Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Time as a test-field: the no-boundary universe in motion and a smooth radiation bounce

Federico Piazza, Siméon Vareilles

TL;DR

This work promotes the proper time of an observer as a test field in minisuperspace quantum cosmology, showing that the time‑dependent wavefunction $\Psi(q^a,t)$ obeys a Schrödinger equation $i\partial_t\Psi=\mathcal{H}\Psi$ with backreaction negligible. It provides two foundations for this view: a path‑integral derivation of a time‑dependent propagator and a clock‑Hamiltonian model where the clock yields $p_t=-i\hbar\partial_t$, together with a probabilistic interpretation via a conserved Schrödinger current. In the semiclassical regime, the framework reproduces classical trajectories and, for a scalar field, the conservation of the curvature perturbation $\zeta$, while the no‑boundary case shows a quantum bounce in global de Sitter space. Remarkably, in a radiation‑dominated minisuperspace the quantum potential acts as an attractive central potential $V(x)\sim -x^{-2/3}$ that stabilizes the bounce, with the smoothness of the bounce controllable by the initial wavepacket spread. The results offer a novel quantum cosmology picture in which a time‑of‑observation clock can give rise to well‑posed dynamics, potentially connecting to early‑universe scenarios with minimal assumptions about UV completion.

Abstract

The proper time of an observer can be introduced as a degree of freedom in quantum cosmology, additional to the existing fields. We review two arguments for using the Schrödinger equation to evolve the corresponding wavefunction. We restrict to solutions in which time acts as a component with negligible backreaction on the metric -- that is, it plays the role of a test field. We apply this idea to various minisuperspace models. In the semiclassical regime we recover expected results: the wavefunction peaks on the classical solution and, in models with a scalar field, the variance of $ζ$ (a mini-superspace analogue of the comoving curvature perturbation) is conserved. Applied to the no-boundary wavefunction, our model recovers the bouncing behavior of classical global de Sitter space, with small corrections associated to the evolving variance of the wavefunction. Other bouncing solutions do not have any classical analogue. This is the case of a radiation dominated universe, which classically leads to a big-bang singularity but corresponds quantum mechanically to an $s$-wave scattering off a central potential of the form $-r^{-2/3}$. As much as the hydrogen atom, this potential is famously made stable by the Heisenberg uncertainty principle. We study the unitary evolution of the wavepacket numerically. During the bounce, the uncertainty and the expectation value of the scale factor become comparable. By selecting a large initial variance, the bounce can be made arbitrarily smooth, the mean value of the Hubble parameter correspondingly soft.

Time as a test-field: the no-boundary universe in motion and a smooth radiation bounce

TL;DR

This work promotes the proper time of an observer as a test field in minisuperspace quantum cosmology, showing that the time‑dependent wavefunction obeys a Schrödinger equation with backreaction negligible. It provides two foundations for this view: a path‑integral derivation of a time‑dependent propagator and a clock‑Hamiltonian model where the clock yields , together with a probabilistic interpretation via a conserved Schrödinger current. In the semiclassical regime, the framework reproduces classical trajectories and, for a scalar field, the conservation of the curvature perturbation , while the no‑boundary case shows a quantum bounce in global de Sitter space. Remarkably, in a radiation‑dominated minisuperspace the quantum potential acts as an attractive central potential that stabilizes the bounce, with the smoothness of the bounce controllable by the initial wavepacket spread. The results offer a novel quantum cosmology picture in which a time‑of‑observation clock can give rise to well‑posed dynamics, potentially connecting to early‑universe scenarios with minimal assumptions about UV completion.

Abstract

The proper time of an observer can be introduced as a degree of freedom in quantum cosmology, additional to the existing fields. We review two arguments for using the Schrödinger equation to evolve the corresponding wavefunction. We restrict to solutions in which time acts as a component with negligible backreaction on the metric -- that is, it plays the role of a test field. We apply this idea to various minisuperspace models. In the semiclassical regime we recover expected results: the wavefunction peaks on the classical solution and, in models with a scalar field, the variance of (a mini-superspace analogue of the comoving curvature perturbation) is conserved. Applied to the no-boundary wavefunction, our model recovers the bouncing behavior of classical global de Sitter space, with small corrections associated to the evolving variance of the wavefunction. Other bouncing solutions do not have any classical analogue. This is the case of a radiation dominated universe, which classically leads to a big-bang singularity but corresponds quantum mechanically to an -wave scattering off a central potential of the form . As much as the hydrogen atom, this potential is famously made stable by the Heisenberg uncertainty principle. We study the unitary evolution of the wavepacket numerically. During the bounce, the uncertainty and the expectation value of the scale factor become comparable. By selecting a large initial variance, the bounce can be made arbitrarily smooth, the mean value of the Hubble parameter correspondingly soft.
Paper Structure (16 sections, 45 equations, 3 figures, 3 tables)

This paper contains 16 sections, 45 equations, 3 figures, 3 tables.

Table of Contents

  1. The Schrödinger equation in quantum cosmology
  2. The path integral point of view
  3. The clock's Hamiltonian point of view
  4. Energy boundedness and probability density
  5. Time as a test field
  6. Comparing $\Psi$ and $\Psi_{WdW}$
  7. The semi-classical regime
  8. Perfect fluid
  9. Scalar field and the conservation of $\zeta$
  10. The no-boundary universe in motion
  11. On quantum de Sitter (dS) space
  12. The radiation bounce
  13. Radiation domination in flat field-space coordinates
  14. Features of the bounce
  15. UV sensitivity
  16. ...and 1 more sections

Figures (3)

  • Figure 1: The path integral on the RHS of \ref{['intro_G']} extends over all four-geometries that interpolate between the two spatial metrics $h$ and $h'$ (in blue) in a given time lapse$\Delta t (\vec{x})$. The latter can be defined as the proper time at which the geodesic observers that leave $h$ in the orthogonal direction (vertical lines) reach $h'$.
  • Figure 2: Top-left panel: representation of the no boundary proposal "in real time". The state \ref{['nbpacket']} (in orange) is numerically evolved backward in time. After bouncing off the potential barrier (in black) it re-expands (dashed, green curve). Dirichelet boundary conditions are imposed at the origin. The behavior of $\langle a \rangle$ is shown in the bottom left panel (for different values of $\alpha$) and in the top right one (for different values of the initial variance). The classical behavior of $a$ in de Sitter space is given by the thick black curve. The bottom right panel follows instead the evolution in time of the variance of the wavepacket.
  • Figure 3: Top-left panel: pictorial representation of the radiation bounce. The behavior of $\langle a \rangle$ is shown in the top right panel for different initial values of $\Delta a$. The evolution of the spread of these solutions is shown in the bottom-left. Bottom-right: the behavior of $\langle H \rangle$ can be made arbitrarily smooth by choosing a large initial variance. The red curve has been chosen to represent a "more singular" solution than in the other panels for pictorial clarity.