Inflation with Gravitationally Enhanced Friction:
Primordial Fluctuations and Non-Gaussian Features
Yuki Watanabe
Arnold Sommerfeld Center, Ludwig Maximilian University of Munich
JCAP 1107(2011)031 [arXiv:1106.0502] with C. Germani
Testing Gravity with Astrophysical and Cosmological Observations, IPMU,
University of Tokyo, Kashiwa, Japan, 27th January 2012
Yuki Watanabe (LMU Munich)
Inflation with GEF
27th January 2012
1 / 28
Introduction
Latest cosmological data agree impressively well
with the Universe that is on large scales
homogeneous,
isotropic
spatially flat
A flat FLRW Spacetime!
Yuki Watanabe (LMU Munich)
Inflation with GEF
1. Introduction
2 / 28
A theoretical puzzle:
A flat FLRW Universe
ds 2 = −dt 2 + a(t)2 d~x · d~x
is extremely fine tuned solution of GR!
A simple idea to solve this puzzle is Inflation:
an exponential (accelerated and homogeneous) expansion
of the early Universe.
Yuki Watanabe (LMU Munich)
Inflation with GEF
1. Introduction
3 / 28
Slow Roll Inflation
A scalar field φ is a good candidate of an Inflaton as
1
1
ρ = φ̇2 +V , p = φ̇2 −V
2
2
By geometrical identity (Raychaudhuri eq.)
ä ∝ −(ρ + 3p) ∝ −(φ̇2 −V )
⇓
φ̇2 V , Inflation happens (“slow roll”)
φ̇2 ∼ V , Inflation ends
Q: How do we achieve the slow roll for a sufficient time?
Yuki Watanabe (LMU Munich)
Inflation with GEF
1. Introduction
4 / 28
Slow Roll Inflation
A scalar field φ is a good candidate of an Inflaton as
1
1
ρ = φ̇2 +V , p = φ̇2 −V
2
2
By geometrical identity (Raychaudhuri eq.)
ä ∝ −(ρ + 3p) ∝ −(φ̇2 −V )
⇓
φ̇2 V , Inflation happens (“slow roll”)
φ̇2 ∼ V , Inflation ends
Q: How do we achieve the slow roll for a sufficient time?
(1) Let V have a non-trivial (positive) minimum,
(2) increase friction, or (3) fine-tune V to be flat.
Yuki Watanabe (LMU Munich)
Inflation with GEF
1. Introduction
4 / 28
Increasing friction
Let us ignore the gravity for a moment. Our goal is to achieve
E ' V , ˜ ≡ − EĖ2 1 and δ̃ ≡
˜˙
E
1
in a non-equilibrium point of V for a long time.
The friction must dominate over the acceleration:
µ̃φ̇ ' −V 0
In order to have an almost constant energy, the friction coefficient must
also be roughly constant. In this case,
˜ '
V 02 1
V 2 µ̃
00
and δ̃ ' −2 VV
1
µ̃
+ 2˜
Slow roll is achieved if µ̃ is large and roughly constant.
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Inflation with GEF
2. GEF in a nutshell
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Increasing friction II
There are two ways to implement the friction:
(1) φ̈ + µ̃φ̇ = −V 0 and
(2) µ φ̈ + 3E φ̇ = −V 0 with µ̃ = 3E µ
The friction dominates over the acceleration if
δ̃˜ ≡
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φ̈
µ̃φ̇
1.
Inflation with GEF
2. GEF in a nutshell
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Increasing friction II
There are two ways to implement the friction:
(1) φ̈ + µ̃φ̇ = −V 0 and
(2) µ φ̈ + 3E φ̇ = −V 0 with µ̃ = 3E µ
The friction dominates over the acceleration if
δ̃˜ ≡
φ̈
µ̃φ̇
1.
As a result,
(1) δ̃˜ ∼ δ̃(E /µ̃) is a new parameter controlling the system
if µ̃ 6∝ E . ⇒ New d.o.f.!
(2) δ̃˜ ∼ δ̃ ⇒ Good starting point to avoid new d.o.f.
Yuki Watanabe (LMU Munich)
Inflation with GEF
2. GEF in a nutshell
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Increasing friction with gravity
Let us now introduce gravity. Since the gravitational Hamiltonian density in the
FLRW Universe is H = 3Mp2 H 2 , we may identify
H ∼ V , E ∼ H.
The slow roll is then achieved by the Friedmann eqn
3Mp2 H 2 ' V and µ̃ = 3Hµ(H) if no new d.o.f. is added.
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Inflation with GEF
2. GEF in a nutshell
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Increasing friction with gravity
Let us now introduce gravity. Since the gravitational Hamiltonian density in the
FLRW Universe is H = 3Mp2 H 2 , we may identify
H ∼ V , E ∼ H.
The slow roll is then achieved by the Friedmann eqn
3Mp2 H 2 ' V and µ̃ = 3Hµ(H) if no new d.o.f. is added.
A typical enhancement of friction could be
µ(H) = 1 +
H2
.
M2
µ φ̈ + 3H φ̇ = −V 0 ⇒ teff '
√t
µ
µ̇
as − µH
1.
If H 2 M 2 during inflation, scalar field’s clock is moving slower than
observer’s clock and friction is enhanced!
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Inflation with GEF
2. GEF in a nutshell
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Gravitationally Enhanced Friction (GEF): Covariant Realization
In order to realize the enhanced friction in a covariant manner, we promote the
rescaling to all coords.
∂µ →
√
µ∂µ , µ = 1 +
H2
M2
By noticing that during slow roll, G µν ' −3H 2 g µν .
The enhanced friction is covariantly realized by shifting the kinetic action
L ∼ g µν ∂µ φ∂ν φ → g µν −
Yuki Watanabe (LMU Munich)
Inflation with GEF
G µν
M2
∂µ φ∂ν φ
2. GEF in a nutshell
8 / 28
Gravitationally Enhanced Friction (GEF): Covariant Realization
In order to realize the enhanced friction in a covariant manner, we promote the
rescaling to all coords.
∂µ →
√
µ∂µ , µ = 1 +
H2
M2
By noticing that during slow roll, G µν ' −3H 2 g µν .
The enhanced friction is covariantly realized by shifting the kinetic action
L ∼ g µν ∂µ φ∂ν φ → g µν −
G µν
M2
∂µ φ∂ν φ
Thanks to Bianchi identities, this action has the properties:
EoM of φ is shift and
Galilean invariant [Germani et al 2011]:
R x curved
α
φ → φ + c + cα
γ,x0
ξ
Propagates only spin 0 (scalar) and spin 2 (graviton) particles (no higher
derivatives, Lapse and Shift are still Lagrange multipliers)
Makes harder for a scalar field to roll down its own potential!
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Inflation with GEF
2. GEF in a nutshell
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Full action of the GEF inflation
S=
R
i
h 2
√
M
d 4 x −g 2p R − 12 ∆αβ ∂α φ∂β φ − V ,
αβ
where ∆αβ ≡ g αβ − GM 2 .
In a FLRW background, the Friedmann and field eqs read
H2 =
1
3Mp2
h
φ̇2
2
i
H2
1+9 M
+
V
,
2
h
i
H2
∂t a3 φ̇ 1+3 M
= −a3 V 0 .
2
During slow roll in the high friction limit (H 2 /M 2 1), the eqs are simplified as
H2 '
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V
3Mp2
,
0
2
V M
.
φ̇ ' − 3H
3H 2
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2. GEF in a nutshell
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Power of the GEF mechanism
Consistency of the eqs requires the slow roll parameters to be small, i.e.
≡ − HḢ2 1 ,
δ≡
φ̈
H φ̇
1.
By explicit calculation, one can show that
'
V 02 Mp2 M 2
,
2V 2 3H 2
δ'−
V 00 Mp2 M 2
V
3H 2
+ 3 = −η + 3 ,
η≡
V 00 Mp2 M 2
.
V
3H 2
We see that, no matter how big the slow roll parameters of GR are
GR ≡
V 02 Mp2
2V 2
and
ηGR ≡
V 00 Mp2
V ,
there is always a choice of scale M 2 3H 2 , during inflation,
such that slow roll parameters are small.
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Inflation with GEF
2. GEF in a nutshell
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Cosmological perturbations in the GEF inflation
ADM form
ds 2 = −N 2 dt 2 + hij (dx i + N i dt)2
Use the gauge δφ = 0
then: hij = a2 [(1 + 2
ζ
|{z}
γij
|{z}
)δij +
curvature perturbation
]
gravitational waves
N, N i ,
Vary wrt the constraints
substitute back into the action and
canonically normalize ζ and γij
N =1+
Γ
H ζ̇,
N i = − HΓ ∂i ζ +
Σ
∂ ∂ −2 ζ̇
H2 i
Γ(φ̇, H, M) ' 1 + 32 , Σ(φ̇, H, M) '
in the high friction limit H M.
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φ̇2
2Mp2
Inflation with GEF
h
1+
3H 2
M2
i
' H 2
3. Cosm. pert. in GEF
11 / 28
Curvature perturbation spectrum
Lζ 2 = 12 [v 02 − cs2 (∂i v )2 +
z” 2
z v ]
with cs2 = 1−O()
2
hζ̂k ζ̂k 0 i = (2π)3 δ (3) (k + k 0 ) 2π
P where Pζ =
k3 ζ
spectral index: ns − 1 =
d ln Pζ
d ln k
running of the spectral index:
H2
8π 2 cs Mp2
≈ −2 − 2δ
dns
d ln k
≈ −6δ − 2δδ 0 + 2δ 2
Matching with the WMAP data, Pζ = 2 × 10−9 , we get a relation
M2
H2
=
109 V 3
8π 2 V 02 Mp6
Note that scalar perturbations are slightly sub-luminal.
Can this lead to observational consequences?
(Any GW or NG due to the new non-linear interaction?)
Yuki Watanabe (LMU Munich)
Inflation with GEF
3. Cosm. pert. in GEF
12 / 28
Gravitational wave spectrum
Lγ 2 =
1 02
λ=±2 2 [vt
P
2 (∂ v )2 +
− cgw
k t
zt ” 2
zt vt ]
2 = 1+O()
with cgw
2
P where Pγ =
hγ̂k γ̂k 0 i = (2π)3 δ (3) (k + k 0 ) 2π
k3 γ
spectral index is red: nt =
d ln Pγ
d ln k
tensor to scalar ratio: r =
Pγ
Pζ
2H 2
π 2 cgw (1+/3)Mp2
≈ −2
= 16 = −8nt
Note that GWs are slightly “super-luminal”, but this does not mean
“acausal” unless a closed timelike curve is formed [Babichev et al 2008].
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Inflation with GEF
3. Cosm. pert. in GEF
13 / 28
GEF saves λφ4 model
The λφ4 model predicts a red spectrum:
V = λ4 φ4 ,
2
M
ns − 1 ' − 40
3 H2
Ne =
Mp2
φ2i
' −5
5
3(1−ns ) .
For ns − 1 = −0.03, one obtains
' 0.0167, Ne ' 56, r ' 0.1 and
φi
Mp
' 7 × 10−2
0.1 1/4
, MHp
λ
' 5 × 10−5 ,
M
Mp
' 2 × 10−7
0.1 1/4
,
λ
The values are compatible with the WMAP data.
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Inflation with GEF
3. Cosm. pert. in GEF
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Is the Lyth bound modified?
The Lyth bound tells us that detectable gravitational waves require
super-Planckian field variation, ∆φ & 2 to 6Mp [Lyth 1997].
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Inflation with GEF
3. Cosm. pert. in GEF
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Is the Lyth bound modified?
The Lyth bound tells us that detectable gravitational waves require
super-Planckian field variation, ∆φ & 2 to 6Mp [Lyth 1997].
Under GEF, this bound reads
r 1/2
0.1
.
H ∆φ 50
20M Mp Ne .
Although at first sight the bound seems to allow detectable GWs for
sub-Planckian values of the field, in fact, it requires the canonically
normalized inflaton [φ̃ ∼ (H/M)φ] to be super-Planckian.
In this sense, the Lyth bound is not modified.
Yuki Watanabe (LMU Munich)
Inflation with GEF
3. Cosm. pert. in GEF
15 / 28
Non-Gaussianity in the GEF inflation
We compute the non-Gaussian feature of the scalar fluctuations from
the cubic action.
We use the comoving gauge variable, ζ, since it is conserved outside
the horizon (at least at order ).
The leading order effect appears in the bispectrum or the three-point
function. Since LGEF
∼ O(2 ), we get fNL ∼ hζ 3 i/hζ 2 i2 ∼ O().
ζ3
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Inflation with GEF
4. NG in GEF inflation
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Bispectrum: three-point correlation function
As in the power spectrum, the bispectrum of ζ is defined by the three-point
correlation function:
hζ̂(τ, k1 )ζ̂(τ, k2 )ζ̂(τ, k3 )i ≡ (2π)3 δ 3 (k1 + k2 + k3 )Bζ (k1 , k2 , k3 ).
One can evaluate the three-point correlator by using the in-in formalism
[Maldacena 2002, Weinberg 2005]. In the lowest order,
hζ̂(0, k1 )ζ̂(0, k2 )ζ̂(0, k3 )i
R0
= −i −∞ dτ ah0|[ζ̂(0, k1 )ζ̂(0, k2 )ζ̂(0, k3 ), Ĥint (τ )]|0i,
where we have set the initial and final times as τi = −∞ and τf = 0, respectively.
The interaction Hamiltonian is given by
R
Ĥint (τ ) = − d 3 x L̂GEF
ζ3 .
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Inflation with GEF
4. NG in GEF inflation
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The 3-pt fn can be calculated from each term of the int Hamiltonian:
R
(1)
Hint (τ ) = −c1 a3 d 3 xζ ζ̇ 2
2 2
k1 k22 k32
k2 k3
(1)
c1 H 4
1
+
+
sym
,
Bζ = 16
3 M 6 (k k k )3
K
K2
1 2 3
s
p
(2)
Hint (τ ) = −c2 a3
(2)
Bζ
=
(3)
(3)
=
d 3 x ζ̇ 3
3c2 H 5
1
,
83s Mp6 k1 k2 k3 K 3
Hint (τ ) = −c3 a
Bζ
R
R
d 3 x∂i2 ζ ζ̇ 2
3c3 H 6
1
,
43s cs2 Mp6 k1 k2 k3 K 3
where k = |k|, K = k1 + k2 + k3 and ”sym” denotes the symmetric terms with
respect to k1 , k2 , k3 .
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Inflation with GEF
4. NG in GEF inflation
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(4)
Hint (τ ) = −c4 a
= 16c34cH2 M 6 (k1 k12 k3 )3
s s
p
h
× (k1 · k2 + k2 · k3 + k3 · k1 ) − K +
(5)
Hint (τ ) = −c5 a
(5)
Bζ
=
(6)
(6)
Bζ
=
(7)
=
d 3 x ζ̇(∂i ζ)2
h 2
R
d 3 x ζ̇∂i ζ∂i χ
h
c6 H 4
1
322s cs2 Mp6 (k1 k2 k3 )3
Hint (τ ) = −c7 a
(7)
R
c5 H 5
1
323s cs2 Mp6 (k1 k2 k3 )3
Hint (τ ) = −c6 a
Bζ
d 3 xζ(∂i ζ)2
4
(4)
Bζ
R
R
(k1 ·k2 )k32
K
1+
2+
k2 +k3
K
+
k1 +k2
K
2k2 k3
K2
+
k1 k2 k3
K2
i
,
i
+ sym ,
i
+ sym ,
d 3 x ζ̇ 2 ∂i2 χ
3c̃7 H 5
1
,
83s Mp6 k1 k2 k3 K 3
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k1 (k2 ·k3 )
K
k1 k2 +k2 k3 +k3 k1
K
c̃7 ≡ c7 .
Inflation with GEF
4. NG in GEF inflation
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By using the Wick’s theorem, we obtain the contribution from field redefinition
ζ → ζ + (/2 + δ/2)ζ 2 :
Bζredef (k1 , k2 , k3 ) =
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(+δ)H 4
162s cs2 Mp4
!
1
k13 k23
Inflation with GEF
+
1
k23 k33
+
1
k33 k13
.
4. NG in GEF inflation
20 / 28
By using the Wick’s theorem, we obtain the contribution from field redefinition
ζ → ζ + (/2 + δ/2)ζ 2 :
Bζredef (k1 , k2 , k3 ) =
(+δ)H 4
162s cs2 Mp4
!
1
k13 k23
+
1
k23 k33
+
1
k33 k13
.
In the squeezed limit,
(1)
Bζ (k1 , k2 → k1 , k3 → 0) =
3
2 Pζ (k1 )Pζ (k3 ),
(4)
Bζ (k1 , k2 → k1 , k3 → 0) = − 3
2 Pζ (k1 )Pζ (k3 ),
Bζredef (k1 , k2 → k1 , k3 → 0) = 2( + δ)Pζ (k1 )Pζ (k3 ).
Other terms are sub-dominant in this limit, and thus we get the consistency
relation:
12
5 fNL
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=
Bζ (k1 ,k2 →k1 ,k3 →0)
Pζ (k1 )Pζ (k3 )
Inflation with GEF
= 1 − ns
4. NG in GEF inflation
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Natural Inflation
In natural inflation, the field φ is a pseudo-Nambu-Goldstone Boson with decay
constant f and periodicity 2πf [Freese et al 1990].
L=
√
−g
h
Mp2
2 R
i
φ
− 12 g αβ ∂α φ∂β φ − me i f ψ̄(1 + γ5 )ψ − ψ̄6Dψ − 21 TrFαβ F αβ ,
where ψ is a fermion charged under the (non-abelian) gauge field with field
strength Fαβ , 6D = γ α Dα is the gauge invariant derivative and m ∼ f is the
fermion mass scale after spontaneous symmetry breaking.
The action is invariant under the chiral (global) symmetry
ψ → e iγ5 α/2 ψ, where α is a constant.
This symmetry is related to the invariance under
shift symmetry of φ, i.e. φ → φ − α f .
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Inflation with GEF
5. UV-protected Inflation
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Natural Inflation II
Suppose the chiral symmetry is broken at energies f >TeV (like in the
QCD axion case) [chiral anomaly induces (φ/f )F F̃ ]
a potential of pNGB φ is produced by the instanton effect:
φ
V (φ) ∼ Λ4 1 ± cos
f
which is protected from QG UV corrections by the restoration of
global shift symmetry φ → φ + c as Λ/Mp → 0
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Inflation with GEF
5. UV-protected Inflation
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Natural Inflation II
Suppose the chiral symmetry is broken at energies f >TeV (like in the
QCD axion case) [chiral anomaly induces (φ/f )F F̃ ]
a potential of pNGB φ is produced by the instanton effect:
φ
V (φ) ∼ Λ4 1 ± cos
f
which is protected from QG UV corrections by the restoration of
global shift symmetry φ → φ + c as Λ/Mp → 0
If Λ ∼ 1016 GeV (GUT scale), inflation is produced with
M2
ns − 1 (∝ ) ' − 8πfp2
so ns − 1 ' −0.04 → f > Mp
⇒ The potential may not be protected from QG UV corrections.
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Inflation with GEF
5. UV-protected Inflation
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GEF saves Natural Inflation
Once again we can increase the friction so that
→
Mp2
old
⇒ ns − 1 ∼ −
µ
8πf µ
For large enough friction µ, we have f Mp
All the coupling scales are sub-Plankian!
(i.e. no UV modifications of the potential)
The new coupling G αβ ∂α φ∂β φ is the unique that
does not introduce any new d.o.f.
is invariant under the global unbroken symmetry φ → φ + c.
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Inflation with GEF
5. UV-protected Inflation
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UV-protected Natural Inflation
Inspired by Natural Inflation, we will consider the following tree-level Lagrangian
for a single pseudo-scalar field φ.
(Germani & Kehagias 2010; Germani & Watanabe 2011)
L=
√
−g
h
Mp2
i φf
1
1 αβ
αβ
2 R − 2 ∆ ∂α φ∂β φ − me ψ̄(1 + γ5 )ψ − ψ̄6D ψ − 2 TrFαβ F
i
,
where ψ is a fermion charged under the (non-abelian) gauge field with field
strength Fαβ , 6D = γ α Dα is the gauge invariant derivative and m ∼ f is the
fermion mass scale after spontaneous symmetry breaking.
The action is invariant under the chiral (global) symmetry
ψ → e iγ5 α/2 ψ, where α is a constant.
This symmetry is related to the invariance under
shift symmetry of φ, i.e. φ → φ − α f .
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Inflation with GEF
5. UV-protected Inflation
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Red spectrum from UV-protected Inflation
Small field branch:
2
V (φ) ' Λ4 2 − 2fφ 2
2
ns − 1 ' − 31 M
H2
Mp2
f2
< 0 ⇒ Red spectrum!
Matching with Pζ = 2 × 10−9 and 1 − ns = 0.04,
Λ2
Mp2
=
√
π √6 φi
,
105 5 f
M
H
√
=
3 f
5 Mp
Consistent with the theoretical hierarchies of scales to protect the
potential:
MM
Mp2
Λ2
f Mp
GW signal is small. Large field branch?
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Inflation with GEF
5. UV-protected Inflation
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Red spectrum from UV-protected Inflation II
Large field branch:
V (φ) ' 12 m2 φ2 ,
m≡
Λ2
f
ns − 1 = − 2N3 e < 0 ⇒ 1 − ns = 0.03 for Ne = 50.
Matching with Pζ = 2 × 10−9 and 1 − ns = 0.03,
φi
Mp
=
√
2π
√ 6
5
× 10−5
Mp
m
=
1
π
q
5
3
M
× 106 M
p
Consistent with the theoretical hierarchies of scales to protect the
potential:
q
Mp M
m φ f Mp and Λ Mp
GW signal is potentially detectable: r ' 16 ' 0.08
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Inflation with GEF
5. UV-protected Inflation
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Non-Gaussianity from UV-protected Inflation
NG can be generated by the inverse decays of gauge fields if the inflaton is
identified as a pseudo-scalar [Barnaby & Peloso 2010]:
6πξ
equil
fNL
' 4.4 × 1010 Pζ3 e ξ9 i , ξi ≡
i
φ̇
2fi H
= ξ ffi , ξ ≡
φ̇
2fH .
equil
At sufficiently large ξi & O(1), fNL
' 8400, which excludes axion-like inflation
models by the observations.
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Inflation with GEF
5. UV-protected Inflation
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Non-Gaussianity from UV-protected Inflation
NG can be generated by the inverse decays of gauge fields if the inflaton is
identified as a pseudo-scalar [Barnaby & Peloso 2010]:
6πξ
equil
fNL
' 4.4 × 1010 Pζ3 e ξ9 i , ξi ≡
i
φ̇
2fi H
= ξ ffi , ξ ≡
φ̇
2fH .
equil
At sufficiently large ξi & O(1), fNL
' 8400, which excludes axion-like inflation
models by the observations.
In the small field branch of the UV-protected inflation,
ξ≡
φ̇
2fH
'
p M Mp
6H f
=
√
√
5 2
' 2 × 10−2 1.
No NG is generated by the gauge field that produces the inflaton potential, but
we also have
ξi ' 2 × 10−2
f
fi ,
which allows a detactable signal for fi ∼ 10−2 f .
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Inflation with GEF
5. UV-protected Inflation
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Conclusions
By increasing the friction, inflation can be obtained with the SM
Higgs or a pNGB.
The friction can be enhanced by nonminimally coupling the Einstein
tensor to the kinetic term of the Inflaton.
This coupling is unique: it does not increase propagating d.o.f.
Consistent with WMAP 7-years result.
NG from single-field inflation models with GEF are generically small.
The parity violating interactions may produce detectable NG.
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Inflation with GEF
6. Conclusions
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