Lev Kofman, Andrei Linde, and Alexei A. Starobinsky explore the theory of reheating after inflation, focusing on effects beyond perturbation theory. They introduce the concept of preheating, a stage of parametric resonance where the inflaton field's oscillations produce particles efficiently. This process occurs when the initial amplitude of the inflaton field is large enough, leading to exponential growth of fluctuations in another scalar field, χ. The theory accounts for the expansion of the universe and backreaction of produced particles, including rescattering. The effective potential's contribution is proportional to |ϕ| rather than ϕ², and preheating has distinct stages. The first stage involves minimal backreaction, while the second increases the inflaton's oscillation frequency, enhancing efficiency. Scattering of χ-particles on the inflaton field terminates the resonance. The number density of χ-particles and their quantum fluctuations are calculated, showing that preheating can produce particles with mass much greater than the inflaton field. The paper discusses the differences between narrow and broad resonance regimes, the role of stochastic resonance, and the implications for reheating temperature and early universe phenomena. It emphasizes that preheating can lead to nonthermal phase transitions, topological defects, and novel baryogenesis mechanisms. The study highlights the importance of considering backreaction and the expansion of the universe in understanding reheating processes.Lev Kofman, Andrei Linde, and Alexei A. Starobinsky explore the theory of reheating after inflation, focusing on effects beyond perturbation theory. They introduce the concept of preheating, a stage of parametric resonance where the inflaton field's oscillations produce particles efficiently. This process occurs when the initial amplitude of the inflaton field is large enough, leading to exponential growth of fluctuations in another scalar field, χ. The theory accounts for the expansion of the universe and backreaction of produced particles, including rescattering. The effective potential's contribution is proportional to |ϕ| rather than ϕ², and preheating has distinct stages. The first stage involves minimal backreaction, while the second increases the inflaton's oscillation frequency, enhancing efficiency. Scattering of χ-particles on the inflaton field terminates the resonance. The number density of χ-particles and their quantum fluctuations are calculated, showing that preheating can produce particles with mass much greater than the inflaton field. The paper discusses the differences between narrow and broad resonance regimes, the role of stochastic resonance, and the implications for reheating temperature and early universe phenomena. It emphasizes that preheating can lead to nonthermal phase transitions, topological defects, and novel baryogenesis mechanisms. The study highlights the importance of considering backreaction and the expansion of the universe in understanding reheating processes.