Standard perturbation theory (SPT) for models with interaction in the dark sector

Abstract

This dissertation investigates the formation and nonlinear evolution of cosmic structures in cosmological models with an interacting dark sector (IDS), extending the standard ΛCDM framework. Starting from the fluid and perturbation equations governing the density and velocity fields, a fully time-dependent perturbative formalism is developed, going beyond the Einstein–de Sitter (EdS) approximation. Evolution equations for second-, third-, and arbitrary-order perturbative kernels are derived, explicitly incorporating the effects of interactions between dark matter and dark energy. A central result of this work is the critical assessment of the commonly employed approximation in which the ratio between the matter density parameter, Ωm, and the square of the perturbation growth rate, f 2 , is assumed to be approximately unity. In the ΛCDM model, this approximation is motivated by the empirical relation f ≈ Ω 0.55 m , which implies that Ωm/f 2 remains close to unity over a significant fraction of cosmic history. This near equality allows the separation of temporal and spatial dependencies in the perturbative equations, rendering nonlinear kernels approximately time-independent and substantially simplifying analytical and numerical calculations. For this reason, the approximation based on the Einstein–de Sitter limit is frequently employed even in cosmologies where it is not strictly valid. It is nevertheless shown that, within the ΛCDM framework, this approximation remains valid even at low redshifts, since deviations from unity remain small in the late Universe. In interacting dark sector models, however, such deviations become significantly larger at late times, indicating that assumptions based on the Einstein–de Sitter limit are no longer reliable when dark sector interactions are present. Using the numerical code SPTIDS, developed and implemented by the author as part of this work, the matter power spectrum at one-loop level and the reduced bispectrum are computed for IDS cosmologies. The results show that dark sector interactions enhance the growth of matter perturbations and amplify higher-order correlations relative to the ΛCDM model, providing distinctive nonlinear signatures. These results demonstrate the necessity of a fully time-dependent perturbative treatment for precision modeling of large-scale structure in interacting dark sector scenarios, while also pointing to possible observational deviations from the standard cosmological model.