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We present a model of (modified) gravity on spacetimes with fractal structurebased on packing of spheres, which are (Euclidean) variants of the Packed SwissCheese Cosmology models. As the action functional for gravity we consider thespectral action of noncommutative geometry, and we compute its expansion on aspace obtained as an Apollonian packing of 3-dimensional spheres inside a4-dimensional ball. Using information from the zeta function of the Diracoperator of the spectral triple, we compute the leading terms in the asymptoticexpansion of the spectral action. They consist of a zeta regularization of adivergent sum which involves the leading terms of the spectral actions of theindividual spheres in the packing. This accounts for the contribution of thepoints 1 and 3 in the dimension spectrum (as in the case of a 3-sphere). Thereis an additional term coming from the residue at the additional point in thereal dimension spectrum that corresponds to the packing constant, as well as aseries of fluctuations coming from log-periodic oscillations, created by thepoints of the dimension spectrum that are off the real line. These terms detectthe fractality of the residue set of the sphere packing. We show that thepresence of fractality influences the shape of the slow-roll potential forinflation, obtained from the spectral action. We also discuss the effect oftruncating the fractal structure at a certain scale related to the energy scalein the spectral action.

Spectral Action Models of Gravity on Packed Swiss Cheese Cosmology