# On almost commutative Friedmann-Lema\^itre-Robertson-Walker geometries

## TL;DR

This paper investigates the spectral action in a noncommutative geometry model with two sheets having different metrics, revealing a nonlinear interaction term that could impact cosmological models.

## Contribution

It introduces the analysis of spectral action for a noncommutative geometry with different metrics on each sheet, including a nonlinear interaction term in the cosmological context.

## Key findings

- Derived the spectral action with different metrics on each sheet.
- Identified a nonlinear interaction term in the cosmological action.
- Explored potential effects of this term on basic cosmological models.

## Abstract

We analyze the leading terms of the spectral action for a model of noncommutative geometry, which is a product of $4$-dimensional Riemannian manifold with a two-point space exploring the previously neglected case when the metrics over each sheet are different. Assuming the Friedmann-Lema\^itre-Robertson-Walker type of the metric for both sheets we obtain the action, which in addition to the the usual cosmological constant terms and the Einstein-Hilbert term involves a nonlinear interaction term. We study qualitative picture of potential consequences of such term in the basic cosmological models.