0. His legacy in this domain are the famous conditions named after him: Baryon number violation, C-symmetry and CP-symmetry violation, and interactions out of thermal equilibrium. Sakharov was also interested in explaining why the curvature of the universe is so small. This lead him to consider cyclic models, where the universe oscillates between contraction and expansion phases. In those models, after a certain number of cycles the curvature naturally becomes infinite even if it had not started this way: Sakharov considered three starting points, a flat universe with a slightly negative cosmological constant, a universe with a positive curvature and a zero cosmological constant, and a universe with a negative curvature and a slightly negative cosmological constant. Those last two models feature what Sakharov calls a reversal of the time arrow, which can be summarized as follows: He considers times t > 0 after the initial Big Bang singularity at t = 0 (which he calls "Friedman singularity" and denotes Φ) as well as times t < 0 before that singularity. He then assumes that entropy increases when time increases for t > 0 as well as when time decreases for t < 0, which constitutes his reversal of time. Then he considers the case when the universe at t < 0 is the image of the universe at t > 0 under CPT symmetry but also the case when it is not so: the universe has a non-zero CPT charge at t = 0 in this case. Sakharov considers a variant of this model where the reversal of the time arrow occurs at a point of maximum entropy instead of happening at the singularity. In those models there is no dynamic interaction between the universe at t < 0 and t > 0. In his first model the two universes did not interact, except via local matter accumulation whose density and pressure become high enough to connect the two sheets through a bridge without spacetime between them, but with a continuity of geodesics beyond the Schwarzschild radius with no singularity, allowing an exchange of matter between the two conjugated sheets, based on an idea after Igor Dmitriyevich Novikov. Novikov called such singularities a collapse and an anticollapse, which are an alternative to the couple black hole and white hole in the wormhole model. Sakharov also proposed the idea of induced gravity as an alternative theory of quantum gravity. Turn