Angular spectrum of quantum fluctuations in causal structure

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 \(c\tau\ll R\) should depend only on \(\tau\), 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 \(\Theta\), \(\langle\delta\tau^2\rangle_\Theta\simeq\tau\:\!t_p\sim\Theta R\:\!t_P/c\), 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.