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T1 — the variational structure of the Zel’dovich/adhesion flow (refutation with derivation)

T1 — the variational structure of the Zel’dovich/adhesion flow (refutation with derivation)

Question (from the scoping post): derive — or refute — a Jacobi-type variational principle for the Zel’dovich/adhesion flow in comoving coordinates: an effective metric $g \propto (E-\Phi)\,g_{\mathrm{Euclid}}$ (or a sub-Riemannian analogue on $\mathbb{R}^3 \times S^2$) whose geodesics the flow follows, with “cheap along the filament axis” anisotropy.

Answer: refuted. The flow has an exact variational principle, but it is optimal transport with quadratic cost, not geodesic flow of a potential-weighted metric. Three short steps.

1. In growth-factor time, Zel’dovich dynamics is free motion

Write the Zel’dovich map with the linear growth factor $D$ as the time variable:

\[\mathbf{x}(\mathbf{q}, D) = \mathbf{q} - D\,\nabla_q \Phi_0(\mathbf{q}), \qquad \frac{d\mathbf{x}}{dD} = -\nabla_q \Phi_0(\mathbf{q}) = \text{const along the trajectory}.\]

Every trajectory is a straight line traversed at constant velocity in $(\mathbf{x}, D)$. Such trajectories extremise the free action

\[S[\mathbf{x}] = \int \left| \frac{d\mathbf{x}}{dD} \right|^2 dD ,\]

i.e. they are geodesics of the flat Euclidean metric. The gravitational potential does not act as a force during the evolution at all — in $D$-time it is entirely absorbed into the initial velocity field $\mathbf{v}0 = -\nabla\Phi_0$. This is the precise reason the Jacobi construction cannot be repaired: the Jacobi metric $2m(E-\Phi)g{\mathrm{Euclid}}$ conformally reweights paths by the potential along the path, but in the variables where the cosmic-web flow is simple there is no potential along the path to reweight. Not only is energy not conserved (the objection raised in the scoping post); there is no force term left to build a conformal factor from.

2. The adhesion model’s variational principle is Hopf–Lax, i.e. optimal transport

The adhesion model regularises multi-streaming with Burgers dynamics, $\partial_D \mathbf{v} + (\mathbf{v}\cdot\nabla)\mathbf{v} = \nu \Delta \mathbf{v}$, $\nu \to 0^+$, with $\mathbf{v} = \nabla \psi$. The Hopf–Cole transformation solves it exactly, and the $\nu \to 0$ limit is the Hopf–Lax formula: the velocity potential evolves as

\[\psi(\mathbf{x}, D) = \min_{\mathbf{q}} \left[ \psi_0(\mathbf{q}) + \frac{|\mathbf{x} - \mathbf{q}|^2}{2D} \right].\]

This is a variational principle — a minimisation over straight-line transport paths with quadratic cost plus initial data — which is exactly the Monge–Kantorovich optimal-transport structure with cost $|\mathbf{x}-\mathbf{q}|^2$ (Brenier; used cosmologically in the MAK reconstruction of Frisch et al.). The “geometry” that governs the cosmic web is therefore: flat metric, straight rays, plus a Legendre-type convexification of the initial potential. Filaments and nodes are the places where the Hopf–Lax minimiser jumps between branches — the shock set (caustics) of an optimal-transport map — not the geodesics of any curved or lifted metric. Structure lives in the singularities of the map, not in curved paths.

3. Why this reproduces E2’s measurements

The derivation predicts the two dynamical signatures E2 measured:

The orientation manifold $\mathbb{R}^3 \times S^2$ retains a legitimate role, but as a descriptor of the shock set — the tangent structure of an already-formed web (which is why the lift wins the alignment statistic M3 on simulations and on the sky) — and not as the state space of transport.

Verdict

References