DUCA: Dynamic Universe Cosmological Analysis: I. The halo mass function in dynamical dark energy cosmologies

Context. The halo mass function (HMF) is fundamental for interpreting the number counts of galaxy clusters and serves as a pivotal theoretical tool in cosmology. With the advent of high-precision surveys such as LSST, eROSITA, DESI, and Euclid, accurate HMF modeling becomes indispensable in order to avoid systematic biases in cosmological parameter estimation from cluster cosmology. Moreover, these surveys aim to shed light on the dark sector and uncover dark energy's (DE) puzzling nature; thus they necessitate models that faithfully capture DE's features to ensure robust parameter inference. Aims. We aim to construct a model for the HMF in dynamical DE cosmologies that preserves the accuracy achieved for the standard Lambda(nu)CDM model of cosmology while meeting the precision requirements necessary for future cosmological surveys. Methods. Our approach models the HMF parameters as functions of the deceleration parameter at the turnaround, a quantity that has been shown to encapsulate essential information regarding the impact of dynamical DE on structure formation. We calibrated the model using results from a comprehensive suite of N-body simulations spanning various cosmological scenarios, which ensured subpercent systematic accuracy. Results. We present a HMF model tailored for dynamical DE cosmologies. The model was calibrated following a Bayesian approach, and its uncertainty is characterized by a single parameter controlling its systematic error, which remains at the subpercent level. This ensures that theoretical uncertainties from our model are subdominant relative to other error sources in future cluster number count analyses.