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arxiv: 2606.16521 · v2 · pith:IH3OF2X3new · submitted 2026-06-15 · 🌌 astro-ph.CO · gr-qc

A First Post-Friedmann Extension of the Schr\"odinger Approach to Cosmic Structure Formation

Yiwen Deng-Huang , Daniele Bertacca , Sabino Matarrese This is my paper

Pith reviewed 2026-06-27 03:18 UTC · model grok-4.3

classification 🌌 astro-ph.CO gr-qc
keywords Schrödinger approachpost-Friedmann approximationcosmic structure formationrelativistic correctionslarge-scale structurecold dark mattervector potential
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0 comments X

The pith

At first post-Friedmann order the Schrödinger description of cosmic structure requires an effective vector potential.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper extends the Schrödinger approach to cosmic structure formation from its Newtonian limit to first post-Friedmann order. The standard Schrödinger-Poisson system captures pressureless fluid dynamics in the leading approximation but omits relativistic corrections important on horizon scales and for precision surveys. Starting from the 1PF continuity and Euler equations in a flat Lambda CDM background, the authors identify a conserved density variable that restores a Newtonian-like conservative form for the continuity equation. Even with vanishing covariant vorticity, however, the spatial velocity field carries a transverse 1PF component, so the full mass flux cannot be written as the gradient of a scalar phase alone and demands an effective vector potential.

Core claim

Starting from the 1PF continuity and Euler equations in a flat Lambda CDM background, the conserved density variable associated with covariant mass conservation leads to a continuity equation in Newtonian-like conservative form. However, the spatial velocity field contains a transverse 1PF component even for vanishing covariant vorticity, so the full 1PF mass flux cannot be represented solely by the gradient of a scalar phase. The Schrödinger-like formulation therefore requires an effective vector potential fixed by this transverse velocity component, which includes the post-Friedmann metric vector perturbation together with nonlinear scalar terms required by the zero-vorticity condition. At

What carries the argument

The effective vector potential in the 1PF Schrödinger-like formulation, fixed by the transverse component of the spatial velocity field and containing the metric vector perturbation plus nonlinear scalar terms.

If this is right

  • The system reduces to the standard Schrödinger-Poisson equations at leading order.
  • The vector potential incorporates the post-Friedmann metric vector perturbation associated with frame-dragging.
  • Nonlinear scalar terms are required to maintain the zero-vorticity condition.
  • When the equation is written in scalar form the corrections enter as an imaginary contribution to the effective potential.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The formulation could be discretized for numerical simulations that capture post-Friedmann corrections without solving the full Einstein equations.
  • It may improve modeling of observables from surveys that reach horizon-scale modes.
  • Similar vector-potential extensions could be explored for other fluid or scalar-field descriptions in cosmology.

Load-bearing premise

The spatial velocity field in the cosmological frame contains a transverse first post-Friedmann component even when the covariant vorticity vanishes.

What would settle it

Direct numerical comparison of density and velocity evolution between the derived 1PF Schrödinger system and the original 1PF fluid equations on scales approaching the horizon.

read the original abstract

We extend the Schr\"odinger approach to large-scale structure formation beyond the Newtonian regime by working at first post-Friedmann (1PF) order. The standard Schr\"odinger--Poisson system gives a useful reformulation of the dynamics of a self-gravitating pressureless fluid, but it corresponds to the leading post-Friedmann, or Newtonian, limit. It therefore misses the relativistic corrections that enter at next-to-leading order and become relevant on horizon scales and for high-precision cosmological surveys. Starting from the 1PF continuity and Euler equations in a flat $\Lambda$CDM background, we identify the conserved density variable associated with covariant mass conservation. In terms of this variable, the continuity equation takes a Newtonian-like conservative form. However, even for vanishing covariant vorticity, the spatial velocity field in the cosmological frame contains a transverse 1PF component. Thus the full 1PF mass flux cannot be represented solely by the gradient of a scalar phase. We show that the Schr\"odinger-like formulation at 1PF order requires an effective vector potential fixed by this transverse velocity component. This vector potential contains the post-Friedmann metric vector perturbation, related to relativistic frame-dragging effects, together with nonlinear scalar terms required by the zero-vorticity condition. Equivalently, when the equation is written in scalar form, these corrections appear as an imaginary contribution to the effective potential. At leading order our system reduces to the usual Schr\"odinger--Poisson formulation, while at 1PF order it provides a relativistic extension of the Schr\"odinger description of cold matter dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the Schrödinger-Poisson formulation of cold dark matter dynamics to first post-Friedmann (1PF) order. Starting from the standard 1PF continuity and Euler equations in a flat ΛCDM background, it identifies a conserved density variable that renders the continuity equation Newtonian-like, then demonstrates that the spatial velocity field in the cosmological frame retains a transverse 1PF component even when covariant vorticity vanishes. This necessitates an effective vector potential (containing the metric vector perturbation plus nonlinear scalar terms) in the Schrödinger-like equation; equivalently, the corrections appear as an imaginary term in the effective potential when written in scalar form. The system reduces exactly to the usual Schrödinger-Poisson equations at leading order.

Significance. If the derivation is correct, the result supplies a parameter-free relativistic extension of the Schrödinger approach that incorporates frame-dragging and other 1PF effects. This could prove useful for precision modeling on horizon scales and for high-precision surveys, while preserving the computational advantages of the Schrödinger method. The construction is grounded in the standard 1PF equations without ad-hoc parameters or fitted quantities.

minor comments (3)
  1. [Abstract] Abstract: the statement that the transverse component 'cannot be represented solely by the gradient of a scalar phase' is central; a parenthetical reference to the explicit decomposition (e.g., the relevant velocity equation in §3 or §4) would help readers locate the supporting algebra.
  2. The manuscript would benefit from a short paragraph comparing the resulting vector potential to existing post-Newtonian or 1PF treatments of vorticity and frame-dragging in the literature (e.g., references to works on relativistic Euler equations or vector modes).
  3. Notation: the distinction between the covariant vorticity and the transverse velocity component in the cosmological frame should be emphasized with a brief reminder of the coordinate choices when first introduced.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation for minor revision. No specific major comments were provided in the report, so there are no individual points to address point-by-point.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins from the standard 1PF continuity and Euler equations in a flat ΛCDM background, identifies the conserved density variable from covariant mass conservation, and decomposes the spatial velocity field to obtain the transverse 1PF component under the zero-vorticity condition. The effective vector potential and imaginary potential term are direct consequences of this decomposition and the post-Friedmann metric perturbations rather than fitted inputs or self-referential definitions. The reduction to the Newtonian Schrödinger-Poisson system at leading order is the expected consistency check of the expansion order and does not force the 1PF result. No load-bearing self-citation chain or ansatz smuggling is required for the central construction, which remains parameter-free and externally falsifiable against the input 1PF equations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the 1PF equations as the starting point and the identification of the transverse velocity component; no free parameters or invented entities with independent evidence are mentioned.

axioms (1)
  • domain assumption The 1PF continuity and Euler equations in a flat ΛCDM background provide the correct starting point for the extension.
    Explicitly stated as the foundation in the abstract.
invented entities (1)
  • effective vector potential no independent evidence
    purpose: To represent the transverse 1PF velocity component that cannot be captured by a scalar phase.
    Introduced in the abstract to complete the Schrödinger-like formulation at 1PF order.

pith-pipeline@v0.9.1-grok · 5836 in / 1258 out tokens · 42257 ms · 2026-06-27T03:18:52.880785+00:00 · methodology

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Reference graph

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