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<p>We uncover a geometric organization of the differential equations for the wave-function coefficients of conformally coupled scalars in power-law cosmologies. To do this, we introduce a basis of functions inspired by a decomposition of the wavefunction into time-ordered components. Representing these basis functions and their singularities by graph tubings, we show that a remarkably simple rule for the merger of tubes produces the differential equations for arbitrary tree graphs (and loop integrands). We find that the basis functions can be assigned to the vertices, edges, and facets of convex geometries (in the simplest cases, collections of hypercubes) which capture the compatibility of mergers and define how the basis functions are coupled in the differential equations. This organization of functions also simplifies solving the differential equations. The merger of tubes is shown to reflect the local properties of bulk physics, in particular the collapse of time-ordered propagators. Taken together, these observations demystify the origin of the <i>kinematic flow</i> observed in these equations [1].</p>

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Geometry of kinematic flow

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年5月19日

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  • Daniel Baumann,
  • Harry Goodhew,
  • Austin Joyce,
  • Hayden Lee,
  • Guilherme L. Pimentel,
  • Tom Westerdijk

摘要

We uncover a geometric organization of the differential equations for the wave-function coefficients of conformally coupled scalars in power-law cosmologies. To do this, we introduce a basis of functions inspired by a decomposition of the wavefunction into time-ordered components. Representing these basis functions and their singularities by graph tubings, we show that a remarkably simple rule for the merger of tubes produces the differential equations for arbitrary tree graphs (and loop integrands). We find that the basis functions can be assigned to the vertices, edges, and facets of convex geometries (in the simplest cases, collections of hypercubes) which capture the compatibility of mergers and define how the basis functions are coupled in the differential equations. This organization of functions also simplifies solving the differential equations. The merger of tubes is shown to reflect the local properties of bulk physics, in particular the collapse of time-ordered propagators. Taken together, these observations demystify the origin of the kinematic flow observed in these equations [1].