  The absence of a physically motivated model for large scale profiles of
cosmic voids limits our ability to extract valuable cosmological information
from their study. In this paper, we address this problem by introducing the
spherically compensated cosmic regions, named CoSpheres. Such cosmic regions
are identified around local extrema in the density field and admit a unique
compensation radius $R_1$ where the internal spherical mass is exactly
compensated. Their origin is studied by extending the standard peak model and
implementing the compensation condition. Since the compensation radius evolves
as the Universe itself, $R_1(t)\propto a(t)$, CoSpheres behave as bubble
Universes with fixed comoving volume. Using the spherical collapse model, we
reconstruct their profiles with a very high accuracy until $z=0$ in N-body
simulations. CoSpheres are symmetrically defined and reconstructed for both
central maximum (seeding haloes and galaxies) and minimum (identified with
cosmic voids). We show that the full non linear dynamics can be solved
analytically around this particular compensation radius, providing useful
predictions for cosmology. This formalism highlights original correlations
between local extremum and their large scale cosmic environment. The
statistical properties of these spherically compensated cosmic regions and the
possibilities to constrain efficiently both cosmology and gravity will be
investigated in companion papers.
