横山順一

(RESCEU & Kavli IPMU, Tokyo)

Jun’ichi Yokoyama

The SM Higgs particle was finally discovered !? 2012/7/4

A candidate event from a Higgs particle observed by ATLAS (© ATLAS experiment, by courtesy of S. Asai.)

Professor Shoji Asai at the press conference together with his view graph on July 4.

ATLAS

126.5GeV 5hm

4

H

H ZZ leptons

CMS

125.3 0.6GeV 4.9hm

4

2 2

H

H ZZ l

H WW l

6% human resources, 10% $\ from Japan

http://aappsbulletin.org/

The discovery of a scalar particle/fieldhas a great impact not only to particlephysics but also to cosmology, sincescalar fields play very important rolesto drive inflation in the early Universeand generate cosmological perturbations.

Here we discuss Cosmology of the Higgs Fieldin several distinct cases:

I. The Higgs field is NOT the inflaton but a subdominant field during inflation.

II. The Higgs field itself is responsible for inflation in the early Universe.

Kunimitsu & JY Phys.Rev. D86 (2012) 083541

Kamada, Kobayashi, Kunimitsu, Yamaguchi & JY 1309.7410Kamada, Kobayashi, Yamaguchi & JY Phys.Rev. D86 (2012) 023504Kamada, Kobayashi, Yamaguchi & JY Phys.Rev. D83 (2011) 083515

Whatever the inflation model is, the Higgs field exists in the Standard Model and plays some roles.

During inflation, short-wave quantum fluctuations are continuously generated for the Higgs field and its wave length is stretched by inflation to yield non-vanishing expectationvalues.

In each Hubble time , quantum fluctuations with an amplitude

and the initial wavelength is generated and

stretched by inflation continuously.

1H

2

H

1H

Brownian motion with step and interval

A well known peculiar property of a practically massless field in the de Sitter(exponentially expanding) spacetime: (Bunchi & Davis 78, Vilenkin & Ford 82…)

2

H

1H

The most dramatic case

Our Universe cannot be reproduced in such a case.

(Espinosa et al 2008)

Hv

[ ]V H

4( )[ ]

4V

: real and neutral component of the Higgs field

Just after the CERN seminars in 2011/12 announcing “hints of the Higgs boson”an analysis of the renormalized Higgs potential was done for m=124 ～ 126GeV.

Elias-Miro et al Phys Lett B709(2012)222

( )

126GeV Hm For there is a finite parameter region within 2σ where the Higgs potential is stable, or its self coupling remain positive even after renormalization.

4 4 2( )[ ] , (10 ) 0.

4 4V O

We consider the case λ is positive (and constant) in theregime of our interest.

coarse-grained modelong wavelength

short-wave quantum fluctuations

: a small parameter

The behavior of a scalar field with such a potential duringinflation has been studied using the stochastic inflation method.

(Starobinsky and JY 1994)

act as a stochastic noise due to cosmic expansion

Decompose the scalar field as3

† *3/ 2

( , ) ( , ) ( ( ) ) ( ) ( )(2 )

i id kt t k a t H a t e a t e

k x k x

k k k kx x

1( , ) ( , )

3t V f t

H x x

3

1 1 2 2 1 2 0 1 22( , ) ( , ) ( ) ( ( ) | |)

4

Hf t f t t t j a t H

x x x x

0

sin( )

zj z

z

Langevin eq. for long wave mode

stochastic noise term

We can derive a Fokker-Planck equation for the one-point (one-domain) probability distribution function (PDF)

Its generic solution can be expanded as

is the complete orthonormal set of eigenfunctions of the Schrödinger-type eq.

If is finite, we find and the equilibrium PDF exists.

Each solution relaxes to the equilibrium PDF at late time which has the de Sitter invariance.

1 1 1[ ( , ) ] [ ( , )] ( , )t t t x x

For we find an equilibrium PDF corresponding to as

First few eigenvalues (numerically obtained)

Equilibrium expectation values obtained using

2H

2H

4H

where

210 H

Two point equilibrium temporal correlation function can be well approximated by

for

The correlation time defined by is given by

The spatial correlation function can be evaluated from it using the de Sitter invariance.

cf de Sitter invariant separation

0( ) Hta t a e

The spatial correlation length defined by therefore reads

Exponentially large !

So we can treat it as a homogeneous field when we discuss its effect on reheating.

The energy density of the Higgs condensation remains constant until the Hubble parameter decreases to .After this epoch it decreases in proportion to since it hasa quartic potential.

4 ( )a t

4 ( )a t

At this time we find

namely, .

tot

condtot

8G PlM m

So the Higgs condensation or its decay products NEVER contributes to the total energy density appreciably in the usual inflation models, because

3( )tot a t 4 ( )tot a t

in the field oscillation regime before reheating,

after reheating.

The story is completely different in inflation modes where the totalenergy density decreases more rapidly than after inflation.4 ( )a t

In some models, inflation may end abruptly without being followed by itscoherent field oscillation and reheating proceeds only through gravitationalparticle creation.

Such inflation models include k-inflation, (original) G-inflation, and quintessential inflation models.

A simple k-inflation example

inf : inflaton

inflation with2

2 1inf 2

212 G

KH

M K

In such models the Higgs condensation contributes to the total energy density appreciably in the end.

[ ]V

Quintessential inflation(Peebles & Vilenkin 99)

(Armendariz-Picon, Damour & Mukhanov 99)

21 inf 2 inf inf inf

1( ) ( ) ,

2K X K X X g L=

k-inflation

Free scalar field

( ) Hta t e

1 3( )a t t

1 2 0K K

1 2 0K K

If this change occurs abruptly, numerical calculation shows that this transition occurs within Hubble time.

Gravitational particle production takes place due to this rapid changeof the expansion law.

kination

1a taking at the end of inflation.

(cf Ford 87)

The energy density of a massless boson created this way is given by

～

Vacuum state in de Sitter space is different fromthat in the power-law expanding Universe.

†0 0 0dS powerlaw powerlaw dSa a k k

time or scale factor

energy density

inflation

kinatic energy of the inflaton

radiation from gravitational particle production

0a6a

4inf

2 4

9

32r

NH

a

4inf

2 4

9

32r

NH

a

Ra

is the tensor-scalar ratio.

inf

time or scale factor

energy density

inflation

kinetic energy of the inflaton

radiation from gravitational particle production

radiation from Higgs condensation

4a

0a6a

0a

4aHiggs condensation

Higgs oscillation

We incorporate the Higgs condensation. When it starts oscillation @ eff( )H t m

inf

inf

If we try to estimate the reheat temperature from the equalitytaking the Higgs condensation into account, we find

which is to be compared with the previous estimate

Thus the Higgs condensate or its decay product contributes to the total energy density appreciably.

Its fluctuation can be important.

Here we analyze properties of fluctuations of the Higgs condensation.

A power-law spatial correlation function like

can be obtained from a power law power spectrummaking use of the formula

We find and

1 inf2 H

It practically behaves as a massless free field.

(Enqvist & Sloth, Lyth & Wands, Moroi & Takahashi 02)

The power spectrum of (potential) energy density fluctuation is given by

Amplitude of fractional density fluctuation on scale 2

rk

(0.1)O

cond cond osc cond

tot cond osc totosc

Hc c

H

Its contribution to the curvature perturbation ζ is unacceptably large.

Hubble parameter at the onset of oscillation osc effH m (Kawasaki, Kobayashi & Takahashi 11)

can be O(1)

1 2 510

Inflation models followed by kination regime with gravitational reheating is incompatible with the Higgs condensation.

f

k-inflation

Standard inflation

Stochastic gravitational wave background from inflation

Models with such anenhanced spectrumare ruled out.

Easy to preach Difficult in practice

Tree level action of the SM Higgs field

Taking with φ being a real scalar field, we find

for .

This theory is the same as that proposed by Linde in 1983 to drive chaotic inflation at .PlM

126GeV Hm 0.13 at low energy scale

In order to realize we must have .

The Higgs potential is too steep as it is.

510T

T

1310

Square amplitude of curvature perturbation on scale 2

rk

2T

T

The spectral index of the curvature perturbation is given by

In potential-driven inflation models

1 6 2s V Vn 22

2, 2

GV V G

M V VM

V V

The Higgs potential is too steep as it is.Several remedies have been proposed to effectively flatten the potential.

1 The most traditional one (original idea: Spokoiny 1984 Cervantes-Cota and Dehnen 1995, ….)

introduce a large and negative nonminimal coupling to scalar curvature R

L = + L

510

2 2 2Pl PlM M

Pl

Effectively flatten thespectrum

Pl GM M

2 New Higgs inflation (Germani & Kehagias 2010)

Introduce a coupling to the Einstein tensor in the kinetic term.

21 1

2 2g g w G

2

1w const

M

During inflation .

If , the normalization of the scalar field changes.

22 00

23

Hw G

M

H M

143

H

M is the canonically normalized scalar field.

24 4

2[ ] [ ]

4 12

MV V

H

effectively reducing the self couplingto an acceptable level.

(Actual dynamics is somewhat more complicated.)

3: Running kinetic inflation (Takahashi 2010, Nakayama & Takahashi 2010)

Introduce a field dependence to the kinetic term.

1 1( )

2 2g f

Canonical normalization of the scalar field changes the potential effectively.

( ) 1f v

( ) ( ) ,X V g X L =

Slow-roll parameters2

2, , , .

'( )

H g g X

H H gH V

, , , 1 we find slow-roll EOMsFor2 23 ( )PlM H V 33 1 ( ) 0gHH V

Background equations of motion2 2

2

3 1 (6 ) ( )

1 (3 )

3 (9 3 6 2 ) (1 2 ) ( ) 0

Pl

Pl

M H gH X V

M H gH X

gH H V

This extra friction term enhances inflation and makesit possible to drive inflation by the standard Higgs field.

Galileon-type coupling

(Kamada, Kobayashi, Yamaguchi & JY 2011)

1

2X g

Higgs G inflation

444

XX

M L

9 1COBE( ) 2.4 10 @ 0.002Mpc ( 60)k k

P N146 134.7 10 10 GeV.PlM M

In fact, all these models can be described in the context ofthe Generalized G-inflation model seamlessly, which is the mostgeneral single-field inflation model with 2nd order field equations.

3

22

4 4

2 33

5 5

,

,

,

1, 3 2

6

X

X

K X

G X

G X R G

G X G G

2

3

4

5

L

L

L

L

Inflation from a gravity + scalar system 5

4

2i

i

S gd x

L

These Lagrangians are obtained by covariantizingthe Galileon theory.

Higher derivative theory with a Galilean shift symmetry in the flat spacetime.const

2

2

22 2

2 32 3

2 2

3 2

1

2

3

4

5

L

L

L

L

L

Field equation containsderivatives up to second-order at most.

(Nicolis, Rattazzi, & Trincherini 2009)

2

3

nonreleativistic limit of 4D probe brane actionin 5D theory (de Rham & Tolley 2010)

Galilean symmetry exists only in flat spacetime.

(Deffayet, Deswer, & Esposito-Farese, 2009)(Deffayet, Esposito-Farese, & Vikman, 2009)

no longer Galilean invariant but field equations remain of second-order.

iiX

GG

X

This theory includes the Einstein action with as well as a nonminimal coupling with .

24 2PlG M2

4 2G Gravity is naturally included by construction.

(Deffayet, Gao, Steer, Zahariade, 2011)

3

22

4 4

2 33

5 5

,

,

,

1, 3 2

6

X

X

K X

G X

G X R G

G X G G

2

3

4

5

L

L

L

L

4 arbitrary functions of and 21

2X

21 1 3 33

8 8 9

2

2 2 6

H X X

X

R R

FR F F X

L

3 3 12 2X XF X [ ]3! with

In fact the most general scalar+gravity theory that yieldssecond-order field eqs. was discovered by Horndeskialready in 1974. (recently revisited by Charmousis et al. 2011)

We have found that the Generalized Galileon is equivalent to Horndeski theory by the following identification.

1 3 8 9, , , ( , )X

(Kobayashi, Yamaguchi & JY 2011)

3

22

4 4

2 33

5 5

,

,

,

1, 3 2

6

X

X

K X

G X

G X R G

G X G G

2

3

4

5

L

L

L

L

Inflation from a gravity + scalar system 5

4

2i

i

S gd x

L

This theory includes potential-driven inflation models k-inflation model Higgs inflation model New Higgs inflation model G-inflation model

5G G

, [ ]K X X V ,K X K X

2 242 2R G

( , ) ( , )K X G X

24 2PlG M gives the Einstein action

3

22

4 4

2 33

5 5

,

,

,

1, 3 2

6

X

X

K X

G X

G X R G

G X G G

2

3

4

5

L

L

L

L

Inflation from a gravity + scalar system 5

4

2i

i

S gd x

L

This theory includes potential-driven inflation models k-inflation model Higgs inflation model New Higgs inflation model G-inflation model

5G G

, [ ]K X X V ,K X K X

2 242 2R G

( , ) ( , )K X G X

24 2PlG M gives the Einstein action

Generalized G-inflation is a framework to study the most general single-field inflation model

with second-order field equations.

2

2 41

2 2 4PlM

R L L

The standard Higgs potential is too steep as it is.Several remedies have been proposed.

(original idea: Spokoiny 1984 Cervantes-Cota and Dehnen 1995)

Higgs G

(Germani & Kehagias 2010)

(Nakayama & Takahashi 2010)

(Kamada, Kobayashi, Yamaguchi & JY 2011)

Running Kinetic

Original2

2R

L

2

1

2 NH

GM

L

2

2

2

n

nRKM

L

HG

XM

L

New Higgs

2

2 41

2 2 4PlM

R L L

All of them are variants of Generalized G-inflation

2

2R

L

2

1

2 NH

GM

L

2

2

2

n

nRKM

L

HG

XM

L

Original

New Higgs

Higgs G

Running Kinetic

( , ) ( ) ( ) ...K X V X K ( , ) ( ) ( ) ...I I IG X g h X

2

2( ) 1

n

nRKM

K

22

4 ( )2 2

PlMg

4 2

1( )

NH

hM

3( )HG

gM

5 ( )RE

hM

Running Einstein

33 10r

0.05 0.16r

0.05 0.13r

0.05r

MCMC analysisBy Popa 2011

So far each model has been analyzed separately.

G original new

In terms of the Generalized G Inflation all the Higgsinflation models can be analyzed seamlessly.

sn

Higgsm

In the generalizedHiggs inflation, r is appreciably larger than the original value .33 10r

spectral index

Viable inflation models must satisfy stability conditionsnot only during inflation but also afterwards.

must be satisfied to avoidghost and gradient instabilities.

Generalized consistency relation

which reduces to the standard one in the appropriate limits.

In fact, as for Higgs G-inflation, it was recently foundthat inclusion of higher order kinetic termis required to make the theory well behaved in thefield oscillation regime after inflation, which makes itpossible to give various predictions for .

( 2)X

( , )sn r

(cf Ohashi & Tsujikawa 2012)

(Kunimitsu et al 2013 in preparation)

The Higgs self coupling must remain positiveto well above the scale of inflation.

Inflation followed by a kination regime withgravitational reheating should be ruled out.

Higgs inflation is possible and subject to observational tests in the context of theGeneralized G inflation.

All the Higgs inflation models are unified in the Generalized G-inflation.

Open issue: Quantum corrections in the presenceof these new interaction terms.

We consider potential-driven inflation in the generalized G-inflation context.So we expand

and neglect higher orders in X.

We find the following identities. (t.d.) ＝ total derivative

As a result we can set in the Lagrangian without loss of generality.

4g g