Gravitational UV-Completion in Cosmology

• Gravitationally induced UV-completion is a framework that integrates renormalization group resummation with non-analytic corrections to control UV divergences in gravitational collapse.
• It refines standard Lagrangian perturbation theory by capturing critical non-analytic behavior near shell-crossing, with a universal exponent of approximately 2/3.
• These techniques yield accurate nonlinear density mappings and support the development of forward-modeling pipelines for large-scale structure analysis.

Gravitationally induced UV-completion refers to mechanisms by which gravitational dynamics or quantum gravitational effects regulate, resolve, or transcend the ultraviolet (UV: short-distance/high-energy) pathologies typical in quantum field theory or cosmological structure formation, such as divergences or breakdowns of perturbative expansions. In astrophysical and cosmological contexts, particularly in gravitational collapse and the non-linear evolution of cosmic structure, standard perturbative techniques (e.g., Lagrangian perturbation theory) are known to converge poorly in the UV regime. Recent work leverages renormalization group (RG) ideas and explicit UV completion constructs to resolve these fundamental issues, both technically and conceptually.

1. Breakdown of Standard Perturbation Theory in Gravitational Collapse

Traditional Lagrangian perturbation theory (LPT) expresses physical quantities—such as the displacement field ψ(a) describing fluid element trajectories via cosmic scale factor a—through Taylor expansions: $\psi(a) = \sum_{n} \psi_n (ka)^n,$ with $k$ encoding local curvature. However, for phenomena involving gravitational collapse (e.g., the spherical collapse model or void evolution), this Taylor series converges only up to the critical "collapse" (shell-crossing) time $a_\star$. Beyond this, the series exhibits secular (unbounded) growth and fails to capture the non-analytic, critical phenomena associated with collapse and multi-streaming.

The behavior near collapse is intrinsically non-analytic. High-order LPT yields coefficients $c_n$ with Domb–Sykes plot behavior indicating a singularity of the form

$$\psi_{\infty}(\mathfrak{a}) \propto (\mathfrak{a}_\star - \mathfrak{a})^{\nu},$$

where $\mathfrak{a} = ak$ and the critical exponent is numerically found to be $\nu \approx 2/3$. LPT by construction is analytic and so cannot represent this, fundamentally limiting its predictive power for late-time cosmic evolution (Rampf et al., 2022).

2. Renormalization Group (RG) Resummation and Fixed-Point Structure

The RG approach, adapted to cosmological structure formation, reformulates the gravitational evolution as a flow in parameter space (via, e.g., the Emden–Fowler equation governing fluid trajectories): $r'^2 = (a/r) - (\varepsilon a/2),$ where $\varepsilon$ is a curvature bookkeeping parameter. One expands $$r = r_0 + \varepsilon r_1 + \varepsilon^2 r_2 + \ldots$$ and observes that higher orders present secular growth. The RG improvement absorbs these contributions back into integration constants ($c_1 \to c_1(1 + \varepsilon A)$) to restructure the solution such that it naturally encodes the non-analytic critical behavior: $$r_{\text{RG,1}}(a) = a\left[1 - \frac{3\varepsilon a}{20}\right]^{2/3},$$ where the exponent $2/3$ emerges analytically, manifesting the universal critical exponent of the phase transition in gravitational collapse. Extended to higher (RG-resummed) orders and with Padé approximants correcting for additional root singularities (particularly in the void regime), this generates RG-improved solutions that are highly accurate even near or beyond shell-crossing (Rampf et al., 2022).

3. Explicit UV Completion via Non-Analytic Extension

The UV completion method explicitly adds a non-analytic term to the truncated LPT series to capture the critical behavior: $$\psi_n^{\text{UV}}(a) = \sum_{s=1}^{n-1} \psi_s (\mathfrak{a})^s + \frac{\psi_n}{c_n}\left[(1 - (\mathfrak{a}/\mathfrak{a}_\star))^{\nu} - \sum_{k=0}^{n-1} c_k (\mathfrak{a})^k\right],$$ with $\nu = 2/3$ as identified from the high-order LPT coefficient scaling: $$c_n / c_{n-1} = \frac{1}{\mathfrak{a}_\star}\left(1 - \frac{1+\nu}{n}\right).$$ This technique ensures the displacement field, and thus all subsequent observables (e.g., nonlinear density $\delta_{\text{NL}}$), exhibit the appropriate non-analytic scaling at late times and strong nonlinearity, as dictated by the physical collapse process rather than artifacts of perturbation theory (Rampf et al., 2022).

4. Improved Nonlinear Density Mapping and Statistical Predictions

These improved methods yield accurate formulas relating the linear (δ_lin) and nonlinear ($\delta_{\rm NL}$) density contrasts. The derived mappings take the general form: $$1 + \delta_{\rm NL} = |x(a)|^{-3}, \quad x(a) = r(a)/a,$$ and template fitting functions, such as

$$(1 + \delta_{\text{fit}}) = (1 - \delta_{\text{lin}}/\alpha)^{-\alpha},$$

with $\alpha$ motivated by the asymptotics (e.g., $3/2$ or $5/3$), now have theoretical justification via the critical exponent $\nu = 2/3$. These relations drive accurate predictions for the one-point PDF of density fluctuations, matching full parametric solutions and capturing the late-time, non-perturbative regime inaccessible to LPT or Eulerian perturbation theory.

5. Extending UV-Completion and RG-Resummed Methods Beyond Symmetry

While original results are derived in spherical symmetry (single-mode, characterized by effective curvature $k$), the underlying philosophy adapts to systems with random field initial conditions. By mapping the fluid equations to Fourier space, one can analyze the evolution of each mode with parameters analogous to single-mode collapse. For multidimensional cosmological fields, the UV completion and RG methods require generalizing to handle paired singularities in the complex-time plane, of the form $$(\mathfrak{a}_\star - \mathfrak{a})^{\nu} + (\bar{\mathfrak{a}}_\star - \mathfrak{a})^{\nu}$$, and then utilize this structure to construct functionals relating initial power spectra to full nonlinear statistics.

A plausible implication is the development of new forward-modeling pipelines for large-scale structure inference, where physical non-analyticities and critical scaling inherited from gravitational collapse are built in at the field level.

6. Comparison and Theoretical Significance

The RG and UV completion approaches both regulate the large-time (UV) failures of LPT and can be summarized as:

Method	Key Principle	Mathematical Structure
RG-Resummation	Absorbs secular divergences, yields fixed-point critical scaling	$r(a) = a [1 - (3a\varepsilon)/20]^{2/3}$, further Padé-improved
UV-Completion	Splices in non-analytic term dictated by collapse	$$\psi_{\rm UV}(a) \propto (\mathfrak{a}_\star - \mathfrak{a})^{\nu}$$ with $\nu \approx 2/3$

Both methods, when appropriately applied and resummed, yield extremely accurate nonlinear mappings and PDFs. Significantly, these techniques not only improve upon the gold standard LPT but also align with critical phenomena and singularity formation unique to collisionless self-gravitating systems.

7. Implications for UV-Completion in Cosmological Modeling

This line of research demonstrates that the UV-completion of gravitational dynamics in structure formation can be realized via a combination of RG fixed-point resummation and explicit non-analytic extension. These methods directly address and resolve UV pathologies of perturbative approaches, enabling robust predictions of nonlinear structure (collapse, void dynamics) in cosmological contexts.

By providing a blueprint for field-level forward modeling that integrates asymptotic critical behavior, this work points to a new paradigm for analytic and semi-analytic approaches in cosmological perturbation theory, potentially impactfully bridging the gap with full N-body simulations. The methodology may be generalized, using RG analysis and non-analytic completion, to other open problems where gravitational nonlinearity and UV sensitivity limit standard perturbative methods (Rampf et al., 2022).