Renormalization group and UV completion of cosmological perturbations: Gravitational collapse as a critical phenomenon

Cornelius Rampf and Oliver Hahn

Submitted on 3 November 2022, last revised on 9 November 2022


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 ultra-violet (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 non-linear 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, formulae to relate linear and non-linear density contrasts. Finally, we chart possible ways on how to adapt our methods to the case of cosmological random field initial conditions.


Comment: 11 pages + appendices and references, 10 figures, v2: added references and fixed some typos, submitted to PRD

Subject: Astrophysics - Cosmology and Nongalactic Astrophysics


LPT series solutions for the comoving trajectory $x(a) = 1 + \psi(a)$ up to 30th order for the case $k\!=\!3/10$ (coloured lines), compared against the exact solution (black dotted line) based on the spherical collapse model. The positive $a$-branch denotes the collapse of a spherical overdensity, while the negative $a$-branch reflects the void evolution with $k=-3/10$. The gray-shaded area indicates the disk of convergence, i.e., the region $-a_\star<a<a_\star$ where $a_\star=(3\pi\sqrt{2})^{2/3} \simeq 5.622$ is the collapse time of the overdensity. Clearly, convergence of the LPT series is lost for $|a|> a_\star$ for both over- and underdense regions.