Moving superfluids in the rotating universe

TL;DR

This work analyzes homogeneous cosmologies with two moving shift-symmetric scalar fields (two superfluids) in 2+1 dimensions, focusing on rotation induced by anisotropy and the momentum constraint that ties spatial gradients to conserved charges. By formulating the mini-superspace action and deriving the coupled equations for the rotation angle $\theta$ and shear $\xi$, the authors show that rotation cannot be neglected: purely non-rotating solutions lie on a saddle surface in phase space and rotation qualitatively alters late-time dynamics, yielding an attractor with $\xi'_{\infty}=2/3$ and $H\propto a^{-4/3}$, and an effective equation of state $w=1/3$ in 2+1D. A key result is the existence of a conserved combination $\mathcal{Q}=Q_4-Q_2$ associated with the residual symmetry, and the demonstration that the zero-rotation limit is non-generic and unstable to rotation. The findings underscore rotation as a crucial ingredient in multi-fluid cosmologies with shift-symmetric scalars and motivate extending the analysis to higher dimensions and more general scalar theories.

Abstract

We study homogeneous cosmological models featuring shift-symmetric scalar fields (or, superfluids) in relative motion. In the presence of anisotropy this universe generally features rotation, in the sense that the principal axes of anisotropic expansion rotate with respect to the cosmic comoving frame. We focus in particular on the minimal case of two superfluids in 2+1 dimensions. The momentum constraint enforces their spatial gradients to be collinear and the dynamics tends to align such a direction with that of maximal expansion at late times. As opposed to the recently studied case of solids, rotation plays a more important role in the present two-superfluids model. The associated energy density does not dilute away but scales as that of anisotropy and affects the total equation of state. We find that purely non-rotating solutions correspond to an unstable surface in phase space in the direction of non-vanishing rotation. This suggests that rotation is a crucial feature of these scenarios that cannot be neglected.

Figures (3)

• Figure 1: Left panel: phase map of the non-rotating solutions. The arrows indicate the direction of growing $x$. The lines at $\bar{\xi}'=0$ and $\bar{\xi}'=-4$ are two separatrices and the white non-shaded area in between represents the physical region with $(a\mathcal{H})^2>0$. The trajectories of this region are attracted towards $\bar{\xi}'=0$ in an expanding phase. Right panels: the evolution of the shear for different initial conditions. We have set them at $x=0$ with $\xi'(0)=0.5$ and $\xi(0)$ ranging between $-10$ and $10$. While the late-time attractor always corresponds to $\xi'=2$ ($\bar{\xi}'=0$), the early-time attractor depends on the sign of $\xi(0)$. Negative values of $\xi(0)$ correspond to early-time solutions with constant derivative different from the separatix $\xi'=-2$, while negative initial values of the shear have the separatrix as early-time attractor. This is in agreement with our analytical findings.
• Figure 2: In this plot we show the shear and angle evolutions obtained by numerically solving the equations. We have chosen initial conditions $\xi(x=0)=1$ and $\xi'(x=0)=0.5$ for the shear, while for the rotation we have selected initial velocities ranging from $0$ (lighter) to $1$ (darker) and $\theta(x=0)=0$. The initial angle has been fixed to $\theta_0=0$ which is always possible by means of a symmetry generated by $\mathcal{Q}$ as explained in the main text. The upper panels show the phase map trajectories and the lower panels correspond to the time evolution. We can see how the angle evolves towards zero, but it leaves an imprint in the evolution of the shear, whose asymptotic evolution has $\bar{\xi}'=-4/3$ ($\xi'=2/3$). This leaves a memory of the rotating phase since the pure non-rotating solution (shown in gray) is attracted towards $\bar{\xi}'=0$ ($\xi'=2$).
• Figure 3: Evolution of the Hubble function (left) and the shear (right) for the same set of initial conditions as in Fig. \ref{['Fig:thetaxiev']}. The gray line corresponds to the non-rotating solution and we see how the presence of rotation crucially changes the asymptotic evolution of the shear.