The Large Hadron Collider (LHC), a 7 + 7 TeV proton-proton collider under completion at CERN, the European Laboratory for Particle Physics in Geneva, will take experiments into a new energy domain beyond the Standard Model of strong and electroweak interactions. As the LHC will unveil the mysteries of the electroweak symmetry breaking, this will also have far-reaching implications for cosmology. The aim of this project is to work out what we may learn about the Early Universe from discoveries at the LHC. This concerns in particular the two fundamental questions of the nature of the Dark Matter and the origin of the matter-antimatter asymmetry of the Universe. The LHC-Cosmology interplay has been a topic of active research in the last years. However, studies have essentially focussed on a single class of models: supersymmetry. The original and innovative directions of this project are: 1) To investigate dark matter particle physics models that have not been explored yet and confront theoretical predictions with existing and upcoming observational constraints. Measuring the properties of the dark matter will require a complementarity between the LHC searches and the other numerous ongoing dark matter experiments such as gamma ray telescopes, neutrino telescopes, cosmic positron detectors ... etc. 2) To work out the details of the electroweak phase transition in extensions of the Standard Model. One of the best-motivated mechanism for generating the baryon asymmetry of the universe relies on a first-order electroweak phase transition. Interestingly, this has strong implications for Gravity Wave physics. We will explore thoroughly how the planned gravity wave detector and space interferometer LISA, which turns out to be a completely independent window on the electroweak scale, could complement the information provided by the LHC. This project will also serve as a solid basis for future research at the Internatinal electron-positron Linear Collider.