Dynamical Renormalization Group applications to Lagrangian Perturbation Theory

Lagrangian Perturbation Theory (LPT) provides a powerful framework for modeling gravitational structure formation in cosmology, avoiding the shell-crossing singularities that plague Eulerian approaches. However, standard LPT suffers from secular growth in perturbative corrections unphysical divergences that invalidate the expansion at late times, particularly in underdense regions. This thesis develops a rigorous Dynamical Renormalization Group (DRG) formalism for LPT using spherical harmonic decomposition to systematically resum secular growth across multipole modes. We derive the complete perturbative hierarchy for the monopole (ℓ=0), dipole (ℓ=1), and quadrupole (ℓ=2) modes, demonstrating that all multipoles satisfy a universal timeevolution ODE with identical homogeneous basis 1, η−3, Wronskian W = -3η−4, and particular solution coefficients (1/10, -1/25). Mode-specific differences appear only through divergence eigenvalues ℓ(ℓ+1) and coupling structure. At perturbative ϵ1 and...Show moreLagrangian Perturbation Theory (LPT) provides a powerful framework for modeling gravitational structure formation in cosmology, avoiding the shell-crossing singularities that plague Eulerian approaches. However, standard LPT suffers from secular growth in perturbative corrections unphysical divergences that invalidate the expansion at late times, particularly in underdense regions. This thesis develops a rigorous Dynamical Renormalization Group (DRG) formalism for LPT using spherical harmonic decomposition to systematically resum secular growth across multipole modes. We derive the complete perturbative hierarchy for the monopole (ℓ=0), dipole (ℓ=1), and quadrupole (ℓ=2) modes, demonstrating that all multipoles satisfy a universal timeevolution ODE with identical homogeneous basis 1, η−3, Wronskian W = -3η−4, and particular solution coefficients (1/10, -1/25). Mode-specific differences appear only through divergence eigenvalues ℓ(ℓ+1) and coupling structure. At perturbative ϵ1 and ϵ2, running constants remain frozen despite cross-mode coupling in source terms. At order ϵ3, true coupled running emerges: three coupled first-order RG equations govern the synchronized evolution of all multipole constants, with coupling mediated by partial derivatives of particular solutions with respect to other modes’ displacement fields.Show less

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Dynamical Renormalization Group applications to Lagrangian Perturbation Theory