A universe from a Lagrangian fixed point

In this paper, we investigate the theoretical possibility that a Lagrangian fixed point, when applied to cosmological models, can drive dynamical evolution towards a bouncing universe. We analyze the physics of a Lagrangian fixed point within the context of a gravitational average effective action featuring scale-dependent couplings. To explore this concept, we develop a toy model set in a four-dimensional, spatially flat spacetime, anchored by a Lagrangian fixed point. Solving the cosmological equations of this model analytically, we identify several non-trivial solution branches. These branches are characterized by a modified scale factor and dynamic gravitational couplings, offering new insights into the behavior of cosmological models under these conditions.