# Lurking systematics in predicting galaxy cold gas masses using dust luminosities and star formation rates

Lurking systematics in predicting galaxy cold gas masses using dust luminosities and star... Abstract We use galaxies from the Herschel Reference Survey to evaluate commonly used indirect predictors of cold gas masses. We calibrate predictions for cold neutral atomic and molecular gas using infrared dust emission and gas depletion time methods that are self-consistent and have ∼20 per cent accuracy (with the highest accuracy in the prediction of total cold gas mass). However, modest systematic residual dependences are found in all calibrations that depend on the partition between molecular and atomic gas, and can over/underpredict gas masses by up to 0.3 dex. As expected, dust-based estimates are best at predicting the total gas mass while depletion time-based estimates are only able to predict the (star-forming) molecular gas mass. Additionally, we advise caution when applying these predictions to high-z galaxies, as significant (0.5 dex or more) errors can arise when incorrect assumptions are made about the dominant gas phase. Any scaling relations derived using predicted gas masses may be more closely related to the calibrations used than to the actual galaxies observed. dust, extinction, galaxies: ISM, infrared: galaxies, radio lines: galaxies 1 INTRODUCTION Key to an understanding of galaxy evolution is a complete census of the baryonic components in galaxies: the stellar populations and the interstellar medium (ISM, gas, and dust). These components are strongly connected to each other, mainly through the process of star formation. Atomic hydrogen gas (H i) is the raw fuel for star formation, but must first cool and condense into molecular hydrogen (H2) before stars can form (e.g. Bigiel et al. 2008; Leroy et al. 2008, but see also Glover & Clark 2012). Once formed, stars eject gas and dust through stellar winds and also significantly pollute the ISM with enriched material. The exquisite interplay between the multi-phase ISM and the star formation cycle has been observed in nearby galaxies in great detail. While star formation rates (SFRs) have been measured in large samples of galaxies for many decades (e.g. Roberts 1963; Kennicutt 1983; Alam et al. 2015), more recently several large observational programmes have quantified galaxy ISM contents in a statistical way, including H i content (Catinella et al. 2010, 2013), H2 content (Saintonge et al. 2011), and dust content (da Cunha et al. 2010; Cortese et al. 2012). These studies and others have shown many connections between star formation and ISM phases, including that H2 is responsible for regulating galaxy SFRs (Tacconi et al. 2013; Saintonge et al. 2016), that the dust-to-gas mass ratio depends on metallicity (Draine et al. 2007; Leroy et al. 2011), and that the dust-to-stellar mass ratio depends on specific SFR (sSFR; da Cunha et al. 2010). Despite the abundance of high-resolution and multi-wavelength observations of galaxies, our understanding of the gas–dust–star cycle is incomplete, even in the local Universe. None the less, as observations of ISM components in galaxies at higher redshifts are becoming increasingly feasible, we are gaining new windows into galaxy evolution. However, the high-redshift view is often limited or partial, and requires careful ‘calibration’ when making comparisons with local galaxies. SFR is the most easily measured piece of the gas–dust–star cycle, and samples of star-forming galaxies have been observed out to high redshift (e.g. Whitaker et al. 2014; Alam et al. 2015). Extensive observations have consistently shown that the star formation history of the Universe reaches a peak at z ∼ 2.5 (Madau & Dickinson 2014), but direct observations of all ISM components are not yet as advanced (e.g. Carilli & Walter 2013). Observational challenges prevent the detection of 21 cm H i emission in samples of galaxies beyond z ∼ 0.2 (Catinella & Cortese 2015, but see recent detection at z = 0.376 from Fernández et al. 2016), but CO (a tracer of H2) observations have begun slowly reaching galaxies at higher redshifts (e.g. Daddi et al. 2010; Aravena et al. 2012; Bauermeister et al. 2013; Hodge et al. 2013; Cybulski et al. 2016). Faced with the great difficulty (or impossibility) of directly observing the gaseous phases of galaxy ISM at high redshift, many observational campaigns have focused on indirect estimates of ISM masses. These estimators are based on correlations observed between properties of local galaxies, and are then applied at higher redshift. These ISM mass predictions fall broadly into two categories: (1) mass-to-light ratios using far-infrared (FIR) luminosities or gas-to-dust ratios (e.g. Eales et al. 2012; Magdis et al. 2012; Scoville et al. 2014); and (2) starting from an integrated star formation law (e.g. the Kennicutt–Schmidt (K-S) law; Schmidt 1959; Kennicutt 1998) and inverting the observed SFR (i.e. a gas depletion time) to estimate a star-forming gas mass (e.g. Tacconi et al. 2013; Genzel et al. 2015; Berta et al. 2016). In the literature, these relations are typically calibrated with samples of local galaxies. Recently, some groups have also explored the theoretical connections between simulated galaxies and different ISM mass predictions. Goz et al. (2017) use a fully cosmological hydrodynamical code to reproduce the multi-wavelength emission (ultraviolet to infrared) of galaxies, and validate some choices of indirect ISM mass indicators. Broadly, star formation can contribute to dust heating (making dust temperature a good tracer of SFR), but longer wavelength dust emission is tightly connected to gas masses. A significant limitation in the development of these indirect relations is that each ISM mass component is usually calibrated independently with an observable quantity, which does not allow an analysis of possible systematic effects or residual dependences on other ISM properties. For example, Scoville et al. (2014) use H i, CO, and FIR observations of a sample of 12 low-z spiral and starburst galaxies in order to calibrate a predictive relationship between FIR dust luminosity and total ISM mass (MISM, the sum of the H i, H2, and dust components), but their sample is too small to test for secondary dependences (e.g. on $$M_{\rm H\,{\small I}}$$/$$M_{\rm H_2}$$ ratio, SFR, etc.). By contrast, Scoville et al. (2016) use the same dust luminosities to calibrate a prediction for molecular gas masses in a larger sample of 70 galaxies. This subtle shift requires an assumption about the relative contributions of H i, H2, and dust to the total ISM mass. In a different work, the same FIR luminosities are used to instead predict total gas mass ($$M_{\rm H\,{\small I}}$$+ $$M_{\rm H_2}$$) for 36 nearby galaxies (Groves et al. 2015). While economical, these indirect estimates may suffer from unknown biases or systematic trends with other ISM properties of galaxies. Before using predicted gas masses to make conclusions about the evolution of high-redshift galaxies, we must first carefully examine these calibrations for any lurking systematic effects that could introduce variations as a function of galaxy properties and mask the true evolutionary trends. We use a sample of galaxies with a complete set of observations of all ISM components to calibrate and test these predictive relations, in order to evaluate how internally consistent these estimators can be. We describe our local sample of galaxies and observations in Section 2. In Section 3, we use mass-to-light and gas-to-dust ratios to predict ISM masses, and Section 4 shows our calibrations using depletion times. In Section 5, we compare these two types of calibrations with each other, and in Section 6 we discuss residual secondary dependences on other quantities. Section 7 includes a discussion of the scientific implications of these systematic effects, and in Section 8 we briefly summarize our main results. Throughout this work, we assume a Λ cold dark matter cosmology with H0 = 70 km s−1 Mpc−1, ΩM = 0.27, and $$\Omega _\Lambda$$ = 0.73. 2 SAMPLE AND DATA Our sample is selected from the Herschel Reference Survey (HRS), which was a volume-limited infrared (IR) imaging survey of 322 galaxies between 15 and 25 Mpc (Boselli et al. 2010). We select our Primary Sample (PS) of N = 68 galaxies to be those HRS targets with a complete set of observations of the ISM components to calibrate and verify relationships between dust and gas content. Required observations include detections in each of the Herschel bands (100–500 μm), detections of atomic and molecular gas (H i and CO), spectroscopic metallicity estimates, and reliable SFRs. Further, we select only galaxies that are not H i  deficient. The H i-deficiency parameter (Def$$_{\rm H\,{\small I}}$$) is the logarithmic difference between the H i content expected based on a galaxy's morphological type and optical diameter and its observed H i mass (Haynes & Giovanelli 1984). We require our PS sample to have Def$$_{\rm H\,{\small I}}$$ ≤ 0.5 in order to exclude galaxies that have likely been stripped of their gas or otherwise affected by dense environments (e.g. Cortese et al. 2016). The 68 galaxies in our PS have been shown to be representative of a volume-limited sample, and do not suffer from significant selection effects (see fig. 1 in Cortese et al. 2016). We next briefly summarize each observed or derived quantity, their uncertainties, and the necessary assumptions. The uncertainties on each quantity typically reflect the errors associated with the measurement and not the calibration or systematic uncertainties, which may be even larger. Stellar mass (M*) and distance (D): M* and D come from Cortese et al. (2012). Estimates of M* are based on the colour-dependent mass-to-light ratios of Zibetti, Charlot & Rix (2009) and measured from Sloan Digital Sky Survey images (Abazajian et al. 2009). Uncertainties on M* include only errors from the broad-band flux measurements and not uncertainties from distances or mass-to-light ratios. Star formation rate (SFR): We use SFRs from Boselli et al. (2015), who compiled and calibrated observations of star formation in the HRS. Specifically we use their ‘SFRMED’ that combines all available SFRs (including those from Hα, FUV, 24 μm, and radio indicators). When available, uncertainties come from the errors on the Hα fluxes or radio continuum fluxes; otherwise, we adopt a 15 per cent error as suggested in section 5.1.1 of Boselli et al. (2015). Metallicity (12+log(O/H)): Spectroscopic gas-phase abundances come from Hughes et al. (2013), who observed HRS galaxies with drift-scan optical spectra and derived oxygen abundances using the calibrations of Kewley & Ellison (2008) to be consistent with the O3N2 scale of Pettini & Pagel (2004). When necessary, all metallicities used in this work (e.g. from Leroy et al. (2011); Berta et al. (2016) in Fig. 1) are converted to the O3N2 scale (Pettini & Pagel 2004) via calibrations from Kewley & Ellison (2008). Uncertainties are given by Hughes et al. (2013) and fully include the errors in converting to a common calibration of metallicity (see section 3 of Hughes et al. 2013). Figure 1. View largeDownload slide Calibration of ξ for each gas phase as a function of metallicity. Top: $$M_{\rm H\,{\small I}}$$; middle: $$M_{\rm H_2}$$; bottom: Mgas. Our best-fitting relations are shown in red. The relations from Leroy et al. (2011) and Berta et al. (2016) are shown in blue and green, respectively, in the bottom panel. Best-fitting lines, Pearson's correlation coefficient (R) and its uncertainty, and the standard deviations (σ, in dex) of points about the best-fitting line are shown in red in each plot. Figure 1. View largeDownload slide Calibration of ξ for each gas phase as a function of metallicity. Top: $$M_{\rm H\,{\small I}}$$; middle: $$M_{\rm H_2}$$; bottom: Mgas. Our best-fitting relations are shown in red. The relations from Leroy et al. (2011) and Berta et al. (2016) are shown in blue and green, respectively, in the bottom panel. Best-fitting lines, Pearson's correlation coefficient (R) and its uncertainty, and the standard deviations (σ, in dex) of points about the best-fitting line are shown in red in each plot. Monochromatic IR luminosities: HRS IR data include imaging from the Photoconductor Array Camera and Spectrometer (PACS) and the Spectral and Photometric Imaging Receiver (SPIRE) instruments. PACS provided images at 100 and 160 μm, and SPIRE at 250, 350, and 500 μm. Photometry based on these images was presented in Cortese et al. (2014) from PACS and in Ciesla et al. (2012) from SPIRE. We convert these flux measurements to luminosities in solar units (L⊙). Uncertainties on the luminosities are propagated from published flux errors in Ciesla et al. (2012) and Cortese et al. (2014). Dust mass (Mdust): We use the dust mass estimates (in M⊙) from Ciesla et al. (2014). They fit spectral energy distributions (SEDs) between 8 and 500 μm using the dust models of Draine & Li (2007), and provide uncertainties on their mass estimates. H i gas mass ($$M_{\rm H\,{\small I}}$$): The atomic gas mass estimates come from Boselli, Cortese & Boquien (2014), who compiled 21 cm observations from the literature for HRS galaxies. When available, we propagate 21 cm flux errors into $$M_{\rm H\,{\small I}}$$ uncertainties; otherwise, we adopt the typical uncertainty of 15 per cent errors on the mass (see section 6 of Boselli et al. 2014). Total gas mass (Mgas): Throughout this work, we use the following definition of total gas mass: Mgas= 1.36 × ($$M_{\rm H\,{\small I}}$$+ $$M_{\rm H_2}$$). This includes a 36 per cent correction for helium in the total gas mass (not included in $$M_{\rm H_2}$$ alone). Uncertainties are propagated from $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$. 2.1 Molecular hydrogen masses ($$M_{\rm H_2}$$) Estimates of molecular hydrogen gas mass come from the CO(1–0) observations and compilation of archival data in Boselli et al. (2014). They use a variable conversion factor between LCO and $$M_{\rm H_2}$$ (XCO), which depends on the near-IR luminosity (measured in the H filter) of each galaxy, following the calibration of Boselli, Lequeux & Gavazzi (2002), not including any contribution or correction from helium. Uncertainties on the mass are propagated from flux errors given in Boselli et al. (2014). The variable XCO from Boselli et al. (2002) relies on a number of assumptions. While a metallicity dependence is expected (Leroy et al. 2009), spectroscopic abundances were not available for all of the galaxies in the calibration sample. Instead, they demonstrated that a luminosity-dependent expression for XCO works well for galaxies without metallicity measurements. Their total molecular gas masses also rely on the assumption that molecular clouds are virialized systems (see section 4 of Bolatto, Wolfire & Leroy 2013 or McKee & Ostriker 2007 for further details). As a simple way to consider the range of effects of different conversion methods, we include three alternative prescriptions for XCO and briefly explore their effects on gas mass predictions. The first, $$M_{\rm H_2,c}$$, is determined using the constant conversion factor from Galactic studies (XCO = 2.3× 1020 cm−2/(K km s−1); Strong et al. 1988). Secondly, we use the recent prescription of Accurso et al. (2017) that depends on metallicity and sSFR to compute $$M_{\rm H_2,acc}$$. Thirdly, $$M_{\rm H_2,bol}$$ comes from Bolatto et al. (2013) and depends on both metallicity and stellar surface density (determined from optical effective radius). While a full discussion of the relative advantages and disadvantages is outside the scope of this work, we re-derive our main calibrations using these alternatives as discussed in Section 4.3. Although the exact values of the relationships between dust luminosity and molecular gas mass change, these alternative prescriptions do not change the underlying systematic trends that are implicit in these relationships. 2.2 Gas-to-dust ratio (ξ) for each phase The ratios of gas to dust mass are computed using each phase (H i, H2, total). As this ratio varies with metallicity (Draine et al. 2007; Leroy et al. 2011), we explore the relationship between ξ (Mdust/Mgas(phase)) and 12+log(O/H) for each phase. Fig. 1 shows these relationships, with uncertainties shown as error bars. Note that the uncertainties on the metallicity are large, as they include both statistical and systematic errors associated with converting many observations to a common metallicity scale. We estimate the uncertainty on Pearson's R correlation coefficient, given in parentheses in Fig. 1 and all subsequent, as the standard deviation of the R values of 10 000 random samples drawn from our data points. We allow for repeated values and require each randomly drawn sample to have the same total number of values as our real sample. As is the case throughout this work, red lines in Fig. 1 show our best fits, which are found through least-squares minimization of differences in the ordinate weighted by their uncertainties. These best-fitting relations are given below:   \begin{eqnarray*} \log \xi ^{\rm d}_{\rm H\,{\small I}} &=& \log \frac{M_{\rm H\,{\small I}}}{M_{\rm dust}} = (10.92\pm 2.00) - (1.03\pm 0.23)\\ && \times {\rm (12+logO/H)}\\ \log \xi ^{\rm d}_{\rm H_2} &=& \log \frac{M_{\rm H_2}}{M_{\rm dust}} = (-4.17\pm 2.23)+(0.67\pm 0.26) \\ &&\times {\rm (12+logO/H)}\\ \log \xi ^{\rm d}_{\rm tot} &=& \log \frac{M_{\rm tot}}{M_{\rm dust}} = (\,8.51\pm 1.65) - (0.72\pm 0.19)\\ && \times {\rm (12+logO/H)}.\\ \end{eqnarray*} Note that the relationship between $$\xi ^{\rm d}_{\rm tot}$$ and 12+logO/H is the tightest (smallest scatter, σ, about the best-fitting line) and also agrees with the relations of Leroy et al. (2011) and Berta et al. (2016). The relationship with $$\xi ^{\rm d}_{\rm H\,{\small I}}$$ is the strongest (greatest absolute value of Pearson's correlation coefficient, R), owing to the fact that our galaxies are H i  dominated (like most in the local Universe). The relationship is very poor for $$M_{\rm H_2}$$, where the correlation is actually the inverse of physical expectations (e.g. Leroy et al. 2011) and the scatter becomes even larger. 3 GAS MASS ESTIMATES FROM LIR AND MDUST We start by considering the simplest possible correlations between cold gas masses (atomic, molecular, and total) and monochromatic IR luminosities, as shown in the first five columns of Fig. 2. We also consider correlations with dust mass, with and without the metallicity-dependent ξ, as shown in the last two columns. In all cases, we carry out least-squares fitting weighted by the uncertainties from both axes. The minimization is only done for differences in the ordinate as these relationships are intended for use as predictions between ‘known’ IR luminosities or dust masses and ‘unknown’ ISM phase masses. Figure 2. View largeDownload slide Correlations between gas-phase masses (H i in top row, H2 in middle row, total Mgas in bottom row) and monochromatic IR luminosities (from 100 to 500 μm) and dust masses (with and without metallicity-dependent ξ for that phase). Light grey lines show 1:1 lines in each plot, and the axes have the same tick sizes in all panels. In the last column (ξ × Mdust), the red line is a unity line and no fitting is performed as ξ was already fitted in Fig. 1 for each phase. The strongest and tightest relationships are indicated by bold boxes. Wide green lines show the agreement with relations from Groves et al. (2015). Figure 2. View largeDownload slide Correlations between gas-phase masses (H i in top row, H2 in middle row, total Mgas in bottom row) and monochromatic IR luminosities (from 100 to 500 μm) and dust masses (with and without metallicity-dependent ξ for that phase). Light grey lines show 1:1 lines in each plot, and the axes have the same tick sizes in all panels. In the last column (ξ × Mdust), the red line is a unity line and no fitting is performed as ξ was already fitted in Fig. 1 for each phase. The strongest and tightest relationships are indicated by bold boxes. Wide green lines show the agreement with relations from Groves et al. (2015). We caution against the blind application of these equations to diverse populations of galaxies. As will be shown in the following sections, calibrations of cold gas mass predictions are susceptible to a number of biases and systematic residual dependences on other physical properties. The best-fitting predictions we derive from our sample may not be appropriate to use across all types of galaxies. Further discussions of these biases and their effects are included in Sections 6 and 7. 3.1 $$M_{\rm H\,{\small I}}$$  estimates from dust Across the M/L estimates of $$M_{\rm H\,{\small I}}$$, the strength and scatter improve continuously with increasing wavelength such that the best monochromatic predictor is L500. Not only does L500 have a value of R = 0.75 ± 0.05 that is statistically larger than the next best value at R = 0.70 ± 0.06, but it also has a scatter that is ∼8 per cent better than the next best. This is expected, as L500 is most closely connected to emission from the diffuse ISM. Multiplying Mdust by the metallicity-dependent $$\xi ^{\rm d}_{\rm H\,{\small I}}$$ gives an even better prediction of $$M_{\rm H\,{\small I}}$$, both in terms of R and σ. Our best three relationships are given below, where $$\xi ^{\rm d}_{\rm H\,{\small I}}$$ was determined in Section 2:   \begin{equation*} \begin{array}{l}\log \frac{{\rm M_{\rm H\,{\small I}}}}{{\rm M_{\odot }}} = (0.72 \pm 0.09) \left( \log \frac{{\rm L_{500}}}{{\rm L_{\odot }}} -9 \right)+ (10.09 \pm 0.08) \\ \log \frac{{\rm M_{\rm H\,{\small I}}}}{{\rm M_{\odot }}} = (0.69 \pm 0.13) \left( \log \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} -9 \right) + (10.50 \pm 0.07) \\ \log \frac{{\rm M_{\rm H\,{\small I}}}}{{\rm M_{\odot }}} = \log \xi ^{\rm d}_{\rm H\,{\small I}} + \log \left( \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} \right). \end{array} \end{equation*} Here and in all subsequent equations, we use mass and luminosity in solar units and use years as the denominator in all SFRs. 3.2 $$M_{\rm H_2}$$ estimates from dust Among the M/L estimates of $$M_{\rm H_2}$$, the 160 μm luminosity is best, because although its R value is not significantly different from that of L100 or L250, its σ is the smallest. The longer wavelength luminosities become increasingly poor estimates of $$M_{\rm H_2}$$. This connection between molecular gas and shorter wavelengths is consistent with the use of LIR as an SFR indicator (e.g. Reddy et al. 2010). Using Mdust instead (with or without ξ) gives an even poorer predictor of $$M_{\rm H_2}$$. For comparison, the relationship between $$M_{\rm H_2}$$ and L160 from Groves et al. (2015) is also shown in Fig. 2 as a wide green line, and agrees with our data points. Note that their best-fitting relation has a steeper slope than ours, as their sample includes a number of dwarf galaxies (M* < 109 M⊙) with small molecular gas masses ($$M_{\rm H_2}$$ ∼ 107 M⊙) that drive this steeper slope. If these dwarf galaxies were removed to match the stellar mass range of our sample, the Groves et al. (2015) best-fitting relationship would be consistent with ours. Our best-fitting relation with L160 is given below, along with the L500 prediction (as it is more commonly used):   \begin{equation*} \begin{array}{l}\log \frac{{\rm M_{\rm H_2}}}{{\rm M_{\odot }}} = (0.92 \pm 0.04) \left( \log \frac{{\rm L_{160}}}{{\rm L_{\odot }}} -9 \right)+ (8.41 \pm 0.04) \\ \log \frac{{\rm M_{\rm H_2}}}{{\rm M_{\odot }}} = (0.99 \pm 0.05) \left( \log \frac{{\rm L_{500}}}{{\rm L_{\odot }}} -9 \right)+ (9.92 \pm 0.06). \end{array} \end{equation*} 3.3 Total Mgas estimates from dust The estimates of Mgas show a similar behaviour to those of $$M_{\rm H\,{\small I}}$$, as H i is the dominant gas component in our galaxies. The best Mgas predictor is L500; although its R value is not significantly better than the ξ-based estimates, it does have smaller scatter in the relationship. This is consistent with its connection to diffuse ISM emission. The Mdust estimates of Mgas are only slightly worse, perhaps due to the additional uncertainties from fitting SEDs to determine dust masses. Regardless of the choice between L500 and Mdust, these correlations with Mgas are the strongest and tightest of any combination of gas phase and indicator discussed thus far. Our three best-fitting Mgas relations are given below, where $$\xi ^{\rm d}_{\rm tot}$$ was determined in Section 2:   \begin{equation*} \begin{array}{l}\log \frac{{\rm M_{\rm gas}}}{{\rm M_{\odot }}} = (0.71 \pm 0.05) \left( \log \frac{{\rm L_{500}}}{{\rm L_{\odot }}} -9 \right)+ (10.39 \pm 0.05) \\ \log \frac{{\rm M_{\rm gas}}}{{\rm M_{\odot }}} = (0.67 \pm 0.08) \left( \log \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} -9 \right) + (10.78 \pm 0.05) \\ \log \frac{{\rm M_{\rm gas}}}{{\rm M_{\odot }}} = \log \xi ^{\rm d}_{\rm tot} + \log \left( \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} \right). \end{array} \end{equation*} Groves et al. (2015) found a similar relationship between L500 and Mgas as ours, as shown by the wide green line in Fig. 2. Their relationship agrees with our data points after it is corrected to include helium to match our convention. Going forward, we prefer the L500 estimates of Mgas, as it gives the best correlation and also does not require any additional assumptions to determine dust mass. For $$M_{\rm H_2}$$ estimates, we prefer L160 if available, but L500 is acceptable. However, there are significant residual trends in these predictions, as discussed in Section 6. 4 DEPLETION TIME ESTIMATES As a second method to estimate gas mass without direct observations, we can use the inverted integrated K-S law (i.e. gas depletion time, tdep). We start by using each galaxy's SFR as a predictor of its amount of (star-forming) gas. In order to do this, we must determine the distance between each galaxy and the so-called star-forming main sequence (SFMS; Noeske et al. 2007; Salim et al. 2007). The SFMS is typically defined as the ridge line in the SFR–M* diagram and is populated by ‘star-forming’ galaxies. We use two different definitions of SFMS. In general, the SFMS can be parametrized as follows:   \begin{equation*} \log \frac{{\rm SFR}_{\rm MS}}{{\rm M_{\odot }}/{\rm yr}} = \alpha \left( \frac{{\rm M_*}}{{\rm M_{\odot }}} - 10.5 \right) + \beta . \end{equation*} Using UV+IR SFRs for ∼22 000 galaxies at 0 < z < 2.5, Whitaker et al. (2012) derived a redshift-dependent SFMS with parameters given below:   \begin{equation*} \alpha _{\rm W12}(z) = 0.70-0.13z, \, \, \, \, \, \, \, \, \, \, \, \, \beta _{\rm W12}(z) = 0.38 + 1.14z - 0.19z^2. \end{equation*} We also considered the SFMS from Catinella et al. (2018, see also Janowiecki et al., in preparation), which is based on UV+IR SFRs of a low-redshift sample of ∼1200 stellar mass selected galaxies from xGASS (Janowiecki et al. 2017), with best-fitting parameters given below:   \begin{equation*} \alpha _{\rm J17} = 0.656, \, \, \, \, \, \, \, \, \, \, \, \, \beta _{\rm J17} = 0.162. \end{equation*} While this is based on a representative sample of local (z∼0) galaxies, the Whitaker et al. (2012) SFMS includes a redshift dependence, making it more useful in extending our predictive relationships for ISM masses to galaxies at higher redshifts. We use only the Whitaker et al. (2012) SFMS going forward, as both give similar results for our sample. Using this definition of SFMS, we compute ΔSFMS, which quantifies each galaxy's distance above or below the SFMS at its M* and redshift:   \begin{equation*} \log \Delta {\rm SFMS} = \log \frac{{\rm SFR}}{{\rm M}_{\odot }\,\mathrm{yr}^{-1}} - \log \frac{{\rm SFR}_{\rm MS}{\rm (M_*,}z{\rm )}}{{\rm M}_{\odot }\,\mathrm{yr}^{-1}}. \end{equation*} In a similar vein, Berta et al. (2016) derive an expression for depletion times based primarily on ΔSFMS, but also including a dependence on M* and redshift (their equation 3). We show this quantity as $$\Delta ^\prime _{\rm B16}$$. Fig. 3 shows our observed depletion times (Mgas/SFR) using different gas masses (H i, H2, total). These are plotted against sSFR (left column), ΔSFMS (centre column), and $$\Delta ^\prime _{\rm B16}$$ (right column). As with Fig. 2, best-fitting lines and quality estimates are shown in each panel. Note that for each predictive indicator shown on the x-axis, the relation with $$M_{\rm H_2}$$ has the tightest scatter and strongest correlation. Unsurprisingly, the SFR-based indicators are less closely related to $$M_{\rm H\,{\small I}}$$. Since our galaxies are H i dominated, the predictions for Mgas are similarly weak. Again we caution against blindly applying our calibrations to populations of galaxies that are significantly different from those in our sample. We next focus on two of these predictions of $$M_{\rm H_2}$$. Figure 3. View largeDownload slide Gas mass divided by SFR (i.e. depletion time) using various gas phases is plotted against SFR-related quantities. Left column x-axis shows sSFR, and our $$M_{\rm H_2}$$/SFR values agree with the average value from Tacconi et al. (2013) and the trend from Saintonge et al. (2011). Centre column shows ΔSFMS using the Whitaker et al. (2012) SFMS. Right column shows $$\Delta ^\prime _{\rm B16}$$ from Berta et al. (2016), a weighted combination of sSFR and M*. Figure 3. View largeDownload slide Gas mass divided by SFR (i.e. depletion time) using various gas phases is plotted against SFR-related quantities. Left column x-axis shows sSFR, and our $$M_{\rm H_2}$$/SFR values agree with the average value from Tacconi et al. (2013) and the trend from Saintonge et al. (2011). Centre column shows ΔSFMS using the Whitaker et al. (2012) SFMS. Right column shows $$\Delta ^\prime _{\rm B16}$$ from Berta et al. (2016), a weighted combination of sSFR and M*. 4.1 $$M_{\rm H_2}$$ estimates from tdep(sSFR) The relations shown in Fig. 3 are consistent with previous results, as seen in the panel showing $$M_{\rm H_2}$$/SFR versus sSFR. Here the constant depletion time estimate from Tacconi et al. (2013) is shown (at the average sSFR of their sample), as well as the relation from Saintonge et al. (2011), both of which are consistent with our observations. Our best-fitting relationship between $$M_{\rm H_2}$$/SFR (in units of years) and sSFR (in inverse years) is given below:   \begin{equation*} \mathrm{log} \frac{{\rm M_{\rm H_2}}}{{\rm SFR}} = -0.46 (\pm 0.74) \times \mathrm{log} {\rm sSFR} + 4.32 (\pm 0.08). \end{equation*} 4.2 $$M_{\rm H_2}$$ estimates from tdep(ΔSFMS) The strongest and tightest correlation is found between $$M_{\rm H_2}$$/SFR and ΔSFMS, as the H2 is directly involved in current star formation. The best-fitting relationship between ΔSFMSW12 and $$M_{\rm H_2}$$/SFR (in years) is given below:   \begin{equation*} \mathrm{log} \frac{{\rm M_{\rm H_2}}}{{\rm SFR}} = -0.46 (\pm 0.03) \times \mathrm{log} \Delta {\rm SFMS}_{\rm W12} + 8.92\, (\pm 0.08). \end{equation*} This relationship has similar scatter and strength as the L500–Mgas and Mdust–Mgas relationships determined in Section 3, but is also affected by the residual trends discussed in the following section. 4.3 Alternative $$M_{\rm H_2}$$ prescriptions We briefly consider the effects of alternative XCO conversion factors to determine $$M_{\rm H_2}$$ from CO observations, as introduced in Section 2.1. In addition to the luminosity-dependent XCO we primarily use, we show here the results of using a constant conversion factor ($$M_{\rm H_2,c}$$), the metallicity-dependent conversion ($$M_{\rm H_2,acc}$$) from Accurso et al. (2017), and the conversion from Bolatto et al. (2013, $$M_{\rm H_2,bol}$$) that depends on metallicity and stellar surface density (determined from optical effective radius). Fig. 4 shows the predictive relations for these determinations of $$M_{\rm H_2}$$ using L500 (top row) and depletion time from ΔSFMS (bottom row). In all cases, the $$M_{\rm H_2}$$ values do not include the contribution from helium. Figure 4. View largeDownload slide Top row: correlations between L500 and $$M_{\rm H_2}$$, using different XCO conversion factors. Wide green line indicates the relationship from Scoville et al. (2016), which uses a constant XCO. Bottom row: correlations between depletion time ($$M_{\rm H_2}$$/SFR) and ΔSFMS, using the same prescriptions. In all panels, points are coloured by their $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio, using the $$M_{\rm H_2}$$ prescription from that panel. Figure 4. View largeDownload slide Top row: correlations between L500 and $$M_{\rm H_2}$$, using different XCO conversion factors. Wide green line indicates the relationship from Scoville et al. (2016), which uses a constant XCO. Bottom row: correlations between depletion time ($$M_{\rm H_2}$$/SFR) and ΔSFMS, using the same prescriptions. In all panels, points are coloured by their $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio, using the $$M_{\rm H_2}$$ prescription from that panel. Using different conversion factors does not dramatically alter these predictive relationships. Adopting the metallicity-dependent conversion from Accurso et al. (2017) moderately improves the strength (and scatter) of the L500 prediction, but it significantly worsens the strength of the relationship with depletion time. Similarly, the Bolatto et al. (2013) prescription marginally improves the strength and scatter of the L500 relation and worsens the trend with ΔSFMS. Also shown in the top row of Fig. 4 is the L500-based $$M_{\rm H_2}$$ prediction from Scoville et al. (2016), which is determined from observations of 70 galaxies between z = 0and2. That calibration adopts the constant Galactic value of XCO, so it is most appropriate to compare it with the middle-left panel. As expected, the Scoville et al. (2016) relationship works better for galaxies with $$M_{\rm H_2}$$≳$$M_{\rm H\,{\small I}}$$; H i-dominated galaxies can fall as much as an order of magnitude below the prediction. This illustrates the difficulty in using FIR dust luminosity to predict $$M_{\rm H_2}$$ alone – without knowledge of the atomic-to-molecular ratio, it is difficult to estimate the mass in each phase from a dust luminosity. 5 CLOSING THE LOOP Fig. 5 visually summarizes the quality of the predictive relationships calibrated thus far. Above each indicator on the x-axis (i.e. monochromatic luminosity, dust mass, sSFR, or depletion time estimate), the strength and scatter of its correlation with each gas mass are shown (for H i, H2, and total gas). The y-axis position shows the absolute value of Pearson's R value, and the size of each point corresponds to the standard deviation of our observations about the best-fitting (or unity) line (σ). Larger symbols show relationships with higher scatter. Figure 5. View largeDownload slide The absolute value of Pearson's R value is shown for each indicator, from the correlations shown in Figs 2 and 3. The size and colour of each point correspond to its scatter (σ, in dex), as indicated in the legend. The best relationships are marked with solid red vertical lines. Uncertainties on |R| are shown as error bars and show the standard deviations from 10 000 random samplings. Figure 5. View largeDownload slide The absolute value of Pearson's R value is shown for each indicator, from the correlations shown in Figs 2 and 3. The size and colour of each point correspond to its scatter (σ, in dex), as indicated in the legend. The best relationships are marked with solid red vertical lines. Uncertainties on |R| are shown as error bars and show the standard deviations from 10 000 random samplings. As is shown in Fig. 5 (and in Sections 3 and 4), $$M_{\rm H\,{\small I}}$$ is better predicted by longer wavelength IR emission (reaching the best prediction at 500 μm), and slightly better still by dust mass and metallicity. As atomic gas is less directly connected to ongoing star formation, predictions based on depletion time are poorer for $$M_{\rm H\,{\small I}}$$. Molecular gas mass is best predicted either by L160 or depletion time estimates, both of which have similar scatter but the M/L method gives a stronger correlation, partly due to its larger dynamic range. Total gas is best predicted by L500, and depletion time estimates perform poorly, as the galaxies in our sample are H i  dominated. We next compare the two methods and evaluate whether these separate predictions are mutually consistent. Fig. 6 shows two different predictions of $$M_{\rm H_2}$$ (left-hand panel, from ΔSFMS and L500) and two different predictions of Mgas (right-hand panel, from Mdust and L500). Since all of these estimates have been calibrated using the same sample of galaxies, their good consistency is not surprising. Figure 6. View largeDownload slide $$M_{\rm H_2}$$ (left) and Mgas (right) comparisons from our calibrations. In both main panels, our predictions for the gas-phase masses are shown, using different indicators on both axes (σ shows the scatter about the unity line, in dex). The residual panels show the differences between the predicted and measured gas-phase masses (Δreal). Also shown (in light grey boxes and best-fitting dashed lines) are the results when using alternative calibrations for the y-axis quantities. In the left-hand panel, small boxes show the effect of assuming a constant depletion time. In the right-hand panel, small boxes show the results of using $$\xi ^{\rm d}_{\rm tot}$$, which depends on metallicity (and assuming Mgas= Mdust × ξ). Note that the two estimates of Mgas (from Mdust and from L500) are better correlated with each other (σ = 0.07) than they are with Mgas, as L500 also tracks the dust content. Figure 6. View largeDownload slide $$M_{\rm H_2}$$ (left) and Mgas (right) comparisons from our calibrations. In both main panels, our predictions for the gas-phase masses are shown, using different indicators on both axes (σ shows the scatter about the unity line, in dex). The residual panels show the differences between the predicted and measured gas-phase masses (Δreal). Also shown (in light grey boxes and best-fitting dashed lines) are the results when using alternative calibrations for the y-axis quantities. In the left-hand panel, small boxes show the effect of assuming a constant depletion time. In the right-hand panel, small boxes show the results of using $$\xi ^{\rm d}_{\rm tot}$$, which depends on metallicity (and assuming Mgas= Mdust × ξ). Note that the two estimates of Mgas (from Mdust and from L500) are better correlated with each other (σ = 0.07) than they are with Mgas, as L500 also tracks the dust content. The two estimates of $$M_{\rm H_2}$$ are observationally independent, as the tdep prediction depends only on SFR and M*, which are separate from the observations of L500. It is possible to calibrate those two predictive relationships so that mutually consistent estimates of $$M_{\rm H_2}$$ are obtained, whether using SFR or FIR luminosity. Note that using a fixed depletion time (e.g. Tacconi et al. 2013) gives a relationship that is ∼15 per cent steeper than the 1:1 line, and that overestimates $$M_{\rm H_2}$$(tdep) at higher H2 masses. The two estimates of total Mgas are less independent, as the L500 emission is closely related to the dust content. As such, the two indicators correlate with each other (σ = 0.07 dex) more tightly than with Mgas (σ = 0.15–0.17 dex). This is reassuring as it means the choice of indicator is not crucial, and both L500 and Mdust are good estimators of Mgas. Note also that adopting a metallicity-dependent $$\xi ^{\rm d}_{\rm tot}$$ (as derived in Section 2) introduces a ∼25 per cent steeper slope to the relation between Mgas(Mdust) and Mgas(L500). This systematic difference implies that $$\xi ^{\rm d}_{\rm tot}$$ scales with galaxy stellar mass (via the mass–metallicity relation), as has been shown by Cortese et al. (2016) and Rémy-Ruyer et al. (2013). In order to eliminate the slope difference between these two estimates, the metallicity-dependent $$\xi ^{\rm d}_{\rm tot}$$ relationship would need a three times steeper slope, which dramatically increases the scatter of the points around the unity line in Fig. 6. These comparisons demonstrate that the choice of calibration method can have significant effects on the indirect predictions of cold gas masses. Even within the same sample of galaxies, there can be systematic deviations when using different assumptions, such as a constant depletion time or a metallicity-dependent ξ. None the less, with appropriate choices, independent predictive methods can be calibrated to produce estimates that are in agreement with each other. 6 RESIDUAL TRENDS IN CALIBRATIONS These predictive relationships perform as expected in our sample of local, star-forming, H i-dominated galaxies. However, we are most interested in applying these relationships to galaxies at higher redshifts, where the partition between atomic and molecular gas is unknown and other physical properties may differ (e.g. M*, sSFR). Across our sample, the $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio varies between ∼3 per cent and ∼300 per cent, which is similar to the range of values observed in galaxies from the xGASS sample (Catinella et al. 2018). Galaxies in xGASS also show a weakly increasing median value of the molecular-to-atomic ratio as a function of stellar mass, from ∼10 per cent at 109 M⊙ to ∼30 per cent at 1011.5 M⊙. 6.1 Quantifying residual systematics Quantifying the strength of any residual secondary dependences requires careful parametrization to avoid being affected by underlying dependences between Mgas and its constituent phases. For example, plotting the accuracy of the L500 prediction of $$M_{\rm H\,{\small I}}$$ as a function of the $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio has a strong intrinsic correlation (from the inclusion of $$M_{\rm H\,{\small I}}$$ on both axes), which makes it difficult to directly interpret the residuals in a physical sense. We explore these intricacies in a Monte Carlo analysis described in Appendix A, and demonstrate that our approach successfully quantifies physical residual dependences without suffering from these effects. 6.2 Residuals between phases We quantify the underlying dependence between predictions in each phase by plotting the accuracy of the predictive relationships for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as functions of $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$, respectively. In this way, we are comparing observationally independent quantities, and the residual trends can be meaningfully interpreted (see Appendix A for more discussion of this method). The top two panels in the left column of Fig. 7 show the differences between the L500-based prediction and the real gas masses for H i and H2. The $$M_{\rm H\,{\small I}}$$ differences are plotted against $$M_{\rm H_2}$$, and the $$M_{\rm H_2}$$ differences against $$M_{\rm H\,{\small I}}$$ (i.e. independently observed quantities). Each residual trend is fitted and their slopes (m), scatters (σ), and correlations (R) are shown, with uncertainties. These fits use least-squares minimization of the ordinate and are weighted by the uncertainties of the observed gas-phase masses only (i.e. they do not include the uncertainties on the predicted gas masses). Figure 7. View largeDownload slide Left-hand panels: residuals of L500 (dust) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. Top panel shows $$M_{\rm H\,{\small I}}$$ divided by its prediction from L500, demonstrating an anti-correlation with $$M_{\rm H_2}$$. Bottom panel shows $$M_{\rm H_2}$$ divided by its prediction from L500, with similarly strong anti-correlation with $$M_{\rm H\,{\small I}}$$. Right-hand panels: residuals of the ΔSFMS (depletion time) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. A stronger residual trend is found when predicting $$M_{\rm H\,{\small I}}$$ from depletion time, while no trend exists in the $$M_{\rm H_2}$$ prediction. Figure 7. View largeDownload slide Left-hand panels: residuals of L500 (dust) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. Top panel shows $$M_{\rm H\,{\small I}}$$ divided by its prediction from L500, demonstrating an anti-correlation with $$M_{\rm H_2}$$. Bottom panel shows $$M_{\rm H_2}$$ divided by its prediction from L500, with similarly strong anti-correlation with $$M_{\rm H\,{\small I}}$$. Right-hand panels: residuals of the ΔSFMS (depletion time) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. A stronger residual trend is found when predicting $$M_{\rm H\,{\small I}}$$ from depletion time, while no trend exists in the $$M_{\rm H_2}$$ prediction. For galaxies with larger $$M_{\rm H_2}$$, L500 underpredicts $$M_{\rm H\,{\small I}}$$, and for galaxies with larger $$M_{\rm H\,{\small I}}$$, L500 underpredicts $$M_{\rm H_2}$$. This behaviour illustrates that L500 is most tightly correlated with the total gas mass (e.g. see Section 3.3); any $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$ prediction based on L500 will have a systematic uncertainty depending on the partition of atomic and molecular gas. These systematic trends are of modest amplitude (slopes of 17–18 per cent per dex, with ≥2σ significance) and large scatter (0.2 dex), but are evident even within our small sample of local star-forming galaxies. At higher redshift and for larger more star-forming systems, extrapolations of these discrepancies could be larger. In a similar way, the top two panels in the right column of Fig. 7 compare the depletion time-based predictions of gas masses (using ΔSFMS) for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ with their true values as a function of $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$, respectively. The discrepancies in $$M_{\rm H\,{\small I}}$$ predictions show steep systematic variations (slope of 40 per cent per dex with ≥4σ significance), which is not unexpected since depletion time is less physically connected to the neutral atomic gas content. Conversely, the errors in the $$M_{\rm H_2}$$ prediction do not depend on $$M_{\rm H\,{\small I}}$$, as this method is best suited to predict $$M_{\rm H_2}$$, the gas which is directly involved in star formation. When using SFR to predict gas mass, there is an implicit assumption about the rate of H2 conversion into stars as well as the conversion between H i and H2. This introduces an underlying dependence on the partition between $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$, as was seen in the M/L estimates. These residual trends serve as a reminder that ISM mass predictions from dust luminosities and SFRs are not equivalent and are sensitive to different gas phases. In the worst cases, using L500 to predict $$M_{\rm H_2}$$ is dependent on $$M_{\rm H\,{\small I}}$$ (R = −0.55) with a modest slope (m = 0.17), giving systematic over/underpredictions of ∼0.15 dex at the extremes of our sample. Similarly, using tdep to predict $$M_{\rm H\,{\small I}}$$ results in an even steeper slope (m = −0.40) and similar correlation (R = −0.48), over/underpredicting by up to ∼0.3 dex. Without prior knowledge of the partition between molecular and atomic gas, it is difficult to apply these predictions. Other studies have shown that the $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio in galaxies can depend on the metallicity and turbulence of their ISM, which affect the conversion between atomic and molecular gas (Krumholz, McKee & Tumlinson 2009; Bialy, Burkhart & Sternberg 2017). 6.3 Residuals with other physical properties We also explored similar residual dependences using other galaxy properties. In particular, we considered galaxy stellar mass and sSFR. The lower four panels of Fig. 7 show, as a function of these physical properties, the ratios between the observations of $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ and their L500- and ΔSFMS-based predictions. As with the top panels, we fit these residual trends to quantify their strengths. For both L500 and ΔSFMS, we see negative (positive) residuals for the $$M_{\rm H\,{\small I}}$$ ($$M_{\rm H_2}$$) predictions with increasing stellar mass. While sometimes weak, these trends are expected, given the moderately increasing $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio observed as a function of stellar mass (Catinella et al. 2018). All of the residual trends with M* are weaker (smaller R values and almost always flatter slopes) than those with $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$, indicating that a stellar mass dependence is not enough to account for the residuals observed between the phases. The sSFR residuals in the predictions of $$M_{\rm H\,{\small I}}$$ suggest that elevated star formation results in L500 and ΔSFMS, underpredicting $$M_{\rm H\,{\small I}}$$, while in more passive galaxies, it may be overpredicted (by ∼0.2 dex, even within our sample's relatively small range of sSFR). These modest systematic trends with M* and sSFR suggest that using L500 or ΔSFMS as ISM mass estimators will be affected by underlying dependences on other galaxy properties. Our calibrations are naturally best suited to predicting cold gas masses for galaxies that are similar to those included in our sample. Any extrapolation or extension of these relationships to significantly different populations of galaxies (e.g. with higher or lower M* or sSFR) may suffer from systematic biases. None the less, these predictions are robust and can reliably generate indirect estimates of cold gas masses. 7 SCIENTIFIC IMPLICATIONS 7.1 Applications to high-z galaxy observations Fig. 8 shows the basic relations between gas masses and L500 for our sample and galaxies at other redshifts where all three observations (21 cm, CO, and 500 μm) are available. In addition to the sample used in this work, we show 26 galaxies from the xGASS sample (z = 0.01–0.05; Catinella et al. 2018) that have L500 observations from the NASA/IPAC Infrared Science Archive,1 survey data from Herschel-ATLAS (Bourne et al. 2016; Valiante et al. 2016), and the Herschel Stripe 82 Survey (Viero et al. 2014). Figure 8. View largeDownload slide All three panels show galaxies in our sample (in grey) and other colours show other galaxies including some at higher redshifts. The dotted line shows the 1:1 relationship. Our best-fitting relationships for $$M_{\rm H\,{\small I}}$$ (left), $$M_{\rm H_2}$$ (centre), and Mgas (right) are shown as red lines. Higher redshift galaxies appear to follow similar behaviour as our local sample, with a tight L500–Mgas relation and correlated scatter in the $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ relations. Figure 8. View largeDownload slide All three panels show galaxies in our sample (in grey) and other colours show other galaxies including some at higher redshifts. The dotted line shows the 1:1 relationship. Our best-fitting relationships for $$M_{\rm H\,{\small I}}$$ (left), $$M_{\rm H_2}$$ (centre), and Mgas (right) are shown as red lines. Higher redshift galaxies appear to follow similar behaviour as our local sample, with a tight L500–Mgas relation and correlated scatter in the $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ relations. We also include two galaxies with observations at higher redshifts (z > 0.1). First, AGC 191728 comes from the HIGHz sample of Catinella & Cortese (2015) at z = 0.176. Its H i observations come from Arecibo and its CO(1–2) observations from the Atacama Large Millimeter Array (Cortese, Catinella & Janowiecki 2017). This galaxy was serendipitously imaged as part of the Herschel-ATLAS observations and released in H-ATLAS DR1 (Valiante et al. 2016). Secondly, COSMOS J100054.83+023126.2 comes from the Cosmic Evolution Survey (COSMOS; Scoville et al. 2007) and is at z = 0.376 (this is the highest redshift detection of H i emission from a galaxy to date). It has recently been observed in 21 cm with the Jansky Very Large Array as part of the COSMOS H i Large Extragalactic Survey (CHILES) and in CO with the Large Millimeter Telescope (Fernández et al. 2016). In another serendipitous observation, DR2 of the Herschel Multi-tiered Extragalactic Survey (Oliver et al. 2012) includes 500 μm observations of this galaxy, with fluxes available through their SUSSEXtractor catalogue. Note that the observed 500 μm emission from these two sources corresponds to rest-frame observations at 425 and 363 μm, respectively. While we use these fluxes in our L500-based relationships, adopting the L350-based predictions would make only a small difference. Within our sample, the L350/L500 ratio is 1.073±0.006. These two higher redshift galaxies provide an illustrative example of the application of the L500-based predictions of cold gas masses. These two galaxies are quite different: while they have very similar $$M_{\rm H\,{\small I}}$$ (and stellar mass estimates from optical photometry), the CHILES galaxy has ∼4 times larger $$M_{\rm H_2}$$ than the HIGHz galaxy. Their $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratios are 50 per cent and 200 per cent, respectively. Remarkably, the L500 prediction for Mgas is very accurate for both! However, L500-based estimates of $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ can be wrong by ∼0.4 dex. The H i-dominated HIGHz galaxy lies along the L500–$$M_{\rm H\,{\small I}}$$ relationship, but falls ∼0.4 dex below L500–$$M_{\rm H_2}$$. Conversely, the H2-dominated CHILES galaxy is consistent with the L500–$$M_{\rm H_2}$$ relation but falls ∼0.3 dex below L500–$$M_{\rm H\,{\small I}}$$. Without additional knowledge of the molecular-to-atomic ratio of these galaxies, L500cannot reliably predict their$$M_{\rm H\,{\small I}}$$ or$$M_{\rm H_2}$$separately. Extending this argument further, observations have suggested that higher redshift galaxies have higher $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratios compared with the H i-dominated galaxies observed at low redshift, but within z < 0.2 this ratio seems to span the same range of values (Cybulski et al. 2016; Cortese et al. 2017). When using these types of relations to predict molecular gas masses at high redshift, corrections may be necessary to account for different mass contributions from ISM phases. When predicting gas masses in high-redshift galaxies, appropriately calibrated relations must be used. While it is possible to separate the atomic and molecular gas phases in our local sample, the distinction becomes difficult at higher redshift. There is inherent uncertainty in extending a low-redshift prediction to higher redshift galaxies, which may have different atomic-to-molecular ratios or other physical differences from local galaxies. 7.2 Consequences of incorrect phase assumptions In Fig. 9, we quantify the possible errors introduced by applying L500-based gas mass predictions to galaxies without knowledge of their molecular-to-atomic partitions. In each panel, we either apply the assumption that galaxies are H i  dominated and use the relationship calibrated for $$M_{\rm H\,{\small I}}$$ (orange circles) or are H2 dominated and use the $$M_{\rm H_2}$$ relationship (purple squares). For completeness, we also show the comparison between total gas mass predictions and the observed $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ (green triangles). Each of these comparisons has a systematic offset from unity: the $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$ predictions scatter around the average molecular and atomic gas fractions for this sample, and the Mgas predictions include the ∼30 per cent correction for the contribution of helium. Figure 9. View largeDownload slide Each panel plots the ratio between the observed $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ and the L500-based predictions of $$M_{\rm H\,{\small I}}$$ ($$M_{\rm H_2}$$) as orange circles (purple squares), as functions of total gas mass, stellar mass, and sSFR. Relations with Mgas are shown as green triangles. Least-squares fits are shown along with best-fitting slopes and uncertainties. Figure 9. View largeDownload slide Each panel plots the ratio between the observed $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ and the L500-based predictions of $$M_{\rm H\,{\small I}}$$ ($$M_{\rm H_2}$$) as orange circles (purple squares), as functions of total gas mass, stellar mass, and sSFR. Relations with Mgas are shown as green triangles. Least-squares fits are shown along with best-fitting slopes and uncertainties. When adopting the H i-dominated assumption (orange circles), the y-axis shows the difference between the true $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ and the L500-predicted $$M_{\rm H\,{\small I}}$$, as $$M_{\rm H_2}$$ is assumed to be negligible. This assumption is most accurate for galaxies with Mgas∼109 M⊙, with large stellar mass, or with passive sSFR, where the L500–$$M_{\rm H\,{\small I}}$$ relationship accurately predicts the total gas mass. Estimates generated from this assumption become significantly worse for more star-forming galaxies or those with larger gas masses (i.e. similar to those observed at higher redshifts) where the predictions can be up to ∼0.5 dex too small. For galaxies with more extreme $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratios, the effects of this incorrect assumption could be even larger. While the H2-dominated assumption (purple squares) is clearly not valid for any of the galaxies in our sample (note that none reach unity in the ratio being plotted), this assumption interestingly shows strong systematic variations with M* and sSFR. These systematic trends are stronger and more significant than those found in Fig. 7, and are a manifestation of the incorrect assumption about the dominant phase. Even the L500–Mgas relationship (green triangles) shows a dependence on sSFR at the ∼2.5σ level, demonstrating the ubiquity of underlying systematic residuals. While it is not unsurprising that incorrect assumptions about the dominant gas phase yield incorrect results, these trends demonstrate quantitatively the nature of errors arising from these assumptions. Most importantly, in addition to the expected systematic offset, Fig. 9 shows that residual trends are present that depend on other galaxy properties. Every application of these indirect gas mass predictions relies on implicit assumptions to produce $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$ estimates. 7.3 The K-S with different calibrations Fig. 10 shows the integrated K-S star formation law (Schmidt 1959; Kennicutt 1983), using different predictors of ISM phase masses. While the top-left panel shows the K-S law from direct observations of SFR and $$M_{\rm H_2}$$, the other panels use different indirect gas mass estimates. Figure 10. View largeDownload slide Integrated star formation law with different calibrations. Top-left panel shows the true relation between the observed $$M_{\rm H_2}$$ and SFR, with the best-fitting relationship (shown in all panels). Other panels plot observed SFR against various estimates of ISM phase masses, with analogous best fits shown as grey-shaded regions. Galaxies are colour-coded based on their position on, above, or below the relation. The full sample of HRS galaxies is included as faint points whenever available. Figure 10. View largeDownload slide Integrated star formation law with different calibrations. Top-left panel shows the true relation between the observed $$M_{\rm H_2}$$ and SFR, with the best-fitting relationship (shown in all panels). Other panels plot observed SFR against various estimates of ISM phase masses, with analogous best fits shown as grey-shaded regions. Galaxies are colour-coded based on their position on, above, or below the relation. The full sample of HRS galaxies is included as faint points whenever available. Note that the slope, width, and offset of the best-fitting relation can all be significantly affected by a different choice of gas mass indicator. For example, when adopting an estimate of $$M_{\rm H_2}$$ based on L500, the K-S relation becomes steeper and tighter than the true relation. Also note the extent to which an individual galaxy can move above/below the K-S relation using different gas estimates. The different coloured points in Fig. 10 demonstrate that >+1σ outliers can become consistent with the relation (using $$M_{\rm H_2}$$ based on L500) and that <−1σ outliers can move above the relation (using $$M_{\rm H_2}$$ based on Mdust). The lower-right panel of Fig. 10 shows what can happen when using total Mgas instead of the star-forming $$M_{\rm H_2}$$. This illustrates the impact of an incorrect assumption that the ISM is dominated by the molecular component when in fact it is mostly atomic. While this assumption is obviously inappropriate for the local H i-dominated galaxies in our sample, higher redshift galaxies have a wide range of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ partitions and may not all be dominated by the molecular component (Cortese et al. 2017). It can be dangerous to use these types of indirect estimates of gas masses to study the K-S relationship or other scaling relations. Any calibration of these gas mass estimates will be linked to the underlying physical properties of the ISM (e.g. atomic-to-molecular hydrogen ratio), and will not necessarily indicate variations in star formation efficiency. There are likely to be systematic effects lurking in these calibrations that will limit any attempt to derive a K-S relation using indirect gas mass predictions. 8 SUMMARY Using a representative sample of N = 68 nearby galaxies from the HRS, we have calibrated a set of relationships between the masses of ISM phases and observable quantities (FIR and SFR). These predictive relationships can estimate ISM masses with ∼20 per cent accuracy and are mutually self-consistent. However, our complete set of observations of all ISM phases show that these predictive relationships suffer from modest systematic residual dependences on the molecular-to-atomic partition and other physical properties. Any application of these relationships to predict gas masses from FIR/SFR observations requires an implicit assumption of the underlying $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio. Incorrect assumptions about the dominant phase of the ISM can yield errors in gas mass predictions as large as 0.5 dex. Furthermore, using these indirect gas estimates to test the evolution of star formation laws or other scaling relations is potentially problematic, as these relations rely on those underlying scaling relations to successfully predict gas masses. Acknowledgements We thank Toby Brown and Katinka Geréb for helpful discussions, and the anonymous referee for their comments that have significantly improved this work and its statistical treatment of residual trends. SJ, BC, and LC acknowledge support from the Australian Research Council's Discovery Project funding scheme (DP150101734). BC is the recipient of an Australian Research Council Future Fellowship (FT120100660). AG acknowledges support from the ICRAR Summer Studentship Programme during which this project was initiated. This research has made use of NASA's Astrophysics Data System, and also the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. 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To simulate and test the relationships between L500 and cold gas masses, we start by generating a random distribution of L500 values (N = 10 000) consistent with that of our sample. We use our relationship between L500 and Mgas to generate estimates for the total gas content, and add random noise to the estimates to match our observed dispersion of σ = 0.15 dex. We next generate a distribution of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ values, which matches that of our sample, ranging from ∼3 per cent to ∼300 per cent, and calculate $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$ for each value of Mgas using this ratio. The top two rows of Fig. A1 show histograms of these quantities and the relationships with L500, where grey points and red lines show our observations and green points and lines show the simulated MC data. Note that we only input the L500–Mgas relation into this MC, and we naturally recover the observed slope, correlation strength, and scatter in the L500–$$M_{\rm H\,{\small I}}$$ relation. However, the resulting MC version of the L500–$$M_{\rm H_2}$$ relation does not agree as well with the observed relation, although the two distributions of points have significant overlap. Figure A1. View largeDownload slide Full summary plots of our Monte Carlo for L500-based predictions of cold gas masses. In all panels, grey points and red lines show data, histograms, and fits to our real observations; green points and lines show the simulated data. Figure A1. View largeDownload slide Full summary plots of our Monte Carlo for L500-based predictions of cold gas masses. In all panels, grey points and red lines show data, histograms, and fits to our real observations; green points and lines show the simulated data. The bottom two rows of Fig. A1 show two different techniques (i.e. quantities on the x-axes) for quantifying residual trends in our L500-based predictions of $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$. The left two panels show the same type of analysis as used in this work: the ratio of $$M_{\rm H\,{\small I}}$$ to its prediction is plotted against $$M_{\rm H_2}$$, and vice versa for the $$M_{\rm H_2}$$ prediction ratio against $$M_{\rm H\,{\small I}}$$. In both cases, the slope of the MC residuals is similar to our observations, although our observed trend with $$M_{\rm H\,{\small I}}$$/$$M_{\rm H\,{\small I}}$$(L500) is slightly larger than the MC value and the scatter about the $$M_{\rm H_2}$$/$$M_{\rm H_2}$$(L500) residual is over twice as large as our observations. The right two panels show a different approach to measure the residuals as a function of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$, which is expected to drive these trends. However, since $$M_{\rm H\,{\small I}}$$ (or $$M_{\rm H_2}$$) appears on both axes, both the real and MC sample show enhanced trends as a function of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$. While not physical, this type of plot is useful when interpreting L500-based predictions for galaxies with different molecular-to-atomic gas mass ratios. None the less, caution is advised when quantifying residual trends in $$M_{\rm H\,{\small I}}$$ predictions as a function of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$. The agreement of this MC analysis with our observations suggests that the strongest predictive power comes from the L500–Mgas relationship. This MC analysis goes one step further and shows that any attempt to use L500 to predict $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$ alone will suffer from a systematic dependence on the (potentially unknown) $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio. We also note that this MC analysis does not produce results that fully agree with our observations, indicating that further residual dependences on other physical properties (e.g. those discussed in Section 6.3) may also play a role in these relationships. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Monthly Notices of the Royal Astronomical Society Oxford University Press

# Lurking systematics in predicting galaxy cold gas masses using dust luminosities and star formation rates

, Volume 476 (1) – May 1, 2018
15 pages

/lp/ou_press/lurking-systematics-in-predicting-galaxy-cold-gas-masses-using-dust-HY6bdLX4oQ
Publisher
The Royal Astronomical Society
ISSN
0035-8711
eISSN
1365-2966
D.O.I.
10.1093/mnras/sty242
Publisher site
See Article on Publisher Site

### Abstract

Abstract We use galaxies from the Herschel Reference Survey to evaluate commonly used indirect predictors of cold gas masses. We calibrate predictions for cold neutral atomic and molecular gas using infrared dust emission and gas depletion time methods that are self-consistent and have ∼20 per cent accuracy (with the highest accuracy in the prediction of total cold gas mass). However, modest systematic residual dependences are found in all calibrations that depend on the partition between molecular and atomic gas, and can over/underpredict gas masses by up to 0.3 dex. As expected, dust-based estimates are best at predicting the total gas mass while depletion time-based estimates are only able to predict the (star-forming) molecular gas mass. Additionally, we advise caution when applying these predictions to high-z galaxies, as significant (0.5 dex or more) errors can arise when incorrect assumptions are made about the dominant gas phase. Any scaling relations derived using predicted gas masses may be more closely related to the calibrations used than to the actual galaxies observed. dust, extinction, galaxies: ISM, infrared: galaxies, radio lines: galaxies 1 INTRODUCTION Key to an understanding of galaxy evolution is a complete census of the baryonic components in galaxies: the stellar populations and the interstellar medium (ISM, gas, and dust). These components are strongly connected to each other, mainly through the process of star formation. Atomic hydrogen gas (H i) is the raw fuel for star formation, but must first cool and condense into molecular hydrogen (H2) before stars can form (e.g. Bigiel et al. 2008; Leroy et al. 2008, but see also Glover & Clark 2012). Once formed, stars eject gas and dust through stellar winds and also significantly pollute the ISM with enriched material. The exquisite interplay between the multi-phase ISM and the star formation cycle has been observed in nearby galaxies in great detail. While star formation rates (SFRs) have been measured in large samples of galaxies for many decades (e.g. Roberts 1963; Kennicutt 1983; Alam et al. 2015), more recently several large observational programmes have quantified galaxy ISM contents in a statistical way, including H i content (Catinella et al. 2010, 2013), H2 content (Saintonge et al. 2011), and dust content (da Cunha et al. 2010; Cortese et al. 2012). These studies and others have shown many connections between star formation and ISM phases, including that H2 is responsible for regulating galaxy SFRs (Tacconi et al. 2013; Saintonge et al. 2016), that the dust-to-gas mass ratio depends on metallicity (Draine et al. 2007; Leroy et al. 2011), and that the dust-to-stellar mass ratio depends on specific SFR (sSFR; da Cunha et al. 2010). Despite the abundance of high-resolution and multi-wavelength observations of galaxies, our understanding of the gas–dust–star cycle is incomplete, even in the local Universe. None the less, as observations of ISM components in galaxies at higher redshifts are becoming increasingly feasible, we are gaining new windows into galaxy evolution. However, the high-redshift view is often limited or partial, and requires careful ‘calibration’ when making comparisons with local galaxies. SFR is the most easily measured piece of the gas–dust–star cycle, and samples of star-forming galaxies have been observed out to high redshift (e.g. Whitaker et al. 2014; Alam et al. 2015). Extensive observations have consistently shown that the star formation history of the Universe reaches a peak at z ∼ 2.5 (Madau & Dickinson 2014), but direct observations of all ISM components are not yet as advanced (e.g. Carilli & Walter 2013). Observational challenges prevent the detection of 21 cm H i emission in samples of galaxies beyond z ∼ 0.2 (Catinella & Cortese 2015, but see recent detection at z = 0.376 from Fernández et al. 2016), but CO (a tracer of H2) observations have begun slowly reaching galaxies at higher redshifts (e.g. Daddi et al. 2010; Aravena et al. 2012; Bauermeister et al. 2013; Hodge et al. 2013; Cybulski et al. 2016). Faced with the great difficulty (or impossibility) of directly observing the gaseous phases of galaxy ISM at high redshift, many observational campaigns have focused on indirect estimates of ISM masses. These estimators are based on correlations observed between properties of local galaxies, and are then applied at higher redshift. These ISM mass predictions fall broadly into two categories: (1) mass-to-light ratios using far-infrared (FIR) luminosities or gas-to-dust ratios (e.g. Eales et al. 2012; Magdis et al. 2012; Scoville et al. 2014); and (2) starting from an integrated star formation law (e.g. the Kennicutt–Schmidt (K-S) law; Schmidt 1959; Kennicutt 1998) and inverting the observed SFR (i.e. a gas depletion time) to estimate a star-forming gas mass (e.g. Tacconi et al. 2013; Genzel et al. 2015; Berta et al. 2016). In the literature, these relations are typically calibrated with samples of local galaxies. Recently, some groups have also explored the theoretical connections between simulated galaxies and different ISM mass predictions. Goz et al. (2017) use a fully cosmological hydrodynamical code to reproduce the multi-wavelength emission (ultraviolet to infrared) of galaxies, and validate some choices of indirect ISM mass indicators. Broadly, star formation can contribute to dust heating (making dust temperature a good tracer of SFR), but longer wavelength dust emission is tightly connected to gas masses. A significant limitation in the development of these indirect relations is that each ISM mass component is usually calibrated independently with an observable quantity, which does not allow an analysis of possible systematic effects or residual dependences on other ISM properties. For example, Scoville et al. (2014) use H i, CO, and FIR observations of a sample of 12 low-z spiral and starburst galaxies in order to calibrate a predictive relationship between FIR dust luminosity and total ISM mass (MISM, the sum of the H i, H2, and dust components), but their sample is too small to test for secondary dependences (e.g. on $$M_{\rm H\,{\small I}}$$/$$M_{\rm H_2}$$ ratio, SFR, etc.). By contrast, Scoville et al. (2016) use the same dust luminosities to calibrate a prediction for molecular gas masses in a larger sample of 70 galaxies. This subtle shift requires an assumption about the relative contributions of H i, H2, and dust to the total ISM mass. In a different work, the same FIR luminosities are used to instead predict total gas mass ($$M_{\rm H\,{\small I}}$$+ $$M_{\rm H_2}$$) for 36 nearby galaxies (Groves et al. 2015). While economical, these indirect estimates may suffer from unknown biases or systematic trends with other ISM properties of galaxies. Before using predicted gas masses to make conclusions about the evolution of high-redshift galaxies, we must first carefully examine these calibrations for any lurking systematic effects that could introduce variations as a function of galaxy properties and mask the true evolutionary trends. We use a sample of galaxies with a complete set of observations of all ISM components to calibrate and test these predictive relations, in order to evaluate how internally consistent these estimators can be. We describe our local sample of galaxies and observations in Section 2. In Section 3, we use mass-to-light and gas-to-dust ratios to predict ISM masses, and Section 4 shows our calibrations using depletion times. In Section 5, we compare these two types of calibrations with each other, and in Section 6 we discuss residual secondary dependences on other quantities. Section 7 includes a discussion of the scientific implications of these systematic effects, and in Section 8 we briefly summarize our main results. Throughout this work, we assume a Λ cold dark matter cosmology with H0 = 70 km s−1 Mpc−1, ΩM = 0.27, and $$\Omega _\Lambda$$ = 0.73. 2 SAMPLE AND DATA Our sample is selected from the Herschel Reference Survey (HRS), which was a volume-limited infrared (IR) imaging survey of 322 galaxies between 15 and 25 Mpc (Boselli et al. 2010). We select our Primary Sample (PS) of N = 68 galaxies to be those HRS targets with a complete set of observations of the ISM components to calibrate and verify relationships between dust and gas content. Required observations include detections in each of the Herschel bands (100–500 μm), detections of atomic and molecular gas (H i and CO), spectroscopic metallicity estimates, and reliable SFRs. Further, we select only galaxies that are not H i  deficient. The H i-deficiency parameter (Def$$_{\rm H\,{\small I}}$$) is the logarithmic difference between the H i content expected based on a galaxy's morphological type and optical diameter and its observed H i mass (Haynes & Giovanelli 1984). We require our PS sample to have Def$$_{\rm H\,{\small I}}$$ ≤ 0.5 in order to exclude galaxies that have likely been stripped of their gas or otherwise affected by dense environments (e.g. Cortese et al. 2016). The 68 galaxies in our PS have been shown to be representative of a volume-limited sample, and do not suffer from significant selection effects (see fig. 1 in Cortese et al. 2016). We next briefly summarize each observed or derived quantity, their uncertainties, and the necessary assumptions. The uncertainties on each quantity typically reflect the errors associated with the measurement and not the calibration or systematic uncertainties, which may be even larger. Stellar mass (M*) and distance (D): M* and D come from Cortese et al. (2012). Estimates of M* are based on the colour-dependent mass-to-light ratios of Zibetti, Charlot & Rix (2009) and measured from Sloan Digital Sky Survey images (Abazajian et al. 2009). Uncertainties on M* include only errors from the broad-band flux measurements and not uncertainties from distances or mass-to-light ratios. Star formation rate (SFR): We use SFRs from Boselli et al. (2015), who compiled and calibrated observations of star formation in the HRS. Specifically we use their ‘SFRMED’ that combines all available SFRs (including those from Hα, FUV, 24 μm, and radio indicators). When available, uncertainties come from the errors on the Hα fluxes or radio continuum fluxes; otherwise, we adopt a 15 per cent error as suggested in section 5.1.1 of Boselli et al. (2015). Metallicity (12+log(O/H)): Spectroscopic gas-phase abundances come from Hughes et al. (2013), who observed HRS galaxies with drift-scan optical spectra and derived oxygen abundances using the calibrations of Kewley & Ellison (2008) to be consistent with the O3N2 scale of Pettini & Pagel (2004). When necessary, all metallicities used in this work (e.g. from Leroy et al. (2011); Berta et al. (2016) in Fig. 1) are converted to the O3N2 scale (Pettini & Pagel 2004) via calibrations from Kewley & Ellison (2008). Uncertainties are given by Hughes et al. (2013) and fully include the errors in converting to a common calibration of metallicity (see section 3 of Hughes et al. 2013). Figure 1. View largeDownload slide Calibration of ξ for each gas phase as a function of metallicity. Top: $$M_{\rm H\,{\small I}}$$; middle: $$M_{\rm H_2}$$; bottom: Mgas. Our best-fitting relations are shown in red. The relations from Leroy et al. (2011) and Berta et al. (2016) are shown in blue and green, respectively, in the bottom panel. Best-fitting lines, Pearson's correlation coefficient (R) and its uncertainty, and the standard deviations (σ, in dex) of points about the best-fitting line are shown in red in each plot. Figure 1. View largeDownload slide Calibration of ξ for each gas phase as a function of metallicity. Top: $$M_{\rm H\,{\small I}}$$; middle: $$M_{\rm H_2}$$; bottom: Mgas. Our best-fitting relations are shown in red. The relations from Leroy et al. (2011) and Berta et al. (2016) are shown in blue and green, respectively, in the bottom panel. Best-fitting lines, Pearson's correlation coefficient (R) and its uncertainty, and the standard deviations (σ, in dex) of points about the best-fitting line are shown in red in each plot. Monochromatic IR luminosities: HRS IR data include imaging from the Photoconductor Array Camera and Spectrometer (PACS) and the Spectral and Photometric Imaging Receiver (SPIRE) instruments. PACS provided images at 100 and 160 μm, and SPIRE at 250, 350, and 500 μm. Photometry based on these images was presented in Cortese et al. (2014) from PACS and in Ciesla et al. (2012) from SPIRE. We convert these flux measurements to luminosities in solar units (L⊙). Uncertainties on the luminosities are propagated from published flux errors in Ciesla et al. (2012) and Cortese et al. (2014). Dust mass (Mdust): We use the dust mass estimates (in M⊙) from Ciesla et al. (2014). They fit spectral energy distributions (SEDs) between 8 and 500 μm using the dust models of Draine & Li (2007), and provide uncertainties on their mass estimates. H i gas mass ($$M_{\rm H\,{\small I}}$$): The atomic gas mass estimates come from Boselli, Cortese & Boquien (2014), who compiled 21 cm observations from the literature for HRS galaxies. When available, we propagate 21 cm flux errors into $$M_{\rm H\,{\small I}}$$ uncertainties; otherwise, we adopt the typical uncertainty of 15 per cent errors on the mass (see section 6 of Boselli et al. 2014). Total gas mass (Mgas): Throughout this work, we use the following definition of total gas mass: Mgas= 1.36 × ($$M_{\rm H\,{\small I}}$$+ $$M_{\rm H_2}$$). This includes a 36 per cent correction for helium in the total gas mass (not included in $$M_{\rm H_2}$$ alone). Uncertainties are propagated from $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$. 2.1 Molecular hydrogen masses ($$M_{\rm H_2}$$) Estimates of molecular hydrogen gas mass come from the CO(1–0) observations and compilation of archival data in Boselli et al. (2014). They use a variable conversion factor between LCO and $$M_{\rm H_2}$$ (XCO), which depends on the near-IR luminosity (measured in the H filter) of each galaxy, following the calibration of Boselli, Lequeux & Gavazzi (2002), not including any contribution or correction from helium. Uncertainties on the mass are propagated from flux errors given in Boselli et al. (2014). The variable XCO from Boselli et al. (2002) relies on a number of assumptions. While a metallicity dependence is expected (Leroy et al. 2009), spectroscopic abundances were not available for all of the galaxies in the calibration sample. Instead, they demonstrated that a luminosity-dependent expression for XCO works well for galaxies without metallicity measurements. Their total molecular gas masses also rely on the assumption that molecular clouds are virialized systems (see section 4 of Bolatto, Wolfire & Leroy 2013 or McKee & Ostriker 2007 for further details). As a simple way to consider the range of effects of different conversion methods, we include three alternative prescriptions for XCO and briefly explore their effects on gas mass predictions. The first, $$M_{\rm H_2,c}$$, is determined using the constant conversion factor from Galactic studies (XCO = 2.3× 1020 cm−2/(K km s−1); Strong et al. 1988). Secondly, we use the recent prescription of Accurso et al. (2017) that depends on metallicity and sSFR to compute $$M_{\rm H_2,acc}$$. Thirdly, $$M_{\rm H_2,bol}$$ comes from Bolatto et al. (2013) and depends on both metallicity and stellar surface density (determined from optical effective radius). While a full discussion of the relative advantages and disadvantages is outside the scope of this work, we re-derive our main calibrations using these alternatives as discussed in Section 4.3. Although the exact values of the relationships between dust luminosity and molecular gas mass change, these alternative prescriptions do not change the underlying systematic trends that are implicit in these relationships. 2.2 Gas-to-dust ratio (ξ) for each phase The ratios of gas to dust mass are computed using each phase (H i, H2, total). As this ratio varies with metallicity (Draine et al. 2007; Leroy et al. 2011), we explore the relationship between ξ (Mdust/Mgas(phase)) and 12+log(O/H) for each phase. Fig. 1 shows these relationships, with uncertainties shown as error bars. Note that the uncertainties on the metallicity are large, as they include both statistical and systematic errors associated with converting many observations to a common metallicity scale. We estimate the uncertainty on Pearson's R correlation coefficient, given in parentheses in Fig. 1 and all subsequent, as the standard deviation of the R values of 10 000 random samples drawn from our data points. We allow for repeated values and require each randomly drawn sample to have the same total number of values as our real sample. As is the case throughout this work, red lines in Fig. 1 show our best fits, which are found through least-squares minimization of differences in the ordinate weighted by their uncertainties. These best-fitting relations are given below:   \begin{eqnarray*} \log \xi ^{\rm d}_{\rm H\,{\small I}} &=& \log \frac{M_{\rm H\,{\small I}}}{M_{\rm dust}} = (10.92\pm 2.00) - (1.03\pm 0.23)\\ && \times {\rm (12+logO/H)}\\ \log \xi ^{\rm d}_{\rm H_2} &=& \log \frac{M_{\rm H_2}}{M_{\rm dust}} = (-4.17\pm 2.23)+(0.67\pm 0.26) \\ &&\times {\rm (12+logO/H)}\\ \log \xi ^{\rm d}_{\rm tot} &=& \log \frac{M_{\rm tot}}{M_{\rm dust}} = (\,8.51\pm 1.65) - (0.72\pm 0.19)\\ && \times {\rm (12+logO/H)}.\\ \end{eqnarray*} Note that the relationship between $$\xi ^{\rm d}_{\rm tot}$$ and 12+logO/H is the tightest (smallest scatter, σ, about the best-fitting line) and also agrees with the relations of Leroy et al. (2011) and Berta et al. (2016). The relationship with $$\xi ^{\rm d}_{\rm H\,{\small I}}$$ is the strongest (greatest absolute value of Pearson's correlation coefficient, R), owing to the fact that our galaxies are H i  dominated (like most in the local Universe). The relationship is very poor for $$M_{\rm H_2}$$, where the correlation is actually the inverse of physical expectations (e.g. Leroy et al. 2011) and the scatter becomes even larger. 3 GAS MASS ESTIMATES FROM LIR AND MDUST We start by considering the simplest possible correlations between cold gas masses (atomic, molecular, and total) and monochromatic IR luminosities, as shown in the first five columns of Fig. 2. We also consider correlations with dust mass, with and without the metallicity-dependent ξ, as shown in the last two columns. In all cases, we carry out least-squares fitting weighted by the uncertainties from both axes. The minimization is only done for differences in the ordinate as these relationships are intended for use as predictions between ‘known’ IR luminosities or dust masses and ‘unknown’ ISM phase masses. Figure 2. View largeDownload slide Correlations between gas-phase masses (H i in top row, H2 in middle row, total Mgas in bottom row) and monochromatic IR luminosities (from 100 to 500 μm) and dust masses (with and without metallicity-dependent ξ for that phase). Light grey lines show 1:1 lines in each plot, and the axes have the same tick sizes in all panels. In the last column (ξ × Mdust), the red line is a unity line and no fitting is performed as ξ was already fitted in Fig. 1 for each phase. The strongest and tightest relationships are indicated by bold boxes. Wide green lines show the agreement with relations from Groves et al. (2015). Figure 2. View largeDownload slide Correlations between gas-phase masses (H i in top row, H2 in middle row, total Mgas in bottom row) and monochromatic IR luminosities (from 100 to 500 μm) and dust masses (with and without metallicity-dependent ξ for that phase). Light grey lines show 1:1 lines in each plot, and the axes have the same tick sizes in all panels. In the last column (ξ × Mdust), the red line is a unity line and no fitting is performed as ξ was already fitted in Fig. 1 for each phase. The strongest and tightest relationships are indicated by bold boxes. Wide green lines show the agreement with relations from Groves et al. (2015). We caution against the blind application of these equations to diverse populations of galaxies. As will be shown in the following sections, calibrations of cold gas mass predictions are susceptible to a number of biases and systematic residual dependences on other physical properties. The best-fitting predictions we derive from our sample may not be appropriate to use across all types of galaxies. Further discussions of these biases and their effects are included in Sections 6 and 7. 3.1 $$M_{\rm H\,{\small I}}$$  estimates from dust Across the M/L estimates of $$M_{\rm H\,{\small I}}$$, the strength and scatter improve continuously with increasing wavelength such that the best monochromatic predictor is L500. Not only does L500 have a value of R = 0.75 ± 0.05 that is statistically larger than the next best value at R = 0.70 ± 0.06, but it also has a scatter that is ∼8 per cent better than the next best. This is expected, as L500 is most closely connected to emission from the diffuse ISM. Multiplying Mdust by the metallicity-dependent $$\xi ^{\rm d}_{\rm H\,{\small I}}$$ gives an even better prediction of $$M_{\rm H\,{\small I}}$$, both in terms of R and σ. Our best three relationships are given below, where $$\xi ^{\rm d}_{\rm H\,{\small I}}$$ was determined in Section 2:   \begin{equation*} \begin{array}{l}\log \frac{{\rm M_{\rm H\,{\small I}}}}{{\rm M_{\odot }}} = (0.72 \pm 0.09) \left( \log \frac{{\rm L_{500}}}{{\rm L_{\odot }}} -9 \right)+ (10.09 \pm 0.08) \\ \log \frac{{\rm M_{\rm H\,{\small I}}}}{{\rm M_{\odot }}} = (0.69 \pm 0.13) \left( \log \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} -9 \right) + (10.50 \pm 0.07) \\ \log \frac{{\rm M_{\rm H\,{\small I}}}}{{\rm M_{\odot }}} = \log \xi ^{\rm d}_{\rm H\,{\small I}} + \log \left( \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} \right). \end{array} \end{equation*} Here and in all subsequent equations, we use mass and luminosity in solar units and use years as the denominator in all SFRs. 3.2 $$M_{\rm H_2}$$ estimates from dust Among the M/L estimates of $$M_{\rm H_2}$$, the 160 μm luminosity is best, because although its R value is not significantly different from that of L100 or L250, its σ is the smallest. The longer wavelength luminosities become increasingly poor estimates of $$M_{\rm H_2}$$. This connection between molecular gas and shorter wavelengths is consistent with the use of LIR as an SFR indicator (e.g. Reddy et al. 2010). Using Mdust instead (with or without ξ) gives an even poorer predictor of $$M_{\rm H_2}$$. For comparison, the relationship between $$M_{\rm H_2}$$ and L160 from Groves et al. (2015) is also shown in Fig. 2 as a wide green line, and agrees with our data points. Note that their best-fitting relation has a steeper slope than ours, as their sample includes a number of dwarf galaxies (M* < 109 M⊙) with small molecular gas masses ($$M_{\rm H_2}$$ ∼ 107 M⊙) that drive this steeper slope. If these dwarf galaxies were removed to match the stellar mass range of our sample, the Groves et al. (2015) best-fitting relationship would be consistent with ours. Our best-fitting relation with L160 is given below, along with the L500 prediction (as it is more commonly used):   \begin{equation*} \begin{array}{l}\log \frac{{\rm M_{\rm H_2}}}{{\rm M_{\odot }}} = (0.92 \pm 0.04) \left( \log \frac{{\rm L_{160}}}{{\rm L_{\odot }}} -9 \right)+ (8.41 \pm 0.04) \\ \log \frac{{\rm M_{\rm H_2}}}{{\rm M_{\odot }}} = (0.99 \pm 0.05) \left( \log \frac{{\rm L_{500}}}{{\rm L_{\odot }}} -9 \right)+ (9.92 \pm 0.06). \end{array} \end{equation*} 3.3 Total Mgas estimates from dust The estimates of Mgas show a similar behaviour to those of $$M_{\rm H\,{\small I}}$$, as H i is the dominant gas component in our galaxies. The best Mgas predictor is L500; although its R value is not significantly better than the ξ-based estimates, it does have smaller scatter in the relationship. This is consistent with its connection to diffuse ISM emission. The Mdust estimates of Mgas are only slightly worse, perhaps due to the additional uncertainties from fitting SEDs to determine dust masses. Regardless of the choice between L500 and Mdust, these correlations with Mgas are the strongest and tightest of any combination of gas phase and indicator discussed thus far. Our three best-fitting Mgas relations are given below, where $$\xi ^{\rm d}_{\rm tot}$$ was determined in Section 2:   \begin{equation*} \begin{array}{l}\log \frac{{\rm M_{\rm gas}}}{{\rm M_{\odot }}} = (0.71 \pm 0.05) \left( \log \frac{{\rm L_{500}}}{{\rm L_{\odot }}} -9 \right)+ (10.39 \pm 0.05) \\ \log \frac{{\rm M_{\rm gas}}}{{\rm M_{\odot }}} = (0.67 \pm 0.08) \left( \log \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} -9 \right) + (10.78 \pm 0.05) \\ \log \frac{{\rm M_{\rm gas}}}{{\rm M_{\odot }}} = \log \xi ^{\rm d}_{\rm tot} + \log \left( \frac{{\rm M_{\rm dust}}}{{\rm M_{\odot }}} \right). \end{array} \end{equation*} Groves et al. (2015) found a similar relationship between L500 and Mgas as ours, as shown by the wide green line in Fig. 2. Their relationship agrees with our data points after it is corrected to include helium to match our convention. Going forward, we prefer the L500 estimates of Mgas, as it gives the best correlation and also does not require any additional assumptions to determine dust mass. For $$M_{\rm H_2}$$ estimates, we prefer L160 if available, but L500 is acceptable. However, there are significant residual trends in these predictions, as discussed in Section 6. 4 DEPLETION TIME ESTIMATES As a second method to estimate gas mass without direct observations, we can use the inverted integrated K-S law (i.e. gas depletion time, tdep). We start by using each galaxy's SFR as a predictor of its amount of (star-forming) gas. In order to do this, we must determine the distance between each galaxy and the so-called star-forming main sequence (SFMS; Noeske et al. 2007; Salim et al. 2007). The SFMS is typically defined as the ridge line in the SFR–M* diagram and is populated by ‘star-forming’ galaxies. We use two different definitions of SFMS. In general, the SFMS can be parametrized as follows:   \begin{equation*} \log \frac{{\rm SFR}_{\rm MS}}{{\rm M_{\odot }}/{\rm yr}} = \alpha \left( \frac{{\rm M_*}}{{\rm M_{\odot }}} - 10.5 \right) + \beta . \end{equation*} Using UV+IR SFRs for ∼22 000 galaxies at 0 < z < 2.5, Whitaker et al. (2012) derived a redshift-dependent SFMS with parameters given below:   \begin{equation*} \alpha _{\rm W12}(z) = 0.70-0.13z, \, \, \, \, \, \, \, \, \, \, \, \, \beta _{\rm W12}(z) = 0.38 + 1.14z - 0.19z^2. \end{equation*} We also considered the SFMS from Catinella et al. (2018, see also Janowiecki et al., in preparation), which is based on UV+IR SFRs of a low-redshift sample of ∼1200 stellar mass selected galaxies from xGASS (Janowiecki et al. 2017), with best-fitting parameters given below:   \begin{equation*} \alpha _{\rm J17} = 0.656, \, \, \, \, \, \, \, \, \, \, \, \, \beta _{\rm J17} = 0.162. \end{equation*} While this is based on a representative sample of local (z∼0) galaxies, the Whitaker et al. (2012) SFMS includes a redshift dependence, making it more useful in extending our predictive relationships for ISM masses to galaxies at higher redshifts. We use only the Whitaker et al. (2012) SFMS going forward, as both give similar results for our sample. Using this definition of SFMS, we compute ΔSFMS, which quantifies each galaxy's distance above or below the SFMS at its M* and redshift:   \begin{equation*} \log \Delta {\rm SFMS} = \log \frac{{\rm SFR}}{{\rm M}_{\odot }\,\mathrm{yr}^{-1}} - \log \frac{{\rm SFR}_{\rm MS}{\rm (M_*,}z{\rm )}}{{\rm M}_{\odot }\,\mathrm{yr}^{-1}}. \end{equation*} In a similar vein, Berta et al. (2016) derive an expression for depletion times based primarily on ΔSFMS, but also including a dependence on M* and redshift (their equation 3). We show this quantity as $$\Delta ^\prime _{\rm B16}$$. Fig. 3 shows our observed depletion times (Mgas/SFR) using different gas masses (H i, H2, total). These are plotted against sSFR (left column), ΔSFMS (centre column), and $$\Delta ^\prime _{\rm B16}$$ (right column). As with Fig. 2, best-fitting lines and quality estimates are shown in each panel. Note that for each predictive indicator shown on the x-axis, the relation with $$M_{\rm H_2}$$ has the tightest scatter and strongest correlation. Unsurprisingly, the SFR-based indicators are less closely related to $$M_{\rm H\,{\small I}}$$. Since our galaxies are H i dominated, the predictions for Mgas are similarly weak. Again we caution against blindly applying our calibrations to populations of galaxies that are significantly different from those in our sample. We next focus on two of these predictions of $$M_{\rm H_2}$$. Figure 3. View largeDownload slide Gas mass divided by SFR (i.e. depletion time) using various gas phases is plotted against SFR-related quantities. Left column x-axis shows sSFR, and our $$M_{\rm H_2}$$/SFR values agree with the average value from Tacconi et al. (2013) and the trend from Saintonge et al. (2011). Centre column shows ΔSFMS using the Whitaker et al. (2012) SFMS. Right column shows $$\Delta ^\prime _{\rm B16}$$ from Berta et al. (2016), a weighted combination of sSFR and M*. Figure 3. View largeDownload slide Gas mass divided by SFR (i.e. depletion time) using various gas phases is plotted against SFR-related quantities. Left column x-axis shows sSFR, and our $$M_{\rm H_2}$$/SFR values agree with the average value from Tacconi et al. (2013) and the trend from Saintonge et al. (2011). Centre column shows ΔSFMS using the Whitaker et al. (2012) SFMS. Right column shows $$\Delta ^\prime _{\rm B16}$$ from Berta et al. (2016), a weighted combination of sSFR and M*. 4.1 $$M_{\rm H_2}$$ estimates from tdep(sSFR) The relations shown in Fig. 3 are consistent with previous results, as seen in the panel showing $$M_{\rm H_2}$$/SFR versus sSFR. Here the constant depletion time estimate from Tacconi et al. (2013) is shown (at the average sSFR of their sample), as well as the relation from Saintonge et al. (2011), both of which are consistent with our observations. Our best-fitting relationship between $$M_{\rm H_2}$$/SFR (in units of years) and sSFR (in inverse years) is given below:   \begin{equation*} \mathrm{log} \frac{{\rm M_{\rm H_2}}}{{\rm SFR}} = -0.46 (\pm 0.74) \times \mathrm{log} {\rm sSFR} + 4.32 (\pm 0.08). \end{equation*} 4.2 $$M_{\rm H_2}$$ estimates from tdep(ΔSFMS) The strongest and tightest correlation is found between $$M_{\rm H_2}$$/SFR and ΔSFMS, as the H2 is directly involved in current star formation. The best-fitting relationship between ΔSFMSW12 and $$M_{\rm H_2}$$/SFR (in years) is given below:   \begin{equation*} \mathrm{log} \frac{{\rm M_{\rm H_2}}}{{\rm SFR}} = -0.46 (\pm 0.03) \times \mathrm{log} \Delta {\rm SFMS}_{\rm W12} + 8.92\, (\pm 0.08). \end{equation*} This relationship has similar scatter and strength as the L500–Mgas and Mdust–Mgas relationships determined in Section 3, but is also affected by the residual trends discussed in the following section. 4.3 Alternative $$M_{\rm H_2}$$ prescriptions We briefly consider the effects of alternative XCO conversion factors to determine $$M_{\rm H_2}$$ from CO observations, as introduced in Section 2.1. In addition to the luminosity-dependent XCO we primarily use, we show here the results of using a constant conversion factor ($$M_{\rm H_2,c}$$), the metallicity-dependent conversion ($$M_{\rm H_2,acc}$$) from Accurso et al. (2017), and the conversion from Bolatto et al. (2013, $$M_{\rm H_2,bol}$$) that depends on metallicity and stellar surface density (determined from optical effective radius). Fig. 4 shows the predictive relations for these determinations of $$M_{\rm H_2}$$ using L500 (top row) and depletion time from ΔSFMS (bottom row). In all cases, the $$M_{\rm H_2}$$ values do not include the contribution from helium. Figure 4. View largeDownload slide Top row: correlations between L500 and $$M_{\rm H_2}$$, using different XCO conversion factors. Wide green line indicates the relationship from Scoville et al. (2016), which uses a constant XCO. Bottom row: correlations between depletion time ($$M_{\rm H_2}$$/SFR) and ΔSFMS, using the same prescriptions. In all panels, points are coloured by their $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio, using the $$M_{\rm H_2}$$ prescription from that panel. Figure 4. View largeDownload slide Top row: correlations between L500 and $$M_{\rm H_2}$$, using different XCO conversion factors. Wide green line indicates the relationship from Scoville et al. (2016), which uses a constant XCO. Bottom row: correlations between depletion time ($$M_{\rm H_2}$$/SFR) and ΔSFMS, using the same prescriptions. In all panels, points are coloured by their $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio, using the $$M_{\rm H_2}$$ prescription from that panel. Using different conversion factors does not dramatically alter these predictive relationships. Adopting the metallicity-dependent conversion from Accurso et al. (2017) moderately improves the strength (and scatter) of the L500 prediction, but it significantly worsens the strength of the relationship with depletion time. Similarly, the Bolatto et al. (2013) prescription marginally improves the strength and scatter of the L500 relation and worsens the trend with ΔSFMS. Also shown in the top row of Fig. 4 is the L500-based $$M_{\rm H_2}$$ prediction from Scoville et al. (2016), which is determined from observations of 70 galaxies between z = 0and2. That calibration adopts the constant Galactic value of XCO, so it is most appropriate to compare it with the middle-left panel. As expected, the Scoville et al. (2016) relationship works better for galaxies with $$M_{\rm H_2}$$≳$$M_{\rm H\,{\small I}}$$; H i-dominated galaxies can fall as much as an order of magnitude below the prediction. This illustrates the difficulty in using FIR dust luminosity to predict $$M_{\rm H_2}$$ alone – without knowledge of the atomic-to-molecular ratio, it is difficult to estimate the mass in each phase from a dust luminosity. 5 CLOSING THE LOOP Fig. 5 visually summarizes the quality of the predictive relationships calibrated thus far. Above each indicator on the x-axis (i.e. monochromatic luminosity, dust mass, sSFR, or depletion time estimate), the strength and scatter of its correlation with each gas mass are shown (for H i, H2, and total gas). The y-axis position shows the absolute value of Pearson's R value, and the size of each point corresponds to the standard deviation of our observations about the best-fitting (or unity) line (σ). Larger symbols show relationships with higher scatter. Figure 5. View largeDownload slide The absolute value of Pearson's R value is shown for each indicator, from the correlations shown in Figs 2 and 3. The size and colour of each point correspond to its scatter (σ, in dex), as indicated in the legend. The best relationships are marked with solid red vertical lines. Uncertainties on |R| are shown as error bars and show the standard deviations from 10 000 random samplings. Figure 5. View largeDownload slide The absolute value of Pearson's R value is shown for each indicator, from the correlations shown in Figs 2 and 3. The size and colour of each point correspond to its scatter (σ, in dex), as indicated in the legend. The best relationships are marked with solid red vertical lines. Uncertainties on |R| are shown as error bars and show the standard deviations from 10 000 random samplings. As is shown in Fig. 5 (and in Sections 3 and 4), $$M_{\rm H\,{\small I}}$$ is better predicted by longer wavelength IR emission (reaching the best prediction at 500 μm), and slightly better still by dust mass and metallicity. As atomic gas is less directly connected to ongoing star formation, predictions based on depletion time are poorer for $$M_{\rm H\,{\small I}}$$. Molecular gas mass is best predicted either by L160 or depletion time estimates, both of which have similar scatter but the M/L method gives a stronger correlation, partly due to its larger dynamic range. Total gas is best predicted by L500, and depletion time estimates perform poorly, as the galaxies in our sample are H i  dominated. We next compare the two methods and evaluate whether these separate predictions are mutually consistent. Fig. 6 shows two different predictions of $$M_{\rm H_2}$$ (left-hand panel, from ΔSFMS and L500) and two different predictions of Mgas (right-hand panel, from Mdust and L500). Since all of these estimates have been calibrated using the same sample of galaxies, their good consistency is not surprising. Figure 6. View largeDownload slide $$M_{\rm H_2}$$ (left) and Mgas (right) comparisons from our calibrations. In both main panels, our predictions for the gas-phase masses are shown, using different indicators on both axes (σ shows the scatter about the unity line, in dex). The residual panels show the differences between the predicted and measured gas-phase masses (Δreal). Also shown (in light grey boxes and best-fitting dashed lines) are the results when using alternative calibrations for the y-axis quantities. In the left-hand panel, small boxes show the effect of assuming a constant depletion time. In the right-hand panel, small boxes show the results of using $$\xi ^{\rm d}_{\rm tot}$$, which depends on metallicity (and assuming Mgas= Mdust × ξ). Note that the two estimates of Mgas (from Mdust and from L500) are better correlated with each other (σ = 0.07) than they are with Mgas, as L500 also tracks the dust content. Figure 6. View largeDownload slide $$M_{\rm H_2}$$ (left) and Mgas (right) comparisons from our calibrations. In both main panels, our predictions for the gas-phase masses are shown, using different indicators on both axes (σ shows the scatter about the unity line, in dex). The residual panels show the differences between the predicted and measured gas-phase masses (Δreal). Also shown (in light grey boxes and best-fitting dashed lines) are the results when using alternative calibrations for the y-axis quantities. In the left-hand panel, small boxes show the effect of assuming a constant depletion time. In the right-hand panel, small boxes show the results of using $$\xi ^{\rm d}_{\rm tot}$$, which depends on metallicity (and assuming Mgas= Mdust × ξ). Note that the two estimates of Mgas (from Mdust and from L500) are better correlated with each other (σ = 0.07) than they are with Mgas, as L500 also tracks the dust content. The two estimates of $$M_{\rm H_2}$$ are observationally independent, as the tdep prediction depends only on SFR and M*, which are separate from the observations of L500. It is possible to calibrate those two predictive relationships so that mutually consistent estimates of $$M_{\rm H_2}$$ are obtained, whether using SFR or FIR luminosity. Note that using a fixed depletion time (e.g. Tacconi et al. 2013) gives a relationship that is ∼15 per cent steeper than the 1:1 line, and that overestimates $$M_{\rm H_2}$$(tdep) at higher H2 masses. The two estimates of total Mgas are less independent, as the L500 emission is closely related to the dust content. As such, the two indicators correlate with each other (σ = 0.07 dex) more tightly than with Mgas (σ = 0.15–0.17 dex). This is reassuring as it means the choice of indicator is not crucial, and both L500 and Mdust are good estimators of Mgas. Note also that adopting a metallicity-dependent $$\xi ^{\rm d}_{\rm tot}$$ (as derived in Section 2) introduces a ∼25 per cent steeper slope to the relation between Mgas(Mdust) and Mgas(L500). This systematic difference implies that $$\xi ^{\rm d}_{\rm tot}$$ scales with galaxy stellar mass (via the mass–metallicity relation), as has been shown by Cortese et al. (2016) and Rémy-Ruyer et al. (2013). In order to eliminate the slope difference between these two estimates, the metallicity-dependent $$\xi ^{\rm d}_{\rm tot}$$ relationship would need a three times steeper slope, which dramatically increases the scatter of the points around the unity line in Fig. 6. These comparisons demonstrate that the choice of calibration method can have significant effects on the indirect predictions of cold gas masses. Even within the same sample of galaxies, there can be systematic deviations when using different assumptions, such as a constant depletion time or a metallicity-dependent ξ. None the less, with appropriate choices, independent predictive methods can be calibrated to produce estimates that are in agreement with each other. 6 RESIDUAL TRENDS IN CALIBRATIONS These predictive relationships perform as expected in our sample of local, star-forming, H i-dominated galaxies. However, we are most interested in applying these relationships to galaxies at higher redshifts, where the partition between atomic and molecular gas is unknown and other physical properties may differ (e.g. M*, sSFR). Across our sample, the $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio varies between ∼3 per cent and ∼300 per cent, which is similar to the range of values observed in galaxies from the xGASS sample (Catinella et al. 2018). Galaxies in xGASS also show a weakly increasing median value of the molecular-to-atomic ratio as a function of stellar mass, from ∼10 per cent at 109 M⊙ to ∼30 per cent at 1011.5 M⊙. 6.1 Quantifying residual systematics Quantifying the strength of any residual secondary dependences requires careful parametrization to avoid being affected by underlying dependences between Mgas and its constituent phases. For example, plotting the accuracy of the L500 prediction of $$M_{\rm H\,{\small I}}$$ as a function of the $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio has a strong intrinsic correlation (from the inclusion of $$M_{\rm H\,{\small I}}$$ on both axes), which makes it difficult to directly interpret the residuals in a physical sense. We explore these intricacies in a Monte Carlo analysis described in Appendix A, and demonstrate that our approach successfully quantifies physical residual dependences without suffering from these effects. 6.2 Residuals between phases We quantify the underlying dependence between predictions in each phase by plotting the accuracy of the predictive relationships for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as functions of $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$, respectively. In this way, we are comparing observationally independent quantities, and the residual trends can be meaningfully interpreted (see Appendix A for more discussion of this method). The top two panels in the left column of Fig. 7 show the differences between the L500-based prediction and the real gas masses for H i and H2. The $$M_{\rm H\,{\small I}}$$ differences are plotted against $$M_{\rm H_2}$$, and the $$M_{\rm H_2}$$ differences against $$M_{\rm H\,{\small I}}$$ (i.e. independently observed quantities). Each residual trend is fitted and their slopes (m), scatters (σ), and correlations (R) are shown, with uncertainties. These fits use least-squares minimization of the ordinate and are weighted by the uncertainties of the observed gas-phase masses only (i.e. they do not include the uncertainties on the predicted gas masses). Figure 7. View largeDownload slide Left-hand panels: residuals of L500 (dust) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. Top panel shows $$M_{\rm H\,{\small I}}$$ divided by its prediction from L500, demonstrating an anti-correlation with $$M_{\rm H_2}$$. Bottom panel shows $$M_{\rm H_2}$$ divided by its prediction from L500, with similarly strong anti-correlation with $$M_{\rm H\,{\small I}}$$. Right-hand panels: residuals of the ΔSFMS (depletion time) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. A stronger residual trend is found when predicting $$M_{\rm H\,{\small I}}$$ from depletion time, while no trend exists in the $$M_{\rm H_2}$$ prediction. Figure 7. View largeDownload slide Left-hand panels: residuals of L500 (dust) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. Top panel shows $$M_{\rm H\,{\small I}}$$ divided by its prediction from L500, demonstrating an anti-correlation with $$M_{\rm H_2}$$. Bottom panel shows $$M_{\rm H_2}$$ divided by its prediction from L500, with similarly strong anti-correlation with $$M_{\rm H\,{\small I}}$$. Right-hand panels: residuals of the ΔSFMS (depletion time) predictions for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ as a function of the mass of the other phase. A stronger residual trend is found when predicting $$M_{\rm H\,{\small I}}$$ from depletion time, while no trend exists in the $$M_{\rm H_2}$$ prediction. For galaxies with larger $$M_{\rm H_2}$$, L500 underpredicts $$M_{\rm H\,{\small I}}$$, and for galaxies with larger $$M_{\rm H\,{\small I}}$$, L500 underpredicts $$M_{\rm H_2}$$. This behaviour illustrates that L500 is most tightly correlated with the total gas mass (e.g. see Section 3.3); any $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$ prediction based on L500 will have a systematic uncertainty depending on the partition of atomic and molecular gas. These systematic trends are of modest amplitude (slopes of 17–18 per cent per dex, with ≥2σ significance) and large scatter (0.2 dex), but are evident even within our small sample of local star-forming galaxies. At higher redshift and for larger more star-forming systems, extrapolations of these discrepancies could be larger. In a similar way, the top two panels in the right column of Fig. 7 compare the depletion time-based predictions of gas masses (using ΔSFMS) for $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ with their true values as a function of $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$, respectively. The discrepancies in $$M_{\rm H\,{\small I}}$$ predictions show steep systematic variations (slope of 40 per cent per dex with ≥4σ significance), which is not unexpected since depletion time is less physically connected to the neutral atomic gas content. Conversely, the errors in the $$M_{\rm H_2}$$ prediction do not depend on $$M_{\rm H\,{\small I}}$$, as this method is best suited to predict $$M_{\rm H_2}$$, the gas which is directly involved in star formation. When using SFR to predict gas mass, there is an implicit assumption about the rate of H2 conversion into stars as well as the conversion between H i and H2. This introduces an underlying dependence on the partition between $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$, as was seen in the M/L estimates. These residual trends serve as a reminder that ISM mass predictions from dust luminosities and SFRs are not equivalent and are sensitive to different gas phases. In the worst cases, using L500 to predict $$M_{\rm H_2}$$ is dependent on $$M_{\rm H\,{\small I}}$$ (R = −0.55) with a modest slope (m = 0.17), giving systematic over/underpredictions of ∼0.15 dex at the extremes of our sample. Similarly, using tdep to predict $$M_{\rm H\,{\small I}}$$ results in an even steeper slope (m = −0.40) and similar correlation (R = −0.48), over/underpredicting by up to ∼0.3 dex. Without prior knowledge of the partition between molecular and atomic gas, it is difficult to apply these predictions. Other studies have shown that the $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio in galaxies can depend on the metallicity and turbulence of their ISM, which affect the conversion between atomic and molecular gas (Krumholz, McKee & Tumlinson 2009; Bialy, Burkhart & Sternberg 2017). 6.3 Residuals with other physical properties We also explored similar residual dependences using other galaxy properties. In particular, we considered galaxy stellar mass and sSFR. The lower four panels of Fig. 7 show, as a function of these physical properties, the ratios between the observations of $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ and their L500- and ΔSFMS-based predictions. As with the top panels, we fit these residual trends to quantify their strengths. For both L500 and ΔSFMS, we see negative (positive) residuals for the $$M_{\rm H\,{\small I}}$$ ($$M_{\rm H_2}$$) predictions with increasing stellar mass. While sometimes weak, these trends are expected, given the moderately increasing $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio observed as a function of stellar mass (Catinella et al. 2018). All of the residual trends with M* are weaker (smaller R values and almost always flatter slopes) than those with $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$, indicating that a stellar mass dependence is not enough to account for the residuals observed between the phases. The sSFR residuals in the predictions of $$M_{\rm H\,{\small I}}$$ suggest that elevated star formation results in L500 and ΔSFMS, underpredicting $$M_{\rm H\,{\small I}}$$, while in more passive galaxies, it may be overpredicted (by ∼0.2 dex, even within our sample's relatively small range of sSFR). These modest systematic trends with M* and sSFR suggest that using L500 or ΔSFMS as ISM mass estimators will be affected by underlying dependences on other galaxy properties. Our calibrations are naturally best suited to predicting cold gas masses for galaxies that are similar to those included in our sample. Any extrapolation or extension of these relationships to significantly different populations of galaxies (e.g. with higher or lower M* or sSFR) may suffer from systematic biases. None the less, these predictions are robust and can reliably generate indirect estimates of cold gas masses. 7 SCIENTIFIC IMPLICATIONS 7.1 Applications to high-z galaxy observations Fig. 8 shows the basic relations between gas masses and L500 for our sample and galaxies at other redshifts where all three observations (21 cm, CO, and 500 μm) are available. In addition to the sample used in this work, we show 26 galaxies from the xGASS sample (z = 0.01–0.05; Catinella et al. 2018) that have L500 observations from the NASA/IPAC Infrared Science Archive,1 survey data from Herschel-ATLAS (Bourne et al. 2016; Valiante et al. 2016), and the Herschel Stripe 82 Survey (Viero et al. 2014). Figure 8. View largeDownload slide All three panels show galaxies in our sample (in grey) and other colours show other galaxies including some at higher redshifts. The dotted line shows the 1:1 relationship. Our best-fitting relationships for $$M_{\rm H\,{\small I}}$$ (left), $$M_{\rm H_2}$$ (centre), and Mgas (right) are shown as red lines. Higher redshift galaxies appear to follow similar behaviour as our local sample, with a tight L500–Mgas relation and correlated scatter in the $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ relations. Figure 8. View largeDownload slide All three panels show galaxies in our sample (in grey) and other colours show other galaxies including some at higher redshifts. The dotted line shows the 1:1 relationship. Our best-fitting relationships for $$M_{\rm H\,{\small I}}$$ (left), $$M_{\rm H_2}$$ (centre), and Mgas (right) are shown as red lines. Higher redshift galaxies appear to follow similar behaviour as our local sample, with a tight L500–Mgas relation and correlated scatter in the $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ relations. We also include two galaxies with observations at higher redshifts (z > 0.1). First, AGC 191728 comes from the HIGHz sample of Catinella & Cortese (2015) at z = 0.176. Its H i observations come from Arecibo and its CO(1–2) observations from the Atacama Large Millimeter Array (Cortese, Catinella & Janowiecki 2017). This galaxy was serendipitously imaged as part of the Herschel-ATLAS observations and released in H-ATLAS DR1 (Valiante et al. 2016). Secondly, COSMOS J100054.83+023126.2 comes from the Cosmic Evolution Survey (COSMOS; Scoville et al. 2007) and is at z = 0.376 (this is the highest redshift detection of H i emission from a galaxy to date). It has recently been observed in 21 cm with the Jansky Very Large Array as part of the COSMOS H i Large Extragalactic Survey (CHILES) and in CO with the Large Millimeter Telescope (Fernández et al. 2016). In another serendipitous observation, DR2 of the Herschel Multi-tiered Extragalactic Survey (Oliver et al. 2012) includes 500 μm observations of this galaxy, with fluxes available through their SUSSEXtractor catalogue. Note that the observed 500 μm emission from these two sources corresponds to rest-frame observations at 425 and 363 μm, respectively. While we use these fluxes in our L500-based relationships, adopting the L350-based predictions would make only a small difference. Within our sample, the L350/L500 ratio is 1.073±0.006. These two higher redshift galaxies provide an illustrative example of the application of the L500-based predictions of cold gas masses. These two galaxies are quite different: while they have very similar $$M_{\rm H\,{\small I}}$$ (and stellar mass estimates from optical photometry), the CHILES galaxy has ∼4 times larger $$M_{\rm H_2}$$ than the HIGHz galaxy. Their $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratios are 50 per cent and 200 per cent, respectively. Remarkably, the L500 prediction for Mgas is very accurate for both! However, L500-based estimates of $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$ can be wrong by ∼0.4 dex. The H i-dominated HIGHz galaxy lies along the L500–$$M_{\rm H\,{\small I}}$$ relationship, but falls ∼0.4 dex below L500–$$M_{\rm H_2}$$. Conversely, the H2-dominated CHILES galaxy is consistent with the L500–$$M_{\rm H_2}$$ relation but falls ∼0.3 dex below L500–$$M_{\rm H\,{\small I}}$$. Without additional knowledge of the molecular-to-atomic ratio of these galaxies, L500cannot reliably predict their$$M_{\rm H\,{\small I}}$$ or$$M_{\rm H_2}$$separately. Extending this argument further, observations have suggested that higher redshift galaxies have higher $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratios compared with the H i-dominated galaxies observed at low redshift, but within z < 0.2 this ratio seems to span the same range of values (Cybulski et al. 2016; Cortese et al. 2017). When using these types of relations to predict molecular gas masses at high redshift, corrections may be necessary to account for different mass contributions from ISM phases. When predicting gas masses in high-redshift galaxies, appropriately calibrated relations must be used. While it is possible to separate the atomic and molecular gas phases in our local sample, the distinction becomes difficult at higher redshift. There is inherent uncertainty in extending a low-redshift prediction to higher redshift galaxies, which may have different atomic-to-molecular ratios or other physical differences from local galaxies. 7.2 Consequences of incorrect phase assumptions In Fig. 9, we quantify the possible errors introduced by applying L500-based gas mass predictions to galaxies without knowledge of their molecular-to-atomic partitions. In each panel, we either apply the assumption that galaxies are H i  dominated and use the relationship calibrated for $$M_{\rm H\,{\small I}}$$ (orange circles) or are H2 dominated and use the $$M_{\rm H_2}$$ relationship (purple squares). For completeness, we also show the comparison between total gas mass predictions and the observed $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ (green triangles). Each of these comparisons has a systematic offset from unity: the $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$ predictions scatter around the average molecular and atomic gas fractions for this sample, and the Mgas predictions include the ∼30 per cent correction for the contribution of helium. Figure 9. View largeDownload slide Each panel plots the ratio between the observed $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ and the L500-based predictions of $$M_{\rm H\,{\small I}}$$ ($$M_{\rm H_2}$$) as orange circles (purple squares), as functions of total gas mass, stellar mass, and sSFR. Relations with Mgas are shown as green triangles. Least-squares fits are shown along with best-fitting slopes and uncertainties. Figure 9. View largeDownload slide Each panel plots the ratio between the observed $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ and the L500-based predictions of $$M_{\rm H\,{\small I}}$$ ($$M_{\rm H_2}$$) as orange circles (purple squares), as functions of total gas mass, stellar mass, and sSFR. Relations with Mgas are shown as green triangles. Least-squares fits are shown along with best-fitting slopes and uncertainties. When adopting the H i-dominated assumption (orange circles), the y-axis shows the difference between the true $$M_{\rm H\,{\small I}}$$+$$M_{\rm H_2}$$ and the L500-predicted $$M_{\rm H\,{\small I}}$$, as $$M_{\rm H_2}$$ is assumed to be negligible. This assumption is most accurate for galaxies with Mgas∼109 M⊙, with large stellar mass, or with passive sSFR, where the L500–$$M_{\rm H\,{\small I}}$$ relationship accurately predicts the total gas mass. Estimates generated from this assumption become significantly worse for more star-forming galaxies or those with larger gas masses (i.e. similar to those observed at higher redshifts) where the predictions can be up to ∼0.5 dex too small. For galaxies with more extreme $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratios, the effects of this incorrect assumption could be even larger. While the H2-dominated assumption (purple squares) is clearly not valid for any of the galaxies in our sample (note that none reach unity in the ratio being plotted), this assumption interestingly shows strong systematic variations with M* and sSFR. These systematic trends are stronger and more significant than those found in Fig. 7, and are a manifestation of the incorrect assumption about the dominant phase. Even the L500–Mgas relationship (green triangles) shows a dependence on sSFR at the ∼2.5σ level, demonstrating the ubiquity of underlying systematic residuals. While it is not unsurprising that incorrect assumptions about the dominant gas phase yield incorrect results, these trends demonstrate quantitatively the nature of errors arising from these assumptions. Most importantly, in addition to the expected systematic offset, Fig. 9 shows that residual trends are present that depend on other galaxy properties. Every application of these indirect gas mass predictions relies on implicit assumptions to produce $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$ estimates. 7.3 The K-S with different calibrations Fig. 10 shows the integrated K-S star formation law (Schmidt 1959; Kennicutt 1983), using different predictors of ISM phase masses. While the top-left panel shows the K-S law from direct observations of SFR and $$M_{\rm H_2}$$, the other panels use different indirect gas mass estimates. Figure 10. View largeDownload slide Integrated star formation law with different calibrations. Top-left panel shows the true relation between the observed $$M_{\rm H_2}$$ and SFR, with the best-fitting relationship (shown in all panels). Other panels plot observed SFR against various estimates of ISM phase masses, with analogous best fits shown as grey-shaded regions. Galaxies are colour-coded based on their position on, above, or below the relation. The full sample of HRS galaxies is included as faint points whenever available. Figure 10. View largeDownload slide Integrated star formation law with different calibrations. Top-left panel shows the true relation between the observed $$M_{\rm H_2}$$ and SFR, with the best-fitting relationship (shown in all panels). Other panels plot observed SFR against various estimates of ISM phase masses, with analogous best fits shown as grey-shaded regions. Galaxies are colour-coded based on their position on, above, or below the relation. The full sample of HRS galaxies is included as faint points whenever available. Note that the slope, width, and offset of the best-fitting relation can all be significantly affected by a different choice of gas mass indicator. For example, when adopting an estimate of $$M_{\rm H_2}$$ based on L500, the K-S relation becomes steeper and tighter than the true relation. Also note the extent to which an individual galaxy can move above/below the K-S relation using different gas estimates. The different coloured points in Fig. 10 demonstrate that >+1σ outliers can become consistent with the relation (using $$M_{\rm H_2}$$ based on L500) and that <−1σ outliers can move above the relation (using $$M_{\rm H_2}$$ based on Mdust). The lower-right panel of Fig. 10 shows what can happen when using total Mgas instead of the star-forming $$M_{\rm H_2}$$. This illustrates the impact of an incorrect assumption that the ISM is dominated by the molecular component when in fact it is mostly atomic. While this assumption is obviously inappropriate for the local H i-dominated galaxies in our sample, higher redshift galaxies have a wide range of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ partitions and may not all be dominated by the molecular component (Cortese et al. 2017). It can be dangerous to use these types of indirect estimates of gas masses to study the K-S relationship or other scaling relations. Any calibration of these gas mass estimates will be linked to the underlying physical properties of the ISM (e.g. atomic-to-molecular hydrogen ratio), and will not necessarily indicate variations in star formation efficiency. There are likely to be systematic effects lurking in these calibrations that will limit any attempt to derive a K-S relation using indirect gas mass predictions. 8 SUMMARY Using a representative sample of N = 68 nearby galaxies from the HRS, we have calibrated a set of relationships between the masses of ISM phases and observable quantities (FIR and SFR). These predictive relationships can estimate ISM masses with ∼20 per cent accuracy and are mutually self-consistent. However, our complete set of observations of all ISM phases show that these predictive relationships suffer from modest systematic residual dependences on the molecular-to-atomic partition and other physical properties. Any application of these relationships to predict gas masses from FIR/SFR observations requires an implicit assumption of the underlying $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio. Incorrect assumptions about the dominant phase of the ISM can yield errors in gas mass predictions as large as 0.5 dex. Furthermore, using these indirect gas estimates to test the evolution of star formation laws or other scaling relations is potentially problematic, as these relations rely on those underlying scaling relations to successfully predict gas masses. Acknowledgements We thank Toby Brown and Katinka Geréb for helpful discussions, and the anonymous referee for their comments that have significantly improved this work and its statistical treatment of residual trends. SJ, BC, and LC acknowledge support from the Australian Research Council's Discovery Project funding scheme (DP150101734). BC is the recipient of an Australian Research Council Future Fellowship (FT120100660). AG acknowledges support from the ICRAR Summer Studentship Programme during which this project was initiated. This research has made use of NASA's Astrophysics Data System, and also the NASA/IPAC Extragalactic Database (NED), which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. 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To simulate and test the relationships between L500 and cold gas masses, we start by generating a random distribution of L500 values (N = 10 000) consistent with that of our sample. We use our relationship between L500 and Mgas to generate estimates for the total gas content, and add random noise to the estimates to match our observed dispersion of σ = 0.15 dex. We next generate a distribution of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ values, which matches that of our sample, ranging from ∼3 per cent to ∼300 per cent, and calculate $$M_{\rm H_2}$$ and $$M_{\rm H\,{\small I}}$$ for each value of Mgas using this ratio. The top two rows of Fig. A1 show histograms of these quantities and the relationships with L500, where grey points and red lines show our observations and green points and lines show the simulated MC data. Note that we only input the L500–Mgas relation into this MC, and we naturally recover the observed slope, correlation strength, and scatter in the L500–$$M_{\rm H\,{\small I}}$$ relation. However, the resulting MC version of the L500–$$M_{\rm H_2}$$ relation does not agree as well with the observed relation, although the two distributions of points have significant overlap. Figure A1. View largeDownload slide Full summary plots of our Monte Carlo for L500-based predictions of cold gas masses. In all panels, grey points and red lines show data, histograms, and fits to our real observations; green points and lines show the simulated data. Figure A1. View largeDownload slide Full summary plots of our Monte Carlo for L500-based predictions of cold gas masses. In all panels, grey points and red lines show data, histograms, and fits to our real observations; green points and lines show the simulated data. The bottom two rows of Fig. A1 show two different techniques (i.e. quantities on the x-axes) for quantifying residual trends in our L500-based predictions of $$M_{\rm H\,{\small I}}$$ and $$M_{\rm H_2}$$. The left two panels show the same type of analysis as used in this work: the ratio of $$M_{\rm H\,{\small I}}$$ to its prediction is plotted against $$M_{\rm H_2}$$, and vice versa for the $$M_{\rm H_2}$$ prediction ratio against $$M_{\rm H\,{\small I}}$$. In both cases, the slope of the MC residuals is similar to our observations, although our observed trend with $$M_{\rm H\,{\small I}}$$/$$M_{\rm H\,{\small I}}$$(L500) is slightly larger than the MC value and the scatter about the $$M_{\rm H_2}$$/$$M_{\rm H_2}$$(L500) residual is over twice as large as our observations. The right two panels show a different approach to measure the residuals as a function of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$, which is expected to drive these trends. However, since $$M_{\rm H\,{\small I}}$$ (or $$M_{\rm H_2}$$) appears on both axes, both the real and MC sample show enhanced trends as a function of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$. While not physical, this type of plot is useful when interpreting L500-based predictions for galaxies with different molecular-to-atomic gas mass ratios. None the less, caution is advised when quantifying residual trends in $$M_{\rm H\,{\small I}}$$ predictions as a function of $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$. The agreement of this MC analysis with our observations suggests that the strongest predictive power comes from the L500–Mgas relationship. This MC analysis goes one step further and shows that any attempt to use L500 to predict $$M_{\rm H\,{\small I}}$$ or $$M_{\rm H_2}$$ alone will suffer from a systematic dependence on the (potentially unknown) $$M_{\rm H_2}$$/$$M_{\rm H\,{\small I}}$$ ratio. We also note that this MC analysis does not produce results that fully agree with our observations, indicating that further residual dependences on other physical properties (e.g. those discussed in Section 6.3) may also play a role in these relationships. © 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society

### Journal

Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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