Effective Relativistic Species Through a Bounce: A Bound on Additional Radiation from the Expansion History

The effective number of relativistic species, conventionally near 3.044 in the standard model, is a compact summary of how much radiation-like energy the early universe contained beyond photons. At a bounce temperature of 3 MeV the equivalent shift is 0.122, giving an effective count of 3.166 against the observed 2.99 with an uncertainty of 0.17; at 10 MeV the shift falls to 0.011. The species count thus independently disfavours a bounce lying within a small factor of the nucleosynthesis temperature, reinforcing the abundance bounds through a channel that depends on the expansion rate alone rather than on any individual nuclear reaction. Computed from stated assumptions, not measured.