Evolution of the discrepancy between a universe and its model

TL;DR

This paper investigates whether cosmological models reliably represent the universe's history by analyzing the spectral distance between a model and a perturbed geometry, finding that the discrepancy tends to decrease over time in linear regimes.

Contribution

It introduces a spectral scheme to quantify the evolution of geometric discrepancies between a universe model and a perturbed geometry, supporting the reliability of models in linear regimes.

Findings

Spectral distance does not increase significantly over time in linear perturbations.

The spectral distance tends to decrease, indicating models remain reliable.

Analysis uses standard linear structure-formation theory.

Abstract

We study a fundamental issue in cosmology: Whether we can rely on a cosmological model to understand the real history of the Universe. This fundamental, still unresolved issue is often called the ``model-fitting problem (or averaging problem) in cosmology''. Here we analyze this issue with the help of the spectral scheme prepared in the preceding studies. Choosing two specific spatial geometries that are very close to each other, we investigate explicitly the time evolution of the spectral distance between them; as two spatial geometries, we choose a flat 3-torus and a perturbed geometry around it, mimicking the relation of a ``model universe'' and the ``real Universe''. Then we estimate the spectral distance between them and investigate its time evolution explicitly. This analysis is done efficiently by making use of the basic results of the standard linear structure-formation theory. We observe that, as far as the linear perturbation of geometry is valid, the spectral distance does not increase with time prominently,rather it shows the tendency to decrease. This result is compatible with the general belief in the reliability of describing the Universe by means of a model, and calls for more detailed studies along the same line including the investigation of wider class of spacetimes and the analysis beyond the linear regime.