  The existence of fluctuations at the early stage of the universe provides
enough confidence to rely on the averaging methods. However, the nonlinearity
of general relativity makes it extremely difficult. There are different methods
to study inhomogeneous cosmology and provide a straightforward solution to the
averaging problem in cosmology, such as Buchert's spatial averaging. I am
attempting to study early cosmic inflation using Buchert equations and the
Mori-Zwanzig projection operator formalism. Buchert equations and Mori-Zwanzig
formalism's coarse-grained description are the geometrical source of early
cosmic inflation through higher-order differential equations. It can be seen
that the observational data and the given theoretical results are in agreement.
