Hence we obtain the complete space-time geodesics from those of three space by solving the following equation that relates the time coordinate t to the angular coordinate φ:

Furthermore, the constant A1 is inessential and can always be fixed to 1 since it can be traded for the constant A2 which does not appear in the equation. The differential (5.3.82) appears rather involved since F 2(t, φ) depends both on time and the angle φ. Yet we can take advantage of the homogeneous character of our space-time and simplify the problem very much. Indeed due to homogeneity it suffices to consider the geodesics whose projection in the xy plane is a circle centered at the origin and of radius R. All other geodesics with center in some point {x0, y0} can be obtained from these ones by a suitable isometry that takes {0, 0} into {x0, y0}. So let us consider geodesics centered at the origin of the xy plane. This corresponds to setting ρ = 0. In this case we obtain:

which depends only on time and the geodesic equations are reduced to quadratures since we get:

The convergence or divergence of the second integral in (5.3.84) determines whether or not there are particle horizons in the considered cosmology. We will discuss the general concept of particle horizons for isotropic cosmologies later on. There the particle horizon appears as a spherical surface and is characterized by a radius. In non-isotropic cosmology as the present one, particle horizon may have a completely different much less intuitive shape. Curiously, in the above geometry horizons appear as an angular deficit. For each chosen radius R one can explore the geodesic (which is a spiral) only up to some maximal angle φmax at each chosen instant of time.

5.4The Standard Cosmological Model: Isotropic and Homogeneous Metrics

Having analyzed the implications of homogeneity without the enforcement of complete isotropy, we turn to the Standard Cosmological Model, by assuming that the candidate cosmological metric is not only homogeneous but also isotropic, namely it admits a 6-parameter group G6 of isometries, generated by space-like Killing vec-

tors. Furthermore isotropy means that G6 includes necessarily an SO(3) rotation subgroup. The classification of three-dimensional homogeneous spaces with such a G6 group of isometries reduces to the classification of three-dimensional Euclidian coset manifolds and we just have three possibilities:

ISO(3)

In the above formula we have emphasized the fact that the three selected coset manifolds, possessing the required isometry type, are metrically equivalent to three group-manifolds of dimension three and therefore fall into the Bianchi classification. In particular we have the following identifications at the level of Lie algebras:

We can describe the geometries of these three spaces simultaneously with a single formula by writing the following 3D-metric:

We show below that when κ takes the two values ±1 the metric (5.4.3) can be identified with the pull-back of the flat R4 metric on either the 3-sphere or the threedimensional hyperboloid and hence just describes the SO(4) or SO(1, 3) invariant metrics on such coset manifolds, respectively. In the first case the variable r is actually compact and takes values in the range [0, 2π ]. In the second case it is non compact and takes values in the infinite interval [−∞, +∞]. In the case κ = 0, the metric (5.4.3) is manifestly identical with the flat Euclidian metric in three dimension, written in polar coordinates and the variable r takes values in the semiinfinite interval [0, +∞]. As such it admits the Euclidian group of isometries ISO(3).

The only parameter in (5.4.3) which is not fixed by isometries is the global scale of the three-dimensional space and this we can take to be time dependent: a(t). Hence we can write the following ansatz for the isotropic and homogeneous cosmological metric

from which we easily calculate the divergence whose vanishing provides a differential equation for the energy density:

Having computed all the ingredients we can finally analyze the Einstein equations, that take the following form:

where Λ is a new constant originally introduced by Einstein and named by him the cosmological constant. It corresponds to the presence in the gravitational action of an additional term of the form 4 Λ√− det g d4x which is allowed by the principle of general covariance.

Inserting the results (5.4.8) and (5.3.51) into (5.4.11) we finally obtain the following two equations:

that are currently known in the literature as Friedman equations.

Obviously these latter have to be supplemented with (5.4.10) which expresses the conservation of the stress energy tensor. It turns out that these three equations are not independent. This is just what should happen, since the Einstein tensor is conserved as a consequence of Bianchi identities. Indeed multiplying the first of (5.4.12) by a2, taking a further derivative and combining it with the second we obtain the following result:

which is nothing else but the already obtained conservation equation (5.4.10). So we can just focus on this latter equation and on the first of Friedman equations (5.4.12).

5.4.1 Viewing the Coset Manifolds as Group Manifolds

Before studying Friedman equations, and in order to better appreciate the role of isotropy versus homogeneity, we reconsider the statement made at the beginning of the present section, namely that each of the three coset manifolds mentioned in (5.4.1) can be also viewed as a group manifold and therefore that each of the