Abstract: The highest energy scales accessible in particle physics are seen at the LHC and ultra high energy cosmic ray detectors such as Pierre Auger Observatory and Telescope Array. First, I consider applying a technique known as integral dispersion relations to extend the reach of the LHC. Integral dispersion relations use analyticity to link experimental values at one energy to those at all energies. This can be used to detect new physics in a model independent fashion at and above a machine's operating energy. Next, to probe physics at even higher energies, I consider the open question of ultra high energy cosmic ray anisotropy. In particular, I look at how problems such as low statistics and partial sky coverage affect an experiment's ability to determine anisotropy.