Dark Radiation from the Fifth Coordinate Bounded by the Effective Number of Relativistic Species

A fifth coordinate in thermal contact with the early universe would contribute dark radiation, conventionally counted as an addition to the effective number of relativistic species. This paper computes that addition for a single extra bosonic or fermionic degree of freedom as a function of the ratio between its temperature and the photon temperature, and applies the published microwave-background bound, which allows an addition of at most about three tenths. The corresponding temperature-ratio ceiling is about 0.61 for one bosonic degree of freedom. A fifth-coordinate sector thermalised at the photon temperature is excluded; it must have decoupled earlier and cooled relative to photons, or remained out of equilibrium, by an amount this paper quantifies. Headline values are: delta Neff boson at equal T = 2.202e+00, xi ceiling boson = 6.076e-01, xi ceiling fermion = 6.282e-01, xi ceiling ten bosons = 3.417e-01, Neff planck = 2.99 +/- 0.17. This study is simulated and computed from stated assumptions; it is not a measurement.