Mutated hybrid inflation

23 February

1995

PHYSICS

Physics

Letters

B 345 (1995)

LETTEhS

B

414-415

Mutated hybrid inflation Ewan D. Stewart ’ Depunmenr

Keceivcd

of Physics.

I I August

Kyo~o

University.

1994; revised manuscript Editor: M. Dine

Kyoro

rec&cd

606, Japan

22 November

1994

Abstract A new model of inflation corresponds to a non-trivial spectral index n = I .- 3/2N

is described. An unusual form for the inflationary potential is obtained path in the configuration space of the two real scalar fields of the model, N 0.97 for the density perturbations and negligible gravitational waves.

1. Introduction

The model of

inflation

[I]

discussed

in this

pa-

Linde’s False Vacuum (or ‘Hybrid’) Inflation idea [2], and shares the important property that chaotic inflation occurs for values of the inflaton field well below the Planck scale However, unlike the usual version of False Vacuum Inflation, this model has a non-trivial inflationary trajectory [ Eq. (4) ] which leads to an unusual form for the inflationary potential [ Eq. (5) 1. I set $,/8’rr = I throughout the paper. per

is

closely

related

to

2. The model

(ti-

JzM)*

+ iA2tp2*2

address:

v=-

Science

2m2 sz A*@

(2)

m2M2

1+a

+4(

l+cu)A*~$~ (+)*

(3)

Now provided cy << I and M 5 I, then # will be constrained to its minimum for fixed 4.

[email protected].

0370-2693/95/$09.50 @ 1995 Elsevier SSDf 0370-2693(94)01646-l

m(d)

(1)

where I& and C$are real scalar fields with canonical kinetic terms. I will assume rri c A 5 I and M s I. ’ E-mail

For chaotic initial conditions A2@2~2/4 will initially be the dominant term giving effective masses rn& = A+/fi and rn+ = A4/JZ. Thus if initially 4 > #, i.e. half the initial condition space, + will decrease much more rapidly than 9 and so the fields will rapidly approach the inflationary trajectory +c$~ = 2d?m2MM/A2 with 4* >> m/A >> fi described below. Henceforth I will assume that 4 < 1 because for & 2 I higher order terms in 4 will become important. It will be convenient to define the quantity

We will see [ Eq. (8) ] that inflation occurs for & > LI and so, as we are assuming 4 CK I, we see that CY< I during inflation. The potential Fq. ( I) can be rewritten as

The effective potential of the model is V = irn*

because the inflaton The model predicts a

B.V. All rights

reserved

E.D. Sfewart /Plrysics

(4)

V”

6a

V

42

(6)

(7)

(8)

The number of e-folds until the end of inflation is given by

(9) Note that cur assumption #J < 1 now implies

The spectral index of the density perturbations is given by 2 V" nczl-3 ; + ‘27 cx 1 - $- 210.97 (11)

, (>

The ratio of gravitational waves to density perturbations is given by - 1o-4

(12)

where I have used Eq. ( 10). The COBE normalisation [3] gives v.712

-

V'

=2N314&M=6

x lCr4

(13)

For example, setting M = I and m = A2 gives A = 4A-‘lL x lOI3 GeV. Also Eq. ( IO) now gives V'!4 < 4 x lOI GeV.

I,

v = A4 If(

+ A@Y&

+ A2pq2pq*

i 14)

(1%

with the 8 fields constrained to zero. Without loss of generality, taking f(0) = 1 and f’(0) = -a < 0, and expanding about q - 0, gives V = A4 (1 - 2aReY + . . .) -!- h2]Q12]\lr]2

cp* 29-n

3

3. Particle physics motivation

where a, v’, BI and 82 are complex scalar fields. The first term might be derived from some non-perturbative mechanism such as gaugino condensation which dynamically generates the scale A < 1, while the second term would already be present at tree-level. This super-potential would then lead to the potentis

and we see that inflation occurs for

< s

415

w = A2f(Y)Z,

.The non-minimal kinetic terms arise because the inflaton is a combination of r$ and #. However, we will see [ Eq. (8) j that the second term in the brackets is small during inflation and so I will neglect it giving canonical kinetic terms. Now -z--

414-415

(5) and the non-minimal kinetic terms

and

B 345 (1995)

This model of inflation might arise from a superpotential of the form

We then get the inflationary potential

!!L2cy v-4

Letters

(16)

During ingation, the A2]@]21q(2 term will dominate the terms quadratic in g derived from f. Then setting ~=&I@Iand~,+=&!Re\IrwegetEq.(l)~ with M = l/a and m = aA2. If .‘. corresponds to the scale at which the hidden sector gauge group that leads to gaugino condensation [4] becomes strong, then one would expect A N lOI - 1Or4GeV. I thank D.H. Lyth, M. Sasaki and the referee for helpful comments on the draft of this paper. I am supported by a JSPS Postdoctoral Fellowship and this work was supported by Monbusho Grant-in-Aid for Encouragement of Young ‘Scientists No. 92062. 1 I ] For a review, sze A.D. Linde, Particle physics and inflationary cosmology (Harwood Academic. Switzerland, 1990). 121 A.D. Linde. Phys. Lett. B,259 (1991) 38; A.R. Liddle and D.H. Lyfh. Phys. Rep. 231 (1993) 1; A.D. Linde, Phys. Rev. D 49 (1994) 748; E.J. Copeland, A.R. Liddle, D.H. Lyth, E.D. Stewan and D. Wands, Phys. Rev. C 43 (1994) 6410. [ 3 1 A.R. Liddle and D.H. Lyth, astm-ph19409077. 141 For reviews, see D. Amati et al.. Rys. Rep. 162 (1988) 169; HI? Nilles. Int. J. Mod. Phys. A 5 (1990) 4199; J. Louis, Status of supersymmety breaking in string theory, in: Pmt. Particles and Fields ‘91 (Vancouver. August 1991) Vol. 2, ed. D. Axen et al. (World Scientific, Singapore, 1992) p. 889. ? Except for the ln2#2/2 term which is r.egligible during Inflation and is just added IO give the model a minimum with zeru potential energy in as simple a way m possible.