Nflation:Nflation:observational predictions from the random matrix mass spectrumSAK and Andrew R. Liddle

Phys. Rev. D 7676, 063515, 2007, arXiv:0707 .1982.

Introduction Nflation• Basic set-up• Multi-field dynamics

Observational predictions Numerical results• The spectral index

Summary

OutlineOutline

Single field (large)

Chaotic inflationMultiple fieldsAssisted inflation

from particle physicsin inflation models

Large number of fields(NNff) Nflation

S.Dimopoulos et al, hep-th/0507205.0507205.

P. Kanti and K. A. Olive, Phys. Rev. D6060, 043502 (1999),

Phys. Lett. B464464, 192 (1999).N. Kaloper and A. R. Liddle, Phys, Rev, D6161, 123513, (2000).

Random initial conditions

Different mass spectrum

Adiabatic perturbations

Density perturbationsThe tensor-to-scalar ratio

IntroductionIntroduction

A. R. Liddle, A. Mazumdar, and F. E. Schunck, Phys. Rev. D5858, 061310 (R).

Initial Initial condition condition problemproblem

NflationNflationⅠⅠS. Dimopoulos et al, hep-th/050205.

R. Easther and L. McAllister, JCAP 06050605, 018 (2006).

SAK and Andrew R. Liddle, Phys. Rev. D7474, 023513 (2006).

Multi-field dynamics• The total potential• Field equations

The number of e-foldings

Nflation Nflation ⅡⅡ

03

231

2

22

j

jjj

jj

j

j

Pl

ddV

H

VM

H

fN

iiiVV )(

endt

t

end HdttatatN)()(ln)(

03

231 2

22

ddVH

VM

HPl

Nflation Nflation ⅡⅡ

03

231

2

22

j

jjj

jj

j

j

Pl

ddV

H

VM

H

fN

iiimV 22

21

2

2

12 41)(

Pl

j jN

ii

i

i

Pl Md

VV

MN

fi

endi

The perturbation spectrum of the curvature perturbations

The tensor-to-scalar ratio

independentindependent of NNf f , of their masses, and of their initial conditions

Observational predictions Observational predictions ⅠⅠ

jPl

k kk

j j

j

Plij

jiR M

mVV

MV

ddN

ddNHP 2

62

222

62

2

96122

NM

VVM

PP

ri i

Pl

i ii

Pl

R

g 832)'/(

82

2

2

2

)(2, 422Plg MHPwhere

D. H. Lyth and A. Riotto, Phys. Rep. 314314, 1 (1999), R. Easther and L. McAllister, JCAP 06050605, 018 (2006).

L. Alabidi and D.H. Lyth, JCAP 06050605, 016 (2006).

M. Sasaki and E. D. Stewart, Prog.Theor.Phys. 9595, 71 (1996).

~0.16(N=5

0)

Observational predictions Observational predictions ⅡⅡ

NM

VVV

VV

VV

Mfi i

Pl

i

ii

i i

i

j j

jPlNL 2

121'5

62

2

22

22

2

22)4(

22,

, ,,,

)()1(

1656

k k

ji ijjiNL N

NNNfrf

fN

fNL 221

56

D. Seery and J. E. Lidsey, JCAP 05090509, 011 (2005),D.H. Lyth and Y. Rodriguez, Phys. Rev. D.7171, 123508

(2005).

SAK and Andrew R. Liddle, Phys. Rev. D7474, 063522 (2006).~O(0.0

1)(N=50)

Observational predictions Observational predictions ⅢⅢ

j jk kk

i iiPl

j jj

l lllPl

j jj

Pl

ki

Pl

RS

m

mM

VVVVV

VM

VVMV

VM

kdPdn

2222

242

2

22

2

22

2

2

14

)/()/(2

)/(2)(

lnln1

M. Sasaki and E. D. Stewart, Prog. Theor. Phys. 9595, 71

(1995).

Y-S. Piao, Phys. Rev. D7474, 047302 (2006).

N

Mn PlS 2

1121 2

2

The relations between the total number of e-foldings Ntotal and the number of fields NNff

NNtotal total NNff /12 /12 indicate more than 600 fields needed

to get enough e-foldings, i.e. more than 50 e-foldings

Numerical resultsNumerical resultsSAK and Andrew R. Liddle, Phys. Rev. D7474, 023513

(2006).

The mass spectrum by the random matrix mi

2 ()• The average mass term

m=10-6Mpl

=0, 0.3, 0.5, 0.9 The spectral index

=0, 0.1, 0.3, 0.5, 0.7, 0.8, 0.9, 0.95

The spectral indexThe spectral index

NNff

ObservationalLower Limit(WMAP3)

nS

i

mi2/MPl

2R. Easther and L. McAllister, JCAP 06050605, 018 (2006).

SAK and Andrew R. Liddle, Phys. Rev. D 7676, 0635156 (2007).

ObservationalLower Limit(WMAP5)

At least 600 fields needed The tensor-to-scalar ratio &

the non-gaussianity parameter

• Completely independentindependent of the model parameters• NNff and the mass spectrum• Also independentindependent of the field initial conditions

Summary Summary ⅠⅠ

fN

fN

r NL 221

56,8

The spectral index• DependsDepends on the model parameters;

nS(NNff, mi)• Also dependsdepends on the initial conditions• Existence of the independentindependent regime

for the initial conditionscalled ‘thermodynamic’ ‘thermodynamic’ regime

• Provided < 0.5, becomes independentindependent•With a large NNff, nS also becomes

independentindependent

Summary Summary ⅡⅡ

Thank Thank you!you!