 © CAMBRIDGE UNIVERSITY PRESS 1999 
3.2 Dynamics of the Expansion
EXPANSION AND GEOMETRY
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 RobertsonWalker 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:
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 RobertsonWalker metric into the field equations [problem 3.1], but the question inevitably arises of whether there is a quasiNewtonian 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 R^{2} is a declining function of R  the potential energy becomes negligible by comparison with the total and tends to a constant. In this massfree 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:
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 [ c^{2} R^{3}] = pd [R^{3}]. We then obtain
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.
Since and H change with time, this defines an epochdependent 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
then the current density of the universe may be expressed as
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
(introducing the normalized scale factor a = R / R_{0}). For some purposes, this separation is unnecessary, since the Friedmann equation treats all contributions to the density parameter equally:
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,
the form of the Friedmann equation says that
which implies q = 3_{m} / 2 + 2_{r} 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:
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 / H_{0}.
SOLUTIONS TO THE FRIEDMANN EQUATION
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):
Because H_{0}^{2} R_{0}^{2} = kc^{2} / (  1), the Friedmann equation becomes
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 distanceredshift 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 H^{2} = 8 G / 3  kc^{2} / R^{2}. Inserting the above dependence of on a gives
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 = R_{0} / (1 + z), this gives
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 = kc^{2} / (H^{2} R^{2}) gives the redshift dependence of the total density parameter:
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 R^{2} > as R > 0, the kc^{2} 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.
MATTERDOMINATED UNIVERSE
At redshifts higher than this, the universal dynamics were dominated by the relativisticparticle content. By a coincidence discussed below, this epoch is close to another important event in cosmological history: recombination. Once the temperature falls below 10^{4} 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 matterdominated 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_{*}:
where we have used the expression for R_{0} in the first step. When only matter is present, the conformaltime version of the Friedmann equation is simple to integrate for R (), and integration of dt = d / R gives t ():
This cycloid solution is a special case of the general solution for the evolution of a spherical mass distribution: R = A [1  C_{k} ()], t = B [  S_{k} ()], where A^{3} = GMB^{2} 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 t^{2/3}.
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.
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:
When we insert the redshift dependences of H (z) and (z),
this gives us the timeredshift relation. An alternative route to this result would have been to use the general differential expression for comoving distance dr / dz; since c dt = [R_{0} / (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
which interpolates between the exact ages of H^{1} for an empty universe and 2/3 H^{1} for a criticaldensity = 1 model.
MATTER PLUS RADIATION BACKGROUND
reflecting the R^{3} and R^{4} dependencies of matter and radiation densities respectively. Now define dimensionless masses of the form y GM / (c^{2} R_{0}), which reduce to y_{m, r} = k _{m, r} / [2(  1)]. The parametric solutions then become
MODELS WITH VACUUM ENERGY
The Friedmann equation itself is independent of the equation of state, and just says H^{2} R^{2} = kc^{2} / (  1), whatever the form of the contributions to . In terms of the cosmological constant itself, we have
STATIC UNIVERSE
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
Since = p for vacuum energy, and this is the only source of pressure if we ignore radiation, this tells us that = 3_{vac} 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 quasiNewtonian 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
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 STEADYSTATE MODEL
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 steadystate model, and may account for the strong reactions generated by the theory. Certainly, the debate between steadystate 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 steadystate model. Since the radiosource 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 steadystate 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
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

(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
where the function f is similar in spirit to C_{k}: 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):
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 highredshift 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
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> R^{2} / (3c^{2}). Since both sides are quadratic in R, this makes it clear that the value of R_{0} 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 matterradiation mix is
which may be integrated to give the time as a function of scale factor:
this goes to 2/3 a^{3/2} for a matteronly model, and to 1/2 a^{2} 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
The whole universe thus always obeys the ruleofthumb for the collapse from rest of a gravitating body: the collapse time 1 / sqrt(G ).
Because _{r} is so small, the deviations from a matteronly model are unimportant for z 1000, and so the distanceredshift relation for the k = 0 matter plus radiation model is effectively just that of the _{m} = 1 Einsteinde 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 H_{0} t_{0} = 2/3, which characterises the Einsteinde Sitter model. Since much observational evidence indicates that H_{0} t_{0} 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
and the t (a) relation is
The x^{4} on the bottom looks like trouble, but it can be rendered tractable by the substitution y = sqrt(x^{3} _{m}  1 / _{m}), which turns the integral into
Here, k in S_{k} 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
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 0.7_{m}  0.3_{v} + 0.3 (Carroll, Press & Turner 1992).
HORIZONS
i.e. just the interval of conformal time. What happens as t_{0} > 0 in this expression? We can replace dt by dR / , which the Friedmann equation says is dR / sqrt( R^{2}) at early times. Thus, this integral converges if R^{2} > as t_{0} > 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 matteronly 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 highredshift 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.