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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.
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