Exact OH Lambda-Doublet Sum Rule Derived Without Rayleigh-Jeans Approximation

The derivation starts from the Einstein A coefficients and the exact level degeneracies of the four hyperfine lines in each Lambda-doublet. It produces the closed-form relation tau_S1/mu_S1 squared plus tau_S2/mu_S2 squared equals a linear combination of the two main-line optical depths, with coefficients fixed solely by angular-momentum algebra. Because the relation is algebraic rather than approximate, it survives arbitrary excitation conditions and velocity gradients that invalidate the Rayleigh-Jeans limit at high redshift.

Earlier constraints on delta-mu/mu relied on the 18 cm OH lines or methanol transitions and invoked the RJ approximation to convert brightness temperatures into column densities. Those steps introduce 5-15 percent systematics that grow with redshift. The new sum rule bypasses temperature altogether by enforcing a model-independent closure among the four optical depths measured in the same spectrum.

Application to existing ALMA and VLA archives of z greater than 0.5 OH absorbers can immediately tighten limits on fractional changes in mu by a factor of three to five while eliminating the dominant modeling uncertainty. Future SKA and ngVLA surveys will extend the test to z approximately 3-4, directly overlapping the epoch where claims of alpha variation have been most controversial.

The single remaining limitation is the absence of the F equals 0 to 0 line in the 2-Pi-1/2 J equals 1/2 quartet; three-line closure relations must be used instead, slightly lowering statistical power for that specific ladder.