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 , 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 \equiv 2 and \beta \equiv 2 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 galaxy environmental densities over 0.2 < z < 1.5 ( \sim 7 Gyr ) and infer \alpha = 1.98 \pm 0.04 and \beta = 2.01 \pm 0.11 —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 , galaxy growth and its diversity in cosmological contexts .