# Big Bounce instead of Big Bang Singularity

For $$\Upsilon > 0$$ GLET predicts a Big Bounce instead of the Big Bang predicted by GR.

In particular, that means that GLET also predicts inflation. Indeed, the Big Bounce requires $$\ddot{a}(\tau)>0$$ at the turning point. But $$\ddot{a}(\tau)>0$$ is the condition which defines inflation. (This is often misrepresented in popular (but not only popular) descriptions of inflation.)

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: $\begin{eqnarray} 3 \left(\frac{a'}{a}\right)^2 &=& -\Upsilon a^{-6} + 3 \Xi a^{-2} + \Lambda + \varepsilon,\\ 2 \frac{a''}{a} + \left(\frac{a'}{a}\right)^2 &=& +\Upsilon a^{-6} + \Xi a^{-2} + \Lambda - k \varepsilon. \end{eqnarray}$ 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 general sense: $$\ddot{a}(\tau) > 0$$

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 a general sense, inflation is simply a regime in the early universe where $$\ddot{a}(\tau) > 0$$, and not, as often misrepresented, a period with very large expansion rate. 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.

This should be distinguished from popular theories about what could cause such an inflation. The usual proposals for such a mechanism of inflation are quite different. In the standard approach to inflation, it is simply a change of the state of the matter fields during the expansion. Thus, there is always some expansion, $$\dot{a}(\tau) > 0$$ all the time. Inflation leads only to some short transition period where the expansion rate increases, $$\ddot{a}(\tau) > 0$$. As a consequence, the evolution before inflation starts follows the same equation, with $$\dot{a}(\tau) > 0$$ when inflation starts. That means, it leads, for the same reasons as the Friedman evolution, to a Big Bang singularity.

## Discussion

The Big Bounce solves two serious problems of the standard cosmology: First, it gets rid of the Big Bang singularity. Second, it solves the horizon problem.

In the standard approach, the horizon problem is solved by inflation theory. But inflation theory does not allow to get rid of the Big Bang singularity. Moreover, it depends on highly speculative theories about particle physics beyond the SM of particle physics.

This suggests to prefer the assumption $$\Upsilon>0$$.