**6.2. The peak-background split**

We now consider the central mechanism of biased clustering, in which a rare high density fluctuation, corresponding to a massive object, collapses sooner if it lies in a region of large-scale overdensity. This `helping hand' from the long-wavelength modes means that overdense regions contain an enhanced abundance of massive objects with respect to the mean, so that these systems display enhanced clustering. The basic mechanism can be immediately understood via the diagram in figure 8; it was first clearly analysed by Kaiser (1984) in the context of rich clusters of galaxies. What Kaiser did not do was consider the degree of bias that applies to more typical objects; the generalization to consider objects of any mass was made by Cole & Kaiser (1989; see also Mo & White 1996 and Sheth et al. 2001).

The key ingredient of this analysis is the mass function of
dark-matter haloes. The universe fragments into virialized
systems such that *f* (*M*) *dM* is the number density of
haloes in the mass range *dM*; conservation of mass requires
that
*M* *f* (*M*) d*M* =
_{0}. A
convenient related dimensionless quantity is therefore the
multiplicity function, *M*^{2}*f* (*M*) /
_{0}, which
gives the fraction of the mass of the universe contained
in haloes of a unit range in ln *M*.
The simplest analyses of the mass function rest on the concept
of a density threshold: collapse to a virialized object
is deemed to have occurred where linear-theory
averaged over a box
containing mass *M* reaches some critical value
_{c}.
Generally, we shall assume the value
_{c} = 1.686
appropriate for spherical collapse in an Einstein-de Sitter universe.
Now imagine that this situation is perturbed, by
adding some constant shift
to
the density perturbations over some large region.
The effect of this is to perturb the threshold:
fluctuations now only need to reach
=
_{c} -
in order to achieve collapse. The number density is therefore modulated:

(104) |

This gives a bias
in the number density of haloes in Lagrangian space:
*f* / *f* =
*b*_{L}
, where the
Lagrangian bias is

(105) |

In addition to this modulation of the halo properties, the large-scale
disturbance will move haloes closer together where
is large, giving a density contrast of
1 + . If
<< 1, the overall
fractional density contrast of haloes is therefore the sum of the
dynamical and statistical effects:
_{halo} =
+
*b*_{L}
.
The overall bias in Eulerian space
(*b* = _{halo}
/ ) is therefore

(106) |

Of course, the field can hardly be imposed by hand; instead, we make the peak-background split, in which is mentally decomposed into a small-scale and a large-scale component - which we identify with . The scale above which the large-scale component is defined does not matter so long as it lies between the sizes of collapsed systems and the scales at which we wish to measure correlations.

To apply this, we need an explicit expression for the mass function.
The simplest alternative is the original expression of
Press & Schechter
(1974),
which can be written in terms of the parameter
=
_{c} /
(*M*):

(107) |

We now use *d* /
*d*_{c} =
(*M*)^{-1}(*d* /
*d*) =
( /
_{c})(*d* /
*d*),
since *M* is not affected by the threshold change, which yields

(108) |

This says that *M*^{*} haloes are unbiased,
low-mass haloes are antibiased and high-mass haloes are positively
biased, eventually reaching the
*b* = /
value expected for high
peaks. The corresponding expression can readily be deduced for more
accurate fitting formulae for the mass function, such as that of
Sheth & Tormen
(1999):

(109) |

We can now understand the observation that Abell clusters are much
more strongly clustered than galaxies in general:
regions of large-scale overdensity contain systematically
more high-mass haloes than expected if the haloes traced
the mass. This phenomenon was dubbed natural bias by
White et al. (1987).
However, applying the idea to galaxies is not
straightforward: we have shown that enhanced clustering is only
expected for massive fluctuations with
1, but
galaxies at *z* = 0 fail this criterion.
The high-peak idea applies will at high redshift, where massive
galaxies are still assembling, but today there has been
time for galaxy-scale haloes to collapse in all environments.
The large bias that should exist at high redshifts is erased
as the mass fluctuations grow:
if the Lagrangian component to the biased density field is kept unaltered,
then the present-day bias will tend to unity as

(110) |

(Fry 1986;
Tegmark & Peebles
1998).
Strong galaxy bias at *z* = 0 therefore requires some form
of selection that locates present-day galaxies preferentially
in the rarer haloes with *M* > *M*^{*}
(Kauffmann, Nusser & Steinmetz 1997).

This dilemma forced the introduction of the idea of high-peak bias: bright
galaxies form only at the sites of high peaks in the initial density field
(Bardeen et al. 1986;
Davis et al. 1985).
This idea is commonly, but incorrectly, attributed to
Kaiser (1984),
but it needs an extra ingredient, namely a non-gravitational threshold.
Attempts were therefore made to argue that the first generation of objects
could propagate disruptive signals, causing
neighbours in low-density regions to be `still-born'.
It is then possible to construct models (*e.g.*
Bower et al. 1993)
in which the large-scale modulation of the galaxy density
is entirely non-gravitational in nature. However, it turned out to be
hard to make such mechanisms operate: the energetics and required scale
of the phenomenon are very large
(Rees 1985;
Dekel & Rees 1987).
These difficulties were only removed when the standard model
became a low-density universe, in which the dynamical
argument for high galaxy bias no longer applied.