## H0

This linear relation is valid only for non-relativistic recessional speeds (v = cz ^ c), i.e., for z ^ 1. At large values there are additional terms, and furthermore one must specify which of the several distance parameters in general relativity is being used. At the small recessional velocities (compared to c) encountered by Hubble, such distinctions were not important. Modern work must take them into account.

Use (33) to calculate the distance of a galaxy that yields a measured redshift of z = 0.002,

which is the same example we just gave above. Equivalently, in astronomical units, 3 x 105 km s—1

At great distances where the recession speeds approach the speed of light, the Hubble law will deviate from the linear expression (33) in a way that depends upon the exact nature of the expansion of the universe. This will change the form of the distance-redshift relation at these large distances. For example, the expansion could be slowing due to the mutual gravitational attraction of the galaxies that make up the universe, or it might be accelerating due to "dark energy" as recent data suggest. In these cases, the starlight left distant galaxies at much earlier times, so the changing speed should become apparent in comparisons of the redshifts of close and distant galaxies. Currently, much effort is being expended to obtain distances independent of the Hubble law, e.g., with galaxies and supernovae as luminous standard candles (see below), in order to determine these deviations.

Size and age of the universe Returning to the simplest model wherein the expansion speed of any given galaxy is constant in time, one can find an approximate "size" of the "observable" universe. We presume this distance to be that where the expansion speed is approaching the speed of light. At this distance, one could not observe a celestial object because, at this recession speed, signals from it would be redshifted to nearly zero frequency. In the limit of recessional speed c, the photons would have no energy. To find this distance, substitute c for v in (29) to find c 3 x 105 km s-1

Ho 20 km s-1 MLY-1 observable universe)

the distance we quote in Table 2.

In this simple model, the material at the 15 GLY distance would be receding at speed c, and hence would have left our location 15 Gyr previously. This is an approximate age of the universe Tuniv. Formally one would write,

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