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

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3. Friedmann equations

The Friedmann-Robertson-Walker universe models obey the Friedmann equation (I use units where [FORMULA], and [FORMULA])

[EQUATION]

A peculiar model is defined by the R-dependence of the (dimensionless) density

[EQUATION]

where [FORMULA], [FORMULA], and [FORMULA] are the present matter density, radiation density and cosmological constant (that I include in the density for convenience), in units of the critical density [FORMULA], respectively (additional terms would be necessary to represent quintessence). The dimensionless quantity [FORMULA].

3.1. The spatially flat case

As an example I consider the case of the spatially flat models ([FORMULA]), where radiation can be neglected ([FORMULA]). The Friedmann equation takes the simple form

[EQUATION]

Since [FORMULA] is arbitrary in this case, I will chose [FORMULA]. I first distinguish two peculiar cases, namely

  • Empty model with cosmological constant: [FORMULA], [FORMULA]: the solution is [FORMULA].

  • The Einstein-de Sitter model, with [FORMULA] and [FORMULA]. The solution is [FORMULA], with [FORMULA], [FORMULA], and [FORMULA]. It follows that

    [EQUATION]

    I show in Fig. 2 a spatial cut of this space-time. The section of space-time in the plane [FORMULA] is an inertial world line: this is the parabola [FORMULA] shown by Fig. 1. A perspective of the whole Einstein-de Sitter space-time in M5 is given by Fig. 3.

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

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

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

Now I consider the general case (spatially flat, assuming [FORMULA]), that I solve by defining [FORMULA]: the equation takes the form [FORMULA], where [FORMULA], with the solution [FORMULA]. Finally, the general solution is

[EQUATION]

All these models have a Big Bang, and I have chosen the integration constant so that [FORMULA] at [FORMULA]. From this, we derive easily

[EQUATION]

and the Hubble parameter [FORMULA]. The present period [FORMULA] corresponds to [FORMULA], or [FORMULA], so that [FORMULA].

The [FORMULA] section of space-time is given by

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] has unfortunately no analytical expression.

I recall

[EQUATION]

[EQUATION]

[EQUATION]

To be more specific, I illustrate (Fig. 4, Fig. 5) the case where [FORMULA], which seems now favored by observational results. Then [FORMULA], with the solution

[EQUATION]

[EQUATION]

and the Hubble parameter

[EQUATION]

The present period [FORMULA] corresponds to [FORMULA].

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

[FIGURE] Fig. 5. RW model with [FORMULA], 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, [FORMULA], so that

[EQUATION]

More specifically, I illustrate (Fig. 7) a spatially closed model, with [FORMULA]. This model (hardly compatible with cosmic observations) has a Big Bang, a maximal expansion at [FORMULA] and a Big Crunch. It appears advantageous to use [FORMULA] as a parameter, with values 0, [FORMULA] and [FORMULA], respectively, for the three events. The parametric equations (8) take the form

[EQUATION]

[EQUATION]

[EQUATION]

In our representation, the space-time is represented by a revolution surface of an arc of parabola [Fig. 6]. Spatial sections ([FORMULA]) are circles (3-spheres in [FORMULA]). The world lines for inertial particles are the arcs of parabola between the Big Bang and the Big Crunch.

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

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

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© European Southern Observatory (ESO) 2000

Online publication: January 29, 2001
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