Phys. Rev. D 107, 065019 (2023) - New formalism to define vacuum states for scalar fields in curved spacetimes

New formalism to define vacuum states for scalar fields in curved spacetimes

Mariana Penna-Lima, Nelson Pinto-Neto, and Sandro D. P. Vitenti

Phys. Rev. D 107, 065019 – Published 30 March 2023

Abstract

The problem of finding a vacuum definition for a single quantum field in curved spacetimes is discussed under a new geometrical perspective. The phase space dynamics of the quantum field modes are mapped to curves in a two-dimensional hyperbolic metric space, in which distances between neighbor points are shown to be proportional to the Bogoliubov coefficients associated with their corresponding mode solutions in phase space. The vacuum state for each mode is then defined as the unique trajectory from which all mapped phase space solutions move within thin annular regions around it. This property implies the stability of the vacuum state: solutions evolved from a point in this trajectory stay close to it as both evolve, and the particle creation is therefore minimized. The new approach is applied to the well-known cases of the time-independent dynamics, where the solutions draw circles around this curve, and to the time-dependent dynamics where adiabatic approximation is valid. Additionally, our analysis is extended to more challenging cases of time-dependent dynamics, where the adiabatic approximation is not possible, such as in the super-Hubble or low frequency regimes. It is shown that stability trajectories can also be found in these situations, and stable quantum vacua can be obtained. This new formalism is applied to two situations: de Sitter space, where the Bunch-Davies vacuum is obtained in a completely different manner through an analysis in the super-Hubble regime, and in the context of cosmological bouncing models, in which the contracting phase is dominated by a cosmological constant in the asymptotic past. A new vacuum state for cosmological perturbations is proposed in this situation.

Received 18 July 2022
Accepted 13 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.065019

Physics Subject Headings (PhySH)

Inflation Quantum fields in curved spacetime

Quantum field theory

Geometry

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Mariana Penna-Lima ^1,*, Nelson Pinto-Neto^2,†, and Sandro D. P. Vitenti ^3,‡

¹Universidade de Brasília, Instituto de Física, Caixa Postal 04455, 70919-970, Brasília, Distrito Federal, Brazil
²COSMO—Centro Brasileiro de Pesquisas Físicas, Xavier Sigaud, 150, Urca, 22290-180, Rio de Janeiro, Brazil
³Departamento de Física, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paraná, Brazil

^*pennalima@unb.br
^†nelsonpn@cbpf.br
^‡Corresponding author. vitenti@uel.br

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 107, Iss. 6 — 15 March 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Numerical solutions for Eq. (45) for the contracting model described in Sec. 5a. All solutions were computed in the frame $^{(1)}$ and transformed to the original frame. The used parameters are $Ω_{w} = 0.3$ , $Ω_{Λ} = 0.7$ , $w = 0.1$ , and $k = 10^{3}$ . Since the solutions are timelike vectors with positive time components [as discussed below Eq. (a42)], the plotted spatial components $x^{1}$ and $x^{2}$ [see Eq. (152)] represent the solution uniquely. In this figure, we show solutions in the original frame. As discussed in Sec. 4b, this frame is dominated by the $1 / m$ term, which can be removed using a canonical transformation generated by $q_{0} L_{2}^{+}$ , leaving a first-order Hamiltonian vector. For this reason, the solutions in this frame for a given initial condition $M_{0}$ are given by Eq. (153), which draw approximate parabolas in the $x_{1} − x_{2}$ plane. We plot here $8 \times 6 + 1$ different solutions. The initial conditions (all set at frame $^{(1)}$ ) are given by points in circles around the origin $G_{0}$ , with six equally log-spaced radius (hyperbolic distance from $G_{0}$ ) in the interval $(10^{- 5}, 4 \times 10^{- 1})$ , and eight equally spaced angles in $(0, 2 π)$ . We also plot the VDC given by $V_{2} (t)$ and the solution $M$ starting at $G_{0}$ and $T = 0$ , i.e., $M^{(1)} (0) = G_{0}$ . In this frame, one cannot notice any discrepancy between them. Furthermore, the divergent term dominates the evolution, and all solutions diverge to infinity at $T = 0$ . Thus we only plot the interval $T \in (10^{- 8}, 10^{- 5})$ . Solutions with the same distance to the VDC are plotted with the same color. We included a zoom at the end of evolution (near $T = 0$ ) to present more clearly the different solutions in this frame.
Reuse & Permissions
Figure 2
The same solutions discussed in Fig. 1 in the original frame are now plotted in frame $^{(1)}$ . The parabolic behavior seen in Fig. 1 has been removed by the canonical transformation. Consequently, we can see the first-order Hamiltonian effect on the solutions. The solutions at radii larger than the solutions’ length are dragged toward the origin by applying a constant rigid motion using a rescale function of the radius $f (r)$ as $(x^{1} (T), x^{2} (T)) \to (x^{1} (T), x^{2} (T)) - f (r) (x^{1} (0), x^{2} (0))$ to improve the readability of the plot. At the frame $^{(1)}$ , the Hamiltonian vector (110) has first-order terms in all components. For this reason, the solutions do not show any clear pattern as the evolution consists of a sequence of small boosts in different directions. Nevertheless, since all solutions keep an almost constant distance from the VDC, we notice that they follow similar trajectories as $V_{2}^{(1)}$ . In this frame, the evolution of the VDC can be read directly from Eq. (144) setting $q = 0$ , yielding $x^{1} \approx 2 r_{1}$ and $x^{2} \approx 2 p_{1}$ . We also included a zoom of the VDC, $M^{(1)}$ , and all nearby solutions. Here we can notice a small discrepancy between the VDC and $M^{(1)}$ at the end of the evolution.
Reuse & Permissions
Figure 3
The same solutions presented in Figs. 1 and 2 are now plotted in frame $^{(2)}$ . Since this frame removes the first-order terms from the Hamiltonian, the solutions (also moved by a rigid translation) present only second-order variations compared to Fig. 2. We obtain the behavior predicted analytically in Eq. (117): solutions with large radii $r$ have their evolution dominated by the first-order rotation describing arcs about $G_{0}$ , which gets affected by second-order terms when the radius becomes sufficiently small. Since we are in a second-order frame, the VDC $V_{2}^{(2)}$ is given by the black dot at the origin, and the solution $M^{(2)}$ only deviates from it by second-order terms. This plot summarizes most of our results about the nonadiabatic VDC: it shows that all points close to the origin provide a reasonable VDC because they stay as close to their starting point as $M^{(2)}$ from $V_{2}^{(2)}$ . That is the stability region discussed in Sec. 4b. As we move away from the origin, solutions start to rotate about $G_{0}$ , and the second-order effects are less and less visible.
Reuse & Permissions
Figure 4
Numerical solutions for Eq. (45) for the contracting model described in Sec. 5b. All solutions were computed at the first adiabatic frame $^{(1)}$ . The used parameters are $Ω_{w} = 1$ , $w = 10^{- 6}$ , and $k = 10^{2}$ . As in Fig. 1, we plotted the components $x_{1}$ and $x_{2}$ representing a solution uniquely. In this figure, we show the solutions in the original frame. As discussed in Sec. 5b, this frame is boosted by $ξ$ . Consequently, the VDC and solutions in this frame move quickly in the $G_{2}$ direction. We plot here four different solutions in addition to the VDC, and the solution $M$ satisfying $M (T_{0}) = V_{4} (T_{0})$ where $T_{0} = 10^{- 10}$ . Initial conditions are given by circles around the VDC with distances from $V_{4}$ for $r = 0.01$ , 0.03, 0.04, and 0.05, and the two angles 0 and $π$ . Solutions with the same distance to $G_{0}$ are plotted with the same color.
Reuse & Permissions
Figure 5
The solutions presented in Fig. 4 are plotted here in the first adiabatic frame. This frame removes the boost generated by $ξ$ . Consequently, the motion in the $G_{2}$ direction is reduced. We predicted this behavior analytically. As we increase the index ( $n$ ) of the adiabatic frame, the VDC stays closer to the origin, and solutions keep rotating around it.
Reuse & Permissions
Figure 6
The same solutions presented in Figs. 4 and 5 are plotted in the second adiabatic frame. The boosts generated by $ξ$ and $ξ_{1}$ are removed. Therefore both movements in the $G_{2}$ and $G_{1}$ directions are reduced. Note that the VDC and $M^{(2)}$ begin to separate only when the adiabatic condition is no longer fulfilled. We can visualize this as the region where the solutions cease to revolve around the stability curve.
Reuse & Permissions

Physical Review D

covering particles, fields, gravitation, and cosmology

New formalism to define vacuum states for scalar fields in curved spacetimes

Mariana Penna-Lima, Nelson Pinto-Neto, and Sandro D. P. Vitenti

Phys. Rev. D 107, 065019 – Published 30 March 2023

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Physical Review D

covering particles, fields, gravitation, and cosmology

New formalism to define vacuum states for scalar fields in curved spacetimes

Mariana Penna-Lima, Nelson Pinto-Neto, and Sandro D. P. Vitenti

Phys. Rev. D 107, 065019 – Published 30 March 2023

Abstract

Physics Subject Headings (PhySH)

Authors & Affiliations

Article Text (Subscription Required)

References (Subscription Required)

Issue

Access Options

Authorization Required

Other Options

Download & Share

Images

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Log In

Search

Article Lookup

Paste a citation or DOI

Enter a citation