Plan of Talk
0. Why inflation?
1. Gravitino problem [thermal production]
2. New gravitino problem [non-thermal]
3. Constraints on inflation models
4. Conclusions
T = 1MeV
0.3eV
std. cosmology
BBN
LSS
Time
inflation
(taken from the WMAP website)
The std. big bang cosmology was firmly
established by the following observations.
Horizon problem
Why do the CMB photons from different directions
have almost the same temperature??
Other problems:
flatness problem, unwanted relics (e.g., monopole,
gravitino), the origin of the density perturbation...
Those problems are related with the initial
condition of the Universe.
How far can we extrapolate the
std. cosmology back in time?
?
: a phase of the exponential expansion.
solves the horizon, flatness and
unwanted-relic problems.
Slow-roll inflation
explains the origin of
the density fluctuations.
V
Inflation is now strongly supported by
observations such as WMAP.
Power-law LCDM model fits
the WMAP data quite well.
CMB temperature anisotropy contains information on
the inflation!!!
Γφ
: the total decay rate of the inflaton.
The reheating occurs when H ∼ Γφ ,
and the reheating temperature is
TR2
H ∼ Γφ ∼
MP
is related to the total decay
rate of the inflaton.
TR
Superpartner of the graviton.
It becomes massive by eating the goldstino
when the local SUSY is spontaneously broken.
(super-Higgs mechanism)
Interactions are very weak, and suppressed
by Mp or F ∼ m3/2 MP
Cosmologically important due to its longevity
(th)
Y3/2
(nt)
Y3/2
=
=
(th)
Y3/2 (TR )
(nt)
Y3/2 (TR , mφ , φ)
Inflaton parameters (mass, vev)
(from thermal scatterings)
Weinberg 82, Krauss 83
For high TR, many gravitinos are abundantly
produced by particle scatterings, leading to
cosmological difficulties.
Boltzmann equation
e.g.)
ṅ3/2 + 3Hn3/2 = σv n2rad
Y3/2
Moroi `95
n3/2
∼
≡
s
TR
∼ 0.01
MP
1
2
MP
2
TR
MP
TR6
· g∗ TR3
Gravitino abundance (from thermal scattering)
Moroi, Murayama, Yamaguchi 93,
Bolz, Brandenburg, Buchmueller 01; Pradler, Steffen 06
(th)
Kawasaki et al, 0804.3745
Note that the abundance of gravitinos produced by
thermal scatterings depends only on the reheating
temperature.
(th)
Y3/2
=
(th)
Y3/2 (TR )
This is NOT the case with non-thermally produced gravitinos.
stable (LSP) gravitino
unstable gravitino
mGe / GeV
(NOTE: precise line positions in this figure may be out-dated.)
Sorry, I drop references.
courtesy of K. Hamaguchi
allowe d
stable (LSP) gravitino
unstable gravitino
d
e
w
allo
allowe d
16 eV
mGe / GeV
(NOTE: precise line positions in this figure may be out-dated.)
Sorry, I drop references.
courtesy of K. Hamaguchi
(nt)
Y3/2
=
(nt)
Y3/2 (TR , mφ , φ)
Inflaton parameters (mass, vev)
(1) Gravitino pair production (direct) :
(induced by the mixing between φ and
z)
(2) Anomaly-induced decay (indirect) :
(decay into the hidden gauge sector)
SUSY breaking scale
inflaton mass
(1) pair production
m
(2) Anomaly-induced decay
2.1. Gravitino Pair-Production
Gravitino Pair-Production
Kawasaki, F.T. and Yanagida, hep-ph/0603265, 0605297
Asaka, Nakamura and Yamaguchi, hep-ph/0604132
Relevant interactions:
−1
e
L
=
1 μνρσ
^
− (Gφ ∂ρ φ + Gz ∂ρ z − h.c.) ψ̄μ γν ψσ
8
1 G/2
− e
(Gφ φ^ + Gz z + h.c.) ψ̄μ [γ μ , γ ν ] ψν ,
8
φ
z
: inflaton field
: SUSY breaking field, w/
z
G Gz 3
G ≡ K + ln |W |2
Taking account of the mixings,
m3/2
Gφ ∼ φ
mφ
for mφ < mz
Gravitino Pair Production Rate:
|Gφ |
288π
2
Γ3/2
m5φ
m23/2 MP2
Endo, Hamaguchi and F.T., hep-ph/0602061
Nakamura and Yamaguchi, hep-ph/0602081
1
32π
φ
MP
2
m3φ
MP2
for mφ < mz
Gravitino pair production is effective for lowscale inflation models.
Gravitino abundance is inversely proportional to
the reheating temperature! [later]
2.2. Anomaly-induced decay
cf. anomaly-mediated SUSY breaking:
The SW anomaly mediates the effects of
the SUSY breaking to the visible sector.
FΦ λ
λ
Φ contains scalar auxiliary field in gravity supermultiplet.
g2
mλ = −
b0 m3/2
2
16π
In a similar way, the anomaly couples
the inflaton to any gauge sectors!
φ
L
XG
φ
x
g, g̃
bμ Φ
g, g̃
2
g2
g
mn
mn
∗
F̃
=
X
φ(F
F
−
iF
)
−
X
m
φ
λλ + h.c.,
G
mn
mn
G
φ
2
2
64π
32π
2TR
= (TG − TR )Kφ +
(log det K|R ),φ ,
dR
Γ
(anomaly)
Ng α 2
2 3
|XG | mφ ,
3
256π
Endo, F.T, Yanagida hep-ph/0701042
(Through Yukawa interactions at tree level)
Through anomalies in SUGRA (at one-loop)
ΓDSB =
(h) 2
Ng αh (h)
(TG
3
256π
−
(h)
TR )2
φ
MP
2
m3φ
MP2
ψμ
φ
φ
x
bμ Φ
The gravitinos are produced from the hidden hadron
decay.
Γ3/2 ⎧
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎩
1
32π
2
α
256π 3
φ
MP
2
φ
MP
m3φ
,
2
MP
2
3
mφ
MP2
for mφ < Λ
,
for mφ > Λ
Gravitino Abundance:
Y3/2
Γ3/2 3 TR
2
,
Γtotal 4 mφ
−1
g − 12 T
∗
R
−14
∼ 10
200
106 GeV
2 2
mφ
φ
×
1015 GeV
1010 GeV
Note: Γtotal
TR2
∼
MP
Gravitino Abundance
Y3/2
non-thermal
thermal
TR
Conservative
Constraints on the inflation models;
1019
<φ > [GeV]
10
C
18
: new(single);1TeV
: new(single);100TeV
: new(multi)
: hybrid
: smooth hyb.
: chaotic (w/o Z 2 )
D
1017
10
16
1015
1014
1013
A
B
1012
10 8 10 9 1010 1011 1012 1013 1014 1015 1016 1017
mφ [GeV]
A: m 3/2 = 1TeV; Bh = 1
-3
B: m 3/2 = 1TeV; Bh = 10
C: m 3/2 = 100TeV
D: m 3/2 = 1GeV
m 3/2 = 1 TeV
10
1012
1010
1010
10 8
10 8
TR [GeV]
TR [GeV]
m 3/2 = 1 GeV
12
10 6
104
10 2
-3
Bh = 1
10 6
104
10 2
excluded
100
10 -2
10 8
Bh = 10
10
10
10
12
10
14
10
excluded
100
16
m φ [GeV]
10 -2
10 8
1010
1012
1014
1016
m φ [GeV]
m 3/2 = 100 TeV
TR cannot be
arbitrarily low!
12
10
TR [GeV]
1010
10 8
10 6
104
10 2
excluded
0
10
10 -2
10 8
1010
1012
1014
m φ [GeV]
1016
φ = 1015 GeV
<φ > = 1015 GeV
1016
1016
1014
1014
mφ [GeV]
mφ [GeV]
<φ > = 1012 GeV
1012
1010
108
106
1012
1010
108
-8
-6
-4
-2
0
2
10 10 10 10 10 10 10
4
m3/2 [GeV]
106
10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4
m3/2 [GeV]
<φ > = 1018 GeV
mφ [GeV]
1016
1014
Smaller mass and vev
are favored, if TR is same.
1012
1010
108
106
10 -8 10 -6 10 -4 10 -2 10 0 10 2 10 4
m3/2 [GeV]
Solutions:
(i) Postulate a symmetry on the inflaton.
e.g.) chaotic inflation
V =
1 2 2
2m φ
w/ φ ↔ −φ
(ii) AMSB, GMSB
cosmological constraints are relaxed.
(iii) late-time entropy production
i j k
W = Yφφ φ
1 G/2
i j k
L = − e Gφijk φ̂ φ χ χ + h.c..
2
Gφijk
Wφ Wijk
Wφijk
= −
+
W W
W
Wijk
Wφijk
Kφ
+
,
W
W
φi
φ
Gφijk
χj
χk
Decay Rate through the Yukawa coupling:
ΓY =
(3)
Cijk 3
m
φ
3
256π
(3)
Cijk
≡ eG/2 Gφijk
The inflaton decays into the visible sector
through the top Yukawa coupling:
W = Yt T QHu ,
The decay rate is given by
ΓT
3
2
=
|Y
|
t
3
128π
φ
MP
2
m3φ
,
2
MP
(minimal Kahler is assumed)
The reheating temperature is
bounded below!!
e.g.) inflaton decay through SU(3)c interactions;
φ
φ
x
g, g̃
bμ Φ
g, g̃
3
m
9
φ
2 φ
αs 2
,
2
3
32π
MP MP
2
ΓSU(3)
(minimal Kahler is assumed)
The decay rate is smaller by one order of magnitude
than that through the top Yukawa coupling.
Lower limit on the reheating temperature:
1019
10
10
18
<φ > [GeV]
10
1017
10
10
12
10
8
16
1015
1014
1013
1012
10
-4
10
-2
10
0
10
2
10
4
10
6
10 8 10 9 1010 1011 1012 1013 1014 1015 1016 1017
mφ [GeV]
: new(single);1TeV
: new(single);100TeV
: new(multi)
: hybrid
: smooth hyb.
: chaotic (w/o Z 2 )
4. Conclusion
We have discovered that gravitinos are generically
produced by the inflaton decay.
The abundance of the non-thermally produced
gravitinos depends on the inflaton parameters,
which enables us to distinguish inflation models.
Finding SUSY mediation mechanism is important for
cosmology, in particular, inflation models !!
Back-up Slides
Potential minimization
i
G
V = e G Gi − 3
Differentiating V w.r.t. φ
φ
z
G ∇ φ Gφ + G ∇ φ Gz + Gφ = 0
mφ
Wφφ
∼
1
∇ φ Gφ ∼
W
m3/2
Wφ Wz
∇ φ Gz ∼
∼ φ
W W
m3/2
Gφ ∼ φ
mφ
Mass Matrix in SUGRA
i
G
V = e G Gi − 3
2
Mij∗
=
2
Mij
=
∂2V
G
k
k ∗
= e ∇i Gk ∇j∗ G − Rij ∗ k∗ G G + gij ∗ ,
i
†j
∂ϕ ∂ϕ
2
V
∂
G
k
2
= e ∇ i Gj + ∇ j Gi + G ∇ i ∇ j Gk ,
Mji =
i
j
∂ϕ ∂ϕ
mφ
Wφφ
∼
1
∇ φ Gφ ∼
W
m3/2
Wφ Wz
∇ φ Gz ∼
∼ φ
W W
2
Mφz̄
= 0
Boson
Fermion
A solution to the gauge hierarchy problem:
SUSY stabilizes the electroweak scale
against the radiative corrections.
The gauge coupling unification.
Relevant interactions for inflation in
supergravity:
K(φ, φ∗ ) : Kahler potential
W (φ)
−1
e
K = |φ|2 + · · ·
: Superpotential
L =
μ ∗
−gij ∗ ∂μ φi ∂ φj
∗
− V (φ)
V = eK Di W g ij (Dj W )∗ − 3|W |2 + VD
gij ∗
∂2K
≡
∂φi ∂φ∗j
Di W = Wi + Ki W
Eta-problem?
The slow-roll parameters must be small enough
for the inflation to last long enough.
1
=
2
V
V
2
η =
V
V
In supergravity, eta tends to be of order unity.
∗
V = eK Di W g ij (Dj W )∗ − 3|W |2 + VD
Fine-tuning (one part in hundred) is needed,
but not so severe...
Supersymmetric inflation models
New inflation model (single & multi-fields)
Hybrid inflation model (F-term)
Smooth Hybrid inflation model
Chaotic inflation model
etc.
New inflation model
k 4
K(φ, φ ) =
|φ| + |φ| ,
4
g
φn+1 .
W (φ)
= v2 φ −
n+1
†
2
Izawa and Yanagida ,`97
Generic superpotential under discrete Z2n R-symmetry.
2
k
g
g
V (ϕ) v 4 − v 4 ϕ2 − n −1 v 2 ϕn + n ϕ2n
2
2
22
V
φ (v 2 /g)1/n
mφ nv 2 / φ
The spectral index
Ibe, Shinbara, Yanagida,`06
The inflation scale
Ibe, Shinbara, Yanagida,`06
Hinf = 106 - 108GeV for n=4.
The gravitino mass is related to the inflaton
parameters in this model.
Λ4SU SY − 3|W (φ0 )|2 0.
2 n1
2
v
nv
m3/2 = W (φ0 ) n+1 g
The single-field new
inflation model favors
a heavy gravitino.
Multi-field new inflation
Asaka, Hamaguchi, Kawasaki,Yanagida ,`97
Senoguz, Shafi, `04
The same dynamics can be realized by the following
Kahler and superpotentials:
n
W = χ(v − gφ )
2
κ1 4
κ3 4
2
2
K = |φ| + |χ| + |φ| + κ2 |φ| |χ| + |χ| + · · ·
4
4
2
2
k 4 2
g
V (ϕ) v − v ϕ − n −1 v 2 ϕn
2
22
4
k ≡ κ2 − 1
Also, the gravitino mass is not related to
the inflaton parameters in this model.
Hybrid inflation model
Copeland et al, Dvali et al, `94 Linde and Riotto, `97
W (φ, ψ, ψ̃) = φ(μ2 − λψ̃ψ),
R-charge:
w/ minimal Kahler
φ(+2), ψ ψ̃(0)
U(1) gauge: φ(0), ψ(1), ψ̃(−1)
√
For |φ| μ/ λ
ψ = ψ̃ = 0
V (ϕ) μ
4
λ
1 + 2 ln
8π
2
ϕ
ϕc
+ ···
The global minimum is located at
ψ
φ
φ = 0
√
ψ = ψ̃ = μ/ λ
Scalar spectral index:
ns 0.98 − 1.0
λ
Bastro-Gil, King, Shafi, `06
Smooth hybrid inflation
Lazarides et al `95
n
(
ψ̃ψ)
2
W (φ, ψ, ψ̃) = φ μ − 2n−2
M
Global minimum is located at
.
ψ
φ = 0
ψ = ψ̃ = (μM n−1 )1/n
The dynamics is similar to hybrid inflation,
but ns is slightly smaller.
ns 0.967 − 0.97
φ
Chaotic Inflation
Kawasaki, Yamaguchi and Yanagida ,`00
∗
V = eK Di W g ij (Dj W )∗ − 3|W |2 + VD
The exponential pre-factor was the obstacle
for the chaotic inflation in supergravity.
Let us impose a shift symmetry on K:
φ → φ + iA
K(φ + φ† ) = c (φ + φ† ) + 12 (φ + φ† )2 + · · ·
Imaginary component does
not appear in the exponent!
If the symmetry is broken by the following W,
W = mφψ
1 2 2
V m ϕ
2
1 2 2
V = m ϕI
2
WMAP normalization:
m = 2 × 1013 GeV
Im[φ]
ns 0.96
Summary of Inflation Models
1019
: new(single);1TeV
: new(single);100TeV
: new(multi)
: hybrid
: smooth hyb.
: chaotic (w/o Z 2 )
1018
<φ > [GeV]
1017
1016
1015
1014
1013
1012 8
10
1010
1012
1014
mφ [GeV]
1016