The existence of fluctuations at the early stage of the universe provides enough confidence to rely on averaging methods. However, the nonlinearity of general relativity makes this process extremely difficult. Several methods have been proposed to study inhomogeneous cosmology and address the averaging problem, such as Buchert's spatial averaging. In this work, early cosmic inflation is investigated using the Buchert equations and the Mori-Zwanzig projection operator formalism. The coarse-grained description derived from these approaches acts as a geometrical source of early cosmic inflation through higher-order differential equations. The theoretical results, while not an exact match, exhibit close agreement with observational data, demonstrating the robustness of the model and its potential for further cosmological applications.

24 pages, 16 figures