Like the actors in ancient Greek tragedy and comedy, neutrinos play more than one role in the drama of the expanding universe. They couple to gravity and contribute to Einstein equations which rule the expansion dynamics. Furthermore, they interact in the primordial plasma with charged leptons and hadrons via electroweak interactions, until the rates for these processes become so low compared with the typical expansion rate that they decouple and start to propagate freely along geodesic lines. Any quantitative description of their role in cosmology thus requires several inputs from the theory of fundamental interactions, as well as a knowledge of their basic properties, such as masses and, in some cases, the features of neutrino flavour oscillations.

Neutrino interactions have been well understood since the first theory of β- decay proposed by Enrico Fermi in 1934, and now are succesfully and beautifully described by the unified picture of electroweak interactions. In the low energy limit the strength of these interactions is encoded in a single coupling, the Fermi coupling constant GF, whose value, combined with the Newton constant, fixes the time of neutrino decoupling. From the strong experimental evidence in favour of neutrino oscillation, we also know that neutrinos are massive particles, and this, as we will see at length in the following, has a strong impact on how structures, i.e., inhomogeneities in the universe, grow on certain length scales.

Our journey in the land of neutrino cosmology is almost over. We have seen how neutrinos have silently influenced the evolution of the universe: how they perhaps may be responsible for the production of the baryon density we observe today, and the way they leave their signature in the nuclear abundances during primordial nucleosynthesis, in the CMB anisotropies and in structure formation.

We cannot leave the patient reader, who has kindly followed us till this point, without offering some final considerations about a simple question he or she might have thought about since the very beginning: can we directly detect the neutrino background in a laboratory experiment as Penzias and Wilson did for the CMB radiation?

There are two major obstacles one must overcome to achieve such a goal. Neutrinos are elusive particles because they interact only weakly. Cross sections are typically small, far smaller than electromagnetic ones, which were exploited by Penzias and Wilson. Furthermore, relic neutrinos today are quite cold particles with an average momentum on the order of c pν ~ 3.15 Tν,0 ~ 5 × 10−4 eV, so reaction rates are further suppressed. For the best interaction process candidate to date, neutrino capture on β-unstable nuclei such as 3H, we have 〈σν/c〈 ~ 10−44 cm2.

To Arianna, Carmen, Isabelle, and María José

When neutrinos first came on the scene in 1930, their father, Wolfgang Pauli, confessed to his colleague, the astronomer Walter Baade, that to save energy conservation in β-decays (quoted in Hoyle, 1967),

This was perhaps the only time Pauli was mistaken. Less than 30 years later, neutrinos were discovered by Reines and Cowan.

Since then, we have learned so many things about neutrinos that Pauli himself would be very surprised. More than this, understanding neutrino properties has always brought new insights into the whole field of fundamental interactions, and new theoretical paradigms.

Today we know quite accurately how to describe their feeble interactions with matter, from the very first attempts of Fermi to the succesful Standard Model of electroweak interactions. Many pieces of information have been collected in laboratory experiments, the traditional setting of particle physics. The study of neutrino interactions has been pursued at accelerators and reactors and, more recently, by sending neutrino beams produced at accelerators to underground laboratories. Accelerator experiments have also confirmed that there are only three generations of light neutrinos which are weakly interacting.

A Tale of Three Numbers:1, 3 and ∞. This might be a good subtitle for this chapter, which is about the problem of baryogenesis in the early universe.

We have already mentioned in Chapter 1 that observations of the light nuclear yields produced during primordial nucleosynthesis and the features of the CMB anisotropies single out a definite value for 1 parameter, the baryon-to-photon density ratio ηB ~ 6 × 10−10 at low temperatures, much below the nucleon mass. If the universe's expansion were starting with symmetric initial conditions, i.e., a zero initial baryon number per comoving volume, and no baryon-violating interactions were at work at all stages of this expansion, ηB would be expected to be much smaller, as we will show in the following. Moreover, we should detect a comparable number of antibaryons in the solar system or on larger scales, such as our galaxy (30 kpc) or the Local Group (3 Mpc). However, this is not what we do observe. Baryogenesis is a collection of several theoretical ideas on how the tiny value of ηB might be dynamically produced at some stage of the universe's history. The number of models which have been proposed is quite large, although maybe not really∞! Despite their different particular properties, they all share some common features, because they should all fullfill three basic conditions which were first put forward by Sakharov quite long ago (Sakharov, 1967).

Why me? Why now? These are the kind of questions that cosmological neutrinos could ask themselves about the strange coincidence of relevant facts at the MeV range of temperatures. The four known forces of Nature all play a role in this very interesting epoch. When the universe was from one-tenth of a second to a few minutes old, neutrinos experienced decoupling from electromagnetic plasma while flavour neutrino oscillations became effective, and they witnessed electron–positron annihilations and in the meantime were involved in the business of fixing the initial conditions for the primordial production of light nuclei. All these processes, which in principle could have occurred in the early universe well separated in time, depend upon the values of a bunch of unrelated parameters such as the Fermi constant, the neutrino mixing angles and squared mass differences, the electron mass and the binding energy of nuclei, in particular that of deuterium. The fact that all these events take place almost simultaneously means that they cannot be understood by a back-of-the-envelope calculation. Once more neutrinos put out a challenge to physicists.

In this chapter we first consider in Section 4.1 the process of relic neutrino decoupling in more detail than in Chapter 2, going beyond the instantaneous decoupling approximation in which neutrinos simply no longer interact with other particles below a certain temperature TνD. To this end, one should solve the Boltzmann integro-differential equations for the neutrino momentum distributions, with essentially no approximations.

The statistical properties of CMB temperature and polarization anisotropy maps encode very precise information on the history and composition of our universe. They depend primarily on the behaviour of inhomogeneities in the photon and baryon medium until photon decoupling, which feels all other species in two ways: through their impact on the cosmological background evolution, and via their contribution to the local gravitational forces. This is why neutrinos play an indirect yet important role in the physics of CMB anisotropies, and why present (and future) data on these observables give us quite remarkable pieces of information on neutrino properties.

To understand this point quantitatively, we need to follow photon decoupling at a much more detailed level than in Section 2.4.1. This is the subject of Section 5.1, where we overview the main features of CMB physics, of cosmological perturbation equations, the different contributions to the spectrum of CMB temperature anisotropies, and the effect of each cosmological parameter on the CMB spectrum. Neutrinos will appear on stage in Section 5.2, where we focus on the evolution of their perturbations until photon decoupling, and in Section 5.3, where we infer the effect of neutrino abundance, masses and properties on CMB anisotropies. Finally, Section 5.4 is a brief summary of recent constraints on neutrino properties, exploiting CMB data alone.

Cosmology is the quantitative study of the properties and evolution of the universe as a whole. Since the discovery of the redshift–distance relationship by Hubble in 1929, observations have supported the idea of an expanding universe, which can be beautifully described in terms of the Friedmann and Lemaître solution of the Einstein equations. The basis of this solution is the empirical observation that on sufficiently large scales, and at earlier times, the universe is remarkably homogeneous and isotropic. This experimental fact has been promoted to the role of a guiding assumption, the Cosmological Principle. Assuming that our observation point is not privileged, in the spirit of the Copernican revolution, one is naturally led to the conclusion that all observations made at different places in the universe should look pretty much the same independent of direction. Homogeneity and isotropy single out a unique form for the spacetime metric, the basic ingredient of Einstein theory. Cosmological models can then be quantitatively worked out after specification of the matter content, which acts as the source for curvature. Results can be then compared with astrophysical data, which in the last decades have reached a remarkable precision.

Actually, the Cosmological Principle works only on scales larger than 100 Mpc, yet it is a powerful assumption. In fact, several observables, such as the distribution in the sky of the cosmic microwave background (CMB), show inhomogeneities which are quite small, so that they can be treated as perturbations of a reference model (i.e., a reference metric) which is homogeneous and isotropic.