mochi_class: Modelling Optimisation to Compute Horndeski In class

We introduce mochi_class, an extension of the Einstein-Boltzmann solver hi_class, designed to unlock the full phenomenological potential of Horndeski gravity. This extension allows for general input functions of time without the need for hard-coded parametrisations or covariant Lagrangians. By replacing the traditional <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>α</mml:mi> </mml:math> -parametrisation with a set of stable basis functions, mochi_class ensures that the resulting effective theories are inherently free from gradient and ghost instabilities. Additionally, mochi_class features a quasi-static approximation implemented at the level of modified metric potentials, enhancing prediction accuracy, especially for models transitioning between a super- and sub-Compton regime. mochi_class can robustly handle a wide range of models without fine-tuning, and introduces a new approximation scheme that activates modifications to the standard cosmology deep in the matter-dominated era. Furthermore, it incorporates viability conditions on the equation of motion for the scalar field fluctuations, aiding in the identification of numerical instabilities. Through comprehensive validation against other Einstein-Boltzmann solvers, mochi_class demonstrates excellent performance and accuracy, broadening the scope of hi_class by facilitating the study of specific modified gravity models and enabling exploration of previously inaccessible regions of the Horndeski landscape. The code is publicly available at this URL (https://github.com/mcataneo/mochi_class_public)