& EARLY UNIVERSE COSMOLOGY

(1)

GRAVITATIONAL WAVES

AS A PROBE OF EARLY UNIVERSE COSMOLOGY

DANIEL G. FIGUEROA

LPPC, EPFL, Lausanne

Rencontres de Moriond: Cosmology 2018 (March 17-24). La Thuile, Italy Rencontres de Moriond, Gravitation 2019, March 24-30, La Thuile, Italy

& EARLY UNIVERSE COSMOLOGY

Invisibles19, Valencia, Spain, Europe, Earth, Solar System, Milky Way, Universe

(soon to be drinking tons of )

(2)

[LIGO & Virgo Scientific Collaborations]

[LIGO & Virgo Scientific Collaborations (arXiv:1602.03841)]

Gravitational Waves (GWs)

detected !

Einstein 1916 … LIGO/VIRGO 2015/16/17

We can observe the Universe

through GWs

Milestone

2015-2019

(3)

* Late Universe: Hubble diagram from Binaries

* Early Universe: High Energy Particle Physics

Cosmology with GWs

Can we really probe High Energy Physics

using Gravitational Waves (GWs) ? How ?

(4)

Particle Production

( t . 1s)

Cosmic Defects Quantum

Fluctuations

Phase Transitions

The Early Universe

(5)

( t . 1s)

Probe of the early Universe

GWs

The Early Universe

Particle Production

Fluctuations

Phase

Transitions

(6)

( t . 1s)

Probe of the early Universe

GWs

The Early Universe

Particle Production

Fluctuations

Phase Transitions

‘Holy Grail’ of Stochastic GW

Backgrounds

(N. Christensen)

(7)

GRAVITATIONAL WAVES (GW): PROBING the EARLY UNIVERSE (t . 1 s)

1

WEAKNESS of GRAVITY:

ADVANTAGE: GW DECOUPLE upon Production DISADVANTAGE: DIFFICULT DETECTION

2

ADVANTAGE: GW ! Probe for Early Universe

!

⇢ Decouple ! Spectral Form Retained Specific HEP , Specific GW

3

Physical Processes:

8 >

> <

> >

:

Inflation Reheating

Phase Transitions Turbulence

GWs: probe of the early Universe

Cosmic Defects

@ Early Universe

Well motivated case !

(8)

0) GW definition

OUTLINE

1) GWs from Inflation

2) GWs from Preheating

3) GWs from Phase Transitions 4) GWs from Cosmic Defects Early

Universe

(9)

Gravitational Waves in Cosmology

1. Gravitational Waves (GWs) [Basics]

• GW: ds

²

= a

²

( d⌘

²

+ (

_ij

+ h

_ij

)dx

ⁱ

dx

^j

), TT :

⇢ h

_ii

= 0 h

_ij

,

_j

= 0

Eom: h

⁰⁰_ij

+ 2 H h

⁰_ij

r

²

h

_ij

= 16⇡ G⇧

^TT_ij

, ⇧

_ij

= T

_ij

h T

_ij

i

_FRW

Transverse-Traceless (TT) dof carry energy out of the source!!!

• GW Source(s): ( SCALARS , VECTOR , FERMIONS )

⇧

^{T T}_ij

/ { @

_i ^a

@

_j ^a

}

^{T T}

, { E

_i

E

_j

+ B

_i

B

_j

}

^{T T}

, { ¯

_i

D

_j

}

^{T T}

FRW:

• GW: ds 2 = a 2 ( d⌘ 2 + ( ij + h ij )dx i dx j ), TT :

⇢ h ii = 0 h ij , j = 0

Eom: h 00 ij + 2 H h 0 ij r 2 h ij = 16⇡ G⇧ TT ij , ⇧ ij = T ij h T ij i

_FRW

Transverse-Traceless (TT) dof carry energy out of the source!!!

⇧ T T ij / { @ i a @ j a } T T , { E i E j + B i B j } T T , { ¯ i D j } T T

• GW: ds

²

= a

²

( d⌘

²

+ (

_ij

+ h

_ij

)dx

ⁱ

dx

^j

), TT :

⇢ h

_ii

= 0 h

_ij

,

_j

= 0

Eom: h

⁰⁰_ij

+ 2 H h

⁰_ij

r

²

h

_ij

= 16⇡G⇧

^TT_ij

, ⇧

_ij

= T

_ij

h T

_ij

i

_FRW

Transverse-Traceless (TT) dof carry energy out of the source!!!

⇧

^{T T}_ij

/ { @

_i ^a

@

_j ^a

}

^{T T}

, { E

_i

E

_j

+ B

_i

B

_j

}

^{T T}

, { ¯

_i

D

_j

}

^{T T}

Source: Anisotropic Stress

Creation/Propagation GWs

Transverse-Traceless (TT)

(10)

0) GW definition

Gravitational Waves as a probe of the early Universe

OUTLINE

1) GWs from Inflation

2) GWs from Preheating

3) GWs from Phase Transitions 4) GWs from Cosmic Defects Early

Universe

(11)

Irreducible GW background from Inflation

⌦

^(o)_GW

(f ) ⌘ 1

⇢

^(o)c

✓ d log ⇢

_GW

d log k

◆

o

= ⌦

^(o)_Rad

24

2h_⇤

(k )

²_h

(k) = 2

⇡

²

✓ H m

_p

◆

2

✓ k aH

◆

n_t

n

_t

⌘ 2✏

-18.0 -14.0 -10.0 -6.0 -2.0 2.0 6.0 10.0

Log[f]

-16.0 -14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0

Log[h 0

2 Ω GW]

Figure 12: The spectrum of amplification of vacuum fluctuations produced by a phase of De Sitter inflation (solid line), with a value of H that saturates the COBE bound. The nucleosynthesis bound (dotted line) and the pulsar bound (triangle shaped) of fig. 11 are also shown for comparison.

63

scale-invariant (RD modes) (MD modes)

⌦

_GW

/ 1/k

²

red tilted (quasi-) scale-invariant (RD modes)

}

T (k) / k

⁰

(RD)

Transfer Funct.:

aLIGO

LISA

energy scale

d⇢

_GW

d log k

(12)

Irreducible GW background from Inflation

⌦

^(o)_GW

(f ) ⌘ 1

⇢

^(o)c

✓ d log ⇢

_GW

d log k

◆

o

= ⌦

^(o)_Rad

24

2h_⇤

(k )

²_h

(k) = 2

⇡

²

✓ H m

_p

◆

2

✓ k aH

◆

n_t

n

_t

⌘ 2✏

-18.0 -14.0 -10.0 -6.0 -2.0 2.0 6.0 10.0

Log[f]

-16.0 -14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0

Log[h 0

2 Ω GW]

63

scale-invariant (RD modes) (MD modes)

⌦

_GW

/ 1/k

²

}

T (k) / k

⁰

(RD)

Transfer Funct.:

aLIGO

LISA

energy scale

Not Obs erva ble !

d⇢

_GW

d log k

(13)

Irreducible GW background from Inflation

⌦

^(o)_GW

(f ) ⌘ 1

⇢

^(o)c

✓ d log ⇢

_GW

d log k

◆

o

= ⌦

^(o)_Rad

24

2h_⇤

(k )

²_h

(k) = 2

⇡

²

✓ H m

_p

◆

2

✓ k aH

◆

n_t

n

_t

⌘ 2✏

-18.0 -14.0 -10.0 -6.0 -2.0 2.0 6.0 10.0

Log[f]

-16.0 -14.0 -12.0 -10.0 -8.0 -6.0 -4.0 -2.0 0.0

Log[h 0

2 Ω GW]

63

scale-invariant (RD modes) (MD modes)

⌦

_GW

/ 1/k

²

}

T (k) / k

⁰

(RD)

Transfer Funct.:

aLIGO

LISA

energy scale

Not Obs erva ble !

except (perhaps)

indir ectly , @ the CMB …

d⇢

_GW

d log k

(14)

INFLATIONARY COSMOLOGY

( initial cond. )

Inflation = {

Scenarios

Primordial perturbations

Tensor Scalar

Irreducible GW Background

Enhanced GWs Enhanced Scalar Pert.

Extra species/symmetries

Modified Gravity, spectator fields, graviton mass, …

{

(15)

INFLATIONARY COSMOLOGY

( initial cond. )

Inflation = {

Scenarios

Primordial perturbations

Tensor Scalar

Enhanced GWs Enhanced Scalar Pert.

Extra species/symmetries

Modified Gravity, spectator fields, graviton mass, …

{

Irreducible GW

Background

(16)

INFLATIONARY MODELS

We focus on a specific scenario, with a natural class of models of inflation and with a mechanism of production / exp ˙

GW signal naturally grows at interferometer scales, while small at CMB scales

• CMB in agreement with simplest models of slow-roll cosmic inflation

r ⌘ P GW

P ⇣ , n s 1 ⌘ d ln P ⇣

d ln k , f NL ⇠ h ⇣ 3 i

h ⇣ 2 i 2 = O (✏ V , ⌘ V )

✏ V ⌘ M p 2 2

⇣ V

,

V

⌘ 2

⌧ 1 ⌘ V ⌘ M p 2 V ,

V ⌧ 1

• Flatness and gaussianity ! small self-couplings Axion (Natural) Inflation

• Shift symmetry on couplings to other fields Freese, Frieman, Olinto ’90; . . .

(review Pajer, MP ’13)

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

has the advantage that

• Smallness of V shift technically natural. V / V shift

• Constrained couplings to matter (predictivity)

We focus on a specific scenario, with a natural class of models of inflation and with a mechanism of production / exp ˙

r ⌘ P GW

P ⇣ , n s 1 ⌘ d ln P ⇣

d ln k , f NL ⇠ h ⇣ 3 i

h ⇣ 2 i 2 = O (✏ V , ⌘ V )

✏ V ⌘ M p 2 2

⇣ V

,

V

⌘ 2

⌧ 1 ⌘ V ⌘ M p 2 V ,

V ⌧ 1

• Flatness and gaussianity ! small self-couplings Axion (Natural) Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

has the advantage that

GW signal naturally grows at interferometer scales, while small at CMB scales

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

technically natural. V / V

_shift

• Constrained couplings to matter (predictivity) We focus on a specific scenario, with a natural class of models of inflation

and with a mechanism of production / exp ˙

• CMB in agreement with simplest models of slow-roll inflation

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏, ⌘)

• Flatness and gaussianity ! small inflaton self-couplings

• Shift symmetry (broken by V ) on couplings to other fields has advantages

Freese, Frieman, Olinto ’90; . . . (review Pajer, MP ’13)

Planck ’15

P

_s

/ k

ⁿ^s ¹

r = P

_t

P

_s

• No observed departure from (primordial) gaussianity, h ⇣

³

i ⌧ h ⇣

²

i

^3/2

Local NG : (x) =

_g

(x) + f

_NL^local

⇥

₂

g

(x) ⌦

₂

g

↵⇤

• Agreement with standard single field slow roll

r, n

_s

1, f

_NL

= O (✏, ⌘)

✏ ⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 , ⌘ ⌘ M

_p²

V

_,

V ⌧ 1

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏, ⌘)

Freese, Frieman, Olinto ’90; . . . (review Pajer, MP ’13)

Planck ’15

P

_s

/ k

ⁿ^s ¹

r = P

_t

P

_s

³

i ⌧ h ⇣

²

i

^3/2

Local NG : (x) =

_g

(x) + f

_NL^local

⇥

₂

g

(x) ⌦

₂

g

↵⇤

r, n

_s

1, f

_NL

= O (✏, ⌘)

✏ ⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 , ⌘ ⌘ M

_p²

V

_,

V ⌧ 1

Axion ⇤ Inflation

• Shift symmetry ! + C on couplings to other fields Freese, Frieman, Olinto ’90; . . .

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

⇤ Not the QCD axion; reference values f ⇠ 10 16 GeV , m ' 10 13 GeV

Axion

^⇤

Inflation

• Shift symmetry on couplings to other fields

Freese, Frieman, Olinto ’90; . . . (review Pajer, MP ’13)

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

⇤

Not the QCD axion; reference values f ⇠ 10

¹⁶

GeV , m ' 10

¹³

GeV

Shi ft s ym me try ! + C . E .g. axi on (na tur al) infl ation

Fre ese , F riem an, Oli nto ’90

L = 1

2 ( @ µ ) 2 + V shif t ( ) + C f

@ µ ¯ µ 5

+

↵ f

F µ ⌫ F ˜ µ ⌫

• Sm allne ss of V shif t

tec hni cal ly na tur al. V / V shif t

• Cons tra ine d c oupl ings to ma tte r (p redi ctiv ity)

! '

C 2

2 ⇡ f 2

m m 2

! AA =

↵ 2

64 ⇡ f 2

m 3

! AA typi cal ly c ont rols rehe ating.

Onl y re cent ly r eal ized

tha t it can pla y a n im por tant role also dur ing infl ation.

— —

Couplings in axion inflation

Freese, Frieman, Olinto ’90 . . .

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

• Constrained couplings to matter (predictivity)

! ' c 2

2 ⇡ f 2 m m 2 ! AA = ↵ 2

64 ⇡ f 2 m 3

! AA typically controls reheating. More recently, realized that it can play an important role also during inflation

Axion ⇤ Inflation

• Shift symmetry ! + C on couplings to other fields

Freese, Frieman, Olinto ’90; . . . (review Pajer, MP ’13)

L = 1

2 ( @ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

• Constrained couplings to matter (predictivity)

⇤ Not the QCD axion; reference values f ⇠ 10 16 GeV , m ' 10 13 GeV We focus on a specific scenario, with a natural class of models of inflation

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

has the advantage that

⇤ Not the QCD axion; reference values f ⇠ 10 16 GeV , m ' 10 13 GeV Slow roll inflation requires very flat potential V , and, generically,

hard to explain why V protected against quantum corrections.

With shift symmetry, V / V shift

We focus on a specific scenario, with a natural class of models of inflation and with a mechanism of production / exp ˙

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

hard to explain why V protected against quantum corrections.

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

has the advantage that

inflaton = pseudo-scalar axion ' V (') + ↵

f F

_µ⌫

F ˜

^µ⌫

r ⌘ P

_GW

P

_⇣

, n

s

1 ⌘ d ln P

_⇣

d ln k , f

NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

• Shift symmetry on couplings to other fields Freese, Frieman, Olinto ’90; . . .

L = 1

2 (@

µ

)

²

+ V

_shift

( ) + c

f @

µ

¯

^µ ₅

+ ↵

f F

µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

• Flatness and gaussianity ! small self-couplings Axion (Natural) Inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

• CMB in agreement with simplest models of slow-roll cosmic inflation

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

µ

¯

^µ ₅

+ ↵

f F

µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

• Constrained couplings to matter (predictivity) We focus on a specific scenario, with a natural class of models of inflation

and with a mechanism of production / exp ˙

r ⌘ PGW

P_⇣ , n_s 1 ⌘ dlnP_⇣

dlnk , fNL ⇠ h⇣³i

h⇣²i² = O (✏,⌘)

Planck ’15

Ps / kⁿ^s ¹ r = Pt

Ps

• No observed departure from (primordial) gaussianity, h⇣³i ⌧ h⇣²i^3/2

Local NG : (x) =

_g

(x) + f

_NL^local

⇥

₂

g

(x) ⌦

₂

g

↵⇤

r, ns 1, f_NL = O (✏,⌘)

✏ ⌘ M_p² 2

⇣

_V

,

V

⌘

²

⌧ 1 , ⌘ ⌘ M_p² V_,

V ⌧ 1

r ⌘ P_GW

P_⇣ , ns 1 ⌘ dlnP_⇣

dlnk , f_NL ⇠ h⇣³i

h⇣²i² = O (✏,⌘)

Planck ’15

P_s / kⁿ^s ¹ r = P_t P_s

• No observed departure from (primordial) gaussianity, h⇣³i ⌧ h⇣²i^3/2

Local NG : (x) = _g (x) + f_NL^local ⇥ ₂

g (x) ⌦ ₂

g

↵⇤

r, n_s 1, f_NL = O (✏,⌘)

✏ ⌘ M_p² 2

⇣_V

,

V

⌘²

⌧ 1 , ⌘ ⌘ M_p² V_,

V ⌧ 1

We focus on a specific scenario, with a natural class of models of inflation and with a mechanism of production / exp ˙

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

Axion^⇤ Inflation

L = 1

2 (@_µ )² + V_shift ( ) + c

f @_µ ¯ ^µ ₅ + ↵

f F_µ⌫ F˜^µ⌫

• Smallness of V_shift technically natural. V / V_shift

⇤ Not the QCD axion; reference values f ⇠ 10¹⁶GeV , m ' 10¹³ GeV

Shi ft s ym me try ! + C . E .g. axi on (na tur al) infl ation

L = 1

2 ( @ µ ) 2 + V shif t ( ) + C f

@ µ ¯ µ 5

+ ↵ f

F µ ⌫ F ˜ µ ⌫

• Cons tra ine d c oupl ings to ma tte r (p redi ctiv ity)

! '

C 2

2 ⇡ f 2

m m 2

! AA =

↵ 2

64 ⇡ f 2

m 3

— —

Couplings in axion inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

!

' c

²

2⇡f

²

m m

² _!_AA

= ↵

²

64⇡ f

²

m

³

Axion

^⇤

Inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

µ

¯

^µ ₅

+ ↵

f F

µ⌫

F ˜

^µ⌫

has the advantage that

• Smallness of V

_shift

⇤

¹⁶

GeV , m ' 10

¹³

GeV We focus on a specific scenario, with a natural class of models of inflation

Axion

^⇤

Inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

⇤

¹⁶

GeV , m ' 10

¹³

GeV Slow roll inflation requires very flat potential V , and, generically,

hard to explain why V protected against quantum corrections.

With shift symmetry, V / V

_shift

Axion

^⇤

Inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

has the advantage that

• Smallness of V

_shift

⇤

¹⁶

GeV , m ' 10

¹³

GeV Slow roll inflation requires very flat potential V , and, generically,

_shift

We focus on a specific scenario, with a natural class of models of inflation and with a mechanism of production / exp ˙

Axion

^⇤

Inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

⇤

¹⁶

GeV , m ' 10

¹³

GeV Slow roll inflation requires very flat potential V , and, generically,

shift

Axion-Inflation ' ! ' + const.

'

(17)

INFLATIONARY MODELS

• CMB in agreement with simplest models of slow-roll cosmic inflation

r ⌘ P GW

P ⇣ , n s 1 ⌘ d ln P ⇣

d ln k , f NL ⇠ h ⇣ 3 i

h ⇣ 2 i 2 = O (✏ V , ⌘ V )

✏ V ⌘ M p 2 2

⇣ V

,

V

⌘ 2

⌧ 1 ⌘ V ⌘ M p 2 V ,

V ⌧ 1

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

GW signal naturally grows at interferometer scales, while small at CMB scales

r ⌘ P GW

P ⇣ , n s 1 ⌘ d ln P ⇣

d ln k , f NL ⇠ h ⇣ 3 i

h ⇣ 2 i 2 = O (✏ V , ⌘ V )

✏ V ⌘ M p 2 2

⇣ V

,

V

⌘ 2

⌧ 1 ⌘ V ⌘ M p 2 V ,

V ⌧ 1

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

• Constrained couplings to matter (predictivity)

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

• Shift symmetry on couplings to other fields Freese, Frieman, Olinto ’90; . . .

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

• Constrained couplings to matter (predictivity) We focus on a specific scenario, with a natural class of models of inflation

• CMB in agreement with simplest models of slow-roll inflation

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏, ⌘)

Planck ’15

P

_s

/ k

ⁿ^s ¹

r = P

_t

P

_s

³

i ⌧ h ⇣

²

i

^3/2

Local NG : (x) =

_g

(x) + f

_NL^local

⇥

₂

g

(x) ⌦

₂

g

↵⇤

r, n

_s

1, f

_NL

= O (✏, ⌘)

✏ ⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 , ⌘ ⌘ M

_p²

V

_,

V ⌧ 1

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏, ⌘)

• Flatness and gaussianity ! small inflaton self-couplings

Planck ’15

P

_s

/ k

ⁿ^s ¹

r = P

_t

P

_s

³

i ⌧ h ⇣

²

i

^3/2

Local NG : (x) =

_g

(x) + f

_NL^local

⇥

₂

g

(x) ⌦

₂

g

↵⇤

r, n

_s

1, f

_NL

= O (✏, ⌘)

✏ ⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 , ⌘ ⌘ M

_p²

V

_,

V ⌧ 1

GW signal naturally grows at interferometer scales, while small at CMB scales

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

Axion

^⇤

Inflation

Freese, Frieman, Olinto ’90; . . . (review Pajer, MP ’13)

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

⇤

¹⁶

GeV , m ' 10

¹³

GeV

Shi ft s ym me try ! + C . E .g. axi on (na tur al) infl ation

L = 1

2 ( @ µ ) 2 + V shif t ( ) + C f

@ µ ¯ µ 5

+

↵ f

F µ ⌫ F ˜ µ ⌫

• Cons tra ine d c oupl ings to ma tte r (p redi ctiv ity)

! '

C 2

2 ⇡ f 2

m m 2

! AA =

↵ 2

64 ⇡ f 2

m 3

— —

Couplings in axion inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

• Constrained couplings to matter (predictivity)

! ' c 2

2 ⇡ f 2 m m 2 ! AA = ↵ 2

64 ⇡ f 2 m 3

Axion ⇤ Inflation

L = 1

2 ( @ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

• Constrained couplings to matter (predictivity)

⇤ Not the QCD axion; reference values f ⇠ 10 16 GeV , m ' 10 13 GeV We focus on a specific scenario, with a natural class of models of inflation

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

has the advantage that

We focus on a specific scenario, with a natural class of models of inflation and with a mechanism of production / exp ˙

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

hard to explain why V protected against quantum corrections.

Axion ⇤ Inflation

L = 1

2 (@ µ ) 2 + V shift ( ) + c

f @ µ ¯ µ 5 + ↵

f F µ⌫ F ˜ µ⌫

has the advantage that

[J. Cook, L. Sorbo (arXiv:1109.0022)] [N. Barnaby, E. Pajer, M. Peloso (arXiv:1110.3327)]

inflaton = pseudo-scalar axion '

Chiral instability

where “h.c.” denotes the Hermitian conjugate of the preceding term, the annihilation/creation operators obey

! a

_λ

(k), a

^†_λ_′

(k

^′

) "

= δ

_λλ^′

δ

⁽³⁾

(k − k

^′

) (2.9) Here ⃗ϵ

_λ

are circular polarization vectors satisfying ⃗ k · ⃗ϵ

_±

#

⃗ k $

= 0, ⃗ k × ⃗ϵ

_±

#

⃗ k $

= ∓ ik⃗ϵ

_±

#

⃗ k $ ,

⃗ϵ

_±

#

− ⃗ k $

= ⃗ϵ

_±

#

⃗ k $

_∗

, and normalized according to ⃗ϵ

_λ

#

⃗ k $

_∗

· ⃗ϵ

_λ^′

#

⃗ k $

= δ

_λλ^′

.

Inserting the decomposition (2.8) into eq. (2.5) results in the equation of motion

% ∂

²

∂τ

²

+ k

²

± 2kξ τ

&

A

_±

(τ , k) = 0, ξ ≡ α φ ˙

2f H (2.10)

for the c-number mode functions A

_±

. During inflation the parameter ξ may be treated as constant, as its time variation is subleading in a slow roll expansion.

From equation (2.10) we see that one of the polarizations of A ⃗

_λ

experiences a tachyonic instability for k/(aH ) ∼

^<

2ξ . Without loss of generality, we assume that ˙ φ > 0 during inflation, so that the mode exhibiting the instability is A

₊

. In appendix A we review the solutions of (2.10) and show that the growth of fluctuations is well described by [15]

A

₊

(τ , k) ∼ = 1

√ 2k

' k

2ξ aH

(

_1/4

e

^πξ⁻²

√

2ξk/(aH)

(2.11)

in the interval (8ξ )

⁻¹

∼

^<

k/(aH ) ∼

^<

2ξ [18] of phase space that accounts for most of the power in the produced gauge fluctuations. The phase space of growing modes is non- vanishing for ξ ∼

^>

O (1), which we assume throughout. Notice the exponential enhancement e

^πξ

in the solution (2.11), which arises due to tachyonic instability, and reflects significant nonperturbative gauge particle production in the regime ξ ∼

^>

1. On the other hand, the production of gauge field fluctuations is uninterestingly small for ξ < 1. Note also that the other polarization state, A

₋

(τ , k), is not produced and can therefore be ignored.

We have thus seen that the motion of the homogeneous inflaton φ(t) leads to production of gauge field quanta δA

_µ

. There are two key physical effects associated with the interactions of these produced quanta with the inflaton. The first effect is the backreaction of the produced quanta on the homogeneous dynamics of φ(t), a(t). In the next subsec- tion we study the conditions under which backreaction effects are negligible. The second key physical effect is the production of inflaton fluctuations via inverse decay ; this is the subject of subsection 2.3.

2.2 Backreaction Eﬀects

Backreaction eﬀects can be accounted for using the mean of the field equations (2.6):

φ ¨ + 3H φ ˙ + V

^′

(φ) = α

f ⟨ E ⃗ · B ⃗ ⟩ (2.12)

3H

²

= 1 M

_p²

% 1

2 φ ˙

²

+ V (φ) + 1

2 ⟨ E ⃗

²

+ B ⃗

²

⟩

&

(2.13) where we have switched to physical time. The expectation values appearing in (2.12,2.13) encode the backreaction of the produced gauge quanta on the homogeneous dynamics of

– 6 –

where “h.c.” denotes the Hermitian conjugate of the preceding term, the annihilation/creation operators obey

! a

_λ

(k), a

^†_λ_′

(k

^′

) "

= δ

_λλ^′

δ

⁽³⁾

(k − k

^′

) (2.9) Here ⃗ϵ

_λ

are circular polarization vectors satisfying ⃗ k · ⃗ϵ

_±

#

⃗ k $

= 0, ⃗ k × ⃗ϵ

_±

#

⃗ k $

= ∓ ik⃗ϵ

_±

#

⃗ k $ ,

⃗ϵ

_±

#

− ⃗ k $

= ⃗ϵ

_±

#

⃗ k $

_∗

, and normalized according to ⃗ϵ

_λ

#

⃗ k $

_∗

· ⃗ϵ

_λ^′

#

⃗ k $

= δ

_λλ^′

.

Inserting the decomposition (2.8) into eq. (2.5) results in the equation of motion

% ∂

²

∂τ

²

+ k

²

± 2kξ τ

&

A

_±

(τ , k) = 0, ξ ≡ α φ ˙

2f H (2.10)

for the c-number mode functions A

_±

. During inflation the parameter ξ may be treated as constant, as its time variation is subleading in a slow roll expansion.

From equation (2.10) we see that one of the polarizations of A ⃗

_λ

experiences a tachyonic instability for k/(aH ) ∼

^<

2ξ. Without loss of generality, we assume that ˙ φ > 0 during inflation, so that the mode exhibiting the instability is A

₊

. In appendix A we review the solutions of (2.10) and show that the growth of fluctuations is well described by [15]

A

₊

(τ , k) ∼ = 1

√ 2k

' k

2ξ aH

(

1/4

e

^πξ⁻²

√

2ξk/(aH)

(2.11)

in the interval (8ξ )

⁻¹

∼

^<

k/(aH ) ∼

^<

2ξ [18] of phase space that accounts for most of the power in the produced gauge fluctuations. The phase space of growing modes is non- vanishing for ξ ∼

^>

O (1), which we assume throughout. Notice the exponential enhancement e

^πξ

in the solution (2.11), which arises due to tachyonic instability, and reflects significant nonperturbative gauge particle production in the regime ξ ∼

^>

1. On the other hand, the production of gauge field fluctuations is uninterestingly small for ξ < 1. Note also that the other polarization state, A

₋

(τ , k), is not produced and can therefore be ignored.

We have thus seen that the motion of the homogeneous inflaton φ(t) leads to production of gauge field quanta δA

_µ

. There are two key physical effects associated with the interactions of these produced quanta with the inflaton. The first effect is the backreaction of the produced quanta on the homogeneous dynamics of φ(t), a(t). In the next subsec- tion we study the conditions under which backreaction effects are negligible. The second key physical effect is the production of inflaton fluctuations via inverse decay ; this is the subject of subsection 2.3.

2.2 Backreaction Eﬀects

Backreaction eﬀects can be accounted for using the mean of the field equations (2.6):

φ ¨ + 3H φ ˙ + V

^′

(φ) = α

f ⟨ E ⃗ · B ⃗ ⟩ (2.12)

3H

²

= 1 M

_p²

% 1

2 φ ˙

²

+ V (φ) + 1

2 ⟨ E ⃗

²

+ B ⃗

²

⟩

&

(2.13) where we have switched to physical time. The expectation values appearing in (2.12,2.13) encode the backreaction of the produced gauge quanta on the homogeneous dynamics of

– 6 – Photon: !

2 helicities

A

₊

/ e

^⇡⇠

, | A | ⌧ | A

₊

| A+ exponentially amplified, !

V (') + ↵

f F

_µ⌫

F ˜

^µ⌫

GW signal naturally grows at interferometer scales, while small at CMB scales

r ⌘ P

_GW

P

_⇣

, n

s

1 ⌘ d ln P

_⇣

d ln k , f

NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

L = 1

2 (@

µ

)

²

+ V

_shift

( ) + c

f @

µ

¯

^µ ₅

+ ↵

f F

µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

• Constrained couplings to matter (predictivity)

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

• Shift symmetry on couplings to other fields Freese, Frieman, Olinto ’90; . . .

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

_µ

¯

^µ ₅

+ ↵

f F

_µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

r ⌘ P

_GW

P

_⇣

, n

_s

1 ⌘ d ln P

_⇣

d ln k , f

_NL

⇠ h ⇣

³

i

h ⇣

²

i

²

= O (✏

_V

, ⌘

_V

)

✏

_V

⌘ M

_p²

2

⇣ V

,

V

⌘

²

⌧ 1 ⌘

_V

⌘ M

_p²

V

_,

V ⌧ 1

• Flatness and gaussianity ! small self-couplings Axion (Natural) Inflation

L = 1

2 (@

_µ

)

²

+ V

_shift

( ) + c

f @

µ

¯

^µ ₅

+ ↵

f F

µ⌫

F ˜

^µ⌫

• Smallness of V

_shift

• Constrained couplings to matter (predictivity) We focus on a specific scenar