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