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:

• When normalized to the small scale, where it fits the data extremely well and in fact is the only viable structure formation model that explains well the small scale clustering of galaxies --> it greatly underpredicts the power on larger scales.

  In other words, standard CDM reaches a "homogeniety horizon" on fairly small scales and thus would produce no power on larger scales because, on those scales, there is no varitaion in the amount of CDM from one region to another (on that scale).

• When normalized to the COBE scale, which is where the normalization now HAS to be done because that is the hard limit; CDM greatly over predicts how much small scale structure we observe.

• To therefore save CDM we must invent ways in which the predicted small scale structure has been greatly supressed, by nature, to give us the observations.

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.

• Low Hubble Constant + Standard CDM: From chapter 1 we have that the critical density of the Universe goes as H². Lowering H hen significantly lowers the matter density which in turn means it takes longer for the Universe to reach the point where the energy density in the radiation field is equal to that in the matter field. This gives the Universe more time to wash out small scale fluctuations and thus reduces the clustering on small scales.

  Furthermore, lowering H makes the Universe older and hence there is more time available for gravitational instability to build the largest structures which are observed. However, for this variant to work, H has to be around 30 and there is no observational evidence for a value this low.

• Mixed Dark Matter: This is a case of fine tuning where the idea is to mix in just enough HDM to allow for the observed power on large scales, while retaining enough CDM to allow for early structure formation on small scales. The required amounts range from 10-30\% of HDM which puts rather stringent limits on the combined mass of the various neutrino species.

• Extra radiation + CDM: Again the goal here is to delay the epoch of matter-radiation energy density equality. The Low $H_o$ model lets this happen by lowering the matter-density. Equivalently we can simply raise the radiation density. Since the observed entropy of the Universe provides a strong constraint on the radiation in the form of CMB photons, we must look towards extra sources. One which has been proposed is an unstable relativistic particle (in particular the tau neutrino) whose main decay channel is radiation.

  But again, some fine tuning is necessary as if this particle decays during the epoch of primordial nucleosynthesis, that would upset one of the more accurate predictions of Big Bang Cosmology. Hence, we need just the right mass range for this particle to allow for a relatively late decay.

• Extra Sources of Anisotropy: In its simplest form, inflation strongly predicts a scale-invariant spectrum of Gaussian density perturbations. In the scale-invariant limit, the spectral index is n = 1, n excellent agreement with the COBE observations. If however, the spectrum is not quite scale invariant and has a spectral index slightly less than 1, then there will be less power on small scales. This deviation from the n = 1 case is called Tilted CDM.

  A similar "fix" can occur if we allow gravitational radiation to be a significant source of the anisotropy observed in the CMB. In this case, the overall amplitude of the density perturbations must also be lower.

• A positive Cosmological Constant: The standard inflationary theory strongly predicts that the Universe has zero spatial curvature at the present day. For most models in the past, this is accomplished by letting OMEGA = 1. However, a broader class of inflationary models reaches zero curvature via a combination of OMEGA and LAMBDA. IF most of the contribution to zero curvature comes from LAMBDA then OMEGA is low and hence the matter density is lower (like the low H case).

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

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The Electronic Universe Project

e-mail: nuts@moo.uoregon.edu