transparencies (pdf) - Rencontres du Vietnam

production of topological defects at the end of inflation 

mairi sakellariadou 

king’s college london 

vi rencontres du vietnam

outline 

 

motivation / introduction 

• hybdrid inflation 

• topological defects – cosmic strings 

o formation 

o evolution 

o predictions on cmb 

 

 

inflation within supersymmetric grand unified theories 

cosmic strings / cosmic 

superstrings 

inflation within braneworld cosmologies 

 

compatibility between predictions and data 

 

conclusions 

production of topological defects at the end of inflation mairi sakellariadou vi th rencontres du vietnam

motivation 

however 

the inflationary scenario is with no doubt extremely succesful 

inflation is still a paradigm in search of a model 

search for an inflationary model inspired from a fundamental theory 

and test its predictions against current data 

high energy 

physics 

astrophysics 

production of topological defects at the end of inflation mairi sakellariadou vi th irencontres du vietnam

hybrid inflation 

1 

chaotic inflation : elegant but needs fine tuning 

2 

hybrid inflation : based on eisntein’s gravity but driven by a false vacuum 

V(S, |φ| ) 

V 0 

S 

Μ 

|φ| 

the inflaton field rolls while another scalar field remains trapped in 

false vacuum 

false vacuum unstable when inflaton field much smaller than a 

critical value and there is a phase transition to the true vacuum 

which signals the end of inflation 

the phase transition may lead to topological defects formation 

non-supersymmetric grand unified theories 

supersymmetric grand unified theories 

1 

linde 1983 

2 

linde 1991, 1994 

3 

S C 

3 

jeannerot 1997; kofman & linde 1997; linde & riotto 1997; 

lyrh & riotto 1999; contaldi, hindmarsh & magueijo 1999; 

battye & weller 2000 

production of topological defects at the end of inflation mairi sakellariadou vi th irencontres du vietnam

topological defect formation and harmless defects 

the universe cools, undergoes phase transitions which may leave 

behind topological defects which are false vacuum remnants 1 1 

ssb: G H , with H ⊂ G by the condensation of a Higgs field 

defect formation during ssb depends on the homotopy group ù k (G/H) of the 

false vacuum if : 

then (2àk)à dim defects appear 

M = G/H 

ù k (G/H)6=I 

þ 

examples : 

SO(3) → O(2) ; ù 1 (SO(3)/O(2)) ' Z 2 strings 

SU(2) → U(1) ; ù 1 (U(1))6=I monopoles 

kibble 1976 


harmless & cosmologically interesting defects: local cosmic strings & global defects 

the scalar field energy density scales like ú TD ∝ 1/(at 2 ) 

ú TD /ú ø 8ùGT 2 c 

(k=2) 

(k =3) 

monopoles or domain walls are incompatible with our universe 

textures are irrelevant : 

1 

(k=0) 

ú textures /ú 

decreases rapidly with 

t 

cosmic strings 

L = D ö öþD ö þ à 1 à õ 4 

(|þ| 2 à þ 2 4 

F ö÷ F ö÷ 

0 )2 

the model obeys local U(1) symmetry which is broken in the low-temperature 

regime by any choice of vacuum |þ | = þ 0 

nambu-goto effective action: 

e.o.m. : 

S 0 [x ö ]=à ö R 

ẍ +2 à á à áà á 

aç 

a 

xç (1 à xç 2 1 x 

)= 0 0 

ï ï 

√ 

à í 

d 2 ð 

in a f.l.r.w. 

abelian higgs model 

2,3 

1 

turok 1989 

2 shellard 1988; moriarty, myers & rebbi 1988; 

laguna matzner 1990 

3 achucarro & de putter 2006 


cosmic string network evolution 

long strings enter the scaling regime and the network is characterised by a single 

1 

length scale, the interstring distance ø which grows with the horizon 

2 

early numerical simulations revealed dynamical processes at scales ü 

ø ø ö ð 

3 

3-scale model : interstring separation , curvature scale , wiggliness 

some of its aspects have been tested numerically in a flat spacetime 

5 

recent simulations have shown the existence of a scaling evolution for string 

loops down to the hundredth of the horizon time 

this result does not rely on gravitational back reaction effect and it is due just 

to string intecommuting mechanism 

4 

1 

kibble 1985 

2 

bennett & bouchet 1988; sakellariadou & vilenkin 1990 

3 

austin, copeland & kibble 1993 

4 

vincent, hindmarsh & sakellariadou 1997 

5 

ringeval, sakellariadou & bouchet 2005; martins & shellard 2005; vanchurin, olum & vilenkin 2005 

ø 


cosmic strings as seeds of structure formation 

DX = S 

for given initial conditions, this eq. can be solved by means of a green’s function: 

X j (ñ 0 , k) = R ñ in 

ñ 0 

G jm (ñ 0 ,ñ,k) 

we want to compute quadratic expectation values of the form: 

hX j (ñ 0 , k)X ? l (ñ 0, k 0 )i = 

R ñ 0 

ñ in 

dñG jm (ñ, k) R ñ 0 

ñ in 

dñ 0 G ? ln (ñ0 , k 0 )âhS m (ñ, k)S ? n (ñ0 , k 0 )i 

1 

hindmarsh 1995 

unequal time 2-pt correlators 

heavy numerical simulations ! 

1 


cmb power spectrum from cosmic strings using field 

1 

evolution simulation of the Abelian Higgs model 

1 

WMAP 


global textures 

a roughly constant slope at low multipoles, rising up 

to a single peak at ` ù 400 , with decay at small 

scales 

` =10 WMAP normalisation 

Gö =2â 10 à6 

1 

ö ø T 2 crit 

1 

bevis, hindmarsh, kunz & urrestilla 2006 

production of topological defects at the end of inflation mairi sakellariadou vi th rencontres de vietnam

inflation in N=1 supersymmetric grand unified theories 

F-term hybrid inflation 

W F infl = ôS(Φ +Φ à à M 2 ) 

M GUT 

M infl 

G GUT à→ H 1 à→ H 2 à→ G GM 

Φ + Φ à 

V = |F Φ + 

| 2 + |F Φ à 

| 2 + |F S | 2 + 2 

1 P a 

F Φi ñ ì ì ∂W ∂Φi 

ò =0 

g 2 a D 2 a 

D a = þ ö i(T a ) i j þj + ø a 

compute one-loop radiative corrections to the scalar potential V along 

the inflationary valley, using the coleman-weinberg expression 


1 

compute one-loop radiative corrections to the scalar potential V 

along the inflationary valley, using the coleman-weinberg expression 

V F eff (|S|) =ô2 M 4è ô 

1+ 2 N 

32ù 

2 

â â |S| 

2ln 2 ô 2 

Λ 

2 +(z +1) 2 ln(1 + z à1 )+(z à 1) 2 ln(1 à z à1 ) ãé 

z = |S| 2 /M 2 

N 

: dimensionality of the representations to which þ + ,þ à 

belong 

N = 126 for SO(10) N = 27 or 351 for E 6 

Λ 

: renormalisation scale 

1 dvali, shafi, schaefer 1994 

lazarides 2001 

rocher & sakellariadou 2005 


emark : 

in general, F-term inflation suffers from the hubble-induced mass problem 

sugra corrections induce contributions of the order unity to the slow roll 

parameter 

any scalar field gets mass of the order of the expansion rate during inflation 

.. 

for minimal kahler potential this problem is lifted by cancellation of the 

problematic mass terms 


D-term hybrid inflation 

in supergravity 

W D infl = õSΦ +Φ à 

M GUT 

M infl 

G GUT â U(1)à→ H â U(1)à→ Hà→ G GM 

Φ + Φ à 

L sugra depends on K(Φ i , Φö i ); W(Φ i ); f ab (Φ i ) 

K(Φ 

G(Φ i , Φö i )= i ,Φö i ) 

M 

2 + ln M 

6 

Pl 

Pl 

2 

superpotential 1 1 halyo 1996 ; binetruy & dvali 1996 

it depends on the combination 

ill-defined formulation at W=0 

|W(Φ i )| 2 

D-term inflation: W=0 at unstable de sitter (and in absolute minkowski) vacuum 

2 

supergravity constructed from superconformal theory 

2 

binetruy, dvali, kallosh & van proyen 2004 


under U(1) gauge transformations in the directions in which there are 

constant fayet-iliopoulos terms, the superpotential must transform as 

gø 

î ë W = à i ë 

M 

2 W(þ) 

Pl 

1 

in sugra with constant fayet-iliopoulos terms, the charge 

assignements for the superfields is non-vanishing, and it should 

1 

be such that the superpotential transforms under local R-symmetry: 

q(S) =à ø M 

2 ú S ; q(Φ æ )=æ 1 à ø 

Pl 

M 

2 ú æ 

Pl 

P 

ú i =1 to avoid ñ -problem : ú S = 0 

i=S,Φ æ 

1 

binetruy, dvali, kallosh & van proyen 2004 


how generic is cosmic strings formation within supersymmetric grand unified theories 

G … SU(3) x SU(2) x U(1) x Z2 

GUT 

C L Y 

consider simple lie groups with: 

rank( SU(3) x SU(2) x U(1) ) = 4 8 ≥ Rank(G) ≥ 4 n ≥ 4 

SM 

subgroup of U(1) B-L 

complex fermionic representations (to keep the nature of EW interactions) 

anomaly free fermionic representation 

SU(5), SU(6), SU(7), SU(8), SU(9) SO(10), SO(14), E6 

• neutrinos get masses via the see-saw mechanism 

1 

SU(5): stable monopoles 

minimal SU(6) and minimal SU(7) 

do not contain U(1) BàL 

• baryogenesis via leptogenesis : (i) non-thermal leptogenesis: U(1) GUT ⊂ G and B-L at the end/after inflation 

B-L 

(ii) thermal leptogenesis: B-L independently of inflation; if B-L before inflation: T BàL ø M ÷R

1 

3 2 2 1 

C 

L 

R 

B-L 

1 

2’(2) 

3 2 1 1 

C 

L 

R 

G SM (Z ) 

2 

B-L 

2 (2) 

G (Z ) 

SM 2 

SO(10) 

1 

4 2 2 

C 

L 

R 

1 

1 

1(1,2) 

4 2 1 

C 

3 2 1 1 G (Z ) 

G (Z ) 

SM 2 

3 2 L 1 1 G (Z ) 

C 

L 

R 

R 

G (Z ) 

SM 2 

B-L 

1 

2’ (2) 

2 (2) 

C 

SM 

L 

2 

R 

B-L 

2 (2) 

SM 

2 

1 

4 2 1 

C L R 

… 

1 

1 

3 2 2 1 … 

C 

3 2 1 1 

C 

1(1,2) 

L 

R 

L R B-L 

2 

SM 2 

G (Z ) 

B-L 

2 (2) 

G (Z ) 

SM 

1 monopoles 

2 cosmic strings 

2’ embedded strings 


E 6 

non-thermal leptogenesis and : 100% string formation 

thermal leptogenesis and Z 2 : ~ 98% string formation 

thermal leptogenesis without Z 2 : ~ 80% string formation 

Z 2 

within supersymmetric grand unified theories, cosmic strings formation is 

generic at the end of F-term hybrid inflation, in ssb schemes from the 

down to the G SM 

F-term strings 

G GUT 

standard D-term inflation ends always with cosmic strings formation 

D-term strings 


simplest D-term inflation in sugra: 

the constant fayet-iliopoulos term gets compensated by a single complex scalar field 

the D-term strings are topologically stable 

ù 1 (M )6= I 

M 

: vacuum manifold of broken U(1) symmetry 

there are models with D-term strings unstable 

• introduce additional matter multiplets (so as to obtain a non-trivial global symmetry), 

leading to a simply connected vacuum manifold and the production of semi-local 

strings 

• embed D-term inflation into some model with large gauge symmetry, leading to 

topologically unstable strings 

1 

1 

urrestilla, achucarro & davis 2004 

binetruy, dvali, kallosh & &van proeyen 2004 


cosmic superstring formation in braneworld cosmological models 

brane inflation is caused by the attraction, and subsequent 

annihilation of a D-brane and a D-anti-brane 

observable brane 

1 2 

examples: D3/D ö 3 D 3 /D7 

higher dimensional bulk 

Dp à D ö p 

brane inflation in IIB string theory: 

as the branes approach, the open string modes between the branes develop a tachyon; 

brane inflation ends by a phase transition mediated by open string tachyons 

brane annihilation leads to lower dimensional branes with D3 and D1-branes favoured ; 

4 

5 

depending on the model, F-strings can also arise and sometimes axionic local strings 

3 

D-strings have been identified in the low-energy sugra with the D-term strings 

6 

1 

2 

3 

kachru, kallosh ,linde, maldacena, mcallister & trivedi 2003 

dasgupta, herdeiro, hiano & kallosh 2002 

majumdar & davis 2002; durrer, kunz & sakellariadou 2005 

4 

5 

6 

copeland, myers & polchinski 2004 

davis, binetruy & davis 2005 

dvali, kallosh & van proeyen 2004 


the cosmic superstrings have many features in common with cosmic strings 

formed in a cosmological phase transition, but there is also a number of 

1 1 

differences 

1 

1 

2 

1 

2 

2 

2 

3 

1 

or 

2 

4 

1 

P 

1 à P 

• FF-strings: 

• DD-strings: 

10 à3 ôPô1 

the network reaches a scaling limit 2, 3 sakellariadou 2005 

0.1 ôPô1 

ø(t) ' 

P 

2 

√ 

t 

3 

• FD-strings: 

0 ô P ô 1 

2 

in r.d.e. : 

ú strings 32ù 

ú total 

= 3 

Gö 

P 

4 

1 

jackson, jones & polchinski 2004 

2 

copeland &shaffin 2005 

tye, wasserman & wyman 2005 

hindmarsh & shaffin 2006 

observational consequences 4 

e.g., damour & vilenkin 2005 


mixed models: anisotropies in the cmb are induced both by quantum 

fluctuations of the inflaton and topological defects, created at the end of inflation 

1 

C` = ëC infl 

` 

+(1à ë)C cs 

` 

taking into account more recent cmb data: 

the upper bound on the fraction of primordial fluctuations sourced by cosmic strings is 

3 

14% and for D-term inflation considering WMAP3 data 7%-11% 

4 

Gö < 2.7 â 10 à7 

2 

1 

bouchet, riazuelo, peter & sakellariadou 2001 

2 

pogosian, wyman & wasserman 2004; wyman, pogosian & wasserman 2005 

3 

fraisse 2006 

4 

spergel et al 2006 


compatibility between predictions and constraints imposed from data 

F-term hybrid inflation 

Cosmic strings contribution % 

100 

80 

60 

40 

20 


inflaton field 

0 

1. 10 7 1. 10 6 0.00001 0.0001 0.001 0.01 Κ 

Gö < 2 â 10 à7 

M ô 2 â 10 15 GeV 


100 

50 

20 

10 

5 

2 

1 

1 

1,2 

ô ô 8 â 10 à3 


100 

50 

20 

10 

CMB 

5 

1 

2 

1 

1. 10 7 1. 10 6 0.00001 0.0001 0.001 0.01 Κ 

κ 

1.5 2 2.5 3 3.5 4 M 1015 GeV 

ô ô 7 â 10 à7 rocher & sakellariadou 2005 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

0000 1111 

Gravitino 

1 

2 in agreement with: kallosh & linde 2003 


curvaton mechanism: 

ψ init ô 5 â 10 13à á ô 

10 

GeV 

à2 

1 

Contribution % 

100 

Κ⩵10 5 , N⩵126 

Contribution % Ψ init⩵10 13 GeV, N⩵126 

100 

80 

80 

60 

40 

60 

40 

ψ 

infl 

20 

20 

cs 

10 13 10 12 10 11 510 10 10 10 Ψ init GeV 

310 3 10 3 10 4 10 5 10 6 Κ 

1 



D-term hybrid inflation 

.. 

• minimal sugra (minimal kahler potential and minimal kinetic function) 

K = |S| 2 + |Φ + | 2 + |Φ à | 2 


100 

10 

1 

0.1 

g=10 −3 

g=10 −2 g=10 −1 

g=2x10 −2 

−4 

g=10 

1. 10 8 1. 10 6 0.0001 

1 

Λ 

1 

Strings contribution % 

100 

50 

10 

5 

0.5 1 

p 

ø ô 2 â 10 15 GeV 

0.5 1 1.5 2 2.5 3 3.5 4 

 

Ξ 10 15 GeV 

for g ô 0.02 : õ ô 3 â 10 à5 

1,2 

1 rocher & sakellariadou 2005 

1 

sugra imposes also lower bound on õ 

2 in agreement with: endo, kawasaki & moroi 2003 


consider a curvaton mechanism: 

Contribution % 

100 

infl 

Λ⩵10 1 , g⩵10 1 

1 

ψ init ô 3 â 10 14à á g 

10 

GeV 

à2 

for õ ∈ [10 à1 , 10 à4 ] 

õ 

for smaller values of , the 

curvaton mechanism is not 

necessary 

80 

60 

40 

20 

cs 

ψ 

510 16 10 16 510 15 10 15 Ψ init GeV 

1 



• higher order terms 

à 

K = |S| 2 + |Φ + | 2 + |Φ à | 2 |S| 

á à 

+ f 2 + |Φ+ | 2 |S| 

á 

+ f 2 à |Φà | 2 

f æ 

ò 

|S| 2 ó 

M 2 PL 

1 

|S| 

= c 2 

æM 2 

Pl 

M 2 Pl 

it was claimed that the contribution of strings gets really suppressed 1 

M 2 Pl 

Cosmic string contribution % 

100 

50 

g⩵10 2 ,c ⩵c ⩵0,1,2,5 

we disagree ! 

2 

strings have a high contribution to the cmb 

data unless the couplings are small 2 

10 

5 

[3 â 10 à8 à 2 â 10 à7 ] ô õ ô [2.5 â 10 à5 à 5.3 â 10 à5 ] 

2 

1 

0.5 

1 seto & yokoyama 2006 


0.1 

1. 10 6 0.00001 0.0001 0.001 

Λ 


• keep all corrections up to the order M à2 

Pl 

à 

K = |S| 2 + |Φ + | 2 + |Φ à | 2 |S| 

á à 

+ f 2 + |Φ+ | 2 |S| 

á 

+ f 2 à |Φà | 2 |S| 

+ b 4 

M 

à2 

M 2 Pl 

1 

M 2 Pl 

Pl 

strings have a high contribution to the cmb data unless the couplings are small 

1 

Cosmic string contribution % 

100 

g⩵10 2 ,c ⩵c ⩵0, b⩵0,0.5,1,2 

10 

1 

0.1 

1. 10 6 0.00001 0.0001 0.001 

Λ 



• inflation with shift symmetry 

K = 1 2 

(S + S ö ) 2 + |þ + | 2 + |þ à | 2 

S = ñ + iþ 0 

V as in D-term infl ation within SUSY 

V as in the minimal SUGRA 

1 


100 

50 

10 

5 

1 

0.5 

õ ô 3 â 10 à5 

0.1 

1. 10 6 0.00001 0.0001 0.001 

Λ 

1 

rocher & sakellariadou 2005,2006 


spectral index for various values of g, in the context of minimal sugra 1 

1 

cosmic strings contributing aroud 5% to 

the CMB on large scales, can increase 

the preferred value of spectral index 

this can be applied to hybrid inflation 

models which predict n s õ 0.98 in 

tension with WMAP3 data 

WMAP: n s =0.951 +0.015 

à0.019 

1 

battye, garbrecht & moss 2006 


7 parameter fit to the CMB data, for minimal F-term hybrid inflation in sugra 

1 

1 

the contours 

are the 68% 

and 95% CL 

1 

battye, garbrecht & moss 2006 


conclusions 

cosmic strings/superstrings form generically at the end of hybrid inflation 

within supersymmetric grand unified theories, or at the end of brane 

inflation within braneworld cosmological models inspired by string/M-theory 

even though cosmic strings can not play a dominant role in structure 

formation, due to the cmb constraints, one has to consider them as a 

sub-dominant partner with tensions of the order of Gö ô 10 à6 à 10 à7

transparencies (pdf) - Rencontres du Vietnam

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?