## Introduction

Observations carried out with the Wide-field Infrared Survey Explorer (WISE)1 recently led to the discovery of a population of hyper-luminous dust-obscured quasars with bolometric luminosities of Lbol 1013−1014L2,3, characterized by hot dust temperatures (Tdust >60 K) and very red mid-infrared (IR) colors. These quasars are thought to be powered by buried active galactic nuclei (AGNs), with accretion rates close to the Eddington limit4. There is a growing consensus that this rare population of extreme objects, often referred to as Hot Dust-Obscured Galaxies (Hot DOGs)2, represents a unique and short-lived stage of galaxy evolution in which both heating and ejection of dusty interstellar medium (ISM) through winds powered by the central AGN are caught in the act5 (the so-called blowout phase6,7), leading to an evolutionary transition from dusty starburst galaxies to ultraviolet (UV)-bright unobscured quasars, and finally to giant elliptical galaxies6.

Models predict that the growth of these massive star-bursting objects is driven by galaxy mergers8,9, which can deliver a significant amount of gas toward their centers, boosting their star formation rate (SFR) up to thousands of Myr−110, and feeding the growth of their supermassive black holes (SMBH), thereby powering the AGN activity8. However, while cumulative number counts in wide-field surveys suggest that Hot DOGs tend to lie in dense large-scale environments11,12, direct observations of the mechanisms responsible for their rapid growth are still sparse or only restricted to scales of a few tens of kpc13. On the other hand, recent works14,15,16 have demonstrated the potential of spectroscopic imaging studies across the environment of luminous quasars and protocluster galaxies, finding companion galaxies out to hundreds of kpc.

In this work, we report rest-frame UV and rest-frame far-infrared (FIR) spectroscopic imaging observations of the Hot DOG WISE J041010.60-0913056.2 (hereafter W0410-0913), obtained with the integral field spectrograph Multi Unit Spectroscopic Explorer (MUSE) at the Very Large Telescope (VLT), and with the Atacama Large Millimeter/Sub-millimeter Array (ALMA), respectively. W0410-0913 was originally selected from a WISE all-sky survey1 and, with a total FIR luminosity of LFIR ~ 2 × 1014 L17, and with large stellar18 (Mstar) and molecular gas19 (Mgas) masses of 1011 M, is one of the brightest, most massive and gas-rich galaxies in the distant Universe19,20. Extensive rest-frame FIR observations, tracing several molecular lines, have shown W0410-0913 to be located at a systemic redshift of z = 3.63118,21 (see details on the redshift determination in the subsection ALMA data in Methods), about only 1.7 Gyr after the Big Bang. Through the combined analysis of the Lyman-α transition of atomic hydrogen (at a rest-frame wavelength of 1215.67 Å) and the rotational transitions of carbon monoxide, these observations allow us to study both the circumgalactic environment surrounding the hyper-luminous galaxy, on scales of 100 kpc, as well as the morphological and kinematic structure of its ISM, on scales of a few kpc, providing information on the multi-scale processes that drive its evolution.

### Results

#### MUSE observations of the circumgalactic environment of W0410-0913

We have obtained deep MUSE observations of W0410-0913 (exploiting a total telescope time of 2 h) as part of an observational campaign aimed at obtaining spatially resolved rest-frame UV spectroscopy observations of a Hot DOG and its halo gas, also called circumgalactic medium (CGM). Due to its wide field of view (FOV) and high sensitivity, MUSE is a suitable instrument to explore the surrounding environment of luminous sources in the distant Universe, and it has carried out observations of both morphology and kinematics of the gas distribution around luminous distant quasars22,23 through direct imaging of the Lyman-α emission, which is mainly produced through recombination radiation following photo-ionization powered by bright sources24.

The spectrum of W0410-0913 observed with MUSE shows highly attenuated rest-frame UV emission. We detect very broad (full widths at half maxima of about FWHM~2800 km s−1) blueshifted components of Nvλ 1240 and C ivλλ 1548,1550, both offset by more than 2000 km s−1 with respect to the systemic redshift, likely tracing fast AGN-driven outflows5,13,25. Also, we find a narrow (FWHM ~ 400 km s−1) Lyman-α line, which, similarly to the narrow component of the optical [OIII] λ 5007 line5, is blueshifted by about 1400 km s−1. We find evidence for a small Lyman-α nebula associated to this line, with a maximum projected linear size of about 30 kpc. This is at odds with recent observations of several Lyman-α nebulae showing sizes up to hundreds of kpc detected with MUSE around UV-bright quasars, with typical Lbol even lower than W0410-0913 and shorter exposure times22,23. Our finding is rather to be ascribed to the large amount of dust in W0410-0913, which prevents Lyman-α and UV photons to escape the galaxy and illuminate the CGM, suggesting that extreme physical conditions of the ISM can have a profound impact on the properties of the CGM accessible through Lyman-α.

The lack of extended diffuse Lyman-α emission dominating the field at wavelengths close to the systemic redshift of W0410-0913 makes it possible to efficiently search for Lyman-α-emitting companion galaxies in the MUSE datacube, and thus provides well-suited conditions to explore the surrounding environment of a Hot DOG on circumgalactic scales, informing on the physics that drives the rapid evolution of this rare population of objects.

Our MUSE observations (reaching a 2σ surface brightness (SB) limit of $$5\times 1{0}^{-19}{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}\,{{{{{{{{\rm{cm}}}}}}}}}^{-2}\,{{{{{{{{\rm{arcsec}}}}}}}}}^{-2}$$ in a $$1\,\,{{{{{{{{\rm{arcsec}}}}}}}}}^{2}$$ aperture and in a single spectral channel) reveal a high overdensity of 24 Lyman-α-emitting satellite galaxies associated with W0410-0913 within the MUSE FOV, in a region of about 400 projected physical kpc (diameter-scale) centered around the central massive luminous object (see Fig. 1). A compilation of their Lyman-α flux maps is shown in Fig. 1a, providing information on their projected spatial distribution in the field. The adopted techniques and the analysis performed to search for these galaxies (as well as a statistical assessment of the reliability of their detections) are discussed in the subsections describing the MUSE data in Methods.

Archival images from the Hubble Space Telescope (HST) obtained with the WFC3 F160W filter (probing rest-frame emission at around 0.3 μm) show that, despite the modest exposure (only 2400 s), ten among these sources (i.e., approximately 40% of the total) have rest-frame optical counterparts detected with a signal-to-noise ratio (SNR) higher than 4σ, and corresponding SFRs ranging between 12−100M yr−1 (computed using standard LUV/optical−SFR calibrations; see subsection Measure of the SFR from HST in Methods). Most of the remaining objects show only tentative individual detections at SNR of 2−3σ, but from a stacking analysis, we can infer an average SFR of about 3 M yr−1.

MUSE data show that the companion galaxies discovered around W0410-0913 have offset line-of-sight velocities, traced by the peak of Lyman-α in their spectra (see Methods, where resonant scattering effects are also discussed), in the broad range of about [−6.000; +10.000] km s−1 with respect to the systemic redshift of the central galaxy (see Fig. 1b, c), traced by its molecular gas. 75% of them (18 out of 24) show offsets in a smaller velocity range of about [−2.500; +1.500] km s−1 (see subsection The circumgalactic environment of W0410-0913 observed with MUSE in Methods; see Fig. 1c), suggesting that these satellites are experiencing a strong attraction by W0410-0913.

To quantify the physical significance of the revealed overdensity we compare the Lyman-α luminosity-weighted number density of the companion galaxies associated with W0410-0913, nLyE, with the expected value for field galaxies according to the Lyman-α luminosity functions measured in wide and deep surveys with MUSE26,27 (see subsection The physical significance of the overdensity in Methods for details on the computation and on the comparison sample). We find that the observed nLyE is a factor of ρCGM ~ 14 higher than the expected value (〈nLyE〉; see Fig. 2). This is higher than other reported observations of overdensities previously discovered through Lyman-α in the redshift range z ~ 2−728,29,30,31, including proto-clusters around radio-galaxies32,33 and multiple-quasar objects34, although a direct comparison with these systems is challenged by the differences in selection criteria, cosmological volumes, and sensitivity limits. Also, we note that the measured overdensity is consistent with cosmological predictions in the proximity of massive, luminous objects like W0410-0913, at the redshift of interest (see subsection The physical significance of the overdensity in Methods).

Altogether, our findings suggest that the circumgalactic environment surrounding W0410-0913 (probed by MUSE on projected physical scales of approximately 400 kpc) is one of the densest regions of the observed Universe. The number of gravitational encounters is expected to be enhanced in such a dense environment6,17,35,36, suggesting that galaxy mergers could be an efficient channel for the mass build-up of the massive luminous galaxy.

However, our MUSE observations cannot firmly confirm this scenario, since we do not detect direct unequivocal signs of mergers. A tentative indication of ongoing dynamical interactions is provided by the morphological and kinematic distribution of gas along the line-of-sight as traced by the Lyman-α emission in and around the companion galaxies, that we study by using commonly adopted flux-extraction techniques that enable one to recover flux of extended sources allowing for local kinematic variations37. Specifically, we find that the three-dimensional (3D) distribution of circumgalactic gas illuminated by the Lyman-α-emitters dynamically associated with the central luminous galaxy (i.e., with an offset velocity range of about [−2.500; +1.500] km s−1) is preferentially distributed over bridge-like shapes on scales up to 100 kpc, some of which appear to be aligned along the common axis with the central massive object (see Fig. 3). While these filamentary structures might be induced by the strong gravitational influence of W0410-0913, projection effects combined with the resonant nature of Lyman-α challenge this interpretation.

Further studies of the morphology and kinematics of the distribution of gas in the halo, tracing e.g., bright rest-frame FIR lines (like [CII]) with ALMA and rest-frame optical/near-IR lines with JWST, are needed to discern whether W0410-0913 is in the process of accreting its neighbors, similarly to what is done in highly interacting dense groups of massive galaxies in the local Universe through maps of atomic hydrogen filaments38.

Furthermore, the merging scenario is at odds with high-resolution spectroscopic imaging data of the internal gas structure in the ISM of the hyper-luminous central object.

### Discussion

ALMA observations of W0410-0913, zooming into the kpc-scale properties of its ISM, interpreted in the context of the CGM-scales studied with MUSE, challenge the scenario, suggested by models, that repeated galaxy encounters in the dense environment might have been efficient in building up the central galaxy. Indeed, multiple mergers in luminous systems are often shown to have a disruptive impact on the kinematics of the ISM17,35,44, which, in the case of W0410-0913, appears to be settled into a smooth rotation-dominated structure.

We speculate that a possible mechanism capable of solving this tension is that the mass growth of W0410-0913 could be sustained by minor mergers with relatively less massive companions, that do not significantly perturb the disk-like structure43,45. In this case, the accretion of discrete small satellites might have played a role in originating the observed high-velocity dispersion in W0410-0913 through gravitational instabilities. Also, minor mergers are usually invoked by models as an efficient channel of bulge formation at the core of massive disk galaxies46. In the case of W0410-0913, the inner peak (<1−2 kpc) of the CO(6-5) vrot(r) and the high central concentration of flux in the intensity map could be interpreted as a tracer of a bulge-like structure, that minor mergers might have contributed to form. However, we note the existence of an alternative (but not exclusive) scenario in which the energy injected by the prominent AGN activity into the surrounding gas might contribute to the observed turbulence throughout the disk, as suggested by similar observations of turbulent ISM revealed through spectroscopic imaging of molecular gas in another distant hyper-luminous obscured quasar13,47.

Altogether, the multi-wavelength and multi-scale observations presented here, obtained with some of the most powerful telescopes available (including the first observations targeting a Hot DOG with MUSE) provide key information on the cosmic assembly of a distant hyper-luminous hot-dust obscured galaxy. Data support that the mass build-up of W0410-0913 takes place in a dense circumgalactic environment, but we cannot identify robust signs of merger that could sustain the growth of the massive galaxy. Instead, we find that gravitational encounters, if any, can have only a modest impact on shaping the morpho-kinematic properties of the ISM. Our results serve as a benchmark to calibrate models of the rare and short-lived populations of Hot DOGs and progress in the understanding of their evolutionary transition towards luminous quasars and giant galaxies at cosmic noon.

## Methods

### Cosmology

Throughout the paper, we adopt a standard Λ cold dark matter (ΛCDM) cosmology, using the following parameters compatible with Planck results48: ΩΛ0 = 0.69, Ωm0 = 0.31, Ωb0 = 0.05, H0 = 67.7 km s−1 Mpc−1. Based on these parameters, at the redshift of W0410-0913 (z = 3.631), the angular scale is 1 = 7.39 kpc.

### MUSE observations

MUSE observations of W0410-0913 were carried out in February 2019 and January 2021, during the ESO Observing Periods P102 (Program 0102.A-0729) and P106 (Program 106.2184.001), respectively (see Data Availability Section). The collection of data in P102 was organized in one observing block (OB) composed of four exposures of 680 s each, for a total exposure time on target of 45 m (1 h of telescope time including overheads). Seeing conditions ranged between FWHM = 0.8−1.1, with airmass values of 1.1−1.3. In P106, the data acquisition was split into three exposures of 900 s each, with time on target (45 m) and telescope time (1 h) similar to the P102 program. Observations were carried out under good seeing conditions, FWHM = 0.8, and airmass  1.1. In both programs the exposures were slightly dithered by ±0.3, and rotated with respect to each other by 90 degrees, to reduce systematic noise.

We have processed the individual exposures from both datasets with standard data reduction techniques using the ESO-MUSE Pipeline Version 2.8.449. In detail, for each exposure, we used the MUSE_BIAS, MUSE_FLAT, and MUSE_WAVECAL pipeline recipes49 to perform bias subtraction, flat fielding, twilight and illumination correction, and wavelength calibration. In order to ensure an accurate relative astronomy, we registered the individual exposures using the position of point sources in the field. This step is needed since the spatial shifts introduced by the derotator wobble50 can cause offsets of a few arcsecs.

We performed the next stages of data reduction using the tools provided by the CubExtractor software package37 (Version 1.8), which has been specifically developed to improve the data quality for the detection of diffuse, low-SB emission in MUSE datacubes22,23,51,52. In detail, we improved the flat fielding by using the tool CubeFix37, which performs an additional correction to homogenize the illumination across the field and as a function of wavelength. We performed the sky subtraction using CubeSharp37, which can reach a better accuracy than the MUSE pipeline22. We then combined the corrected and sky-subtracted individual exposures from both datasets using an average 3σ-clipping. The final datacube samples the instrument FOV of $$1^{\prime} \times 1^{\prime}$$ in pixels of size 0.2. Each pixel contains a spectrum covering the wavelength domain 4750−9350 Å, and individual channels have a spectral resolution of 1.25 Å.

Our combined MUSE observations, comprising a total telescope time of 2h, reach a (2σ) SB limit of $$5\times 1{0}^{-19}\,\,{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}\,{{{{{{{{\rm{cm}}}}}}}}}^{-2}\,{{{{{{{{\rm{arcsec}}}}}}}}}^{-2}$$ in $$1\,{{{{{{{{\rm{arcsec}}}}}}}}}^{2}$$ aperture and in a single spectral channel (1.25 Å), at wavelengths adjacent to 5632 Å, corresponding to the expected Lyman-α in W0410-0913 (systemic redshift of z = 3.631; see next Sections). For reference, the SB limit on a pseudo-narrow band (NB) image of 30 Å (a common size for filters used in past NB surveys), obtained by collapsing 24 wavelength layers close to the location of Lyman-α, is $$2.8\times 1{0}^{-18}\,{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}\,{{{{{{{{\rm{cm}}}}}}}}}^{-2}\,{{{{{{{{\rm{arcsec}}}}}}}}}^{-2}$$ in $$1\,{{{{{{{{\rm{arcsec}}}}}}}}}^{2}$$ (2σ).

### The rest-frame UV spectrum of W0410-0913

In Fig. 5a, we show the observed rest-frame UV spectrum of W0410-0913, covering the whole MUSE wavelength domain (~1000−2000 Å rest-frame), extracted from a circular aperture with a radius of 1 (which maximizes the SNR) centered at the position of the luminous galaxy, based on its MUSE continuum image (obtained collapsing the full datacube). The spectrum shows a very faint rest-frame UV emission, with an average flux of 4.0(±3.2) × 10−19 erg s−1 cm−2 Å−1 computed at the rest-frame wavelength of 1300 Å. This corresponds to a monochromatic luminosity of $$\log (\lambda {L}_{\lambda }/[{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}])=43.{8}_{-0.7}^{+0.3}$$, which is orders of magnitudes lower than the expected value $$\log (\lambda {L}_{\lambda }/[{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}])=47.{3}_{-1.1}^{+1.1}$$, estimated through a standard bolometric correction relation for rest-frame UV luminosities53, using Lbol = 1.68 × 1014 L for W0410-0913, as computed from multiwavelength photometry20. This faint UV emission can be explained through dust attenuation caused by the large amount of obscuring ISM in the system17,18,19.

At the systemic redshift of W0410-0913, z = 3.631, traced by its molecular gas emission, we do not find any Lyman-α emission. The average flux, computed within a spectral window of Δv = 1500 km s−1 centered around the expected line center at 5632 Å, is consistent with zero within the uncertainties, ~3(±30) × 10−20 erg s−1 cm−2 Å−1 (see Fig. 5b), and by producing pseudo-NB images at the expected position of Lyman-α (collapsing the continuum-subtracted datacube along the wavelength axis using different spectral widths, from 5 to 30 Å) we do not find any sign of emission. However, a narrow emission line (FWHM~400(±80) km s−1), emitted at the same location of W0410-0913, is detected in the spectrum with a blueshift of about −1500(±150) km s−1 with respect to z = 3.631 (see Fig. 5). We interpret this line as a Lyman-α emission from W0410-0913, emitted at different velocities than the systemic redshift traced by the molecular gas. This interpretation is supported by the blueshift of about −1400 km s−1 (with respect to the systemic redshift adopted in this work) of the narrow component of the non-resonant line [OIII] λ 5007 from a rest-frame optical spectrum of W0410-0913 observed with KECK/NIRES5. These findings suggest that the rest-frame UV and optical narrow emission lines in W0410-0913 are emitted at different velocities than the molecular gas. An alternative, but less likely (because of the reasons discussed above) interpretation is that the narrow rest-frame UV emission line that we find in the MUSE spectrum traces Lyman-α emission produced by a nearby companion galaxy that is part of the overdense environment (see subsection The circumgalactic environment of W0410-0913 observed with MUSE), and is located along the line of sight of W0410-0913.

We compute the spatial extension of such emission using the tool CubEx (from the CubExtractor package37). CubEx is extensively used to perform automatic detection and extraction of extended sources in MUSE datacubes22,23, and it is based on a 3D extension of the connected-labeling-component algorithm with union finding, usually used in classical binary image analysis22. We feed CubEx with a sub-cube that we extract to have a wavelength range of Δλ~100 Å centered around the peak of Lyman-α emission (4388 Å), and from which we remove continuum sources by adopting a median-filtering approach. We also apply a spatial Gaussian filtering of 2 pixels (0.4) and a spectral smoothing with a Gaussian filter size of 1 layer (1.25 Å) to bring out extended but narrow features. Running CubEx with a SNR threshold for detection of SNR > 3 in individual datacube elements (voxels) we detect a nebula consisting of approximately 1600 connected voxels and a maximum spectral width, as defined in the 3D-segmentation-mask, of Δλ ~ 15 Å. From the optimally extracted image of the nebula, derived by applying the 3D-segmentation-mask to the datacube (see subsection A study of the 3D distribution of gas around W0410-0913, for details on the advantage of this method), we measure a spatial extension, defined as the maximum projected linear size, of about 30 kpc (see Fig. 6). This indicates that, under the reasonable assumption that such emission could be considered a Lyman-α nebula associated with the Hot DOG, it would be smaller than several nebulae observed with MUSE (usually with lower exposure times) around UV-bright quasars with lower Lbol22,23, pointing toward a significant effect of a dusty ISM on the detectability of Lyman-α nebulae.

In the MUSE spectrum of W0410-0913 we also clearly detect the Nvλ 1240 line emission and the Civλλ 1548,1550 doublet, both showing very broad line-profiles with FWHM~2800(±200) km s−1, computed through a non-linear least-squares fit of a Gaussian profile. Similarly to other luminous quasars and other Hot DOGs13,54, by computing the center of their Gaussian profiles, we measure large blueshifts for both Nv and Civ of about 2000(±150) km s−1 (see Fig. 5), corresponding to a redshift of z = 3.600 ± 0.005. This value is consistent with the first published estimate of the W0410-0913 optical redshift, based on shallow observations of Civ and Heiiλλ 1640 emission obtained with Keck/LRIS2. We note that the MUSE spectrum does not show the low SNR Heii emission present in the Keck/LRIS, as the line falls in a noisy region of the spectrum. Furthermore, similarly to the Nv and Civ broad lines in the MUSE spectrum, the optical [OIII] observed with KECK/NIRES (see above) has a broad component that is also very blueshifted, showing an offset velocity of 3000(±1000) km s−1 with respect to the narrow component5. Large blueshifts in the rest-frame UV and optical broad emission lines, like the ones observed in W0410-0913, are often interpreted as an indication of fast nuclear outflows55.

### The circumgalactic environment of W0410-0913 observed with MUSE

To study the environment of W0410-0913 over an area of about 450 kpc × 450 kpc (corresponding to the MUSE $$1^{\prime} \times 1^{\prime}$$ FOV), we search for Lyman-α-emitting companion galaxies in a continuum-subtracted sub-cube (produced through median-filtering) of about 380 Å centered around the expected Lyman-α emission from W0410-0913, corresponding to offset velocities of [−10.000; + 10.000] km s−1 with respect to its systemic redshift.

Our method can be summarized as follows. We inspect the whole wavelength domain by producing mini-cubes with spectral widths ranging from a minimum allowed number of 2 channels (2.5 Å; to avoid single-channel spikes of noise) to a maximum of 20 channels (about 25 Å; to allow for ~3 × typical FWHMs of Lyα-emitters at this redshift). We collapse each mini-cube along the wavelength direction to obtain pseudo-NB images that we scan through their whole spatial domain by measuring the aperture photometry of 1−sized regions centered around each pixel. We consider as candidate detections the regions with a significance of their measured flux of SNR > 3 (where the noise is extracted from the local background), we extract a spectrum, and we save their position (RA and DEC of the central pixel), central wavelength and spectral width. For multiple candidate detections arising from the same physical source, namely two or more outputs found at about the same location (within 0.5, corresponding to about half of the size of the PSF) and wavelength (within 4 Å), we keep the candidate that has higher SNR in the spectrum. We then perform a visual inspection of the pseudo-NB images and spectra of the candidate detections to make sure that there is no contamination by sky residuals. Also, for each candidate detection, we visually inspect the original non-continuum subtracted full datacube (covering the whole MUSE spectral domain) to check whether the candidate detections show the clear presence of multiple (at least two) emission lines at wavelengths not included in the sub-cube used for studying the environment of W0410-0913. In these cases, the sources would be line emitters at lower redshifts (emitting e.g., [OII] λ 3727,3729, Hβ, [OIII] λ 4959,5007, and Hα) and would not be relevant for our analysis. Through this analysis, we exclude from the list two sources emitting [OII] at z = 0.495 (Hβ and the [OIII] doublet are detected at redder wavelengths) and Hβ at z = 0.167 (with [OII], [OIII] and Hα detected in the full spectrum). We assume that the remaining sources, showing a single emission line in the sub-cube centered around the redshift of W0410-0913 are Lyman-α-emitting galaxies in the surroundings of W0410-0913.

We find 24 Lyman-α emitters within a box with a diameter of about 400 kpc centered around W0410-0913. This result is confirmed by an independent analysis that we perform using CubEx. Although CubEx is usually exploited to extract diffuse extended emission, it can be used to mimic a line finder by decreasing the threshold of the minimum allowed number of connected voxels and by increasing the threshold of SNR for the detection of individual voxels. In detail, by adopting a SNR > 4 threshold over a lightly smoothed cube (0.2 pixels in the spatial domain, and no filtering on the wavelength direction), we find the same number of sources of the method described above, with only tiny differences (0.3) in the spatial location of the peak of a few objects, likely due to the different pre-processing of the cube.

In Supplementary Figs. 1 and 2, we show the spectra of these sources (extracted from a region with a radius of 1) centered around the central wavelength of their Lyman-α, that we compute through a non-linear least-squares Gaussian fit. A selection of these spectra is shown in Fig. 7a. From the modeled Lyman-α peaks we measure the offset velocities with respect to the redshift of W0410-0913, and find values in the range [−6.000; + 10.000] km s−1, spatially distributed in the MUSE FOV as shown in Fig. 1b. We note that radiative transfer effects are known to shift the peak of Lyman-α emission in the spectra of star-forming galaxies even by a few hundreds of km s−1 compared to the systemic velocity56, though this effect is unlikely to have a dominant impact on the suggested interpretation of the observations, since we measure offset velocities up to thousands of km s−1.

We also produce pseudo-NB images of each object by collapsing the datacube across the wavelength direction within spectral ranges defined by [λcen − FWHM:λcen + FWHM], where the central wavelength λcen and FWHM are defined by the spectra shown in Supplementary Figs. 1 and 2. A composition of the derived pseudo-NB images, showing the projected spatial distribution of the Lyman-α-emitting galaxies around W0410-0913, is shown in Fig. 1a.

As shown in Fig. 1c, the distribution of offset velocities is centered around W0410-0913, confirming that the massive luminous galaxy is located at the dynamic center of its dense environment and dominates its gravitational potential. Most of the companions (18 out of 24) show offset velocities in a smaller range of about [−2.500; +1.500] km s−1. With the caveat of partial information provided by the measurement of projected line-of-sight velocities, it is likely that these galaxies are more closely interacting with the central massive object.

We dub the Lyman-α-emitters using alphabetically ordered letters from #A to #X, according to their velocity offsets, from the largest negative to the largest positive. We compute the Lyman-α fluxes of the companion galaxies by integrating the Gaussian model of their spectra, and convert them to Lyman-α luminosities (LLyα), obtaining values in the range $$\log ({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/[{{{{{{{\rm{erg}}}}}}}}\,{{{{{{{{\rm{s}}}}}}}}}^{-1}]) \sim$$41.8–42.65. A summary of the measured properties of the Lyman-α-emitters is reported in Table 1.

### A statistical assessment of the overdensity

To provide a statistical assessment of our finding, we perform the same analysis discussed in the previous subsection on other four 380 Å-sized sub-cubes adjacent to the sub-cube used to study the environment of W0410-0913, in the large wavelength range 4750−6500 Å (corresponding to offset velocities of [−50.000; +50.000] km s−1). This wavelength range is optimal because the sensitivity of the instrument varies only by a factor less than 25%, and it is not strongly affected by sky lines emission (see a description of MUSE performances at https://www.eso.org/sci/facilities/paranal/instruments/muse/inst.html). After checking for contamination from sky-residuals and multiple line emitters at intermediate redshifts (the number of the latter is found to be between 0 and 2 per sub-cube), we find that the number of detected candidate Lyman-α emitters is either 1 and 2, which confirms the overdense nature of the environment in the surrounding of the Hot DOG.

As an additional sanity check, using the same method, we scan through the negative version (obtaining by changing the sign to each pixel) of the sub-cubes defined above. In this case, the visual inspection following the line finder algorithm is limited to a check of the contamination by sky-residuals. We find that the number of detections is 0 in all the sub-cubes. This excludes the possibility that a fraction of the positive sources could be resulting from spikes of noise and thus be spurious.

### The physical significance of the overdensity

To quantify the significance of the discovered overdensity we compute the Lyman-α luminosity-weighted number density (Φ) of Lyman-α-emitting galaxies surrounding W0410-0913 and compare it with the value predicted by the most updated Lyman-α luminosity functions measured for field galaxies in wide and deep surveys with MUSE26,27.

Given the non-uniform and highly peaked distribution of offset velocities (see Fig. 1c), we derive two values of Φ, distinguishing between the whole volume containing 24 Lyman-α emitters (including W0410-0913) in the velocity range [−6.000; +10.000] km s−1whole) and a smaller (but denser) volume containing 18 Lyman-α emitters in the velocity range [−2.500; +1.500] km s−1dense).

The $$1^{\prime} \times 1^{\prime}$$ projected area (i.e., the MUSE FOV) over which we find the companion galaxies corresponds to a comoving area of 4.2 Mpc2. In the whole volume case, the redshift-side has a comoving size of about 190Mpc, which yields to a cosmological comoving volume of 800 Mpc3 and a number density of Lyman-α emitters of nLyE ~ 3 × 10−2 Mpc−3. In the dense volume case, the redshift-side has a smaller comoving size of 50Mpc, yielding to a cosmological comoving volume of about 210 Mpc3 and a number density of Lyman-α emitters of nLyE ~ 8.5 × 10−2 Mpc−3. To convert nLyE into Φ we divide by the range of Lyman-α luminosities of galaxies in both volumes, namely $${{\Delta }}\log ({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/{{{{{{{{\rm{L}}}}}}}}}_{\odot })=0.86$$ for the whole volume and $${{\Delta }}\log ({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/{{{{{{{{\rm{L}}}}}}}}}_{\odot })=0.75$$ for the dense volume. We obtain Lyman-α luminosity-weighted number densities of $${{{\Phi }}}_{{{{{{{{\rm{whole}}}}}}}}}=3.5\times 1{0}^{-2}\,{{{{{{{\rm{dlog}}}}}}}}{({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/{{{{{{{{\rm{L}}}}}}}}}_{\odot })}^{-1}\,\,{{{{{{{{\rm{Mpc}}}}}}}}}^{-3}$$ and $${{{\Phi }}}_{{{{{{{{\rm{dense}}}}}}}}}=1.1\times 1{0}^{-1}\,{{{{{{{\rm{dlog}}}}}}}}{({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/{{{{{{{{\rm{L}}}}}}}}}_{\odot })}^{-1}\,\,{{{{{{{{\rm{Mpc}}}}}}}}}^{-3}$$. These values are higher than the expected values for field galaxies (see Fig. 2).

In the following we focus on the dense volume, which is more likely to contain companion galaxies gravitationally associated with W0410-0913. In detail, at the median Lyman-α luminosity of galaxies in the dense volume, $$\log ({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/{{{{{{{{\rm{L}}}}}}}}}_{\odot })=42.2$$, the MUSE-wide survey27, which observed Lyman-α-emitters over a large area of $$> 20\,{{{{{{{{\rm{arcmin}}}}}}}}}^{2}$$, yields an expected $${{{\Phi }}}_{{{{{{{{\rm{field}}}}}}}}}=1{0}^{-(2.{1}_{-0.47}^{+0.37})}\,{{{{{{{\rm{dlog}}}}}}}}{({L}_{{{{{{{{\rm{Ly}}}}}}}}\alpha }/{{{{{{{{\rm{L}}}}}}}}}_{\odot })}^{-1}\,\,{{{{{{{{\rm{Mpc}}}}}}}}}^{-3}$$, where the uncertainty represents the errors on the best-fitting Schechter function parameters27. Comparing this value with Φdense, we find that the circumgalactic environment around W0410-0913 has a denser concentration of Lyman-α-emitting galaxies than the average volume, with an estimated overdensity factor of $${\rho }_{{{{{{{{\rm{CGM}}}}}}}}}=\frac{{{{\Phi }}}_{{{{{{{{\rm{dense}}}}}}}}}}{{{{\Phi }}}_{{{{{{{{\rm{field}}}}}}}}}}=1{4}_{-8}^{+16}$$. We note that this calculation is conservative since it is based on the assumption that offset velocities with respect to the central object are tracing relative distances, while it is more likely that the observed dispersion of offset velocities (at least for objects in the dense volume) is affected by gravitational attraction (see Article and previous subsections). Thus the effective values of Φ in our field are likely to be higher, e.g., up to a factor of about 20 times higher in the dense volume case under the extreme assumption that the 18 Lyman-α emitters are contained in a cube with a side of 60 centered around W0410-0913.

Despite its high value, we find that the measured overdensity of Lyman-α emitters in the dense volume, $${\rho }_{{{{{{{{\rm{CGM}}}}}}}}}=1{4}_{-8}^{+16}$$, is consistent with cosmological predictions in the proximity of extreme, luminous objects like W0410-0913.

To check this, we first estimate how rare the fluctuation of matter density associated with ρCGM would be. The first step is to consider that in general the spatial clustering of observable galaxies does not precisely mirror the clustering of the bulk of matter in the Universe. This caveat is often addressed through the definition of a galaxy clustering bias, which describes the relationship between the spatial distribution of galaxies and the underlying dark matter density field57. Thus, by using a standard assumption of linear bias58, we convert the observed overdensity of Lyman-α emitters ρCGM into an estimate of the underlying matter overdensity, ρM, by dividing it by the clustering bias parameter of Lyman-α emitters, b. By using an updated value of b = 2.8 ± 0.4, recently computed through a 3D clustering analysis of hundreds of Lyman-α emitters at z 4 in the large MUSE-Wide survey59, and consistent with previous similar studies58,60, we obtain an estimate of the matter overdensity in the surroundings of W0410-0913 of $${\rho }_{{{{{{{{\rm{M}}}}}}}}}={5}_{-3}^{+7}$$. To estimate the rarity of such a fluctuation, we start from the computation of the linear rms density fluctuation61, $${\sigma }_{{{{{{{{\rm{LN}}}}}}}}}$$, predicted by the standard ΛCDM cosmology. For the dense volume over which we compute ρM, i.e., about 210 Mpc3, we find a value of $${\sigma }_{{{{{{{{\rm{LN}}}}}}}}} \sim 1.6$$. The next step is to compute the probability associated with observing an overdensity ρM given a certain density fluctuation. On the physical scales of our system, the density field is non-linear, and a possible approximation for the true density distribution P(ρ) is a log-normal model62, for which the $${\sigma }_{{{{{{{{\rm{LN}}}}}}}}} \sim 1.6$$ can be interpreted as the rms in $${{{{{{{\rm{\ln }}}}}}}}(\rho )$$, with a mean in this quantity of $$\,-\frac{{\sigma }_{{{{{{{{\rm{LN}}}}}}}}}^{2}}{2}$$. Under this assumption, a matter overdensity of $${\rho }_{{{{{{{{\rm{M}}}}}}}}}={5}_{-3}^{+7}$$ corresponds to a probability of P(ρM) = 10−1.4±0.6. Given the considered volume of 210 Mpc3 and the estimated P(ρM), we find that the comoving density of objects like the central galaxy would need to be about $${n}_{{{{{{{{\rm{halo}}}}}}}}} \sim \frac{P({\rho }_{{{{{{{{\rm{M}}}}}}}}})}{210\,\,{{{{{{{{\rm{Mpc}}}}}}}}}^{3}} \sim 1{0}^{-3.7\pm 0.6}\,\,{{{{{{{{\rm{Mpc}}}}}}}}}^{-3}$$. This values is consistent, within the uncertainties, to the predicted comoving density of halos hosting central massive galaxies like W0410-0913, with maximum velocities of $${v}_{\max }=500\pm 70\,{{{{{{{\rm{km\,{s}}}}}}}^{-1}}}$$ (see the radial profile of the rotational velocity in Fig. 4b) at z = 3.631, which we find to be nhalo ~ 10−3.8±0.5 Mpc−3, as we compute integrating the halo mass function provided by the large N-body simulation MDPL263, included in the open-access CosmoSim database (https://www.cosmosim.org/cms/simulations/mdpl2/).

### Measure of the SFR from HST

We estimate the SFR of the Lyman-α-emitting companion galaxies around W0410-0913 using archival HST data, that we retrieve from the public Barbara A. Mikulski Archive for Space Telescopes (MAST) portal (https://archive.stsci.edu/). A $$2^{\prime} \times 2^{\prime}$$ field around our object was observed for a total exposure time of 2400 s in October 2012 during the HST Cycle 20 (Program ID: 12930) exploiting the filter F160W on the Wide Field Camera 3 (WFC3), centered at about 15436 Å, which corresponds to ~0.3 μm at the redshift of interest.

We extract the fluxes in the HST WFC3/F160W image from circular apertures with 1-sized diameters centered around the peak emission, that we search within a radius of 0.5 from the Lyman-α peak (obtained through a 2D Gaussian fit of the pseudo-NB images; see Fig. 1a), to allow for slight spatial offsets between Lyman-α and the rest-frame UV/optical continuum64. We then derive the luminosity density Lν and convert it in SFR by using a standard rest-frame UV/optical Lν−SFR calibration and assuming a Chabrier initial mass function65. Although the available HST data are shallow (with a 3σ flux limit corresponding to a SFR3σ ~ 9M yr−1), we detect 10 galaxies at SNR > 4, with star formation rates ranging between SFR ~ 12−100M yr−1.

In Supplementary Fig. 3, we show the HST cutouts for the companion galaxies detected with the WFC3/F160W filter, along with the Lyman-α contours from MUSE. A selection of HST maps is shown in Fig. 7b. The SFRs are reported in Table 1.

To obtain a better constraint of the average star formation rate in the 14 companion galaxies not detected by HST we perform a median stacking of their 5 × 5 cutouts centered at the Lyman-α peaks. Prior to stacking, we run a sigma-clipping algorithm (with a 3σ threshold on individual pixels) over each cutout to mask possible continuum sources outside the central circular area of 1 diameter. This is necessary to guarantee an uncontaminated estimate of the noise level in the final image, that would be otherwise affected by bright field objects. The noise level in the stacked image decreases down to a corresponding 3σ SFR limit of SFR3σ ~ 2.5M yr−1, enabling us to measure with a SNR ~ 3.6 an average SFR ~ 3M yr−1, suggesting that a non-negligible star formation is taking place in the companion galaxies not detected by HST too.

### A study of the 3D distribution of gas around W0410-0913

Extended diffuse Lyman-α emission, mainly produced through recombination radiation following photoionization (often referred to as fluorescence) powered by UV-bright sources24, is a commonly used tracer to observe the cool gas (with temperatures of approximately T = 104 K) in the CGM around galaxies. At z 2−3, where the Lyman-α is redshifted into the optical domain, great advances have been recently made with wide-field integral field spectrographs at large telescopes (like MUSE, or the Keck Wide-Field Imager), which have identified ubiquitous Lyman-α nebulae around quasars, extending up to hundreds of kpc22,23, and Lyman-α halos around star-forming galaxies, extending up to tens of kpc, between factors of 5−20 larger than the rest-frame UV sizes66.

We search for extended Lyman-α emission around the companion galaxies of W0410-0913 to study the morphological and kinematic distribution of gas in the circumgalactic environment of W0410-0913. We run CubEx over the sub-cube in which the 24 Lyman-α-emitters are found, using the same pre-processing and setup of parameters adopted for analyzing the Lyman-α nebula (see previous subsections), namely a SNR > 3 threshold for voxel-detection in a smoothed cube with a Gaussian spatial filter of 0.4 and a Gaussian spectral filter of 1.25 Å. This setup is different from that exploited in the line-emitters searches, and it is tailored at extracting faint and narrow features. We produce optimally extracted Lyman-α flux maps of the sources by selecting the voxels detected in their 3D-segmentation-masks and integrating their fluxes along the wavelength axis. This technique is operationally equivalent to combining different pseudo-NB images in which the size of the spectral filter is optimized for each pixel. The advantage of this approach, enabled by integral field spectrographs, is that one can maximize the SNR of the whole emitting region by revealing both kinematically narrow and broad features of a source at the same time. In contrast, imaging through classical NB filters with a fixed width would produce either a noise-dilution or an underestimation of the signal emitted.

A collection of the optimally extracted images of the companion galaxies is shown in Fig. 3a. We find that some of the luminous companion galaxies in the denser dynamic volume around W0410-0913, with offset velocities in the range [−2.500; +1.500] km s−1, show extended diffuse Lyman-α envelopes (not revealed, or barely detected, in the pseudo-NB images; see Fig. 1a) that are suggestive of connecting filamentary structures on scales up to 100 kpc (see Fig. 3). We use the 3D-segmentation-masks provided by CubEx to produce also the 2D maps of the first moment (moment-1) of the flux distribution in its spectral domain. These maps give an indication of the centroid offset velocities with respect to the systemic redshift of W0410-0913 at each spatial location. A collage of such kinematics maps is showed in Fig. 3b.

### ALMA data

In this work, we use archival ALMA observations of W0410-0913 targeting the emission-line spectra arising from rotational transitions of carbon monoxide CO(4-3), CO(6-5), and CO(7-6), and the atomic carbon transition [CI]3P23P1 (hereafter [CI]). CO(4-3) 461GHz, redshifted to 99.5GHz at z = 3.631, was observed with ALMA in Band 3 on January 2018 (Program ID: 2017.1.00123.S), with a total integration time of $$11.5\,\min$$. Deep observations of CO(6-5) 691.4GHz, redshifted to 149.3GHz, were taken in Band 4 in December 2017 (Program ID: 2017.1.00908.S) exploiting a total integration time of about 3.6h. CO(7-6) 806.6GHz and the adjacent [CI] 809.3GHz, redshifted to 174.2GHz and 174.75GHz respectively, were observed in Band 5 on September 2018 (Program ID: 2017.1.00123.S), with a total integration time of about $$10\,\min$$. See the Data Availability Section for information on the data repositories. We performed a standard calibration of the data using the Common Astronomy Software Applications package (CASA67), running different versions of the software per each dataset, according to the Quality Assurance report provided by the ALMA staff. The pipeline calibration outputs did not show any issue, and thus we did not perform any additional flagging. We obtained continuum-subtracted visibilities using the CASA task uvcontsub, and we generated the continuum-subtracted datacubes of the spectral windows containing the emission lines by running the CASA cleaning algorithm clean with 500 iterations and a SNR threshold of 3. In order to maximize the sensitivity, we adopted a natural weighting of the visibilities for all the three datasets. The resulting synthesized beam in the CO(4-3) datacube is 0.7 × 0.5 (position angle, PA = 83°), 0.22 × 0.18 (PA = −69°) for CO(6-5), and 0.6 × 0.4 (PA = 80°) in the datacube containing CO(7-6) and [CI]. We adopted pixel sizes of 0.1, 0.03, and 0.1, and spectral bins of 50 km s−1, 60 km s−1, and 50 km s−1 (which enable us to have both a good sampling of the lines and enough SNR per individual channel) for CO(4-3), CO(6-5) and CO(7-6) + [CI] respectively. The average sensitivities reached in the cubes (in spectral regions close to the line frequencies) are 0.7 mJy beam−1 in a spectral channel of 50 km s−1 for CO(4-3), 0.08 mJy beam−1 in a spectral channel of 60 km s−1 for CO(6-5) and 0.8 mJy beam−1 in a spectral channel of 50 km s−1 for CO(7-6) + [CI].

In Fig. 8 we show the CO(4-3), CO(6-5), CO(7-6) and [CI] spectra of W0410-0913. These spectra are extracted from a spatial region defined by the connected pixels with SNR > 2 in the moment-0 maps, that we first obtain by collapsing the cube spectral channels in the range [fcen−FWHM: fcen + FWHM], where fcen (the central frequency) and FWHM are derived by a Gaussian fit of a spectrum extracted from a 1-sized circular aperture centered at the expected position of the object. Such a 2-iterations procedure helps in case of resolved observations to ensure that the whole line-emitting region is accounted for in the spectrum while avoiding signal dilution. Through a non-linear least-squares Gaussian fit of the CO(6-5) line, which has the highest SNR among the available data, we derive a systemic redshift of zCO(6−5) = 3.631 ± 0.001. This value is consistent with an independent measurement of zCO(6−5) recently reported18 (z = 3.6301), and the tiny difference is probably due to a slightly different definition of the extracting aperture. The velocity axes of the spectra in Fig. 8 are aligned to the common systemic redshift of zCO(6−5) = 3.631. From our Gaussian fit we derive large FWHMs of about 800 km s−1 (with average associated uncertainties of 70 km s−1) in CO(4-3), CO(6-5) and CO(7-6). [CI] appears to be narrower (FWHM~600 ± 90km s−1), although this might be an effect of spectral blending with the more luminous CO(7-6). Broad molecular line profiles in W0410-0913 were also reported in previously published ALMA observations of CO(4-3)19, and [CII]18, although in both cases the instrument spectral setup was tuned on the optical redshift2, which is blueshifted with respect to the systemic redshift as traced by the molecular gas, and the lines, close to the edge of the spectral window, were only partially detected. Such broad emission lines in the integrated spectra extracted from the whole galaxy reflect the rapidly spinning nature of its disk (see subsection Kinematic modeling of the CO(6-5) emission in the ISM of W0410-0913).

### Study of the CGM environment with ALMA

Unfortunately, none of these datasets, designed to observe the cold gas emission from the hyper-luminous W0410-0913, provides information about the rest-frame FIR line properties of the companion Lyman-α emitters that we discover with MUSE in its CGM. However this is not surprising, and it is mainly due to a combination of insufficient sensitivity and too high resolution (that dilutes the flux over several beams), as we test by running simulations of the estimated telescope time on the ALMA Observing Tool (OT; version released for Cycle 8). In detail, we first estimate the total molecular gas mass (Mmol) from the SFR measured with HST by using standard Mmol-SFR scaling relations calibrated at high-z68. We then convert Mmol into total CO fluxes by adopting a conservative Milky-Way-like conversion factor between Mmol and the CO(1-0) luminosity, $${\alpha }_{{{{{{{{\rm{CO}}}}}}}}}^{{{{{{{{\rm{MW}}}}}}}}}=4.3$$, and standard Milky-Way-like intensity ratios between CO(1-0) and the rotational transitions observed by ALMA69. We note that the adopted value of αCO has been calibrated at solar metallicities, and should be considered conservative in our case since αCO typically increases at lower metallicities70, and high-z galaxies tend to be less metal-rich at a fixed stellar mass71. Assuming projected sizes of about 1 and line-widths of 300 km s−1, we find that the estimated total ALMA time to observe the integrated CO spectrum in a galaxy with SFR = 30 M yr−1 (the median SFR of the HST-detected companions), with at least 3σ per channel in at least 3 spectral bins, at the same angular resolution of the available observations, would be about 1.6h for CO(4-3) and CO(7-6), and about 24.5h for CO(6-5), between a factor of 6−8 longer than the observed datasets. Other additional limitations derive from the ALMA smaller fields of view (the primary beams range from about 30 in Band 5 to about 45 in Band 3), and the narrower spectral coverage (sidebands ranging from 5000 km s−1 in Band 5 to 9000 km s−1 in Band 3) with respect to MUSE data. The combination of these effects reduces the number of companion galaxies that could be possibly detected (up to 60%, in the best case) and, together with the sensitivity issue discussed above, limits the information that can be derived through stacking analyses. Indeed we find signals that are consistent with noise in the stacked images that we produce, for each dataset, by combining moment-0 maps centered at the expected positions of the galaxies (using the peak of Lyman-α emission as a prior), obtained by collapsing three different velocity ranges, 300, 400, 500 km s−1, centered around the expected systemic frequencies (using the redshifts derived from the Lyman-α as a prior).

### Kinematic modeling of the CO(6-5) emission in the ISM of W0410-0913

The deep and highly resolved CO(6-5) observations of W0410-0913 are best suited to study the molecular gas kinematics in W0410-0913, that we model using the 3D tilted ring model fitting code 3DBarolo72 (3D-Based Analysis of Rotating Objects from Line Observations). This technique has the advantage that it directly models the 3D cube taking beam-smearing effects into account and it has been extensively applied to observations of high-redshift (z > 2) galaxies73. The fit routine is performed on the continuum-subtracted CO(6-5) datacube and can be summarized as follows.

First, we isolate the signal by producing a 3D signal mask of the emission line through the 3DBarolo SEARCH tool (based on DUCHAMP74). In detail, this algorithm identifies pixels in the datacube with intensities above a certain SNR (SNRupper = 3.5, in our case), and then it searches around these signal peaks for emission above a second SNR threshold (i.e., SNRlower = 2.0, in our case). The 3DBarolo best-fit results are highly dependent on the overall geometry of the model (i.e., number and width of the rings) and on the initial estimates of the parameters. Thus, once the signal is isolated we create the moment maps needed for the initial guesses. Using the CASA toolkit task image.moments we collapse the cube over the spectral channels containing the line and produce a moment-0 map, which we then fit with a 2D Gaussian (using the CASA task image.fitcomponents) to obtain the FWHM of the major and minor axes, the position angle and the central position. We also produce a velocity dispersion map (i.e., moment-2) including only the pixels that belong to the 3D mask identified before. Then, we fit a tilted ring model to the line emission by using the 3DBarolo key function 3DFIT. We adopt a value of twice the FWHM of the major axis of the restoring beam as a maximum model radius. As the value for the width of each ring, we adopt the FWHM of the minor axis, in order to not over-resolve the data73. This corresponds to a number of 12 model rings. The model central position is fixed to be coincident with the centroid of a 2D Gaussian fit of the moment-0 map, and we use the maximum value in the moment-2 map as the initial guess of the velocity dispersion. A thin disk, with a scale height of 0.01, is assumed. We adopt an initial guess for the inclination of 45°, and we allow it to vary between 10 and 80°. Starting from this set of guesses, 3DBarolo produces a model of the first ring by randomly populating the volume with discrete emitting gas clouds in order to reproduce the position angle, the rotational velocity, etc of the initial estimates. The resulting model datacube is then convolved with the ALMA synthesized beam and rescaled so that the velocity-integrated intensity of each spaxel in the data and model cubes are equal. Then, the values of the pixels in the data and model datacubes are compared, and each parameter is varied until the absolute residual (i.e., model-observation) is minimized. When this process reaches a minimum, the next ring is analyzed. During the first run 3DFIT provides best-fit results for the rotational velocity, velocity dispersion, systemic velocity, inclination, and position angle. The three latter variables are averaged across all the rings, and the whole fit routine is run again, with only the rotational velocity and velocity dispersion as variables. Among the final 3DBarolo outputs, we obtain morphological parameters (e.g., central position), and kinematic parameters such as inclination, position angle, velocity dispersion, and rotation.

In Fig. 9 we show that our model is able to reproduce the kinematic properties of the target to an excellent level, with small residuals in the moment maps and a good agreement between the model and data position-velocity diagram (PVDs). The galaxy has a well-defined rotation, with rotational velocities that range between vrot ~ ± 500 km s−1, as can be seen from the moment-1 map, showing the spatial distribution of line-of-sight velocities, and from the PVDs extracted along the major and minor axes (see Fig. 9). Despite the fast coherent rotation, the velocity dispersion is still large, in analogy with other recent ALMA observations of Hot DOGs reporting turbulent ISM18. Nevertheless, as discussed in the Article, we find that vrot/σ > 1 at each ring in our model (see Fig. 4). We find morphological and kinematic position angles of 133 ± 5° and 133.8 ± 4.9° respectively, and morphological and kinematic inclination on the plane of the sky of 49.5 ± 3.3° and 50.1 ± 13.1°.