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.

 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.

(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.

 Figure 3.5. This plot shows the different possibilities for the cosmological expansion as a function of matter density and vacuum energy. Models with total > 1 are always spatially closed (open for < 1), although closed models can still expand to infinity if v 0. If the cosmological constant is negative, recollapse always occurs; recollapse is also possible with a positive v if m >> v. If v > 1 and m is small, there is the possibility of a ``loitering'' solution with some maximum redshift and infinite age (top left); for even larger values of vacuum energy, there is no big bang singularity.

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).

 Figure 3.6. The age of the universe versus the matter density parameter. The solid line shows open models; the dashed line shows flat models with m + v = 1. For values of a few tenths for m, the flat models allow a significantly greater age for a given H0, and so a non-zero can be inferred if t0 and H0 can be measured sufficiently accurately.

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.