HarrisonZeldovichs

Harrison–Zel’dovich approximation, often referred to as the Harrison–Zel’dovich formalism, is a theoretical framework in cosmology used to describe the evolution of small perturbations in the early universe. Developed independently by American physicist Harrison and Soviet physicist Zel’dovich in the 1970s, this approach provides a simplified yet powerful way to analyze the growth of density fluctuations, which are fundamental to understanding the formation of large-scale structures like galaxies and galaxy clusters.

The approximation assumes that the universe is dominated by a homogeneous and isotropic background, with small deviations from this uniformity described by linear perturbations. These perturbations are typically modeled as scalar fields, meaning they are described by a single function rather than vector or tensor fields. The key idea is that the growth of these perturbations can be tracked using a Lagrangian approach, where the positions of particles are treated as evolving in time rather than their densities.

A central concept in the Harrison–Zel’dovich framework is the "top-hat" model, where perturbations are approximated as spherical regions with uniform density contrasts. This allows for analytical solutions to the equations governing structure formation, particularly in the context of cold dark matter and baryonic matter interactions. The approximation is particularly useful in the early stages of perturbation growth, where nonlinear effects are negligible.

The Harrison–Zel’dovich formalism has been instrumental in predicting the primordial power spectrum of density fluctuations, which is often assumed to follow a scale-invariant form (the Harrison–Zel’dovich spectrum). This spectrum serves as a key input for numerical simulations, such as N-body simulations, which further refine our understanding of cosmic structure formation. While the approximation breaks down at later stages when nonlinearities dominate, it remains a cornerstone of modern cosmological theory.