| © CAMBRIDGE UNIVERSITY PRESS 1999
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3.2 Dynamics of the Expansion
EXPANSION AND GEOMETRY
The equation of motion for the scale factor
can be obtained in a quasi-Newtonian fashion.
Consider a sphere about some arbitrary point,
and let the radius be R (t) r,
where r is arbitrary. The motion of a point
at the edge of the sphere will, in Newtonian gravity,
be influenced only by the interior mass.
We can therefore write down immediately a differential
equation Friedmann's equation) that expresses conservation of energy:
r)2 / 2 - GM
/ (Rr) = constant. In fact, to get this far, we do require general
relativity: the gravitation from mass shells at large distances is not
Newtonian, and so we cannot employ the usual argument about
their effect being zero. In fact, the result that the
gravitational field inside a uniform shell is zero
does hold in general relativity, and is known as Birkhoff's
theorem (see chapter 2).
General relativity becomes even more vital in giving us the constant
of integration in Friedmann's equation [problem 3.1]:
(3.24)
Note that this equation covers all contributions
to 
, i.e. those from
matter, radiation and vacuum; it is independent
of the equation of state.
A common shorthand for relativistic cosmological models,
which are described by the Robertson-Walker metric and
which obey the Friedmann equation, is to speak of FRW models.
The Friedmann equation shows that a universe that is spatially closed
(with k = +1) has negative
total ``energy'': the expansion will eventually be halted
by gravity, and the universe will recollapse.
Conversely, an unbound model is spatially open
(k = -1) and will expand forever.
This is marvelously simple: the dynamics of the entire
universe are the same as those of a cannonball fired
vertically against the Earth's gravity.
Just as the Earth's gravity defines an escape velocity for projectiles,
so a universe that expands sufficiently fast will
continue to expand forever.
Conversely, for a given rate of expansion there is a
critical density that will bring the
expansion asymptotically to a halt:
(3.25)
This connection between the rate of expansion of the
universe and its global geometry is an astonishing
and deep result. The proof of the equation quoted
above is ``only'' a question of inserting the Robertson-Walker
metric into the field equations [problem 3.1], but the
question inevitably arises of whether there is a
quasi-Newtonian way of seeing that the result must be true;
the answer is ``almost''.
First note that any open model will evolve towards
undecelerated expansion provided its equation of
state is such that
R2 is a declining function of R -
the potential energy becomes negligible by comparison with the total
and tends to a constant. In this
mass-free limit, there
can be no spatial curvature and the open RW metric must
be just a coordinate transformation of Minkowski spacetime.
We will exhibit transformation later in this chapter
and show that it implies
R = ct for this model, proving the k = -1 case.
An alternative line of attack is to rewrite the Friedmann
equation in terms of the Hubble parameter:
(3.26)
Now consider holding the local observables H and
fixed but increasing
R without limit. Clearly, in
the RW metric this corresponds to going to the k = 0 form:
the scale of spatial curvature goes to infinity and
the comoving separation for any given proper separation
goes to zero, so that the comoving
geometry becomes indistinguishable from the Euclidean form.
This case also has potential
and kinetic energy much greater than total energy, so that the
rhs of the Friedmann equation is effectively zero.
This establishes the k = 0 case, leaving the closed
universe as the only stubborn holdout against Newtonian arguments.
It is sometimes convenient to work with the time derivative
of the Friedmann equation, for the same reason that
acceleration arguments in dynamics are sometimes
more transparent than energy ones. Differentiating
with respect to time requires a knowledge of
,
but this can be eliminated by means of conservation of energy:
d [
c2 R3] = -pd
[R3]. We then obtain
(3.27)
Both this equation and the Friedmann equation in fact
arise as independent equations from different components
of Einstein's equations for the RW metric [problem 3.1].
DENSITY PARAMETERS ETC.
The ``flat'' universe with k = 0 arises for a particular
critical density.
We are therefore led to define a density parameter
as the ratio of density to critical density:
(3.28)
Since and H
change with time, this defines an epoch-dependent density parameter. The
current value of the parameter should
strictly be denoted by 
0. Because this is
such a common symbol, we shall keep the formulae
uncluttered by normally dropping the subscript;
the density parameter at
other epochs will be denoted by (z).
The critical density therefore just depends on the
rate at which the universe is expanding.
If we now also define a dimensionless (current) Hubble parameter as
(3.29)
then the current density of the universe may be expressed as
(3.30)
A powerful approximate model for the energy content of the
universe is to divide it into pressureless matter
( 
R-3),
radiation ( R-4) and vacuum energy
( constant).
The first two relations just say that the number density
of particles is diluted by the expansion, with photons also
having their energy reduced by the redshift; the third relation
applies for Einstein's cosmological constant.
In terms of observables, this means that the density is written as
(3.31)
(introducing the normalized scale factor a = R / R0).
For some purposes, this separation is unnecessary, since the
Friedmann equation treats all contributions to the density parameter equally:
(3.32)
Thus, a flat k = 0 universe requires
i = 1 at all times,
whatever the form of the contributions to the density, even if the
equation of state cannot be decomposed in this simple way.
In terms of the deceleration parameter,
(3.33)
the 
form of the Friedmann
equation says that
(3.34)
which implies q = 3m / 2 + 2r -1 for a flat universe.
One of the classical problems of cosmology is to
test this relation experimentally.
Lastly, it is often necessary to know the present
value of the scale factor, which may be read directly from the
Friedmann equation:
(3.35)
The Hubble constant thus sets the curvature length,
which becomes infinitely large as
approaches unity from
either direction. Only in the limit of zero density
does this length become equal to the other common measure
of the size of the universe - the Hubble length, c /
H0.
SOLUTIONS TO THE FRIEDMANN EQUATION
The Friedmann equation is so named because Friedmann
was the first to appreciate, in 1922, that Einstein's
equations admitted cosmological solutions
containing matter only (although it was
Lemaître who in 1927 both obtained the solution
and appreciated that it led to a linear distance-redshift
relation). The term Friedmann model is therefore
often used to indicate a matter-only cosmology,
even though his equation includes contributions from all equations of state.
The Friedmann equation may be solved most simply in
``parametric'' form, by recasting it in terms of the
conformal time d 
= c dt / R (denoting derivatives
with respect to by
primes):
(3.36)
Because H02
R02 = kc2 / ( - 1), the Friedmann equation becomes
(3.37)
which is straightforward to integrate provided v = 0.
Solving the Friedmann equation for R (t) in this way is
important for determining global quantities such as the
present age of the universe, and explicit solutions for
particular cases are considered below.
However, from the point of view of observations, and in
particular the distance-redshift relation, it is
not necessary to proceed by the direct route of
determining R (t).
To the observer, the evolution of the scale factor is
most directly characterised by the change with redshift
of the Hubble parameter and the density parameter;
the evolution of H (z) and (z) is given immediately
by the Friedmann equation in the form
H2 = 8 
G
/ 3 -
kc2 / R2. Inserting the above dependence of
on a gives
(3.38)
This is a crucial equation, which can be used to obtain
the relation between
redshift and comoving distance. The radial equation of motion
for a photon is R dr = c dt = c dR / = c dR / (RH).
With R = R0 / (1 + z), this gives
(3.39)
This relation is arguably the single most important equation in
cosmology, since it shows how to relate comoving distance
to the observables of redshift, Hubble constant and density
parameters. The comoving distance determines the apparent
brightness of distant objects, and the comoving volume
element determines the numbers of objects that are observed.
These aspect of observational cosmology are discussed in
more detail below in section 3.4.
Lastly, using the expression for H (z) with (a) - 1 =
kc2 / (H2 R2) gives
the redshift dependence of the total density parameter:
(3.40)
This last equation is very important. It tells us that, at
high redshift, all model universes apart from those
with only vacuum energy will tend to look like
the = 1 model. This is not
surprising given the
form of the Friedmann equation: provided R2 -> 
as R -> 0, the -kc2 curvature term will become
negligible at early times.
If 
1, then in the distant
past (z) must
have differed from unity by a tiny amount: the density and
rate of expansion needed to have been finely balanced for the
universe to expand to the present. This tuning of the initial
conditions is called the flatness problem and is one of
the motivations for the applications of quantum theory to
the early universe that are discussed in later chapters.
MATTER-DOMINATED UNIVERSE
From the observed temperature of the microwave background
(2.73 K) and the assumption of three species of neutrino
at a slightly lower temperature (see later chapters), we deduce
that the total relativistic density
parameter is r h2 
4.2 x 10-5, so at present
it should be a good approximation to ignore radiation.
However, the different redshift dependences of matter and
radiation densities mean that this assumption fails at
early times: m / r (1
+ z)-1.
One of the critical epochs in cosmology is therefore
the point at which these contributions were equal:
the redshift of matter-radiation equality
(3.41)
At redshifts higher than this, the universal dynamics were
dominated by the relativistic-particle content.
By a coincidence discussed below, this epoch is close
to another important event in cosmological history:
recombination. Once the temperature falls below 104 K,
ionized material can form neutral hydrogen. Observational
astronomy is only possible from this point on, since
Thomson scattering from electrons in ionized
material prevents photon propagation. In practice, this
limits the maximum redshift of observational interest to
about 1000 (as discussed in detail in chapter 9); unless
is very low or vacuum energy
is important, a matter-dominated model is therefore a good approximation
to reality.
By conserving matter,
we can introduce a characteristic mass M*,
and from this a characteristic radius R*:
(3.42)
where we have used the expression for R0 in the first step.
When only matter is present, the conformal-time version of the
Friedmann equation is simple to integrate for R (), and
integration of dt = d / R gives t ():
(3.43)
This cycloid solution
is a special case of the general solution for the evolution of
a spherical mass distribution:
R = A [1 - Ck ()],
t = B [
- Sk ()], where A3 = GMB2 and
the mass M need not be the mass of the universe.
In the general case, the variable is known as the
development angle; it is only equal to the conformal
time in the special case of the solution to the Friedmann equation.
We will later use this solution to
study the evolution of density inhomogeneities.
The evolution of R (t) in this solution is plotted in
figure 3.4. A particular
point to note is that the behaviour at early times is
always the same: potential and kinetic
energies greatly exceed total energy and we always have the
k = 0 form R
t2/3.
At this point, we have reproduced one of the great
conclusions of relativistic cosmology: the universe is of
finite age, and had its origin in a mathematical
singularity at which the scale factor went to zero,
leading to a divergent spacetime curvature.
Since zero scale factor also implies infinite density (and
temperature), the inferred picture of the early universe is one
of unimaginable violence. The term big bang was
coined by Fred Hoyle to describe this beginning, although it was intended
somewhat critically. The problem with the singularity is
that it marks the breakdown of the laws of physics;
we cannot extrapolate the solution for R (t) to t < 0, and
so the origin of the expansion becomes an unexplained
boundary condition. It was only after about 1980 that a
consistent set of ideas became available
for ways of avoiding this barrier, as discussed in chapter 11.
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Figure 3.4. The time dependence of the
scale factor for open, closed
and critical matter-dominated cosmological models.
The upper line corresponds to k = -1, the middle line to the flat
k = 0
model, and the lowest line to the recollapsing closed k = +1 universe.
The log scale is designed to bring out the early-time behaviour, although it
obscures the fact that the closed model is a symmetric cycloid
on a linear plot of R against t.
|
The parametric solution cannot be rearranged to give
R (t), but it is clearly possible to solve for t (R).
This is most simply expressed
in terms of the density parameter and the age of the universe
at a given stage of its development:
(3.44)
When we insert the redshift dependences of H (z) and (z),
(3.45)
this gives us the time-redshift relation.
An alternative route to this result would have been
to use the general differential expression for comoving distance dr /
dz; since c dt = [R0 / (1 + z)]
dr, this gives the age of the universe as an integral over z.
An accurate and very useful approximation to the above exact result is
(3.46)
which interpolates between the exact ages of H-1 for
an empty universe and 2/3 H-1 for a critical-density
= 1 model.
MATTER PLUS RADIATION BACKGROUND
The parametric solution can
be extended in an elegant way for a universe containing a mixture
of matter and radiation. Suppose we write the mass inside R as
(3.47)
reflecting the R-3 and R-4
dependencies of matter and radiation densities respectively. Now define
dimensionless masses of the form y 
GM / (c2 R0), which
reduce to ym, r = k m, r / [2( - 1)].
The parametric solutions then become
(3.48)
MODELS WITH VACUUM ENERGY
The solution of the Friedmann equation becomes more complicated if we allow a
significant contribution from vacuum energy - i.e. a non-zero
cosmological constant. Detailed discussions of the problem are
given by Felten & Isaacman (1986) and Carroll, Press & Turner (1992);
the most important features are outlined below.
The Friedmann equation itself is independent of the
equation of state, and just says H2
R2 = kc2 / ( - 1),
whatever the form of the contributions to . In terms of the cosmological constant itself, we have
(3.49)
STATIC UNIVERSE
The reason that the cosmological constant was first introduced
by Einstein was not simply because there was no general reason to expect
empty space to be of zero density, but because it allows
a non-expanding cosmology to be constructed. This is perhaps not so
obvious from some forms of the Friedmann equation, since now H = 0
and = ; if we
cast the equation in its original form without defining
these parameters, then zero expansion implies
(3.50)
Since 
can have either sign,
this appears not to constrain k. However, we also want to have
zero acceleration for this model, and so need the
time derivative of the Friedmann equation:
= -4 GR
( + 3p) / 3. A
further condition for a static model is therefore that
(3.51)
Since = -p
for vacuum energy, and this is the only
source of pressure if we ignore radiation, this tells
us that = 3vac and hence that the
mass density is twice the vacuum density. The total
density is hence positive and k = 1; we have a closed model.
Notice that what this says is that a positive
vacuum energy acts in a repulsive way, balancing
the attraction of normal matter. This is related
to the idea of +
3p as the effective source density for gravity. This insight
alone should make one appreciate that the static model cannot
be stable: if we perturb the scale factor by a small positive amount,
the vacuum repulsion is unchanged whereas the ``normal'' gravitational
attraction is reduced, so that the model will tend to expand further (or
contract, if the initial perturbation was negative). Thinking along these
lines, a tidy history of science would have required Einstein to predict
the expanding universe in advance of its observation. However, it is
perhaps not so surprising that this prediction was never clearly made,
despite the fact that expanding models were studied by Lemaître
and by Friedmann in the years prior to Hubble's
work. In those days, the idea of a quasi-Newtonian
approach to cosmology was not developed; the common
difficulty of obtaining a clear physical interpretation
of solutions to Einstein's equations obscured the
meaning of the expanding universe even for its creators.
DE SITTER SPACE
Before going on to the general case, it is worth looking
at the endpoint of an outwards perturbation of Einstein's
static model, first studied by de Sitter and named after him.
This universe is completely dominated by vacuum energy,
and is clearly the limit of the unstable expansion, since
the density of matter redshifts to zero while the
vacuum energy remains constant. Consider again the Friedmann
equation in its general form 2 - 8 G
R2
/ 3 = -kc2:
since the density is constant and R will increase without
limit, the two terms on the lhs must eventually become
almost exactly equal and the curvature term on the rhs will
be negligible. Thus, even if k 0, the universe will have a density that differs only
infinitesimally from the critical, so that we can solve the equation by
setting k = 0, in which case
(3.52)
An interesting interpretation of this behaviour
was promoted in the early days of cosmology by Eddington:
the cosmological constant is what caused the
expansion. In models without , the expansion is merely
an initial condition: anyone who asks why the universe
expands at a given epoch is given the unsatisfactory
reply that it does so because it was expanding at
some earlier time, and this chain of reasoning
comes up against a barrier at t = 0. It would be more satisfying to
have some mechanism that set the expansion into
motion, and this is what is provided by vacuum repulsion.
This tendency of models with positive to end up undergoing an exponential phase of expansion
(and moreover one with = 1) is
exactly what is used in inflationary cosmology to
generate the initial conditions for the big bang.
THE STEADY-STATE MODEL
The behaviour of de Sitter space is in some ways reminiscent of the
steady-state universe, which was popular in the 1960s.
This theory drew its motivation from the philosophical problems of
big-bang models - which begin in a singularity at t = 0, and
for which earlier times have no meaning. Instead, Hoyle, Bondi and Gold
suggested the perfect cosmological principle in which
the universe is homogeneous not only in space, but also in time: apart
from local fluctuations, the universe appears the same to
all observers at all times. This tells us that the Hubble constant
really is constant, and so the model necessarily has exponential
expansion, R
exp(Ht), exactly as for de Sitter space. Furthermore, it is necessary
that k = 0, as may be seen by considering the
transverse part of the Robertson-Walker metric:
d 
2 = [R
(t) Sk (r) d 
]2. This has
the convention that r is a dimensionless
comoving coordinate; if we divide by R0
and change to physical radius r', the metric becomes
d 2 = [a
(t) R0 Sk (r' / R0)
d ]2.
The current scale factor R0 now plays the role
of a curvature length, determining the distance over which the model is
spatially Euclidean. However, any such curvature radius must be constant
in the steady-state model, so the only possibility is that it is infinite
and that k = 0. We thus see that de Sitter space is a steady-state
universe: it contains a constant vacuum energy density, and has an
infinite age, lacking any big-bang singularity. In this sense, some
aspects of the steady-state model have been resurrected in inflationary
cosmology. However, de Sitter space is a rather uninteresting
model because it contains no matter. Introducing matter into a
steady-state universe violates energy conservation, since matter does
not have the p = - c2 equation of state that allows the
density to remain constant. This is the most radical
aspect of steady-state models: they require
continuous creation of matter. The energy to
accomplish this has to come from somewhere, and
Einstein's equations are modified by
adding some ``creation'' or ``C-field'' term to the energy-momentum
tensor:
(3.53)
The effect of this extra term must be to cancel
the matter density and pressure, leaving just the overall
effective form of the vacuum tensor, which is required to
produce de Sitter space and the exponential expansion.
This ad hoc field and the lack of any physical motivation for
it beyond the cosmological problem it was designed to solve
was always the most unsatisfactory feature of the steady-state
model, and may account for the strong reactions generated by the theory.
Certainly, the debate between steady-state supporters and protagonists
of the big bang produced some memorable displays of vitriol in the
1960s. At the start of the decade, the point at issue was whether the
proper density of active galaxies was constant as predicted by the
steady-state model. Since the radio-source count data were in a somewhat
primitive state at that time, the debate remained inconclusive until the
detection of the microwave background in 1965. For many, this spelled
the end of the steady-state universe, but doubts lingered on about
whether the radiation might originate in interstellar dust.
These were perhaps only finally laid to rest in 1990,
with the demonstration that the radiation was
almost exactly Planckian in form (see chapter 9).
BOUNCING AND LOITERING MODELS
Returning to the general case of models with a mixture of energy in the
vacuum and normal components, we have to distinguish three cases. For
models that start from a big bang (in which case radiation dominates completely
at the earliest times), the universe will either recollapse or expand
forever. The latter outcome becomes more likely for low densities
of matter and radiation, but high vacuum density. It is however also
possible to have models in which there is no big bang: the universe was
collapsing in the distant past, but was slowed by the repulsion of a
positive term
and underwent a ``bounce'' to reach its present state
of expansion. Working out the conditions for these different events is a
matter of integrating the Friedmann equation.
For the addition of , this can only
in general be done numerically. However, we can
find the conditions for the different behaviours
described above analytically, at least if we simplify
things by ignoring radiation. The equation in the
form of the time-dependent Hubble parameter looks like
(3.54)
and we are interested in the conditions under which
the lhs vanishes, defining a turning point in the expansion.
Setting the rhs to zero yields a cubic equation, and it
is possible to give the conditions under which this has a
solution (see Felten & Isaacman 1986), which are as
follows.
always
implies recollapse, which is intuitively reasonable (either the mass causes
recollapse before dominates,
or the density is low enough that comes to dominate, which cannot lead to infinite expansion
unless is positive.
| (2)
|
If is positive and m < 1, the model always
expands to infinity.
|
| (3)
|
If m > 1, recollapse
is only avoided if v
exceeds a critical value
(3.55)
| (4)
|
If is large enough, the
stationary point of the expansion is at a < 1 and we have a
bounce cosmology. This critical value is
(3.56)
where the function f is similar in spirit to Ck:
cosh if m < 0.5,
otherwise cos. If the universe lies exactly on the critical line,
the bounce is at infinitely early times and we have a solution that is
the result of a perturbation of the Einstein static model. Models that almost
satisfy the critical v (m) relation
are known as loitering models, since they
spend a long time close to constant scale factor.
Such models were briefly popular in the early 1970s,
when there seemed to be a sharp peak in the quasar
redshift distribution at z 2. However, this no longer seems a reasonable explanation for what
is undoubtedly a mixture of evolution and observational selection.
| |
In fact, bounce models can be ruled out quite strongly.
The same cubic equations that define the critical
conditions for a bounce also give an inequality for
the maximum redshift possible (that of the bounce):
(3.57)
A reasonable lower limit for m of 0.1 then
rules out a bounce once objects are seen at z > 2.
The main results of this section are summed up in
figure 3.5..
Since the radiation density is very small today,
the main task of relativistic cosmology is to work out
where on the matter
- vacuum plane
the real universe lies. The existence of high-redshift
objects rules out the bounce models, so that the idea
of a hot big bang cannot be evaded. As subsequent
chapters will show, the data favour a position somewhere
near the point (1,0), which is the worst possible
situation: it means that the issues of recollapse
and closure are very difficult to resolve.
FLAT UNIVERSE
The most important model in cosmological research is
that with k = 0 -> total = 1; when dominated by matter,
this is often termed the Einstein-de Sitter model.
Paradoxically, this importance arises because it is
an unstable state: as we have seen earlier, the universe
will evolve away from = 1,
given a slight perturbation.
For the universe to have expanded by so many e-foldings
(factors of e expansion)
and yet still have ~ 1 implies
that it was very close to being spatially flat at early times. Many
workers have conjectured that it would be contrived if this flatness
was other than perfect - a prejudice raised to the status
of a prediction in most models of inflation.
Although it is a
mathematically distinct case, in practice
the properties of a flat model can usually
be obtained by taking the limit -> 1
for either open or closed universes with k = ± 1.
Nevertheless, it is usually easier to start again from the
k = 0 Friedmann equation, 2 = 8 G
img src="../GIFS/rho2.gif" alt="rho" align=middle> R2
/ (3c2). Since both sides are quadratic in R,
this makes it clear that the value of R0 is arbitrary,
unlike models with 1:
the comoving geometry is Euclidean, and there is no natural curvature scale.
It now makes more sense to work throughout in terms of the normalized
scale factor a (t), so that the Friedmann equation for a
matter-radiation mix is
(3.58)
which may be integrated to give the time as a function of scale factor:
(3.59)
this goes to 2/3 a3/2 for a matter-only model,
and to 1/2 a2 for radiation only.
One further way of presenting the model's dependence on time
is via the density. Following the above, it is easy to show that
(3.60)
The whole universe thus always obeys the rule-of-thumb for the
collapse from rest of a gravitating body:
the collapse time 1 /
sqrt(G ).
Because r is so
small, the deviations from a matter-only model are unimportant for
z 
1000, and
so the distance-redshift relation for the
k = 0 matter plus radiation model is effectively
just that of the m =
1 Einstein-de Sitter model.
An alternative k = 0 model of greater observational interest has a
significant cosmological constant, so that m + v = 1
(radiation being neglected for simplicity).
This may seem contrived, but once k = 0 has been established,
it cannot change: individual contributions to must adjust to keep in balance.
The advantage of this model is that it is the only way of retaining
the theoretical attractiveness of k = 0 while changing the
age of the universe from the relation H0
t0 = 2/3, which characterises the
Einstein-de Sitter model. Since much observational
evidence indicates that H0 t0 
1 (see chapter 5),
this model has received a good deal of interest in recent years.
To keep things simple we shall neglect radiation, so that
the Friedmann equation is
(3.61)
and the t (a) relation is
(3.62)
The x4 on the bottom looks like trouble, but it can be
rendered tractable by the substitution y =
sqrt(x3 |m - 1| / m), which turns the integral into
(3.63)
Here, k in Sk is used to mean sin if m > 1,
otherwise sinh; these are still k = 0 models.
This t (a) relation is compared to models without
vacuum energy in figure 3.6.
Since there is nothing special about the current era, we can clearly
also rewrite this expression as
(3.64)
where we include a simple approximation that is accurate to a few % over the
region of interest (m 
0.1). In the general case of significant but k 0,
this expression still gives a very good approximation to the exact result,
provided m is
replaced by 0.7m -
0.3v + 0.3
(Carroll, Press & Turner 1992).
HORIZONS
For photons, the radial equation of motion is just
c dt = R dr. How far can a photon get in a given time?
The answer is clearly
(3.65)
i.e. just the interval of conformal time.
What happens as t0 -> 0 in this expression? We can
replace dt by dR / , which the Friedmann equation says is dR / sqrt( R2) at
early times. Thus, this integral converges if
R2
-> as
t0 -> 0, otherwise it diverges.
Provided the equation of state is such that changes faster than R-2, light
signals can only
propagate a finite distance between the big bang and the present;
there is then said to be a particle horizon.
Such a horizon therefore exists in conventional
big bang models, which are dominated by radiation at early times.
A particle horizon is not at all the same thing as an event horizon:
for the latter, we ask whether 
r diverges as t -> . If it does, then
seeing a given event is just a question of waiting long enough.
Clearly, an event horizon requires R (t) to increase more quickly than
t, so that distant parts of the universe recede ``faster than light''.
This does not occur unless the universe is dominated by vacuum energy at
late times, as discussed above.
Despite this distinction, cosmologists usually
say the horizon when they mean the particle horizon.
There are some unique aspects to
the idea of a horizon in a closed universe, where
you can in principle return to your starting point by
continuing for long enough in the same direction. However, the
related possibility of viewing the back of your head (albeit
with some time delay) turns out to be more difficult once dynamics
are taken into account. For a matter-only model,
it is easy to show that the horizon
only just reaches the stage of allowing a photon to circumnavigate
the universe at the point of recollapse - the ``big crunch''.
A photon that starts at r = 0 at t = 0 will return to its initial
position when r = 2 , at which
point the conformal time = 2 also (from above)
and the model has recollapsed. Since we live in an expanding
universe, it is not even possible
to see the same object in two different directions, at
radii r and 2 -
r. This requires a horizon size
larger than ; but conformal time = is attained
only at maximum expansion, so antipodal pairs of high-redshift
objects are visible only in the collapse phase.
These constraints do not apply if
the universe has a significant cosmological constant;
loitering models should allow one to see antipodal pairs
at approximately the same redshift. This effect
has been sought, but without success.
