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Abstract The identification and description of point sources is one of the oldest problems in astronomy, yet even today the correct statistical treatment for point sources remains one of the field’s hardest problems. For dim or crowded sources, likelihood-based inference methods are required to estimate the uncertainty on the characteristics of the source population. In this work, a new parametric likelihood is constructed for this problem using compound Poisson generator (CPG) functionals that incorporate instrumental effects from first principles. We demonstrate that the CPG approach exhibits a number of advantages over non-Poissonian template fitting (NPTF)—an existing method—in a series of test scenarios in the context of X-ray astronomy. These demonstrations show that the effect of the point-spread function, effective area, and choice of point-source spatial distribution cannot, generally, be factorized as they are in NPTF, while the new CPG construction is validated in these scenarios. Separately, an examination of the diffuse-flux emission limit is used to show that most simple choices of priors on the standard parameterization of the population model can result in unexpected biases: when a model comprising both a point-source population and diffuse component is applied to this limit, nearly all observed flux will be assigned to either the population or to the diffuse component. A new parameterization is presented for these priors that properly estimates the uncertainties in this limit. In this choice of priors, CPG correctly identifies that the fraction of flux assigned to the population model cannot be constrained by the data. 

To answer NASA’s call for a sensitive X-ray observatory in the 2030s, we present the High Energy X-ray Probe (HEX-P) mission concept. HEX-P is designed to provide the required capabilities to explore current scientific questions and make new discoveries with a broadband X-ray observatory that simultaneously measures sources from 0.2 to 80 keV. HEX-P’s main scientific goals include: 1) understand the growth of supermassive black holes and how they drive galaxy evolution; 2) explore the lower mass populations of white dwarfs, neutron stars, and stellar-mass black holes in the nearby universe; 3) explain the physics of the mysterious corona, the luminous plasma close to the central engine of accreting compact objects that dominates cosmic X-ray emission; and 4) find the sources of the highest energy particles in the Galaxy. These goals motivate a sensitive, broadband X-ray observatory with imaging, spectroscopic, and timing capabilities, ensuring a versatile platform to serve a broad General Observer (GO) and Guest Investigator (GI) community. In this paper, we present an overview of these mission goals, which have been extensively discussed in a collection of more than a dozen papers that are part of this Research Topic volume. The proposed investigations will address key questions in all three science themes highlighted by Astro2020, including their associated priority areas. HEX-P will extend the capabilities of the most sensitive low- and high-energy X-ray satellites currently in orbit and will complement existing and planned high-energy, time-domain, and multi-messenger facilities in the next decade.