Silent initial conditions for cosmological perturbations with a change of space-time signature [Cross-Listing]
Recent calculations in loop quantum cosmology suggest that a transition from a Lorentzian to an Euclidean space-time might take place in the very early Universe. The transition point leads to a state of silence, characterized by a vanishing speed of light. This behavior can be interpreted as a decoupling of different space points, similar to the one characterizing the BKL phase. In this study, we address the issue of imposing initial conditions for the cosmological perturbations at the transition point between the Lorentzian and Euclidean phases. Motivated by the decoupling of space points, initial conditions characterized by a lack of correlations are investigated. We show that the "white noise" initial conditions are supported by the analysis of the vacuum state in the Euclidean regime adjacent to the state of silence. Furthermore, the possibility of imposing the silent initial conditions at the trans-Planckian surface, characterized by a vanishing speed for the propagation of modes with wavelengths of the order of the Planck length, is studied. Such initial conditions might result from a loop-deformations of the Poincar\’e algebra. The conversion of the silent initial power spectrum to a scale-invariant one is also examined.