Cosmic-ray interactions with the nuclei of the Earth's atmosphere produce a flux of neutrinos in all directions with energies extending above the TeV scale. However, the Earth is not a fully transparent medium for neutrinos with energies above a few TeV. At these energies, the charged-current neutrino-nucleon cross section is large enough so that the neutrino mean-free path in a medium with the Earth's density is comparable to the Earth's diameter. Therefore, when neutrinos of these energies cross the Earth, there is a non-negligible probability for them to be absorbed. Since this effect depends on the distance traveled by neutrinos and on their energy, studying the zenith and energy distributions of TeV atmospheric neutrinos passing through the Earth offers an opportunity to infer the Earth's density profile. Here we perform an Earth tomography with neutrinos using actual data, the publicly available one-year through-going muon sample of the atmospheric neutrino data of the IceCube neutrino telescope. We are able to determine the mass of the Earth, its moment of inertia, the mass of the Earth's core and to establish the core is denser than the mantle, using weak interactions only, in a way completely independent from gravitational measurements. Our results confirm that this can be achieved with current neutrino detectors. This method to study the Earth's internal structure, complementary to the traditional one from geophysics based on seismological data, is starting to provide useful information and it could become competitive as soon as more statistics is available thanks to the current and larger future neutrino detectors.

16 pages, 10 figures and 1 table