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

4. The renormalization group equations

The starting point for the subsequent analysis is the KPZ equation for the velocity potential, eq. (13). Because this is a nonlinear equation, and because there are fluctuations, as represented by the stochastic noise term, it is to be expected on general grounds that renormalization effects will modify the coefficients appearing in this equation. Due to the existence of fluctuations, the coefficients in Eq. (13) do not remain constant with scale: as is well known (Ma 1976; Amit 1984; Binney 1992) there are renormalization effects that take place and modify these coefficients (or "coupling" constants); the modifications are calculable and can be computed using dynamical renormalization group (DynRG) techniques. The renormalization group is a standard tool developed for revealing in a systematic way, how couplings change with scale in any physical system under the action of (coarse)-graining. Moreover, it predicts that all correlation functions go as power laws when the system is near a fixed, or critical point, i.e., they exhibit self-similar behavior. However, the value of the scaling exponent changes from one fixed point to another. In fact, the values for the couplings at some reference scale establish the fixed point to which the system will be attracted to or repelled from, because these reference values will belong to a specific "basin of attraction". We will look for the IR stable fixed points since these reflect the system behavior characteristic of long times and large distances and ceases to change as we look at the system at ever larger scales.

Indeed, even before fluctuations are accounted for, one can easily obtain the so-called canonical scaling laws for the parameters appearing in (13); these follow by performing the simultaneous scaling transformation , and and requiring the resultant equation to be form-invariant , that is, that the KPZ equation transforms to a KPZ equation at the new, larger () length and time scale. This simple requirement leads to scale dependence in the parameters appearing in the original equation of motion as well as in the correlation functions built up from the KPZ field . When fluctuations and noise are "turned on", renormalization effects change the scaling behavior of the couplings and correlation functions away from their canonical forms, in a way which depends on the basins of attraction for the fixed points of the DynRG equations of the couplings.

4.1. Calculation of the RGEs

The RG transformation consists of an averaging over modes with momenta k in the range where is the scale factor for the transformation, followed by a dilatation of the length scale in order to bring the system back to its original size ². Here, plays the rôle of an UV-momentum, or short distance, cut-off characterizing the smallest resolvable detail whose physics is to be described by the dynamical equations. In classical hydrodynamics, this scale would be identified with the scale in the fluid at and below which the molecular granularity of the medium becomes manifest and the hydrodynamic limit breaks down. The renormalization group transformation has the important property that it becomes an exact symmetry of the physical system under study whenever that system approaches or is near a critical point, because it is near the critical point (or points) where the system exhibits scale-invariance or, equivalently, where the system can be described in terms of a conformal field theory. The hallmark for a system near criticality is that its correlation functions display power law behavior. This means that the statistical properties of the system remain the same, except possibly up to a global dilatation or change of unit of length. We can calculate the power law exponents by requiring that the dynamical equations remain invariant under the above RG transformation and under the further change of scale

where z and are numbers which account for the response to the re-scaling. By eliminating s from the above, we arrive at the fact that the two-point correlation function for the velocity potential scales as

where is the roughness exponent, z the dynamical exponent, and the scaling function has the following asymptotic behavior (see, e.g., Barabàsi & Stanley 1995):

Notice that because of eq. (18), large s means going to the large distance or infrared (IR) and (for ) long time limit, while small s corresponds to the short distance or ultraviolet (UV) and short time limit.

Making use of the relations Eq. (14) and Eq. (10), one can immediately obtain the scaling behavior of the 2-point correlation function for the density contrast, and thus study the asymptotic behavior of this function in different regimes. In the following we will implement this procedure. First, we obtain the scaling or RG equations for the couplings by imposing form-invariance on the Eq. (6), then by using the property of constancy of the couplings near fixed points, we obtain and calculate the fixed points themselves and the corresponding values of the exponents.

We now characterize the Gaussian noise, . This is done by choosing the noise correlation function. Here we will use colored or correlated noise, whose Fourier transform satisfies

where and are two couplings describing the noise strength, and the , exponents characterize the noise power spectrum in the momentum and frequency domains. White noise corresponds to . The explicit functional form of the noise amplitude reflects the fact that correlated noise has power law singularities of the form written above (Medina el al. 1989).

The solution of (13) can be carried out in Fourier space where iterative and diagrammatic techniques may be developed (Medina et al. 1989). A standard perturbative expansion of the solution to Eq. (13) coupled with the requirement of form-invariance and the property of renormalizability, lead to the following RG equations for the couplings:

The full details of the calculation of these equations will be presented elsewhere (Martín-García & Pérez-Mercader 1998). The calculation of the RGE's for the massless KPZ equation in the presence of colored noise is given in (Medina et al. 1989). Here, s is the scale factor of Eq. (18), and is a dimensionless geometric factor proportional to the surface area of the d-sphere , where d is the spatial dimension. This set of equations describes how the coupling constants evolve as one varies the scale at which the system is studied, a process commonly referred to in the condensed matter literature as "graining". In the present context we will be interested in the coarse-graining behavior, since we seek to uncover the behavior of our cosmological fluid as one goes to larger and larger scales. In actual practice we will investigate the coarse-graining flow in a two-dimensional (white noise) or three-dimensional (correlated noise) parameter space. This is because one may cast the above set of RG equations in terms of a smaller, yet equivalent set, by employing the dimensionless couplings defined as , , and . Doing so, one arrives at the following reduced set of renormalization group equations:

The dimensionless couplings measure the strength of the "roughening" effect due to the combined action of noise and the non-linearity against the "smoothing" tendency of the diffusive term in Eq. (13): these couplings grow when either or the noise intensity (as given by the parameters and ) grow, and they become smaller when the diffusive action (measured by ) grows. The dimensionless coupling V measures the competition between the diffusion and mass terms. Indeed, neglecting for the moment the noise and nonlinear terms, one can write (13) in Fourier space as

Now, if (i.e., ), the perturbations in the field are damped. But if on the other hand, (), there exists a length scale below which the perturbations are damped, but above which they grow. The case corresponds to a length scale : this scale is smaller than the minimum resolvable length scale in the problem at hand, and therefore the diffusive term becomes unimportant. We remark in passing that the limiting behavior in the RG flow as leads to a technical requirement: one must assume that the diffusive term is non-negligible (i.e., ) in order to compute the RG equations.

From the above equations one calculates their fixed points to which the graining flow drives the couplings. The corresponding fixed point exponents z and follow from substituting the fixed point solutions so obtained back into the previous set of RG equations (23). These exponents control and determine the calculated asymptotic behavior of the correlation functions. The system will be attracted to fixed points in the IR or UV regimes depending on whether they are IR-attractive or UV-attractive although, as with any autonomous set of differential equations, other possibilities exist.

© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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