| |
*Astron. Astrophys. 344, 27-35 (1999)*
## 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
inhomogeneity (,
) to homogeneity
(,
). But
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
helpdesk.link@springer.de |