אוניברסיטת בן-גוריון
Ram Brustein
Introduction: The case for
closed string moduli as inflatons Cosmological stabilization of moduli
Designing inflationary potentials (SUGRA, moduli)
The CMB as a probe of model parameters
with I. Ben-Dayan, S. de AlwisTo appear
============Based on:
hep-th/0408160with S. de Alwis, P. Martens
hep-th/0205042 with S. de Alwis, E. Novak
Models of Modular Inflation
Stabilizing closed string moduli• Any attempt to create a deSitter like phase will induce a
potential for moduli
• A competition on converting potential energy to kinetic energy, moduli win, and block any form of inflation
241
2/14)4( ))(( AFRgxdS
3214)4(
eRgxdS
Inflation is only ~ 1/100 worth of tuning away!(but … see later)
Generic properties of moduli potentials:
• The landscape allows fine tuning• Outer region stabilization possible• Small +ve or vanishing CC possible • Steep potentials• Runaway potentials towards
decompactification/weak coupling• A “mini landscape” near every stable mininum +
additional spurious minima and saddles
Proposed resolution:role of other sources
• The 3 phases of evolution
– Potential push: jump
– Kinetic : glide
– Radiation/other sources : parachute opens
Previously:Barreiro et al: tracking
Huey et al, specific temp. couplings Inflation is only ~ 1/100 worth of tuning away!
Using cosmological stabilization for designing models of inflation:
• Allows Inflation far from final resting place
• Allows outer region stabilization
• Helps inflation from features near
the final resting place
(My) preferred models of inflation: small field models
– wall thickness in spaceInflation H > 1 > mp H2~1/3 mp
2
2
-1
mp-4 -2 0 2 4
• “Topological inflation”: inflation off a flat featureGuendelman, Vilenkin, Linde
Enough inflation V’’/V<1/50
Results and Conclusions: preview
• Possible to design fine-tuned models in SUGRA and for string moduli
• Small field models strongly favored• Outer region models strongly disfavored
• Specific small field models
– Minimal number of e-folds– Negligible amount of gravity waves: all models
ruled out if any detected in the foreseeable future
Predictions for future CMB experiments
Designing flat features for inflation
• Can be done in SUGRA• “Can” be done with steep exponentials alone• Can (??) be done with additional (???) ingredients
(adding Dbar, const. to potential see however ….. ) • Lots of fine tuning, not very satisfactory• Amount of tuning reduces significantly towards
the central region
Designing flat features for inflation in SUGRA
Take the simplest Kahler potential and superpotential
Always a good approximation when expanding in a small region (< 1)
For the purpose of finding local properties V can be treated as a polynomial
Design a wide (symmetric) plateau with polynomial eqs.
A simple solution: b2=0, b4=0, b1=1,b3=/6,
b5 determined approximately by (*)
(*)
In practice creates two minima @ +y,-y
Designing flat features for inflation in SUGRA
The potential is not sensitive to small changes in coefficientsIncluding adding small higher order terms, inflation is indeed 1/100 of tuning away
-0.6 -0.4 -0.2 0.2 0.4 0.6
-0.5
0.5
1
1.5A numerical example:
b2=0, b4=0, b1=1,b3=/6,
b5 y4(y2+5) + y2+1=0
b1 b3 – 2(b0)2
Need 5 parameters:V’(0)=0,V(0)=1,V’’/V=DTW(-y), DTW(+y) = 0
• An example of a steep superpotential Tai
ieATW )(
1:,)()( TasteepeTaATWT iTa
iiTi
)ln(3)( TTTK
TTTKT
3
)()()( TKTW TT
• An example of Kahler potential
Similar in spirit to the discussion of stabilization
WKKWWWKeTTV TTTTTTTTK ),(
3
2
)(canonical eTVV
Designing flat features for inflation for string moduli: Why creating a
flat feature is not so easy
• extrema• min: WT = 0• max: WTT = 0• distance T
Why creating a flat feature is not so easy (cont.)
WKKWWWKeTTV TTTTTTTTK ),(
Example: 2 exponentials
WT = 0
WTT = 0
0212211 TaTa eaAeaA
021 222
211 TaTa eaAeaA
211
222
12max ln
1
aA
aA
aaT
11
22
12min ln
1
aA
aA
aaT
1
2
12minmax ln
1
a
a
aaTTT
21
11
aaT For (a2-a1)<<a1,a2
21max
aaV
V
T
TT
1))(( max2max1can
''can
max
TaTaV
V
11
max1max
TaT
T
For (a2-a1)<<a1,a2
1))(( max2max1can
''can
max
TaTaV
V
11
max1max
TaT
T
Amount of tuning
To get ~ 1/100 need tuning of coefficients @ 1/100 x 1/(aT)2
The closer the maximum is to the central area the less tuning.Recall: we need to tune at least 5 parameters
Designing flat features with exponential superpotentials
Need N > K+1 (K=5N=7!) unless linearly dependent
Trick: compare exponentials to polynomials by expanding about T = T2
Linear equations for the coefficients of
Numerical examples
2.5 2.75 3.25 3.5 3.75
2
4
6
8
10
2 4 6 8 10 12 14
10
20
30
40 7 (!) exponentials + tuning ~ (aT)2 = UGLY
Lessons for models of inflation
• Push inflationary region towards the central region
• Consequences:– High scale for inflation– Higher order terms are important, not
simply quadratic maximum 2.4 2.6 2.8 3.2 3.4 3.6
0.94
0.95
0.96
0.97
0.98
0.99
2.5 2.75 3.25 3.5 3.75
2
4
6
8
10
Phenomenological consequences
• Push inflationary region towards the central region
• Consequences:– High scale for inflation– Higher order terms are important, not
simply quadratic maximum
Models of inflation: Background
de Sitter phase + p << const.
Parametrize the deviation from constant H22
2)(
V
Vmpl )()( 2
V
Vmpl
Or by the number of e-folds
ei
eieiei
dm
dH
HdttadNpl
t
t
t
t )(
1
2
1)(log)(
by the value of the field
Inflation ends when = 1
Models of inflation:Perturbations• Spectrum of scalar perturbations
• Spectrum of tensor perturbations
aHkm
HkP
pl
12)(
2
aHkm
HkP
plT
2
2)(
Tensor to scalar ratio (many definitions)r is determined by PT/PR , background cosmology, & other effectsr ~ 10 (“current canonical” r =16 )
Spectral indices
r = C2Tensor/ C2
Scalar (quadropole !?)
CMB observables determined by quantities ~ 50 efolds before the end of inflation
Wmapping Inflationary Physics W. H. Kinney, E. W. Kolb, A. Melchiorri, A. Riotto,hep-ph/0305130
See also Boubekeur & Lyth
The “minimal” model: – Quadratic maximum
– End of inflation determined by higher order terms
2max3
2max2
4 1~)( aaV
Minimal tuning minimal inflation, N-efolds ~ 60 “largish scale of inflation” H/mp~1/100
Sufficient inflation enoughsmallmax init
6/1)('' max2max VHinit Qu. fluct. not too large
For example:
The “minimal” model: – Quadratic maximum
– End of inflation determined by higher order terms
2.4 2.6 2.8 3.2 3.4 3.6
0.94
0.95
0.96
0.97
0.98
0.99
2max3
2max2
4 1~)( aaV
Unobservable!
'' ( ).92 .08(25 | | 1)
( )
0.01
CMBS
CMB
Vn
V
r
CMB
CMBCMBS
r
n
16
241
2 4( '/ ) ~ 0.01
~ 1/ 25 0CMB
CMB
V V
Expect for the whole class of models
1/2 < 25|V’’/V| < 1 .92 .96
0.01Sn
r
WMAP
Detecting any component of GW in the foreseeable future will rule out this whole class of models !