Phase Transitions in the Early Universe

Aaron Klaus

Research Advisor: Harsh Mathur

CWRU Physics REU 2009

Background

Fields pervade all space.

• A familiar example:

€

L =1

2ε0 ∇V +

∂A

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟

2

−1

μ0

∇ × A( )2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

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In the Beginning …• Big bang: very hot• Universe cooled, fields underwent phase

transitions as they lost energy and tried to find their ground state

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Finding Evidence for Phase Transitions

• Cannot replicate on Earth (1016 GeV)

• Can (hopefully some day) observe its effects in the form of gravitational radiation imprinted on the Cosmic Microwave Background

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CMB, Gravitational Radiation• CMB: faint source of

microwaves, originated at last scattering

• Gravitational Radiation: created when massive bodies accelerate, creating fluctuations in the curvature of spacetime that propagate as waves

• Phase transitions should have created gravitational radiation that we can observe

Fig. from Weiss et. al.

Scalar Fields

• Simplest type of field that can undergo phase transitions

• First, we looked at the following scalar field:

€

L =1

2

∂φ

∂t

⎛

⎝ ⎜

⎞

⎠ ⎟2

−1

2

∂φ

∂x

⎛

⎝ ⎜

⎞

⎠ ⎟2

−1

2m2φ2 ⇔

∂ 2φ

∂t 2−∂ 2φ

∂x 2+ m2φ = 0

Scalar Field with a Potential Term

Double-Well Potential Mexican-hat potential

€

V (φ) = −λ

4(φ2 −η 2)2

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V (φ1,φ2) = −λ

4(φ1

2 + φ22 −η 2)2

Spontaneous symmetry breaking phase transitions

Scalar Fields at Higher Energy

• Need (x) ± as x ± so that the field will have finite energy

• Four cases: (x) + as x + (x) - as x - (x) + as x - (x) - as x +

• Cases 3 and 4 contain what are known as topological defects, where the lowest energy state for the particular case is not the true ground state of the system

Topological Defects• The particular type of

topological defect that shows up is called a “kink”

• Clearly does not have the lowest energy for the system, just the lowest energy consistent with the given boundary conditions

• Will not go away on its own• According to some GUTs,

phase transitions in the early universe were accompanied by the formation of topological defects

Making the Field Relax• Add in friction term to make it lose energy, so our full

equation thus far is:

• With random initial conditions, the system will relax to ± everywhere, but first develop a kink and an anti-kink

A Simulation:€

∂2φ

∂t 2−∂ 2φ

∂x 2= −λ

4φ2 −η 2

( )2

−∂φ

∂t

Ising Model• Easier way to simulate the relaxation on a scalar

field at larger scales and in higher dimensions• The field is considered to be an array of discrete

points, each with a “spin” value of +1 or -1• Start with a random array of spins, and evolved

it according to the following rules– If the majority of the four spins adjacent to a

particular spin is +1 or -1, then that spin has to switch to whatever the value of the majority is

– If there is an equal number of both types, then the point chooses +1 or -1 at random

• Can calculate the stored energy as well

Conclusions• We showed that when fields are subjected to

damping, they relax to their lowest energy state• However, they do not relax immediately; rather,

they first develop kinks and anti-kinks which annihilate each other

• In this process, they inevitably give off energy, which we hope to observe some day in the form of gravitational radiation

• It turns out that the longer fields take to relax in the universe, the more gravitational radiation they give off