Two-field Inflation with Non-minimal Coupling 1

Tonnis ter Veldhuis

Department of Physics and AstronomyMacalester College

Saint Paul, MN 55105

terveldhuis@Macalester.edu

August 26, 2013

1Work in progress in collaboration with Ruby Byrne

Introduction

• Primordial non-Gaussianity is reflected in departures from aGaussian distribution of the Cosmic Microwave Background(CMB) anisotropies.

• Measurements of the non-Gausianity of the CMB thereforeput constraints on models of inflation that complementconstraints based on measurements of the power spectrum.

• In particular, non-Gaussianity in the CMB anisotropies carriesinformation about interactions of the fields responsible forinflation and the primordial curvature perturbations.

• The dimensionless non-linearity parameter fnl provides ameasure for the amplitude of primordial non-Gaussianity; itprovides a bridge between the observations of the CMB andmodeling of the physics of the early Universe.

• Inspired by the Higgs/Dark Matter inflation model, weinvestigate fnl in an inflation model with two real scalar fields.

• Depending on the frame in which it is viewed, the scalar fieldseither have non-minimal couplings to the curvature scalar ornon-canonical kinetic terms.

Model

We consider a model with two real scalar fields, h and s,non-minimally coupled to the curvature scalar. The Jordan frameaction is

ΓJ =

∫d4x

√−g[

1

2f (h, s)R

+1

2∂µs∂

µs +1

2∂µh∂

µh − V (h, s)

],

with

f (h, s) = m2Pl + ξh2 + ξss

2

V (h, s) =1

2m2

hh2 +

1

2m2

s s2 +

1

4λh4 +

1

2κh2s2 +

1

4λss

4

A conformal transformation gµν = f (h, s)gµν removes thenon-minimal couplings but introduces a non-trivial field-spacemetric. The Einstein frame action is

ΓE =

∫d4x√−g[

1

2m2

PlR

+Ghh1

2∂µh∂

µh + Gsh∂µs∂µh +

1

2Gss∂µs∂

µs − V (h, s)

],

with

Gss =1

f (h, s)(1 + 6

ξ2s s2

f (h, s))

Gsh = 6ξξs

f (h, s)2

Ghh =1

f (h, s)(1 + 6

ξ2h2

f (h, s))

V (h, s) =V (h, s)

f (h, s)2.

Approximate SO(2) symmetry

• The model has an exact rotation symmetry if λ = κ = λs , andξ = ξs .

• The parameters δ1, δ2, and α quantify the amount and typeof symmetry breaking and are defined as

δ1 ≡ 1

2(λ− λs)

δ2 ≡ 1

4(2κ− λ− λs)

α ≡ 1

2(ξ − ξs)

• The potential is flat in the “radial” direction due to thenon-minimal coupling of the scalar fields.

• If the symmetry breaking parameters are small, then thepotential is also flat in the “azimuthal” direction.

Illustration of the potential

• Parameters: λ = 1 and ξ = 10.

• Symmetry breaking parameters: δ2 = 0 and α = 0.

-5

0

5

h

-5

0

5

s0

2

4

1000 VHh,sL

δ1 = 0

-5

0

5

h

-5

0

5

s0

2

4

1000 VHh,sL

δ1 = 0.4 (exaggerated value)

• Symmetry breaking creates valleys and ridges in the potential.

Time evolution

• The time evolution of the Universe is determined by theEinstein equation and the field equations for h and s.

• The scalar fields are expanded around their classicalbackground values

h(xµ) = h(t) + δh(xµ)

s(xµ) = s(t) + δs(xµ)

• Similary, the metric is expanded around the FRW metric.

• The time evolution of the background fields is then given by

h = −Γhhhh

2 − 2Γhhs hs − Γh

ss s2 − 3Hh − G−1

hh

∂V

∂h− G−1

hs

∂V

∂s

s = −Γsss s

2 − 2Γssh s h − Γs

hhh2 − 3Hs − G−1

ss

∂V

∂s− G−1

sh

∂V

∂h

H =

√1

3(

1

2Ghhh2 + Ghs hs +

1

2Gss s2 + V )

Trajectories

Results for h0 = 2.7978, λ = 1, ξ = 10.4.Symmetry breaking parameters: δ1 = 0.004, δ2 = 0, and α = 0.

0 1000 2000 3000 4000-4

-2

0

2

4

Τ

hHΤ

L,sH

ΤL

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

h

s

s0 = 0

0 1000 2000 3000 4000-4

-2

0

2

4

ΤhH

ΤL,

sHΤ

L

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

h

s

s0 = 0.00001

0 1000 2000 3000 4000-4

-2

0

2

4

Τ

hHΤ

L,sH

ΤL

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

h

s

s0 = 0.1

Calculation of fnl

• The parameter fnl characterizes the bi-spectrum of the gaugeinvariant curvature perturbations. It can be calculated as

fnl = −5

6

DANDBNDADBN

(DCNDCN)2, with A,B,C = h, s.

• Twenty-five neighboring trajectories for the evolution of theUniverse are calculated numerically.

• The end of inflation for each trajectory is established as thetime when the first slow roll parameter first exceeds unity.

• The number of e-folds is determined for each trajectory. Afinite differences method is employed to calculate thecovariant derivatives of N.

fnl as a function of s0

Results for λ = 1.0.Symmetry breaking parameters: δ1 = 0.004, δ2 = 0, and α = 0.

0.0000 0.0002 0.0004 0.0006 0.0008-100

-50

0

50

100

s0

f nl

Red: ξ = 9.6, h0 = 2.9101, N = 60.Blue: ξ = 10.0, h0 = 2.8523, N = 60.Green: ξ = 10.4, h0 = 2.7978, N = 60.

fnl as a function of s0, continued

Results for λ = 1.0.Symmetry breaking parameters: δ1 = 0, δ2 = 0.001, and α = 0.

0.0000 0.0001 0.0002 0.0003 0.0004-10

-5

0

5

10

s0

f nl

ξ = 10.0, h0 = 2.8523, N = 60.

Scaling of fnl(ξ, s0)

The function fnl(ξ, s0) is observed to obey the approximate scalingrelation

fnl(ξ + δξ, s0) = (1 + δx1)fnl(ξ, (1 + δx2)s0).

This scaling relation implies that fnl satisfies the differentialequation

∂fnl∂ξ

=δx1δξ

fnl + s0δx2δξ

∂fnl∂s0

.

The solution to this differential equation takes the form

fnl(ξ, s0) = eδx1δξ

ξFnl(eδx2δξ

ξs0).

How well does this scaling relation work?

δx1δξ

= 3.52

δx2δξ

= 0.935

0 1 2 3 4 5 6

-8. ´ 10-15

-6. ´ 10-15

-4. ´ 10-15

-2. ´ 10-15

0

2. ´ 10-15

4. ´ 10-15

u

Fnl

HuL

0.0000 0.0001 0.0002 0.0003 0.0004-80

-60

-40

-20

0

20

40

s0

f nlHΞ

=10

.4,s

0L

Green: directly calculatedOrange: via scaling relation

Extrapolation of scaling behavior

6 8 10 12 14 16 18 20

10-6

0.01

100

106

1010

1014

Ξ

Èf n

lm

ax

6 8 10 12 14 16 18 20

10-7

10-6

10-5

10-4

0.001

0.01

Ξ

Ds 0

• ∆s0 is the difference between the values of s0 for which fnl ismaximal and minimal (large in magnitude but negative). It isa measure of the range in s0 for which non-Gaussianity issignificant.

• Increasing ξ produces an exponentially increasing maximumvalue of fnl .

• At the same time, the range of s0 for which a significant valueof fnl is obtained is exponentially suppressed.

Implications for Higgs-Dark Matter inflation

• In a very minimal scenario, a combination of the StandardModel Higgs field and a singlet scalar dark matter field candouble as the inflaton fields of slow roll inflationary models.

• Consistency with the observed power spectrum requires largenon-minimal couplings (ξ ≈ 104) of the Higgs and darkmatter scalars to the gravitational scalar curvature.

• Unitarity violation provides a challenge to this scenario andmay require the introduction of additional fields or higherdimension interaction terms for its resolution.

• Applying our preliminary analysis of fnl to this scenario hintsthat non-Gaussianity will be small except in highly fine-tunedslices of parameter space.