Analog cosmology with two-fluid systems in a strong gradient magnetic field

We propose an experiment combining fluid dynamics and strong magnetic field physics to simulate cosmological scenarios. Our proposed system consists of two immiscible, weakly magnetized fluids moved through a strong gradient magnetic field. The diamagnetic and paramagnetic forces thus generated amount to a time-dependent effective gravity, which allows us to precisely control the propagation speed of interface waves. Perturbations on the interface therefore experience a nonstationary effective metric. In what follows, we demonstrate that our proposed system is capable of simulating a variety of cosmological models. We then present a readily realizable experimental setup which will allow us to capture the essential dynamics of standard inflation, wherein interface perturbations experience a shrinking effective horizon and are shown to transition from oscillatory to frozen and squeezed regimes at horizon crossing.

Figure 1

Schematics of the two-liquid system. Two immiscible liquids, with densities ${\rho }_{1,2}$, magnetic susceptibility ${\chi }_{1,2}$, and height $h_{1,2}$, separated by the equilibrium interface at $z_0$ (dotted line). Gravity interface waves distort the interface layer (solid line) where each point is characterized by its amplitude $\xi$ and velocity $v_x$ ($v_y$). $\partial S$ are the boundaries (here depicted for fluid 2) given by a hard wall (upper), an interface boundary (right), and a moving boundary at the interface of the two liquids (lower).

Figure 2

(a) depicts the effective gravity $g_{\mathrm{eff}}$ in the bore of the magnet for the butanol-aqueous solution (${\ddot{z}}_0=0$). The vertical axis gives the vertical position in the magnet $z$, and the horizontal axis the radial position $r$. The magnitude is given by the colorbar. The solid white lines are contours of the effective gravity, with the dashed line depicting the region over which $g_{\mathrm{eff}}$ changes sign. (b) is the effective gravity along the axis of the solenoid ($r=0$). The inset rescales the horizontal axis to better demonstrate the sign change of $g_{\mathrm{eff}}$.

Figure 3

(a) shows the evolution of the vertical position $z_0\left(t\right)$ (dashed red) and acceleration ${\ddot{z}}_0\left(t\right)$ (solid purple) of the vessel in the magnet corresponding to an exponentially inflating analog universe with Hubble parameter $H=3{\text{s}}^{-1}$. (b) depicts the solution to the wave equation (12). The solid (dashed) line is the real part (absolute value) of the velocity potential ${\varphi }_k$ (purple) and of the height field ${\xi }_k$ (red). The black dash-dotted lines represent different number of e-folds $N=0,1,2$, and 4 for the mode. The shaded region indicates where the mode is outside the Hubble horizon. (c) depicts the maximal two-mode squeezing of the system projected onto the instantaneous eigenbasis at the times (equivalently number of e-folds) indicated. The solid line indicates the theoretical full width at half maximum for each distribution. The intensity plots represent the probability of a given measurement of $X_k$ and $X_{-k}$ after 2000 simulated experimental runs.