the expansion should be slowing down. Instead it speeds up. What can be the cause of this repulsion? We shall return to this question repeatedly through the rest of this chapter.

12.3 Co s m o l o g i c a l d y n a m i c s : u n d e r s t a n d i n g t h e ex p a n d i n g u n i v e r s e

In the last section we saw how to describe a homogeneous and isotropic universe and how to measure its expansion. In order to study the evolution of the universe, to understand the creation of the huge variety of structures that we see, and indeed to make sense of the accelerating expansion measured today, we have to apply Einstein’s equations to the problem, and to marry them with enough physics to explain what we see. In this section we will study Einstein’s equations, with relatively simple perfect-fluid physics and with the cosmological constant that seems to be implied by the expansion. In the following sections we will study more and more of cosmological physics.

Dynamics of Robertson–Wal ker universes : Big B ang

and dark energy

We have seen that a homogeneous and isotropic universe must be described by one of the three Robertson–Walker metrics given by Eq. (12.13). For each choice of the curvature parameter k = (−1, 0, +1), the evolution of the universe depends on just one function of time, the scale factor R(t). Einstein’s equations will determine its behavior.

As in earlier chapters, we idealize the universe as filled with a homogeneous perfect fluid. The fluid must be at rest in the preferred cosmological frame, for otherwise its velocity would allow us to distinguish one spatial direction from another: the universe would not be isotropic. Therefore, the stress-energy tensor will take the form of Eq. (4.36) in the cosmological rest frame. Because of homogeneity, all fluid properties depend only on time: ρ = ρ(t), p = p(t), etc.

First we consider the equation of motion for matter, Tμν ;ν = 0, which follows from the Bianchi identities of Einstein’s field equations. Because of isotropy, the spatial components of this equation must vanish identically. Only the time component μ = 0 is nontrivial. It is

where R(t) is the cosmological expansion factor. This is easily interpreted: R3 is proportional to the volume of any fluid element, so the left-hand side is the rate of change of its total energy, while the right-hand side is the work it does as it expands (−p d V).

There are two simple cases of interest, a matter-dominated cosmology and a radiationdominated cosmology. In a matter-dominated era, which includes the present epoch, the main energy density of the cosmological fluid is in cold nonrelativistic matter particles, which have random velocities that are small and which therefore behave like dust: p = 0. So we have

In a radiation-dominated era (as we shall see, in the early universe), the principal energy density of the cosmological fluid is in radiation or hot, highly relativistic particles, which have an equation of state p = 13 ρ (Exer. 22, § 4.10). Then we get

The Einstein equations are also not hard to write down for this case. Isotropy will guarantee that Gtj = 0 for all j, and also that Gjk gjk. That means that only two components are independent, Gtt and (say) Grr. But the Bianchi identity will provide a relationship between them, which we have already used in deriving the matter equation in the previous paragraph. (The same happened for the spherical star.) Therefore we only need compute one component of the Einstein tensor (see Exer. 16, § 12.6):

Therefore, besides Eqs. (12.47) or (12.49), we have only one further equation, the Einstein equation with cosmological constant

Physicists today hope that they will eventually be able to compute the value of the cosmological constant from first principles in a consistent theory where gravity is quantized along with all the other fundamental interactions. From this point of view, the cosmological constant will represent just another contribution to the whole stress-energy tensor, which can be given the notation

From this point of view, the energy density and pressure of the cosmological constant ‘fluid’ are

Cosmologists call ρ the dark energy: an energy that is not associated with any known matter field. Its associated dark pressure p has the opposite sign. Physicists generally expect that the dark energy will be positive, so that most discussions of cosmology today are in the framework of ≥ 0. We will return below to the implications of the associated negative value for the dark pressure. Notice that, as the universe expands, the dark energy density and dark pressure remain constant. In these terms the tt-component of the Einstein equations can be written

where now we write ρm for the energy density of the matter (including radiation), to distinguish it from the dark energy density. This equation makes it easy to understand the observed acceleration of our universe. It appears (see below) that k = 0, or at least that the k-term is negligible. Then, since we are in a matter-dominated epoch, the term R2ρm decreases as R increases, while the term R2ρ increases rather strongly. Since today, as we

˙

shall see below, ρ > ρm, the result is that R increases as R increases. This trend must continue now forever, provided the acceleration is truly propelled by a cosmological constant, and not by some physical field that will go away later.

How, physically, can a positive dark energy density drive the universe into accelerated expansion? Is not positive energy gravitationally attractive, so would it not act to slow down the expansion? To answer this it is helpful to look at the spatial part of Einstein’s equations,

¨

where there acceleration R explicitly appears. Rather than derive this from the Christoffel symbols, we can use the fact that (as remarked above) it follows from the two basic equations we have already written down: Eq. (12.46) and the time-derivative of Eq. (12.54). In Exer. 17, § 12.6 we show that the combination of these two equations implies the following simple ‘equation of motion’ for the scale factor:

where ρ and p are the total energy density and pressure, including both the normal matter and the dark energy.

The acceleration is produced, not by the energy density alone, but by ρ + 3p. We have met this combination before, in Exer. 20, § 8.6, where we called it the active gravitational mass. We showed there that, in general relativity, when pressure cannot be ignored, the source of the far-away Newtonian field is ρ + 3p, not just ρ. In the cosmological context, the same combination generates the cosmic acceleration. It is clear that the negative pressure associated with the cosmological constant can, if it is large enough, make this sum negative, and that is what drives the universe faster and faster. Einstein’s gravity with a cosmological constant has a kind of in-built anti-gravity!

Notice that a negative pressure is not by any means unphysical. Negative stress is called tension, and in a stretched rubber band, for example, the component of the stress tensor along the band is negative. Interestingly, our analogy using a balloon to represent the expanding universe also introduces a negative pressure, the tension in the stretched rubber. What is remarkable about the dark energy is that its tension is so large, and it is isotropic. See Exer. 18, § 12.6 for a further discussion of the tension in this ‘fluid’.

Eq. (12.54) is written in a form that suggests studying the expansion of the universe in a way analogous to the energy methods physicists use for particle motion (as we did for orbits in Schwarzschild in Ch. 11). The left-hand side looks like a ‘kinetic energy’ and the right-hand side contains a constant (−k/2) that plays the role of the ‘total energy’ and a potential term proportional to R2(ρm + ρ ), which depends on R explicitly and through ρm. The dynamics of R will be constrained by this energy equation.

We can use this constraint to explore what might happen to our universe in the far distant future, assuming of course the Cosmological Principle, that nothing significantly new comes over our particle horizon. If ρ ≥ 0 (see above) and if the matter content of the universe also has positive energy density, then one conclusion from Eq. (12.54) is immediate: an expanding hyperbolic universe (k = −1) will never stop expanding. For the flat universe (k = 0), an expanding universe will also never stop if ρ > 0; however, if ρ = 0, then it could asymptotically slow down to a zero expansion rate as R approaches infinity, since the matter density will decrease at least as fast as R−3. An expanding closed universe (k = 1) will, if ρ = 0, always reach a maximum expansion radius and then turn around and re-collapse, again because R2ρm decreases with R. A re-collapsing universe eventually reaches another singularity, called the Big Crunch! But if ρ > 0, then the ultimate fate of an expanding closed universe depends on the balance of ρm and ρ .

We can ask similar questions about the history of our universe: was there a Big Bang, where the scale factor R had the value zero at a finite time in the past? First we consider for simplicity ρ = 0. Then Eq. (12.54) shows that, as R gets smaller, the matter term gets more and more important compared to the curvature term −k/2. Again this is because R2ρm is proportional either to R−1 for matter-dominated dynamics or, even more extremely, to R−2 for the radiation-dominated dynamics of the very early universe. Therefore, since our

˙ =

universe is expanding now, it could not have been at rest with R 0 at any time in the past. The existence of a Big Bang, i.e. whether we reach R = 0 at a finite time in the past, depends only on the behavior of the matter; the curvature term is not important, and all three kinds of universes have qualitatively similar histories.

Let us do the computation for a universe that is radiation-dominated, as ours will have been at an early enough time, and that has = 0 for simplicity. We write ρ = BR−4 for

where T is a constant of integration. So, indeed, R = 0 was achieved at a finite time in the past, and we conventionally adjust our zero of time so that R = 0 at t = 0, which means we redefine t so that T = 0.

Note that we have found that a radiation-dominated cosmology with no cosmological constant has an expansion rate where R(t) t1/2. If we had done this computation for a matter-dominated cosmology with ρ = 0, we would have found R(t) t2/3 (see Exer. 19, § 12.6).

What happens if there is a cosmological constant? If the dark energy is positive, there is no qualitative change in the conclusion, since the term involving ρ simply increases the

˙ =

value of R at any value of R, and this brings the time where R 0 closer to the present epoch. If the matter density has always been positive, and if the cosmological constant is non-negative, then Einstein’s equations make the Big Bang inevitable: the universe began with R = 0 at a finite time in the past. This is called the cosmological singularity: the curvature tensor is singular, tidal forces become infinitely large, and Einstein’s equations do not allow us to continue the solution to earlier times. Within the Einstein framework we cannot ask questions about what came before the Big Bang: time simply began there.

How certain, then, is our conclusion that the universe began with a Big Bang? First, we must ask if isotropy and homogeneity were crucial; the answer is no. The ‘singularity theorems’ of Penrose and Hawking (see Hawking and Ellis 1973) have shown that our universe certainly had a singularity in its past, regardless of how asymmetric it may have been. But the theorems predict only the existence of the singularity: the nature of the singularity is unknown, except that it has the property that at least one particle in the present universe must have originated in it. Nevertheless, the evidence is strong indeed that we all originated in it. Another consideration however is that we don’t know the laws of physics at the incredibly high densities (ρ → ∞) which existed in the early universe. The singularity theorems of necessity assume (1) something about the nature of Tμν , and (2) that Einstein’s equations (without cosmological constant) are valid at all R.

The assumption about the positivity of the energy density of matter can be challenged if we allow quantum effects. As we saw in our discussion of the Hawking radiation in the previous chapter, fluctuations can create negative energy for short times. In principle, therefore, our conclusions are not reliable if we are within one Planck time tPl = GMPl/c2 10−43 s of the Big Bang! (Recall the definition of the Planck mass in Eq. (11.111).) This is the domain of quantum gravity, and it may well turn out that, when we have a quantum theory of the gravitational interaction, we will find that the universe has a history before what we call the Big Bang.

Philosophically satisfying as this might be, it has little practical relevance to the universe we see today. We might not be able to start our universe model evolving from t = 0, but we can certainly start it from, say, t = 100tPl within the Einstein framework. The primary uncertainties about understanding the physical cosmology that we see around us are, as we will discuss below, to be found in the physics of the early universe, not in the time immediately around the Big Bang.