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Phase Transitions in the Early Universe
Aaron Klaus
Research Advisor: Harsh Mathur
CWRU Physics REU 2009
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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.
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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
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Scalar Field with a Potential Term
Double-Well Potential Mexican-hat potential
€
V (φ) = −λ
4(φ2 −η 2)2
€
V (φ1,φ2) = −λ
4(φ1
2 + φ22 −η 2)2
Spontaneous symmetry breaking phase transitions
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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
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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
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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
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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
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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