  We present a first-principles analysis of the noise spectra of
alkali-metal-metal vapors in and out of the spin-exchange-relaxation-free
(SERF) regime, and we predict non-intuitive features with a potential to
further improve the sensitivity of SERF media, and which must be taken into
account in their use in quantum optical applications. Studying the process of
spin-noise spectroscopy (SNS), we derive analytic formulas for the observable
noise spectra, and for the correlation functions among different hyperfine
components, which give additional insight into the spin dynamics. The analytic
results indicate a variety of distortions of the spin-noise spectrum relative
to simpler models, including a broad spectral background that mimics optical
shot noise, interference of noise contributions from the two ground-state
hyperfine levels, noise reduction at the spin-precession frequency, and
``hiding'' of spin-noise power that can introduce a systematic error in
noise-based calibrations, e.g., for spin-squeezing experiments.
