On surface brightness fluctuations: probabilistic and statistical bases

Aims. This work aims to provide a theoretical formulation of Surface Brightness Fluctuations (SBF) in the framework of probabilistic synthesis models, in which there are no deterministic relations between the different stellar components of a population but only relations on average, and to distinguish between the different distributions involved in the SBF definition. Methods. By applying the probabilistic theory of stellar population synthesis models, we estimate the shape (mean, variance, skewness, and kurtosis) of the distribution of fluctuations across resolution elements, and examine the implications for SBF determination, definition and application. Results. We propose three definitions of SBF: (i) stellar population SBF, which can be computed from synthesis models and provide an intrinsic metric of fit for stellar population studies; (ii) theoretical SBF, which include the stellar population SBF plus an additional term that takes into account the distribution of the number of stars per resolution element ψ(N); theoretical SBF coincide with Tonry & Schneider (1998) definition in the very particular case that ψ(N) is assumed to be a Poisson distribution. However, the Poisson contribution to theoretical SBF is around 0.1% of the contribution due to the stellar population SBF, so there is no justification to include any reference to Poisson statistics in the SBF definition; (iii) observational SBF, which are those obtained in observations that are distributed around the theoretical SBF. Finally, we show alternative ways to compute SBF and extend the application of stellar population SBF to defining a metric of fitting for standard stellar population studies. Conclusions. We demostrate that SBF are observational evidence of a probabilistic paradigm in population synthesis, where integrated luminosities have an intrinsic distributed nature, and they rule out the commonly assumed deterministic paradigm of stellar population modeling.