Introduction

According to the hierarchical growth of cosmic structures, dense regions at high redshift collapse and merge into the mostmassive gravitationally bound objects in the local universe, i.e., galaxy clusters, which are characterized by more evolved galaxy populations than the field environment (e.g. Alberts et al. 2014). Therefore, galaxy evolution must have been accelerated in their ancestors, i.e., protoclusters (e.g. Overzier 2016;Chiang et al. 2017). In these regions, star formation is efficiently fueled by large amounts of gas infalling from the forming cosmic web (e.g. Umehata et al. 2019), and is likely boosted by the high rates of ⋆ fabio.vito@inaf.it galaxy interactions and mergers in these dense and unvirialized systems (e.g. Liu et al. 2023).

Radiative and mechanical feedback produced by gas accretion onto supermassive black holes (SMBHs) observed as active galactic nuclei (AGN) plays a fundamental role in regulating, and eventually hindering, further galaxy and SMBH growth in cluster members (e.g. Fabian 2012;Gilli et al. 2019;Gaspari et al. 2020). However, the effect of a dense environment on the triggering of nuclear activity at high redshift is still not well understood. X-ray observations are the best tools to investigate the incidence and physical properties of the AGN population in protoclusters, as bright X-ray emission is a reliable and nearly complete tracer of nuclear activity, even in the pres-ence of heavy obscuration (e.g., Brandt & Alexander 2015;Ivison et al. 2019). Dedicated X-ray programs with Chandra generally find enhanced AGN activity in protoclusters with respect to the field environment at similar redshift and local galaxy clusters (e.g., Lehmer et al. 2009bLehmer et al. , 2013;;Digby-North et al. 2010;Tozzi et al. 2022b, but see also Yang et al. 2018a;Macuga et al. 2019). These results support a scenario in which the large reservoirs of gas and the high rate of galaxy interactions promote the growth of SMBHs in the protocluster galaxy members, in addition to boosting the star-formation activity. Theoretical models (e.g. Hopkins et al. 2006) predict that these conditions favor fast, efficient, and possibly heavily obscured nuclear accretion. Most protocluster AGN are indeed characterized as being heavily obscured (e.g. Vito et al. 2020;Monson et al. 2023).These properties are typical of the peak phases of SMBH mass building, after which AGN feedback hampers additional galaxy and SMBH growth, eventually impacting the entire cluster's evolution. In addition, the AGN enhancement may also be an effect of galaxies in protoclusters being typically more massive than in the field environment (e.g., Monson et al. 2021), as luminous AGN are typically found in galaxy with large stellar masses (e.g. Yang et al. 2017Yang et al. , 2018b)).

Protocluster candidates are identified via detections of overdensities of galaxies, selected in many different ways (see Overzier 2016 and references therein). The identification of protocluster candidates as overdensities of dusty star-forming galaxies (DSFGs), Lyα emitters (LAEs),or Lyman-break galaxies (LBGs) are among the most efficient techniques up to z ≈ 8 (e.g. Laporte et al. 2022;Morishita et al. 2023). Recently, the high angular resolution and sensitivity of ALMA allowed the identification of two extremely massive and star-forming overdensities of DSFGs,the Distant Red Core (DRC) at z = 4.0 (e.g. Oteo et al. 2018;Ivison et al. 2020), and SPT 2349-56 at z = 4.3 (e.g. Miller et al. 2018;Hill et al. 2020Hill et al. , 2022)), discovered originally by Herschel and the South Pole Telescope, respectively (Vieira et al. 2010;Ivison et al. 2016) . The cores of these structures extend to a few hundred kpc in projection, and are unique in terms of overdensity, total gas mass, and SFR density. Based on cluster evolutionary models and simulations, Oteo et al. (2018) and Hill et al. (2020) argued that DRC and SPT 2349-56 are the likely progenitors of ≈ 10 15 M ⊙ Coma-like clusters in the local universe. DRC consists of at least 13 spectroscopically identified z ≈ 4.0 DSFGs with individual SFRs in the range of 50 -3000 M ⊙ yr -foot_0 , for a total SFR ≈ 6500 M ⊙ yr -1 . A total molecular gas mass M H 2 > 10 12 M ⊙ was estimated from the [C I](1-0) emission lines detected from the protocluster members.

SPT 2349-56 was identified with ALMA observations as an overdensity of 30 galaxies spectroscopically confirmed at z ≈ 4.3 via the detection of [C II] and CO(4-3) emission lines (Miller et al. 2018;Hill et al. 2020;Rotermund et al. 2021). Among them, 21 objects are detected in sub-mm/mm continuum emission (850 µm, 1.1 mm, or 3.2 mm; Hill et al. 2020), and in this paper we refer to them as DSFGs. The protocluster members are located in a massive core with radius R ≲ 20 arcsec (≲ 100 kpc), and two smaller components at ≈ 0.9 arcmin (≈ 0.75 Mpc) and ≈ 3.8 arcmin (≈ 1.5 Mpc) from the center of the main overdensity. Numerical simulations predict that the galaxies in the central core of SPT2349-56 will eventually merge into the brightest cluster galaxy of the descendant structure (Rennehan et al. 2020). Similar arguments as those used for DRC return total values of S FR ≈ 8000 M ⊙ yr -1 and M vir ≈ 10 13 M ⊙ . Notably, the derived SFR density is a factor of ten larger than the most extreme values found in simulations at the same redshift (Hill et al. 2020). A total gas mass of M gas ≳ 3 × 10 11 M ⊙ is estimated from the luminosity of the carbon monoxide emission lines (Hill et al. 2020).

Using deep optical/near-IR observations, Apostolovski et al. (2023) and Rotermund et al. (2021) identified additional members of the SPT2349-56 core as LAEs (9 spectroscopically confirmed galaxies, although 6 of them are marked as "tentative" in that work) and LBGs (4 galaxies), 1 respectively. One of the tentative LAE is the counterpart of a sub-mm continuum detected galaxy with neither [C II] nor CO(4-3) emission in Hill et al. (2020), increasing the number of spectroscopically identified DSFGs in the structure to 22. Among the four LBGs in the protocluster core, two are likely counterparts of the DSFGs, named C2 and C17 (Hill et al. 2022), one has been found by Rotermund et al. (2021) to be a weak [C II] emitter not included in the Hill et al. (2020) sample, and one is the counterpart of a LAE. Thus, accounting for the few galaxies selected with multiple methods, the total number of individual and spectroscopically confirmed protocluster members is 38. Chapman et al. (2023) detected bright radio emission (L 1.4GHz,rest = (2.2 ± 0.2) × 10 26 W Hz -1 ) with spectral index α = -1.45 ± 0.16, where F ν ∝ ν α , from the inner region of the SPT2349-56 core, likely of AGN origin. The spatial resolution of the radio observations prevented the secure identification of the optical/IR couterpart, as a few protocluster members are consistent with being the host of the radio source, and none of them shows clear AGN features at other wavelengths. Still, mainly based on its large mass, Chapman et al. (2023) proposed that one such galaxies, referred to as C6 in our work following the Hill et al. (2020) naming convention, is the AGN host.

Due to their extreme properties, these two protoclusters, at similar redshift and identified via similar selection techniques, are unique testbeds to study the link between the availability of huge reservoirs of gas in high-redshift overdense environments, and SMBH growth in the galaxy members. Vito et al. (2020) used Chandra observations (140 ks) to investigate the AGN content of DRC, and identified two obscured AGN among 13 DSFGs. These are the two brightest, gas-rich, most strongly star-forming members of the protocluster, and are possibly in a merger phase, as derived from a high angular-resolution ALMA observation of one of them. In particular, the X-ray brightest AGN (L 2-10 keV = 2.7 +8.9 -1.8 × 10 45 erg s -1 ), namely DRC-2, has remarkable properties. It is as X-ray luminous as optically selected QSOs at all redshifts, but, in contrast to those, obscured by Compton-thick gas column densities, similar to, but even more extreme than, other populations of luminous obscured QSOs such as hot dust-obscured galaxies (e.g., Vito et al. 2018a).

In this work, we present new Chandra observations (200 ks) of SPT2349-56. Our goals are to probe the population of AGN and their physical properties in the extremely gas-rich and dense environment of SPT 2349-56, and to study more generally the effect of an overdense environment on SMBH growth in the early universe by comparing the AGN content in z ≈ 4 gas-rich protoclusters with lower redshift structures and blank fields. Errors are reported at 68% confidence levels, while limits are given at 90% confidence levels. We refer to the 0.5 -2 keV, 2 -7 keV, and 0.5 -7 keV energy ranges as the soft band, hard band, and full band, respectively. We assume solar metallicities and abun-F. Vito et al.: Fast and obscured SMBH accretion in the SPT2349-56 protocluster at z = 4.3 dances (Anders & Grevesse 1989), and adopt a flat cosmology with H 0 = 67.7 km s -1 and Ω m = 0.307 (Planck Collaboration et al. 2016).

Data analysis

In this section, we describe the reduction of the Chandra observations ( § 2.1) and the source detection procedure ( § 2.2).

Data reduction

We observed the SPT2349-56 protocluster with Chandra for a total of 200 ks, split among 9 pointings (see Tab. 1). The protocluster core was placed at the aimpoint of the ACIS-S detector, where the Chandra sensitivity is maximum, and all of the confirmed or candidate protocluster members are covered by the back-illuminated S3 chip. We reprocessed the Chandra observations with the chandra_repro script in CIAO 4.15 (Fruscione et al. 2006),foot_2 using CALDB v4.10.4,foot_3 and setting check_vf_pha=yes, since observations were taken in Very Faint mode.

In order to correct the astrometry of the Chandra observations, we performed source detection on each pointing with the wavdetect script with a significance threshold of 10 -6 , and then used the wcs_match and wcs_update tools to compute and apply the astrometric offsets with respect to a reference catalog. First, we applied a relative astrometric correction to each pointing using OBSID 25267 (i.e., the pointing with the longest exposure, see Tab. 1) as reference. Only point-sources with PSF size ≤ 3 arcsec and with ≥ 5 detected counts are considered. Then, we mapped the individual observations onto OBSID 25267 and merged all of the observations with the reproject_obs tool. We repeated the source-detection procedure on the merged dataset, but this time we matched the detected point sources to the GAIA DR3 catalog (Gaia Collaboration et al. 2023) 4 , in order to derive and apply the absolute astrometric correction factors. Only two X-ray point sources could be matched with GAIA objects on the ACIS-S3 chip, and the average spatial offset is 0.15 arcsec. This value can be considered the systematic spatial uncertainty of the X-ray dataset.

We obtained images and exposure maps with the repro-ject_obs tool, while we extracted spectra, response matrices, and ancillary files of the detected sources (see § 2.2) from individual pointings using the specextract tool and added them using the mathpha, addrmf, and addarf HEASOFT toolsfoot_5 , respectively, weighting by the individual exposure times. Ancillary files, which are used to derive fluxes and luminosities, were aperture corrected by setting via the correctpsf parameter in specextract.

Source detection and X-ray photometry

We assessed the detection of the SPT2349-56 protoclusters member candidates (Hill et al. 2020(Hill et al. , 2022;;Rotermund et al. 2021) in the soft, hard, and full bands using the binomial nosource probability (Weisskopf et al. 2007;Broos et al. 2007) where S is the total number of counts in the source region in the considered energy band, B is the total number of counts in the background region, N = S + B, and p = 1/(1 + BACKS CAL), with BACKS CAL being the ratio of the background and source region areas. The source counts are extracted from circular regions with R = 1 ′′ , whereas the background counts are measured from nearby regions free of evident X-ray sources. We checked that reasonably different choices of extraction regions returned consistent results. We defined (1 -P B ) > 0.99 as the detection threshold, a value often used to assess the X-ray detections of objects with pre-determined positions (Vito et al. 2019(Vito et al. , 2020)). Due to the small projected distances between the protocluster members, especially in the core of the structure, the extraction regions may overlap. In such cases, we assign each detected photon to the nearest galaxy, to avoid double counting. Two protocluster members are detected significantly in Xrays: sources C1 (1 -P B > 0.999 in both the hard and full bands) and C6 (1 -P B = 0.999 and 0.994 in the hard and full band, respectively), which are among the spectroscopically identified DSFGs in the core region of SPT2349-56 (Hill et al. 2020). Fig. 1 and 2 present X-ray cutouts of these two sources.

Net counts, which are reported in Tab. 2, were computed in the same R = 1 ′′ region used for the detection for C6, whereas we used a larger region with R = 1.5 ′′ for C1, given its relatively bright emission. We also derived the hardness ratios HR = (H-S ) H+S , where S and H are the net counts in the soft and hard band, respectively, and the corresponding effective power-law photon indices, following the procedure of Vito et al. (2019), which are also reported in Tab. 2.

We visually inspected all of the remaining secure protocluster members or member candidates presented by Hill et al. (2020Hill et al. ( , 2022)); Rotermund et al. (2021); Apostolovski et al. (2023). Three X-ray photons clustered on three contiguous pixels on top of the galaxy named LBG2 in Rotermund et al. (2021) are detected in the full band, suggesting that this object might be a sub-threshold X-ray source.

Results

In this section, we report the results obtained from the X-ray observations of SPT2349-56. In § 3.1 and § 3.2 we present the results of a spectral analysis of the two detected X-ray sources, C1 and C6, respectively. In § 3.3, we used the X-ray emission of these two galaxies together with the available optical-to-mm photometry to estimate the physical parameters of the AGN host galaxies via a spectral energy distribution (SED) fitting procedure. In § 3.4 we investigate possible evidence of low-rate SMBh accretion in the individually undetected galaxies of the structure via an X-ray stacking analysis.

A&A proofs: manuscript no. SPT2349_Xray Table 2. Positions and X-ray photometric properties of the two members of the protocluster detected with Chandra (see § 2.2).

) (11) C1 23:49:42.65 -56:38:19.4 < 4.8 19.1 +4.8 -4.1 20.5 +5.0 23:49:42.84 -56:38:25

X-ray spectral analysis of C1

The relatively bright hard-band detection of C1 with no associated soft-band emission and the resulting flat effective photon index (Tab. 2) are strongly indicative of the presence of a large column density of gas (N H ) obscuring this high-redshift galaxy. We performed a basic spectral analysis with XSPEC v.12.13 (Ar-naud 1996) 6 to measure physical quantities, such as the N H , the observed fluxes, and the intrinsic luminosity. Following the anal-ysis of DRC-2 (Vito et al. 2020), we used the MYTorus model (Murphy & Yaqoob 2009), accounting for the Galactic absorption (Kalberla et al. 2005), and fixing Γ = 1.9, the normalizations of the scattered and line components to that of the transmitted component, and the inclination angle Θ = 90 degrees. Therefore, the only two parameters left free to vary were N H and the intrinsic powerlaw normalization. Fig. 3 reports the observed and response-corrected spectrum and best-fitting model, highlighting the hardness of the source, as typically found for heavily obscured AGN.

Our best-fitting model returns N H = 2.4 +2.3 -1.2 × 10 24 cm -2 , implying that C1 is a Compton-thick AGN. The observed flux F 0.5-7 keV = 2.4 +4.9

-1.5 × 10 -15 erg cm -2 s -1 corresponds to an absorption-corrected, rest-frame luminosity L 2-10 keV = 2.2 +4.5 -1.4 × 10 45 erg s -1 , that is, ≈ 1 dex larger than the break luminosity of the X-ray luminosity function at that redshift (e.g., Ueda et al. 2014;Aird et al. 2015;Vito et al. 2018b). In Fig. 4, we compare the column density and luminosity of C1 from the best-fitting model with other populations of AGN over wide redshift ranges. C1 has an X-ray luminosity typical of bright optically selected blue and red QSOs, which, however, are typically unobscured or, at most, obscured by Compton-thin column densities of gas (e.g. Just et al. 2007;Martocchia et al. 2017;Lansbury et al. 2020). C1 is even more obscured than hot dust-obscured galaxies (Hot DOGs; e.g., Stern et al. 2014;Vito et al. 2018a), which are often considered as representative of an extreme phase of galaxy and SMBH growth, while X-ray selected Compton-thick AGN and DSFGs have significantly lower luminosities. Intriguingly, C1 and DRC-2, both selected as luminous and obscured AGN in z ≈ 4 gas-rich protoclusters, share the same position in Fig. 4. Only two other AGN with similar X-ray luminosities have been discovered in other protoclusters (Ivison et al. 2019;Tozzi et al. 2022b), but they are unobscured or, at most, mildly obscured, sources. In particular, Ivison et al. (2019) detected broad Hα emission from the luminous AGN hosted in HATLAS J084933.4+021443, a DSFG in a z = 2.41 protocluster, which is surprising considering the expected high extinction in such a dust-rich galaxy.

The X-ray luminosity of C1 translates into a bolometric luminosity L bol ≈ 8 × 10 46 erg s -1 according to the bolometric correction of Duras et al. (2020). Assuming that the SMBH powering C1 is accreting at the Eddington limit, we can place a lower limit on its mass of M SMBH ≈ 6.5 × 10 8 M ⊙ (but see the caveats raised by King 2024 about this approach). Therefore, either C1 is accreting at super-Eddington rate, or it has already accumulated a significant fraction of the SMBH mass characterizing AGN in the centers of local galaxy clusters. Similar conclusions were also drawn for DRC-2 by Vito et al. (2020).

We note that the spectrum of C1 can be fitted equally well with a pure reflection model by multiplying the intrinsic powerlaw continuum by a constant equal to zero. In this case, we obtain a similar value of N H and a factor of ≈ 2 higher luminosity. We also tried leaving Γ free to vary, but the fit cannot constrain its value.

A note on the possible foreground nature of the X-ray emission

Rotermund et al. ( 2021) reported the spectroscopic identification of a foreground galaxy at z = 2.54 along the line of sight of C1. Based on considerations on the blue optical colors, magnitudes, and [O III] 5007Å emission-line width, they provide an upper limit on its stellar mass of 1.6 × 10 9 M ⊙ . If this were the host galaxy, the spectral analysis would return N H = 1.1 +0.6 -0.5 × 10 24 cm -2 and L 2-10 keV = 4.1 +5.4 -2.2 × 10 44 erg s -1 . Given the well-known relation between AGN actvity and stellar mass of the host galaxies (Xue et al. 2010;Lusso et al. 2011;Wang et al. 2017;Yang et al. 2017Yang et al. , 2018b, e.g.), e.g.), it is highly unlikely that such a small object host a moderately luminous AGN. Moreover, its rest-frame UV spectrum present no indications of AGN activity (Rotermund et al. 2021). Therefore, in this paper we assume that the X-ray source is hosted by the DSFG at z = 4.3.

X-ray spectral properties of C6

Due to the low number of detected X-ray counts, we did not attempt to perform a spectral fit of C6. Instead, we assumed a simple Γ = 1.9 power-law as intrinsic spectrum, and estimated N H > 4.0×10 23 cm -2 in order to reproduce the observed value of hardness ratio, accounting for the proper instrumental response and PSF correction by using the ancillary and response file extracted at the position of C6 (see Sec. 2.1). We used this model to estimate the observed flux and the intrinsic luminosity of C6 (Tab. 2). Based on these loose constraints, we conclude that C6 confidently is a heavily obscured AGN, although possibly not as extreme as C1.

The X-ray detection of C6 strongly supports that this galaxy is the host of the radio AGN discovered by Chapman et al. (2023) with . The ATCA 2.2 GHz contours presented by Chapman et al. (2023) are reported in Fig. 2 in cyan. The limit on the X-ray luminosity of C6 is consistent with the relation between radio and X-ray emission from radio-loud AGN (e.g., Fan & Bai 2016;D'Amato et al. 2020;Mazzolari et al. 2024).

SED fitting

We used CIGALE v2022.1 (e.g. Boquien et al. 2019;Yang et al. 2022) to fit the SEDs of the two X-ray selected AGN in SPT2349-56. Fig. 5 presents the best-fitting SED models, and the resulting physical parameters are reported in Tab. 3. CIGALE produces a SED model for every combination of the input parameters, convolves it with the filters corresponding to the utilized photometric points, and computes the likelihood exp(-χ 2 /2) of every model in a Bayesian framework. Then, it computes the marginalized probability distribution function of each physical parameter based on the likelihood of all models, and returns the mean and the standard deviation, which can be considered as the estimated value and associated uncertainty.

We used the optical-to-mm photometric points from Gemini-GMOS, FLAMINGOS-2, HST-WFC3, Spitzer-IRAC, and ALMA presented by Hill et al. (2020Hill et al. ( , 2022)); Chapman et al. (2023). Upper limits at 3σ are adopted here for filters in which the sources are not detected. We fitted the SEDs using simple stellar populations from Bruzual & Charlot (2003) and a delayed star-formation history with optional late burst and a Chabrier (2003) initial mass function. We also accounted for nebular emission, dust attenuation with a modified Calzetti et al. (2000) law, and dust thermal emission using the Draine et al. (2014) templates. In particular, CIGALE treats consistently dust attenuation and re-emission, thus conserving the total energy. A summary of the grid values used for SED fitting is reported in Appendix A.

Since we are fitting X-ray selected AGN SEDs, we also included the CIGALE AGN module based on Stalevski et al. (2016) and the X-ray module of Yang et al. (2020Yang et al. ( , 2022)). The Xray module is especially useful to constrain the AGN component in the SEDs. In fact, bright X-ray emission is largely dominated by the AGN flux, and CIGALE uses the well-known correlation between the AGN intrinsic luminosities at rest-frame UV and X-ray wavelengths (e.g., Just et al. 2007) as a prior to constrain the AGN intrinsic optical/UV luminosity. This procedure is particularly valuable in the presence of obscuration, as in this case the AGN optical/UV emission is strongly suppressed. The X-ray module includes the X-ray emission from binaries and hot gas that depends on the SFR and stellar mass, although it does not include shocks that a starburst can have. These contributions are expected to be negligible at the observed X-ray luminosities (e.g. Lehmer et al. 2016Lehmer et al. , 2019)). The fitted X-ray fluxes used in the fitting procedure correspond to the X-ray models discussed in Sec. 3.1 and 3.2, and have been corrected for absorption, as required by CIGALE.

The optical photometry of C1 is contaminated by a spectroscopically identified foreground galaxy at z = 2.54 (Rotermund et al. 2021). Following Hill et al. (2022), we thus consider the optical fluxes as upper limits for source C1. The SED-fitting procedure for this object returns an AGN bolometric luminosity L bol = (1.9 ± 0.7) × 10 47 erg s -1 , which is slightly higher than the value estimated in Sec. 3.1. The large uncertainties are probably due the fact that only the X-ray point provides an anchor for the AGN component, as only upper limits are used for the optical photometry and the rest-frame mid-IR emission is not sampled by the available datasets. CIGALE returns a star-formation rate averaged over the last 100 Myr of SFR 100Myr = 228 ± 140 M ⊙ yr -1 . This value is significantly lower than the SFR reported by Hill et al. (2020), who estimated it from the FIR luminosity and thus on similar timescales, but did not account for the AGN component, which can contribute significantly to the total IR luminosity (e.g. Di Mascia et al. 2021;McKinney et al. 2021). The stellar mass M * = (3.5 ± 2.8) × 10 10 M ⊙ yr -1 is nominally lower than the estimate of Hill et al. (2022) , but consistent within the large uncertainties.

Considering C6, we corrected the observed X-ray flux assuming the lower limit we could place on N H in Sec. 3.2. Thus, the contribution of the AGN component to the best-fitting model for this source might have been underestimated, as higher intrinsic X-ray fluxes would correspond to the observed ones for higher values of N H . We further assumed that the radio emission detected by Chapman et al. (2023) in the core region of SPT2349-56 is entirely associated with the X-ray AGN hosted in C6, and thus when fitting this source we considered also the radio fluxes and upper limits from ASKAP, MeerKAT, and ATCA observations presented in that work. This addition required us to employ the CIGALE radio module, that models non-thermal radio emission from star formation and AGN. The CIGALE fit requires a moderately luminous AGN (L bol = (3.3 ± 0.2) × 10 45 erg s -1 ), which is sub-dominant at all frequen- cies, except for the X-ray and radio bands. In particular, C6 is a radio-loudfoot_7 AGN, as also discussed by Chapman et al. (2023), with R = 98 ± 9. We obtain SFR 100Myr = 263 ± 77 M ⊙ yr -1 and M * = (3.6 ± 1.3) × 10 10 M ⊙ yr -1 . Both of these values are significantly lower than those reported by Hill et al. (2020) and Hill et al. (2022). In particular, we note that the stellar-mass value that we find is more consistent with the dynamical mass of ≈ (2 -5) × 10 10 M ⊙ estimated by Chapman et al. (2023).

The stellar masses of these two galaxies correspond roughly to the break mass of the galaxy stellar mass function at z ≈ 4 (e.g. Song et al. 2016;Weaver et al. 2023). Comparing to the SED fitting results of Monson et al. (2023) on X-ray selected AGN in the SSA22 protocluster at z = 3.09, the AGN in SPT2349-56 on average are hosted in slightly less massive, but significantly more star-forming galaxies. In fact, Monson et al. (2023) found that most of the AGN in SSA22 are located below the main sequence, while few of them are consistent with it within the uncertainties on mass and SFR. Instead, according to our findings, both C1 and C6 are consistent with being main-sequence or even starbursting galaxies (e.g. Khusanova et al. 2021;Popesso et al. 2023). The AGN in SPT2349-56 might be in an earlier stage of galaxy evolution than those in SSA22, and are probably still in the peak phase of stellar and BH mass assembly. We also note that the SFRs of C1 and C6 averaged on a shorter timescale, i.e. 10 Myr, and the instantaneous SFRs returned by CIGALE are even higher than the values reported above (see Tab. 3). This is due to the SFHs of these galaxies favoring a recent and short burst of star formation.

X-ray stacking analysis

We performed an X-ray stacking analysis on the protocluster members to check for sub-threshold X-ray emission and constrain the average X-ray luminosity. We used the CIAO wavdetect and dmfilth tools to identify detected X-ray sources, and replace them with Poisson noise sampled from nearby regions. We

Table 3. Best-fitting physical parameters of C1 and C6 obtain via SED fitting.

(1.9 ± 0.7) × 10 47 (3.3 ± 0.2) × 10 45 Radio Loudness -98 ± 9

Notes. The rows report the best-fitting SFRs averaged over 100 Myr and 10 Myr, the instantaneous SFRs, the stellar masses, and the AGN bolometric luminosities.

then used the dmcopy tool to cut thumbnails of the X-ray images and exposure maps centered at the positions of the protocluster galaxies, excluding C1 and C6 that are detected individually. We summed them separately in the soft, hard, and full bands. We note that all of the protocluster members are within ≈ 3 arcmin from the average aim point of the observations, such that we do not expect a strong variation of the PSF at their positions. The sum of the counts in a R = 2 pixel (i.e., ≈ 1 arcsec) region around the centers of the stacked images divided by the average values of the stacked exposure maps in the same regions returns the stacked count rates in the three bands. We estimated the background level from nearby regions in the stacked images. We assessed detection significance and net-count numbers following the procedure in § 2.2. We did not detect significant stacked X-ray emission from the protocluster members in any energy band. We repeated the procedure considering first all of the possible members (i.e., 97 objects), then only those identified spectroscopically (i.e., 36 objects), and finally only the DSFGs (i.e., 20 objects). Assuming obscured powerlaw emission with Γ = 1.9 and N H = 10 24 cm -2 , the lack of significant stacked signal for the sample of DSFGs corresponds to an average intrinsic X-ray luminosity < 2 × 10 43 erg s -1 . We estimate similar values for the other samples of stacked galaxies. We stress that the stacked exposure times are in the range 5 × 10 6 s to 1.9 × 10 7 s, depending on the stacked sample. Based on these results, we did not find evidence for widespread low-rate SMBH accretion in the structure, although we cannot exclude that some of the protocluster members host faint AGN.

Discussion

In this section we discuss some implications of the results presented in § 3. In particular, in § 4.1 we estimate the contribution of the diffuse gas in the AGN host galaxies, i.e., the interstellar medium, to the nuclear obscuration. In § 4.2 we estimate the incidence of AGN among the protocluster member galaxies, and we compare it with other protoclusters and expectations in the field environment. In § 4.3 we quantify the enhancement of luminous AGN discovered in z ≈ 4 gas-rich protocluster cores.

ISM obscuration

In the past decades, several works based on X-ray surveys established observationally that the fraction of obscured AGN increases significantly from the local universe up to at least z ≈ 4 -5 (e.g. La Franca et al. 2005;Treister & Urry 2006;Buchner et al. 2015;Lanzuisi et al. 2018;Vito et al. 2018b;Iwasawa et al. 2020;Peca et al. 2023). A possible explanation for that behaviour is that the contribution of the diffuse gas in the host galaxies (i.e., the interstellar medium; ISM) to the nuclear obscuration of AGN increases strongly from the local to the highredshift universe, due to the larger gas content and smaller sizes characterizing galaxies at early cosmic epochs that almost automatically produce larger gas column densities. This scenario has been tested both observationally and via numerical simulations (e.g. Gilli et al. 2014;Circosta et al. 2019;Trebitsch et al. 2019;D'Amato et al. 2020;Ni et al. 2020;Lupi et al. 2022;Vito et al. 2022). In particular, Gilli et al. (2022) developed an analytical model for the ISM obscuration that accounts for the ISM clumpiness with a distribution of cloud sizes, masses, and densities, and showed that it predicts well the measured evolution of the obscured AGN fraction up to z ≈ 4.

We tested whether the ISM in the two AGN discovered in SPT2349-56 can contribute significantly to their heavy nuclear obscuration, under simple geometrical assumptions. Following Sec. 4.1 of Gilli et al. (2022), we assumed a smooth distribution of the ISM and that the [C II] surface brightness traces the ISM density, which is distributed in a disk with an exponential profile. This is consistent with the ALMA high-resolution imaging of C1 presented by Hill et al. (2022), in which the galaxy appears as a nearly edge-on disk, and the [C II] light profile is best fitted with a Sersic model with index n = 1.07 ± 0.02, that is, close to an exponential profile. Gilli et al. (2022) showed that, in that case, the gas column density for a line-of-sight inclined by an angle θ is

where r 0 is the scale radius, which for a pure exponential disk can be expressed in terms of the half-light radius r hl = 1.678r 0 , ρ 0 = 2.8M gas 4hπr 2 hl is the central gas density, and 2h is the disk thickness. It is further assumed a typical thickness h = 0.15r hl (Gilli et al. 2022, and references therein) and no vertical gradient for the gas density. 2020), we consider M gas = 6 5 M H 2 to account for the mass of atomic gas in the galaxy. Therefore, assuming an edge-on configuration for C1, we obtain N θ=90 • H ≈ 1.2 × 10 24 cm -2 , while assuming the average viewing angle for random inclinations we get N θ=57.3 • H ≈ 4 × 10 23 cm -2 . Smaller inclinations would not be consistent with the nearly edge-on [C II] image of this galaxy, under the assumption of a disk geometry, which is consistent with recent ALMA observations of 4 < z < 5 DSFGs (e.g. Rizzo et al. 2021). These order-of-magnitude values are close to the column density measured for C1 from our X-ray spectral analysis (Tab. 2), confirming that the gas distributed in the host galaxy can contribute significantly to the nuclear obscuration. We note that the use of the [C II] profile as a proxy of the molecular gas extension is conservative, as that transition traces also atomic gas, which is typically more extended than the dense molecular phase.

The molecular gas mass and [C II] half-light radius of C6 are M H 2 = 3.7 × 10 10 M ⊙ and r hl = 1.30 kpc (Hill et al. 2020(Hill et al. , 2022)). Although the [C II] morphology is best fitted with a Sèrsic profile n = 0.78 (Hill et al. 2022), we considered an exponential profile for simplicity. C6 appears close to face-on in the [C II] imaging (Hill et al. 2022), and thus we use θ = 57.3 and θ = 0 as boundary values of its inclination, finding N θ=57.3 • H ≈ 1.1 × 10 24 cm -2 and N θ=0 • H ≈ 7 × 10 23 cm -2 . Therefore, also for C6 the nuclear obscuration observed in the X-ray band can be due partly or totally to the gas in the host galaxy. In particular, we note that the ISM in C6 can reach higher column densities than C1 at a given inclination angle because of its compactness, that only high-resolution ALMA imaging could probe.

We stress that in this section we do not aim to measure the ISM contribution to the nuclear obscuration estimated via the X-ray observations, but rather to check whether such contribution can in principle be significant. One key assumption of the computations above is that the ISM is distributed smoothly in the galaxies, while it is known to be clumpy. Thus, depending on the geometry and physical properties (e.g., size and mass distribution) of the individual clouds, the ISM column density can be significantly different from, and possibly much lower than, the values estimated in this section. However, due to the large total gas masses estimated for C1 and C6 and depending on the assumed sizes and masses of molecular gas clouds (e.g., Miville-Deschênes et al. 2017;Dessauges-Zavadsky et al. 2019), the cloud filling factors in these galaxies can be as large as 100%. Therefore, the assumption of a smooth ISM is reasonable for the order-of-magnitude computations of this section.

Incidence of AGN activity in SPT2349-56

We detected two X-ray AGN out of 38 known members, implying an overall AGN fraction of 0.05 +0.05 -0.03 , where uncertainties are based on the Jeffrey Bayesian credible interval for binomial proportions (e.g. Brown et al. 2001). However, different populations of galaxies are intrinsically characterized by different AGN incidences, such that any comparison with other protoclusters or blank fields should take the specific galaxy selection into account (e.g. Vito et al. 2023). Therefore, in the following we consider separately galaxies selected as DSFGs, LAEs, and LBGs.

Two out of the 22 galaxies selected as DSFGs in SPT2349-56 are X-ray selected AGN, corresponding to an X-ray AGN -0.07 +0.06 -0.03 -0.08 +0.11 -0.05 -USS 1558-003 2.53 < 0.19 ----0.04 +0.03 -0.02 0.02 +0.02 -0.01 SSA22 3.09 0.50 +0.14 -0.14 -0.22 +0.09 -0.07 -0.03 +0.02 -0.01 --DRC 4.002 0.15 +0.12 -0.07 ------SPT2349-56 4.3 0.09 +0.04 -0.08 < 0.36 -< 0.21 ---

Notes. Uncertainties (upper limits) are at the 68% (90%) confidence level, and have been computed as the Jeffrey Bayesian Intervals for binomial proportions (e.g. Brown et al. 2001).

Fig. 6. Fraction of X-ray selected AGN in a sample of protoclusters as a function of redshift. The different symbol shapes and colors correspond to different structures and selection methods of the parent galaxy population, as marked in the figure. Different symbols for the same structure are slightly shifted in redshift for clarity. We consider the parent samples of spectroscopically confirmed protocluster members when available (filled symbols), otherwise we consider all member candidates (empty symbols). We refer to Sec. 4.2 and Appendix B for the computation of these values and the relevant citations. For comparison with low-redshift, virialized systems, the filled and hashed yellow stripes represent the X-ray AGN fractions in galaxy clusters presented by Martini et al. (2009) and Bufanda et al. (2017), respectively.

fraction in such a galaxy population of 0.09 +0.08 -0.04 . This value is remarkably close to the AGN fraction in DRC (Vito et al. 2020) down to similar FIR luminosity limits, adding the AGN content to other physical properties in common between the two protoclusters, thus suggesting that they have been caught during a similar phase of their galaxy and SMBH evolution. Since SPT2349-56 and DRC share a similar selection and are located at similar redshifts, we use both of them jointly to improve the number statistics and estimate the fraction of AGN among DS-FGs in z ≈ 4 gas-rich protoclusters by considering all of their DSFG members together (35, four of which are X-ray AGN), finding an X-ray AGN fraction of 0.11 +0.06 -0.04 . These results are consistent with the typical ranges found for DSFGs (e.g. Alexander et al. 2005;Georgantopoulos et al. 2011;Wang et al. 2013;Shanks et al. 2021). However, such samples have typically lower redshift (i.e., z = 2 -3, where the cosmic AGN activity peaks; e.g., Aird et al. 2015) than SPT2349-56, such that a direct comparison may be misleading. This is due to the quite strong evolution of the cosmic AGN and DSFG populations, which is also dependent on the considered luminosity regimes (e.g. Ueda et al. 2014;Aird et al. 2015;Traina et al. 2024). For example, if the space density of the X-ray AGN selected AGN decreases from z = 2 to z = 4 more strongly than the density of DSFGs, as it appears from observational works, it will cause generally a decreasing AGN fraction at increasing redshift. In this case, observing similar AGN fractions at z = 4 as at z = 2 would suggest stronger positive environmental effects at high redshift, but this is currently quite speculative.

In order to factor out the cosmic evolution of the AGN population, the comparison should be made with a sample of DSFGs at similar redshift as SPT2349-56. We collected a total of 54 sub-mm-selected galaxies in the E-CDFS, COSMOS, and UDS fields from da Cunha et al. (2015), Scoville et al. (2016), andDudzevičiūtė et al. (2020), respectively, with photometric redshift 3.5 < z < 4.5 and overlapping with the available X-ray coverage in those fields (Civano et al. 2016;Xue et al. 2016;Kocevski et al. 2018), with sensitivities similar to, or deeper than, the Chandra observations covering SPT2349-56 and DRC. We found no match with the X-ray catalogs, corresponding to an observed AGN fraction among field DSFGs < 0.04. To test quantitatively if the AGN fraction in z ≈ 4 gas-rich protoclusters is consistent with the field value, we ran the Boschloo's exact test for a 2x2 contingency table. The null hypothesis is that the intrinsic AGN fractions in protoclusters and in the field are equal, with the observed difference due only to statistical fluctuations. Considering SPT2349-56 only, the test returns a probability of ≈ 5% to obtain only by chance a case at least as extreme as the observed ones (i.e., no AGN out 54 DSFGs in the field, and ≥ 2 AGN out of 22 DSFGs in SPT2349-56). Considering SPT2349-56 and DRC together (i.e., ≥ 4 AGN out of a total of 35 DSFGs), such probability decreases to ≈ 1%. This comparison points toward a higher AGN incidence among DSFGs in z ≈ 4 protoclusters than in the field at similar redshift, although the AGN content of additional and similar structures should be investigated to obtain a definitive proof. Moreover, we caution that the reference samples have been selected differently from observations performed at different sub-mm/mm frequencies and with different depths, and thus might have different flux, luminosity, and mass distributions from the DSFGs population in protoclusters.

Considering the entire sample of 30 protocluster members spectroscopically identified with ALMA via detection of the [C II] and CO(4-3) emission lines, thus including both DSFGs and sources undetected in sub-mm/mm continuum, the AGN fraction decreases slightly to 0.07 +0.06 -0.03 . None of the 8 spectroscopically confirmed Lyα emitter galaxies (LAEs) discovered by Apostolovski et al. (2023) in the SPT2349-56 structure is significantly detected in the X-rays, resulting in an upper limit on AGN fraction among that galaxy population of < 0.21, consistent with results from blank fields (e.g. Lehmer et al. 2009b;Digby-North et al. 2010;Zheng et al. 2010). We obtain no X-ray detection also of the 4 LBGs in the SPT2349-56 core (Rotermund et al. 2021), implying an AGN fraction of < 0.36.

In Fig. 6, we compare the X-ray AGN fractions of SPT2349-56 with those of a collection of other protoclusters covered with sensitive X-ray observations, as computed in Appendix B and reported in Tab. 4. The AGN incidence among DSFGs in SPT2349-56 is consistent with the results for other z > 2 protoclusters, although a couple of structures have significantly higher AGN fractions. Instead, the AGN fraction drops dramatically in virialized clusters at lower redshift. This behavior might be connected to the virialization of the structures or strong AGN feedback hindering the infall of large amounts of cold gas into galaxies, and thus the triggering of luminous nuclear activity. We note that the points in Fig. 6 are observed fractions, and we refer to Appendix B for a discussion of the caveats. A more in-depth investigation of the possible cosmic evolution of the AGN incidence in protoclusters would require taking several effects into account, among which are the different sensitivities of the multiwavelength observations, the dependence of SMBH accretion on the host-galaxy stellar mass, and the intrinsic cosmic evolution of the AGN population (e.g., Aird et al. 2015Aird et al. , 2018;;Yang et al. 2017Yang et al. , 2018b;;Zou et al. 2024), which can be controlled for by comparing with the field AGN incidence. Such analysis require, among other things, a proper assessment of the multi-band observation sensitivities across multiple extragalactic fields and of the different specific selections applied on such fields, as well as a consistent SED fitting analysis of the resulting large samples of SMGs. These tasks are beyond the scope of this paper, and we reserve it for a dedicated future work.

Enhancement of fast SMBH growth in z ≈ 4 overdensities of DSFGs

The obscuration level and luminosity of C1 are remarkably similar to those of DRC-2 (Fig. 4). To our knowledge, AGN with similar X-ray luminosities in protocluster environments have been detected only in two structures at z = 2.16 -2.41 by (Ivison et al. 2019;Tozzi et al. 2022a, see Fig. 4), and those are unobscured or at most mildly obscured objects. The detection of a luminous, Compton-thick AGN in the core regions of the only two z ≈ 4 protoclusters selected as overdensities of dusty starforming galaxies and covered by sensitive X-ray observations suggests that gas-rich and dense regions of the Universe at those epochs may promote the triggering of extremely fast SMBH growth in heavily obscured conditions.

To estimate the level of enhancement of luminous AGN in SPT2349-56 and DRC, we compare their space density with that of AGN in the field environment with similar luminosity and redshift. We assume that the two protoclusters are enclosed in spherical volumes with radii equal to the projected distances between the observed centers of the structures and the farthest spectroscopically confirmed members, which are ≈ 8.8 comoving Mpc ( cMpc) and ≈ 3.8 cMpc, respectively (Ivison et al. 2020;Hill et al. 2022). Accounting for the uncertainties on their estimated luminosities, we consider the two luminous AGN in SPT2349-56 and DRC as representative of the AGN population in the range log L X erg s -1 = 45 -46 in z ≈ 4 gas-rich protoclusters. The space density of such a population is then two divided by the sum of the volumes computed above; i.e., Φ prot AGN = 6.4 +8.5 -4.2 × 10 -4 cMpc -3 dex -1 , where the uncertainties account for the statistical errors on the number of objects (Gehrels 1986). The space density of log L X erg s -1 = 45 -46 at z = 4.15 in the field environment is Φ field AGN ≈ 5 × 10 -8 cMpc -3 dex -1 (Fig. 7; e.g., Gilli et al. 2007;Ueda et al. 2014;Aird et al. 2015;Vito et al. 2018b), corresponding to an expected number of 1.5×10 -4 luminous AGN in the considered volume. The Poisson probability of instead finding two AGN with such luminosity by chance only is negligible. This simple computation suggests that the triggering of luminous AGN in gas-rich protocluster environments at z ≈ 4 is enhanced by about four orders of magnitude with respect to the field environment at similar redshift with high significance.

The most uncertain quantities that enter in the estimate are the volumes of the two protoclusters. As a second and more conservative estimate, we assumed that these structures extend up to R = 28 cMpc. According to Muldrew et al. (2015), this is the average radius that encloses 90% of the stellar mass of protoclusters at z = 4 that form massive galaxy clusters at z = 0. In this case, we estimate Φ prot AGN = 1.0 +1.4 -0.6 × 10 -5 cMpc -3 dex -1 , which is still > 2 dex higher than the field value. The Poisson probability of finding two luminous AGN while expecting the number predicted by the field environment is negligible also in this case.

As a comparison, Tozzi et al. (2022b) found for the Spiderweb protocluster at z ≈ 2 an AGN enhancement of a factor of tens, depending on the AGN luminosity. This lower value might be due to the fact that z ≈ 2 protoclusters have often already consumed most of their gas, as suggested by the large fraction of passively evolving galaxies in the Spiderweb structure (Shimakawa et al. 2024), and to the overall cosmic evolution of the AGN space density, that peaks close to that epoch. The much higher space density of luminous AGN in SPT2349-56 and DRC than in the field at similar redshift can in principle be driven by the space density of the underlying galaxy population, which in protoclusters is enhanced with respect to the field by definition. For instance, Miller et al. (2018) estimated a SMG overdensity in SPT2349-56 of a factor > 1000. However, they considered only the central R = 130 kpc region, where their selection is complete at S 1.1 mm > 0.5 µJy. In that same region we detected C1, i.e., 1.0 +2.3 -0.8 X-ray luminous AGN, which is at least several million times larger than the expected number of 4 × 10 -8 similar objects in the field, assuming a spherical volume with that radius. This enhancement largely exceeds and thus is hardly driven by the overdensity level of the underlying galaxy population in the considered protoclusters. We stress that C1 is not an extreme galaxy in terms of stellar mass (see § 3.3), and thus the comparison with the overdensity estimated by Miller et al. (2018) for the entire population of DSFGs in the structure is fair. This result is only in apparent contrast with the lower significance found for the enhanced AGN fraction in protoclusters with respect to the field environment discussed in § 4.2, since in this section we focused on the high-luminosity regime only, and compared space densities rather than AGN fractions. Given the shape of the X-ray luminosity function (Fig. 7), such luminous AGN are extremely rare in the field, and finding two of them in small volumes corresponds to a large enhancement factor. In fact, the small enhancement of the overall AGN fraction coupled with the large enhancement of the space density of highluminosity AGN suggests that the AGN X-ray luminosity function in these protoclusters is flatter than in the field, as also found by Tozzi et al. (2022b) for the Spiderweb protocluster. These computations suggest that gas-rich overdensities of DSFGs at z ≈ 4 promote extremely luminous and obscured AGN activity. A larger sample of similar protoclusters is needed to confirm it with better statistics.

We note that Yang et al. (2018a) investigated the possible dependence of the average SMBH accretion rate density in samples of galaxies in the COSMOS field as a function of their environments up to 10 Mpc, finding no significant difference in the SMBH accretion power between overdense and field regions at a fixed stellar mass. However, they analysis is limited to z < 3 and the environments that they probed are not as dense as SPT2349-56.

The luminous AGN in the z ≈ 4 protocluster cores represent the phase of fast SMBH growth required to explain the masses of SMBHs in the centers of low-redshift galaxy clusters. Such AGN are likely caught just before the "blow-out" phase, when AGN feedback clears the line of sight of most of the obscuring material (e.g. Ivison et al. 2019), and eventually hinders star formation and further SMBH growth. X-ray observations of a larger sample of similar environments are required to investigate the AGN population in such structures and prove this scenario securely. The specific selection that led to the identification of SPT2349-56 and DRC appears to be key to identifying highredshift structures hosting such luminous AGN. In fact, other protoclusters selected as overdensities of Lyman-break galaxies via optical observations do not present strong evidence for the presence of such an AGN population, although they host an unusual large number of rest-frame UV bright galaxies (Toshikawa et al. 2024).

Summary and conclusions

We presented new Chandra observations of the z = 4.3 SPT2349-56 protocluster, which was identified as an extreme erg s -1 = 45 -46) and obscured AGN in gas-rich protocluster cores at z = 4.0 -4.3 (red circles) computed under two assumptions for the volumes of the structures (see Sec. 4.3), compared with the predictions of AGN X-ray luminosity functions at z = 4.15 in blank fields (Ueda et al. 2014;Georgakakis et al. 2015;Vito et al. 2018b). Gas-rich overdense environments at high redshift enhance the triggering of luminous AGN by 3-5 orders of magnitude. This is likely a physical effect, as it appears not to be simply driven by the large number of galaxies in the structures (see § 4.3). overdensity of DSFGs (Miller et al. 2018;Hill et al. 2020;Rotermund et al. 2021;Hill et al. 2022;Chapman et al. 2023). We summarize here our main results.

-We identified two X-ray detected AGN among the SPT2349-56 member galaxies, namely C1 and C6, which are among the most gas-rich galaxies in the system (Hill et al. 2020). We did not detect significant emission by stacking the X-ray data of the individually undetected galaxies, implying an average X-ray luminosity < 2 × 10 43 erg s -1 . See § 2 and § 3.4. -C1 is an extremely luminous (L 2-10 keV = 2 × 10 45 erg s -1 ), Compton-thick (N H = 2 × 10 24 cm -2 ) AGN. The X-ray luminosity translates into a bolometric power ≈ 10 47 erg s -1 , which is also confirmed via SED fitting. Assuming that the SMBH accretion is capped at the Eddington limit, we place a lower limit on its mass of ≈ 7 × 10 8 M ⊙ . Both its luminosity and obscuration level are similar to those of another AGN previously detected in the central region of DRC, a similar protocluster at z = 4 (Vito et al. 2020). Both of these AGN might have already accreted a significant fraction of the typical mass of SMBHs in the centers of local clusters at much later (> 10 Gyr) cosmic times. See § 3.1. -Due to the low number of detected X-ray photons, we can only place lower limits on the luminosity (L 2-10 keV > 2 × 10 43 erg s -1 ) and column density (N H = 4 × 10 23 cm -2 ) of C6, which is also a radio-loud AGN. See § 3.2. -Both C1 and C6 are hosted in galaxies with stellar masses ≈ 3 × 10 10 M ⊙ , which is close the break value of the galaxy stellar mass function at z = 4.3, and have star formation rates consistent with, or in excess of, the expectation of the main sequence of star-forming galaxies at that redshift. See § 3.3.

-Under simple, but reasonable, assumptions on the geometries of the host galaxies, we conclude that the ISM can in principle contribute significantly to the observed nuclear obscuration of both AGN, in agreement with previous works on high-redshift AGN, although that contribution can be lower in the case of highly clumpy medium. See § 4.1. -The X-ray AGN fraction among DSFGs in SPT2349-56 is about 10%, consistent with other z > 2 protoclusters, and in particular with DRC. The fraction is higher than the X-ray AGN incidence in DSFGs in the field environment at z ≈ 4. We could place only loose upper limits on the AGN incidence in LBGs and LAEs in SPT2349-56, due to their small number. See § 4.2. -Both SPT2349-56 and DRC, which share similar selection and physical properties, host highly luminous Comptonthick AGN, indicating the existence of a tight link between vigorous phases of star formation, fed by the availability of huge gas reservoirs, and high SMBH accretion rates in the densest environments at high redshift. Such luminous AGN probably represent the period of fast SMBH growth required to explain the presence of 10 9 -10 10 M ⊙ SMBHs in the central galaxies of local clusters. Under different assumptions about the volumes of these structures and comparing with the predictions of the X-ray luminosity function z = 4, we suggest that gas-rich and dense protoclusters at z ≈ 4 enhance the triggering of extremely fast SMBH accretion by a factor of 3-5 dex with respect to the field environment. This factor exceeds the galaxy overdensity level of the protoclusters, and thus is probably not merely driven by the large number of galaxies in the structures. Further X-ray observations of similar structures are needed to confirm this result. See § 4.3.

Our results demonstrate that sensitive X-ray observations with high angular resolution are crucial to identify AGN in highredshift protoclusters, which are characterized by large amounts of dust and gas, and thus heavy nuclear obscuration. In the next years, Chandra will play a leading role in this respect, by increasing the samples of high-redshift gas-rich protoclusters with the deep X-ray coverage required to investigate their AGN content. Future X-ray missions will then be crucial to obtain a complete view of the relation between overdense environments and SMBH growth at high redshift (e.g., Vito et al. 2023).