In Section 25.2 we considered the standard paradigm, according to which the curvature perturbation is generated by the vacuum fluctuation of the inflaton field during slow-roll inflation. We saw that observation gives considerable information about the potential when the pivot scale leaves the horizon. The normalization of the spectrum requires (V/∈)1/4 = 6.6 × 1016 GeV and the spectral index determines 2η − 6∈. Also, the bound on the running n′ constrains ξ, while the upper bound on the tensor fraction r constrains V.

If instead the curvature perturbation is generated after inflation, there is no particular reason to consider slow-roll inflation. If one does consider it, the only constraints are ∈ « 1 and (V/∈)1/4 « 6.6 × 1016 GeV (implying a small and probably negligible tensor fraction). In this chapter, we assume the standard paradigm and see how different models of slow-roll inflation are then constrained by observation.

By a model of inflation, we mean an effective field theory that is supposed to apply during inflation. In the end, it should be part of an effective field theory that takes us all the way from inflation to the present. A lot of work has been done in this direction, but so far no preferred model has emerged.

As we write, the Large Hadron Collider (LHC) is beginning operation, and may or may not find evidence for supersymmetry.

The question whether gravitational radiation exists is of great interest, both theoretically and experimentally. In the weak field approximation Einstein's field equations lead to a wave equation, which, in analogy with the situation with Maxwell's equations of electrodynamics, clearly suggests that gravitational waves exist; and, it would be hoped, again in analogy with the electrodynamic case, that they carry energy. We recall the crucial discovery by Hertz of electromagnetic waves, which convinced him, as well as the general public, of the reality of the field. From the 1960s Weber pioneered experiments to search for gravitational waves, but they have not yet been found. On the other hand there is some very convincing, though indirect, evidence that the gravitational field may radiate energy, which comes from the discovery that the period of the binary pulsar PSR 1913+16 is decreasing. One may feel justified in taking an optimistic view that gravitational radiation exists and might soon be discovered.

On the theoretical side there is a problem which is totally absent in the electromagnetic case. General Relativity is a non-linear theory, and this has the physical consequence that the gravitational field itself carries energy – witness the pseudo-energy-momentum tensor discussed earlier – which then acts as a source for more gravitational field. In contrast the electromagnetic field carries no electric charge so is not a source of further field. In the language of quantum theory, there is a graviton–graviton coupling but no photon–photon coupling.

If we take seriously – as of course we must – the notion that Einstein's theory of gravity has a status equal in validity to that of other major theories of physics, for example Maxwell's electrodynamics or the more modern gauge field theories of particle physics, then we shall want to ask how General Relativity may be formulated at a fundamental level. In Chapter 5 the field equations were introduced on a more or less ad hoc basis, arguing that what was wanted were equations relating space-time curvature to the energy and momentum of the source, that they should therefore involve second rank tensors; and that the equations also reduced to Newton's law in the non-relativistic limit. This approach is fine as far as it goes, but recall that Maxwell's equations, for example, may be derived from a principle of least action; a Lagrangian formulation. May Einstein's equations also be derived from a Lagrangian formulation? Indeed they may, and this is the subject of the first part of this chapter. We go on to investigate the tricky topic of conservation laws in General Relativity, a subject which has complications resulting from the fact that the (matter) energy-momentum tensor is covariantly conserved, whereas ‘true’ conservation laws involve simply partial, rather than covariant, differentiation. The chapter finishes with a consideration of the Cauchy, or initial value, problem. Einstein's field equations are second order differential equations, whose solutions will therefore involve specifying ‘initial data’ on a spacelike hypersurface.

The recent excitement in cosmology has been due to the emergence of a precision description of the present Universe and its recent past, with the standard model of cosmology able to accurately reproduce a wide range of sensitive observations. The ambitions of this effort are to determine the material composition of the present Universe, to establish the basic properties of perturbations in the Universe, and to verify that standard physical laws can be applied on a Universal scale. All of these now appear to be well in hand, as we have described in the earlier parts of this book. In particular, the successful theoretical reproduction of the observed cosmic microwave anisotropies is a tour de force of general relativity, particle interactions, and detailed modelling of the Universe's composition, that leaves little room for doubt that the fundamentals of cosmology are in place and secure.

Our book has also followed a broader ambition – to use that knowledge to tell us about the very early Universe and about the nature of fundamental physical laws. According to standard belief, almost every quantity measured in the present Universe has its origin in the Universe's earliest stages, when the primordial perturbations were created, perhaps by inflation, and when various as-yet uncertain particle physics processes established the densities of baryons, of dark matter, and perhaps also dark energy.

We have seen how the vacuum fluctuation of each light scalar field is converted to a classical perturbation at the time of horizon exit. One or more of these perturbations should in turn generate the curvature perturbation ζ, that is probed by observation when cosmological scales begin to enter the horizon.

In this chapter we are going to consider the simplest scenario, where the curvature perturbation already achieves its final value a few Hubble times after horizon exit. In other words, we are going to consider the scenario in which ζ is constant during the entire era when the smoothing scale is outside the horizon. According to Section 5.4.2, this means that the locally defined pressure is a unique function of the locally defined energy density throughout that era. This is achieved in a single-field inflation model, if at each position the field value a few Hubble times after horizon exit determines the subsequent pressure and energy density. During inflation, this is the same thing as saying that the trajectory φ(x, t) is independent of x, up to a shift in t. In other words, it is the same as saying that the inflationary trajectory is an attractor. After inflation though, it would be possible for some other light field to play a significant role. We shall assume in this chapter that such is not the case.

In Part III we deal with some essential aspects of field theory. Except in the last two sections of the present chapter, the entire discussion is in flat spacetime. After a brief introduction to field theory in general, in this chapter we deal with the classical theory of scalar fields.

Field theory

Basic concepts

A quantum field theory generally yields various species of elementary particle, each corresponding to one of the fields. In particular the photon corresponds to the electromagnetic field. Fields are classified as bosonic or fermionic according to the spin of the particle. One generally considers only fields corresponding to particles with spin-0 (scalar fields) and spin-1/2 particles (spinor fields), plus a special type of field corresponding to particles with spin-1 (vector field) known as a gauge field. The particles corresponding to gauge fields are called gauge bosons. One doesn't generally consider fields of higher spin, because these are difficult to accommodate within the theory and the corresponding elementary particles are not observed. The only exceptions are the spin-2 graviton that is supposed to correspond to a quantized weak gravitational wave, and the spin-3/2 gravitino that will be its partner in a supergravity theory.

At the quantum level, each field corresponds to an operator. Bosonic fields can be regarded as classical in a suitable regime. The fields live in curved spacetime described by the metric tensor gµν.

Although cosmology can trace its beginnings back to Einstein's formulation of his general theory of relativity in 1915, which enabled the first mathematically consistent models of the Universe to be constructed, for most of the following century there was much uncertainty and debate about how to describe our Universe. Over those years the various necessary ingredients were introduced, such as the existence of dark matter, of the hot early phase of the Universe, of cosmological inflation, and eventually dark energy. In the latter part of the last century, cosmologists and their funding agencies came to realize the opportunity to deploy more ambitious observational programmes, both on the ground and on satellites, which began to bear fruit from 1990 onwards. The result is a golden age of cosmology, with the creation and observational verification of the first detailed models of our Universe, and an optimism that that description may survive far into the future. The objective, often described as precision cosmology, is to pin down the Universe's properties as best as possible, in many cases at the percent or few percent level. In particular, the landmark publication in 2003 of measurements of the cosmic microwave background by the Wilkinson Microwave Anisotropy probe (WMAP), seems certain to be identified as the moment when the Standard Cosmological Model became firmly established.

The key tool in understanding our Universe is the formation and evolution of structure in the Universe, from its early generation as the primordial density perturbation to its gravitational collapse to form galaxies.

We have seen how the inhomogeneities of the matter and photon densities evolve, from early times to the present epoch. Now we will do the same thing for the anisotropy of the photon distribution function. This anisotropy is observed today as the cosmic microwave background (CMB) anisotropy.

Photons in the early Universe are in thermal equilibrium, with the blackbody distribution of momenta. As the epoch of photon decoupling is approached, the distribution begins to fall out of equilibrium, developing anisotropy which is different for the two polarization states. After decoupling at z ~ 1000, the redshifting of the photons through the inhomogeneous gravitational field generates more anisotropy, without affecting the polarization. Then reionization at z ~ 10 generates further anisotropy and polarization. Finally, the photons are observed at the present epoch, as the CMB anisotropy. The anisotropy is characterized by a perturbation in the intensity, which corresponds to a perturbation in the temperature of the blackbody distribution, and by two polarization parameters. In this chapter we study the temperature perturbation.

We are going to see that the CMB anisotropy is observable only on rather large scales, corresponding to comoving wavenumber k ≲ (10 Mpc)−1. In this regime, first-order (linear) cosmological perturbation theory is almost always a good approximation, failing on the smallest scales and only around the present epoch. In the latter regime, the dominant effect is the thermal Sunyaev–Zel'dovich effect, which occurs when a galaxy cluster is in the line of sight.

In this chapter we study thermal equilibrium in the early Universe. Then we look at possible mechanisms for the creation of baryon number (baryogenesis). We pay particular attention to baryogenesis mechanisms that directly involve a scalar field, because they offer the best chance of a primordial isocurvature perturbation.

Thermal equilibrium before the electroweak phase transition

In this section we show that electroweak symmetry is likely to be restored in the early Universe, with every particle of the Standard Model in thermal equilibrium.

Electroweak symmetry is restored if the temperature is bigger than a critical temperature TEW. The critical temperature is of order a few hundred GeV, the precise value depending on the parameters of the Standard Model, or an extension like the Minimal Supersymmetric Standard Model (MSSM).

A particle species is in thermal equilibrium if the rate per particle for all relevant interactions exceeds the Hubble rate H. The dominant interactions are (i) decays and two-body scattering and (ii) the sphaleron transitions that violate B and L conservation. As we are interested in the case that electroweak symmetry is restored, particle masses vanish except for the masses of two Higgs particles which are roughly of order 100 GeV and hence less than the temperature. As a result, T is the only relevant dimensionful parameter.