BURGERS-EQUATION, DEVILS STAIRCASES AND THE MASS-DISTRIBUTION FOR LARGE-SCALE STRUCTURES

Work initiated by Zeldovich (Astran. & Astrophys. 5, 84, 1970) and the Russian school indicates that the formation of large-scale structures in the Universe can be analyzed via the adhesion model, a multi-dimen sional form of the Burgers equation. Initial conditions frequently con sidered for the cosmological problem are Gaussian with a power-law spe ctrum (fractional Brownian motion). Already in the simpler one-dimensi onal case, with ordinary Brownian motion as initial condition, numeric ally supported conjectures by She et al. (Comm. Math. Phys. 148, 623, 1992) have led to a proof by Sinai (ibidem, p. 601) of the following r esult: there is a Devil's staircase of dimension 1/2 in the Lagrangian map for the solution of the Burgers equation in the limit of vanishin g viscosity. The Lagrangian map is the correspondence between initial (Lagrangian) position and present (Eulerian) position. A Devil's stair case is a non-decreasing function which varies only on a set of zero m easure and fractal dimension D. Such a Devil's staircase is not direct ly observable on Eulerian data, but its signature is a power-law in th e mass function (distribution of masses) at small masses, with an expo nent related to the fractal dimension D. After critical examination of the steps leading from the microscopic Jeans-Vlasov-Poisson for self- gravitating dust to the adhesion model, the main goal of this paper is to give a comprehensive introduction to these recent theoretical deve lopments and to extend them to initial power-law spectra with a wide r ange of exponents and to more than one dimension. The necessary geomet ric and probabilistic tools, due mostly to Sinai, are here presented i n a detailed but rather elementary way, intended for a readership of g eneral physicists. The extensions of Sinai's theory presented here oft en include heuristic elements. Most of the predictions are however tes ted by accurate numerical experiments in one and two dimensions, using a new ''Fast Legendre Transform'' algorithm which exploits a monotoni city property and has very low storage requirements. Predictions of th e present theory for the mass function are compared to those of Press and Schechter (Ap. J. 187, 425, 1974). Their expression for the mass f unction is found to agree with the adhesion model at large masses in a ny dimension. At small masses, there are discrepancies in dimensions h igher than one. In one dimension the scaling behavior at small masses is correctly given by the Press-Schechter theory. This may however be fortuitous: Sinai's theory, recast in the language of gravitational co llapse, tells us that the scaling originates not from the condition of collapse of a given region of small size, but from a condition of non -collapse of an extended halo around that region.