2.1. Systematic Effects on the HI and H₂ -to-Stellar Mass Correlations

To reduce potential systematic effects that can bias how we derive the HI and H₂-to-stellar mass correlations we homogenize all the compiled observations to the same basis. Following, we discuss some potential sources of bias/segregation and the calibration that we apply to the observations. It is important to stress that to infer scaling correlations, as those of the gas fraction as a function of stellar mass, it is important to have a statistically representative and unbiased population of galaxies in each mass bin. Thus, there is no need to have mass limited volume-complete samples (see also § 4.1). However, a volume-complete sample assures that possible biases of the measure in question due to selection functions in galaxy type, color, environment, surface brightness, etc., are not introduced. The main expected bias in the gas content at a given stellar mass is due to the galaxy type/color; this is why we need to separate the samples at least into two broad populations, LTGs and ETGs.

2.2. The Compiled HI Sample

Appendix A presents a summary of the HI samples compiled in this paper (see also Table 2). Table 3 lists the total numbers and fractions of compiled galaxies with detection and non-detection for each galaxy population. Table 4 lists the number of detected and non-detected galaxies for the golden, silver, and bronze categories listed above (§ 2.1.5).

Figure 1 shows the mass ratio R_HI ≡ M_HI /M_∗ vs. M_∗ for the compiled samples. Note that we have applied some corrections to the reported samples (see above) to homogenize all the data. The upper and bottom left panels of Figure 1 show, respectively, the compilations for LTGs and ETGs. The different symbols indicate the source reference of the data and the downward arrows are the corresponding upper limits of the HI-flux for non-detections. We also reproduce the mean and standard deviation of different mass bins as reported in ^{Maddox et al. (2015)} for a cross-match of the ALFALFA and SDSS surveys. As mentioned in the Introduction, the ALFALFA survey is biased to high R_HI values, specially towards the low mass side. Note that the small ALFALFA subsample of dwarf galaxies by ^{Huang et al. (2012b, dark purple dots)} was selected mainly as an attempt to take into account low-HI mass galaxies at the low-mass end.

Fig. 1 Atomic gas-to-stellar mass ratio as a function of M_∗. Upper panels: Compiled and homogenized data with information on R_HI and M_∗ for LTGs (the different sources are indicated inside the left panel; see Appendix A for the acronyms and authors); downward arrows show the reported upper limits for non-detections. The blue triangles with thin error bars are mean values and standard deviations from the v.40 ALFALFA and SDSS crossmatch according to ^{Maddox et al. (2015)}; the ALFALFA galaxies are biased toward high values of R_HI (see text). Right panel is the same as left one, but with the data separated into three categories: Golden, Silver, and Bronze (yellow, gray, and brown symbols, respectively). The red and blue lines are Buckley-James linear regressions (taking into account non-detections) for the high- and low-mass regions, respectively; the dotted lines show extrapolations from these fits. Squares with error bars represent the mean and standard deviation of the data in different mass bins, taking into account non-detections by means of the Kaplan-Meier estimator. Open circles with error bars show the corresponding median and 25-75 percentiles. Estimates of the observational uncertainties are shown in the panel corners (see text). Lower panels: Same as upper panels but for ETGs. In the right panel, we have corrected for distance the galaxies with upper limits from GASS to make them consistent with the distances of the ATLAS3D sample (see text); the upper limits from the latter were increased by a factor of two to homogenize them to the ALFALFA instrument and signal-to-noise criteria. For the bins where more than 50% of the data are upper limits, the median and percentiles are not calculated. The color figure can be viewed online. 

2.3. The Compiled H₂ Sample

Since the emission of cold H₂ in the ISM is extremely weak, a tracer of the H₂ abundance should be used. The best tracer from the observational point of view is the CO molecule due to its relatively high abundance and its low excitation energy. The H₂ mass is related to the CO luminosity through a CO-to-H₂ conversion factor: MH2 = α_CO L_CO. This factor has been determined in molecular clouds in the Milky Way (MW), αCO,MW = 3.2 (K km s⁻¹ pc⁻¹)⁻¹, with a systematic uncertainty of 30%. It was common to assume that this conversion factor was the same for all galaxies. However, several pieces of evidence show that αCO is not constant, and that it depends mainly on the gas-phase metallicity, increasing as the galaxy metallicity decreases (e.g., ^{Boselli et al. 2002}; ^{Schruba et al. 2012}; ^{Narayanan et al. 2012}; ^{Bolatto et al. 2013}, and references therein). At firstorder, αCO changes slowly for metallicities larger than 12 + log₁₀(O/H) ≈ 8.4 (approximately half the solar one) and increases considerably as the metallicity decreases. Here, we combine the dependence of αCO on metallicity given by ^{Wolfire et al. (2010)} and the observed mass-metallicity relation to obtain an approximate estimation of the dependence of αCO on M_∗ for LTGs; see Appendix C for details. We are aware that the uncertainties involved in any metallicity-dependent correction remain substantial (Bolatto et al. 2013). Note, however, that our aim is to introduce and explore at a statistical level a reasonable mass-dependent correction to the CO-to-H₂ factor, which must be better than ignoring it. In any case, we present results both for α_CO = α_CO,MW and our inferred mass-dependent α_CO factor. In fact, the mass-dependent factor is important only for LTGs with M_∗ ≲ 3 × 10¹⁰ M_⊙; for larger masses and for all ETGs, α_CO ≈ α_CO,MW¹.

Appendix B presents a description of the CO (H₂) samples that we utilize in this paper. Table 3 lists the number of galaxies with detections and upper limits in the compilation sample in terms of morphology. Table 4 lists the number of detections and upper limits for the golden, silver, and bronze categories mentioned above (§ 2.1.5).

Figure 2 shows the mass ratio RH2≡ MH2 /M_∗ vs. M_∗ for the compiled samples. Similarly to the R_HI vs. M_∗ relation, we applied some corrections to observations in order to homogenize our compiled sample and to allow a more consistent comparison between the different samples. The upper and bottom left panels of Figure 2 show, respectively, the compiled datasets for LTGs and ETGs.

Fig. 2 Molecular gas-to-stellar mass ratio as a function of M_∗. Upper panels: Compiled and homogenized data with information on RH2 and M_∗ for LTGs (see inside the panels for the different sources; see Appendix B for the acronyms and authors); downward arrows show the reported upper limits for non-detections. Right panel is the same, but with the data separated into three categories: Golden, Silver, and Bronze (yellow, gray, and brown symbols, respectively). The red and blue lines are Buckley-James linear regressions (taking into account non-detections). The dotted lines show extrapolations from these fits. The green dashed line shows an estimate for the RH2 -M_∗ relation inferred from combining the empirical SFR MH2 and SFR-M_∗ correlations for blue/star-forming galaxies (see text for details). Squares with error bars are the mean and standard deviation of the data in different mass bins, taking into account non-detections by means of the Kaplan-Meier estimator. Open circles with error bars are the corresponding median and 25-75 percentiles. Estimates of the observational/calculation uncertainties are shown in the panel corners (see text). Lower panels: The same as upper panels but for ETGs. In the right panel, we have corrected for distance the galaxies with upper limits from COLD GASS to make them consistent with the distances of the ATLAS^3D sample (see text). For the bins where more than 50% of the data are upper limits, the median and percentiles are not calculated. The color figure can be viewed online. 

3.1. R_HI vs. M_∗

In Figure 3, results for log R_HI vs. log M_∗ are shown for LTGs (upper panels) and ETGs (lower panels). From left to right, the regressions for samples in the Golden, Silver, and Bronze categories are plotted. The error bars correspond to the 1σ scatter of the regression. Each line covers the mass range of the corresponding sample. The blue/red dashed lines and shaded regions in each panel correspond to the mean and standard deviation values calculated with the Kaplan-Meier estimator in mass bins for all the compiled LTG and ETG samples, previously plotted in Figures 1 and 2, respectively. However, the yellow, gray, and brown dots connected with thin solid lines in each panel are the mean values in each mass bin calculated only for the Golden, Silver, and Bronze samples, respectively. The standard deviations are plotted as dotted lines. In the following, we discuss the results shown in Figure 3.

Fig. 3 Atomic gas-to-stellar mass ratio as a function of M_∗ for the Golden, Bronze, and Silver LTGs (upper panels) and ETGs (lower panels). The mean and standard deviation in different mass bins, taking into account upper limits by means of the Kaplan-Meier estimator, are plotted for each case (filled circles connected by a dotted line and dotted lines around, respectively). For comparison, the mean and standard deviation (dashed lines and shaded area) from all the LTG (ETG) samples are reproduced in the corresponding upper (lower) panels. For each sample compiled and homogenized from the literature, the Buckley-James linear regression is applied, taking into account upper limits. The lines show the result, covering the range of the given sample; the error bars show the corresponding standard deviations obtained from the regression. When the data are too scarce and dominated by upper limits, the linear regression is not applied but the data are plotted. The numbers of LTG and ETG objects in each category are indicated in the respective panel. The color figure can be viewed online. 

Golden category: For LTGs, the three samples grouped in this category agree well among themselves in the mass ranges where they overlap; even the 1σ scatter of each sample does not differ significantly⁴. Therefore, as expected, these samples provide unbiased information for determining the R_HI -M_∗ relation of LTGs from log(M_∗/M_⊙ )≈ 7.3 to 11.4. For ETGs, the deviations of the Golden linear regressions among themselves and compared to all galaxies are within the 1σ scatter, which is actually large. If no corrections to the upper limits of the GASS and ATLAS^3D are applied, then the regression for the former would be significantly above the regression for the latter. Within the large scatter, the three Golden samples of ETGs seem not to be particularly biased, and they cover a mass range from log(M_∗/M_⊙)≈ 8.5 to 11.5. At smaller masses, the Updated Nearby Galaxy Catalog (UNGC) sample provides mostly only upper limits to R_HI.

Silver category: The LTG and ETG samples in this category, as expected, show more dispersed distributions in their respective R_HI-M_∗ planes than those from the Golden category. However, the deviations of the Silver linear regressions among themsleves and compared to all the galaxies are within the corresponding 1σ scatter. If any, there is a trend of the Silver samples to have mean R _HI values above the mean values of all galaxies especially for ETGs. Since the samples in this category are volume-complete, (they were specially constructed to study HI gas content), a selection effect towards objects with non-negligible or higher than the mean HI content can be expected. In any case, the biases are small. Thus, we decided to include the Silver samples to infer the R_HI-M_∗ correlations in order to slightly increase the statistics (the number of galaxies in this category is actually much smaller than in the Golden category), specially for ETGs of masses smaller than log(M_∗/M_⊙)≈ 9.7 (see Table 4).

Bronze category and effects of the environment: The very isolated LTGs (from the UNAMKIAS and Analysis of the interstellar Medium of Isolated GAlaxies -AMIGAsamples) have HI contents higher than the mean of all the galaxies, especially at lower masses: log R_HI is 0.1 − 0.2 dex larger than the average at log(M_∗/M_⊙)>∼ 10 and these differences increase up to 0.6 − 0.3 dex for 8 < log(M_∗/M_⊙) < 9, though the number of galaxies at these masses is very small. The HI content of the ^{Bradford et al. (2015)} isolated dwarf galaxies is also larger than the mean of all the galaxies but not by a factor larger than 0.4 dex. For isolated ETGs, the differences can attain an order of magnitude and are at the limit of the upper standard deviations around the means of all the ETGs. Thus, while isolated LTGs have somewhat larger R_HI ratios on average than galaxies in other environments, in the case of isolated ETGs, this difference is very large; isolated ETGs can be almost as gas rich as LTGs. In the Bronze group we have included also galaxies from the central regions of the Virgo Cluster, as reported in HRS and ATLAS^3D (only ETGs for the latter). According to Figure 3, the LTGs in this high-density environment are clearly HI deficient with respect to LTGs in less dense environments. For ETGs, the HI content is very low but only slightly lower on average than the HI content of all ETGs. It should be noted that ETGs, in particular the massive ones, tend to be located in high-density environments.

We conclude that the HI content of galaxies is affected by the effects of extreme environments. The most remarkable effect occurs for ETGs, which in a very isolated environment can be as rich in HI as LTGs. Therefore, we decided not to include galaxies from the Bronze category to determine the R_HI-M_∗ correlation of ETGs. In fact, our compilation in the Golden and Silver categories includes galaxies from a range of environments (for instance, in the largest compiled catalog, UNGC, 58% of the galaxies are members of groups and 42% are field galaxies, see ^{Karachentsev et al. 2014}) in such a way that the R_HI-M_∗ correlation determined below should represent an average of different environments. Excluding the Bronze category for the ETG population, we avoid biases due to effects of the most extreme environments. For LTGs, the inclusion of the Bronze category does not introduce significant biases in the R_HI-M_∗ correlation of all galaxies but it helps to improve the statistics. The mean values of R _HI in mass bins above ≈ 10⁹ M_⊙ are actually close to the mean values of the entire sample (compare the brown solid and blue dashed lines); at smaller masses the deviation increases, but the differences are well within the 1σ dispersion.

3.2. RH2 vs. M_∗

In Figure 4, we present plots similar to Figure 3 but for log RH2 vs. log M_∗. The symbol and line codes are the same in both figures. In the following, we discuss the results shown in Figure 4.

Fig. 4 Same as Figure 3 but for the molecular gas-to-stellar mass ratio. The color figure can be viewed online. 

Golden category: For LTGs, the two samples grouped in this category agree well among themsleves and with the overall sample, though for masses < 10¹⁰ M_⊙, where the Golden galaxies are only those from the HRS sample, the average RH2 values are slightly larger than those from the overall LTG sample (compare the solid yellow and dashed blue lines), but still well within the 1σ scatter (shaded area). For ETGs, the deviations of the linear regressions of the Golden samples among themselves, and compared to all ETGs, are within the respective 1σ scatters, which are actually large. If no corrections to the upper limits of the GASS and ATLAS^3D were applied, then the regression for the former would be significantly above the regression for the latter. Summarizing, the Golden samples of LTGs and ETGs do not show particular shifts in their respective RH2 - M_∗ correlations. Therefore, the combination of them is expected to provide reliable information for determining the respective RH2 - M_∗ correlations: for LTGs in the ≈ 10^8.5 − 10^11.5 M_⊙ mass range, and for ETGs, only for M_∗ ≳ 10¹⁰ M_⊙.

Silver category: The LTG samples present a dispersed distribution in the log RH2 -logM_∗ plane but well within the 1σ scatter of the overall sample (shaded area). The mean values in mass bins from samples of the Silver category are in reasonable agreement with the mean values from all the samples (compare the gray solid and blue dashed lines). Therefore, the Silver samples, though scattered and not complete in any sense, seem not to exhibit a clear systematical shift in their H₂ content. We include these samples to infer the RH2 - M_∗ correlation of LTGs. For ETGs, the two Silver samples provide information for masses below M_∗ ≈ 10¹⁰ M_⊙, and both are consistent with each other. Therefore, we include these samples to infer the ETG RH2 -M_∗ correlation down to M_∗ ≈ 10^8.5 M_⊙.

Bronze category and the effects of environment: The isolated (from the AMIGA sample) and Virgo central (from the HRS catalog) LTGs have H₂ contents similar to the mean in different mass bins of all the galaxies. If any, the Virgo LTGs have on average slightly higher values of RH2 than the isolated LTGs, especially at masses smaller than M_∗ ≈ 10¹⁰ M_⊙. Given that LTGs in extreme environments do not segregate from the average RH2 values at different masses of all galaxies, we include them for calculating the RH2 -M_∗ correlation of LTGs. For ETGs, the AMIGA isolated galaxies have on average values of RH2 significantly higher than the mean of other galaxies, while those ETGs from the Virgo central regions (from HRS and ATLAS^3D; mostly upper limits), seem to be on average consistent with the mean of all the galaxies, though the scatter is large. Given the strong deviation of isolated ETGs from the mean trend, we prefer to exclude galaxies from the Bronze category to determine the ETG RH2 -M_∗ correlation. We conclude that the H₂ content of LTGs is weakly dependent on the environment of galaxies, but in the case of ETGs, very isolated galaxies have systematically higher R_H2 values than galaxies in more dense environments.

4.1. Strategy for Constraining the Correlations

In spite of the diversity in the compiled samples and their different selection functions, the exploration presented in the previous section shows that the HI and H₂ contents as a function of M_∗ for most of the samples compiled here do not segregate significantly among them. The exception are the Bronze samples for ETGs. Therefore, the Bronze ETGs are excluded from our analysis. The strong segregation is actually by morphology (or color, or star formation rate), and this is why we have separated from the beginning the compiled data into two broad galaxy groups, LTGs and ETGs.

To determine gas-to-stellar mass ratios as a function of M_∗ we need (1) to take into account the upper limits of undetected galaxies in radio, and (2) to evaluate the correlation independently of the number of data points in each mass bin. If we have many data points in some mass bins and only a few ones in other mass bins (as would happen if we use, for instance, a mass-limited volume-complete sample, with many more data points at smaller masses than at larger masses), then the overall correlation of R_HI or RH2 with M_∗ would be dominated by the former, probably giving incorrect values of R_HI or RH2 at other masses. In view of these two requirements, our strategy to determine the logR _HI-logM_∗ and log RH2 -logM_∗ correlations is as follows:

4.2. The HI-to-Stellar Mass Correlations

In the upper left panel of Figure 5, along with the data from the Golden, Silver, and Bronze LTG samples, the mean and standard deviation (squares and black error bars) calculated for each mass bin with the Kaplan-Meier method are plotted. In the lower left panel, the same is plotted but for the Golden and Silver ETG samples (recall that the Bronze samples are excluded in this case). We see that the total standard deviations in log R_HI, σ_dat, do not evidence a systematical dependence on mass both for LTGs and ETGs. Then, we can use a constant value for each case. For LTGs, the standard deviations have values around 0.45-0.65 dex with an average of σ_dat ≈ 0.53 dex. For ETGs, the standard deviations are much larger and more dispersed than for LTGs (see § 4.4 below for a discussion on why this could be). We assume an average value of σ_dat = 1 dex for ETGs.

Fig. 5 Left panels: The R_HI-M_∗ correlation for LTGs (upper panel) and ETGs (lower panel). Dots are detections and arrows are upper limits for non-detections (for ETGs the Bronze sample was excluded). The squares and error bars show the mean and standard deviation in different mass bins calculated by means of the Kaplan-Meier estimator for censored and uncensored data. The thin error bars correspond to our estimate of the intrinsic scatter after taking into account the observational errors (shown in the panel corners). The solid and long-dashed lines in each panel are respectively the best double- and single-power law fits. The shaded areas show the intrinsic scatter; to avoid overcrowding, for the single power-law fit, the intrinsic scatter is plotted only at one point. The dotted lines are extrapolations of the correlations to low masses, where the data are scarce and dominated by upper limits. Middle panels: Same as in the left panels but for RH2. For the ETG population, the double power-law fit was performed with the conservative constraint that below M_∗ = 10⁹ M_⊙, the low-mass slope is zero. Right panels: The R_gas-M_∗ correlations for LTGs and ETGs as calculated from combining the respective double- and single-power law R_HI-M_∗ and RH2 -M_∗ correlations and taking into account helium and metals (see text). The shaded area and error bar are the (1σ) intrinsic scatter obtained by error propagation of the intrinsic scatter around the corresponding R _HI-M _∗ and RH2 -M_∗ relations. For completeness, the data from our compilation that have determinations of both HI and H₂ masses are also plotted (the obtained correlations are not fits to these data). Dotted lines are extrapolations of the inferred relations to smaller masses. The short dashed lines show the best fits using the double power-law function. The color figure can be viewed online. 

The intrinsic standard deviation (scatter) can be estimated as σintr2≈σdat2-σerr2 (this is valid for normal distributions), where σ_err is the mean statistical error in the log R _HI determination due to the observational uncertainties. In Appendix E we present an estimate of this error, σ_err ≈ 0.14 dex. Therefore, σ_intr ≈ 0.52 and 0.99 dex for LTGs and ETGs, respectively. These estimates should be taken only as indicative values given the assumptions and rough approximations involved in their calculations. For example, we will see in § 5 that the distributions of log R _HI (detections and non-detections) in different mass bins tend to deviate from a normal distribution, in particular for ETGs.

Next, we propose that the HI-to-stellar mass relations can be described by the general function:

where y = R_HI, C is the normalization factor, a and b are the low and high-mass slopes of the function and M*tr is the transition mass. This function is continuous and differentiable. If a = b, then equation (1) describes a single power law, or a linear relation in logarithmic scales. In this case, the equation remains as y(M_∗) = C′(M_∗/M_⊙)^−a. For a ≠ b, the function corresponds to a double power law.

We fit the logarithm of function, equation (1), to the mean values of logR_HI as a function of mass (squares in the left panels of Figure 5) with the corresponding (constant) intrinsic standard deviation as estimated above (thin blue/red error bars). For LTGs, the fit is carried out in the range 7.3 ≲ log(M_∗/M_⊙ )≲ 11.2, and for ETGs in the range 8.5 ≲ log(M_∗/M_⊙)≲ 11.5. The Levenberg Marquardt method is used for the fit (^{Press et al. 1996}). First, we perform the fits to the binned LTG and ETG data using a single power law, i.e., we fix a = b. The dashed orange and green lines with an error bar in the left panels of Figure 5 show the results. The fit parameters are given in Table 5. We note that these fits and those of the Buckley-James linear regression for all the data (not binned) in logarithm are very similar.

Then, we fit to the binned data the logarithm of the double power-law function given in equation (1). The corresponding best-fit parameters are presented in Table 6. We note that the fits are almost the same if the total mean standard deviation, σ_dat, is used instead of the intrinsic one. The reduced χred2 are 0.01 and 0.03, respectively. The fits are actually performed to a low number of points (the number of mass bins) with large error bars; this is why the χred2 are smaller than 1. Note, however, that the error bars are not related to measurement uncertainties but correspond to the population scatter of the data. Therefore, in this case χred2 < 1 implies that while the best fit is good, other fits could be also good within the scatter of the correlations. In the case of the single power-law fits, the χred2 are 0.03 and 0.01, respectively for LTG and ETG.

The double power-law R_HI-M_∗ relations and the estimated intrinsic (1σ) scatter for the LTG (ETG) population are plotted in the left upper (lower) panel of Figure 5 with solid lines and shaded areas, respectively. From the fits, we find for LTGs a transition mass M*tr = 1.74 ×10⁹ M_⊙, with R ∝ M_∗ ^−0.21 and M_∗ ^−0.67 at masses much smaller and larger than this, respectively. For ETGs, M*tr = 1×10⁹ M_⊙, and R_HI ∝ M_∗ ^0.0 and M_∗ ^-0.58, at masses much smaller and larger than this, respectively.

Both the double and single power laws describe well the HI-to-stellar mass correlations. However, the former could be more adequate than the latter. In Figure 1 we plot the Buckley-James linear regressions to the R_HI vs. M_∗ data for the low and high mass regions (below and above log(M_∗/M_⊙)≈ 9.7; for ETGs the regression is applied only for masses above 10⁸ M _⊙); the dotted lines show the extrapolation of the fits. The slope at low masses for LTGs, −0.36, is shallower than the one at high masses, −0.55. For ETGs, there is even evidence of a change in the slope sign at low masses. A flattening of the overall (late + early type galaxies) correlation at low masses has been also suggested by ^{Baldry et al. (2008)}, who have used the empirical mass- metallicity relation coupled with a metallicity-to-gas mass fraction relation (which can be derived from a simple chemical evolution model) to obtain a gasto-stellar mass correlation in a large mass range. Another evidence that at low masses the R_HI-M_∗ relation flattens is shown in the work by ^{Maddox et al. (2015)} already mentioned (see also ^{Huang et al. 2012a}). While the sample used by these authors does not allow to infer the R_HI-M_∗ correlation of galaxies due to its bias towards high R _HI values (see above), the upper envelope of this correlation can be actually constrained; the high-R_HI envelope does not suffer from selection limit effects. As seen for the data from ^{Maddox et al. (2015)} reproduced in the left upper panel of our Figure 1, this envelope tends to flatten at M_∗ ≲ 2 × 10⁹ M_⊙,⁵ which suggests (but does not demonstrate) that the mean relation can also exhibit such a flattening. Another piece of evidence in favor of the flattening can be found in ^{Huang et al. (2012b)}, and more recently in ^{Bradford et al. (2015)} for their sample of low-mass galaxies combined with larger mass galaxies from the ALFALFA survey.

4.3. The H₂ -to-Stellar Mass Correlations

In the upper middle panel of Figure 5, along with the data from the Golden, Silver, and Bronze LTG samples, the mean and standard deviation (error bars) calculated in each mass bin with the KaplanMeier method are plotted. In the lower panel, the same is plotted but for the Golden and Silver ETG samples (recall that the Bronze samples are excluded in this case). The poor observational information at stellar masses smaller than ≈ 5 × 10⁸ M_⊙ does not allow us to constrain the correlations at these masses, both for LTG and ETGs. Regarding the total standard deviations, for both LTGs and ETGs, they vary from mass bin to mass bin but without a clear trend. Then we can use a constant value for both cases. For LTGs, the total standard deviations have values around 0.5-0.8 dex with an average of σ_dat ≈ 0.58 dex. For ETGs, the average value is roughly 0.8 dex. As in the case of HI (previous subsection), we further estimate indicative values for the intrinsic population standard deviations (scatter). For this, we present in Appendix E an estimate of the the mean observational error of the log RH2 determination, σ_err ≈ 0.34 dex. Therefore, the estimated mean intrinsic scatters in log RH2 are σ_intr ≈ 0.47 and 0.72 dex for LTGs and ETGs, respectively. Given the assumptions and approximations involved in these estimates, they should be taken with caution. For example, we will see in § 5 that the distributions of log RH2 (detections and non-detections) in different mass bins tend to deviate from a normal distribution, in particular for the ETGs.

We fit the logarithm of function equation (1), y = RH2, to the mean values of log RH2 as a function of mass (squares in the left panels of Figure 5) with their corresponding scatter as estimated above (thin blue/red error bars), assumed to be the individual standard deviations for the fit. Again, the Levenberg-Marquardt method is used to perform the fit. The fits extend only down to M_∗ ≈ 5 × 10⁸ M_⊙. First, the fits are performed for a singe power law, i.e., we fix a = b. The dashed orange and green lines in the middle panels of Figure 5 show the results. The parameters of the fit and their standard deviations are given in Table 5. The fits are very similar to those obtained using the Buckley-James linear regression to all (not binned) logarithmic data.

Then, we fit the binned LTG and ETG data to the double power-law function equation (1). In the case of the ETG population, we impose an extra condition to the fit: that the slope of the relation at masses below ≈ 10⁹ M_⊙ be flat. The few data at these masses clearly show that RH2 does not increase for smaller M_∗; it is likely that it even decreases, so that our assumption of a flat slope is conservative. The corresponding best-fit parameters are presented in Table 6. As in the case of the R_HI −M_∗ correlations, the reduced χred2 are smaller than 1 (0.04 and 0.10, respectively), which implies that while the best fits are good, other fits could describe reasonably well the scattered data. In the case of the single power-law fits, χred2 were 0.04 and 0.07, respectively for LTG and ETG. The double power-law RH2 -M_∗ relations and their (1σ) intrinsic scatter for the LTG (ETG) population are plotted in the middle upper (lower) panel of Figure 5 with solid lines and shaded areas, respectively. We note that the fits are almost the same if the total mean standard deviation, σ_dat, is used instead of the intrinsic one.

From these fits, we find for LTGs, M*tr = 1.74 × 10⁹ M_⊙, with RH2 ∝ M_∗ ^−0.07 and M_∗ ^−0.47 at much smaller and larger masses than this, respectively. For ETGs, M*tr = 1.02 × 10⁹ M_⊙, with RH2 ∝ M_∗ ^0.00 and M_∗ ^−0.94 at much smaller and larger masses than this, respectively. In the middle upper panel of Figure 5, we plot also the best double power-law fit to the RH2 -M_∗ correlation of LTGs when the αCO factor is assumed constant and equal to the MW value (purple dashed line).

Both the single and double power-law functions describe equally well the RH2 -M_∗ correlations for the LTG and ETG population, but there is some evidence of a change of slope at small masses. In Figure 2, the Buckley-James linear regressions to the RH2 vs. M_∗ data below and above log(M_∗/M_⊙)≈ 9.7 are plotted (in the former case the regressions are applied for masses only above 10⁸ M_⊙); the dotted lines show the extrapolation of the fits. The slopes in the small mass range at low masses for LTGs/ETGs are shallower than those at high masses. Besides, in the case of ETGs, if the single power-law fit shown in Figure 5 is extrapolated to small masses, ETGs of M_∗ ≈ 10⁷ M_⊙ would be dominated in mass by H₂ gas. Red/passive dwarf spheroidals are not expected to contain significant fractions of molecular gas. Recently, ^{Accurso et al. (2017)} have also reported a flattening in the H₂-to-stellar mass correlation at stellar masses below ≈ 10¹⁰ M_⊙.

4.4. The Cold Gas-to-Stellar Mass Correlations

Combining the R_HI-M_∗ and RH2 -M_∗ relations presented above, we can obtain now the R_gas-M_∗ relation, for both the LTG and ETG populations. Here, R_gas = M_gas/M_∗ = 1.4(R_HI + RH2), where M_gas is the galaxy cold gas mass, including helium and metals (the factor 1.4 accounts for these components). The intrinsic scatter around the gas-tostellar mass relation can be estimated by propagating the intrinsic scatter around the HI and H₂ to-stellar mass relations. Under the assumption of null covariance, the logarithmic standard deviation around the logR_gas -logM_∗ relation is given by

The obtained cold gas-to-stellar mass correlations for the LTG and ETG populations are plotted in the right panels of Figure 5. The solid lines and shaded bands (intrinsic scatter given by the error propagation) were obtained from the double powerlaw correlations, while the solid green lines and the error bars were obtained from the single power-law correlations. For completeness, we also plot in Figure 5 those galaxies from our compilation that have determinations for both the HI and H₂ masses. Note that a large fraction of our compilation has no determinations for both quantities at the same time. We fit the results obtained for the single (double) power-law fits, taking into account the intrinsic scatter, to the logarithm of the single (double) power law function given in equation (1) with y = R _gas and report in Table 5 (Table 6) the obtained parameters for both the LTGs and ETGs. The fits for the double power-law are shown as dotted lines in Figure 5. The standard deviations σlog R_gas change slightly with mass; we report an average value for them in Tables 5 and 6. Both for LTGs and ETGs, the mass at which the R_gas-M_∗ correlations change slope is M*tr ≈ 1.7 × 10⁹ M_⊙, the mass that roughly separates dwarf from normal galaxies.

According to Figure 5, the LTG and ETG R_gas- M_∗ correlations are significantly different. The gas content in the former is at all masses larger than in the latter, the difference being maximal at the largest masses. For the LTG population, M_gas ≈ M_∗ on average at log(M_∗/M_⊙) ≈ 9, and at smaller masses, these galaxies are dominated by cold gas; at stellar masses around 2 × 10⁷ M_⊙ , M_gas is on average three times larger than M_∗. For ETGs, there is a hint that at ≈ 10⁹ M_⊙, R_gas changes from increasing as M_∗ is smaller to decreasing at larger masses.

6.1. The Mock Galaxy Mass Functions

6.1.1. Stellar Mass Function

The mock GSMF is plotted in panel (a) of Figure 13 along with the Poisson errors given by the thickness of the gray line; except for the highest masses, the Poisson errors are actually thinner than the line. The mock GSMF is an excellent realization of the empirical GSMF (compare it with Figure 12). We also plot the corresponding contributions to the mock GSMF of the LTG and ETG populations (blue and red dashed lines). As expected, LTGs dominate at small stellar masses and ETGs dominate at large stellar masses. The contribution of both populations is equal (f_early = f_late = 0.5) at M*cross = 10^10.20 M_⊙ (recall that the fraction fearly used here comes from ^{Moffett et al. (2016)}, who included Sa galaxies as ETGs; if consider Sa galaxies as LTGs, then M*cross would likely be higher). In order to predict accurate gas and baryonic mass functions, the present analysis will be further refined in Rodriguez-Puebla et al. (in prep.), where several sources of systematic uncertainty in the GSMF measurement and in the definition of the LTG/ETG fractions will be taken into account. Our aim here is only to test whether the empirical correlations derived in § 4 are roughly consistent with the total HI and H₂ empirical mass functions.

Fig. 13 Panel (a): Total GSMF from the mock catalog that reproduces the empirical GSMF of Figure 12 (solid line). The gray shadow represents the Poisson errors (except for large masses, these errors are thinner than the line thickness). The GSMF from the mock catalog samples very well the empirical GSMF used as input. The blue/red dotted lines and shadows correspond to the LTG/ETG mass function components, using the empirical ETG fraction as a function of M_∗ shown in Figure 12. Panel (b): Same as Panel (a) but for atomic gas, using the mean R _HI-M_∗ relation and its scatter distribution as given in § 5. Several observational GHIMF’s from blind HI samples, and the ETG GHIMF from ATLAS^3D and HIPASS surveys are reproduced (see labels inside the panel). Panel (c): Same as Panel (a) but for molecular gas. The GH₂MF calculated from the ^{Keres et al. (2003)} L_CO function is reproduced. The dotted purple line is the total GH2MF from the mock catalog using a RH2 -M_∗ correlation obtained from our compilation, but assuming that α_CO=α_CO,MW=const., as done in ^{Keres et al. (2003)}. The color figure can be viewed online. 

7.1. The H₂ -to-HI Mass Ratio

The global H₂-to-HI mass ratio of a galaxy characterizes its global efficiency of converting atomic into molecular hydrogen. This efficiency is tightly related to the efficiency of large-scale SF in the galaxy (see e.g., ^{Leroy et al. 2008}). From the empirical correlations inferred in § 4, we can calculate MH2 /M _HI as a function of M_∗ for both the LTG and ETG populations. We do this by using our double power-law fits to the data. The left panel of Figure 14 presents the obtained MH2 /M_HI -M_∗ relations and their 1σ H2 HI scatter calculated by propagating the dispersions in the assumption of null covariance. In this sense, the plotted scatters are upper limits, since there is evidence of some (weak) correlation between the HI and H2 content of galaxies, in particular among those deficient in HI and H₂ (^{Boselli et al. 2014c}). We can plot the same correlations for the mock catalog presented in § 6, which samples the observed GSMF, the LTG and ETG fractions as a function of M_∗, and the empirical correlations inferred by us. The middle panels of Figure 14 present our measures from the mock catalog for LTG (blue), ETG (red), and all galaxies (gray). The lines are the logarithmic means in small mass bins and the shaded regions are the corresponding standard deviations. At small masses, LTGs dominate, so the correlation of all galaxies is practically the one of LTGs. At large masses, ETGs become more important.

Fig. 14 Left panels: Molecular-to-atomic mass ratio, MH2 /M_HI, for LTGs (upper panel) and ETGs (lower panel) inferred from our double power-law fits to the R _HI-M_∗ and RH2 -M_∗ correlations. The shaded areas are the 1σ scatter obtained by error propagation of the scatter around the R_HI-M_∗ relations. Middle panels: Same as left panels, but from our mock catalog generated to sample the empirical GSMF, volume-complete ETG/LTG fractions as a function of mass, and R_HI-M_∗ and RH2 -M_∗ correlations. The dotted line surrounded by the gray area is the total MH2 /M_HI ratio and the 1σ scatter as a function of stellar mass. Right panels: Molecular-to-atomic mass ratio as a function of the cold gas mass, M_gas, from the mock catalog for LTGs (upper panel) and ETGs (lower panel). We plot available detected and undetected cold gas observational data as gray unfilled circles and downward arrows respectively. The color figure can be viewed online. 

According to Figure 14, the molecular-to-atomic mass ratio of LTGs increases with M_∗, albeit with a large scatter. On average, MH2 /M_HI increases from ≈ 0.1 to ≈ 0.8 for masses ranging from M_∗ = 10⁸ M_⊙ to 3 × 10¹¹ M_⊙. Given that the surface density of LTGs correlates significantly with M _∗, one can expect this dependence of MH2 /M_HI on M_∗ at least from two arguments: (1) Disk instabilities, which drive the formation of molecular clouds (e.g., the ^{Toomre criterion Toomre 1964}), are more probable to occur as the disk surface density is higher. (2) The H₂-to-HI mass ratio in galaxies has been shown to be directly related to the hydrostatic gas pressure (^{Blitz & Rosolowsky 2006}; ^{Krumholz et al. 2009}), and this pressure depends on the (gas and stellar) surface density (^{Elmegreen 1989}). In fact, the physics of H₂ condensation from HI is very complex and it is expected to be driven by local parameters of the ISM (see e.g., ^{Blitz & Rosolowsky 2006}; ^{Krumholz et al. 2009}; ^{Obreschkow & Rawlings 2009}). Therefore, the dependence of the H₂-to-HI mass ratio on M_∗ should be understood as consequence of the correlations of these parameters (their mean values along the galaxy) with M_∗, introducing this actually large scatter in the dependence of MH2 /M_HI on M_∗. Indeed, several authors have shown that MH2/M_HI correlates better with the mean gas-phase metallicity or mean stellar surface density than with M_∗ (e.g., ^{Saintonge et al. 2011}; ^{Boselli et al. 2014a}).

For ETGs, the trend of the H₂-to-HI mass ratio is inverse to the one of LTGs and shows a very large scatter. The ETGs more massive than ≈ 10¹¹ M_⊙ have mean ratios around 0.15 and a 1-σ scatter of ≈ ±1 dex; for intermediate masses, this ratio increases on average, and for ETGs with masses M∗ ≈ 109 M⊙, which are actually very rare, their mean H2-to-HI mass ratios are ≈ 1 with the same scatter of ≈ ±1 dex. Even though the gas fraction in ETGs is much smaller than in LTGs at all masses (see Figure 5), the former are also typically more compact than the latter, resulting probably in similar or higher gas pressures on average, and consequently similar or even higher MH2 /M_HI ratios, especially at masses smaller than M_∗ ≈ 10¹⁰ M_⊙. In fact, given the large scatter in MH2 /M_HI for ETGs, this ratio depends likely on many other internal and external factors (mergers, environment, etc.) that do not correlate significantly with M_∗.

Regarding MH2/M_HI vs. M_gas, for LTGs, which for M_∗ > 10⁷ M_⊙ have mostly gas masses > 108 M_⊙, there is no significant dependence, while for ETGs, which are almost inexistent with M_gas ≳ 10⁹ M_⊙, MH2/M_HI is larger on average for lower values of M_gas. This can be seen in the right panel of Figure 14, where the mock catalog has been used. Basically, for a given Mgas, in the mass range M_gas ≈ 10⁷ − 10⁹ M_⊙, ETGs have typically larger H₂-to-HI mass ratios than LTGs. In combination, the H₂-to-HI ratio appears to be larger for lower values of M _gas. Such a dependence has been reported by ^{Obreschkow & Rawlings (2009)} for their compiled sample of galaxies, and predicted by these authors for a physical model.

The dependences of the H₂-to-HI mass ratio on M_∗, M_gas, and morphological type discussed above are in qualitative agreement with several previous observational works, which actually are part of our compilation (^{Leroy et al. 2008}; ^{Obreschkow & Rawlings 2009}; ^{Saintonge et al. 2011}; ^{Boselli et al. 2014a}; ^{Bothwell et al. 2014}). However, our results extend to a larger mass range and separate explicitly the two main populations of galaxies.

7.2. The Role of Environment

There are several pieces of evidence that the atomic gas fraction of galaxies is lower in higherdensity environments (e.g., ^{Haynes & Giovanelli 1984}; ^{Gavazzi et al. 2005}; ^{Cortese et al. 2011}; ^{Catinella et al. 2013}; ^{Boselli et al. 2014c}). The fact that the ETG population has lower HI gas fractions than the LTG one (§ 4), the former being commonly found in higher-density environments, agrees with the mentioned environment trends. Thus, due to the morphology-density relation, our determinations of the R_HI-M_∗ (as well as RH2-M_∗ and R_gas-M_∗) correlations for the LTG and ETG populations, partially account for the dependence of these correlations on environment. Moreover, for the very isolated LTGs and for the subsample of LTGs in the Virgo cluster central regions, we confirm higher and lower HI-to-stellar mass ratios than the average of the overall LTG sample, respectively (see § 3.1). However, this systematic difference according to the environment is within the 1σ scatter of the R_HI-M_∗ correlation of LTGs (see Figure 3). Instead, in the case of ETGs, the isolated galaxies have much larger RHI values than the mean of all ETGs, above the 1σ scatter; isolated ETGs are almost as HI gas rich as the mean of LTGs.

Regarding the molecular gas fraction, the observational results are controversial in the literature. Recent studies seem to favor the the fact that galaxies in clusters are actually H₂-deficient compared to similar galaxies in the field. However, the deficiencies are smaller than in the case of HI (^{Boselli et al. 2014c}, and references therein). Here, for isolated and Virgo-center LTGs, we do not see any systematical segregation of RH2 from the rest of our compiled LTGs (Figure 4), but in the case of ETGs, the isolated galaxies have on average larger values of RH2.

In summary, the results from our compilation point out that the HI content of LTGs has a (weak) dependence on environment, mainly due to the fact that at high densities LTGs are HI deficient. However, the H₂ content of LTGs seems not to change on average with the environment. In the case of ETGs, those very isolated are significantly more gas rich (both in HI and H₂) than the average at a given mass.

An important aspect related to the environment is whether a galaxy is central or satellite. The local environmental effects once a galaxy becomes a satellite inside a halo. Ram pressure, viscous stripping, starvation, harassment and tidal interactions work in the direction of lowering the gas content of the galaxy, likely more the more massive the halo is (^{Boselli & Gavazzi 2006}; ^{Brown et al. 2017}). Part of the scatter in the gas-to-stellar mass correlations is probably due to the external processes produced by these local-environment mechanisms. A result in this direction has been recently shown for the R_HI-M_∗ correlation by ^{Brown et al. (2017)}. These authors have found that the HI content of satellite galaxies in more massive halos has, on average, lower HI-tostellar mass ratios at fixed stellar mass and specific SFR. According to their analysis, the systematic environmental suppression of the HI content at both fixed stellar mass and fixed specific SFR in satellite galaxies begins at halo masses typical of the group regime (> 10¹³ M_⊙), and fast-acting mechanisms such as ram-pressure stripping are suggested to explain their results. In a future study, we will attempt to characterize the central/satellite nature of our compiled galaxies, as well as calculate a proxy to their halo masses, in order to clarify this question.

7.3. Comparisons with Previous Works

In Figure 15 we compare our results with those of previous works. When necessary, the data are corrected to a ^{Chabrier (2003)} IMF. Most of the previous determinations of the HI- and H2-to-stellar mass correlations are not explicitly separated for the two main galaxy populations as done here, and in several cases non-detections are assumed to have the values of the upper limits, or are not taken into account.

Fig. 15 Upper panel: Our empirical HI-to-stellar mass correlations for LTGs and ETGs (blue and red shaded areas, respectively) compared with some previous determinations (see labels inside the panel and details of each determination in the text). Previous determinations are based on compilations typically biased to late-type, blue galaxies, and/or do not take into account non-detections. The blue and red arrows correspond to estimates of the difference between the logarithm of the mean (the stacking technique provides the equivalent of the mean value) and the logarithmic mean (our determinations are for this case) for standard deviations of 0.52 and 0.99 dex, respectively (see text for more details). Lower panel: Our empirical molecular H₂-to-stellar mass correlations for LTGs and ETGs (blue and red shaded areas, respectively) compared with very rough previous determinations not separated into LTGs and ETGs (see labels inside the panel and details of each determination in the text). The color figure can be viewed online. 

In the upper panel, our empirical R_HI-M_∗ correlations for LTGs and ETGs are plotted along with the linear relations given by ^{Stewart et al. (2009)} (cyan line, the dashed lines show the 1σ scatter) and ^{Papastergis et al. (2012)} (gray line). The former authors used mainly the observational data presented in ^{McGaugh (2005)} for disk-dominated galaxies, and the latter authors used samples from ^{Swaters & Balcells (2002)}, ^{Garnett (2002)}, ^{Noordermeer et al. (2005)}, and ^{Zhang et al. (2009)}, which refer mostly to late-type galaxies. Their fits are slightly above the mean of our LTG R_HI-M_∗ correlation. This is likely due to their ignoring non detections. We also plot the logarithmic average values in mass bins reported by ^{Catinella et al. (2013)} for GASS (green open circles). Since ETGs progressively dominate in number as the mass increases, our total (density-weighted) R_HI-M_∗ correlation would fall below the one by ^{Catinella et al. (2013)}, especially at the largest masses. Note that for the data plotted from ^{Catinella et al. (2013)}, the HI masses of non-detections were set equal to their upper limits. Therefore, the plotted averages are biased to high values of RHI, specially for ETGs which are dominated by non-detections. Furthermore, recall that we have corrected the upper limits of GASS for distance to make them compatible with those of the nearer ATLAS^3D survey.

More recently, ^{Brown et al. (2015)} have used the HI spectral stacking technique for a volume-limited, stellar mass selected sample from the intersection of SDSS DR7, ALFALFA, and GALEX surveys. With this technique the stacked signal of co-added raw spectra of detected and non-detected galaxies (about 80% of the ALFALFA selected sample) is converted into a (lineal) average HI mass. The authors have excluded from their analysis HI-deficient galaxies - typically found within clusters- because of their significant offset to lower gas content. The black dots connected by a dotted line show the logarithm of the average R_HI values reported at different stellar mass bins in ^{Brown et al. (2015)}. Since HI-deficient galaxies -which typically are ETGs- were excluded, the ^{Brown et al. (2015)} correlation should be compared with our correlation for LTGs. Note that with the stacking technique it is not possible to obtain the population scatter in R _HI because the reported mean values come from stacked spectra instead of from averaging individual values of detections and nondetections. However, the stacking can be applied to subsets of galaxies, for example, selected by color. ^{Brown et al. (2015)} have divided their sample into three groups by their NUV−r colors: [1,3), [3,5), and [5,8]. The average RHI values at different masses corresponding to the bluest and reddest groups are reproduced in Figure 15 with the blue and red symbols, respectively. Note that the logarithmic mean is lower than the logarithm of the mean. For a lognormal distribution, log⁡x=log⁡x-0.5×σlog⁡x2 ln⁡10 (see e.g., ^{Rodríguez-Puebla et al. 2017}). Then, for the typical scatter of 0.44 dex corresponding to LTGs, the logarithm of the stacked values of R_HI should be decreased by ≈ 0.2 dex to compare formally with our reported values of logarithmic means; this is shown with a black arrow in Figure 15. If the reddest galaxies in the ^{Brown et al. (2015)} stacked sample are associated with ETGs (which is true only partially), then for them the correction to a logarithmic mean is of ≈ 1 dex, shown with a red arrow.

Finally, recently ^{van Driel et al. (2016)} reported the results for HI observations at the Nancay Radio Telescope (NRT) of 2839 galaxies selected evenly from SDSS. The authors present a Buckley-James linear regression to their data (long-dashed green line in Figure 15), taking into account this way upper limits for non-detections (though their upper limits are quite high given the low sensitivity of NRT). They fitted the entire sample, that is, they do not separate into LTG/ETG or blue/red groups. In a subsequent paper (^{Butcher et al. 2016}), the authors obtained ≈ 4 times more sensitive follow-up HI observations at Arecibo for a fraction of the galaxies that were either not detected or marginally detected; 80% of them were detected with HI masses ≈ 0.5 dex smaller than the upper limits in ^{van Driel et al. (2016)}, and the rest, mostly luminous red galaxies, were not detected. If this trend is representative of the rest of the NRT undetected galaxies, ^{Butcher et al. (2016)} expect the fit plotted in Figure 15 to be offset toward lower RHI values by about 0.17 dex and even more at the highest masses. This fit lies between a density-weighted fit to our two correlations when taking into account that at large masses the fraction of ETG/red galaxies increases and at small masses LTG/blue galaxies dominate.

The lower panel of Figure 15 is similar to the upper panel, but for the RH2-M_∗ correlations. Regarding the molecular gas content, there are only a few attempts in the literature to determine the relation between MH2 and M_∗. In fact, those works that report approximate correlations are included in our compilation: ^{Saintonge et al. (2011)} for COLD GASS, and ^{Boselli et al. (2014a)} for HRS. The former authors report a linear regression to their binned data assuming H₂ masses for non-detection set equal to their upper limits. The latter authors present a bisector fit using only detected, late-type gas-rich galaxies. Therefore, in both cases the reported relations are clearly biased to LTGs and toward high RH2 values.

The differences we find between our correlations and those plotted in Figure 15, as discussed above, can be understood on the basis of the different limitations that are present in each of the previous works. Bearing in mind these limitations we can conclude that the correlations presented here are in rough agreement with previous ones, but compared to them (i) they extend the correlations to a larger mass range, (ii) they separate explicitly galaxies into their two main populations, and (iii) they take into account adequately the non-detections.