FLRW embeddings in $\mathbb{R}^{n+2}$, differential geometry and conformal photon propagator

TL;DR

This paper develops differential-geometric techniques to embed conformally flat spaces like FLRW models into higher-dimensional Euclidean spaces, deriving explicit formulas that simplify the photon propagator in four dimensions.

Contribution

It introduces new embedding formulas for FLRW spaces in higher dimensions and derives simplified expressions for the photon propagator using these embeddings.

Findings

Explicit embedding formulas for FLRW spaces in $R^{n+2}$

Simplified expressions for the photon propagator in four dimensions

Relations between intrinsic and ambient geometric quantities

Abstract

This paper introduces differential-geometric methods to study $n$-dimensional locally conformally flat spaces as submanifolds in $\mathbb{R}^{n+2}$. We derive explicit formulas relating intrinsic and ambient differential-geometric objects, including curvature tensors, the codifferential and laplacian operators. We apply this approach to Friedmann-Lema\^itre-Robertson-Walker (FLRW) spaces using newfound embedding formulas, obtaining new and simplified expressions for the photon propagator in four dimensions.