There are two additional parameters to measure:

= ratio of actual to critical density.

= 1 is spatially flat Universe predicted by inflation

T_glob = ages of the oldest stars in the galaxy, which provides a lower limit on the expansion age.

Here are the choices of cosmological models that are in agreement with the observations in one state of another:

Choices of Cosmological Models (Five Options):

1. The Sandage Model:
   • H = 50
   • The ages of globular clusters are T = 16+/- 1 Gyr
   • = 0.1
   • All Baryonic Universe

2. The de Vaucouleurs/Aaronson Model
   • H = 100
   • The ages of globular clusters are not well known but must be around 10 billion yars
   • The Universe has to be open
   • All the Mass is in Galaxies

3. The Overlapping Error Bar Model:
   • H = 75
   • The age of globular clusters is 12--15 years
   • is 0.1--1.0
   • There may or may not be lots of dark matter

4. The Inflationary Model:
   • H = 50
   • The ages of globular clusters are 12-13 Gyr
   • = 1.0000000000000
   • Dark Matter Dominated Universe

5. The 90's Data Driven Cosmology
   • H = 80--90
   • Globular Cluster Ages: 15--17 Gyr
   • --> 0.1 --0.3
   • Quasi-dark matter dominated
   • Inflation is satisified with non-zero
   The Power Spectrum of the Galaxy Distribution
   • We assume Guassian fluctuations
   • We assume scale invariant fluctuations (since we don't have a theory for their origin) and express this as a power law:
     

     where K is the wave number.

   • The phases of are random
   • The spectral index n determines the relative distribution of power on large and small scales
   • If we assume that the density contrast within a horizon is a constant (constant-curvature fluctuations) then n= 1 is predicted. This is the so called "Harrison-Zeldovich" scale invariant spectrum and is a prediction of inflation.

   • The observed spectrum today has n less than 1 because differently sized structures enter the horizon at different times and thus are exposed to different growing conditions. One can specify various transfer functions that operate to reshape the initial power spectrum.

   • This reshaping occurs within one horizon and is what we can measure today. This is the complex galaxy distribution:
     

     The COBE fluctuations are measured over many horizons and thus the scale invariant spectrum n = 1 should be recovered. The COBE data has measured n = 1.1 +/- 0.2.

     

     So now here is the data on the Power Spectrum of the Galaxy Distribution where we may have reached turnover so the n less than 0 portion meets the predicted n = 1 portion.

     

     Exotic Variations in CDM to Salvage the Model:

     The basic problem is that CDM can not simultaneously satisfy the large and small scale power constraints. Since the large scale constraint comes from COBE and hence is rather difficult to get around, can we be clever and think of some physics that would supress the small-scale fluctuations which are over predicted by CDM normalized at the COBE Scale:

     Why Hell Yes We Can

     The trick is to increase the ratio of radiation to matter density so as to extend the era of radiation dominance so that more of the small scale flucutations can be washed out.

     How to do this?

   • Lower H_o this automatically lower the matter density. If H_o = 30 we have no worries.

   • 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: Let's just raise the radiation density! But 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 necessaryi.

   • 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 n = 1 - in 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 \app 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.

   • Non-zero Lambda: The inflationary theory strongly predicts that the Universe has zero spatial curvature at the present day. A broad class of inflationary models reaches zero curvature via a combination of and . If most of the contribution to zero curvature comes from , then the lower leads to lower matter density, as in the case of low $H_o$.

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     The Physics of a Cosmological Constant