# Emergent metric and geodesic analysis in cosmological solutions of (torsion-free) Polynomial Affine Gravity

## TL;DR

This paper explores a gravity model based on affine connections without a fundamental metric, demonstrating how an emergent metric arises from Ricci curvature and analyzing geodesics in cosmological solutions.

## Contribution

It introduces a metric-free affine gravity model with a cosmological ansatz, showing emergent metrics and geodesic behavior similar to standard cosmology.

## Key findings

- Emergent metric from Ricci curvature in affine gravity.
- Geodesic analysis reproduces Friedman--Robertson--Walker model.
- Distinguishes particle trajectories based on Ricci curvature.

## Abstract

Starting from an affinely connected space, we consider a model of gravity whose fundamental field is the connection. We build up the action using as sole premise the invariance under diffeomorphisms, and study the consequences of a cosmological ansatz for the affine connection in the torsion-free sector. Although the model is built without requiring a metric, we show that the nondegenerated Ricci curvature of the affine connection can be interpreted as an \emph{emergent} metric on the manifold. We show that there exists a parametrization in which the \((r,\varphi)\)-restriction of the geodesics coincides with that of the Friedman--Robertson--Walker model. Additionally, for connections with nondegenerated Ricci we are able to distinguish between space-, time- and null-like self-parallel curves, providing a way to differentiate \emph{trajectories} of massive and massless particles.