Abstract. Holonomy corrections to scalar perturbations are investigated in the loop quantum cosmology framework. Due to the effective approach, modifications of the algebra of constraints generically lead to anomalies. In order to remove those anomalies, counter-terms are introduced. We find a way to explicitly fulfill the conditions for anomaly freedom and we give explicit expressions for the counter-terms. Surprisingly, theμ-scheme naturally arises in this procedure. The gauge invariant variables are found and equations of motion for the anomaly-free scalar perturbations are derived. Finally, some cosmological consequences are discussed qualitatively.

No abstract

Abstract. Quantum gravity is sometimes considered as a kind of metaphysical speculation. In this review, we show that, although still extremely difficult to reach, observational signatures can in fact be expected. The early universe is an invaluable laboratory to probe "Planck scale physics". Focusing on Loop Quantum Gravity as one of the best candidate for a non-perturbative and background-independant quantization of gravity, we detail some expected features.Invited topical review for Classical and Quantum Gravity.

In the recent paper by Mielczarek \emph{et al.} (JCAP {\bf 1007} (2010) 004) an idea of the method which can be used to put some constraint for the reheating phase was proposed. Another method of constraining the reheating temperature has been recently studied by Martin and Ringeval (Phys.\ Rev.\ D {\bf 82} (2010) 023511). Both methods are based on observations of the cosmic microwave background (CMB) radiation. In this paper, we develop the idea introduced in this first article to put constraint on the reheating after the slow-roll inflation. We restrict our considerations to the case of a massive inflaton field. The method can be, however, easily extended to the different inflationary scenarios. As a main result, we derive an expression on the reheating temperature $T_{\text{RH}}$. Surprisingly, the obtained equation is independent on the unknown number of relativistic degrees of freedom $g_*$ produced during the reheating. Based on this equation and the WMAP 7 observations, we find $T_{\text{RH}}=3.5\cdot 10^6$ GeV, which is consistent with the current constraints. The relative uncertainty of the result is, however, very high and equal to $\sigma(T_{\text{RH}})/T_{\text{RH}} \approx 53$. As we show, this uncertainty will be significantly reduced with future CMB experiments.Comment: 6 pages, 3 figures. Matches version published in Phys. Rev.

Loop quantum cosmology provides an efficient framework to study the evolution of the Universe beyond the classical Big Bang paradigm. Because of holonomy corrections, the singularity is replaced by a "bounce." The dynamics of the background is investigated into the details, as a function of the parameters of the model. In particular, the conditions required for inflation to occur are carefully considered and are shown to be generically met. The propagation of gravitational waves is then investigated in this framework. By both numerical and analytical approaches, the primordial tensor power spectrum is computed for a wide range of parameters. Several interesting features could be observationally probed.

The Wick rotation is commonly considered only as an useful computational trick. However, as it was suggested by Hartle and Hawking already in early eighties, Wick rotation may gain physical meaning at the Planck epoch. While such possibility is conceptually interesting, leading to no-boundary proposal, mechanism behind the signature change remains mysterious. We show that the signature change anticipated by Hartle and Hawking naturally appear in loop quantum cosmology. Theory of cosmological perturbations with the effects of quantum holonomies is discussed. It was shown by Cailleteau et al. (Class. Quant. Grav. 29 (2012) 095010) that this theory can be uniquely formulated in the anomalyfree manner. The obtained algebra of effective constraints turns out to be modified such that the metric signature is changing from Lorentzian in low curvature regime to Euclidean in high curvature regime. Implications of this phenomenon on propagation of cosmological perturbations are discussed and corrections to inflationary power spectra of scalar and tensor perturbations are derived. Possible relations with other approaches to quantum gravity are outlined. We also propose an intuitive explanation of the observed signature change using analogy with spontaneous symmetry breaking in "wired" metamaterials.

Signature change at high density has been obtained as a possible consequence of deformed space-time structures in models of loop quantum gravity. This article provides a conceptual discussion of implications for cosmological scenarios, based on an application of mathematical results for mixed-type partial differential equations (the Tricomi problem). While the effective equations from which signature change has been derived are shown to be locally regular and therefore reliable, the underlying theory of loop quantum gravity may face several global problems in its semiclassical solutions.

We investigate vector perturbations with holonomy corrections in the framework of loop quantum cosmology. Conditions to achieve anomaly freedom for these perturbations are found at all orders. This requires the introduction of counterterms in the hamiltonian constraint. We also show that anomaly freedom requires the diffeomorphism constraint to hold its classical form when scalar matter is added although the issue of a vector matter source, required for full consistency, remains to be investigated.The gauge-invariant variable and the corresponding equation of motion are derived. The propagation of vector modes through the bounce is finally discussed.