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A homogeneous and isotropic universe is characterized by the Friedmann-Robertson-Walker line element

Equation 2 (2)

In this metric the Einstein equations (1) with matter in the form of a perfect fluid acquire the following simple form

Equation 3 (3)


Equation (4) can be recast to look like the equation of motion of a point particle on the surface of a sphere of radius R ident a and mass M, setting c = 1 we obtain

Equation 5 (5)

The total `gravitating mass' M = 4pi / 3 R3(rho + 3P) reflects the fact that `pressure carries weight' in Einstein's theory of gravity. From (5) we find that a particle on the sphere feels both attractive and repulsive forces. The force of repulsion Frep = Lambda/3 R is caused by the cosmological constant and increases with distance if Lambda > 0. (For negative Lambda this becomes a force of `attraction', formally resembling the force of confinement between quarks which binds them within the nucleus.)

The opposite signs of the forces of attraction and repulsion in (5) allow for a large number of new solutions to the Einstein equations. As pointed out in the previous section, Einstein himself used the repulsive effect of the cosmological constant to balance the attraction of matter resulting in a static closed universe which Einstein felt was in agreement with Mach's principle. A quantitative analysis of solutions to (3) & (4) can be gained by eliminating rho in these equations and combining them into a single equation for the evolution of the scale factor in the presence of a Lambda-term

Equation 6 (6)

which is also valid if Lambda is a function of time (i.e. if Tik = Lambda(t) gik). (We have assumed that matter has an equation of state P = w rho c2.) A comprehensive quantitative analysis of (6) has been carried out in [62] for a cosmological constant, and in [147] for a time varying cosmological term Lambda(t). For our purpose it will be sufficient to note that the qualitative behaviour of the universe in the presence of a cosmological term which is either constant or time varying, can be understood very simply by rewriting (3) in the suggestive form (we assume c = 1 for simplicity)

Equation 7 (7)


Equation 8 (8)

Since rho = rho0(a0 / a)3(1 + w), we find, substituting w = 0 for dust

Equation 9 (9)

where A = 4piG/3 rho0 a03. (We assume for simplicity that matter is pressureless so that w = 0, however the qualitative analysis given below remains valid for matter possessing more general equations of state.) Equation (7) reminds one of classical motion with conserved energy E in a one dimensional potential V(a) whose generic form is shown in Figure 2 for w = P / rho geq 0. From the form of V(a) several things can be said about the behaviour of the expansion factor a(t). We shall first examine the case kappa = 1 (E < 0) since it provides us with the largest variety of qualitatively different solutions to the Einstein equations.

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