  Cosmological perturbation theory is known to converge poorly for predicting
the spherical collapse and void evolution of collisionless matter. Using the
exact parametric solution as a testing ground, we develop two asymptotic
methods in spherical symmetry that resolve the gravitational evolution to much
higher accuracy than Lagrangian perturbation theory (LPT), which is the current
gold standard in the literature. One of the methods selects a stable
fixed-point solution of the renormalization-group flow equation, thereby
predicting already at the leading order the critical exponent of the phase
transition to collapsed structures. The other method completes the truncated
LPT series far into the UV regime, by adding a non-analytic term that captures
the critical nature of the gravitational collapse. We find that the UV method
most accurately resolves the evolution of the nonlinear density as well as its
one-point probability distribution function. Similarly accurate predictions are
achieved with the renormalization-group method, especially when paired with
Pad\'e approximants. Further, our results yield new, very accurate, formulas to
relate linear and nonlinear density contrasts. Finally, we chart possible ways
on how to adapt our methods to the case of cosmological random field initial
conditions.
