Local H~{\sc i} emissivity measured with the {\it Fermi}-LAT and implications for cosmic-ray spectra

Cosmic-ray (CR) electrons and nuclei interact with the Galactic interstellar gas and produce high-energy $\gamma$ rays. The $\gamma$-ray emission rate per hydrogen atom, called emissivity, provides a unique indirect probe of the CR flux. We present the measurement and the interpretation of the emissivity in the solar neighborhood for $\gamma$-ray energy from 50~MeV to 50~GeV. We analyzed a subset of 4 years of observations from the Large Area Telescope (LAT) aboard the {\it Fermi Gamma-ray Space Telescope} ({\it Fermi}) restricted to absolute latitudes $10^o