We present the tensor computer algebra package xPert for fast construction and manipulation of the equations of metric perturbation theory, around arbitrary backgrounds. It is based on the combination of explicit combinatorial formulas for the n-th order perturbation of curvature tensors and their gauge changes, and the use of highly efficient techniques of index canonicalization, provided by the underlying tensor system xAct, for Mathematica. We give examples of use and show the efficiency of the system with timings plots: it is possible to handle orders n = 4 or n = 5 within seconds, or reach n = 10 with timings below 1 hour.

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We continue our study on corrections from canonical quantum gravity to the power spectra of gauge-invariant inflationary scalar and tensor perturbations. A direct canonical quantization of a perturbed inflationary universe model is implemented, which leads to a Wheeler-DeWitt equation. For this equation, a semiclassical approximation is applied in order to obtain a Schrödinger equation with quantum-gravitational correction terms, from which we calculate the corrections to the power spectra. We go beyond the de Sitter case discussed earlier and analyze our model in the first slowroll approximation, considering terms linear in the slow-roll parameters. We find that the dominant correction term from the de Sitter case, which leads to an enhancement of power on the largest scales, gets modified by terms proportional to the slow-roll parameters. A correction to the tensorto-scalar ratio is also found at second order in the slow-roll parameters. Making use of the available experimental data, the magnitude of these quantum-gravitational corrections is estimated. Finally, the effects for the temperature anisotropies in the cosmic microwave background are qualitatively obtained.

When quantum back-reaction by fluctuations, correlations and higher moments of a state becomes strong, semiclassical quantum mechanics resembles a dynamical system with a high-dimensional phase space. Here, systematic computational methods to derive the dynamical equations including all quantum corrections to high order in the moments are introduced, together with a (deparameterized) quantum cosmological example to illustrate some implications. The results show, for instance, that the Gaussian form of an initial state is maintained only briefly, but that the evolving state settles down to a new characteristic shape afterwards. Remarkably, even in the regime of large high-order moments, we observe a strong convergence within all considered orders that supports the use of this effective approach.

In Ref. [1, 2] a formalism to deal with high-order perturbations of a general spherical background was developed. In this article, we apply it to the particular case of a perfect fluid background. We have expressed the perturbations of the energy-momentum tensor at any order in terms of the perturbed fluid's pressure, density and velocity. In general, these expressions are not linear and have sources depending on lower order perturbations. For the second-order case we make the explicit decomposition of these sources in tensor spherical harmonics. Then, a general procedure is given to evolve the perturbative equations of motions of the perfect fluid for any value of the harmonic label. Finally, with the problem of a spherical collapsing star in mind, we discuss the high-order perturbative matching conditions across a timelike surface, in particular the surface separating the perfect fluid interior from the exterior vacuum.Comment: 21 page