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:
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).
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 [116
, 123
] (see also [21
] 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.
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 = w
.
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
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
= 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 = 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,
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 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].