The mass function of HI clouds

I will start by estimating the baryonic mass-function of collapsed halos, assuming that baryons and dark matter are distributed equally over the sky (i.e. there is no biasing). The results are fairly insensitive to the precise form of the power spectrum. I will take parameters appropriate for the standard Cold Dark Matter (CDM) cosmology (other hierarchical cosmologies would give similar results): density parameter, ; Hubble parameter, kmsMpc; baryon fraction, ; normalisation, .

An analytic estimate for the number density of halos as a function of mass was first provided by Press & Schechter (1974). To estimate the proportion of the Universe which is contained in structures of mass m at redshift z, the density-field is first smoothed with a top-hat filter of radius R, where and is the mean density of the Universe. F(m,z) is then defined to be the fractional volume where the smoothed density exceeds some critical density . Assuming a gaussian distribution, then

where is the root-mean-square fluctuation within the top-hat filter and is the complementary error function. The key step was to realize that fluctuations on different mass-scales are not independent. In fact, to a first approximation Press & Schechter assumed that high-mass halos were entirely made up of lower-mass ones with no underdense matter mixed in. Then F must be regarded as a cumulative mass fraction and it can be differentiated to obtain the fraction of the universe contained in structures of a given mass,

To convert this to a number density of halos per logarithmic mass-interval we simply multiply by : dn/d. The main drawback of this approach is that, because of the above assumption of crowding together of low-mass halos into larger ones, it seems to undercount the number of objects. However modern techniques (e.g. Bond et al. 1992) give the same analytic form, simply scaled by a factor of two in normalization. The modified formula gives good agreement with numerical simulations (e.g. Lacey & Cole 1994) for (as is appropriate for a spherical, top-hat collapse of density peaks).

Figure 1 shows the predicted mass function for the baryonic content of halos and contrasts it with the observed galactic mass function. I have assumed a Schechter luminosity function,

where Mpc, , , and a mass-luminosity ratio of m/L=15h.

  
Figure 1: The mass function of halos in the standard CDM cosmology (solid line) compared with the observed mass function of galaxies (dashed line) assuming . Also shown is the HI content of normal spiral galaxies (dotted line).

It is apparent that the model predicts approximately the correct number density of normal () galaxies. However, it gives far too many halos at both higher and lower masses. The reason for the former discrepancy was first explained by Rees & Ostriker (1977) and by Silk (1977). They compared the ratio of the cooling time of the gas in proto-galactic halos to the dynamical time of the halo. In low-mass objects () the cooling time is always shorter than the dynamical time, thus gas can cool to form stars and hence a visible galaxy. In larger systems, however, the cooling time exceeds the dynamical time. Then mergers may shock-heat the gas before it has time to cool, thus preventing significant star-formation. For this reason clusters of galaxies have little on-going star-formation except perhaps in a cooling flow deep within the cluster core. The dividing mass between these two regimes is highly sensitive to the gas fraction and cooling function but covers the range corresponding to the exponential cut-off in the luminosity function.

The predicted excess of low-mass halos is harder to explain. It would seem that not all the cooled gas in proto-galaxies has formed visible stars. Until I researched this paper, it seemed possible to me that the missing gas resided inside the halos of low-mass galaxies in the form of HI. The HI content of normal spiral galaxies is insufficient (see, for example, the mass function shown by the dotted line in Figure 1 which is taken from the model in Briggs 1990) but I thought it possible that there may be a significant population of gas-rich dwarfs. However, the observations of the number counts of HI clouds, described below, seem to rule this out. This is consistent with the optical luminosity function which, although it may miss many low-surface-brightness, predominantly low-mass galaxies is unlikely to have a faint-end slope as steep as the required value of (see, however, Driver et al. 1994 and references therein). A more realistic explanation for the missing gas is that much of the proto-galactic interstellar medium was heated by an early generation of supernovae (and/or strong galactic winds) and expelled from the halo. If only a small fraction of HI remains, however, or if it has fallen back into the galaxy, then it should be visible with the Multibeam Survey.

Figure 2 shows the observed number density of HI clouds in various surveys. The solid data points are taken from Briggs (1990). They all come from pointed observations towards different clusters, but including different proportions of foreground and background objects: solid squares, Leo group (Schneider et al. 1989); circles, Virgo (Hoffman et al. 1989); triangle, Hercules (Salpeter & Dickey 1985). Unfortunately these surveys are all highly biased. Blind surveys are harder to come by--see the article by Schneider in this volume for a review. One of the most sensitive to date is that of Kerr & Henning (1987) which has recently been analysed by Henning (1995). This gives a lower space-density as shown by the open squares in the figure. A realistic estimate of the true space-density is probably given by the lower locus of the points in the figure, some one to two orders-of-magnitude below the predicted curve if all the missing matter were in the form of HI. As the observations must be shifted by about two orders-of-magnitude to the right to give good agreement with the predictions, this suggests that about one percent of the original baryonic mass of the halo persists in the form of HI.

  
Figure 2: The detected density of HI clouds (i.e. galaxies) of different mass, together with the CDM predicted mass function of halos (solid line) and the expected sensitivity from the Multibeam Survey (dashed line). For details see the text.

The dashed line in Figure 2 shows the space density of HI clouds at which the Multibeam Survey will detect 10 objects of a given mass. The sensitivity is almost three orders-of-magnitude higher than any previous survey thanks mainly due to the large volume of space which is covered. In making this prediction I have have assumed that for a 5 sigma detection

where D is the distance to the cloud and the data have been binned to match its velocity width. Supposing one per cent of the baryons to be in the form of HI, then the Tully-Fisher relation is

which gives a minimum density for the detection of N clouds of

The flat part of the sensitivity curve arises for HI clouds which are detectable to the survey limit of 13500 kms.

On the basis of Figure 2, I predict that the Multibeam Survey will detect several hundred HI clouds per decade of mass. I would expect most of these to be associated with dwarf galaxies, but there may be a few truly intergalactic gas clouds. The mass and spatial distribution of these objects will be a major constraint on models of galaxy formation.

© Copyright Astronomical Society of Australia 1997