How Is the Local-scale Gravitational Instability Influenced by the Surrounding Large-scale Structure Formation?

We develop the formalism to investigate therelation between the evolution of the large-scale(quasi) linear structure and that of the small-scalenonlinear structure in Newtonian cosmology within theLagrangian framework. In doing so, we first derive thestandard Friedmann expansion law using the averagingprocedure over the present horizon scale. Then thelarge-scale (quasi) linear flow is defined by averaging the full trajectory field over a large-scaledomain, but much smaller then the horizon scale. Therest of the full trajectory field is supposed todescribe small-scale nonlinear dynamics. We obtain the evolution equations for the large-scale andsmall-scale part of the trajectory field. These arecoupled each other in most general situations. It isshown that if the shear deformation of fluid elements is ignored in the averaged large-scaledynamics, the small-scale dynamics is described byNewtonian dynamics in an effectiveFriedmann-Robertson-Walker (FRW) background with a localscale factor. The local scale factor is defined by the sum of theglobal scale factor and the expansion deformation of theaveraged large-scale displacement field. This means thatthe evolution of small-scale fluctuations is influenced by the surrounding large-scale structurethrough the modification of FRW scale factor. The effectmight play an important role in the structure formationscenario.