stringpheno2014.ictp.itstringpheno2014.ictp.it/parallels/wednesday/Cosmology... · 2014-11-24 · Higher-Curvature Corrections in String Theory Stringy tree-level (!’) and 1-loop

Higher Curvature Corrections and Non-Minimally Coupled Scalars

Mahdi Torabian Institute for Research in Fundamental Sciences (IPM), Tehran

String Phenomenology 2014 ICTP, Trieste

1

Higher-Curvature Corrections in String Theory

✔ Stringy tree-level (!’) and 1-loop level (gs) corrections to 10 dimensional SUGRA has been computed in type II theories.[Green-Schwarz 1982] [Gross-Witten 1986] [Grisaru-Zanon 1986] [Freeman-Pope 1986]

✔ Corrections are at 8-derivative level, Quartic in the Riemann Tensor ✔ Corrections are uplifted to 11 dimensional SUGRA

[Green-Gutperle-Vanhofe 3x1997] [Antoniadis-Ferrara etal 1997] [Russo-Tseytlin 1997]

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R

a1a2 ∧Ra3a4 ∧Ra5a6 ∧Ra7a8 ∧ ea9 ∧ ea10 ∧ ea11

− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1

4Rab ∧Rba ∧Rcd ∧Rdc

'$%,

(2)

ttRRRR =+ 12(RmnpqRmnpq)2 + 24RmnpqRmnrsRtupqRturs − 96RmnpqRmnrsRtupsRturq

− 192RmnpqRmnqrRtursRtusp + 192RmnpqRntqrRtursRumsp + 384RmnpqRtuqrRntrsRumsp,(3)

ϵϵRRRR = −3!8!δm1m2m3m4m5m6m7m8

[n1n2n3n4n5n6n7n8]Rm1m2n1n2Rm3m4n3n4Rm5m6n5n6Rm7m8n7n8 , (4)

AϵtRRRR = 24Am1m2m3ϵm1m2m3···m11

#Rm4m5npRm6m7pqRm8m9qsRm10m11sn

− 1

4Rm4m5npRm6m7pnRm8m9qsRm10m11sq

$,

(5)

[1] A. A. Starobinsky, Phys. Lett. B 91 (1980) 99.[2] A. A. Starobinsky, Sov. Astron. Lett. 9 (1983) 302.

2

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R


− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1


'$%,

(2)







− 1


$,

(5)


2

2

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R


− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1


'$%,

(2)







− 1


$,

(5)


2

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R


− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1


'$%,

(2)







− 1


$,

(5)


2

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R


− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1


'$%,

(2)







− 1


$,

(5)


2

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R


− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1


'$%,

(2)







− 1


$,

(5)


2

I. INTRODUCTION

S = (2π)−8m911

!g1/2d11x

"R− 1

2 · 4!FF − g−1/2

(6 · 4!)2 ϵAFF +π2

9 · 211m611

#tt− 1

4!ϵϵ− 1

6Aϵt

$RRRR

%, (1)

S = (2π)−8m911

! "Rab ∧ ∗(ea ∧ eb)− 1

2F ∧ ∗F − 1

6A ∧ F ∧ F

+π2

9 · 211m611

#∗ 1ta1···a8tb1···b8R

b1b2a1a2

Rb3b4a3a4

Rb5b6a5a6

Rb7b8a7a8

− 2

3ϵa1···a11R


− 3 · 27 A ∧&Rab ∧Rbc ∧Rcd ∧Rda − 1


'$%,

(2)







− 1


$,

(5)


2


3

✔ KK Reduction on compact manifolds to get 4D minimal SUGRA +Corrections

11D N=1 SUGRA

3D N=2 SUGRA5D N=1 SUGRA

4D N=1 SUGRA

G2

CY4CY3

UpliftReduce

✔ The Road Map:

[Antoniadis-Ferrara etal 1997] [Grimm-Savelli etal 2013]


4

✔ A glimpse of what we get


5

✔ Note we also have

The Electroweak Symmetry Breaking and the Vacuum Metastabilityof a Non-minimally Coupled Higgs in the Curvature-driven Inflation

Mahdi TorabianSchool of Particles and Accelerators, Institute for Research in Fundamental Sciences (IPM), 19395-5531, Tehran, Iran and

International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151, Trieste, Italy

The LHC-I measurements of the Higgs and top quark mass and the strong coupling constantindicates that the electroweak vacuum is metastable. In fact, the Standard Model Higgs potentialis unbounded from below; the global minimum is orders of magnitude deeper than the local oneand the barrier separating the two is miniscule. Although quite long-lived on cosmological scales,it is not clear how did the Higgs entrap in the local minimum stick to it during a rather high-scaleinflation. In this note it is shown that the curvature-driven inflation and a non-minimally coupledHiggs to gravity solve the problems. Radiative breaking is also possible.....

Introduction The Higgs field is non-minimally cou-pled to a general theory of gravity

S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1

2gµν∂µϕ∂νϕ−V (ϕ)

#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)

By a suitable Weyl transformation of the metric

gEµν = (1 + ξm−2Pl ϕ

2 + 2αm−2Pl R)gµν ≡ eχgµν , (4)

one can move to the Einstein frame with the followingaction

SE =

!d4x(−gE)

1/2"12m2

PlRE − 1

2gµνE ∂µχ∂νχ

−1

2e−χgµνE ∂µϕ∂νϕ− VE(ϕ,χ)

#,(5)

where now

VE(ϕ,χ)=e−2χV (ϕ)+1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)

and the canonically normalized dimensionfull Weyl scalarfield is χ = (3/2)1/2mPlχ.

χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2

Ple−2χ0(1 + ξϕ2

0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2(λ+ ξ2/2α)ϕ20, (10)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

The 4-dimensional Effective Action






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)


gEµν = fRgµν = (1 + ξm−2Pl ϕ

2 + 2αm−2Pl R)gµν ≡ eχgµν ,

(4)one can move to the Einstein frame with the followingaction

SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

✔ A Weyl transformation from the Jordan to the Einstein frame:

A phenomenological view:

✔ A scalar field non-minimally coupled to a higher-derivative gravity:

If not at tree-level, will be loop-induced anyway

6






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2(λ+ ξ2/2α)ϕ20, (10)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2(λ+ ξ2/2α)ϕ20, (10)

The 4-dimensional Effective Action

non-canonical kinetic term

✔ The action in the Einstein frame: Weyl scalar

✔ The scalar potential:

I. INTRODUCTION

II. THE SETUP

A. The Action

The action in the Jordan frame takes the following form

S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1

2gµν∂µϕ∂νϕ, (4)




one can move to the Einstein frame with the following action

S =

!d4x(−gE)

1/2"12m2

PlRE−1

2gµνE ∂µχ∂νχ− VE(χ) + PE(X,ϕ)

#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)

PE(X,ϕ) = −e−2χf−1(ϕ)$(1 + feχgµνE ∂µϕ∂νϕ)

1/2 − 1%− e−2χV (ϕ), (8)

and the canonically normalized dimensionfull Weyl scalar field is χ = (3/2)1/2mPlχ.

B. The Equations of Motion

We find the equations of motion for the metric, the DBI field and the Weyl field in a isotropic/homogenousbackground

ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)

The boost factor in the homogenous background field is

γ =&1− f(ϕ)ϕ2eχ

'−1/2. (10)

The equations of motion for the metric in a Friedman-Robertson-Walker background can be read as the Friedmannequation

3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)

and the time variation of the Hubble scale as follows

− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)

The equation of motion for the homogenous DBI field is

ϕ+ 3Hγ−2ϕ+ e−χγ−3Vϕ(ϕ) +1

2(1− 3γ−2)ϕχ+

1

2e−χf−2(ϕ)fϕ(ϕ)

&1− 3γ−2 + 2γ−3

'= 0, (13)

Finally the equation of motion for the Weyl field can be found as

χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

( )

7






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2χϕ+ V Eϕ = 0, (12)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

The Equations of Motion

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

✔ EOMs for the scalars and metric in an isotropic/homogenous b.g.






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

✔ Numerically solve the equations for varieties of potentials…..

✔ Embed this model in a cosmological setup

8

No Scalar Potential in the Jordan Frame

dominated






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(15)

✔ A potential is induced in the Jordan frame:






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(15)

χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (16)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (17)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (18)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (19)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(15)

χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (16)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (17)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (18)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (19)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(15)

χ0 = ln&(1 + ξϕ2

0)', (16)

m2χ =

1

6αm2


0)2, (17)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (18)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (19)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(15)

χ0 = ln&(1 + ξϕ2

0)', (16)

m2χ =

1

6αm2

Pl, (17)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (18)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (19)

0 00

eχ0 = 2(1 + (m/mPl)2α/ξ), (30)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (31)

m2χ =

m2Pl

6α(1 + (m/mPl)2α/ξ), (32)

λ(mPl) + ξ(mPl)2/2α > 0, (33)

ξ(mPl) ! 50000, (34)

ξϕ20 " m2

Pl, (35)

ξ " 1032, (36)

VE(χ,ϕ) = V (χ)− 1

4αe−2χ(eχ − 1)ξϕ2 +

1

8αe−2χξ2ϕ4, (37)

I.

The Higgs field is non-minimally coupled to a general theory of gravity

S =

!d4x(−g)1/2

"12m2

Plf(ϕi, R)− 1

2gµν

#

i

∂µϕi∂νϕi −#

i

Vi(ϕi)$, (38)

where

f(ϕi, R) = (1 +m−2Pl

#

i

ξiϕ2i )R+ αm−2

Pl R2, (39)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (40)


gEµν = fRgµν = (1 +m−2Pl

#

i

ξiϕ2i + 2αm−2

Pl R)gµν ≡ eχgµν , (41)


SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1

2e−χgµνE

#

i

∂µϕi∂νϕi − VE(ϕi,χ)$, (42)

where now

VE(ϕi,χ) = e−2χ#

i

Vi(ϕi)+1

8α−1m4

Pl

"1− e−χ

%1 +m−2

Pl

#

i

ξiϕ2i

&$2,

(43)


4

dominated






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ)+1

2e−2χ

$m2−ξ(e−χ−1)m2

Pl/2α%ϕ2+

1

4e−2χ

$λ+ξ/2α

%ϕ4,

(7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ

1− ξ(m2/λ)m−2Pl

∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (17)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (18)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (19)

− 2Hm2Pl = χ2 + e−χϕ2, (20)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(21)

9






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ)+1

2e−2χ

$m2−ξ(e−χ−1)m2

Pl/2α%ϕ2+

1

4e−2χ

$λ+ξ/2α

%ϕ4,

(7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ


∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (17)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (18)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (19)

− 2Hm2Pl = χ2 + e−χϕ2, (20)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(21)

CM

B

end

[Starobinsky ’80]

Starobinsky Curvature-Driven Inflation

✔ The simplest model used to be fascinating in the pre-BICEP era!

10






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ)+1

2e−2χ

$m2−ξ(e−χ−1)m2

Pl/2α%ϕ2+

1

4e−2χ

$λ+ξ/2α

%ϕ4,

(7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ


∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (17)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (18)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (19)

− 2Hm2Pl = χ2 + e−χϕ2, (20)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(21)

CM

B

end

[Starobinsky ’80]

Starobinsky Curvature-Driven Inflation

11

✔ Some variations are consistent with large tensor modes

Massive Scalar in the Jordan Frame

χ0 = ln!(1 + ξϕ2

0)", (22)

m2χ =

1

6αm2

Pl, (23)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (24)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (25)

V (ϕ) =1

2m2ϕ2, (26)

2

χ0 = ln!(1 + ξϕ2

0)", (22)

m2χ =

1

6αm2

Pl, (23)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (24)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (25)

V (ϕ) =1

2m2ϕ2, (26)

ϕ20 = ξ−1m2

Pl, (27)

2

χ0 = ln!(1 + ξϕ2

0)", (22)

m2χ =

1

6αm2

Pl, (23)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (24)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (25)

V (ϕ) =1

2m2ϕ2, (26)

ϕ20 = ξ−1m2

Pl, (27)

eχ0 = 2(1 + (m/mPl)2α/ξ), (28)

2

χ0 = ln!(1 + ξϕ2

0)", (22)

m2χ =

1

6αm2

Pl, (23)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (24)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (25)

V (ϕ) =1

2m2ϕ2, (26)

ϕ20 = ξ−1m2

Pl, (27)

eχ0 = 2(1 + (m/mPl)2α/ξ), (28)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (29)

2

✔ The minimum of the potential:

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (27)

V (ϕ) =1

2m2ϕ2, (28)

ϕ20 = ξ−1m2

Pl, (29)

eχ0 = 2(1 + (m/mPl)2α/ξ), (30)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (31)

m2χ =

m2Pl

6α(1 + (m/mPl)2α/ξ), (32)

λ(mPl) + ξ(mPl)2/2α > 0, (33)

ξ(mPl) ! 50000, (34)

ξϕ20 " m2

Pl, (35)

ξ " 1032, (36)

2

12






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2(λ+ ξ2/2α)ϕ20, (10)

✔ The scalar potential:






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


χ0 = ln&(1 + ξϕ2

0) + 8α(1 + ξϕ20)

−1V (ϕ0)', (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (11)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (12)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (13)

− 2Hm2Pl = χ2 + e−χϕ2, (14)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(15)

χ0 = ln&(1 + ξϕ2

0)', (16)

m2χ =

1

6αm2

Pl, (17)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (18)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (19)

Higgs-like Field in the Jordan Frame

13

Higgs dominated

Inflaton dominated






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

ϕ20 ≈ −m2/λ

1− ξ(m2/λ)∼ (246 GeV)2, (11)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (12)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (13)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (14)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (15)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (16)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (17)

− 2Hm2Pl = χ2 + e−χϕ2, (18)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(19)

χ0 = ln&(1 + ξϕ2

0)', (20)

m2χ =

1

6αm2

Pl, (21)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

ϕ20 ≈ −m2/λ

1− ξ(m2/λ)∼ (246 GeV)2, (11)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (12)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (13)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (14)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (15)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (16)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (17)

− 2Hm2Pl = χ2 + e−χϕ2, (18)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(19)

χ0 = ln&(1 + ξϕ2

0)', (20)

m2χ =

1

6αm2

Pl, (21)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

ϕ20 ≈ −m2/λ

1− ξ(m2/λ)∼ (246 GeV)2, (11)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (12)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (13)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (14)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (15)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (16)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (17)

− 2Hm2Pl = χ2 + e−χϕ2, (18)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(19)

χ0 = ln&(1 + ξϕ2

0)', (20)

m2χ =

1

6αm2

Pl, (21)

Higgs-like Fields in the Jordan Frame

✔ The minimum of the potential:






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ)+1

2e−2χ

$m2−ξ(e−χ−1)m2

Pl/2α%ϕ2+

1

4e−2χ

$λ+ξ2/2α

%ϕ4 ,

(7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ

1− (m/mPl)2ξ/λ∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (17)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (18)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (19)

− 2Hm2Pl = χ2 + e−χϕ2, (20)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(21)

✔ 2 parameters get determined






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ)+1

2e−2χ

$m2−ξ(e−χ−1)m2

Pl/2α%ϕ2+

1

4e−2χ

$λ+ξ2/2α

%ϕ4 ,

(7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ

1− (m/mPl)2ξ/λ∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

−m(mZ)2/λ(mZ) ∼ (246 GeV)2, (17)

λ(mZ) + ξ(mZ)2/2α ∼ (125 GeV)2, (18)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (19)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (20)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (21)

− 2Hm2Pl = χ2 + e−χϕ2, (22)






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ)+1

2e−2χ

$m2−ξ(e−χ−1)m2

Pl/2α%ϕ2+

1

4e−2χ

$λ+ξ2/2α

%ϕ4 ,

(7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2

Ple−2χ0(1+ ξϕ2

0)2+

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ

1− (m/mPl)2ξ/λ∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

−m(mZ)2/λ(mZ) ∼ (246 GeV)2, (17)

λ(mZ) + ξ(mZ)2/2α ∼ 0.13, (18)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (19)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (20)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (21)

− 2Hm2Pl = χ2 + e−χϕ2, (22)

14

246 GeV

10 GeV10

(10 GeV)10

10 GeV18

4

- Huge GeV4

h[Degrassi etal 2012]

The Metastable Electroweak Vacuum

✔ Fine-tuning of the Higgs value!

✔ The Standard Model extrapolated to HE:

[see Lebedev-Westphal 2013]

Stable enough, however How Inflation began at all?

How come we end up exactly here? Howe come not kicked off from here?15

✔ The Higgs potential

Introduction The Higgs field is non-minimally coupled to a general theory of gravity

S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)− 1

2gµν∂µϕ∂νϕ− V (ϕ)

#, (1)

where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#, (5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


VE(χ,ϕ) = V (χ) +1

2e−2χ

$m2 − ξ(e−χ − 1)m2

Pl/2α%ϕ2 +

1

4e−2χ

$λ+ ξ2/2α

%ϕ4 , (7)

V (χ) =1

8α

$1− e−χ

%2m4

Pl, (8)

eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (9)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (10)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (11)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (12)

ϕ20 ≈ −m2/λ


∼ (246 GeV)2, (13)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (14)

2

Higgs-like Field in the Jordan Frame

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (27)

V (ϕ) =1

2m2ϕ2, (28)

ϕ20 = ξ−1m2

Pl, (29)

eχ0 = 2(1 + (m/mPl)2α/ξ), (30)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (31)

m2ϕ =

m2Pl

6α(1 + (m/mPl)2α/ξ), (32)

λ(mPl) + ξ(mPl)2/2α > 0, (33)

2

✔ Boundedness from below implies that

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (27)

V (ϕ) =1

2m2ϕ2, (28)

ϕ20 = ξ−1m2

Pl, (29)

eχ0 = 2(1 + (m/mPl)2α/ξ), (30)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (31)

m2ϕ =

m2Pl

6α(1 + (m/mPl)2α/ξ), (32)

λ(mPl) + ξ(mPl)2/2α > 0, (33)

ξ(mPl) ! 50000, (34)

2

Q. Is that consistent with collider phenomenology?

✔ Certainly consistent with cosmology

Q. Is it too big a number?

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (27)

V (ϕ) =1

2m2ϕ2, (28)

ϕ20 = ξ−1m2

Pl, (29)

eχ0 = 2(1 + (m/mPl)2α/ξ), (30)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (31)

m2ϕ =

m2Pl

6α(1 + (m/mPl)2α/ξ), (32)

λ(mPl) + ξ(mPl)2/2α > 0, (33)

ξ(mPl) ! 50000, (34)

ξϕ20 " m2

Pl, (35)

ξ " 1032, (36)

2

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

m2ϕ = e−2χ0m2

Pl(eχ0 − 1)ξ/α = e−2χ0(ξ2/α)ϕ2

0 , (27)

V (ϕ) =1

2m2ϕ2, (28)

ϕ20 = ξ−1m2

Pl, (29)

eχ0 = 2(1 + (m/mPl)2α/ξ), (30)

m2ϕ =

ξm2Pl

4α(1 + (m/mPl)2α/ξ), (31)

m2ϕ =

m2Pl

6α(1 + (m/mPl)2α/ξ), (32)

λ(mPl) + ξ(mPl)2/2α > 0, (33)

ξ(mPl) ! 50000, (34)

ξϕ20 " m2

Pl, (35)

ξ " 1032, (36)

2

[Atkins-Calmet PRL’13]

At the LHC, the Higgs boson production and decay will be effected by the above sup-

pression. At each vertex involving the Higgs boson coupled to standard model particles, a

factor of 1/√1 + β will be introduced. Clearly if β ≫ 1 the Higgs boson would simply not

be produced in a large enough abundance to be observed.

In the following we will make the assumption that the Higgs boson like particle recently

observed at the LHC is the standard model Higgs boson and that there are no other degrees

of freedom beyond those present in the standard model and Einstein gravity. We will refer

to the usual standard model total cross section for Higgs boson production and decay with

β = 0 as σSM. If the cross section including a non-zero β is given by σ, we are interested

in the ratio σ/σSM. The LHC experiments produce fits to the data assuming that all Higgs

boson couplings are modified by a single parameter κ [19] which in our model corresponds

to κ = 1/√1 + β. Using the narrow width approximation, the cross section for Higgs

production and decay from any initial i to final state f is given by

σ(ii → H) · BR(H → ff) = σSM(ii → H) · BRSM(H → ff) · κ2 . (10)

One might naively expect the cross section to be proportional to κ4, but in the narrow

width approximation this is not the case. The presence of the branching fraction, which

is independent of a universal suppression of the couplings, leads to the cross section being

proportional to κ2. For a 125 GeV Higgs the narrow width limit is an excellent approximation

and is used in the determination of the signal strength at the LHC.

The ATLAS detector has currently measured the global signal strength µ = σ/σSM =

1.4 ± 0.3 [9] and CMS has measured this as µ = 0.87 ± 0.23 [8]. Combining these results

gives µ = 1.07± 0.18. This excludes |ξ| > 2.6× 1015 at the 95% C.L.

Reference [20] estimates the expected reach in the accuracy of the measurement of the

Higgs boson couplings in a large number of processes in future runs at the LHC and the

proposed ILC. We combine these results to give an estimated uncertainty in the global

signal strength µ. We assume a central value of µ = 1. At a 14 TeV LHC with an integrated

luminosity of 300 fb−1, the uncertainty in the measurement of µ is expected to be 0.07 which

would lead to a bound on |ξ| < 1.6×1015. At the ILC with a center of mass of 500 GeV and

an integrated luminosity of 500 fb−1, the expected uncertainty on µ is 0.005, which gives

a bound of |ξ| < 4 × 1014. Despite expected measurements of the total cross section to an

accuracy better than 1% at future high energy runs at the ILC, we cannot expect to push

the constraints on |ξ| below about 1014.

Given a large non-minimal coupling to gravity, one might also expect to have decreased

observable rates for Higgs decays at the LHC arising from unobserved decays to gravitons.

The effect is in fact very small as we will now discuss. The lowest order vertex in ξ is a three

4

CMS+Atlas exclude

16

No Scalar Potential in the Jordan Frame: Revisited

✔ A non-zero vev and mass is induced by non-zero Weyl vev

17

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

−m(mZ)2/λ(mZ) ∼ (246 GeV)2, (17)

λ(mZ) + ξ(mZ)2/2α ∼ 0.13, (18)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (19)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (20)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (21)

− 2Hm2Pl = χ2 + e−χϕ2, (22)

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

ϕ20 ≈ ξ−1mPlχ0 ≈ ξ−1ξHiggsv

2, (27)

m2ϕ = (ξ2/α)ϕ2

0 ≈ ξξHiggsv2/α, (28)

V (ϕ) =1

2m2ϕ2, (29)

3

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (15)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (16)

−m(mZ)2/λ(mZ) ∼ (246 GeV)2, (17)

λ(mZ) + ξ(mZ)2/2α ∼ 0.13, (18)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (19)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (20)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (21)

− 2Hm2Pl = χ2 + e−χϕ2, (22)

VE(ϕ,χ) =1

8α−1m4

Pl

!1− e−χ(1 + ξm−2

Pl ϕ2)"2,

(23)

χ0 = ln#(1 + ξϕ2

0)$, (24)

m2χ =

1

6αm2

Pl, (25)

ϕ20 = ξ−1m2

Pl(eχ0 − 1), (26)

ϕ20 ≈ ξ−1mPlχ0 ≈ ξ−1ξHiggsv

2, (27)

m2ϕ = (ξ2/α)ϕ2

0 ≈ ξξHiggsv2/α, (28)

V (ϕ) =1

2m2ϕ2, (29)

3






S =

!d4x(−g)1/2

"12m2

Plf(ϕ, R)−1


#,

(1)where

f(ϕ, R) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

V (ϕ) =1

2m2ϕ2 +

1

4λϕ4, (3)





SE =

!d4x(−gE)

1/2"12m2

PlRE − 1


−1


#,(5)

where now


8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(6)


eχ0 = (1 + ξϕ20) + 8α(1 + ξϕ2

0)−1V (ϕ0), (7)

m2χ =

1

6αm2


0)2 +

4

3m−2

Pl e−2χ0V (ϕ0), (8)

ϕ20 =

−m2 +m2Plξ(e

χ − 1)/2α

λ+ ξ2/2α, (9)

m2ϕ = 2e−2χ0

$−m2 +m2

Plξ(eχ − 1)/2α

%

= 2e−2χ0(λ+ ξ2/2α)ϕ20, (10)

ϕ20 ≈ −m2/λ

1− ξ(m2/λ)∼ (246 GeV)2, (11)

m2ϕ ≈ 2ϕ2

0(λ+ ξ2/2α) ∼ (125 GeV)2, (12)

χ0 ≈ ξϕ20m

−1Pl ∼ 10−5ξ eV, (13)

m2χ ≈ 1

6αm2

Pl ∼ (1013 GeV)2, (14)

χ+ 3Hχ+ 6−1/2e−χϕ2 + V Eχ = 0, (15)

ϕ+ 3Hϕ− (2/3)1/2m−1Pl χϕ+ V E

ϕ = 0, (16)

3H2m2Pl =

1

2χ2 +

1

2e−χϕ2 + VE , (17)

− 2Hm2Pl = χ2 + e−χϕ2, (18)

VE(ϕ,χ) =1

8α−1m4

Pl

$1− e−χ(1 + ξm−2

Pl ϕ2)%2,

(19)

χ0 = ln&(1 + ξϕ2

0)', (20)

m2χ =

1

6αm2

Pl, (21)

✔ Higgs potential induces non-zero vev for Wely field

Starobinsky meets DBI in the SkyI. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + 2ξm−2Pl ϕ

2)R+ 8αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1



gEµν = (1 + 2ξm−2Pl ϕ



S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) = α−1m4Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

[Kaviani-MT, to appear]

✔ The effect of the higher-order correction and non-minimal coupling on DBI inflation.

✔ An interesting brane-inflation model in string theory. Not consistent with observation though: too much non-gaussianities.

✔ Scalar with non-minimal coupling and non-canonical kinetic term

18

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

Starobinsky meets DBI in the Sky

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

✔ A Weyl transformation from the Jordan to the Einstein frame:

✔ The Einstein frame action:

19

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

I. INTRODUCTION

II. THE SETUP

A. The Action


S =

!d4x(−g)1/2

"12m2

Plf(R,ϕ) + P (X,ϕ)#, (1)

where

f(R,ϕ) = (1 + ξm−2Pl ϕ

2)R+ αm−2Pl R

2, (2)

P (X,ϕ) = −f−1(ϕ)$(1− 2f(ϕ)X)1/2 − 1

%− V (ϕ), (3)

and

X = −1






S =

!d4x(−gE)

1/2"12m2

PlRE−1


#, (6)

where now

VE(χ) =1

8α−1m4

Pl

&1− e−χ

'2, (7)


1/2 − 1%− e−2χV (ϕ), (8)




ds2 = −dt2 + a(t)dx, ϕ = ϕ(t), χ = χ(t). (9)



'−1/2. (10)


3H2m2Pl =

1

2χ2 + α−1m4

Pl(1− e−χ)2 + e−2χV (ϕ)− e−2χf−1(1− γ), (11)


− 2Hm2Pl = χ2 + e−2χf−1(γ − γ−1) (12)



2(1− 3γ−2)ϕχ+

1


&1− 3γ−2 + 2γ−3

'= 0, (13)


χ+ 3Hχ+ (8/3)1/2α−1m3Ple

−χ(1− e−χ)− 2e−2χV (ϕ) +1

2e−2χf−1(ϕ)

&4− γ − 3γ−1

'= 0, (14)

2

Starobinsky meets DBI in the Sky

✔ The dynamics on the FRW geometry

✔ A double-inflation model

1.¥106 7.¥106 1.3¥1071

2

4610

t

f@tD,c@tD

1.¥106 7.¥106 1.3¥107

10-40.01

1

100

t

N

125102050100

t

gHtL

20

Summary

✔ The non-minimal coupling of scalar fields to an f(R) theory of gravity changes the behavior of both the scalar and the gravity sectors.

✔ If the Standard Model is to be extrapolated to high scale, this coupling can stabilize the electroweak vacuum. Besides it explains why the Higgs ended up here. ✔ All (non-minimally coupled) scalars get stabilized even if they have no potential in the Jordan frame. !✔ This framework also changes the dynamics of scalars with non-canonical kinetic terms (DBI, K-essence).

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A phenomenological study:

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stringpheno2014.ictp.itstringpheno2014.ictp.it/parallels/wednesday/Cosmology... · 2014-11-24 · Higher-Curvature Corrections in String Theory Stringy tree-level (!’) and 1-loop