The branch-cut quantum gravity with a self-coupling inflation scalar field: Dynamical equations

This article focuses on the implications of the recently developed commutative formulation based on branch-cutting cosmology, the Wheeler-DeWitt equation, and Horava-Lifshitz quantum gravity. Assuming a mini-superspace of variables, we explore the impact of an inflaton-type scalar field phi(t)$$ \phi (t) $$ on the dynamical equations that describe the trajectories evolution of the scale factor of the Universe, characterized by the dimensionless helix-like function ln-1[beta(t)]$$ {\ln}<^>{-1}\left[\beta (t)\right] $$. This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the big bang and big crunch singularities. Taking the Horava-Lifshitz action as our starting point, which depends on the scalar curvature of the branched Universe and its derivatives, with running coupling constants denoted as gi$$ {g}_i $$, the commutative quantum gravity approach preserves the diffeomorphism property of General Relativity, maintaining compatibility with the Arnowitt-Deser-Misner formalism. We investigate both chaotic and nonchaotic inflationary scenarios, demonstrating the sensitivity of the branch-cut Universe's dynamics to initial conditions and parameterizations of primordial matter content. The results suggest a continuous connection of Riemann surfaces, overcoming primordial singularities and exhibiting diverse evolutionary behaviors, from big crunch to moderate acceleration.