Physics of Solitons Leandros Perivolaropoulos Demokritos Research Center WWW of Talk: leandros.chem.demokritos.gr/solitons

Structure of Talk 1. Introduction: Solitons in Experimental Data Solitons in Cosmological Data? 2. Formation of Solitons in Phase Transitions: Walls, Vortices (Strings) 3. Evolution - Interactions: Walls, Vortices (Strings) 4. Dynamics of Non-topological Solitons: Q Balls, Q Rings 5. Summary - Relevant Papers What are Solitons? Solitons are localized energy concentrations appearing in systems described by non-linear field equations. Their stability is due to the conservation of quantities called ‘charges’. Charges can be of topological or non-topological origin?

Strings in Liquid Crystals Phase transition Nematic Phase (High T>150 C) (Low T) Defect in Nematic Phase String Network Evolution 2 String Interaction Scattering (Intercommutation) Isotropic Phase Phase Transition (Strings from merging bubbles)

Strings in Superconductors and 3 He Abrikosov Flux Lattice (magnetic field through superconductor) String Formation during 3 He Phase Transitions

Cosmological Data Maximum Distance Matter has Moved since the Big Bang: Sheets of Galaxies Voids Superclusters CfA Survey Gives a Direct Window to the Primordial Fluctuations What Caused these Fluctuations? Typical Scale: Two physically motivated mechanisms: Quantum fluctuations of Scalar Field during rapid universe expansion (Inflation) Network of Strings formed after a phase transition.

Cosmic Microwave Background COBE Satellite Thermal Black Body Spectrum as detected from the FIRAS instrument of COBE Isotropy and Homogeneity CMB Photons Earth Universe yrs old (galactic coordinates)

Predictions of Cosmological Models Predicted Maps of CMB Temperature Fluctiations Simulation of Inflation induced CMB Fluctuations (continous random field) Simulation of String induced CMB Fluctuations (temperature discontinuities) Measured all-sky maps of Temperature Fluctuations Question: What is a good statistical test that can optimally identify the signatures of each model in the observed maps? Answer: A specially designed statistical test can identify temperature discontinuities hidden in high resolution temperature maps. L. Perivolaropoulos, N. Simatos Phys. Rev. D63:025018, 2001 astro-ph/

What Broke the Homogeneity Symmetry? Wall Formation Potential Energy of Scalar Field Φ Energy Trapped due to boundary conditions (topological defect) Network of trapped energy walls formed after a phase transition (Φ=0 |Φ|=Φ ) 0 Wall: Simplest Form of Topological Soliton

String Formation Potential of Complex Scalar Field Φ. Trapped Energy due to Boundary Conditions Network of trapped energy Vortices (Strings in 3D) after a phase transition (Φ=0 |Φ|=Φ ) 0 Φ0Φ0

Strings From Bubble Collisions I A. Melfo and L. Perivolaropoulos, Phys. Rev. D, 52, 992 (1995). First Order Phase Transition Bubble Wall Collisions: Large Phase Difference leads to oscillating phase before merging.

Strings From Bubble Collisions II Three Bubble Collisions without String Formation Different Nucleation Time Strings from Three Bubble Collisions: Simultaneous Nucleation

Geometric Suppression Geometric Suppression Factor (Monte Carlo Simulation): Not all ‘three bubble’ configurations lead to string formation Vortex Formation Probability:

String Network Formation Plaquettes of Cubic Space Lattice: (a)Circle in field space fully covered as we span square of real space plaquette. String goes through plaquette. (b)Circle in field space not fully covered as we span square of real space plaquette. String does not go through plaquette. (c)Join strings through plaquettes to create string network.. Assign a random field direction (0, 1 or 2) at each vertex of the cubic lattice (Monte Carlo) Initial String Network Vachaspati – Vilenkin, Phys. Rev D30,

String Network: Formation - Evolution String Interaction (Intercommuting): Initial Distribution Monte Carlo Evolved Distribution Numerical Simulation (Expanding Background) Primordial Fluctuations Individual String Segment: Action (Surface spanned in spacetime)

Intercommuting: A Result of Right Angle Scattering L. Perivolaropoulos, Nucl. Phys. 375B, 665 (1992) Vortices Scatter at Right Angles Model Lagrangian: Dynamical Field Equations: Initial Conditions: Intercommutation is a result of Right Angle Scattering

Nontopological Solitons: Q Balls Dynamically Stable Time Dependent Solutions with a rotating internal phase. Their stability is due to the conservation of Noether charge. Field equation for time evolution Conserved Noether Charge Axenides, Floratos, Komineas, Perivolaropoulos, Phys.Rev.D62:047301,2000 t=0t=t Stationary Q Ball (2+1 dim). Static Vortex (String) Φ Φ Effective Potential of Virtual Particle

Q Balls on the Line: Existence - Stability Energy: For Stability: Find B, ω for 0 eigevalue Ground State x x..

Q Balls on the Plane Fusion or Fision I

Q Balls on the Plane Fusion or Fision II Particle Behavior (pass through) Particle Behavior (merge) Soliton Behavior (fision)

Scattering in 3D Head on Collision: Unstable Q loop formation

Stable Q Rings Axenides, Floratos, Komineas, Perivolaropoulos, Submitted to Phys. Rev. Lett. (2001) L=2πR 0 ρ0ρ0 z=0 z=z Identify Ring Approximation:Stack of phase shifted planar Q Balls along z axis Ansatz: Tension (E increases with L): I 2, I 4 Pressure (E decreases with L): I 1, I 3 There is Stable Configuration Q Ring

Q Rings: Numerical Analysis Full 3D Energy Minimization + Time Evolution Stable Q Rings (N=1) Pressure supports Tension x-z plane Q contours Collapsing Q Ring (N=0) No pressure to support Tension Ansatz: Q Ball pair moving along z-axis

Conclusions Topological and Nontopological Solitons can form in a wide variety of physical systems in Condenced Matter and in Cosmology The Formation Mechanisms and the Dynamical Properties of these Solitons can be studied by a combination of Analytical and Numerical Methods. Predictions of Theoretical Models based on Topological and Non-topological Solitons can be compared with Experiment and Observations to test the theoretical models. Related Papers: Q RINGS., Axenides, Floratos, Komineas, Perivolaropoulos. Jan Submitted to Phys. Rev. Lett., hep-ph/ SEARCHING FOR LONG STRINGS IN CMB MAPS. L. Perivolaropoulos, N. Simatos, Phys.Rev.D63:025018,2001 DYNAMICS OF NONTOPOLOGICAL SOLITONS: Q BALLS Axenides, Floratos, Komineas, Perivolaropoulos, Phys.Rev.D61:085006,2000, hep-ph/ FORMATION OF VORTICES IN FIRST ORDER PHASE TRANSITIONS. Melfo, Perivolaropoulos, Phys.Rev.D52: ,1995 hep-ph/ INSTABILITIES AND INTERACTIONS OF GLOBAL TOPOLOGICAL DEFECTS. Perivolaropoulos, Nucl.Phys.B375: ,1992 FORMATION AND EVOLUTION OF COSMIC STRINGS. Vachaspati, Vilenkin Phys.Rev.D30:2036,1984 COSMIC STRINGS AND OTHER TOPOLOGICAL DEFECTS Vilenkin and Shellard, Cambridge Univ. Press. (1994).

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