# How do I create a galaxy stellar mass function?

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Suppose that I have an array of stellar masses of tens of thousands of galaxies, and that I also already know what the total "survey volume" is (in Mpc^3) within which all of those galaxies are contained. For context, my data is from a simulation of galaxy formation at redshift ~ 0, rather than from an observational simple -- so in principle I don't think I need to worry about incompleteness corrections at the faint (low-mass) end.

Given my array of stellar masses and the total survey volume in Mpc^3, how do I create and plot the "galaxy stellar mass function" that I always see people talking about? Isn't it just basically a histogram of the stellar mass array, and then divide the # of galaxies in every bin by the survey volume (it's almost just like a normalization constant)?

(Of course this question could equally apply to galaxy luminosity functions.)

Your approach is completely correct, just note three things:

### Logarithmic distribution

First, since the distribution of masses is logarithmic in nature (as is most other things), be sure to bin them logarithmically. Otherwise you will oversample (undersample) the bins at the low-(high-)mass end.

### Comoving densities

Second, to be able to compare mass functions at different redshifts, ones uses the comoving volume rather than the physical volume, such that the expansion of the Universe is factored out. The two are related as $$V_mathrm{com} = V_mathrm{phys}(1+z)^3$$.

### Damn you, little $h$!

Finally, observers and modellers tend to use a slightly different definition of the unit volume. Whereas observers usually use $$mathrm{Mpc}$$ for distances, and hence $$mathrm{Mpc}^{-3}$$ for number densities, if your galaxies come from a cosmological simulation where the cosmological parameters can when tuned at will, it is custom to factor out the Hubble constant $$H_0$$. In simulations, masses and distances are then measured in $$h^{-1}M_odot$$ and $$h^{-1}mathrm{Mpc}$$, respectively, so number densities are measured in $$h^3 mathrm{Mpc}^{-3}$$. Here $$hequiv H_0,/,(100,mathrm{km},mathrm{s}^{-1},mathrm{Mpc}^{-1})$$.

This is probably a reminiscence of the time when the Hubble constant was rather uncertain. Nowadays, in my opinion there is no need to do this, but since everybody does, it's difficult to go against the stream. For a discussion of this issue, see Croton (2013).

### Python code

Since you've tagged the question with python, I wrote this little snippet that should do the work (I randomly chose $$10^5,mathrm{Mpc}$$ as your survey volume; note also that in this example I don't factor out $$h$$):

import numpy as np import matplotlib.pyplot as plt M = np.loadtxt('Mstar.dat') #Read stellar masses in Msun logM = np.log10(M) #Take logarithm nbins = 10 #Number of bins to divide data into V = 1e5 #Survey volume in Mpc3 Phi,edg = np.histogram(logM,bins=nbins) #Unnormalized histogram and bin edges dM = edg[1] - edg[0] #Bin size Max = edg[0:-1] + dM/2. #Mass axis Phi = Phi / V / dM #Normalize to volume and bin size plt.clf() plt.yscale('log') plt.xlabel(r'$$log(M_star,/,M_odot)$$') plt.ylabel(r'$$Phi,/,mathrm{dex}^{-1},mathrm{Mpc}^{-3}$$') plt.plot(Max,Phi,ls='steps-post')

## The Galactic Stellar Halo

Originally it was thought that all globular clusters were part of the halo. Now, however, it is realized that two distinct populations of globulars exist. Old, metal-poor clusters ([Fe/H] < -0.8) are part of an extended, spherical halo, while (slightly) younger clusters with [Fe/H] > -0.8 are in a more concentrated and flattened distribution.

The less-metal-poor clusters have a scale height similar to that of the thick disk, and they may be associated. Other ideas have them related to the Galaxy's bulge instead.

Globular clusters are generally old, with ages ranging from 9-12 billion years. However, there's still a lot of uncertainty about absolute ages here.

How do we find field halo stars? Look at the space velocities of stars with respect to the Sun. If they are low, they are probably disk stars (like the Sun). If they are high, they are usually associated with the halo.

Like the GCs, the halo field stars are also metal-poor . This tells us something about the formation of the Galaxy!

If we add up all the mass in the field stars and the metal-poor GCs, we can come up with a rough density distribution for the Galaxy's stellar halo: Where n0 is about 0.2% of the thin disk's central density.

The total mass of the halo is about 10 8 - 10 9 M sun, about 1% of which is the globular clusters, and the rest in field stars. So there's not a lot of stuff in the stellar halo, but what is there holds a lot of information about the early history of the Galaxy!

Spectra of extreme metal-poor stars (solar, -4, -5.3, zero metals), courtesy ESO:

### Halo substructure

The distribution of stars in the Milky Way's halo is not smooth, but shows evidence for stellar streams.

Below: A plot showing the distribution of halo stars on the sky. In this plot, color is not the color of the star, but distance (blue nearer, red further).

On larger scales, these streams can be overlaid on other measurements of structure in the galactic halo, showing how the big SDSS stream continues around the whole galaxy:

(RA and dec are coordinates on the sky, like latitude and longitude on Earth)

## How Plural are Singularities?

As the joy from yesterday’s Nobel Prize announcement continues to linger far and wide, I’d like to lend words to my delight by talking a bit about black holes. And LIGO. Well, black holes seen(heard?) by LIGO.

Only a short while ago, in September 2015, the Laser Interferometer Gravitational-Wave Observatory (LIGO) made its maiden detection of the ripples travelling in spacetime, known as gravitational waves. The detection of these waves, a key prediction made by the Einstein’s Theory of General Relativity, has since revolutionalised the field of astronomy by presenting a novel avenue for us to understand the workings of the ever-so-elusive-and-vast cosmos.

In the discovery that LIGO made, it sensed the gravitational waves created by the merger of two distant black holes, each about as massive as 30 Suns. For scientists specializing in black hole physics, as well as others with a keen excitement about the field, this event immediately triggered the question: how many more events will we see, and how often?

Motivated by this curiosity, the authors of today’s paper have analyzed the gravitational wave signals from the first three binary black hole mergers detected by LIGO, and developed an understanding of their characteristics in the context of our current knowledge about galaxy formation. Based on what we know about the formation of stars, the relationship between their ages/metallicities and the mass of their host galaxy, as well the overall number density of galaxies in our universe, the authors infer the age and mass distribution of black holes in different types of galaxies. In their study, however, they assume that the extant population of black holes is attained solely from the death of massive stars, and does not feature any primordial black holes, allowing them to make confident estimates from reasonably well quantified galaxy observables.

How many black holes in store?

Through their study, Elbert et al. found out that the total number of black holes of all masses increases linearly with host galaxy mass for galaxies with stellar masses ≲10 10 solar masses (M) see Fig 1. For instance, the Milky Way would host up to 100 million black holes, 10 million of which would weigh about 30 M such as those detected by LIGO, whereas dwarf satellite galaxies like Draco orbiting the Milky Way, may be home to just about 100 black holes.

Figure 1. The number of stellar remnant black holes of different masses per galaxy increases linearly as a function of galaxy stellar mass for masses upto 10 10 M. (Figure 2. in the original paper)

Further incorporating the relationship between the mass and metallicity of a galaxy in their analysis, the authors predict that most low-mass black holes (

10 M) should reside in massive galaxies like our own. This is because larger galaxies have more metal-rich stars that undergo vigorous mass-loss over their lifetimes, thereby ending up as low mass black holes. Contrarily, dwarf galaxies predominantly host metal-poor massive stars that do not shed as much of their mass, and hence have more mass ending up into the black holes (

How often do they collide?

In the case of binary black hole mergers, it can often be difficult to say whether the pair of merging black holes was created way back in the past and took a long time to merge, or whether it was recently minted and merged soon after. To demystify this situation, the authors, in addition to the original black hole census, also sought to develop a framework for predicting the frequency of future merger events.

Depending on the probability that certain black holes will occur in binary systems and an estimate of the typical timescale of their merger, their results indicate that a range of merger efficiencies (0.1-1%), albeit very low, is needed to explain the characteristics of existing LIGO detections. Conversely, fixing the value of merger efficiency would necessitate a range of merger timescales to accommodate the data. Constraining the value of both the merger efficiency and merger timescales would require knowledge about the size of host galaxy population (massive vs. dwarf).

For a nominal efficiency of 1% applied to the current detections, the results from this study indicate a merger density of 12–213 per cubic gigaparsec, or a merger timescale of ≲5 Gyrs, for

50 M black holes. Such a high merger density suggests that mergers involving

50 M black holes should be detected by LIGO within a decade.

With another merger already under LIGO’s belt since this study was published, impending detections seem galore. There is no doubt that enthusiastic scientists all over the world will have a lasting opportunity to shine light on the exact physics driving the exotic phenomena that are binary black hole mergers.

## What Is the Initial Mass Function? (with picture)

The initial mass function (IMF) was first derived in 1955 by Edwin Saltpeter, an Austrian astrophysicist, and is a method of calculating the range of different masses for stars that will form out of condensing gasses in space. It is a form of probability distribution that uses complex mathematics and physics equations with a base value of one solar mass, which represents the mass of Earth's Sun as a stepping off point for the range of other stars that will form. The premise of the initial mass function in stellar astronomy is that it is much more common and likely for stars of low mass to form in space than it is for stars of high mass, with stars that have approximately 0.5 solar masses being the most common in the Milky Way galaxy as of 2011. Despite this fact, the rarest of stars, at about 60 solar masses in size or higher, contribute most of the visible light to the Milky Way galaxy.

According to most astronomy estimates as of 2011, there exists somewhere between 200,000,000,000 and 400,000,000,000 stars in the Milky Way galaxy. The initial mass function predicts that the probability for the majority of these stars is that they are 0.9 solar masses or less, while less than 1% of them make up sizes ranging from 8 to 120 solar masses. The IMF calculates masses based on when each star first formed, and most stars begin as dwarf stars of only 0.085 to 0.8 solar masses. As these main sequence stars age, they tend to lose mass and gain volume.

Despite widely varying conditions in substellar regions of space where stars form, the power laws of the initial mass function have proven to be true. This means that, whether star formation is occurring in small molecular clouds of gas or in dense star clusters, the same distribution of star ranges arises regardless. These observations conflict with star formation theories as of 2011 due to conditions such as the fact that, in a metal dense region of space, the star distribution should include a greater percentage of massively large stars.

It is estimated that, in about 5,000,000,000 years, the Sun itself will expand as it burns away its hydrogen fuel and begins to fuse helium to heavier elements. At this stage, the Sun will fill a volume of space reaching to the orbit of the Earth for about 20% of its total life span, and retain 50% of its previous mass as a red giant. As small stars like the Sun age and lose mass in the process, they increasingly bias the initial mass function more towards the small mass end of the spectrum, in large part because there are far more small stars in existence.

## Acknowledgements

F.S. thanks D. Weinberg, D. Baron, P. Behroozi, G. Calderone, B. Davis, F. Fiore, P. Gandhi, S. Hoenig, C. Knigge, C. Li, J. Miralda-Escudé, B. Moster, M. Powell, R. Vasudevan, C. Villforth and G. Yang for discussions and input, and acknowledges partial support from a Leverhulme Trust Research Fellowship and the European Union’s Horizon 2020 programme under the AHEAD project (grant agreement no. 654215). V.A. acknowledges funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement no. 749348. M.B. acknowledges partial support from NSF grant AST-1816330. A.L. is supported by PRIN MIUR 2017 prot. 20173ML3WW_002 ‘Opening the ALMA window on the cosmic evolution of gas, stars and supermassive black holes’. M.K. acknowledges support from DLR grant 50OR1904. L.Z. and P.J.G. acknowledge funding from the Science and Technology Facilities Council (STFC).

## 3. ANALYSIS

### 3.1. Estimate of Redshift and Stellar Mass

In order to estimate the photometric redshift and stellar mass of the sample galaxies, we performed SED fitting of the multiband photometry described above (UBVizJHK, 3.6 μm, 4.5 μm, and 5.8 μm) with population synthesis models. We adopted the standard minimum χ 2 method for the fitting procedure. The resulting best-fit redshift (i.e., photometric redshift) is used for objects without spectroscopic identifications and the best-fit stellar mass-to-luminosity (M/L) ratio is used to calculate the stellar mass. We adopted spectroscopic redshifts (if available) from the literature (Cohen et al. 2000 Cohen 2001 Dawson et al. 2001 Wirth et al. 2004 Cowie et al. 2004 Treu et al. 2005 Chapman et al. 2005 Reddy et al. 2006a Barger et al. 2008), including those from our NIR spectroscopic observation of

20 star-forming BzK galaxies (Daddi et al. 2004) at z

2 with Subaru/MOIRCS (T. Yoshikawa et al. 2009, in preparation). For these objects, the SED fitting was performed fixing the redshift to each spectroscopic value.

In this study, three models, i.e., GALAXEV (Bruzual & Charlot 2003), PEGASE version 2 (Fioc & Rocca-Volmerange 1997), and Maraston (2005) model, were used as SED templates. Comparisons among the results with different SED template sets allow us to check the systematic effects of these models on the estimate of redshift and stellar mass. We also used a public photometric redshift code, EAZY (Brammer et al. 2008) for an independent check of the photometric redshift. In all SED models, Salpeter IMF (Salpeter 1955) with lower and upper mass limits of 0.1 and 100 M is adopted for easy comparison among the results with the models and those in other studies. The details of the model templates from each population synthesis model are given in the following.

In Figure 1, we compare the photometric redshifts with a sample of 2102 spectroscopic redshifts from the literature described above. The photometric redshift accuracy and the fraction of the catastrophic failure with δz/(1+zspec)>0.5 for each SED model are (δz/(1+zspec) = −0.009 ± 0.077, 4.0% outliers) for GALAXEV, (−0.002 ± 0.103, 4.0%) for PEGASE2, (−0.007 ± 0.082, 3.9%) for the Maraston model, and (+0.003 ± 0.098, 4.7%) for EAZY, respectively. Although the photometric redshift accuracy is relatively good in all cases, there are some systematic differences among the models especially at 1.5 < z < 3.0 in Figure 1. We check the effects of these differences of the photometric redshifts on the SMF in Section 3.4.

Figure 1. Photometric redshift vs. spectroscopic redshift for galaxies with spectroscopic redshifts from the literature. Four panels show the different population synthesis models used as SED templates in the photometric redshift estimate.

Table 1 lists the number of objects in each redshift bin for different SED models. These redshift bins, of which width is sufficiently larger than the typical photometric redshift errors, were defined so as to include a reasonable number of galaxies in the bin for calculating the SMF. The comoving volumes are 8.5 × 10 4 Mpc 3 (2.5 × 10 4 Mpc 3 for the deep field) at 0.5 < z < 1.0, 1.4 × 10 5 Mpc 3 (3.7 × 10 4 Mpc 3 ) at 1.0 < z < 1.5, 3.4 × 10 5 Mpc 3 (9.2 × 10 4 Mpc 3 ) at 1.5 < z < 2.5, and 3.4 × 10 5 Mpc 3 (9.3 × 10 4 Mpc 3 ) at 2.5 < z < 3.5, respectively. The parenthesis in Table 1 represents the number of objects with spectroscopic redshift. The fraction of spectroscopically identified sources is relatively high, thanks to the extensive spectroscopic surveys in this field.

Table 1. Sample Size in Each Redshift Bin

Redshift Model Templates Wide (K < 23, 103 arcmin 2 ) a Deep (K < 24, 28 arcmin 2 ) a
z = 0.5–1.0 GALAXEV 1945 (859) 902 (311)
PEGASE2 1860 (859) 836 (311)
Maraston 2118 (859) 1023 (311)
EAZY 2141 (859) 991 (311)
z = 1.0–1.5 GALAXEV 1426 (353) 635 (105)
PEGASE2 1319 (353) 608 (105)
Maraston 1403 (353) 573 (105)
EAZY 1141 (353) 464 (105)
z = 1.5–2.5 GALAXEV 1306 (209) 666 (75)
PEGASE2 1380 (209) 677 (75)
Maraston 1015 (209) 488 (75)
EAZY 1334 (209) 774 (75)
z = 2.5–3.5 GALAXEV 487 (95) 366 (57)
PEGASE2 507 (95) 370 (57)
Maraston 427 (95) 302 (57)
EAZY 494 (95) 356 (57)

Note. a Number in the parenthesis indicates objects with spectroscopic redshift.

As discussed in previous studies (e.g., Papovich et al. 2001 Kajisawa & Yamada 2005 Shapley et al. 2005), the uncertainty of the stellar mass is smaller than other parameters such as stellar age, star formation timescale, metallicity, and dust extinction. While the broadband SEDs are degenerated with respect to these parameters of the stellar populations (stellar age, star formation timescale, metallicity, and dust extinction), the stellar M/L ratio (and therefore stellar mass) is much less affected by the degeneracy because the effects of these parameters on the stellar M/L ratio tend to be canceled out with each other. Figure 2 shows the uncertainty of the stellar mass estimated from a χ 2 map in SED fitting with the GALAXEV model for each object. For sources without spectroscopic identifications, we also varied redshift as a free parameter in the calculation of the χ 2 map to take into account the photometric redshift error. The uncertainty increases with decreasing stellar mass and increasing redshift. The stellar mass errors become

0.3–0.4 dex at the limiting stellar mass (the vertical dashed lines in the figure) described in the following subsection.

Figure 2. Uncertainty of the estimated stellar mass as a function of stellar mass in each redshift bin. Red circles represent the wide sample and blue circles show the deep sample. Vertical dashed lines show the limiting stellar mass described in the text for the wide (red) and deep (blue) samples, respectively. Open squares represent the median values at each stellar mass for all sample. For objects without spectroscopic redshift, the photometric redshift error is taken into account in the estimate of stellar mass uncertainty (see the text).

### 3.2. Stellar Mass-limited Sample

The K-band magnitude-limited sample does not have a sharp limit in stellar mass even at a fixed redshift, because the stellar M/L ratio at the observed K band varies with different stellar populations. We used the rest-frame UV color distribution as a function of stellar mass in each redshift bin to estimate the limiting stellar mass above which most of the galaxies are expected to be brighter than the magnitude limits and detected in the K-band image. Since rest-frame UV is correlated with stellar M/L ratio (e.g., Rudnick et al. 2003 Marchesini et al. 2007), one can predict the mass dependence of stellar M/L and then estimate the effect of the magnitude limit on the stellar mass distribution.

Figure 3 shows the rest-frame UV distribution in each redshift bin for the wide sample (top panels) and the deep sample (middle panels). A dashed line in each panel represents the K-band magnitude limit (K = 23 for the wide sample and K = 24 for the deep one). All objects with stellar mass larger than this line (on the right side of the line in the figure) at each UV value are brighter than the magnitude limit. In order to calculate the line, we used the stellar M/L ratio and the rest-frame color of the GALAXEV model with various SFHs, dust extinction, and metallicity. The maximum mass was selected from the possible range of the stellar mass for the models with K = 23 (or 24 for the deep sample) and each UV color in order to depict the dashed line in the figure. Dashed-dot lines in Figure 3 represent the 90 percentile of UV color at each stellar mass. We adopted the point where the lines of the magnitude limit and 90 percentile of UV color cross each other as the limiting stellar mass. Above this limiting mass, more than 90% of objects are expected to be brighter than the K-band magnitude limit.

Figure 3. Top and middle: rest-frame UV color distributions of wide (top) and deep (middle) samples for each redshift bin. Dashed lines represent the K-band magnitude limits (K = 23 for the wide sample and K = 24 for the deep one) at the central redshift of each bin. Dashed-dot curves show the 90 percentile of UV color at each stellar mass. Bottom: comparison of the 90 percentile of UV color between the wide (solid line) and deep (short dashed line) samples. Long-dashed line and dashed-dot line show the K-band magnitude limits for the wide and deep samples.

The bottom panels in Figure 3 show the comparison of the 90 percentiles of UV color for the wide and deep samples. The 90 percentiles for both the samples agree well with each other even near the limiting mass of the wide sample. This suggests that the incompleteness near the magnitude limit of the wide sample does not strongly affect the UV color distribution, although some red galaxies might be missed on the left-hand side of the long-dashed line.

Figure 4 shows the calculated limiting stellar mass as a function of redshift for the wide and deep samples. We also estimated these mass limits for the cases with PEGASE2 and Maraston models as well as the case with GALAXEV. As seen in Figure 3 and other previous studies (e.g., Kajisawa & Yamada 2005 Kajisawa & Yamada 2006 Labbé et al. 2005 Taylor et al. 2009), less massive galaxies tend to have bluer rest-frame color even at high redshift. Such a mass-dependent color distribution can be seen well above the K-band magnitude limit up to at least z

2.5 in Figure 3. Since the bluer color of low-mass galaxies indicates a lower M/L ratio, we can detect galaxies down to the relatively lower mass limit with high completeness compared with, for example, the mass limit based on the M/L ratio of the passively evolving models, which is used in other previous studies (e.g., Dickinson et al. 2003 Fontana et al. 2004). On the other hand, our "wide" field data are relatively shallow for galaxies at z > 3 (top right panel in Figure 3) and the completeness is relatively low even at high mass, where galaxies tend to have red rest-frame colors (high M/L ratios). We use objects with the stellar mass larger than these mass limits at each redshift to estimate and discuss the SMF in the following.

Figure 4. Stellar mass limit as a function of redshift for the wide and deep samples. Three panels show the results with the different SED models. Solid lines show the wide sample and dashed lines show the deep one.

### 3.3. Deriving the Stellar Mass Function

The SMF of galaxies was derived with the non-parametric 1/Vmax formalism and the parametric STY method (Sandage et al. 1979). Both methods are commonly used to estimate the luminosity function and SMF of galaxies.

In the Vmax method, Vmax was calculated with the best-fit model SED template for each galaxy. For each best-fit SED, we estimated the K-band apparent magnitude as a function of redshift, taking into account both the luminosity distance and K correction. Then we determined the maximum redshift, zmax above which the object becomes fainter than the K-band magnitude limit (K = 23 for the wide sample or K = 24 for the deep sample). Vmax is a comoving volume integrated from the lower limit of each redshift bin to zmax or the upper limit of the bin (the smaller of these two). Then 1/Vmax estimates were used to calculate the number density of galaxies in each mass bin.

In the STY method, assuming the Schechter function form (Schechter 1976) for the SMF, we estimated best-fit values of the Schechter parameters (α, M*, *). The limiting stellar mass Mlim(z) (described in the previous subsection) for the redshift of each object was used to calculate the probability that the object has the observed stellar mass M as

Here (M) is the SMF represented by the Schechter function. We searched for the values of the Schechter parameters (α, M*, *) which maximize the likelihood L = ∏p, the products of the probability densities for the objects with the stellar mass larger than Mlim(z) in each redshift bin. Both the wide and deep samples were used simultaneously in the maximum likelihood technique.

### 3.4. Evolution of the Stellar Mass Function

Figure 5 shows the SMF of galaxies in the different redshift bins for the different SED models. The results of the Vmax method and the best-fit Schechter function estimated with the STY method are plotted in each panel. Error bars are based on the Poisson statistics. Dashed lines show the local SMF derived from the 2dF and Two Micron All Sky Surveys (Cole et al. 2001) with the small correction for the "maximum age" method as described in Fontana et al. (2004). In Figure 6, we plot the combined wide and deep complete data (same as the solid symbols in Figure 5) for the different SED models with different symbols in the same panel.

Figure 5. Evolution of the stellar mass function of galaxies in the MODS field. From top to bottom the panels show the results with the different population synthesis models. Circles and squares show the SMF calculated with the 1/Vmax formalism for the wide and deep samples. Open symbols indicate data points located below the limiting stellar mass, where the incompleteness could be significant. Solid symbols show data points above the limiting stellar mass. The results of the deep sample are plotted by shaded symbols at stellar mass where the wide sample is also above the limiting mass. Error bars are based on the Poisson statistics. The solid lines show the results calculated with the STY method for the all (wide and deep) samples. The best-fit Schechter parameters are also shown in each panel. For reference, the local SMF of Cole et al. (2001) is shown as the dashed line.

Figure 6. Evolution of stellar mass function for different SED models. For each case, only data points above the limiting stellar mass are plotted. At stellar mass where the wide and deep samples are above the limiting mass, the data points for the wide sample are plotted (the same ones as solid symbols in Figure 5). The dashed line shows the local SMF of Cole et al. (2001).

Figures 5 and 6 show that the SMFs obtained from different samples and SED models are in good agreement, although there are some systematic differences among the SMFs. The number densities for the deep sample are systematically larger (by

0.1 dex) than those for the wide sample at 0.5 < z < 1.0 for all SED models. Our deep field is centered at the HDF-N field where the extensive spectroscopic surveys revealed the large-scale structures at z = 0.85 and z = 1.02 (Cohen et al. 2000 Wirth et al. 2004). These filaments or clumps around the HDF-N could cause the slightly larger number density in our deep field, and such differences can be considered as the possible field-to-field variance. Slighter excess in the deep field can also be seen at 2.5 < z < 3.5 in most SED-model cases, and this may also be due to large-scale structures. On the other hand, the systematic differences among the different SED models in Figure 6 seem to be larger. The number density of galaxies for the Maraston model is systematically smaller by

0.5 dex at > 10 11 M) than those for the GALAXEV and PEGASE2 models especially at z > 1.5. Since the differences are also seen between the EAZY+Maraston and EAZY+GALAXEV/PEGASE2, where the same redshifts of EAZY are used, the differences of the estimated stellar M/L ratio among the SED models cause the systematic differences of the number density. For example, Maraston et al. (2006) performed the broadband SED fitting of relatively young (

0.2–2 Gyr stellar age) galaxies at 1.4 < z < 2.7 with the Maraston and GALAXEV models and reported that the Maraston model gives systematically younger age and lower stellar mass (

60%) than the GALAXEV model. Such difference of the estimated stellar mass seems to explain the differences seen in Figure 6. If the fraction of young galaxies becomes larger at high redshift, the larger differences in the SMFs at z > 1.5 could also be explained because the contribution of TP-AGB stars is expected to be significant in the relatively young ages (

0.2–2 Gyr, Maraston 2005 Maraston et al. 2006).

We can see general evolutionary features in Figures 5 and 6 in spite of the field variance and the systematic differences among the SED models mentioned above. We note, first, that the overall number density decreases gradually with redshift in all cases. While the number density of galaxies at 0.5 < z < 1.0 is similar with that in the local universe, the number density at 2.5 < z < 3.5 is about an order of magnitude smaller than the local value. Figure 7 and Table 2 show the best-fit Schechter parameters (α, M*, *) estimated with the STY method. The redshift evolution of the overall number density can be seen as the decrease of the normalization of the SMF *. * decreases down to

50% of that in the local universe at z

3. Similar evolution of the SMF is also seen in previous studies of general fields (e.g., Fontana et al. 2006 Pérez-González et al. 2008 Marchesini et al. 2009).

Figure 7. Best-fit Schechter parameters as a function of redshift. Different symbols represent the results with different population synthesis models. Data points for different SED models are plotted with small horizontal offsets for clarity. Those of the local SMFs of Cole et al. (2001) are also shown for reference.

Table 2. Best-fit Schechter Parameters Obtained with the STY Method

SED Model Redshift Bin α log10M*(M) log10* (Mpc −3 )
GALAXEV z = 0.5–1.0 −1.26 +0.03 −0.03 11.33 +0.10 −0.07 −2.79 +0.07 −0.08
z = 1.0–1.5 −1.48 +0.04 −0.04 11.48 +0.16 −0.13 −3.40 +0.13 −0.15
z = 1.5–2.5 −1.52 +0.06 −0.06 11.38 +0.14 −0.12 −3.59 +0.14 −0.16
z = 2.5–3.5 −1.75 +0.15 −0.13 11.42 +0.40 −0.24 −4.14 +0.34 −0.51
PEGASE2 z = 0.5–1.0 −1.21 +0.03 −0.02 11.31 +0.07 −0.08 −2.73 +0.07 −0.06
z = 1.0–1.5 −1.32 +0.04 −0.04 11.36 +0.13 −0.10 −3.16 +0.10 −0.11
z = 1.5–2.5 −1.45 +0.06 −0.06 11.32 +0.13 −0.10 −3.51 +0.12 −0.15
z = 2.5–3.5 −1.59 +0.13 −0.14 11.39 +0.32 −0.20 −3.98 +0.26 −0.40
Maraston z = 0.5–1.0 −1.33 +0.02 −0.03 11.43 +0.12 −0.10 −3.04 +0.08 −0.10
z = 1.0–1.5 −1.42 +0.04 −0.04 11.31 +0.14 −0.12 −3.36 +0.12 −0.13
z = 1.5–2.5 −1.35 +0.07 −0.08 10.98 +0.12 −0.10 −3.40 +0.12 −0.14
z = 2.5–3.5 −1.58 +0.19 −0.18 11.03 +0.31 −0.21 −3.93 +0.30 −0.43
EAZY + GALAXEV z = 0.5–1.0 −1.30 +0.03 −0.03 11.37 +0.10 −0.08 −2.84 +0.07 −0.08
z = 1.0–1.5 −1.35 +0.04 −0.05 11.43 +0.15 −0.11 −3.28 +0.11 −0.14
z = 1.5–2.5 −1.58 +0.05 −0.06 11.50 +0.18 −0.13 −3.73 +0.15 −0.20
z = 2.5–3.5 −1.63 +0.14 −0.15 11.33 +0.32 −0.20 −3.99 +0.28 −0.42
EAZY + PEGASE2 z = 0.5–1.0 −1.30 +0.03 −0.02 11.45 +0.09 −0.09 −2.87 +0.08 −0.07
z = 1.0–1.5 −1.29 +0.04 −0.05 11.41 +0.13 −0.10 −3.19 +0.09 −0.13
z = 1.5–2.5 −1.48 +0.04 −0.05 11.48 +0.14 −0.12 −3.62 +0.14 −0.15
z = 2.5–3.5 −1.57 +0.15 −0.14 11.37 +0.32 −0.22 −4.00 +0.28 −0.39
EAZY + Maraston z = 0.5–1.0 −1.34 +0.02 −0.03 11.45 +0.13 −0.09 −3.06 +0.07 −0.11
z = 1.0–1.5 −1.38 +0.04 −0.04 11.39 +0.17 −0.12 −3.42 +0.12 −0.14
z = 1.5–2.5 −1.52 +0.06 −0.07 11.12 +0.14 −0.11 −3.55 +0.13 −0.17
z = 2.5–3.5 −1.59 +0.18 −0.17 10.98 +0.27 −0.20 −3.84 +0.28 −0.38

Second, we found the mass-dependent evolution of the SMF. The evolution of the number density of low-mass galaxies with Mstar

10 9 –10 10 M is smaller than that of the massive galaxies with stellar mass

10 11 M. While the number density of galaxies with

10 11 M at 2.5 < z < 3.5 is smaller by a factor of

20 than the local value, that of the galaxies with

5 × 10 9 M is smaller by only a factor of

6 at the same redshift. The shape of the SMF at 0.5 < z < 1.0 is similar with that in the local universe, and it becomes steeper with redshift at z > 1. This can be seen in Figure 7 and Table 2 as a steepening of the low-mass slope α with redshift. α decreases with redshift gradually from α = −1.29 ± 0.03(±0.04) at 0.5 < z < 1.0 to α = −1.48 ± 0.06(±0.07) at 1.5 < z < 2.5 and α = −1.62 ± 0.14(±0.06) at 2.5 < z < 3.5. The quoted errors are statistical errors estimated from the maximum likelihood method. The values in parenthesis show the uncertainty due to the different SED models.

The uncertainty of the Schechter parameters becomes larger with redshift, especially at 2.5 < z < 3.5, because of the larger limiting mass at higher redshift and of the small number of galaxies even at relatively high mass due to the evolution of the overall number density mentioned above. Nonetheless, since the uncertainty at z < 2.5 is rather small by virtue of our deep and wide NIR data, the evolution of the low-mass slope of the SMF between 0 z 3 is found to be significant. Figure 8 shows the best-fit Schechter parameters and their uncertainty for the different SED models in the M*–α plane, which represents the evolution of the shape of the SMF. In all cases, the evolution of α is significant, although there is degeneracy between M* and α especially at 2.5 < z < 3.5, where we can reach only to the relatively high stellar mass.

Figure 8. Evolution of the Schechter parameters in M*–α plane for the different SED models. Crosses show the best-fit values determined with the STY method. 1σ (solid) and 2σ (dashed) error contours are also shown.

On the other hand, the characteristic mass M* shows no significant evolution except for the results with the Maraston model. The M* values at 0.5 < z < 3.5 are similar with or slightly larger than those in the local universe. No significant evolution for M* is also seen in previous studies (Fontana et al. 2006 Pozzetti et al. 2007 Marchesini et al. 2009). For the Maraston model, M* becomes smaller by a factor of

2–2.5 at z > 1.5. As in the above discussion of the overall number density, this can be explained by the systematically lower stellar M/L ratio of the Maraston model because the same result is also seen when the same photometric redshifts (EAZY) are used. While passively evolving galaxies dominate the massive end (10 11 M) of the SMF at z 1 (e.g., Juneau et al. 2005 Borch et al. 2006 Vergani et al. 2008 Ilbert et al. 2009), many massive star-forming (i.e., relatively young) galaxies have been found at z

2 (e.g., Daddi et al. 2007 Papovich et al. 2006 Borys et al. 2005 Shapley et al. 2004). It is possible that TP-AGB stars contribute to the SED of massive galaxies significantly only at z > 1.5. Therefore, the Maraston model would give systematically lower stellar mass for these massive galaxies.

### 3.5. Possible Biases for the Evolution of the Low-mass Slope

Here we investigate possible biases for the results in the previous subsection, in particular, systematic effects which could cause the steepening of the low-mass slope at high redshift.

The larger limiting mass at higher redshift causes the degeneracy between M* and α as mentioned in the previous section. Since low-mass galaxies near the limiting mass tend to be faint in each band, the photometric errors are larger, which results in the large uncertainty in their photometric redshift. The large errors of the photometric redshifts of faint objects might lead to the systematic increase of the low-mass galaxies at high redshift because the redshift distribution of the K-selected sample has a peak around z

1 and a tail to higher redshift. In order to evaluate the effect on the low-mass slope, we performed a Monte Carlo simulation, assuming no evolution of the shape of the SMF (i.e., constant M* and α). At first, we constructed mock catalogs with the observed M* and α at 0.5 < z < 1.0. For the normalization of the SMF, we assumed * evolving as *(z) ∝ (1 + z) −2 so that the redshift distribution of the mock sample is consistent with our observation. Even if we assume *(z) ∝ (1 + z) −1 or *(z) ∝ (1 + z) −3 , the results shown in the following do not change significantly. Stellar mass and redshift of mock objects were randomly selected from the ranges of 10 8 M< Mstar < 10 12 M and 0 < z < 6, using the probability distribution estimated from the assumed SMF and the corresponding comoving volume of our survey at each redshift. For each mock object, we randomly extract an object from the observed sample with similar mass and redshift (allowing duplicate) and adopted its observed multiband photometry. The observed multiband photometry was extracted from the catalog that contains all sources detected on the K-band image (including K > 23 or K > 24 objects) in order to take into account the scattering of objects fainter than the magnitude limits. Then we added random offsets to the multiband photometry according to the measured photometric errors, and adopted the mock object if the resulting K-band magnitude of the object was brighter than the magnitude limits (K < 23 for the wide sample and K < 24 for the deep sample). We repeated this procedure and made the mock catalogs with the same sample sizes as the observed wide and deep samples. The same SED-fitting procedure as for the observed one was performed in order to estimate the photometric redshift and stellar mass of the mock objects. For objects with spectroscopic identification, redshifts are fixed to the spectroscopic values. We performed 200 simulations and calculated the best-fit Schechter parameters in each simulation with the STY method.

Figure 9 shows the results of the simulation in the case with GALAXEV (the results for the other models are similar). A relatively large scatter of α is seen in the highest redshift bin, which probably reflects the large degeneracy between M* and α due to the large limiting mass. Furthermore, the simulated α distributes around a systematically steeper (by

0.1–0.15) value than the assumed one at 2.5 < z < 3.5, while the simulated values tend to be slightly flatter (by

0.1) in lower redshift bins. However, since the observed evolution of α is much stronger than the systematic effects in Figure 9, the observed steepening of the low-mass slope at high redshift is significant, especially at 1 < z < 2.5.

Figure 9. Monte Carlo simulation for the effects of photometric redshift uncertainty on the shape (M* and α) of the stellar mass function with the GALAXEV model. The large square in each redshift bin shows the observed values at 0.5 < z < 1.0, and these are assumed not to evolve with redshift in the simulation. Small circles show the results of 200 simulations (see the text for details). Observed M*–α values (cross) and 1σ (solid) and 2σ (dashed) error contours are also shown for each redshift bin.

In the simulation, the random offsets added to the multiband photometry and the recalculation of the photometric redshift include the effect of the catastrophic failure. We extracted the mock objects whose photometric redshift was changed catastrophically (δz/(1 + z)>0.5) by the random offsets and checked the effect of these objects on the resulting SMF. Figure 10 shows the fractional increase and decrease of the number of galaxies due to the catastrophic failure of the photometric redshift as a function of stellar mass in each redshift bin. At 0.5 < z < 1.5, there is

10%–20% decrease near the limiting stellar mass, while there is only negligible effect of the catastrophic failure at Mstar 10 10 M. Most contamination from z < 0.5 or z > 1.5 occurs only at the stellar mass lower than limiting mass and it is not plotted in the figure. About 10%–20% decrease near the limiting stellar mass might cause a slightly flatter low-mass slope seen in Figure 9, but the effect is relatively small (

0.1 dex decrease in the number density). At z > 1.5, the effect of the catastrophic failure is similar or even smaller than that at low redshift. Furthermore, the contamination from lower redshift and the dropout from the redshift bin tend to be canceled out with each other, which results in the negligible effect on the SMF.

Figure 10. Effect of the catastrophic failure of the photometric redshift on the SMF in the Monte Carlo simulation. Upward and downward triangles show the fractional increase and decrease of the number of galaxies as a function of stellar mass in each redshift bin due to the objects whose photometric redshift was changed catastrophically (δz/(1 + z)>0.5) by the random offsets of the multiband photometry. Upward (downward) triangles represent the fraction of the objects which enter into (drop out from) the redshift bin due to the catastrophic failure. Data points and error bars represent the median value and 68 percentile interval of the 200 simulations. Vertical lines show the limiting stellar mass for the wide (solid) and deep (dashed) samples.

In Figure 11, we also show the results of the same simulation on the α–* plane to investigate the effect on the normalization. The systematic effect on the normalization of the SMF due to the photometric redshift errors seems to be relatively small, although the degeneracy between the parameters makes direct comparison difficult.

Figure 11. Effect of photometric redshift uncertainty on the normalization (*) of stellar mass function. Large squares and small circles are the same as those in Figure 9, but different colors represent different redshift bins. The normalization is assumed to evolve with redshift as *(z) ∝ (1 + z) −2 (see the text).

Next, we discuss a possibility of the over-deblending of faint objects with relatively low S/N ratio near the detection limit. Although the K band, where we performed the source detection, corresponds to the rest-frame B band even at z = 3.5, the morphological K correction could enhance the over-deblending at high redshift because galaxies tend to show the patchy appearance in shorter wavelengths due to the dominance of young stars and the dust extinction (e.g., Kuchinski et al. 2001 Rawat et al. 2009). Figure 12 shows the fraction of the objects with the deblending flag by SExtractor as a function of stellar mass in each redshift bin for the wide and deep samples. We cannot see the mass nor redshift dependence of the fraction of the deblended objects. We conclude that the effect of deblending does not cause the evolution of the low-mass slope.

Figure 12. Fraction of deblended objects as a function of stellar mass in each redshift bin for the wide (top) and deep (bottom) samples. The solid line shows all K-selected galaxies in each redshift bin and the shaded histogram represents deblended objects. The dashed-dot line shows the fraction of the deblended objects. Vertical long-dashed line shows the limiting stellar mass. The results with the GALAXEV model are shown.

Finally, we estimated the stellar mass of galaxies in the highest redshift bin with the GALAXEV templates with two-component (old and young) SFHs. If the old stellar population is hidden by recent star formation, the stellar mass could be underestimated especially for relatively blue low-mass galaxies at high redshift, where the S/N ratio tends to be low (e.g., Papovich et al. 2001 Drory et al. 2005). We used the exponentially decaying star formation models (young component) with an old population component. For the young population, free parameters are stellar age, star formation timescale τ, color excess, metallicity (same as for one-component SFH). For the old component, we limited the star formation timescale and stellar age to shorter and older values than the young population, respectively, and assumed no dust extinction. Figure 13 shows the comparison of the stellar masses estimated with one- and two-component models for galaxies at 2.5 < z < 3.5. No significant systematic difference of the stellar mass can be seen. The scatter is consistent with the uncertainty of the stellar mass of these galaxies shown in Figure 2, although the stellar mass with the two-component model is slightly larger for a small fraction of galaxies at low-mass end. The long wavelength data with Spitzer/IRAC, which sample the rest-frame NIR region even at high redshift, could make the systematic uncertainty relatively small (Fontana et al. 2006 Elsner et al. 2008).

Figure 13. Comparison between the stellar masses estimated with the one-component SFH model and two-component (old and young) model for galaxies at 2.5 < z < 3.5. The result with the GALAXEV model is shown.

## 2 Answers 2

Ironically, it's actually harder to measure the mass of the Milky Way than that of other galaxies. You'd think that with it being RIGHT THERE it would be easy, but alas. Most of the difficulty comes from (1) the galaxy spans a huge part of the sky, so it takes an extremely long time to observe any particular feature in detail across the whole thing (say mapping the strength of an emission line, for instance), and (2) it's hard to get an overall picture of the galaxy because parts of it get in the way of seeing other parts - there's a lot of dust in the galactic disk that obscures our view of the more distant parts of the disk, and the disk is where most of the stars are.

Stellar mass is actually the easiest mass to measure in astronomy, because you can see it much more directly than other mass components. All that needs to be done is measure the intrinsic (rather than apparent) luminosity of a galaxy, assume a "mass-to-light ratio" and multiply to get the stellar mass. Mass to light ratios are on the order of $Upsilonsim1< m M>_odot/< m L>_odot$ So a galaxy with a luminosity a billion times solar has a stellar mass of about a billion solar masses. More accurate estimates get complicated quickly, as you need to account for the initial distribution of stars in the stellar population(s) involved (the initial mass function: IMF), the age of the populations, dust extinction, etc. etc.

Gas mass is not too bad either. Depending on the phase of the gas - whether it's ionized, molecular or atomic (neutral) hydrogen it may be possible to measure line emission. Neutral hydrogen shows up in the radio at 21cm from the hyperfine transition (spin flip). Most of the gas mass is in neutral hydrogen. Depending on conditions, the Lyman or Balmer series lines may be visible (the first Balmer line is called $< m H>alpha$ in astronomy jargon, it's a common one to observe). Molecular hydrogen - the stuff that stars are made from directly, think Pillars of Creation, is tougher to measure as it has no strong emission lines. What's usually done is to measure emission from other molecular species - $< m CO>$ is a common one - and assume something about what fraction of the gas mass that species makes up.

Dark matter mass is inferred from things like galactic rotation curves or gravitational lensing, which both probe the total mass of the system. When we get a total mass from one of these tracers, we always seem to come up about an order of magnitude short (I'm using "always" very loosely here). This, coupled with cosmological observations that seem to imply there is a lot of matter ("dust" in cosmology jargon) that is not "baryonic", but is rather something else that outguns baryons a little less than 10:1 in mass.

As to the Milky Way, there are a number (about 10 that I know of) of ways you can try to measure the mass. I've co-authored a paper which uses several methods. One fairly well known measurement of the total (not just stellar) mass of the MW and M31 is this one, which is more than a factor of 2 bigger than the one you quote. Other sources are more in line with your number. the uncertainty is still rather large. Here's another paper that does the total mass with a different methodology (and get about $1.26 imes10^<12>< m M>_odot$), and also models the stellar mass, finding about $6.43 imes10^<10>< m M>_odot$, which is about the same ballpark as most estimates for the Milky Way.

If you're adventurous and want to get your hands dirty, stellar mass estimates for at least several hundred thousand galaxies from the SDSS are readily available. These are based on the luminosity of the galaxies, more or less as I've described above. Total mass estimates also exist, but I can't recall where they're easily obtained right now, and they're more uncertain.

Jerry Schirmer mentioned black holes in the comments, so I may as well add a note. The MW black hole is thought to be about $10^<6>< m M>_odot$, so less than one part in ten thousand of the stellar mass, and perhaps a millionth of the total mass. This is more or less typical, though some particularly large black holes get up to perhaps a hundredth of the mass of their galaxy, at most. SMBH's are not thought to be the dominant mass component in any known galaxy (though of course they do dominate in the very central regions).

## Dark halo response and the stellar initial mass function in early-type and late-type galaxies

We investigate the origin of the relations between stellar mass and optical circular velocity for early-type galaxies (ETGs) and late-type galaxies (LTGs) – the Faber–Jackson (FJ) and Tully–Fisher (TF) relations. We combine measurements of dark halo masses (from satellite kinematics and weak lensing), and the distribution of baryons in galaxies (from a new compilation of galaxy scaling relations), with constraints on dark halo structure from cosmological simulations. The principal unknowns are the halo response to galaxy formation and the stellar initial mass function (IMF). The slopes of the TF and FJ relations are naturally reproduced for a wide range of halo response and IMFs. However, models with a universal IMF and universal halo response cannot simultaneously reproduce the zero-points of both the TF and FJ relations. For a model with a universal Chabrier IMF, LTGs require halo expansion, while ETGs require halo contraction. A Salpeter IMF is permitted for high-mass (σ≳ 180 km s −1 ) ETGs, but is inconsistent for intermediate masses, unless Vcirc(Re)/σe≳ 1.6. If the IMF is universal and close to Chabrier, we speculate that the presence of a major merger may be responsible for the contraction in ETGs while clumpy accreting streams and/or feedback leads to expansion in LTGs. Alternatively, a recently proposed variation in the IMF disfavours halo contraction in both types of galaxies. Finally we show that our models naturally reproduce flat and featureless circular velocity profiles within the optical regions of galaxies without fine-tuning.

## The Ks-band luminosity and stellar mass functions of galaxies in z∼ 1 clusters

We present the near-infrared (Ks-band) luminosity function of galaxies in two z∼ 1 cluster candidates, 3C 336 and Q1335+28. A third cluster, 3C 289, was observed but found to be contaminated by a foreground system. Our wide-field imaging data reach to Ks= 20.5 (5σ) , corresponding to ∼M*+ 2.7 with respect to passive evolution. The near-infrared luminosity traces the stellar mass of a galaxy due to its small sensitivity to the recent star formation history. Thus the luminosity function can be transformed to the stellar mass function of galaxies using the JKs colours with only a small correction (factor ≲2) for the effects of ongoing star formation. The derived stellar mass function spans a wide range in mass from ∼3 × 10 11 M down to ∼6 × 10 9 M (set by the magnitude limit). The form of the mass function is very similar to lower-redshift counterparts such as that from 2MASS/LCRS clusters (given by Balogh et al.) and the z= 0.31 clusters (given by Barger et al.). This indicates little evolution of galaxy masses from z= 1 to the present day. Combined with colour data suggesting that star formation is completed early (z≫ 1) in the cluster core, it seems that the galaxy formation processes (both star formation and mass assembly) are strongly accelerated in dense environments and have been largely completed by z= 1 . We investigate whether the epoch of mass assembly of massive cluster galaxies is earlier than that predicted by the semi-analytic hierarchical galaxy formation models. These models predict the increase of characteristic mass by more than a factor of ∼3 between z= 1 and the present day. This seems incompatible with our data.

## Title: EVOLUTION OF GALAXY STELLAR MASS FUNCTIONS, MASS DENSITIES, AND MASS-TO-LIGHT RATIOS FROM z 7 TO z 4

We derive stellar masses from spectral energy distribution fitting to rest-frame optical and UV fluxes for 401 star-forming galaxies at z 4, 5, and 6 from Hubble-WFC3/IR camera observations of the Early Release Science field combined with the deep GOODS-S Spitzer/IRAC data (and include a previously published z 7 sample). A mass-luminosity relation with strongly luminosity-dependent M/L ratios is found for the largest sample (299 galaxies) at z 4. The relation ML0.2)> has a well-determined intrinsic sample variance of 0.5 dex. This relation is also consistent with the more limited samples at z 5-7. This z 4 mass-luminosity relation, and the well-established faint UV-luminosity functions at z 4-7, are used to derive galaxy mass functions (MFs) to masses M10 at z 4-7. A bootstrap approach is used to derive the MFs to account for the large scatter in the M- relation and the luminosity function uncertainties, along with an analytical cross-check. The MFs are also corrected for the effects of incompleteness. The incompleteness-corrected MFs are steeper than previously found, with slopes -1.4 to -1.6 at low masses. These slopes are, however, still substantially flatter thanmore » the MFs obtained from recent hydrodynamical simulations. We use these MFs to estimate the stellar mass density (SMD) of the universe to a fixed M < - 18 as a function of redshift and find an SMD growth (1 + z)0.8> from z 7 to z 4. We also derive the SMD from the completeness-corrected MFs to a mass limit M10 M. Such completeness-corrected MFs and the derived SMDs will be particularly important for comparisons as future MFs reach to lower masses. « less