3. Friedmann equations
The Friedmann-Robertson-Walker universe models obey the Friedmann equation (I use units where , and )
A peculiar model is defined by the R-dependence of the (dimensionless) density
where , , and are the present matter density, radiation density and cosmological constant (that I include in the density for convenience), in units of the critical density , respectively (additional terms would be necessary to represent quintessence). The dimensionless quantity .
3.1. The spatially flat case
As an example I consider the case of the spatially flat models (), where radiation can be neglected (). The Friedmann equation takes the simple form
Since is arbitrary in this case, I will chose . I first distinguish two peculiar cases, namely
Now I consider the general case (spatially flat, assuming ), that I solve by defining : the equation takes the form , where , with the solution . Finally, the general solution is
All these models have a Big Bang, and I have chosen the integration constant so that at . From this, we derive easily
and the Hubble parameter . The present period corresponds to , or , so that .
The section of space-time is given by
where has unfortunately no analytical expression.
and the Hubble parameter
The present period corresponds to .
3.1.1. The matter dominated universes
As an other example, I consider the models where only non-relativistic matter governs the cosmic evolution, , so that
More specifically, I illustrate (Fig. 7) a spatially closed model, with . This model (hardly compatible with cosmic observations) has a Big Bang, a maximal expansion at and a Big Crunch. It appears advantageous to use as a parameter, with values 0, and , respectively, for the three events. The parametric equations (8) take the form
In our representation, the space-time is represented by a revolution surface of an arc of parabola [Fig. 6]. Spatial sections () are circles (3-spheres in ). The world lines for inertial particles are the arcs of parabola between the Big Bang and the Big Crunch.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001