^{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.

We study the fluctuations and the correlations between spatial regions generated in the primordial quantum gravitational era of the universe. We point out that these can be computed using the Lorentzian dynamics defined by the Loop Quantum Gravity amplitudes. We evaluate these amplitudes numerically in the deep quantum regime. Surprisingly, we find large fluctuations and strong correlations, although not maximal. This suggests the possibility that early quantum gravity effects might be sufficient to account for structure formation and solve the cosmological horizon problem.

Standard cosmology–with or without inflation–requires an initial state that exhibits fluctuations and correlations between distinct regions of space. These play a key role, in particular as seeds for structure formation. Here we investigate how these fluctuations and correlations can emerge from a primordial quantum gravitational cosmological phase, using Loop Quantum Gravity (LQG) and a simple model of the early universe.

We consider the quantum transition from an empty state to a 3-geometry. The amplitude of this transition may be relevant in a Big Bang cosmology (

We truncate the degrees of freedom of the gravitational field to a small finite number in addition to the scale factor [cfr (

All this indicates that the universe emerging from an early quantum era includes fluctuations, homogeneity properties, and large scale correlations, due to the common quantum origin of spatially separated regions. These can be studied theoretically and appear to be compatible with the observed universe. In particular, inflation or a bounce might not be strictly necessary to circumvent the horizon problem. If the initial quantum phase is taken into account, our result suggests that distant regions may have not been causally disconnected in the past, as in classical cosmology.

We discretize a closed cosmological 3-geometry into five tetrahedra glued to one another, giving an _{3} topology. This is a regular triangulation of a topological 3-sphere, and it corresponds to the boundary of a 4-simplex. The geometry of a flat 4-simplex has twenty degrees of freedom, which capture the gravitational field in this truncation. The result of the transition from nothing to a 3-geometry is described by its covariant LQG Lorentzian amplitude (

The LQG Hilbert space for this truncation is _{
l
}’s are spins (half-integer values labeling _{
n
}’s are a basis in the corresponding intertwiner space _{
j
} is the spin-

The transition amplitude from an empty state to a state

The spin-network basis states can be viewed as a collection of quantum tetrahedra (

Say we use the recoupling basis that pairs links

The Lorentzian EPRL vertex amplitude

Analytical results show that in the large spin limit this amplitude is generally exponentially suppressed except in two cases (

The form of the amplitude

The sum over spins

All the computations of the present work were carried out using the ^{1}
^{2}

The results below are given for increasing values of the scale parameter

1.The expectation value of the angle operator

1.The spread

2.The correlations between angle operators on different nodes depend on the pairing. We write

3.To quantify the degree of correlation between operators we computed the entanglement entropy between different tetrahedra, viewed as quantum subsystems. A result by Page (

The computed average external dihedral angle of boundary tetrahedra as function of the scale factor. The gray line shows the dihedral angle of a regular tetrahedron.

Quantum spread of the cosine of the external dihedral angle of boundary tetrahedra as function of the scale factor.

Left: correlations of angle operator

We studied the degree of non-typicality of the primordial state

The entanglement entropy of a boundary node with respect to the rest of the graph. Gray continuous line shows the maximum entropy attainable as function of the scale factor parameter. Gray diamonds show the result of

Summarizing, the quantum state for the primordial universe predicted by the dynamics of Loop Quantum Gravity can be computed in a kinematical truncation and at first order in the vertex expansion. It describes the fluctuating metric of a topologically closed universe in its early quantum regime. Its degrees of freedom encode the shapes of neighboring spatial regions. Their size (area), taken to be equal, is related to the scale factor. We have found that the mean geometry of this state is that of a (truncated) 3-sphere, as we expected by symmetry, but the fluctuations are large. Neighboring regions are correlated and correlations do not vanish as the scale factor increases. This opens the possibility that an inflationary phase may not be needed in order to circumvent the horizon problem, as the primordial quantum phase may introduce stochastic correlations in otherwise causally-independent spatial regions. We also computed the entanglement entropy of a single region viewed as a quantum subsystem of the whole universe. We found that the cosmological state is highly non-typical, showing an entanglement entropy that is apparently reaching an asymptotic value as the scale factor increases. Our work is one of the first explorations of the purely quantum regime of LQG—without resorting to the high-spin semiclassical limit of the theory—and one of the first applications to a concrete physical model of the numerical tools that are recently being developed for covariant Loop Quantum Gravity (

We thank for discussions Pietro Doná, Carlo Rovelli, Giorgio Sarno and Simone Speziale. We thank the Department of Theoretical Physics at UPV/EHU where part of this research was carried, supported by the grant IT956-16 of the Basque Government and by the grant FIS2017-85076-P (MINECO/AEI/FEDER, UE). We acknowledge the Anishinaabek, Haudenosaunee, Lūnaapéewak, and Attawandaron peoples, on whose traditional lands Western University is located.

The raw data supporting the conclusion of this article will be made available upon request by the authors, without undue reservation.

FG contributed to this work writing an original code, performing all the numerical computations, and contributing to the analytical aspects of the computation. FV contributed with the original idea of this paper, devising the computations, supervising their completions, and analysing the results.

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.

Gozzini (2021). High performance lorentzian spin foam numerics. In preparation.

Gozzini (2021). Numerical simulation of the quantum cosmological vacuum with many spin foam vertices. In preparation.