H0 = H (t0)

The Hubble constant actually is not a constant, rather it is a function of the cosmic time and it encodes information about the first derivative of the scale factor, namely about the velocity of the expansion of the Universe. The parameter H0 originally measured by Hubble and determined with increasing precision in subsequent observations is the value at the present time of the Hubble function H (t).

The first who introduced the notion of Big Bang, namely the theory according to which the present Universe evolved by expansion in the course of time starting from an initial state of enormous density, very tiny and extremely hot was Monsegneur Georges Lemaitre, on the basis of the solution of Einstein equations that bears his name together with those of Friedman, Robertson and Walker. He never used the wording Big Bang, rather referred to his own hypothesis as to that of the primeval atom. As far as we know the name Big Bang was invented as a despising joke by Fred Hoyle during a radio interview in 1949. Hoyle, like Einstein did not like expanding universes.

It is almost a historical nemesis that this ironical nickname of a serious but audacious theory became the official scientific name of the standard cosmological model. The idea of the primeval explosion has so much penetrated the common language and has become so popular that when my daughter, now twenty-four of age, started studying history at the elementary school, her textbook started no longer with the Great Flood rather with the Big Bang.

4.4 The Cosmological Principle

The mathematical basis needed to postulate the type of metric which now goes under the name of Friedman, Lemaitre, Robertson and Walker (FLRW), arriving at those conclusions that Einstein so much hated and from which emerges the Big Bang scenario, are provided by the Cosmological Principle.

This latter assumes two properties that are supposed to characterize the structure of space-time on very large scales, namely:

1.Isotropy.

2.Homogeneity.

Isotropy means invariance against rotations, namely in whatever direction it is pointed, our telescope should reveal approximately the same panorama. Homogeneity, on the other hand means invariance against translations. In other words what we

Fig. 4.13 The hierarchy of cosmic distances. First step in the ladder: 100.000 light years, the Galaxy

see from our own galaxy should be the same landscape observed by any other observer placed in any other galaxy no matter how far from us.

There is no a priori reason to assume the Cosmological Principle and at first sight no empirical basis for it appears to exist, given the granular structure of our universe made of stars grouped into galaxies that are, in turn, grouped into galaxy clusters. Cosmology, however, aims at studying the history of the Universe, analyzing its evolution at so large scales that we can consider galaxies as the point particles of a cosmic dust.

Let us then consider the scale-hierarchy.

Indeed the Universe appears granular only at short distant scales.

•100.000 light-years is the typical dimension of medium size galaxies like our own, the Milky Way. See Fig. 4.13.

•10 millions of light years is the scale of galactic clusters. See Fig. 4.14.

•100 millions of light years is the scale of galactic super clusters. See Fig. 4.15.

•At the scale of one billion of light years, our Universe appears as a homogeneous soup of galaxies and it may be modeled as a perfect fluid. See Fig. 4.16.

The first basis for the Cosmological Principle is this matter of fact evidence on the homogeneous distribution of galactic clusters at very large scales.

Mathematically the Cosmological Principle is enforced by assuming that the space-time metric possesses a set of isometries, namely a set of continuous trans-

Fig. 4.14 The hierarchy of cosmic distances. Second step in the ladder: 10 millions of light years, the Local Group and its neighbors

formations that leave it invariant and form among themselves a Lie group. Isotropy requires that all rotations contained in SO(3) should act as isometries on the cosmological metric. Similarly homogeneity requires that there should be three translational isometries namely as many as the spatial dimensions of the Universe. Imposing such conditions is equivalent to selecting the geometry of constant time sections of space-time. The proper mathematical treatment of isometries is encoded in the theory of coset manifolds and symmetric spaces which is a chapter of Lie algebra and Lie group theory. The geometry of coset manifolds will be summarized in an appropriate mathematical section in next chapter.

At each instant of time the Universe is a three-dimensional space. Assuming the Cosmological Principle means that such a space should admit the maximal possible number of isometries. The mathematical theory of coset manifolds shows that in dimension d = n such maximal number is 12 n(n + 1), namely precisely 6 for d = 3. Furthermore the same mathematical theory shows that there are just only three maximally symmetric manifolds in d = 3. In these spaces the curvature is constant over the manifold and we just have to decide its sign, namely the sign of the constant curvature scalar, positive, negative or null. This choice is encoded in a parameter κ whose possible values are κ = 1, −1, 0, corresponding to the three possibilities we just mentioned. The three maximally symmetric spaces in d = 3 are S3, namely the three-dimensional generalization of the sphere, corresponding to κ = 1, the three-

Fig. 4.15 The hierarchy of cosmic distances. Third step in the ladder: 100 millions of light years, the galactic superclusters

dimensional analogue H3 of the pseudo-sphere, corresponding to κ = −1, and R3, corresponding to κ = 0, which is the standard three-dimensional Euclidian space.

As we will discuss in great detail later on, having imposed such conditions the four-dimensional line-element which encodes the cosmic gravitational field, takes an extremely simple form which is indeed that of the FLRW metric. Naming t the time coordinate and collectively x the spatial coordinates that label the points of the chosen three-dimensional manifold, we can write:

where dΩκ2(x) denotes the line-element of the three-dimensional maximally symmetric space selected by the value of κ. Substituting the ansatz (4.4.1) in Einstein equations and modeling the matter content of the universe as a perfect fluid one obtains certain differential equations for the scale factor that are named Friedman equations and have radically different solutions for different signs of the scale fac-

Fig. 4.16 The hierarchy of cosmic distances. Fourth step in the ladder: 1 billion of light years: homogeneous distribution of super-clusters

tor. In all cases there is a fast initial expansion, later, however, the expansion velocity decreases and the deceleration is stronger and stronger as the curvature increases.

The case of negative curvature κ = 1 is named the open universe. As we will derive in the sequel from Friedman equations the expansion of the open universe continues indefinitely, yet it slows down until it reaches a linear behavior. When the open universe is very old, the scale factor grows like a(t) ≈ t .

The case κ = 0 is named the flat universe. Also here the expansion is endless, yet, as we will see, it tends asymptotically to a weaker growth than linear. When the flat universe is old, its scale factor grows as: a(t) ≈ t2/3.

The case κ = 1 is named the closed universe. For positive curvature the scale factor growth slows down up to zero velocity in a finite time. After that the expansion reverts into a contraction. The galaxies no longer recede from each other rather they begin to come together and the further apart they are the faster they approach each other. The redshift is turned into a blueshift. The universe becomes progressively smaller and smaller, hotter and hotter. In a finite time the closed universe collapses