Close encounters with attractors of the third kind

TL;DR

This work demonstrates the existence of a hydrodynamic attractor for a conformal MIS fluid in Grozdanov's hyperbolic dS3×R geometry, extending the study of attractors beyond Bjorken and Gubser flows. By solving the MIS equations in this background and introducing the dimensionless attractor variable f=\frac{\dot\varepsilon}{\varepsilon}, it shows rapid convergence of diverse initial conditions to a universal curve by ρ≃1, with a late-time attractor value determined by $9 C_τ f^2+48 C_τ f+64 C_τ−16 C_η=0$ giving $f_\infty=\frac{4(\sqrt{C_η}-2\sqrt{C_τ})}{3\sqrt{C_τ}}$. The analysis reveals a non-monotonic approach to the attractor, a diverging Knudsen number yet a vanishing inverse Reynolds number at late times, indicating hydrodynamization despite large gradients; gradients are shown to yield a factorially divergent expansion, highlighting the need for resummation techniques. The study also compares to Weyl-transformed Bjorken flow and discusses uplifted flat foliation results, outlining potential microscopic realizations in kinetic theory and holography and suggesting avenues for phenomenological applications in high-energy collisions.

Abstract

We report on the existence of a hydrodynamic attractor in the Mueller-Israel-Stewart framework of a fluid living in the novel geometry discovered recently by Grozdanov. This geometry, corresponding to a hyperbolic slicing of dS$_3\times\mathbb{R}$, complements previous analyses of attractors in Bjorken (flat slicing) and Gubser (spherical slicing) flows. The fluid behaves like a sharply localized droplet propagating rapidly along the lightcone, reminiscent of wounded nuclei in the CGC picture. Typical solutions approach the hydrodynamic attractor rapidly at late times despite a Knudsen number exceeding unity, suggesting that the inverse Reynolds number captures hydrodynamization more faithfully since the shear stress vanishes at late times. This is in stark contrast to Gubser flow, which has both the Knudsen and inverse Reynolds number becoming small for intermediate times. We close with a comparison to Weyl-transformed Bjorken flow and discuss possible phenomenological applications.

Figures (4)

• Figure 1: Top: Arbitrary initial conditions approach the same universal curve, for fixed initial energy density, $\varepsilon_0=1,$ with varying initial time variation of the energy density, $\dot\varepsilon_0.$ The late time behavior \ref{['eq:latetime']} is shown as a dashed line. Bottom: Dimensionless shear stress tensor for the same initial conditions.
• Figure 2: Large order behavior of the hydrodynamic series \ref{['eq:expansion']} diverges factorially for all $\rho$.
• Figure 3: Top: Constant proper time slices viewed in the $(\rho, \theta)$ plane. The lines indicate increasing values of $\tau.$ Bottom: Temperature profile of the viscous fluid described in text in Milne coordinates subject to the initial condition $\dot\varepsilon(\rho_0=0.1)=2$, with red-blue corresponding to hot-cold. For both plots, the same initial spacetime point is marked by an orange star.
• Figure 4: Attractor of the flat foliation of dS$_3\times \mathbb{R}$ defined by the metric \ref{['eq:flat-metric']}. Left: the time derivative of the logarithm of the temperature tends to $\frac{2}{3}$ in this space. Right: at the same time, the dissipation drops to zero.