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 Hořava–Lifshitz quantum gravity. Assuming a mini-superspace of variables, we explore the impact of an inflaton-type scalar field $\varphi \left(t\right)$ on the dynamical equations that describe the trajectories evolution of the scale factor of the Universe, characterized by the dimensionless helix-like function ${\ln }^{-1}\left[\beta \left(t\right)\right]$. This scale factor characterizes a Riemannian foliated spacetime that topologically overcomes the big bang and big crunch singularities. Taking the Hořava–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 $g_i$, 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.