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

158 Phase Transitions and Inflation

shift the Φ field over the barrier and down into the true vacuum. Let us indicate by Φb the value assumed by the order parameter at this event. The dynamics of this process depends on the shape of the potential. If the potential is such that the transition is first order (as in Figure 7.2), the new phase appears as bubbles nucleating within the false vacuum background; these then grow and coalesce so as to fill space with the new phase when the transition is complete. If the transition is second order, one generates domains rather than bubbles, like the Weiss domains in a ferromagnet. One such region (bubble or domain) eventually ends up including our local patch of the Universe.

The energy–momentum tensor of the whole system, Tij, also contains, in addition to terms due to the Φ field, terms corresponding to interacting particles, which can be interpreted as thermal excitations above the minimum of the potential, with an energy density ρ and pressure p; in this period we have p = ρ/3. The Friedmann equations therefore become

The evolution of Φ is obtained from the equation of motion for a scalar field:

This equation is similar to that describing a ball moving under the action of the force −∂V/∂Φ against a source of friction described by the viscosity term proportional to 3a/a˙ ; in the usual language, one talks of the Φ field ‘rolling down’ the potential towards the minimum at Φ0. Let us consider potentials which have a large interval (Φi, Φf) with Φb < Φi Φ Φf < Φ0 in which V(Φ; T) remains

roughly constant; this property ensures a very slow evolution of Φ towards Φ0,

˙2 usually called the slow-rolling phase because, in this interval, the kinetic term Φ /2

¨

is negligible compared with the potential V(Φ; T) in Equation (7.10.3 b) and the Φ term is negligible in Equation (7.10.5). One could say that the motion of the field is in this case dominated by friction, so that the motion of the field resembles the behaviour of particles during sedimentation.

In order to have inflation one must assume that, at some time, the Universe contains some rapidly expanding regions in thermal equilibrium at a temperature T > Tc which can eventually cool below Tc before any gravitational recollapse can occur. Let us assume that such a region, initially trapped in the false vacuum phase, is su ciently homogeneous and isotropic to be described by a Robertson–Walker metric. In this case the evolution of the patch is described by

Equation (7.10.3 a). The expansion rapidly causes ρ and K/a2 to become negligible with respect to ρΦ, which is varying slowly. One can therefore assume that Equation (7.10.3 a) is then

which is of order 10−34 s in typical models. Let us now fix our attention upon one such region, which has dimensions of order 1/H(tb) at the start of the slow-rolling phase and is therefore causally connected. This region expands by an enormous factor in a very short time τ; any inhomogeneity and anisotropy present at the initial time will be smoothed out so that the region loses all memory of its initial structure. This e ect is, in fact, a general property of inflationary universes and it is described by the so-called cosmic no-hair theorem. The number of e-foldings of the inflationary expansion during the interval (ti, tf) depends on the potential:

if this number is su ciently large, the horizon and flatness problems can be solved. The initial region is expanded by such a large factor that it encompasses our present observable Universe.

Because of the large expansion, the patch we have been following also becomes practically devoid of particles. This also solves the monopole problem (and also the problem of domain walls, if they are predicted) because any defects formed during the transition will be drastically diluted as the Universe expands so that their present density will be negligible. After the slow-rolling phase the field Φ falls rapidly into the minimum at Φ0 and there undergoes oscillations: while this happens there is a rapid liberation of energy which was trapped in the term V V(Φf; Tf), i.e. the ‘latent heat’ of the transition. The oscillations are damped by the creation of particles coupled to the Φ field and the liberation of the latent heat thus raises the temperature to some value Trh Tc: this phenomenon is called reheating, and Trh is the reheating temperature. The region thus acquires virtually all the energy and entropy that originally resided in the quantum vacuum by particle creation.

Once the temperature has reached Trh, the evolution of the patch again takes the character of the usual radiative Friedmann models without a cosmological constant; this latter condition is, however, only guaranteed if V(Φ0; 0) = 0 because

160 Phase Transitions and Inflation

Φ

Φ0

Φb

Figure 7.6 Evolution of Φ inside a ‘patch’ of the Universe. In the beginning we have the slow-rolling phase between ti and tf, followed by the rapid fall into the minimum at Φ0, representing the true vacuum, and subsequent rapid oscillations which are eventually smeared out by particle creation leading to reheating of the Universe.

any zero-point energy in the vacuum would play the role of an e ective cosmological constant. We shall return to this question in the next section.

It is important that the inflationary model should predict a reheating temperature su ciently high that GUT processes which violate conservation of baryon number can take place so as to allow the creation of a baryon asymmetry.

As far as its global properties are concerned, our Universe is reborn into a new life after reheating: it is now highly homogeneous, and has negligible curvature. This latter prediction may be a problem for, as we have seen, there is little strong evidence that Ω0 is very close to unity.

Another general property of inflationary models, which we have not described here, is that fluctuations in the quantum field driving inflation can, in principle, generate a primordial spectrum of density fluctuations capable of seeding the formation of galaxies and clusters. We shall postpone a discussion of this possibility until Section 14.6.

7.11 Types of Inflation

We have already explained that there are many versions of the inflationary model which are based on slightly di erent assumptions about the nature of the scalar field and the form of the phase transition. Let us mention some of them here.

7.11.1Old inflation

The first inflationary model, suggested by Guth (1981), is usually now called old inflation. This model is based on a scalar field theory which undergoes a first-order phase transition. The problem is that, being a first-order transition, it occurs by a process of bubble nucleation. It turns out, however, that these bubbles would be too small to be identified with our observable Universe and would be carried apart by the expanding phase too quickly for them to coalesce and produce a large

bubble which one could identify in this way. The end state of this model would therefore be a highly chaotic universe, quite the opposite of what is intended. This model was therefore abandoned soon after it was suggested.

7.11.2 New inflation

The successor to old inflation was new inflation (Linde 1982a,b; Albrecht and Steinhardt 1982). This is again a theory based on a scalar field, but this time the potential is qualitatively similar to Figure 7.1, rather than 7.2. The field is originally in the false vacuum state at Φ = 0, but as the temperature lowers it begins to roll down into one of the two degenerate minima. There is no potential barrier, so the phase transition is second order. The process of spinodal decomposition which accompanies a second-order phase transition usually leaves one with larger coherent domains and one therefore ends up with relatively large space-filling domains.

The problem with new inflation is that it su ers from severe fine-tuning problems. One such problem is that the potential must be very flat near the origin to produce enough inflation and to avoid excessive fluctuations due to the quantum field. Another is that the field Φ is assumed to be in thermal equilibrium with the other matter fields before the onset of inflation; this requires that Φ be coupled fairly strongly to the other fields. But the coupling constant would induce corrections to the potential which would violate the previous constraint. It seems unlikely therefore that one can achieve thermal equilibrium in a self-consistent way before inflation starts under the conditions necessary for inflation to happen.

7.11.3 Chaotic inflation

One of the most popular inflationary models is chaotic inflation, due to Linde (1983). Again, this is a theory based on a scalar field, but it does not require any phase transitions. The basis of this model is that, whatever the detailed shape of the e ective potential, a patch of the Universe in which Φ is large, uniform and static will automatically lead to inflation. For example, consider the simple quadratic potential

where m is an arbitrary parameter describing the mass of the scalar field. Assume that, at t = ti, the field Φ = Φi is uniform over a scale H−1(ti) and that

The equation of motion of the scalar field then simply becomes

which, with the slow-rolling approximation, is just

162 Phase Transitions and Inflation

Since H V1/2 Φ, this equation is easy to solve and it turns out that, in order to get su cient inflation to solve the flatness and horizon problems, one needs Φ > 3mP in the patch.

In chaotic inflation one assumes that at some initial time, perhaps just after the Planck time, the Φ field varied from place to place in an arbitrary manner. If any region satisfies the above conditions it will inflate and eventually encompass our observable Universe. While the end result of chaotic inflation is locally flat and homogeneous in our observable ‘patch’, on scales larger than the horizon the Universe is highly curved and inhomogeneous. Chaotic inflation is therefore very di erent from both old and new inflationary models. This is reinforced by the fact that no mention of GUT or supersymmetry theories appears in this analysis. The field Φ which describes chaotic inflation at the Planck time is completely decoupled from all other physics.

7.11.4Stochastic inflation

The natural extension of Linde’s chaotic inflationary model is called stochastic inflation or, sometimes, eternal inflation (Linde et al. 1994). The basic idea is the same as chaotic inflation in that the Universe is globally extremely inhomogeneous. The stochastic inflation model, however, takes into account quantum fluctuations during the evolution of Φ. One finds in this case that the Universe at any time will contain regions which are just entering into an inflationary phase. One can picture the Universe as a continuous ‘branching’ process in which new ‘miniuniverses’ expand to produce locally smooth Hubble patches within a highly chaotic background Universe. This picture is like a Big Bang on the scale of each miniuniverse, but globally is reminiscent of the steady-state universe. The continual birth and rebirth of these miniuniverses is often called, rather poetically, the ‘Phoenix Universe’ model.

7.11.5 Open inflation

In the mid-1990s there was a growing realisation among cosmologists that evidence for a critical matter density was not forthcoming (e.g. Coles and Ellis 1994). This even reached inflation theorists, who defied the original motivation for inflation and came up with versions of inflation that would homogeneous but curved universes. Usually inflation stretches the curvature as well as smoothing lumpiness, so this seems at first sight a very di cult task for inflation.

Open inflation models square the circle by invoking a kind of quantum tunnelling from a metastable false vacuum state immediately followed by a second phase of inflation, an idea originally due to Gott (1982). The tunnelling creates a bubble inside which the space–time resembles an open universe.

Although it is possible to engineer an inflationary model that produces Ω0 0.2 at the present epoch, it certainly seems to require more complexity than models that produce flat spatial sections. Recent evidence from microwave background

observations that the Universe seems to be flat even if it does not have a critical density have reduced interest in these open inflation models too; see Chapter 18.

7.11.6 Other models

At this point it is appropriate to point out that there are very many inflationary models about. Indeed, inflation is in some sense a generic prediction of most theories of the early Universe. We have no space to describe all of these models, but we can briefly mention some of the most important ones.

Firstly, one can obtain inflation by modifying the classical Lagrangian for gravity itself, as mentioned in Chapter 6. If one adds a term proportional to R2 to the usual Lagrangian, then the equations of motion that result are equivalent to ordinary general relativity in the presence of a scalar field with some particular action. This ‘e ective’ scalar field can drive inflation in the same way as a real field can.

An alternative way to modify gravity might be to adopt the Brans–Dicke (scalar– tensor) theory of gravity described in Section 3.4. The crucial point here is that an e ective equation of state of the form p = −ρc2 in this theory produces a powerlaw, rather than exponential, inflationary epoch. This even allows ‘old inflation’ to succeed: the bubbles which nucleate the new phase can be made to merge and fill space if inflation proceeds as a power law in time rather than an exponential (Lucchin and Matarrese 1985). Theories based on Brans–Dicke modified gravity are usually called extended inflation.

Another possibility relies on the fact that many unified theories, such as supergravity and superstrings, are only defined in space–times of considerably higher dimensionality than those we are used to. The extra dimensions involved in these theories must somehow have been compactified to a scale of order the Planck length so that we cannot perceive them now. The contraction of extra spatial dimensions can lead to an expansion of the three spatial dimensions which must survive, thus leading to inflation. This is the idea behind so-called Kaluza–Klein theories.

There are many other possibilities: models with more than one scalar field, with modified gravity and a scalar field, models based on more complicated potentials, on supersymmetric GUTs, supergravity and so on. Inflation has led to an almost exponential increase in the number of inflationary models since 1981!

7.12Successes and Problems of Inflation

As we have explained, the inflationary model provides a conceptual explanation of the horizon problem and the flatness problem. It may also rescue grand unified theories which predict a large present-day abundance of monopoles or other topological defects.

We have seen how inflationary models have evolved to avoid problems with earlier versions. Some models are intrinsically flawed (e.g. old inflation) but can be salvaged in some modified form (extended inflation). The density and gravitational

164 Phase Transitions and Inflation

wave fluctuations they produce may also be too high for some parameter choices, as we discuss in Chapter 14. For example, the requirement that density fluctuations be acceptably small places a strong constraint on m in Equation (7.11.1) corresponding to the chaotic inflation model. This, however, requires a fine-tuning of the scalar field mass m which does not seem to have any strong physical motivation. Such fine-tunings are worrying but not fatal flaws in these models.

There are, however, much more serious problems associated with these scenarios. Perhaps the most important is one we have mentioned before and which is intimately connected with one of the successes. Most inflationary models predict that spatial sections at the present epoch should be almost flat. In the absence of a cosmological constant this means that Ω0 1. However, evidence from galaxyclustering studies suggests this is not the case: the apparent density of matter is less than the critical density. It is possible to produce a low-density universe after inflation, but it requires very particular models. On the other hand, one could reconcile a low-density universe with apparently more natural inflationary models by appealing to a relic cosmological constant: the requirement that spatial sections should be (almost) flat simply translates into Ω0 + Ω0Λ 1. This seems to be that a potentially successful model of structure formation, as well as allowing accounting for the behaviour of high-redshift supernovae (Chapter 4) and cosmic microwave background fluctuations (Chapter 18).

One also worries about the status of inflation as a physical theory. To what extent is inflation predictive? Is it testable? One might argue that inflation does predict that Ω0 1. This may be true, but one can have Ω0 close to unity without inflation if some process connected with quantum gravity can arrange it. Likewise one can have Ω0 < 1 either with inflation or without it. Inflationary models also produce density fluctuations and gravitational waves. If these are observed to have the correct properties, they may eventually constitute a test of inflation, but this is not the case at present. All we can say is the COBE fluctuations in the microwave background do indeed seem to be consistent with the usual inflationary models. At the moment, therefore, inflation has a status somewhere between a theory and a paradigm, but we are still a long way from being able to use these ideas to test GUT scale physics and beyond in any definite way.

7.13 The Anthropic Cosmological Principle

We began this book with a discussion of the importance of the Cosmological Principle, which, as we have seen in the first two chapters, has an important role to play in the construction of the Friedmann models. This principle, in light of the cosmological horizon problem, has more recently led to the idea of the inflationary universe we have explored in this chapter. The Cosmological Principle is a development of the Copernican Principle, asserting that, on a large scale, all spatial positions in the Universe are equivalent. At this point in the book it is worth mentioning an alternative Cosmological Principle – the Anthropic Cosmological Principle – which seeks to explore the connection between the physical structure of the Universe and the development of intelligent life within it. There

are, in fact, many versions of the Anthropic Principle. The Weak Anthropic Principle merely cautions that the fact of our own existence implies that we do occupy some sort of special place in the Universe. For example, as noted by Dicke (1961), human life requires the existence of heavy elements such as Carbon and Oxygen which must be synthesised by stars. We could not possibly have evolved to observe the Universe in a time less than or of order the main sequence lifetime of a star, i.e. around 1010 years in the Big Bang picture. This observation is itself su cient to explain the large-number coincidences described in Chapter 3 which puzzled Dirac so much. In fact, the Weak Anthropic Principle is not a ‘principle’ in the same sense as the Cosmological Principle: it is merely a reminder that one should be aware of all selection e ects when interpreting cosmological data.

It is important to stress that the Weak Anthropic Principle is not a tautology, but has real cognitive value. We mentioned in Chapter 3 that, in the steady-state model, there is no reason why the age of astronomical objects should be related to the expansion timescale H0−1. In fact, although both these timescales are uncertain, we know that they are equal to within an order of magnitude. In the Big Bang model this is naturally explained in terms of the requirement that life should have evolved by the present epoch. The Weak Anthropic Principle therefore supplies a good argument whereby one should favour the Big Bang over the steady state: the latter has an unresolved ‘coincidence’ that the former explains quite naturally.

An entirely di erent status is held by the Strong Anthropic Principle and its variants. This version asserts a teleological argument (i.e. an argument based on notions of ‘purpose’ or ‘design’) to account for the fact that the Universe seems to have some properties which are finely tuned to allow the development of life. Slight variations in the ‘pure’ numbers of atomic physics, such as the finestructure constant, would lead to a world in which chemistry, and presumably life, as we know it, could not have developed. These coincidences seem to some physicists to be so striking that only a design argument can explain them. One can, however, construct models of the Universe in which a weak explanation will su ce. For example, suppose that the Universe is constructed as a set of causally disjoint ‘domains’ and, within each such domain, the various symmetries of particle physics have been broken in di erent ways. A concrete implementation of this idea may be realised using Linde’s eternal chaotic inflation model which we discussed earlier. Physics in some of these domains would be similar to our Universe; in particular, the physical parameters would be such as to allow the development of life. In other domains, perhaps in the vast majority of them, the laws of physics would be so di erent that life could never evolve in them. The Weak Anthropic Principle instructs us to remember that we must inhabit one of the former domains, rather than one of the latter ones. This idea is, of course, speculative but it does have the virtue of avoiding an explicitly teleological language.

The status of the Strong Anthropic Principle is rightly controversial and we shall not explore it further in this book. It is interesting to note, however, that after centuries of adherence to the Copernican Principle and its developments,

166 Phase Transitions and Inflation

cosmology is now seeing the return of a form of Ptolemaic reasoning (the Strong Anthropic Principle), in which man is again placed firmly at the centre of the Universe.

Bibliographic Notes on Chapter 7

More detailed treatments of elementary particle physics can be found in Chaichian and Nelipe (1984); Collins et al. (1989); Dominguez-Tenreiro and Quiros (1987); Hughes (1985); Kolb and Turner (1990) and Roos (1994). A more technical treatment of particle cosmology can be found in Barrow (1983). Weinberg (1988) gives an authoritative review of the cosmological constant problem. A nice introductory account of inflation can be found in Narlikar and Padmanabhan (1991) of Linde (1990); a more technical review is Linde (1984). The definitive treatment of the anthropic principles is Barrow and Tipler (1986).

Problems

The following problems all concern a simplified model of the history of a flat universe involving a period of inflation. The history is split into four periods: (a) 0 < t < t3 radiation only; (b) t3 < t < t2 vacuum energy dominates, with an e ective cosmological constant Λ = 34 t32; (c) t2 < t < t1 a period of radiation domination; and (d) t1 < t < t0 matter domination.

1. Show that in epoch (c) ρ(t) = ρr(t) = 323 πGt2, and in (d) ρ(t) = ρm(t) = 16 πGt2.

2.Give simple analytical formulae for a(t) which are approximately true in these four phases.

3.Show that, during the inflationary phase (b) the universe expands by a factor

a(t2) = exp t2 − t3 . a(t3) 2t3

4.Derive an expression for Λ in terms of t2, t3 and ρ(t2).

5.Show that

ρr(t0) = 9 t1 2/3. ρm(t0) 16 t0

6.If t3 = 10−35 s, t2 = 10−32 s, t1 = 104 years and t0 = 1010 years, give a sketch of log a against log t marking any important epochs.

8

The Lepton Era

8.1 The Quark–Hadron Transition

At very high temperatures, the matter in the Universe exists in the form of a quark–gluon plasma. When the temperature falls to around TQH 200–300 MeV the quarks are no longer free, but become confined in composite particles called hadrons. These particles are generally short lived (with the exception of the proton and neutron), so there is only a brief period in which the hadrons flourish. This period is often called the hadron era, but that is a somewhat misleading term because the hadrons even in this era do not dominate the energy density of the Universe. At the energy corresponding to a temperature TQH, the Universe – which was composed of photons, gluons, lepton–antilepton pairs and quark–antiquark pairs before – undergoes a (probably first-order) phase transition through which the quark–antiquark pairs join together to form the hadrons, including pions and nucleons. In this period pion–pion interactions are very important and, consequently, the equation of state of the hadron fluid becomes very complicated: one can certainly not apply the ideal gas approximation (Section 7.1) to hadrons in this era. The end of this era occurs when T 130 MeV at which point the pions annihilate.

At a temperature just a little greater than 100 MeV the Universe comprises three types of pion (π+, π−, π0); small numbers of protons, antiprotons, neutrons and antineutrons (these particles are no longer relativistic at this temperature); charged leptons (muons, antimuons, electrons, positrons – the tau leptons will have annihilated at this stage) and their respective neutrinos (νµ, ν¯µ, νe, ν¯e, ντ , ν¯τ ); and photons. At a temperature of T 130 MeV the π+–π− pairs rapidly annihilate and the neutral pions π0 decay into photons. This is the last act of the brief era of the hadrons. After this, there remain only leptons, antileptons, photons and the small excess of baryons (protons and neutrons) that we discussed in relation to the radiation entropy per baryon in Chapter 5; this, as we have explained, is probably due to processes which violated baryon number conservation while the