Implications of the Curvaton on Inflationary Cosmology

Tomo TakahashiICRR, University of Tokyo

March 4, 2005 @ KEK, TsukubaKEK theory meeting 2005

“Particle Physics Phenomenology”

Collaborators: Takeo Moroi (Tohoku) Yoshikazu Toyoda (Tohoku)

1. Introduction

Inflation (a superluminal expansion at the early universe) is one of the most promising ideas to solve the horizon and flatness problem.

[Guth 1980, Sato 1980]

The potential energy of a scalar field (inflaton) causes inflation.

Inflation can also provide the seed of cosmic density perturbations today.

Quantum fluctuation of the inflaton field during inflation can be the origin of today’s cosmic fluctuation.

★

From observations of CMB, large scale structure, we can obtain constraints on inflation models.

However, fluctuation can be provided by some sources other than fluctuation of the inflaton.

What if another scalar field acquires primordial fluctuation?

Such a field can also be the origin of today’s fluctuation.

Curvaton field (since it can generate the curvature perturbation.)

Generally, fluctuation of inflaton and curvaton can be both responsible for the today’s density fluctuation.

★

What is the implication of the curvaton for constraints on models of inflation?

From the viewpoint of particle physics, there can exist scalar fields other than the inflaton.

(late-decaying moduli, Affleck-Dine fields, right-handed sneutrino...)

[Enqvist & Sloth, Lyth & Wands, Moroi & TT 2001]

This talk ➡

Primordial fluctuation: the standard case

The quantum fluctuation of the inflaton

2. Generation of fluctuations: Standard case

δχ ∼

H

2π

The curvature perturbation R ∼ −H

χδχ

The power spectrum of .

PR ∝ kns−1

Almost scale-invariant spectrum

Generally, it can be written as a power-law form;

InflationInflation RDRD MDMD

reheatingreheating

Inside the horizonInside the horizon

Scal

e

Scale factor a

horizon scale

R(The initial power spectrum)

H ≡

a

a:Hubble parameter

horizon crossingPR(k) ∼< R

2 >∼

(H

χ

)2 (H

2π

)2∣∣∣k=aH

χ ≡

dχ

dt

depends on models of inflation ns★

What is the discriminators of models of inflation?

The initial power spectrum ( ) depends on models.PR ∝ kns−1

The observable quantities

During inflation, the gravity waves are also produced.

The size of gravity waves (tensor mode) also depends on models.

Scalar spectral index

Tensor-scalar ratio

ns

r ≡

Pgrav

PR

Constraints on models of inflation.

These parameters can be determined from observations.(such as CMB, large scale structure and so on.)

Overall normalization of PR( )

(ns − 1 ≡

d lnPR

d ln k

∣∣∣k=aH

)

Constraints from current observations [Leach, Liddle 2003 ]

ns!1

r

!0.08 !0.04 0 0.040

0.1

0.2

0.3

0.4

0.5

r

ns − 1

95%99%

[Data from CMB (WMAP, VSA, ACBAR) and large scale structure (2dF) are used.]

V ∼ χ4

V ∼ χ6

V ∼ χ2

Some inflation models are already excluded!

★

Example: chaotic inflation

V (χ) = λM4pl

(χ

Mpl

)α

α = 4, 6, · · ·The cases with ★ are almost excluded.

Slow-roll parameters, ε ≡1

2M2pl

(V ′

V

)2

η ≡

1

M2pl

V ′′

V

The scalar spectral index ns − 1 = 4η − 6ε

The tensor-scalar ratio

The slow-roll approximation is valid when .

The observable quantities can be written with the slow-roll parameters.

Equation of motion for a scalar field: where

When it slowly rolls down the potential,

χ + 3Hχ +dV (χ)

dχ= 0 χ ≡

dχ

dt

V ′(χ) ≡dV (χ)

dχ

3Hχ +dV (χ)

dχ! 0

(Slow-roll approximation)

ε, |η| ! 1

r =Pgrav

PR

= 16ε

where

★

3. Curvaton mechanism [Enqvist & Sloth, Lyth & Wands, Moroi & TT 2001]

a light scalar field which is partially or totally responsible for the density fluctuation.

Curvaton:

Roles of the inflaton and the curvaton

Inflaton ➡ causes the inflation,

not fully responsible for the cosmic fluctuation

Curvaton ➡ is not responsible for the inflation,

(Usually, quantum fluctuation of the inflaton is assumed to be totally responsible for that.)

is fully or partially responsible for the cosmic fluctuation

Constraints on inflation models can be relaxed.

Thermal history of the universe with the curvaton

Inflation RD1Curvatondominated

RD2

!BBN

Ener

gy d

ensi

ty

Timereheating reheating

Vcurvaton =1

2m2

φφ2The potential of the curvaton is assumed as

The oscillating field behaves as matter.

reheating

(For simplicity, an oscillating inflaton period is omitted.)

Inflation RD

(Standard case)

Ener

gy d

ensi

ty

Time

!BBN

inflaton

curvaton

Thermal history of the universe with the curvaton

!BBN

Ener

gy d

ensi

ty

Timereheating reheating

Vcurvaton =1

2m2

φφ2The potential of the curvaton is assumed as

The oscillating field behaves as matter.

(For simplicity, an oscillating inflaton period is omitted.)

V (φ)

φ

V (φ)

φ

H > mφ

H ∼ mφ

Inflation RD1Curvatondominated

RD2

inflaton

curvaton

Density Perturbation

Fluctuation of the inflaton Curvature perturbation

R ∼ −H

χδχ

Fluctuation of the curvaton No curvature perturbation(The curvaton is subdominant component during inflation.)

However, isocurvature fluctuation can be generated.

Sφχ ∼ δχ − δφ =2δφinit

φinit

Inflaton

curvaton

Curvaton becomes dominant

At some point, the curvaton dominates the universe.

the isocurvature fluc.becomes adiabatic (curvature) fluc.

δi ≡

δρi

ρi

where

(δφ ∼

H

2π

)

(δχ ∼

H

2π

)

After the decay of the curvaton, the curvature fluctuation becomes

From inflaton From curvaton

X = φinit/Mpl

where

f(X): represents the size of contribution from the curvaton

The power spectrum (scalar mode)

f = (3/√

2)fwhere

The tensor mode spectrum is not modified.

The scalar spectral index and tensor-scalar ratio with the curvaton

ns − 1 = 4η − 6ε(Notice: the standard case , )

r =16ε

1 + f2εns − 1 = −2ε +

2η − 4ε

1 + f2ε

r = 16ε

,

[Langlois, Vernizzi, 2004]

RRD2 =1

M2pl

Vinf

V ′

inf

δχinit +3

2f(X)

δφinit

Mpl

PR =9

4

[1 + f2(X)ε

] Vinf

54π2M4plε

Pgrav =2

M2pl

(H

2π

)2

f(X) !

4

9X: φinit " Mpl

1

3X : φinit # Mpl

0

0.5

1

1.5

2

2.5

3

3.5

4

0 1 2 3 4 5 6 7 8 9 10

f(!

*)

!*/MplX(= φinit/Mpl)

f(X

)

Contribution from the curvaton

[Langlois, Vernizzi, 2004]

f(X) can be obtained solving a linear perturbation theory.

For small and large limit,

Small and large initial amplitudegive large curvaton contributions.

★

RRD2 =1

M2pl

Vinf

V ′

inf

δχinit +3

2f(X)

δφinit

Mpl

Relaxing constraint on inflation models with curvaton

Vinf = λχ4

for φinit = 0.1Mpl

With the curvaton,

The model becomes viable due to the curvaton!

ns!1

r

!0.08 !0.04 0 0.040

0.1

0.2

0.3

0.4

0.5

r

ns − 1

95%99%

V ∼ χ4

ns − 1 ∼ −0.04, r ∼ 0.08★

An example: the chaotic inflation with the quartic potential

[Langlois & Vernissi 2004; Moroi, TT & Toyoda 2005]

Without the curvaton (i.e., the standard case)

ns − 1 ∼ −0.05, r ∼ 0.27

(Assuming N=60)

Model parameter dependence

Whether inflation models can be relaxed or not depends on the model parameters in the curvaton sector.

★

Initial amplitude of :

Mass of :

Decay rate of :

φ

mφ

φinit

Γφ

φ

φ

Model parameters in the curvaton sector

affects the amplitude of fluctuation and the background evolution

affects the background evolution

InflationInflation RDRD MDMD

reheatingreheating

Inside the horizonInside the horizon

InflationRDI RD2

MDφD

Scal

e

Scale factor a

horizon scale

horizon crossing

(standard case)

[Moroi, TT & Toyoda 2005]

Model parameter dependence: Case with the chaotic inflation

X(=

init /

Mpl)

102

103

104

105

106

107

108

109

1010

m (GeV)

10-2

10-1

100

101

Vinf = λχ4Potential:

99 % C.L.

The case with large initial amplitude cannot liberate the model,although contribution from the curvaton fluc. is large.

★

★

Γφ = 10−10

GeV

Γφ = 10−18

GeV

The region where the curvaton causesthe second inflation.

[Moroi, TT & Toyoda 2005]

r

ns − 1

99%

★

w/o curvaton

★

Allowed Not allowed

The criterion for the alleviation of the model

Case with the natural inflationX

(=in

it /

Mpl )

5 6 7 8 9 10

F (1018

GeV)

10-2

10-1

100

101

Vinf = Λ4

[1 − cos

( χ

F

)]

0.75

0.8

0.85

0.9

0.95

1

4 5 7 10 20

ns(kCOBE)

(1018GeV)F

Potential:

95 % C.L.

The scalar spectral index

[Freese, Friemann, Olint, Adams,Bond, Freese, Frieman,Olint,1990]

ε =1

2

(Mpl

F

)2 1

tan2(χ/2F )

η =1

2

(Mpl

F

)2 [1

tan2(χ/2F )− 1

]

[Moroi, TT 2000]

[Moroi, TT & Toyoda 2005]

Case with the new inflation

X(=

init /

Mpl )

1018

1019

v(GeV)

10-2

10-1

100

101

Vinf = λ2v4

[1 − 2

(χ

√2v

)n

+

(χ

√2v

)2n

]Potential:

ns − 1 = −2n − 1

n − 2

1

Ne

ε = n2

(Mpl

v

)2[

1

n(n − 2)

(v

Mpl

)2 1

Ne

] 2(n−1)n−2

ε =1

9N4e

(v

Mpl

)6

n = 3for ,

99 % C.L.

The scalar spectral index

The tensor-scalar ratio

is small for r v ! Mpl

Curvaton cannot liberate the model,when is very small.

ns − 1 = −2ε +2η − 4ε

1 + f2ε

Case with Curvaton

ε

[Linde; Albrecht, Steinhardt 1982]

[Kumekawa, Moroi, Yanagida 1994; Izawa, Yanagida1997]

[Moroi, TT & Toyoda 2005]

5. Summary

Current cosmological observation such as CMB, LSS give severe constraints on inflation models.★

★ However, if we introduce the curvaton, primordial fluctuation can be also provided by the curvaton field.

★ As a consequence, constraints on inflation models cannot be applied in such a case.

★ We studied this issue in detail using some concrete inflation models, and showed that how constraints on inflation models can be liberated.

★ We found that in what cases inflation models are made viable by the curvaton.