Confidence limits of evolutionary synthesis models. IV Moving forward to a probabilistic formulation

TL;DR

This paper introduces a probabilistic framework for stellar population synthesis, addressing the distribution of integrated properties and providing tools to assess when statistical modeling is necessary.

Contribution

It develops a formalism linking stellar luminosity distributions to population properties, enabling interpretation of synthesis models and Monte Carlo simulations within a unified probabilistic approach.

Findings

Derived the population luminosity distribution function (pLDF) from the stellar luminosity distribution function (sLDF).

Provided diagnostic tools to evaluate the statistical robustness of observed clusters.

Reconciled standard synthesis models with Monte Carlo simulations through a probabilistic formalism.

Abstract

Synthesis models predict the integrated properties of stellar populations. Several problems exist in this field, mostly related to the fact that integrated properties are distributed. To date, this aspect has been either ignored (as in standard synthesis models, which are inherently deterministic) or interpreted phenomenologically (as in Monte Carlo simulations, which describe distributed properties rather than explain them). We approach population synthesis as a problem in probability theory, in which stellar luminosities are random variables extracted from the stellar luminosity distribution function (sLDF). We derive the population LDF (pLDF) for clusters of any size from the sLDF, obtaining the scale relations that link the sLDF to the pLDF. We recover the predictions of standard synthesis models, which are shown to compute the mean of the sLDF. We provide diagnostic diagrams and a simplified recipe for testing the statistical richness of observed clusters, thereby assessing whether standard synthesis models can be safely used or a statistical treatment is mandatory. We also recover the predictions of Monte Carlo simulations, with the additional bonus of being able to interpret them in mathematical and physical terms. We give examples of problems that can be addressed through our probabilistic formalism. Though still under development, ours is a powerful approach to population synthesis. In an era of resolved observations and pipelined analyses of large surveys, this paper is offered as a signpost in the field of stellar populations.