At any given time, we can define a Hubble parameter

which is obviously related to the Hubble constant, because it is the ratio of an increase in scale factor to the scale factor itself. In fact, the Hubble constant HIf = 0, we can derive the kinematics of the Universe quite simply from the first Friedman equation. For a spatially flat Universe k = 0, and we therefore have

where is known as the critical density. For Universes whose densities are less than this critical density, k 0 and space is negatively curved. For such Universes it is easy to see from the first Friedman equation that we require 0, and therefore the Universe must carry on expanding for ever. For positively curved Universes (k 0), the right hand side is negative, and we reach a point at which = 0. At this point the expansion will stop and thereafter go into reverse, leading eventually to a Big Crunch as becomes larger and more negative.For the global history of the Universe in models with a cosmological constant, however, we need to consider the term as providing an effective acceleration. If the cosmological constant is positive, the Universe is almost bound to expand forever, unless the matter density is very much greater than the energy density in cosmological constant and can collapse the Universe before the acceleration takes over. (A negative cosmological constant will always cause recollapse, but is not part of any currently likely world model). [21] provides further discussion of this point.

We can also introduce some dimensionless symbols for energy densities in the cosmological constant at the current time, , and in “curvature energy”, . By rearranging the first Friedman equation we obtain

The density in a particular component of the Universe X, as a fraction of critical density, can be written as

where the exponent represents the dilution of the component as the Universe expands. It is related to the w parameter defined earlier by the equation = –3(1 + w); Equation (7) holds provided that w is constant. For ordinary matter = –3, and for radiation = –4, because in addition to geometrical dilution the energy of radiation decreases as the wavelength increases, in addition to dilution due to the universal expansion. The cosmological constant energy density remains the same no matter how the size of the Universe increases, hence for a cosmological constant we have = 0 and w = –1. w = –1 is not the only possibility for producing acceleration, however; any general class of “quintessence” models for which w will do. Moreover, there is no reason why w has to be constant with redshift, and future observations may be able to constrain models of the form w = wIn the simple case,

by definition, because = 0 implies a flat Universe in which the total energy density in matter together with the cosmological constant is equal to the critical density. Universes for which is almost zero tend to evolve away from this point, so the observed near-flatness is a puzzle known as the “flatness problem”; the hypothesis of a period of rapid expansion known as inflation in the early history of the Universe predicts this near-flatness naturally.We finally obtain an equation for the variation of the Hubble parameter with time in terms of the Hubble constant (see e.g. [114]),

where represents the energy density in radiation and the energy density in matter. We can define a number of distances in cosmology. The most important for present purposes are the
angular diameter distance D_{A}, which relates the apparent angular size of an object to its proper size, and
the luminosity distance D_{L} = (1 + z)^{2} D_{A}, which relates the observed flux of an object to its intrinsic
luminosity. For currently popular models, the angular diameter distance increases to a maximum as z
increases to a value of order 1, and decreases thereafter. Formulae for, and fuller explanations of, both
distances are given by [56].

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