Cosmology. The Origin and Evolution of Cosmic Structure - Coles P., Lucchin F

148 Phase Transitions and Inflation

with the existence of the initial singularity. As we saw in Chapter 2, all the Friedmann models with equation of state in the form p = wρc2, with w 0, possess a particle horizon. This result can also be extended to other equations of state with p 0 and ρ 0. If the expansion parameter tends to zero at early times like tβ (with β > 0), then the particle horizon at time t,

This demonstrates that the condition for the existence of the Big Bang singularity, a¨ < 0, requires that 0 < β < 1 and that there must therefore also be a particle horizon.

The existence of a cosmological horizon makes it di cult to accept the Cosmological Principle. This principle requires that there should be a correlation (a very strong correlation) of the physical conditions in regions which are outside each other’s particle horizons and which, therefore, have never been able to communicate by causal processes. For example, the observed isotropy of the microwave background implies that this radiation was homogeneous and isotropic in regions on the last scattering surface (i.e. the spherical surface centred upon us which is at a distance corresponding to the look-back time to the era at which this radiation was last scattered by matter). As we shall see in Chapter 9, last scattering probably took place at an epoch, tls, corresponding to a redshift zls 1000. At that epoch the last scattering surface had a radius

because zls 1. The radius of the particle horizon at this epoch is given by Equation (2.7.3) with w = 0,

at zls the microwave background was homogeneous and isotropic over a sphere with radius at least ten times larger than that of the particle horizon.

Various routes have been explored in attempts to find a resolution of this problem. Some homogeneous but anisotropic models do not have a particle horizon at all. One famous example is the mix-master model proposed by Misner (1968), which we mentioned in Chapters 1 and 3. Other possibilities are to invoke some kind of isotropisation process connected with the creation of particles at the Planck epoch, or a modification of Einstein’s equations to remove the Big Bang singularity and its associated horizon.

150 Phase Transitions and Inflation

Imagine that there exists a period ti < t < tf sometime during the expansion of the Universe, in which the comoving scale l0, which has already been causally connected, somehow manages to escape from the horizon, in the sense that any physical processes occurring in this interval can no longer operate over the scale l0. We stress that it is not possible to ‘escape’ in this way from a particle horizon (or event horizon), but the cosmological horizon is not a true horizon in the formal sense explained in Section 2.7. Such an escape occurs if

This inequality can only be valid if the comoving horizon ca0/a˙ decreases with time, which requires an accelerated expansion, a¨ > 0. After tf we suppose that the Universe resumes the usual decelerated expansion. The behaviour of rc in such a model is shown graphically in Figure 7.4. The scale l0 is not causally connected before t1. It becomes connected in the interval t1 < t < t2; at t2 it leaves the horizon; in the interval t2 < t < t3 its properties cannot be altered by (causal) physical processes; at t3 it enters the horizon once more, in the sense that causal processes can a ect the physical properties of regions on the scale l0 after this time. An observer at time t3, who was unaware of the existence of the period of accelerated expansion, would think the scale l0 was coming inside the horizon for the first time and would be surprised if it were homogeneous. This observer would thus worry about the horizon problem. The problem is, however, non-existent if there is an accelerated expansion and if the maximum scale which is causally connected is greater than the present scale of the horizon, i.e.

To be more precise, unless we accept it as a coincidence that these two comoving scales should be similar, a solution is only really obtained if the inequality (7.8.8) is strong, i.e. rc(t0) rc(ti).

In any case the solution is furnished by a period ti < t < tf of appropriate duration, in which the universe su ers an accelerated expansion: this is the definition of inflation. In such an interval we must therefore have p < −ρc2/3; in particular if p = wρc2, with constant w, we must have w < −13 . From the Friedmann equations in this case we recover, for tf > t > ti,

(this solution is exact when the curvature parameter K = 0). For H(ti)t 1 one has

the exponent q is greater than one in the first case and negative in the last case; τ = (a/a)˙ t=ti and ta = ti − [2/(3(1 + w))]H(ti)−1 > ti. The types of expansion

152 Phase Transitions and Inflation

7.9 The Cosmological Flatness Problem

7.9.1The problem

In the Friedmann equation without the cosmological constant term

when the universe is radiation dominated so that ρ T4, there is no obvious characteristic scale other than the Planck time

From a theoretical point of view, in a closed universe, one is led to expect a time of maximum expansion tm which is of order tP followed by a subsequent rapid collapse. On the other hand, in an open universe, the curvature term Kc2/a2 is expected to dominate over the gravitational term 8πGρ/3 in a time t tP. In this second case, given that, as one can deduce from Equation (2.3.9), for t > t we have

The Universe has probably survived for a time of order 1010 years, corresponding to around 1060tP, meaning that at very early times the kinetic term (a/a)˙ 2 must have di ered from the gravitational term 8πGρ/3 by a very small amount indeed. In other words, the density at a time t tP must have been very close to the critical density.

As we shall see shortly, we have

The kinetic term at tP must have di ered from the gravitational term by about one part in 1060. This is another ‘fine-tuning’ problem. Why are these two terms tuned in such a way as to allow the Universe to survive for 1010 years? On the other hand the kinetic and gravitational terms are now comparable because a very conservative estimate gives

This problem is referred to as the age problem (how did the Universe survive so long?) or the (near) flatness problem (why is the density so close to the critical density?).

There is yet another way to present this problem. The Friedmann equation, divided by the square of the constant Ta = T0ra0, becomes

this constant can be rendered dimensionless by multiplying by the quantity ( /kB)2. We thus obtain

the dimensionless constant we have introduced remains constant at a very small value throughout the evolution of the Universe. The flatness problem can be regarded as the problem of why |H(T)| is so small. Perhaps one might think that the correct resolution is that H(T) = 0 exactly, so that K = 0. However, one should bear in mind that the Universe is not exactly described by a Robertson–Walker metric because it is not perfectly homogeneous and isotropic; it is therefore di cult to see how to construct a physical principle which requires that a parameter such as H(T) should be exactly zero.

It is worth noting that H(T) is related to the entropy Sr of the radiation of the Universe. Supposing that K ≠ 0, the dimensionless entropy contained inside a sphere of radius a(t) (the curvature radius) is

Given that the entropy of the matter is negligible compared with that of the radiation and of the massless neutrinos (Sν is of order Sr), the quantity σU can be defined as the dimensionless entropy of the Universe (a0 is often called the ‘radius of the universe’). This also represents the number of particles (in practice, photons and neutrinos) inside the curvature radius. What is the explanation for this enormous value of σU? This is, in fact, just another statement of the flatness problem. It is therefore clear that any model which explains the high value of σU also solves this problem. As we shall see, inflationary universe models do resolve this issue; indeed they generally predict that Ω0 should be very close to unity, which may be di cult to reconcile with observations.

It is now an appropriate time to return in a little more detail to Equation (7.9.5). From the Friedmann equation

one easily finds that during the evolution of the Universe we have

154 Phase Transitions and Inflation

The standard picture of the Universe (without inflation) is well described by a radiative model until zeq and by a matter-dominated model from then until now. From Equation (7.9.11) and the usual formulae

we can easily obtain the relationship between Ω, corresponding to a time t teq when the temperature is T, and Ω0:

If we accept that |Ω0−1 −1| 1, this implies that Ω must have been extremely close to unity during primordial times. For example, at tP we have |ΩP−1 − 1| 10−60, as we have already stated in Equation (7.9.5).

7.9.2The inflationary solution

Now we suppose that there is a period of accelerated expansion between ti and tf. Following the same philosophy as we did in Section 7.8, we divide the history of the Universe into the same three intervals (ti, tf), (tf, teq) and (teq, t0), where ρ a−3(1+wij ), with wij = w < −13 , wij = 13 and wij = 0, respectively. We find, from Equation (7.9.11),

(Ωi−1 − 1)ρia2i = (Ωf−1 − 1)ρfa2f = (Ωeq−1 − 1)ρeqa2eq = (Ω0−1 − 1)ρ0a20, (7.9.14)

so that

which gives, in a similar manner to Equation (7.8.15),

After some further manipulation we find

One can assume that the flatness problem is resolved as long as the following inequality is valid:

1 − Ω−1

1 − Ωi−1 1, (7.9.18)

0

Ω(t)

(b)

1

(a)

Figure 7.5 Evolution of Ω(t) for an open universe (a) and closed universe (b) characterised by three periods (0, ti), (ti, tf), (tf, t0). During the first and last of these periods p/ρc2 = w > −13 (decelerated expansion), while in the second w < −13 (accelerated expansion). If the inflationary period is su ciently dramatic, the later divergence of the trajectories from Ω = 1 is delayed until well beyond t0.

in other words Ω0 is no closer to unity now than Ωi was. The condition (7.9.18), expressed in terms of the number of e-foldings N , becomes

For example, in the case where w −1 the solution of the horizon problem

N pNmin = p30[2.3+ 301 ln(Tf/TP)], with p > 1, implies a relationship between Ωi and Ω0

In general, therefore, an adequate solution of the horizon problem (p 1) would imply that Ω0 would be very close to unity for a universe with |1 − Ωi−1| 1. In other words, in this case inflation would automatically take care of the flatness problem as well.

This argument may explain why Ω is close to unity today, but it also poses a problem of its own. If Ω0 1 to high accuracy, what is the bulk of the matter made from, and why do dynamical estimates of Ω0 yield typical values of order 0.2? If it turns out that Ω0 is actually of this order, then much of the motivation for inflationary models will have been lost. We should also point out that inflation does not predict an exactly smooth Universe; small-amplitude fluctuations appear in a manner described in Chapter 14. These fluctuations mean that, on the scale of our observable Universe, the density parameter would be uncertain by the amount of the density fluctuation on that scale. In most models the fractional fluctuation is of order 10−5, so it does not make sense to claim that Ω0 is predicted to be unity with any greater accuracy than this.

156 Phase Transitions and Inflation

7.10 The Inflationary Universe

The previous sections have given some motivation for imagining that there might have been an epoch during the evolution of the Universe in which it underwent an accelerated expansion phase. This would resolve the flatness and horizon problems. It would also possibly resolve the problem of topological defects because, as long as inflation happens after (or during) the phase transition producing the defects, they will be diluted by the enormous increase of the scale factor. Beginning in 1982, various authors have also addressed another question in the framework of the inflationary universe which is directly relevant to the main subject of this book. The idea here is that quantum fluctuations on microscopic scales during the inflationary epoch can, again by virtue of the enormous expansion, lead to fluctuations on very large scales today. It is possible that this ‘quantum noise’ might therefore be the source of the primordial fluctuation spectrum we require to make models of structure-formation work. In fact, as we shall see in Section 14.6, one obtains a primordial spectrum which is slightly dependent upon the form of the inflationary model, but is usually close to the so-called Harrison–Zel’dovich spectrum which was proposed, for di erent reasons, by Harrison, Zel’dovich and also Peebles and Yu, around 1970.

Assuming that we accept that an epoch of inflation is in some sense desirable, how can we achieve such an epoch physically? The answer to this question lies in the field of high-energy particle physics, so from now until the end of this chapter we shall use the language of natural units with c = = 1.

The idea at the foundation of most models of inflation is that there was an epoch in the early stages of the evolution of the Universe in which the energy density of the vacuum state of a scalar field ρv V(φ) is the dominant contribution to the energy density. In this phase the expansion factor a grows in an accelerated fashion which is nearly exponential if V const. This, in turn, means that a small causally connected region with an original dimension of order H−1 can grow to such a size that it exceeds the size of our present observable Universe, which has a dimension of order H0−1.

There exist many di erent versions of the inflationary universe. The first was formulated by Guth (1981), although many of his ideas had been presented previously by Starobinsky (1979). In Guth’s model inflation was assumed to occur while the universe is trapped in a false vacuum with Φ = 0 corresponding to the first-order phase transition which characterises the breaking of an SU(5) symmetry into SU(4)×U(1). This model was subsequently abandoned for reasons which we shall mention below.

The next generation of inflationary models shared the characteristics of a model called the new inflationary universe, which was suggested independently by Linde (1982a,b) and Albrecht and Steinhardt (1982). In models of this type, inflation occurs during a phase in which the region which grows to include our observable ‘patch’ evolves slowly from a ‘false’ vacuum with Φ = 0 towards a ‘true’ vacuum with Φ = Φ0. In fact, it was later seen that this kind of inflation could also be achieved in many di erent contexts, not necessarily requiring the existence of a

phase transition or a spontaneous symmetry breaking. Anyway, from an explanatory point of view, this model appears to be the clearest. It is based on a certain choice of parameters for an SU(5) theory which, in the absence of any experimental constraints, appears a little arbitrary. This problem is common also to other inflationary models based on theories like supersymmetry, superstrings or supergravity which have not yet received any experimental confirmation or, indeed, are likely to in the foreseeable future. It is fair to say that the inflationary model has become a sort of ‘paradigm’ for resolving some of the di culties with the standard model, but no particular version of it has received any strong physical support from particle physics theories.

Let us concentrate for a while on the physics of generic inflationary models involving symmetry breaking during a phase transition. In general, gauge theories of elementary particle interactions involve an order parameter Φ, determining the breaking of the symmetry, which is the expectation value of the scalar field which appears in the classical Lagrangian LΦ

As we mentioned in Section 6.1, the first term in Equation (7.10.1) is called the kinetic term and the second is the e ective potential, which is a function of temperature. In Equation (7.10.1) for simplicity we have assumed that the expectation value of Φ is homogeneous and isotropic with respect to spatial position. As we have already explained in Section 6.1, the energy–momentum tensor of a scalar field can be characterised by an e ective energy density ρΦ and by an e ective pressure pΦ given by

respectively. The potential V(Φ; T) plays the part of the free energy F of the system, which displays the breaking symmetry described in Section 7.3; in particular, Figure 7.2 is a useful reference for the following comments. This figure refers to a first-order phase transition, so what follows is relevant to the case of Guth’s original ‘old’ inflation model. The potential has an absolute minimum at Φ = 0 for T Tc, this is what will correspond to the ‘false’ vacuum phase. As T nears Tc the potential develops another two minima at Φ = ±Φ0, which for T Tc have a value of order V(0; Tc): the three minima are degenerate. We shall now assume that the transition ‘chooses’ the minimum at Φ0; at T Tc this minimum becomes absolute and represents the true vacuum after the transition; at these energies we can ignore the dependence of the potential upon temperature. We also assume, for reasons which will become clear later, that V(Φ0; 0) = 0. In this case the transition does not occur instantaneously at Tc because of the potential barrier between the false and true vacua; in other words, the system undergoes a supercooling while the system remains trapped in the false vacuum. Only at some later temperature Tb < Tc can thermal fluctuations or quantum tunnelling e ects