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The Hubble constant for the present epoch is determined from the dependence of luminosity distance [d.sub.L] versus the redshift parameter z, which for the populated Milne-type model universe is given by
where [d.sub.L] is the luminosity distance and [r.sub.0] is the radius of radiation area, which is the area of the horizon broken.
where z is the redshift and -[D.sub.L](z) is the luminosity distance depending on the cosmological model.
This translates to 7.5 billion years travel time to Earth, or 20 billion light-years in its luminosity distance (see "Distances in the Ever-Expanding Universe," page 40).
We are interested in a special class of cosmological models: cosmologies with a Hubble constant that does not vary over time to conform to the linear relationship between the luminosity distance and the redshift observed for Type Ia supernovae [1].
The luminosity distance that we can calculate from measured brightness and known luminosity represents how far away such an object would be in a static universe, one that is neither expanding nor contracting.
As a matter of fact, in the present cosmological model, the luminous portion of the Universe is expanding at a constant rate as in the de Sitter cosmology in a flat Universe; this is also the condition required in order for the model to match the luminosity distance versus redshift relationship of supernovae Ia.
The luminosity distance is an important concept in cosmology as this is the distance measure obtained from supernovae data using the distance modulus.
Combining the previous equations we get the luminosity distance R as a function of the redshift z of distant astronomical objects:
There are five possible ways to explain the luminosity distance ([D.sub.L]) and redshift (Z) measurements of type Ia supernovae (SNeIa) according to the general relativity (GR), which derives the Friedmann equation (FE) with the FriedmannLemaitre-Robertson-Walker (FLRW) metric of the 4D spacetime (Figure 1).
A dimensionless luminosity distance is given by (see appendix)
which is essentially smaller than the luminosity distances. This means that the angular distances of objects at large z values depend as [d.sub.[[theta]] [varies] [z.sup.-1].