Persistence and Transition Varieties in Scalar Field Cosmology

TL;DR

This paper uses bifurcation theory to analyze scalar field cosmologies, identifying key loci and transitions in phase space for exponential and quadratic potentials, revealing invariant structures and regimes.

Contribution

It develops a bifurcation-theoretic framework for scalar field cosmologies, including stratifications and normal forms, to understand persistence and transitions in different potential models.

Findings

Identified five key loci organizing phase portraits in exponential potential models.

Derived normal forms and invariant structures governing transitions near critical loci.

Revealed invariant gates and thresholds affecting hyperbolicity and regime changes.

Abstract

We develop a bifurcation-theoretic description of Friedmann--Robertson--Walker cosmologies with a scalar field $\phi$, a barotropic fluid of index $\gamma$, and spatial curvature. For the strict exponential potential $V(\phi)=V_{0}e^{\lambda\phi}$, with $a=\sqrt{3/2}\,\lambda$, the local phase portrait is organised by five loci in the $(\gamma,a)$-plane: $|a|=3$, $a^{2}=3$, $a^{2}=9\gamma/2$, $\gamma=2/3$, and $\gamma=2$. Near these loci we compute translated jets, centre(-like) reductions, and normal forms governing persistence and transitions. For the quadratic potential $V(\phi)=(1/2)m^{2}\phi^{2}$, the effective slope $\lambda$ is dynamical. Using the bounded variable $\zeta=\arctan\lambda$, we obtain a regular autonomous $4$-dimensional system in $(X,Y,\Omega_{k},\zeta)$, where $\Omega_{k}$ is the curvature variable. This reveals invariant gates, robust equilibrium continua, and vertical $\gamma$-thresholds for loss and recovery of normal hyperbolicity. We then construct an explicit stratification for the exponential class and a pull-back stratification for the massive case, together with the corresponding physical path maps into unfolding space. The resulting framework also organises slow-roll, ultra slow-roll, and oscillatory regimes.