A Stringy Mechanism for A Small Cosmological Constant

7/30/2012
A Stringy Mechanism
for A Small
Cosmological Constant
• X. Chen, Shiu, Sumitomo, Tye,
arxiv:1112.3338, JHEP 1204 (2012) 026
• Sumitomo, Tye,
arXiv:1204.5177
• Sumitomo, Tye, in preparation
1
Yoske Sumitomo
IAS, The Hong Kong University of
Science and Technology

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Contents
Motivation
Moduli stabilization ~random approach~
Moduli stabilization ~concrete models~
Statistical approach
More on product distribution
Multi-moduli analyses
Summary & Discussion

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Motivation

4
Dark Energy
Late time expansion
Awarded Nobel Prize in 2011!
What can be a source for this

5
Acceleration
The universe is accelerating if
or pressure-density ratio:
Cosmological scale
EOM (Friedmann eq.)
for flat background
Observationally
DE domination

Two possibilities
6
• For cosmological constant
WMAP+BAO+SN suggests
for a flat universe
Cosmological constant
• For time-varying DE
WMAP+BAO+H0+DΔt+SN suggests
Time varying DE
e.g. Stringy Quintessence models
[Kiwoon, 99], [Svrcek, 06], [Kaloper, Sorbo, 08],
[Panda, YS, Trivedi, 10], [Cicoli, Pedro, Tasinato, 12]…

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dS
Landscape
dS
Metastable vacua
in moduli space
• Inflation
AdS
rolling down
(& tunneling)
• dS vacua
tunneling
Low energy
• AdS vacua
We may stay here for a while.
But how likely with tiny CC

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Stringy Landscape
There are many types of vacua in string theory, as a result of
a variety of (Calabi-Yau) compactification.
ds 2 10 = ds 2 2
4 + ds 6
A class of Calabi-Yau gives Swiss-cheese type of volume.
V 6 = γ 1 T 1 + T 1 − γ i T i + T i
E.g. workable models:
i=2
,
[Denef, Douglas, Florea, 04]
4
• P 1,1,1,6,9
: h 1,1 = 2, h 2,1 = 272
• F 11 : h 1,1 = 3, h 2,1 = 111
• F 18 : h 1,1 = 5, h 2,1 = 89
All can be stabilized
(a la KKLT),
but in various way.
Any implication of multiple vacua

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Keys in this talk
Product distribution
Assuming products of random variables:
z = y 1 y 2 y 3 ⋯
Many terms
Correlation
through stabilization
z = y 1 y 2 y 3 ⋯ f(y 1 , y 2 , y 3 , ⋯ )
still peaked
We apply this mechanism for cosmological constant (CC)

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Before proceeding…
I have to say
we don’t solve cosmological constant problem
completely.
But here,
we introduce a tool to make cosmological
constant smaller, maybe up to a certain value.
“A Stringy Mechanism for A Small Cosmological Constant”

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Moduli stabilization
~random approach~

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Gaussian suppression on stability
Various vacua in string landscape
Mass matrix given randomly at extrema
How likely stable minima exist
Positivity of mass matrix
Positivity of Hessian
∂ φi ∂ φj V
min
Real/complex symmetric matrix
• Gaussian Orthogonal Emsemble
[Aazami, Easther, 05], [Dean, Majumdar, 08], [Borot, Eynard, Majumdar, Nadal, 10]
Z =
dM ij e −1 2 tr M2 , M = M T
Gaussian term dominates even at lower N.
ln 3
4
∼ 0.275,
ln 2 3−3
2
∼ −0.384

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Hierarchical setup
• Assuming hierarchy between diag. and off-diag. comp.
Actual models are likely to have minima at AdS.
+ uplifting term toward dS vacua.
Hessian = A + B where A: diagonal positive definite with σ A
B: GOE with σ B
Still Gaussianly suppressed, but a chance for dS
P = a e −bN2 −cN
[X. Chen, Shiu, YS, Tye, 11]
When applying a model in type IIA,
quite tiny chance remains.
Larger
hierarchy
• Assuming more randomness in SUGRA at SUSY AdS
P = e −bN2
[Bachlechner, Marsh, McAllister, Wrase, 12]

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Moduli stabilization
~concrete models~

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Type IIB
Sources: H 3 , F 1 , F 3 , F 5 , dilaton, localized sources
Metric: ds 2 10 = e 2A ds 2 4 + e −2A ds 6 2
Calabi-Yau
Then EOM becomes
[Giddings, Kachru, Polchinski, 02]
∇ 2 e 4A − α = e2A
6 Im τ iG 3 −∗ 6 G 3 2 + e −6A ∂ e 4A − α 2 + (local sources)
LHS=0 when integrating out
positive contributions
e 4A = α, iG 3 =∗ 6 G 3 : imaginary self-dual condition
where α is a function in F 5 , G 3 = F 3 − τ H 3 , τ = C 0 + i e −φ

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No-scale structure
Take a scaling: g mn → λ g mn
e 4A = α, iG 3 =∗ 6 G 3 : invariant
The other equations are also unchanged.
No-scale structure
superpotential W 0 = ∫ G 3 ∧ Ω is independent of Kahler
4D effective potential with K = −3 ln T + T , W 0 = const
V = e K M P 2 K IJ D I W 0 D J W 0 − 3 M P
2 W 2 = 0
Kahler directions remain flat.

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A bonus in type IIB
Hierarchical structure of mass matrix/potential helps to
stabilize moduli at positive cosmological constant.
Moduli stabilization with positive cosmological constant
• Fluxes Complex structure & dilaton
[X. Chen, Shiu, YS, Tye, 12]
• Non-perturbative effect, α ′ -correction, localized branes
Kahler
[KKLT, 03], [Balasubramanian, Berglund, Conlon, Quevedo, 05],
[Balasubramanian, Berglund, 04]…
V = V Flux + V NP + V α′ + ⋯
No scale structure
Complex
Kahler
Hierarchy
between Kahler and Complex

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KKLT
Non-trivial potential for Kahler is generated by NP-corrections.
E.g.
Gluino condensation on D7-branes
D7-branes wrapping the four cycle: W NP = A e −a 8π2 g D7 = A e−a T
Together with the superpotential from fluxes: W = W 0 + W NP
Supersymmetric vacuum
D T W = 0 existes.
But exponentially small W 0 is
required.
|W 0 | ∼ A e −a T , naturally realized
|W 0 | ∼ 10 −4

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Large Volume Scenario
[Balasubramanian, Beglund, Conlon, Quevedo, 05]
α ′ -corrections can break no-scale structure too.
O α ′ 3 -correction in type II action [Becker, Becker, Haack, Louis, 02]
K = −2 ln V + ξ 2 −i τ + τ 3 2
− ln(−i τ + τ ) + ⋯
4
E.g. P 1,1,1,6,9
scales differently
model (assuming complex sector is stabilized)
V = 1
9 2 t 3 2 3
1 − t 2 2 , W = W 0 + A 1 e −a 1T 1 + A 2 e −a 2T 2
Solution: W 0 ∼ −20, A 1 ∼ 1, t 1 ∼ 10 6 , t 2 ∼ 3
V min ∼ −10 −25 : AdS vacua
|W 0 | ≫ |W NP |, V ≫ ξ: naturally realized

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Kahler uplifting
Same setup as that of LVS
[Balasubramanian, Berglund, 04],
[Westphal, 06], [Rummel, Westphal, 11],
[de Alwis, Givens, 11]
K = −2 ln V + ξ 2 + ⋯ , V = γ 1 T 1 + T 1 − γ i T i + T i
,
i=2
W = W 0 + A 1 e −a 1T 1 +
A i e −a iT i
i=2
Interested in a region
where this term plays a roll.
less large volume than LVS, but still |W 0 | ≫ |W NP |, V ≫ ξ
E.g. single modulus
[Rummel, Westphal, 11]
V ∼ − W 0a 1 3 A 1
2 γ 1
2
2C
9x 1
9 2 − e−x 1
x 1
2
, C = −27 W 0 ξ a 1
3 2
64 2γ 1 A 1
, x 1 = a 1 t 1
When W 0 A 1 < 0, the C ∝ ξ term contributes the uplifting.

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KKLT vs Kahler uplifting
• KKLT
Add an uplifting potential by hand
V = V SUGRA + V D3−D3
V D3−D3 = 2T 3 d 4 x −g 4
Backreaction of D3
Safe or not
• Kahler uplifting
V = V SUGRA
A singularity exists, but finite action
[DeWolfe, Kachru, Mulligan, 08], [McGuirk, Shiu, YS, 09],
[Bena, Giecold, Grana, Halmagyi, Massai, 09-12], [Dymarsky, 11],…
Owing to |W 0 | ≫ |W NP |
SUGRA + α′-correction
No fine-tuning for W 0

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Statistical approach

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Further approximation
V
4
M = − W 0a 3 1 A 1 C
P
2 γ 1 9x
9 2 − e−x 1
2
, C = −27W 3
0ξa 1
2
1 x 1 64 2γ 2 , x 1 = a 1 t 1
1 A 1
Further focusing on smaller CC region: C ∼ 3.65
[Rummel, Westphal, 11]
The stability constraint with positive CC at stationery points:
V ≥ 0 3.65 ≤ C < 3.89
∂ 2 x V > 0
V
M P
4 ∼ 1 9
2
5
9
2 −W 0 a 1 3 A 1
γ 1
2
C − 3.65
Neglecting the parameters a 1 , γ 1 , ξ, the model is simplified to be
Λ = w 1 w 2 c − c 0 , c 0 ≤ c = w 1
w 2
< c 1
(w 1 = −W 0 , w 2 = A 1 , c ∝ C)

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Stringy Random Landscape
Starting with the simplified potential:
[YS, Tye, 12]
Λ = w 1 w 2 c − c 0 , c 0 ≤ c = w 1
w 2
< c 1
Since W 0 , A 1 are given model by model (various ways of
stabilizing complex moduli), here we impose reasonable
randomness on parameters.
w 1 , w 2 ∈ [0, 1], uniform distribution (for simplicity)
Probability distribution function
P Λ = N 0 dc dw 1 dw 2 δ w 1 w 2 c − c 0 − Λ δ w 1
w 2
− c
N 0 : normalization constant

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Divergence in product distribution
When z = w 1 w 2 ,
P z = dw 1 dw 2 δ w 1 w 2 − z = 1 2 ln 1 z
log divergence at z = 0
With constraint Λ = w 1 w 2 c − c 0 , c 0 ≤ c = w 1
< c
w 1
2
P Λ = c 1
ln c positivity stability
1 − c 0
c 1 − c 0 c 1 Λ still diverging!!
Comparison to the full-potential (randomizing W 0 , A 1 without approx.)
Good agreement
at smaller Λ

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Zero-ness of parameters
We assumed the parameters W 0 , A 1 passing through zero value,
but is it true
• E.g. T 6 model: W 0 = c 1 + d i U i − c 2 + e i U i S
SUSY condition
W 0 = 2 c 1 + c 2 s
easy to be zero
i
k
d i + e i s
d k − e k s
(d j − e j s)
j≠i
s = Re(S)
• Brane position dependence of A 1
[Baumann, Dymarsky, Klebanov,
Maldacena, McAllister, Murugan, 06]
A 1 = A 1 U i f X i
1/n
, f Xi = X i
p i
− μ q
f X i
= 0 when D3-brane hits D7-brane (divisor, at μ)
known as Ganor zero

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Comments on sum distribution
Sum distribution smooths out the divergence and moves the peak.
E.g. z = x 1
n 1
+ x 2
n 2
+ ⋯ + x p
n p
• Each has divergent peak: P w i = x i
n i ∝ wi
−1+ 1 n i
• Independent of each other, no correlations.
But uncorrelated summation gives P z ∝ z −1+ 1
n i .
When all n i = 2, and x i ∈ normal distribution,
P z = e−p 2 z −1+p 2
2 p 2 Γ(p 2)
p = 1
p =2
known as Chi-squared distribution
p =3

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Bousso-Polchinski
4-form quantization
S = d 4 x −g 1 M P
2 R − Λ bare − Z
2 × 4! F 4 2
Λ = Λ bare + 1 2
n i 2 q i
2
Assume randomness in Bousso-Polchinski;
J
n i : random integer, 0 ≤ q i ≤ 1: uniform,
−100 ≤ Λ bare ≤ 0: uniform
Λ = 0
But…
Moduli fields couple each term
Λ ∼ −W 0 A 1
C
9x
9 2 − e−x 1
2
1 x 1
correlation generated via stabilization

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Multi-moduli analyses

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Multi-moduli stabilization
Again, we work in the region: |W 0 | ≫ |W NP |, V ≫ ξ.
Assuming stabilization of complex structure moduli and dilaton
at higher energy scale,
V
4
M = − A 3
1W 0 a 1 2C
P
2 γ 1 9V 3 − x 1e −x 1
V 2
3
V = x 2 3 2
1 − δ i x i
,
i=2
−
i=2
• Now we have N K × N K mass matrix.
B i x i e −x i
V 2
• Stability at positive CC requires B i > 0.
x i = a i t i , C = −27W 3 2
0ξa 1
, B i = A i
, δ
64 2γ 1 A 1
A i = γ 3 2
ia i
3 2
1 γ 1 a 1
N K extremal equations + N K stability constraints
Uplifting is controlled by the first term.
[Sumitomo, Tye, in preparation]
All upper-left sub-determinants are positive (Sylvester’s criteria).
,

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Multi-Kahler statistics
Still complicated system
V
4
M = − A 3
1W 0 a 1 2C
P
2 γ 1 9V 3 − x 1e −x 1
V 2 −
i=2
B i x i e −x i
We just randomize W 0 , A i obeying uniform distribution,
while keeping other parameters fixed.
Solve for t i (or x i ) −15 ≤ W 0 ≤ 0, 0 ≤ A i ≤ 1
V 2
N K = 1: blue
N K = 3: red
(neglecting N K =1)
More moduli bring shaper peak.
(though mild suppression)
Λ ∼ 1.1 × 10 −3 N K 0.23 e −0.027 N KM P
4

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Cosmological moduli problem
Reheating for BBN: T r ≥ O 10 MeV T r ∼ M P Γ φ , Γ φ ∼ m φ 3
m φ ≥ O 10 TeV ∼ 10 −15 M P
What happens in lightest (physical) moduli mass
M P
(neglecting N K =1)
2
m min
= 0.031 N K 1.0 e −0.10 N KM P
2
: also suppressed
Suppression of mass is relatively faster than Λ.
2
m min
∼ 10 −30 M P
2
is likely met earlier than Λ ∼ 10 −122 M P
4

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More peaked parameters
So far we assumed uniform distribution for W 0 , A i . But realistic
models have a number of complex moduli and others.
Different distributions for W 0 , A i
Consider the effect of multiple independent parameters.
W 0 = −w 1 w 2 ⋯ w n ,
A i = y 1 i y 2
i
⋯ y n
i
0 ≤ w i ≤ 15 1 n , 0 ≤ yj
i
≤ 1, all obey uniform distribution.
Now,
P W0 =
P A i =
1
15 n − 1 !
1
n − 1 !
ln 15
ln 1 A i
W 0
n−1
n−1
,
n=3
n=2
n=1
See how CC is affected by “n”

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Cosmological constant
We cannot simply consider effect of the coefficient.
V
4
M = − A 3
1W 0 a 1 2C
P
2 γ 1 9V 3 − x 1e −x 1
V 2
−
i=2
B i x i e −x i
V 2
Dynamics also affects.
The result:
Λ NK =1 = 4.7 × 10 −3 n 0.080 e−1.40 n
Red: N K = 1
Blue: N K = 2
Green: N K = 3
Λ NK =2 = 3.7 × 10 −3 n 0.97 e−1.49 n
Λ NK =3 = 3.4 × 10 −3 n 1.5 e−1.55 n
More than the effect of
the coefficient!
A 1 W 0
∼ 15 e−1.39 n

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Moduli mass
We worry about the cosmological moduli problem.
Red: N K = 1
Blue: N K = 2
Green: N K = 3
2
m min
2
m min
2
m min
= 0.18 N K =1 n0.14 −1.40 n
e
= 0.061 N K =2 n0.73 −1.56 n
e
= 0.039 N K =3 n1.2 −1.66 n
e
Compare with CC
Λ ∝ e −1.40 n , e −1.49 n , e−1.55 n
Suppression in mass is getting
larger as increasing N K .
also suggests

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Estimation
Using the estimated functions, we get
N K (= h 1,1 ) 1 2 3
Λ ∼ 10 −122 4
M P n ∼ 197 n ∼ 188 n ∼ 182
m 2 ∼ 10 −30 2
M P n ∼ 48 n ∼ 44 n ∼ 42
n: number of
product in W 0 , A i
Rather considerable number, e.g.
4
• P 1,1,1,6,9
: h 1,1 = 2, h 2,1 = 272
• F 11 : h 1,1 = 3, h 2,1 = 111
and the other moduli (e.g. brane position, open string) come
in a complicated way, like
• A 1 = A 1 U i f X i
1/n
, f Xi = X i
p i
− μ q
While, without help of product distribution in W 0 , A i
N K ∼ 10100 for Λ ∼ 10 −122 M P 4 , N K ∼ 1350 for m 2 ∼ 10 −30 M P
2

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Mass matrix
Physical mass matrix is a linear combination of ∂ xi ∂ xj V| min .
Assuming uniformly distributed −15 ≤ W 0 ≤ 0, 0 ≤ A i ≤ 1,
∂ xi ∂ xj V min
∼ 10 −3 ×
x 1 x 2 ⋯ x NK
7 4 ⋯ ⋯ 4
4 60 1 ⋯ 1
⋮ 1 ⋱ ⋯ ⋮
⋮ ⋮ ⋮ ⋱ 1
4 1 ⋯ 1 60
some
hierarchical
structures
Though off-diagonal comp. are relatively suppressed,
eigenvalue repulsion gets more serious when increasing N K .
e.g. 2 × 2 matrix: a
b
b
c
λ ± = 1 2 a + c ± a − c 2 + 4b 2
The lowest mass eigenvalue is generically suppressed
more than CC.

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Summary & Discussion

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Summary & Discussion
• Stringy Random Landscape
We may expect that stringy motivated models have the
following properties:
• Product of parameters
• Correlation of each term by dynamics
Both works for smaller CC.
• A number of Kahler moduli
Correlation makes CC smaller. But the effect is modest.
• A number of complex moduli and other moduli
Those are likely to produce more peakiness in parameters
Interesting to see detailed effect in concrete models

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Summary & Discussion
• A potential problem
Lightest moduli mass is suppressed simultaneously.
cosmological moduli problem
before reaching Λ ∼ 10 −122 M P 4 .
Other than “product” and “correlation” effect,
“eigenvalue repulsion” also makes the value smaller.
This is presumably a generic problem
when taking statistical approach without fine-tuning.
Once finding a way out, the stringy mechanism
naturally explain why CC is so small.
Thermal inflation, coupling suppression to SM,
or some other corrections may help