Méthodes Mathématiques pour la Relativité Générale LUTh - Meudon 2 octobre 2009 Courants multiples dans les cordes cosmiques Patrick Peter Institut d Astrophysique de Paris
GUT SUSY SUSY
A simple example: SO(10) SO(10) 4 C 2 L 2 R 1 1 1 1 1 2 3 C 2 L 2 R 1 3 B-L C 2 L 1 R 1 B-L G SM Z 2 2 G SM Z 1 2 2 3 C 2 L 1 R 1 B-L G Z SM 2 4 C 2 L 1 R 2 G SM Z 2 3 C 2 L 1 R 1 B-L 2 G SM Z 2 G SM Z 2 1: Monopoles 2: Cosmic strings INFLATION SO(10) : 34 possible schemes E6 : 1024 R. Jeannerot, J. Rocher & M. Sakellariadou, PRD 68, 104514 (2003) 3
Higgs field and potential V (φ) T T crit T T crit φ
General symmetry breaking G H Vacuum manifold: G/H Defect classification π 0 (G/H) (T. Kibble) Domain walls Ω 10 8 for E 100 GeV π 1 (G/H) Cosmic strings Ω 10 6 for E E GUT π 2 (G/H) Monopoles Inflation... π 3 (G/H) Textures ΔT/T...
Phase transition ξ Correlation length 0 φ( x)φ( x + r) 0 e r /ξ
Abelian Higgs model L a.h = D µ φd µ φ 1 4 C µνc µν V (φ) φ = f(r)e inθ (r)
Localized energy / axis Interaction...
Reconnexion (intercommutation) Simulations : P. Shellard (DAMTP - Cambridge)
Radiation Oscillations Axions, h +,... h
Not so exotic objects...
Initial conditions: random phases + Evolution... l αt Scaling! ρ = ζ U t 2 α ρ l = ΓG N U ρ
String simulation C. Ringeval et al.
End of radiation dominated era
End of matter dominated era
Inflation + Strings! Mandatory (peaks,,...) Ω 1 Linearized gravity C l = αc inf l + (1 α)c CS l Predicted (GUT...) Best fit α < 0.1
(Old) CMB data... Bouchet et al. (2000) 17
Pogosian et al. (2004)
depends on theoretical model (SUGRA here)
Currents: Witten bosonic model L W = L a.h (φ,c µ ; q φ,m φ, λ φ ) +L (Σ,A µ ; q σ,m σ, λ σ ) +V (φ, Σ) One (abelian) current Σ =e iψ(ξ a) σ(x ) 20
Generalisation (1): L W = L a.h (φ,c µ ; q φ,m φ, λ φ ) + N i=1 L ( (i) Σ i,a (i) µ ; q(i) σ,m(i) σ, λ(i) σ + V (φ, Σ i ) ) Many abelian currents Σ i =e iψ(i) (ξ a ) σ i (x ) 21
Generalisation (2): L W = L a.h (φ,c µ ; q φ,m φ, λ φ ) +L ( Σ,A a µ; q σ,m σ, λ σ ) + V (φ, Σ) Nonabelian currents Σ =e iψ(ξ a) T σ u (x ) 22 0. σ
Typical string configuration (neutral limit) 3.0 2.5 ~ g 2 = ~ g 3 2 = 0.5 γφ γ σ 2.0 Y( ) 1.5 Z( ) 1.0 0.5 Q( ) X( ) 0 0 5 10 15 20 Distance to the string core 23
Equation of state (B. Carter) 5.2 U/ T µν = g µν L 2 δl δg µν 5.1 Timelike u µ and spacelike v µ eigenvectors T µν = d 2 x T µν = Uu µ u ν Tv µ v ν 5.0 T/ Diagonalisation & Integration 4.9-0.02-0.01 0 ~ 0.01 0.02 0.03 State parameter w η µν µ ψ ν ψ 24
State parameters χ ij η µν µ ψ i ν ψ j symmetric matrix + Diagonalisation & Integration 25
State parameters χ ij η µν µ ψ i ν ψ j symmetric matrix + Diagonalisation & Integration 26
Single current: Macroscopic modelling { L magn (w) = m 2 1 ( 2 w 1+ w m 2 ) 1 w 0 L elec (w) = m 2 1 ( 2 m2 ln 1+ w ) w 0 m 2 5.2 5.1 U/ Many currents V (φ, Σ i ) ( φ 2 η 2) i f (i) Σ i 2 + i,j λ (i,j) Σ i 2 Σ j 2 5.0 T/ w 0 w 0 4.9-0.02-0.01 0 ~ 0.01 0.02 0.03 each condensate acts as a positive mass term for all the others λ (i,j) i j λ (i,i) 27
Small coupling additive model ( ) L (χ) = m 2 + i L i (w i ) T µν = m 2 η µν + i T µν (i) with T µν (i) = U iu µ i u ν i T iv µ i v ν i The dynamics essentially depends on w i but not on x
Non abelian current(s)? Σ n Arbitrary gauge group G Parameters such that there exists a condensate ( Minimum energy state σ ) 0 x G exitations in the worldsheet 1.4 1.2 Σ =e iψa (ξ)t a σ 0 1 currents: Generator for G Jµ a = δl δ µ ψ a Integrated charge: Q a = d 2 x J 0 a 0.8 0.6 0.4 0.2 0 0 5 10 15 20 25 29
( + quantization = current algebra J a µ = σ 0 {T a,t b } σ 0 µ ψ b + i ( µ σ 0T a σ 0 σ 0T ) a µ σ 0 0 Q a = d 2 x π a where π a = δl(2) δ ψ a EQCR [ψ a (l 1,t), π b (l 2 (,t)] = iδb a δ (l 1 l 2 ) (worldsheet quantization) = [Q a,q b ] = 0 ( String (wall, brane,...) current [U(1)] n 30
add gauge fields... J a µ = σ 0 {T a,t b } σ 0 µ ψ b + i ( µ σ 0T a σ 0 σ 0T ) a µ σ 0 + B B 2 1.8 1.6 Only ONE finite energy configuration... 1.4 1.2 B B 0 1 0.8 0.6 0.4 0.2 0-0.2 0 10 20 30 40 50 31
Conclusions Cosmic strings are a generic prediction of GUT/string/high energy theories Various cosmological consequences Often superconducting One current = well-defined one parameter worldsheet model Many current = Sum over one-current models Non abelian current? Not possible as embedding in higher dimensions... Non abelian current observed = not a 4D field theory! 32