Star formation efficiency and AGN feedback in narrow-line Seyfert 1 galaxies with fast X-ray nuclear winds

We present the first systematic study of the molecular gas and star formation efficiency in a sample of ten narrow-line Seyfert 1 galaxies selected to have X-ray Ultra Fast Outflows and, therefore, to potentially show AGN feedback effects. CO observations were obtained with the IRAM 30m telescope in six galaxies and from the literature for four galaxies. We derived the stellar mass, star formation rate, AGN and FIR dust luminosities by fitting the multi-band spectral energy distributions with the CIGALE code. Most of the galaxies in our sample lie above the main sequence (MS) and the molecular depletion time is one to two orders of magnitude shorter than the one typically measured in local star-forming galaxies. Moreover, we found a promising correlation between the star formation efficiency and the Eddington ratio, as well as a tentative correlation with the AGN luminosity. The role played by the AGN activity in the regulation of star formation within the host galaxies of our sample remains uncertain (little or no effect? positive feedback?). Nevertheless, we can conclude that quenching by the AGN activity is minor and that star formation will likely stop in a short time due to gas exhaustion by the current starburst episode.


INTRODUCTION
It is now well-established that active galactic nuclei (AGN) and their hosts galaxies are evolving together, influencing each other (see the review by Heckman & Best 2014). In particular, AGN feedback is often invoked as a key element for star formation and galaxy evolution (Di Matteo et al., 2005;Hopkins & Elvis, 2010). However, the mechanisms governing AGN feedback are still not understood. Galaxy outflows and winds driven by the AGN have the potential to impact star formation. Several models (e.g. Faucher-Giguère & Quataert 2012; King & Pounds 2015) have proposed that quasar feedback is initiated via a subrelativistic wind launched at the scale of the accretion disc with velocity v > 10 4 km s −1 and observed in X-ray spectra as ultra-fast outflows (UFO; Tombesi et al. 2010Tombesi et al. , 2012. While traveling outward, the ultra-fast wind interacts with the surrounding interstellar medium (ISM), producing large scale outflows that are commonly observed in the optical band (Marasco et al., 2020;Robleto-Orús et al., 2021) and at millimetric frequencies (Feruglio et al., 2010(Feruglio et al., , 2015Cicone et al., 2012Cicone et al., , 2014Tombesi et al., 2015;Bischetti et al., 2019;Zanchettin et al., 2021;Longinotti et al., 2023). At the contact discontinuity produced in the shock of the wind with the ISM, the pressure and momentum of the wind are conserved. The interaction also produces an inner reverse shock front where the nuclear wind slows, and an outer forward shock which accelerates the ISM (see the review by King & Pounds 2015). If the inner and outer shock fronts are as- Table 1. Sample of NLSy1 galaxies with known X-ray UFO. The columns correspond to: (1) Name of the galaxy; (2-3) central coordinates; (4) spectroscopic redshift; (5) luminosity distance; (6) AGN bolometric luminosity estimated by fitting the multi-band SED with the code CIGALE (see Section 3.3 and Table  5); (7) mass of the central black hole; (8) Eddigton ratio λ Edd = L AGN /L Edd where L Edd is the Eddington luminosity; (10) velocity of the X-ray UFO,and (11) references for the black hole mass and UFO velocity. The demarcation line separates the sources observed with the IRAM 30m from those with ancillary CO data. Notes. (a) CO-estimated redshift (see Salomé et al. 2021  sociated with cooling, a large fraction of the kinetic energy of the nuclear wind is dissipated (momentum-conserving outflow). Conversely, if the shocked regions are expanding semi-adiabatically, most of the kinetic energy is transmitted to the large-scale outflow (energy-conserving outflow) and the momentum flux at large scale is boosted (King & Pounds, 2015).
A common method to study the coupling of the nuclear wind with the galaxy-scale outflow is to look at the energetics of the X-ray and of the molecular phases (e.g. Feruglio et al. 2015;Longinotti et al. 2023). Few sources show the presence of both an X-ray UFO and a molecular outflow (e.g. Mrk 509; Tombesi et al. 2010;Zanchettin et al. 2021). In four cases, the momentum flux of both phases shows that the outflows seem to be driven by an energy-conserving wind, while for the others sources they are consistent with the momentum-conserving scenario. Marasco et al. (2020) suggested that these two well-defined regimes trace an evolutionary path from highly accreting sources, which are still growing their black holes, to a phase where the outflows can finally eject most of the ISM by reaching galaxy scales, quenching both star formation processes and black hole growth.
In this regard, narrow-line Seyfert 1 galaxies (NLSy1) represent an interesting class of sources where to explore feedback properties. NLSy1 are young AGN characterised by a high accretion rate (Eddington ratio larger than 0.1; Boroson & Green 1992;Mathur 2000b,a) on supermassive black holes with generally small masses (M BH < 10 8 M ⊙ ; Peterson 2011). Recent findings of Xray fast outflowing gas (Gupta et al., 2013;Longinotti et al., 2015;Parker et al., 2017;Krongold et al., 2021) and relativistic jets (Lähteenmäki et al., 2017 suggest that nuclear winds may be a common feature of this AGN class possibly due to their high accretion rate. Although the bolometric luminosity of NLSy1 is rather moderate compared to the typical dust and gas-rich sources where feedback processes are expected (ULIRGs and quasars; Cicone et al. 2014;Järvelä et al. 2015), the presence of X-ray UFO may play a role in triggering galaxy scale outflows capable of producing efficient feedback.
In this paper, we seek for any possible effect indicating that the presence of nuclear fast outflow winds may imprint in the properties of the host galaxy at large scale. We therefore characterise the molecular gas content (as traced by the CO emission) and study the star formation efficiency (as traced by the molecular depletion time) in a sample of NLSy1 for which a X-ray UFO was clearly observed. We aim to establish whether the molecular gas is affected or not by the presence of the AGN-driven winds. The sample and the observations are presented in Section 2 and analysed in Section 3. We then discuss the relation between the molecular gas reservoir, the stellar mass and the star formation in Section 4, as well as the possible impact of the AGN activity in Section 5. Finally, Section 6 will summarise the results. Throughout this paper, we assume H 0 = 70, Ω M = 0.3, Ω vac = 0.7.

Sample
We compiled our sample with all the known NLSy1 for which nuclear X-ray fast winds are well established. We also limit our sample to the sources observable from the IRAM 30m telescope (declination above -15 degrees). We selected the following sources: PG 1211+143, Mrk 205, Mrk 766 from positive detections in the sample of Tombesi et al. (2010); Ark 564, I Zw 1, PG 1448+273, IRAS 13349+2438, Mrk 1044, IRAS 17020+4544 (hereafter IRAS 17020) all from the literature on individual objects; and Mrk 110 from our ongoing work on X-ray winds. The selected sample consists of 10 sources. The references for the UFO in each of them are given in Table 1.
With the IRAM 30m, we observed the five northern sources with no previous CO observations: IRAS 13349+2438 (hereafter IRAS 13449), Ark 564, Mrk 110, Mrk 205 and Mrk 1044. We also re-observed IRAS 17020 as a reference target to see whether molecular outflows can be detected or not. However, the data are not sensitive enough to detect the broad emission observed with the LMT and NOEMA by Longinotti et al. (2018Longinotti et al. ( , 2023.
Global properties of the galaxies in our sample are presented in Table 1. Although limited in size, our sample spans two orders of magnitude in bolometric luminosity and in black hole mass.

Observational setup
Six NLSy1 galaxies of our sample were observed in CO(1-0) and CO(2-1) simultaneously with the IRAM 30m telescope, with the exception of Mrk 205 for which the redshift did not allow us to find a configuration to observe both lines simultaneously. Observations were made in May-June 2020 using the EMIR receiver 1 with the FTS (bandwidth of 2 × 4 GHz; resolution of 195 kHz) and WILMA backends (bandwidth of 4 GHz; resolution of 2 MHz). At the redshift of the galaxies, the lines are observable at frequencies of 104-113.5 GHz and 208-227 GHz, which leads to a beam of about 22 ′′ − 24 ′′ and 11 ′′ − 12 ′′ , respectively. The beams of the IRAM 30m thus translate into spatial scales for our sources that go from 3.7 − 7.4 kpc in Mrk 1044 to 24 − 48 kpc in IRAS 13349. Along the observing nights, the system temperature varied between 100-300 K at 3mm and 200-700 K at 1mm. During observations, the pointing was monitored by observing standard continuum sources tuned to the frequency corresponding to the redshifted CO(1-0) emission line. Observations were obtained using wobbler switching with a rate of ∼ 0.5 Hz. Six-minute scans were taken, and a calibration was made every three scans. The observing time on-source varies between 60-120 min, providing a noise level of ∼ 0.9 − 1.7 mK in CO(1-0) and ∼ 1.5 − 3.7 mK in CO(2-1) in channels of 20 − 30 km s −1 (see Table 2 for the details). Pointing was checked every few hours and was generally determined to be accurate to within a few arcseconds.

CO emission from IRAM 30m
The data were reduced and analysed using the IRAM package CLASS 2 . The average spectra were first smoothed to a spectral resolution of 20 − 30 km s −1 (refer to Table 2 for the details). The baseline was then subtracted by fitting the channels with no signal with a linear function or a degree 2 polynomial. In the case of the CO(1-0) spectrum in Mrk 1044, a platform is observed with a turnover velocity ∼ 320 km s −1 , outside the velocity range of the CO emission. This was corrected by fitting a baseline for each platform separately. The resulting spectra are plotted in Appendix A. The CO(1-0) emission is detected in all six galaxies with a signalto-noise ratio ≥ 3.5 for the peak temperature. Regarding the CO(2-1) line, it was not detected in Mrk 110. The CO emission presents a double-horn profile in IRAS 13349, IRAS 17020 and Mrk 1044, suggesting the presence of molecular gas disc. On the other hand, only one gaussian-like profile is observed in the other galaxies. We fitted the CO emission using one or two Gaussians. The characteristics of the emission lines are summarised in Table 2. IRAS 17020+4544 -The CO(1-0) emission was previously observed with the LMT (Longinotti et al., 2018) and NOEMA (Salomé et al., 2021). With the IRAM 30m, we found a CO luminosity in the host galaxy, in agreement with both the LMT and NOEMA measurements. We thus confirm the conclusion of Salomé et al. (2021) that all the CO emission was recovered by the NOEMA observations, with no spatial filtering by the interferometer. However, the data from the IRAM 30m show no evidence for the molecular outflows detected with the LMT (Longinotti et al., 2018) and NOEMA (Longinotti et al., 2023). The rms noise of 3.9 mJy we reached with the IRAM 30m for CO(1-0) is higher than the peak temperature of 1mJy for the resolved outflows with NOEMA and 1.1 mJy for the broad component observed with the LMT. Therefore, the non detection of the outflow components is simply due to a lack of sensitivity.
IRAS 13349+2438 -The CO(1-0) spectrum presents a double-horn profile, while the CO(2-1) emission does not. A plausible explanation is that it is a result of the difference in the beam size. In particular, the beam of the IRAM 30m for the CO(2-1) is about half the optical diameter of the galaxy and the CO(2-1) emission may not be fully encompassed with one beam. Therefore, a significant fraction of the CO(2-1) can potentially be missed by the IRAM 30m. However, we cannot exclude that the gas close to the centre is more excited and presents brighter CO(2-1) emission than the outskirt region.
We note that the two peaks in the CO(1-0) spectrum of IRAS 13349 are redshifted compared with the optically estimated redshift (z opt = 0.107641 based on the Hα, Hβ and [OIII] lines; Kim et al. 1995). Using the profile of the CO(1-0) emission, we find a new systemic velocity (corresponding to the centre of the doublepeak) redshifted by 240 km s −1 from the nominal frequency. This corresponds to a redshifted frequency of 103.98596 GHz. We thus get a CO-estimated redshift z CO = 0.10852, which corresponds to a luminosity distance of 502.4 Mpc. Differences between the systemic velocity of the CO lines and optical lines are commonly observed. Such difference may be produced by outflowing gas within the narrow-line region or dynamical perturbations due to galaxy interactions (e.g. Davies et al. 2020). We note that the velocity difference between the optical and CO emission in IRAS 13349 is similar to the velocity difference observed in IRAS 17020 (Salomé et al., 2021). This suggests that the same process may be at play in both galaxies.
Interferometric observations of the molecular gas and integral field spectroscopy of IRAS 13349 will be necessary to investigate both the different profiles of the CO(1-0) and CO(2-1) emission, and the velocity difference between the CO and optical emission.

CO line ratio
In this section, we look at the integrated intensities line ratio when both CO lines were observed. In the case of a double peak profile, we consider the two peaks at the same time to derive the total intensities. For the galaxies observed at IRAM 30m, the intensities were derived in units of the main beam temperature. To derive accurate line ratios, we must consider units of the brightness temperature and correct them from the beam difference. We derived the line ratio using R 21 = I CO21 Ω CO21 I CO10 Ω CO10 (see Wilson et al. 2009). This equation assumes that in either line the sources are unresolved by the 30m beams. This is unlikely the case here (see below) but, in absence of spatial information on the CO emission, we use these line ratios as an indication.
The line ratio for the galaxies in our sample is smaller than unity (see Table 3), which indicates that the CO emission is optically thick. If the gas was thermally excited, the excitation temperature should be smaller than 10 K to explain the line ratios (see Braine & Combes 1992). However, Ocaña Flaquer et al. (2010); Table 2. Properties of the CO emission as observed with the IRAM 30m observations: (1) Short name; (2) on-source observing time; (3) CO transition; (4) central redshifted frequency; (5) noise rms; (6) channel width for which the rms was derived; (7) main beam peak temperature; (8) full-width at half maximum; (9) peak velocity; (10) integrated intensity. If the line emission presents a double-peak profile, we characterised each peak with a Gaussian and give the values for each peak. We have considered a systematic uncertainty of 10% on the absolute flux calibration. Notes. For the upper limit of the CO(2-1) emission in Mrk 110, we assumed a full-width at half maximum FWHM = 80 km s −1 . Table 3. Derived properties of the CO emission: (1) CO luminosity corrected for aperture effect; (2) line ratio in the hypothesis of unresolved emission; (3) molecular gas mass derived from the CO luminosity with the CO-to-H 2 conversion factor in Section 3.4. When the spectrum presents a double-horn profile, the CO luminosity is the sum of the two Gaussians. The demarcation line separates the sources observed with the IRAM 30m from those with ancillary CO data.
In the case of IRAS 17020, the molecular gas content is fully covered by the CO(1-0) beam size. However, it extends further than the FWHM of the CO(2-1) beam size: 15 ′′ versus 11.6 ′′ (17.7 vs 13.7 kpc; Salomé et al. 2021). In 14 galaxies from the HERACLES survey, Leroy et al. (2009) found that the CO emission extends to about half the radius of the B-band 25 mag arcsec −2 isophote. For IRAS 13349, the molecular gas is thus expected to extend on the same scale as the FWHM of the beam of the CO(2-1). However, the distribution of the molecular gas in IRAS 17020 extends to a radius about 15% larger than half r 25 therefore, some CO(2-1) emission may have been missed. In Ark 564, Mrk 1044, Mrk 110 and Mrk 766, the molecular gas is predicted to extend to about 80% of the beam of the CO(1-0) emission. Therefore, a large fraction of the CO(2-1) emission is missing, implying that the line ratio is a lower limit.
With the present observations, it is impossible to determine precisely the influence of the fields of view. To derive accurate line ratios and study the excitation of the CO emission in the galaxies of our sample, we therefore need to resolve the molecular gas in both lines with interferometric observations. We cannot exclude that the CO(1-0) emission extends further than the CO(1-0) beam. However, the radial profile of the molecular gas in star-forming galaxies shows an exponential decrease (Leroy et al., 2009;Saintonge et al., 2011), indicating that the majority of the CO(1-0) emission has been recovered by our observations. Saintonge et al. (2011) derived an aperture correction based on the optical diameter D 25 which we applied to our IRAM 30m galaxies. In the case of IRAS 17020, we see that this can also correct the CO luminosity calculated using the formula of Solomon et al. (1997) from the distribution of the emission within the beam. The corrected luminosities and molecular gas masses are reported in Table  3.

Multi-band SED fitting
To study star formation efficiency, we will compare the molecular gas mass reservoir as traced by the CO emission with global properties of the galaxies of our sample (e.g. stellar mass, star formation rate, AGN bolometric luminosities). To estimate these properties, we fitted the multi-band spectral energy distribution (SED) with the version 2020.0 of the Code Investigating GALaxy Emission (CIGALE; Noll et al. 2009;Boquien et al. 2019). The code considers several distinct emission components: (i) stellar emission, dominating the wavelength range 0.3−5µm; (ii) the FIR emission by cold dust; (iii) the emission from a central AGN, as direct  . The blue, green and red shaded areas show the contribution of the stellar emission, the AGN emission and the dust thermal emission, respectively. Those shaded areas are compiled using the most probable input parameters returned by CIGALE and correspond to the galaxy properties reported in Table 5. Right: Comparison between the SFR derived by CIGALE from the stellar population modelling and the SFR derived from the dust FIR emission for the galaxies in our sample. The dotted line shows the identity relation. energy coming from the accretion disc and reprocessed emission by the dusty torus; (iv) radio synchrotron emission. After assembling the models, given a range of input parameters, the code computes the model-expected fluxes that are compared to the observed photometry through a Bayesian statistical analysis.
To build the SED, we compiled the multi-band photometry from the far ultraviolet (FUV) to the far infrared (FIR). The ultraviolet from GALEX (Martin et al., 2005) comes from the NED archive 3 . In the optical, we used the photometry in the u',g',r',i',z' filters of the SDSS 4 (York et al., 2000;Eisenstein et al., 2011). For Mrk 205, there is no ancillary data from SDSS therefore we used the photometry from the Hubble Source Catalog 5 (Whitmore et al. 2016 (2003) initial mass function (IMF), assuming a delayed exponentially declining star formation history (SFH) for the main population and a decaying exponential for a late starburst event. We used a fixed metallicity Z = 0.02, the solar value. The modelled stellar emission is attenuated by dust, following a modified version of the Calzetti et al. (2000) curve. A reduction factor E(B − V) old /E(B − V) young = 0.44 (Jarvis et al., 2019) is applied to account a differential reddening for the old stellar population (> 10 Myr). The FIR dust emission is reproduced using the library of Dale et al. (2014), without contribution from the AGN. Therefore, the library only accounts for the contribution from star formation. Finally, the AGN emission is treated separately using the torus models from Fritz et al. (2006). We only considered the AGN templates which are compatible with type 1 AGN (Ψ ≥ 40 ∘ ; see Table 4).
For each set of parameters, CIGALE models the associated components of the SED. It can also derive the flux in given bands and estimate physical quantities, like the stellar mass, the star formation rate (SFR) or the AGN bolometric luminosity. The code then uses a Bayesian-like approach by weighting the models depending on "their goodness-of-fit" (Boquien et al., 2019). This returns the most probable values for the output parameters, along with their uncertainties. We report the main physical quantities in Table 5 and show an example of SED fitting in Figure 1 (left panel). The SEDs of the whole sample are presented in Appendix B. When GALEX or Herschel data are missing, the flux in those band is modelled by CIGALE, which provides a prediction of the real flux for future observations. The associated errorbar is associated to the Bayesian-like approach.
Two estimates of the SFR are reported in Table 5: (1) (Chomiuk & Povich, 2011;Boissier, 2013;Boselli et al., 2015). We see that the two estimates of the SFR are consistent with each other (Figure 1 -right panel). In the following we will consider the SFR derived from the FIR emission, for consistency with the different samples we use as reference. For Mrk 1044 and PG 1448, this provides a much better estimate of the SFR than the stellar emission, which has relative uncertainties larger than unity.
Only six galaxies in our sample were observed with Herschel/PACS and SPIRE with bands at 70, 100, 160 µm and 250, 350, 500 µm, respectively. For the other four sources, the FIR emission is only covered up to 70 − 100 µm by IRAS. However, the cold dust emission follows a modified black body which typically peaks around 100 − 160 µm therefore this component is only well constrained when the source is observed and detected by Herschel. For the six galaxies with Herschel observations, we run the CIGALE code twice: (1) using the Herschel/PACS and SPIRE data to model the dust emission, and (2) limiting the observational constraints of the FIR emission to the IRAS data only. The properties computed by CIGALE typically varies by a factor up to 2-3, as indicating by the quartiles boxes. Nevertheless, we observed that in the case of PG 1211, the SFR derived from the stellar emission and the dust emission vary by a factor of 10, highlighting the importance of the Herschel data. We thus conclude that some of the properties derived with CIGALE for IRAS 17020, Mrk 205, Ark 564 and Mrk 1044 may have been under/overestimated, in particular in the case of Mrk 205 for which the predictions of the FUV and FIR fluxes deviate significantly from the SED fitting ( Figure B1). However, with the current data, it is not possible to say whether this is the case or not.
Most of our sample galaxies have stellar masses M * 4 × 10 10 M ⊙ . This is roughly compatible with the median value reported by Koutoulidis et al. (2022) for their sample of type 1 galaxies, for which masses down to few 10 9 M ⊙ are derived. Only three galax- Table 5. Main properties of our sample galaxies as derived by CIGALE: (1) Short name; (2) Stellar mass of the host galaxy; (3) Average SFR over 100 Myr; (4) Bolometric luminosity of the AGN; (5) AGN-corrected FIR luminosity in the range 8 − 1000 µm; (6) SFR derived from the dust emission following Kennicutt & Evans (2012) Zhao et al. (2021). For consistency with the rest of our sample, we decided to use the estimation from CIGALE. We note that the stellar mass is not the focus of this paper, neither is critical for the present analysis. In particular, in the following we use the SFR derived from the FIR luminosity. Nevertheless, the stellar masses reported in Table 5 are model-dependent as a different set of input parameters may give a different mass. They should therefore be considered with caution in further studies and would need to be confirmed using an independent method.

Molecular gas mass
It is possible to derive the total molecular gas mass from the CO emission by applying a CO-to-H2 conversion factor α CO = M H 2 /L ′ CO . The CO luminosity L ′ CO was calculated from the CO(1-0) emission using the formula of Solomon et al. (1997), then corrected for aperture effects (see Section 3.2 and Table 3). There is a strong uncertainty about the value of this conversion factor (we refer to the review by Bolatto et al. 2013 for a complete discussion). While the conversion factor may vary by more than one order of magnitude at solar metallicity, typical values are commonly accepted for local galaxies. For normal star-forming galaxies, Bolatto et al. (2013) recommend to use a conversion factor of 4.3 M ⊙ (K km s −1 pc 2 ) −1 , which corresponds to the typical X CO observed in the Milky Way. In starburst galaxies and LIRG/ULIRG, the molecular gas experiences very different conditions. The gas volume and column densities are much higher than in typical of normal disks and the molecular gas might not be virialized (Downes & Solomon, 1998), which results in a lower conversion factor. While the conversion factor presents a large dispersion, a value of 0.8 M ⊙ (K km s −1 pc 2 ) −1 is commonly adopted in the literature (e.g. Papadopoulos et al. 2012;Cicone et al. 2012Cicone et al. , 2014Veilleux et al. 2017). Sargent et al. (2014) predicted variations of the α CO factor with the stellar mass M * and SFR. The M * -SFR plane shows the inhomogeneity of galaxy populations, in particular the bimodality between the blue star-forming galaxies and the red passive galaxies (Baldry et al., 2004;Daddi et al., 2007;Elbaz et al., 2007;Wuyts et al., 2011). According to Sargent et  The grey contours are those of the xCOLDGASS sample (Saintonge et al., 2011(Saintonge et al., , 2017. The color points correspond to additional sample of galaxies with CO observations: PG quasars (blue, Shangguan et al. 2020a,b), LIRG/ULIRG (green, Gao & Solomon 2004;Graciá-Carpio et al. 2008;Armus et al. 2009;Tacconi et al. 2018) and BAT AGN (magenta, Rosario et al. 2018;Koss et al. 2021). The dashed line shows the definition of the MS from Whitaker et al. (2012); Accurso et al. (2017). version factor suddently drops when the specific star formation rate sS FR = S FR/M * reaches about 3-4 times the value sS FR MS in the so-called "main sequence" of star-forming galaxies (hereafter MS). This provides a good observational diagnostic to separate starburst galaxies from "normal" star-forming galaxies. Figure 3 shows the position of the objects studied here in the M * -SFR plane, based on the SED fitting with CIGALE (see Sec. 3.3). Six out of ten NLSy1 fall in the regime of starbursts, while Mrk 110, Mrk 1044 and PG 1448 are located along the MS. The last galaxy (Mrk 205) lie in between the two regimes. We therefore used a CO-to-H 2 conversion factor of 0.8 M ⊙ (K km s −1 pc 2 ) −1 for the six galaxies which lie high above the MS, and 4.3 M ⊙ (K km s −1 pc 2 ) −1 for Mrk 110, Mrk 205, Mrk 1044 and PG 1448. The galaxies of our sample host a molecular gas mass between 7 × 10 7 and 5 × 10 9 M ⊙ (see Table 3).
It is well established that the α CO factor is also dependent on the gas metallicity: a lower metallicity means a lower CO abundance, resulting in a higher α CO value (Bolatto et al., 2013;Accurso et al., 2017). However, in absence of information on the metal-  licity in our sample, we do not take into account the metallicitydependence of the α CO factor. Integral field unit (IFU) observations of optical emission lines can provide information about the metal content in the host galaxies but this is out the scope of the present paper.
Molecular gas fraction -Several studies comparing the molecular gas fraction between active and inactive galaxies present opposite conclusions, with the active galaxies being more gas-rich (Scoville et al., 2003;Bertram et al., 2007;Vito et al., 2014;Koss et al., 2021) or, on the contrary, containing less molecular gas (Fiore et al., 2017; Kakkad et al., 2017) than inactive galaxies. Finally, Maiolino et al. (1997) and Rosario et al. (2018) found no difference in the molecular gas mass between active and inactive galaxies, at a given stellar mass. In the literature, various definitions are used for the molecular gas fraction. Here, we adopt the convention f H 2 = M H 2 /M * . The NLSy1 of our sample are in agreement with the general trend of the increasing molecular gas mass with increasing stellar mass (Figure 4 -left). However, they tend to present small molecular gas fractions in comparison to the other surveys described above: f H 2 ∼ 0.02 − 0.04, except for Mrk 205 ( f H 2 ∼ 0.45) and I Zw 1 ( f H 2 ∼ 0.24).
Star formation efficiency -To study the global star formation efficiency in galaxies, the most commonly used method is the Kennicutt-Schmidt relation (Kennicutt, 1998) between the surface densities of the molecular gas mass and the star formation rate. However we have no information about the star formation distribution and most of the molecular gas is not or barely resolved. As an alternative, we therefore plotted the relation between the molecular gas mass M H 2 and the SFR (Figure 4 -right). Except Mrk 205, the NLSy1 galaxies in our sample lie above the relation of the starforming galaxies with molecular depletion times down to a few 10 Myr, 1-2 order of magnitude shorter than the typical ∼ 2 Gyr in star-forming disc . Previous analysis comparing active and non active galaxies also found that AGN host galaxies are more efficient in forming stars (Maiolino et al., 1997;Fiore et al., 2017;Kakkad et al., 2017;Shangguan et al., 2020b). On the contrary, Rosario et al. (2018) and Koss et al. (2021) did not find any evidence that the star formation efficiency is affected by the presence of a central AGN.

IMPACT OF THE AGN ACTIVITY
In this section, we discuss the impact of the AGN activity on the molecular gas reservoir and star formation in the galaxies in our sample. In the following, all the Pearson's coefficient r and the pvalues are derived for our sample of 10 galaxies only. We observe that the CO luminosity and the molecular gas mass correlate well with the AGN bolometric luminosities (Pearson r = 0.78 − 0.87, p-value ≤ 0.01; Figure  appears as an outlier. Note that the correlation is less clear when PG 1211 is considered (r = 0.33 − 0.55, p-value of 0.1 − 0.4). Conversely, the molecular gas mass fraction M H 2 /M * is not correlated with the AGN luminosity (Pearson's coefficient r = 0.17; p-value: 0.67). Note that we observe no correlation between the molecular gas reservoir and the Eddington ratio λ Edd = L AGN L Edd , contrary to Koss et al. (2021) who report a correlation for the gas mass fraction The SFR shows a clear correlation with the AGN luminosity (r = 0.66, p-value of 0.04; Figure 5 -top right). This suggests that the SFR could be regulated by the AGN activity and not by the molecular gas reservoir, contrary to the conclusion of Baker et al. (2022) for 46 local non-active galaxies. Moreover, we observe a correlation of the star formation efficiency S FE = S FR/M H 2 = 1/t mol dep with the Eddington ratio (Pearson r = 0.64; p-value∼ 0.05; Figure 5 -lower right), as well as a tentative correlation with the AGN luminosity (Pearson r = 0.54; p-value∼ 0.13; Figure 5 lower left) if we do not consider Mrk 205 for which the SFE is not higher than star-forming galaxies. The results differ from Shangguan et al. (2020b) who found no dependence in their sample. Bischetti et al. (2021) studied the cold gas properties within the host-galaxies of a sample of 8 high-redshift quasi-stellar ob-jects (QSO) and observed both low molecular gas fractions and short depletion times which they interpreted as the result of AGN feedback. The depletion time in our sample of NLSy1 is of the same order of that observed by Bischetti et al. (2021), suggesting that AGN feedback might be at work in the sources of our sample. This hypothesis is also supported by the nuclear wind activity that characterizes all of them, and by the recent findings reported in Longinotti et al. (2023), where energy-conservation was confirmed for the galaxy scale molecular outflow and the X-ray UFO. In the following, we explore different scenarios which could explain the low depletion times we observe. Nevertheless, the current observations and ancillary data do not enable to disentangle between the different scenarios.
Positive AGN feedback? -AGN feedback is often invoked as a quenching mechanism (negative feedback). However, there is evidence of AGN positive feedback which favour or trigger star formation (e.g. Zinn et al. 2013). While positive feedback is mostly observed by the presence of recent star formation along radio jets (Klamer et al., 2004;Emonts et al., 2014;Salomé et al., 2015Salomé et al., , 2016Zovaro et al., 2020), enhanced star formation is also observed around cavities produced by AGN-driven outflows (Cresci et al., 2015b(Cresci et al., ,a, 2023. Therefore, the AGN activity in our sample might be boosting star formation by compressing the gas. The M H 2 −L AGN correlation suggests that the AGN activity might also regulate the molecular gas reservoir and fulfill the observed starburst episode. In this scenario, the short depletion times indicate that the AGN activity will quench star formation by gas exhaustion. Weak or no AGN feedback? -Several studies proposed a scenario where the AGN activity is ignited by a supply of gas from stellar feedback and supernovae explosions in the central region (Chen et al., 2009;Dahmer-Hahn et al., 2022;Tillman et al., 2022). Moreover, recent observations and simulations favour a scenario where AGN feedback is delayed (see Cresci & Maiolino 2018 and references therein). In this scenario, the observed starburst event may have ignited the AGN activity without having a major impact on the star formation in the host of the NLSy1.
As to the effect of the morphology of the galaxies on their SFR, we note that in a sample of Seyfert 2 galaxies, Maiolino et al. (1997) concluded that the high SFR they observed is the result of morphology perturbations. Krongold et al. (2002) proposed an evolutionary sequence between galaxy interactions, starbursts episodes and AGN activity. While NLSy1 mostly reside in late-type galaxies (Krongold et al., 2001;Ohta et al., 2007), some are found in disturbed or interacting systems Salomé et al., 2021). Another process which has the potential to ignite the nuclear activity is the presence of stellar bars (Simkin et al., 1980;Shlosman et al., 1990;Maiolino et al., 2000;García-Burillo & Combes, 2012). In particular, Crenshaw et al. (2003) found that the presence of a large-scale stellar bar is common in NLSy1.
A detailed study of the morphology of the galaxies in our sample is out of the scope of this paper. However, we performed a visual inspection of SDSS and HST images. A stellar bar is clearly present in at least half of our sample. Conversely, only two galaxies show evidence of galaxy interaction: Mrk 110 with the presence of large-scale tidal tails, and I Zw 1 (Scharwächter et al., 2003). This suggests that the nuclear activity in our sample is regulated by the molecular gas reservoir which fuels the nucleus via the stellar bar. Nevertheless, the presence of galaxy interaction in more sources cannot be excluded. For example, using resolved CO observations, Salomé et al. (2021) discovered that the host galaxy of IRAS 17020 is interacting with a smaller mass companion, while the optical images do not show any evidence of interaction.
Early phase of AGN feedback? -García-Burillo et al. (2021) observed the molecular gas in the centre of 13 Seyfert galaxies at a resolution of ∼ 7 − 10 pc with ALMA. These observations allowed them to resolve the dusty molecular tori which extend to diameters ≤ 100 pc and reveal clear evidence of AGN feedback at small scales. In particular, García-Burillo et al. (2021) highlighted that the imprint of AGN feedback at small scales is likely to be more extreme in higher luminosity and/or higher Eddington ratio Seyfert galaxies, contrary to what we observe at larger scales in this paper. However, AGN activity occurs on rather short timescales (few Myr; Martini 2004;Hickox et al. 2014 and NLSy1 have been dubbed as a class of young AGN in terms of the black hole mass (see Section 1). Moreover numerical simulations predict that the super-Eddington accretion is short (Li, 2012;Smith & Bromm, 2019) and does not affect star formation on short timescales (Massonneau et al., 2023). All together, this suggests that the current AGN activity we observe in our sample may not have had time yet to affect galaxy-scale properties like the mass of stars and gas or the SFR, which extend to several kpc. This is in agreement with the scenario of Cresci & Maiolino (2018) where the so-called "negative feedback" does not affect the full gas reservoir but is rather delayed.

CONCLUSION
In this paper, we studied the molecular gas reservoir of a sample of NLSy1 for which a relativistic UFO has been detected in X-ray. Six galaxies were observed in CO with the IRAM 30m telescope, including the galaxy IRAS 17020 which was previously observed with NOEMA (Salomé et al., 2021;Longinotti et al., 2023). We complemented our sample with four NLSy1 detected in CO by NOEMA or ACA (Cicone et al., 2014;Shangguan et al., 2020a;Domínguez-Fernández et al., 2020). We compared the molecular gas mass estimated from the CO observations with properties of the host galaxies and the AGN which were derived with the multiband SED fitting code CIGALE.
The CO molecule is detected in the ten galaxies considered in this article. Seven sources were observed in CO(1-0) and CO(2-1), and five are detected in both CO lines. The intensity line ratio I CO21 /I CO10 suggests that the lines are not thermalised. We used the commonly adopted α CO values based on the position of the galaxies in the M * − S FR plane (see Section 3.4) and we derived molecular gas masses covering almost two orders of magnitude between 7 × 10 7 and 4 × 10 9 M ⊙ , typical of the molecular gas mass observed in local galaxies by Saintonge et al. (2017).
By fitting the multi-band SED with CIGALE, we computed the stellar mass and star formation rate of the host galaxies, as well as the AGN bolometric and the dust FIR luminosities. These properties also spans two orders of magnitude. The NLSy1 in our samples tend to host less molecular gas that local galaxies of the same M * . However, the star formation rate is significantly higher, resulting in molecular depletion times 1-2 orders of magnitude shorter than in typical star-forming galaxies (except in Mrk 205). Moreover, we found that the SFR increases with increasing AGN luminosity and the molecular depletion time is anticorrelated with the AGN luminosity and the Eddington ratio. This suggests that the AGN activity has little or no quenching feedback on star formation and conversely might be enhancing star formation (positive feedback?).
To further investigate the possible relation between the molecular gas content and the AGN activity, additional observations would be necessary. Firstly, interferometric observations enable to resolve the CO emission and study the distribution of the molecular gas. In particular, NOEMA data enabled Salomé et al. (2021) to discover that IRAS 17020 is currently interacting with a small companion. This interaction was not known before that as the host galaxy is observed as a typical undisturbed spiral barred galaxy in optical images (Ohta et al., 2007;Olguín-Iglesias et al., 2020). Secondly, HI observations will trace the atomic phase of the ISM and enable to get a more complete view of the gas reservoir. Finally, CO observations of a larger sample of NLSy1 will provide more statistic to resolve the impact of the AGN activity in NLSy1. For instance, the IBISCO survey contains 8 NLSy1 (Molina et al., 2018) with unpublished CO observations , including some ALMA data (mentionned by Zanchettin et al. 2021). Figure A1. Spectra of the CO(1-0) and CO(2-1) emission as observed with the IRAM 30m. For Mrk 205, it was not possible to simultaneously observe both CO(1-0) and CO(2-1). The vertical line in the spectra of IRAS 13349 correspond to our new estimate of the redshift.