The James Webb Space Telescope JWST has unveiled numerous massive black holes BHs in faint broadline active galactic nuclei AGNs. The discovery highlights the presence of dustreddened AGN populations referred to as little red dots LRDs more abundant than Xrayselected AGNs which are less influenced by obscuration. This finding indicates that the cosmic growth rate of BHs within this population does not decrease but rather increases at higher redshifts beyond z  6. The BH accretion rate density deduced from their luminosity function is remarkably higher than that from other AGN surveys in Xray and infrared bands. To align the cumulative mass density accreted to BHs with the observed BH mass density at z  45 as derived from the integration of the BH mass function the radiative efficiency must be doubled from the canonical 10 value achieving significance beyond the gt3 confidence level. This suggests the presence of rapid spins with 96 of the maximum limit among these BHs under the thindisk approximation maintained by prolonged mass accretion instead of chaotic accretion with randomly oriented inflows. Moreover we derive an upper bound for the stellar mass of galaxies hosting these LRDs ensuring consistency with galaxy formation in the standard cosmological model where the host stellar mass is limited by the available baryonic reservoir. Our analysis gives a lower bound for the BHtogalaxy mass ratio that exceeds the typical value known in the nearby universe and aligns with that for JWSTdetected unobscured AGNs. Accordingly we propose a hypothesis that the dense dustrich environments within LRDs facilitate the emergence of rapidly spinning and overmassive BH populations during the epoch of reionization. This scenario predicts a potential association between relativistic jets and other highenergy phenomena with overmassive BHs in the early universe.
['Galaxy formation', 'High-redshift galaxies', 'Quasars', 'Supermassive black holes', '595', '734', '1319', '1663', 'Astrophysics - Astrophysics of Galaxies']
Birth of Rapidly Spinning Overmassive Black Holes in the Early Universe
{'Birth of Rapidly Spinning, Overmassive Black Holes in the Early Universe': '<!-- image -->  \n1 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China  \n- 2 Global Center for Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan\n- 3 Department of Physics, School of Advanced Science and Engineering, Faculty of Science and Engineering, Waseda University, 3-4-1, Okubo, Shinjuku, Tokyo 169-8555, Japan', 'ABSTRACT': "The James Webb Space Telescope (JWST) has unveiled numerous massive black holes (BHs) in faint, broad-line active galactic nuclei (AGNs). The discovery highlights the presence of dust-reddened AGN populations, referred to as 'little red dots (LRDs)', more abundant than X-ray selected AGNs, which are less influenced by obscuration. This finding indicates that the cosmic growth rate of BHs within this population does not decrease but rather increases at higher redshifts beyond z ∼ 6. The BH accretion rate density deduced from their luminosity function is remarkably higher than that from other AGN surveys in X-ray and infrared bands. To align the cumulative mass density accreted to BHs with the observed BH mass density at z ≃ 4 -5, as derived from the integration of the BH mass function, the radiative efficiency must be doubled from the canonical 10% value, achieving significance beyond the > 3 σ confidence level. This suggests the presence of rapid spins with 96% of the maximum limit among these BHs under the thin-disk approximation, maintained by prolonged mass accretion instead of chaotic accretion with randomly oriented inflows. Moreover, we derive an upper bound for the stellar mass of galaxies hosting these LRDs, ensuring consistency with galaxy formation in the standard cosmological model, where the host stellar mass is limited by the available baryonic reservoir. Our analysis gives a lower bound for the BH-to-galaxy mass ratio that exceeds the typical value known in the nearby universe and aligns with that for JWST-detected unobscured AGNs. Accordingly, we propose a hypothesis that the dense, dust-rich environments within LRDs facilitate the emergence of rapidly spinning and overmassive BH populations during the epoch of reionization. This scenario predicts a potential association between relativistic jets and other high-energy phenomena with overmassive BHs in the early universe.  \nKeywords: Galaxy formation (595); High-redshift galaxies (734); Quasars (1319); Supermassive black holes (1663)", '1. INTRODUCTION': "The cosmic evolution of massive black hole (BH) populations is predominantly driven by mass accretion, powering active galactic nuclei (AGNs) (e.g., LyndenBell 1969) with a certain level of merger contributions to BHs harbored in nearby massive ellipticals (e.g., McWilliams et al. 2014; Kulier et al. 2015). Multiwavelength observations have consistently shown that AGN activity peaks around z ∼ 2 and declines toward higher redshifts (e.g., Ueda et al. 2014; Delvecchio et al.  \n2014). The analysis of AGN activity offers insights into the radiative efficiency of accreting BHs by comparing it with the local mass density of relic BHs (Soltan 1982; Yu & Tremaine 2002).  \nRecent observations by the James Webb Space Telescope (JWST) have revealed a new category of dustreddened, broad-line AGNs, often referred to as 'little red dots' (hereafter LRDs, Matthee et al. 2024). These AGNs are characterized by their compact morphology and moderate dust obscuration ( A V ≈ 3) in the spectra (Barro et al. 2024; Kocevski et al. 2023; Harikane et al. 2023; Labbe et al. 2023). Investigations into the AGN luminosity function of LRDs at z = 4-8 found an abundance of Φ ∼ 10 -5 -10 -4 cMpc -3 mag -1 in the ob-  \nFigure 1 presents the bolometric luminosity functions at z ≃ 5 (left) and z ∼ 7 (right), i.e., the number den-  \n<!-- image -->  \nFigure 1. Bolometric AGN luminosity functions at 4 . 5 < z < 6 (left) and 6 . 5 < z < 8 . 5 (right). The luminosity function data obtained from different surveys are shown: the rest-UV-selected quasars (Niida et al. 2020; Matsuoka et al. 2023), the X-ray selected AGNs (Ueda et al. 2014), and dust-reddened AGNs reported as 'little red dots (LRDs)' identified with JWST photometry and slitless spectroscopy (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024). The bright end slope of the LRD luminosity function is consistent with Φ ∝ L -1 bol at both redshifts.  \n<!-- image -->  \nd UV absolute magnitude range of -22 ≲ M UV ≲ -18 (e.g., Kokorev et al. 2024), significantly exceeding those predicted by extrapolations from the unobscured AGN luminosity function in ground-based surveys (e.g., Niida et al. 2020). Moreover, spectroscopic analysis of LRDs, facilitating direct measurement of broad H α emissions - a tracer of AGN activity (Greene & Ho 2005) - has led to the construction of the AGN bolometric luminosity function for LRDs. This result suggests that in contrast to expectations based on AGN surveys in the pre-JWST era, the cosmic growth rate of BHs within this AGN population does not decline but appears to increase at higher redshifts ( z > 6).  \nIn this Letter , compiling data from extensive highz AGN and LRD surveys, we constrain the radiative efficiency of BHs in dominant LRD populations by comparing the observed BH mass density at z ≃ 4 -5 to the mass accreted to BHs calculated from the BH growth rate, i.e., the Soltan-Paczy'nski argument at the epoch of reionization. This analysis suggests a radiative efficiency higher than the canonical 10% value and favors rapid spins of these BHs under the thin-disk approximation (e.g., Shakura & Sunyaev 1973; Novikov & Thorne 1973). Furthermore, we establish an upper limit for the stellar mass of galaxies harboring these LRDs, and give a lower bound for the BH-to-galaxy mass that exceeds the typical value known in the nearby universe and aligns with that for JWST-detected unobscured AGNs. Consequently, we propose a hypothesis that the dust-rich environments in LRDs promote the emergence of rapidly spinning and overmassive BH populations.  \nThroughout this paper, we assume a flat Λ cold dark matter (CDM) cosmology consistent with the constraints from Planck (Planck Collaboration et al. 2020); h = 0 . 6732, Ω m = 0 . 3158, Ω Λ = 1 -Ω m , Ω b = 0 . 04938, and σ 8 = 0 . 8102. It is important to note that observational studies referenced in our work have adopted a different set of cosmological parameters, h = 0 . 7 and Ω m = 0 . 3. However, the differences in parameter choice have a negligible impact on the results.", '2. SOLTAN ARGUMENT AT Z ≥ 5': 'Early studies of LRDs have indicated that the observed values of M UV are much fainter than the intrinsic UV magnitudes due to dust extinction (e.g., Kocevski et al. 2023; Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024). To address the underlying AGN activity, these studies estimated bolometric AGN luminosity from rest-optical emissions. Matthee et al. (2024) and Greene et al. (2024) conducted JWST/NIRSpec observations on LRDs and converted the measured H α luminosity to bolometric luminosity, while Kokorev et al. (2024) and Akins et al. (2024) relied on continuum luminosity L 5100 and L 3000 for a bolometric luminosity estimator, respectively, by assuming that all the continuum flux originates from the AGN. The AGN bolometric luminosity derived from the H α luminosity, particularly its broad-line component emitted from fast-moving clouds near the AGN, is more accurate than using dustdereddened continuum (see Section 4.3).  \nsity per unit comoving volume per log L bol in units of cMpc -3 dex -1 , combining data from various sources: rest-UV-selected unobscured quasars (Niida et al. 2020; Matsuoka et al. 2023), X-ray selected AGNs (Ueda et al. 2014), and LRDs identified through JWST photometry and spectroscopy (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024) 1 . For UV and X-ray selected AGNs, we adopt the bolometric correction factors calibrated by Duras et al. (2020). The abundance of unobscured quasars at z ∼ 5 aligns closely with that of X-ray detected AGNs for L bol ≳ 4 × 10 45 erg s -1 , while the X-ray AGN abundance further increases at lower luminosity regimes 2 . This suggests that the obscured AGN fraction increases toward the fainter end (Vito et al. 2018). In contrast, LRDs exhibit a substantially greater abundance at L bol ≲ 10 46 erg s -1 , nearly one order of magnitude above that of X-ray AGNs. While the LRD abundance data show some variations owing to differences in sample size and methods for estimating bolometric luminosity across the referred studies, the trend of over-abundance relative to the other AGN populations is consistently observed. The bright end slope of the LRD luminosity function is consistent with Φ ∝ L -1 bol (Kokorev et al. 2024; Akins et al. 2024). Therefore, it is ensured that a large fraction of the production of radiation (i.e., the amount of material accreted to BHs) is dominated by these bright populations. The over-abundance of LRDs and the bright-end slope of Φ ∝ L -1 bol hold even at z ≳ 7.  \nIn our analysis below, we consider abundance data from luminosity bins where the sample size is N ≥ 2, but exclude bins with a sample size of N = 1. This approach aligns with Poisson statistical error estimates, where a single occurrence ( N = 1) is statistically indistinguishable from zero. Additionally, for the COSMOS-Web luminosity function data, we exclude the data at the faint end of L bol < 10 46 erg s -1 from Akins et al. (2024) since the COSMOS-Web survey is not deep enough to accurately measure the abundance of these faint populations and thus completeness correction matters.  \nFigure 2 illustrates the BH accretion rate density (BHAD) across various redshifts, with each data point  \nFigure 2. The cosmic BH accretion rate density (BHAD) as a function of redshift. Each data point and curve represent BHADs estimated under the assumption of a 10% radiative efficiency ( ϵ rad = 0 . 1) for the three different populations, including LRDs (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024), X-ray selected AGNs including Compton-thick populations (Aird et al. 2015; Ananna et al. 2019; Pouliasis et al. 2024), and mid-infrared selected AGNs (Delvecchio et al. 2014). For comparison, the cosmic SFRD scaled by a factor of 3,000 is overlaid (Harikane et al. 2022). The BHAD attributed to LRDs remains significantly dominant at z > 6.  \n<!-- image -->  \nand curve representing BHADs estimated under the assumption of a 10% radiative efficiency ( ϵ rad = 0 . 1). These include LRDs (Matthee et al. 2024; Kokorev et al. 2024; Greene et al. 2024; Akins et al. 2024), as well as Xray selected AGNs (Aird et al. 2015; Ananna et al. 2019; Pouliasis et al. 2024), and mid-infrared selected AGNs (Delvecchio et al. 2014). The BHAD estimated from X-ray sources including Compton-thick AGN contributions agrees well to that of mid-infrared AGNs at z ∼ 3. With the same value of ϵ rad = 0 . 1, the BHAD inferred from the bolometric luminosity function of LRDs indicates a persistent or even increasing trend toward higher redshifts (4 < z < 8 . 5), opposite to the declining trend of X-ray selected AGNs. Note that completeness corrections for some LRD samples we adopt (e.g., Matthee et al. 2024; Kokorev et al. 2024) have not been fully implemented in estimating the abundance of faint sources, thereby potentially leading to an increased abundance at the faint sources. Nevertheless, our finding is unlikely to alter because the brighter LRD populations dominantly contribute to the BHAD (i.e., d log Φ / dlog L bol ≃ -1). Note that the BHAD estimated from the COSMOSWeb result increases by < 20% when faint AGNs with L bol < 10 46 erg s -1 are included in our analysis.  \n<!-- image -->  \nFigure 3. Left : Cosmic evolution of the BH mass density in a comoving volume. At z ∼ 5, the BH mass density is derived from the integration of the BHMF for LRDs (red symbols). From this point, the mass density grows toward lower redshifts following the BHAD deduced from known AGN populations with a 10% radiative efficiency (solid curve; Ueda et al. 2014) and reaches the density of relic BHs in the nearby universe (Shankar et al. 2009). At z > 5, the cumulative mass accreted to BHs during the LRD phase, ∆ ρ · ≡ BHAD × ∆ t inferred from their bolometric luminosity function over a time span ∆ t for given redshift range based on the COSMOS-Web (magenta) and the other surveys (blue), assuming a 10% radiative efficiency, substantially exceed the observed mass density at z ≃ 5 as well as the predictions from a BH growth model calibrated with UV and X-ray selected AGN luminosity function (dotted curve; Li et al. 2024a). Data for LRDs are derived from luminosity functions and BH mass estimates provided in the literature (open symbols, Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024; Akins et al. 2024) and the mean values for each group (filled symbols). Right : Summary of the BH mass density and the cumulative mass density during the LRD phase assuming a 10% radiative efficiency. Shaded areas indicate the BH mass density ρ · at z ≃ 5 (red) and the cumulative mass density accrued during the LRD stage calculated from the COSMOS-Web (magenta) and the other surveys (blue). The total sum of ∆ ρ · over the entire redshift range in each data group is shown with a star symbol.  \n<!-- image -->  \nFor comparison, we overlay the cosmic star-formation rate density (SFRD) scaled by a factor of 3,000 (Harikane et al. 2022). The scaled SFRD matches well the BHAD based on the mid-infrared selected AGNs at z ≲ 3 and appears to be consistent with the BHAD of LRDs at z ≃ 5 -6. On the other hand, the BHAD attributed to LRDs remains significantly dominant at z > 6. This finding, based on the assumption of ϵ rad = 0 . 1, indicates that rapid growth of BHs at these earlier epochs established a trend of overmassive BH in terms of the M · /M ⋆ ratio, as observed in recent JWST AGN studies at z > 6 (e.g., Harikane et al. 2023; Maiolino et al. 2023; Pacucci et al. 2023).  \nIn the left panel of Figure 3, we show the evolution of the BH mass density within a comoving volume throughout cosmic time. The solid curve represents the cumulative BH mass density deduced from the AGN bolometric luminosity functions (primarily X-ray selected populations) at 0 < z < 5, under an assumed 10% radiative efficiency (Ueda et al. 2014). This projection agrees closely with the observed BH mass density at z ≃ 0 (Shankar et al. 2009), concluding the plausibility of the pre-assumed radiative efficiency ( ϵ rad = 0 . 1; Soltan 1982; Yu & Tremaine 2002). Additionally, we  \npresent the BH mass density directly derived from the integration of the BHMF for LRDs at z ≃ 5, estimated to be ρ · ≃ 2 . 8 +2 . 2 -1 . 2 × 10 3 M ⊙ cMpc -3 (red symbols; Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024) 3 . This estimate remarkably aligns with expectations based on the Soltan argument assuming the conventional radiative efficiency of 10% at 0 < z < 5, further reinforcing the consistency across the three distinct physical measures.  \nNext, we extend our analysis to the universe at z > 5, focusing on the cumulative mass of BHs accreted during the LRD phase, ∆ ρ · ≡ BHAD × ∆ t , calculated from their bolometric luminosity function over a redshift interval. We categorize the LRD samples into two groups: those identified through the COSMOS-Web survey (magenta, Akins et al. 2024) and LRDs from other observational programs (blue, Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024). This classification is  \nTable 1. Significance of the difference between ρ · ( z ≃ 5) and ∆ ρ · ( z ≳ 5) for different radiative efficiencies.  \nNote - Column (1): Survey. Column (2): Redshift ranges. Column (3): Cumulative mass density of BHs accreted during the LRD phase (in units of M ⊙ cMpc -3 ) with a 10% radiative efficiency. Column (4)-(7): the p -value evaluated in the t -test for the null hypothesis between ρ · ( z = 5) and ∆ ρ · at z > 5 for different values of the radiative efficiency (and the corresponding BH spin parameters). Here, we consider two cases with LRD data based on the COSMOS-Web survey (Akins et al. 2024) and other LRD surveys (Matthee et al. 2024; Greene et al. 2024; Kokorev et al. 2024) for calculating the total cumulative mass. The COSMOS-Web result requires ϵ rad ≥ 0 . 2 beyond the > 3 σ confidence level, while the confidence level is ≳ 2 σ with LRD samples from other surveys.  \nbased on two considerations: (1) the redshift intervals different among the samples in the literature, requiring a uniform redshift bin size for comparative analysis; and (2) the need to evaluate the impact of wide-area surveys such as COSMOS-Web on the LRD Soltan argument. As seen in the left panel of Figure 3, one can find the cumulative mass of BHs at z > 5 substantially exceeds the observed mass density at z ≃ 5 as well as the predictions from a BH growth model calibrated with UV and X-ray selected AGN luminosity functions (dotted curve; Li et al. 2024a). This discrepancy raises concerns about a potential violation of the BH mass conservation law; namely, ρ · ( z ≃ 5) ≳ ∆ ρ · ( z > 5) needs to be hold. Thus, the possible inapplicability of ϵ rad = 0 . 1 is suggested for the early universe beyond z > 5. Adjusting the radiative efficiency upwards impacts the inferred BHAD, which follows ∝ (1 -ϵ rad ) /ϵ rad . For instance, adopting the theoretical upper limit of ϵ rad = 0 . 42 for an extreme Kerr BH with a spin parameter a · = 1 (e.g., Kerr 1963; Novikov & Thorne 1973) resolves the discrepancy between the integrated BHMF values and the cumulative mass derived from the BHAD. The relationship between the radiative efficiency and BH spin is well understood for geometrically-thin accretion disks, where a thermal equilibrium is maintained through efficient radiative cooling that balances with viscous heating (Shakura & Sunyaev 1973). However, this scenario changes in low-accretion-rate states, where the disk becomes geometrically thick due to inefficient cooling (e.g., Yuan & Narayan 2014). In such cases, the radiative efficiency substantially decreases from the values in the thin-disk approximation (Inayoshi et al. 2019). This reduction in ϵ rad enlarges the discrepancy between ρ · ( z ≃ 5) and ∆ ρ · ( z > 5), rather than mitigating it.  \nThe right panel of Figure 3 provides a detailed quantitative comparison of BH mass density values from several studies, using the least-squares method for fitting  \nρ · and ∆ ρ · at each redshift interval. The cumulative values of ∆ ρ · across the entire redshift range are denoted by star symbols for each LRD sample (see also Table 1). For the case without the COSMOS-Web survey (blue symbols), the accreted mass density at z ∼ 5 is found to be lower than that at z ∼ 7. This difference is primarily due to the finding of Kokorev et al. (2024), where N = 9 luminous LRDs with L bol ≃ 10 47 erg s -1 were identified at 6 . 5 < z < 8 . 5 but only one was reported at 4 . 5 < z < 6 . 5 within a large sample set of LRDs from multiple survey fields. We note that luminosity function bins with a sample size of N = 1 are excluded in our analysis as a single occurrence is statistically indistinguishable from zero. Therefore, the inclusion of these luminous populations substantially influences the BHAD estimate. Using only the COSMOSWeb result (magenta symbols), we consistently observe higher values of ∆ ρ · at the two redshift ranges, owing to the wide-area survey designed to identify more luminous and rarer populations. As a result, the total sum in each case reaches as high as ∆ ρ · ≃ 3 . 0 +2 . 2 -1 . 2 × 10 4 and 6 . 6 +6 . 3 -2 . 3 × 10 4 M ⊙ cMpc -3 , respectively (star symbol). With the mean values, the cumulative mass densities during the LRD phases over 5 < z < 9 appear to be ≳ 10 times higher than the BH mass density at z ≃ 5. However, there is a concern regarding the classification of both the LRD samples of Kokorev et al. (2024) and Akins et al. (2024), where all photometrically selected LRDs are considered as AGNs due to the lack of spectroscopic observations (see also Section 4.3). Due to these concerns, the cumulative mass density of BHs and their difference from the BH mass density are considered to be upper bounds.  \nTo understand the influence of each contribution of ∆ ρ · on this analysis and the need for a radiative efficiency beyond the standard 10% value, we explore two scenarios: one considering the contribution from the  \nCOSMOS-Web survey (magenta) and another one compiling LRD samples from other observational programs (blue). In this work, to assess the statistical difference between ρ · ( z ≃ 5) and ∆ ρ · for each of the two cases, we employ the t -test, which serves as an appropriate statistical method to determine whether there is a statistically significant difference in the mean values between two groups with unequal sample variances. The p -values, as summarized in Table 1, indicate that the hypothesis of agreement between the two quantities at ϵ rad = 0 . 1 is statistically rejected in both the two cases with a confidence level of > 99 . 7% ( p < 0 . 003). For the analysis with the COSMOS-Web survey result, a radiative efficiency greater than ϵ rad ≥ 0 . 3 is concluded with a confidence level of > 98%, while the case with other LRD survey data suggest ϵ rad ≥ 0 . 2 with a similar confidence level. This finding suggests that the majority of BHs within LRDs or similarly gas/dust-rich environments are likely to process rapid spins with an average ϵ rad ≥ 0 . 2 -0 . 3 (the corresponding BH spin is a · ≃ 0 . 96 -0 . 996), indicating a prevalent condition of rapid angular momentum in BH growing environments in the early universe.  \nThis result suggests that BH growth at these high redshifts, especially in LRDs, is likely dominated by prolonged accretion episodes with coherent angular momentum directions or a modest degree of anisotropy (e.g., Volonteri et al. 2005; Dotti et al. 2013), unlike shortlived chaotic accretion with randomly oriented inflows that tend to spin BHs down (e.g., King et al. 2008). Our conclusion on rapid spins of the early BH population will be directly testable through future gravitationalwave observations with the space-based detectors such as LISA, TianQin, and Taiji (e.g., Amaro-Seoane et al. 2023; Torres-Orjuela et al. 2024).  \nIntriguingly, clustering analyses of quasars and galaxies at z ≳ 6 suggest that the duty cycle of UV-bright quasars is as low as ≲ 1% (Eilers et al. 2024; Pizzati et al. 2024), corresponding to a quasar lifetime of 1 -10 Myr, which is significantly shorter than the e -folding time assuming the Eddington accretion rate. This finding implies that most of the BH mass growth would have occurred in highly (UV-)obscured environments and/or through episodic super-Eddington phases with a lower radiative efficiency (Davies et al. 2019; see also Inayoshi et al. 2016, 2022b). The first implication is consistent with the hypothesis that LRDs are moderately obscured AGNs (e.g., Li et al. 2024b). The second implication, concerning the potential mass contribution from radiatively inefficient super-Eddington growth, would be constrained by our findings in this work and is left for future investigations.  \nFigure 4. Stellar mass density in galaxies hosting LRDs at various redshifts, calculated using Eq. (1) and assuming F ( ≡ f IMF f L ) = 1 . 0 (open symbols) at two redshift ranges of 4 . 5 < z < 6 . 5 and 6 . 5 < z < 8 . 5. For comparison, the stellar mass function derived from the DM halo mass function at 5 ≤ z ≤ 8 is shown with a 100% star formation efficiency. An upper bound of the stellar mass constrains F < 0 . 3 f ⋆ for 4 . 5 < z < 6 . 5 and F < 0 . 04 f ⋆ for 6 . 5 < z < 8 . 5 (filled symbols).  \n<!-- image -->', '3. POTENTIAL OVERMASSIVE BH TRENDS IN LITTLE RED DOTS': "In this section, we explore the possibility that BHs within LRDs are overmassive relative to the mass correlation with their host mass, as implied from the BHADto-SFRD ratio shown in Figure 2. In general, estimating the stellar mass of dust-obscured sources poses a significant challenge in the absence of rest-frame near-infrared data provided by JWST MIRI (e.g., Williams et al. 2024; P'erez-Gonz'alez et al. 2024). Instead of examining the detailed spectral energy distribution fitting analysis, we focus on putting an upper bound for the stellar mass. This approach ensures that the observed abundance of LRDs does not exceed the theoretical upper bound in the standard ΛCDM model with a 100% conversion efficiency from gas to stars (e.g., Boylan-Kolchin 2023).  \nThe stellar continuum for LRDs can be constrained by the dust-corrected continuum flux at 5100 ˚ A. Given that broad H α emission indicates AGN dominance in the continuum (see Section 4.3), we adjust L ⋆, 5100 = f L L 5100 , where f L is significantly less than unity. To estimate an upper bound of stellar mass, we employ the STARBURST99 population synthesis code (version 7.0.1; Leitherer et al. 1999), adopting a Kroupa IMF (Kroupa 2001; 0 . 1 -100 M ⊙ ), Padova isochrone models, constant star formation, and solar metallicity. This approach  \nFigure 5. M · -M ⋆ distribution for high-redshift AGNs, including LRDs, JWST-detected unobscured ANGs at z = 4 -8 (purple, Maiolino et al. 2023; green, Harikane et al. 2023; cyan, Stone et al. 2024; and blue, Ding et al. 2023), and quasars identified in ground-based surveys (Izumi et al. 2021). For the LRDs at 4 . 5 < z < 6 . 5 (red) and 6 . 5 < z < 8 . 5 (orange), we derive the upper bound of the stellar mass based on the dust-corrected continuum flux measured by Greene et al. (2024) using Eqs. (1) and (2). Additionally, a z = 8 . 5 LRD with broad H β emission, for which the stellar mass is constrained by ALMA non-detections, is overlaid (blue, Kokorev et al. 2023). Two different mass correlations are overlaid: the local relationship (solid, Kormendy & Ho 2013) and the JWST-detected AGNs (dashed, Pacucci et al. 2023).  \n<!-- image -->  \nyields a galaxy mass -luminosity relation:  \nM ⋆ 10 9 M ⊙ ≃ 1 . 3 f IMF ( L ⋆, 5100 10 43 erg s -1 )( t age 1 Gyr ) , (1)  \nwhich is applicable for stellar ages of t age ∼ 0 . 3 -3 Gyr. This estimate is sensitive to the low-mass end of the stellar IMF ( f IMF = 1 for m ⋆, min = 0 . 1 M ⊙ ); for instance, setting the minimum mass up to m ⋆, min = 1 . 0 M ⊙ decreases the factor to f IMF ∼ 0 . 3.  \nFigure 4 presents the stellar mass function (in units of M ⊙ cMpc -3 ) in galaxies hosting LRDs at various redshifts, calculated by using Eq. (1) and assuming F ( ≡ f IMF f L ) = 1 . 0 at two redshift ranges of 4 . 5 < z < 6 . 5 and 6 . 5 < z < 8 . 5 (open circles), based on the LRD luminosity function obtained by Kokorev et al. (2024). Since the luminosity function of LRDs follows Φ ∝ L -1 bol , the stellar mass density becomes flatter at the high-mass end when the stellar mass is translated from the luminosity with Eq. (1). We compare the results to the stellar mass function derived from the halo mass function at  \n5 ≤ z ≤ 8, assuming M ⋆ = f ⋆ f b M h , where f b = 0 . 16 is the cosmic baryon fraction and f ⋆ is the star formation efficiency (see more details in Inayoshi et al. 2022a). Setting f ⋆ = 1 . 0 offers a theoretical upper limit on stellar mass in galaxies ( f ⋆ > 1; the forbidden region), highlighting a mismatch between the stellar mass density contained in LRDs with F = 1 . 0 and the ΛCDM upper limit. As a result, we deduce a stringent constraint denoted with filled circles,  \nF < 0 . 3 f ⋆ at 4 . 5 < z < 6 . 5 , F < 0 . 04 f ⋆ at 6 . 5 < z < 8 . 5 . (2)  \nFigure 5 shows the M · -M ⋆ distribution for high-redshift AGNs, including LRDs (square), JWSTdetected unobscured AGNs (cross), and quasars identified in ground-based surveys (circle). For the LRDs at 4 . 5 < z < 6 . 5 (red) and 6 . 5 < z < 8 . 5 (orange), we derive the stellar mass of those with broad H α emission from Greene et al. (2024), using Eq. (1). This calculation incorporates the upper limit for the stellar continuum ratio F to the observed continuum (see Eq. 2 and Figure 4), ensuring consistency with the theoretical upper bound of stellar mass density in the ΛCDM universe. The M · -M ⋆ values for LRDs tend to be overmassive compared to the local relationship (solid line; Kormendy & Ho 2013), and aligns well with other AGNs detected by JWST and follows the mass correlation inferred from JWST AGN data, excluding quasars from ground-based surveys (dashed line; Pacucci et al. 2023). Moreover, their distribution is consistent with the locus of a JWST/NIRSpec confirmed z = 8 . 5 LRD that exhibits broad H β emission, for which an upper limit on the stellar mass has been constrained by non-detection in ALMA observations (blue; Kokorev et al. 2023).  \nThe mass ratios for these LRDs are also consistent with the BHAD/SFRD values at z > 6 shown in Figure 2. This suggests a model in which transient rapid growth phases during the LRD stages elevate these BHs into an overmassive state. This hypothesis is supported both theoretically (e.g., Inayoshi et al. 2022b; Hu et al. 2022) and observationally (Fujimoto et al. 2022), providing a comprehensive insight into the BH growth dynamics in the early universe.", '4.1. Missing X-ray radiation from LRDs': 'X-ray AGN surveys are generally effective in identifying obscured AGNs. However, in the case of LRDs observed with JWST, no X-ray counterparts have been reported in the early studies (e.g., Kocevski et al. 2023; Furtak et al. 2023; Matthee et al. 2024). X-ray weak-  \nness has been consistently observed in LRDs as demonstrated by stacking analyses (Yue et al. 2024), and this phenomenon extends beyond LRDs to a more general category of unobscured broad-line AGNs (Maiolino et al. 2024). Further emphasizing the rarity of X-ray emissions, Kocevski et al. (2024) have identified only two X-ray detected LRDs at z = 3 . 1 and 4 . 66 among 341 examined objects, resulting in a detection fraction of less than 0 . 6 %. The optical continuum extinction measurements suggest a gas column density of N H ∼ 3 . 3 × 10 22 ( A V / 3 . 0) cm -2 (Maiolino et al. 2001), indicating that the column density outside the broadline region is too low to obscure X-rays. The column density estimate is broadly consistent with those measured from the X-ray spectral analysis for the two X-ray detected LRDs, N H ∼ (5 -20) × 10 22 cm -2 (Kocevski et al. 2024).  \nConsidering the absence of X-ray counterparts for LRDs, we explore the possibility that their X-ray emission is intrinsically weak, as compared to typical Xray selected AGNs. Figure 6 presents the critical Xray luminosity for each bolometric luminosity, so that Φ X ( L X , crit ) ≥ Φ LRD ( L bol ), where we adopt the X-ray AGN luminosity function Φ X from Ueda et al. (2014) and the LRD bolometric luminosity function Φ LRD from Kokorev et al. (2024), respectively. This condition requires that LRDs must have a bolometric correction factor to X-rays that prevents them from being classified as X-ray AGNs and contributing to their abundance (red horizontal lines with arrows). The critical X-ray luminosity is limited below those derived from comparison between the X-ray and optical-based AGN luminosity functions at lower redshifts of z ≲ 2 (Ueda et al. 2003).  \nTo quantify this intrinsic X-ray faintness in LRDs, we utilize the optical to X-ray spectral index, defined as α OX = log( L ν, 2keV /L ν, 2500 ) / log( ν 2keV /ν 2500 ), where L ν, 2keV and L ν, 2500 are the extinction corrected luminosity density at 2 keV and 2500 ˚ A. Our findings support a constant value of α OX lower than -1 . 8, rather than the luminosity-dependent α OX values observed in unobscured quasars across 0 ≲ z ≲ 6 (e.g., Steffen et al. 2006; Duras et al. 2020). The upper bound of α OX ≃ -1 . 8 is seen in the most luminous quasar populations with L bol ≳ 10 47 erg s -1 (Nanni et al. 2017), which does not apply to most LRDs studied in this paper.  \nThe extensive LRD samples by Kokorev et al. (2024), consisting of 260 dust-reddened AGN candidates, are compiled from deep JWST/NIRCam fields totaling ∼ 340 arcmin 2 . This includes observations from the CEERS field, which spans ∼ 51 . 9 arcmin 2 and falls within the coverage area of the Chandra AEGIS-XD survey coverage (Nandra et al. 2015). The survey detection  \nFigure 6. The critical X-ray luminosities for LRDs at z ∼ 5, determined by the requirement for LRDs not to be classified as X-ray AGNs nor contributing to the abundance of X-ray AGNs (red horizontal lines with arrows). For comparison, several models for bolometric correction to X-rays are shown (Ueda et al. 2003; Nanni et al. 2017; Duras et al. 2020), as well as the cases with constant optical to X-ray spectral indices of α OX = -1 . 5 (dashed) and -1 . 8 (solid). The L bol -L X values of the two LRDs detected in X-rays are shown with orange symbols; JADES 21925 (square) and PRIMER-COS 3982 (diamond) (Kocevski et al. 2024). The shaded area denotes X-ray luminosities below the detection threshold of current Chandra observations for JWST fields.  \n<!-- image -->  \nlimit reaches 1 . 5 × 10 -16 erg s -2 cm -1 at 0.5-10 keV, which corresponds to L X ≃ 1 . 1 × 10 43 erg s -1 at the restframe 2-10 keV for z ∼ 5 -7 sources assuming a photon index of Γ = -1 . 7 and a Compton-thin limit ( N H < 10 24 cm -2 ). Given the average surface density of these LRDs ( ≃ 0 . 77 arcmin -2 ), ∼ 40 LRDs in the CEERS field show no X-ray detection above this threshold. This suggests that non-detection of X-rays among LRDs can be explained by the intrinsic faintness of X-rays as shown in Figure 6. Nevertheless, the most luminous LRDs with L bol ≳ 10 47 erg s -1 might still be observed in X-rays unless classified as Compton-thick AGNs. Additionally, we note that the two LRDs detected in X-rays with a modest hydrogen column density of N H ∼ 10 23 cm -2 , show obscuration-corrected X-ray luminosities of L X ≃ 5 . 4 × 10 43 erg s -1 (for JADES 21925 at z photo = 3 . 1) L X ≃ 5 . 0 × 10 44 erg s -1 (for PRIMER-COS 3982 at z spec = 4 . 66), respectively (Kocevski et al. 2024). The bolometric luminosities calculated from the rest-optical fluxes are L bol ≃ 6 . 5 × 10 46 erg s -1 and 3 . 1 × 10 47 erg s -1 after dust attenuation correction. These findings also suggest a lower value of α OX ≃ -1 . 8 as shown in Figure 6 (orange symbols).  \nAlternatively, if the X-ray emission was not intrinsically faint (except the two X-ray detected LRDs), most  \nof these LRDs would be embedded in Compton-thick gas with N H ≫ 10 24 cm -2 , concealing the LRDs from deep X-ray observations. However, such high densities are not expected in the LRD rest-optical spectra, which show a modest extinction A V ∼ 3 mag, equivalent to N H ∼ 3 × 10 22 cm -2 . This discrepancy is also consistent with the lack of expected absorption features in their NIRSpec rest-frame UV spectra of LRDs with broademission lines (e.g., Greene et al. 2024).', '4.2. Dusty young starburst galaxies mimicking LRD-like AGNs?': 'The classification of LRDs as AGNs has relied on the detection of broad H α emission. However, such highvelocity gas can also originate from stellar processes, such as stellar winds or supernova explosions. WolfRayet (WR) galaxies, characterized by young and massive stellar populations, can exhibit broad H α emission as well as other high-ionization lines (e.g., Ho et al. 1995; Schaerer et al. 1999). Therefore, to conclusively confirm the AGN nature of LRDs, it is essential to perform line diagnostics that extend beyond simply identifying broad H α emission, e.g., detection of broad He II λ 4686 emission, a signature frequently associated with WR galaxies (e.g., NGC 4214 discussed in Sargent & Filippenko 1991).  \nSpectroscopic observations provide valuable insights into the characteristics of emission lines in LRDs. Initial studies by Kocevski et al. (2023) and Greene et al. (2024) have found that the He II λ 4686 line, commonly associated with WR stellar activity, is absent in these LRD sources. Additionally, the spectral signatures typically linked to WR stars have not been observed within the LRD samples. The lack of He II and other WR indicative spectral lines suggests that the broad H α emissions detected in LRDs are not of stellar origin (note that detection of He II λ 4686 does not necessarily exclude the AGN possibility because the emission is also observed in AGNs). This finding supports the hypothesis that AGNs are responsible for these emissions. More detailed spectroscopic analyses of multiple emission lines would strength this conclusion (Greene et al. 2020; Reines 2022).', '4.3. Bolometric luminosity estimates': 'Our study is motivated by intriguing discoveries on the high abundance of LRDs, but an important consideration about the AGN luminosity estimate needs to be noted. While the UV luminosity functions show similar shapes across different studies, significant variations in the bolometric luminosity function arise due to the different methods used for bolometric luminosity estimates  \n(see Figure 1). Since the observed rest-frame UV flux is heavily attenuated, earlier studies have employed either the rest-frame continuum flux at 5100 ˚ A (Kokorev et al. 2024), or the direct H α emission luminosity, when available, as a proxy for the AGN bolometric luminosity (Matthee et al. 2024; Greene et al. 2024).  \nThe approach that relies on continuum flux introduces uncertainties of determining the AGN contribution to the total flux, which might result in overestimated luminosity. Furthermore, the selection of LRDs based solely on photometric criteria might include non-AGN sources such as Galactic brown dwarfs (Langeroodi & Hjorth 2023), thereby possibly overestimating the AGN abundance. Greene et al. (2024) reported the identification success rate of AGNs among LRDs in the UNCOVER field to be approximately 60%. Kokorev et al. (2024) studied LRDs based on the photometric data from multiple JWST survey areas, using color selection conditions provided by Greene et al. (2024), which effectively remove contaminants of Galactic brown dwarfs. To date, other types of low-z interlopers, such as Balmer break galaxies, for LRDs have not been reported via spectroscopic studies.  \nIn contrast, spectroscopic data particularly with measurements of broad H α emission facilitate confirmation of the AGN presence and a more accurate determination of L bol . This method adopts an empirical relationship derived from local AGN observations (Greene & Ho 2005). For LRD sources with detected H α emissions, the continuum fluxes at 5100 ˚ A calculated through the two methods yield ratios of L 5100 , H α /L 5100 , c ≃ 1 . 2 ± 0 . 2 (Greene et al. 2024). This result supports the scenario that the continuum emission at 5100 ˚ A is substantially AGN-origin, not from dust-reddened stellar continuum of the host galaxy (i.e., f L ≪ 1).  \nOne limitation in the work by Greene et al. (2024) is the use of low-resolution PRISM spectra for analyzing H α emissions, which complicates the decomposition of broad and narrow H α emissions in some cases. To address this challenge, we can use an empirical relationship obtained through higher-resolution spectroscopic observations by Matthee et al. (2024). This relationship examines the flux ratio between narrow and broad H α emissions in LRDs; namely, the ratio F H α, broad /F H α, tot exceeds 0.6 with a positive rest-frame optical spectral index, with the ratio approaching unity as the spectral index increases. Therefore, the potential systematic errors in estimating broad H α line luminosity due to incomplete spectral line decomposition could be alleviated by employing this relationship.  \nDespite these complexities found in current analyses, the central conclusion of our discussion remains valid  \non a qualitative level. Nevertheless, the development of more quantitative arguments will benefit from further observational explorations that refine the understanding of the AGN characteristics of LRDs and provide a more precise estimate of their cosmic abundance (e.g., Li et al. 2024b).', '4.4. Multi-messenger Counterparts': 'In this study, we propose a scenario where the dustrich environments within LRDs lead to the emergence of rapidly spinning and overmassive BH. The high spins of these BHs may yield a strong correlation between the presence of relativistic jets (e.g., Blandford & Znajek 1977), high-energy emissions and particles (e.g., Dai & Fang 2017; Murase et al. 2020), and transient bursts such as stellar tidal disruption events (e.g., Inayoshi et al. 2024) with early BHs being overmassive compared to the mass correlation observed in the nearby universe.  \nMulti-wavelength surveys including radio, optical, and X-ray bands have reported that the radio powers (or radio loudness) of obscured AGNs initially identified by the VLA/FIRST survey as bright radio sources tend to increase with redshifts at 0 . 5 < z < 3 . 5, with jet powers exceeding P jet ≳ 10 46 erg s -1 (Ichikawa et al. 2021, 2023). The inferred jet production efficiency calculated from η jet ∼ ϵ rad P jet /L bol approaches unity, implying rapid spins of these nuclear BHs (e.g., Tchekhovskoy et al. 2011). These types of radio AGNs at intermediate redshifts may offer valuable insights into understanding the characteristics of LRDs at higher redshifts, thus further supporting our hypothesis of rapid BH spins.', '4.5. Implications to BH growth mechanisms at z ≳ 5': 'The spin of a massive BH is influenced by a range of physical processes including BH mergers and mass accretion. The coalescence of two non-spinning BHs results in a remnant with a significant spin, approximately a · ≃ 0 . 69 for an equal-mass merger (Berti et al. 2007), but the spin diminishes if the merging BHs have non-zero, misaligned spins relative to the orbital angular momentum (e.g., Berti & Volonteri 2008). Gas accreting onto BHs through a disk aligned with the angular momentum direction of the BH is expected to enhance the spin during mass accumulation (Bardeen et al. 1972; Thorne 1974). However, chaotic accretion characterized by short-lived episodes with random orientations tends to dampen the BH spin toward average values of a · ≃ 0 . 2 (e.g., King et al. 2008).  \nChaotic accretion and related feeding mechanisms with low angular momentum gas are considered to promote the efficiency of BH mass growth in the early uni-  \nverse, due to moderate centrifugal support (Eisenstein & Loeb 1995) and a lower radiative efficiency as a result of spin down (King et al. 2008). However, we show that the majority of the early BH populations growing in dust-rich environments tend to exhibit high spins with a · ≃ 0 . 9, corresponding to ϵ rad ≳ 0 . 2. This suggests that BH growth is likely dominated by prolonged accretion episodes with coherent angular momentum, consistent with an idea pioneered by Volonteri et al. (2005), or a modest degree of anisotropy (Dotti et al. 2013; Dubois et al. 2014). Cosmological simulations focused on galaxy assembly suggest that BHs retain high spins through coherent accretion modes. This occurs once BHs are settled down to the centers of their host galaxies and then the BH spin direction is well aligned with the angular momentum of the hosts (Peirani et al. 2024). As a consequence, a significant fraction of AGNs are expected to launch radio jets and influence the BH-galaxy coevolution (e.g., Beckmann et al. 2024, see also Section 4.4).  \nOur conclusion regarding the rapid spins of BHs can be directly testable through future gravitational-wave observations with the space-based detectors such as LISA (e.g., Amaro-Seoane et al. 2023). Moreover, if these BHs frequently merge during galaxy coalescences leading to the LRD phase or similar activities observed in ultra-luminous infrared galaxies, such events could significantly contribute to a stochastic gravitationalwave background, detectable by pulsar-timing array experiments (Inayoshi et al. 2018; see also Agazie et al. 2023). Therefore, further exploration of the rapidly spinning, overmassive BHs in LRDs is needed in multiple aspects.', 'ACKNOWLEDGMENTS': "We greatly thank Luis C. Ho for emphasizing the significance of the Soltan-Paczy'nski argument and for dedicating time to an inspiring discussion during the spring festival holiday. We also thank Dale D. Kocevski for sharing their SED fit results for the two LRDs detected in X-rays, which are used in Figure 6. We also wish to thank Xiaoyang Chen, Anna-Christina Eilers, Joseph F. Hennawi, Haojie Hu, Kazumi Kashiyama, Vasily Kokorev, Masafusa Onoue, Elia Pizzati, and Yasushi Suto for constructive discussions. K. Inayoshi acknowledges support from the National Natural Science Foundation of China (12073003, 12003003, 11721303, 11991052, 11950410493), and the China Manned Space Project (CMS-CSST-2021-A04 and CMS-CSST-2021A06). This work is also supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (20H01939; K. Ichikawa)."}