Probing the Scale Invariance of the Inflationary Power Spectrum in Expanding Quasi-Two-Dimensional Dipolar Condensates

We consider an analogue de Sitter cosmos in an expanding quasi-two-dimensional Bose-Einstein condensate with dominant dipole-dipole interactions between the atoms or molecules in the ultracold gas. It is demonstrated that a hallmark signature of inflationary cosmology, the scale invariance of the power spectrum of inflaton field correlations, experiences strong modifications when, at the initial stage of expansion, the excitation spectrum displays a roton minimum. Dipolar quantum gases thus furnish a viable laboratory tool to experimentally investigate, with well-defined and controllable initial conditions, whether primordial oscillation spectra deviating from Lorentz invariance at trans-Planckian momenta violate standard predictions of inflationary cosmology.

Figure 1

(a) Squared Bogoliubov excitation energy in units of ${\hslash }^4/4m^2{d}_{z,0}^4$, for DDI domination, $R=\sqrt{\pi /2}$. Counting from bottom to top at small $\zeta$, $A$ in (6) is $A_c/10$, $A_{\min }$, $A_c$, and $1.1A_c$. For $A>A_{\min }=1.249$, the spectrum develops a roton minimum and becomes unstable for $A>A_c=3.4454$. (b) Time evolution of the Bogoliubov spectrum in the course of expansion at criticality $A=A_c$. Initially, a roton minimum occurs, disappearing at late times.

Figure 2

(a) Freezing process of the inflaton mode function $k{h}_k$ in units of $\sqrt{\hslash HV/\pi m{c}_0^2}=\sqrt{HV/\pi A{\omega }_{z,0}}$ in terms of the wave number dependent logarithmic conformal time $k\eta$. The blue solid line represents the imaginary part of $k{h}_k$ and the black dashed line the absolute value of $k{h}_k$. (a) Lorentz-invariant relativistic regime. (b)–(d) demonstrate that when the trans-Planckian spectrum is taken into account, solving Eq. (16), freezing still occurs ($A=A_c/10$ and $R=\sqrt{\pi /2}$).

Figure 3

(a) Coordinate space representation of the (real) field $\delta \tilde{\varphi }(\mathbf{x},\tau )$ in units of $\sqrt{\hslash HV/\pi m{c}_0^2{d}_{z,0}^2}$ after the completion of the freezing process, where the 2D volume of the system is $V=(2{\kappa }_0{d}_{z,0}{)}^2$, with initial aspect ratio ${\kappa }_0$, and the wave vector separation is chosen to be $\mathrm{\Delta }k=2\pi /2{\kappa }_0{d}_{z,0}$. The statistical self-similarity reveals itself by the same degree of “wrinkliness” on each scale. (b) The field obtained from numerical implementation of the full Bogoliubov equations ($A=A_c/10$, $R=0$). Plots (c) through (f) are for increasing $A$ and dominating DDI ($R=\sqrt{\pi /2}$).

Figure 4

${\mathrm{\Delta }}^2(k)=k^{2}P(k)$ as a function of in-plane momentum $\zeta$, for 2.5 $e$-folds. The black dashed line represents SIPS. The black solid line corresponds to contact interaction, $R=0$ ($A=A_c/10$). The other lines correspond to DDI domination ($R=\sqrt{\pi /2}$), with values of $A$ as specified in the inset. In the long-wavelength limit, they all converge to SIPS. The slope of the $R=0$ curve decreases for increasing number of $e$-folds, asymptotically yielding SIPS for pure contact interactions.