Population synthesis of pulsar wind nebulae and pulsar halos in the Milky Way -- Predicted contributions to the very-high-energy sky

The discovery of extended gamma-ray emission toward a number of middle-aged pulsars suggests the possibility of long-lived particle confinement beyond the classical pulsar wind nebula (PWN) stage. How this emerging source class can be extrapolated to a Galactic population remains unclear. We aim to evaluate how pulsar halos fit in existing TeV observations, under the assumption that all middle-aged pulsars develop halos similar to those observed toward the J0633+1746 or B0656+14 pulsars. We modeled the populations of supernova remnants, PWNe, and pulsar halos in the Milky Way. The PWN-halo evolutionary sequence is described in a simple yet coherent framework, and both kinds of objects are assumed to share the same particle injection properties. We then assessed the contribution of the different source classes to the very-high-energy emission from the Galaxy. The synthetic population can be made consistent with the flux distribution of all known objects, including unidentified objects, for a reasonable set of parameters. The fraction of the populations predicted to be detectable in surveys of the Galactic plane with HESS. and HAWC is then found to be in good agreement with their actual outcome, with a number of detectable halos ranging from 30 to 80% of the number of detectable PWNe. Prospects for CTA involve the detection of 250-300 sources in the Galactic Plane Survey, including 170 PWNe and up to 100 halos. The extent of diffusion suppression in halos has a limited impact on such prospects but its magnitude has a strong influence. The level of diffuse emission from unresolved populations in each survey is found to be dominated by halos and comparable to large-scale interstellar radiation powered by cosmic rays above 0.1-1TeV. Pulsar halos are shown to be viable counterparts to a fraction of the currently unidentified sources if they develop around most middle-aged pulsars (abridged)