Integrated Galactic Archaeology: An Inverse-Problem Framework for Galaxy Evolution

Integrated-light spectral energy distribution modeling is widely used to infer the star-formation and assembly histories of galaxies that cannot be resolved into individual stars. However, existing approaches are often discussed primarily in terms of particular fitting codes, star-formation-history parameterizations, or inference algorithms. In this Review, we formulate the recovery of galaxy evolution histories from integrated spectral energy distributions as a unified inverse problem. We separate the physical spectral-generation operator from the observational operator and examine the resulting information loss through non-identifiability, singular-value structure, null directions, effective resolution, regularization, and model discrepancy. We then classify parametric and nonparametric star-formation histories, PCA, MOPED, VESPA, non-negative matrix factorization, deep learning, and simulation-based inference within a common five-component framework consisting of the representation space, forward operator, physical or statistical constraints, inference method, and uncertainty assessment. On this basis, we introduce information-driven adaptive representation as a general design principle in which the complexity of the recovered history is matched to the information supported by the observations. Finally, we extend the framework from star-formation histories to coupled galaxy-evolution states involving chemical enrichment, dust evolution, interstellar-medium conditions, and radiative transfer, and outline a three-layer research program linking controlled mock experiments, inverse-problem theory, and physical forward modeling.