  In scalar-tensor Horndeski theories, non-singular cosmological models -
bounce and genesis - are problematic because of potential ghost and/or gradient
instabilities. One way to get around this obstacle is to send the effective
Planck mass to zero in the asymptotic past ("strong gravity in the past"). One
may suspect that this feature is a signal of strong coupling problem at early
times. However, the classical treatment of the cosmological background is
legitimate, provided that the strong coupling energy scale remains at all times
much higher than the scale associated with the classical evolution. We
construct various models of this sort, namely (i) bouncing Universe which
proceeds through inflationary epoch to kination (expansion within GR, driven by
massless scalar field); (ii) bouncing Universe with kination stage immediately
after bounce; (iii) combination of genesis and bounce, with the Universe
starting from flat space-time, then contracting and bouncing to the expansion
epoch; (iv) "standard" genesis evading the strong coupling problem in the past.
All these models are stable, and perturbations about the backgrounds are not
superluminal.
