The high entropy of the UHECR arrival direction distribution favors a light composition
We analyze the constraints on the composition of ultra-high-energy cosmic rays (UHECRs), and on the density and distribution of their sources, that may be inferred from their arrival-direction distribution, using a novel semi-analytical description of the propagation energy loss of atomic nuclei ($A>4$) with energy $>2\times10^{19}~\text{eV}$, that allows generating UHECR arrival maps faster than by using detailed propagation simulations and yields insights to the impact of propagation energy loss. We show that the anisotropy of the UHECR arrival direction distribution due to the large-scale structure (LSS) of matter distribution is larger for heavy nuclei composition compared to protons, despite their larger deflections by magnetic fields, due to their shorter propagation distance and weaker dependence of rigidity on observed energy. Identifying the LSS anisotropy signal is hampered for heavy nuclei due to their large deflections by the uncertain Galactic magnetic field (GMF). We introduce a new measure of anisotropy, an "entropy" of the arrival-direction distribution, that is largely independent of the GMF configuration and has strong discriminating power between heavy- and light-composition models. Analyzing the public $>3.2\times10^{19}$~eV Auger data, we show that the correlation with the LSS on large angular scales is weak and requires a low source density, $s_0\le10^{-4}{\rm Mpc}^{-3}$, to allow masking the LSS signature by "cosmic-variance" ($s_0=10^{-2}{\rm Mpc}^{-3}$ is ruled out at $>99\%$ confidence level (CL)). The high entropy of the distribution is consistent with proton models and inconsistent with heavy nuclei models at $>96\%$ CL for $s_0\ge10^{-5}{\rm Mpc}^{-3}$. Reducing the absolute energy calibration uncertainty may allow detection of the LSS correlation for proton models (increased exposure alone will not suffice due to the dominance of cosmic variance).