Edward J. Barlow
Member of National Academy of Engineering
Recipient of NASA Public Service Award
Previous member Report Review Committee of the National Research Council
Retired Vice President, Research & Development, Varian Associates

Now, let the story unfold

NOTE: If you don't have the "Symbol" font, you won't be able to view some of the Greek letters correctly. On some older browsers the greek letter rho sometimes looks like the letter "r."
When astronomers look out at the heavens on a clear night with large telescopes, they see many distant galaxies of different sizes and shapes and different brightness. They also see supernovae and quasars. For each such object its brightness or apparent luminosity and the red shift of its spectral lines can be measured. When a galaxy (or supernova or quasar) is moving away from us, the wavelengths of its spectral lines are shifted toward the red end of the spectrum. This red shift is related to the velocity of recession of the galaxy (blue shift if the galaxy is coming toward us) and the apparent luminosity is an indication of its distance. It was noted in early observations that although some nearby galaxies had slight blue shifts all the remote ones had redshifts.

Edwin Hubble made many measurements and found that the velocity of recession, v, as determined from the redshift, was approximately proportional to the distance from us, d, determined from the apparent luminosity. The velocities he measured were small compared to the velocity of light, c. This observation created the Hubble constant, H, as in the equation v=Hd. Later measurements extended the range of this relationship but still supported the form v=Hd. This simple equation suggests that if we run time backwards and assume a constant velocity for each galaxy, there was a point in time when all the galaxies in a particular finite region of space such as the presently visible universe were closer together in the past and the universe was more dense.

If we think of this contraction as occurring at a constant rate then there was a time in the past when the space represented by that presently visible universe was extremely dense and, in fact, reached a condition of such extreme matter-energy density and temperature and pressure that the present laws of physics no longer held. If we arbitrarily continue the extrapolation backward to a point of apparent infinite density, we can take this time as t=0 and think of the rapid expansion of space itself after that time as the "big bang". The time after the big bang, the present time, can be called to. This time is simply 1/H in suitable units for the case where the velocity of expansion has been taken to be constant during this period.

Cosmologists generally think in terms of an extremely short time like 10-43 seconds for the initial period when the density was too high for present physics to hold followed by a very rapid expansion period called inflation. All of this taking place in something like 10-32 seconds. Following this, came a period of a very dense, rapidly expanding universe dominated by radiation rather than matter. This period lasted some 300,000 years or so. After this period radiation and matter decoupled and the radiation left over from that period now stretched in wavelength by something like a factor of 1000 is the cosmic microwave background radiation (CMBR) of such interest to cosmologists today. At about the same epoch the universe became matter dominated and we can neglect radiation density compared to matter density for significantly later periods.

The present age of the universe is generally taken to be something like 14 billion years. 300,000 years is thus 0.002% of this total age. Although these early periods are of great interest to cosmologists we will consider in this paper only models that apply after this phase or for more than 99.99% of the history of the universe as we are interested in questions of the present and future rate of expansion and such things as lookback times and distances of distant galaxies. It should be noted in this picture however, that if our present universe is finite in extent it would have been squeezed to a minute speck at the beginning, but if the universe now is infinite in extent it could be infinitely dense and still infinite in extent at the beginning even though any particular finite volume such as the presently visible universe would be just a speck.

While H may be a constant at a given time over galaxies at many distances, it is not a constant over time itself during the period of interest in this paper for the commonly used models considered. Since H=v/d, if we track a given galaxy as space expands (and neglect its own proper motion which is a good approximation except for close-in galaxies) we will see that as time goes on for those models in which the recession velocity, v, is a constant or decreases with time and since d is increasing with time for an expanding universe, H will decrease with time. A recent set of measurements of distant supernovae indicates that perhaps the expansion rate is now actually increasing somewhat but this set of measurements still indicates insufficient increase with v in time to offset the increase in d as galaxies move away, hence H is decreasing at the present time in this case also.

For several models of this paper H in fact varies as 1/t. For the last model, Model V, H approaches a constant value eventually. The present value of H is called Ho. Recent measurements suggest a value in the units usually used of between 50 and 80 kilometers per second per megaparsec for this constant ( a parsec is 3.26 light years). This would give a value for to of 19.6 to 12.2 billion years if galaxies have receded at constant velocities during this time. This is called the Hubble time or 1/Ho in suitable units. More about this later as specific models are discussed.

A cosmological model as discussed in this paper is a mathematical model used by cosmologists to relate, among other things, redshift and luminosity to distances and look-back times and velocities of recession and to realize that there might be two distances, the distance a galaxy is away now as we receive its light, Dr, and the distance away it was when the light was emitted that we see now, De. There are also two times to be considered, the time since the big bang when the light was emitted that we see now, te, and the time when the light is received, tr (which is also called to when it is the present time). There are several different cosmological models in use by cosmologists and they give somewhat different results for these distances and times for any given observation of luminosity and redshift. They also give somewhat different results for the future development of the universe. New observational results continue to refine and select the models and suggest new ones.

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