FLRW models are spatially homogeneous, but they are a very restricted subclass of such models because of their isotropy. Why are spatially homogeneous anisotropic models interesting? Basically, because they are tractable solutions of the full non-linear equations since there is only one essential variable, time, so the equations become ordinary differential equations, but they allow investigation of much more general behaviour than the FLRW models. They can represent anisotropic modes, including rotation and global magnetic fields, which could occur in the real universe (indeed, must do so, if the universe is indeed generic, as some claim): here an anisotropic but not necessarily inhomogeneous model is required (see e.g. Thorne (1967)). They allow new classes of singularities, and modification of the BBN–baryon relation in the early universe. They may also be good approximations in regions where there is inhomogeneity but spatial gradients are small, see Section 19.9. They have been explored in various quantum cosmology contexts (see Chapter 20) as well as in GR.

In particular, the tilted cases provide the only tractable cosmological solutions we have which involve rotation: rotation is ubiquitous in the universe, and, because of the vorticity conservation theorems discussed in Chapter 6, this suggests there always was and always will be rotation. Thus it is valuable to have solutions where we can investigate its effects on, for example, the CMB, where we find new classes of anisotropy patterns, and on nucleosynthesis.

The primordial seeds of inhomogeneity, whose imprint is seen at last scattering in the CMB anisotropies, may be generated by quantum fluctuations during inflation in the very early universe. These seeds subsequently evolve from linear to nonlinear fluctuations via gravitational instability, and produce the large-scale matter distribution that is observed at lower redshifts. The previous chapter dealt with the CMB anisotropies. In this chapter we provide brief overviews of the primordial fluctuations from inflation, and then of the evolution of large-scale structure, as described via the power spectrum of matter. A key probe of the total matter (dark and baryonic) and its distribution is weak gravitational lensing by the large-scale structure of light from distant sources. We develop the theoretical framework for gravitational lensing and briefly describe how this is applied in cosmology. The following chapter will draw on this chapter and its predecessors to show how current observations constrain and describe the standard model of cosmology. We start with a summary of the statistical description of perturbations.

Correlation functions and power spectra

Perturbations on an FLRW background are treated as random variables in space at each time instant, and observations determine the statistical properties of these random distributions. (See Durrer (2008) for a more complete discussion.) A perturbative variable A(x) at some fixed time is associated with an ensemble of random functions, each with a probability assigned to it. We define the 2-point correlation function 〈A(x)A(x′)〉 as the average over the ensemble (incorporating the probability distribution).

Although the observations appear to be well fitted by perturbed FLRW models, as described above, more general models need to be considered. One major reason is that the appropriateness of the perturbed FLRW models cannot be said to have been tested unless the consequences of alternatives have been calculated and compared with observation. In particular, there could be drastic changes to the models for the very early universe, since what may now be small and decaying perturbations in the standard picture would have been non-negligible earlier, and could give very different dynamics. Local observations can bound, but could not be sure to detect, such perturbations, so their testable consequences, if any, must arise from effects in the early universe.

We also need to consider the possibility of large-scale anisotropies, for example arising from a cosmic magnetic field aligned on a supergalactic scale, and of large-scale inhomogeneities (advanced as a possible explanation, which we discussed in Chapter 15, of the apparent acceleration seen in the supernova data).

This chapter considers the space of all models and the definition of classes of cosmological models wider than the FLRW models (compare e.g. Ellis (2005)). There are many ways of classifying spacetimes, of which the most common are by symmetry and by Petrov type (see Stephani et al. (2003)). In the cosmological case, symmetries are the more relevant and we consider that here. (Some models characterized by other covariant properties are described in Sections 19.6 and 19.7.)

In standard cosmology, gravity is modelled by GR. In this chapter we review how, in GR, gravity is represented by a curved spacetime, with matter moving on timelike geodesics and photons on null geodesics. There is no definition of gravitational force or gravitational energy. Thus although GR has a good Newtonian limit, it has totally different conceptual foundations. It is only in restricted circumstances that gravity will be well represented by Newtonian theory. GR also has its limits: it can only be a good description if quantum gravity effects are negligible. Then it is very good: there are no data requiring us to alter it in such contexts, which include all of cosmology except the very earliest times.

This chapter discusses the Einstein field equations of GR, after a short discussion of physics in a curved spacetime and the energy–momentum tensor.We give a brief introduction to the physical foundations of GR such as the equivalence principle and the motivation of the form adopted for the field equations but do not cover the experimental tests (for which see Will (2006); note that except for the binary pulsar data, these tests are essentially tests of the weak field slow motion regime).

Equivalence principles, gravity and local physics

Using our understanding of spacetime geometry, we now consider how to describe local physics in a curved spacetime. Two principles underlie the way we do this: namely, use of tensor equations, and minimal coupling based on covariant differentiation.

The standard approach to cosmology is a model-based approach: find the simplest possible model of spacetime that can accommodate the observational data. An alternative is a direct observational approach. The first method determines observational relations and parameters from a model; the second attempts to determine a model from observational relations. We introduce the latter method in this chapter, and it will also feature in Chapter 15; the former is essentially used in the rest of this book.

As mentioned before, a fundamental feature of cosmology is that there is only one universe, which we cannot experiment on: we can only observe it, and moreover, on a cosmological scale, only from one specific spacetime event. Observations therefore give direct access only to our past light-cone, at one cosmological time. How can we then devise and test suitable cosmological models?

Model-based approach

In the standard approach, one chooses a family of models first, characterized by as few free parameters and free functions as possible. Then one fixes these parameters and functions in order to reproduce astronomical observations as accurately as possible. Therefore this is in fact a form of light-cone best-fitting procedure: one is obtaining a best-fit of the chosen family of models to the real universe via comparison of observational relations predicted by the model with actual observations.

Traditionally, this is applied almost exclusively to the FLRW models. The merit of the approach is that it has good explanatory power, which serves as a vindication of the chosen models.

FLRW cosmological models are those universes which are everywhere isotropic about the fundamental velocity (technically: there is a G3 group of isotropies acting about every spacetime point which leaves the fundamental velocity invariant). This will be the case if and only if the observations of every fundamental observer are isotropic at all times. This implies further symmetries of these universes: as well as being isotropic about each event, they are spatially homogeneous: all physical properties are the same everywhere on spacelike surfaces orthogonal to the fluid flow (technically: there is a G3 group of isometries acting simply transitively on these surfaces). This will be proved in the sequel, but geometrically the result is clear: spheres of constant density centred on one point P are only consistent with spheres of constant density centred on other points Q and R if the density is constant.

Because of these exact symmetries, these spacetimes cannot themselves be realistic models of the observed universe: they do not represent any of the inhomogeneities associated with the astronomical structures we see all around us. Realistic models of the observed universe are provided by perturbed FLRW universes, which are almost isotropic about every point, and hence are almost spatially homogeneous (they are inhomogeneous on small scales but homogeneous on large scales). The ‘almost FLRW’ models are the standard models of cosmology at the present time (considered in the following chapter).

In 1934, two astronomers, Walter Baade and Fritz Zwicky, proposed the existence of a new form of star, the neutron star, which would be the end point of stellar evolution. They wrote:

These prophetic remarks seemed at the time to be beyond any possibility of actual observation, since a neutron star would be small, cold and inert, and would emit very little light. More than 30 years later the discovery of the pulsars, and the realisation a few months later that they were neutron stars, provided a totally unexpected verification of the proposal.

The physical conditions inside a neutron star are very different from laboratory experience. Densities up to 1014 g cm−3, and magnetic fields up to 1015 gauss (1011 tesla), are found in a star of solar mass but only about 20 kilometres in diameter. Again, predictions of these astonishing conditions were made before the discovery of pulsars. Oppenheimer & Volkoff in 1939 used a simple equation of state to predict the total mass, the density and the diameter; Hoyle, Narlikar & Wheeler in 1964 argued that a magnetic field of 1010 gauss might exist on a neutron star at the centre of the Crab Nebula; Pacini in 1967, just before the pulsar discovery, proposed that the rapid rotation of a highly magnetised neutron star might be the source of energy in the Crab Nebula.