Boost-invariant early time dynamics from AdS/CFT

TL;DR

This work uses the AdS/CFT correspondence to study the early-time dynamics of a boost-invariant, strongly coupled conformal plasma, demonstrating that unlike late times there is no universal scaling and that boundary evolution is determined by initial bulk data encoded in the Fefferman-Graham metric. It shows that a bulk metric singularity is inevitable at all times, constraining allowed initial conditions, and establishes a mapping between the early-time energy-density expansion and the initial bulk geometry. To access large proper times, the authors employ a Padé resummation of the early-time series, illustrating a transition toward local equilibrium and the hydrodynamic regime, and they discuss implications for holographic thermalization. The results emphasize how initial bulk data and horizon-like structures influence thermalization in strongly coupled gauge theories and provide a framework to connect early-time dynamics to late-time hydrodynamics via controlled resummation techniques.

Abstract

Boost-invariant dynamics of a strongly-coupled conformal plasma is studied in the regime of early proper-time using the AdS/CFT correspondence. It is shown, in contrast with the late-time expansion, that a scaling solution does not exist. The boundary dynamics in this regime depends on initial conditions encoded in the bulk behavior of a Fefferman-Graham metric coefficient at initial proper-time. The relation between the early-time expansion of the energy density and initial conditions in the bulk of AdS is provided. As a general result it is proven that a singularity of some metric coefficient in Fefferman-Graham frame exists at all times. Requiring that this singularity at tau = 0 is a mere coordinate singularity without the curvature blow-up gives constraints on the possible boundary dynamics. Using a simple Pade resummation for solutions satisfying the regularity constraint, the features of a transition to local equilibrium, and thus to the hydrodynamical late-time regime, have been observed. The impact of this study on the problem of thermalization is discussed.

Figures (5)

• Figure 1: Description of QGP formation in heavy ion collisions. The kinematic landscape is defined by ${\tau = \sqrt{(x^0)^2-(x^3)^2}\ ;\ {y=\frac{1}{2} \log \frac{x^0+x^3}{x^0-x^3}}\ ;\ {x_\bot\!=\!\{x^1,x^2\}}}\ ,$ where the coordinates along the light-cone are $x^0 \pm x^1,$ the transverse ones are $\{x^1,x^2\}$ and $\tau$ is the proper time, $y$ the "space-time rapidity".
• Figure 2: Approximate value of $s$ obtained from the logarithmic derivative and Pade approximation for A) $N_{cut} = 32$ (dotted line), B) $N_{cut} = 40$ (dashed line) and C) $N_{cut} = 48$ (solid line) for initial profile $v_{+}^{\left( 1\right)} \left( z^2 \right)$. Two horizontal lines denote $s = 1$ (free streaming scenario) and $s = 4/3$ (perfect fluid case).
• Figure 3: A) Energy density $\epsilon_{1}\left( \tau \right)$ as a function of proper-time $\tau$ obtained from Pade approximation for cut-off $N_{cut} = 46$ and initial profile $v_{+}^{\left( 1 \right)} \left( z^2 \right)$ in the bulk; B) Energy density $\epsilon_{2}\left( \tau \right)$ for the second profile. C) Energy density $\epsilon_{3}\left( \tau \right)$ as a function of proper-time $\tau$ obtained from Pade approximation for cut-off $N_{cut} = 34$ and initial profile $v_{+}^{\left( 3 \right)} \left( z^2 \right)$ in the bulk.
• Figure 4: Relative difference between the energy densities for the first profile for cut-offs $N_{cut} = 16$ and $N_{cut} = 46$ does not exceed 10%.
• Figure 5: Relative difference in pressures for the first (left) and second (right) profiles -- for $\tau \approx 1$ there is a rapid fall-off but (perhaps, see text) does not reach yet a complete izotropization for the first profile.