Tomo Takahashi- Implications of the Curvaton on Inflationary Cosmology

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Implications of the Curvatonon Inflationary Cosmology

Tomo Takahashi ICRR, 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 andflatness problem.

[Guth 1980, Sato 1980]

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

Inflation can also provide the seed of cosmic densityperturbations today.

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

★

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



However, fluctuation can be provided by some sources other thanfluctuation 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 forconstraints 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 .

P R ∝ kns−1

Almost scale-invariant spectrum

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

Inflationnflation RDD MDD

reheatingeheating

Inside the horizonnside the horizon

S c

a l e

Scale factor a

horizon scale

R

(The initial power spectrum)

H ≡a

a:Hubble parameter

horizon crossingP R(k) ∼< R2 >∼

H

χ

2H

2π

2k=aH

χ ≡dχ

dt

depends on models of inflationns★



What is the discriminators of models of inflation?

The initial power spectrum ( ) depends on models.P R ∝ 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 ≡P grav

P R

Constraints on models of inflation.

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

Overall normalization of P R( )

ns − 1 ≡

d lnP R

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 arealready excluded!

★

Example: chaotic inflation

V (χ) = λM 4pl

χ

M pl

α

α = 4, 6, · · ·The cases with★

are almost excluded.



Slow-roll parameters, ≡

1

2M 2pl

V V

2

η ≡1

M 2pl

V

V

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

The tensor-scalar ratio

The slow-roll approximation is valid when .

The observable quantities can be written with the slow-rollparameters.

Equation of motion for a scalar field: where

When it slowly rolls down the potential,

χ + 3H χ +dV (χ)

dχ= 0 χ ≡

dχ

dt

V (χ) ≡

dV (χ)

dχ

3H

˙χ

+

dV (χ)

dχ

0(Slow-roll approximation)

, |η| 1

r =P grav

P R= 16

where

★



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

a light scalar field which is partially or totallyresponsible 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 RD1 Curvatondominated RD2

!BBN

E n e r g y

d e n s i t y

Timereheating reheating

V curvaton =1

2m

2

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)

E n e r g y

d e n s i t y

Time

!BBN

inflaton

curvaton



Thermal history of the universe with the curvaton

!BBN

E n e r g y

d e n s i t y

Timereheating reheating

V curvaton =1

2m

2

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 RD1 Curvatondominated 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.

S φχ ∼ δχ − δφ =2δφinit

φinit

Inflaton

curvaton

Curvaton becomes dominant

At some point, the curvatondominates the universe.

the isocurvature fluc.becomesadiabatic (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/M pl

where

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

The power spectrum (scalar mode)

f = (3/√

2)f where

The tensor mode spectrum is not modified.

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

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

r =16

1 + f 2ns − 1 = −2 +

2η − 4

1 + f 2

r = 16

,

[Langlois, Vernizzi, 2004]

RRD2 =

1

M 2pl

V inf

V inf δχ

init +

3

2f (X )

δφinit

M pl

P R =9

4

1 + f 2(X ) V inf

54π2M 4pl

P grav =2

M 2pl

H

2π

2



f (X )

4

9X : φinit M pl

1

3X : φinit M pl

00.5

1

1.5

2

2.5

33.5

4

0 1 2 3 4 5 6 7 8 9 10

f ( ! *

)

!*/ M lX (= φinit/M pl)

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

M 2pl

V inf

V inf δχ

init +

3

2f (X )

δφinit

M pl



Relaxing constraint on inflation models with curvaton

V inf = λχ4

for φinit = 0.1M pl

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 fluctuationand the background evolution

affects the background evolution

Inflationnflation RDD MDD

reheatingeheating

Inside the horizonnside the horizon

InflationRDI RD2

MDφD S c a l e

Scale factor a

horizon scale

horizon crossing

(standard case)

[Moroi, TT & Toyoda 2005]



Model parameter dependence: Case with the chaotic inflation

X ( = i n i t

/ M p l )

10210

310

410

510

610

710

810

910

10

m (GeV)

10-2

10

-1

100

101

V inf = λχ4Potential:

99 % C.L.

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

★

★

Γφ = 10−10GeV

Γφ = 10−18

GeV

The region where the curvaton causesthe second inflation.


r

s−1

99%

★

w/o curvaton

★

AllowedNot allowed

The criterion for the alleviation of the model



Case with the natural inflation

X ( = i n i t

/ M p l

)

5 6 7 8 9 10

F (1018

GeV)

10-2

10-1

100

101

V inf = Λ

4 1−cos

χF

0.75

0.8

0.85

0.9

0.95

1

4 5 7 10 20

n s

( k C O B E )

(1018

GeV)F

Potential:

95 % C.L.

The scalar spectral index

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

=1

2

M plF

21

tan2(χ/2F )

η =1

2M pl

F 2 1

tan2(χ/2F ) − 1

[Moroi, TT 2000]




Case with the new inflation

X ( = i n i t

/ M p l )

1018

1019

v(GeV)

10-2

10-1

100

101

V inf

=λ

2v4 1− 2 χ

√2vn + χ

√2v2n

Potential:

ns − 1 = −2n− 1

n− 2

1

N e

= n2

M pl

v

2

1

n(n− 2)

v

M pl

21

N e

2(n−1)n−2

=1

9N 4e

v

M pl

6

n = 3for ,

99 % C.L.

The scalar spectral index

The tensor-scalar ratio

is small forr v M pl

Curvaton cannot liberate the model,when is very small.

ns − 1 = −2 +2η − 4

1 + f 2

Case with Curvaton

[Linde; Albrecht, Steinhardt 1982]

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




5. Summary

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

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

★ As a consequence, constraints on inflation modelscannot be applied in such a case.

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

★

We found that in what cases inflation models are madeviable by the curvaton.

Documents

Tomo Takahashi- Implications of the Curvaton on Inflationary Cosmology