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)).

This book is designed for final year undergraduates or beginning graduate students in physics or theoretical physics. It assumes an acquaintance with Special Relativity and electromagnetism, but beyond that my aim has been to provide a pedagogical introduction to General Relativity, a subject which is now – at last – part of mainstream physics. The coverage is fairly conventional; after outlining the need for a theory of gravity to replace Newton's, there are two chapters devoted to differential geometry, including its modern formulation in terms of differential forms and coordinate-free vectors, then the Einstein field equations, the Schwarzschild solution, the Lense–Thirring effect (recently confirmed observationally), black holes, the Kerr solution, gravitational radiation and cosmology. The book ends with a chapter on field theory, describing similarities between General Relativity and gauge theories of particle physics, the Dirac equation in Riemannian space-time, and Kaluza–Klein theory.

As a research student I was lucky enough to attend the Les Houches summer school in 1963 and there, in the magnificent surroundings of the French alps, began an acquaintance with many of the then new aspects of this subject, just as it was entering the domain of physics proper, eight years after Einstein's death. A notable feature was John Wheeler's course on gravitational collapse, before he had coined the phrase ‘black hole’. In part I like to think of this book as passing on to the community of young physicists, after a gap of more than 40 years, some of the excitement generated at that school.

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.

Important milestones in the early history of General Relativity were the Einstein field equations, Schwarzschild's solution to them and the observational consequences of this solution. The Schwarzschild solution describes the space-time in the vicinity of a static, spherically symmetric mass, like the Sun, and the observational tests of this solution include the precession of the perihelion of planetary orbits – in particular the orbit of Mercury – and the bending of light in a gravitational field. A more recent test is the so-called radar echo delay of a signal sent from one planet (Earth) and reflected back from another one. An additional test of General Relativity, which depends only on the Equivalence Principle and not on the field equations, is the gravitational red-shift of light. The successful passing of these tests established General Relativity as the ‘correct’ theory of gravity. A feature of the Schwarzschild solution, not emphasised in the early days but given great prominence since, is the presence of the ‘Schwarzschild’ radius, which is the signature for the phenomenon of black holes. These matters are the concerns of this chapter. We begin with a comparison of the geodesic equation and the Newtonian limit of a weak, static gravitational field.

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.

We have already explored some features of the Schwarzschild solution, including the tests of General Relativity that it allows. In the Schwarzschild solution the Sun is taken to be static, that is, non-rotating. In fact, however, the Sun does rotate, and this suggests the question, is there another exact solution, a generalisation of the Schwarzschild solution, describing a rotating source? And, if there is, does it suggest any additional tests of General Relativity? It turns out that a generalisation of the Schwarzschild solution does exist – the Kerr solution. This is a rather complicated solution, however; it will be discussed further in the next chapter. In this chapter we shall find an approximate solution for a rotating source (which of course will also turn out to be an approximation of the Kerr solution). The tests for this solution include a prediction for the precession of gyroscopes in orbit round the Earth (which of course also rotates). This is a tiny effect, but in April 2004 a satellite was launched to look for this precession, which goes by the names of Lense and Thirring. We shall see that there is a parallel between the Lense–Thirring effect and magnetism, just as there is between ‘ordinary’ gravity (not involving rotations) and electricity – hence the name ‘gravitomagnetism’. After a discussion of these matters the chapter finishes with a more theoretical look at the nature of the distinction between ‘static’ (Schwarzschild) and ‘stationary’ (Kerr) space-times.

In this chapter we introduce the mathematical language which is used to express the theory of General Relativity. A student coming to this subject for the first time has to become acquainted with this language, which is initially something of a challenge. Einstein himself had to learn it (from his friend Marcel Grossmann). The subject is widely known as tensor calculus; it is concerned with tensors and how to define and differentiate them in curved spaces. In more recent times tensor calculus has been recast using a more sophisticated formalism, based on coordinate-free notation and differential forms. At first physicists were disinclined to learn this higher grade language, since it involved more work, without, perhaps, any reward in terms of mathematical or physical insight. Eventually, however, sceptical minds became convinced that there were indeed pay-offs in learning this new formalism, and knowledge of it is now almost essential to read research papers in the field of General Relativity.

Brief description of the Universe

Our Sun is one star in a collection of about 1011 stars forming our Galaxy. The Galaxy is shaped roughly like a pancake – approximately circular in ‘plan’ and with thickness much less than its radius – and the Sun is situated towards the outside of this distribution, not far from the central plane. The Galaxy is about 100 000 light years (ly) across. Almost all the stars visible to the naked eye at night belong to our Galaxy, and looking at the Milky Way is looking into its central plane, where the density of stars is greatest. The Andromeda Nebula, also visible to the naked eye, is a separate galaxy about 2 million light years away, and in fact is a member of the Local Group of galaxies. The construction of large telescopes in the first decades of the twentieth century led to the discovery of many galaxies and groups of galaxies and it is now known that there are about 1011 galactic clusters in the visible Universe. Considering these clusters as the ‘elementary’ constituents of the Universe, on scales larger than that of the clusters their distribution in space appears to be homogeneous and isotropic. This is the first – and very remarkable – feature of the Universe.

Special relativity assumes the existence of Minkowski coordinates, such that the line element takes the form (2.1). In that case one says that spacetime is flat, otherwise one says that spacetime is curved. Spacetime is flat only insofar as gravity can be ignored.

In this chapter we first see how to write the equations of special relativity using generic coordinates. Then we consider curved spacetime, the equivalence principle and the Einstein field equation. All of these things taken together are called general relativity. We end the chapter by giving a basic description of the particular curved spacetime that corresponds to a homogeneous and isotropically expanding universe.

Special relativity with generic coordinates: mathematics

To handle curved spacetime we have to learn how to use generic coordinates. It is helpful to do this first in the familiar context of special relativity, where Minkowski coordinates do exist.

Once a coordinate choice xµ has been made, it defines a threading of spacetime into lines (corresponding to fixed xi) and a slicing into hypersurfaces (corresponding to fixed x0), as shown in Figure 3.1. The threads are chosen to be timelike, so that they are the worldlines of possible observers, and the slices are chosen to be spacelike. The coordinate choice uniquely defines the threading and slicing, but the reverse is not true. Given a slicing and threading there is still freedom in choosing the coordinates which label the slices and threads.