JORGE NORONHA

Colloquium at IFT/UNESP, June 2016Colloquium at IFT/UNESP, June 2016

Weak x strong coupling non-equilibrium dynamics in an expanding universe

Based on Phys. Rev. Lett. 116 (2016) 2, 022301, arXiv:1507.07834 [hep-ph]

And arXiv:1603.05344 [hep-th]

OUTLINE

I) The ubiquitousness of fluid dynamics

II) Expanding universe as the simplest setup to study thermalization

III) Toy model at weak coupling: Boltzmann equation

IV) Toy model at strong coupling: N=2* gauge theory

III) Conclusions

The ubiquitousness of fluid dynamics

Across vastly different length scales

Cosmological ~ Planetary ~ Human ~

Based on conservations laws + large separation of length scales

macroscopic: microscopic:Separation of scales →

Knudsen numberexpansion:

FLUID

~ ~ 1 m ~ 1 m ~

Macroscopic: Gradient of velocity field

Example of microscopic scale:

In microfluids(blood cells)

Smaller “macro” scales L bring fluid dynamics to the realm ofparticle physics

Fluid dynamics = low energy description of interacting systems

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Standard Model of Particle Physics

Current Status (~4.9% of the Universe)Cosmic Pie Chart

Dark matter ?Dark energy?Quantum gravity?

ESA/Planck

2012

7

Quantum Electrodynamics

+ special relativity+ quantum mechanics

= QED

In many-body systems, new rules “emerge” from simple fundamental laws

Ex: Superconductivity

(“more is different”)

8

What if the microscopic theory is Quantum Chromodynamics (QCD)?

= QCD

gluon self-interactions

2004

Here the microscopic theory resolvesmolecules and their interactions

Non-Abelian gauge theory

9

Confinement of quarks and gluons = fundamental property of standardmodel

How can we study many-body aspects of QCD away from equilibrium?

Non-equilibriumphenomena ???

Lattice QCD

What are the new properties that appear in QCD out-of-equilibrium?

How does confinement (strong coupling) affect many-bodycollective behavior?

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0 ~1 ~15

fm/c ~

Heavy Ion Collisions in a Nutshell

Initial state

Pre-equilibrium

QGP as a relativistic fluid Hadronization Freeze-out

~ 30 fm

Au, Cu, Pb, Unucleitraveling withgamma factorper nucleon

Hot enough to “melt” hadrons

QCD out of equilibrium

T ~ 400 MeV

RHIC, CERN

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Quark-gluon plasma: The smallest fluid ever made

gluon self-interactions

QCD = confinement + asymptotic freedom Quark-Gluon Plasma

?

Ex: Schenke, Jeon, Gale, PRL 2011

QGP perfect fluidity: → emergent property of QCD.

This seems to appear even in elementary proton+proton collisions.

Fluid dynamics at length scales of the size of a proton.

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Q: How does such a fluid dynamic behavior emerge from the quantum fields of the Standard Model?

A: Not known.

Q: How can this system (nearly) thermalize within such shorttimescales?

A: Not known.

Q: How can you prove the QGP's perfect fluidity property from first principles?

A: Not known.

A research area with many known unknowns ...

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The expanding universe provides a much simpler case to study.

More symmetries, slower relative expansion, though thereis only one event to analyze (this talk is multiverse free).

Figure from D. Baumann'slectures

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1907 - 1912

Geodesic equation

Christoffel symbols

“Spacetime is curved and it tells matter how to move”

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1915

But how does matter curve spacetime?

General theory of relativity

Geometry Matter

Differential geometryis now “part” of physics

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Friedmann-Lemaitre-Robertson-Walker spacetime

Spatial isotropy + homogeneity

Isotropic and homogeneous expanding FLRW spacetime

(zero spatial curvature)

Ex: metric

Determined from Einstein's equations

Universe

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Friedmann-Lemaitre-Robertson-Walker spacetime

We consider an isotropic and homogeneous expanding FRW spacetime

Cosmological scale factor(e.g., radiation)

(zero spatial curvature)

Hubble parameter

Distances get stretched

metric

Primordial matter microseconds after the Big Bang

A thermal history of the Universe

Thermalization in an expanding Universe

Pros:

- Applications in cosmology

- Spatial isotropy + homogeneity = strong constraining symmetries

- Expansion of the Universe is “simple”

Cons:

- Inclusion of general relativistic effects: numerics more involved

- Why would the inclusion of something “hard” (GR) help anybodyhere???

If your goal is to understand the thermalization process of rapidlyexpanding systems:

Nothing is easier than studying how a locally static system thermalizes(or not) in an expanding Universe.

I will show you in the following two “toy models” that may be useful to understand thermalization in rapidly expanding systems

First toy model: General relativistic Boltzmann equation

Boltzmann

Molecular chaos General relativity

+

The non-equilibrium dynamics of a gas with

- Ultrarelativistic particles in an expanding universe

time

is identical to that of a non-relativistic gas of Maxwellian molecules

(repulsive interactions)

Expanding box

Fixed box

Relativistic Boltzmann Equation

- Dilute gases display complex non-equilibrium dynamics.

- The Boltzmann equation has been instrumental in physics and mathematics (e.g., 2010 Fields Medal).

Collision termSpace-time variation

Relativistic Boltzmann equation

- It describes how the particle distribution function varies in time and spacedue to the effects of collisions (and external fields).

The relativistic Boltzmann equation has been applied in:

- Cosmology.

- Neutrino transport in supernovae.

- Non-equilibrium processes involving quarks and gluons at sufficiently large temperatures. - Numerical models (BAMPS, MPC, ZPC, AMPT, URQMD).

- Calculation of transport coefficients.

- Determine the regime of validity of relativistic hydrodynamics in rapidly expandingsystems.

And also in heavy ion collisions:

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Boltzmann Equation in FLRW spacetime

We consider an expanding FLRW spacetime(zero spatial curvature)

metric

Simplest toy model:

- Massless particles, classical statistics, constant cross section:

- This already captures most of the physics we want

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We want to find solutions for the distribution function

Given an initial condition: and

This equation includes general relativistic effects + full nonlinear collision dynamics

How does one solve this type of nonlinear integro-differential equation?

Our Boltzmann equation:

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The moments method

- Originally introduced by Grad (1949) and used by Israel and Stewart (1979) in therelativistic regime.

- Used more recently in Phys. Rev. Lett. 116 (2016) 2, 022301

The idea is simple

Instead of solving for the distribution function itself directly, one uses the Boltzmann eq. to find equations of motion for the moments of the distribution function.

Ex: The particle density is a scalar moment

with equation

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Ex: The energy density is a scalar moment

with equation

Clearly, due to the symmetries, only scalar moments can be nonzero.

Thus, if we can find the time dependence of the scalar moments

via solving their exact equations of motion, one should be able to recover

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It is convenient to define the scale time

(constant mean free path)

And the normalized moments which obey the exact set of eqs:

GR effect Simple recursive nonlinearity

Conservation laws require

PRL 2016, arXiv:1507.07834 [hep-ph]

ALL THE NONLINEAR BOLTZMANN DYNAMICS IS ENCODED HERE

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Defining the time variable to account for the expansion of theuniverse, one finds

- Equation is identical to BKW non-relativistic Boltzmann equation for Maxwellian molecules found nearly 40 years ago.

- Underlying symmetry of the non-relativistic Boltzmann (Galilean invariance) is very different than our equation that was derived within GR.

- Different physical systems can be identical from a dynamical systems perspective and they evolve thermalize in a universal manner.

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Full Analytical Solution

Using the moments equations in this form

One can show that

is an analytical solution of the moments equations !

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Full Analytical Solution

1st analytical solution of the Boltzmann equation for an expandingsystem (since 1872)

= fugacity

PRL (2016)arXiv:1507.07834 [hep-ph]

Initial condition

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Full Analytical Solution

Time evolution Momentum dependence

For radiation dominated universe higher order moments will certainly not erase the info about initial conditions.

The approach to equilibrium here depends on the occupancy of each moment.

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Non-equilibrium entropy

One can prove that H-theorem is valid. Entropy production solelyfrom non-hydrodynamic modes.

Even though energy-momentum tensor always the same as in equilibrium.

Expansion is never truly adiabatic.

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This is all very nice but QCD is a non-Abelian gauge theory.

Around the QCD phase transition, QCD is strongly coupled.

Boltzmann description (based on weak coupling) not valid.

How do we study thermalization for T ~ QCD phase transitionin the early universe?

Lattice QCD is useless here (need real time dynamics).

The only thing left to do is to jump into a black hole (brane)

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“Any strongly interacting quantum many-body system at finite density and temperature with sufficiently many d.o.f / volume is predicted to behave at low energies as a perfect fluid”

Holography

Strong coupling limit of QFT in 4 dimensions

String Theory/Classical gravity in d>4 dimensions

Holography (gauge/string duality)

Maldacena 1997; Witten 1998; Gubser, Polyakov, Klebanov 1998

Universality and perfect fluidity

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The holographic correspondence at finite temperatureand density

Near-equilibrium fluctuations in the plasma ~ black brane fluctuations !!!!

Fluid dynamics from black hole physics

“Bulk”

T, μ

4d QFT

Quasiparticle dynamics replaced by geometry

M, Q

Holographic coordinate “r”

(t,x,y,z)

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SU(Nc) Supersymmetric Yang-Mills in d=4

- 16 + 16 supercharges - SU(4) R-symmetry - SO(6) global symmetry

Fields in the adjoint rep. of SU(Nc)

STANDARDEXAMPLE

Maldacena, 1997: This gauge theory is dual to Type IIB string theory on AdS_5 x S_5

Strongly-coupled, large Nc gauge theory

CFT !!!!

Weakly-coupled, low energy string theory

t'Hooft coupling inthe gauge theory

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Universality and perfect fluidity

in QFT → string theory in weakly curved backgrounds

d.o.f. / vol. in QFT → vanishing string coupling

in QFT → spatially isotropic black brane

Kovtun,Son,Starinets, 2005

Universality of black hole horizons

HOLOGRAPHY Universality of transportcoefficient in QFT

Universality of shear viscosity

For anisotropic models there is violationsee arXiv:1406.6019

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Given that heavy ion data shows that T ~ QCD transition the QGP is a perfect fluid …

There must have been perfect fluidity in the early universe.

Phenomenological consequences of that are not yet known.

Given that around those temperatures QCD is not conformal, wewould like to use a nonconformal gravity dual in a FLRW spacetime

This was done by A. Buchel, M. Heller, JN in arXiv:1603.05344 [hep-th]

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Holography2nd toy model: N=2* gauge theory

N = 4 SYM theory +

Pilch, Warner, Buchel, Peet, Polchinski, 2000

Bosonic mass

Fermionic mass

A relevant deformation of SYM: Breaking of SUSY

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Holography2nd toy model: N=2* gauge theory

Pilch, Warner, Buchel, Peet, Polchinski, 2000

Classical gravity dual action:

Scalar potential

- Well defined stringy origin

- Non-conformal strongly interacting plasma:

- Used in tests of holography in non-conformal setting

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HolographyN=2* gauge theory in a FLRW Universe

Characteristic formulation of gravitational dynamics in asymptoticaly AdS5 spacetimes

Assuming spatial isotropy and homogeneity leads to

Chesler,Yaffe, 2013

Encode non-equilibriumdynamics in an expandingUniverse !!!

Buchel, Heller, JN, arXiv:1603.05344 [hep-th]

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HolographyN=2* gauge theory in a FLRW Universe

Conformal limit

Analytical solution for SYM in FLRW spacetime

Temperature Energy density Pressure

Conformal anomaly!!!!

Buchel, Heller, JN, arXiv:1603.05344 [hep-th]

First studied by P. S. Apostolopoulos, G. Siopsis, and N. Tetradis, PRL, (2009)

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HolographyDivergence of the hydrodynamic series

Viscous hydrodynamics → Knudsen series expansion

macroscopic: microscopic:Separation of scales →

Knudsen numbergradient expansion:

FLUID

- Used in kinetic theory (Chappman-Enskog)

- Within the gauge/gravity duality (Minwalla, Hubeny, Rangamani, etc)

Buchel, Heller, JN, arXiv:1603.05344 [hep-th]

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HolographyDivergence of the hydrodynamic series

In our FLRW case, such a Knudsen gradient series gives

equilibrium dissipation

Energy-momentumtensor

In terms of the energy density and pressure out-of-equilibrium

Bulk viscosity

Buchel, Heller, JN, arXiv:1603.05344 [hep-th]

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HolographyDivergence of the hydrodynamic series

Entropy production

For single component cosmologies

Factorial growth!!!

Buchel, Heller, JN, arXiv:1603.05344 [hep-th]

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HolographyDivergence of the hydrodynamic series

1st analytical proof of the divergence of hydro series:

→ Knudsen gradient series has zero radius of convergence !!!

→ There must be a new way to define hydrodynamics beyond the gradient expansion

→ Such a theory may involve the mathematics of resurgence

Buchel, Heller, JN, arXiv:1603.05344 [hep-th]

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Conclusions

- The early Universe may be the simplest “way” to study how Standard Modelquantum fields thermalize.

- The perfect fluidity of the early Universe (microsecs after big bang) is an emergingproperty of the Standard Model out-of-equilibrium.

- Exactly solvable nonlinear kinetic models in a FLRW can be studied (led to the1st analytical solution of the Boltzmann equation for expanding gas).

- Due to strong coupling near the QCD phase transition in the early Universe, non-equilibrium dynamics can only be studied using the gauge/gravity duality.

- Toy model of QCD, N=2* gauge theory, behaves as a perfect fluid but thehydrodynamic expansion has zero radius of convergence.

- New ideas are needed to make further progress in this field.