From the Friedmann equation
(5) (where henceforth we take the effects of a
cosmological constant into account by including the vacuum energy
density _{} into the total density ), for any
value of the Hubble parameter *H* there is a critical value
of the energy density such that the spatial geometry is flat
(*k* = 0):

It is often most convenient to measure the total energy density in terms of the critical density, by introducing the density parameter

One useful feature of this parameterization is a direct connection between the value of and the spatial geometry:

[Keep in mind that some references still use ``'' to refer strictly to the density parameter in matter, even in the presence of a cosmological constant; with this definition (23) no longer holds.]

In general, the energy density
will include
contributions from various distinct components.
From the point of view of cosmology, the relevant feature of
each component is how its energy density evolves as the universe
expands. Fortunately, it is often (although not always) the
case that individual components *i* have very simple equations
of state of the form

with *w*_{i} a constant. Plugging this equation of state into
the energy-momentum conservation equation
_{µ}
*T*^{µ}
= 0, we find that the energy density has a power-law dependence
on the scale factor,

where the exponent is related to the equation of state parameter by

The density parameter in each component is defined in the obvious way,

which has the useful property that

The simplest example of a component of this form is a set
of massive particles with negligible relative velocities,
known in cosmology as ``dust'' or simply ``matter''.
The energy density of such particles is given by their
number density times their rest mass; as the universe
expands, the number density is inversely proportional to the
volume while the rest masses are constant, yielding
_{M}
*a*^{-3}. For
relativistic particles,
known in cosmology as ``radiation'' (although any relativistic
species counts, not only photons or even strictly massless
particles), the energy density is the number density times
the particle energy, and the latter is proportional to *a*^{-1}
(redshifting as the universe expands); the radiation energy
density therefore scales as
_{R}
*a*^{-4}.
Vacuum energy does not change as the universe expands, so
_{}
*a*^{0};
from (26) this implies a
negative pressure, or positive tension, when the vacuum energy is
positive. Finally, for some purposes it is
useful to pretend that the *-ka*^{-2}
*R*_{0}^{-2} term in (5)
represents an effective ``energy density in curvature'',
and define _k
-(3*k* /
8
*GR*_{0}^{2}) *a*^{-2}. We can define a
corresponding density parameter

this relation is simply (5) divided by *H*^{2}. Note that the
contribution from _{k}
is (for obvious reasons) not included in the definition of
. The usefulness of
_{k} is that it
contributes to the expansion rate
analogously to the honest density parameters
_{i}; we can write

where the notation _{i(k)}
reflects the fact that the
sum includes _{k} in
addition to the various components
of =
_{i}
_{i}. The most popular
equations of
state for cosmological energy sources can thus be summarized
as follows:

The ranges of values of the
_{i}'s which are allowed in
principle (as opposed to constrained by observation) will depend
on a complete theory of the matter fields, but lacking that we
may still invoke energy conditions to get a handle on what
constitutes sensible values. The most appropriate condition
is the dominant energy condition (DEC), which
states that *T*_{µ}
*l*^{µ}
*l*^{}
0, and
*T*^{µ}_{}
*l*^{µ} is non-spacelike, for any
null vector *l*^{µ}; this implies that energy does not
flow faster than the speed of light
[17].
For a perfect-fluid energy-momentum
tensor of the form (4), these two requirements
imply that +
*p* 0 and
||
|*p*|, respectively.
Thus, either the density is positive and greater in magnitude
than the pressure, or the density is negative and equal in
magnitude to a compensating positive pressure; in terms of the
equation-of-state parameter *w*, we have either positive
and |*w*|
1 or negative and
*w* = -1. That is,
a negative energy density is allowed only if it is in the form
of vacuum energy. (We have actually modified the conventional
DEC somewhat, by using only null vectors *l*^{µ} rather than
null or timelike vectors; the traditional condition would
rule out a negative cosmological constant, which there is no
physical reason to do.)

There are good reasons to believe that the energy density in
radiation today is much less than that in matter. Photons,
which are readily detectable, contribute
_{}
~ 5 x 10^{-5}, mostly in the
2.73 °K cosmic microwave background
[18,
19,
20].
If neutrinos are sufficiently low mass as to
be relativistic today, conventional
scenarios predict that they contribute approximately the same
amount [11].
In the absence of sources which are even more exotic,
it is therefore useful to parameterize the universe today by
the values of _{M} and
_{}, with
_{k} = 1 -
_{M} -
_{}, keeping
the possibility of surprises always in mind.

One way to characterize a specific
Friedmann-Robertson-Walker model is by the values of
the Hubble parameter and the various energy densities
_{i}. (Of
course, reconstructing the history of such a
universe also requires an understanding of the microphysical
processes which can exchange energy between the different
states.) It may be difficult, however, to directly measure the
different contributions to
, and it is therefore useful
to consider extracting these quantities from the behavior of
the scale factor as a function of time. A traditional measure
of the evolution of the expansion rate is the deceleration
parameter

where in the last line we have assumed that the universe is
dominated by matter and the cosmological constant.
Under the assumption that _{} = 0,
measuring
*q*_{0} provides a direct measurement of the current density
parameter _{M0}; however, once
_{}
is admitted as a possibility there is no single parameter
which characterizes various universes, and for most purposes
it is more convenient to simply quote experimental results
directly in terms of _{M} and
_{}.
[Even this parameterization, of course, bears a certain theoretical
bias which may not be justified; ultimately, the only unbiased
method is to directly quote limits on *a(t)*.]

Notice that
positive-energy-density sources with *n* > 2 cause the universe to
decelerate while *n* < 2 leads to acceleration; the more rapidly
energy density redshifts away, the greater the tendency towards
universal deceleration. An empty universe
( = 0,
_{k} = 1)
expands linearly with time; sometimes called the ``Milne
universe'', such a spacetime is really flat Minkowski space
in an unusual time-slicing.