Exact solutions in bouncing cosmology

We discuss the effects of a (possibly) negative $(1+z{)}^6$ type contribution to the Friedmann equation. No definite answer can be given as to the presence and magnitude of a particular mechanism, because any test using the general relation $H(z)$ is able to estimate only the total of all sources of such a term. That is why we describe four possibilities: 1) geometric effects of loop quantum cosmology, 2) braneworld cosmology, 3) metric-affine gravity, and 4) cosmology with spinning fluid. We find the exact solutions for the models with ${\rho }^2$ correction in terms of elementary functions, and show all evolutional paths on their phase plane. Instead of the initial singularity, the generic feature is now a bounce.