In the loop quantum gravity there is a bounce because at the lowest level spacetime is discrete and can't be compressed. The big bounce theory of relating to big bang in the loop quantum theory might be similar to the black hole/white hole theory.

In Penrose's theory it's the gravity/mass that is creating time and clumping things together between the big bang and the heat death of the universe (you need mass to create clock and time).

At the beginning of the universe temperature was so high that masses did not matter, at the end of the universe there is no mass, just massless particles (after googol or so when the biggest black holes have evaporated). Massless particles (photon, gluon, graviton) don't experience time so they in a sense fly into the the end of the eternity instantly.

Then there is something geometrical I don't understand and the ends meet and the new universe matches the end of the previous across timeless eon.

Penrose thinks you could perceive the ripples caused by the massive black holes from the last universe in the background radiation of our universe.