Stability and instability of expanding solutions to the Lorentzian constant-positive-mean-curvature flow

We study constant mean curvature Lorentzian hypersurfaces of R^{1,d+1} from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric solutions exhibit one of three (future) asymptotic behaviours: (i) finite time collapse (ii) convergence to a time-like cylinder isometric to some RxS^d and (iii) infinite expansion to the future converging asymptotically to a time translation of the de Sitter solution. For class (iii) we examine the future stability properties of the solutions under arbitrary (not necessarily spherically symmetric) perturbations. We show that the usual notions of asymptotic stability and modulational stability cannot apply, and connect this to the presence of cosmological horizons in these class (iii) solutions. We can nevertheless show the global existence and future stability for small perturbations of class (iii) solutions under a notion of stability that naturally takes into account the presence of cosmological horizons. The proof is based on the vector field method, but requires additional geometric insight. In particular we introduce two new tools: an inverse-Gauss-map gauge to deal with the problem of cosmological horizon and a quasilinear generalisation of Brendle's Bel-Robinson tensor to obtain natural energy quantities.

Scale-Invariant Models of Gravity and Particle Physics and their Cosmological Implications

Daniel Zenhäusern

Currently, the best theoretical description of fundamental matter and its gravitational interaction is given by the Standard Model (SM) of particle physics and Einstein's theory of General Relativity (GR). These theories contain a number of seemingly unrelated scales. While Newton's gravitational constant and the mass of the Higgs boson are parameters in the classical action, the masses of other elementary particles are due to the electroweak symmetry breaking. Yet other scales, like ΛQCD associated to the strong interaction, only appear after the quantization of the theory. We reevaluate the idea that the fundamental theory of nature may contain no fixed scales and that all observed scales could have a common origin in the spontaneous break-down of exact scale invariance. To this end, we consider a few minimal scale-invariant extensions of GR and the SM, focusing especially on their cosmological phenomenology. In the simplest considered model, scale invariance is achieved through the introduction of a dilaton field. We find that for a large class of potentials, scale invariance is spontaneously broken, leading to induced scales at the classical level. The dilaton is exactly massless and practically decouples from all SM fields. The dynamical break-down of scale invariance automatically provides a mechanism for inflation. Despite exact scale invariance, the theory generally contains a cosmological constant, or, put in other words, flat spacetime need not be a solution. We next replace standard gravity by Unimodular Gravity (UG). This results in the appearance of an arbitrary integration constant in the equations of motion, inducing a run-away potential for the dilaton. As a consequence, the dilaton can play the role of a dynamical dark-energy component. The cosmological phenomenology of the model combining scale invariance and unimodular gravity is studied in detail. We find that the equation of state of the dilaton condensate has to be very close to the one of a cosmological constant. If the spacetime symmetry group of the gravitational action is reduced from the group of all diffeomorphisms (Diff) to the subgroup of transverse diffeomorphisms (TDiff), the metric in general contains a propagating scalar degree of freedom. We show that the replacement of Diff by TDiff makes it possible to construct a scale-invariant theory of gravity and particle physics in which the dilaton appears as a part of the metric. We find the conditions under which such a theory is a viable description of particle physics and in particular reproduces the SM phenomenology. The minimal theory with scale invariance and UG is found to be a particular case of a theory with scale and TDiff invariance. Moreover, cosmological solutions in models based on scale and TDiff invariance turn out to generically be similar to the solutions of the model with UG. In usual quantum field theories, scale invariance is anomalous. This might suggest that results based on classical scale invariance are necessarily spoiled by quantum corrections. We show that this conclusion is not true. Namely, we propose a new renormalization scheme which allows to construct a class of quantum field theories that are scale-invariant to all orders of perturbation theory and where the scale symmetry is spontaneously broken. In this type of theory, all scales, including those related to dimensional transmutation, like ΛQCD, appear as a consequence of the spontaneous break-down of the scale symmetry. The proposed theories are not renormalizable. Nonetheless, they are valid effective theories below a field-dependent cut-off scale. If the scale-invariant renormalization scheme is applied to the presented minimal scale-invariant extensions of GR and the SM, the goal of having a common origin of all scales, spontaneous breaking of scale invariance, is achieved.

EPFL2011

Cosmological Perturbation Theory

Sylvain Bréchet

Firstly, a dynamical analysis of adiabatic perturbations of a perfect fluid is performed to first-order about a general FLRW background using the 1 + 3 covariant and gauge-invariant formalism. The 1 + 3 covariant analog of the Mukhanov-Sasaki and Grishchuk variables needed to quantise the scalar and tensor perturbations respectively about a spatially-curved FLRW background space-time are identied. The dynamics of the vector perturbations is also discussed. Secondly, a dynamical analysis of an inhomogeneous and anisotropic effective Weyssenhoff fluid, which is a perfect fluid with spin in the Einstein-Cartan theory, is performed in a gauge-invariant manner using the 1+3 covariant and gauge-invariant approach. A verication of the dynamical equations is performed for the special case of irrotational flow with zero peculiar acceleration by evolving the constraints. Thirdly, a dynamical analysis of an effective homogeneous and irrotational Weyssenhoff fluid in general relativity is performed using the 1 + 3 covariant and gauge-invariant approach. The spin contributions to the field equations produce a bounce that averts an initial singularity, provided that the spin density exceeds the rate of shear. At later times, when the spin contribution can be neglected, a Weyssenhoff fluid reduces to a standard cosmological fluid in general relativity. Numerical solutions for the time evolution of the generalised scale factor in spatially-curved models are presented, some of which exhibit eternal oscillatory behaviour without any singularities. In spatially- at models, analytical solutions for particular values of the equation-of-state parameter are derived. Although the scale factor of a Weyssenhoff fluid generically has a positive temporal curvature near a bounce, it requires unreasonable fine tuning of the equation-of-state parameter to produce a sufficiently extended period of inflation to fit the current observational data. Fourthly, in order to determine numerically the background dynamics of general in inflationary models, suitable classical initial conditions have to be found from which to start the integration of the equations of motion. The method proposed by Boyanovsky, de Vega and Sanchez assumes a spatially-flat model with an inflaton potential typical of new inflation and determines the initial conditions at a time when the inflaton field is at the local maximum of the potential by assuming equipartition of the kinetic and potential energies of the inflaton field. This leaves the normalisation of the solutions undetermined. The procedure followed by Lasenby and Doran assumes a spatially closed model with a chaotic inflaton potential and determines the initial conditions as the model emerges from the initial singularity by performing a series expansion. We note that quite generically immediately after the initial singularity, the pre-inflationary dynamics of the universe is dominated by the kinetic energy of the inflaton, which naturally yields analytic solutions for any spatial curvature - provided there is no bounce - with no dependence on the potential. This therefore suggests a new generic way of setting initial conditions for inflation. Using this new procedure, we determine the initial conditions for chaotic and new inflation potentials and study both the subsequent background evolution and the spectrum of scalar perturbations produced in the spatially-flat case.

University of Cambridge2009

Scale-Invariant Models of Gravity and Particle Physics and their Cosmological Implications

Daniel Zenhäusern

EPFL2011

Cosmological Perturbation Theory

Sylvain Bréchet

University of Cambridge2009