## 3. Friedmann equationsThe Friedmann-Robertson-Walker universe models obey the Friedmann equation (I use units where , and ) A peculiar model is defined by the where ,
, and
are the ## 3.1. The spatially flat caseAs an example I consider the case of the Since is arbitrary in this case, I will chose . I first distinguish two peculiar cases, namely -
Empty model with cosmological constant: , : the solution is . -
The Einstein-de Sitter model, with and . The solution is , with , , and . It follows that I show in Fig. 2 a spatial cut of this space-time. The section of space-time in the plane is an inertial world line: this is the parabola shown by Fig. 1. A perspective of the whole Einstein-de Sitter space-time in M5 is given by Fig. 3.
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. I recall To be more specific, I illustrate (Fig. 4, Fig. 5) the case where , which seems now favored by observational results. Then , with the solution and the Hubble parameter The present period corresponds to .
## 3.1.1. The matter dominated universesAs 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 |