# Origin of time before inflation from a topological phase transition

## TL;DR

This paper proposes a model where the universe's origin involves a topological phase transition from a Euclidean to a hyperbolic manifold, introducing complex time to explain the quantum-to-classical transition of fluctuations.

## Contribution

It introduces a novel topological phase transition framework with complex time to explain the universe's origin and the quantum-classical transition of geometric fluctuations.

## Key findings

- Topological phase transition from Euclidean to hyperbolic manifold.
- Complex time formalism explains quantum to classical transition.
- Natural resolution of the origin of time and fluctuations.

## Abstract

We study the origin of the universe (or pre-inflation) by suggesting that the primordial space-time in the universe suffered a global topological phase transition, from a 4D Euclidean manifold to an asymptotic 4D hyperbolic one. We introduce a complex time, $\tau$, such that its real part becomes dominant after started the topological phase transition. Before the big bang, $\tau$ is a space-like coordinate, so that can be considered as a reversal variable. After the phase transition is converted in a causal variable. The formalism solves in a natural manner the quantum to classical transition of the geometrical relativistic quantum fluctuations: $\sigma$, which has a geometric origin.