  The PTOLEMY project aims to develop a scalable design for a Cosmic Neutrino
Background (CNB) detector, the first of its kind and the only one conceived
that can look directly at the image of the Universe encoded in neutrino
background produced in the first second after the Big Bang. The scope of the
work for the next three years is to complete the conceptual design of this
detector and to validate with direct measurements that the non-neutrino
backgrounds are below the expected cosmological signal. In this paper we
discuss in details the theoretical aspects of the experiment and its physics
goals. In particular, we mainly address three issues. First we discuss the
sensitivity of PTOLEMY to the standard neutrino mass scale. We then study the
perspectives of the experiment to detect the CNB via neutrino capture on
tritium as a function of the neutrino mass scale and the energy resolution of
the apparatus. Finally, we consider an extra sterile neutrino with mass in the
eV range, coupled to the active states via oscillations, which has been
advocated in view of neutrino oscillation anomalies. This extra state would
contribute to the tritium decay spectrum, and its properties, mass and mixing
angle, could be studied by analyzing the features in the beta decay electron
spectrum.
