The Radiation Density of the Early Universe as Bookkeeping: What the Measured N_eff Leaves Room For, Priced in Decoupling Temperatures

The effective number of relativistic species, N_eff, summarises the radiation density of the early universe in a single number. The standard prediction is N_eff = 3.044. The Planck 2018 measurement, combined with baryon acoustic oscillation data, is N_eff = 2.99 ± 0.17, and primordial abundances give a consistent value. This paper converts the room left by those measurements into a price list in decoupling temperature. For one additional species that was in thermal equilibrium and decoupled while relativistic, the contribution ΔN_eff is computed from entropy conservation at five representative decoupling temperatures, for a real scalar, a Weyl fermion, a Dirac fermion, and a vector boson. The contributions range from 0.0268 for a scalar decoupling at 200 GeV to 2.0 for a Dirac fermion decoupling at 3 MeV. The Planck 95% allowance for extra radiation is ΔN_eff < 0.2792, which requires an effective entropy count at decoupling above 18.4 for a scalar, 28.0 for a Weyl fermion, 47.1 for a Dirac fermion, and 30.9 for a vector boson; in the discrete entropy count used here, every one of the four cases must therefore have decoupled above the QCD transition. A projected sensitivity of σ(N_eff) ≈ 0.03 for CMB-S4, labelled throughout as a projection and not a measurement, would reduce the 95% allowance to 0.0588 and would leave no decoupling temperature at which a fully thermalised Dirac fermion satisfies it. A conversion to the present-day density is given for the case in which the same species later becomes non-relativistic, together with an explicit statement of where that conversion ceases to apply. No new species is proposed.