Relaxing Constraints on Inflation Models with the Curvaton February - - PowerPoint PPT Presentation

Relaxing Constraints on Inflation Models with the Curvaton

Tomo Takahashi

ICRR, University of Tokyo

February 15, 2005 @ Sendai Int. Workshop “Windows to New Paradigm in Particle Physics”

• Ref. Moroi, T.T. , Toyoda, [hep-ph/0501007]

• 1. Introduction

Inflation : a superluminal expansion at the early universe

• ne of the most promising ideas to solve

the horizon problem and the 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 can be the

• rigin of today’s cosmic fluctuation.

Details of fluctuation (such as scale dependence) depend on inflation models (or the form of the potential).

★

Current cosmological observations such as In particular, after WMAP , observations can give severe constraints on inflation models.

★

Usually, the scale dependence of the fluctuation is expressed with the initial power spectrum: PR ∝ kns−1

(the spectral index) depends on models.

ns

has information of the model.

ns

Cosmic microwave background (CMB) Large scale structure

・・・

can give constraints on .

ns

Information on models.

(The gravity wave is also generated. The size of it can give information of 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 2002]

This talk ➡

Primordial fluctuation: the standard case

The quantum fluctuation of the inflaton

• 2. Constraints on models of inflation: Standard case

δχ ∼ H 2π

The curvature perturbation R ∼ −H

˙ χ δχ

The power spectrum of .

PR ∝ kns−1

During inflation, the Hubble parameter is almost constant Almost scale-invariant spectrum Generally, it can be written as a power-law form;

Inflation Inflation RD RD MD MD reheating reheating Inside the horizon Inside the horizon

Scale Scale factor a horizon scale

R

(The initial power spectrum)

H ≡ ˙ a a :Hubble parameter

horizon crossing

PR(k) ∼< R2 >∼ H ˙ χ 2 H 2π 2

• k=aH

˙ χ ≡ dχ dt

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 ln PR

d ln k

• k=aH

Slow-roll parameters,

≡ 1 2M 2

pl

V V 2 η ≡ 1 M 2

pl

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 using the slow-roll approximation.

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

(Slow-roll approximation)

, |η| 1

r = Pgrav PR = 16

where

Constraints from current observations

[Leach, Liddle 2003 ] ns1 r

0.08 0.04 0.04 0.1 0.2 0.3 0.4 0.5

(The reference scale )

k = 0.01Mpc−1

r

ns − 1

95% 99% WMAP+VSA+ACBAR+2dF

[Data from CMB (WMAP

, VSA, ACBAR) and large scale structure (2dF) are used.]

A worked example: the chaotic inflation [Linde 1983] Potential:

V (χ) = λM 4

pl

χ Mpl α

χ∗ = χ|k=aH

where

χ∗ can be written with the number of e-foldings during inflation. The number of e-foldings: Ne ≡ ln aend

a∗

a∗

aend

:Scale factor at horizon crossing :Scale factor at the end of inflation

Using the slow-roll approximation,

Ne = 1 M 2

pl

χ∗

χend

Vinf V

inf

dχ 1 2αM 2

pl

χ2

∗.

= 1 2M 2

pl

V V 2 = 1 2α2 M 2

pl

χ2

∗

η = M 2

pl

V V = α(α − 1) M 2

pl

χ2

∗

The scalar spectral index: ns − 1 = −α + 2

2N

r = 4α N

The tensor-scalar ratio:

(−0.04, 0.16)

α = 4

α = 2

(−0.05, 0.27)

r

(Assuming N=50)

For For

(Assuming N=60)

Almost excluded However, the curvaton can liberate the model!

ns1 r

0.08 0.04 0.04 0.1 0.2 0.3 0.4 0.5

r

ns − 1

95% 99%

V ∼ χ2

V ∼ χ4

V ∼ χ6

ns − 1

• 3. Curvaton mechanism [Enqvist & Sloth, Lyth & Wands, Moroi & TT 2002]

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

Curvaton: Curvaton scenario

Inflaton ➡ causes the inflation, not fully responsible for the cosmic fluctuation Curvaton ➡ is not responsible for the inflation, is fully or partially responsible for the cosmic fluctuation

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

Thermal history of the universe with the curvaton

Inflation RD1 Curvaton dominated RD2

→BBN

Energy density Time reheating reheating

Vcurvaton = 1 2m2

φφ2

The 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)

Energy density Time

Thermal history of the universe with the curvaton

Inflation RD1 Curvaton dominated RD2

→BBN

Energy density Time reheating reheating

Vcurvaton = 1 2m2

φφ2

The 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

Density Perturbation Fluctuation of the inflaton Curvature perturbation

R ∼ −H ˙ χ δχ

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)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 2

ns − 1 = −2 + 2η − 4 1 + ˜ f 2

r = 16

,

[Langlois, Vernizzi, 2004]

RRD2 = 1 M 2

pl

Vinf V

inf

δχinit + 3 2f(X)δφinit Mpl

PR = 9 4

• 1 + ˜

f 2(X)

• Vinf

54π2M 4

pl

Pgrav = 2 M 2

pl

H 2π 2

f(X)        4 9X : φinit Mpl 1 3X : φinit Mpl

0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7 8 9 10 f(!*) !*/Mpl

X(= 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 amplitude give large curvaton contributions.

★

RRD2 = 1 M 2

pl

Vinf V

inf

δχinit + 3 2f(X)δφinit Mpl

• 4. Relaxing constraints on models of

inflation with curvaton mechanism

[Moroi, TT & Toyoda, hep-ph/0501007]

Fluctuation of the curvaton can be fully or partly responsible for density fluctuation today. Affects constraints on inflation models. We investigated this subject for various inflation models.

★

Chaotic inflation with several monomials. Natural inflation New inflation

Case with the chaotic inflation

Vinf = λχ4

Potential: For φinit = 0.1Mpl Including curvaton

The model becomes viable due to the curvaton!

ns1 r

0.08 0.04 0.04 0.1 0.2 0.3 0.4 0.5

r

ns − 1

95% 99% V ∼ χ4

ns − 1 ∼ −0.04, r ∼ 0.08

★

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 = λχ4

Potential:

99 % C.L.

0.5 1 1.5 2 2.5 3 3.5 4 1 2 3 4 5 6 7 8 9 10 f(!*) !*/Mpl

f(X)

X(= φinit/Mpl)

The case with large initial amplitude cannot liberate the model, although is large.

f(X)

★

Model parameter dependence

★

Initial amplitude of : Mass of : Decay rate of :

φ mφ

φinit

Γφ

φ φ

Γφ = 10−10GeV Γφ = 10−18GeV

X(=!init /Mpl)

Ne=40 Ne=50 Ne=55 Ne=57.5

102 103 104 105 106 107 108 109 1010 m! (GeV) 10-2 10-1 100 101

The number of e-folding during inflation

For large initial curvaton amplitudes, the curvaton can cause the 2nd inflation. The second inflation reduces .

Ne

modify the spectral index.

Notice:

n(inf)

s

− 1 = −α + 2 2N

N(k)(curvaton) = N(k)(standard) − N2 + 1 12 ln 3 − 1 6 ln mφ Γφ − 1 6 ln φinit Mpl

For other chaotic inflation models

Vinf = λ χ6 M 2

pl

Potential:

X(=

init /Mpl )

102 103 104 105 106 107 108 109 1010 m (GeV) 10-2 10-1 100 101 99 % C.L.

Γφ = 10−18GeV Γφ = 10−10GeV

The situation is almost the same as the quartic case.

★

ns1 r

0.08 0.04 0.04 0.1 0.2 0.3 0.4 0.5

r

ns − 1

V ∼ χ6 (w/o curvaton)

For larger monomial models

Even if case,

f(X) → ∞

(Pure curvaton case)

the spectral index becomes

V (χ) = λM 4

pl

χ Mpl α

(For )

Large cases cannot be made viable even even with the curvaton.

α

ns − 1 = −2 + 2η − 4 1 + ˜ f 2

Remind that the spectral index is modified as

ns − 1 ∼ −2 = α 2Ne

Case with the natural inflation

X(=

init /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]

= 1 2 Mpl F 2 1 tan2(χ/2F)

η = 1 2 Mpl F 2 1 tan2(χ/2F) − 1

• [Moroi, TT 2000]

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

= n2 Mpl v 2 1 n(n − 2) v Mpl 2 1 Ne 2(n−1)

n−2

= 1 9N 4

e

v Mpl 6

n = 3

for ,

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 = −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 give severe constraints on inflation models.

★ ★ However, if we introduce the curvaton, a late-decaying

scalar condensate, primordial fluctuation can be 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

• n inflation models can be liberated.

★ We found that in what case inflation models are made

viable by the curvaton.