**6.5. The halo model - I: mass**

The formation of galaxies must be a non-local process to some extent, and the modern paradigm was introduced by White & Rees (1978): galaxies form through the cooling of baryonic material in virialized haloes of dark matter. The virial radii of these systems are in excess of 0.1 Mpc, so there is the potential for large differences in the correlation properties of galaxies and dark matter on these scales. The `halo model' addresses this by creating a density field in which dark-matter haloes are superimposed. The key feature that allows bias to be included is to encode all the complications of galaxy formation via the halo occupation number: the number of galaxies found above some luminosity threshold in a virialized halo of a given mass.

To some extent, this is a very old idea:
one of the earliest suggested models for the
galaxy correlation function was to consider a density
field composed of randomly-placed independent clumps
with some universal density profile
(Neyman, Scott & Shane
1953;
Peebles 1974).
Since the clumps are placed at random (with number density *n*),
the only excess neighbours to a given mass point arise from points in
the same clump, and the correlation
function is straightforward to compute in principle.
For the case where the clumps have a power-law density profile,

(112) |

truncated at *r* = *R*, the small-*r* behaviour
of the correlation function is
*r*^{3-2},
provided 3/2 < <
3. For smaller values of
,
(*r*)
tends to a constant as *r* 0. In the isothermal
= 2 case, the
correlation function for *r* << *R* is

(113) |

where *N* is the total number of particles per clump
(Peebles 1974).

The general result is that the correlation function is
less steep at small *r* than the clump density profile, which
is inevitable because an autocorrelation function involves convolving
the density field with itself.
A long-standing problem for this model is therefore that the
predicted correlation function is much flatter than is
observed for galaxies:
*r*^{-1.8} is the canonical slope, apparently requiring
clumps with very steep density profiles,
= 2.4.
This is not in agreement with the profiles of dark-matter
haloes as `observed' in numerical simulations.

Traditionally, virialized systems have been found by a criterion based on percolation (`friends-of-friends'), such that the mean density is about 200 times the mean. Sometimes, the criterion is taken as a density of 200 times the critical value. We shall use the former definition:

(114) |

Thus *r*_{v} is related to the Lagrangian radius containing
the mass via *r*_{v} = *R* / 200^{1/3}.
Of course, the density contrast used to define the
boundary of an object is somewhat arbitrary. Fortunately, much of the
mass resides at smaller radii, near a `core radius'. These core radii
are relatively insensitive to the exact definition of virial radius.

The simplest model for the density structure of the
virialized system is the singular isothermal sphere:
=
_{v}^{2} /
(2*Gr*^{2}), or

(115) |

A more realistic alternative is the profile proposed by Navarro, Frenk & White (1996; NFW):

(116) |

The parameter
_{c} is
related to the core radius and the virial radius via

(117) |

(we change symbol from NFW's
_{c} to avoid
confusion with the linear-theory density threshold for collapse, and
also because our definition
of density is relative to the mean, rather than the critical density).
NFW showed that
_{c} is
related to collapse redshift via

(118) |

An advantage of the definition of virial radius used here is that
_{c} is
independent of (for
given *z*_{c}), whereas
NFW's _{c} is
.

The above equations determine the concentration,
*c* = *r*_{v} / *r*_{c} implicitly, hence
in principle giving *r*_{c} in terms of
*r*_{v} once
_{c} is known.
NFW give a procedure for determining *z*_{c}. A simplified
argument would suggest a typical formation era determined by
*D*(*z*_{c}) = 1 /
,
where *D* is the linear-theory growth factor between
*z* = *z*_{c} and the present, and
is the dimensionless fluctuation
amplitude corresponding to the system in units of the rms:
_{c} /
(*M*),
where _{c}
1.686. For very massive
systems with >> 1, only
rare fluctuations have collapsed by the present, so *z*_{c}
is close to zero. This suggests the interpolation formula

(119) |

The NFW formula is actually of this form, except that the 1 /
term
is multiplied by a spectrum-dependent coefficient of order unity.
It has been claimed by Moore et al.
(1999; M99)
that the NFW density profile is in error at small *r*. M99 proposed the
alternative form

(120) |

It is straightforward to use this in place of the NFW profile.

We now compute the power spectrum for the halo model.
Start by distributing point seeds throughout
the universe with number density *n*, in which case the power
spectrum of the resulting density field is just shot noise:

(121) |

Here, we use a dimensionless notation for the power spectrum:
^{2} is the
contribution to the fractional density variance
per unit interval of ln *k*. In the convention of
Peebles (1980),
this is

(122) |

(*V* being a normalization volume), and the relation to the
correlation function is

(123) |

The density field for a distribution of clumps is produced by convolution of the initial field of delta-functions, so the power spectrum is simply modified by the squared Fourier transform of the clump density profile:

(124) |

where

(125) |

For a practical calculation,
we should also use the fact that hierarchical models are expected to
contain a distribution of masses of clumps.
If we use the notation *f* (*M*) *dM* to denote the
number density of haloes in the mass range *dM*, the effective
number density in the shot noise formula becomes

(126) |

The window function also depends on mass, so the overall power spectrum is

(127) |

The normalization term
*M* *f* (*M*) d*M* just gives the total background
density,
_{b},
so there is only a single numerical integral to perform.
Using this model, it is then possible to calculate the correlations
of the nonlinear density field, neglecting only the
large-scale correlations in halo positions. The
power spectrum determined in this way is shown in
figure 13,
and turns out to agree very well with the exact nonlinear result on
small and intermediate scales. The lesson here is that a good deal of the
nonlinear correlations of the dark matter field can be understood as a
distribution of random clumps, provided these are given the correct
distribution of masses and mass-dependent density profiles.

So far, we have ignored any spatial correlations in the halo positions. A simple guess for amending this is to add the linear power spectrum to the power generated by the halo structure:

(128) |

The justification for this is that the extra small-scale power introduced
by nonlinear evolution is associated with the internal structure of the
haloes. In practice, this model
works extremely well, giving an almost perfect description of the
power spectrum on all scales.
This is a novel way of looking at the features in the
nonlinear spectrum, particularly the steep rise between
*k* 0.5 *h*
Mpc^{-1} and
*k* 5 *h*
Mpc^{-1}, and the flattening on
smaller scales. According to the ideas presented here, the
flat small-scale spectrum arises because haloes have central
density profiles rising as *r*^{-1.5}, but not much faster.
The sharp fall in power at smaller *k* reflects the cutoff at the
virial radii of the haloes that dominate the correlation signal.

It might be objected that this model is still not completely
realistic, since we have treated haloes as smooth objects
and ignored any substructure. At one time, it was generally believed that
collisionless evolution would lead to the destruction
of galaxy-scale haloes when they are absorbed into the
creation of a larger-scale nonlinear system such as a group
or cluster. However, it turns out that this `overmerging problem' was
only an artefact of inadequate resolution (see e.g.
van Kampen 2000).
When a simulation is carried out
with ~ 10^{6} particles in a rich cluster, the cores of
galaxy-scale haloes can still be identified after many crossing times
(Ghigna et al. 1998).
This substructure must have
some effect on the correlations of the density field, and indeed
Valageas (1999)
has argued that the high-order correlations of the density field seen in
*N*-body simulations are inconsistent with a model where the
density file is composed of smooth virialized haloes.
Nevertheless, substructure seems to be unimportant at the
level of two-point correlations.

The existence of substructure is important for the obvious next step of this work, which is to try to understand galaxy correlations within the current framework. It is clear that the galaxy-scale substructure in large dark-matter haloes defines directly where luminous galaxies will be found, giving hope that the main features of galaxy formation can be understood principally in terms of the dark-matter distribution. Indeed, if catalogues of these `sub-haloes' are created within a cosmological-sized simulation, their correlation function is known to differ from that of the mass (e.g. Klypin et al. 1999; Ma 1999). The model of a density field consisting of smooth haloes may therefore be a useful description of the galaxy field, and this is explored in the following section.