  We report a comprehensive study of stochastic electron spin fluctuations --
spin noise -- in lightly doped ($n$-type) bulk GaAs, which are measured using
sensitive optical magnetometry based on off-resonant Faraday rotation.
Frequency spectra of electron spin noise are studied as a function of electron
density, magnetic field, temperature, probe laser wavelength and intensity, and
interaction volume. Electron spin lifetimes $\tau_s$ are inferred from the
width of the spin noise spectra, and are compared with direct measurements of
$\tau_s$ using conventional Hanle effect methods. Both methods reveal a strong
and similar dependence of $\tau_s$ on the wavelength and intensity of the probe
laser, highlighting the undesired influence of sub-bandgap absorption effects
on the nominally `non-perturbative' spin noise measurements. As a function of
temperature, the spin noise power increases approximately linearly from 1.5 K
to 30 K, as expected for degenerate electrons obeying Fermi-Dirac statistics,
but with an additional zero-temperature offset. Finally, as the cross-sectional
area of the probe laser shrinks and fewer electrons are probed, the measured
Faraday rotation fluctuations due to electron spin noise are shown to increase,
as expected.
