FAR-FROM-EQUILIBRIUM ISOTROPISATION, QUASI-NORMAL

MODES AND RADIAL FLOWComparing numerical evolution with

linearisation

Wilke van der Schee

Supervisors: Gleb Arutyunov, Thomas Peitzmann, Koenraad Schalm and Raimond Snellings

Workshop Holographic Thermalization, LeidenOctober 11, 2012

Work with Michał Heller, David Mateos, Michał Spalinski, Diego Trancanelli and Miquel Triana

References: 1202.0981 (PRL 108) and 1210.xxxx

Outline

Simple set-up for anisotropy

Quasi-normal modes and linearised evolution

Radial flow (new results, pictures only)

2/19

Simplest set-up: Pure gravity in AdS5

Background field theory is flat Translational- and SO(2)-invariant field theory

We keep anisotropy: Caveat: energy density is constant so final state is

known

Holographic context3/19

P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma (2008)

The geometry4/19

Symmetry allows metric to be:

A, B, are functions of r and t B measures anisotropy

Einstein’s equations simplify Null coordinates Attractive nature of horizon

Key differences with Chesler, Yaffe (2008) are Flat boundary Initial non-vacuum state

The close-limit approximation5/

19

Early work of BH mergers in flat space

Suggests perturbations of an horizon are always small

Linearise evolution around final state (planar-

AdS-Schw):

Evolution determined by single LDE:

R. H. Price and J. Pullin, Colliding black holes: The Close limit (1994)

Quasi-normal mode expansion6/

19

Expansion:

Solution possible for discrete Imaginary part always positive

G.T. Horowitz and V.E. Hubeny, Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium(1999)J. Friess, S. Gubser, G. Michalogiorgakis, and S. Pufu, Expanding plasmas and quasinormal modes of anti-de Sitter black holes (2006)

7

First results (Full/Linearized/QNM)

Bouncing off the boundary8/19

IR, normal, UV9/

19

Statistics of 2000 profiles10/19

Recent additions11/19

Same linearised calculations with a boost-invariant direction Subtlety: final state is not known initially Add-on: non-homogeneous and includes

hydrodynamics Works well

Second and third order corrections The expansion seems to converge Works quite well

Radial flow12/19

Calculation incorporating longitudinal and radial expansion

Numerical scheme very similar to colliding shock-waves: Assume boost-invariance on collision axis Assume rotational symmetry (central collision) 2+1D nested Einstein equations in AdS

P.M. Chesler and L.G. Yaffe, Holography and colliding gravitational shock waves in asymptotically AdS5 spacetime (2010)

Radial flow – initial conditions

13/19

Two scales: T and size nucleus Energy density is from Glauber model

(~Gaussian) No momentum flow (start at ~ 0.05fm/c) Scale solution such that Metric functions ~ vacuum AdS (not a solution

with energy!)

fm/catMeV 6.0506 T

H. Niemi, G.S. Denicol, P. Huovinen, E. Molnár and D.H. Rischke, Influence of the shear viscosity of the quark-gluon plasma on elliptic flow (2011)

Radial flow – results14/19

Radial flow - acceleration15/19

Velocity increases rapidly:

Acceleration is roughly with R size nucleus Small nucleus reaches maximum quickly

g3110

Radial flow – energy profile16/19

Energy spreads out:

Radial flow - hydrodynamics17/19

Thermalisation is quick, but viscosity contributes

Radial flow - discussion18/19

Radial velocity at thermalisation was basically unknown

Initial condition is slightly ad-hoc, need more physics? We get reasonable pressures Velocity increases consistently in other runs Results are intuitive

Input welcome

Conclusion19/19

Studied (fast!) isotropisation for over 2000 states UV anisotropy can be large, but thermalises fast

(though no bound)

Linearised approximation works unexpectedly well Works even better for realistic and UV profiles

Numerical scheme provides excellent basis Radial flow, fluctuations, elliptic flow What happens universally? What is the initial state?