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Description
Summary
The cosmological bounce is an alternative paradigm to big-bang cosmology, in which the universe never hits the singularity, but a contracting universe contracts to a certain minimum size and is then followed by an expanding universe. It is well established that in GR bounce can occur only for k=+1 case of the FRW solution, and that too in presence of some form of matter(the most familiar is a scalar field driven bounce). However, it was noted that in higher order gravity theories, bounce may be possible for k=0 also.
The bounce conditions for the general FRW solutions ($H=0, \dot{H}>0$) can be worked out for from the f(R) field equations. We are specially interested in the case $k=0$ as this is notably different from GR. It is seen that in this case a matterless bounce is possible whenever $Rf^{\prime}-f$ has a positive root. In particular for a form of $f(R)=R+\alpha R^{n}$ a matterless bounce is impossible, and also for a matter bounce $\alpha<0$.
For a quantitative analysis it is helpful to go to the Einstein frame via the conformal transformation $g_{\mu\nu}\rightarrow f^{\prime}(R)g_{\mu\nu}$ , where the theory can be mapped into a GR with an extra minimally coupled scalar field $\phi=\sqrt{\frac{3}{2\kappa}}\ln f^{\prime}(R)$ in a potential $V(\phi)=\frac{Rf^{\prime}-f}{2\kappa f^{2}}$.
We form a system of coupled differential equations in the Einstein frame and solve it numerically for $R^{2}$ gravity by choosing suitable initial conditions. It is seen that though we are considering a bounce, the bounce is absent in the Einstein frame. A bounce in one frame does not imply a bounce in other, and only on specific conditions we get bounce in both the frames simultaneously.
Next we consider the evolution of scalar metric perturbation through such a scenario. In the Jordan frame there are two Bardeen potentials(gauge invariant scalar perturbation d.o.f.), whereas in Einstein frame there is only one. Also, in Einstein frame the theory is ordinary GR so it is easier to derive the perturbation equations. We have solved the perturbation equations in the Einstein frame by choosing suitable initial conditions, and then went back to the Jordan frame via well established relations between Jordan frame and Einstein frame Bardeen potentials. A number of interesting cases, both for symmetric and asymmetric bounce, is considered.
