The ALMA Spectroscopic Survey in the Hubble Ultra Deep Field: Constraining the Molecular Content at log(M_*/M_⊙) ∼ 9.5 with CO Stacking of MUSE-detected z ∼ 1.5 Galaxies

The detailed survey strategy and data reduction of the ASPECS Pilot and Large Program surveys are presented in Walter et al. (2016) and González-López et al. (2019), respectively. The observational setups of the LP survey were the same as the Pilot survey, except its coverage is extended to $\sim 5\,{\mathrm{arcmin}}^{2}$. In this work, we used the combined data of the ASPECS Pilot and LP surveys taken with ALMA Band 3. ASPECS LP carried out a full frequency scan in Band 3 (84−115 GHz) over the XDF, which resides in the HUDF. With a total of 17 pointings centered at (R.A., decl.) = (03:32:38.5, −27:47:00), which completely cover the Pilot region, the total area with a primary beam response $\gt 50 \%$ in the LP survey is $4.6\,{\mathrm{arcmin}}^{2}$ at $\approx 99.5\,\mathrm{GHz}$ (the central frequency of Band 3).

The Common Astronomy Software Applications (CASA) software was used to calibrate and image the data. With the C40-3 array configuration, we obtained a synthesized beam size of $1.75^{\prime\prime} \times 1.49^{\prime\prime}$ with a position angle $91.5\,\deg$ at $\approx 99.5\,\mathrm{GHz}$ by using natural weighting in CASA when imaging the data. The frequency channel was rebinned to 7.813 MHz ($23.5\,\mathrm{km}\,{{\rm{s}}}^{-1}$ at 99.5 GHz). The sensitivity for this channel bin size was $\sim 0.2\,\mathrm{mJy}\,{\mathrm{beam}}^{-1}$ throughout the entire scanned frequency range. The 5σしぐま CO(2–1) line sensitivity is $\gt 1.4\times {10}^{9}\,{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$ assuming a line width of $200\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Based on assumptions made in this work (see Section 5.1), the corresponding molecular gas (H₂) limit is $\gtrsim 6.8\times {10}^{9}\,{M}_{\odot }$. For our main target emission line, CO(2–1), the ALMA Band 3 scan offers redshift coverage of $z=1.0059-1.7387$.

Using the same data cube, González-López et al. (2019) reported CO emission line detections from an unbiased blind search without prior knowledge of source positions and observed CO frequencies. They found 16 high significance CO emitting sources, among which 11 were identified as CO(2–1) emission. We refer the readers to González-López et al. (2019) for comprehensive discussions on the detection methods. Aravena et al. (2019) assessed the molecular gas properties of these sources, which will be used for comparison in this paper.

The MUSE–HUDF deep survey was conducted as a two layer survey of different depths (Bacon et al. 2017). The $3^{\prime} \times 3^{\prime}$ deep survey observed the entire HUDF region, whereas the $1^{\prime} \times 1^{\prime}$ ultra-deep survey was carried out near the center of the deep survey area. The $\approx 10\,\mathrm{hr}$ and $\approx 31\,\mathrm{hr}$ exposure times, respectively, reached 3σしぐま emission line flux limits of 3.1 and $1.5\times {10}^{-19}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{cm}}^{-2}$ at $\sim 7000\,\mathring{\rm A}$.

The MUSE spectra were extracted with two methods: prior extractions based on the Ultraviolet Hubble Ultra Deep Field catalog (UVUDF; Rafelski et al. 2015) and a blind search for emission lines in the data cubes. In the former case, the coordinates for the source extraction were from UVUDF, ²⁹ whereas for the latter, the coordinates were determined from the MUSE data. For a more detailed description of the survey and data treatments, see Bacon et al. (2017). Following Dunlop et al. (2017), we corrected the known systematic offset of the Hubble Space Telescope (HST) positions to match the radio astrometric reference frame by applying $+0.279^{\prime\prime}$ in decl. and $-0.076^{\prime\prime}$ in R.A.

The MUSE spectroscopic redshifts were measured from the spectral features in the extracted spectra. Each redshift has an associated confidence level of 3 (secure redshift, determined by multiple spectral lines), 2 (secure redshift, determined by a single spectral line), or 1 (possible redshift, determined by a single spectral line whose spectral identification remains uncertain). The typical MUSE redshift uncertainty is ${\sigma }_{z}\,=0.00012(1+z)$ or ${\sigma }_{v}\approx 40\,\mathrm{km}\,{{\rm{s}}}^{-1}$, which is smaller than the typical line width of CO emission. We refer to Inami et al. (2017) for details about the redshift determination and redshift catalogs of the MUSE–HUDF field. In this work, we only use the reliable MUSE redshifts of confidence levels 2 and 3.

The MUSE–HUDF survey obtained 1338 reliable redshifts in total, a factor of eight increase over the previously available spectroscopic redshifts in this field. The simultaneous wavelength coverage of $4650-9300\,\mathring{\rm A}$ and the spectral resolution of $R\sim 3000$ of MUSE allowed detections and unambiguous identifications of major rest-frame ultraviolet and optical emission lines, including Hαあるふぁ, [O ii], [O iii], and Lyαあるふぁ. These lines were used to determine redshifts over the range $0\lt z\lt 6.5$. Although more difficult to probe, the redshift range $z\sim 1.5-3$ can be covered by C iii] and absorption features (e.g., C iv, Fe ii, Mg ii; see Figure 13 of Inami et al. 2017).

Among the spectroscopic redshifts assessed in the MUSE–HUDF field, 503 sources with a confidence level of 2 or higher lie within the ASPECS survey region (LP primary beam response $\gt 50 \%$). Out of these sources, 107 sources are [O ii] emitters covering $z=0.25-1.5$ and 363 sources are Lyαあるふぁ emitters covering $z=2.8-6.6$. For the analysis presented in the main part of this paper, we took advantage of the prevalent [O ii] line detections and their redshift overlap with the CO(2–1) selection function in ALMA Band 3 (Figure 1) to perform a stacking analysis. In the redshift range where CO(2–1) can be detected, the MUSE spectroscopic redshift sources with absorption features and C iii] emission also contributed to the stacking, although the number was small (Figure 1). We also attempted to stack spectra to detect higher-J CO lines. The MUSE redshifts used in higher-J CO stacked spectra were mostly measured using the Lyαあるふぁ line, which is known to be offset from the systemic redshift for a few hundred $\mathrm{km}\,{{\rm{s}}}^{-1}$ (e.g., Shapley et al. 2003). We have applied a correction to this offset (Section 4.2).

Owing to the same HUDF coverage of the MUSE and ASPECS observations, there are abundant ancillary data available. We assembled optical and near-infrared photometric data from Skelton et al. (2014). These photometry catalogs and data include ultraviolet to infrared from the HST, various ground-based telescopes, and all of the Spitzer IRAC channels.

Based on this photometric data set, we inferred physical parameters such as SFR and stellar mass (M_*) via modeling with the high-redshift extension of the spectral energy distribution (SED) fitting code MAGPHYS (da Cunha et al. 2008, 2015). In addition to Spitzer/MIPS and Herschel/PACS, the ALMA 1.2 and 3 mm photometry from the ASPECS data (González-López et al. 2019, 2020) were also used for the SED fitting for a subset of the sources whose CO or continuum emission was detected. The same procedure was carried out in the other ASPECS work (Aravena et al. 2020; Boogaard et al. 2020). The SED fits were computed based on the MUSE spectroscopic redshift. See Boogaard et al. (2019) for the detailed process of the SED fitting.

From the ALMA data, we first extracted a sub-cube centered at the position of each MUSE source with a secure spectroscopic redshift. This sub-cube has a size of $11^{\prime\prime} \times 11^{\prime\prime} \,\times 3000\,\mathrm{km}\,{{\rm{s}}}^{-1}$. The primary beam correction using the combined Pilot and LP primary beam response was applied after the sub-cube extraction.

The uncertainty of the extracted spectra was calculated based on the region in the data cube where the combined beam response is $\gt 99 \%$ of the peak sensitivity at the phase center. The uncertainty for each spectrum is then scaled by the combined primary beam response of the location of the objects under consideration.

To perform stacking, the channel frequencies of the extracted spectra were converted to the rest frame, then the spectra were resampled onto a common frequency grid. In the rest frame, a CO line detected at the lower frequency end of the spectrum (i.e., galaxies at higher redshift) has a coarser spectral sampling due to the $(1+z)$ correction of the redshift. Thus, the common frequency grid was given at the coarsest sampling, corresponding to the CO line detected at the lowest frequency end of Band 3. In the case of CO(2–1), we set the final spectral range and the channel width of the extracted spectra to be $3000\,\mathrm{km}\,{{\rm{s}}}^{-1}$, centered on the CO(2–1) rest-frame frequency ($230.538\,$ GHz), and $27.8\,\mathrm{km}\,{{\rm{s}}}^{-1}$ ($21.4\,\mathrm{MHz}$), respectively. This resulted in 108 channels in the extracted spectra. The final extracted rest-frame frequency range is $229.395-231.682\,$ GHz ($\pm 1500\,\mathrm{km}\,{{\rm{s}}}^{-1}$). The same procedure is adopted for stacking high-J CO lines.

This conversion was implemented by first shifting the channel frequencies of the spectra to the rest frame, according to the MUSE redshift. We then applied a Gaussian decimation filter to the rest-frame spectra via the Fourier plane. This filter removes fine scale (high sampling frequency) noise in the spectrum which would otherwise be aliased into spurious noise when sub-sampled to the lower target resolution (Lyons 2010). A better signal-to-noise ratio (S/N) was thus obtained when we binned all spectra onto the coarser common frequency grid in the later step. In the Fourier plane, the width of the Gaussian filter was set to give an attenuation of 40 dBでしべる at the Nyquist folding frequency of half a cycle per channel of the new channel width (0.5 cycles per $27.8\,\mathrm{km}\,{{\rm{s}}}^{-1}$). This value is a reasonable compromise between the desirable suppression of aliased noise and the corresponding (minor) reduction in the resolution of the sub-sampled spectra. The Gaussian kernel was scaled to be unity at the origin of the Fourier plane to guarantee flux conservation. In addition, the kernel was given an odd number of elements, placed symmetrically around the origin, to prevent spectral shifts during the convolution. The effect of this filter was equivalent to performing a linear convolution of the spectrum with an area-normalized Gaussian of FWHM $63.2\,\mathrm{km}\,{{\rm{s}}}^{-1}$.

Next, we interpolated the filtered spectrum onto the frequency grid of the stacked spectra. The mean of the resulting spectra was then calculated to obtain the final stacked spectrum. ³⁰ We did not apply any weighting when performing the stack.

We also carried out two different random extractions following the same procedure described above, to produce random stacked spectra for comparisons. One random extraction involved assigning random redshifts to each spectrum before combining them at the location of the known MUSE position. The other involved using the correct MUSE redshifts, but extracting the spectra at random positions within the region where the LP primary beam response is $\gt 50 \%$, instead of at the known source position. The number of randomized extractions is the same as the number of CO emission samples. Neither of these methods should produce stacked spectra with real features. This random spectral extraction highlights which features in the real stacked spectra should be discounted as noise.

In Figure 2, we show the standard deviation of randomly stacked spectra against the number of the stacks. For this plot, we randomize both redshift and position for the spectral extractions. The standard deviation of stacked spectra roughly decreases with $1/\sqrt{N}$.

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We took advantage of the significant redshift overlap between the ASPECS and MUSE surveys to carry out a stacking analysis on CO line emission: we focused on the ASPECS Band 3 data where the full MUSE redshift coverage can be exploited. In particular, over the range $1.0059\leqslant z\,\leqslant 1.7387$, the ASPECS survey has the highest sensitivity for detecting CO(2–1) among all of the observable CO emission features (Figure 9 of Boogaard et al. 2019). In addition, this redshift range is where MUSE is efficient at detecting spectral features (up to $z\sim 1.5$ for [O ii]) as shown in Figure 1.

We first selected a subset of the MUSE sources for stacking CO(2–1) emission in the ASPECS Band 3 data cube. The criteria were the following: (1) $1.0059\leqslant \mathrm{MUSE}\ \mathrm{spec}-z\leqslant 1.7387$, (2) MUSE spec $-z$ confidence levels $\geqslant 2$, and (3) location within ALMA Band 3 LP primary beam response $\geqslant 50 \%$. For high-J CO emission, we used the same selection criteria, except for the redshift range. The ranges were $z=2.0088-3.1080$ for CO(3–2), $z=3.0115-4.4771$ for CO(4–3), $z=4.0142\,-5.8460$ for CO(5–4), and $z=5.0166-7.2146$ for CO(6–5).

The resulting number of MUSE sources for stacking CO(2–1) was 111 in total. ³¹ For 104 sources, the MUSE spectroscopic redshift identifications were based on [O ii], five sources used absorption features, one source used C iii], and one quasar had strong Mg ii emission. Among these 111 sources, 10 sources ³² were identified with CO(2–1) emission by the ASPECS blind search: MUSE IDs 8 (ASPECS-LP-3mm.06), 16 (3mm.11), 924 (3mm.14), 925 (3mm.16), 996 (3mm.02), 1001 (3mm.05), 1011 (3mm.10), 1117 (3mm.04), 6415 (3mm.08), and 6870 (3mm.15) (Boogaard et al. 2019; Aravena et al. 2019).

We performed the stacking in each group of the sample galaxies classified by the stellar mass and specific SFR (${SSFR}={SFR}/{M}_{* }$) derived from the MAGPHYS SED fits (see Section 2.3). The ${SFR}-{M}_{* }$ relation of our sample is shown in Figure 3. The specific star formation rate (SSFR) classification was based on the MS relation from Equation 11 of Boogaard et al. (2018), using the mean redshift of the CO(2–1) line detectable in Band 3 ($z=1.43;$ Walter et al. 2016). Galaxies lying within, above, and below the intrinsic scatter (0.44 dex ³³ ) of this relation are referred to as the "MS," "above" the MS, and "below" the MS, respectively, throughout this paper. We here use the MS relation from Boogaard et al. (2018) because their ${SFR}-{M}_{* }$ correlation is assessed with the objects whose spectral features and redshifts were measured based on the MUSE data. Compared with earlier studies such as Whitaker et al. (2014) and Schreiber et al. (2015), whose samples are mostly massive galaxies, Boogaard et al. (2018) better constrained the low-mass end of the relation. The numbers of galaxies in each group for stacking are presented in Table 1.

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Table 1.  Numbers of Galaxies in the Stellar Mass and SSFR Bins for the CO(2–1) Stacking Sample

	$\mathrm{log}({M}_{* }/{M}_{\odot })$
	$8.0-9.0$	$9.0-10.0$	$10.0-11.0$	$11.0-12.0$
Above the MS	11 (11)	4 (3)	1 (1)	0 (0)
On the MS	38 (38)	32 (32)	12 (7)	3 (0)
Below the MS	0 (0)	5 (5)	2 (2)	3 (2)

Note. The numbers in parentheses are after excluding the galaxies which have the CO(2–1) line identified by the blind search (see Section 3.2).

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The MS relation of Boogaard et al. (2018) was also adopted for classifying galaxies which expected to have CO $J\gt 2$ lines. Similar to CO(2–1), we used the mean redshifts of z = 2.61, 3.80, 4.99, and 6.18 for CO(3–2), CO(4–3), CO(5–4), and CO(6–5) lines, respectively (Walter et al. 2016).

For each stacked spectrum, we performed a best fit on the 2D image (moment-0) with a 2D Gaussian function to estimate the line flux or upper limit. The x- and y-positions are allowed to vary in the vicinity of the MUSE positions within a radius of 1/3 of the ALMA beam size. The widths were fixed to the beam size. We considered a CO emission line to have been detected when the amplitude of the fitted Gaussian function was higher than 3σしぐま of the local fluctuations in the 2D image. The line flux was estimated by integrating the fitted Gaussian function. When the CO line was not detected, the standard deviation in the central area of the image was used to evaluate the 3σしぐま upper limit, assuming a point source.

Uncertainties in MUSE redshifts can cause some flux losses when we stack spectra. We assumed an emission line with a width of $300\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (full width at half maximum; FWHM) and performed a bootstrap simulation to assess the flux loss resulting from ${\sigma }_{z}=0.00012(1+z)$ (Section 2.2). The total flux was measured within ${\rm{\Delta }}v=528\,\mathrm{km}\,{{\rm{s}}}^{-1}$ (corresponding to 19 slices in frequency; see Section 4.1.1 for this choice of ${\rm{\Delta }}v$) centered at the rest frequency of CO(2–1). This velocity range was the same as the one we used to create coadded 2D images for flux measurements (see Section 4). With 10,000 realizations, we found that 92% of them resulted in less than 3% flux loss. We do not scale up our measured fluxes in this work to take account of this flux loss, because its impact on our line flux measurements is not significant.

4.1.1. Stacking the Entire Sample

4.1.2. Stacking in Stellar Mass and SSFR Bins

We carried out the stacking on both the entire sample and the sample that excluded the direct CO detections. To investigate the dependence of the molecular gas content on fundamental properties of galaxies, we divided the sample into ranges of stellar mass and SSFR to perform the stacking. As shown in Figure 3, we set the bins to have steps of $\mathrm{log}({M}_{* }/{M}_{\odot })=1.0$ over the range $8.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 12.0$ (four bins) with stellar mass, and galaxies "above" the MS, on the "MS", and "below" the MS based on the MS relation at z = 1.43 discussed in Section 3.2. In the redshift range where CO(2–1) can be detected with the ASPECS Band 3 survey ($z=1.0-1.7$), the MS relation does not evolve significantly.

The stacked 2D and 1D spectra are shown in Figure 4 and the CO line flux measurements are summarized in Table 2. When the directly detected CO(2–1) emission is included, there are solid detections ($\gt 3\sigma$) at $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10.0$, regardless of their SSFRs (except in the $10.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 11.0$ bin where there is only a single source). On the other hand, at $\mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10.0$, we only obtain a signal in the bin above the MS with $9.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10.0$. When we exclude the known CO(2–1) emission from the stacks, a significant detection is seen in the MS bin with $10.0\,\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 11.0$ (seven sources).³⁴

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Table 2.  Summary of Measured Stacked CO(2–1) Line Fluxes (${F}_{\mathrm{line}}$)

	$\mathrm{log}({M}_{* }/{M}_{\odot })$
	$8.0-9.0$	$9.0-10.0$	$10.0-11.0$	$11.0-12.0$
Above the MS	$\lt 0.032$ ($\lt 0.032$)	0.110 ± 0.027 ($\lt 0.041$)	$\lt 0.081$ ($\lt 0.081$)	⋯ (⋯)
On the MS	$\lt 0.022$ ($\lt 0.022$)	$\lt 0.019$ ($\lt 0.019$)	0.130 ± 0.021 (0.095 ± 0.024)	0.453 ± 0.042 (⋯)
Below the MS	⋯ (⋯)	$\lt 0.051$ ($\lt 0.051$)	0.119 ± 0.039 (0.119 ± 0.039)	0.152 ± 0.027 ($\lt 0.054$)

Note. The units are in $\mathrm{Jy}\,\mathrm{km}\,{{\rm{s}}}^{-1}$. Values in parentheses are after excluding the galaxies which have the CO(2–1) line identified by the blind search (see Section 3.2).

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Although the bins of the MS galaxies in the ranges of $8.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 9.0$ and $9.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10.0$ contain the largest numbers of galaxies, we do not detect a CO(2–1) line. To attempt to detect stacked CO emission for galaxies in these low-mass bins, we adopt a ${\rm{\Delta }}\mathrm{log}({M}_{* }/{M}_{\odot })=0.5$ stellar mass bin size in case some hidden emission from a small number of sources has been diluted by averaging too many sources without any emission. This smaller bin size results in 14, 24, 23, and 9 MS sources from $8.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 8.5$ to $9.5\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10.0$ in steps of $\mathrm{log}({M}_{* }/{M}_{\odot })=0.5$. No detection is identified even with these finer bins. The estimated 3σしぐま upper limits with the 2D data are 0.028, 0.024, and 0.017, $0.030\,\mathrm{Jy}\,\mathrm{km}\,{{\rm{s}}}^{-1}$, from lower to higher stellar mass bins, respectively.

Among the bins with stacked detections, although the censoring fraction is unity for the bin of below the MS with stellar mass $10.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 11$, all of the remaining bins are $\leqslant 0.75$. In particular, the fraction is $\leqslant 0.6$ for the MS galaxies.

4.1.3. The CO(2-1) Lines Detected in Individual Galaxies

Along with the CO(2–1) stacking analysis, we attempt to detect higher excitation CO emission, up to $J=6-5$, with the same stacking method. The census of the stacked sources is shown in Table 3.

Table 3.  Total Number of Galaxies Used for Stacking High-J CO Emission

	$\mathrm{log}({M}_{* }/{M}_{\odot })$
	CO	$7.0-8.0$	$8.0-9.0$	$9.0-10.0$	$10.0-11.0$	$11.0-12.0$
Above the MS	3-2	1 (1)	5 (5)	5 (4)	1 (-)	⋯
	4-3	4 (4)	4 (4)	5 (5)	1 (1)	⋯
	5-4	⋯	⋯	⋯	⋯	⋯
	6-5	⋯	⋯	⋯	⋯	⋯
On the MS	3-2	4 (4)	21 (21)	9 (9)	4 (3)	⋯
	4-3	63 (63)	79 (79)	29 (29)	4 (4)	⋯
	5-4	40 (40)	46 (46)	26 (26)	11 (11)	2 (2)
	6-5	2 (2)	16 (16)	14 (14)	7 (7)	3 (3)
Below the MS	3-2	⋯	⋯	⋯	⋯	⋯
	4-3	⋯	⋯	2 (2)	⋯	⋯
	5-4	1 (1)	⋯	1 (1)	1 (1)	⋯
	6-5	2 (2)	⋯	⋯	⋯	⋯

Note. The numbers in parentheses are after excluding the galaxies which have the CO line identified by the blind search (see Section 3.2).

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For CO(3–2), among our sample selection (Section 3.2), MUSE IDs 35 and 1124 ³⁵ are the only sources that were already known from the blind search (ASPECS-LP-3mm.01 and 3mm.12). The former is a galaxy above the MS and the latter is on the MS with $\mathrm{log}({M}_{* }/{M}_{\odot })=10.39$ and 10.64, respectively. In addition, we consider MUSE ID 35's pair galaxy MUSE ID 24 ($\mathrm{log}({M}_{* }/{M}_{\odot })=9.45$) to have a CO(3–2) detection in our analysis. This is because the CO spatial extent, which peaks at the location of ID 35, covers ID 24 and it is difficult to distinguish the flux contributions from these two sources. When the stacking was performed in the same way as for CO(2–1), these two sources dominated the detections in their bins (Appendix B). If we discard them, no significant emission remains. This result agrees with the finding of Uzgil et al. (2019) who also did not find additional CO(3–2) emission with a masked auto-power spectrum using all MUSE positions with LP primary beam response $\geqslant 20 \%$. There is also no stack detection for ${J}_{\mathrm{up}}\geqslant 4$ CO emission in our sample with the blind search,³⁶ nor the stacking.

We note that the MUSE spectroscopic redshifts of galaxies at $z\gtrsim 3$, which allow detections of ${J}_{\mathrm{up}}\gt 4$ CO emission, were mostly determined using the Lyαあるふぁ line. Its redshift was measured using the peak emission, which can have a few hundred $\mathrm{km}\,{{\rm{s}}}^{-1}$ offset from the systemic redshift (e.g., Shapley et al. 2003). When the redshifts of the stacked sources were measured with Lyαあるふぁ, we tried to recover their systemic redshifts based on an empirical correlation between the Lyαあるふぁ emission peak and the Lyαあるふぁ line width (Verhamme et al. 2018). The intrinsic scatter of the relation is $\pm 73\,\mathrm{km}\,{{\rm{s}}}^{-1}$ measured with 13 sources that have both Lyαあるふぁ and C iii] detections. We do not find any detection for ${J}_{\mathrm{up}}\gt 4$ CO emission either. This is likely either due to dilution of the stacked signal, owing to uncertainties in the velocity offset, or to Lyαあるふぁ emitters on average having smaller SFRs and molecular gas content.

We also visually investigate individual CO spectra with the MUSE redshifts as priors to search for possible CO emission which is washed out by stacking. Two potential CO(5–4) emission lines were identified as shown in Appendix C. If confirmed, they may be one of the first cases of high-J CO detection at $z\gt 4.5$ for Lyαあるふぁ emitters. Deeper observations are needed to confirm these detections.

We employed the following equation to obtain the CO luminosities from the measured CO(2–1) emission (Solomon et al. 1997; Solomon & Vanden Bout 2005):

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{COJ}-[{\rm{J}}-1]}^{{\prime} }}{{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}}=3.25\times {10}^{7}\,\displaystyle \frac{{F}_{\mathrm{line}}}{\mathrm{Jy}\,\mathrm{km}\,{{\rm{s}}}^{-1}}\,\displaystyle \frac{{D}_{L}^{2}}{{\left(1+z\right)}^{3}{\nu }_{\mathrm{obs}}^{2}},\end{eqnarray} \tag{ 1 }$$

where ${F}_{\mathrm{line}}$ is the integrated line flux density, D_L is the luminosity distance in Mpc, and ${\nu }_{\mathrm{obs}}$ is the observed frequency in GHz. For the redshift (z), we used the mean redshift of the galaxies in each stacking bin.

We then adopted the following equation and the same conditions as Decarli et al. (2016b) to infer molecular gas masses (${M}_{\odot }$) from the line luminosities (${\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$):

$$\begin{eqnarray}&&{M}_{\mathrm{gas}}=\displaystyle \frac{{\alpha }_{\mathrm{CO}}}{{r}_{J1}}\,{L}_{\mathrm{COJ}-[{\rm{J}}-1]}^{{\prime} },\end{eqnarray} \tag{ 2 }$$

where J is the upper level of the CO excitation and ${\alpha }_{\mathrm{CO}}$ is the CO luminosity to gas mass conversion factor. We assume ${r}_{21}=0.76\pm 0.09$ following Daddi et al. (2015). Based on galaxies detected with the ASPECS survey, Boogaard et al. (2020) estimated ${r}_{21}=0.75\pm 0.11$, which is comparable to the value from Daddi et al. (2015). Here we adopt the value from the former to be consistent with the molecular mass measurements published in Aravena et al. (2019), which are used for comparison in this paper. For ${\alpha }_{\mathrm{CO}}$, we used $3.6\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ for galaxies in the stellar mass bins of $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$ which is one of the best estimates for high-redshift MS star-forming galaxies (Daddi et al. 2010). Among the ASPECS directly detected CO sources with metallicity estimates available from the MUSE spectra, all have roughly solar metallicity (Boogaard et al. 2019), which further justifies our choice of ${\alpha }_{\mathrm{CO}}$ (Aravena et al. 2019). For the galaxies with $\mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10.0$, because they are likely to have lower metallicity, ${\alpha }_{\mathrm{CO}}$ could be higher (e.g., Bolatto et al. 2013). In this work, we assume that the metallicities of galaxies with $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 9.5$ and ∼8.5 are $\sim 2/3{Z}_{\odot }$ and $\sim 1/2{Z}_{\odot }$ (e.g., Savaglio et al. 2005; Erb et al. 2006; Mannucci et al. 2009; Zahid et al. 2011; Sargent et al. 2014; Sanders et al. 2020). Thus, ${\alpha }_{\mathrm{CO}}$ would increase by a factor of ∼2 and ∼3, respectively, compared to galaxies with $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10.0$ (e.g., Genzel et al. 2012; Schruba et al. 2012). We adopt ${\alpha }_{\mathrm{CO}}=7.2$ and $10.8\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ for galaxies with $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 9.5$ and $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 8.5$, respectively. In these lower stellar mass bins, the ${M}_{\mathrm{gas}}$ estimates with a constant ${\alpha }_{\mathrm{CO}}=3.6\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ are also shown as dotted symbols in figures for reference. The calculated line luminosities and molecular gas masses from CO(2–1) are summarized in Table 4.

Table 4.  Summary of CO(2–1) Line Luminosities (${L}_{\mathrm{CO}2-1}^{{\prime} }$) and Molecular Gas Masses (${M}_{\mathrm{gas}}$)

		$\mathrm{log}({M}_{* }/{M}_{\odot })$
		$8.0-9.0$	$9.0-10.0$	$10.0-11.0$	$11.0-12.0$
Above the MS	(1)	$\lt 0.62$ ($\lt 0.62$)	2.47 ± 0.61 ($\lt 1.02$)	$\lt 1.59$ ($\lt 1.59$)	⋯ (⋯)
	(2)	$\lt 8.78$ ($\lt 8.78$)	23.42 ± 5.89 ($\lt 9.70$)	$\lt 7.53$ ($\lt 7.53$)	⋯ (⋯)
On the MS	(1)	$\lt 0.43$ ($\lt 0.43$)	$\lt 0.38$ ($\lt 0.38$)	2.51 ± 0.40 (1.85 ± 0.47)	10.57 ± 0.97 (⋯)
	(2)	$\lt 6.11$ ($\lt 6.11$)	$\lt 3.62$ ($\lt 3.62$)	11.87 ± 1.95 (8.74 ± 2.26)	50.07 ± 5.00 (⋯)
Below the MS	(1)	⋯ (⋯)	$\lt 1.03$ ($\lt 1.03$)	2.12 ± 0.69 (2.12 ± 0.69)	3.10 ± 0.55 ($\lt 1.04$)
	(2)	⋯ (⋯)	$\lt 9.71$ ($\lt 9.71$)	10.03 ± 3.31 (10.03 ± 3.31)	14.67 ± 2.68 ($\lt 4.95$)

Note. The units are in ${10}^{9}\,{\rm{K}}\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{pc}}^{2}$ and ${10}^{9}\,{M}_{\odot }$ for (1) CO(2–1) line luminosities (the first row in each MS bin) and (2) molecular masses (${M}_{\mathrm{gas}};$ the second row), respectively. The CO–to–H₂ conversion factors are assumed to be ${\alpha }_{\mathrm{CO}}=3.6$, 7.2, and 10.8 $\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ for galaxies in the bins of $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$, $9\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10$, and $8\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 9$, respectively (see Section 5.1). The values in parentheses are after excluding the galaxies which have the CO(2–1) line identified by the blind search (see Section 3.2).

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Table 5.  Summary of Molecular Gas to Stellar Mass Ratios (${\mu }_{\mathrm{gas}}={M}_{\mathrm{gas}}/{M}_{* }$) and Depletion Times (${t}_{\mathrm{depl}}={M}_{\mathrm{gas}}/{SFR}$)

		$\mathrm{log}({M}_{* }/{M}_{\odot })$
		$8.0-9.0$	$9.0-10.0$	$10.0-11.0$	$11.0-12.0$
Above the MS	(1)	$\lt 22.01$ ($\lt 22.01$)	9.94 ± 2.50 ($\lt 4.52$)	$\lt 0.51$ ($\lt 0.51$)	⋯ (⋯)
	(2)	$\lt 5.95$ ($\lt 5.95$)	1.82 ± 0.46 ($\lt 1.84$)	$\lt 0.05$ ($\lt 0.05$)	⋯ (⋯)
On the MS	(1)	$\lt 14.12$ ($\lt 14.12$)	$\lt 1.25$ ($\lt 1.25$)	0.49 ± 0.08 (0.46 ± 0.12)	0.24 ± 0.02 (⋯)
	(2)	$\lt 11.27$ ($\lt 11.27$)	$\lt 1.46$ ($\lt 1.46$)	0.69 ± 0.11 (0.67 ± 0.17)	1.13 ± 0.11 (⋯)
Below the MS	(1)	⋯ (⋯)	$\lt 2.82$ ($\lt 2.82$)	0.18 ± 0.06 (0.18 ± 0.06)	0.08 ± 0.01 ($\lt 0.02$)
	(2)	⋯ (⋯)	$\lt 31.44$ ($\lt 31.44$)	1.75 ± 0.58 (1.75 ± 0.58)	3.13 ± 0.57 ($\lt 3.56$)

Note. The first and second rows in each MS bin are (1) molecular-to-stellar mass ratios (${M}_{\mathrm{gas}}/{M}_{* }$) and (2) depletion times (${M}_{\mathrm{gas}}/{SFR}$ in $\mathrm{Gyr}$), respectively. The CO–to–H₂ conversion factors are assumed to be ${\alpha }_{\mathrm{CO}}=3.6$, 7.2, and 10.8 $\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ for galaxies in the bins of $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$, $9\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10$, and $8\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 9$, respectively (see Section 5.1). The values in parentheses are after excluding the galaxies which have the CO(2–1) line identified by the blind search (see Section 3.2).

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In the following subsections, we discuss the gas content of galaxies at $z\sim 1.5$, together with their stellar masses and SFRs derived from the SED fitting (Section 2.3). The CO(2–1) emission used here stems from the measurements based on the 2D Gaussian fits in the images presented in Section 4.1.2 and Figure 4. Because we are interested in global properties of the molecular gas, we use the measurements that include the direct CO(2–1) detections from the blind search.

Previous observational and theoretical studies of molecular gas in high-redshift galaxies have established a set of scaling relations that relate the molecular gas content to galaxy properties such as stellar masses, SFRs, and source sizes (e.g., Magdis et al. 2012; Somerville et al. 2012; Tacconi et al. 2013, 2018; Santini et al. 2014; Genzel et al. 2015; Scoville et al. 2016; Davé et al. 2017; Aravena et al. 2019; Freundlich et al. 2019; Popping et al. 2019). These relations have been key to understanding how the galaxy growth process has taken place through cosmic time. We first compare average molecular gas properties to the stellar mass of distant galaxies.

In conjunction with the MUSE deep survey in the same field, the stacking analysis facilitates the exploration of the gas scaling relations below stellar masses of $\mathrm{log}({M}_{* }/{M}_{\odot })=10$, a regime that is rather uncharted in CO emission for individual detections at $z\gt 1$ (see also Appendix D). We focus on discussing the MS galaxies, which have the largest numbers of stacked spectra in our CO(2–1) sample.

In Figure 5 (left), we plot the molecular gas mass from the stacking analysis against stellar mass of the MS galaxies, along with individual ASPECS CO detections (González-López et al. 2019; Aravena et al. 2019). The majority of directly detected galaxies have $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$. At $\mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10$, there is only one direct detection. Although stacking also leads to no detection, the stacks of 38 and 32 spectra provide tight constraints on the average gas mass in the range of $8\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10$ at $z\sim 1.5$. Taking upper limits into account, an increase of gas mass is found with increasing stellar mass from at least $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 9.0$ to 12.0.

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For comparison, we also show the contours of the molecular gas measurements from the PHIBSS2 survey (included both the CO- and dust-based measurements; Tacconi et al. 2013, 2018; Freundlich et al. 2019). For further comparison with the literature, we show the distributions of the PHIBSS2 galaxies at lower redshifts in Appendix D. The gas masses of the PHIBSS2 sources were derived via a metallicity-based prescription for this parameter (Genzel et al. 2012), resulting in ${\alpha }_{\mathrm{CO}}\sim 4-7\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ at $z\sim 1.5$.

We also compare our measurements with cosmological galaxy formation model calculations of molecular gas mass presented in Popping et al. (2019). In the same diagram we show three different model predictions: the IllustrisTNG hydrodynamical simulations with the "3.5arcsec" and "Grav" apertures and the SC SAM. The former corresponds to all the H₂ within a radius of $3.5^{\prime\prime}$ of the source center (similar to ASPECS) and the latter corresponds to all the H₂ gravitationally bound to the galaxy. The H₂ properties of galaxies were derived based on the molecular hydrogen fraction recipe of Gnedin & Kravtsov (2011).³⁷

We use the gas mass model predictions at z = 1.43 (Popping et al. 2019), which is the mean redshift of the CO(2–1) line detectable in Band 3. At the higher stellar mass end, all of the three models predicted lower gas mass than the observed gas mass. At $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$, about half of the sources with ASPECS direct detections lie within the 2σしぐま scatter of the predictions. At $\mathrm{log}({M}_{* }/{M}_{\odot })\lesssim 10.0$, where the predicted gas mass is below the ASPECS detection limit, the only CO constraint is from stacking. The upper limit at $\mathrm{log}({M}_{* }/{M}_{\odot })\,\sim 9.5$ derived from the stacking is consistent with the models. This result also implies that at least a factor of 10 increase in the sample size is needed to confirm or rule out the model predictions.

We depict this plot with a different presentation in Figure 5 (right) to show a more common presentation of molecular gas mass normalized by the stellar mass (molecular gas-to-stellar mass ratio, ${\mu }_{\mathrm{gas}}\equiv {M}_{\mathrm{gas}}/{M}_{* }$; Table 5) as a function of stellar mass. In the literature, a decrease of the gas-to-stellar mass ratio with increasing stellar mass at $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10.5$ has been reported at $z\sim 1.5$ (e.g., Tacconi et al. 2013). However, no observational CO constraints exist at lower stellar masses below ${10}^{10}\,{M}_{\odot }$. Our stacking analysis facilitates exploration of this lower stellar mass regime.

For galaxies with $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$ and 11.5, we estimate ${\mu }_{\mathrm{gas}}\,=$ 0.49 ± 0.08 and 0.24 ± 0.02, respectively. Similarly to Figure 5 (left), these values are in agreement with the measurements of the PHIBSS2 galaxies lying on the MS relation of Boogaard et al. (2018; gray contours; see Appendix D) for a version using the MS relation of Speagle et al. (2014). In the lower stellar mass bin, $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 9.5$, we constrain ${\mu }_{\mathrm{gas}}$ to have a 3σしぐま upper limit of $\lt 1.25$ with ${\alpha }_{\mathrm{CO}}=7.2\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ (or $\lt 0.63$ if ${\alpha }_{\mathrm{CO}}\,=3.6\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$).

Beyond $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$, a decrease of the gas-to-stellar mass ratio is discerned with increasing stellar mass in our stack results, as well as in previous work reporting both CO-based and dust-based gas estimates (Magdis et al. 2012; Tacconi et al. 2013, 2018; Genzel et al. 2015; Scoville et al. 2017; Aravena et al. 2019; Liu et al. 2019). As an example, we show in Figure 5 the result from PHIBSS2 with a steady decrease of ${\mu }_{\mathrm{gas}}$ with an increase of stellar mass (black line).

This decline of ${\mu }_{\mathrm{gas}}$ does not seem to be as steep at $\mathrm{log}({M}_{* }/{M}_{\odot })\lt 10.0$. If a constant ${\alpha }_{\mathrm{CO}}$ is also adopted for these lower-mass bins, the gas-to-stellar mass ratio is consistent with being constant from $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 9.5$ to 10.5, then turning down around $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$. We note that the models from Popping et al. (2019), despite a discrepancy with the observed results at the high mass side, show a similar constant gas-to-stellar mass ratio up to $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$. This is consistent with our finding if we assume a constant conversion factor.

Given the limitations of our data, and the unknown dependence of the ${\alpha }_{\mathrm{CO}}$ conversion factor, we cannot determine at which stellar mass a possible downturn of the gas-to-stellar mass ratio occurs. We, however, note that such a "plateau" in the low stellar mass regime has also been identified for local galaxies with direct detections of CO emission: here ${\mu }_{\mathrm{gas}}$ stays constant from at least $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 9.0$ and starts decreasing around $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$ (e.g., Saintonge et al. 2017; Bothwell et al. 2014). A similar result is found in Tacconi et al. (2018) who compared local star-forming galaxies, where CO measurements for low-mass galaxies are available (including Saintonge et al. 2017), to distant galaxies after removing assumed redshift effects on their gas mass content (see also Appendix D).

The declining ${\mu }_{\mathrm{gas}}$ at the high stellar mass end can be attributed to stellar feedback (e.g., Davé et al. 2011; Tacconi et al. 2013; Genzel et al. 2015). In contrast, the flatter ${\mu }_{\mathrm{gas}}$ at $\mathrm{log}({M}_{* }/{M}_{\odot })\lesssim 10.5$ may imply that the effects of feedback are weaker. The drop in gas-to-stellar mass ratio seems to appear around $\mathrm{log}({M}_{* }/{M}_{\odot })\sim 10.5$, which is relevant to the characteristic stellar mass of star-forming galaxies (e.g., Duncan et al. 2014) where mass-quenching is becoming dominant (Peng et al. 2010).

Furthermore, the gas-to-stellar mass ratio below ∼1 at 3σしぐま of low stellar mass galaxies suggests that we may have retrieved most of the CO emission at $z\sim 1.5$. As shown in Decarli et al. (2019), CO luminosity functions derived from the ASPECS data are assumed to have a fixed faint-end slope, which is consistent with the faint-end slope of the stellar mass function of star-forming galaxies. Thus, the value of ${\mu }_{\mathrm{gas}}\lesssim 1$ indicates that the assumed faint-end slope of the CO luminosity function is at least consistent or could be flatter than that of the stellar mass function. If this is the case, most of the CO emission at $z\sim 1.5$ has been recovered (see also Uzgil et al. 2019).

The estimated molecular gas density $\rho ({{\rm{H}}}_{2})$ at $z\sim 1.5$, based on our CO(2–1) stacking measurements, is $(0.49\pm 0.09)\,\times {10}^{8}\,{{\rm{M}}}_{\odot }\,{\mathrm{Mpc}}^{-3}$. This value is lower than, but formally consistent with, the value derived in Decarli et al. (2019) with a different CO selection. The CO emission used in Decarli et al. (2019) included sources that did not enter the present analysis because of the lack of a counterpart with high-quality MUSE redshift. Our total CO flux estimate from stacking is also in line with the result from the CO auto-power spectrum analysis using the MUSE positions (Uzgil et al. 2019).

In the previous section we only considered MS galaxies; in the remaining discussion, we will include galaxies above and below the MS relation of Boogaard et al. (2018) to discuss the dependence of molecular gas content on SSFR. Below, we keep the same assumed ${\alpha }_{\mathrm{CO}}$ conversion factors as above: 3.6, 7.2, and $10.8\,{M}_{\odot }{\left({\rm{K}}\mathrm{km}{{\rm{s}}}^{-1}{\mathrm{pc}}^{2}\right)}^{-1}$ for galaxies with $\mathrm{log}({M}_{* }/{M}_{\odot })\gt 10$, ∼9.5, and 8.5, respectively (Section 5.1). We include the constraints using a constant ${\alpha }_{\mathrm{CO}}=3.6$ throughout the stellar mass bins in the figures for reference.

5.4.1. Gas-to-stellar Mass Ratios

As shown earlier, the gas-to-stellar mass ratio (${\mu }_{\mathrm{gas}}$) provides information on the supply and depletion of gas reservoirs in galaxies. We compare ${\mu }_{\mathrm{gas}}$ against SSFRs in the left panel of Figure 6. We find that ${\mu }_{\mathrm{gas}}$ increases with increasing SSFRs, which is also seen in earlier studies of high-redshift galaxies, including the dust-based measurements of the molecular gas mass (e.g., Magdis et al. 2012; Tacconi et al. 2013, 2018; Genzel et al. 2015; Scoville et al. 2016; Liu et al. 2019). With the stacking analysis, we show that this trend holds even for galaxies below the MS at $z\sim 1.5$.

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In the right panel of Figure 6, we show ${\mu }_{\mathrm{gas}}$ of stacked sources color-coded by different SSFRs to compare with their evolution (Geach et al. 2011; Magdis et al. 2012). Because our data points are derived from the CO(2–1) line, all of them lie between $z=1.0-1.5$. A wide spread is related to the variations of ${\mu }_{\mathrm{gas}}$ across the MS relation seen in the left panel (see also e.g., Tacconi et al. 2018). The lower ${\mu }_{\mathrm{gas}}$ of low SSFR sources is also found by Spilker et al. (2018) who investigated galaxies lying below the MS with $\mathrm{log}({M}_{\mathrm{gas}}/{M}_{\odot })\sim 11$ at $z\sim 0.7$. The depletion times of our sample below the MS are on average comparable with Spilker et al.'s sample. Following the same assumption as Spilker et al. (2018), if these galaxies continuously consume the existing gas with the observed SFR, then the ${\mu }_{\mathrm{gas}}$ of the galaxies with $\mathrm{log}({M}_{\mathrm{gas}}/{M}_{\odot })\sim 10.5$ (11.5) would reduce to 1/10 at $z\approx 0.5$ (0.2) and 1/100 at $z\approx 0.1$ (0.0). Hence, by z = 0 their ${\mu }_{\mathrm{gas}}$ would be comparable to those of passive galaxies at the current epoch.

5.4.2. Gas Depletion Time

The gas depletion time, ${t}_{\mathrm{depl}}={M}_{\mathrm{gas}}/{SFR}$, is another way of examining the gas content in galaxies. It estimates the time taken for the gas to be fully consumed at the current SFR without accounting for additional fueling. We show the gas depletion time as a function of SSFR in Figure 7. Our results from the stacking analysis are in good agreement with previous studies. Both the stacked and directly detected sources show decreasing ${t}_{\mathrm{depl}}$ with increasing SSFR, except that the $9.0\lt \mathrm{log}({M}_{* }/{M}_{\odot })\leqslant 10.0$ bin above the MS shows an elevated ${t}_{\mathrm{depl}}$ value. The data points from these two sets of measurements occupy the same range in the diagram, following the constant gas-to-stellar mass ratio (${M}_{\mathrm{gas}}/{M}_{* }$) from around 0.1 to 1.0. The decreasing gas depletion timescale with SSFR demonstrates that galaxies with more extreme star formation consume their gas at a higher rate. This may be the major cause of the scatter as shown in Figure 6.

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