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
Top Related