Non-oscillating power spectra in loop quantum cosmology

Abstract

We characterize in an analytical way the general conditions that a choice of vacuum state for the cosmological perturbations must satisfy to lead to a power spectrum with no scale-dependent oscillations over time. In particular, we pay special attention to the case of cosmological backgrounds governed by effective loop quantum cosmology and in which the Einsteinian branch after the bounce suffers a pre-inflationary period of decelerated expansion. This is the case more often studied in the literature because of the physical interest of the resulting predictions. In this context, we argue that non-oscillating power spectra are optimal to gain observational access to those regimes near the bounce where loop quantum cosmology effects are non-negligible. In addition, we show that non-oscillatory spectra can indeed be consistently obtained when the evolution of the perturbations is ruled by the hyperbolic equations derived in the hybrid loop quantization approach. Moreover, in the ultraviolet regime of short wavelength scales we prove that there exists a unique asymptotic expansion of the power spectrum that displays no scale-dependent oscillations over time. This expansion would pick out the natural Poincaré and Bunch–Davies vacua in Minkowski and de Sitter spacetimes, respectively, and provides an appealing candidate for the choice of a vacuum for the perturbations in loop quantum cosmology based on physical motivations.

Non-Oscillatory Power Spectrum From States of Low Energy in Kinetically Dominated Early Universes

“…We have applied this reasoning to the NO vacuum state, that had previously been shown to be at least of fourth order (Elizaga Navascués et al, 2018b). We find that, in the ultraviolet limit of large wave numbers, the asymptotic expansion that the NO vacuum satisfies (found in (Elizaga Navascués et al, 2020)) agrees exactly with that of the SLE (Banerjee and Niedermaier, 2020) (no matter the test function chosen to define it). As a consequence, the β coefficients of the transformation between the two will be identically zero in the ultraviolet.…”

“…In this context, there is no notion of a unique natural vacuum. Indeed, several proposals have been made of initial vacua within the LQC framework (Agullo et al, 2015;de Blas and Olmedo, 2016;Ashtekar and Gupt, 2017;Elizaga Navascués et al, 2020) that result in primordial power spectra compatible with observations. In these analyses, initial conditions are set at the LQC bounce where the scale factor of the geometry reaches a minimum, and then it starts expanding.…”

“…where ∼ means the behavior in the large k regime. The NO vacuum state has recently been analysed analytically in (Elizaga Navascués et al, 2020). In particular, that work has found that the state admits the UV asymptotic expansion (18) with:…”

“…It turns out that this minimization in time is reflected in a minimization of oscillations in the k domain of the power spectra. This NO prescription has been motivated as well as a good candidate for the vacuum of the perturbations (Elizaga Navascués et al, 2020). One question that so far remained unanswered is whether this NO vacuum is or not of Hadamard type.…”

“…In this work, by comparing with SLEs, we show that indeed this is the case. To do so, we resort to their UV expansions, obtained for SLEs in (Banerjee and Niedermaier, 2020) and for the NO vacuum in (Elizaga Navascués et al, 2020).…”