Cosmological perturbations including matter loops. A study in de Sitter

Zusammenfassung

In this thesis we studied quantum effects in de Sitter space coming from the interaction of gravitons and matter. We derived the lowenergy effective action for metric perturbations which includes corrections from matter loops, in a flat background for scalar fields of arbitrary mass and minimal curvature coupling, and in a de Sitter background for massless scalar fields of minimal and conformal coupling. From this action we derived the semiclassical Einstein equations, which in this case give a small correction to the relation between the cosmological constant and the Hubble constant. To study the stability of the de Sitter background, we derived the equations satisfied by general linear metric perturbations, using the order reduction method which in contrast to a strictly perturbative treatment produces solutions that are reliable for extended periods of time. We solved these equations for an initial vacuum state and for general initial states. In both cases, the induced changes in the Riemann tensor, which is a gauge invariant and local observable, are small and vanish in the infinite future. In this way, we extended the classical nohair theorems of de Sitter space to the quantum case. We calculated also the two-point function of these perturbations, using a generalization of the flat-space i? prescription that permitted us to define an interacting vacuum state ? in the infinite past. From this correlation function, we obtained a cosmological observable, the power spectrum of tensorial perturbations. The size of the quantum corrections is too small to be measured, being appreciable only if the Hubble constant were of a magnitude comparable to the Planck scale, where the effective theory ceases to be valid. As a local observable, we calculated the two-point function of the Riemann tensor. The result was decomposed into correlation functions of the Weyl and Ricci tensors and the Ricci scalar, which are de Sitter invariant, showing that there is no physical breaking of this invariance. The twopoint function decay exponentially for large separations, showing that the background curvature acts as an effective mass for the graviton. To generalize these calculations to the interaction with other kinds of matter, we exploited the Bianchi identities to show that the two-point function of the Riemann tensor always is de Sitter invariance if this is the case for the stress-energy tensor. We gave explicit formulas to calculate it, and saw that the result is completely determined except for an integration constant. This constant can naturally be interpreted as the strength of free gravitons which propagate in the background spacetime, and depends on the unknown coefficients that multiply terms in the gravitational action which involve squares of curvature tensors.