, 1975) and the precession of the orbit will lead to additional intricate variability, a fact that was used by Franchini et al. (2023) to explain some of the temporal properties of the QPEs in GSN 069, eRO-QPE1, eRO-QPE2, and RX J1301.9+2747.

5Conclusion

EMRI observations through the LISA GW detector will provide a unique probe for fundamental physics in our Universe. Environment-rich formation channels of EMRIs would potentially give rise to their EM counterparts. These may be observable as soft X-ray ( ∼ 10 − 1 keV) band, low-frequency phenomena known as QPEs, and/or the less-abrupt QPOs, which emit semi-regular X-ray signals from both quiescent and active galactic nuclei. In this paper, we estimated the frequency band, in which a population of EMRIs in the Universe, as described in the M1 EMRI catalog of Babak et al. (2017), will most likely emit a QPE or QPO signal in the soft X-ray band “today” (in 2024). After establishing this frequency band to be ≈ 0.46 ± 0.22 mHz, we did a converse analysis on the well-known QPO source, RE J1034+396, treating it as an EMRI to estimate its CO’s mass to be ≈ 46 ⁢ 𝑀 ⊙ , and calculated a prior-predictive distribution on its LISA-band SNR with a best-fit value of 𝜌 ≈ 14 , highlighting it as a potential multi-messenger EMRI source. Given the estimate on the frequency band, we explored the possibility of observing LISA-EMRI relevant QPEs and QPOs in the EM spectrum through a variety of current and upcoming detectors in the X-ray, UV, and even radio bands. The impact of several EMRI parameters currently unaccounted for in our estimates, such as smaller masses of the CO, and the effect of eccentricity and inclination on the QPE/QPO flares, were also discussed. Our study thus attempted to strengthen the connection between EMRIs and QPEs/QPOs, highlighting them as potential multimessenger inference targets. If such observations become feasible in the future, it would greatly enhance the science output of the recently adopted LISA observatory.

Acknowledgements

SK acknowledges the computational resources made accessible by the National University of Singapore’s IT Research Computing group and the support of the NUS Research Scholarship (NUSRS). MZ acknowledges the financial support of the GAČR Junior Star grant no. GM24-10599M. Fig. 9 was generated with the help of the KerrGeodesics package within the BHPToolkit (bhptoolkit.org). VW and SK are grateful to the organizers for their invitation to the 1st Trieste meeting on the physics of gravitational waves where the idea of this paper was conceived.

Data Availability

The results of this study are fully reproducible and are made available on a Github repository called EMRI_EM_Counterparts9. Babak et al. (2017)’s M1 EMRI catalog as modified by Pozzoli et al. (2023) can be obtained by contacting the authors10, but we have also made available the back evolution trajectories data for our analysis in Section 2 on Zenodo11.

References AXIS Time-Domain et al. (2023) ↑ AXIS Time-Domain T., et al., 2023, arXiv e-prints, p. arXiv:2311.07658 Abbott et al. (2022) ↑ Abbott R., et al., 2022, Phys. Rev. Lett., 129, 061104 Alston et al. (2014) ↑ Alston W. N., Markevičiūtė J., Kara E., Fabian A. C., Middleton M., 2014, Monthly Notices of the Royal Astronomical Society: Letters, 445, L16–L20 Alston et al. (2015) ↑ Alston W. N., Parker M. L., Markevič iūtė J., Fabian A. C., Middleton M., Lohfink A., Kara E., Pinto C., 2015, Monthly Notices of the Royal Astronomical Society, 449, 467 Amaro-Seoane (2018) ↑ Amaro-Seoane P., 2018, Living Reviews in Relativity, 21 Amaro-Seoane (2019) ↑ Amaro-Seoane P., 2019, Phys. Rev. D, 99, 123025 Amaro-Seoane et al. (2007) ↑ Amaro-Seoane P., Gair J. R., Freitag M., Miller M. C., Mandel I., Cutler C. J., Babak S., 2007, Classical and Quantum Gravity, 24, R113 Amaro-Seoane et al. (2015) ↑ Amaro-Seoane P., Gair J. R., Pound A., Hughes S. A., Sopuerta C. F., 2015, Journal of Physics: Conference Series, 610, 012002 Amaro-Seoane et al. (2017) ↑ Amaro-Seoane P., et al., 2017, Laser Interferometer Space Antenna (arXiv:1702.00786) Amaro Seoane et al. (2021) ↑ Amaro Seoane P., et al., 2021, General Relativity and Gravitation, 54 Arcodia et al. (2021) ↑ Arcodia R., et al., 2021, Nature, 592, 704 Arcodia et al. (2024) ↑ Arcodia R., et al., 2024, arXiv e-prints, p. arXiv:2401.17275 Babak et al. (2010) ↑ Babak S., et al., 2010, Classical and Quantum Gravity, 27, 084009 Babak et al. (2017) ↑ Babak S., et al., 2017, Physical Review D, 95 Baghi et al. (2019) ↑ Baghi Q., Thorpe J. I., Slutsky J., Baker J., Canton T. D., Korsakova N., Karnesis N., 2019, Phys. Rev. D, 100, 022003 Balbus & Hawley (1991) ↑ Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214 Balbus & Hawley (1998) ↑ Balbus S. A., Hawley J. F., 1998, Reviews of Modern Physics, 70, 1 Barack & Cutler (2004) ↑ Barack L., Cutler C., 2004, Physical Review D, 69 Barausse (2012) ↑ Barausse E., 2012, Mon. Not. Roy. Astron. Soc., 423, 2533 Barausse et al. (2014) ↑ Barausse E., Cardoso V., Pani P., 2014, Phys. Rev. D, 89, 104059 Bardeen & Petterson (1975) ↑ Bardeen J. M., Petterson J. A., 1975, ApJ, 195, L65 Bellovary et al. (2016) ↑ Bellovary J. M., Mac Low M.-M., McKernan B., Ford K. E. S., 2016, ApJ, 819, L17 Ben-Ami et al. (2022) ↑ Ben-Ami S., et al., 2022, in den Herder J.-W. A., Nikzad S., Nakazawa K., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 12181, Space Telescopes and Instrumentation 2022: Ultraviolet to Gamma Ray. p. 1218105 (arXiv:2208.00159), doi:10.1117/12.2629850 Berry et al. (2019) ↑ Berry C. P. L., et al., 2019, The unique potential of extreme mass-ratio inspirals for gravitational-wave astronomy (arXiv:1903.03686) Bian & Huang (2010) ↑ Bian W.-H., Huang K., 2010, Monthly Notices of the Royal Astronomical Society, 401, 507 Bird et al. (2023) ↑ Bird S., et al., 2023, Physics of the Dark Universe, 41, 101231 Boller, Th. et al. (2001) ↑ Boller, Th. Keil, R. Trümper, J. O’Brien, P. T. Reeves, J. Page, M. 2001, A&A, 365, L146 Bonetti & Sesana (2020) ↑ Bonetti M., Sesana A., 2020, Physical Review D, 102 Braun et al. (2019) ↑ Braun R., Bonaldi A., Bourke T., Keane E., Wagg J., 2019, arXiv e-prints, p. arXiv:1912.12699 Brenneman (2013) ↑ Brenneman L., 2013, Measuring the Angular Momentum of Supermassive Black Holes. Springer New York, doi:10.1007/978-1-4614-7771-6, https://doi.org/10.1007%2F978-1-4614-7771-6 Britzen et al. (2018) ↑ Britzen S., et al., 2018, MNRAS, 478, 3199 Britzen et al. (2023) ↑ Britzen S., Zajaček M., Gopal-Krishna Fendt C., Kun E., Jaron F., Sillanpää A., Eckart A., 2023, ApJ, 951, 106 Brunner et al. (2008) ↑ Brunner H., Cappelluti N., Hasinger G., Barcons X., Fabian A. C., Mainieri V., Szokoly G., 2008, A&A, 479, 283 Burrows et al. (2005) ↑ Burrows D. N., et al., 2005, Space Sci. Rev., 120, 165 CHIME Collaboration et al. (2022) ↑ CHIME Collaboration et al., 2022, ApJS, 261, 29 Chakraborty et al. (2021) ↑ Chakraborty J., Kara E., Masterson M., Giustini M., Miniutti G., Saxton R., 2021, The Astrophysical Journal Letters, 921, L40 Chakraborty et al. (2024) ↑ Chakraborty J., et al., 2024, arXiv e-prints, p. arXiv:2402.08722 Chaudhury et al. (2018) ↑ Chaudhury K., et al., 2018, Monthly Notices of the Royal Astronomical Society, 478, 4830 Chen et al. (2022) ↑ Chen X., Qiu Y., Li S., Liu F. K., 2022, The Astrophysical Journal, 930, 122 Chua & Cutler (2022) ↑ Chua A. J., Cutler C. J., 2022, Physical Review D, 106 Chua et al. (2021) ↑ Chua A. J. K., Katz M. L., Warburton N., Hughes S. A., 2021, Phys. Rev. Lett., 126, 051102 Collin & Zahn (1999) ↑ Collin S., Zahn J.-P., 1999, Ap&SS, 265, 501 Colpi et al. (2024) ↑ Colpi M., et al., 2024, arXiv e-prints, p. arXiv:2402.07571 Cornish & Crowder (2005) ↑ Cornish N. J., Crowder J., 2005, Phys. Rev. D, 72, 043005 Czerny et al. (2016) ↑ Czerny B., et al., 2016, A&A, 594, A102 Damour & Schäfer (1988) ↑ Damour T., Schäfer G., 1988, Nuovo Cimento B Serie, 101B, 127 Del Pozzo (2012) ↑ Del Pozzo W., 2012, Phys. Rev. D, 86, 043011 Dey et al. (2018) ↑ Dey L., et al., 2018, The Astrophysical Journal, 866, 11 Dey et al. (2021) ↑ Dey K., Karnesis N., Toubiana A., Barausse E., Korsakova N., Baghi Q., Basak S., 2021, Physical Review D, 104 Eckart et al. (2017) ↑ Eckart A., et al., 2017, Foundations of Physics, 47, 553 Franchini et al. (2023) ↑ Franchini A., et al., 2023, A&A, 675, A100 Fujita & Shibata (2020) ↑ Fujita R., Shibata M., 2020, Physical Review D, 102 Gair et al. (2013) ↑ Gair J. R., Vallisneri M., Larson S. L., Baker J. G., 2013, Living Reviews in Relativity, 16 Gaskin et al. (2019) ↑ Gaskin J. A., et al., 2019, Journal of Astronomical Telescopes, Instruments, and Systems, 5, 021001 Gierliński et al. (2008) ↑ Gierliński M., Middleton M., Ward M., Done C., 2008, Nature, 455, 369 Giustini et al. (2020) ↑ Giustini M., Miniutti G., Saxton R. D., 2020, A&A, 636, L2 Gourgoulhon et al. (2019) ↑ Gourgoulhon E., Le Tiec A., Vincent F. H., Warburton N., 2019, A&A, 627, A92 Gray et al. (2023) ↑ Gray R., et al., 2023, Joint cosmological and gravitational-wave population inference using dark sirens and galaxy catalogues (arXiv:2308.02281) Greene et al. (2020) ↑ Greene J. E., Strader J., Ho L. C., 2020, ARA&A, 58, 257 Guolo et al. (2023) ↑ Guolo M., et al., 2023, arXiv e-prints, p. arXiv:2309.03011 Gupta et al. (2018) ↑ Gupta A. C., Tripathi A., Wiita P. J., Gu M., Bambi C., Ho L. C., 2018, A&A, 616, L6 Hasinger et al. (2001) ↑ Hasinger G., et al., 2001, A&A, 365, L45 Hills (1988) ↑ Hills J. G., 1988, Nature, 331, 687 Hils & Bender (1995) ↑ Hils D., Bender P. L., 1995, ApJ, 445, L7 Holz & Hughes (2005) ↑ Holz D. E., Hughes S. A., 2005, ApJ, 629, 15 Ingram & Motta (2019) ↑ Ingram A. R., Motta S. E., 2019, New Astronomy Reviews, 85, 101524 Ingram et al. (2021) ↑ Ingram A., Motta S. E., Aigrain S., Karastergiou A., 2021, MNRAS, 503, 1703 Ivanov et al. (1998) ↑ Ivanov P. B., Igumenshchev I. V., Novikov I. D., 1998, ApJ, 507, 131 Jin et al. (2020) ↑ Jin C., Done C., Ward M., 2020, Monthly Notices of the Royal Astronomical Society, 495, 3538 Karas & Šubr (2001) ↑ Karas V., Šubr L., 2001, A&A, 376, 686 Karas et al. (2021) ↑ Karas V., Svoboda J., Zajaček M., 2021, in RAGtime: Workshops on black holes and netron stars. p. E1 (arXiv:1901.06507), doi:10.48550/arXiv.1901.06507 Katz et al. (2021) ↑ Katz M. L., Chua A. J. K., Speri L., Warburton N., Hughes S. A., 2021, Phys. Rev. D, 104, 064047 Kejriwal et al. (2023) ↑ Kejriwal S., Speri L., Chua A. J. K., 2023, Impact of Correlations on the Modeling and Inference of Beyond Vacuum-GR Effects in Extreme-Mass-Ratio Inspirals (arXiv:2312.13028) King (2023a) ↑ King A., 2023a, Monthly Notices of the Royal Astronomical Society: Letters, 523, L26 King (2023b) ↑ King A., 2023b, Monthly Notices of the Royal Astronomical Society: Letters, 526, L31 Kley & Nelson (2012) ↑ Kley W., Nelson R., 2012, Annual Review of Astronomy and Astrophysics, 50, 211–249 Kocsis et al. (2011) ↑ Kocsis B., Yunes N., Loeb A., 2011, Phys. Rev. D, 84, 024032 Kormendy & Ho (2013) ↑ Kormendy J., Ho L. C., 2013, ARA&A, 51, 511 Krtička et al. (2024) ↑ Krtička J., et al., 2024, Space Sci. Rev., 220, 24 Kulkarni et al. (2021) ↑ Kulkarni S. R., et al., 2021, arXiv e-prints, p. arXiv:2111.15608 Laghi et al. (2021) ↑ Laghi D., Tamanini N., Pozzo W. D., Sesana A., Gair J., Babak S., Izquierdo-Villalba D., 2021, Monthly Notices of the Royal Astronomical Society, 508, 4512 Lehto & Valtonen (1996) ↑ Lehto H. J., Valtonen M. J., 1996, ApJ, 460, 207 Levin (2007) ↑ Levin Y., 2007, MNRAS, 374, 515 Levin & Beloborodov (2003) ↑ Levin Y., Beloborodov A. M., 2003, ApJ, 590, L33 Li et al. (2023) ↑ Li R.-N., Zhao Z.-Y., Gao Z., Wang F.-Y., 2023, The Astrophysical Journal Letters, 956, L2 Lin et al. (2013) ↑ Lin D., Irwin J. A., Godet O., Webb N. A., Barret D., 2013, The Astrophysical Journal, 776, L10 Linial & Metzger (2023) ↑ Linial I., Metzger B. D., 2023, ApJ, 957, 34 Linial & Metzger (2024) ↑ Linial I., Metzger B. D., 2024, ApJ, 963, L1 Littenberg & Cornish (2023) ↑ Littenberg T. B., Cornish N. J., 2023, Phys. Rev. D, 107, 063004 Lops et al. (2023) ↑ Lops G., Izquierdo-Villalba D., Colpi M., Bonoli S., Sesana A., Mangiagli A., 2023, MNRAS, 519, 5962 Metzger et al. (2022) ↑ Metzger B. D., Stone N. C., Gilbaum S., 2022, The Astrophysical Journal, 926, 101 Miller et al. (2005) ↑ Miller M. C., Freitag M., Hamilton D. P., Lauburg V. M., 2005, ApJ, 631, L117 Miniutti et al. (2019) ↑ Miniutti G., et al., 2019, Nature, 573, 381 Miniutti et al. (2023a) ↑ Miniutti G., Giustini M., Arcodia R., Saxton R. D., Read A. M., Bianchi S., Alexander K. D., 2023a, A&A, 670, A93 Miniutti et al. (2023b) ↑ Miniutti G., Giustini M., Arcodia R., Saxton R. D., Chakraborty J., Read A. M., Kara E., 2023b, A&A, 674, L1 Moore et al. (2019) ↑ Moore C. J., Gerosa D., Klein A., 2019, Monthly Notices of the Royal Astronomical Society: Letters, 488, L94–L98 Mushotzky et al. (2019) ↑ Mushotzky R., et al., 2019, in Bulletin of the American Astronomical Society. p. 107 (arXiv:1903.04083), doi:10.48550/arXiv.1903.04083 Nandra et al. (2013) ↑ Nandra K., et al., 2013, arXiv e-prints, p. arXiv:1306.2307 Narayan (2000) ↑ Narayan R., 2000, ApJ, 536, 663 Ostriker (1999) ↑ Ostriker E. C., 1999, ApJ, 513, 252 Paardekooper et al. (2010) ↑ Paardekooper S. J., Baruteau C., Crida A., Kley W., 2010, MNRAS, 401, 1950 Paczynski (1978) ↑ Paczynski B., 1978, Acta Astron., 28, 91 Pan & Yang (2021) ↑ Pan Z., Yang H., 2021, Physical Review D, 103 Pan et al. (2016) ↑ Pan H.-W., Yuan W., Yao S., Zhou X.-L., Liu B., Zhou H., Zhang S.-N., 2016, ApJ, 819, L19 Pasham et al. (2019) ↑ Pasham D. R., et al., 2019, Science, 363, 531 Pasham et al. (2024a) ↑ Pasham D. R., et al., 2024a, arXiv e-prints, p. arXiv:2402.10140 Pasham et al. (2024b) ↑ Pasham D. R., et al., 2024b, The Astrophysical Journal Letters, 963, L47 Peters & Mathews (1963) ↑ Peters P. C., Mathews J., 1963, Physical Review, 131, 435 Pihajoki (2016) ↑ Pihajoki P., 2016, MNRAS, 457, 1145 Pozzoli et al. (2023) ↑ Pozzoli F., Babak S., Sesana A., Bonetti M., Karnesis N., 2023, Computation of stochastic background from extreme mass ratio inspiral populations for LISA (arXiv:2302.07043) Predehl et al. (2021) ↑ Predehl P., et al., 2021, A&A, 647, A1 Rauch (1995) ↑ Rauch K. P., 1995, MNRAS, 275, 628 Ray et al. (2019) ↑ Ray P. S., et al., 2019, arXiv e-prints, p. arXiv:1903.03035 Remillard & McClintock (2006) ↑ Remillard R. A., McClintock J. E., 2006, ARA&A, 44, 49 Remillard et al. (2022) ↑ Remillard R. A., et al., 2022, AJ, 163, 130 Reynolds (2013) ↑ Reynolds C. S., 2013, Classical and Quantum Gravity, 30, 244004 Robertson (1938) ↑ Robertson H. P., 1938, Annals of Mathematics, pp 101–104 Robson et al. (2019) ↑ Robson T., Cornish N. J., Liu C., 2019, Classical and Quantum Gravity, 36, 105011 Schmidt (2002) ↑ Schmidt W., 2002, Classical and Quantum Gravity, 19, 2743 Schutz (1986a) ↑ Schutz B. F., 1986a, Nature, 323, 310 Schutz (1986b) ↑ Schutz B. F., 1986b, Nature, 323, 310 Secunda et al. (2020) ↑ Secunda A., et al., 2020, The Astrophysical Journal, 903, 133 Semerák et al. (1999) ↑ Semerák O., Karas V., de Felice F., 1999, PASJ, 51, 571 Shakura & Sunyaev (1973) ↑ Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337 Shapiro & Teukolsky (1983) ↑ Shapiro S. L., Teukolsky S. A., 1983, Black holes, white dwarfs and neutron stars. The physics of compact objects (A Wiley-Interscience Publication, New York: Wiley), doi:10.1002/9783527617661. Shvartzvald et al. (2023) ↑ Shvartzvald Y., et al., 2023, arXiv e-prints, p. arXiv:2304.14482 Sigl et al. (2007) ↑ Sigl G., Schnittman J., Buonanno A., 2007, Phys. Rev. D, 75, 024034 Sigurdsson & Rees (1997) ↑ Sigurdsson S., Rees M. J., 1997, MNRAS, 284, 318 Sikora et al. (2007) ↑ Sikora M., Stawarz Ł., Lasota J.-P., 2007, ApJ, 658, 815 Speri et al. (2023) ↑ Speri L., Antonelli A., Sberna L., Babak S., Barausse E., Gair J. R., Katz M. L., 2023, Phys. Rev. X, 13, 021035 Štolc et al. (2023) ↑ Štolc M., Zajaček M., Czerny B., Karas V., 2023, MNRAS, 522, 2869 Suková et al. (2021) ↑ Suková P., Zajaček M., Witzany V., Karas V., 2021, The Astrophysical Journal, 917, 43 Syer & Clarke (1995) ↑ Syer D., Clarke C. J., 1995, Mon. Not. Roy. Astron. Soc., 277, 758 Syer et al. (1991) ↑ Syer D., Clarke C. J., Rees M. J., 1991, MNRAS, 250, 505 Tagawa & Haiman (2023) ↑ Tagawa H., Haiman Z., 2023, MNRAS, 526, 69 Tanaka & Ward (2004) ↑ Tanaka H., Ward W. R., 2004, ApJ, 602, 388 Tchekhovskoy et al. (2010) ↑ Tchekhovskoy A., Narayan R., McKinney J. C., 2010, ApJ, 711, 50 Vallisneri (2009) ↑ Vallisneri M., 2009, Classical and Quantum Gravity, 26, 094024 Valtonen et al. (2016) ↑ Valtonen M. J., et al., 2016, ApJ, 819, L37 Vaughan (2005) ↑ Vaughan S., 2005, A&A, 431, 391 Volonteri et al. (2005) ↑ Volonteri M., Madau P., Quataert E., Rees M. J., 2005, The Astrophysical Journal, 620, 69 Wang et al. (2014) ↑ Wang J.-Z., Lei W.-H., Wang D.-X., Zou Y.-C., Zhang B., Gao H., Huang C.-Y., 2014, The Astrophysical Journal, 788, 32 Werner et al. (2024) ↑ Werner N., et al., 2024, Space Sci. Rev., 220, 11 Wevers et al. (2022) ↑ Wevers T., Pasham D. R., Jalan P., Rakshit S., Arcodia R., 2022, A&A, 659, L2 Xian et al. (2021) ↑ Xian J., Zhang F., Dou L., He J., Shu X., 2021, ApJ, 921, L32 Yuan & Narayan (2014) ↑ Yuan F., Narayan R., 2014, ARA&A, 52, 529 Yuan et al. (2022) ↑ Yuan W., Zhang C., Chen Y., Ling Z., 2022, in , Handbook of X-ray and Gamma-ray Astrophysics. p. 86, doi:10.1007/978-981-16-4544-0_151-1 Zajaček (2021) ↑ Zajaček M., 2021, ApJ, 915, 111 Zajaček et al. (2023) ↑ Zajaček M., et al., 2023, arXiv e-prints, p. arXiv:2306.15082 Zhang et al. (2017) ↑ Zhang P., Zhang P.-f., Yan J.-z., Fan Y.-z., Liu Q.-z., 2017, ApJ, 849, 9 Zhang et al. (2023) ↑ Zhang H., Yang S., Dai B., 2023, ApJ, 946, 52 Zhou et al. (2024) ↑ Zhou C., Huang L., Guo K., Li Y.-P., Pan Z., 2024, Probing orbits of stellar mass objects deep in galactic nuclei with quasi-periodic eruptions (arXiv:2401.11190) Zwick & Mayer (2023) ↑ Zwick L., Mayer L., 2023, MNRAS, 526, 2754 eLISA Consortium et al. (2013) ↑ eLISA Consortium T., et al., 2013, The Gravitational Universe (arXiv:1305.5720) van der Klis (2000) ↑ van der Klis M., 2000, ARA&A, 38, 717 Appendix AAdditional priors on RE J1034+396 parameters

Here, for the QPO source RE J1034+396, we determine the priors on the MBH dimensionless spin parameter, 𝑎 , and the eccentricity and inclination of the CO’s orbit, i.e. 𝑒 0 and 𝜄 respectively. We use a right skewed-normal prior on 𝑎 with a mean of 0.9. This is based on evidence suggesting that MBHs in AGNs are generally expected to be endowed with a large spin (see, for example, Volonteri et al., 2005; Reynolds, 2013; Brenneman, 2013). Our model assumes prograde orbits, i.e. orbits in the same direction as the MBH spin. This is expected for EMRI sources, in which CO is either captured by the accretion disk surrounding the MBH or forms in-situ within the disk and migrates towards tighter orbits (see references for formation channels (iii)-(iv) as described in Section 1, and Bellovary et al., 2016). For the eccentricity and inclination of the CO’s orbit, we choose exponentially decaying distributions with means 0.05 and 0.01 for eccentricity and inclination respectively, allowing for only slightly inclined and eccentric orbits. This builds on the evidence that small objects orbiting a massive body in the presence of a disk lose their energy and angular momentum through interactions with the disk, circularizing their orbits and decreasing their inclination (Tanaka & Ward, 2004; see also Kley & Nelson, 2012 for a review). Energy and angular momentum loss through GW radiation in EMRIs further assists this circularization, such that environment-rich EMRIs can be expected to have small eccentricities and inclinations (Pan & Yang, 2021). As noted in Section 3, the correlations of the estimated parameters ( 𝜇 , 𝜌 ) with the set { 𝑎 , 𝑒 0 , 𝜄 } is small ( | 𝐶 𝑖 ⁢ 𝑗 | ≲ 0.02 ) , and thus the choice of prior does not change our results significantly. The full 8-dimensional prior-predictive estimates on the RE J1034+396 parameter set { 𝑀 , 𝑎 , 𝑒 0 , 𝜄 , 𝑓 orb , 2007 , 𝑓 ˙ , 𝜇 , SNR } are provided in Fig. 11.

Figure 11:Distributions of { 𝑀 , 𝑎 , 𝑒 0 , 𝜄 , 𝑓 orb , 2007 , 𝑓 ˙ orb , obs , 𝜇 , SNR } of RE J1034+396 modeled as an extreme-mass-ratio inspiral. { 𝑀 , 𝑓 orb , 2007 , 𝑓 ˙ orb , obs } are estimated from current observational constraints on RE J1034+396 and distributions of { 𝑎 , 𝑒 0 , 𝜄 } are based on expected properties of EMRIs as described in the text. The thus obtained prior-predictive distribution on 𝜇 is described in the seventh panel, and the corresponding LISA-band SNR in the final panel. Report Issue Report Issue for Selection Generated by L A T E xml Instructions for reporting errors

We are continuing to improve HTML versions of papers, and your feedback helps enhance accessibility and mobile support. To report errors in the HTML that will help us improve conversion and rendering, choose any of the methods listed below:

Click the "Report Issue" button. Open a report feedback form via keyboard, use "Ctrl + ?". Make a text selection and click the "Report Issue for Selection" button near your cursor. You can use Alt+Y to toggle on and Alt+Shift+Y to toggle off accessible reporting links at each section.

Our team has already identified the following issues. We appreciate your time reviewing and reporting rendering errors we may not have found yet. Your efforts will help us improve the HTML versions for all readers, because disability should not be a barrier to accessing research. Thank you for your continued support in championing open access for all.

Have a free development cycle? Help support accessibility at arXiv! Our collaborators at LaTeXML maintain a list of packages that need conversion, and welcome developer contributions.