7. Discussion and conclusions
In this paper we have presented an analytical calculation of the density-density correlation function for non-relativistic matter in a FRW background cosmology. The calculation hinges on blending two essential theoretical frameworks as input. First, we take and then reduce the complete set of non-relativistic hydrodynamic equations (Euler, Continuity and Poisson) for a self-gravitating fluid to a single stochastic equation for the peculiar-velocity potential . The noise term models in a phenomenological but powerful way different stochastic processes on various length and time scales. Second, we apply the well-established techniques of the dynamical Renormalization Group to calculate the long-time, long-distance behavior of the correlation function of the velocity potential (which relates directly, as we have seen, and under the conditions we have specified, to the density-density correlation) as a function of the stochastic noise source in the cosmological KPZ equation.
When we carry out a detailed RG fixed point analysis for our KPZ equation, we find that simple white noise alone is not sufficient to account for the scaling exponent inferred from observations of the galaxy-galaxy correlation function. However, we can get close to this range for the observed exponent if the cosmic hydrodynamics is driven by correlated noise. In fact, we need only adjust the degree of spatial correlation of the noise to achieve this, since the calculated value of the exponent associated with is independent of the degree of temporal correlations, as encoded in . Moreover, is the only fixed point with this property, and is simultaneously IR-attractive, both desirable properties from the phenomenological point of view. This last property is very important, since it means that the self-similar behavior of correlations is a generic outcome of the dynamical evolution, rather than an atypical property that the system exhibits only under very special conditions.
Thus we have learned that colored noise seems to play an important rôle in the statistics of large scale structure. Now, why it is that a non-zero and a are the relevant noise exponents is a question one still has to ask. To answer it, one would in principle have to derive the noise source itself starting from the relevant physics responsible for generating the fluctuations. Here, we content ourselves with the phenomenological approach. Nevertheless, at some future point, it may be possible to say more about the spectrum of noise fluctuations.
It must be pointed out that the Renormalization Group allows us to compute only asymptotic correlations, i.e., after sufficient time has elapsed so that the effect of noise completely washes out any traces of the initial conditions in the velocity field or initial density perturbations. Hence, there will be a time-dependent maximum length scale, above which noise has not become dominant yet and correlations therefore remember the initial conditions, as well as a transition region between these two regimes. The length scale at which this transition occurs depends on noise intensity and on the amplitude of initial perturbations. Therefore, a physical model of the origin of noise would enable us to predict this scale, but such a task is well beyond the scope of the present paper.
It is also interesting that the Renormalization Group allows us to
compute the proportionality coefficient of the matter density
correlations, i.e., the quantity in
, by means of the so-called
improved perturbation expansion (Weinberg 1996). This length
scale marks the transition from
) to homogeneity
is also dependent on the noise
intensity and other free parameters of our model, so that a prediction
is impossible without a deeper knowledge of the noise sources. At this
point, we should mention the recent debate about the observational
value of : on the one hand there is
the viewpoint that homogeneity has been reached well within the
maximum scale reached by observations (Davis 1996)
(); on the other hand there is an
opposing viewpoint as expressed by Sylos Labini et al. 1998who argue
that . Either viewpoint can be
easily incorporated into our model, by suitably choosing the noise
intensity. But if the latter value of
turns out to be correct, this would
indicate within our model that the noise intensity is really large and
therefore, that noise and stochastic processes played a much more
important rôle in the History of the Universe than has been
thought to date.
© European Southern Observatory (ESO) 1999
Online publication: March 10, 1999