Superluminal modes in a quantum field simulator for cosmology from analog Transplanckian physics

Abstract

The quantum-field-theoretic description for the U(1)-Goldstone boson of a scalar Bose-Einstein condensate with time-dependent contact interactions is developed beyond the acoustic approximation in accordance with Bogoliubov theory. The resulting effective action is mapped to a relativistic quantum field theory on a dispersive (or rainbow) cosmological spacetime which has a superluminal Corley-Jacobson dispersion relation. Time-dependent changes of the s-wave scattering length to quantum-simulate cosmological particle production are accompanied by a time-dependent healing length that can be interpreted as an analog Planck length in the comoving frame. Non-adiabatic transitions acquire a dispersive character, which is thoroughly discussed. The framework is applied to exponentially expanding or power-law contracting $(2+1)$-dimensional spacetimes which are known to produce scale-invariant cosmological power spectra. The sensitivity of these scenarios to the time-dependence of the Bogoliubov dispersion is investigated: We find a violation of scale-invariance via analytically trackable Transplanckian damping effects if the cut-off scale is not well separated from the horizon-crossing scale. In case of the exponential expansion, these damping effects remarkably settle and converge to another scale-invariant plateau in the far ultraviolet regime where non-adiabatic transitions are suppressed by the high dispersion. The developed framework enables quantitative access to more drastic analog cosmological scenarios with improved predictability in the ultraviolet regime that ultimately may lead to the observation of a scale-invariant cosmological power spectrum in the laboratory.

Figures (12)

• Figure 1: Modified dispersion relation in the quantum field simulator (dashed line) compared to the non-dispersive lightlike case (solid line).
• Figure 2: Non-adiabaticity for time-dependent Bogoliubov dispersion (absolute values, upper image) compared to the time-indepedent case (relative values, central image) and the non-dispersive (acoustic) case (relative values, lower image) as a function of conformal time $\eta / \eta_\mathrm{f}$ and wavenumbers $k \xi_\mathrm{f}$ in case of the linearly expanding scenario discussed in \ref{['sec:analogParticleProduction']}. Blue regions in the upper image show non-adiabaticity to be highly localized in time along a single throat-like trajectory. As expected, one has an increasing adiabaticity towards the ultraviolet regime at all times. Dark regions in the central and lower image indicate similarity regarding (non)-adiabaticity whereas blue regions demonstrate a trend towards the Bogoliubov dispersion. To numerically evaluate $\alpha$ including the singular contributions to the effective mass (to be discussed in \ref{['sec:analogParticleProduction']}), we smoothed the delta-distributions via the Lorentzian profile $\delta_\epsilon(x) = (\epsilon/\pi)/(x^2 + \epsilon^2)$ where $\epsilon = 0.1$ models a relative switching sensitivity of ten percent.
• Figure 3: Upper image: Power spectrum $\mathcal{P}_\psi/H$ after an exponential expansion by $N = 10$ e-foldings compared to the initial vacuum spectrum. A constant (scale-invariant) value $\mathcal{P}_\psi / H = 1/2\pi^2$ occurs for modes $1 \ll k \eta_0 \ll \mathrm{e}^N$, where the boundaries become more accurate as $N$ increases. In the limit $\eta_0 \to \infty$, the left boundary is shifted to arbitrary large scales, resulting in the well-known Bunch-Davies spectrum which would furthermore extend to arbitrary small scales in the limit $\eta_\mathrm{f} \to \eta_0$ (or $N \to \infty$). Lower image: Time-dependence of the power spectrum as parametrized by the number of e-foldings $N$. The scale-invariant plateau grows with time towards smaller comoving scales.
• Figure 4: Sensitivity of scale-invariant power spectra produced by an exponential expansion to the dispersive effects described in cases (i) - (iii) using the reference values \ref{['eq:TypicalParamsScalInv']} that can be combined into the initial scale-separation parameter $\sigma_\mathrm{i} = 30.68$ (more details in the main text and in context of \ref{['fig:ScalSep']}). The time-independent dispersion preserves the scale-invariant value $\mathcal{P}_\psi/H = 1/2\pi^2$ whereas the time-dependent case violates it on moderate ultraviolet scales due to Transplanckian damping but converges to another plateau in the far ultraviolet given by $\mathcal{P}_\psi / H = \sigma_\mathrm{i} \mathrm{e}^{-2N} / 2 \pi^2$.
• Figure 5: Sensitivity of scale-invariant power spectra produced by a power-law contraction to the dispersive effects described in cases (i) - (iii) using the reference values \ref{['eq:TypicalParamsScalInvCont']}. The inset shows the time-indepedent Bogoliubov case (ii) where the spectrum is orders of magnitude lower and exhibits an oscillatory pattern.