KASHz+SUPER: Evidence for cold molecular gas depletion in AGN hosts at cosmic noon

The sphere of gravitational influence of supermassive black holes (SMBHs) is spatially limited to the innermost regions of galaxies ($\sim 10$pc for a $10^{8}\,{{\rm M}_{\odot}}$ black hole; Alexander & Hickox 2012). Yet, SMBH accretion can release large amounts of energy that, if efficiently coupled to the surrounding material, allows active galactic nuclei (AGN) to shape galaxy growth by heating, exciting and/or removing the gas of the interstellar medium (ISM). Such an interplay between AGN activity and the life of galaxies is referred to as AGN feedback (e.g., Silk & Rees 1998; Fabian 2012). Astronomers found multiple observational evidence for a connection between observational and physical properties of the central AGN and its host galaxy (e.g., Strateva et al. 2001; McConnell et al. 2011; Rodighiero et al. 2015), as also predicted by theoretical models (e.g., Somerville et al. 2008; Lapi et al. 2014; Costa et al. 2018), leading to the so-called “AGN/galaxy co-evolution” scenario. However, we still lack a full understanding of AGN feedback despite its importance (e.g., Kormendy & Ho 2013; Harrison 2017; Harrison et al. 2018; Ward et al. 2022), for instance whether galaxy evolution is influenced by AGN activity as a whole or whether AGN-driven winds and jets are key properties in driving AGN feedback effects or whether we are using sensitive observational tracers for AGN feedback. AGN feedback is believed to be highly dominant when both cosmic SMBH growth and star formation rate (SFR) were at their peaks (the “cosmic noon”, $z\simeq 1-3$; e.g., Madau & Dickinson 2014; Aird et al. 2015). Hence, sources at the cosmic noon are the most promising and most informative targets to investigate AGN feedback, both in terms of “causes” (e.g., AGN-driven winds and jets) and “effects” (e.g., reduced gas fraction and disturbed gas kinematics in AGN hosts).

A way to test the impact of AGN feedback on the host galaxy is to observe its molecular gas content and conditions. In particular, the Schmidt-Kennicutt law (Schmidt 1959; Kennicutt 1989) is a fundamental relation that links the molecular gas content and the level of star formation (SF) in galaxies, that is often used in its integrated form, i.e., global SFR of galaxies vs. their total molecular mass (e.g., Carilli & Walter 2013; Sargent et al. 2014; Perna et al. 2018; Bischetti et al. 2021; Circosta et al. 2021). If AGN influence the evolution of galaxies as a whole (e.g., Fabian 2012; Harrison 2017), they will reasonably impact the molecular gas reservoir first, and then, consequently, SF on longer timescales. Besides addressing AGN feedback as “caught in the act” (i.e., AGN-driven outflows or radio jets), comparative studies of the molecular gas properties in AGN host galaxies and matched non-active galaxies (i.e., galaxies not hosting an AGN) can shed light on the impact of AGN on galaxy evolution, regardless of whether outflows are concurrently detected in a given source.

A powerful proxy to measure the molecular content of galaxies is represented by the carbon monoxide (CO), the second most abundant molecule after H₂, and hence the primary tracer of molecular gas (e.g., Carilli & Walter 2013; Hodge & da Cunha 2020). The molecular gas mass can be derived from the luminosity of the ground state transition CO(1-0) ($M_{\rm H_{2}}=\alpha_{\rm CO}L^{\prime}_{\rm CO(1-0)}$). Yet, the value of $\alpha_{\rm CO}$ has to be assumed and it is very uncertain as it depends on different quantities, for instance the distance from the main sequence (MS, Elbaz et al. 2007; Noeske et al. 2007) and the metallicity (e.g., Accurso et al. 2017). The value calibrated on the Milky Way ($\alpha_{\rm CO}=4\,{{\rm M}_{\odot}}\rm\,(K\ km\ s^{-1}\ pc^{2})^{-1}$, Bolatto et al. 2013) can be a good approximation for regular star-forming galaxies (Carilli & Walter 2013), while a lower value ($\alpha_{\rm CO}=0.8-1\,{{\rm M}_{\odot}}\rm\,(K\ km\ s^{-1}\ pc^{2})^{-1}$, Downes & Solomon 1998; Solomon & Vanden Bout 2005; Calistro Rivera et al. 2018; Amvrosiadis et al. 2023 or $\alpha_{\rm CO}=1.8-2.5\,{{\rm M}_{\odot}}\rm\,(K\ km\ s^{-1}\ pc^{2})^{-1}$, Cicone et al. 2018; Herrero-Illana et al. 2019; Montoya Arroyave et al. 2023), could be more representative for high-z sub-mm galaxies, which usually have an extremely dense ISM and often intense SF activity (e.g., Birkin et al. 2021). Observations of massive MS galaxies at cosmic noon resulted in an $\alpha_{\rm CO}=3.6\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$ (e.g., Daddi et al. 2010a; Genzel et al. 2015).

Measuring the ground transition of CO is not always possible, especially at high redshift. However, observations of CO transitions from higher $J$ levels can be converted to the CO(1-0) flux by assuming a CO Spectral Line Energy Distribution (CO-SLED), which expresses the relative strength of the CO rotational lines as a function of the quantum number $J$ (e.g., Pozzi et al. 2017; Mingozzi et al. 2018; Kirkpatrick et al. 2019; Boogaard et al. 2020; Valentino et al. 2021; Pensabene et al. 2021; Esposito et al. 2022; Molyneux et al. 2024). Even though AGN mostly contribute to high-$J$ levels, CO ladders of AGN hosts and non-active galaxies are seen to vary also at $J<5$, despite the large associated uncertainties, with AGN showing larger ratios that increase for increasing AGN luminosity (Carilli & Walter 2013; Kirkpatrick et al. 2019; Vallini et al. 2019; Boogaard et al. 2020).

Results regarding the effects of AGN on the molecular gas phase are controversial and seem to hint at a dichotomy between low-z and high-z scenarios. For what concerns the integrated properties of galaxies, AGN hosts in the local Universe are usually similar to non-active galaxies (e.g., Rosario et al. 2018; Koss et al. 2021; Salvestrini et al. 2022, but see also Mountrichas et al. 2024) or even gas richer, in the sense that more powerful AGN are hosted in gas richer host galaxies (e.g., Vito et al. 2014; Husemann et al. 2017). Yet, significant effects of negative AGN feedback are seen in local AGN hosting molecular outflows (Fiore et al. 2017). Additionally, some results hint at enhanced SF efficiencies in both local and high-z AGN hosts possibly linked to large molecular gas reservoirs (e.g., Shangguan et al. 2020a; Jarvis et al. 2020; Bischetti et al. 2021).Spatially-resolved studies of local galaxies unveiled gas depletion in the nuclear regions of AGN hosts, and not on galactic scales (Sabatini et al. 2018; Rosario et al. 2019; Fluetsch et al. 2019; Feruglio et al. 2020; Ellison et al. 2021; Zanchettin et al. 2021), with indications that the central SF activity (within few 100 pc) is indeed reduced due to the presence of AGN, while the galaxy-wide SFR is unaffected (e.g., Sánchez et al. 2018; Garc´ıa-Burillo et al. 2021; Lammers et al. 2023, but see also Molina et al. 2023, who find enhanced SF activity in the proximity of the AGN). At high redshift, the observed impact of AGN on the molecular ISM of their hosts is debated. Several studies find that AGN hosts are significantly CO-depleted when compared to control samples of non-active galaxies, either in terms of gas fraction¹¹1The definition of gas fraction varies within different studies. Throughout this work, we assume the definition $f_{\rm gas}=M_{\rm gas}/M_{\ast}$, also dubbed $\mu_{\rm mol}$ in the literature. or of depletion timescales (e.g., Brusa et al. 2016, 2018; Kakkad et al. 2017; Perna et al. 2018; Bischetti et al. 2021, but see also Herrera-Camus et al. 2019; Spingola et al. 2020). Yet, other studies find limited (Circosta et al. 2021, e.g.,, hereafter, C21) or no significant difference between the gas fraction of non-active galaxies and AGN hosts (e.g., Valentino et al. 2021) and some other works find no clear link between gas content and AGN power (e.g., Kirkpatrick et al. 2019). An additional issue in comparing the properties of AGN hosts and non-active galaxies resides in the intrinsic variability of AGN and their flickering activity (Harrison 2017), which can also affect the distinction of galaxies that host and that do not host an AGN based on when the system was targeted, for instance, in the X-rays.

Predictions from cosmological simulations seem to be likewise controversial. In particular, Ward et al. (2022) recently analyzed the output of three key cosmological simulations (Illustris-TNG, hereafter TNG, Springel et al. 2018; Pillepich et al. 2018; Naiman et al. 2018; Nelson et al. 2018; Marinacci et al. 2018; EAGLE, Crain et al. 2015; Schaye et al. 2015; Simba, Davé et al. 2019) applying the same methods employed to analyze observations of galaxies and AGN hosts. Considering samples of local and $z=2$ targets, the authors conclude that none of the selected simulations predicts strong negative correlations between AGN power and molecular gas fraction or specific SFR (sSFR $=$ SFR/$M_{\ast}$). Conversely, powerful AGN seem to preferentially reside in gas-rich, highly-star-forming galaxies, whereas gas-depleted and quenched fractions are higher in the control samples of regular galaxies than for the AGN hosts. Ward et al. (2022), however, argue that there is a quantifiable difference among the predictions of the three selected simulations and that the bolometric luminosity range covered by simulations and observational efforts hardly overlap, especially at cosmic noon. This is mainly due to the difficulty in reproducing sizable samples of high-luminosity AGN (${L_{\rm bol}}\gtrsim 10^{43.5}$ erg s^-1 at $z=0$ and ${L_{\rm bol}}\gtrsim 10^{45}$ erg s^-1 at $z=2$) in cosmological simulations, since they are rare and too short-lived to be captured in the simulated volumes, and to the observational cost of observing sizable samples of less luminous AGN (e.g., ${L_{\rm bol}}\lesssim 10^{44}$ erg s^-1 at $z>1$).

The dichotomy between results on local and high-z targets, and between different samples of high-z sources, could arise from selection effects. High-z studies usually rely on few, high-luminosity ($\log(L_{\rm Bol}/\rm erg\leavevmode\nobreak\ s^{-1})>46$) and heterogeneous targets, both because high-power AGN are the best candidates to produce efficient feedback (e.g., Fiore et al. 2017) and because of the challenging observations required for distant objects.

Two surveys of X-ray selected AGN were recently conceived to provide multi-wavelength studies of AGN samples blindly selected with respect to the presence of ionised outflows or jets at cosmic noon: KASHz (KMOS AGN Survey at High redshift, PI: D. Alexander; Harrison et al. 2016, Scholtz et al., in prep) at $z=0.6-2.6$ and SUPER (SINFONI Survey for Unveiling the Physics and Effect of Radiative feedback, PI: V. Mainieri; Circosta et al. 2018, hereafter, C18) at $z=2-2.5$. Both surveys aim at studying the interplay between AGN and galaxy evolution in terms of AGN-driven ionized winds, traced through [OIII] emission lines, and properties of the host galaxy (see Sect. 2). SUPER relied on adaptive-optics-assisted, spatially-resolved studies of ionized gas, which require longer exposures, leading to a reduced sample size and sparser sampling of AGN with $\log({L_{\rm bol}}/$erg s^-1$)<45$ compared to the KASHz survey, with its larger sample size and lower bolometric luminosity cut. In this context, the different observational capabilities and selection criteria make KASHz and SUPER complementary surveys, thus using them jointly allows us to capitalize on their strengths.

As part of the SUPER survey, C21 provided the first systematic analysis of the molecular gas content of AGN host galaxies at $z\simeq 2$ through a dedicated follow-up in CO(3-2) with ALMA (Atacama Large Millimeter Array). With robust measurements of physically relevant quantities (SFR, $M_{\rm gas}$) traced through observational proxies ($L_{\rm FIR}$, $L^{\prime}_{\rm CO(3-2)}$), the authors find hints for negative AGN feedback effects in their sample. Yet, the sample size of SUPER ALMA AGN did not allow to discriminate whether such hints of gas depletion were indicative of less prominent AGN feedback effects in moderate luminosity AGN or due to a large fraction of upper limits combined with a limited sample size. As Figure 1 shows, combining SUPER targets with KASHz AGN allows for a robust sampling of moderate-luminosity AGN ($44<\log({L_{\rm bol}}/$erg s^-1$)<46$), still covering a wide range in AGN bolometric luminosity by means of the different sample selection criteria (see Fig. 1, but also Fig. 1 of C18).

The aim of this work is to investigate the molecular gas properties of a statistically-robust, consistently-selected, coherently-analyzed sample of AGN at cosmic noon ($z=1-2.6$) available in the ALMA archive, improving the sampling of the $\log({L_{\rm bol}}/$erg s^-1$)<44-46$ bolometric range. We build upon the methods and analysis of C21, and build our sample combining SUPER ALMA AGN with targets from the KASHz survey (see Sect. 2). The control sample of non-active galaxies is presented in Sect. 3. The ALMA data selection and reduction of KASHz targets is presented in Sect. 4 and Appendix C, while the SED fitting and photometry collection of both SUPER and KASHz AGN are described in Appendix D. We then apply a quantitative analysis in the Bayesian framework developed by C21 to compare our enlarged AGN sample with the control sample of non-active galaxies in Sect. 5. In Sect. 6, we compare our observed sample with the output of cosmological simulations. Results are discussed and summarized in Sects. 7 and 8. We assume a flat ${\Lambda}$CDM cosmology (Planck Collaboration et al. 2020), with $H_{\rm 0}=67.7\ {\rm km\ s^{-1}\ Mpc^{-1}}$ and $\Omega_{\rm m,0}=0.31$ throughout the paper.

To build a robust sample of moderate-to-high-luminosity AGN ($\log({L_{\rm bol}}$/erg s^-1) $\simeq 44-47$), we considered all the SUPER ALMA targets presented in C21 and complement them with archival ALMA observations targeting the CO emission with $J<5$ (CO(1-0), CO(2-1), CO(3-2), CO(4-3)) of KASHz AGN (see Sects. 2.1 and 4.1, Appendix C and Table 4). The other needed parameters for our analysis (mainly $M_{\ast}$ and $L_{\rm FIR}$) are estimated from SED fitting performed with CIGALE (Boquien et al. 2019; Yang et al. 2020) using the most up-to-date multi-wavelength, broadband photometry released by the deep fields’ collaborations (see Appendix D). Figure 1 shows the $L_{\rm X}$ vs. z distribution of the full KASHz and SUPER survey samples. We also show the ${L_{\rm bol}}$ vs. z distribution of the AGN used in this work, i.e., ALMA targets from KASHz and SUPER, as derived from SED fitting (see Appendix C).

For the analysis presented in Sect. 5, we discard 6 AGN from the SUPER and KASHz ALMA samples because they are missing at least one of the parameters of interest needed for our analysis ($M_{\ast}$ or $L_{\rm FIR}$; we mark these targets with an asterisk in Table 5 and in the bottom panel of Fig. 1). These six targets are extremely bright Broad Line AGN with sparse or missing FIR photometry coverage. A reasonable decoupling of AGN and galaxy emission in the SED fitting is thus not possible, hampering also the determination of meaningful upper limits for stellar mass and FIR luminosity of the host galaxy (\al@circosta2018,circosta2021; \al@circosta2018,circosta2021).

We build the comparison sample using literature measurements of non-active galaxies. We select star-forming, non-active galaxies from the Plateau de Bure high-z Blue Sequence Survey catalog (PHIBSS, Tacconi et al. 2018), which aims at assessing the gas properties of galaxies across cosmic time, employing a sample of 1444 targets placed at $z=0-4.4$. Each target is complemented with estimates of the molecular gas mass, as traced by CO emission (including CO non-detections), and FIR luminosity, as measured either from SED fitting or from the dust continuum luminosity. We include also the ALMA/NOEMA survey of sub-$mm$ galaxies in the COSMOS, UDS, and extended CDFS (ECDFS) fields by Birkin et al. (2021) and the non-active galaxies of ASPECS (Boogaard et al. 2020, and references therein), to better sample the higher-end and the lower-end, respectively, of the stellar mass range spanned by our AGN host galaxies. The PHIBSS project includes also the pilot ALMA program of ASPECS that was presented in Decarli et al. (2016); for those galaxies, we use the latest estimates provided by the ASPECS survey, as reported in Boogaard et al. (2020).

We include in the control sample all the non-active galaxies observed in CO(3-2) and CO(2-1), as obtained by ALMA and NOEMA, and retrieve the CO(1-0) using $r_{21}=0.6$ and $r_{31}=0.5$, i.e., excitation ratios commonly found for star-forming galaxies in the same redshift range of our AGN (Daddi et al. 2010b; Tacconi et al. 2013; Kakkad et al. 2017). CO fluxes were retrieved from the literature (Boogaard et al. 2020; Birkin et al. 2021) or provided by the PHIBSS collaboration (L. Tacconi, private communication) since Tacconi et al. (2018) only report the final molecular gas masses. We also checked the nature of the galaxies in the control sample and excluded those that are flagged as AGN. We do not include such targets in our AGN sample because their stellar masses and SFRs were not corrected accounting for the AGN contribution and could thus be overestimated, and because they are identified as AGN with methods different than ours.

All the galaxies in the comparison sample have available estimates of stellar mass and FIR luminosity. However, the SFR of a small subsample of PHIBSS galaxies was estimated from H$\alpha$ fluxes. Based on the reasonable agreement between the SFRs derived from H$\alpha$ and from the FIR for MS galaxies at cosmic noon (within $\simeq$0.4 dex; Rodighiero et al. 2014; Puglisi et al. 2016; Shivaei et al. 2016), we convert such SFRs to FIR luminosity applying the Kennicutt (1998) relation corrected for a Chabrier (2003) IMF (i.e., reduced by 0.23 dex).

We build the control sample of non-active galaxies as follows. First, we divide our AGN sample into two redshift bins ($z=1-1.8$ and $z=1.8-2.55$), which we use as redshift constraints for the match in stellar mass and SFR to take into account the evolution of the MS with redshift (e.g., Genzel et al. 2015; Tacconi et al. 2018). For each redshift bin, we only consider non-active galaxies within the same stellar mass range covered by our AGN host galaxies within the uncertainties ($\log M_{\ast}/\,{{\rm M}_{\odot}}\simeq 10-12$), and we further divide the samples in bins of stellar mass of 0.5 dex width. For each stellar mass bin, we then select all those non-active galaxies with a SFR consistent with (within $+0.2$dex to consider the uncertainties) or lower than that of AGN hosts in a given mass (and redshift) bin, so to also account for the SFR upper limits in the AGN sample. We show in Fig. 2 the comparison of our AGN (in red) and control sample (in blue) in terms of SFR vs. $M_{\ast}$. Figure 12 in Appendix B shows the same comparison in the low-z and high-z bins. Figures 2 and 12 also show the distribution of $M_{\ast}$ and SFR both for AGN hosts and non-active galaxies for illustration purposes, since the upper limits are considered, for these plots only, at face value. In fact, a robust comparison of the distribution of SFR and $M_{\ast}$ applying the hierarchical method described in Sect. 5.2 and the Kolmogorov-Smirnov (KS) test is presented in Appendix B. We find that AGN hosts and non-active galaxies selected as presented in this Section are consistent with being drawn from the same distribution for what concerns the stellar mass in all redshift bins (low-z, high-z, total) and for what concerns the SFR in the low-z bin and total redshift range. However, they are significantly different for what concerns SFR distribution of the high-z bin (p-value$\simeq$1%) . In fact, the galaxy database used to build the control sample is missing non-active galaxies with CO observations that fall below the MS at $z>1.8$ ($\Delta\rm MS<-0.5dex$; see Fig. 12, bottom) due to the rareness of low-SFR star-forming galaxies not hosting AGN and the time-consuming observations required to target their molecular phase. Since not even dust-continuum observations cover non-active galaxies in such a region of the MS (e.g., Tacconi et al. 2020), we run our comparative analysis twice: we compare the full AGN and non-active galaxy samples built as described in this section and in Sect. 2, and then we compare them again excluding those AGN lacking a match in the control sample in the high-z bin (i.e., $z>1.8$, $\log(M_{\ast}/\,{{\rm M}_{\odot}})>11$, $\rm\log(SFR/\,{{\rm M}_{\odot}}yr^{-1})<2$; see Fig. 12) and those galaxies in the low-z bin that do not have a match in the AGN sample (i.e., $z<1.8$, $\log(M_{\ast}/\,{{\rm M}_{\odot}})$¡$10.5$, $\rm\log(SFR/\,{{\rm M}_{\odot}}yr^{-1})<1.5$; see Fig. 12) as a sanity check of our results.

We present in this Section our comparative, quantitative analysis of the cold molecular gas content of AGN host galaxies and non-active galaxies at $z=1-2.6$. The AGN sample selection is described in Sect. 2 and the control sample of non-active galaxies is built as described in Sect. 3 (see Fig. 2). We address the effects of AGN feedback on the properties of host galaxies at $z=1-2.6$ by assessing whether they differ from the respective control sample in terms of: i) CO vs. FIR luminosity, ii) distribution of molecular gas fraction, for which we consider the observational proxy $f_{\rm gas}=$$L^{\prime}_{\rm CO}$/$M_{\ast}$, iii) CO luminosity vs. stellar mass, and iv) molecular gas fraction (i.e., $L^{\prime}_{\rm CO}$/$M_{\ast}$) vs. stellar mass. Moreover, we also run the analysis removing the AGN at $z>1.8$ that fall below the MS and do not have a match in the control sample, and galaxies in the low redshift bin without a counterpart in the AGN sample (i.e., below the MS; see Sect. 3 and B). We anticipate that the results obtained excluding such AGN and non-active galaxies are consistent with those of the full samples.

The total AGN sample totals 46 AGN and presents a CO detection rate of $\simeq$50%. Roughly half of the AGN have a constrained $L_{\rm FIR}$ and about $\simeq$44% have an upper limit for both quantities. All but five AGN have constrained stellar mass, of which three (cid_178, cid_1605, cid_467) are undetected in CO and thus do not allow to derive a limit on their gas fraction. We summarize the properties of KASHz and SUPER ALMA AGN in Table 5. The control sample totals 98 galaxies, with a CO detection rate of $\simeq$90%.

We performed our analysis within a Bayesian framework as presented in C21, which allows us to take into account the upper (and lower) limits on both dependent and independent variables. There is only one main difference in this work with respect to C21: KASHz and literature AGN are heterogeneous in terms of targeted CO transition, while SUPER AGN were all observed in CO(3-2). We relied on the CO(1-0) luminosity by assuming a CO SLED that we uniformly applied to the CO measurements of KASHz and SUPER AGN (see Sect. 4.2) and a second one for the star-forming galaxies of the control sample (see Sect. 3). Our analysis remains free of the additional uncertainties inherent to the $\alpha_{\rm CO}$ conversion factor and the conversion of FIR luminosity into SFR. Figures of this Section also show the molecular mass axis, derived assuming $\alpha_{\rm CO}=3.6\,{{\rm M}_{\odot}}/(\rm K\,{\rm km\,s}^{-1}pc^{2})$, yet this serves for illustration purposes only, since we only consider $L^{\prime}_{\rm CO}$ in our quantitative analysis.

Ward et al. (2022) recently analyzed the output of three cosmological simulations (TNG, Springel et al. 2018; Pillepich et al. 2018; Naiman et al. 2018; Nelson et al. 2018; Marinacci et al. 2018, data release: Nelson et al. 2019; EAGLE, Crain et al. 2015; Schaye et al. 2015, data release: McAlpine et al. 2016; SIMBA, Davé et al. 2019) with approaches similar to those taken by observers, to assess the predictions regarding the effects of AGN feedback on the properties of AGN host galaxies and how these relate with observational constraints and results. Ward et al. interestingly found that simulations predict that powerful AGN tend to be located in gas rich galaxies and that the gas-depleted fraction of AGN hosts is lower than that of non-active galaxies.

The molecular gas phase cannot be directly traced in cosmological simulations because, due to computational limitations, the ISM is poorly resolved. However, the amount of molecular hydrogen can be measured through models that are calibrated in the post-processing phase (e.g., Lagos et al. 2015; Diemer et al. 2018, 2019; Davé et al. 2019, using for instance the models by Gnedin & Kravtsov 2011). When tested against observational data, such models are in good agreement with the galaxy populations observed in the local Universe (Lagos et al. 2015; Diemer et al. 2019), while they slightly underestimate the amount of molecular gas observed at $z\simeq 2$ (by a factor of 1.5 in EAGLE, Lagos et al. 2015, and by a factor of 2–3 in TNG, Popping et al. 2019).

Following Ward et al. (2022), we compare the predictions for the molecular gas fraction of galaxies at cosmic noon of TNG, EAGLE and SIMBA tuned to the observed range of $M_{\ast}$, SFR and $L_{\rm bol}$ spanned by our sample. We thus filter the simulations’ output at $z=2$ to select galaxies with stellar masses larger than $10^{10}\,{{\rm M}_{\odot}}$ and lower than $10^{11.7}\,{{\rm M}_{\odot}}$ and $\rm\log(SFR/\,{{\rm M}_{\odot}}\leavevmode\nobreak\ yr^{-1})$ within 0.7 and 2.8 (the lower and upper bounds of the observed sample; see Fig. 2). We then identify AGN hosts in the simulations by combining bolometric luminosity and Eddington ratio ($\lambda_{\rm Edd}$) selection criteria. We use $\log{L_{\rm bol}}>44.3$ erg s^-1to span the same bolometric luminosity range as our observations, and we also consider $\lambda_{\rm Edd}>1\%$ to remove those AGN accreting in a radiatively inefficient fashion. We identify non-active galaxies in the simulations as those systems with $\lambda_{\rm Edd}<0.1\%$, yet we note that results are identical if relaxing the selection ($\lambda_{\rm Edd}<1\%$). Similarly to the analysis of Sect. 5.2, we remove from the observed sample those three AGN (cid_178, cid_1605, cid_467) for which both stellar mass and CO luminosity are upper limits.

Simulations only allow to compute the molecular gas fraction as $f_{\rm H_{2}}=M_{\rm H_{2}}/M_{\ast}$, thus we have to assume an $\alpha_{\rm CO}$ value to convert the observed $f_{\rm gas}$ in $f_{\rm H_{2}}=\alpha_{\rm CO}f_{\rm gas}$. One of the backbones of our work is not to elect a value of $\alpha_{\rm CO}$ since it is not trivial to discriminate whether each galaxy is more MS- ($\alpha_{\rm CO}\simeq 3.6\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$; e.g., Daddi et al. 2010a; Genzel et al. 2015) or more starburst-like ($\alpha_{\rm CO}=0.8\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$; e.g., Downes & Solomon 1998; Solomon & Vanden Bout 2005; Calistro Rivera et al. 2018; Amvrosiadis et al. 2023) or somewhere in the middle ($\alpha_{\rm CO}=1.8-2.5\,{{\rm M}_{\odot}}\rm\,(K\ km\ s^{-1}\ pc^{2})^{-1}$, Cicone et al. 2018; Herrero-Illana et al. 2019; Montoya Arroyave et al. 2023). Yet, a good fraction of the AGN sample corresponds to MS host galaxies (see Fig. 2, thus we present the comparison between observed data and simulations assuming $\alpha_{\rm CO}=3.6\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$ and show the shift corresponding to the lowest $\alpha_{\rm CO}$ value ($\alpha_{\rm CO}=0.8\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$) in the plots.

Figure 9 shows the $f_{\rm H_{2}}$ vs. $L_{\rm bol}$ distribution of AGN in the simulations and in our observed sample. Density contours show the distribution of simulated sources in TNG (red), EAGLE (purple) and SIMBA (orange), while observed targets are shown as grey pentagons for ALMA KASHz AGN (presented in this work) and grey circles for SUPER AGN from C21 and this work. Figure 10 shows the comparison of simulations and observations in the $f_{\rm H_{2}}$ vs. SFR plane. We show AGN and non-active galaxies of the simulations computing their median molecular gas fraction in SFR bins of 0.3dex width, and use the 18th, 64th percentiles as errorbars. We also show the un-binned distribution of AGN host galaxies as hexagons, color-coded based on the mean stellar mass of the galaxies in each bin. Figure 11 compares the gas fraction distributions (violin plots) and mean values (diamonds) of AGN hosts and non-active galaxies from cosmological simulations and the total gas fraction distributions and mean values of observed AGN and non-active galaxies as derived in Sect. 5.2.

While the KASHz survey in theory spans a very broad range of X-ray and bolometric luminosity ($\log(L_{\rm X}/$erg s^-1$)=41-45.3$; $\log({L_{\rm bol}}/$erg s^-1$)\simeq 42-47$ using the bolometric correction of Duras et al. 2020), the ALMA KASHz AGN have bolometric luminosity $\log({L_{\rm bol}}/$erg s^-1$)\geq 44.3$, comparable to the lower bound of the observed AGN used in Ward et al. (2022). As a consequence, despite the better sampling of the intermediate-luminosity regime, the $L_{\rm bol}$ range spanned by observed and simulated AGN in Fig. 9 hardly overlap, as seen by Ward et al. (2022). To bridge such a gap, both observations and simulations need to take a leap forward. On the one hand, we need to gather observational information of AGN with lower bolometric luminosity, possibly through targeted programs. On the other hand, simulations still struggle in reproducing the high-${L_{\rm bol}}$ population of AGN (Fig. 9), due to their small box size that prevents to reproduce rare and short lived objects like AGN with ${L_{\rm bol}}>10^{46}$ erg s^-1. We note that this mismatch in ${L_{\rm bol}}$ is still present even by relaxing the SFR and $M_{\ast}$ filtering, as seen in Ward et al. (2022).

Additionally, cosmological simulations suggest that AGN hosts should match or exceed non-active galaxies in gas fraction, which is the opposite of the observed trend (see Fig. 11). We note that also this result still holds even by relaxing the mass and SFR filtering of the simulations’ output, as can be seen in the left column panels of Fig. 7 in Ward et al. (2022). EAGLE and Simba also suggest that AGN host galaxies show similar molecular gas mass for fixed SFR compared to non-active galaxies (see Fig. 10), which is at odds with our result in Sect. 5.1. Yet, AGN hosts in TNG show molecular gas depletion for fixed SFR, even though with an opposite trend compared to the observations: the discrepancy increases for increasing SFR in the simulations, while in the observations it increases for decreasing SFR (see Fig. 4). Figure 10 also shows that simulations do not fully reproduce the observed properties of AGN host galaxies. In fact, SIMBA is the only cosmological simulation that covers the full $M_{\ast}$ and SFR range, while TNG and EAGLE are basically missing AGN hosts with $M_{\ast}>10^{11}\,{{\rm M}_{\odot}}$ and $\rm\log(SFR/\,{{\rm M}_{\odot}}\leavevmode\nobreak\ yr^{-1})\gtrsim 2$. Yet, a direct comparison to the simulations is difficult, as their limited box sizes and subgrid models for AGN feedback and molecular gas estimation reduce the overlap with our observations, especially at the high-mass, high-${L_{\rm bol}}$ end.

Another pressing issue that arises from the comparisons in Figures 9–11 is related to the choice of $\alpha_{\rm CO}$. On the one hand, the gas fraction distributions of AGN hosts in the simulations are in fair agreement with the observed distributions of KASHz and SUPER AGN in the case of $\alpha_{\rm CO}=3.6\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$ (see Fig. 11), in the sense that their mean values are consistent, but simulations seem to lack more gas-rich systems, as is evident also as a function of bolometric luminosity in Figs. 9. On the other hand, simulations fail to reproduce the observed $f_{\rm H_{2}}$ distributions in the case of $\alpha_{\rm CO}=0.8\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$ (see Fig. 11) but, at the same time, would show a better agreement with the observed $f_{\rm H_{2}}$ vs. $L_{\rm bol}$  distribution (see Fig. 9), at least in the range of $L_{\rm bol}$, $M_{\ast}$ and SFR used to filter the simulations’ output. The clear impact of the adopted value of $\alpha_{\rm CO}$ on the observed results in Figs. 11 and 9 further stresses the importance of measuring tracers that can be used to derive or set a prior for $\alpha_{\rm CO}$, for instance metallicity and distance from the MS (e.g., Elbaz et al. 2007; Noeske et al. 2007; Accurso et al. 2017), two quantities that are not available for the full sample of AGN used in this work.

Our results show that AGN at $z=1-2.6$, uniformly selected and analyzed, without prior knowledge on feedback tracers (e.g. ionised outflows or radio jets), and non-active galaxies, matched in z, $M_{\ast}$, SFR, are statistically different in terms of $L^{\prime}_{\rm CO(1-0)}$ vs. $L_{\rm FIR}$ (3$\sigma$ level; see Sect. 5.1), and $\log($$L^{\prime}_{\rm CO}$$/M_{\ast})$ distribution (proxy of the molecular gas fraction distribution; p-value $<10^{-7}$; see Sect. 5.2). Yet, even though there are hints of gas depletion even at fixed stellar mass (see Fig. 7-8), AGN and non-active galaxies are statistically consistent in the $L^{\prime}_{\rm CO(1-0)}$ vs. $M_{\ast}$, $f_{\rm gas}$ vs. $M_{\ast}$ and $\log($$L^{\prime}_{\rm CO}$$/M_{\ast})$ vs. $M_{\ast}$ two-dimensional spaces (see Sect. 5.3).

The control sample of non-active galaxies is built to match the properties shown by AGN host galaxies, thus in principle the sample of selected non-active galaxies is an analog of our AGN hosts with the only difference of not hosting an AGN. Moreover, we robustly controlled for consistency in SFR and $M_{\ast}$ between the two samples (see Sect. 3) and that our results stay unvaried when considering a subsample of AGN and non-active galaxies. Additionally, we trace the SF level in our galaxies through $L_{\rm FIR}$, i.e. corresponding to SF averaged over 100 Myr. Thus, we interpret our comparisons of AGN hosts and non-active galaxies in terms of differences (or absence of) in the molecular gas content due to the presence of the AGN.

In particular, we find that AGN hosts are, on average, more CO depleted for fixed FIR luminosity (see Fig. 4). In other words, AGN hosts show a reduced amount of cold molecular gas compared to matched non-active galaxies, and such reduced CO luminosities may convert into reduced SF in AGN hosts in the future. We stress that this difference holds also when we exclude AGN and non-active galaxies that are not well matched, thus does not depend on biases in building the control sample. Moreover, CO depletion is differential in $L_{\rm FIR}$, meaning that it increases for decreasing $L_{\rm FIR}$. CO content and molecular gas fraction for fixed stellar mass of AGN and non-active galaxies are different at the 2$\sigma$ level (see Figs. 7–8). Such a reduced difference for fixed stellar mass could be driven by the differential CO depletion level for fixed SFR (see Fig. 4): at fixed stellar mass, we are sampling galaxies that span a broad range of SF levels, which differentially depend on stellar mass (e.g., see Fig. 8 of Saintonge & Catinella 2022). In this sense, the mixing of host galaxies with different SF levels and the differential CO depletion observed for fixed $L_{\rm FIR}$ (see Fig. 4) likely dilute the differences between AGN and non-active galaxies for fixed stellar mass. Yet, we do not find more enhanced differences considering only targets on the MS, possibly due to the small sample size (14 AGN host galaxies on the MS). Differences in the one-dimensional distributions of $f_{\rm gas}$ are more prominent because the contribution of CO detections and upper limits in the AGN sample is better decoupled: in fact, the mean value of the distribution is mainly driven by CO detections, while CO upper limits (and in this case, $f_{\rm gas}$ upper limits) shape the left wing of the distribution, producing the skewness at lower values that is missing in the distribution of non-active galaxies.

We confirm the hints observed by C21 in SUPER ALMA AGN, using a uniformly selected and homogeneously analyzed AGN sample that is twice in size: molecular gas depletion is significant also when considering moderate-luminosity AGN ($44\lesssim\log({L_{\rm bol}}/{\rm erg\leavevmode\nobreak\ s^{-1}})\lesssim 47$). This work thus supports the scenario for which AGN play a key role in shaping the life of galaxies, in particular by altering the molecular gas reservoir of their hosts, as was observed in higher-${L_{\rm bol}}$ source ($46\lesssim\log({L_{\rm bol}}/{\rm erg\leavevmode\nobreak\ s^{-1}})\lesssim 48$; e.g., Harrison et al. 2012; Brusa et al. 2018; Vietri et al. 2018; Perna et al. 2018; Bischetti et al. 2021). This is also coherent with the recent results in Frias Castillo et al. (2024) in moderate-luminosity, lensed AGN ($43\lesssim\log({L_{\rm bol}}/{\rm erg\leavevmode\nobreak\ s^{-1}})\lesssim 46$) at end of cosmic noon ($z=1.2-1.6$), which are seen as CO depleted for fixed FIR luminosity compared to star forming galaxies. We also investigated the possibility of a dependency of the molecular gas fraction with properties of the AGN, like the bolometric luminosity, yet found no significant correlation, as already seen in previous studies (e.g., Shangguan et al. 2020b; C21).

In this work, we assumed a CO SLED at $J<5$ for AGN hosts that is consistent with that of non-active galaxies. This is a conservative assumption: works focused on determining the CO ladder of AGN found them to show higher $r_{J,1}$ values compared to non-active galaxies even at $J<5$, despite the large error bars (e.g., Kirkpatrick et al. 2019; Boogaard et al. 2020). Moreover, the backbone of our analysis, and other previous studies, is to compare the gas content of AGN host galaxies and non-active galaxies in terms of $L^{\prime}_{\rm CO}$, thus without assuming an $\alpha_{\rm CO}$ value. In this way, the difference found between the two samples in molecular gas mass and fraction represents a lower limit of the real discrepancy: in fact, this would be unchanged if we were to assume the same $\alpha_{\rm CO}$ value for both AGN hosts and non-active galaxies, while the difference would increase in the case of the reasonable assumption of AGN hosts showing lower $\alpha_{\rm CO}$ values than non-active galaxies (e.g., see Bolatto et al. 2013, and references therein). Given all our conservative assumptions, our results indeed provide evidence for AGN reducing the molecular gas reservoir of their hosts and thus their evolution.

Bischetti et al. (2021) recently investigated the effects of AGN feedback on the molecular gas reservoir of WISSH quasars. Among the various tests, the authors compared the gas fraction of WISSH AGN with that of star-forming galaxies after correcting for the evolution of the gas fractions with redshift and distance from the MS ($\rm\Delta MS=sSFR/sSFR_{MS}$). In fact, analyses focused on assessing the molecular gas content of star-forming galaxies up to $z\simeq 4$ (e.g., Genzel et al. 2015; Tacconi et al. 2018) pointed out a dependency of the gas fraction on redshift and distance from the MS. Our analysis already accounts for the redshift evolution of the MS when building the control sample by matching AGN and galaxies in two redshift bins ($z\simeq 1-1.8$, $z\simeq 2-2.6$), within which the MS evolution is not large ($\simeq 0.2$dex; e.g., Whitaker et al. 2012). However, not correcting for the distance from the MS could still affect the comparison of non-active galaxies and AGN hosts. Unfortunately, such a correction is not viable for the AGN sample used in this work: about half of our targets have upper limits on $L_{\rm FIR}$, making impossible to constrain how much they deviate from the MS. Nonetheless, we tentatively measure the $\rm\Delta MS=sSFR/sSFR_{\rm MS}$ parameter of the AGN sample considering $L_{\rm FIR}$ at face value and compute the gas fractions corrected for the distance from the MS ($f_{\rm gas}^{\rm corr}$) using Equation 15 of Genzel et al. (2015) in which $\rm sSFR_{\rm MS}$ is parameterized as in Whitaker et al. (2012). We conservatively assume $\alpha_{\rm CO}=3.6\,{{\rm M}_{\odot}}\rm\ (K\ km\ s^{-1}\ pc^{2})^{-1}$ for both AGN and non-active galaxies. Applying the analysis of Sect. 5.2, we find that the $f_{\rm gas}^{\rm corr}$ total distributions of AGN and non-active galaxies are still significantly different according to the KS test (p value¡0.001), with consistent mean values, both using the full sample and excluding those AGN and non-active galaxies without a good match (see Sect. 3). Following the analysis of Sect. 5.3, we also find that AGN and non-active galaxies are consistent with each other within 2$\sigma$ in the $f_{\rm gas}^{\rm corr}$ vs. $M_{\ast}$ parameter space.

Yet, there are still some effects that we are not fully considering, or cannot fully consider, when studying AGN feedback. First, we are comparing two phenomena that occur on very different timescales (e.g., Harrison 2017; Harrison et al. 2018; Harrison & Ramos Almeida 2024): the timescales of visible AGN activity do not necessarily mirror the AGN duty cycle, plus the timescales of feeding and feedback in general are uncertain. For instance, the AGN might have become less luminous or non-visible by the time the effects of its impact have become observable, or even the galaxy might have accreted more gas from its surroundings (e.g., from the CGM) reducing the overall depletion effect of AGN activity. Additionally, we have no means of understanding how long the currently observed AGN has been active or if it is the $n$-th active phase in a cycle of flickering accretion. In other words, we cannot know how much energy the AGN has input in the surrounding medium before being observed, and thus on the total energy input in the system on timescales that are relevant for SF activity.

Second, we are comparing phenomena and effects possibly happening on spatial scales that are very different: high-resolution studies in the local Universe unveiled AGN that create regions of CO depletion in their surroundings (within the central few 100 pc to kpc; e.g., Sabatini et al. 2018; Rosario et al. 2019; Fluetsch et al. 2019; Feruglio et al. 2020; Ellison et al. 2021), where the bulk of the molecular gas is possibly in a warmer phase due to AGN heating (e.g., through accretion radiation in the form of X-ray photons, Fabbiano et al. 2019). Studies regarding SF activity in local AGN confirm such a scenario where AGN have a significant impact only in their surroundings, either reducing (e.g., Molina et al. 2023) or enhancing (e.g., Cresci et al. 2015; Garc´ıa-Burillo et al. 2021; Lammers et al. 2023) the central SF without affecting it on larger scales. Yet, this work supports the scenario in which AGN at cosmic noon are indeed affecting the total molecular gas mass of galaxies (e.g., total molecular gas mass, future total SFR), and thus a significant portion of the full system. An additional possibility is that AGN excite CO higher up in the rotational ladder without making the host molecular gas poor (e.g., Paraficz et al. 2018; Fogasy et al. 2020; Spingola et al. 2020). However, such a scenario cannot be probed with the present sample since many AGN were targeted in only one CO transition.

Third, the SUPER and KASHz surveys have selected AGN only based on their X-ray luminosity and redshift, with no priors regarding AGN-driven outflows and AGN feedback effects. Here and in C21, we investigated AGN feedback by studying the overall properties of the AGN sample compared to those of non-active galaxies. Literature results finding AGN hosts to be significantly gas depleted, or with reduced depletion timescales, are usually associated with sources showing massive AGN-driven outflows on kpc-scales in at least one gas phase and that are often caught at the peak of their power (e.g., Carniani et al. 2017; Brusa et al. 2018; Loiacono et al. 2019, but see also Scholtz et al. 2021; Lamperti et al. 2021). One might then infer that gas depletion is significant and widespread only in those sources where feedback is “caught in the act”. However, gas depletion in AGN host galaxies could be also due to earlier AGN event(s), so that the reduction in gas content and/or SF activity is a product of the integrated past activity of the AGN (e.g., Piotrowska et al. 2022). A detailed comparison between [OIII] outflows and CO properties of KASHz and SUPER targets is out of the scope of the present study and will be the subject of a future work.

Lastly, this work provides a uniformly-selected and -analyzed, sizable sample to compare observations and predictions of cosmological simulations. We compared the observed cosmic noon AGN and sample of non-active galaxies with the properties of AGN and galaxies from three cosmological simulations (TNG, EAGLE and SIMBA, following Ward et al. 2022), by filtering their output at $z=2$ to match the properties of the observed sample of AGN (see Sect. 6). One of the issues encountered in Ward et al. (2022), and also in this work (see Fig. 9), is that simulations and data hardly overlap, especially in terms of bolometric luminosity. Our observed sample provides a better coverage of the $44<\log({L_{\rm bol}}/{\rm erg\leavevmode\nobreak\ s^{-1}})<46$ than previously available. However, a robust sampling of observed AGN at $\log({L_{\rm bol}}/{\rm erg\leavevmode\nobreak\ s^{-1}})<44$, where the bulk of simulated AGN lies, is still lacking. Moreover, the three simulations predict that AGN hosts should match non-active galaxies in molecular gas mass for fixed SFR (see Fig. 10) and that they should even exceed non-active galaxies in gas fraction (see Fig. 11). The only exception is TNG, which predicts AGN to be gas depleted for fixed SFR but with a trend (increasing gas depletion for increasing SFR) that is opposite to what seen in the observations (see Fig. 4). Another issue encountered in these comparisons is that only SIMBA covers the full parameter range of observed AGN hosts, while TNG and EAGLE are missing AGN hosts with $M_{\ast}>10^{11}\,{{\rm M}_{\odot}}$ and $\log(\rm SFR/\,{{\rm M}_{\odot}}\leavevmode\nobreak\ yr^{-1}\gtrsim 2$. Lastly, We also highlight another (well-known) issue: the selection of the “right” $\alpha_{\rm CO}$ value. In fact, how well (or not well) the simulations agree with the observed properties strongly depends on the chosen value for $\alpha_{\rm CO}$ (see Sect. 6 and Figs. 11-9). For the present sample, for instance, additional observations tailored at measuring the metallicity and better constrain the distance from the MS could allow to take advantage of the continuous $\alpha_{\rm CO}(z,\Delta MS)$ function of Accurso et al. (2017) and solve this issue.

This work addresses the effects of AGN activity on the molecular gas content of host galaxies with a carefully-selected, sizable sample of AGN at cosmic noon, to shed light on whether AGN have any impact on the fuel of future star formation.

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  We collected all the available ALMA observations of CO ($J<5$) emission lines of KASHz AGN at cosmic noon, a survey built to study AGN feedback in X-ray selected AGN as traced by the ionized gas component. We complement our KASHz ALMA sample with SUPER ALMA targets from C21 (observed in CO(3-2)). Being heterogeneous in the CO transitions observed (CO(2-1), CO(3-2), CO(4-3)), we assumed a reasonable CO excitation ladder (from Kirkpatrick et al. 2019) to convert our measurements to the CO(1-0). Other key quantities of the AGN+galaxy system were retrieved through SED fitting with the CIGALE code (Boquien et al. 2019; Yang et al. 2020), collecting the most up-to-date multi-wavelength broad-band photometry as provided by the deep-field survey collaborations (CANDELS/GOODS-S, COSMOS, HELP collaboration).

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  The AGN sample thus totals 46 AGN at $z=1-2.6$, i.e. a sample size that is twice that of the SUPER AGN presented in C21 and with a larger CO detection rate ($\simeq 50$% vs. 40%). We built the control sample of non-active galaxies (98 galaxies), matched in z, $M_{\ast}$, SFR to our AGN sample, by capitalizing on the PHIBSS project (Tacconi et al. 2018, and references therein), on the ASPECS survey (Boogaard et al. 2020, and references therein) and on the ALMA/NOEMA survey of sub-$mm$ galaxies in the COSMOS, UDS, and ECDFS fields by Birkin et al. (2021).

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  We then quantitatively compared AGN and non-active galaxies in a Bayesian framework, as developed by C21, robustly considering upper and lower limits in any of the considered quantities. The samples were compared in terms of i) $L^{\prime}_{\rm CO(1-0)}$ vs. $L_{\rm FIR}$ (see Sect. 5.1), ii) $\log($$L^{\prime}_{\rm CO}$$/M_{\ast})$ distribution (proxy of the gas fraction distribution; see Sect. 5.2), iii) $L^{\prime}_{\rm CO(1-0)}$ vs. $M_{\ast}$, and iv) $\log($$L^{\prime}_{\rm CO}$$/M_{\ast})$ vs. $M_{\ast}$ (see Sect. 5.3). Our results confirm the hints observed by C21 on SUPER AGN: AGN hosts at cosmic noon, selected without priors regarding AGN feedback, are significantly gas depleted compared to non-active galaxies. Differences in the $\log($$L^{\prime}_{\rm CO}$$/M_{\ast})$ vs. $M_{\ast}$ and $L^{\prime}_{\rm CO(1-0)}$ vs. $M_{\ast}$ two-dimensional spaces are possibly washed out by the rather uniform distribution of CO upper limits with respect to stellar mass (see Figs. 7-8), and the differential CO depletion observed for fixed $L_{\rm FIR}$ (see Fig. 4).

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  Building on the study of Ward et al. (2022), we also compared our observed samples of AGN and non-active galaxies with the output of three cosmological simulations (TNG, EAGLE, SIMBA), filtered to match the properties of the observed AGN, AGN hosts and non-active galaxies (see Sect. 6). We find that SIMBA is the only simulation that can well reproduce the mass and SFR of host galaxies, TNG is the only simulation predicting gas depletion for fixed SFR in AGN hosts, but with a trend opposite to that of observations (see Fig. 10), and that all three equally fail at reproducing the observed distribution of molecular gas fractions (see Fig. 11) and bolometric luminosity (see Fig. 9).

The latest (and upcoming) facilities have opened (and will broaden) new windows to assess AGN feedback in the high-z Universe, especially regarding high-spatial-resolution studies. ALMA has proven its outstanding capabilities over the past decade but high-resolution observations, matching the spatial resolution of NIR data, are still highly time-consuming, yet necessary to fully understand how AGN affect the molecular gas distribution within their hosts and bridge the gap between the local Universe and cosmic noon. Nonetheless, JWST/MIRI will allow us to probe molecular transitions previously inaccessible at cosmic noon, like the roto-vibrational $\rm H_{2}$ lines, which can shed new light on the effects of AGN activity on the molecular gas phase. Three SUPER targets (one of which is shared with KASHz), selected based on their prominent [O III] outflows, will be observed by JWST/MIRI/MRS (PI: V. Mainieri; wavelength coverage: $\lambda\simeq 6.5-27.9\leavevmode\nobreak\ \mu$m) to map the warm-molecular gas phase, thus completing their multi-phase characterization. We will be able to determine the dynamics of the molecular gas and to derive the total (ionized+molecular) mass outflow rate and kinetic energy for these outflows, some of the most difficult wind parameters to measure with accuracy, thus providing a key constrain for current models of AGN feedback (e.g., Costa et al. 2020).