Thermal Axion-Like Particle Dark Radiation: Decoupling Temperatures Implied by the Photon Coupling

At a coupling of 10^-10 per GeV the decoupling temperature is 1.93e+04 GeV and the residual contribution is 2.68e-02. At 10^-12 per GeV decoupling moves to 1.93e+08 GeV, still above the electroweak scale, so the contribution sits at the same floor value of 2.68e-02, the smallest a thermalised scalar can contribute. At 10^-9 per GeV the decoupling temperature is 1.73e+02 GeV. At the weakest coupling scanned, 10^-13 per GeV, the formal decoupling temperature of 1.93e+10 GeV exceeds the ceiling of 2.82e+09 GeV implied by the bound density of 9.72e+54 g per cubic centimetre: within this history such a particle never reaches equilibrium at all, and its thermal contribution is absent rather than merely small. This study is computed from stated assumptions and published comparison values; it is not a measurement.