The study of the flux of atmospheric neutrino crossing the Earth can provide useful information not only on the matter density of the different layers that make up the planet but also on their chemical composition. The key phenomenon that makes this possible is flavor oscillations and their dependence on the electron density along the neutrino baseline. To extract the relevant information, we simulate the energy and azimuth angle distribution of events produced in a generic neutrino telescope by atmospheric neutrinos passing through the deepest parts of the Earth. Changes in the densities of the outer core and the mantle are implemented by varying the location of the boundary between these layers so that the restrictions on the mass of and the moment of inertia of the Earth are both satisfied. This allows us to examine the effect of simultaneous changes in composition and density of the outer core, unlikely other works on the subject, where only one of these quantities was varied.

Knowledge of the composition of the Earth’s interior is highly relevant to many geophysical and geochemical problems. Neutrino oscillations are modified in a non-trivial way by the matter effects and can provide valuable and unique information not only on the density but also on the chemical and isotopic composition of the deep regions of the planet. In this paper, we re-examine the possibility of performing an oscillation tomography of the Earth with atmospheric neutrinos and antineutrinos to obtain information on the composition and density of the outer core and the mantle, complementary to that obtained by geophysical methods. Particular attention is paid to the D$$^{\prime \prime }$$ ″ layer just above the core-mantle boundary and to the water (hydrogen) content in the mantle transition zone. Our analysis is based on a Monte-Carlo simulation of the energy and azimuthal angle distribution of $$\mu $$ μ -like events generated by neutrinos. Taking as reference a model of the Earth consisting of 55 concentric layers with constant densities determined from the PREM, we evaluate the effect on the number of events due to changes in the composition and density of the outer core and the mantle. To examine the capacity of a detector like ORCA to resolve such variations, we construct regions in planes of two of these quantities where the statistical significance of the discrepancies between the reference and the modified Earth are less than $$1\sigma $$ 1 σ . The variations are implemented in such a way that the constraint imposed by both the total mass of the Earth and its moment of inertia are verified.

In this note we give a direct proof of the F. Riesz representation theorem which characterizes the linear functionals acting on the vector space of continuous functions defined on a set K. Our start point is the original formulation of Riesz where K is a closed interval. Using elementary measure theory, we give a proof for the case K is an arbitrary compact set of real numbers. Our proof avoids complicated arguments commonly used in the description of such functionals. RESUMEN En esta nota, damos una demostración directa del teorema de representación de F. Riesz que caracteriza los funcionales lineales actuando en el espacio vectorial de funciones continuas definidas en un conjunto K. Nuestro punto de partida es la formulación original de Riesz, donde K es un intervalo cerrado. Usando teoría elemental de la medida, damos una demostración para el caso en que K es un conjunto arbitrario compacto de números reales. Nuestra demostración evita argumentos complicados comúnmente usados en la descripción de dichos funcionales.