For the flat homogeneous universe
ds2 = dτ2 - a(τ)2( dx2+ dy2+ dz2)
we obtain the equations
| 3(a' ⁄ a)2 | = | -Υa-6 + 3Ξa-2 + Λ + ε |
| 2(a" ⁄ a) + (a' ⁄ a)2 | = | +Υa-6 + Ξa-2 + Λ - kε |
The Ξ-term defines a candidate for Instead, the Υ-term becomes important in the early universe, for small values of a(τ).
For Υ <0 this increases the speed of the expansion during the early big bang.
But much more interesting things happen if we assume Υ >0. In this case, the big bang does not have a singularity. Instead, we obtain a minimal value a0 for a(τ):
Υa0-6
= 3Ξa0-2 + Λ + ε
The equation is symmetric in time, we have a minimal value for a, and for this value the second derivative a'' is positive. This leads to time-symmetric solutions, with a big collapse before the big bang.
For example, for some special assumptions (no matter and Ξ=0) we have the analytical solution
a(τ) = a0cosh1/3((3Λ)1/2 τ)
The property to have no big bang singularity, but a symmetric collapse before, is not only nice (it is always nice to have no singularities). It also solves a serious cosmological problem: the horizon problem.
This suggests to prefer the assumption Υ >0.
Modifications for the early universe
Time-symmetric solutions
Discussion