Angular spectrum of quantum fluctuations in causal structure

Abstract

Scaling arguments are used to constrain the angular spectrum of distortions on boundaries of macroscopic causal diamonds, produced by Planck-scale vacuum fluctuations of causally-coherent quantum gravity. The small-angle spectrum of displacement is derived from a form of scale invariance: the variance and fluctuation rate of distortions normal to the surface of a causal diamond of radius $R$ at transverse physical separation $cτ\ll R$ should depend only on $τ$, with a normalization set by the Planck time $t_P$, and should not depend on $R$. For measurements on scale $R$, the principle leads to universal scaling for variance on angular scale $Θ$, $\langleδτ^2\rangle_Θ\simeqτ\:\!t_p\simΘR\:\!t_P/c$, and angular power spectrum $C_\ell\sim (R\:\!l_P)/\ell^3$ at $\ell\gg1$. This spectrum is consistent with a relational model of holographic noise based on causally coherent virtual null gravitational shocks, a general picture conjectured for all $\ell$. The high $\ell$ scaling is contrasted with that predicted in some other quantum models, which differ by one power of angular wavenumber $\ell$ and are shown to predict excessive blurring of images from distant sources.