1. A Study of the Early Universe Alexander Stameroff James O’Brien
Cosmic Strings
A Study of the Early Universe
Alexander Stameroff
James O’Brien

2. Symmetrical Groups
E(3) Hamiltonian Mechanics xyz translations and rotations, conservation is momentum in translation and angular momentum in rotation
E(3) + R with time, conservation is energy
Galilean transformations (space-time begins here)
Yet to account for more symmetries in Maxwell's equations we need Weyl transformation. Mass sets length scale, M = hb/c
conformal group SU(...)

3. Phase Transitions in Early Universe
Early universe changed phase several times to present universe.
Like ice changing into water into steam during the process of heating.
Each transition from to a new symmetrical group leads to the change of the universe.
Symmetry is also lost.
One single force separated into several forces, em electroweak, strong... constitutes a loss of symmetry.

4. G → H → ... → SU(3) · SU(2) · U(1) → SU(3) · U(1)
U(1) ≡ electromagnetism
SU(2) · U(1) ≡ electroweak
SU(3) ≡ strong nuclear force
Gravity not presented here because there is no theory (Theory Of Everything) to link gravity to others.

5. Types of Phase Transitions
First Order
Expansion and Collision of Bubbles continue until old phase is gone.
collision leads to superposition of bubbles.

6. Second Order Phase Transition
Smoother continues phase transition process

7. Topological Defects
Stable forms of matter which remain from a phase transition of the old phase.
These topological defects persist trough the phase transition into the next phase.
Considered remnants of the old universe.
Studying them leads to theories and understanding of the composition of the early phases of the universe.

8. Types of Topological Defects
Several types including:
Monopoles - single magnetic charges
Domain walls - two dimensional causal separation between space such is seen in ferromagnets.
Textures - when more complicated symmetry groups are completely broken.
Cosmic Strings - one dimensional defects formed when axial or cylindrical symmetry is broken.

9. Empirical Evidence
Though cosmological defects are a theory and to produce them would require extremely high energy particle accelerators and complex systems similar defects can be viewed in condensed matter systems.
Nematic Liquid crystals.
Heating a sample to cause it to change phase one can view defects akin to cosmic strings in the early universe.

10. Formation of Cosmic Strings
Causally Separated Phases
Different Phase Cells separated by Domain Walls
Invariant Point at Node of Cells
Kibble Mechanism

11. Evolution of Cosmic Strings
Intercomuting
Loop Radiationon
Radiation
Cosmological Expansion
The expansion of the universe will stretch the string because it is gravitationally bound, much like the inflating balloon example.

12. Intercomuting
When strings touch they intercomute or exchange ends. Namely they connect together at the point and separate from their parent string.
A string can intercomute on itself forming a loop.

13. Radiation
Strings emit gravitational radiation usually, but can also emit electromagnetic. Loops will disappear because they will emit all their radiation.

14. Two loops intercomutiong and then decaying.

15. Evolution of Cosmic Strings in Radiation Era

16. Evolution in Matter Era

17. Numerical Simulations to Study Strings
Use string code by Bruce Allen and Paul Sheoard (20000 lines of C code).
Simulates evolution of strings
Takes into account known equations governing mostion and interaction of string networks.
Can be calibrated for expanding or static universe.
Can be calibrated for Radation era or Matter era.
Runs for CPU hours on COSMOS, 10 years real-life. Models in time steps.

18. Time Travel
Princeton physicist J. Richard Gott has calculated that cosmic strings warp spacetime sufficiently for a spaceship to outrace a light ray, and that two strings moving past one another in opposite directions would change the shape of spacetime to such an extent that, "a spacecraft looping around the pair of strings could return to its starting point before it had left."(Roberts)

19. Works Cited
Kibbles, T.W.B. Hindsmarsh, M.B. “Cosmic Strings”. Blackett Laboratory, Imperial College, London. hep-ph/
Gangui, Alejandro. “Superconducting Cosmic Strings”. American Scientist. May-June 2000:
Morris, Richard. Cosmic Questions. New York: John wiley & Sons, Inc
Chase, Scott. Gravitational Radiation. 27 April, <
Roberts, Sherrill. Quantum Strangeness and Space-Time. April 27, <
Cambridge Cosmology. Cosmic Strings and Other Topological Defects. 27 April, <
Cosmology in the Labarotory. 27 April,