Influence of inhomogeneous stochasticity on the falsifiability of mean-field theories and examples from accretion disc modeling

TL;DR

This paper develops a new error propagation method for mean-field theories, accounting for nonlinearities and inhomogeneous fluctuations, and applies it to accretion disc models and telescope data to assess their falsifiability.

Contribution

It introduces a generalized error propagation formula that relaxes linearity and homogeneity assumptions, enhancing the testing of mean-field theories against observations.

Findings

Error propagation formula accounts for nonlinear and inhomogeneous fluctuations.

Application to accretion disc models shows non-monotonic precision dependence on frequency.

Analysis of telescope spectral data reveals how resolution affects fluctuation detection.

Abstract

Despite spatial and temporal fluctuations in turbulent astrophysical systems, mean-field theories can be used to describe their secular evolution. However, observations taken over time scales much shorter than dynamical time scales capture a system in a single state of its turbulence ensemble. Comparing with mean-field theory can falsify the latter only if the theory is additionally supplied with a quantified precision. The central limit theorem provides appropriate estimates to the precision only when fluctuations contribute linearly to an observable and with constant coherent scales. Here we introduce an error propagation formula that relaxes both limitations, allowing for nonlinear functional forms of observables and inhomogeneous coherent scales and amplitudes of fluctuations. The method is exemplified in the context of accretion disc theories, where inhomogeneous fluctuations in the surface temperature are propagated to the disc emission spectrum--the latter being a nonlinear and non-local function of the former. The derived precision depends non-monotonically on emission frequency. Using the same method, we investigate how binned spectral fluctuations in telescope data change with the spectral resolving power. We discuss the broader implications for falsifiability of a mean-field theory.