Reheating as a Geometric Selector of Axion-Like Dark Matter Contours*

This preprint develops an analytic framework for understanding how reheating shapes the geometry of axionlike dark-matter parameter space. Focusing on sourced effective misalignment during a constant-www reheating epoch, it derives a closed-form expression for the slope of constant-θeffθeffθeff contours in the (log⁡mϕ,log⁡fϕ)(log m_φ,log f_φ)(logmϕ,logfϕ) plane. In the minimal case, the slope is (1−w)/[2(1+w)](1-w)/[2(1+w)](1−w)/[2(1+w)], showing that the leading contour orientation is determined directly by the reheating equation-of-state parameter. A generalized formula further incorporates model-dependent reheating and source-sector scaling through the exponent quartet (α,β,u,v)(α,β,u,v)(α,β,u,v), providing a compact map from reheating microphysics to observable contour geometry. Limited numerical calculations confirm that the analytic formulas capture the leading tangent structure across representative reheating histories. The main message is that reheating acts as a geometric selector: different expansion histories rotate the family of allowed ALP contours in a predictable way, offering a scan-light organizing principle for axion and ALP dark-matter phenomenology. Version 2 changes Notation unified: Hend/aendHend/aend Hend/aend → Hreh/arehHᵣₑₕ/aᵣₑₕ Hreh/areh throughout. Appendix B rewritten: exponents 2w/(1+w)2w/(1+w) 2w/(1+w) and (1−w)/(1+w)(1-w)/(1+w) (1−w)/(1+w) now derived correctly from the scalar field EOM and oscillation condition. New Sec. 4.1: slope formula for representative reheating models (inflaton decay α=0α=0 α=0, modulus reheating α=3α=3 α=3) with summary Table 1. New Sec. 5.4: experimental slope comparison (ADMX, ABRACADABRA, CASPEr) via analytic transformation Eq. (5.2). Sec. 5.3 expanded with three numerical examples for the inverse problem. Expression adjustments: overstated claims softened; "in the minimal case" added to inverse-problem statements in Sec. 5.3 and Conclusions. v3:This version clarifies the sourcing-epoch formulation underlying the geometric-selector picture. The analysis is now consistently written in terms of a localized effective sourcing epoch a∗a_*a∗ and its associated Hubble scale H∗=H(a∗)H_*=H(a_*)H∗=H(a∗), rather than conflating the source scale with the end-of-reheating scale. The derivation of the effective-misalignment scaling has been rewritten accordingly, with the dilution factor between sourcing and oscillation stated consistently throughout the main text and Appendix B. The generalized parametrization has also been clarified as an effective scaling of the sourcing dynamics, and the discussion of representative microscopic models has been softened to reflect this interpretation. In addition, notation, equation references, appendix numbering, and several phenomenological statements were tightened for internal consistency and readability, while leaving the main result unchanged: the leading contour slope remains analytically controlled by the reheating equation of state and its generalized exponent structure.