Gauge-field inflation andthe origin of the matterantimatter asymmetryPeter Adshead
University of Illinois at Urbana-Champaign
YITP International Molecule Workshop, May 14 2019
PRL: 108,261302 (2012), with M. Wyman
PRD: 88, 021302 (2013), with E. Martinec & M. Wyman
JHEP: 09, 087 (2013), with E. Martinec & M. Wyman
JHEP: 12, 137 (2016), with E. Martinec, E. Sfakianakis & M. Wyman
JHEP: 08, 130 (2017), with E. Sfakianakis
PRD: 98, 043525 (2018), with A. Long and E. Sfakianakis

Invitation• Another model of inflation - why do I care?
-Rather than a new model, a new class
Replace:
With:

00
V
2
⌘V = MPl
⌧1
V

00
V
2
MPl
⇠1
V

and (e.g.)f (Fµ⌫ , F̃µ⌫ )• Definite (potentially) testable predictions!
e.g. Parity violation (EB, TB correlations), large-amplitude
chiral gravitational waves, large tensor non-Gaussianities,
gravitational leptogenesis• Important to understand implications of potential
B-mode measurements⇣ r ⌘1/4
p?
⇤inf ⇠ HMPl = 1.04 ⇥ 1016 GeV
0.01

Scalar-field Inflation—parametrization ofignorance
scalar drives inflation• Potential of a slowly-rolling

L=

p

2
MPl
g
R
2

(@ )2
2

V( )

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2✓

M2Pl
M
✏ = Pl

✏= 2
2

◆
0 22
V
⌧1 1
⌧
VV

V0

0000
V
V
22
=MPl
⌧1 1
⌘ =M
⌧
Pl
VV

<latexit sha1_base64="+3EY5kDhto8bphAlH58bOSSLS38=">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</latexit><latexit sha1_base64="yn1vEwAs/PLShMARB3/BvSeXXjs=">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</latexit><latexit sha1_base64="6gbl4bmpCobuO2iPPPU5ZJsB4cM=">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</latexit>• Fluctuation spectra<latexit sha1_base64="rbenglV49qIrmsUVF782cDLRkXg=">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</latexit>

reheating<latexit sha1_base64="0qJrCIKy4uMlzREtr0fOVjgqZQs=">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</latexit>

0

1
V
2
10
=
⇠
10
R
2 ✏
24⇡ 2 MPl
2
2
V ⇠?
h =
2
2
3⇡ MPl• Tensor-to-scalar ratio<latexit sha1_base64="uB9Hitg0TSG9R+BV9IWaOtgYf3I=">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</latexit>

end<latexit sha1_base64="cIwmGTqseuRqPTitlItcKzpNoXk=">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</latexit><latexit sha1_base64="go+mXovE9S0E7TeMZ8lKqmNkR7I=">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</latexit>=
rr =<latexit sha1_base64="bMg/wlrxhiCHtKT2xdF2Zszpby4=">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</latexit><latexit sha1_base64="V3nYlUQqw3z6P2HrXX1JpqcIAPk=">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</latexit>2
h = 16✏
2 = 16✏R
R

CMB

The ‘eta’ problem

We need a flat potential to sustain slow roll.
2 V⌘V ⌘ MPl00

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potential
energyslowroll⌘V ⌧ 10

field value,
Linde; Steinhardt & Albrecht (1982)

But this introduces an inflationary hierarchy problem.
The eta problem:

00
V2⌘V ⌘ MPl⌧1
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potential
energyfastroll!

⌘V ⇠ 1“Quantum effects”⌘V ⌧ 10
field value,Today, a way to avoid this hierarchy problem.

potential
energy‘magnetic’ friction
via Chern-Simonsinteraction

0field value,

The new physics is analogous to
the Lorentz force:
potentialforce

Y

V HX, YL
m2

ordinary
frictionX

magneticforce

Analysis: Normal modes• 2-mode characterized by orientation of angular momentum
relative to magnetic field:

At low B field strength...
Y

V HX, YL
m2

X

Turning up the B-fieldY

V HX, YL
m2

X

This is magnetic drift.
Y

V HX, YL
m2

Fast mode rapidly damped
leaving slow magnetic drift
mode.
XLong slow spiral down the potential!

Magnetic Friction@V(X, Y )Ẍ + H Ẋ +=B Ẏ@X
@V(X, Y )Ÿ + H Ẏ += B Ẋ
@Y• In the slow2 roll 2limit, diagonalize velocities(H + B )Ẋ =HV,X
2

BV,Y

2

(H + B )Ẏ =BV,X + HV,Y

Magnetic Friction• In magnetic drift limit B >> H, gradient flowbalanced by magnetic friction

Building a new theory:
L=

p



R
g
2

1
(@X )2
2

V (X )

1
Tr [F µ⌫ Fµ⌫ ]
2

✏µ⌫↵
X p
Tr [Fµ⌫ F↵ ]4f
g

Start with the basics...
L=

p



R
g
2

1
(@X )2
2

V (X )

1
Tr [F µ⌫ Fµ⌫ ]
2

✏µ⌫↵
X p
Tr [Fµ⌫ F↵ ]4f
g

Usual inflationary action.

• with something likeV (X ) = µ

4✓

1 + cos

✓

X
f

◆◆

“Natural Inflation” - Freese, Frieman and Olinto ’90

add gauge fields,
L=

p



R
g
2

1
(@X )2
2

V (X )

1
Tr [F µ⌫ Fµ⌫ ]
2

✏µ⌫↵
X p
Tr [Fµ⌫ F↵ ]4f
g

Action for a vector (gauge) field theory.
e.g. Maxwell’s E&M: U(1)Weak: SU(2)L
QCD: SU(3)and let them interact.
L=

p



R
g
2

1
(@X )2
2

V (X )

Interaction
X✏

µ⌫↵

h

~ ·B
~
Tr [Fµ⌫ F↵ ] = X Tr E

• Dimension 5 operator• Chern-Simons term
• Topological

✏µ⌫↵
X p
Tr [Fµ⌫ F↵ ]4f
g

1
Tr [F µ⌫ Fµ⌫ ]2iKey: a single time derivative.
• Chern-Simons terms are total derivatives,
X✏

↵ µ⌫

Tr [F↵ Fµ⌫ ] = X ✏

↵ µ⌫



Tr @↵

✓

1
A @ µ A⌫ + A Aµ A⌫
2◆integrate action by parts

L

Ẋ ✏

ijk

Tr

✓

1
Ai @ j Ak + Ai Aj Ak
2

Like Lorentz force!

◆Cosmological Gauge fields• Homogeneity and isotropy of FRW at first appear toprohibit cosmological vector fields• On second thought, need at least three fields with anadditional (gauge) symmetry• Rotations map to gauge-equivalent configurations
Gauge symmetry

Spatial rotation

Isotropic, homogeneous vector fields?
•

Flavor-space locked configuration:aA0 = 0,•aAi = a(t)

⇠ 10

2

MPl

How can a classical vector field be consistent with thesymmetries of FRW?
•

Under rotations:
a
a
k
a
~
A ! Rij (✓)A = ( ij + ✏ijk ✓ )A
i

•

j

j

Residual (large) gauge transformationsaAi ! (U (

•

a
(t) i

)Ai U

1

( ))

a

a
a c
b
= ( b + ✏bc )Aj

Rotations map to gauge-equivalent configurations

We call this
Chromo-Natural Inflation.
potential
force
Equationsofmotion:•2 ˙✓◆Ẍ + 3H Ẋ + 4V (X ) = 3g̃
( +
H ) ⌘
⇣
µ
X
f
2
˙ +H
Ẍ + 3H Ẋ
sin
= 3g̃
f
f
f
0

ordinarymagnetic¨ + 3H ˙ + (Ḣ + 2H 2 ) +2g̃ 2 3 = g̃force ffriction

2Ẋ ,PA and M. Wyman, PRL: 108,261302 (2012), PRD 86, 043530 (2012)

Magnetic drift leads to slow roll.
simplifies.
• In the slow-roll, large✓ λ limit, system
◆
Ẋ 2f
=
H
˙=

g
H
+
H
g
f V 0 (X )
H +
2
3g

• Axion equation of motion independent of V’!
fixed-point• Gauge˙ field evolves to (approximate)✓◆<latexit sha1_base64="Rbkgm4lpJJefEZF5TMrB9VFHuZ8=">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</latexit>

H

⌧1

⇡

0

f V (X )3g H

1/3• Axion drives slow-roll inflation independent of V
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2<latexit sha1_base64="tLW+32cA/E3cpvbKuwEcJp1dHI8=">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</latexit>

3

Chromo-Natural Drift
0.10yFast `Larmor’oscillations

0.05

0.00
-0.05
-0.10

2.0
V H c, yL

Magnetic drift
1.5

m4
1.0

0.5

0.0
0
1
cêf

2
3

The axion is not required….
• Can integrate out the axion—get Gauge-flation
L=



2
MPl
g
R
2

p

⇣ h
i⌘2
1

Tr [Fµ⌫ F µ⌫ ] +Tr Fµ⌫ F̃ µ⌫4
192(Maleknejad and Sheikh-Jabbari, 2011)

1
PYM = ⇢YM
3P =<latexit sha1_base64="w0P4cCbBwLPD201yPwh1vB3579k=">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</latexit>⇢<latexit sha1_base64="1mgaSJ2LYEjXb2CUMq5RnNsKO7g=">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</latexit>• Inflation requires⇢YM ⌧ ⇢<latexit sha1_base64="pIQkuP1Hh59Jq4AuGkv9r+EUcRg=">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</latexit>

Inflation without a scalar field**Perturbations sold separately

Fluctuations

Fluctuations are… complicated•Gravitation introduces 10 degrees of freedom subject to
8 constraints•

SU(2) gauge field introduces 12 degrees of freedom
subject to 6 constraints

10-dof-8-dof12-dof-6-dof
1-dof
9-physical dof•

Physical fluctuations decompose as usual:
3 scalar, 2 vector, 4 tensor

Even though it looks very complicated...
Time
^
|X|

10

2mχ/HΛ

mψΛ

mψ

Amplitude

|z^|
1

|δφ^|

0.1

IV

0.01
0.001

III

II

0.1

10

−kτ

I
1000

...we can isolate the adiabatic curvature
mode:
Time
^
|X|

10

2mχ/HΛ

mψΛ

mψ

Amplitude

|z^|
1

|δφ^|

0.1

IV

0.01
0.001

III

II

0.1

10

−kτ

I
1000

Scalar spectral index is a strong
function of gauge field effective mass.
g2 2
=
H2
a
a
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g2 2
=
2
2
H

The tensor sector has new features.

2

ds =

2

2

i

j

a
a
(t) i + ti (t, x))adt + a e

Aµ = (0, a(t)

ij

dx dx

2

Gauge and gravity tensors mix
usual equation
ˆ

t̂

±00

✓

±00

+ k

2+2
+ k +
⌧2
2

✓

2
⌧2

2

◆

◆

k
t̂ ±
⌧
±

±

p

1+2
p

◆

ˆ = O( ✏)t̂

✓

±

±

p

t̂ = O( ✏)±Gauge tensors split; one is amplified.

• Parity is spontaneously broken by the background.
[P.A, E. Martinec, Mark Wyman 2013]

The gauge field sources gravitational waves
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GW spectrum grows due
to t± instability

g2 2
=
2
2
H

CNI is incompatible with dataA positive mass for the gauge field could help...

ω- � (�/��� γ)��γ��=��
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Higgs mechanism gives positive gauge-field mass!

Massive Gauge-field inflation•

Add a Higgs sector:
1|Dµ |2
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PlPl• Anisotropic stress appears at higher order in perturbations⇡ij ⇠ @i<latexit sha1_base64="/bE3qCUyDSfMLe4FpQqfqeAtIzs=">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</latexit>@j• GWs and scalars sourced at one-loophhR<latexit sha1_base64="oagMSI9QcvIw6r9IEjxhuaqEkmw=">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</latexit>

R<latexit sha1_base64="oagMSI9QcvIw6r9IEjxhuaqEkmw=">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</latexit>• Generically:rsourced ⇠ ✏2 < rvac ⇠ ✏(Mirbabayi, Silverstein, Senatore, Zaldarriaga )Gravitational Waves from Inflation<latexit sha1_base64="uUN7nwl8p48uAOn9Pps9s4Bpmj8=">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</latexit>

2
H
hk ⇠
⇥
⇠
MPl
2⇡

Canonical quantization

✓

V
MPl

◆1/4

Massless free-field in de Sitter⇣ r ⌘1/4
16⇤inf ⇠ 10 GeV
0.01

Tilt inherited from evolution of H,
guaranteed to be red!

• Therefore observation of B-modes implies:
Inflation happened at the GUT scale
Gravity is quantized(The inflaton moved over a Planck sized region in fieldspace — Lyth-Turner bound)Gauge-field inflation loophole• Background vector field sidesteps thisby allowing

Aai Aaj
⇡ij ⇠
2
MPl

a
Ai = a<latexit sha1_base64="y68zEBauGSE2BLxK0vrkeZw0xRs=">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</latexit>

a
i,

⇠ 10

• Linear sourcing of gravitational waves:(p)hij (x) ⇠<latexit sha1_base64="hcqVwrxVz5roVwQrRPk056LiaNU=">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</latexit>Z

d4 y Glm
ij (x

y)

Aal

Aam (y)
2
MPl

h

Aai

Aai
<latexit sha1_base64="lc0moOy8G0QCQLN9qdaWw6Lz8Zo=">AAACq3icbVHJTsMwEHXDXnY4comokDhUUQJl6Y3lwhEkAoi2VI47CRaOHdkOqIr6DYgbfBp/g5OUNYxk6/nNG88WJIwq7brvNWticmp6ZnauPr+wuLS8srp2pUQqCfhEMCFvAqyAUQ6+pprBTSIBxwGD6+DhNPdfP4JUVPBLPUygF+OI05ASrA3lH9/hPu2vNFzHLcyuAm8MGmhs5/3V2kt3IEgaA9eEYaU6npvoXoalpoTBqN5NFSSYPOAIOgZyHIPqZUW1I3vLMAM7FNIcru2C/RmR4VipYRwYZYz1vfrry8n/fJ1Uh4e9jPIk1cBJmShMma2FnbduD6gEotnQAEwkNbXa5B5LTLQZ0K8sZaX17gBCM9filcXDJJWJ6S6TUTDKXGev1TRT8g7y+7BdUUcSgH+Km66z33QrmkCKpy+N0yq+2inv3YpYSMyj7/RtL9e1vLKIkVmi93dlVeDvOG3Hu2g1jk7G25xFG2gTbSMPHaAjdIbOkY8IougZvaI3y7EurVurW0qt2jhmHf0yCz4AzbHPnw==</latexit>• Analogous linear sourcing of scalars prohibited bydecomposition theorem• Gauge field fluctuations ‘oscillate’ into GWs• GW spectra (can be) set by gauge-field tensor
fluctuations

2

MPl

h
Aai
<latexit sha1_base64="lc0moOy8G0QCQLN9qdaWw6Lz8Zo=">AAACq3icbVHJTsMwEHXDXnY4comokDhUUQJl6Y3lwhEkAoi2VI47CRaOHdkOqIr6DYgbfBp/g5OUNYxk6/nNG88WJIwq7brvNWticmp6ZnauPr+wuLS8srp2pUQqCfhEMCFvAqyAUQ6+pprBTSIBxwGD6+DhNPdfP4JUVPBLPUygF+OI05ASrA3lH9/hPu2vNFzHLcyuAm8MGmhs5/3V2kt3IEgaA9eEYaU6npvoXoalpoTBqN5NFSSYPOAIOgZyHIPqZUW1I3vLMAM7FNIcru2C/RmR4VipYRwYZYz1vfrry8n/fJ1Uh4e9jPIk1cBJmShMma2FnbduD6gEotnQAEwkNbXa5B5LTLQZ0K8sZaX17gBCM9filcXDJJWJ6S6TUTDKXGev1TRT8g7y+7BdUUcSgH+Km66z33QrmkCKpy+N0yq+2inv3YpYSMyj7/RtL9e1vLKIkVmi93dlVeDvOG3Hu2g1jk7G25xFG2gTbSMPHaAjdIbOkY8IougZvaI3y7EurVurW0qt2jhmHf0yCz4AzbHPnw==</latexit>

Chiral GWs — observational
impact• Parity violating GWs

generate TB, EB spectra• GW helicity potentiallyobservable

TB
BB
EB
r = 0.35,= 0.9(Caldwell 2017)

=

2
R
2 +
R

2
L
2
L

• Typically| | & 0.9

(Gluscevic and Kamionkowski 2010)

Tensor non-G
hh(k1 )h(k2 )h(k3 )i = (2⇡)3 3

3
X
i=1

ki

!

Bhhh (k1 , k2 , k3 )• 3-pt function from vacuum modeshBhhh (k, k, k)⇠
O(1)P 2 (k)(Maldacena 2003)hh• n-pt statistics inherited from gauge field
Bhhh (k, k, k)⇠OP 2 (k)
1 a a⌦A = (Ei Ei + Bia Bia )
2

✓

10
⌦A

◆

h

1

(Agrawal, Fujita, Komatsu 2017)

Aai

h

h

Summary I•

Classical non-Abelian gauge fields allow for theconstruction of novel inflationary scenarios•

Symmetry breaking in models of inflation with nonAbelian gauge fields can bring them into agreementwith data•

Requires very weakly coupled gauge fields•

Chiral, possibly blue-tilted GW spectra•

Observable GW spectra at sub-GUT energies⇣ r ⌘1/4
p
16⇤inf ⇠ HMPl 6= 1.04 ⇥ 10 GeV
0.01

Implications for B-mode searches• Observation of B-modes does not imply• Inflation happened at the GUT scale• The inflaton moved over a Planck sized region infield space• Gravity is quantized
Unless:• Small Tensor Gaussianity• Vanishing of parity odd correlators (TB, EB)• Scale invariance…and the origin of thematter-antimatterasymmetry
Why is there stuﬀ?

Andrew Long

Evangelos Sfakianakis

PRD: 98, 043525 (2018), with A. Long and E. Sfakianakis

There is a matter-antimatter asymmetry• How do we know there are not vast anti-matter domains in the
Universe?

• Non-observation of gamma-ray emission excludes antimatter domains from within our horizon[Steigman (1976); Cohen, De Rujula, & Glashow (1998)]• If global universe is symmetric, our matter pocket is larger
than ~100 Gly

Quantitatively(Wayne Hu)⌦b h2 = 0.02205 ± 0.00028 ! YBnB
10
05 ± 0.00028 ! YB == (0.861 ± 0.008) ⇥ 10s<latexit sha1_base64="+/BiQyrpXQI3emRStOhJWSjXHSw=">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</latexit>• For every 1010 anti-protons there are ~ 1010 +1 protons

Inflation requires Baryogenesis!
• Inflation dilutes relics to unobservable small densities
Same is true of baryon density• Baryon asymmetry must be generated dynamically after inflation
Sakharov conditions:(Sakharov, 1967)

Baryon number violation
CP violation
Out of equilibrium(history.aip.org)

Can we baryogenesis happen during inflation?

Cosmological non-conservation offermions• Fermion number is not conserved cosmologically• Chiral fermion currents are anomalous<latexit sha1_base64="Fxn40ADnD5gIhKCB0VFdpJK5MdU=">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</latexit>T↵1µ
rµ jL,R = ±RR̃
24 16⇡ 2

T
Eguchi, Gilkey, Hanson (1980)• Standard model lepton-current is anomalousµrµ j ` =<latexit sha1_base64="nXP9j6rniBNSyXISzbSV4YkFFsk=">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</latexit>

3
X
i=1

rµ

⇣

µ
µ
µ
j ei + j ⌫ i + j ei
L
L
R

⌘

3 RR̃
=
24 16⇡ 2• Whenever RR̃ is non-zero, the lepton current changes<latexit sha1_base64="xE1YuDqdiduRL7mwQAbzjoQ/1gg=">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</latexit>

Alexander, Peskin, Sheikh-Jabbari (2004)

T↵

Gravitational Leptogenesis
T• Chiral gravitational waves generates asymmetries in chiral leptons
1
NL̄,R̄ = ±24(16⇡ 2 )NL,R<latexit sha1_base64="nNA2R6Y10qs1ijnsP1OLZa88L48=">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</latexit>

Z

4

d x

p

gRR̃

• At the end of inflation, we have in SM (for left-chiral GWs)
uR

dR

eR

uL

dL

eL

Nc Nf
Nf

⌫L

Small scale chiral GWs
L
R(Caldwell ’17)

• Blue-tilted spectra generates large gravitational
wave density on small scales• Fairly easy to generate suﬃcient net lepton number

How do (net) leptons getturned into (net) baryons?

• Hot electroweak sphaleron violates B+L, conserves B-L
L

B

L

B

L

B

28
YB =
YB L
79

but neutrino mass…
• But neutrinos are massive…
• Requires adding new degrees of freedom to SM
uR

dR

eR

⌫R

uL

dL

eL

⌫L

• Neutrinos are Dirac:• RH neutrinos are sterile, sequester lepton number
Result unchanged!

but neutrino mass…
• But neutrinos are massive…
• Requires adding new degrees of freedom to SM
uR

dR

eR

⌫R

uL

dL

eL

⌫L

• Neutrinos are Majorana:• Theory contains explicit lepton number violation• Neutrinos are Majorana:• Theory contains explicit lepton number violation…
LΦ

BaryonAsymmetry:
Asymmetry:YBY=
on
Asymmetry:/ s/ sBaryon
nBn/B s/ sB=
BnRelicryon
Asymmetry:YBY=B n=Relic
B

10-10
10-10

L

L

L

Φ

⌫

-12
10NR
10-12

⌫RN
Φ

Φ

L

⌫RN
L

Φ

Φ

⌫RN

L

Φ

L

Φ

10-14
10-14

• BUT! Small Yukawas sequester lepton number in right chiral leptons
eL

! eR

µL

10-16
-16
10
10-10
10-10

! µR

10-18
-18
10
10-12
10-12
10-20
-20
10
10-14 104
10-14 104

10-16
10-16

106
106

108
108

1010
1010

1012
1012

⌧L

! ⌧R

positive
positive
negative
negative
1014
1016
1014
1016

Majorana Mass Scale: mN [GeV]
Majorana Mass Scale: mN [GeV]

A definite connection between GW helicity and neutrino mass scale!

A lower bound on r?
• Generating a suﬃcient lepton number puts (so far, model
dependent) lower bounds on r:

Chromo-Caldwell inflation(Caldwell and Devulder 2017)• Can a more general lower bound be imposed?

Summary II• Observation of chiral GWs could imply• Inflationary origin of baryon asymmetry• Left Chiral GW implies Dirac ν, or high-scale see-saw• Right Chiral GW implies Majorana mass scale
6

12

10 GeV < mN < 10 GeV• Non-observation of B-modes likely rules out inflation as
origin of baryon asymmetry

Summary• Slow roll inflation is in excellent agreement for scalars:• Super-Horizon, isotropic, adiabatic, and red-tilted• But, “extraordinary claims require extraordinary evidence”
We seek gravitational waves with wavelengths ofbillions of light years

Summary I• Observation of B-modes does not imply• Inflation happened at the GUT scale• The inflaton moved over a Planck sized region infield space• Gravity is quantized
Unless:• Small Tensor Gaussianity• Vanishing of parity odd correlators (TB, EB)• Scale invarianceThanks!