Annu. Rev. Astron. Astrophys. 1998. 36: 599-654
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5.5. Self-Similar Clustering in Scale-Free Models

Hierarchical clustering from scale-free initial conditions - initial power spectra P(k) propto kn in an Einstein-de Sitter universe (Omega = 1) - is expected to evolve in a self-similar way, with a unique length scale growing in comoving coordinates as [a(t)]alpha, alpha = 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 gamma = (9 + 3n) / (5 + n). Higher-order correlation functions are expected to vary as xiN propto r-(N-1) gamma, in agreement with the measured hierarchical scaling of correlation functions (Peebles 1980). The tempting conclusion is that the observed correlation function, with gamma = 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 xi (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 Omega < 1 and Lambda neq 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.

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