  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.
