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It is shown how self-reproduction can be easily avoided in the inflationary universe, even when inflation starts at Planck scales. This is achieved by a simple coupling of the inflaton potential with a mimetic field. In this case, the problem of fine-tuning of the initial conditions does not arise, while eternal inflation and the multiverse with all their widely discussed problems are avoided., 19 pages, 1 figure

We identify an infinite class of unbounded potentials for which the Coleman instantons do not exist in D-dimensional spacetime. For such potentials, the decay of a false vacuum is described by the new instantons introduced in [5,6,4]. For each spacetime dimension D, we also construct the two different classes of potentials for which the instanton equations are exactly solvable., 5 pages, LaTeX

We consider the decay of a false vacuum in circumstances where the methods suggested by Coleman run into difficulties. We find that in these cases quantum fluctuations play a crucial role. Namely, they naturally induce both an ultraviolet and infrared cutoff scales, determined by the parameters of the classical solution, beyond which this solution cannot be trusted anymore. This leads to the appearance of a broad class of new $O(4)$ invariant instantons, which would have been singular in the absence of an ultraviolet cutoff. We apply our results to a case where the potential is unbounded from below in a linear way and in particular show how the problem of small instantons is resolved by taking into account the inevitable quantum fluctuations., 16 pages, LaTeX, 3 figures

International audience; We consider the Schwarzschild black hole and show how, in a theory with limiting curvature, the physical singularity “inside it” is removed. The resulting spacetime is geodesically complete. The internal structure of this nonsingular black hole is analogous to Russian nesting dolls. Namely, after falling into the black hole of radius $r_{g}$ , an observer, instead of being destroyed at the singularity, gets for a short time into the region with limiting curvature. After that he re-emerges in the near horizon region of a spacetime described by the Schwarzschild metric of a gravitational radius proportional to $r_{g}^{1/3}$ . In the next cycle, after passing the limiting curvature, the observer finds himself within a black hole of even smaller radius proportional to $r_{g}^{1/9}$ , and so on. Finally after a few cycles he will end up in the spacetime where he remains forever at limiting curvature.

International audience; We find a simple modification of the longitudinal mode in General Relativity which incorporates the idea of limiting curvature. In this case the singularities in contracting Friedmann and Kasner universes are avoided, and instead, the universe has a regular bounce which takes place during the time inversely proportional to the square root of the limiting curvature. Away from the bounce, corrections to General Relativity are negligible. In addition the non-singluar modification of General Relativity delivers for free a realistic candidate for Dark Matter.

We find a rather unique extension of inflationary scenario which avoids selfreproduction and thus resolves the problems of multiverse, predictability and initial conditions. In this theory the amplitude of the cosmological perturbations is expressed entirely in terms of the total duration of inflation.

We explore Sakharov’s seminal idea that gravitational dynamics is induced by the quantum corrections from the matter sector. This was the starting point of the view that gravity has an emergent origin, which soon gained impetus due to the advent of black hole thermodynamics. In the generalized framework of Riemann–Cartan spacetime with both curvature and torsion, the induced gravitational action is obtained for free nonminimally coupled scalar and Dirac fields. For a realistic matter content, the induced Newton constant is obtained to be of the magnitude of the ultraviolet cutoff, which implies that the cutoff is of the order of the Planck mass. Finally, we conjecture that the action for any gauge theory of gravity at low energies can be induced by Sakharov’s mechanism. This is explicitly shown by obtaining the Poincaré gauge theory of gravity.