Cosmic and superconducting strings

Transcription

1 UFES Vitória, Brasil & Jacobs University Bremen, Germany 22 Janeiro 2014

2 In collaboration with... Yves Brihaye - Université de Mons, Belgium Brandon Carter - Observatoire de Paris, Meudon, France Florent Michel - École Normale Supérieure, Paris, France Patrick Peter - Institut d Astrophysique de Paris, France Jon Urrestilla - University of the Basque Country, Bilbao, Spain

3 Outline Introduction 1 Introduction

4 (as field theoretical solutions)... form whenever a U(1) symmetry is spontaneously broken (Kibble mechanism)

5 topological solitons, i.e. possess a topological charge width inversely proportional to mass ruled out as (main) source of primordial density perturbations (by CMB data) have analogues in condensed matter: flux tubes in type-ii superconductors, vortices in superfluid helium space-time has deficit angle δ = 8πGU, U: energy per unit length gravitational lensing =8 G Numerical simulation for the Hubble deep field

6 Cosmic string loops form in the evolution of cosmic string networks decay due to string tension under emission of gravitational radiation important energy loss mechanism contribution of strings to total energy density of the universe is kept constant ( scaling regime ) BUT: loops could be stable if they carry additional internal structure (bosonic or fermionic currents) superconducting strings

7 Fundamental (F-) strings (of string theory)... have zero width have tension close to the Planck scale end on D-branes D1-brane = D-string Connection between cosmic strings and fundamental strings??? NO: perturbative strings as cosmic strings ruled out (Witten (1985))

8 YES: cosmic superstrings are formed in inflationary models originating from string theory D-, F- and bound states of p F-strings and q D-strings (p-q-strings) are formed in brane inflation interbrane distance inflaton brane U(1) antibrane U(1) brane-anti-brane collide and annihilate e.g. D3-brane e.g. D3-anti-brane D-strings, F-strings form Jones, Stoica, Tye (2002); Sarangi, Tye (2002)

9 ... and also: Hybrid inflation (Linde (1994)) Φ Ψ two scalar fields inflation ends due to spontaneous symmetry breaking cosmic strings form generically at the end of hybrid inflation in Supersymmetric Grand Unified Theories (Jeannerot, Rocher, Sakellariadou (2003))

10 In agreement with PLANCK 2013? Inflationary models Data favours "simple", single-field models BUT: where do these models come from? String inflationary models are not ruled out (Burgess, Cicoli, Quevedo, JCAP 11 (2013)) Abelian-Higgs strings (Ade et al. [Planck Collaboration], arxiv: ) GU/c , f 10 = U: energy per unit length f 10 : fraction of contribution to power spectrum at l = 10

11 U(1) U(1) gauge field model with Lagrangian density L = D µ φ(d µ φ) 1 4 F µνf µν + D µ ξ(d µ ξ) 1 4 H µνh µν V (φ, ξ) with D µ φ = µ φ ie 1 A µ φ, D µ ξ = µ ξ ie 2 B µ ξ F µν = µ A ν ν A µ, H µν = µ B ν ν B µ where φ, ξ: complex scalar fields A µ, B µ : U(1) gauge fields e 1, e 2 : gauge couplings

12 The potential Introduction V (φ, ξ) = λ ( ) 1 2 φφ η1 2 λ ( 2 + ξξ η2 2 4 ) ( 4 ) + λ 3 (φφ η1 2 ξξ η2 2 ) 2 η 1 0, η 2 0 λ 1 > 0, λ 2 > 0: self interaction couplings λ 3 > 1 2 λ1 λ 2 : interaction coupling

13 The potential Introduction 1 4λ 2 3 < λ 1λ 2 Minimum at (φ 2, ξ 2 ) = (η 2 1, η2 2 ) spontaneous symmetry breaking of both U(1)s describes two cosmic strings for λ 3 = 0: two non-interacting Nielsen-Olesen strings (Nielsen, Olesen (1973)) λ 3 < 0: toy model for bound states of p F-strings and q D-strings, i.e. p-q-strings (Saffin (2005)) 2 4λ 2 3 > λ 1λ 2 Minimum at (φ 2, ξ 2 ) = (0, η λ 3 λ 2 η1 2) or (φ 2, ξ 2 ) = (η λ 3 λ 1 η2 2, 0) spontaneous symmetry breaking of only one of the U(1)s describes superconducting strings

14 (Interacting) Cylindrically symmetric Ansatz φ(ρ, ϕ) = η 1 h(ρ)e inϕ, ξ(ρ, ϕ) = η 2 f (ρ)e imϕ A µ dx µ = 1 e 1 (n P(ρ)) dϕ, B µ dx µ = 1 e 2 (m R(ρ)) dϕ n, m: winding numbers magnetic fields B i = B i,z e z, i = 1, 2 with B 1,z = 1 e 1 dp/dρ ρ quantized magnetic fluxes Φ 1 = 2πn e 1, B 2,z = 1 e 2 dr/dρ ρ, Φ 2 = 2πm e 2

15 λ 3 = 0: Nielsen-Olesen strings scalar core width (Higgs mass) 1 = M 1 H,i = ( 2λ i η i ) 1, i = 1, 2 width of flux tubes (gauge boson mass) 1 = M 1 W,i = (e i η i ) 1, i = 1, 2 T 0 0 = T z z : energy per unit length U = string tension T energy per unit length U = U 1 + U 2 where U 1 = 2πη 2 1 n g 1(M 2 H,1 /M2 W,1 ), U 2 = 2πη 2 2 m g 2(M 2 H,2 /M2 W,2 ) g i = O(1) and weak dependence on M 2 H,i /M2 W,i

16 λ 3 = 0: Nielsen-Olesen strings for λ 1 = 2e1 2, λ 2 = 2e2 2 BPS limit: width of flux tube = width of scalar core saturate energy bound U 1 = 2πη1 2n and U 2 = 2πη2 2m Equations of motion: 1.order Typical solution: BPS and n = 1, m = 2

17 λ 3 < 0: p-q-strings Introduction (Saffin (2005)) p n, q m interaction of n-string with m-string no longer BPS binding increases with decreasing 1 < λ 3 < 0 increasing n and m formation of bound states effective energy loss mechanism to reach scaling regime for (p, q) networks

18 Gravitational effects Introduction Action S = Ansatz for the metric d 4 x ( ) R g 16πG + L ds 2 = N 2 (ρ)dt 2 dρ 2 L 2 (ρ)dϕ 2 N 2 (ρ)dz 2 weak gravitational fields ds 2 = dt 2 d ρ 2 ρ 2 d ϕ 2 dz 2 where ϕ = (1 4GU)ϕ with 0 ϕ < 2π(1 4GU) locally identical to flat space-time globally: deficit angle δ = 8πGU

19 Gravitational effects Introduction globally regular for δ < 2π, i.e. G (8πη 2 1 n) 1 String solutions N(ρ ) c 1, L(ρ ) c 2 ρ + c 3 where c 1, c 2, c 3 constants and δ = 2π(1 c 2 ) Melvin solutions (have no flat space-time counterpart): N(ρ ) a 1 ρ 2/3, L(ρ ) a 2 ρ 1/3 where a 1, a 2 constants

20 Gravitational effects Introduction For G > (8πη 2 1 n) 1 : singular solutions supermassive string solutions (Garfinkle, Laguna (1989); Ortiz (1991)) δ > 2π with L(ρ = ρ s ) = 0, N(ρ = ρ s ) finite Kasner solutions (Christensen, Larsen, Verbin (1999)) L(ρ = ρ k ) =, N(ρ = ρ k ) = 0 for λ 1 = 2e 2 1, λ 2 = 2e 2 2, λ 3 = 0 are still BPS (Linet (1987))

21 Gravitating p-q-strings (B.H. & J. Urrestilla (2008)) G fixed singular solutions can be made regular by decreasing λ 3 δ decreases for decreasing λ 3, i.e. increasing interaction λ 3 fixed δ increases for G increasing G=1/(16πη 1 2 ) λ 3 = (n,m) (n,m) -λ 3 8πGη 1 2

22 in de Sitter (B.H. & Y. Brihaye (2008)) only one string: R = f 0, m = 0, λ 2 = λ 3 = η 2 = 0 de Sitter background space-time ds 2 = (1 r 2 ) dt 2 (1 r 2 ) 1 dr 2 r 2 (dθ 2 +sin 2 θdφ 2 ) l 2 de Sitter radius l = 3/Λ, Λ positive cosmological constant For Λ > 24M 2 W, Λ > 2/3M2 H l 2 P(ρ >> 1) = P 0 ρ a f (ρ >> 1) = 1 f 0 ρ b P 0, F 0, a, b : parameter dependent constants

23 in de Sitter (B.H. & Y. Brihaye (2008)) For Λ < 24MW 2, Λ < 2/3M2 H ( ) P(ρ >> 1) = P 0 ρ 1/2 sin R1 /2 log ρ + φ 1 ( ) f (ρ >> 1) = 1 f 0 ρ 3/2 sin R2 /2 log ρ + φ 2 P 0, f 0, R 1, R 2, φ 1, φ 2 : parameter dependent constants oscillating deficit angle ( ) δ 8πGU Λ 9 increases for Λ increasing M 2 H

24 let e 2 = 0, only one U(1) is gauged Ansatz for standard string φ(ρ, ϕ) = η 1 h(ρ)e inϕ, A µ dx µ = 1 e 1 (n P(ρ)) dϕ neutral current carrier field ξ(ρ, z, t) = η 1 f (ρ)e i(kz ωt), k integer f (ρ ) 0 unbroken U(1) f (ρ = 0) 0 condensate inside the string

25 Conserved quantities Conserved Noether current in z-direction J z = k dx 3 f 2 Conserved Noether charge Q = ω dx 3 f 2 Conserved momentum P z = ωj z = ωk dx 3 f 2 conserved momentum z conserved angular momentum take finite conserved current superconducting string piece of string and identify ends current loop of string stabilized against collapse by angular momentum Vorton

26 Parameter restrictions only one U(1) broken 4λ 2 3 > λ 1λ 2 minimum of potential at (φ 2, ξ 2 ) = (η λ 3 λ 1 η 2 2, 0) λ 1 η 4 1 > λ 2η 4 2 New restriction (B.H. & B. Carter (2008)): non-vanishing condensate inside string, i.e. f (0) 0 only for cλ 3 η 4 1 < λ 2η 4 2 where c = O(1) and c 2.8

27 Stability of superconducting strings Stability criterion (B. Carter (1989)): speed of transversal (T) & longitudinal (L) perturbations real c 2 T = T U > 0, c2 L = dt du > 0 energy per unit length with T 00 = 2ω 2 f 2 L tension with T zz = 2k 2 f 2 + L U = 2π T = 2π ρdρt 00 ρdρt zz

28 Stability of superconducting strings Question: how does L = 2π ρdρl depend on ω and k? Suggestion (P. Peter (1992)) L = m m2 ln(1 + δ 2 w), w = k 2 ω 2 δ = 1/m ξ with m ξ = 2λ 3 η1 2 λ 2η2 2 mass of carrier field m 2 = L w=0

29 A logarithmic equation of state Confirmation and evaluation of parameters (B.H. & B. Carter, PRD (2008)) ( ) m 2 = πη1 2 ln MH, m 2 = πmξ 2 2MW valid for M W M H λ 2 η 2 λ 1 λ 2 η 1

30 Semilocal strings Introduction in a SU(2) global U(1) local model (T. Vachaspati & A. Achucarro (1991)) L = D µ Φ(D µ Φ) 1 4 F µνf µν λ 4 (Φ Φ η 2 ) 2 Φ = (φ, ξ) T complex doublet invariant under global SU(2) transformations D µ Φ = µ Φ iea µ Φ with A µ : U(1) gauge field

31 Stability of semilocal strings (B.H. & P. Peter, PRD (2012)) c 2 L dt du 0 semilocal strings in SU(2) global U(1) local are always unstable

32 Gravitational effects Introduction For weak gravitational fields ds 2 = (1 + 2ψ( ρ)) dt 2 (1 2ψ( ρ)) dz 2 d ρ 2 (1 2G(U + T )) 2 ρ 2 dϕ 2 where ψ( ρ) = 4G(U T ) ln( ρ/ ρ 0 ) deficit angle δ = 4πG(U + T ) attractive force towards string G(U T )/ ρ

33 Gravitational effects Introduction Full gravitating system (B.H. & F. Michel, PRD (2012)) No boost symmetry along z-axis ds 2 = N 2 (ρ)dt 2 dρ 2 L 2 (ρ)dϕ 2 K 2 (ρ)dz 2, K (ρ) N(ρ) bigger currents possible for gravitating superconducting strings κ = κ = 8πGη² 1 current current = k² - ω² = k² - ω²

34 Summary Introduction Link between cosmic strings fundamental strings to understand observational effects it is important to understand gravitational effects (gravitational lensing,...)... know which type of strings form and how string networks evolve (CMB data, gravitational waves,...) in view of this gauge and Higgs fields describing a cosmic string oscillate for all accepted values of the (positive) cosmological constant... gravitating p-q-strings can be made regular by increasing the interaction between the p- and the q-string... superconducting strings can be described by a logarithmic equation of state... semilocal strings are always unstable cannot form stable vortons

35 And also... Introduction Detection of cosmic strings? CMB data (power and polarization spectra) (e.g. P. Ade et al. [Planck Collaboration], arxiv: ) motion of test particles in the space-time of cosmic strings (V. Diemer, B. H., E. Hackmann, C. Lämmerzahl, F. Michel, P. Sirimachan ( ))

36 Selected References B. Hartmann, B. Carter: Logarithmic equation of state for superconducting cosmic strings, Phys. Rev. D 77 (2008) B. Hartmann, J. Urrestilla: Gravitating (field theoretical) cosmic (p,q)-superstrings, JHEP 0807 (2008) 006 Y. Brihaye, B. Hartmann: in a space-time with positive cosmological constant, Phys. Lett. B 669 (2008) 119 B. Hartmann, P. Peter: Can type II Semi-local cosmic strings form?, Phys. Rev. D 86 (2012) B. Hartmann, F. Michel: Gravitating superconducting strings with timelike or spacelike currents, Phys. Rev. D 86 (2012)

37 Beware: vortices are everywhere! Muito obrigada pela sua atenção!