What is a residual speckle? An answer with analytical modeling of contrast stability

In this paper we discuss the statistics of systematic noise (aka speckles) in high-contrast imaging. We start from first principles and derive the probability density function of speckles stemming from various misalignments (wavefront errors or others) in an astronomical observatory equipped with a coronagraph instrument in various cases.We show that this theoretical formalism matches very well diffractive end-to-end simulations of a few possible instrument configurations of the Habitable Worlds Observatory. We then consider the case of normally distributed wavefront errors and show how this formalism simplifies into a concise analytical model that ties together contrast stability, coronagraph sensitivities (calculated using said diffractive models), and wavefront variance. We demonstrate that this analytical model matches very well end-to-end simulations of hypothetical HWO exoplanet observations that use differential imaging for signal discrimination.