General Relativity represents the pillar of our understanding of gravity and it is well tested in the Solar System. It is also able to explain the observed large-scale phenomenology as long as there is a Cosmological Constant and Cold Dark Matter. Nevertheless one needs to understand why the observed Cosmological Constant is so small if compared with the expected theoretical value, and of course to detect Dark Matter. Moreover, General Relativity shows theoretical problems in the ultraviolet regime because it is not renormalizable. Therefore, in the last decades a lot of theories and toy models have been proposed in order to solve some of the most intriguing open problems at the scales where gravity plays a fundamental role. An important constraint in this endeavor is Lovelock's theorem, which does single out General Relativity and does not leave much room for modifying it. My work has focused on two directions. On the latter I have considered Horava gravity, which breaks Lorentz invariance. On the other I have explored the possibility to circumvent Lovelock's theorem by adding extra auxiliary fields, which are nondynamical extra fields.
High Quality Content by WIKIPEDIA articles! In mathematics, a toy theorem is a simplified version of a more general theorem. For instance, by introducing some simplifying assumptions in a theorem, one obtains a toy theorem. Usually, a toy theorem is used to illustrate the claim of a theorem. It can also be insightful to study proofs of a toy theorem derived from a non-trivial theorem. Toy theorems can also have education value. After presenting a theorem (with, say, a highly non-trivial proof), one can sometimes give some assurance that the theorem really holds, by proving a toy version of the theorem. For instance, a toy theorem of the Brouwer fixed point theorem is obtained by restricting the dimension to one. In this case, the Brouwer fixed point theorem follows almost immediately from the intermediate value theorem.
In this thesis, the author considers quantum gravity to investigate the mysterious origin of our universe and its mechanisms. He and his collaborators have greatly improved the analyticity of two models: causal dynamical triangulations (CDT) and n-DBI gravity, with the space-time foliation which is one common factor shared by these two separate models.In the first part, the analytic method of coupling matters to CDT in 2-dimensional toy models is proposed to uncover the underlying mechanisms of the universe and to remove ambiguities remaining in CDT. As a result, the wave function of the 2-dimensional universe where matters are coupled is derived. The behavior of the wave function reveals that the Hausdorff dimension can be changed when the matter is non-unitary.In the second part, the n-DBI gravity model is considered. The author mainly investigates two effects driven by the space-time foliation: the appearance of a new conserved charge in black holes and an extra scalar mode of the graviton. The former implies a breakdown of the black-hole uniqueness theorem while the latter does not show any pathological behavior.
This twelfth volume in the Poincaré Seminar Series presents a complete and interdisciplinary perspective on the concept of Chaos, both in classical mechanics in its deterministic version, and in quantum mechanics. This book expounds some of the most wide ranging questions in science, from uncovering the fingerprints of classical chaotic dynamics in quantum systems, to predicting the fate of our own planetary system. Its seven articles are also highly pedagogical, as befits their origin in lectures to a broad scientific audience. Highlights include a complete description by the mathematician É. Ghys of the paradigmatic Lorenz attractor, and of the famed Lorenz butterfly effect as it is understood today, illuminating the fundamental mathematical issues at play with deterministic chaos, a detailed account by the experimentalist S. Fauve of the masterpiece experiment, the von Kármán Sodium or VKS experiment, which established in 2007 the spontaneous generation of a magnetic field in a strongly turbulent flow, including its reversal, a model of Earth's magnetic field, a simple toy model by the theorist U. Smilansky - the discrete Laplacian on finite d- regular expander graphs - which allows one to grasp the essential ingredients of quantum chaos, including its fundamental link to random matrix theory, a review by the mathematical physicists P. Bourgade and J.P. Keating, which illuminates the fascinating connection between the distribution of zeros of the Riemann zeta-function and the statistics of eigenvalues of random unitary matrices, which could ultimately provide a spectral interpretation for the zeros of the zeta-function, thus a proof of the celebrated Riemann Hypothesis itself, an article by a pioneer of experimental quantum chaos, H-J. Stöckmann, who shows in detail how experiments on the propagation of microwaves in 2D or 3D chaotic cavities beautifully verify theoretical predictions, a thorough presentation by the mathematical physicist S. Nonnenmacher of the "anatomy" of the eigenmodes of quantized chaotic systems, namely of their macroscopic localization properties, as ruled by the Quantum Ergodic theorem, and of the deep mathematical challenge posed by their fluctuations at the microscopic scale, a review, both historical and scientific, by the astronomer J. Laskar on the stability, hence the fate, of the chaotic Solar planetary system we live in, a subject where he made groundbreaking contributions, including the probabilistic estimate of possible planetary collisions. This book should be of broad general interest to both physicists and mathematicians.