χ

χ

χ

χ

χ

χ

χ

χχ

χ

χχ

RESearch Center for the Early UniverseThe University of Tokyo

Scalar fields could be the origin of everything!

Large Homogeneous, Isotropic, & Flat Universe

Density fluctuations & CMB Anisotropy

Radiation

Baryon Asymmetry

Dark Matter

Inflation driven by a scalar fieldcalled the Inflaton

Quantum fluctuations of the Inflaton field

Reheating by Inflaton’s decay

Affleck-Dine scalar fields in SUSY ？

Q-balls ？

But some other scalar fields could be harmful: Moduli

Today’s talk

• Behavior of an oscillating scalar field in a thermal bath

• The case decay products of the oscillating scalar field (in particular, the inflaton) has a larger thermal mass than the oscillation frequency (or the inflaton mass).

• Application to the cosmological moduli problem

The Origin of the Hot Big Bang Universe

Reheating Processes After Inflation ＝ Entropy Production through the decay of the Inflaton, a scalar field which drives inflation.

New/Topological inflation

Hybrid inflation

φ

V

Ψ

eff

V[ ]

v

eff

“Energy history” of the inflationary universe

Exponential expansion

Potential energy Slow rollover

Kinetic + potential energy

Rapid field oscillation Preheating (parametric resonance) Reheating (perturbative decay)

Radiation dominated stage

Reheating temperature RT

The maximum temperature after inflation is much higher than the reheating temperature in general.

The inflaton decays in a thermal bath.

Chaotic inflationφ

V[φ]

eff

１

An interesting possibility

If the would-be decay product of the inflaton acquires a thermal masswhich is larger than the inflaton’s mass, its decay is temporarily suspended.

(Linde 1985, Kolb, Notari, & Riotto 2003)

other scalar particles

fermions

or

( )m T gT thermal mass

Phase space is closed and the scalar field cannot decay if would-bedecay products have a thermal mass larger than !?2M

The decay rate of the inflaton to two massive particles with mass . m1

2 2

0 2

41

m

M

Decay rate to two massless particles.

M : the inflaton mass

thermal mass

Finite-Temperature Effective Potential

2222 22 41

[ ]2 4

...4

Tg

Vg

m

φdominant

radiation dominant

conventional reheating

φdecays completely at

1T a

3 8T a

1t

If so, thermal history after inflation is drastically changed.

(3 )

4 rr

dH

dtd

Hdt

3

( )

4

( )( ) ( )

( )

( )( ) ( )

( )

e

e

t te

e

t

ret

a tt t e

a t

a tt dt t

a t

field oscillation

radiation

Reheat temperature

,R convT

Thermal history after preheating in conventional theory with const

If inflaton’s decay rate vanishes at high temperature , thermal history is drastically changed.

conventional reheating

,R convT

ρφ=ρr

This would affect the relic abundances of gravitinos, superheavy particles etc.

constT

new reheating scenario

φ decays gradually, keeping .,R newT M g

huge discrepancy

discrepancy

T M g

Thermal mass is different from intrinsic mass.

Coherent field oscillation is different from a collection of particles.

Here, we consider① Nonequilibrium field theory for the oscillating scalar field② The case decay products do not have any thermal masses.③ The case decay products have a large thermal mass.

Assumptions & Conditions: ① Neglect cosmic expansion ② The would-be decay products of the oscillating field are

in thermal equilibrium at a fixed temperature. ③ The oscillating field is in nonequilibrium and oscillating. ④ Parametric resonance ineffective (after the preheating stage, if any).

Fromalism to analyze behavior ofoscillating scalar fields

in a thermal bath

4 4

0 0

( ) exp (0)expt t

t i d x i d x H H

Time ordered & anti-ordered product

4 4

0 0

in ( ) in in exp (0)exp int t

t i d x i d x H H

Heisenberg picture

time flow

Coherent field oscillation behaves almost classically.But its decay is of course a quantum process.

Derive an effective equation of motion for the expectation value ofthe scalar field φ by calculating its effective action Γ(φ).

cf Quantity calculated in ordinary quantum field theory: Transition Amplitude

What fraction of the initial state goes to the final state?

time flow Time ordered product

4out ( )exp inT t i d x

O L

oscillating scalar field (inflaton)

φ→χχ φ→ψψinteracting field χ in a thermal state with temperature

Model Lagrangian

1T

Generating functional

+branch

-branch

Effective Action in terms of the Legendre Transform

Field variables also have suffices ±, and + fields interact with – fields, although they should be regarded as the same fieldin the end.

timei

e H

Calculate the Effective action perturbatively.

f f M2h

2h 2h2h 2h

M

using finite-temperature propagatorsin the closed-time path formalism

represent interactions between and .

Since and are identified in the end, it is more convenient to define

and set in the end.0

f f M2h

2h 2h2h 2h

M

0

0

Equation of motion

M M

induced by the interaction . From now on, I concentrate on the diagram related with the decay process

Since φ is a real scalar field, we cannot obtain a sensible equation of motion by the variation of such a complex-valued effective action Γ.

Its contribution is complex-valued which is a manifestation of the dissipativenature of this interaction.

imaginary part

Its real part and imaginary part are mutually related.

Effective Action

2

As we often encounter a complex-valued effective action or effective potential even for a real scalar field, there is a known prescription to obtain a real-valued equation of motion.

instability,d

issipa

tion

① 　 Introduce a real-valued random Gaussian auxiliary field 　　　 and rewrite the effective action as

including a path integral of .

② 　　　　 is a probability distribution function defined by

Gaussian with a dispersion

： the imaginary part of the effective action

（ N.B. ）　 If we performed path integral over using Gaussian integral,

we would recover the original complex-valued effective action.

The expectation value of the scalar field evolves according to the above Langevin equation.

③ 　 Here we take variation of the effective action as it is.

Manifestly real-valued equation of motion!

Auxiliary field is treated as a random Gaussian noise with a dispersion .

④ 　 Equation of motion: a Langevin equation auxiliary stochasticfield

quantum correction Memory term depending on the past

Real part of the effective action: Deterministic terms in EOMImaginary part of the effective action governs a Stochastic Noise term.

,

multiplicative noise

f f 2h 2h2h

2h 2h2h 2h

(N.B.) If we incorporateother diagrams, theLangevin equation hasboth additive and multiplicative noises anddissipation terms.

The Langevin eq. can easily be solved via Fourier transform.

spatial Fourier transform

General solution

The memory of the initialcondition is erased after　　　　　　　　　1( )k kt M

temporal Fourier transform pure imaginary

real

1( )k kt M

and that the mean square amplitude of each Fourier mode

averages to zero

noise correlation relaxation time, inverse dissipation rate

relaxes to a constant.

Next we calculate

From the solution we find only the following term survives at late time

Inflaton’s dissispation rate is given by evaluated at .

So we calculate

using

kM

0

cos( )p k pdt t

( )p B pn n

DR CRannihilation terms creation terms

k p

p p

k p

First line

k p

pk p

p

Second line

ー

( )

(1 )(1 )p k pp k pC

D p k p

n nRe e

R n n

( )(1 )

(1 )p k pk p pC

D p k p

n nRe e

R n n

( )p k p

( )p k p

The ratio takes a constant .e C DR R

Detailed balance relation

The dissipation rate of the homogeneous mode ( ) has a simple form

One particle decayrate in the vacuum

Induced emission

This dissipation rate vanishes if .2m M

at hightemperature

This δfunction vanishesif .2m M

Decay rate throughYukawa coupling

Pauli blocking

Cf The decay rate to fermions is suppressed by Pauli blocking.

f f

Incorporation of the thermal massof the decay products

In our scheme, thermal mass is included in , if we incorporate the finite-temperature self energy of χ, Σ(T ), in its propagator, because is determined by the pole of the propagator ….

( )m T

Σ＝　　　　　＋　　　　　　　＋　　　　　　　　　　＋　　　　　　　　　　　　　……　　Σ Σ Σ Σ Σ

Full or ‘dressed’propagator

originalpropagator

Resummation

Does this apply to the large thermal mass as well?

Does the dissipation rate vanish if ?

( ) 2m T M

( ) 2m T M

Full propagator in the Matsubara representation

Σ＝　　　　　＋　　　　　　　＋　　　　　　　　　　＋　　　　　　　　　　　　　……　　Σ Σ Σ Σ Σ

self energy due to χ’s interaction

2 2 2

1( , )M n

n

Gm

k

k

1 2 2 2

....

1 1 1

1 ( ) ( )

M M M M M M M M M M

MM M n

G G G G G G G G G G

GG G T m T

k

MG

( )T 2 2g Tincludes a thermal mass term such as which depends on the nature of χ’s interaction.

Apparently, high-temperature effect closes the phase space of φ’s decay.

2 2 2 2 2 ( ) ...m m T m g T

0 ?

2 2 2 2, , 2

Ip R R p

p

m g T

p

Then the δ function is replaced by

the Breit-Wigner form

It is nonvanishing even when . ( ) 2m T M

However, contains an imaginary part as well,and the full propagator has a complex phase.

( )T

0

cos( 2 ) ( 2 )p pdt M t M

2

2 20

2cos( 2 ) .

( 2 ) (2 )pt p

pp p

dte M tM

( ) R IT i

Dissipation rate of the zero mode coherent field oscillation of the inflaton φ

0 2 pM 0p for

0 for 0p

When , the dissipation rate reads( )M m T

It is nonvanishing and proportional to . p

Imaginary part of self energy dissipation rate of the decay product χ, not the inflaton φ.It depends on interaction of χ which thermalizes it.

dominant

For example, if χ thermalizes through interaction,

we find for the imaginary part of χ’s self energy.

24

4

g 4 23

( )128p p

p

g T

2g

As a result, the dissipation rate of the inflaton φ is given by

2

0 2 2

3( )

2 24

T gM

M

2M

( )2

gTM m T for

dissipation rate to massless particlesat high temperature

suppression factordepending on the form of p

p p p m coupling constantsin general.

The dissipation rate of the inflaton is finite even when its decay product, χ, acquires a larger mass than the inflaton in a high temperature plasma.

12 2

2 2 2*

90 3

2 24G

R

M gT

g M

2M

reheat temperature when inflaton decays to massless particles

suppression factor

conventional

in case thermal mass prohibits decay

,R convT

,R actualTM

suppressed by couplingconstants of the decay product

actual thermal history

1 2T a

3 8T a

Oscillating scalar fields can dissipate their energy even if thermal massesof the decay products are larger than the oscillation frequency.

Not only the thermal mass, namely real part of the self energy, but also its imaginary part of the would-be decay product, ,plays an important role.

When , the reheat temperature is suppressed by a powerof coupling constants which thermalizes the decay product χ.

( )M m T p

conventional

in case thermal massprohibits decay

actual thermal history

,R convT

,R actualTM g

gravitino abundance dependson the physics of decay products.

supermassive particles could be created.M

Application to the cosmological moduli* problem

*Here we mean scalar fields typically with weak or TeV mass scales which have Planck-suppressed interactions with other fields

Modulus field does not move due to a largeHubble friction until the Hubble parameterdecreases to its mass scale .

Evolution of moduli fields in the early Universe

φ

H m

Hm

It has a very long lifetime because it interacts with other fields only with the

gravitational strength,1 33

1 82 2

10 sec10 GeVG

m m

M

It starts oscillation around apotential minimum when .

φ

H m H m

This is a coherent oscillation of zero-modescalar field condensate.

181 8 2.4 10 GeVGM G

21 2

82

1010 GeV

initial

G

n m

s M

3 2 1210

1010 GeV

Rn T

s

much severer than the gravitino problem

demolishes primordial nucleosynthesis

1610n

s must satisfy

What all the previous studies have neglected….

φ

H m

H m12

102

10 GeV10 GeV

mT

The cosmic temperature at the onset of moduli oscillation

was very high .

3

2G

m

M is the decay rate in the vacuum, modified at finite temperature.

2 2

G

mM intL

3 3

2 2

4( ) 1 2

2BG G

m m m TT n

M M m

2

1G

m

H TM

e.g. Decay into two bosons through the interaction

If was in thermal equilibrium, the decay rate is enhanced by theinduced emission.

But it does not help, because .

3

2G

m

M

2

G GM M

intLG

F FM

intL

2m

In the thermal background, acquires a large thermal mass .

If we could replace by , the coupling of moduli with could be significantly enhanced, leading a much larger decay rate.

Tm gT T

2 2g T

Moduli are coupled with kinetic terms as well such as

It has been concluded, however, that such couplings lead to the

decay rate similar to using equation of motion .

GMe

Taking we find .

Moduli decay right after they start oscillation if ?

2 2 2

G

g TM

intL

2 4 4

2

4(suppression factor)

8 G

g T T

M m m

2 1010 GeV, 10 GeVm T 910 GeV (suppression factor) 210 GeVH m

In the presence of thermal background, moduli fields decay as soon as they start oscillation ??

due to the fact that the decay product has a large thermal mass.

GM

intLDecay rate through

would be given by

Calculate the modulus dissipation rate with the same method

Oscillating scalar field Moduli

Standard field interacting with

Simple Model

GM

GM

One loop effective action relevant to dissipation

φ

χ

int+L

thermalizing interaction

One-loop Effective Action in terms of

Real part: Dissipation

Imaginary part: Fluctuation

Langevin type Equation of motion auxiliary stochasticfield

quantum correction Memory term depending on the past

Moduli’s dissipation rate is given by the Fourier transformof the memory kernel which is related to theimaginary part of ’s self energy.

In order to take thermal effects on into account correctly, we should usethe dressed propagator of in the loop calculation.The dressed propagator is obtained by resummation.

GM

GM

Σ＝　　　　　＋　　　　　　　＋　　　　　　　　　　＋　　　　　　　　　　　　　……　　Σ Σ Σ Σ Σ

Full or ‘dressed’propagator

originalpropagator

Resummation (Matsubara representation)

self energy

,which aredetermined by ’s interaction

Temperature-corrected moduli dissipation rate (to the first order in )

: themal mass of

The last term could be larger than the other terms.

number of decay modes

3

2G

T

M

much larger than the vacuum value but yet insufficient

N

p

in the current version of the paper, I have incorrectly neglected in the denominatorand got a very large dissipation rate in proportionto .

22 p

2m

This point was indicated by Boedeker recently,who inappropriately put and obtained a dissipation rate .

0m 3 2

GT M

number of decay modesIn reality,…

This factor is bounded by unityand is maximalwhen .2 p m

This gives the largest contribution if , when we find2 p m 4

2G

T

M m

enhanced by rather than .T

m

2

2

T

m

we require when . Putting , we find2GH T M m H Tm gT

2 3

1650N N

g

Modulus may be dissipated due to thermaleffects.

the modulus oscillation can be dissipated through the proposed mechanism.If this dissipation rate is larger than the cosmic expansion rate,

Several comments:

The proposed mechanism works well only at relatively high temperature.1 2

92

10 GeV10 GeV f

mH T T

Gravitino problem may still be a problem.Entropy production of several order of magnitudeis required.

So the reheat temperature after inflation must be sufficiently high.

In this mechanism, the moduli fields decay into only particles in a high-temperature thermal state. This means that gravitinos are not produced practically in this mechanism. This may be helpful to the “new gravitino problem” in the heavy moduli scenarios.

Several comments:

Thermal effects may also shift the potential minimum.

GM

2 2 41[ ]

2effG

V m O TM

4

min 2( )

G

TT

m M

large shift at high temperature

The proposed mechanism anchors the modulus at the finite-temperature minimum. After that, traces adiabatically as the universe cools down, because theHubble parameter is much smaller than by that time. So this does not cause any serious problem.

min ( )Tm

Thermal effects may also induce a mass term proportional to .

2 22 2

42

41[ ]

2 2eG G

ffV m O O TM

TM

22 2

2H

2 4G GM Me e T

2H

Then the modulus field oscillates with the frequency and our mechanismmay work when .

H( ) 2 pH t

Thermal effects can dissipate the dominant part of thecosmological moduli problem, provided and . 2 p m t 50N

Thermal effects in the early Universeis nontrivial and could be very important !