*Astron. Astrophys. 344, 27-35 (1999)*
## 6. Scaling properties of matter density correlations
We can now begin to put together the results we have obtained in
relation with the scaling properties *predicted* by the DynRG.
Using equations (10) and (14) we can write for the 2-point correlation
function of the density contrast in comoving coordinates that
We see that the scaling behavior of
is determined once we know the
scaling behavior for the equal time two-point correlation function of
the density contrast for the velocity potential
,
From Eqns. (19) and (21), we have
Introducing (for convenience and notational consistency with the
literature in critical phenomena) ,
we have
which on using Eq. (26) above for equal times gives
for the correlation function, and
for its Fourier transform, the power spectrum.
Eqs. (30) and (31) are now suitable for comparison with
observations ^{3},
since they are written in comoving coordinates, and therefore
correspond to quantities directly inferred from observations and
provided we assume that light traces mass. Otherwise we would need to
correct for bias, a strategy we leave for a future publication. From
(30) and the definition of the exponent
, we see that the exponent
measured from large scale galaxy
surveys, where , is calculable in
term of the roughening exponent and
is given by ^{4}
Using our fixed point analysis we have computed all the exponents
and *z* for all the fixed
points (fourth column in Table 1). The predicted values for
are tabulated in the table as
shown. Thus, at each fixed point, we derive a power law for the
density correlation function . In
the case of white noise alone, we see that none of the three
corresponding fixed points are capable of reproducing any of the
inferred values of
() from observations. It is of
interest to point out, however, that the point
yields the exponent
, which means that the correlation
function is strictly constant at this point, i.e., the matter
distribution is perfectly homogeneous. However,
is an IR unstable saddle point and
a fine-tuning in the initial values of
and *V* would be required in
order to have the system flow into it under coarse-graining.
For colored noise, the RG behavior becomes significantly richer
since we now must map out RG equation flow trajectories in a three
dimensional space of couplings (see Fig. 2). A glance at the tabulated
calculated values of the exponent
shows that is the only fixed point
capable of yielding values of
within the currently accepted range of values inferred from
observations, by choosing and any
, since
for
. Although
is independent of
, a non-zero value of
in the above interval must be chosen
in order that exist (this pair of
fixed points vanish identically when
). It is also the *only*
IR-stable fixed point in the three-dimensional coupling space spanned
by the couplings (*V*, ,
). This implies the following
behavior: any point initially in the basin of attraction of
inevitably ends up at this fixed
point as larger and larger scales are probed: *no fine tuning is
required* .
© European Southern Observatory (ESO) 1999
Online publication: March 10, 1999
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