^{1}

^{1}

^{2}

This article was submitted to Cosmology, a section of the journal Frontiers in Astronomy and Space Sciences

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Recently, States of Low Energy (SLEs) have been proposed as viable vacuum states of primordial perturbations within Loop Quantum Cosmology (LQC). In this work we investigate the effect of the high curvature region of LQC on the definition of SLEs. Shifting the support of the test function that defines them away from this regime results in primordial power spectra of perturbations closer to those of the so-called Non-oscillatory (NO) vacuum, which is another viable choice of initial conditions previously introduced in the LQC context. Furthermore, through a comparison with the Hadamard-like SLEs, we prove that the NO vacuum is of Hadamard type as well.

In a previous work (

In this work, we investigate the effect of shifting the test function away from the high curvature regime. Firstly, this provides a more complete analysis of the SLEs and the ambiguity of the test function. Secondly, this allows us to distinguish in the primordial power spectra the consequences coming directly from LQC corrections and those related to having a period of kinetic dominance prior to inflation, which can also be obtained in a classical scenario. We will show that if the test function ignores the Planckian region, the effect in the resulting SLE is appreciable. Furthermore, in the power spectra the oscillations that were previously found for lower wave numbers are now dampened.

This motivates us to compare our results with those found in the LQC literature that adopts as initial conditions for the perturbations the so-called non-oscillatory (NO) vacuum state (

This manuscript is organized as follows. In

Throughout we adopt Planck units

In this section we will briefly review the dynamics of cosmological perturbations in LQC through its hybrid approach, as well as the definition of SLEs in this context, as exposed in (_{
k
} depends on the quantization. It is common to work with the rescaled fields _{
k
} and _{
k
} respectively, to be.^{(s)} (^{(t)} (

To simplify notation, in the following we will use ^{
I
}. It is easy to find that

Generally, there are no analytical solutions to such equations of motion, and results have to be obtained numerically, given initial conditions _{
k
} (0), _{
k
} is a positive function and _{
k
} any real function. Once defined, the perturbations can be evolved until a time _{end} during inflation when all the scales of interest have crossed the horizon. The primordial power spectra of the comoving curvature perturbation _{end}. The choice of initial conditions amounts to a choice of vacuum state for the perturbations. 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 (

Note that these quantities carry a dependence on the test function

In (

In this section we explore the consequences of not including the bounce in the support of the test function. This way we will be able to study also the effect of the shape of the test function when supported only on the expanding branch away from the high-curvature regime. This will allow us to provide a comparison with an analogous classical scenario of an FLRW model with a period of kinetic dominance prior to inflation.

Let us start by considering the smooth step function ^{2} plotted in

The smooth step function defined in _{
i
} and _{
f
} respectively, and

_{0} = 0, 1, 10 and 100 Planck seconds after the bounce, with _{
f
} fixed at the onset of inflation, and for a sharp step of _{0} = 0 corresponds to the one analysed in (_{0} = 100 we see some convergence. The corresponding figure for tensor modes is omitted since the initial conditions are essentially the same, as discussed in (

Initial conditions in terms of _{
k
} and _{
k
}, as constructed in _{0} = 0 (solid gray line), _{0} = 1 (dashed red line), _{0} = 10 (dotted green line) and _{0} = 100 (dotted-dashed blue line). The scale of ^{2}
^{2}/2, with ^{–6} and with the value of the scalar field at the bounce fixed to _{
B
} = 1.225 (toy value). For tensor modes, the resulting SLE at the bounce shows no significant qualitative differences.

_{0} = 100 will essentially correspond to that obtained by using the SLE as the vacuum state of primordial perturbations in a classical FLRW model with a period of kinetic dominance prior to inflation. However, for smaller _{0}, SLEs show oscillations in

Power spectra of the comoving curvature perturbation _{0} = 0 (solid gray line), _{0} = 1 (dashed red line), _{0} = 10 (dotted green line) and _{0} = 100 (dotted-dashed blue line). The scale of ^{2}
^{2}/2, with ^{–6} and with the value of the scalar field at the bounce fixed to _{
B
} = 1.225 (toy value).

Comparison between the power spectra of the comoving curvature perturbation _{0} = 100 (dotted-dashed blue line) and to the NO vacuum (solid black line). The scale of ^{2}
^{2}/2, with ^{−6} and with the value of the scalar field at the bounce fixed to _{
B
} = 1.225 (toy value).

For completion, we added an appendix where we apply the SLE and NO vacuum prescriptions in a classical Universe dominated by the kinetic energy of the scalar field. We discuss the situations in which they agree with the natural choice for vacuum state considered in (

One remarkable property of the SLEs that is usually not explicitly proven for other vacua proposals is that they are of Hadamard type (^{−l−2}, where ^{−2−n
} at least up to

To simplify the comparison, let us write the UV expansions of both the SLE and the NO state as_{
n
} coefficients are given by the iterative relation_{0} = _{−n
} = 0 for all _{
n
} are determined recursively by_{0} = 1. Remarkably, in (

Therefore, the

SLEs have recently been proposed as a suitable choice for the vacuum state of perturbations in LQC (

We have found that whether the support of the test function includes the high curvature regime or not has a greater influence on the resulting SLE than any other parameter of the test function that has been studied previously. Indeed, in (

Finally, the fact that SLEs are proven to be Hadamard is a great advantage that most proposals don’t enjoy. Typically, this property is difficult to prove explicitly. One strategy, that may be enough for practical purposes, is to compare a state with an adiabatic one of increasing (finite) order, and show, through the

The original contributions presented in the study are included in the article/

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

This work is supported by the Spanish Government through the projects FIS2017-86497-C2-2-P and PID2019-105943GB-I00 (with FEDER contribution). RN acknowledges financial support from Fundação para a Ciência e a Tecnologia (FCT) through the research grant SFRH/BD/143525/2019. JO acknowledges the Operative Program FEDER2014-2020 Junta de Andalucía-Consejería de Economía y Conocimiento under project E-FQM-262-UGR18 by Universidad de Granada.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

The authors would like to thank Beatriz Elizaga Navascués and Guillermo Mena Marugán for useful discussions.

The Supplementary Material for this article can be found online at: