Evolution of TopologicalEvolution of TopologicalDefectsDefects in Field Theory in Field Theory

Jon PearsonJon PearsonJodrell Jodrell Bank Centre for AstrophysicsBank Centre for Astrophysics

www.www.jpofflinejpoffline.com.com

Jon PearsonJon Pearson

Introductory PhD ClassesIntroductory PhD ClassesNov 2009Nov 2009

Outline

• Fields & potentials– Lagrangian field theory

• Phase transitions• A toy model• Topological defects• Kinky vortons• The cosmological motivation (dark energy)

Fields

• (2nd year stuff)• At every point in spacetime there is a

value of a “field”– Temperature (scalar)– Velocity of water in stream (vector)

42

159

x

y

!

"(x,y,z,t)

(work in 2Dfor visualsimplicity)

Potentials• Field “lives on” a potential

– Particle in gravitational field• Drop ball & position “evolves” to minimise its

energy

• Potential like parabola gives SHM• Every point in space has field &

location on potential

Parabolic Potential

• Single minimum at origin• Field will evolve to that minimum

• A pendulum will eventually come to rest atthe minimum, after oscillating about thepotential– A pendulum is not a field

• A field would be a set of pendula(??) at every point inspace

• This makes for a very boring system & wasstudied to death at undergrad level!

x

y

Field points don’t have to occupythe same point on the potential throughout space

CRUCIALPOINT:

Evolving to the minimum

• Different locations start off with the fieldat different locations in the potential

• Hence, different locations have the fieldbeing in the minimum at different times

• Eventually, the field everywhere will bein the minimum

Now to complicate thepotential

• Rather than a single minimum, let’shave two minima

When the field evolves,which minima does it choose?

• If the minima of the potential aredegenerate (i.e. more than just a singlepoint) defects form– Jargon: vacuum manifold topologically non-

trivial• Field over space evolves to occupy

different minima– Makes “clumps” of field in same minima

Topological Defects(called the vacuum manifold)

A bit of Lagrangian field theoryThis is the “action”(like the energy)Things follow paths φ(x) whichminimise this number SThis is the “Lagrangian” & hasall the information about thesystem at hand

This is the equation of motionthat gives the minimum valueof S

Example

Klein-Gordon equation:

Massive

Evolution

• The field “knows how to move” via equationsof motion (Euler-Lagrange)

• Lagrangian field theory• Minimises energy “automatically”

• Imagine a balloon: squish it & it will wobble back toits un-squished state: minimise its surface area!

• Similar to bubbles, or how a rope hangs under gravity– Optimum shape given forces involved

Phase Transitions- Formation of Defects

(where the physics comes in)Field couples to heat bath

That means potentialis a function of temperature

As temperature drops stabilitychanges

“old” minimum = “new” maximum

Two “new” minima createdThigh

Tlow

Vacuum manifold changes

Like photons in expanding universe

Was invariant under reflection about old minimum … not any more: symmetry broken

minimum minimummaximum

Universe cooling

minimum

Early universe

Now

Tc

Schematic view of a phase transition…… one “old” minimum becomes two “new”

A toy model

• Z2 Discrete Goldstone Model

φ

V(φ)

-η +η+η

-η

φ

xSolution to equations of motion: interpolates between minima

Simulations

• Can simulate these things• Start off “just after” the transition• Field randomly occupies domains

• Press play!(Zoom in)

Red & bluetwo minima

Initial configuration (a mess!)

x

y

Simulations

t = 200 t = 400 t = 800 t = 1600

See domains have formed- “like” found “like”

Evolved to reduce the length of domain wall- eventually all one domain

(remember the balloon!)VIDEO!

Early Universe Cosmology(quickly)

• Universe was hot, but cooled as expanded• Lots of fields around (inflaton, Higgs)• Fields live in potentials, which are

temperature dependant• As temperature drops the vacuum manifold

of potential changes– If in “such a way” then topological defects form

Similar idea to Grand Unified Theory symmetry breaking - one “big” thing breaks into lots of little

things as temperature falls

Complications- Getting more interesting evolution

• Add in “conserved charges”• Like have energy conservation: it must go

somewhere in a closed system• We use “electric charge”

– Kinky vortons• Start off with charge homogeneous

• Evolves to align with domain walls– As conserved means that domain walls don’t

evolve in a similar way!

The kinky vorton model

Kinetic terms

Global U(1) x Z2 Symmetry broken in Z2 vacuumU(1) symmetry retained

U(1) symmetry has conserved Noether charge

Mexican hat(looks like that bit welooked at before)Bit that breaks symmetry

Bit with electric Charge: U(1) How each field

knows about eachother (interaction)

Kinky vortons

Solutions toequations ofmotion

kink solution

condensate Construct “ring solutions”

k = N/RN winding number

Stable kink solution with charged condensate- stable radii computed for given N & charge Q- charge flows along kink

Kinky Vortons, Battye & Sutcliffe, 2008

Videos of evolution

Evolve from P = 10242, with ρQ = 0.09

φ ρQ

Red/blue: positive/negativeGrey: less than 10% maximum valueBlack lines: domain walls50 time-steps per second

Re(σ)

Slow-motion(25fps)

Watch blue/redflow along wall

Grey: less than 40% maximum value

Images of φ

Time

Initial charge density

(colours = each domain, P = 4096)

0

0.01

0.09

0.25

80 160 320 640 1280

ρQ(0)

Other things we can do…

• Study stability of networkThis initial configis stable…

… can it be formedfrom “natural” initialconditions?

If a pattern stabilises, can work as model of dark energy- this does not happen with “normal” models

Cycle 4 different vacua(completely different model)…decay …. unstable

Cosmology

Acceleration equation(Raychaudhuri)

Requirement for acc’n

Equation of state

Equation of state for domain walls … Works as dark energy IF v = 0(domain walls “freeze in”… don’t evolve)

Evolution of TopologicalEvolution of TopologicalDefects in Field TheoryDefects in Field Theory

Jon PearsonJon PearsonJodrell Jodrell Bank Centre for AstrophysicsBank Centre for Astrophysics

www.www.jpofflinejpoffline.com.com

Jon PearsonJon Pearson

Introductory PhD ClassesIntroductory PhD ClassesNov 2009Nov 2009