namely a flat Universe with an exponentially growing scale factor. The corresponding Hubble function and acceleration parameter are:

The important lesson told by this analysis is that there are cases of cosmological homogeneous and isotropic metrics where, irrespectively from the sign of the spatial curvature, the universe expands indefinitely, even exponentially. The question is which kind of energy filling of the universe can yield such solutions, in particular de Sitter space. The answer will be vacuum-energy. To address such a question and similar ones we ought to consider the general properties and consequences of Friedman equations.

5.6 General Consequences of Friedman Equations

Let us reconsider the differential equation (5.4.13) and inspect its solution for a class of equations of state of the form

where w is a constant coefficient. We already saw that w = 0 corresponds to baryonic matter (dust universe), while w = 13 provides the equation of state of relativistic radiation. Other notable cases will be met soon.

Inserting (5.6.1) into (5.4.13) we get:

where ρ0 and a0 are, respectively, the energy density and the scale factor at a reference instant of time, which we choose to be our own.

Evaluating the first of Friedman equations at current time, we obtain:

where it is proper to recall that H0, the Hubble constant, is an experimentally evaluated parameter. It follows that the sign of the space curvature of the Universe depends on whether the present energy density is bigger, equal or less than the critical

Evaluating (5.6.12) at the present time we obtain:

which is to be paired with (5.6.10).

Let us now consider the possible energy filling of the Universe at the present time. Radiation density decays very fast because of the 1/a4 law. Hence its contribution is certainly negligible and we can forget it. As for matter we can divide it into two parts:

(a)the visible baryonic matter composed of galaxies and their clusters, whose contribution we name Ω0B ( the corresponding coefficient is wB = 0),

(b)the invisible dark matter composed of possibly existing stable massive particles predicted in unified theories of particle interactions and/or by other non-

radiating conventional matter filling galactic interstellar space, whose contribution we name Ω0D (the corresponding coefficient is wD = 0).

In addition to that we envisage the possible presence of:

(c)vacuum energy, whose contribution we name Ω0Λ and whose defining equation of state is characterized by wΛ = −1.

As we are going to see in next sections, the equation of state p = −ρ describes the contribution to the overall stress-energy tensor of a cosmological constant or better of the potential energy of scalar fields. On the other hand, it is evident from (5.6.13) and (5.6.14) that an accelerating expansion of the universe (a¨ > 0) is possible if and only if there are components of its energy filling that have w < − 13 and if they are dominant.

With this assumption we obtain the following two equations for the four cosmological parameters Ω0B , Ω0D , Ω0Λ and Ω0κ :

Up to the end of the second millennium the only known parameter was Ω0B estimated to be Ω0B 0.06 by the observation and counting of galaxies. After 1999, the measure of the deceleration parameter q0 and the discovery that it is negative (the universe actually accelerates) revealed that Ω0Λ > 0 and provided a constraint on the remaining parameters that could be completely solved when, both from the observation of the anisotropies in the Cosmic Background Radiation and from the supernova project, it was established that Ω0 1, namely that our universe is spatially flat.

where the numerical evaluation depends on the experimental result for q0, whose determination was the motivation for the award of the 2011 Nobel Prize in Physics.

Let us now reconsider the general form of the Friedman Lemaitre Robertson Walker metric (5.4.4). Introducing the following functions:

R−2 1(χ ) = sinh2 χ

and denoting the volume element of the two-sphere by

the metric (5.4.4) can be rewritten as follows:

where we have performed the coordinate change r = Rκ (χ ). It is also convenient to introduce a further coordinate change to the so named conformal time:

upon which (5.6.20) transforms into:

Using such coordinates the radial light-like geodesics are very easily characterized by the following equation:

which is immediately integrated to:

Relying on this result we can now introduce the concepts of particle and event horizons.