  We study turn-by-turn fluctuations in the number of spontaneously emitted
photons from an undulator, installed in the Integrable Optics Test Accelerator
(IOTA) electron storage ring at Fermilab. A theoretical model is presented,
showing the relative contributions due to the discrete nature of light emission
and to the incoherent sum of fields from different electrons in the bunch. The
model is compared with a previous experiment at Brookhaven and with new
experiments we carried out at IOTA. Our experiments focused on the case of a
large number of longitudinal and transverse radiation modes, a regime where
photon shot noise is significant and the total magnitude of the fluctuations is
very small. The experimental and data analysis techniques, required to reach
the desired sensitivity, are detailed. We discuss how the model and the
experiment provide insights into this emission regime, enable diagnostics of
small beam sizes, and improve our understanding of beam lifetime in IOTA.
