Gary Shiu University of Wisconsin & HKUST Outline of these - - PowerPoint PPT Presentation

Searching for de Sitter String Vacua

Gary Shiu

University of Wisconsin & HKUST

Outline of these Lectures

Lecture 1: No-go theorems for dS and explicit model building

S.S. Haque, GS, B. Underwood, T . Van Riet, Phys. Rev. D79, 086005 (2009). U.H. Danielsson, S.S. Haque, GS, T . Van Riet, JHEP 0909, 114 (2009). U.H. Danielsson, S.S. Haque, P . Koerber, GS, T . Van Riet, T . Wrase, Fortsch.

• Phys. 59, 897 (2011).

GS, Y. Sumitomo, JHEP 1109, 052 (2011).

Lecture 2: T wo roles of tachyons in String Cosmology Random (Super)gravities & Implications to the Landscape

• X. Chen, GS, Y. Sumitomo, H. T

ye, JHEP 1204, 026 (2012)+work in progress

Detectable Primordial Gravity Waves in Small Field Inflation

• N. Barnaby, J. Moxon, R. Namba, M. Peloso, GS, P

. Zhou, arXiv:1206.6117.

STRING THEORY LANDSCAPE

• Many perturbative formulations:
• In each perturbative limit, many topologies:
• For a fixed topology, many choices of fluxes.

A Flux Landscape

• Quantized fluxes contribute to vacuum energy:
• A finely spaced discretuum [Bousso, Polchinski]:
• # solutions ~ (# flux quanta)#moduli ~mN~10500

n q n q

2 2 1 1 q2

1/2

bare

Λ

q

1

= 0

Λ = Λbare + 1 2 X

i

n2

i q2 i

Λ=0 Λ

Z

Σ

F ∈ Z

Explicit Constructions

KKLT, LVS, ... Classical dS

A Mini Landscape

✤# of unipotent 6D group spaces ~ O(50). Among them, only a handful have de Sitter critical points that are compatible with

• rbifold/orientifold symmetries.

✤Each of these group spaces has O(10) left-invariant modes. Tadpole constraints restrict flux quanta on each cycle ≤ O(10). ✤A sample space of O(1010) solutions, no dS that is tachyon free. ✤Flux quantization: Pictorially

• For SU(2)xSU(2) examples,

can explicitly check flux quantization demands solutions outside SUGRA.

Probability Estimate

• Consider
• Then

Vmin(ϕ) = ∑j Vj,min(ϕj). If Vj has nj minima, then there are ∏ nj classical minima. For nj ~ n, # minima = nN [Susskind].This is implicit in BP .

• Say

Vj has 2nj extrema, roughly half of which are minima.

• Probability for an extremum to be a minimum is
• Still, there are P x (# extrema) = eN ln n minima.

V (φ) =

N

X

j=1

Vj(φj)

P = 1/2N = e−Nln2

Probability for de Sitter Vacua

• We are interested in dS vacua from string theory.
• The various Φj interact with each other. It is difficult to

estimate how many minima there are.

• Explicit form of

V is typically very complicated, e.g., in IIA:

J =kiY (2−)

i

Ω =FKY (3−)

K

+ iZKY (3+)

K

K = − 2 ln ✓ −i Z e−2φΩ ∧ Ω∗ ◆ − ln ✓4 3 Z J ∧ J ∧ J ◆ √ 2W = Z ⇣ Ωc ∧ (−iH + dJc) + eiJc ∧ ˆ F ⌘ fαβ = − ˆ κiαβti, Dα = − eφ4 √2vol6 ˆ rK

α FK,

√

Jc =J − iB = tiY (2−)

i

Ωc =e−φIm(Ω) + iC3 = N KY (3+)

K

V = eK ⇣ KijDtiWDtjW + KKLDNKWDNLW − 3|W|2⌘ + 1 2 (Ref)−1αβ DαDβ

ˆ κiαβ = Z Y (2−)

i

∧ Y (2+)

α

∧ Y (2+)

β

,

, dY (2+)

α

= ˆ rα

KY (3+) K

.

Stability of Extrema

• The Hessian mass matrix H=

Vij at an extremum Vi =0 must be positive definite for (meta)stability.

• We can use Sylvester’s criterion to check whether there

are tachyons, but time-consuming for a large Hessian H (c.f. last lecture).

• If the Hessian is large and complicated, how do we

estimate the probability of an extremum to be a min.?

Random

Random Matrix Theory

• A tool to study a large complicated matrix statistically

[Wigner, Tracy-Widom, ....]

• Given a random H, the theory of fluctuation of extreme

eigenvalues allows one to compute the probability of drawing a positive definite matrix from the ensemble.

• Eigenvalue repulsion: probability for H to have no

negative eigenvalue is Gaussianly suppressed.

• Some initial foray in applying these RMT results to

cosmology was made [Aazami, Easther (2005)].

Wigner Ensemble

• 2
• 1

1 2

M = A + A† ,

Dyson

Wigner’s semi-circle

Elements of A are independent identically distributed variables drawn from some statistical distribution.

ρ(λ) λ

Tracy-Widom & Beyond

(2N)

1/2

(2N)1/2

−

ρ (λ, Ν)

sc

N−1/6 TRACY−WIDOM WIGNER SEMI−CIRCLE λ SEA

Study of the fluctuations of the smallest (largest) eigenvalue was initiated by Tracy-Widom, and generalized to large fluctuations by Dean and Majumdar (cond-mat/0609651).

Probability of Stability

If the probability is Gaussianly suppressed, while # extrema goes like ecN (recall 10500), unlikely to find metastable vacua. The large N analytic result of Dean & Mujumdar and further refinement by Borot et al:

2 3 4 5 6 7 8 N 107 105 0.001 0.1 P

a b N2c N

P = a e−bN2−cN

Probability of the form: seems to work well, and agrees with: Consider a Gaussian orthogonal ensemble

[Chen, GS, Sumitomo, Tye]

P ≈ e− ln 3

4 N 2

P = exp " ln 3 4 N 2 + ln(2 p 3 3) 2 N 1 24 ln N 0.0172 #

Random Supergravities

• Consider the SUGRA potential:

and its Hessian, which is a function of DAW, DADBW, and DADBDCW, as well as W.

• Instead of randomizing elements of H, one can randomize

K, W, and its covariant derivatives [Denef, Douglas];[Marsh,

McAllister, Wrase]

• This approach is applicable to F-term breaking, but not to

D-term breaking, and models with explicit SUSY breaking.

• Also a different ansatz was used. Quantitative

details differ, but 𝒬 ¡less likely than exponential also found. V = eK DAWDAW − 3|W|2 P = ae−bN c

Random Supergravities

• 2
• 1

1 2 1 2 3 4

Figure 1: The eigenvalue spectra for the Wigner ensemble (left panel), and the Wishart ensem- ble with N = Q (right panel), from 103 trials with N = 200.

M = A + A†

M = AA†

The Hessian is well approximated by a sum of a Wigner matrix and two Wishart matrices.

IIA Flux Vacua

• An infinite family of AdS vacua are known to arise from flux

compactifications of IIA SUGRA [Derendinger et al;

Villadoro et al; De Wolfe et al; Camara et al].

• Attempts to construct IIA dS flux vacua often start with

similar setups as SUSY AdS ones and then introduce new ingredients to uplift (e.g., negative curvature of internal space).

• We can model the Hessian as H = A + B where A= diagonal

mass matrix at AdS min., B is uplift contribution.

• A does not have to be positive definite for stability, as long as

the BF bound is satisfied. To play it safe, we start with a SUSY AdS vacuum with A=positive definite diagonal matrix.

IIA Flux Vacua

• We take B to be a randomized real symmetric matrix.
• A and B have variances σA and σB. The relative ratio y=

σB/σA determines the amount of uplift.

• The ansatz works well when the mass

matrix is not completely random, but has a hierarchy:

0.02 0.04 0.06 0.08 0.10 y 0.001 0.01 0.1 1 b 0.02 0.04 0.06 0.08 0.10 y 0.10 1.00 0.50 0.20 0.30 0.15 1.50 0.70 bêc

Chen, GS, Sumitomo, Tye Gaussianly suppressed when y ~ 0.025 for N=10

b = 0.000395y + 1.05y2 − 2.39y3, b c = 0.0120 + 2.99y − 12.2y2 + 1650y3.

P = a e−bN2−cN

A Type IIA Example

Chen, GS, Sumitomo, Tye

• Return to the SU(2)xSU(2) group manifold studied earlier

in the systematic search of [Danielsson, Haque, Koerber, GS, Van

Riet, Wrase]

• This model evades the no-goes for dS extrema and stability

in the universal moduli subspace. There are 14 moduli.

• Evaluating the variance: >> 0.025.
• There is no surprise that tachyon appears.
• Tachyon appears in a 3x3 sub-Hessian.
• In this model, η =

V’’/V ≲ -2.4 at the extremum, so the tachyon becomes more tachyonic as the CC increases.

y ⇠

1 14×13/2

P

A<B M 2 AB 1 14

P14

A=1 M 2 AA

!1/2 = 0.274.

CC and Stability

• As we lift the CC, the off-diagonal terms become bigger

and the extremum becomes unstable.

• In general, we expect some moduli to be very heavy and

essentially decouple from the light sector, so N= NH + NL.

• The # of extrema is controlled by N, while the fraction of

stable critical points is controlled by NL.

• Example: a 2-sector SUGRA where some moduli have

very large SUSY masses while SUSY is broken in a decoupled sector involving only the light moduli.

• As we go to higher energies, more moduli come into play

(larger eff. N) ➱ probability more Gaussianly suppressed.

Less Democratic Landscape

CC = 0 Before After Stabilization:

[Bousso, Polchinski, 00]

Raising the CC destabilizes the classically stable vacua.

Implications to the Landscape?

Detectable Primordial Gravity Waves without Large Field Inflation

[when having tachyons is a good thing]

Gravitational Waves

✦ A consequence of General Relativity, predicted by

Einstein in 1916.

✦ Remains a Holy Grail of Observational Cosmology. ✦ Indirect evidence: e.g., Hulse-Taylor binary. ✦ Direct detection has so far come up empty. Present

and future interferometers include LIGO, VIRGO, Einstein Telescope, LISA, TAMA, KAGRA, Decigo, ...

Primordial Gravity Waves

• Besides astrophysical sources of GW, even more

interesting are perhaps “echoes” from the Big Bang.

• Spacetime metric undergoes quantum fluctuations

during inflation leading to production of GW ➪ B-mode polarization of CMB.

Tensor Modes & Large Field Inflation

• Tensor to scale ratio: r=PT/PS
• Current bound (WMAP+SPT+H0+BAO): r<0.17, while

r≃0.01 may be detectable in future missions.

• Lyth Bound:
• Detectable tensors ⇔ Large field inflation
• Need to control an infinite set of operators in the

EFT of inflation; a UV completion is necessary.

∆φ MP = 1 √ 8 Z Nend dN r1/2 & 1.06 × ⇣ r∗ 0.01 ⌘1/2

PT ∼ H2 M 2

P

, PS ∼ H2 ✓H ˙ φ ◆2

⇒ r = 16✏

Tensor Modes

WMAP7

Tensor Modes

WMAP7

Tensor Modes

WMAP7

Large field inflation

∆φ Mpl & 1.06 × ⇣ r? 0.01 ⌘1/2

[Lyth]

Tensor Modes

WMAP7

Large field inflation

∆φ Mpl & 1.06 × ⇣ r? 0.01 ⌘1/2

[Lyth]

Needs a symmetry respected by Planck scale physics to have control on EFT.

Tensor Modes

WMAP7

Large field inflation

∆φ Mpl & 1.06 × ⇣ r? 0.01 ⌘1/2

[Lyth]

Needs a symmetry respected by Planck scale physics to have control on EFT.

[e.g., N-flation & Axion-Mondoromy inflation]

Challenges for Large Field Inflation

• N-flation [Dimopoulos, Kachru, McGreevy, Wacker]

assisted inflation with N~O(500) axions; backreaction on MP.

• Axion-monodromy inflation [McAllister, Silverstein, Westphal]
• Issues on backreaction and entropy bound [Conlon]
• Seems to be statistically disfavored [Westphal]

V ({φn}) =

N

X

n=1

Vn(φn) =

N

X

n=1

Λ4

n cos

✓2πφn fn ◆

anti 5B

5B

• C(2) = c

Summary of Our Work

• Tachyonic production of gauge fields induced by axion

couplings during (even small field) inflation can lead to detectable tensors [Barnaby, Moxon, Namba, Peloso, GS, Zhou]

• Our scenario may find a natural home in string theory.
• Studied also scalar, fermion, vector particle production

during inflation due to a non-adiabatic change of mass.

• Particle production sources not only GW but scalar

spectrum; only axion model leads to detectable tensors while consistent with constraints on scalar spectrum.

Vector Production by Axion

• A simple model (φ=inflaton, ¡𝜔=axion):
• Time dependence of axion leads to particle production:
• One helicity (with tachyonic mass) gets copiously produced:
• Vector particles produced source GW & scalar spectrum.

S = Z d4xpg  M 2

p

2 R 1 2(∂ϕ)2 V (ϕ) | {z }

inflaton sector

1 2(∂ψ)2 U(ψ) 1 4F 2 ψ 4f F ˜ F | {z }

hidden sector

• 

∂2

τ + k2 ± 2kξ

τ

• A±(τ, k) = 0 ,

ξ ⌘ ˙ ψ(0) 2Hf .

A+(τ, k) ⇡ ✓ τ 8ξk ◆1/4 eπξp2ξkτ , A0

+(τ, k) ⇡

✓2ξk τ ◆1/2 A+(τ, k) .

Background Dynamics

• Energy density of gauge fields & axion must be subdominant:

which can be satisfied if the axion decay constant:

• The parameter ξ is adiabatically evolving if

1 2h ~ E2 + ~ B2i ⌧ ˙ (0)2 2

1 2 ⇣ ˙ (0)⌘2 + U ⇣ (0)⌘ ⌧ 3H2M 2

p .

• 0.074

p ✏Peπξ ⇠5/2 ⌧ f Mp ⌧ 1.2 ⇠ s 1 U ( ) V (')

¨ (0) H ˙ (0) ⌧ 1 , mψ ⌧ 3H 2

Scalar Spectrum

• Particle production contribute additionally to power spectrum:

where and

• Super-horizon regime: -kτ<<1, and for ξ≿O(1),
• Numerically evaluating integral gives:

Pζ,s(k) = ⇠e4πξH4 128⇡2M 4

p

Z d3q (2⇡)3 q1/2|ˆ k ~ q|1/2 h 1 (q |ˆ k ~ q|)2i2 " 1 ~ q · (ˆ k ~ q) q|ˆ k ~ q| #2 I2 h q, |ˆ k ~ q| i ,

~ q ≡ ~ p/k, ˆ k ≡ ~ k/k

⌘ ⌘ I [a, b] ⌘ Z 1

kτ

dz sin z z cos z z1/2 e2p2ξz[

pa+ p b]

at z ⌘ k⌧ 0) particle prod

I [a, b] ⇡ Z 1 dz z5/2 3 e2p2ξz[

pa+ p b] =

15 32 p 2 ⇣pa + p b ⌘7 ⇠7/2 Pζ,s(k) ⇡ 4 · 1010 H4 M 4

p

e4πξ ⇠6 .

GW and Scalar Spectrum

• Power Spectrum:

☞ Particle production contribution is subdominant.

• Gravity Wave:
• Standard “consistency condition” is violated
• r interpolates between 16ε and 218, observable signal can be
• btained for any ε in this model.

P+ = P+,v + P+,s ' H2 ⇡2M 2

p

 1 + 8.6 · 107 H2 M 2

p

e4πξ ⇠6

• P = P,v + P,s '

H2 ⇡2M 2

p

 1 + 1.8 · 109 H2 M 2

p

e4πξ ⇠6

• Pζ ⇡ P

 1 + 2.5 · 10−6 ✏2P e4πξ ⇠6

• ,

P ⌘ H2 8⇡2✏M 2

p

r ⌘ P

λ Pλ

Pζ ⇡ 16✏ 1 + 3.4 · 10−5✏P e4πξ

ξ6

1 + 2.5 · 10−6✏2P e4πξ

ξ6

.

Non-Gaussianity

• Non-Gaussianity (bispectrum) is peaked at equilateral limit

because the source at any moment is dominated by modes with wavelength comparable to horizon at that moment.

• To quantify the size of non-Gaussianity:
• Current bound:

f equil.eff

NL

⇡ 1.5 · 10−9✏3 P3 P 2

ζ

e6πξ ⇠9

Bζ (k1 = k2 = k3 ⌘ k) ⇡ 2.6 · 1013 H6 M 6

p

e6πξ ⇠9 1 k6

D ⇣ ⇣ ~ k1 ⌘ ⇣ ⇣ ~ k2 ⌘ ⇣ ⇣ ~ k3 ⌘E = Bζ ⇣ ~ ki ⌘ (3) ⇣ ~ k1 + ~ k2 + ~ k3 ⌘

s 214 < f equil

NL

< 266 plitude of density fluc

Signatures & Constraints

0.0001 0.001 0.01 0.1 1 10 100 1000 3 3.5 4 4.5 5 5.5 r ξ ε = 10-3 ε = 10-4 ε = 10-5 r=0.1

50 100 150 200 250 300 350 3.8 4 4.2 4.4 4.6 4.8 5 5.2 fNL

equil

ξ ε = 10-3 ε = 10-4 ε = 10-5 WMAP

3 3.5 4 4.5 5 1e-05 0.0001 0.001 ξ ε fNL

equiv = 266

r=0.1 r=0.01

10−4 << f MP . 10−2

For detectable tensor, consistency of model requires:

0.1 1 0.001 0.01 0.1 r P S C CV = 10-5 = 10-4 = 10-3 |∆| ⌘

• P+ P−

P+ + P−

• =

3.4 · 10−5✏P e4πξ

ξ6

1 + 3.4 · 10−5✏P e4πξ

ξ6

' 1 16 ✏ r ,

Parity Violating Effects in GW

P=Planck, S=SPIDER, C=CMB-Pol, CV=cosmic-variance limited expt.

Comments

• Many axion-like fields in string theory, e.g.,
• Typically f ~ MGUT, sits comfortably in our allowed window.
• Pseudoscalar coupling comes from, e.g.,
• ∃ inflation candidates that couple only gravitationally to

hidden sector, e.g., another axion with:

• r D-brane moduli that do not couple to the axion and the

D-brane on which the U(1) lives.

a = C0, ai = Z

Σi

2

C2, aI = Z

ΣI

4

C4

Z ωa ∧ ∗ωb = 0 , Z ωa ∧ ˜ ωb = 0

ωa,b= p-form associated with each axion

Z

D5

C2 ∧ F ∧ ˜ F

Comments

• Detection of GW probes not the scale of inflation, but the

nature and dynamics of axions.

• GW sourced by scalar particle production during inflation

has recently been explored ([Cook, Sorbo];[Senatore,

Silverstein, Zaldarriaga]), though constraints from backreaction

• n scalar spectrum and bispectrum were not known.

Thus, effects on LIGO scales were instead considered.

• We computed the backreaction due to particle production
• n the power spectrum and bispectrum of such models.

Comments

• We derived a universal formula for GW sourced by scalar,

fermion, vector production due to non-adiabatic change of mass: where s=spin of source, gs= # of dof of a spin-s field

• We found the GW signal too small on CMB scales.

Pλ,s ' 2 gs k3 15π4a2M 4

p

˜ T 2

k

Z dp p6|βp|2 ⇣ |αp|2 + (1)2s |βp|2⌘

• Tk ' a (τ) ˙

m⇤ 27 H3

2F3

✓3 2, 3 2; 5 2, 5 2, 5 2; k2 4H2 ◆

m (τ) = ˙ m⇤ (t t⇤) = ˙ m⇤ H ln ✓τ⇤ τ ◆ = ˙ m⇤ H ln (Hτ)

Aλ (k) = ↵k (⌧) fk (⌧) + k (⌧) f ⇤

k (⌧)

, fk ⌘ ei

R τ dτ 0 ω(τ 0)

p 2! (⌧)

↵k (t t⇤) ' q 1 + e πk2

˙ m⇤

, k (t t⇤) ' e πk2

2 ˙ m⇤

御清聴有難う御座います

THANKS