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Cosmic rays that strike the top of the Earth’s atmosphere generate a shower of secondary particles that move toward the surface with relativistic speeds. Water Cherenkov detectors (WCDs) on the ground can detect charged muons, which are one of the many particles generated in the shower, with the Cherenkov imaging technique. A large number of these muons travel in WCD tanks near the speed of light in a vacuum, faster than the speed of light in water, and so trigger isotropic Cherenkov radiation, which is detected by the photomultiplier tubes (PMTs) placed inside the tanks. When the radial component of the speed of the muon toward a PMT drops from superluminal to subluminal, the PMT records Cherenkov light from an optical phenomenon known as relativistic image doubling (RID), which causes two Cherenkov images of the same muon to appear suddenly, with both images moving in geometrically opposite directions on the original muon track. The quantities associated with the RID effect can be measured experimentally with a variety of detector types and can be used to find various points on the original trajectory of the muon. In this paper, a detailed study of reconstructing the trajectory of a muon entering a WCD using the RID technique has been presented. It is found that the measurements of standard RID observables enables a complete reconstruction of the trajectory of the muon to a high degree of accuracy with less than 1% error.

We train convolutional neural networks to correct the output of fast and approximate N-body simulations at the field level. Our model, Neural Enhanced COLA --NECOLA--, takes as input a snapshot generated by the computationally efficient COLA code and corrects the positions of the cold dark matter particles to match the results of full N-body Quijote simulations. We quantify the accuracy of the network using several summary statistics, and find that NECOLA can reproduce the results of the full N-body simulations with sub-percent accuracy down to $k\simeq1~h{\rm Mpc}^{-1}$. Furthermore, the model, that was trained on simulations with a fixed value of the cosmological parameters, is also able to correct the output of COLA simulations with different values of $\Omega_{\rm m}$, $\Omega_{\rm b}$, $h$, $n_s$, $\sigma_8$, $w$, and $M_\nu$ with very high accuracy: the power spectrum and the cross-correlation coefficients are within $\simeq1\%$ down to $k=1~h{\rm Mpc}^{-1}$. Our results indicate that the correction to the power spectrum from fast/approximate simulations or field-level perturbation theory is rather universal. Our model represents a first step towards the development of a fast field-level emulator to sample not only primordial mode amplitudes and phases, but also the parameter space defined by the values of the cosmological parameters., Comment: 9 pages, 3 figures and 1 table

Cosmic gamma-ray photons incident on the upper atmosphere create air showers that move to the Earth's surface with superluminal speed, relative to the air. Even though many of these air showers remain superluminal all along their trajectories, the shower's velocity component toward a single Imaging Atmospheric Cherenkov Telescope (IACT) may drop from superluminal to subluminal. When this happens, an IACT that is able to resolve the air shower both in time and angle should be able to document an unusual optical effect known as relativistic image doubling (RID). The logic of RID is that the shower appears to precede its own Cherenkov radiation when its speed component toward the IACT is superluminal, but appears to trail its own Cherenkov radiation when its speed component toward the IACT is subluminal. The result is that the IACT will see the shower start not at the top of the atmosphere but in the middle -- at the point along the shower's path where its radial velocity component drops to subluminal. Images of the shower would then be seen by the IACT to go both up and down simultaneously. A simple simulation demonstrating this effect is presented. Clear identification of RID would confirm in the atmosphere a novel optical imaging effect caused not by lenses but solely by relativistic kinematics, and may aid in the accuracy of path and speed reconstructions of the relativistic air shower., 8 pages, 3 figures, comments welcome