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
there is a critical value of the energy density such that the spatial geometry is flat
(
):
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
have very simple equations of state of the form
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
. 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
(redshifting as
the universe expands); the radiation energy density therefore scales as
. Vacuum
energy does not change as the universe expands, so
; 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
term in (5
) represents an effective “energy density in
curvature”, and define
. We can define a corresponding density parameter
The ranges of values of the ’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
, and
is non-spacelike, for any
null vector
; this implies that energy does not flow faster than the speed of light [117]. For
a perfect-fluid energy-momentum tensor of the form (4
), these two requirements imply that
and
, 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
, we have either positive
and
or negative
and
. 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
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 , mostly in the
cosmic
microwave background [211, 87, 225]. If neutrinos are sufficiently low mass as to be relativistic today,
conventional scenarios predict that they contribute approximately the same amount [149
]. In the absence of
sources which are even more exotic, it is therefore useful to parameterize the universe today by the values
of
and
, with
, 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 . (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
Notice that positive-energy-density sources with cause the universe to decelerate while
leads to acceleration; the more rapidly energy density redshifts away, the greater the tendency towards
universal deceleration. An empty universe (
,
) expands linearly with time;
sometimes called the “Milne universe”, such a spacetime is really flat Minkowski space in an unusual
time-slicing.
In the remainder of this section we will explore the behavior of universes dominated by matter and vacuum
energy, . According to (33
), a positive cosmological constant accelerates the
universal expansion, while a negative cosmological constant and/or ordinary matter tend to decelerate it.
The relative contributions of these components change with time; according to (28
) we have
Given , the value of
for which the universe will expand forever is given by
The dynamics of universes with are summarized in Figure 1
, in which the arrows
indicate the evolution of these parameters in an expanding universe. (In a contracting universe they would
be reversed.) This is not a true phase-space plot, despite the superficial similarities. One important
difference is that a universe passing through one point can pass through the same point again
but moving backwards along its trajectory, by first going to infinity and then turning around
(recollapse).
Figure 1 includes three fixed points, at
equal to
,
, and
. The
attractor among these at
is known as de Sitter space – a universe with no matter density,
dominated by a cosmological constant, and with scale factor growing exponentially with time.
The fact that this point is an attractor on the diagram is another way of understanding the
cosmological constant problem. A universe with initial conditions located at a generic point on
the diagram will, after several expansion times, flow to de Sitter space if it began above the
recollapse line, and flow to infinity and back to recollapse if it began below that line. Since our
universe has expanded by many orders of magnitude since early times, it must have begun at a
non-generic point in order not to have evolved either to de Sitter space or to a Big Crunch.
The only other two fixed points on the diagram are the saddle point at
,
corresponding to an empty universe, and the repulsive fixed point at
, known as the
Einstein–de Sitter solution. Since our universe is not empty, the favored solution from this combination
of theoretical and empirical arguments is the Einstein–de Sitter universe. The inflationary
scenario [113
, 159
, 6
] provides a mechanism whereby the universe can be driven to the line
(spatial flatness), so Einstein–de Sitter is a natural expectation if we imagine that some unknown
mechanism sets
. As discussed below, the observationally favored universe is located on
this line but away from the fixed points, near
. It is fair to conclude
that naturalness arguments have a somewhat spotty track record at predicting cosmological
parameters.
The lookback time from the present day to an object at redshift is given by
In a generic curved spacetime, there is no preferred notion of the distance between two objects.
Robertson–Walker spacetimes have preferred foliations, so it is possible to define sensible notions of the
distance between comoving objects – those whose worldlines are normal to the preferred slices. Placing
ourselves at in the coordinates defined by (2
), the coordinate distance
to another comoving
object is independent of time. It can be converted to a physical distance at any specified time
by
multiplying by the scale factor
, yielding a number which will of course change as the universe
expands. However, intervals along spacelike slices are not accessible to observation, so it is typically more
convenient to use distance measures which can be extracted from observable quantities. These include the
luminosity distance,
The proper-motion distance between sources at redshift and
can be computed by using
along a light ray, where
is given by (2
). We have
The comoving volume element in a Robertson–Walker universe is given by
which can be integrated analytically to obtain the volume out to a distanceThe introduction of a cosmological constant changes the relationship between the matter density and expansion rate from what it would be in a matter-dominated universe, which in turn influences the growth of large-scale structure. The effect is similar to that of a nonzero spatial curvature, and complicated by hydrodynamic and nonlinear effects on small scales, but is potentially detectable through sufficiently careful observations.
The analysis of the evolution of structure is greatly abetted by the fact that perturbations start out very small (temperature anisotropies in the microwave background imply that the density perturbations were of order 10–5 at recombination), and linearized theory is effective. In this regime, the fate of the fluctuations is in the hands of two competing effects: the tendency of self-gravity to make overdense regions collapse, and the tendency of test particles in the background expansion to move apart. Essentially, the effect of vacuum energy is to contribute to expansion but not to the self-gravity of overdensities, thereby acting to suppress the growth of perturbations [149, 189].
For sub-Hubble-radius perturbations in a cold dark matter component, a Newtonian analysis suffices. (We may of course be interested in super-Hubble-radius modes, or the evolution of interacting or relativistic particles, but the simple Newtonian case serves to illustrate the relevant physical effect.) If the energy density in dynamical matter is dominated by CDM, the linearized Newtonian evolution equation is
The second term represents an effective frictional force due to the expansion of the universe, characterized by a timescale
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