# Classical transitions with the topological number changing in the early Universe

## TL;DR

This paper investigates classical scalar field dynamics with saddle points, focusing on the formation of topological solitons that can change winding numbers, potentially impacting early universe structure formation and black hole genesis.

## Contribution

It introduces a simple expression for the winding number of closed loops in field space and analyzes conditions for topological transitions in the early universe.

## Key findings

- Topological solitons can form even with a single minimum potential.
- Transitions can change the winding number of field configurations.
- Non-trivial configurations may influence primordial black hole formation.

## Abstract

We consider classical dynamics of two real scalar fields within a model with the potential having a saddle point. The solitons of such model are field configurations that have the form of closed loops in the field space. We study the formation and evolution of these solitons, in particular, the conditions at which they could be formed even when the model potential has only one minimum. These non-trivial field configurations represent domain walls in the three-dimensional physical space. The set of these configurations can be split into disjoint equivalence classes. We provide a simple expression for the winding number of an arbitrary closed loop in the field space and discuss the transitions that change the winding number. We also show that non-trivial field configurations could be responsible for the energy density excess that could evade the CMB constraints but could be important at scales which are responsible for the formation of galaxies and the massive primordial black holes.