Astron. Astrophys. 344, 27-35 (1999)

2. From hydrodynamics to the KPZ equation

Structure 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 ¹. The connection with observations and the subsequent phenomenology is made by writing these equations in terms of the comoving coordinates and coordinate time t, and are (Peebles 1993)

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 T are introduced to carry dimensions. One can think of them as typical values of the corresponding time-dependent coefficients during the epoch we are interested in. The reason why we introduce them here explicitly (instead of setting them equal to unity) will become clear in Sect. 4. The -term arises from the pressure term in the Euler equation which we have expanded to lowest order about the zero-pressure limit. Note that higher-order terms in this Taylor expansion will yield higher-derivative terms involving quadratic and higher powers of the field . The -term is simply the convective term from the original Euler equation (2) written in the new variables, while the -term entails the competition between the damping of perturbations due to the expansion and the enhancement due to self-gravity .

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 coarse-graining length (indeed, the very fact that we are using continuous fields to describe the dynamics of a system of discrete particles means that we are implicitly introducing a coarse-graining length, such that the details below this resolution length are not resolvable. (The relevance of this length-scale will become clear in Sect. 4.) There exist a number of physical processes on various length and time scales that contribute to an effective stochastic source in the Euler equation. Indeed, any dissipative or frictional process leads to a stochastic force, by virtue of the fluctuation-dissipation theorem. So, fluid viscosity and turbulence should be accountable to some degree by adding a noise term in the dynamical equations. Early and late Universe phase transitions, the formation of cosmic defects such as strings, domain walls, textures, are sources for a noisy fluctuating background, as are also the primordial gravitational waves and gravitational waves produced during supernovae explosions and collapse of binary systems. Another way to visualize the noise is to imagine a flow of water in a river; there are stones and boulders in the river-bed which obviously perturb the flow and have an impact on its nature. The noise can be thought of as a means for modelling the distribution of obstacles in the river bed. In cosmology the obstacles in the river-bed represent, e.g., density fluctuations and the fluid is the matter flowing in this "river". The noise is phenomenologically characterized by its average value and two-point correlation function as follows,

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" t into a "conformal-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
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