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Effective Sourcing Epochs and the Geometry of Reheating-Selected Axion Contours

Authors: Shioiri, Hiroyuki;

Effective Sourcing Epochs and the Geometry of Reheating-Selected Axion Contours

Abstract

This paper develops an effective‑sourcing framework for reheating‑selected axion contours, clarifying how microscopic sourcing histories reduce to geometric data that control contour orientation in the (log⁡ma,log⁡fa) plane. Instead of relying on a detailed model‑dependent sourcing kernel, we classify sourcing histories into three effective classes—peak‑type, threshold‑type, and multi‑stage—and show that each class admits a reduction to an effective sourcing epoch a∗, an associated Hubble scale H∗=H(a∗), and an effective source amplitude Y1. At leading order, these effective quantities may be represented by the exponent map H∗∝fa αma β,Y1∝fa uma v, which directly determines the contour slope through the generalized formula dlog⁡fadlog⁡ma=2βw+v(1+w)+(1−w)2αw+(2−u)(1+w). This framework separates genuine geometric effects—tilt deformations and departures from a single effective slope—from pure normalization or position shifts. It also identifies multi‑stage sourcing as a natural origin of curvature or piecewise‑tilt behavior in reheating‑selected axion contours. The result is a model‑independent geometric layer between microscopic reheating dynamics and observable contour structure, clarifying which features of the sourcing history survive coarse‑graining and which do not.

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