Analytic discrete self-similar solutions of Einstein-Klein-Gordon at large D

TL;DR

This work constructs an infinite family of discretely self-similar solutions to the Einstein–massless Klein–Gordon system in the large-$D$ limit using a controlled $1/D$ expansion. The leading order (LO) solution is analytic and parameterized by a time-dependent function $\beta(\tau)$, yielding closed forms for $\Pi_{\text{LO}}$, $\Omega_{\text{LO}}$, and $f_{\text{LO}}$, with DSS distinguished by periodic $\beta(\tau)$. Systematic corrections at next-to-leading order (NLO) and next-to-next-to-leading order (NNLO) fix several geometric features, impose an echoing-period constraint $\Delta=|\beta''|/(3|\beta'|)$ at zeros of $\beta$, and bring the large-$D$ solutions into qualitative agreement with finite-$D$ critical collapse data, notably in the behavior of the NEC lines and the SSH maxima. The paper also provides explicit example computations and a supplemental material set that details CSS branches, scalar-field reconstruction, and higher-order NEC analyses, offering a path to improving finite-$D$ matches and exploring the convergence of the $1/D$ expansion.

Abstract

Discretely self-similar solutions govern critical gravitational collapse and have been known only numerically since Choptuik's pioneering work. We construct, in closed analytic form, an infinite family of such solutions of the Einstein-massless-Klein-Gordon system using the large-D expansion. We characterize their structure and compare them with numerical critical solutions at finite D, identifying both universal features and distinctly large-D behavior.

Figures (5)

• Figure 1: Illustration of the past patch of the Choptuik spacetime with DSS. The closer one moves towards the top, the more rapidly the Ricci scalar oscillates until the singularity is reached. The inset zooms into one of the fundamental domains to highlight the Ricci contours, i.e., the magnitude of the Ricci scalar is bigger in the center and smaller at the SSH.
• Figure 2: NNLO solution for \ref{['eq:example']} with $D=300$.
• Figure 3: NEC lines for \ref{['eq:example']} with $D=300$.
• Figure 4: NEC lines for \ref{['eq:example']} with $D=100,300,500$.
• Figure 5: Maxima of the SSH function $f$ at NNLO for different values of $D$ for the example \ref{['eq:example']}.