## 2. From hydrodynamics to the KPZ equationStructure formation in the Universe in the range of a few Mpc up to
several hundred Mpc can be modelled as the dynamical evolution of an
ideal self-gravitating fluid and studied within the framework of
General Relativity. However, we will restrict our treatment to the
case of Newtonian gravity for a number of physically justifiable
reasons. First and foremost, the length scales involved in large scale
structure formation by matter after decoupling from radiation are
smaller than the Hubble radius (of order
Mpc at the present epoch), so that
general relativistic effects are negligible. Secondly, it is known
observationally that matter flow velocities are much smaller than the
speed of light and that this non-relativistic matter plays a dominant
rôle over radiation in the epoch of structure formation. Thus,
the evolution in space and time of large scale structure in the
Universe will be described adequately by the Newtonian hydrodynamic
equations for a fluid whose components are in gravitational
interaction in an expanding background
metric where and are the peculiar velocity and acceleration fields of the fluid, respectively. In the above equations, is the Hubble parameter, is the scale factor for the underlying cosmological background and is the (time-dependent) average cosmological density. Thus, the local density is and the dimensionless density contrast is . In the linear regime and for dust (), the vorticity of the velocity field damps out rapidly by virtue of the background expansion, and therefore can be derived from a velocity potential , , in the long-time limit. Furthermore, parallelism between and holds, i. e., where is the long-time limit solution of the following Ricatti equation Following the Zel'dovich approximation, we make the assumption that in the weakly non-linear regime, the parallelism condition (4) continues to hold, with given as the solution of (5) but we now employ the fully non-linear Euler equation (2) with pressure term included. That is, we assume that the non-linearities and pressure have not yet had enough time to destroy the alignment in the acceleration and velocity fields of the cosmic fluid. Assuming also that the pressure is a function of the density, , one arrives at the following equation for the velocity potential (Buchert et al. 1997) where we have also added a stochastic source or noise (see a few lines below for a discussion). The three dimensionless functions of time, , and are The positive constants and
The Zel'dovich approximation (4) together with the Poisson equation yield an important relationship between the density contrast and , the Laplacian of the velocity potential, namely which shows that the density contrast tracks the divergence of the peculiar velocity. Because of this identity, we are able to calculate density correlation functions in terms of (derivatives of) velocity potential correlations. This is why it is worthwhile investigating the scaling properties of the KPZ equation in detail. The noise appearing in (6)
represents the effects of random forces acting on the fluid particles,
including the presence of dynamical friction, and also of degrees of
freedom whose size is smaller than the with a given function of its arguments (see below). All higher cumulants vanish. The noise is thus Gaussian. However, the velocity field need not be (and in general, will not be) Gaussian as a consequence of coarse-graining of the dynamics. The stochastic hydrodynamic equation (6) can be further cast into by means of the following simple change of variables and
redefinition of the "physical-time" The linear (mass) term in Eq. (13) is given by and the noise transforms into Finally, Eq. (13) is a generalization to a cosmological setting of the Kardar-Parisi-Zhang (KPZ) equation for surface growth (Kardar et al. 1986), and differs from the standard KPZ equation in the linear (mass) term which, as seen from Eq. (16), originates in the expansion present in the background cosmology. Here, plays the rôle of a diffusion constant while is proportional to the average "speed of growth" of . © European Southern Observatory (ESO) 1999 Online publication: March 10, 1999 |