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!
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
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
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