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Revisiting the Cosmic Cooling Crisis

Recent measurements of the K-band luminosity function now provide us with strong, reliable constraints on the fraction of baryons which have cooled. Globally, this fraction is only about 5%, and there is no strong evidence that it is significantly higher in clusters. Without an effective sub-grid feedback prescription, the cooled gas fraction in any numerical simulation exceeds these observational constraints, and increases with increasing resolution. This compromises any discussion of galaxy and cluster properties based on results of simulations which include cooling but do not implement an effective feedback mechanism.


INTRODUCTION
Gas cooling in the expanding Universe is an intrinsically unstable process because cooling acts to increase the density of the gas, which in turn increases the cooling rate. The consequence of this is that as soon as the gas within a system is able to cool at all it tends to do so catastrophically, only regulated by the speed at which the gas can respond to the new configuration. Systems which collapse at low redshift have a low mean density and, thus, long cooling times which generally exceed their dynamical times. Systems collapsing at higher redshifts have a higher mean density and, because the cooling rate depends strongly upon the gas density, they cool much more rapidly. In a Universe dominated by cold dark matter (CDM), the hierarchical growth of structure consequently results in almost all of the baryons cooling by the present day, unless additional physics is considered (Cole 1991;White & Frenk 1991;Blanchard et al. 1992).
Unfortunately, the naive theoretical prediction that all the baryonic material in the Universe should have cooled into galaxies and formed stars conflicts with the observation of X-ray emission from galaxy clusters. This emission demonstrates that large amounts of hot gas persist in the Universe. This material can arise from two possible sources: either it was never contained within a collapsed halo in which cooling was efficient, or energy injection (for instance feedback of energy due to supernovae) has either reheated or prevented the cooling of the gas. The commonly used theory of Press & Schecter (1974) (see also Bower 1991) implicitly assumes that all the matter in the Universe is contained within haloes.
The best numerical models (Jenkins et al. 2000) indicate that the fraction of material contained within haloes is certainly high, and only leave room for a small uncollapsed fraction. Therefore, we appeal to feedback mechanisms (Larson 1974;White & Rees 1978;Cole 1991;White & Frenk 1991) to reduce the amount of cold gas within collapsed haloes.
In this paper we review the observational constraints on the density of cold and hot baryons, both globally and in rich clusters ( §2). These data establish the global cooled gas fraction at f c,global ≈ 0.073h, and there is no convincing evidence that it is much higher in clusters. We then examine cosmological simulation results in §3, and show that they can easily cool far too much gas to be consistent with these data. Finally, in §4 we discuss the necessity of combining simulations with analytic, sub-grid physics models, and the merits of various initial attempts to do this.
as long as 1) we do not miss a significant contribution from low surface brightness galaxies; and 2) the massto-light ratio is well understood. Fortunately, surface brightness is tightly correlated with total luminosity, and there does not appear to be a significant contribution to the baryon density from low surface brightness galaxies (Driver 1999;. Although there is an uncomfortably large range in the Schechter parameters of the local optical luminosity function (due to the large degeneracy between φ * and L * ), Fukugita, Hogan & Peebles (1998, hereafter FHP) demonstrate that the total integrated B−band luminosity is constrained to within about 15%. However, the use of the optical luminosity function requires large and uncertain corrections for the M/L B ratio, which is sensitive to stellar populations and, thus, must be determined separately for different types of galaxy.
A more robust estimate is obtained from the K−band luminosity function, since M/L K is less sensitive to star formation history, varying by only a factor of about two due to stellar population age differences (Bell & de Jong 2000). Recently, Cole et al. (2000b) have computed the mass function using over 17000 galaxies with redshifts from the 2dF galaxy redshift survey and magnitudes from the 2-micron all sky survey (2MASS). From the galaxy colours and stellar population synthesis models they determine M/L K for each galaxy (independent of h). Their (ΛCDM ) result is most sensitive to the assumed initial mass function (IMF): for a Kennicutt (1983) IMF, Ω stars = 0.0014h −1 , while for the Salpeter (1955) IMF, Ω stars = 0.0026h −1 .
The average stellar M/L K from which Cole et al. (2000b) derive Ω stars is 0.73 and 1.32 for the Kennicutt and Salpeter IMFs, respectively (independent of h). A substantially higher M/L K would require a large contribution from low mass stars and brown dwarfs. Although these objects are likely to be a significant component of the mass budget, there is no strong evidence that they are more abundant than expected from standard IMFs (Fuchs et al. 1998;Gizis et al. 2000;Lucas & Roche 2000). In particular, Reid et al. (1999) claim that brown dwarfs contribute no more than 15% of the mass of the Galactic disk. Earlier microlensing results which suggested that there might be a large population of dark objects in the Galaxy halo are no longer compelling, following a reanalysis of the MACHO project data (Alcock et al. 2000).
Dynamical measurements of early type galaxies (van der Marel 1991; Bell & de Jong 2000) favour M/L K ∼ < 1.1h; this is an upper limit to the global average because later type galaxies have lower M/L K . We therefore prefer the estimate of Ω stars = 0.0014h −1 based on the Kennicutt (1983) IMF with a mean M/L K = 0.73.
Cooled gas is also present in the form of neutral and molecular gas, but these make only small contributions. The density of neutral gas is Ω atomic ≈ 0.000188h −1 , as recently calculated by the HI Parkes All Sky Survey (HIPASS) team (Kilborn et al. 1999). For the amount of gas in molecular form, we adopt the relation ρ H2 /ρ HI = 0.81 used by FHP, gleaned from the CO-based compilation of Young & Scoville (1991). To-gether, the neutral and molecular gas only represent ∼ 10% of the stellar mass, which is the fraction we will adopt, so that Ω cold = 1.1Ω stars .
The total baryon content of the universe is unknown observationally, because baryons in the warm plasma phase believed to occupy normal galaxy haloes are very difficult to see . However, if nucleosynthesis calculations are correct, the baryon density Ω b can be determined from deuterium abundances. The best current estimate, Ω b = 0.019h −2 (Burles & Tytler 1998;Burles et al. 2000), therefore implies f c,global = Ω cold /Ω b = 0.073h. There are indications, from recent analysis of the combined BOOMERANG and MAXIMA data, that Ω b is much larger than this, Ω b = 0.039h −2 (Jaffe et al. 2000), which would reduce f c,global by a factor of two.
In summary, only about 5% of the available baryons in the universe have cooled (for h ≈ 0.7 and a Kennicutt IMF). This result is slightly lower than the minimum of the range 0.062 < f c,global < 0.167 found by FHP from optical data, but much more certain as it does not rely on knowing the relative abundances of different galaxies (disks, spheroids and irregulars) with very different (and uncertain) M/L B ratios. As we will show in §3, this low value is in stark contrast to the results obtained in numerical simulations which do not employ an adequate feedback model.

The Abundance of Hot Baryons
In the previous section we claimed that most of the baryons in the Universe are in a warm or hot phase, which is difficult to observe because it is not hot enough to be observable in X-ray radiation (Blanchard et al. 1992;Davé et al. 2000;Benson et al. 2000). The existence of such a warm component is compatible with constraints on the anisotropy of the microwave background, a long as the gas temperature satisfies the constraint T < 4h × 10 7 K (at z < 1) (Wright et al. 1994).
In clusters, however, the same gas is much hotter, and is directly observable. As originally shown by White et al. (1993), the total baryon content of rich clusters, including this plasma, is fully consistent with Ω b from element abundance determinations, if Ω • ≈ 0.3. FHP compile data on the baryon contributions in clusters due to stars, cold gas and plasma, and demonstrate that Ω b /Ω • = 0.112 ± 0.05 (h = 0.7). In this case, Ω b is a sum of all observed baryons in clusters. Hence, for Ω • = 0.3 ± 0.1, we have that Ω b = 0.034 ± 0.019 (h = 0.7) which is consistent with the baryon fraction deduced from deuterium abundances (Burles & Tytler 1998). Thus, the fact that Ω stars ≪ Ω b in the Universe is almost certainly a reflection of the fact that the warm gas has not been directly observed.

The Dependence on Halo Mass
Although we have demonstrated that f c,global ≈ 0.073h, it is possible that the cooled fraction in some environments, rich clusters for example, could be very different. In particular, Bryan (2000) claims that the efficiency of galaxy formation is dependent on halo mass, with more gas cooling in small haloes, relative to large ones. Most of the cluster data in the literature has been obtained at optical wavelengths, so the correction for M/L variations is not as robust as for the global value computed from the K-band field luminosity function in §2.1. We will use the optimal values from FHP, who compile data from various sources, including dynamical measurements and population synthesis model estimates. The M/L B depends strongly on morphological type, and FHP find M/L B = 6.5 +1.8 −2.0 for E/S0 galaxies; M/L B = 1.5 ± 0.4 for spiral galaxies; and M/L B = 1.1 ± 0.25 for irregular galaxies. For the morphological composition of a "typical" cluster, they determine M/L B = 4.5 ± 1, and we will use this number. Assuming a pure elliptical population would increase stellar mass estimates by ∼ 45%.
For low mass clusters and groups, the X-ray data only extend out to a small fraction of the virial radius. This can lead to a strong bias, since the gas distribution is less concentrated than the stars (Roussel et al. 2000). We will consider the cluster and group data recently compiled by Roussell et al. (2000) specifically for the purpose of addressing this issue. In that paper, care was taken to ensure that stellar and gas masses are computed to the same radius, and only those clusters for which the X-ray data extend to at least 25% of the virial radius are considered here. Nonetheless, we warn that the data for groups with kT < 5 keV must be treated with caution, as their gas masses still require a substantial extrapolation to the virial radius. In addition to the standard analysis of the X-ray data, Roussel et al. also consider hot gas mass estimates based on simulation-motivated scaling laws, which do not require an assumption of isothermal, β−model profiles. We use the measurements made based on the scaling laws of Bryan & Norman (1998). In addition, we consider the 15 rich clusters with reliable velocity dispersions from the CNOC sample (Carlberg et al. 1996;Carlberg et al. 1997), with X-ray data taken from Lewis et al. (1999).
The stellar masses are computed from the integrated stellar luminosity function, assuming M/L B = 4.5, and corrected for undetected galaxies by extrapolating the luminosity function with an assumed faint end slope of α ≈ −1.2. Finally, we assume that the mass of cold baryons is 10% larger than the stellar mass, to account for the presence of neutral and molecular gas (see § 2.1).
We consider two approaches to compute f c,cluster from M cold in the clusters. In the first case, we compute where M hot is determined from the observable X-ray emission from the hot plasma, extrapolated to the virial radius. This will be valid if the extrapolation is accurate (less likely for lower mass clusters), and will be an overestimate of f c,cluster if there is an unobserved, warm component which does not contribute to the Xray emission (as is the case in galactic mass haloes). We show f c,cluster for the two samples in the bottom panel of Figure 1, for a ΛCDM cosmology with h = 0.7. Both M cold and M hot are distance dependent measurements, Most of the f c,cluster values in the bottom panel of Figure 1 are higher than the global fraction f c,global computed in §2.1, shown as the horizontal line. There is only weak evidence for a trend, in the sense that f c,global is lower in higher temperature systems. However, this trend is much weaker than suggested by Bryan (2000) and, we caution that the kT < 5 keV groups may still be biased toward high fractions because of the limited extent of the X-ray data; none of these groups have X-ray emission detected beyond half the virial radius. Also, both the random and systematic uncertainties in observationally determining M hot , especially for the low temperature groups, are large. This is emphasized in the top panel of Figure 1, in which the same data are plotted, but f c,cluster is computed as where M total is the total dynamical mass of the cluster, and we take Ω b from Burles & Tytler (1998). This does not depend on the difficult measurement of M hot , but does require an estimate of Ω • , which is 0.3 in our ΛCDM cosmology. Note that, in this case, f c,cluster ∝ Ω • h. Not only has the apparent trend in f c,cluster with temperature now virtually disappeared, but the scatter has been reduced considerably. In this case the f c,cluster values are in much closer agreement with f c,global ; the remaining difference is not very compelling, since a slightly lower Ω • ≈ 0.2 would result in concordance between the cluster and global fractions. These results differ from those of Bryan (2000), who claims that f c,cluster increases from cluster to group scales. However, this trend could well be the result of a bias due to the limited radial extent of the X-ray observations, R x , in the group samples (Roussel et al. 2000). For example, from the Mulchaey et al. (1996) sample considered by Bryan (2000), the groups with the smallest R x have among the highest f c,cluster values. On the other hand, the two groups for which R x > 0.5 Mpc have the lowest f c,cluster . This emphasizes the importance of ensuring the X-ray data extend out to at least a sizable fraction of the virial radius, as we have done when considering the Roussel et al. compilation. We have shown that there is no discernable trend of f c,cluster with temperature in this sample, particularly if the baryon fraction is assumed from nucleosynthesis arguments, rather than relying on the measurement of M hot .

NUMERICAL SIMULATIONS
Hydrodynamical numerical simulations which attempt to follow the effects of gas cooling have proved very difficult to perform across the full range of the gravitational mass hierarchy. Allowing gas to dissipate its energy via radiative cooling generates enormous density contrasts and a huge range of interesting spatial scales within the model, even before the intrinsically sub-resolution pro- cesses of star formation and the feedback of energy due to supernovae and stellar winds are considered.
The earliest attempt at such a simulation was made by Thomas & Couchman (1992), though the model which included radiative cooling was only briefly discussed. This model had a very high mass resolution threshold, allowing only the largest galaxies to form, and only a small fraction of the gas to cool. This was quickly followed by the work of Katz & White (1993) who produced a model that was several years ahead of its time. They demonstrated that the formation of a Virgo-like cluster, including the effects of cooling, could be followed by a computer simulation. However, the model had some problems: in particular, as the authors make clear, much more gas cools than is observed. These results are shown as the large, filled circles in Figure  2. The lower point shows the actual fraction of cold baryons in the simulation; the higher point to which it is connected shows this fraction corrected by extrapolating the simulated luminosity function to include the additional cooled gas within unresolved objects, as described in Katz & White. The simulation of Katz & White was stopped at z = 0.13, so the redshift zero fraction will be even higher.
Recently there has been renewed interest in this problem. Suginohara & Ostriker (1998) also produced a cluster model, and noted that additional physical processes (such as star formation and feedback) would be required to prevent the cooling of an excessively large fraction of the gas; unfortunately, the fraction of cooled baryons was not published in their paper so we cannot consider their results in Figure 2. Their work was followed by that of Pearce et al. (1999;, Lewis et al. (2000) and Davé et al. (2000). Lewis et al. used a more advanced code to improve upon the earlier simulation of Katz & White (1993), and they also found that around 40% of the gas within the cluster cools. This is shown in Figure 2 as filled triangles; the upper triangle includes a correction for galaxies below the resolution limit, by extrapolating the resultant luminosity function assuming a faint end slope α = −0.96. Davé et al. used both a parallel treecode and a Eulerian grid based code in several cosmologies and, although the cluster-by-cluster fractions are not quoted, find a universal cooling fraction of around 30% to 40% for their high resolution simulations, in both cases. This is shown as a lower limit in Figure 2, since it does not include a correction for galaxies formed in haloes below the resolution limit. Simulations like those discussed above which cool too much gas cannot be expected to model galaxy formation in a physically correct way. An alternative, heuristic approach can be taken to force the simulations to satisfy the constraint on f c,global , by limiting the resolution (see § 3.1). This is the method used by Pearce et al. (2000), who deliberately set their mass resolution to cool ∼15% of the gas within their simulation volume; the cooled fractions for the twenty most massive of these clusters are shown in Figure 2 as the filled squares (both before and after correction for sub-resolution galaxies). By construction, the cooled gas fractions of these simulated haloes are closer to the observed values, though they are still probably too high. The obvious problem with this simulation is that it does not allow the formation of small galaxies.

Resolution
It has long been theorised that progessively increasing the resolution within a simulation leads to a continual increase in the fraction of material which cools (Cole 1991;White & Frenk 1991;Suginohara & Ostriker 1998), a situation commonly referred to as the cooling catastrophe. We demonstrate this in Figure 3, by presenting the results of standard cold dark matter simulations, at different resolutions (the mass of the smallest object that can effectively cool). This figure shows how the fraction of gas particles in the cold phase increases as resolution is increased. At the low resolution end, the relation is too steep, due to the artificial heating problems described by Steinmetz & White (1997). However, the trends discussed below are qualitatively retained, even when this problem is overcome (Kay 2000). Apart from this effect, the shape of the curve will be approximately independent of the simulations. However, the scale along the abscissa will vary; that is, the curves may be translated laterally, for example, by changing Ω b . More subtly, in a model with a positive cosmological constant, structure forms earlier and so more gas cools at the same threshold mass. This causes the curves plotted on Figure 3 to move to the right by about a factor of two. Changes in metallicity can affect the result in the same way, though even primoridal abundances cannot prevent the cooling catastrophe (Davé et al. 2000). In addition, differences in the numerical implementation can also shift the curve by factors of order two. Figure 3 makes it clear why some groups find cooled gas fractions around 40% whilst others find much lower fractions. Values around 40% will be achieved for a wide range of simulation resolutions, as this is the value obtained when the resolution mass is less than the characteristic mass of the halo mass function. More gas will cool as the resolution is increased but a large increase is required to see a significant change, since the slope of the mass function is shallow in this regime. Any required cold gas fraction below 40% can be obtained by choosing the resolution appropriately, although care must be taken to account for artificial heating (Steinmetz & White 1997). However, in order to get a good match to the observed gas fraction within a standard cold dark matter simulation the resolution limit must be set to ∼ 10 11 M ⊙ or higher, which is clearly unphysically high. This is the situation we referred to as the "cooling crisis"; it is a crisis because its resolution will Figure 3. Fraction of baryons in three phases (representing shocked gas, cooled gas [galaxies] and unshocked, diffuse baryons) against the baryonic mass of the smallest resolvable galaxy for standard CDM simulations with radiative cooling. The simulations use a box-size of 10h −1 Mpc and Ω b = 0.06. The two points with the largest mass threshold are too low, due to artificial heating effects. Further details are contained in Kay (2000).
result in a significant improvement in the utility of simulation results.

DISCUSSION
We have demonstrated that it is easy to exceed the observed cooled gas fraction with numerical simulations of sufficient resolution. The failure to match this constraint reflects the well-known need for some method of reheating the gas in order to prevent excessive star formation at high redshift (Larson 1974;White & Rees 1978;White & Frenk 1991). Until an effective feedback model is implemented in simulations, it will not be possible to obtain physically illuminating results.
Present attempts to incorporate feedback within cosmological simulations all suffer from the same drawback, that the physical processes responsible for the redistribution of energy occur on scales well below those that can presently be resolved. Hence, an accurate treatment of the propagation of energy from the local ISM to super-galactic scales is currently an intractable problem. However, several groups have attempted to include feedback by injecting the energy into regions at the resolution limit of the simulation. For example, an approach that has been taken in smoothed particle hydrodynamics (SPH) simulations is simply to add available energy from newly-formed "star" particles to the thermal energy of the surrounding gas (Katz et al. 1996;Lewis et al. 2000;Davé et al. 2000). However, the majority of this gas is cold and dense, and so any excess energy is rapidly re-radiated. It is plausible that this problem is due to the failure of the SPH simulations to resolve a multiphase medium, in which the feedback would be able to create hot bubbles of diffuse gas, with significantly longer cooling times than the cold dense gas from which they were spawned. Several groups have attempted to "correct" for this by preventing reheated gas from radiatively cooling over a finite timescale (e.g. Gerritsen 1997;Springel 2000;Thacker & Couchman 2000). Another method is to supply energy in kinetic form (e.g. Kay 2000), which is efficient at limiting the fraction of cooled gas. However, the effectiveness of this implementation may be artificial since the SPH simulations fail to resolve shocks that would efficiently thermalize the reheated material (Katz et al. 1996).
An alternative approach to feedback is that of Cen & Ostriker (1999), who use a grid-based cosmological code, and implement analytic rules, similar to those of semi-analytic models (Somerville & Primack 1999;Cole et al. 2000a;Kauffmann et al. 1999;Somerville et al. 2001) to regulate the formation of galaxies within a single cell. Unfortunately, the resolution of grid-based methods such as this is still too coarse (∼ 100h −1 kpc in the highest resolution simulation of Cen & Ostriker) to provide useful results on galaxy scales. It is not clear that the analytic rules implemented to model the subgrid physics in this code are robust to changes in resolution, as changes in the cell size can result in large changes in the gas densities, for which the analytic formalism may not appropriately compensate. A more promising route may be the use of adaptive-mesh refinement codes (Bryan & Norman 1998;Bryan & Voit 2001), which will allow the implementation of an analytic treatment of gas cooling within cells of sufficient resolution.
The goal now is to find a physically motivated feedback model which is able to sufficiently reduce f c . It is therefore useful to consider, qualitatively, what is required of such a model. The amount of gas which can cool in a given halo cannot be computed by considering a halo structure at a single redshift and treating it as if it had existed unchanged over a Hubble time. For example, the cooling time in clusters today is much longer than the Hubble time, and this has been used to explain why not all of the gas in these environments has cooled (Binney 1977;Silk 1977;Rees & Ostriker 1977). However, this argument does not hold in hierarchical models, where cluster progenitors at earlier epochs were low mass haloes in which the cooling time was short, and simulations of hierarchical cluster formation demonstrate that a large fraction of the baryons cool by the present day. To treat the problem analytically, it is necessary to consider the hierarchical growth of a halo through the plane of Figure 4. The smooth curved lines represent the evolution of structure, for a standard CDM model (σ 8 = 0.8, Γ = Ω • h = 0.5); the central line shows the median mass halo which virializes as a function of redshift, and the two others bracket the region in which 50% of the mass in the Universe collapses into haloes. In a manner similar to that of Rees & Ostriker (1977), we can define an efficient cooling mass, M cool , as the mass which can cool all the gas out to its virial radius, within a Hubble time at redshift z. The cooling rate, which depends on halo density and temperature, is computed using the models of Raymond et al. (1976), with one-third solar metallicity. This cooling mass, for the standard CDM model described above, is shown in Figure 4, and is given by M cool ∼ 10 12 M ⊙ , almost completely independent of redshift for z < 10. This non-trivial result is a consequence of the fact that, when cooling is dominated by line emission, the increase in the cooling rate with increasing redshift is closely balanced by the decrease in the Hubble time. This forms an approximate upper mass limit to efficient cooling; haloes with masses smaller than this limit can cool all of their gas, out to the virial radius, while to the right of this line, only a rapidly decreasing fraction of the gas is able to cool.
At low redshifts, most of the haloes which are collapsing are more massive than M cool ; therefore, cooling plays a minor role today. By z ≈ 1, however, the characteristic mass crosses the M cool threshold, and cooling becomes a dominant process, leading to the cooling catastrophe. In simulations, the resolution threshold imposes a mass limit, M th , which prevents smaller mass haloes from cooling. Efficient cooling is therefore restricted to the shaded region in Figure 4. As M th is lowered, more and more mass cools, and will quickly exceed the tight observational constraints. If the threshold is high enough to prevent efficient cooling at high redshift, no low mass galaxies are formed.
The final fraction of cooled gas then depends on the trajectory a particular halo takes through this plane, as it grows. This growth will occur via both smooth accretion, and discrete jumps due to mergers. The resolution threshold imprints a scale on the accumulation of cooled baryons, as the final fraction f c in a given halo will depend on its trajectory only after it becomes sufficiently resolved. This could have an effect on the dependence of f c on halo mass in simulations (but see Kay 2000).
It is clear that any analytic feedback model which behaves in a manner similar to a sharp, fixed mass threshold will fail to simultaneously match the observed f c,global and still produce a reasonable galaxy luminosity function. A physical feedback scheme, however, does not provide a hard mass threshold, but gradually alters the cooling efficiency of haloes as a function of mass and redshift; thus the shaded region in Figure 4 becomes a "fuzzy" region without a well defined lower boundary. There are many models for such feedback, including supernovae or AGN reheating (Larson 1974;White & Rees 1978;Lacey & Silk 1991;Cole 1991;Wu et al. 1998;Bower et al. 2000;Mac Low & Ferrara 1999;Efstathiou 2000), ionizing radiation from the ultraviolet background (Efstathiou 1992;Weinberg et al. 1997), and an entropy threshold established at high redshift (Kaiser 1991;Blanchard et al. 1992;Ponman et al. 1999;Balogh et al. 1999). A very successful implementation of a feedback scaling law is that used by semianalytic models of galaxy formation (Cole et al. 2000a;Somerville & Primack 1999;Kauffmann et al. 1999;Somerville et al. 2001), in which the mass of reheated gas is assumed to scale in proportion with the halo circular velocity raised to some power α. Though this scal-  Figure 4. The region of efficient gas cooling in the Universe, as a function of halo mass and redshift. The smooth curved lines delineate the formation of structure in the Universe. The central line represents the median halo mass which virialises at the corresponding redshift; 50% of the mass in the Universe collapses between the two lines which bracket this one. M cool is the mass threshold below which haloes can cool out to their virial radius. M th is the mass threshold within the numerical model, generally given by the resolution, below which cooling cannot occur. The shaded region therefore denotes the region within which cooling is important.
ing is constructed to allow the models to match the observed luminosity function, it also succeeds in limiting f c to more reasonable values (though generally still high compared with the constraints described here). Furthermore, there may be some theoretical motivation for this scaling relation from supernova-driven wind models (Efstathiou 2000). The next step in this work is to combine these models with realistic halo merger trees to determine, analytically, how they affect the cooling rate as a function of halo mass and redshift, for implementation into numerical simulations.

CONCLUSIONS
In this paper, we have presented an up-to-date review of the cooling catastrophe. Although the issues we have discussed have been known for a long time, recent advances in observations and simulation techniques warrant a reevaluation of the situation, which can be summarized in two key points: • The global fraction of baryons in the form of stars and cold gas is now well constrained observationally from the K−band luminosity function to be f c,global ≈ 0.073h. Although there is some uncertainty about how this may vary with halo mass, there is no convincing evidence for a significant difference from the global value.
• The amount of gas which cools in simulations is directly related to the resolution, unless a working reheating scheme is implemented. No current simulation is able to produce both (1) the correct f c,global and (2) a population of low mass galaxies, without implementing analytic models of the sub-grid physics on uncomfortably large scales.
The use of simulations to aid our understanding of galaxy properties and of cluster scaling laws therefore awaits the implementation of a sub-grid, physical feedback scheme which allows the models to satisfy observational constraints on f c,global in a resolutionindependent way.