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.