A comparison theorem for cosmological lightcones

Abstract Let ( M , g ) denote a cosmological spacetime describing the evolution of universe which is isotropic and homogeneous on large scales, but highly inhomogeneous smaller scales. We consider two past lightcones, first, $${{\mathcal {C}}_{L}^{-}}(p, g)$$ CL-(p,g) associated with physical observer $$p\in \,M$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">p∈M who describes actual geometry at length scale L whereas second, $${\mathcal {C}_{L}^{-}}(p, \hat{g})$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">CL-(p,g^) an idealized version p who, notwithstanding presence local inhomogeneities given wish to model member $$(M, xmlns:mml="http://www.w3.org/1998/Math/MathML">(M,g^) family Friedmann–Lemaitre–Robertson–Walker spacetimes. In such framework, we discuss number mathematical results that allows rigorous comparison between lightcones . particular, introduce scale-dependent lightcone-comparison functional, defined by harmonic type energy, natural map FLRW reference lightcone This functional has remarkable properties, in particular it vanishes iff, length-scale, corresponding surface sections (the celestial spheres) are isometric. detail its variational analysis prove existence minimum characterizes distance lightcones. also indicate how possible extend our case when caustics develop Finally, exploiting causal diamond theory, show related (to leading order scalar curvature briefly illustrate applications.