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2.1. Expansion

Probably the most important characteristic of the space in which we live is that it is expanding. The expansion rate, given by the Hubble Constant, sets the overall scale for most other observables in cosmology. Thus it is of vital importance to pin down its value if we hope to seriously constrain other cosmological parameters.

Fortunately, over the past five years tremendous strides have been made in our empirical knowledge of the Hubble constant. I briefly review recent developments and prospects for the future here.

HST-KEY Project:

This is the largest scale endeavor carried out over the past decade with a goal of achieving a 10% absolute uncertainty in the Hubble constant. The goal of the project has been to use Cepheid luminosity distances to 25 different galaxies located within 25 Megaparsecs in order to calibrate a variety of secondary distance indicators, which in turn can be used to determine the distance to far further objects of known redshift. This in principle allows a measurement of the distance-redshift relation and thus the Hubble constant on scales where local peculiar velocities are insignificant. The four distance indicators so constrained are: (1) the Tully Fisher relation, appropriate for spirals, (2) the Fundamental plane, appropriate for ellipticals, (3) surface brightness fluctuations, and (4) Supernova Type 1a distance measures.

The HST-Key project has recently reported Hubble constant measurements for each of these methods, which I present below [1]. While I shall adopt these as quoted, it is worth pointing out that some critics of this analysis have stressed that this involves utilizing data obtained by other groups, who themselves sometimes report different values of the Hubble constant for the same data sets.

Equation 0a

In the weighted average quoted above, the dominant contribution to the 9% one sigma error comes from an overall uncertainty in the distance to the Large Magellanic Cloud. If the Cepheid Metallicity were shifted within its allowed 4% uncertainty range, the best fit mean value for the Hubble Constant from the HST-Key project would shift downard to 68 ± 6.

S-Z Effect:

The Sunyaev-Zeldovich effect results from a shift in the spectrum of the Cosmic Microwave Background radiation due to scattering of the radiation by electgrons as the radiation passes through intervening galaxy clusters on the way to our receivers on Earth. Because the electron temperature in Clusters exceeds that in the CMB, the radiation is systematically shifted to higher frequencies, producing a deficit in the intensity below some characteristic frequency, and an excess above it. The amplitude of the effect depends upon the Thompson scattering scross section, and the electron density, integrated over the photon's path:

Equation 0b

At the same time the electrons in the hot gas that dominates the baryonic matter in galaxy clusters also emits X-Rays, and the overall X-Ray intensity is proportional to the square of the electron density integrated along the line of sight through the cluster:

Equation 0c

Using models of the cluster density profile one can then use the the differing dependence on ne in the two integrals above to extract the physical path-length through the cluster. Assuming the radial extension of the cluster is approximately equal to the extension across the line of sight one can compare the physical size of the cluster to the angular size to determine its distance. Clearly, since this assumption is only good in a statistical sense, the use of S-Z and X-Ray observations to determine the Hubble constant cannot be done reliably on the basis of a single cluster observation, but rather on an ensemble.

A recent preliminary analysis of several clusters [2] yields:

Equation 0d

Type 1a SN (non-Key Project):

One of the HST Key Project distance estimators involves the use of Type 1a SN as standard candles. As previously emphasized, the Key Project does not perform direct measurements of Type 1a supernovae but rather uses data obtained by other gorpus. When these groups perform an independent analysis to derive a value for the Hubble constant they arrive at a smaller value than that quoted by the Key Project. Their most recent quoted value is [3]:

Equation 0e

At the same time, Sandage and collaborators have performed an independent analysis of SNe Ia distances and obtain [4]:

Equation 0f

Surface Brightness Fluctuations and The Galaxy Density Field:

Another recently used distance estimator involves the measurement of fluctuations in the galaxy surface brightness, which correspond to density fluctuations allowing an estimate of the physical size of a galaxy. This measure yields a slightly higher value for the Hubble constant [5]:

Equation 0g

Time Delays in Gravitational Lensing:

One of the most remarkable observations associated with observations of multiple images of distant quasars due to gravitational lensing intervening galaxies has been the measurement of the time delay in the two images of quasar Q0957 + 561. This time delay, measured quite accurately to be 417 ± 3 days is due to two factors: The path-length difference between the quasar and the earth for the light from the two different images, and the Shapiro gravitational time delay for the light rays traveling in slightly different gravitational potential wells. If it were not for this second factor, a measurement of the time delay could be directly used to determine the distance of the intervening galaxy. This latter factor however, implies that a model of both the galaxy, and the cluster in which it is embedded must be used to estimate the Shapiro time delay. This introduces an additional model-dependent uncertainty into the analysis. Two different analyses yield values [6]:

Equation 0h

where kappa is a parameter which accounts for a possible deviation in cluster parameters governing the overall induced gravitational time delay of the two signals from that assumed in the best fit. It is assumed in the analysis that kappa is small.


It is difficult to know how to best incorporate all of the quoted estimates into a single estimate, given their separate systematic and statistical uncertainties. Assuming large number statistics, where large here includes the nine quoted values, I perform a simple weighted average of the individual estimates, and find an approximate average value:

Equation 1   (1)

2.2. Geometry

It has remained a dream of observational cosmologists to be able to directly measure the geometry of space-time rather than infer the curvature of the universe by comparing the expansion rate to the mean mass density. While several such tests, based on measuring galaxy counts as a function of redshift, or the variation of angular diameter distance with redshift, have been attempted in the past, these have all been stymied by the achilles heel of many observational measurements in cosmology, evolutionary effects.

Recently, however, measurements of the cosmic microwave background have finally brought us to the threshold of a direct measurement of geometry, independent of traditional astrophysical uncertainties. The idea behind this measurement is, in principle, quite simple. The CMB originates from a spherical shell located at the surface of last scattering (SLS), at a redshift of roughly z approx 1000):

Figure 1

Figure 1. A schematic diagram of the surface of last scattering, showing the distance traversed by CMB radiation.

If a fiducial length could unambigously be distinguished on this surface, then a determination of the angular size associated with this length would allow a determination of the intervening geometry:

Figure 2

Figure 2. The geometry of the Universe and ray trajectories for CMB radiation.

Fortunately, nature has provided such a fiducial length, which corresponds roughly to the horizon size at the time the surface of last scattering existed. The reason for this is also straightforward. This is the largest scale over which causal effects at the time of the creation of the surface of last scattering could have left an imprint. Density fluctuations on such scales would result in acoustic oscillations of the matter-radiation fluid, and the doppler motion of electrons moving along with this fluid which scatter on photons emerging from the SLS produces a characteristic peak in the power spectrum of fluctuations of the CMBR at a wavenumber corresponding to the angular scale spanned by this physical scale. These fluctuations should also be visually distinguishable in an image map of the CMB, provided a resolution on degree scales is possible.

Recently, two different ground-based balloon experiments, one launched in Texas and one launched in Antarctica have resulted in maps with the required resolution [7, 8]. Shown below is a comparison of the actual boomerang map with several simulations based on a gaussian random spectrum of density fluctuations in a cold-dark matter universe, for open, closed, and flat cosmologies. Even at this qualitative level, it is clear that a flat universe provides better agreement to between the simulations and the data than either an open or closed universe [7].

Figure 3

Figure 3. Boomerang data visually compared to expectations for an open, closed, and flat CDM Universe.

On a more quantitative level, one can compare the inferred power spectra with predicted spectra [9]. Such comparisions, for both the Boomerang and Maxima results yields a constraint on the density parameter:

Equation 2   (2)

For the first time, it appears that the longstanding prejudice of theorists, namely that we live in a flat universe, may have been vindicated by observation! However, theorists can not be too self-satisfied by this result, because the source of this energy density appears to be completely unexpected, and largely inexplicable at the present time, as we will shortly see.

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