Phenomenology for the Large Hadron Collider

The search for physics Beyond the Standard Model is the underlying motivation for the physics programme of the Large Hadron Collider at CERN. In this thesis we will present studies into the Large Hadron Collider phenomenology of dark matter inspired extensions to the Standard Model, di-Higgs + 2 jet (hhjj) production, CP violating effects in the Higgs sector, and the use of shape information from top polarisation measurements in searches for new physics. Full scans of the parameter space of Simplified Dark Matter models are demonstrated to be viable and allow for intriguing comparisons to relic density constraints. Strongly self-interacting dark sectors are shown to leave a potentially measurable imprint on the energy scaling of mediator production. The weak boson fusion induced production of hhjj turns out to be sensitive to a number of new physics effects, including CP violation in the Higgs sector. The use of top polarisation measurements to discover new physics which predicts polarised resonances is shown to be an effective way to ameliorate the loss of shape information for kinematically challenging regions of parameter space.