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1.2 A little cosmology

The expanding Universe is a consequence, although not the only possible consequence, of general relativity coupled with the assumption that space is homogeneous (that is, it has the same average density of matter at all points at a given time) and isotropic (the same in all directions). In 1922 Friedman [47] showed that given that assumption, we can use the Einstein field equations of general relativity to write down the dynamics of the Universe using the following two equations, now known as the Friedman equations:
a˙2 − 1-(8πG ρ + Λ)a2 = − kc2, (2 ) 3 ¨a 4 2 1 a-= − 3-πG (ρ + 3p ∕c ) + 3Λ. (3 )
Here a = a(t) is the scale factor of the Universe. It is fundamentally related to redshift, because the quantity (1 + z) is the ratio of the scale of the Universe now to the scale of the Universe at the time of emission of the light (a0 / a). Λ is the cosmological constant, which appears in the field equation of general relativity as an extra term. It corresponds to a universal repulsion and was originally introduced by Einstein to coerce the Universe into being static. On Hubble’s discovery of the expansion of the Universe, he removed it, only for it to reappear seventy years later as a result of new data [116Jump To The Next Citation Point123Jump To The Next Citation Point] (see also [21Jump To The Next Citation Point] for a review). k is a curvature term, and is –1, 0, or +1, according to whether the global geometry of the Universe is negatively curved, spatially flat, or positively curved. ρ is the density of the contents of the Universe, p is the pressure and dots represent time derivatives. For any particular component of the Universe, we need to specify an equation for the relation of pressure to density to solve these equations; for most components of interest such an equation is of the form p = wρ. Component densities vary with scale factor a as the Universe expands, and hence vary with time.

At any given time, we can define a Hubble parameter

H (t) = ˙a∕a, (4 )
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 H0 is just the value of H at the current time.

If Λ = 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

2 ρ = ρ ≡ 3H--, (5 ) c 8πG
where ρ c 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 a˙> 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 a˙ = 0. At this point the expansion will stop and thereafter go into reverse, leading eventually to a Big Crunch as a˙ becomes larger and more negative.

For the global history of the Universe in models with a cosmological constant, however, we need to consider the Λ 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, 2 ΩΛ ≡ Λ ∕(3H 0), and in “curvature energy”, 2 2 Ωk ≡ kc ∕H 0. By rearranging the first Friedman equation we obtain

H2 ρ --2 = --− Ωka −2 + Ω Λ. (6 ) H 0 ρc

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

ρX ∕ρc = ΩX a α, (7 )
where the exponent α represents the dilution of the component as the Universe expands. It is related to the w parameter defined earlier by the equation α = –3(1 + w); Equation (7View Equation) holds provided that w is constant. For ordinary matter α = –3, and for radiation α = –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 α = 0 and w = –1. w = –1 is not the only possibility for producing acceleration, however; any general class of “quintessence” models for which w < − 1 3 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 = w0 + w1z. The term “dark energy” is usually used as a general description of all such models, including the cosmological constant; in most current models, the dark energy will become increasingly dominant in the dynamics of the Universe as it expands.

In the simple case,

∑ ΩX + Ω Λ + Ωk = 1 (8 ) X
by definition, because Ωk = 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 Ωk 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]),

H2 = H2 (Ω Λ + Ωma −3 + Ωra −4 − Ωka− 2), (9 ) 0
where Ωr represents the energy density in radiation and Ωm 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 DA, which relates the apparent angular size of an object to its proper size, and the luminosity distance DL = (1 + z)2 DA, 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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