  A key obstacle to developing a satisfying theory of galaxy evolution is the
difficulty in extending analytic descriptions of early structure formation into
full nonlinearity, the regime in which galaxy growth occurs. Extant techniques,
though powerful, are based on approximate numerical methods whose Monte
Carlo-like nature hinders intuition building. Here, we develop a new solution
to this problem and its empirical validation. We first derive closed-form
analytic expectations for the evolution of fixed percentiles in the real-space
cosmic density distribution, {\it averaged over representative volumes
observers can track cross-sectionally\}. Using the Lagrangian forms of the
fluid equations, we show that percentiles in $\delta$---the density relative to
the median---should grow as $\delta(t)\propto\delta_{0}^{\alpha}\,t^{\beta}$,
where $\alpha\equiv2$ and $\beta\equiv2$ for Newtonian gravity at epochs after
the overdensities transitioned to nonlinear growth. We then use 9.5 sq. deg. of
Carnegie-Spitzer-IMACS Redshift Survey data to map {\it galaxy\} environmental
densities over $0.2<z<1.5$ ($\sim$7 Gyr) and infer $\alpha=1.98\pm0.04$ and
$\beta=2.01\pm0.11$---consistent with our analytic prediction. These
findings---enabled by swapping the Eulerian domain of most work on density
growth for a Lagrangian approach to real-space volumetric averages---provide
some of the strongest evidence that a lognormal distribution of early density
fluctuations indeed decoupled from cosmic expansion to grow through
gravitational accretion. They also comprise the first exact, analytic
description of the nonlinear growth of structure extensible to (arbitrarily)
low redshift. We hope these results open the door to new modeling of, and
insight-building into, the galaxy growth and its diversity in cosmological
contexts.
