Implications of the Curvaton on Inflationary Cosmology

Implications of the Curvaton on Inﬂationary Cosmology

March 4, 2005 @ KEK, Tsukuba KEK theory meeting 2005 “Particle Physics Phenomenology”

Tomo Takahashi ICRR, University of Tokyo

Collaborators: Takeo Moroi (Tohoku) Yoshikazu Toyoda (Tohoku) 1. Introduction

Inﬂation (a superluminal expansion at the early universe) [Guth 1980, Sato 1980] is one of the most promising ideas to solve the horizon and ﬂatness problem.

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. From the viewpoint of particle physics, there can exist scalar fields other than the inflaton. (late-decaying moduli, Affleck-Dine fields, right-handed sneutrino...)

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

★Generally, fluctuation of inflaton and curvaton can be both responsible for the today’s density fluctuation. What is the implication of the curvaton for This talk ➡ constraints on models of inflation? 2. Generation of fluctuations: Standard case Primordial fluctuation: the standard case a˙ H H ≡ :Hubble parameter The quantum fluctuation of the inflaton δχ ∼ a 2π dχ χ˙ ≡ H dt The curvature perturbation R ∼ − δχ χ˙ The power spectrum of R . Inflation RD MD

(The initial power spectrum) Scale horizon scale 2 2 2 H H horizon crossing PR(k) ∼< R >∼ ! χ˙ " !2π " #k=aH # # Inside the horizon Almost scale-invariant spectrum Scale factor a reheating

ns−1 Generally, it can be written as a power-law form; PR ∝ k ★ ns depends on models of inﬂation What is the discriminators of models of inﬂation?

ns−1 The initial power spectrum ( P R ∝ k ) depends on models.

During inﬂation, the gravity waves are also produced. The size of gravity waves (tensor mode) also depends on models.

The observable quantities d ln P n n − 1 ≡ R Scalar spectral index s s d ln k ! "k=aH # P " Tensor-scalar ratio r ≡ grav " PR ( Overall normalization of PR )

These parameters can be determined from observations. (such as CMB, large scale structure and so on.) Constraints on models of inﬂation. Constraints from current observations [Leach, Liddle 2003 ]

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

0.5 ★ Some inﬂation models are 6 V ∼ χ already excluded! 0.4

V ∼ χ4 Example: chaotic inﬂation 0.3

α r r 4 χ ∼ 2 V (χ) = λMpl V χ M 0.2 ! pl " 99% 95% ★ The cases with α = 4, 6, · · · are almost excluded. 0.1

0 !0.08 !0.04 0 0.04 nn −!11 s s The observable quantities can be written with the slow-roll parameters.

1 V ! 2 1 V !! where ! ≡ η ≡ ! dV (χ) Slow-roll parameters, 2M 2 V 2 V (χ) ≡ pl ! " Mpl V dχ

dV (χ) dχ χ¨ + 3Hχ˙ + = 0 where χ˙ ≡ Equation of motion for a scalar ﬁeld: dχ dt dV (χ) When it slowly rolls down the potential, 3Hχ˙ + ! 0 dχ (Slow-roll approximation)

★ The slow-roll approximation is valid when ! , | η | ! 1 .

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

P The tensor-scalar ratio r = grav = 16! PR 3. Curvaton mechanism [Enqvist & Sloth, Lyth & Wands, Moroi & TT 2001]

Curvaton: a light scalar field which is partially or totally responsible for the density fluctuation. (Usually, quantum fluctuation of the inflaton is assumed to be totally responsible for that.)

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, is fully or partially responsible for the cosmic ﬂuctuation

Constraints on inﬂation models can be relaxed. Thermal history of the universe with the curvaton

Curvaton Inflation RD Inflation RD1 RD2 dominated (Standard case) inflaton y density y density

curvaton Energ !BBN Energ

reheating Time

!BBN

reheating reheating Time (For simplicity, an oscillating inﬂaton period is omitted.) 1 = 2 2 The potential of the curvaton is assumed as Vcurvaton 2mφφ The oscillating ﬁeld behaves as matter. Thermal history of the universe with the curvaton

H > mφ V (φ) Inﬂation Curvaton RD1 dominated RD2

inﬂaton

y density curvaton φ Energ

V (φ) H ∼ mφ

!BBN

reheating reheating Time φ (For simplicity, an oscillating inﬂaton period is omitted.) 1 = 2 2 The potential of the curvaton is assumed as Vcurvaton 2mφφ The oscillating ﬁeld behaves as matter. Density Perturbation

Fluctuation of the inflaton Curvature perturbation H δχ ∼ H ! 2π " R ∼ − δχ χ˙ Fluctuation of the curvaton No curvature perturbation H δφ ∼ (The curvaton is subdominant component during inflation.) ! 2π " However, isocurvature fluctuation can be generated.

2δφinit Sφχ ∼ δχ − δφ = φinit Curvaton becomes dominant δρi Inflaton where δi ≡ ρi At some point, the curvaton curvaton dominates the universe. the isocurvature fluc.becomes adiabatic (curvature) fluc. After the decay of the curvaton, the curvature fluctuation becomes [Langlois, Vernizzi, 2004] R 1 Vinf 3 δφinit RD2 = 2 ! δχinit + f(X) Mpl Vinf 2 Mpl where X = φinit/Mpl From inflaton From curvaton f(X): represents the size of contribution from the curvaton

9 2 Vinf PR f˜ X ! The power spectrum (scalar mode) = !1 + ( ) " 2 4 4 54π Mpl!

where f˜ = (3/√2)f 2 H 2 P = The tensor mode spectrum is not modiﬁed. grav M 2 2π pl ! " The scalar spectral index and tensor-scalar ratio with the curvaton 2 − 4 16! η ! r = ns − 1 = −2! + , ˜2 1 + f˜2! 1 + f !

(Notice: the standard case n s − 1 = 4 η − 6 " , r = 16 ! ) Contribution from the curvaton

R 1 Vinf 3 δφinit RD2 = 2 ! δχinit + f(X) Mpl Vinf 2 Mpl f(X) can be obtained solving a linear perturbation theory. For small and large limit, [Langlois, Vernizzi, 2004] 4 4 : φinit " Mpl  9X !  3.5 f(X)  1 X : φ # M 3  init pl  3

) 2.5 ) X * ( ! 2 ( f f 1.5 1 ★ Small and large initial amplitude 0.5 give large curvaton contributions. 0 0 1 2 3 4 5 6 7 8 9 10 ! /M X(= φ*initpl/Mpl) Relaxing constraint on inﬂation models with curvaton [Langlois & Vernissi 2004; Moroi, TT & Toyoda 2005] 4 An example: the chaotic inﬂation with the quartic potential Vinf = λχ

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

0.4 ns − 1 ∼ −0.05, r ∼ 0.27 4 V ∼ χ (Assuming N=60) 0.3 r r With the curvaton, 0.2 for φinit = 0.1Mpl 99% 95% 0.1 ns − 1 ∼ −0.04, r ∼ 0.08 ★

0 !0.08 !0.04 0 0.04 nn −!11 s s The model becomes viable due to the curvaton! Model parameter dependence [Moroi, TT & Toyoda 2005]

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

Initial amplitude of φ : φinit affects the amplitude of ﬂuctuation and the background evolution Mass of φ : mφ affects the background evolution Decay rate of φ : Γφ

RDI RD2 Inﬂation φD MD (standard case) Scale horizon scale Inflation RD MD horizon crossing

Inside the horizon

reheating Scale factor a Model parameter dependence: Case with the chaotic inﬂation 4 ★ Potential: Vinf = λχ The region where the curvaton causes the second inﬂation. [Moroi, TT & Toyoda 2005] 101 The criterion for the alleviation of the model ) pl 0 10 −18 /M Γφ = 10 GeV init F =

X( 10-1 −10 w/o curvaton Γφ = 10 GeV

99 % C.L. r 10-2 2 3 4 5 6 7 8 9 10 10 10 10 10 10 10 10 10 10 ★ Not allowed ★ mF (GeV) Allowed

99%

ns − 1

★ The case with large initial amplitude cannot liberate the model, although contribution from the curvaton ﬂuc. is large. Case with the natural inﬂation [Freese, Friemann, Olint, Adams,Bond, Freese, Frieman,Olint,1990]

4 χ Potential: Vinf = Λ 1 − cos The scalar spectral index ! "F #$ 1

0.95

) 0.9 E COB (k s

n 0.85

0.8 101

0.75 4 5 7 10 20 18 F (10 GeV) [Moroi, TT 2000] ) 0 pl 10

/M 2

init 1 Mpl 1 F ! = 2 = 10-1 2 ! F " tan (χ/2F ) X(

2 95 % C.L. 1 Mpl 1 -2 − 10 η = 2 1 5 6 7 8 9 10 2 ! F " #tan (χ/2F ) $ F (1018 GeV) [Moroi, TT & Toyoda 2005] Case with the new inﬂation [Linde; Albrecht, Steinhardt 1982]

n 2n 2 4 χ χ Potential: Vinf = λ v 1 2 + [Kumekawa, Moroi, Yanagida 1994; − √2v √2v Izawa, Yanagida1997] ! " # " # $ 101 The scalar spectral index n − 1 1 n − 1 = −2 ) 0 s pl 10 n − 2 Ne /M init

The tensor-scalar ratio F

2(n−1) = -1 M 2 v 2 n−2 10 1 1 X( ! = n2 pl v n(n − 2) M Ne ! " # ! pl " $ -2 99 % C.L. 1 v 6 10 n = 3 ! = 18 19 for , 4 10 10 9Ne !Mpl " v(GeV) r is small for v ! Mpl [Moroi, TT & Toyoda 2005] Case with Curvaton Curvaton cannot liberate the model, 2η − 4! ns − 1 = −2! + when ! is very small. 1 + f˜2! 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.