The ALMA-ALPINE [CII] survey: Kennicutt-Schmidt relation in four massive main-sequence galaxies at z ∼ 4.5

Aims. The Kennicutt-Schmidt (KS) relation between the gas and the star formation rate (SFR) surface density ( Σ gas − Σ SFR ) is essential to understand star formation processes in galaxies. To date, it has been measured up to z ∼ 2.5 in main-sequence galaxies. In this Letter our aim is to put constraints at z ∼ 4 . 5 using a sample of four massive main-sequence galaxies observed by ALMA at high resolution. Methods. We obtained ∼ 0.3 (cid:48)(cid:48) -resolution [CII] and continuum maps of our objects, which we then converted into gas and obscured SFR surface density maps. In addition, we produced unobscured SFR surface density maps by convolving Hubble ancillary data in the rest-frame UV. We then derived the average Σ SFR in various Σ gas bins, and estimated the uncertainties using a Monte Carlo sampling. Results. Our galaxy sample follows the KS relation measured in main-sequence galaxies at lower redshift, and is slightly lower than the predictions from simulations. Our data points probe the high end both in terms of Σ gas and Σ SFR , and gas depletion timescales (285–843Myr) remain similar to z ∼ 2 objects. However, three of our objects are clearly morphologically disturbed, and we could have expected shorter gas depletion timescales ( (cid:46) 100Myr) similar to merger-driven starbursts at lower redshifts. This suggests that the mechanisms triggering starbursts at high redshift may be di ﬀ erent than in the low-and intermediate-z Universe.

The second and third authors are master's students, who had a similar contributions to this letter (analysis of the first three sources and initial results).
galaxies (e.g., Noeske et al. 2007;Elbaz et al. 2007).The normalization of this main sequence increases rapidly with increasing redshift (e.g., Schreiber et al. 2015) together with the gas fraction (e.g., Magdis et al. 2012;Saintonge et al. 2013;Béthermin et al. 2015), suggesting that the larger gas reservoirs are driving the higher specific SFR (sSFR=SFR/M ) observed at high z.However, a small population of starbursts with a Σ SFR excess compared to the KS relation of main-sequence galaxies was found by Genzel et al. (2010) and Daddi et al. (2010), among others.Resolved studies of high-z starbursts confirmed the Σ SFR excess at sub-galactic scale in these systems (e.g., Freundlich et al. 2013;Rawle et al. 2014;Hodge et al. 2015).Another population of starbursts with an SFR excess was also identified above the main sequence (e.g., Rodighiero et al. 2011).The starburst populations observed in both relations are suspected to be driven by mergers (e.g., Sargent et al. 2014;Cibinel et al. 2019).
To date, the KS relation in z>2.5 main-sequence galaxies has remained unexplored.However, the Atacama Large Millimeter Array (ALMA) has opened new perspectives to explore earlier times.In particular, the bright 158 µm rest-fame [CII] line is now easily observable from the ground at z 4, and can be used as a gas tracer (e.g., Zanella et al. 2018).Recently, Vallini et al. (2024) published a study on the KS relation in five z∼7 bright Lyman beak galaxies (LBGs), but the main-sequence nature of these galaxies remains unclear.The ALMA large program to investigate [CII] at early times (ALPINE, Le Fèvre et al. 2020;Béthermin et al. 2020;Faisst et al. 2020) built a sample of 118 main-sequence galaxies at 4<z<6, observed in [CII] and continuum at low angular resolution (∼1", marginally or not resolved).
In this sample, Dessauges-Zavadsky et al. (2020) found a flattening of the evolution of the gas fraction with redshift, similar to that observed for the sSFR (Khusanova et al. 2021).In contrast, Jones et al. (2021) and Romano et al. (2021) estimated that a high fraction (∼40 %) of these objects exhibits morphokinematical signatures of mergers despite being on the main sequence, suggesting that the mechanisms driving star formation in the z 4 Universe may differ from lower redshifts.
In this letter we explore the KS relation at z∼4.5 using a sample of four ALPINE galaxies followed up at higher resolution (∼ 0.3 arcsec, 2 kpc) by ALMA.In Sect. 2 we describe our observations and the data analysis.We then present our new results on the KS relation in Sect.3. Finally, we discuss them and conclude in Sect. 4. We assume a flat ΛCDM cosmology (h=0.7, 3) and a Chabrier initial mass function (IMF).
The data were calibrated by the standard observatory pipeline, and imaged using the common astronomy software applications for radio astronomy (CASA; CASA Team et al. 2022).We produced continuum maps after excluding the [CII]-contaminated channels and the [CII] moment-0 maps (i.e., velocity-integrated line flux maps) after subtracting the continuum in Fourier space.For DC818760, DC873756, and VC5101218326 we also included the compactconfiguration visibilities from the initial ALPINE observations to recover the large-scale components.This was not necessary for VC5110377875 since the full ALPINE integrated flux is already recovered by the new high-resolution observations alone.The data reduction is described in detail in a companion paper focusing on the morpho-kinematical analysis of these objects (Devereaux et al. 2023).The source properties and the achieved ALMA performance are listed in Table 1.
The [CII] flux is then used to derive the gas mass, while the rest-frame far-infrared continuum provides the dust obscured SFR.While initially considered to be an SFR tracer (e.g., De Looze et al. 2014;Capak et al. 2015), recent theoretical and observational studies pointed out that [CII] is more tightly connected to the molecular gas mass (e.g., Zanella et al. 2018;Madden et al. 2020;Vizgan et al. 2022;D'Eugenio et al. 2023;Ramambason et al. 2023), and correlates with SFR through the KS relation (e.g., Ferrara et al. 2019).The [CII] line also comes from the neutral and ionized medium, but their contribution is expected to be small in massive high-redshift galaxies (e.g., Vizgan et al. 2022).In addition, starbursting systems with short de-pletion timescales (M gas /SFR∼100 Myr) tend to have low [CII]-110 to-IR luminosity ratios (Díaz-Santos et al. 2013;Gullberg et al. 111 2015), which should not be the case for an SFR tracer.A the-112 oretical discussion about this [CII] deficit in starbursts can be 113 found in Vallini et al. (2021).Finally, integrated [CII]-based 114 gas masses of the ALPINE sample agree with dust-based mea-115 surements and dynamical estimates after subtracting the stellar 116 masses (Dessauges-Zavadsky et al. 2020).A systematic compar-117 ison of the various tracers (CO, [CI], [CII], and dust) in high-z 118 lensed dusty star-forming galaxies, with higher SFRs than our 119 targets by a factor of 2-50, also found a good agreement between 120 them (Gururajan et al. 2023).Contrary to CO, the [CII] line lu-121 minosity has a weak dependence on metallicity and is sensitive 122 to CO-dark gas as shown by studies in nearby low-metallicity 123 dwarfs (Madden et al. 2020;Ramambason et al. 2023).

124
The gas surface density (Σ gas ) is derived from the [CII] 125 moment-0 map m [CII] in Jy km s −1 beam −1 using 126 where α [CII] is the [CII]-to-gas conversion factor (31 M /L , 127 Zanella et al. 2018).This conversion factor has been cross-128 calibrated using CO up to z∼2 to measure the molecular gas 129 mass.If the small expected fractional contribution from other 130 phases to the [CII] luminosity in their sample and in our ob-131 jects are similar, we should thus obtain directly a molecular gas 132 mass corrected from the contribution of the rest of the interstel-133 lar medium.The parameter D A is the angular diameter distance 134 and Ω beam the solid angle of the synthesized beam defined as the 135 integral of a beam after normalizing its peak to unity, D 2 A Ω beam 136 being the physical area associated with the synthesized beam.137 The following factors in Eq. 1 correspond to the conversion from 138 line flux to luminosity (Solomon et al. 1992) with D L being the 139 luminosity distance and ν obs the observed frequency.

140
The SFR surface density (Σ SFR ) is the second quantity in-141 volved in the KS relation.In galaxies with a non-negligible dust 142 content as ALPINE galaxies (Fudamoto et al. 2020), we can es-143 timate the total SFR by combining the obscured SFR probed by 144 the far-infrared (SFR IR ) and unobscured SFR seen in the UV 145 (SFR UV ).The total SFR is the sum of these two values, and can 146 be derived using (Madau & Dickinson 2014) 147 where L UV is the rest-frame 1500 Å luminosity and L IR the total 148 8-1000 µm IR luminosity; κ UV and κ IR are the conversion factors 149 with a value of 1.02 × 10 −10 M /L and 1.47 × 10 −10 M /L after 150 converting to the Chabrier IMF. 151 The 158 µm rest-frame continuum map (m 158 ) in Jy/beam is 152 converted into obscured SFR surface density (Σ SFR IR ) using where ν cont is the rest-frame continuum frequency, and L IR L 158 is the 154 ratio of the total luminosity to the 158 µm rest-frame monochro-155 matic luminosity.We use the value of 1/0.113 computed by 156 Béthermin et al. (2020) based on Herschel stacking.This con-157 stant conversion factor assumes implicitly a constant dust tem-158 perature in our objects, and variations could lead to systematic 159 effects on Σ SFR IR estimates (Cochrane et al. 2022).Unfortunately, 160 high-resolution observations at higher frequency to constrain the 161 Table 1.Summary of the observations and achieved performance.The σ cont and σ [CII] column are the noise of the continuum map and the [CII] moment-0 map, respectively.The 5σ Σ gas limit is derived from the gas surface density map obtained after applying the conversion from Eq. 1.To produce UV rest-frame maps, we started from cutouts of the HST COSMOS mosaics (Koekemoer et al. 2007) in the F814W filter (∼ 1480 Å rest-frame), and converted the instrumental units into physical units (M yr −1 kpc −2 ) using κ UV , the luminosity distance, and the physical area at z∼4.5 corresponding to the HST pixel.To match the ALMA resolution, we then convolved the HST map by an elliptical Gaussian kernel with a major-axis width σ ker maj = (σ ALMA maj ) 2 − σ 2 HST (same formula for the minor axis), while ensuring that the normalization preserves the surface density.We tested this procedure on the HST PSF and found that the recovered beam is similar to the ALMA beam with an accuracy better than 10 %.

Source
In Fig. 1 we present the surface density maps of DC873756 as an example, while the other sources are shown and briefly discussed in Appendix A. We note that the gas map has a very high S/N, and only the galaxy core is detected in the obscured SFR map.We also note that the obscured and unobscured SFR surface density maps exhibit different morphologies with the UV star formation coming mainly from the diffuse gas extension in the northwest and not the IR-bright core.The southwest bright UV blob has a robust photo-z of 3.4±0.1 (Weaver et al. 2022), and is thus probably not related to our target.This highlights the necessity to have access to both the obscured and unobscured star formation in our analysis.This source is thus asymmetric with a dusty star-forming core slightly offset in the southeast direction and a diffuse and less obscured extension in the northwest, illustrating the complexity of the morphologies at this redshift (see Devereaux et al. 2023 for more details).

Results
We first investigated the KS relation on a pixel-per-pixel basis (0.06" in VC5110377875 and 0.04" in the other galaxies).Since our Σ gas map is by far the deepest, we restricted our study to the regions were the gas is detected in [CII] at better than 5 σ (see Table 1) to robustly avoid working on noise spikes in the source outskirts.We then summed the Σ SFR IR and Σ SFR UV maps to obtain the Σ SFR maps, and assumed a quadratic combination of the noise.In our objects, the obscured SFR contributes 75-99 % of the total SFR.In Fig. 2 we show our results for each source.
While in DC873756 most of the lines of sight have a Σ SFR signal above 3 σ, this is not the case in the three other sources.The risk of bias is very high in the Σ gas regime where a small fraction of pixels are detected in Σ SFR .In addition, in interferometry, the data points from the neighboring pixels are not independent because the noise is correlated at the scale of the synthesized beam.In this letter we thus only focus on deriving unbiased average quantities, which unfortunately prevents us from studying the scatter.For each source, we defined two different regions based on 211 their Σ gas .We cut the range between the 5σ limit and the maxi-212 mum in two bins of the same logarithmic size, building two dif-213 ferent regions with a lower and a higher Σ gas .The high-density 214 region tends to be in the center, while the lower-density region 215 forms a ring around it (see Fig. 1).However, the disturbed mor-216 phologies of our systems lead to rather complex shapes, which 217 justifies a posteriori not using radial profiles in this object type.218 Finally, we computed the mean Σ gas and Σ SFR in each region of 219 each object.In Appendix C we describe a simulation validating 220 our method.This simulation shows that we can recover with-221 out bias the intrinsic relation in the presence of noise.However, 222 this simulation does not include the potential complex system-223 atic effects from spatial variations of the conversion factors used 224 in Eqs. 1, 2, and 3.

225
To derive uncertainties on these quantities, we moved ran-226 domly the two regions in a noisy area of the map using the same 227 offset for both regions, and measured our observables using the 228 same method as previously.This process was repeated 10 000 229 times, and the standard deviation of the results provides the un-230 certainty.The results are summarized in Table 2.We also com-231 puted the typical correlation between the noise realizations in 232 the two regions and found a Pearson correlation coefficient of 233 0.5±0.1.This is expected since the two regions have a common 234 border, and the interferometric noise is correlated at the scale of 235 the synthesized beam.In Fig. 2 these average values are located 236 at the middle of the cloud of points when most of the pixels are 237 detected (DC873756 and high-density bin of DC818760), but lie 238 significantly below otherwise.This is not surprising since in the 239 regime of low average S/N the detections are biased toward pos-240 itive outliers of the KS relation and the instrumental noise.

241
In Fig. 3 we present a synthesis of our results together 242 with a compilation of previous works.All our data points but 243 one are consistent with a 0.5-1 Gyr depletion timescale ex-244 pected for massive main-sequence galaxies at z∼4.5 ( ).The small black filled circles correspond to lines of sight with a >3 σ signal on the SFR surface density, and the small red downward triangles to 3 σ upper limits.The large filled symbols (same as in Fig. 3) correspond to the average value for all the pixels in a given gas surface density bin (see Sect. 3 and Table 2).
(  2023) at the scale of 1.6 kpc (compared to ∼2 kpc in our analysis) in two strongly lensed z∼1 galaxies have significantly lower Σ SFR at a given Σ gas than other analyses; this is discussed in their paper without converging on a final explanation.Since their Cosmic Snake measurements agree with those of Pessa et al. ( 2021) and with our results, but not their measurements in the A521 galaxy, it could suggest that the the A521 galaxy is an outlier.In contrast, our sample of main-sequence galaxies have significantly lower Σ SFR at a given Σ gas than typical high-redshift starbursts, which have an average depletion timescale of 100 Myr (Sharon et al. 2013;Rawle et al. 2014;Hodge et al. 2015).The z∼7 LBGs studied by Vallini et al. (2024) exhibit a similar behavior, suggesting that they could also be starbursting systems.Finally, we compared our results with the KS relation of Kraljic et al. (2023, fit for z=4 and 10 8 < M /M < 10 9.5 ) from the hydrodynamical simulation NewHorizon (Dubois et al. 2021).Their results suggest a slightly higher normalization (by a factor of ∼1.5) than our measurements.However, objects as massive as our targets are not found in the limited volume of their simulated box, and low-and high-mass high-z galaxies could exhibit different behaviors.

Discussion and conclusion
Our study suggests that main-sequence galaxies follow the same KS relation from z=0 to z=4.5, but exhibit a strong increase in their gas density with increasing redshift driving a rise in their SFR density.This universality of the KS relation for mainsequence galaxies might seem surprising considering how different local and z>4 galaxies are, for example in term of gas fraction or sSFR.Since a significant fraction of our sample (Devereaux et al. 2023), and more globally of z∼5 massive mainsequence galaxies (e.g., Romano et al. 2021), are identified as possible mergers by morpho-kinematical analyses, this opens the question of a possible decoupling of the merging and starburst events in high-redshift galaxies, but also of the timescale of the SFR enhancement and the associated merger stage.Theoretical works suggest that high-z mergers could often be inefficient in increasing star formation (e.g., Fensch et al. 2017)  noise smoothed by our convolution kernel to the Σ SFR UV map after matching its level to the observed value.
We then applied the same measurement process as described in Sect. 3 on both noiseless and noisy maps.The results are presented in Fig. C.1.The KS relation in the noiseless case is our reference (light blue small squares).Since there is no noise associated with these maps, we used the noise level of the Σ gas noisy map to define the region above the 5σ density limit in which we analyzed individual pixels.The low and high gas density regions are defined in a similar way, and we derived the mean Σ gas and Σ SFR in each of them (dark blue large squares).As expected, there is a near perfect agreement between the individual pixels and the mean measurements in the two simulated objects.
We also analyzed the noisy maps.The various regions used in our analysis can slightly change compared to the noiseless case, since some pixels can pass above or below a threshold because of the noise.The individual detected pixels (small black filled circles) agree with the noiseless relation in case 1 where the S/N is high.In case 2 the detections agree only at the high-Σ gas end, but are systematically higher than the noiseless case at the low-Σ gas end.Since only pixels with Σ SFR above 2.5 M yr −1 kpc −2 are detected, only the positive noise outliers are detected in the low-Σ gas regime for which the average Σ SFR is below this limit.In contrast, the mean values measured in the low and high gas density regions (dark green pentagons) agree at 1σ with the noiseless relation.To check for a possible bias, we computed their average positions over 1000 noise realizations (light green hexagons), and they agree at better than 3 % with the noiseless case.
Our simulation confirms that mean measurements over several well-selected regions are more effective at recovering mean Σ SFR than individual pixel measurements, which are strongly biased in the low S/N regime.It is not surprising, since these regions encompass several synthesized beams and averaging them improves the S/N by approximately a factor of √ N sb , where N sb is the number of synthesized beams in the region. 2However, this method works only because we benefit from deep Σ gas maps to select these regions.

Fig. 1 .Fig. 2 .
Fig.1.Surface density maps of DC873756: Gas (left, traced by [CII]), obscured FIR SFR (center, traced by the 158 µm rest-frame continuum), and unobscured UV SFR (right, traced by the F814W observer-frame continuum).The black contours are the (3+2k) σ levels (k≥0) of the gas surface density.The thicker solid and dashed lines, respectively, are used to highlight the 5σ Σ gas limit and the border between the low-and high-density regions used in our analysis (see Sect. 3).The ALMA synthesized beam size is shown in the lower left corner.The unobscured SFR surface density map based on HST data is convolved by a Gaussian kernel to match the ALMA resolution.
Myr) is the central high-density region of DC873756, 247 which has the highest obscured SFR fraction (SFR IR /SFR tot = 248 98.7±0.3 %) of the sample, and could thus hide a mild but heav-249 ily obscured starburst.250 We compared our results with CO measurements of the 251 global KS relation from the nearby COLD GASS sample (Sain-252 tonge et al. 2012) and the z 2.5 PHIBBS and PHIBBS2 sam-253ples(Tacconi et al. 2013;Freundlich et al. 2019).We find that 254 our objects are located in the scatter of these previous studies 255 (∼0.3 dex corresponding to a factor of 2), but with a higher gas 256 surface density than most of the nearby sample by almost two 257 orders of magnitude.Similarly, our sources are within a factor 258 of 2 of the global KS relation fits by de los Reyes & Kennicutt 259 (2019) at low redshift and by Wang et al. (2022) at high redshift, 260 and the resolved low-z KS fit by Pessa et al. (2021).In contrast, most of the measurements of Nagy et al. (

Table 2 .
Average gas ( Σ gas ) and SFR ( Σ SFR ) surface density measured in two Σ gas bins for our four objects.The uncertainties are derived using the Monte Carlo simulations described in Sect.3. The uncertainties are based only on the instrumental noise.
Nagy et al. (2023)24)19 relation between the SFR and gas surface density.The averaged data points from DC818760, DC873756, VC5101218326, and VC5110377875 are shown as violet squares, blue filled circles, yellow hexagons, and green diamonds, respectively.Our results are compared with the CO measurements of the global KS relation from the low-z COLD GASS sample(Saintonge et al. 2012,  crosses)and at redshifts up to z∼2.5 from the PHIBBS(Tacconi et al. 2013, plus signs) and PHIBBS2(Freundlich et al. 2019, three-branch stars) programs, together with the resolved KS relation in z∼1 lensed main-sequence galaxies(Nagy et al. 2023, five-branch stars).Also shown are the global (de los Reyes & Kennicutt 2019, dashed line) and resolved(Pessa et al. 2021, two-dot-dashed line) KS relations measured in the local Universe, the high-redshift global KS relation obtained by ALMA CO stacking(Wang et al. 2022, dotted line), and the global KS relation from simulations at z∼4(Kraljic et al. 2023, three-dot-dashed  line).The thin gray lines indicate the various depletion timescales.The red points are resolved measurements in high-z starbursts byHodge et al. (2015).The brown six-branch stars are the measurements in z∼7 LBGs byVallini et al. (2024).The error bars include only the uncertainties from the instrumental noise, but not from the calibration (∼10 % for ALMA) or from the conversion factors used in Sect. 2. They are often smaller than the symbols.theKSrelation breaks at smaller scales in mainsequence galaxies, as discussed inNagy et al. (2023)at z∼1.
317push toward lower masses with potentially less mature systems 318 and lower metallicities.Deeper ALMA observations would also 319 allow us to detect the dust continuum in most lines of sight 320 providing a local estimate of Σ SFR IR , and measure the scatter 321 on the KS relation.Finally, higher-resolution observations will 322 test whether