# Big Bounce instead of Big Bang Singularity

For the flat homogeneous universe we have the following ansatz:
\[ds^2 = d\tau^2 - a(\tau)^2(dx^2+dy^2+dz^2) = a(\mathfrak{t})^6 d\mathfrak{t}^2 - a(\tau)^2(dx^2+dy^2+dz^2)\]

This gives (in terms of the usual Friedmann variables, \(a' = \frac{da(\tau)}{d\tau}\), the following variant of the Friedmann equations:
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For \(\Upsilon>0\) this leads to a qualitative change of the global evolution: Given that the left hand side of the first equation is non-negative, the right hand side has to be non-negative too. But for very small values of a, the right hand side would become negative. As a consequence, we obtain some minimal possible value \(a_0\) defined by the equation
\[\Upsilon a_0^{-6} = 3 \Xi a_0^{-2} + \Lambda + \varepsilon. \]

This would lead to a time-symmetric big bounce picture: A big crunch, down to a minimal value of \(a_0\), following by a bouncing back leading to a big bang.

## Time-symmetric solutions

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 \(\Xi=0\)) we have the analytical solution
\[ a(\tau) = a_0 \cosh^{\frac13}(\sqrt{3\Lambda}\tau)\]

## Inflation in a technical sense: \(a''>0\)

The main advantage of the time-symmetric big bounce picture is that it gives what is reached in the standard big bang scenario with inflation. In some technical sense, inflation is simply a regime in the early universe where \(a''(\tau) > 0\). Such a regime is necessary because of the horizon problem: Else, without it, the regions which are possible causally connected via a common cause after the big bang would be far too small, smaller than the inhomogeneities of the background radiation.

## Discussion

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 \(\Upsilon>0\).