Efimovian Phonon Production for an Analog Coasting Universe in Bose-Einstein Condensates

Abstract

Efimov effects arise from scale invariance, a fundamental symmetry with universal implications. While spatial Efimov physics has been extensively studied, realizing its temporal counterpart remains challenging, as it requires a dynamical system that breaks time-translation symmetry yet preserves the essential time-scaling symmetry. Analog cosmology offers a powerful platform to address this challenge, bridging the domains of Efimov physics and cosmology. Here, we predict a temporal Efimov effect in an analog linearly expanding universe realized with a quasi-two-dimensional Bose-Einstein condensate. The invariance of phonon mode equations under time rescaling leads to particle production with two distinct dynamics: power-law growth and log-periodic oscillations, with the latter being the hallmark signature of the Efimov effect. Furthermore, these dynamics map directly onto sub- and super-horizon cosmological modes. Our predictions can be directly verified through time-averaged measurements of the density-fluctuation spectrum $S_{k}(t)$ in current experiments.

Figures (5)

• Figure 1: Schematic of the scale factor $a(t)$ and the $s$-wave scattering length $a_{s}(t)$. (a): The scale factor exhibits linear growth $a(t)= t/l$ during the interval $t_{\rm i}<t<t_{\rm f}$. (b) The required scattering length follows the form $a_{s}(t)\propto 1/t^{2}$ in this regime. Red and blue curves correspond to fast (small $l$) and slow (large $l$) expansion scenarios, respectively. Note that the initial time is defined as $t_{\rm i}\equiv la_{\rm i}$. Consequently, $t_{\rm i}$ for fast expansion (red) is smaller than that for slow expansion (blue). Similarly, for a fixed target scale factor $a_{\rm f}$, the final time $t_{\rm f}$ occurs earlier for fast expansion. However, the expansion ratio $t_{\rm f}/t_{\rm i}$ remains identical for both cases.
• Figure 2: Post-expansion phonon production of the $(2+1)$ dimensional analog coasting universe for varying $kl$ and $\ln(t_{\rm f}/t_{\rm i})$. The color scale represents $\ln(N_{k}+1)$. According to the behaviors of $N_{k}$, two distinct regimes are identified: In the sub-horizon regime ($kl>1/2$), $N_{k}$ is a log-periodic function of $t_{\rm f}/t_{\rm i}$. In the super-horizon regime($kl<1/2$), $N_{k}$ exhibits symmetric power-law dependence on $t_{\rm f}/t_{\rm i}$. The purple-dashed line at $kl=1/2$ defines the boundary between these two regimes.
• Figure 3: Phonon number densities in the sub- and super-horizon regimes as a function of $\ln( t_{\rm f}/t_{\rm i})$. The black solid and blue dashed curves represent the phonon number density in the super-horizon regime ($n_{\rm super}$) and the sub-horizon regime ($n_{\rm sub}$), respectively. The red vertical line separates the early-time, sub-horizon-dominated phase from the late-time, super-horizon-dominated phase. The black and blue dotted lines indicate the asymptotic behaviors of $n_{\rm super}$ and $n_{\rm sub}$, respectively. The ultraviolet cutoff $\Lambda=200/l$. We adopt $l$ as the unit of length.
• Figure 4: Density-fluctuation spectrum $S_{k}(t)$ and post-expansion phonon number $N_{k}$ in (a) sub-horizon regime and (b) super-horizon regime. The time-dependent spectra $S_{k}(t)$ are represented by colored curves, while the extracted $N_{k}$ values are shown as black curves with corresponding colored markers. In the sub-horizon regime, both $N_{k}$ and the amplitude of $S_{k}(t)$ are oscillatory as a function of $k$, while in the super-horizon regime, they exhibit monotonic growth. In our calculation, we set $\ln(t_{\rm f}/t_{\rm i})=8$ and zero temperature.
• Figure 5: Amplitude of Sakharov oscillations as a function of the scattering length ratio $a_{s,{\rm i}}/a_{s,{\rm f}}$ in the (a) super-horizon and (b, c) sub-horizon regimes. The arrows indicate the specific values of $a_{s,{\rm i}}/a_{s,{\rm f}}$ where the amplitude ${\cal A}_k$ vanishes. These zero-crossing points form a geometric sequence with a common ratio of $\exp(2\pi/\sqrt{(kl)^2-1/4})$. The parameters are set to $kl=0.4$ for (a), $kl=4$ for (b), and $kl=6$ for (c).