A comparison theorem for cosmological lightcones

Abstract

Let $(M, g)$ denote a cosmological spacetime describing the evolution of a universe which is isotropic and homogeneous on large scales, but highly inhomogeneous on smaller scales. We consider two past lightcones, the first, $\mathcal{C}^-_L(p, g)$, is associated with the physical observer $p\in\,M$ who describes the actual physical spacetime geometry of $(M, g)$ at the length scale $L$, whereas the second, $\mathcal{C}^-_L(p, \hat{g})$, is associated with an idealized version of the observer $p$ who, notwithstanding the presence of local inhomogeneities at the given scale $L$, wish to model $(M, g)$ with a member $(M, \hat{g})$ of the family of Friedmann-Lemaitre-Robertson-Walker spacetimes. In such a framework, we discuss a number of mathematical results that allows a rigorous comparison between the two lightcones $\mathcal{C}^-_L(p, g)$ and $\mathcal{C}^-_L(p, \hat{g})$. In particular, we introduce a scale dependent ($L$) lightcone-comparison functional, defined by a harmonic type energy, associated with a natural map between the physical $\mathcal{C}^-_L(p, g)$ and the FLRW reference lightcone $\mathcal{C}^-_L(p, \hat{g})$. This functional has a number of remarkable properties, in particular it vanishes iff, at the given length-scale, the corresponding lightcone surface sections (the celestial spheres) are isometric. We discuss in detail its variational analysis and prove the existence of a minimum that characterizes a natural scale-dependent distance functional between the two lightcones. We also indicate how it is possible to extend our results to the case when caustics develop on the physical past lightcone $\mathcal{C}^-_L(p, g)$. Finally, we show how the distance functional is related to spacetime scalar curvature in the causal past of the two lightcones, and briefly illustrate a number of its possible applications.