So far, we demanded that only one scalar field varies during inflation. Now we allow two or more fields to vary. In the first section we generalize the slow-roll paradigm, and in the second section we give a more general discussion which applies even if the fields are not responsible for inflation.

Multi-field slow-roll inflation

Let us suppose that more than one field varies during inflation, corresponding to an inflationary trajectory φ1(t), φ2(t), …. We continue to assume Einstein gravity, with the energy density dominated by the scalar field potential of canonically normalized fields.

We recover the single-field case if the inflationary trajectory is a straight line in field space, because we can then perform a rotation of the field basis so that just a single field φ has significant variation. Even if the trajectory is not straight, we recover the single-field case provided that the trajectory lies in a steep-sided valley of the potential. The inflaton field then just corresponds to the distance in field space along the valley bottom.

To get something non-trivial, we suppose that the inflationary trajectory is not straight, and that it is one of a continuous family of possible slow-roll trajectories. The family is obtained by displacing the original trajectory sideways in field space. We then say that there is multi-field slow-roll inflation.

The field theory action is constructed to be invariant under Lorentz transformations and spacetime translations. These don't mix different fields. Now we consider internal symmetries, corresponding to transformations which mix different fields but don't involve spacetime.

Internal symmetries come in two kinds, called global symmetry and gauge symmetry. A gauge symmetry must be exact, but a global symmetry may only be approximate.

What is the motivation for considering a symmetry? One might first choose an action, guided by observation and/or aesthetics, and then look to see what is its symmetry group. That was the case when Quantum Electrodynamics was first formulated. Nowadays, the more usual procedure is to first choose a symmetry group (guided again by observation and/or aesthetics) and then to write down the most general action consistent with the symmetry group, which turns out to be quite strongly constrained. Often, the action arrived at in this way turns out to have additional global symmetries which were not imposed from the start, called accidental symmetries.

In this chapter we look at some examples of internal symmetry, as it applies to scalar fields at the classical level. Although we won't generally write down formulas involving spin-1/2 fields, it will be important to remember their existence.

Symmetry groups

To say that the action is invariant under some transformation is to say that it is the same before and after the transformation. It follows that the action is invariant also under the inverse of the transformation.

The beginning of the twenty-first century stands a good chance of being identified in history as the time when humankind first came to grips with the Universe. In a rush of observational progress, the charge led by the 1992 discovery of cosmic microwave background anisotropies by the Cosmic Background Explorer (COBE) satellite, the elements required to build accurate cosmological models were assembled. Complementing this, development of theoretical methods allowed accurate predictions to be made to confront those observations.

The landmark was the 2003 announcement of precision cosmic microwave measurements from the team operating the Wilkinson Microwave Anisotropy Probe (WMAP). Ironically, this success lay in a kind of failure – a failure to uncover anything new and unexpected. In the words of astrophysicist John Bahcall at NASA's announcement press conference, “the biggest surprise is that there are no surprises”. Instead, then, the power of the observations became fully focussed on determining the properties of the cosmological model, and for the first time many of its components were determined to a satisfying degree of accuracy: the percent level for quantities such as the geometry of the Universe, its age, and the density of the baryonic material, and the ten percent level for many other aspects.

In 2000, we published a graduate-level textbook, Cosmological Inflation and Large-Scale Structure, written during the late 1990s and which described many of the ideas underpinning the modern cosmology.

The time dependence of each perturbation is well defined, being determined by laws of physics. Viewed instead as a function of position at fixed time, the perturbations have random distributions. It is the statistical properties of these distribution that we wish to uncover via observation, and relate to fundamental physics models for the origin of perturbations. Those are usually referred to as stochastic properties.

The inherent randomness means that one shouldn't aim to predict things like the precise location of particular galaxies. Questions should refer to stochastic properties only. Don't ask ‘how far is it to the nearest large galaxy?’; instead ask ‘what is the typical separation between large galaxies?’. This randomness echoes simple quantum mechanics, e.g. one shouldn't hope to predict the precise position of a single particle in a closed box, but could compute the typical distance of the particle from its centre averaged over many such boxes. Indeed, we will see that in the inflationary cosmology the randomness of cosmological perturbations does have its origin in quantum uncertainty.

To describe the stochastic properties of the perturbations, one invokes the mathematical concept of a random field. In this chapter we describe the relevant aspects of that concept, without tying ourselves at this stage to any particular perturbation.

Random fields

Consider just one perturbation, evaluated at some instant, which we denote by g(x). We take g(x) to be associated with what is called a random field.

In the previous two chapters we considered the cosmological perturbations before horizon entry. As no causal processes can then operate, the description of these perturbations is simple, but they are not directly observable, except on the very largest scales via the cosmic microwave anisotropy. Over the next few chapters we follow the evolution of the cosmological perturbations after horizon entry, under the influence of causal processes.

The first section of this chapter is devoted to an overview of the evolution. In the rest of the chapter we see how to calculate the evolution in the regime of Newtonian gravity.

Free-streaming, oscillation, and collapse

As summarized in Table 7.1, the cosmic fluid at T < 1MeV has four components: baryons, cold dark matter (CDM), photons, and neutrinos. The CDM and neutrinos have negligible interaction. Until the temperature falls below its mass each neutrino species has relativistic motion, so that it behaves as radiation rather than matter up to that point.

The essence of what happens after horizon entry can be stated very simply. For each component, there is a competition between gravity, which tries to increase the density perturbation by attracting more particles to the overdense regions, and random particle motion which drives particles away from those regions. The relative importance of these effects depends on which component we are considering, and it may depend also on what era we are talking about.

In Part II of this book, we described the perturbations at the ‘primordial’ epoch T ~ 1 MeV, when they first become directly accessible to observation. At that stage the dominant perturbation is the curvature perturbation ζ. There may also be a tensor perturbation hij, and isocurvature perturbations Si (with i = c, b or ν).

Now we broaden the definition of ‘primordial’ and ask about perturbations at earlier times. In this chapter we see how the perturbations of light scalar fields are generated during inflation.

The idea is quite simple. Let us focus on some comoving wavenumber k. Well before horizon exit the curvature of spacetime is negligible and we are dealing with flat spacetime field theory where the particle concept should be valid. The particle number is assumed to be negligible, so that each field is in the vacuum state.

The crucial point now is that the vacuum fluctuation of a light field will ‘freeze in’ at horizon exit, to become a classical perturbation. The process was understood in the 1970s, before inflation was proposed as a physical reality, and has nothing to do with gravity. It occurs simply because the timescale a/k of the would-be vacuum fluctuation becomes bigger than the Hubble time H−1. We will see in some detail how this intuitive picture can emerge from a proper calculation.

In this chapter, we describe one ultimate consequence of the evolution of the primordial perturbation, namely the observable matter distribution in the Universe. It is not our intention to provide a detailed account of how observers probe the matter distribution, something which is now carried out with impressive precision. Instead, we focus on the outcome of those observations and their comparison with theory. We will explain the fundamental theoretical strategy, and highlight some simple physical arguments which account for the broad features of the data.

To use linear perturbation theory after galaxies begin to form we need to smooth the density contrast as described in Section 5.1.2. After looking in more detail at the description of smoothing, we will see in a qualitative way how the theory describes ‘bottom-up’ structure formation. We go on to give a more quantitative description, which applies to the formation of objects with a given mass as long as they are rare. In this way we obtain an estimate of the abundance of such objects, which may be compared with observation.

Next we come to the issue of comparing the calculated density perturbation directly with observation. More precisely, we seek to compare with observation the spectrum of the density contrast, and higher correlators which may signal nongaussianity. For this purpose we have to remember that the galaxy number density won't precisely trace the matter density, since the latter includes dark matter (both baryonic and cold dark matter (CDM)).

In this and the following two chapters, we consider scenarios for the history of the Universe from the end of inflation to neutrino decoupling at T ~ 1 MeV. Einstein gravity is assumed in all cases. A viable scenario must lead to a radiationdominated Universe at T ~ 1MeV, with properties not too different from those described in Section 4.5. In particular, the abundance of relic particles such as gravitinos or moduli should be below observational bounds. As seen in Section 24.7.3, the detection of a cosmic gravitational wave background may in the far future offer powerful discrimination between different scenarios.

This chapter begins with the reheating process, which establishes thermal equilibrium after inflation and initiates the Hot Big Bang. Then we see how the spontaneous breaking of symmetries may lead to the creation of solitons of various kinds, including in particular cosmic strings and other topological defects. Finally, we discuss the possibility of a short burst of late inflation known as thermal inflation.

Reheating

Initial reheating

At the end of inflation, the entire energy density of the Universe remains locked in the scalar fields. Everything else has presumably been diluted away by the inflationary period. We have to free this energy density by converting it into other forms, with the ultimate goal of creating the Hot Big Bang radiation that is certainly present when the run-up to nucleosynthesis begins at T ~ 1 MeV.

In this chapter and the next, we consider the perturbation in the energy density of the Universe, as it exists when the run-up to nucleosynthesis begins at temperature T ~ 1MeV. Although it is not essential, we will take T to actually be a bit below 1MeV so that the positrons have annihilated leaving just the cold dark matter (CDM), baryons, photons and neutrinos, with the last decoupled.

Perturbations existing at this epoch may be called primordial perturbations because they provide a simple initial condition for the subsequent evolution of the perturbed Universe. That evolution and its contact with observation will occupy us for the rest of Part II.

We will also study what is called the curvature perturbation. It is a powerful quantity, because on superhorizon scales it is conserved provided that the pressure of the cosmic fluid depends only on its energy density. The curvature perturbation determines the perturbation in the total energy density, defined on a given slicing of spacetime. We consider also the isocurvature perturbations which determine the distribution of energy density between different components of the cosmic fluid.

The strategy will be to study the perturbations themselves in this chapter, and their stochastic properties in the next. The study is important both in its own right, and because it introduces basic concepts that will be used throughout the book.