Early universe inverted harmonic-oscillator model over ultrametric space

One of the interesting models we occasionally come across in physics is the inverted harmonic oscillator. This paper demonstrates that the dynamics of the flat Friedmann minisuperspace cosmological model, in both classical and quantum scenarios, can be reduced to analyzing the dynamics of the one-dimensional inverted harmonic oscillator with a time-dependent frequency. Early in the paper, following the standard approach, the Lagrangian for the flat Friedmann model is constructed, which takes the form of the Lagrangian for the inverted harmonic oscillator with a time-dependent frequency. The classical action and the Feynman propagator are determined in spaces with both non-ultrametric and ultrametric geometries. For the quantum mechanical form of the model over ultrametric space, vacuum p-adic states and the conditions of their existence have been determined. At the end of the paper, the adelization of the model pointed to the necessity of discretization of the minisuperspace in the early stages of dynamic evolution.