Annu. Rev. Astron. Astrophys. 1998. 36:
599-654
Copyright © 1998 by . All rights reserved |

**5.5. Self-Similar Clustering in Scale-Free Models**

Hierarchical clustering from scale-free initial conditions - initial
power spectra *P*(*k*)
*k*^{n}
in an Einstein-de Sitter
universe ( = 1) - is
expected to evolve in a self-similar
way, with a unique length scale growing in comoving coordinates as
[*a*(*t*)]^{},
= 2 / (3 + *n*) (e.g.
Press & Schechter
1974,
Efstathiou et al 1979,
Peebles 1980,
1985).
Although scale-free
initial conditions differ from realistic models with a physical
transfer function (Equation 10), they provide a theoretical
laboratory for understanding nonlinear gravitational instability and
have therefore been studied extensively.

One of the few analytical approaches to the strongly nonlinear regime
is provided by self-similar solutions of the BBGKY hierarchy governing
the growth of clustering
(Peebles 1980). Making
several approximations
to close the hierarchy and render it tractable,
Davis & Peebles
(1977)
obtained power-law solutions for the nonlinear correlation functions.
Their key assumption was that on small scales, the mean proper velocity
between pairs vanishes so that, on average, each particle has a fixed
number of neighbors per unit volume. Under this assumption, known as
stable clustering, the logarithmic slope of the nonlinear two-point
correlation function is predicted to be
= (9 +
3*n*) / (5 + *n*). Higher-order correlation functions are
expected to vary as
_{N}
*r*^{-(N-1)
}, in
agreement with the measured hierarchical scaling of correlation functions
(Peebles 1980).
The tempting conclusion is that the observed correlation function, with
= 1.8,
could be explained by stable clustering starting after recombination
from white noise (*n* = 0;
Peebles 1974).
However, *N*-body simulations show that scale-free
initial conditions result in a varying slope of the nonlinear correlation
function, approaching the predicted slope asymptotically only at very
high values of
(Efstathiou & Eastwood
1981,
Efstathiou et al 1988,
Bertschinger & Gelb
1991).

The assumption of stable clustering has come under increased scrutiny
recently.
Padmanabhan et al
(1996)
concluded from their simulations that stable clustering is violated, while
Jain (1997)
concluded that it holds.
Colombi et al (1996a)
tested stable clustering and the
predicted scaling of the *N*-point correlation functions as determined
from the cumulants of counts in cells, and they found a departure from the
predicted scaling but also showed that higher-resolution simulations
are needed for a definitive test.

Stable clustering is one of the ingredients of a remarkable linear to
nonlinear mapping of the correlation function introduced by
Hamilton et al (1991).
Guided by the simulation results of
Efstathiou et al (1988),
they showed how the initial linear correlation function may be deduced
from the nonlinear evolved one and vice versa. Their method has been
modified and tested with high-resolution *N*-body
simulations by
Jain et al (1995),
Padmanabhan (1996)
and extended to the power spectrum and to universes with
< 1 and
0 by
Peacock & Dodds
(1994,
1996).
This body of work is important in enabling deduction of the initial
power spectrum of fluctuations from the observed nonlinear spectrum
(Peacock & Dodds
1994,
1996,
Baugh & Gaztañaga
1996,
Peacock 1997).
However, its theoretical basis is not yet fully understood.