Quantum cosmological background superposition and perturbation predictions

TL;DR

The paper develops a quantum cosmology framework in which the background evolution is described by quantum trajectories arising from a superposition of semiclassical states, implemented via affine quantization and an eikonal approach. Tensor perturbations are evolved on this quantum background using a Born–Oppenheimer-like separation, revealing that while the background becomes asymptotically classical, the resulting gravitational-wave power spectrum can differ substantially from single-state predictions, especially for multi-bounce biverse trajectories. The main finding is that background quantum structure can generate observable signatures in the primordial tensor spectrum, including amplitude shifts and oscillatory features depending on the state parameters, with potential implications for early-un-Universe physics and gravitational-wave cosmology. The work highlights the distinct predictions of trajectory-based quantum cosmology versus standard projection-based methods and motivates extension to scalar modes and direct links to observational data.

Abstract

Predictions from early universe cosmology typically concern primordial perturbations generated during epochs where effects arising from the quantum nature of gravity may be important; quantum vacuum fluctuations being stretched to cosmological scales during a phase of inflation. Quantizing the background is then done by assuming a single close-to-classical state over which perturbations grow, as well as a Born-Oppenheimer factorization throughout the relevant phase. We present a scenario in which although the latter factorization remains valid at all times, we allow the background state to be very non-classical by defining quantum trajectories through an eikonal approximation. We find that these trajectories asymptotically reproduce an almost classical behavior for the background, but the predictions for the power spectrum of perturbations can significantly differ.

Figures (11)

• Figure 1: Time development of a trajectory for the two-component state \ref{['PhiB2']} with parameters $r=2$, $\Delta\tau=50$, same amplitude contribution $\rho=1$, and no phase $\delta=0$. The reference semi-classical trajectory $q_0(\tau)$ appears as the dashed line, while $q_1(\tau)$ is the dotted line; the full line shows the actual trajectory $x (\tau)$, chosen with initial condition $x (\tau_\text{i}) = q_0(\tau_\text{i})$. We set initial conditions at $\tau_\text{i} = - 300$, where the trajectory evolution is close to the single state case, but already feels some effects of the other wave function. The insert shows a zoom on the time interval separating the two semi-classical bounces. Although not obvious on the figure because of the scale chosen to emphasize the long time behavior, all trajectories are regular with a minimum value given by $q_0(\tau_{\textsc{b},0}) = q_1(\tau_{\textsc{b},1}) = \xi_1 = 9\pi/16 \simeq 1.77$. The trajectory $x (\tau)$ follows the bounces of the two semiclassical trajectories and exhibits the typical oscillatory behavior characteristic of quantum trajectories and emphasized in Fig. \ref{['p_r2tau50rho1delta0']}.
• Figure 2: Time development of the momentum for the two-component state \ref{['PhiB2']} with the same parameters as in Fig. \ref{['x_r2tau50rho1delta0']} and the same conventions. The oscillations along the trajectory translate into large but finite peaks in the momentum (they appear sharp in the figure for reasons pertaining to image resolution). The insert zooms on the central region showing again oscillating but smooth transitions.
• Figure 3: Phase space evolution of the trajectory for the two-component state \ref{['PhiB2']} with the same parameters as in Figs. \ref{['x_r2tau50rho1delta0']} and \ref{['p_r2tau50rho1delta0']}, plotted with the same conventions. The quantum trajectory starts from the reference phase trajectory $(q_0,p_0)$, bounces once and subsequently connects to the second trajectory $(q_1,p_1)$ to bounce a second time before finally asymptotically getting back to the $(q_0,p_0$) trajectory.
• Figure 4: Phase space evolution of multiple trajectories for the two-component state \ref{['PhiB2']} with a tiny energy difference $r=1.2$, small bouncing time delay $\Delta\tau=2$, small but not entirely negligible relative contribution $\rho=0.2$ of the second wave function, and different phases $\delta \in \qty{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}}$. This diagram shows that the relative phase $\delta$ impacts the quantum trajectory. All trajectories return to their initial $|p|$ value at later times, which are outside the scope of this plot ($q \sim 40$). We set initial conditions at $\tau_{\text{i}} = -30$.
• Figure 5: Effective potential \ref{['Veff']} needed to solve the mode equation \ref{['EqMode']} obtained from the trajectory of Fig. \ref{['x_r2tau50rho1delta0']} and the same underlying parameters. The individual effective potentials, calculated through $q_0"/q_0$ and $q_1"/q_1$, are shown as dashed and dotted lines, respectively. Additionally, the bounce times are respectively marked at locations $\tau=0$ and $\tau=50$.