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

128 The Very Early Universe

topology). Such a space is usually called a superspace. To include matter in this formulation then one would have to write Ψ[hµν , Φ], where Φ labels the matter field at x. Notice that, unlike (6.4.1), there is no need for an explicit time labelling beside hµν in the argument of Φ: a generic 3-geometry will actually only fit into a generic 4-geometry in at most only one place, so hµν carries its own labelling of time. The integral is taken over appropriate 4-geometries consistent with the 3- geometry hµν . The usual quantum-mechanical wavefunction ψ evolves according to a Schrödinger equation; our ‘wavefunction of the Universe’, Ψ, evolves according to a similar equation called the Wheeler–de Witt equation (de Witt 1967). It is the determination of what constitutes the appropriate set of histories over which to integrate that is the crux of the problem and it is easy to see that this is nothing other than the problem of initial conditions in quantum cosmology (by analogy with the single-particle problem discussed above). This problem is far from solved.

One suggestion, by Hartle and Hawking (1983), is that the sum on the right-hand side of (6.4.2) is over compact Euclidean 4-geometries. This essentially involves making the change t → −iτ with respect to the usual Lorentzian calculations. In this case the 4-geometries have no boundary and this is often called the noboundary conjecture. Amongst other advantages, the relevant Euclidean integrals can be made to converge in a way in which the Lorentzian ones apparently cannot. Other choices of initial condition have, however, been proposed. Vilenkin (1984, 1986), amongst others, has proposed a model wherein the Universe undergoes a sort of quantum-tunnelling from a vacuum state. This corresponds to a definite creation, whereas the Hawking proposal has no ‘creation’ in the usual sense of the word. It remains to be seen which, if any, of these formulations is correct.

6.5 String Cosmology

Recent years have seen a radically di erent approach to the problem of quantum gravity which has led to a di erent idea of the possible structure of a quantum gravity theory. One of the most exciting ideas is that the fundamental entities upon which quantum operations must be performed are not point-like but are one dimensional. Such objects are usually known as strings, or more often superstrings (because they are usually discussed within so-called supersymmetric theories that unite fermions and bosons; see Chapter 8). Many physicists feel that string theory holds the key to the unification of all four forces of nature (gravity included) in a single over-arching theory of everything. Such a theory does not yet exist, but there is much interest in what its possible consequences might be.

String cosmology entered the doldrums in the early 1990s after a period of initial excitement. However it has since seen a resurgence largely because of the realisation that string theories can be thought of in terms of a more general class of theories known as M-theories. It is a property of all these structures that in order to be mathematically consistent they must be defined in space–times having more dimensions than the (3+1)-dimensional one with which we are familiar. One of the consequences of such theories is that fundamental constants ‘live’ in the higher-dimensional space and can vary in the 4-dimensional subspace we inhabit.

They therefore lead naturally to models like those we discussed in Chapter 3 in which fundamental constants may vary with time.

The idea that there may be more than four space–time dimensions is not itself new. Kaluza (1921) and Klein (1926) examined a (4 +1)-dimensional model which furnished an intriguing geometrical unification of gravity and electromagnetism. In the Kaluza–Klein theory the extra space dimension was compactified on a scale of order the Planck length, i.e. wrapped up so small as to be unobservable.

String theories have to hide many dimensions, not just one, and until recently it was assumed that they would all have to be compactified. However, a radically new idea is called the braneworld scenario in which at least one of the extra dimensions might be large. In this picture we are constrained to live on a threedimensional brane inside a higher-dimensional space called the bulk. Gravity is free to propagate in the bulk, and the gravity we see on the brane is a kind of projection of this higher-dimensional force. In the simplest braneworld model, known as the Randall–Sundrum model (Randall and Sundrum 1999) this theory results in a modification of the low-energy form of gravity such that the Newtonian potential becomes

where k corresponds to a very small length scale, of order the Planck length. At higher energies, however, there are interesting e ects. In a particular case of the Randall–Sundrum model the high-energy behaviour of the Friedmann equation is modified:

where λ is the tension in the brane.

A further development of the braneworld scenario is the notion that what we think of as the Big Bang singularity may in fact be the result of a collision between two branes. This has been dubbed the Ekpyrotic universe. Interest in this model stems from the fact that the impact of two branes may lead to e ects that appear to be acausal when viewed from one of them. It remains to be seen whether this model can be developed to the point where it stands as a rival to the Big Bang.

Bibliographic Notes on Chapter 6

Two classic compilations on fundamental gravity theory are Hawking and Israel (1979, 1987). Du and Isham (1982) is also full of interesting thoughts on quantum gravity, and Hartle (1988) is readable as well as authoritative.

Problems

1.In natural units = c = 1. Show that in such a system all energies, lengths and times can be expressed in terms of the Planck mass mP.

130The Very Early Universe

2.Show that, in natural units, an energy density may be expressed as the fourth power of a mass. If the vacuum energy contributed by a cosmological constant is now of order the critical density, what is the mass to which this density corresponds?

3.Obtain a formula relating the Hawking temperature (6.3.5) to the radius of the event horizon of a Black Hole. In a de Sitter universe the scale factor increases exponentially with time such that a/a˙ = H is constant. Show that in this model there is an event horizon with radius c/H. Assuming the Hawking formula also works for this radius, calculate the temperature of the event horizon in de Sitter space. How do you interpret this radiation?

4.Find a solution of Equation (6.5.2) with λ constant.

7

Phase

Transitions

and Inflation

7.1 The Hot Big Bang

We shall see in the next chapter that, if cosmological nucleosynthesis is the correct explanation for the observed light-element abundances, the Universe must have been through a phase in which its temperature was greater than T 1012 K. In this chapter we shall explore some of the consequences for the Universe of phases of much higher temperature than this. Roughly speaking, we can define ‘Hot’ Big Bang models to be those in which the temperature increases as one approaches t = t0. We assume that, after the Planck time, the temperature follows the law:

we shall give detailed justification for this hypothesis later on.

Travelling backward in time, so that the temperature increases towards TP, the particles making up the contents of the present Universe will all become relativistic and all the interactions between them assume the character of a long-range force such as electromagnetism. One can apply the model of a perfect ultrarelativistic gas of non-degenerate (i.e. with chemical potential µ = 0) particles in thermal equilibrium during this stage. The equilibrium distribution of a particle species i depends on whether it is a fermionfermions or a boson and upon how many spin or helicity states the particle possesses, gi. The quantity gi is also

132 Phase Transitions and Inflation

sometimes called the statistical weight of the species i. The number-density of particles can be written

where the integrand includes a ‘+’ sign for fermions and a ‘−’ sign for bosons producing a factor of 34 or 1 in these respective cases. In (7.1.2) ζ is the Riemann zeta function which crops up in the integral; ζ(3) 1.202. Similarly, the energy density of the particles is

in which we have used the definition of the radiation density constant σr. The total energy density is therefore given by

in which B stands for bosons and F for fermions; the sums are taken over all the bosons and fermions with their respective statistical weights giB and giF. The quantity g (T) is called the e ective number of degrees of freedom. To obtain the total density of the Universe one must add the contribution ρd(T), coming from those particles which are no longer in thermal equilibrium (i.e. those which have decoupled from the other particles, such as neutrinos after their decoupling) and the contribution ρnr(T), coming from those particles which are still coupled but no longer relativistic, as is the case for the matter component in the plasma era. There may also be a component ρnt(T) due to particles which are never in thermal equilibrium with the radiation (e.g. axions). As we shall see, for the period which interests us in this chapter, the contributions ρd(T), ρnr(T) and ρnt(T) are generally negligible compared with ρ(T).

The number-densities corresponding to each degree of freedom (spin state) of a boson, nB, and fermionfermions, nF, are

As we shall see later, g (T) < 200 or so. This means that the average separation of the particles is

¯

so that d practically coincides with the ‘thermal wavelength’ of the particles,c/kBT, which is in some sense analogous to the Compton radius.

The cross-section of all the particles is, in the asymptotic limit T → TP,

with α of the order of 1/50, so that the collision time is

This time is to be compared with the expansion timescale τH = a/a˙:

The hypothesis of thermal equilibrium is consequently well founded.

One can easily verify that the assumption that the particles behave like a perfect gas is also valid. Given that asymptotically all the interactions are, so to speak, equivalent to electromagnetism with the same coupling constant, one can verify this hypothesis for two electrons: the ratio r between the kinetic energy, Ec kBT,

to a good approximation r is the inverse of the fine-structure constant.

In Equations (7.1.4) and (7.1.5) there is an implicit hypothesis that the particles are not degenerate. We shall see in the next chapter that this hypothesis, at least for certain particles, is held to be the case for reasonably convincing reasons.

7.2 Fundamental Interactions

The evolution of the first phases of the hot Big Bang depends essentially on the physics of elementary particles and the theories that describe it. For this reason in this section we will make some comments on interactions between particles. It is known that there are four types of fundamental interactions: electromagnetic, weak nuclear, strong nuclear and gravitational. As far as the first three of these are concerned, quantum describes them in terms of the exchange of bosonic particles which play the role of force carriers.

The electromagnetic interactions are described classically by Maxwell’s equations and in the quantum regime by quantum electrodynamics (QED). These forces are mediated by the photon, a massless boson: this implies that they have a long range. The coupling constant, a quantity which, roughly speaking, measures the strength of the interaction, is given by gQED = e2/ c 1/137. From the point of view of group theory the Lagrangian describing electromagnetic interactions is invariant under the group of gauge transformations denoted U(1) (by gauge

134 Phase Transitions and Inflation

transformation we mean a transformation of local symmetry, i.e. depending upon space–time position).

In the standard model of particle physics the fundamental interacting objects are all fermions, either quarks or leptons. There are three families of leptons in this model, each consisting of a charged lepton and an associated neutrino. The charged leptons include the electron e− but there are also µ− and τ− particles. Each of these has an accompanying neutrino designated νe, νµ and ντ . The leptons are therefore arranged in three families, each of which contains a pair of related particles. There are also antiparticles of the charged leptons (i.e. e+, µ+, τ+), and antineutrinos of each type (ν¯e, ν¯µ, ν¯τ ).

The other fundamental fermions are the quarks, which also occur in three families of pairs mirroring the leptons. Quarks have fractional electronic charge but also possess a property known as colour which plays a role in strong nuclear reactions. The six quarks are denoted ‘up’ (u), ‘down’ (d), ‘strange’ (s), ‘charmed’ (c), ‘bottom’ (b) and ‘top’ (t). Each quark comes in three di erent colours, red, green and blue: these are denoted ur, ug, ub and so on. The three families are arranged

¯

as (u, d), (s, c) and (b, t). There are also antiquarks for each quark (i.e. u,¯ d) possessing opposite electrical charge and also opposite colour. Note that ‘anti-up’ is not the same thing as ‘down’!

Basic properties of the quarks and leptons are shown in Appendix A.

The weak nuclear interactions involve all particles, but are generally of most interest when they involve the leptons. These interactions are of short range because the bosons that mediate the weak nuclear force (called W+, W− and Z0) have masses mW 80 GeV and mZ0 90 GeV. It is the mass of this boson that makes the weak interactions short range. The weak interactions can be described from a theoretical point of view by a theory developed by Glashow, Salam and Weinberg around 1970. According to this theory, the electromagnetic and weak interactions are di erent aspects of a single force (the electroweak force) which, for energies greater than EEW 102 GeV, is described by a Lagrangian which is invariant under the group of gauge transformations denoted SU(2)×U(1). At energies above EEW the leptons do not have mass and their electroweak interactions are mediated by four massless bosons (W1, W2, W3, B), called the intermediate vector bosons, with a coupling constant of order gQED. At energies lower than EEW the symmetry given by the SU(2) × U(1) transformation group is spontaneously broken; the consequence of this is that the leptons (except perhaps the neutrinos) and the three bosons acquire masses (W+, W− and Z0 can be thought of as ‘mixtures’ of quantum states corresponding to the W1, W2, W3 and B). The only symmetry that remains is, then, the U(1) symmetry of electromagnetism.

The strong nuclear interactions involve above all the so-called hadrons. These are composite particles made of quarks, and are themselves divided into two classes: baryons and mesons. Baryons consist of combinations of three quarks of di erent colours (one red, one green, one blue) in such a way that they are colourless. Mesons are combinations of a quark and an anti-quark and are also colourless. Familiar examples of hadrons are p, p,¯ n, and n,¯ while the most relevant mesons are the pions π+, π−, π0. All hadronic states are described from a

quantum point of view by a theory called quantum chromodynamics (QCD). This theory was developed at a similar time to the theory which unifies the electromagnetic and weak interactions and, by now, it has gained considerable experimental support. According to QCD, hadrons are all made from quarks. There are various types of quark. These have di erent weak and electromagnetic interactions. The characteristic which distinguishes one quark from another is called flavour. The role of the bosons in the electroweak theory is played by the gluons, a family of eight massless bosons; the role of charge is replaced by a property of quarks and gluons called colour. At energies exceeding of the order of 200–300 MeV quarks are no longer bound into hadrons, and what appears is a quark–gluon plasma. The symmetry which the strong interactions respect is denoted SU(3).

The success of the unification of electromagnetic and weak interactions (Weinberg 1967; Salam 1968; Georgi and Glashow 1974) by the device of a restoration of a symmetry which is broken at low temperatures – i.e. SU(2) ×U(1) – has encouraged many authors to attempt the unification of the strong interactions with the electroweak force. These theories are called GUTs (Grand Unified Theories); there exist many such theories and, as yet, no strong experimental evidence in their favour. In these theories other bosons, the superheavy bosons (with masses around 1015 GeV), are responsible for mediating the unified force; the Higgs boson is responsible for breaking the GUT symmetry. Amongst other things, such theories predict that protons should decay, with a mean lifetime of around 1032– 1033 years; various experiments are in progress to test this prediction, and it is possible that it will be verified or ruled out in the not too distant future. The simplest version of a GUT respects the SU(5) symmetry group which is spontaneously broken at an energy EGUT 1015 GeV, so that SU(5) → SU(3) × SU(2) × U(1) (even though the SU(5) version seems to be rejected on the grounds of the mean lifetime of the proton, we refer to it here because it is the simplest model). For a certain choice of parameters the original SU(5) symmetry breaks instead to SU(4)×U(1) around 1014 GeV, which disappears around 1013 GeV giving the usual SU(3)×SU(2)×U(1). The possible symmetry breaking occurs as a result of a firstorder phase transition (which we shall discuss in the next section) and forms the basis of the first version of the inflationary universe model produced by Guth in 1981. It is, of course, possible that there is more than one phase transition between 1015 GeV and 100 GeV. At energies above EGUT, in the simplest SU(5) model, the number of particle types corresponds to g (T) 160.

The fourth fundamental interaction is the gravitational interaction, which is described classically by general relativity. We have discussed some of the limitations of this theory in the previous chapter. The boson which mediates the gravitational force is usually called the graviton. It is interesting in the context of this chapter to ask whether we will ever arrive at a unification of all four interactions. Some attempts to construct a theory unifying gravity with the other forces involve the idea of supersymmetry; an example of such a theory is supergravity. This theory, amongst other things, unifies the fermions and bosons in a unique multiplet. More recently, great theoretical attention has been paid to the idea of superstrings, which we mentioned in the previous chapter. Whether these ideas

136 Phase Transitions and Inflation

will lead to significant progress towards a ‘theory of everything’ (TOE) remains an open question.

7.3 Physics of Phase Transitions

In certain many-particle systems one can find processes which can involve, schematically, the disappearance of some disordered phase, characterised by a certain symmetry, and the appearance of an ordered phase with a smaller degree of symmetry. In this type of order–disorder transition, called a phase transition, some macroscopic quantity, called the order parameter and denoted by Φ in this discussion, grows from its original value of zero in the disordered phase. The simplest physical examples of materials exhibiting these transitions are ferromagnetic substances and crystalline matter. In ferromagnets, for T > Tc (the Curie temperature), the stable phase is disordered with net magnetisation M = 0 (the quantity M in this case represents the order parameter); at T < Tc a non-zero magnetisation appears in di erent domains (called the Weiss domains) and its direction in each domain breaks the rotational symmetry possessed by the disordered phase at T > Tc. In the crystalline phase of solids the order parameter is the deviation of the spatial distribution of ions from the homogeneous distribution they have at T > Tf, the melting point. At T < Tf the ions are arranged on a regular lattice. One can also see an interesting example of a phase transition in the superconductivity properties of metals.

The lowering of the degree of symmetry of the system takes place even though the Hamiltonian which describes its evolution maintains the same degree of symmetry, even after the phase transition. For example, the macroscopic equations of the theory of ferromagnetism and the equations in solid-state physics do not pick out any particular spatial position or direction. The ordered states that emerge from such phase transitions have a degree of symmetry which is less than that governing the system. In fact, one can say that the solutions corresponding to the ordered state form a degenerate set of solutions (solutions with the same energy), which has the same degree of symmetry as the Hamiltonian. Returning to the above examples, the magnetisation M can in theory assume any direction. Likewise, the positioning of the ions in the crystalline lattice can be done in an infinite number of di erent ways. Taking into account all these possibilities we again obtain a homogeneous and isotropic state. Any small fluctuation, in the magnetic field of the domain for a ferromagnet or in the local electric field for a crystal, will pick out one preferred solution from this degenerate set and the system will end up in the state corresponding to that fluctuation. Repeating the phase transition with random fluctuations will produce randomly aligned final states. This is a little like the case of a free particle, described in Newtonian mechanics by v˙ = 0, which has both translation and rotational symmetries. The solutions r = r0 +v0t, with r0 and v0 arbitrary, form a set which respects the symmetry of the original equation. But it is really just the initial conditions r0 and v0 which, at one particular time, select a solution from this set and this solution does not have the same degree of symmetry as that of the equations of motion.

A symmetry-breaking transition, during which the order parameter Φ grows significantly, can be caused by external influences of su cient intensity: for example, an intense magnetic field can produce magnetisation of a ferromagnet even above Tc. Such phenomena are called induced symmetry-breaking processes, to distinguish them from spontaneous symmetry breaking. The spontaneous breaking of a symmetry comes from a gradual change of the parameters of the system itself. On this subject, it is convenient to consider the free energy of the system F = U − TS (U is internal energy; T is temperature; S is entropy). Recall that the condition for existence of an equilibrium state of a system is that F must have a minimum. The free energy coincides with the internal energy only at T = 0. At higher temperatures, whatever the form of U, an increase in entropy (i.e. disorder) generally leads to a decrease in the free energy F, and is therefore favourable. For systems in which there is a phase transition, F is a function of the order parameter Φ. Given that Φ must respect the symmetry of the Hamiltonian of the system, it must be expressible in a manner which remains invariant with respect to transformations which leave the Hamiltonian itself unchanged. Under certain conditions F must have a minimum at Φ = 0 (disordered state), while in others it must have a minimum with Φ ≠ 0 (ordered state).

Let us consider the simplest example. If the Hamiltonian has a reflection symmetry which is broken by the appearance of an order parameter Φ or, equivalently in this case, −Φ, the free energy must be a function only of Φ2 (in this example Φ is assumed to be a real, scalar variable). If Φ is not too large we can develop F in a power series

where the coe cients α and β depend on the parameters of the system, such as its temperature. For α > 0 and β > 0 we have a curve of the type marked ‘1’ in Figure 7.1, while for α < 0 and β > 0 we have a curve of type ‘2’.

Curve 1 corresponds to a disordered state; the system is in the minimum at Φ = 0. Curve 2 has two minima at Φm = ±(−α/2β)1/2 and a maximum at Φ = 0; in this case the disordered state is unstable, while the minima correspond to ordered states with the same probability: any small external perturbation, which renders one of the two minima Φm slightly deeper or nudges the system towards it, can make the system evolve towards this one rather than the other with Φ = −Φm. In this way one achieves a spontaneous symmetry breaking. If there is only one parameter describing the system, say the temperature, and the coe cient α is written as α = a(T − Tc), with a > 0, we have a situation represented by curve 2 for T < Tc. While T grows towards Tc the order parameter Φ decreases slowly and is zero at Tc. This type of transition, like its inverse, is called a second-order phase transition and it proceeds by a process known as spinodal decomposition: the order parameter appears or disappears gradually and the di erence ∆F between T > Tc and T < Tc at T Tc is infinitesimal.

There are also first-order phase transitions, in which at T Tc the order parameter appears or disappears rapidly and the di erence ∆F is finite. This di erence