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Conditional KPZ reduction in a one-dimensional model of bosonic dark matter

Rin Takada

Abstract

Wave-like dark matter described by a high-occupancy self-gravitating bosonic field provides a microscopic setting in which both amplitude and phase are dynamical. We study a one-dimensional Gross--Pitaevskii--Poisson toy model and ask which coarse-grained variable, if any, can be meaningfully compared with the 1+1-dimensional Kardar--Parisi--Zhang (KPZ) fixed point. We show that the relevant field is not the raw microscopic phase but a branch-resolved coarse-grained phase built from the sound sector. Above the Jeans scale and below the microscopic cutoff, self-gravity acts as a weak deformation of local sound dynamics. In this window the exact linear modes admit a local sound form, and a weakly nonlinear projection yields a nonvanishing same-chirality Burgers self-coupling. Under one-branch dominance together with a local Markov closure, the dominant branch reduces conditionally to a KPZ-type equation. We also formulate a dictionary from microscopic initial data to the canonical curved, flat, and stationary KPZ benchmarks. Our results do not establish KPZ universality for self-gravitating bosonic dark matter, but they identify the proper comparison field and the controlled regime in which an exact fixed-point test can be posed.

Conditional KPZ reduction in a one-dimensional model of bosonic dark matter

Abstract

Wave-like dark matter described by a high-occupancy self-gravitating bosonic field provides a microscopic setting in which both amplitude and phase are dynamical. We study a one-dimensional Gross--Pitaevskii--Poisson toy model and ask which coarse-grained variable, if any, can be meaningfully compared with the 1+1-dimensional Kardar--Parisi--Zhang (KPZ) fixed point. We show that the relevant field is not the raw microscopic phase but a branch-resolved coarse-grained phase built from the sound sector. Above the Jeans scale and below the microscopic cutoff, self-gravity acts as a weak deformation of local sound dynamics. In this window the exact linear modes admit a local sound form, and a weakly nonlinear projection yields a nonvanishing same-chirality Burgers self-coupling. Under one-branch dominance together with a local Markov closure, the dominant branch reduces conditionally to a KPZ-type equation. We also formulate a dictionary from microscopic initial data to the canonical curved, flat, and stationary KPZ benchmarks. Our results do not establish KPZ universality for self-gravitating bosonic dark matter, but they identify the proper comparison field and the controlled regime in which an exact fixed-point test can be posed.

Paper Structure

This paper contains 51 sections, 389 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Phase-sector dynamics and the origin of the KPZ nonlinearity without self-gravity
  3. From the Gross--Pitaevskii equation to KPZ
  4. Elimination of delta rho and a formal solution for the height-field equation
  5. A minimal model of self-gravitating bosonic dark matter
  6. Model definition and hydrodynamic form
  7. Linearization, exact sound modes, and the Jeans scale
  8. Weakly nonlinear reorganization around the linear sector
  9. Comparison window and scope of the KPZ question
  10. Sound-mode projection and conditional KPZ reduction in the coarse-grained phase sector
  11. Starting point: sound sector and coarse-grained phase
  12. Comparison window and its operational definition
  13. Exact linear sound modes and their local approximation
  14. Weakly nonlinear projection and same-chirality self-coupling
  15. One-branch dominance and Burgers form
  16. ...and 36 more sections

Figures (3)

  • Figure 1: Representative TASEP initial conditions. (a) step initial condition for the curved subclass, (b) alternating initial condition for the flat subclass, and (c) a typical Bernoulli initial condition for the stationary subclass.
  • Figure 2: Comparison of the cumulative distribution functions $F_2$, $F_1$, and $F_0$, corresponding respectively to the curved, flat, and stationary KPZ subclasses. This plot is included only as a visual reference for the one-point benchmarks summarized in Table \ref{['tab:5.1']}.
  • Figure 3: Comparison of the probability densities $p_2(s)=\partial_s F_2(s)$, $p_1(s)=\partial_s F_1(s)$, and $p_0(s)=\partial_s F_0(s)$. The plot highlights the differences in center, width, and skewness among the three canonical one-point laws.