After more than 80 years from its first evidence in the Coma galaxy cluster, dark matter represents one of the deepest mystery in current physics. Even though its existence is strongly suggested by several gravitational effects such as the anomalies of galactic rotation curves, the gravitational lensing, the bullet clusters and the cosmological observations, the nature of dark matter still remains unknown. The most attractive and simplest scenario is the one dubbed as Weakly Interacting Massive Particle (WIMP) paradigm in which dark matter particles typically have a mass in the GeV-TeV range and interactions of the order of weak processes. However, the lacking detection of a signal related to dark matter has led to very stringent constraints to the WIMP paradigm, encouraging physicists to look for alternatives to it. During the last decades, different interesting schemes have been investigated in elementary particle physics in order to allocate viable dark matter candidates. In the so-called dark matter zoo, the mass is spread over many orders of magnitude, ranging from about $10^{-32}$ GeV up to $10^{18}$ GeV. For instance, alternatives to WIMPs, with a mass smaller than the GeV energy scale, are represented by axions at about $10^{-6}\div10^{-3}$ eV or keV sterile neutrinos. On the other hand, WIMPzilla generally have a mass of the order of $10^{12}$ GeV. Up to now all direct, indirect and colliders searches, especially dedicated to the GeV-TeV energy range, have not provided any clear evidence for dark matter. In this context, the only viable way to look for very massive dark matter candidates, with a mass larger than TeV, is based on exploiting indirect searches in astrophysical observations. Indeed, the astrophysical measurements of high energy neutrinos and gamma-rays have opened a new window to the cosmos, giving the chance to explore very high-energy phenomena that can be potentially linked to new physics. The recent discovery of a diffuse neutrino flux at the TeV-PeV range by the IceCube Collaboration has ushered us into a new era for astroparticle physics, since it provides an important diagnostic tool for physics and astrophysics. The IceCube Neutrino Observatory is a neutrino telescope located at the Amundsen-Scott South Pole Station, able to observe highly energetic neutrinos reaching the Earth’s surface. In six year of data-taking (2010-2016), the IceCube detector has collected 82 High Energy Starting Events (HESE) with energies larger than about 10 TeV. Moreover, for the first time, three events fully contained in the detector with energy larger than PeV (Ernie (1.14 PeV), Bert (1.04 PeV), and Big Bird (2.2 PeV)) have been observed. In the other hemisphere, from 2007 to 2015, the ANTARES Neutrino Telescope has additionally observed 33 events with energies above 20 TeV. All these events correspond to the most energetic neutrinos ever measured and offer the possibility to study neutrino physics at energies where phenomena beyond the Standard Model (SM) can be relevant. The origin of such high energy neutrinos still remains unclear: they can be produced by a variety of galactic and extragalactic astrophysical sources, or they could be intriguingly related to dark matter. In this context, this thesis aims to investigate the relation of the dark matter paradigm to neutrino and gamma-ray telescopes. Indeed, depending on the interaction with SM particles, dark matter can decay or annihilate producing high-energy neutrinos and gamma-rays. Hence, one can infer the properties of dark matter particles that are able to provide a detectable signal by comparing the predicted neutrino and gamma-ray fluxes with the corresponding astrophysical observations. Such studies based on characterizing at the same time the neutrino and gamma-ray fluxes are defined as multi-messenger analyses. In this thesis, we deeply analyze the scenario where the diffuse TeV-PeV neutrino flux is explained in terms of a two-component flux, one of which is related to dark matter. The two-component flux is indeed suggested by the tension of neutrino data, taken by IceCube and ANTARES telescopes, with the simple assumption of a single power-law, behavior that is expected in case of standard astrophysical sources once a correlation with hadronic cosmic-rays is reasonably considered. Indeed, since one would expect a hard power-law (spectral index smaller than 2.3) according to the gamma-ray and up-going muon neutrino observations, the diffuse neutrino flux shows a $2$-$3\sigma$ excess at 10-100 TeV energies, pointing towards a two-component neutrino flux. Remarkably, such a low-energy excess is present in different IceCube and ANTARES data samples. Once such a tension is statistically characterized, we focus on the scenario where the low-energy excess is intriguingly due to dark matter. For this purpose, we phenomenologically scrutinize several dark matter models characterized by distinct interactions with the Standard Model particles and by different halo density distributions of our galaxy. Moreover, we show that the dark matter models proposed to explain the diffuse neutrino flux are further constrained once the gamma-ray observations, measured for instance at Fermi-LAT, are taken into account. This strongly underlines the paramount importance of multi-messenger analysis in the attempt to unveil the nature of dark matter. The thesis is organized as follows. In the first Chapter we briefly introduce the Standard Model and, in particular, we discuss theoretical open problems and experimental observations (like the ones related to the neutrino physics) that underline the need to go beyond. The second Chapter is devoted to dark matter: all its evidence and its properties are presented, different production mechanisms are described, a list of dark matter candidates is reported and the status of dark matter searches is reviewed. In the third Chapter, we introduce all the ingredients required to compute the flux of neutrinos and gamma-rays that are produced by decaying/annihilating dark matter particles. The fourth Chapter is dedicated to review the physics and the observations of Neutrino Telescopes. Moreover, we also report the first combined analysis of IceCube and ANTARES measurements, highlighting the tension of data with the assumption of a single power-law explaining the diffuse neutrino flux. In the fifth Chapter, the low-energy excess is statistically characterize and its interpretation in terms of a dark matter signal is investigated by performing analyses on the angular distribution and the energy spectrum of the observed neutrinos. In the sixth Chapter, we propose a complete theoretical framework allocating a very heavy leptophilic dark matter candidate. Here, the model is constrained by examining its predictions for the neutrino flux and by requiring a viable production mechanism in the early Universe. Finally, the last Chapter is devoted to the conclusions.
