2.1. Generalities

As discussed above, a CPDF, $P_j\text{(M}_j| M_{*}\text{)}$ , determines the chances that a galaxy of mass $M_{*}$ possess a specific galaxy property $M_j$ , with $ j = {\rm HI}$ , $\rm H_{2}$ , cold gas, or baryonic mass. Note that the units of $P_j$ is per $\text{M}_{\odot}$ . The relation between the distribution $P_j$ in bins per $\text{M}_{\odot}$ to dex-1, $\mathcal{P}_{j} $ , is given by

(1) \begin{equation}\mathcal{P}_{j} \text{(}M_j| M_{*}\text{)} = P_{j}\text{(}M_j| M_{*}\text{)} \times \frac{M_j}{\log e}.\end{equation}

The advantage of using $P_j\text{(}M_j|M_{*}\text{)}$ is that it contains information about all the moments of the distribution, in particular the mean $M_j-M_{*}$ relation and its standard deviation.

The joint distribution function of $M_{*}$ and $M_j$ , hereafter referred as the bivariate distribution function, is defined as:

(2) \begin{equation}\Phi(M_{j},M_{*}) = \frac{d^2 N\text{(}M_{j}| M_{*}\text{)}} {V d\log M_{j} d\log {M_{*}}}=\mathcal{P}_{j}\text{(}M_{j}| M_{*}\text{)} \phi_* \text{(}M_{*}\text{)},\end{equation}

where $d^2 N$ is the bivariate number of galaxies within the mass range $\log {M_{*}}\pm d\log{M_{*}}/2$ and $\log M_{j}\pm d\log M_{j}/2$ in a given volume V, and $\phi_* \text{(}M_{*}\text{)}$ is the GSMF in units of ${\rm Mpc}^{-3} {\rm dex}^{-1}$ . The integration (marginalisation) of $\Phi(M_{j},{M_{*}})$ over $M_{*}$ results in the total MF for $M_j$ , $\phi_j(M_j)$ , that is,

(3) \begin{align}\phi_{j} (M_j) &=\int \Phi(M_{j},{M_{*}})d\log{M_{*}}\nonumber\\ &= \int \mathcal {P}_{j}(M_j|{M_{*}}) \phi_* \text{(}M_{*}\text{)} d\log{M_{*}}.\end{align}

The above equation shows how the CPDFs are projected into a number density function via the GSMF. Note that integration of $\Phi(M_{j},{M_{*}})$ over $M_j$ gives the total GSMFFootnote ^c.

As discussed previously, when studying the properties of galaxies, it is useful to separate them into, at least, two morphological components such as early types, or spheroid-dominated galaxies, and late types, or disc-dominated galaxies. Thus, the total GSMF can be formally represented as the contribution of these two types

(4) \begin{equation}\phi_*\text{(}M_{*}\text{)} = \phi_{*,E}\text{(}M_{*}\text{)} + \phi_{*,L}\text{(}M_{*}\text{)},\end{equation}

denoted respectively by $\phi_{*,E}$ , and $\phi_{*,L}$ . In terms of the fraction of early- and late-type galaxies ( $f_E$ and $f_L$ ), their corresponding galaxy stellar MFs are given respectively by $\phi_{*,E} = f_E\times\phi_*$ , and $\phi_{*,L} = f_L\times\phi_*$ , with $f_E+f_L = 1$ .

Early- and late-type galaxies are different in their HI- and ${\rm H_{2}}$ -to-stellar mass distributions. Thereby, Equation (3) can be generalised in terms of the distribution $\mathcal {P}_{i,j}(M_j|{M_{*}})$ , where the subscripts indicate $i = $ early or late type, and $ j = $ $\rm HI$ , $\rm H_{2}$ , cold gas, or baryonic mass. Then, the generalisation of Equation (3) to galaxies with morphological type i and mass component j is

(5) \begin{equation}\phi_{j, i} (M_j) = \int f_i\text{(}M_{*}\text{)} \mathcal {P}_{i,j}(M_j|{M_{*}}) \phi_* \text{(}M_{*}\text{)} d\log{M_{*}}.\end{equation}

Finally, the total CPDFs are calculated from the respective conditional distributions of early- and late-type galaxies as:

(6) \begin{align}\mathcal {P}_{j}\text{(}M_{j}| M_{*}\text{)} &= f_{E}\text{(}M_{*}\text{)}\times \mathcal {P}_{E,j} \text{(}M_{j}| M_{*}\text{)}\nonumber\\& \quad +f_{L}\text{(}M_{*}\text{)}\times \mathcal {P}_{L,j} \text{(}M_J| M_{*}\text{)},\end{align}

with $ j = {\rm HI}, {\rm H_{2}},$ cold gas, or baryonic mass.

2.2. The $\text{\em HI}$ and $\text{\em H}_{\text{\em 2}}$ conditional distribution functions

As shown in Equations (2) and (5), the conditional or bivariate distribution functions are useful to statistically determine the MFs. Evidently, in the case of atomic and molecular gas, we are assuming that for every galaxy that is optically selected, there must exist $\rm HI$ and $\rm H_{2}$ counterparts. The discussion on the possible existence of pure $\rm HI$ or $\rm H_{2}$ galaxies, those that will not be observed in optically selected samples but rather in radio blind surveys, is out of the scope of this paper. Note that if they exist, the chance of observing those galaxies is very low over the mass ranges that we will derive the MFs. For example, in the case of pure $\rm HI$ galaxies, the Arecibo Legacy Fast ALFA (ALFALFA) survey has found $\sim\!1.5\%$ of $\rm HI$ sources that were not clearly associated with an optical counterpart. Of those, $\sim\!75\%$ are likely tidal in origin (Haynes et al. Reference Haynes2011). Thus, $\sim\!0.4\%$ of HI source observed in the ALFALFA survey are purely gaseous galaxies candidates, most of them at the mass range $10^{7}<M_{\rm HI}/\text{M}_{\odot}< 10^{10}$ (Cannon et al. Reference Cannon2015). As we will show, our completeness limit for the $\rm HI$ MF is $M_{\rm HI}\sim 10^{8}\text{M}_{\odot}$ . The above fraction could be considered as an upper limit as some of these sources have already detected optical counterparts revealing unusual high HI mass-to-light ratios (Cannon et al. Reference Cannon2015). Thus, we conclude that our results are unlikely to be affected by excluding pure gas galaxies in our analysis.

2.2.2. The functional forms of the $\text{\em HI}$ and $\text{\em H}_{\text{\em 2}}$ conditional distribution functions

For the $\rm HI$ and $\rm H_{2}$ CPDFs of late-type galaxies, in Paper I, we found that they are described by a Schechter function. In the case of early-type galaxies, the CPDFs are better described by a (broken) Schechter function plus a uniform distribution at the low $-\mathcal{R}_j$ values. Following, we describe in more detail these functional forms.

We begin by introducing the following Schechter-type probability distribution function for the $\rm HI$ - or $\rm H_{2}$ -to-stellar mass ratios, $ \mathcal{R}_j=M_j/M_{*}$ , in the range $\log \mathcal{R}_j \pm d \log \mathcal{R}_j / 2$ :

(7) \begin{equation}\mathcal{S}_{i,j} (\mathcal{R}_j) = \frac{\ln(10)}{\mathcal{N}_{i,j}}\left(\frac{\mathcal{R}_{j}}{\mathcal{R}^*_{i,j}}\right)^{\alpha_{i,j} + 1}\exp\left({-}\frac{\mathcal{R}_{j}}{\mathcal{R}^*_{i,j}}\right),\end{equation}

where the morphology is represented with $i = $ early or late type, and the galaxy property is represented with $ j = $ $\rm HI$ or $\rm H_{2}$ . The parameters are the characteristic gas-to-stellar mass ratio, $\mathcal{R}^*_{i,j}$ , the normalisation parameter, $\mathcal{N}_{i,j}$ , which constrains the probability to be between zero and one,Footnote ^d and the power law slope $\alpha_{i,j}$ for the part of the distribution of galaxies with low gas-to-stellar mass ratio.

• Late-type galaxies:

For late-type galaxies, that is $i = L$ , in Paper I, we found that the HI-CPDF and H2-CPDF are described by the Schechter-type distribution function given by (7) with the parameters $\alpha_{L,j}$ and $\mathcal{R}^*_{L,j}$ functions of $M_{*}$ as follows:

(8) \begin{equation}\alpha_{L,j} = \alpha_{0;L,j} \log {M_{*}} + \alpha_{1;L,j} ,\end{equation}

and

(9) \begin{equation}\mathcal{R}^*_{L,j} = \frac{\mathcal{R}^*_{0;L,j}}{\left(\frac{{M_{*}}}{\mathcal{M}^*_{L,j}} \right)^{\beta_{L,j}} + \left(\frac{{M_{*}}}{\mathcal{M}^*_{L,j}} \right)^{\gamma_{L,j}}}.\end{equation}

Consider that $\mathcal{S}_{L,j} (\log\mathcal{R}_j) d\log \mathcal{R}_j = \mathcal{S}_{L,j} (\log M_j - \log{M_{*}}) d\log (M_j / {M_{*}})$ . By definition $M_{*}$ is fixed, thus the $\rm HI$ and $\rm H_{2}$ CPDFs of late-type galaxies are given by:

(10) \begin{equation}\mathcal {P}_{L,j}(M_j | {M_{*}}) d\log M_j = \mathcal{S}_{L,j} (\log M_j - \log{M_{*}}) d\log M_j.\end{equation}

The above explicitly shows that the integration over conditional distribution functions can also be interpret as convolutions in Equation (3).

• Early-type galaxies:

In the case of early-type galaxies, $i = E $ , we showed in Paper I that both for the HI-CPDF and H2-CPDF are described as the sum of two distribution functions; the Schechter-type distribution function, $\mathcal{S}_{E,j}$ , and a uniform function, $\mathcal {U}_{0,j}$ , are

(11) \begin{equation}\mathcal {E}_{j} (\mathcal{R}_j) = \left\{\begin{array}{l@{\quad}l}\mathcal {U}_{0,j} & \mathcal{R}_{0,j} \le\mathcal{R}_j < \mathcal{R}_{1,j} \\A \times \mathcal{S}_{E,j} (\mathcal{R}_j) & \mathcal{R}_{1,j} \le \mathcal{R}_j\end{array},\right.\end{equation}

where $\mathcal{R}_{0,j} = \mathcal{R}_{1,j} / 10,$ Footnote ^e and $ \log \mathcal{R}_{1,j} = r_{0,j} \log{M_{*}} + r_{1,j}$ , while the uniform distribution is given by

(12) \begin{equation}\mathcal {U}_{0,j} \text{(}M_{*}\text{)} = \frac{p_{0,j} \log{M_{*}} + p_{1,j}}{\Delta},\end{equation}

and

(13) \begin{equation}A = \left( 1 - \mathcal {U}_{0,j} \times \Delta\right) \times \frac{\mathcal{N}_{i,j}}{\eta_{i,j}(\mathcal{R}_{1,j})},\end{equation}

where in Paper I we assumed that $\Delta = \log_{10} = 1$ dex, the symbol $\eta_{i,j}(\mathcal{R}_{1,j})$ takes into account the fraction of galaxies in the Schechter-type mode for galaxies with gas ratio above $\mathcal{R}_{1,j}$ .Footnote ^f The HI-CPDF and $\rm H_{2}$ -CPDF of early-type galaxies are

(14) \begin{equation}\mathcal {P}_{E,j}(M_j | {M_{*}}) d\log M_j = \mathcal{E}_{j}(\log M_j - \log{M_{*}}) d\log M_j.\end{equation}

2.3. The cold gas and baryonic conditional distribution functions

Once we have constructed the HI-CPDF and $\rm H_{2}$ -CPDF, we can now define the conditional distributions for the cold gas and baryon masses, $M_{\rm gas}$ and $M_{\rm bar}$ .

The total cold gas content in a galaxy is composed of $\rm HI$ , $\rm H_{2}$ , helium, and metals; helium and metals account for roughly 30% of the cold gas, $M_{\rm He} + M_{\rm Z}\approx0.3 M_{\rm gas}$ Therefore, $M_{\rm gas} = M_{\rm HI} + M_{H_{2}} + M_{\rm He} + M_{\rm Z} = 1.4 \times (M_{\rm HI} + M_{H_{2}}\text{)}$ . For simplicity, let $M_{\rm HI}$ and $M_{H_{2}}$ be two independent random variables. Section 5 discusses the validity of this assumption. Then, $M_{\rm gas}$ is a random variable with the conditional distribution function:

(15) \begin{align}P_{\rm gas}(M_{\rm gas}|{M_{*}}) &=\frac{1}{1.4} \int P_{\rm HI}\left(0.71 M_{\rm gas} - M_{H_{2}}|{M_{*}}\right) \nonumber\\[2pt] & \quad \times P_{\rm H_2}(M_{H_{2}}|{M_{*}}) d M_{H_{2}}, \nonumber\\[2pt] &=\frac{1}{1.4} \int P_{\rm HI}(M_{\rm HI}|{M_{*}}) \nonumber\\[2pt] & \quad \times P_{\rm H_2}(0.71 M_{\rm gas} - M_{\rm HI}|{M_{*}}) d M_{\rm HI},\end{align}

or after some algebra, the same distribution function but per bin in log space is:

(16) \begin{align}\mathcal{P}_{\rm gas}({M_{\rm gas}}|{M_{*}}) &= \int \frac{\mathcal{P}_{\rm HI}(0.71{M_{\rm gas}} - {M_{H_{2}}}|{M_{*}})}{1 - 1.4\ {M_{H_{2}}}/{M_{\rm gas}}} \nonumber\\[2pt] & \quad \times \mathcal{P}_{\rm H_2}({M_{H_{2}}}|{M_{*}}) d \log {M_{H_{2}}}, \nonumber\\[3pt] &= \int \mathcal{P}_{\rm HI}({M_{\rm HI}}|{M_{*}}) \nonumber\\[3pt] &\quad\times \frac{\mathcal{P}_{\rm H_2}(0.71 {M_{\rm gas}} -{M_{\rm HI}}|{M_{*}})}{1 - 1.4\ {M_{\rm HI}}/{M_{\rm gas}}} d \log {M_{\rm HI}}.\end{align}

For the baryonic conditional distribution functions, we again assume that $M_{\rm gas}$ and $M_{*}$ are two independent random variables. Thus, $M_{\rm bar} = M_{\rm gas} + M_{*}$ is a random variable with a distribution function given by

(17) \begin{align}P_{\rm bar}(M_{\rm bar}|{M_{*}}) &= \int P_{\rm gas} (M_{\rm bar} - \mathcal{M}_*|{M_{*}}) \nonumber \\[2pt] &\quad\times \delta(\mathcal{M}_* - {M_{*}}) d \mathcal{M}_*\\[3pt] &= P_{\rm gas} (M_{\rm bar} - {M_{*}}|{M_{*}}),\nonumber\end{align}

where $P_{\rm gas}$ is the conditional distribution function for gas, Equation (15), and the Dirac- $\delta$ function appears explicitly for the $M_{*}$ term. Similarly as above, we find that

(18) \begin{equation}\mathcal{P}_{\rm bar}(M_{\rm bar}|{M_{*}}) = \frac{\mathcal{P}_{\rm gas}(M_{\rm bar} - {M_{*}}|{M_{*}})}{1 - {M_{*}} / M_{\rm bar}}.\end{equation}

Finally, we derive the gas and baryon MFs using Equations (5), (16), and (18), the last two valid for early- and late-type galaxies.

3.1. The galaxy samples and the GSMF

To estimate the GSMF over a large dynamical range, we use two galaxy samples. Next, we shortly describe the procedure and our determinations.

(1) For masses above $M_{*} = 10^9\ \text{M}_{\odot}$ , we use the SDSS DR7 based on the photometric catalogue from Meert et al. (Reference Meert, Vikram and Bernardi2015) and Meert et al. (Reference Meert, Vikram and Bernardi2016)Footnote ^g at the redshift interval $0.005<z<0.2$ . Previous studies have concluded that the measurements of the apparent brightnesses based on the standard SDSS pipeline photometry are underestimated due to sky subtraction problems, particularly, in crowded fields (Bernardi et al. Reference Bernardi, Shankar, Hyde, Mei, Marulli and Sheth2010; Blanton et al. Reference Blanton, Kazin, Muna, Weaver and Price-Whelan2011; Simard et al. Reference Simard, Mendel, Patton, Ellison and McConnachie2011; Bernardi et al. Reference Bernardi, Meert, Sheth, Vikram, Huertas-Company, Mei and Shankar2013; He et al. Reference He, Xia, Hao, Jing, Mao and Li2013; Mendel et al. Reference Mendel, Simard, Palmer, Ellison and Patton2014; Kravtsov, Vikhlinin, & Meshscheryakov Reference Kravtsov, Vikhlinin and Meshscheryakov2014; Meert et al. Reference Meert, Vikram and Bernardi2015; D’Souza, Vegetti, & Kauffmann Reference D’Souza, Vegetti and Kauffmann2015; Bernardi et al. Reference Bernardi, Meert, Sheth, Huertas-Company, Maraston, Shankar and Vikram2016; Meert et al. Reference Meert, Vikram and Bernardi2016. New determinations of the GSMF based on the new algorithms for obtaining more precise measurements of the sky subtraction, and thus to improve the photometry, have concluded that the bright end of the luminosity/MF has been systematically underestimated (Bernardi et al. Reference Bernardi, Meert, Sheth, Fischer, Huertas-Company, Maraston, Shankar and Vikram2017). While there are various groups working in improving the determination of galaxy apparent brightnesses, see references above, Bernardi et al. (Reference Bernardi, Meert, Sheth, Fischer, Huertas-Company, Maraston, Shankar and Vikram2017) showed that all those studies agreed up to 0.1 dex in the GSMF. In this paper, we use the apparent Sérsic r, g, and i band luminosities reported in Meert et al. (Reference Meert, Vikram and Bernardi2015) and (2016) derived for the SDSS DR7 based on the PyMorph software pipeline (Vikram et al. Reference Vikram, Wadadekar, Kembhavi and Vijayagovindan2010; Meert, Vikram, & Bernardi Reference Meert, Vikram and Bernardi2013). This software has been extensively tested in Meert et al. (Reference Meert, Vikram and Bernardi2013) and shows that it does not suffer from sky subtraction problems. All magnitudes and colours are K+E corrected at a redshift rest frame $z=0$ , see Appendixes A and B. As described in Appendix A, for every galaxy, we estimate $M_{*}$ from five colour-dependent mass-to-light ratios but we define as our fiducial $M_{*}$ the geometric mean of all the determinations. Using the $1/{V}_{\rm max}$ method, we derive six $\mbox{GSMF}$ s based on the mass definitions described above. Consistent with Bernardi et al. (Reference Bernardi, Meert, Sheth, Fischer, Huertas-Company, Maraston, Shankar and Vikram2017), we find that the differences in mass-to-light ratios introduce large discrepancies in the GSMF, especially at the high-mass end. In Figure 11 from Appendix A, we find that a shift of $\sim\pm0.15$ dex in the $M_{*}$ axis recovers systematic errors in the GSMF due to different mass-to-light ratios.

(2) For masses below $M_{*} =10^9\ \text{M}_{\odot}$ , we use the SDSS DR4 NUY-VAGC low-z sample,Footnote ^h at the redshift interval $0.0033<z<0.005$ , and ideal to study the low mass/luminosity galaxies (Blanton et al. Reference Blanton2005a; Blanton et al. Reference Blanton, Lupton, Schlegel, Strauss, Brinkmann, Fukugita and Loveday2005b). As before, all absolute magnitudes and colours were K+E corrected at a redshift rest frame $z=0$ . Also, we derive $M_{*}$ from five colour-dependent mass-to-light ratios and, again, we define our fiducial $M_{*}$ as the geometric mean of all the determinations. We construct the GSMF using the $1/V_{\rm max}$ method and include missing galaxies due to surface brightness incompleteness, as described in Appendix C. For surface brightness incompleteness, we follow closely the methodology described in Blanton et al. (Reference Blanton, Lupton, Schlegel, Strauss, Brinkmann, Fukugita and Loveday2005b). The latter correction is relevant for the low-mass end. Based on the conclusions from Baldry et al. (Reference Baldry2012), we use a simple correction for the low-mass end in order to correct for the local flow model distances from Willick et al. (Reference Willick, Courteau, Faber, Burstein, Dekel and Strauss1997) to the one by Tonry et al. (Reference Tonry, Blakeslee, Ajhar and Dressler2000).

Our final GSMF is the result of combining the SDSS NUY-VAGC low-z sample, for galaxies with masses $M_{*}\approx 3\times 10^{7}\text{M}_{\odot}$ to $M_{*}\approx10^{9}\text{M}_{\odot}$ , and the SDSS DR7 sample for galaxies with $M_{*}\gtrsim 10^{9}\text{M}_{\odot}$ , based on our fiducial $M_{*}$ determination. Figure 1 presents our final GSMF with the black solid circles and error bars. The black solid line shows the best fit to the data (described below), and the grey shaded area shows a shift in the $M_{*}$ axis of $\pm 0.15$ dex. As discussed above, in Appendix A, we find that this is a good approximation to the systematic errors in the GSMF due to differences in the mass-to-light ratios. In the same figure, we include comparisons to previous works. In order to account for differences in cosmologies, we scale previous studies to our cosmology using the following relations:

(19) \begin{equation}\phi_{*, {\rm us}} = \phi_{*, {\rm lit}}\left(\frac{h_{\rm us}}{h_{\rm lit}}\right)^3,\end{equation}

and

(20) \begin{equation}M_{*, {\rm us}} = M_{*, {\rm lit}}\left(\frac{h_{\rm lit}}{h_{\rm us}}\right)^2,\end{equation}

where $h_{\rm us} = 0.678$ and $h_{\rm lit}$ is the respective value reported in the literature. Nonetheless, the impact of accounting for different cosmologies is small.

Figure 1.

Observed GSMF when combining the SDSS NYU-VAGC low-redshift sample and the SDSS DR7 sample, black filled circles with error bars. We reproduce our results in the upper and the middle panels. The best-fit model composed of a Schechter function with a sub-exponential slope and a double power law function is shown as the black solid line. The shaded area shows an estimate of the systematic errors with respect to the best-fitting model. The bottom panel shows the residuals for our best-fitting model as a function of $M_{*}$ . We include comparisons to some previous observational determinations of the GSMF: in the upper panel we show determinations that are complete down to $\sim\! 10^9\ \text{M}_{\odot}$ , mostly based on the SDSS DR7, while in the middle panel we show determinations based on the GAMA survey, which are complete down to $\sim\! 3{-}5\times 10^7\ \text{M}_{\odot}$ , but suffer from cosmic variance at high masses due to the small volume.

In the upper panel of Figure 1, we reproduce the $\mbox{GSMF}$ s from previous determinations with stellar mass completeness above $M_{*}\,{\sim}\,10^9\ \text{M}_{\odot}$ . The violet triangles with error bars are the determinations from Moustakas et al. (Reference Moustakas2013), who used a spectroscopic sample of SDSS DR7 galaxies from the New York University Value Added Galaxy Catalog (NYU-VAGC) with redshifts $0.01<z<0.2$ combined with observations from GALEX. The red squares with error bars are the estimation obtained in Bernardi et al. (Reference Bernardi, Meert, Sheth, Vikram, Huertas-Company, Mei and Shankar2013) from a sample of SDSS DR7 galaxies with photometry based on the PyMorph software pipeline at $z\,{\sim}\,0.1$ . Here, we reproduce their result based on Sérsic luminosities. Additionally, we compute the GSMF using the stellar mass estimates from Sérsic photometry from Mendel et al. (Reference Mendel, Simard, Palmer, Ellison and Patton2014) who used the Simard et al. (Reference Simard, Mendel, Patton, Ellison and McConnachie2011) SDSS DR7 sample of g and r band photometry and extended to u, i, and z bands, blue filled circles with error bars. We show the best-fitting model from D’Souza et al. (Reference D’Souza, Vegetti and Kauffmann2015), who estimated the GSMF by stacking images of galaxies with similar stellar masses and concentrations to correct Model magnitudes from the SDSS DR7, dark green solid line. Finally, we compare our result to Thanjavur et al. (Reference Thanjavur, Simard, Bluck and Mendel2016), who derived the GSMF using the analysis from Mendel et al. (Reference Mendel, Simard, Palmer, Ellison and Patton2014).

Our GSMF agrees well with previous determinations at the $\sim 10^{9.3}-10^{11}\ \text{M}_{\odot}$ range. At the high-mass end, it is shallower than previous determinations (e.g., Moustakas et al. Reference Moustakas2013) except to Bernardi et al. (Reference Bernardi, Meert, Sheth, Vikram, Huertas-Company, Mei and Shankar2013), who use Sérsic photometry from the SDSS DR7. As extensively discussed in Bernardi et al. (Reference Bernardi, Meert, Sheth, Fischer, Huertas-Company, Maraston, Shankar and Vikram2017), there are two systematic effects that could lead to differences when comparing to previous determinations from the literature; assumptions on mass-to-light ratios and estimations of galaxy surface brightness. In the case of Moustakas et al. (Reference Moustakas2013) and D’Souza et al. (Reference D’Souza, Vegetti and Kauffmann2015), who used cmodel and Model magnitudes, the comparison is not obvious due to systematic effects in both mass-to-light ratios and photometry (Bernardi et al. Reference Bernardi, Meert, Sheth, Fischer, Huertas-Company, Maraston, Shankar and Vikram2017). In the case of Mendel et al. (Reference Mendel, Simard, Palmer, Ellison and Patton2014) and Bernardi et al. (Reference Bernardi, Meert, Sheth, Vikram, Huertas-Company, Mei and Shankar2013), effects on photometry are not the dominant ones but mass-to-light ratios. Nonetheless, those differences are within the expected systematic effect, especially at the massive end, (Bernardi et al. Reference Bernardi, Meert, Sheth, Fischer, Huertas-Company, Maraston, Shankar and Vikram2017, see also Figure 11). We therefore conclude that when comparing to other previous determinations, the differences that we observe are consistent with the differences expected from systematic effects. Indeed, Figure 1 shows that most of the previous determinations are within our region of systematic errors. Thus, hereafter we will assume that our shift of $\pm0.15$ dex in the $M_{*}$ axis approximately captures systematics not only from stellar population models but also from photometry.

The middle panel of Figure 1 presents comparisons to some previous determinations from deep but small-volume samples. The purple dots with error bars are from Baldry et al. (Reference Baldry, Glazebrook and Driver2008), who used the SDSS NYU-VAGC low-z sample but did not include missing galaxies due to surface brightness incompleteness. In addition, we compare to Baldry et al. (Reference Baldry2012), who used the Galaxy And Mass Assembly (GAMA) survey for galaxies at $z<0.06$ , and complete down to $r=19.4$ mag for two-thirds of the galaxy sample and to $r=19.8$ for one-third of the sample. Finally, we reproduce the observed GSMF from Wright et al. (Reference Wright2017), who also used the GAMA survey to estimate the GSMF.

At low masses, our results are in excellent agreement with the GAMA $\mbox{GSMF}$ s. This is encouraging since the GAMA survey does not suffer from surface brightness incompleteness, at least within the stellar mass range that we are comparing our results. This is an indication that the surface brightness corrections described in Appendix C are able to recover the slope of the GSMF at low masses. Consistent with the values reported in Baldry et al. (Reference Baldry2012) and Wright et al. (Reference Wright2017), we find that the faint-end slope of the GSMF is $\alpha \approx -1.4$ , below we describe in more detail the fitting model for the GSMF. The above is also in good agreement with Sedgwick et al. (Reference Sedgwick, Baldry, James and Kelvin2019) who recently determined the low-mass end of the GSMF by identifying low surface brightness galaxies based on data of core-collapse supernovae. The authors used the IAC Stripe 82 legacy project (Fliri & Trujillo Reference Fliri and Trujillo2016) and the SDSS-II Supernovae Survey (Frieman et al. Reference Frieman2008).

At the massive end we notice, however, some apparent tension between our and the GAMA results. Effects due to cosmic variance (due to the small redshift and angular coverage of the GAMA sample) could explain those differences as well as systematics in the mass-to-light ratios. Indeed, we see that some of the data are within the systematic errors. In addition, note that Figure C5 from Appendix C shows that using the mass-to-light ratios from Taylor et al. (Reference Taylor2011), utilised in the Baldry et al. (Reference Baldry2012) GSMF, tend to underestimate the high-mass end of the GSMF.

3.2. Best-fitting model to the GSMF

To provide an analytic form to our GSMF, we choose to use a function composed of a Schechter function with a sub-exponential decreasing slope and a double power law function. Note that the resulting high-mass end of our GSMF is shallower than an exponential function, and, thus, better fitted to a power law (see also Tempel et al. Reference Tempel2014). The Schechter sub-exponential function is given by:

(21) \begin{equation}\phi_{*, {\rm S}}(M_{*})=\phi^{*}_S\ln 10 \left(\frac{M_{*}}{\mathcal {M}_S}\right)^{1+\alpha_S} \exp\left[{-\left(\frac{M_{*}}{\mathcal {M}_S}\right)^{\beta}}\right],\end{equation}

where $\phi^{*}_S$ is the normalisation parameter in units of Mpc^–3 dex^–1, $\alpha$ is the slope at the low-mass end, $\mathcal {M}_S$ is the characteristic mass, and $\beta$ is the parameter that controls the slope at the massive end; note that $\beta=1$ corresponds to a Schechter function. The double power law function is given by:

(22) \begin{equation}\phi_{*, {\rm D}}(M_{*}) =\phi^{*}_D\ln 10\left(\frac{M_{*}}{\mathcal {M}_D}\right)^{1+\alpha_D}\left[1 + \left(\frac{M_{*}}{\mathcal {M}_D}\right)^{\gamma}\right]^{\frac{\delta-\alpha_D}{\gamma}},\end{equation}

where $\phi^{*}_D$ is the normalisation parameter in units of Mpc^–3 dex^–1, $\alpha$ and $\delta$ control the slope at low and high masses, respectively, while $\gamma$ determines the speed of the transition between the low- and high-mass regimes; and $\mathcal {M}_D$ is the characteristic mass of the transition. Finally, the analytic form for fitting the observed GSMF is given by

(23) \begin{equation}\phi_{*, {\rm model}}(M_{*}) = \phi_{*, {\rm S}}(M_{*}) + \phi_{*, {\rm D}}(M_{*}),\end{equation}

where we assumed that $\mathcal {M}_S =\mathcal {M}_D $ .

We find the best-fit parameters $\vec{p}_{\rm GSMF} = (\phi_{S}^{*}, \alpha_{S}, \mathcal{M}_{S}, \beta, \phi_{D}^{*}, \alpha_{D}, \delta,\gamma)$ that maximise the likelihood function $\mathcal{L}\propto \exp({-\chi^2/2})$ using the Markov chain Monte Carlo (MCMC) method algorithm described in Rodríguez-Puebla et al. (2013). Here

(24) \begin{equation}\chi^2 = \sum_{i=1}^{N_{\rm obs}}\left(\frac{\phi^{i}_{*,{\rm SDSS}} - \phi_{*, {\rm model}}^i }{\sigma^{i}_{\rm SDSS}}\right)^2,\end{equation}

with $N_{\rm obs} $ as the number of observational data points of the GSMF each with an ith value of $\phi^{i}_{*,{\rm SDSS}}$ and an error of $\sigma^{i}_{\rm SDSS}$ . The ith value of our model is given by $\phi_{*, {\rm model}}^i$ .

We sample the best-fit parameters by running a set of 10 chains with $1\times 10^{5}$ MCMC models each. Table 1 lists the best-fit parameters. For our best-fitting model, we find that $\chi^2=85.42$ from a number of $N_{\rm obs} = 50$ observational data points. Our model consist of $N_{\rm p} = 8$ free parameters, thus the reduced $\chi^2$ is $\chi^2 / {\rm d.o.f.} = 2.03$ . The upper and middle panels of Figure 1 show our best-fitting model as the black solid line and the bottom panel shows the residuals as a function of $M_{*}$ . Our best-fitting model has an error of $\sim\!2\%$ at the range $M_{*}\sim\!2\times10^{9}-5\times10^{11}\text{M}_{\odot}$ and an error lower than $\sim\!10\%$ at the mass range $M_{*}\sim\!7\times10^{8}-1\times10^{12}\text{M}_{\odot}$ . For lower masses, errors can be up to $\sim\!20\%$ .

Table 1.

Best-fitting parameters for the GSMF Equations (21)–(23)

A valid question is how much we improve the analytic prescription when using a Schechter sub-exponential plus a double power law function model confronted to a double Schechter function model, commonly employed by previous authors (see e.g., Baldry et al. Reference Baldry2012; Wright et al. Reference Wright2017). We have explored this possibility but assuming a Schechter function, $\beta=1$ in Equation (21), and a Schechter sub-exponential function, that is, we are adding a extra degree of freedom due to the shallow decay at the high-mass end. Based on this alternative, we repeat our fitting procedure but this time finding that $\chi^2=662.817$ from a number of $N_{\rm obs} = 50$ observational data points. Now, our model consists of $N_{\rm p} = 6$ free parameters resulting in a reduced $\chi^2$ of $\chi^2 / {\rm d.o.f.} = 15.06$ . This is considerably worse when combining Schechter sub-exponential and double power law functions. Thus, hereafter, we will consider only the latter model.

3.3. The GSMFs of early- and late-type galaxies

Our main goal for this paper is to construct bivariate distributions as well as MF based on the observed gas mass CPDFs and the $\mbox{GSMF}$ s of early- and late-type galaxies. In this section, we determine the GSMF of early- and late-type galaxies from the SDSS DR7 spectroscopic sample with the public automated morphological galaxy classification by Huertas-Company et al. (Reference Huertas-Company, Aguerri, Bernardi, Mei and Sánchez Almeida2011).Footnote ⁱ The morphological classification in Huertas-Company et al. (Reference Huertas-Company, Aguerri, Bernardi, Mei and Sánchez Almeida2011) was determined based on support vector machine algorithms. Here we use their tabulated probabilities for each SDSS galaxy as being classified as an early type, P(E). For masses below the completeness of the SDSS DR7 sample, we use an extrapolation of the observed fraction of early-type galaxies. We will come back to this point later in this section.

From a catalogue of galaxies with visual morphological classification (UNAM-KIAS; Hernández-Toledo et al. Reference Hernández-Toledo, Vázquez-Mata, Martínez-Vázquez, Choi and Park2010), we find that galaxies with types $T\le0$ are mostly those with $P(E)>0.65$ , and those with $P(E)\le0.65$ correspond mostly to $T>0$ ; here T is the Fukugita et al. (Reference Fukugita2007) notation.Footnote ^j Based on the above, we consider as early-type galaxies those with a probability $P(E)>0.65$ , while late-type galaxies those with $P(E)\le0.65$ . We checked that our morphology definition between early- and late-type galaxies is consistent with the morphological classification based on the concentration parameter $c=R_{90}/R_{50}$ . That is, the division between early- and late-types is approximately at $c=2.85$ (see below and also, Hyde & Bernardi Reference Hyde and Bernardi2009; Bernardi et al. Reference Bernardi, Shankar, Hyde, Mei, Marulli and Sheth2010).

We calculate the SDSS DR7 GSMF of early- and late-type galaxies using the $1/\text{V}_{\rm max}$ method described in Appendix A. Figure 2 shows the corresponding $\mbox{GSMF}$ s of early and late types in the upper left and right panels, respectively. For comparison, we show the $\mbox{GSMF}$ s for red and blue galaxies based on a colour cut limit in the ${(g-r)^{0.0}-M_{*}}$ diagram. In this diagram, we find that a rough division criteria from blue to red galaxies is given by the colour limit of ${(g-r)^{0.0}=0.66}$ .Footnote ^k In the same figure, we compare our results to different determinations from the literature as we describe below. All the data in this figure have been renormalised to our cosmology.

Figure 2.

SDSS DR7 GSMFs for early- and late-type galaxies, left and right upper panels, respectively. Early- and late-type galaxies are defined as those with $P(E)>0.65$ ( $P(E)\le0.65$ ) from the tabulated probabilities of Huertas-Company et al. (Reference Huertas-Company, Aguerri, Bernardi, Mei and Sánchez Almeida2011). This is equivalent to morphological types that comprises E and S0 galaxies or $T\le0$ (Sa to Irr galaxies or $T>0$ ). We compare to various previous determinations from the literature as indicated by the legends, see also the text for details. Our determinations are in general in good agreement with previous determinations from SDSS spectroscopic samples, while a tension is evident with determinations from the GAMA survey. We also present our resulting GSMFs for blue and red galaxies. These GSMFs follow closely those by morphology from the GAMA survey. The bottom panel shows our number density-weighted fractions of early-type and red galaxies as a function of $M_{*}$ . Their corresponding best-fit models Equation (25) are shown with solid and dashed lines, respectively.

Recently, Moffett et al. (Reference Moffett2016a) visually classified morphologies in the GAMA survey and reported the GSMF for different morphologies. Here, we reproduce their GSMF from E to Sa galaxies as early types and the complement as late types. Contrary to our definition, Sa galaxies are included in the early-type group; this is because the authors report S0 and Sa galaxies as one morphology group. As shown in Figure 2, the GSMF of early-type galaxies from Moffett et al. (Reference Moffett2016a) results in an overabundance of low-mass galaxies compared to other studies. We reproduce the results from Thanjavur et al. (Reference Thanjavur, Simard, Bluck and Mendel2016) with bulge-to-total ratios of $B/T>0.8$ as early types and $B/T\leq0.8$ as late types. Thanjavur et al. (Reference Thanjavur, Simard, Bluck and Mendel2016) used the bulge-to-disc decomposition from Simard et al. (Reference Simard, Mendel, Patton, Ellison and McConnachie2011) SDSS DR7 spectroscopic sample, and stellar masses derived from Mendel et al. (Reference Mendel, Simard, Palmer, Ellison and Patton2014). We also include results from Kelvin et al. (Reference Kelvin2014). Similarly to Moffett et al. (Reference Moffett2016a), Kelvin et al. (Reference Kelvin2014) visually classified morphologies in the GAMA survey. We again use their GSMF from E to Sa galaxies for early types since the authors combined S0–Sa galaxies as in Moffett et al. (Reference Moffett2016a). The filled triangles with error bars show the GSMF from Bernardi et al. (Reference Bernardi, Shankar, Hyde, Mei, Marulli and Sheth2010) for galaxies with concentration parameter $c>2.86$ for early types and $c\leq2.86$ for late types.Footnote ^l

Finally, using the Nair & Abraham (Reference Nair and Abraham2010) morphology catalogue, who visually classified 14,034 spectroscopic galaxies from the SDSS DR4, we derive the GSMF for early-type galaxies.Footnote ^m We utilise their morphological notation and define early-type galaxies as those objects with $-5\le T\le0$ (E-S0s), equivalent to $T\le1$ in the Fukugita et al. (Reference Fukugita2007) notation. We additionally derive the GSMF with morphologies between $-5\le T\le1$ in the Nair & Abraham (Reference Nair and Abraham2010) notation, which include Sa galaxies.

In general, our results agree with previous determinations, especially with those from the SDSS spectroscopic samples. In contrast, the GSMF of early-type galaxies from the visual classification of the GAMA survey are systematically above our results at the low-mass end, $M_{*}\lesssim2\times10^{10}\text{M}_{\odot}$ , but closer to our classification based on galaxy colour. While it is not clear the reason of the differences outlined above (the inclusion of Sa galaxies as early types, environment, etc.), in Appendix C.4, we will discuss the impact of using galaxy colour instead of morphology when deriving the $\rm HI$ , $\rm H_{2}$ , cold gas, and baryonic MFs separated into two main galaxy populations.

Finally, the bottom panel of Figure 2 shows the resulting fraction of early-type galaxies as a function of stellar mass, $f_E\text{(}M_{*}\text{)}$ . In addition, we show the fraction of red galaxies when using our $g-r$ colour cut limit, $f_r\text{(}M_{*}\text{)}$ . We find that the fraction at which early-type galaxies is 50% is at $M_{*}\sim\!10^{11}\text{M}_{\odot}$ , while at $M_{*}\sim\!1.6\times10^{10}\text{M}_{\odot}$ and $M_{*}\sim\!8\times10^{11}\text{M}_{\odot}$ the fractions are 16% and 84%, respectively. For red galaxies, the fraction of 50% is at $M_{*}\sim\! 10^{10}\text{M}_{\odot}$ , while at $M_{*}\sim\!3\times10^{9}\text{M}_{\odot}$ and $M_{*}\sim\!10^{11}\text{M}_{\odot}$ the fractions are 16% and 84%, respectively. Note that the characteristic mass at which the fraction of early-type galaxies is 50% is a factor of $\sim\!10$ larger than for red galaxies. In general, $f_E\text{(}M_{*}\text{)}$ rises slower than the fraction $f_r\text{(}M_{*}\text{)}$ . In the same figure, we present the best-fit model to the data. After exploring different functions, we find that two sigmoid functions accurately describe the functionality of $f_E\text{(}M_{*}\text{)}$ or $f_r\text{(}M_{*}\text{)}$ :

(25) \begin{equation}f_{k}\text{(}M_{*}\text{)} = \frac{1-A}{1+e^{-\gamma_1(x_{C,1}+x_{0,1})}}+ \frac{A}{1+e^{-\gamma_2(x_{C,2}-x_{0,2})}},\end{equation}

where $k=E$ or r, $x_{C,i} = {M_{*}} / \mathcal{M}_{C,i}$ , with $i = 1,2$ . The best-fit parameters for the two fractions are listed in Table 2.

To derive the analytic model for the GSMF of early- and late-type galaxies, we use the best-fit model to our GSMF, Section 3.2, and the best-fit model for $f_E\text{(}M_{*}\text{)}$ . For masses below $5\times 10^8\ \text{M}_{\odot}$ we extrapolate $f_E\text{(}M_{*}\text{)}$ . This is an acceptable approximation since as seen in Figure 2, the fraction $f_E\text{(}M_{*}\text{)}$ tends to $\sim\!0$ below $M_{*}=10^9\ \text{M}_{\odot}$ . Recall that our main goal in this paper is to derive the MFs for $\rm HI$ , $\rm H_{2}$ , cold gas, and baryons by combining the observed $\rm HI$ and $\rm H_{2}$ CPDFs with the GSMF, both for early- and late-type galaxies, over a large mass range. Thus, at this point, we are in a position to determine these MFs.