In this work, we elaborate on the finite action in the framework of Hořava-Lifshitz gravity. Assuming the finite action principle, we show that the beginning of the Universe is flat and homogeneous. Depending on the version of the theory, different cosmological scenarios are possible. Furthermore, we show that the Hořava-Lifshitz gravity action selects only the regular black-hole spacetimes since the singular black holes possess infinite action. We also comment on the possibility of traversable wormholes in theories with higher-curvature invariants. The possible cosmological solutions in Hořava-Lifshitz and quadratic gravity are similar, proving that the finite action principle is not model sensitive.

Recently, based on swampland considerations in string theory, the (no) eternal inflation principle has been put forward. The natural question arises whether similar conditions hold in other approaches to quantum gravity. In this article, the asymptotic safety hypothesis is considered in the context of eternal inflation. As exemplary inflationary models the SU(N) Yang-Mills in the Veneziano limit and various RG-improvements of the gravitational action are studied. The existence of UV fixed point generically flattens the potential and our findings suggest no tension between eternal inflation and asymptotic safety, both in the matter and gravitational sector in contradistinction to string theory. Moreover, the eternal inflation cannot take place in the range of applicability of effective field quantum gravity theory. We employ the analytical relations for eternal inflation to some of the models with single minima, such as Starobinsky inflation, alpha-attractors, or the RG-improved models and verify them with the massive numerical simulations. The validity of these constraints is also discussed for a multi-minima model.

We study the non-linear stability of fixed-point solutions to the α′-exact equations from O(d, d) invariant cosmology, with and without matter perturbations. Previous non-linear analysis in the literature is revisited, and its compatibility with known linear perturbation results is shown. Some formal aspects of cosmological perturbations in duality invariant cosmology are discussed, and we show the existence of time-reparameterization invariant variables for perturbations.

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The swampland criteria in string theory assert the no eternal inflation scenario. This work studied the impact of generic gravitational quantum corrections on eternal inflation. In particular, we find that the Starobinsky model of inflation should receive higher-order corrections stemming from quantum gravity. In this work, we studied the effect of the R3/2 and R4 corrections on the eternal inflation conditions for the Starobinsky model.