# Turnaround size of non-spherical structures

## TL;DR

This paper extends the concept of turnaround radius from spherical to slightly non-spherical structures in an accelerating universe, using gauge-invariant methods and first-order perturbation theory.

## Contribution

It demonstrates that the gauge-invariant characterization of turnaround size remains valid for small deviations from spherical symmetry, advancing the understanding of non-spherical cosmic structures.

## Key findings

- Turnaround size characterization applies to small deviations from sphericity.
- Gauge-invariant methods are effective for non-spherical structures.
- First-order perturbation theory suffices for initial non-spherical corrections.

## Abstract

The turnaround radius of a large structure in an accelerating universe has been studied only for spherical structures, while real astronomical systems deviate from spherical symmetry. We show that, for small deviations from spherical symmetry, the gauge-invariant characterization of the turnaround size using the Hawking-Hayward quasi-local mass and spherical symmetry still applies, to first order in the cosmological perturbation potentials and in the deviations from sphericity. This is the first step to include non-spherical systems in the physics of turnaround.