Reheating of the Universe after Inflation with f()R Gravity:

Spontaneous Decay of Inflatons to Bosons, Fermions, and Gauge Bosons

Eiichiro Komatsu & Yuki WatanabeUniversity of Texas at Austin

Caltech High Energy Physics Seminar, March 28, 2007

Reference: Phys. Rev. D 75, 061301(R) (2007)

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Why Study Reheating?•The universe was left cold and empty after inflation.

•But, we need a hot Big Bang cosmology. •The universe must reheat after inflation.

Successful inflation must transfer energy in inflaton to radiation, and heat the universe to at least ~1 MeV for successful nucleosynthesis.

…however, little is known about this important epoch….

Outstanding Questions•Can one reheat universe successfully/naturally? •How much do we know about reheating? •What can we learn from observations (if possible at all)?•Can we use reheating to constrain inflationary models?•Can we use inflation to constrain reheating mechanism?

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Standard Picture )(φV

φinflaton

Oscillation Phase: around the potential minimum at the end of inflati

on

Slow-roll Inflation: potential shape is arbitrary here, as long as it is flat.

inf2

2inf

24inf

4

4

~

~)(~

~

HMgT

HMgVg

T

Plrh

Pl

rhrad

⇒

φ

ρEnergetics:

What determines “energy-conversion efficiency factor”, g?

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φ

φ

Perturbative ReheatingDolgov & Linde (1982); Abbott, Farhi & Wise (1982); Albrecht et al. (1982)

Thermal medium effect

Inflaton can decay if allowed kinematically with the widths given by

g g

⎟⎠

⎞⎜⎝

⎛ +++−= L42int 4

1λφφχψψφ χψ ggL

Pauli blocking

Bose condensateφ

φφ

φφ

φ

Inflaton decays and thermalizes through the tree-level interactions like:

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Reheating Temperature from Energetics

Coupling constants determine the decay width, But, what determines coupling constants?

0)3( 2 =+++ φφφ σmH tot&&&

density.energy thedominate productsDecay 3

density.energy thedominatesInflaton 3

⇒<⇒>

totosc

totosc

HH

4*

22222 )(

3033)( rhrh

totPloscPlrhrad TTg

MHMt

πρ =

==

4/1

*4/12 100

)(

)10(

−

⎟⎠

⎞⎜⎝

⎛= rhtotPl

rhTgM

Tπ

2/3inf

−∝>> aHH oscφ

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.10~then

, 10 and 10 If

~

7

6inf

10

inf2

−

−− ==

g

MHMT

HMgT

PlPlrh

Plrh

Fine-tuning Problem?

inf2

2inf

24inf

4

4

~

~)(~

~

HMgT

HMgVg

T

Plrh

Pl

rhrad

⇒

φ

ρ

To relax fine-tuning, one needs:(a)High reheat temperature

-> unwanted relics (e.g., gravitinos), (b)Very low-scale inflation (H~10-18 Mpl~10 GeV) -> worse fine-tuning, or

(c)Natural explanation for the smallness of g.

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What are coupling constants? Problem: arbitrariness of the nature of inflaton fields

•Inflation works very well as a concept, but we do not understand the nature (including interaction properties) of inflaton.

•Arbitrariness of inflaton = Arbitrariness of couplings•Can we say anything generic about reheating? Universal reheating? Universal coupling?

e.g. Higgs-like scalar fields, Axion-like fields, Flat directions, RH sneutrino, Moduli fields, Distances between branes, and many more…

Gravitational coupling is universal

What happens to “gravitational decay channel”, when GR is modified?

-> however, too weak to cause reheating with GR.In the early universe, however, GR would be modified.

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Conventional Einstein gravity during inflation

( )L+++−= 222int χλφφχψψφ χψ ggL

Conventionally one had to introduce explicit couplings between inflaton and matter fields by hand.

Einstein-Hilbert term generates GR.Inflaton minimally couples to gravity.

mattLg−2PlM φφ νμ ∇∇

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2)( PlMvf =

Modifying Einstein gravity during inflation

( )L+++−= 222int χλφφχψψφ χψ ggL

Instead of introducing explicit couplings by hand,

Non-minimal gravitational coupling: common in effective Lagrangian from e.g., extra dimensional theories.In order to ensure GR after inflation,

Matter (everything but gravity and inflaton) completely decouples from inflaton and minimally coupled to gravity as usual.

mattLg−φφ νμ ∇∇

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Field equations: GR

Linearized field equation:

Wave modes are gravitational waves. To identify the wave modes, we usually define

Harmonic (Lorenz) gauge:

2PlM φφ νμ ∇∇

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Field equations: f()R gravity

Linearized field equation during coherent oscillation

Wave modes are mixed up. To extract “true” gravitational degrees of freedom, we define

Harmonic (Lorenz) gauge:

φ

σ

v

φφ νμ ∇∇

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New decay channel through “scalar gravity waves”

Fermionic (spinor) matter field:

Bosonic (scalar) matter field:

Three-legged interaction

Yukawa interaction

g

σ

g

σ

g

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Spontaneous emergence of Yukawa interaction: analog of spontaneous symmetry breaking

.4

1

2

1

2

1 422 constL +⎟⎠

⎞⎜⎝

⎛ +−∂∂−= λφφμφφ μμ

⎟⎠

⎞⎜⎝

⎛ ++−∂∂−= 4322

4

1

2

1ληηληληη μ

μ vvL

)()( xvx ηφ +=

vλμ

φφ

λμ2

2

0)(

0 ,0

−=⇒=

∂

∂

><

vV

Expanding around the vev, one gets

-v

New term appeared.

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)()( xvx σφ +=

( ) mLvvRvvgL +⎥⎦

⎤⎢⎣

⎡⎟⎠

⎞⎜⎝

⎛++−∇∇−++−= 432222

4

1

2

12

2

1λσσλσλσσξσσξξ μ

μ

( ) mLvRgL +⎥⎦

⎤⎢⎣

⎡−−∇∇−−=

2222

42

1

2

1φ

λφφξφ μ

μ

Wave mode mixing in the linear perturbations(appearance of scalar gravity waves)

σg σg

Spontaneous emergence of Yukawa interaction: analog of spontaneous symmetry breaking

Yukawa interactionsare induced.

Expanding around the vev, one gets

2)( ξφφ =fe.g.

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Magnitude of Yukawa coupling

For f()=ξg=ξ(v/Mpl)(m/Mpl) Natural to obtain g~10-7 for e.g., m~10-7 Mpl

The induced Yukawa coupling vanishes for massless fermions: conformal invariance of massless fermions.

Massless, minimally-coupled scalar fields are not conformally invariant. Therefore, the three-legged interaction does not vanish even for massless scalar fields:

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The Results So Far…

After inflation with f()R gravity, inflatons decay spontaneously into: Massive fermions, Massive scalar bosons, or Massless scalar bosons with non-conformal coupling.

The smallness of coupling can be explained naturally. Inflaton decay is “built-in” and the coupling constrants can be c

alculated explicitly from a single function, f(). Rates of decay to fermions and bosons are related.

This mechanism allows inflatons to decay into any fields that are not conformally coupled. Other possibilities?

Phys. Rev. D 75, 061301(R) (2007)

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Breaking of conformal invariance by anomaly

Example: decay to massless gauge bosons, F

Conformally coupled fields at the tree-level may not be conformally invariant when loops are included.

g

F

F

(c.f.) two-photon decay of the Higgs

€

α

€

α

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Conformal anomaly: Lowest order decay channel to massless gauge fields

Inflaton -> 2 gauge fields

∞→→→→

xxI

xxI

as 0)(

0 as 3/1)(

g

F

F€

α

€

α

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Decay Width Summary

Fermions

€

=f '(v)[ ]

2mψ

2mσ

32πM pl4

Scalar Bosons

€

=f '(v)[ ]

2mχ

2 +mσ

2

2

⎛

⎝ ⎜

⎞

⎠ ⎟

2

32πM pl4 mσ

Gauge Bosons

€

=α 2 f '(v)[ ]

2mσ

3

256π 3M pl4

× # of internal fermions etc( )

Probably the most dominant decay

channels

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Constraint on f()R gravity models from reheating

4/1

*4/12 100

)(

)10(

−

⎟⎠

⎞⎜⎝

⎛= rhtotPl

rhTgM

Tπ

e.g.Constraints from chaotic inflation

999)Futamase(1&Komatsu

105||

)Maeda(1989&Futamase

10

4

3

λξ

ξ

×>

−> −

v

Trh

μμσ aa AAΓ+

2)( ξφφ =f

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Connection to Supergravity?

Similar effects have been pointed out by Endo, Takahashi and Yanagida (2006; 2007) in the context of supergravity Inflatons decay into any fields even if inflatons are not coupl

ed directly with these fields in the superpotentialA correspondence may be made as

f()R gravity <-> Kahler potentialConformal anomaly <-> Super-Weyl anomaly

Our model is simpler and does not require explicit use of supergravity -- hence more general. It may also give physical (rather than mathematical) insight i

nto their effects.

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Conclusions A natural mechanism for reheating after inflation with f()R gra

vity: Why natural? Inflaton quanta decay spontaneously into any matter fields (spin-0, ½,

1) without explicit interactions in the original Lagrangian Conformal invariance must be broken at the tree-level or by loops Reheating spontaneously occurs in any theories with f()R gravity

Predictability All the decay widths are related through a single function, f().

A constraint on f() from the reheat temperature can be found A possible limit on the reheat temperature can constrain the form of f(),

or vice versa. These constraints on f() are totally independent of the other constraints

from inflation and density fluctuations

Further Study… Preheating? F(,R) gravity?