C. Germani - Sub-Planckian Higgs and Axionic Inflations

Introduction

Slow roll

The failure ofHiggs boson

New HiggsInflation

GravitationallyEnhanced Friction

Introducing theSlotheon

Uniqueness

Unitarity

UV ProtectedInflation

Sub-Planckian Higgs and Axionic Inflations

Cristiano GermaniBased on different collaborations: Martucci, Moyassari, Kehagias, Watanabe

LMU, ASC, Munich, Germany

BW2011, August 2011, Donji Milanovac, Serbia

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

Latest cosmological data agree impressively well with the aUniverse which is at large scales

homogeneous,

isotropic

spatially flat

A flat FRW Spacetime!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

homogeneous,

isotropic

spatially flat

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

homogeneous,

isotropic

spatially flat

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

homogeneous,

isotropic

spatially flat

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

homogeneous,

isotropic

spatially flat

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

A theoretical puzzle:

A flat FRW Universe

ds2 = −dt2 + a(t)2d~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 ofthe Early Universe

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

A theoretical puzzle:

A flat FRW Universe

ds2 = −dt2 + a(t)2d~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 ofthe Early Universe

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

By geometrical identities (Raychaudhuri equation)

a ∝ −( ρ︸︷︷︸effective energy density

+ 3 p︸︷︷︸effective pressure

ρ+ 3p < 0 , during inflationρ+ 3p > 0 , after inflation

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Slow Roll

A scalar field φ is a good candidate as

2φ2+V , p =

2φ2−V

ρ+ 3p ∝ φ2−V

φ2 V , Inflation happens (“slow roll”)φ2 ∼ V , Inflation ends

Q: Do we know any scalar field?

Higgs Boson!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Slow Roll

2φ2+V , p =

2φ2−V

ρ+ 3p ∝ φ2−V

Higgs Boson!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Slow Roll

2φ2+V , p =

2φ2−V

ρ+ 3p ∝ φ2−V

Higgs Boson!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Slow Roll

2φ2+V , p =

2φ2−V

ρ+ 3p ∝ φ2−V

Higgs Boson!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Slow Roll

2φ2+V , p =

2φ2−V

ρ+ 3p ∝ φ2−V

Higgs Boson!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Slow Roll

2φ2+V , p =

2φ2−V

ρ+ 3p ∝ φ2−V

Higgs Boson!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

The failure of Higgs boson in GR

The Higgs Lagrangian is

∫d4x√−g

2− DµH†DµH− λ

(H†H− v 2

All fields but H are subdominant during Inflation

Unitary gauge HT = (0, v+Φ√2

In order to slow roll Φ v

∫d4x√−g

2κ2− 1

2∂µΦ∂µΦ− λ

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∫d4x√−g

(H†H− v 2

∫d4x√−g

2κ2− 1

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∫d4x√−g

(H†H− v 2

∫d4x√−g

2κ2− 1

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∫d4x√−g

(H†H− v 2

∫d4x√−g

2κ2− 1

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∫d4x√−g

(H†H− v 2

∫d4x√−g

2κ2− 1

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Inflation:

H = a/a ' const (exponential expansion)

Φ 3HΦ (slow roll)

ε ≡ − HH2 1

Φ Mp !!!!

R M2p for the Standard Model values of λ ∼ 10−1 !

Inflation happens during the Quantum Gravity regime!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Inflation:

Φ 3HΦ (slow roll)

ε ≡ − HH2 1

Φ Mp !!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Inflation:

Φ 3HΦ (slow roll)

ε ≡ − HH2 1

Φ Mp !!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Inflation:

Φ 3HΦ (slow roll)

ε ≡ − HH2 1

Φ Mp !!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Lowering the curvature during InflationRecapitulate:

R ∼ H2 ∝ V (Φ)

∝ Φ4

Φ ' −MpΦ

ε = − H

H2∝ Φ2

Φ2 1⇒ Φ Mp

Solution: Increase the friction!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

R ∼ H2 ∝ V (Φ)

∝ Φ4

Φ ' −MpΦ

ε = − H

H2∝ Φ2

Φ2 1⇒ Φ Mp

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

R ∼ H2 ∝ V (Φ)

∝ Φ4

Φ ' −MpΦ

ε = − H

H2∝ Φ2

Φ2 1⇒ Φ Mp

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

R ∼ H2 ∝ V (Φ)

∝ Φ4

Φ ' −MpΦ

ε = − H

H2∝ Φ2

Φ2 1⇒ Φ Mp

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

R ∼ H2 ∝ V (Φ)

∝ Φ4

Φ ' −MpΦ

ε = − H

H2∝ Φ2

Φ2 1⇒ Φ Mp

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

New Higgs Inflation

How to increase the friction:

if Φ→ Ω2Φ with Ω 1

ε ∝ Φ2

Φ2Ω4 1→ M2

p R M2

Ω8(for large enough Ω)

Quantum Gravity regime is avoided during Inflation!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

New Higgs Inflation

ε ∝ Φ2

Φ2Ω4 1→ M2

p R M2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

New Higgs Inflation

ε ∝ Φ2

Φ2Ω4 1→ M2

p R M2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

New Higgs Inflation

ε ∝ Φ2

Φ2Ω4 1→ M2

p R M2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

New Higgs Inflation

ε ∝ Φ2

Φ2Ω4 1→ M2

p R M2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Gravitationally Enhanced Friction (GEF)

The friction should only be efficient at high energies:

Ω2 ∼ 3µH = f (H), dfdH ≥ 0

A typical (positive) enhancement could be

µ = 1 + H2

If no new d.o.f. are added, the scalar e.o.m. can only be

Φ + 3HΦ)

= −V ′ → teff ' t√µ as µ

If H M the scalar field clock is frozen w.r.t. the observerand friction is enhanced.

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Ω2 ∼ 3µH = f (H), dfdH ≥ 0

µ = 1 + H2

Φ + 3HΦ)

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Ω2 ∼ 3µH = f (H), dfdH ≥ 0

µ = 1 + H2

Φ + 3HΦ)

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Ω2 ∼ 3µH = f (H), dfdH ≥ 0

µ = 1 + H2

Φ + 3HΦ)

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Gravitationally Enhanced Friction: Realization

We promote the rescaling to all coords.

∂µ →√µ∂µ, µ = 1 + H2

Gαβ ' −H2gαβ during Inflation

gµν∂µΦ∂νΦ→(gµν − Gµν

)∂µΦ∂νΦ

New Higgs Inflation Lagrangian

∫d4x√−g

2κ2− 1

(gαβ − Gαβ

)∂αΦ∂βΦ− λ

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∂µ →√µ∂µ, µ = 1 + H2

)∂µΦ∂νΦ

∫d4x√−g

2κ2− 1

(gαβ − Gαβ

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∂µ →√µ∂µ, µ = 1 + H2

)∂µΦ∂νΦ

∫d4x√−g

2κ2− 1

(gαβ − Gαβ

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

∂µ →√µ∂µ, µ = 1 + H2

)∂µΦ∂νΦ

∫d4x√−g

2κ2− 1

(gαβ − Gαβ

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

The New Higgs inflation is a Slotheonic theory

LK = −1

(gαβ − Gαβ

)∂αΦ∂βΦ

The name comes to the fact that on a given metric

HK ∼ Φ2

)≥ Φ2

⇓Given a HK the Slotheon is slower than the canonical cousin!

But this is another story...

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

LK = −1

(gαβ − Gαβ

)∂αΦ∂βΦ

HK ∼ Φ2

)≥ Φ2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

LK = −1

(gαβ − Gαβ

)∂αΦ∂βΦ

HK ∼ Φ2

)≥ Φ2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

LK = −1

(gαβ − Gαβ

)∂αΦ∂βΦ

HK ∼ Φ2

)≥ Φ2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Uniqueness

We found a realization of the ”New Higgs Inflation” ideaIs this unique?

Require only a spin-2 and a spin-0 degrees of freedomto propagate (in general background)

Modify only the kinetic term

The unique action is the New Higgs Inflation action!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Uniqueness

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Uniqueness

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Uniqueness

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

By Bianchi identities: δΦGαβ∂αΦ∂βΦ→ Gαβ∇α∇βΦNo higher derivatives!

In ADM language: no. of time derivatives inGαβ∂αΦ∂βΦ never exceed the number of fields!

Any intercation with Φ2 (for example G ttΦ2) isdangerous! (because of higher derivatives)

However: G tt is special in ADM as it is the Hamiltonianconstraint (only 1 derivative)!N.B.: Any other curvature interaction would bring higher

derivatives increasing the no. of propagating modes !!!

In ADM language: Lapse and Shift are still Lagrangemultipliers → graviton propagates only 2 polarization

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

However: G tt is special in ADM as it is the Hamiltonianconstraint (only 1 derivative)!

N.B.: Any other curvature interaction would bring higher

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Proof:

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Does the non-renormalizable operator

I =Gαβ

M2∂αΦ∂βΦ

violates unitarity during Inflation since H2 M2?

Check list:

Expand the fields at linear level

Canonically normalize the Higgs: Φ = Φ0 + M√3Hφ

(the non-standard normalization comes from M−2G tt φ2)

Canonically Normalize the metric: gµν = g 0µν + 1

Mphµν

Read out the scale at which I ∼ O(1):No additional constraint than QG constraint H Mp!

Unitarity is not violated up to the Quantum Gravity scales!!!!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

I =Gαβ

M2∂αΦ∂βΦ

violates unitarity during Inflation since H2 M2?Check list:

Mphµν

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

I =Gαβ

M2∂αΦ∂βΦ

Mphµν

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

I =Gαβ

M2∂αΦ∂βΦ

Mphµν

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

I =Gαβ

M2∂αΦ∂βΦ

Mphµν

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

I =Gαβ

M2∂αΦ∂βΦ

Mphµν

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

I =Gαβ

M2∂αΦ∂βΦ

Mphµν

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

UV Protected Inflation

In large field scenariosΦ Λcut−off

The (unknown) UV completed theory may spoil the effectiveInflaton potential by higher powers of Φ2/Λ2

cut−off , unless...

Some symmetries protect the potential

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Suppose some global symmetry is broken at energiesf >TeV (like in the QCD axion case)

a Pseudo Nambu-Goldstone Boson Φ is produced with a(one loop) potential

V (Φ) ' 2Λ4

(1− Φ2

)which is protected by the restoration of global shiftsymmetry Φ→ Φ + c at Λ→ 0

With Λ Mp, Inflation predicts

ns − 1 ∝ ε ' − M2p

8πf 2

so ns ≤ 1→ f > Mp ⇒ the model cannot be trusted!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

V (Φ) ' 2Λ4

(1− Φ2

ns − 1 ∝ ε ' − M2p

8πf 2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

V (Φ) ' 2Λ4

(1− Φ2

ns − 1 ∝ ε ' − M2p

8πf 2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

V (Φ) ' 2Λ4

(1− Φ2

ns − 1 ∝ ε ' − M2p

8πf 2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Resolution

Once again we can increase the friction so that

ε→ εold

Ω2⇒ ns − 1 ∼ −

8πf Ω2

Then for large enough Ω, f Mp!!!!

The model is Natural!!!(i.e. no UV modifications of the potential)

The new coupling Gαβ∂αΦ∂βΦ is the unique that

Does not introduce new degrees of freedom

Is invariant under the global unbroken symmetryΦ→ Φ + c

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Resolution

ε→ εold

Ω2⇒ ns − 1 ∼ −

8πf Ω2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Resolution

ε→ εold

Ω2⇒ ns − 1 ∼ −

8πf Ω2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Resolution

ε→ εold

Ω2⇒ ns − 1 ∼ −

8πf Ω2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Resolution

ε→ εold

Ω2⇒ ns − 1 ∼ −

8πf Ω2

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Quantum gravity involvement to the one loop potential?

Since the symmetry is broken to a discrete group, Quantumgravity respect the symmetry

⇓The only effect is to shift Λ→ Λ0 + ΛQG

⇓The form of the potential is unchanged, Λ fixed by

observations!

Introduction

Slow roll

New HiggsInflation

Uniqueness

Unitarity

Hvala!

C. Germani - Sub-Planckian Higgs and Axionic Inflations

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