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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.