The origin of scaling in the galaxy distribution

I present an analytic model for non-linear clustering of the luminous (baryonic) material in a universe in which the gravitational field is dominated by dark matter. The model is based on a two-component generalization of the adhesion approximation in which the gravitational potential of the dark component is determined by the standard Zel'dovich approximation or one of its variants, or by an N-body simulation. The baryonic matter flow is dissipative and is driven by this dark matter gravitational potential. The velocity potential of the matter is described by a generalization of the Burgers equation: the random heat equation ('RH equation') with a spatially correlated Gaussian driving potential. The properties of the RH equation are well understood: it is closely related to the equation for the Anderson model and to Brownian motion in a random potential: the solution can be expressed in terms of path integrals. Using this it is possible to derive the scaling properties of the solution and, in particular, those of the resultant velocity field. Even though the flow is non-linear, the velocity field remains Gaussian and inherits its scaling properties from the gravitational potential. This provides an underlying dynamical reason why the density field in the baryonic component is lognormally distributed and manifests multifractal scaling. By explicitly putting dark and luminous matter on different footings, the model provides an improved framework for considering the growth of large-scale cosmic structure. It provides a solution for the velocity potential of the baryonic component in closed form (albeit a path integral) from which the statistical properties of the baryonic flow can be derived.