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

Technical Summary: Analytic Discrete Self-Similar Solutions of Einstein–Klein–Gordon at Large D

Problem Statement
Discretely self-similar (DSS) solutions govern the critical threshold of gravitational collapse, a phenomenon first discovered numerically by Choptuik. These solutions exhibit an emergent discrete symmetry and possess a self-similar horizon (SSH) terminating in a naked singularity. Despite their physical importance, closed-form analytic solutions for the Einstein-massless-Klein–Gordon system have remained elusive; prior to this work, the critical DSS solution could only be constructed numerically. The challenge lies in the lack of a small dimensionless parameter in four-dimensional Einstein gravity to facilitate perturbative expansions.

Methodology
The authors employ the large-$D$ expansion, treating $1/D$ (specifically $\epsilon = 1/(D-1)$) as a small dimensionless parameter. By generalizing the field equations of spherically symmetric Einstein gravity coupled to a massless scalar field $\psi$ to arbitrary spacetime dimensions $D > 3$, they reformulate the system using a specific coordinate transformation ($\tau = -\ln(-t)$, $x = -r/t$) and field redefinitions.

The core methodology involves a systematic perturbative expansion of the metric functions ($\Omega, f$) and matter variables ($\Pi, \Psi$) in powers of $\epsilon$:
$$\text{Field} = \text{Field}_{\text{LO}} + \epsilon \text{Field}_{\text{NLO}} + \epsilon^2 \text{Field}_{\text{NNLO}} + \dots$$
Crucially, the authors demonstrate that in the chosen gauge, the coupled non-linear partial differential equations (PDEs) decouple at each order of the expansion. This allows for the algebraic determination of higher-order terms from lower-order solutions without solving new coupled systems.

Key Contributions and Results

1. Leading Order (LO) Analytic Solution:
   The authors construct an infinite family of exact DSS solutions at leading order ($O(1)$). The solution is characterized by a single arbitrary periodic function $\beta(\tau)$ with an echoing period $\Delta$.

   • The metric and matter fields are expressed in closed form (Eqs. 7–13).
   • The solution is regular between the center and the SSH ($x=1$) and exhibits a naked singularity at the center, consistent with the structure of Choptuik's critical solution.
   • At LO, the Null Energy Condition (NEC) lines appear as straight horizontal lines, and the function $f$ (related to the metric) maintains a maximum of 1 for all $x$, differing from finite-$D$ numerical results.

2. Next-to-Leading Order (NLO) Constraints:
   At $O(\epsilon)$, the authors derive corrections to the fields. A significant finding is that not all periodic functions $\beta(\tau)$ are admissible at this order.

   • To satisfy convexity conditions (specifically $f \leq 1$) near the SSH, the echoing period $\Delta$ is constrained by the derivatives of $\beta$ at its zeros:
     $$\Delta = \left| \frac{\beta''}{3\beta'} \right|_{\beta=0}$$
   • This condition excludes simple trigonometric functions where derivatives vanish coincident with the function itself.

3. Next-to-Next-to-Leading Order (NNLO) and Finite-$D$ Features:
   The authors extend the expansion to $O(\epsilon^2)$ (NNLO). This order is required to recover specific qualitative features observed in numerical simulations at finite $D$:

   • Decay of the SSH function: At NNLO, the maximum of $f$ decays from 1 at the SSH to a value proportional to $1/(D-1)^2$ at the center, matching the behavior seen in $D=4$ simulations.
   • NEC Line Curvature: The downward bending of NEC lines near the center, a feature absent at LO and NLO, emerges at NNLO.
   • Convergence: Numerical tests with a specific example function $\beta(\tau)$ suggest that the perturbative series converges for sufficiently large $D$ (e.g., $D \gtrsim 500$ for the chosen example), while convergence is slow or non-existent for lower dimensions (e.g., $D \lesssim 100$).

4. Scalar Field Reconstruction:
   The paper provides the reconstruction of the scalar field $\psi$ up to NNLO, showing that it involves a leading term of order $1/\sqrt{\epsilon}$ and subleading terms determined by the derivatives of $\beta(\tau)$.

Significance and Claims
The paper claims to provide the first closed-form analytic expressions for DSS solutions in the Einstein-massless-Klein–Gordon system. By utilizing the large-$D$ expansion, the authors transform a problem previously solvable only numerically into an analytically tractable one.

The significance of the work lies in:

• Analytic Accessibility: It demonstrates that the critical DSS solution is not inherently non-analytic but can be captured systematically via $1/D$ expansion.
• Systematic Improvement: It establishes a framework where features of finite-$D$ critical collapse (such as the specific profile of the SSH function and the curvature of NEC lines) can be recovered by including higher-order corrections (NLO, NNLO).
• Universality vs. Large-$D$ Behavior: The work identifies universal features captured by the expansion while also highlighting distinct large-$D$ behaviors, such as the specific constraints on the echoing period $\Delta$ and the convergence properties of the series.

The authors remain modest regarding the convergence of the series, noting that while NNLO captures key qualitative features, the radius of convergence and the behavior of the echoing period $\Delta$ as $D \to \infty$ remain open questions. They also note that the Choptuik exponent $\gamma$ is not addressed, though they acknowledge theoretical arguments suggesting $\gamma \to 1/2$ in the large-$D$ limit.