Astron. Astrophys. 364, 894-900 (2000)

## 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

• 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.

 Fig. 1. An inertial world line, i.e., a section of Einstein-de Sitter space-time, embedded in

 Fig. 2. Flat Euclidean space , embedded in the three-dimensional manifold of described by the coordinates , , w.

 Fig. 3. The Einstein-de Sitter model (with flat spatial sections) embedded in a flat Lorentzian space. Both spatial sections () and inertial world lines () are parabolas.

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 .

 Fig. 4. A world line of the RW model, embedded in a flat Lorentzian two-dimensional space

 Fig. 5. RW model with , embedded in a flat Lorentzian space

#### 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.

 Fig. 6. An inertial world line of a closed RW model, purely matter dominated, with , is an arc of a parabola.

 Fig. 7. Closed RW model (purely matter dominated, with ) embedded in a flat Lorentzian space. Spatial sections () are circles. Inertial world lines () are the arcs of a parabola illustrated in Fig. 6.

© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001