Power Spectrum of Galaxies

Statistical Characterization of Structure:

Structure as a function of physical scale size (wave number) is usually described in terms of a power spectrum:

is the Fourier transform of the primeval density fluctuations which are amplified by gravity to produce the observed structure. These density fluctuations give rise to the observed CMB anisotropy.

The spectral index n determines the relative distribution of power on various scales. n is not necessarily constant over the whole range of wave numbers

P(K) itself is most correctly considered as the functional representation of the power per unit volume in k-space. Observations reveal the power (or correlation function) per unit volume in physical space.

The mapping between physical space and k-space can only really be done under the hypothesis that the phases of are random . Fortunately, the random-phase hypothesis is directly predicted from inflation and in fact, would hold in any Universe which is isotropic.

We thus assume that the density fluctuations, , are Gaussian in nature. When windowed by some physicaly process, these guassian fluctuations produce a power spectrum of structure formation.

We attempt to uncover this power spectrum by preforming galaxy redshift surveys and compute the clustering or galaxies as a function of scale size. This produces a set of correlation functions which essentially define the probability of another galaxy occuring within a radius of X from a given galaxy.

A probability of unity defines the correlation length scale for that sample.

On the very largest scales, the cobe data have shown that the spectral index n is 1.1 +/- 0.1 ( where n = 1 is a prediction of inflation).

On the smallest scales, we have known for a long time that the galaxy correlation function has a slope of -1.8. A negative slope indicates that the clustering is stronger has the scale size decreases.

To connect the large scale to the small scale requires a power spectrum which "turns-over" at some characteristic spatial scale or wave number (k). This is a feature of the Cold Dark Matter Model.

Comparisons with the Observations:

The basic problem with standard CDM is simple:

This general dilemma appears on all different redshift surveys. Another examples is shown below (from Schueker etal 1996).

Note also that there is still som difficulty in establishing what the galaxy power specturm is on the largest scales. This is shown below.

The LCRS redshift survey data seem to indicate a stronger turnover at large scales than the CFA survey does.

How to Save CDM: ???

Again, we want to save it because its the only viable structure formation scenario, coupled with gravitational instability, that can actually form small scale structure early on in the Universe.

To save CDM requires some variations of the basic model. In general, these variations are designed to "fix" CDM so that it produces the correct shape and normalization of the power spectrum at both large and small scales. From both observational and physical points of view, some of these modifications should best be viewed as "desperate" or at least rather complex.

Over the next few lectures we will look at what the observational constraints are for these various fixes to standard CDM.

The Electronic Universe Project
e-mail: nuts@moo.uoregon.edu