Exploring Low-mass Black Holes through Tidal Disruption Events in the Early Universe: Perspectives in the Era of the JWST, Roman Space Telescope, and LSST Surveys

The James Webb Space Telescope (JWST) has recently uncovered the presence of low-luminosity active galactic nuclei (AGNs) at z = 4–11. Spectroscopic observations have provided estimates of the nuclear black hole (BH) masses for these sources, extending the low-mass boundary down to M • ∼ 106–7 M ⊙. Despite this breakthrough, the observed lowest mass of BHs is still ≳1–2 orders of magnitude heavier than the predicted mass range of their seed population, thereby leaving the initial mass distribution of massive BHs poorly constrained. In this paper, we focus on UV-to-optical (in the rest frame) flares of stellar tidal disruption events (TDEs) embedded in low-luminosity AGNs as a tool for exploring low-mass BH populations with ≲104–6 M ⊙. We provide an estimate of the TDE rate over z = 4–11, associated with the properties of JWST-detected AGN host galaxies, and we find that deep and wide survey programs with JWST and the Roman Space Telescope (RST) can detect and identify TDEs up to z ≃ 4–7. The predicted detection numbers of TDEs at z > 4 in 1 yr are NTDE∼2–10(0.2–2) for the JADES-Medium (and COSMOS-Web) survey with JWST and NTDE∼2–10(8–50) for the deep (and wide) tiers of the High Latitude Time Domain Survey with RST. We further discuss survey strategies for hunting for transient high-redshift TDEs in wide-field surveys with RST, as well as a joint observation campaign with the Vera C. Rubin Observatory for enhancing the detection number. The high-redshift TDE search will give us a unique opportunity to probe the mass distribution of early BH populations.


Introduction
Observations of high-redshift quasars within the first few billion years of cosmic history have benefited from wide-field surveys such as the Sloan Digital Sky Survey (SDSS; Jiang et al. 2016;Wu et al. 2022), Pan-Starrs 1 (Morganson et al. 2012;Bañados et al. 2023), the Dark Energy Spectroscopic Instrument (Yang et al. 2023), and the Hyper Suprime-Cam (HSC) Subaru Strategic Program (Matsuoka et al. 2016(Matsuoka et al. , 2018;;Akiyama et al. 2018;Niida et al. 2020).Extensive efforts in spectroscopic follow-up observations have revealed that supermassive black holes (SMBHs) have formed and grown in mass to M •  10 9 M e within a short amount of cosmic time (1 Gyr at z  6; Wu et al. 2015;Shen et al. 2019;Wang et al. 2021; see also a recent review by Fan et al. 2023).This rapid growth of SMBHs represents one of the most intriguing unsolved mysteries in modern astrophysics.
The presence of such monster SMBHs has stimulated numerous ideas for their quick assembly mechanisms (e.g., Inayoshi et al. 2020;Volonteri et al. 2021): for instance, rapid gas feeding into the nuclei of early protogalaxies (Volonteri & Rees 2005;Inayoshi et al. 2016;Toyouchi et al. 2021), the formation of massive seed black holes (BHs) through primordial star formation (Omukai 2001;Bromm & Loeb 2003;Inayoshi et al. 2014;Regan et al. 2014;Wise et al. 2019), and runaway stellar collisions in dense clusters (Devecchi & Volonteri 2009;Chon & Omukai 2020).Previous studies have often focused on individual seeding scenarios within specific mass ranges and discussed their impact on subsequent evolution toward lower redshifts (Bhowmick et al. 2021;Spinoso et al. 2023).However, in reality, the mass distribution of seed BHs is expected to be more continuous, ranging from so-called "light seeds" with M • ∼ 10 2 M e to "heavy seeds" exceeding ∼10 5 M e , due to the combination of various seeding mechanisms (e.g., Sassano et al. 2021;Toyouchi et al. 2023).Recent studies by Li et al. (2021) and Lupi et al. (2021) have found that the formation of heavy seed BHs occurs more efficiently in rare, overdense regions of the Universe, where nonlinear galaxy clustering creates the peculiar external environments required for BH seeding (e.g., H 2 dissociating radiation and dynamical heating via halo mergers).Consequently, the black hole mass function (BHMF) for these seed populations is well approximated with a distribution of and extends the upper mass to M •  10 5 M e (Li et al. 2023a).
The subsequent evolution of the BHMF is primarily influenced by the quasar luminosity function (QLF), which contains key information about the underlying physics of the radiative and accretion processes.The shapes of both the BHMF and QLF have been extensively investigated by various approaches.Semi-analytical calculations have been employed to incorporate BH growth mechanisms, including gas accretion and BH mergers in the framework of hierarchical structure formation (Shankar et al. 2009;Ricarte & Natarajan 2018;Kim & Im 2021;Yung et al. 2021;Trinca et al. 2022;Li et al. 2023b).A recent work by Li et al. (2023b) presented a theoretical model describing BH growth starting from initial seeding at z  20 to z ∼ 4, taking into account the episodical accretion nature of quasars.This model successfully explains the cosmic evolution of the observed z 4 QLFs at UV absolute magnitude brighter than −24 mag (see Akiyama et al. 2018;Matsuoka et al. 2018;Niida et al. 2020) and furthermore offers predictions for the BHMF with masses ranging from the low-mass end at M • ∼ 10 5−6 M e to ∼10 10 M e at 4 z 11.
Recently, the unprecedented sensitivity of the James Webb Space Telescope (JWST) has opened a new window to explore the low-luminosity active galactic nuclei (AGNs) at z ∼ 4-7 (Onoue et al. 2023).Spectroscopic observations have confirmed the broad components of emission lines and the presence of AGNs (e.g., Harikane et al. 2023;Kocevski et al. 2023;Larson et al. 2023;Maiolino et al. 2023;Matthee et al. 2024).The BH masses for those JWST-detected faint AGNs are M • ∼ 10 6−8 M e , which is 1-2 orders of magnitude lower than those of previously known quasars at z > 4. Intriguingly, some of the sources have extremely red colors in the rest-frame optical bands, while the rest-frame UV color seems fairly consistent with those of known Lyman-break galaxies at similar redshifts (e.g., Harikane et al. 2023;Kocevski et al. 2023;Labbe et al. 2023;Barro et al. 2024;Greene et al. 2024;Matthee et al. 2024).The so-called "little red dot" sources are interpreted to be in a transition stage from a dust-obscured starburst to an unobscured luminous quasar due to the expelling of gas and dust (Fujimoto et al. 2022).These findings demonstrate the capability of JWST to extend high-z BH studies to masses approaching the BH seed population, while a significant 1-2 dex mass gap remains.
In this paper, we focus on UV-to-optical flares (in the rest frame) resulting from stellar tidal disruption events (TDEs) caused by BHs with masses around ∼10 4−6 M e (Rees 1988;Guillochon & Ramirez-Ruiz 2013;Stone et al. 2013; see also the review papers by van Velzen et al. 2020 andGezari 2021).Stellar TDEs can be triggered by low-mass BHs, where the BH gravitational force on the stellar surface dominates over the self-gravity of plunging stars before they reach the event horizon.These transient sources can be brighter than the persistent AGN emission, alleviating the restrictions imposed by the sensitivity of telescopes and thus improving the detectability of such low-mass BHs.Moreover, it is worth noting that a large fraction of AGNs discovered with JWST exhibit extremely red colors and compact morphologies.These characteristics suggest that the nuclear BHs are likely embedded in active star-forming regions, presumably dense stellar clusters, where TDEs occur frequently due to the short relaxation time.The rate of TDEs, calculated based on the twobody relaxation time, is sensitive to the low-mass end of the BHMF around M •  10 6 M e (Magorrian & Tremaine 1999;Wang & Merritt 2004;Stone & Metzger 2016).Therefore, TDEs caused by low-mass BHs can be used as powerful tools to explore the underlying BH populations in the early Universe (e.g., Kashiyama & Inayoshi 2016).
Motivated by the recent discovery of high-z AGNs in the JWST surveys, we consider two types of transient TDEs, triggered within unobscured and obscured AGNs.TDEs occur more frequently in the dense gaseous and stellar environments within dust-reddened obscured AGNs.However, the emission from these TDEs is attenuated by dust grains, requiring deep JWST surveys for detection.Since these surveys cover limited areas (<1 deg 2 ), the TDEs produced by abundant BH populations with M • ∼ 10 4−5 M e are detectable.On the other hand, the wide-field surveys of the Roman Space Telescope (RST; Spergel et al. 2015) and Galaxy Reionization EXplorer and PLanetary Universe Spectrometer (GREX-PLUS; Inoue et al. 2023) are capable of identifying rarer TDEs in unobscured AGNs.Assuming the High Latitude Time Domain Survey with RST planned for supernova (SN) cosmology (Rose et al. 2021), we expect to detect ∼O(10) TDEs per year originating from AGNs with M • ∼ 10 4−5 M e at z  4.
This paper is organized as follows.In Section 2, we begin by presenting an overview of the current status of AGN observations with JWST.We also provide a theoretical model to characterize the properties of galactic nuclei on scales of 100 pc for JWST-detected AGNs at high redshifts.In Section 3, we give an estimate of the TDE rate led by BHs with M • ; 10 4−6 M e in those AGNs.By convolving the TDE rate per galaxy with a theoretical BHMF model calibrated using the most up-to-date Subaru-HSC AGN data (Li et al. 2023a(Li et al. , 2023b)), we make a prediction of the cosmic TDE rate as a function of BH mass and host galaxy properties, including dust obscuration.In Section 4, we discuss the detectability of high-z TDEs by the JWST, RST, and GREX-PLUS surveys, as well as larger sky-coverage surveys, such as Euclid and the Legacy Survey of Space and Time (LSST) by the Vera C. Rubin Observatory, and propose survey strategies aimed at identifying transient high-redshift TDEs.We finally summarize our findings and present some discussion in Section 5.

Populations
In this paper, we categorize AGNs into three groups based on the properties of their spectral energy distributions (SEDs), as summarized in Figure 1.
1.The first group comprises unobscured AGNs with broad emission lines (Type 1 AGNs).High-z unobscured AGNs observed with JWST exhibit SED shapes consistent with a low-redshift composite spectrum of quasars (e.g., Vanden Berk et al. 2001), while their bolometric luminosities L bol ∼ 10 43−45 erg s −1 are 2 dex fainter than those of the bright quasars previously known at similar redshifts (Onoue et al. 2023).Broadband photometry and measurements of the narrowline Balmer decrement suggest minimal to modest dust attenuation for this AGN population, with an average extinction value of A V ; 0.3 mag (e.g., Maiolino et al. 2023).2.An intriguing subset consists of the dust-reddened AGNs (referred to as "little red dots"), characterized by moderate obscuration in the SED and compact morphology (Harikane et al. 2023;Kocevski et al. 2023;Labbe et al. 2023;Barro et al. 2024;Matthee et al. 2024).This class of AGNs shows a distinctive spectral shape, which can be attributed to a dust-reddened AGN continuum coupled with excess emission in the rest-frame UV produced either by the unobscured stellar continuum of the host galaxy or scattered light from the buried AGN (e.g., Zakamska et al. 2005;Polletta et al. 2006;Noboriguchi et al. 2023).Despite significant extinction levels, typically with A V ∼ a few mag, these reddened sources show broad Balmer line emission, indicating the presence of AGNs. 3. The third group includes AGNs lacking broadline emission due to substantial extinction (commonly referred to as Type 2 AGNs).These AGNs are distinguished from ordinary galaxies by the presence of strong narrowline emission features (e.g., [O III])9 or high-ionization emission lines (e.g., [Ne IV] λ2424).
Deep NIRSpec observations have confirmed the presence of broadline components in high-z AGNs preselected or identified by JWST observations.Based on measurements of the width and luminosity of the Balmer emission lines, as calibrated in the local Universe, the virial masses of the BHs powering these faint AGNs are estimated as M • ∼ 10 6−7 M e for unobscured AGNs and M • ∼ 10 7−8 M e for dust-reddened AGNs (e.g., Harikane et al. 2023;Kocevski et al. 2023;Maiolino et al. 2023;Übler et al. 2023;Matthee et al. 2024).These AGN populations, particularly those categorized into the first and second groups in Figure 1, place stringent constraints on the BHMF at early cosmic epochs and, consequently, the rate of TDEs.However, Type 2 AGNs do not provide a robust measurement of BH mass, despite their potential prevalence in the overall AGN population.In the following discussion, we focus on two classes of AGNs: unobscured AGNs (Type 1s) and obscured AGNs (dust-reddened AGNs).Note that our analysis overlooks the contribution of TDE production in Type 2 AGNs, where the TDE emission might be heavily obscured as well.Regardless of the level of obscuration, the arguments presented below give a lower bound of the TDE rate and conservative detection forecasts.

Abundances
The first abundance measurement for high-z faint AGNs was conducted by Onoue et al. (2023), using a single AGN candidate (CEERS-AGN-z5-1 or CEERS 2782), which has been confirmed as a broadline AGN by Kocevski et al. (2023).The AGN at z ∼ 5 indicates an abundance as high as Φ L ∼ 10 −5 Mpc −3 mag −1 at M UV ; − 19.5 mag within four pointings covering a survey area of 34.5 arcmin 2 .This abundance is more than 1 order of magnitude higher than what was expected from the extrapolation of the QLFs derived from the ground-based surveys, such as Subaru/HSC+SDSS (Niida et al. 2020)   (2) broadline AGNs, but with dust extinction with A V ∼ a few; and (3) AGNs without broad emission lines, but heavily obscured by dust tori with A V ?O(1).The left and right populations are classified as Type 1 and 2 AGNs in the classical unified model picture (e.g., Antonucci 1993;Urry & Padovani 1995).The middle population (or "little red dots") shows a unique spectral shape that can be explained by a dust-reddened AGN continuum (dashed line) coupled with excess emission in the rest-frame UV (indicated by the arrow).The SEDs for each population are taken from (1) the composite quasar spectrum of Vanden Berk et al. (2001); (2) the Torus model of Polletta et al. (2006); and (3) the Type 2 Seyfert template of Francis et al. (1991), respectively.The total AGN abundance at z = 4-11 adopted in this paper is calibrated with that for the dust-reddened AGN population recently reported by JWST observations, which is 1-2 dex higher than the extrapolation of the UV luminosity function of more luminous quasars based on ground-based surveys (e.g., Niida et al. 2020).galaxies at 4 < z < 6 (specifically, within JWST NIRSpec galaxy samples from each survey program) exhibit signs of broadline AGNs.This high AGN fraction implies a prevalence of unobscured AGNs, corresponding to an abundance of Φ L ∼ f AGN Φ L,å = 10 −4 -10 −3 Mpc −3 mag −1 , where Φ L,å denotes the UV galaxy luminosity function at the same redshift (e.g., Finkelstein et al. 2015;Bouwens et al. 2021;Harikane et al. 2022).However, it is worth emphasizing that the selection function of the NIRSpec targets can be quite complex and may vary across different observation tiers, particularly for slit spectroscopy modes influenced by presample selection.As a result, constructing QLFs based on the AGN fractions within limited samples could introduce a level of unknown systematics.In contrast, estimating the abundance of these faint AGNs based on the observed volume density yields a lower limit of Φ L ∼ 10 −5 Mpc −3 mag −1 , which aligns with other studies at similar redshifts and UV magnitudes.
Dust-reddened AGNs (or "little red dots") have been identified in several survey fields (e.g., Labbe et al. 2023;Barro et al. 2024).Among them, 31 sources have been confirmed as broadline AGNs (Harikane et al. 2023;Kocevski et al. 2023;Greene et al. 2024;Matthee et al. 2024).The large number of detections highlights the significant abundance of these red and compact sources at 4 < z < 6; Φ L ; 10 −5 Mpc −3 mag −1 over −18 < M UV < − 20 (Greene et al. 2024;Matthee et al. 2024).Moreover, BH mass measurements with spectroscopic observations of their broad emission lines put constraints on the BHMF at z ; 4-5.5 (Matthee et al. 2024),  F ´-5 10 M 5 • , and 10 −5 Mpc −3 dex −1 at M • ; 10 7.5 and 10 8.1 M e , respectively.Notably, these abundances are approximately ten times higher than that for the Type 1 AGNs selected by SDSS and Subaru-HSC (He et al. 2024), but seem consistent with the BHMF calculated with the continuity equation based on X-rayselected obscured samples that include the Type 2 AGNs with heavy extinction by dust and gas (Ueda et al. 2014; see also Small & Blandford 1992).This argument suggests that the obscured AGN samples with very red optical continua and a broad Hα line could represent a bulk population of massive BHs at the end of cosmic reionization (Li et al. 2023b).
The recent theoretical models of the QLF and BHMF evolution have also suggested a high AGN abundance at fainter ends (e.g., Trinca et al. 2022;Li et al. 2023aLi et al. , 2023b)).In Figure 2, we show the BHMFs for the obscured (solid) and unobscured (dashed) population at 4 z 11 constructed by Li et al. (2023aLi et al. ( , 2023b)), 10 where the BH growth model is calibrated to reproduce the redshift-dependent QLFs for unobscured populations (e.g., Akiyama et al. 2018;Matsuoka et al. 2018Matsuoka et al. , 2023;;Niida et al. 2020).The BHMF evolves to the high-mass end between the BH seeding epochs of z > 15 and the end of cosmic reionization at z ∼ 5. Toward a lower redshift of z ∼ 4, the growth of substantially heavy BHs with M • > 10 9 M e is stunted and makes the shape of the distribution consistent with the observational results.The abundance at the low-mass end of 10 5  M • /M e  10 7 , where massive BHs trigger stellar TDEs (see Section 3), monotonically increases and reaches  F -10 M 4 • and 10 −3 Mpc −3 dex −1 for the unobscured and obscured BH populations, respectively.The number difference between the two populations reflects the nature of obscuration due to dust and gas in the galactic nuclei.In this work, we adopt the obscuration fraction fitted by Ueda et al. (2014), based on X-ray AGN observations up to z ∼ 5.
It is worth emphasizing that the BHMF model we adopt in this paper considers BH populations formed in relatively biased regions of the Universe with mass variance of 3σ, where the quasar progenitor halos are likely irradiated by intense H 2 photodissociating radiation from nearby star-forming galaxies and the interior gas is heated by successive mergers (e.g., Wise et al. 2019;Li et al. 2021;Lupi et al. 2021), so the mass distribution of the seed BHs is thus extended to M • ∼ 10 5 M e (Li et al. 2023a;Toyouchi et al. 2023).However, their model neglects contributions from substantially low-mass BH populations and their mass growth in less massive halos formed in the typical regions of the early Universe, leading to the underestimation of the BH abundance at the low-mass end of M • < 10 7 M e .Despite this caveat, our results as discussed in the following sections manifest the importance of transient observations with JWST and RST to probe the low-mass BH population and thus to constrain their seeding mechanisms (e.g., Inayoshi et al. 2020;Volonteri et al. 2021).

Dusty Nuclear Star Clusters
The unprecedented infrared sensitivity and spatial resolution of JWST have revealed the remarkable fact that JWST-detected obscured AGNs produce a sufficient amount of dust grains within a compact region with the size of R e ∼ 100 pc (Labbe et al. 2023;Matthee et al. 2024).In particular, those red sources found in the UNCOVER field are well resolved down to R e ∼ 10 pc because of the high magnification in the gravitational lensing field and the median value of the half-light radius of the sample being 〈R e 〉 ∼ 50 pc (Labbe et al. 2023).From the SED fitting analysis, the visual extinction to explain the continuum slope is considered to be as high as A V ∼ 2-4 mag (Harikane et al. 2023;Labbe et al. 2023;Greene et al. 2024)-for instance, a sample of red sources spectroscopically confirmed as AGNs indicate a strong dust attenuation with an average 〈A V 〉 ; 3 mag and a range spanning 1.2-4.6 (Furtak et al. 2023;Matthee et al. 2024).Adopting a dust-to-gas mass ratio of = 0.01  , the  (2023a, 2023b).The cosmic evolution of the BHMF for high-z quasars is modeled so that the observed unobscured QLFs at 4  z  6 are well described.
hydrogen column density along the line of sight is calculated as 1 .Thus, the dust mass spherically distributed within R e is given by where m H is the hydrogen mass, μ is the mean molecular weight, and the gas density is assumed to follow ∝r − γ , with γ < 1 and thus g Dust production in galaxies is led by SN explosions of shortlived massive stars and stellar wind from long-lived asymptotic giant branch stars (e.g., Asano et al. 2013).In galaxies at z  6 with ages of 1 Gyr, the dust-to-stellar mass ratio is expected to be where the upper and lower values correspond to the cases with continuous and instantaneous star formation, respectively (Valiante et al. 2009).We note that the exact value, in general, depends on the metallicity and shape of the stellar initial mass function (IMF), but the scatter becomes smaller at t  1 Gyr.In the following, we adopt a ratio of f dust,å ∼ 1.5 × 10 −5 for the continuous star formation history and estimate the stellar mass in the nuclear region as This estimate also leads to a stellar surface density of Σ å,nuc ; 5.2 × 10 4 M e pc −2 , which is comparable to or lower than an upper-stellar-density limit for the densest star clusters or the densest elliptical galaxy progenitors (Hopkins et al. 2010; see also Baggen et al. 2023;Vanzella et al. 2023).
Assuming the stellar mass density to follow a power-law profile of r −2 , it is expressed by where the value of ζ is defined by as this combination appear in the following equations.Note that in this assumption, the stellar mass distribution is concentrated toward the center, compared to the distribution of gas and dust (see Equation (3)).If we approximate the stellar density profile as a singular isothermal sphere, r =  ˜( ) s pGr 2 2 2 , the stellar velocity dispersion of the cluster is given by  .For our fiducial case, the BH gravitational influence radius is given by We note that the BH influence radius is smaller than the half-light radius (i.e., r h < R e ) when the BH mass is as low as ( )( ) 2 .It is also worth pointing out that Equations (3) and (6) imply a correlation between the stellar mass and velocity dispersion as is consistent with that seen for nearby massive galaxies (e.g., Kormendy & Ho 2013), and the normalization of the relation agrees with the mass of nuclear star clusters in the local Universe (see the observed data summarized in Figure 1 of Stone et al. 2017a).

Timescales
Let us consider the fate of a population of ordinary low-mass stars, massive main-sequence stars, and stellar remnant BHs in the dense nuclear star cluster surrounding a massive BH in a red obscured AGN.Here, we assume a spherical shape for the cluster and estimate the two-body relaxation time for the lowmass stars as where the stellar velocity dispersion in the nucleus at r < r h is , 〈m å 〉 is the mean stellar mass, and the effective mass,  , is the ratio of the mean-squared stellar mass to the mean stellar mass.Figure 3 presents the mean mass, the effective mass, and their ratio as a function of the maximum mass of a stellar mass distribution with a power-law slope of α; . Therefore, the presence of massive perturbers in the cluster region accelerates the relaxation process compared to the case where all stars have a single mass.
Figure 4 shows the dynamical friction timescale as a function of the effective mass m of stars surrounding a massive BH with M • = 5 × 10 5 , 10 6 , 2 × 10 6 , and 4 × 10 6 M e (from the bottom to the top; black lines).The timescale is evaluated at the BH gravitational influence radius of r = r h .We also overlay the age of a main-sequence star as a function of mass for Z = 0 taken from Marigo et al. (2001;   for M •  3 × 10 5 M e , as the stellar age hardly depends on the stellar mass at m å > 50 M e .In conclusion, for a dense cluster in a galactic nucleus hosting a BH with M •  5 × 10 5 M e , massive OB stars can contribute to the occurrence of TDEs without undergoing SN explosions or gravitational collapse to BHs, unlike TDEs observed in the low-redshift Universe.The role of low-mass BHs in the high-z Universe as factories of massive stellar TDEs has been proposed in Kashiyama & Inayoshi (2016), in the context of direct-collapse BH formation during the rapid assembly of infalling gas and stars within protogalaxies in overdense regions of the Universe.

TDE Rates
An SMBH tidally disrupts a main-sequence star of mass with m å if the pericenter distance of the stellar orbit is close or located inside the tidal radius r t , defined as where r å is the stellar radius of interest.The quantity of η characterizes the ratio between the duration of the periapsis passage at the tidal radius and the hydrodynamical timescale of the star, and thus depends on the internal stellar structure.For Sun-like stars and massive stars, the equation of state at the interior is approximated with a polytropic law, and thus we set η ; 0.84.A star at a distance of r from the cluster center will be removed if the orbital angular momentum is small enough, owing to tidal disruption.The angular size θ lc of a stellar velocity vector that populates itself into the "loss cone" is given by at r  r h , and the angular size decreases outward as θ lc ∝ r −1 at r > r h , where the stellar gravity dominates over the BH gravity.
Using the simplified formulation of Syer & Ulmer (1999), we estimate the stellar TDE rate as , where the two terms are the stellar consumption rates in the emptyand full-loss-cone regimes, respectively.At the scale of interest at r ∼ r h , the total rate is dominated by the first term and the rate per star is given by with the integrated rate being expressed by where k = 0.34 is a numerical prefactor from the relaxation time and ( ) q L º L ln ln ln 2 lc lc .The critical radius r 0 is the location where the per-star consumption rate is equally partitioned between the full-and empty-loss-cone regimes, and it is approximately given by solving the equation as Since the scaling relations we have described remain valid at r  r h , we evaluate  < N at r = r h and neglect the contribution from the full-loss-cone regime at r > r 0 .Therefore, we give an  estimate of the TDE rate per galaxy as , corresponding to the case for a Salpeter mass distribution with a mass range of 0.1-10 M e (see Figure 3).Note that this ratio can be higher if the mass of the nuclear BH is below ;5 × 10 5 M e (see the discussion in Section 3.1).While the presence of massive stellar perturbers in the cluster region enhances the TDE rate, the fraction of TDEs caused by those massive stars would be reduced from the estimate in Equation (15) by a factor of t age /t AGN ∼ 0.1, where t age ∼ 10 Myr represents the typical age of massive stars and t AGN ; 100 Myr is the typical AGN lifetime (e.g., Martini 2004).
The TDE rate depends on the properties of the obscured nuclear region as 5 3 .The rate estimated for high-redshift dust-obscured AGNs is indeed consistent with the rate expected in low-redshift ultraluminous infrared galaxies (Tadhunter et al. 2017).Furthermore, the expected TDE rate for less obscured systems is reduced to 10 −5 -10 −4 yr −1 galaxy −1 for A V ; 0.1-0.4mag, which generally agrees with the rate for the field galaxy and post-starburst galaxy population (e.g., van Velzen & Farrar 2014;French et al. 2016; see also Gezari 2021).
It is also worth noting that two-body relaxation in spherical star clusters is generally considered to give a conservative floor for the true TDE rate.Other effects may contribute to the relaxation rate in galactic nuclei and potentially enhance the TDE rate; for instance, the nonspherical potential of the stellar cluster and resonant relaxation (e.g., Alexander 2012; Merritt 2013 and references therein).Of particular significance is the interaction between AGN disks and stars-involving mechanisms such as gas dynamical friction, orbital migration, and the excitation of multibody stellar scattering-which plays an essential role in triggering TDEs (Prasad et al. 2023;Ryu et al. 2023;Wang et al. 2024).Additionally, strong gravitational interactions induced by a massive binary BH can accelerate the process and enhance the TDE rate compared to the standard relaxation process around a single BH (e.g., Chen et al. 2011).

Cosmic TDE Rates
To estimate the TDE-rate density in a given survey area, we convolve the rate per galaxy with the BHMFs at various redshifts (see Figure 2) as where ΔV c,deg is the comoving volume between z − 0.5 and z + 0.5 within a 1 deg 2 solid angle and the (1 + z) factor is required to convert the rate in the galaxy rest frame to the observer frame.In Figure 5, we show the TDE-rate density (in units of yr −1 deg −2 ) as a function of BH mass for the obscured BH population (A V = 3 mag; left) and the unobscured one (A V = 0.3 mag; right).Overall, the TDE rate is higher at lower BH masses because: (1) the event rate per galaxy is higher following ; and (2) lower-mass SMBHs are more abundant (see Figure 2).At higher redshifts (z  7), the TDE rate increases toward lower mass, following As the redshift is lowered to z ∼ 4, the distribution is extended to the higher-BH-mass regime, following the cosmic evolution of the entire BH population.Furthermore, the TDE rate for the obscured BH population is approximately 2 orders of magnitude higher than that for the unobscured BH population, reflecting dust-reddened, obscured AGNs being more abundant and leading to TDEs more frequently.
The horizontal lines represent the inverse of the survey areas within a 1 yr timeline for individual programs with JWST, RST, GREX-PLUS, Euclid, and LSST.In this paper, we consider the following four deep-imaging surveys using JWST NIRCam: the JWST Advanced Deep Extragalactic Survey (JADES; Eisenstein et al. 2023), the CEERS survey (Finkelstein et al. 2023) (Laureijs et al. 2011;Euclid Collaboration et al. 2022a). 11The LSST 10 yr survey will involve more than five million exposures, collecting data to produce a deep, time-dependent, panorama of ∼20,000 deg 2 of the sky (Ivezić et al. 2019). 12n the case of obscured AGN populations, the TDE rate is sufficiently high, and thus a substantial number of events are expected within the surveyed area.However, it is crucial to note that when considering the detection limit, observations of TDEs within obscured AGN populations should be restricted to a relatively small field of view achievable through deep JWST observation programs, where the limiting magnitude should be at or deeper than 28 mag, to ensure reliable detection.Conversely, for unobscured AGN populations, the occurrence rate of TDEs is low.Therefore, wide-field (2 deg 2 ) surveys with imaging with RST and GREX-PLUS deeper than 25-26 mag are required to effectively identify TDEs in these populations (see the further details in Section 4).

Observational Signatures
The accretion luminosity of TDEs occurring in high-z AGNs can be sufficiently high and thus the associated emission would be detectable even from z  4 to 7. The fallback accretion rate is estimated as where the fallback time t fb is calculated as (Stone et al. 2013) and its analytical expression 1 2 fits well the relation between the light-curve decay timescale and the BH mass for optical-/UV-selected TDEs (e.g., van Velzen et al. 2020).
For a wide range of the parameters, the peak accretion rate exceeds the Eddington value significantly and reaches . Despite a large gas supply rate to the nuclear scale, powerful outflows are launched from the vicinity of the BH horizon and carry a large fraction of the inflowing mass and momentum outward (e.g., Ohsuga et al. 2009;Jiang et al. 2014;McKinney et al. 2015;Saḑowski et al. 2015;Hu et al. 2022).As a result, the inflow rate decreases toward the center as ( )  µ M r r q in (q > 0), and thus the BH feeding rate is reduced to a mildly super-Eddington accretion rate of - ~M 2 10 Edd (Hu et al. 2022, q ∼ 1).The reduction factor is not well understood, either in theory or through observations.However, it could potentially be determined by examining the ratio between the inflow radius (e.g., the radius where the rapid circularization of stellar debris occurs following a disruption) and the radius of the innermost stable circular orbit around the central BH.Furthermore, the source of TDE emission in the optical-to-UV bands has been debated to be either the reprocessing of the X-ray emission from the accretion disk (e.g., Guillochon & Ramirez-Ruiz 2013;Roth et al. 2016) or emission from outer shocks between the debris streams when they collide (e.g., Piran et al. 2015), or perhaps some combination of both (see more references in van Velzen et al. 2020).
In this paper, we adopt the fitting result of TDE light curves in the optical-to-UV bands summarized in van Velzen et al. (2020), instead of modeling the TDE light curves based on the theoretical frameworks.This approach allows us to avoid numerous uncertainties in the modeling of the BH feeding, mass loading into outflows, circularization of stellar debris, and optical-UV emission mechanisms.Among the 33 TDEs listed in Table 2 of van Velzen et al. (2020), we consider the light-curve fit for PS1-10jh, which was the first TDE detected with a wellsampled rise to peak in optical survey data (Gezari et al. 2012).The spectral shape is fitted by a diluted blackbody spectrum with an effective temperature of T 0 ; 10 4.59 K, and the bolometric luminosity is estimated as L peak ; 10 44.47 erg s −1 at the peak, which subsequently decays following a ∝t −5/3 power law.van Velzen et al. (2019) estimate the mass of the SMBH that causes the TDE PS1-10jh using the M • -σ relation as M • ; 10 6.06 M e .They also find that the light curves of TDEs from such low-mass BHs (M • < 10 6.5 M e ) show significant late-time flattening at several years after the peaks.The observed late-time emission is Figure 5.The predicted rate density (in units of yr −1 deg −2 ) of stellar TDEs within obscured (solid) and unobscured (dashed) AGNs within the redshift range of 4 z 11.The horizontal lines represent the inverse of the survey areas within a 1 yr period planned for the individual JWST missions (COSMOS-Web, CEERS, JADES, and NGDEEP), the High Latitude Time Domain Survey for SN cosmology with RST, GREX-PLUS, Euclid, and LSST.The TDE rate in the obscured AGNs (A V ; 3) is approximately 2 orders of magnitude higher than that for unobscured BH AGNs (A V ; 0.3), where we fix R e = 50 pc and ¯á ñ = m m 4   (for a Salpeter stellar mass function with 0.1 m å /M e 10).These findings highlight the capability of deep JWST observations in detecting TDEs in obscured AGNs with lowmass BHs, while the wider coverage of the RST, Euclid, GREX-PLUS, and LSST surveys is pivotal for identifying TDEs in unobscured AGNs.
Motivated by those observational facts, we model the TDE light curve with a power-law decay and a plateau component as .We take into account dust attenuation by the extinction law of starburst galaxies (Calzetti et al. 2000) and apply it both to the AGN and TDE spectra in the form of , where τ ν is calculated from the extinction law.For unobscured systems, we fix the level of dust extinction to A V ; 0.3 mag, the average value of extinction for JWSTdetected broadline AGNs with M • < 10 7 M e presented in Table 3 of Maiolino et al. (2023).On the other hand, we consider a higher level of extinction with A V = 3.0 mag for obscured systems, motivated by the reddened rest-optical continuum spectra seen in dust-reddened, obscured broadline AGNs (Kocevski et al. 2023;Labbe et al. 2023;Matthee et al. 2024).The observed spectra of those red sources also show the blue excess at shorter wavelengths of λ rest < 0.3-0.4μm, requiring additional blue components.Although the origin of the blue excess remains uncertain, in analogy with the SEDs for the blue-excess dust-obscured galaxies, a small fraction f scatt of the radiation flux of the intrinsic AGN spectrum would be scattered to our line of sight and develop the blue component (Zakamska et al. 2005;Polletta et al. 2006;Alexandroff et al. 2018;Noboriguchi et al. 2022Noboriguchi et al. , 2023)).In this paper, we adopt f scatt = 0.03 to reproduce the spectral shapes of the dustreddened AGNs reported in the JWST UNCOVER field (e.g., Greene et al. 2024), leading to a luminosity density of ( ) In our analysis, we do not consider the contribution of the underlying host galaxy to the observed SED for the following reason.The rest-frame UV luminosity (L UV,å ) to host stellar mass (M å ) relationship can be approximated as Therefore, as long as the M • /M å ratio is consistent with the local values (∼0.5%;Kormendy & Ho 2013) or above, as suggested by JWST-identified AGNs (see also Pacucci et al. 2023), the UV luminosity from the AGN dominates over the host galaxy contribution, assuming similar levels of extinction for both.

Deep-field Surveys for Obscured TDEs
We first consider stellar TDEs occurring within dustreddened, obscured AGNs.In this scenario, we assume dust extinction of A V = 3 mag and a scattering fraction of f scatt = 0.03 for both the TDE and AGN components, as described in Equation ( 19).Due to dust extinction, the restframe UV light on the total SED is significantly attenuated.Consequently, deep surveys with JWST are required to meet the detection threshold.However, as a trade-off for achieving such depths in observations, the survey area becomes severely limited (<0.6 deg 2 ), given the limited amount of observing times.Therefore, we need to consider a more abundant population of BHs providing TDEs in an efficient way; specifically those with low masses, with M •  10 6 M e (see the solid curves in Figure 5).
In Figure 6, we present the SEDs of stellar tidal disruption in an obscured AGN with M • = 10 6 M e at z = 4 (left) and z = 5 (right), respectively.Each curve indicates the time series of the SED at each epoch since the TDE peak in the observer frame; t − t peak = 0 days, 30 days, 60 days, 0.5 yr, 1.0 yr, and ∞ (TDE emission is neglected), from the top to the bottom.We overlay the 5σ point-source imaging depths in each NIRCam filter used for the JWST survey programs; COSMOS-Web, CEERS, and JADES-Medium/Deep (see also the summary of the depths for each filter in Table 1).In Figure 7, we show the colormagnitude diagram for this obscured TDE+AGN at z = 4 (left) and z = 5 (right).Each curve presents the F277W-F444W (blue) and F115W-F277W (purple) colors, respectively, and the circle symbols indicate the elapsed time since the peak brightness of the TDEs.
At the lower redshift (z = 4), the SED enables the robust detection of the red-continuum component using the longerwavelength filters (F277W, F356W, and F444W) in the three surveys: COSMOS-Web, CEERS, and JADES.As the TDE light curve decays over time, the F277W flux density declines by ∼1 mag in 1 yr, resulting in a reddening of the F277W-F444W color.Furthermore, the scattered light from the TDE introduces a blue-excess component (i.e., rest-frame UV bands), which is observable within 1-2 months of the luminosity peak in the CEERS and JADES surveys.The dominance of the TDE-originating light on the blue side of the spectrum allows for the direct tracing of the t −5/3 decay rate using the F115W-F277W color in the JADES-Deep survey.
At the higher redshift (z = 5), the overall trend of the SED shape still holds.However, due to the far distance, the F277W flux density becomes too low for detection in the CEERS and JADES-Medium survey sensitivities after ∼0.5 yr from the TDE peak.The blue excess in the spectrum can only be detectable with the JADES-Deep survey until the same elapsed time, when the F115W-F277W color reaches ;0.6 mag.In the subsequent epochs, the TDE is identified only with longerwavelength filters and the F277W-F444W color becomes redder with time.
Toward higher redshifts (z > 5), only deep surveys such as JADES-Deep (Eisenstein et al. 2023) and NGDEEP (Bagley et al. 2024) achieve limiting magnitudes of ∼31 mag, but cover smaller areas of <0.01 deg 2 .In this scenario, we consider a further abundant population of BHs with M • = 10 5 M e .Even within such limited fields of view, we expect to detect ∼O(1) TDEs triggered by the low-mass BH. Figure 8 illustrates the SEDs of the stellar tidal disruption occurring in a dustreddened, obscured AGN with M • = 10 5 M e at various redshifts from z = 4 to 11.Each panel represents the SED shapes at two different time points: immediately after the TDE peak time (left) and 1 yr after the peak (right).Figure 9 shows the color-magnitude diagram for the TDE with longer-wavelength NIRCam filters.The JADES-Medium survey is capable of identifying the characteristic colors of the TDE in the early stages at z  7, while the detection horizon extends as far as z ∼ 11 in the JADES-Deep and NGDEEP surveys.Even within 1 yr of the TDE peak, the deepest two surveys capture the redcontinuum component at wavelengths around ∼3-5 μm.

Wide-field Surveys for Unobscured TDEs
Next, we consider TDEs occurring within unobscured AGNs.In this scenario, we adopt dust extinction of A V = 0.3 mag for both the TDE and AGN components.While this low level of extinction exerts negligible influence on the total SED, the intrinsic TDE rate is approximately 2 orders of magnitude lower than that for the obscured cases (see Section 4.1).As a result, the detection of unobscured TDEs requires wide-field surveys with RST and the COSMOS-Web survey program with JWST.Specifically, to detect more than one TDE at z  4, the planned surveys need to target AGN populations with BH masses with M •  10 5 M e , 10 6 M e , and 3 × 10 7 M e for survey areas larger than ∼2.5 deg 2 , 50 deg 2 , and 1700 deg 2 , respectively (see Figure 5).
In Figure 10, we present the SEDs of the stellar tidal disruption in an unobscured AGN with M • = 10 6 M e at various redshifts from z = 4 to z = 7, respectively.Each curve indicates the time series of the SED at each epoch since the TDE peak; t − t peak = 0 days, 30 days, 60 days, 0.5 yr, 1.0 yr, and ∞ (TDE emission is neglected), from the top to the bottom.To assess the detectability of these events, we overlay the 5σ point-source imaging depths in each filter of the RST and JWST (COSMOS-Web) observations.For the RST observations, we consider single-epoch exposure times of 1 hr (=3.6 ks), 1.0 ks, and 0.5 ks. 13 In the cases of unobscured TDEs, the observed flux at <2 μm is as bright as 26-27 mag at the peak time, depending on the redshifts and decays within 1 yr, down to 27-28, which remains detectable with RST in an exposure time of ∼1 ks.Similar observations can be conducted using JWST in the COSMOS-Web survey, which offers the additional advantage of covering the TDE spectrum with two longer-wavelength filters, F277W and F444W, and monitoring the characteristic decay of the TDE emission.19)).Each curve indicates the time series of the SEDs since the peak time in the observer frame: t − t peak = 0 days, 30 days, 60 days, 0.5 yr, 1.0 yr, and ∞ (only the AGN spectrum), from the top to the bottom.We overlay the 5σ point-source imaging depths in each NIRCam filter of the JWST survey programs; COSMOS-Web (gray), CEERS (purple), and JADES-Medium/Deep (blue solid and dashed).Note that we only show the depths of four short-wavelength filters in JADES-Deep for illustrative purposes.Figure 11 shows a color-magnitude diagram for TDEs in an unobscured AGN with M • = 10 6 M e over z = 4-7.Each curve presents the F087-F129 colors as a function of the F129 flux density, assuming the RST filter transmissions.The circle symbols indicate the elapsed time since the peak brightness of the TDEs.This particular filter combination, F087 and F129, is chosen in alignment with the initial survey design for the High Latitude Time Domain Survey for SN cosmology with RST (e.g., Rose et al. 2021).This survey employs a two-tier strategy: wide and deep.The wide tier aims to cover the RZYJ bands (F062, F087, F106, and F129) across a survey area of ;20 deg 2 , while the deep tier focuses on the YJHF bands (F106, F129, F158, and F184), with a survey area of ;4 deg 2 .The entire survey spans six months of observing time spread over 2 yr, with 30 hr visits occurring every 5 days.The expected 5σ point-source imaging depths are 26.3 mag in the F087 filter and 26.1 mag in the F129 filter for each singleepoch exposure with 100 s, as summarized in Table 2. Therefore, to capture the early-stage TDE emission (∼90 days in the observed frame, equivalent to 18 visits), it is necessary to cover two to three visits for TDEs at z = 4-5 and at least five to seven visits for TDEs at z = 6-7.This allows for the construction of a robust light curve with three to nine data points during each stacking period.
In practice, there still remains potential contamination from nearby and lower-redshift astrophysical objects.The High Latitude Time Domain Survey is primarily targeting Type Ia SNe at redshifts of z > 0.5, as key components of transient survey programs aimed at measuring the expansion history of the Universe.The reference survey described in Rose et al. (2021) is designed so that a substantial number, the order of 10 4 , of Type Ia SNe with a high signal-to-noise ratio (S/N >10) will be detected in the survey.In Figure 11, we also overlay the colors and magnitudes of Type Ia SNe at lower redshifts, specifically in the range of z = 0.7-1.2(gray curves).We adopt the Type Ia SNe spectral templates that span the time between the peak luminosity and 85 days after the peak in the rest frame, based on the work of Hsiao et al. (2007).The template luminosity is set to −19.3 mag at the peak in the R band, which is the average of standard Type Ia SNe (Yasuda & Fukugita 2010).The black arrows indicate the direction of time evolution for the colors and magnitudes, with the length of the arrows corresponding to a duration of 10 days in the observed frame.As clearly demonstrated, the colors and magnitudes of Type Ia SNe at z ; 0.7-1.0 in the later stages exhibit similar values to those of TDEs at z ; 4-6 in the earlier stages.This similarity makes it challenging to distinguish high-z TDEs from low-z Type Ia SNe with only one or two exposures.However, it is worth noting that the colors and magnitudes of these objects evolve by ∼0.1-0.2 mag within a duration of 10 days (equivalent to three visits) in the observed frame, as indicated by the black arrows.Importantly, the direction of the colormagnitude evolution in this diagram is nearly perpendicular to that of high-z TDEs.This distinct evolution pattern plays an essential role in identifying high-z TDEs among lower-z Type Ia SNe.In addition to the color and magnitude criteria, Lyman dropout selection with the F062 and F087 bands can be used to distinguish z > 5.3 transients from those low-z Type Ia SNe.
Other types of energetic explosions occurring in the highredshift Universe are also intriguing targets for the RST transient survey.Moriya et al. (2022) have explored the detectability of superluminous SNe (SLSNe) or pair-instability SNe (PISNe) with luminosities comparable to the brightest TDEs.Their analysis indicates that a significant number of SLSNe and PISNe at 5 < z < 6 can be detectable and identified using the F158 and F213 filters,14 covering longer wavelengths than those considered for the TDE searches in this paper.Our filter combination with F087 and F129 can also be employed to detect these energetic SNe at lower redshifts (z ∼ 2-3).However, it is crucial to note that these explosions should be substantially brighter (24.5 mag) than TDEs.

Other Surveys for TDEs at z  4
We briefly discuss the capabilities of other wide-field surveys, such as LSST and Euclid, for the exploration of high-z TDEs.The expected 5σ point-source depths for the LSST and Euclid bands are summarized in Table 2.
LSST is designed for high-cadence optical transient exploration, which is particularly suitable for detecting TDEs (or their candidates) at high redshifts.In Figure 12, we present the SEDs of a TDE occurring in an unobscured AGN with M • = 10 6 M e at z ∼ 4. The blue parts of the spectra at λ obs  0.7 μm can be detectable through image stacking of 10-20 visits in the g and r bands up to ∼60 days (in the observer frame) after the TDE emission peak.However, it could be challenging to distinguish and separate TDEs from other astrophysical sources using the single g − r color.To address this limitation, a synergistic joint observation with LSST and RST, utilizing multiple-band photometry, is expected to enhance the TDE search efficiency.For instance, follow-up RST wide-survey observations of LSST-detected TDE candidates can improve high-z TDE selection (e.g., the F087-F129 color in RST; see Figure 11).Toward lower redshifts, a few visits at low cadence are adequate for TDE identification.Detailed quantitative arguments about the photometric selection criteria and observation strategies for detecting z ∼ 2-3 TDEs are left for future work.
The Euclid deep survey will cover ∼53 deg 2 , three times wider than the area of the RST wide survey.The expected 5σ point-source depths are I E = 28.2 and Y E J E H E ; 26.4 mag.Those coadded depths will be achieved through ∼40-50 visits over 6 yr of operations (Euclid Collaboration et al. 2022a, 2022b), i.e., I E = 25.3 and Y E J E H E ; 24 mag for each individual visit (e.g., Laureijs et al. 2011).For the given survey parameters, detecting TDEs at z ∼ 4 (Y E  25.5; see Figure 10) would require ∼20 visits within 30 days of the TDE emission peak.However, designing such a high-cadence observation program may pose challenges.

Detection Numbers of High-z TDEs
Finally, we give an estimate of the expected detection numbers of TDEs from obscured and unobscured AGNs identified in the survey as where the observation time is set to ΔT = 1 yr, A survey is the survey area, the survey efficiency is defined as , and Δt obs is the time duration (in the observer frame) when the TDEs are sufficiently bright for the given detection thresholds.We summarize the detection numbers of TDEs for each survey program in Table 3.
The deep JWST imaging surveys, excluding COSMOS-Web, exhibit sufficient sensitivity to detect TDEs occurring within obscured AGNs at z ∼ 4-7.These surveys cover relatively limited areas where abundant populations of BHs with masses M •  10 6 M e are prevalent, as shown in Figure 5.For such obscured sources, we consider two TDE rates, TDE  , integrated for BH masses M • 10 4 M e and 10 5 M e .Since the TDE emission dominates over the steady AGN emission when M •  10 5 M e , we assess the detectability based on the F277W flux density in comparison to the survey's limiting magnitude.Additionally, we estimate the observable time window, Δt obs , from the color-magnitude diagram shown in Figure 9.
The COSMOS-Web survey, due to its relatively shallow depth, can only reach the TDE flux densities when the underlying AGN emission significantly contributes to the total SED, requiring BH masses of M •  10 6 M e .However, the observable time window is limited to Δt obs  10 days.As a  result, the survey efficiency is constrained to f obs  0.03, as demonstrated in Figures 6 and 7.
The High Latitude Time Domain Survey with RST proves to be sensitive to TDEs occurring within unobscured AGNs at z ; 4-7.Both the wide and deep survey tiers cover sufficiently large areas, enabling detections of even rarer unobscured TDEs in AGNs with masses M •  10 6 M e .For these unobscured sources, we consider two TDE rates integrated for BH masses M • 10 4 M e and 10 5 M e .To assess the detectability, we analyze the F129 flux density in relation to the survey's limiting magnitude and calculate the observable time window, Δt obs , from the color-magnitude diagram shown in Figure 11.Since the depth of the RST surveys can reach the expected TDE flux density through multiple-epoch observations, we can maximize the survey efficiency, with f obs ; 1, for most redshift ranges of z = 4-7.Given these survey setups, we predict the detection of N TDE ∼ 8 (40) TDEs originating from AGNs with M •  10 5 (10 4 )M e in 1 yr of operation.The number of detections decreases with increasing redshift, but remains N TDE ∼ 1 (30) even at z ∼ 7 for the wide-tier survey for the same BH mass range.
It is also worth noting that the COSMOS-Web survey shows the capability of detecting TDEs in unobscured AGNs at z ∼ 4-7, regardless of the BH mass at the nuclei, as described in Figure 10.However, even with the largest survey area planned for JWST surveys, covering ;0.6 deg 2 , the detection is primarily limited to TDEs occurring in AGNs with the lowestmass BHs of M • ; 10 4 M e .To expand the sample size of detectable TDEs in the COSMOS-Web survey, it might be beneficial to consider a transient survey plan that incorporates multiple visits, as in the RST High Latitude Time Domain Survey.Such an observational strategy has the potential to push the boundary for BH detection down to another order of magnitude, allowing us to probe the underlying shape of the mass function of seed BHs in a more comprehensive way.
The detection number of TDEs at z  4 through a joint observation campaign with LSST and RST will depend on the design of the synergy observation programs.As a showcase, we provide a reference number for the detections, considering a scenario where all TDE candidates selected by LSST (t obs = 60 days is adopted) are confirmed as true high-z TDEs through follow-up observations with RST.The upper bounds  19)).Each curve indicates the time series of the SEDs since the peak time: t − t peak = 0 days, 30 days, 60 days, 0.5 yr, 1.0 yr, and ∞ (only the AGN spectrum), from the top to the bottom.We overlay the 5σ point-source imaging depths in each filter of the RST and JWST (COSMOS-Web) observations.For the RST observations, we consider 1 hr, 10 3 s, and 500 s of exposure time.
of the detection numbers in this optimistic scenario are N TDE ∼ 51, 970, and 5100 for z ∼ 4 TDEs originating from AGNs with BH masses greater than M •  10 6 , 10 5 , and 10 4 M e , respectively.A quantitative discussion on the synergy design is beyond the scope of this work and left for future investigation.

JWST Spectroscopy of High-z TDEs
JWST spectroscopic follow-up observations of high-z TDEs detected through JWST NIRCam surveys, RST deep-and wide-tier surveys, or RST-LSST joint campaigns promise to provide rich insights into these phenomena.Primarily, the PRISM mode of JWST facilitates the detection of emission lines such as Lyα (for z > 4) and Hα (for z < 6.5), determining the redshift of the source.Furthermore, the confirmation of broadline emission from those TDEs is a definitive signature of massive BHs in their nuclei.Medium-or high-resolution spectroscopy allows for the estimation of the nuclear BH masses via single-epoch methods validated in the low-redshift Universe.
The spectral decomposition of narrow and broad components of Balmer lines (e.g., Hβ and Hα) for unobscured TDEs is feasible down to M UV ∼ − 18 mag for the underlying AGN brightness, particularly when the BH mass exceeds M •  0.3-1.0 × 10 6 M e (see Maiolino et al. 2023 on the JADES survey).Such follow-up observations, targeting unobscured TDEs preselected by RST and LSST, can directly constrain the TDE-rate mass function (see Figure 5).
For obscured TDEs, deep NIRSpec observations can reach down to M • ∼ 10 7 M e or lower (e.g., the dust-reddened AGN, CEERS 746, discussed in Kocevski et al. 2023).Even without direct BH mass measurements, the TDE rate, primarily dominated by lower-mass BHs, provides constraints on the BHMF beyond z  4 for 10 4  M • /M e  10 6 , analogous to the discussions on local TDEs (z ∼ 0) in Stone & Metzger (2016).
In cases where broadline emission is undetected, possibly due to smaller BH masses, the existence of AGNs can still be inferred through several methods (e.g., see Maiolino et al. 2023).These include: (1) identifying emission lines with high ionization potential, such as He II λ1640 and λ4686 (E ion = 54.4 eV) and Ne IV λ2424 (E ion = 63.5 eV); (2) observing semi-forbidden nebular lines that indicate highdensity gas consistent with a broadline region of an AGN (e.g., [N IV] λ1483/N IV] λ1486 and N III] λ1754/N III] (total); and (3) noting broader permitted/semi-forbidden lines in comparison to forbidden lines, as observed in local narrowline Seyfert 1 galaxies.

Summary and Discussion
The unprecedented sensitivity of JWST has enabled the discovery of low-luminosity AGNs at z ; 4-11, which were hidden in the pre-JWST era.Spectroscopic follow-up observations have provided estimates of the nuclear BH masses for these sources and pushed the low-mass boundary down to M • ∼ 10 6-7 M e .Despite this progress, the observed lowest mass of BHs remains 1-2 orders of magnitude heavier than the typical mass of seed BHs with 10 6 M e , leaving the mass distribution of early BHs poorly constrained.
In this paper, we focus on UV-to-optical (in the rest frame) flares of stellar TDEs embedded in low-luminosity AGNs as a tool to explore low-mass BH populations with 10 5−6 M e .  1 , suggesting that lower-mass BHs preferentially trigger TDEs.The parameter ζ is characterized by the level of dust extinction and the size of the nuclear stellar cluster embedded in lowluminosity AGNs (see the discussion in Section 2.3).For dustreddened AGNs showing compact morphology reported by JWST observations, we find ζ ∼ O(1), while the value decreases for unobscured AGNs.The cosmic TDE rate is calculated by convolving  N TDE with the BHMF model over the redshift range 4 z 11, which aligns well with the QLF shapes observed in populations at 4 z 6 (Li et al. 2023a(Li et al. , 2023b)).The TDE rate is higher at lower BH masses due to the higher event rates per galaxy and greater abundance of lower-mass BHs (see Figures 2 and 5).
For obscured AGN populations that require deep JWST observations, a substantial number of TDE events are expected within the surveyed areas of the planned missions.Conversely, for unobscured AGN populations, the occurrence rate of TDEs is low.Therefore, large survey coverage through RST and GREX-PLUS is needed to identify TDEs in these populations.for the deep (and wide) tiers of the High Latitude Time Domain Survey with RST, in 1 yr of observations (see Table 3).Follow-up observations with RST for TDE candidates preselected in the larger LSST sky coverage of ∼20,000 deg 2 are expected to significantly increase the detection number of high-z TDEs.
We conduct SED modeling for high-redshift TDEs embedded in the underlying host AGN continuum and line emission.Our analysis reveals that the UV-to-optical TDE emission dominates at λ rest  5000 Å within 1 yr of the emission peak in the observed frame.Based on the SED modeling, we discuss the selection strategies for detecting transient high-redshift TDEs with the ongoing JWST and forthcoming RST observations.Specifically, we propose colormagnitude selection for high-redshift TDEs in RST transient surveys, utilizing the F087-F129 color and the F129 flux density, to distinguish them from low-redshift Type Ia SNe with only one or two exposures.A long-term (∼1 yr) observation campaign with RST will enable us to monitor the TDE light curve characterized by a power-law form of ∝t −5/3 .This paper focuses on the UV-to-optical emissions originating from high-redshift TDEs.A considerable fraction of TDEs may accompany relativistic jets, which can produce prompt gamma-ray and X-ray emissions along with radio/submillimeter afterglow emissions.Specifically, in cases involving the disruption of massive OB stars by relatively low-mass BHs, these emissions from jetted TDEs could serve as a viable target for multiwavelength transient surveys aimed at uncovering high-redshift BH populations (e.g., Kashiyama & Inayoshi 2016).
Furthermore, dense environments, such as nuclear star clusters and AGN disks, can also serve as potential sources for gravitational waves (GWs) induced by mergers of binary BHs (e.g., O'Leary et al. 2016;Stone et al. 2017b;Bartos et al. 2017;McKernan et al. 2018;Tagawa et al. 2020).Constraints on the low-mass end of the BHMF through high-z TDE detections are crucial for accurately estimating the detection rates of GWs using space-based interferometers, such as LISA, Tianqin, and Taiji (Sesana et al. 2008;Luo et al. 2016;Mei et al. 2021;Amaro-Seoane et al. 2023).This not only allows for independent validations of models, but also provides insights from a distinct perspective of multimessenger observations.

Figure 1 .
Figure 1.Flow chart of the AGN classification in this paper: (1) unobscured AGNs with broad components of Balmer emission lines;(2) broadline AGNs, but with dust extinction with A V ∼ a few; and (3) AGNs without broad emission lines, but heavily obscured by dust tori with A V ?O(1).The left and right populations are classified as Type 1 and 2 AGNs in the classical unified model picture (e.g.,Antonucci 1993;Urry & Padovani 1995).The middle population (or "little red dots") shows a unique spectral shape that can be explained by a dust-reddened AGN continuum (dashed line) coupled with excess emission in the rest-frame UV (indicated by the arrow).The SEDs for each population are taken from (1) the composite quasar spectrum of VandenBerk et al. (2001); (2) the Torus model ofPolletta et al. (2006); and (3) the Type 2 Seyfert template ofFrancis et al. (1991), respectively.The total AGN abundance at z = 4-11 adopted in this paper is calibrated with that for the dust-reddened AGN population recently reported by JWST observations, which is 1-2 dex higher than the extrapolation of the UV luminosity function of more luminous quasars based on ground-based surveys (e.g.,Niida et al. 2020).

Figure 2 .
Figure2.BHMFs for the obscured (solid) and unobscured (dashed) population at 4 z 11, constructed by a semi-analytical model inLi et al. (2023aLi et al. ( , 2023b)).The cosmic evolution of the BHMF for high-z quasars is modeled so that the observed unobscured QLFs at 4  z  6 are well described.

.
While the mean mass weakly depends on m max for α 2, the effective mass increases with m max following a α < 3.For a Salpeter mass distribution with α = 2.35 and


and t rel = 150 Myr.Therefore, in the cluster, B-type stars with m å  3.5 M e can sink to the center and lead to TDEs within the main-sequence stage.The upper mass rapidly increases as the BH mass decreases; for instance, for M • = 5 × 10 5 M e and 

Figure 3 .
Figure3.The mean mass 〈m å 〉, the effective mass m , and their ratio as a function of the maximum mass of a stellar mass distribution with a power-law slope of α.

Figure 4 .
Figure 4. Dynamical friction timescales as a function of the effective mass m of stars surrounding a massive BH with M • = 5 × 10 5 , 10 6 , 2 × 10 6 , and 4 × 10 6 M e (from the bottom to the top; black lines).The dynamical friction timescale is evaluated at the BH influence radius of r h for ζ = 1 and scales with ζ −3/2 (see Equation (9)).The stellar age of a star with m å (magenta curve) and the Hubble time at z = 5 are shown, respectively.
= c/λ 0 = 2 × 10 15 Hz, t 0 is the characteristic decay timescale, p(= − 5/3) is the power-law index of the light-curve decay, B ν (T 0 ) is the Planck function, and σ SB is the Stefan-Boltzmann constant.The UV luminosity of the AGN, as a steady source, is estimated as l correction factor for the monochromatic UV band.The AGN spectral shape f ν is taken from the composite spectrum of low-redshift quasars (see VandenBerk et al. 2001) and its normalization is set so that =

Figure 6 .
Figure6.SEDs of stellar tidal disruption occurring in a dust-reddened, obscured AGN with M • = 10 6 M e at z = 4 (left) and z = 5 (right), respectively.Dust extinction with A V = 3 mag and a scattering fraction of f scatt = 3% are assumed both for the TDE and AGN components (see Equation (19)).Each curve indicates the time series of the SEDs since the peak time in the observer frame: t − t peak = 0 days, 30 days, 60 days, 0.5 yr, 1.0 yr, and ∞ (only the AGN spectrum), from the top to the bottom.We overlay the 5σ point-source imaging depths in each NIRCam filter of the JWST survey programs; COSMOS-Web (gray), CEERS (purple), and JADES-Medium/Deep (blue solid and dashed).Note that we only show the depths of four short-wavelength filters in JADES-Deep for illustrative purposes.

Figure 7 .
Figure 7.The color-magnitude diagram for TDEs in a red obscured AGN with M • = 10 6 M e at z = 4 (left) and z = 5 (right), respectively.Each curve presents the F277W-F444W (blue) and F115W-F277W (purple) colors, respectively, and the circle symbols indicate the elapsed time since the TDE brightness peak.

Figure 8 .
Figure 8. SEDs of the stellar tidal disruption occurring in a dust-reddened, obscured AGN with M • = 10 5 M e over z = 4-11, at the elapsed time of t − t peak = 0 days (left) and 1 yr (right) since the TDE peak time, respectively.We overlay the 5σ point-source imaging depths in each NIRCam filter of the JWST survey programs; COSMOS-Web (gray), CEERS (purple), JADES-Medium/Deep (blue solid and dashed), and NGDEEP (black).

Figure 9 .
Figure 9.The color-magnitude diagram for TDEs in a red obscured AGN with M • = 10 5 M e over z = 4-11.Each curve presents the F227W-F444W (blue) and F115W-F277W (purple) colors, respectively, and the circle symbols indicate the elapsed time since the TDE brightness peak.

Figure 10 .
Figure 10.SEDs of the stellar tidal disruption occurring in an unobscured AGN with M • = 10 6 M e at z = 4 (top left), z = 5 (top right), z = 6 (bottom left), and z = 7 (bottom right), respectively.Dust extinction with A V = 0.3 mag is assumed both for the TDE and AGN components (see Equation (19)).Each curve indicates the time series of the SEDs since the peak time: t − t peak = 0 days, 30 days, 60 days, 0.5 yr, 1.0 yr, and ∞ (only the AGN spectrum), from the top to the bottom.We overlay the 5σ point-source imaging depths in each filter of the RST and JWST (COSMOS-Web) observations.For the RST observations, we consider 1 hr, 10 3 s, and 500 s of exposure time.

Figure 11 .
Figure11.The color-magnitude diagram for TDEs in an unobscured with M • = 10 6 M e over z = 4-7.Each curve presents the F087-F129 colors as a function of the F129 flux density, assuming the RST filter transmissions, and the circle symbols indicate the elapsed time since the TDE brightness peak.The gray curves are the colors and magnitudes of Type Ia SNe at z = 0.7-1.2, while the black arrows indicate the direction of the time evolution, with the length corresponding to a duration of 10 days in the observed frame.The limiting magnitude of the F129 filter is used for multiple exposures (N = 1, 2, 5, 10, and 20), where a single exposure time is assumed to be 100 s.
The predicted detection numbers for TDEs caused by massive BHs with M •  10 4-6 M e at z = 4 and the CFHT Legacy Survey (McGreer et al. 2018).Subsequently, Harikane et al. (2023) and Maiolino et al. (2023) reported six and 12 newly detected unobscured AGNs at z ; 4-7 from the Cosmic Evolution Early Release Science (CEERS; eight NIRSpec pointings covering 72 arcmin 2 ) and JADES survey fields (175 arcmin 2 ), respectively.They found that f AGN ∼ 5%-10% of star-forming note that the stellar age becomes longer as the stellar metallicity increases)., one can calculate the upper mass of the stars that can feed the nuclear BH within its lifetime.In a stellar cluster hosting a BH with M • ; 10 6 M e , the upper mass is estimated as  (Casey et al. 2023)rvey(Casey et al. 2023), and the Next Generation Deep Extragalactic Exploratory Public (NGDEEP) survey (Bagley et al. 2024).For the RST survey, we adopt the initial survey design for the High Latitude Time Domain Survey for SN cosmology (e.g., Rose et al. 2021), which employs the wide and deep tiers with survey areas of 19.04 and 4.20 deg 2 , respectively.For the GREX-PLUS survey, a deep-imaging survey with an area of 10 deg 2 is considered (Inoue et al. 2023).Additionally, Euclid, which launched successfully in 2023 July, is planned to conduct two major surveys: a wide survey of ∼15,000 deg 2 and a deep survey of ∼50 deg 2 , covering substantially larger sky areas compared to the survey programs mentioned above

Table 2
Rose et al. 2021022)1), anduture Wide-field Surveys Average 5σ point-source depths for each filter in the RST wide tier of the High Latitude Time Domain Survey for SN cosmology(Rose et al. 2021), Euclid(Laureijs et al. 2011), and LSST (Bianco et al. 2022).Synergy between the LSST and Roman wide survey for detecting a TDE occurring in an unobscured AGN (A V = 0.3 mag) with M • = 10 6 M e at z = 4.We overlay the 5σ point-source imaging depths in each filter of LSST(Bianco et al. 2022)and RST (the wide survey;Rose et al. 2021) for multiple exposures.Using the AGN host galaxy properties inferred from JWST observations, we estimate the rate of TDEs led by two-body relaxation in a dense stellar cluster.The TDE rate per galaxy is

Table 3
The Detection Numbers of TDEs at z = 4-7 Note.Column (1): telescope.Column (2): survey name.Column (3): survey area in square degrees.Columns (4)-(11): expected detection numbers of TDEs from obscured and unobscured AGNs at z ∼ 4, 5, 6, and 7 identified in the survey, assuming Δz = 1.The values indicate the detection numbers for TDEs occurring in AGNs with masses greater than M • = 10 5 M e (and 10 4 M e ).The observable duty cycle is taken into account (see the text for details).For each survey program, we refer to the limiting magnitudes of the F277W filter for JWST, the F129 filter for RST, and the F232 filter for GREX-PLUS, respectively.Note that the TDE detection numbers listed above are calculated by assuming ¯á ñ = m m 4   for a Salpeter stellar mass distribution with 0.1 m å /M e 10, and the value increases as μ For obscured TDE searches in the COSMOS-Web survey, we consider TDEs occurring in AGNs with M • 10 6 M e ; otherwise, the emission in the F277W filter is undetectable.b For the LSST survey, we consider a scenario where all TDE candidates selected by LSST are confirmed as true high-z TDEs through follow-up observations with RST.The three values are the detection numbers of TDEs triggered by BHs with masses greater than M • = 10 6 , 10 5 , and 10 4 M e .The actual number will depend on the design of a joint observation campaign with LSST and RST. a