SLIDE 1
de Sitter Vacua
Bret Underwood
McGill University Pheno 2009 Does dS space exist classically from a top‐down construction?
- S. Haque, G. Shiu, BU, T
. Van Riet, Phys. Rev. D79:086005 (2009), arXiv:0810.5328
de Sitter Vacua Bret Underwood McGill University Pheno 2009 S. - - PowerPoint PPT Presentation
de Sitter Vacua Bret Underwood McGill University Pheno 2009 S. - - PowerPoint PPT Presentation
de Sitter Vacua Bret Underwood McGill University Pheno 2009 S. Haque, G. Shiu, BU , T . Van Riet, Phys. Rev. D79:086005 (2009), arXiv:0810.5328 Does dS space exist classically from a top down construction? dS Space Bottom Up 1 1 H 2 = H 2
SLIDE 2
SLIDE 3
dS Space
Bottom Up
To construct classical dS solutions, just add cosmological constant
H2 = 1 3M 2
p
(½ + ¤cc) H2 = 1 3M 2
p
(½ + ¤cc)
SLIDE 4
dS Space
Bottom Up
To construct classical dS solutions, just add cosmological constant
Top Down
Constructions require moduli stabilization, quantum effects. [KKLT, 03], [LARGE volume, 05],… Is nature telling us that dS space is inherently quantum, or do classical top‐down constructions exist? Cosmological constant comes from scalar potential with positive energy local minimum.
¢V ¢V
H2 = 1 3M 2
p
(½ + ¤cc) H2 = 1 3M 2
p
(½ + ¤cc)
Other interesting features:
¢V » M 2
pm2 3=2; ) Hinf < m3=2
¢V » M 2
pm2 3=2; ) Hinf < m3=2
mÁ » O(few £ TeV) mÁ » O(few £ TeV)
- Typically
- Typically
non‐thermal decay into DM
SLIDE 5
No‐Go Theorems
Dimensional reduction of 10d supergravity potential in 4D: Two scalar fields always present in any compactification
Internal curvature, H3 flux, KK5‐branes, NS5‐ branes, (indefinite sign) D‐branes O‐planes
(indefinite sign)
p‐form fluxes
(always positive)
No‐Go: [Maldacena, Nunez, 00]
“If no sources, then no dS” Always has minimum with V<0
V (½; ¿) = a(½)¿ ¡2 ¡ b(½)¿ ¡3 + c(½)¿ ¡4 V (½; ¿) = a(½)¿ ¡2 ¡ b(½)¿ ¡3 + c(½)¿ ¡4
SLIDE 6
No‐Go Theorems
H3 flux, (always positive) D‐branes O‐planes (indefinite sign) p‐form fluxes (always positive)
No‐Go: [Tegmark et al, 07]
“If IIA, fluxes, O6/D6, then no dS vacua” Finding dS vacua is as easy as “a,b,c”: dS vacua Minimizing the quantity
[Silverstein, 08]
a(½) = ANSNS ½3 a(½) = ANSNS ½3 b(½) = nO6AO6 ¡ nD6AD6 b(½) = nO6AO6 ¡ nD6AD6
c(½) = X
p
½3¡pARR
p
c(½) = X
p
½3¡pARR
p
4ac b2 = (const) X
p
½¡pARR
p
4ac b2 = (const) X
p
½¡pARR
p
Cannot be minimized in ρ direction: No dS vacua!
SLIDE 7
Minimal dS vacua [Underwood et al 08]
V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4 V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4
Acurvature » ¡ Z R6 Acurvature » ¡ Z R6
Zero Curvature:
Acurvature » ¡ Z R6 = 0 Acurvature » ¡ Z R6 = 0
Positive Curvature:
Acurvature » ¡ Z R6 < 0 Acurvature » ¡ Z R6 < 0
Negative Curvature:
Acurvature » ¡ Z R6 > 0 Acurvature » ¡ Z R6 > 0
SLIDE 8
Minimal dS vacua [Underwood et al 08]
V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4 V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4
Acurvature » ¡ Z R6 Acurvature » ¡ Z R6
Zero Curvature:
Acurvature » ¡ Z R6 = 0 Acurvature » ¡ Z R6 = 0
Positive Curvature:
Acurvature » ¡ Z R6 < 0 Acurvature » ¡ Z R6 < 0
Negative Curvature:
Acurvature » ¡ Z R6 > 0 Acurvature » ¡ Z R6 > 0
SLIDE 9
Minimal dS vacua [Underwood et al 08]
V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4 V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4
Acurvature » ¡ Z R6 Acurvature » ¡ Z R6
Zero Curvature:
Acurvature » ¡ Z R6 = 0 Acurvature » ¡ Z R6 = 0
Positive Curvature:
Acurvature » ¡ Z R6 < 0 Acurvature » ¡ Z R6 < 0
Negative Curvature:
Acurvature » ¡ Z R6 > 0 Acurvature » ¡ Z R6 > 0
Minimal dS vacua:
“If IIA, fluxes, O6/D6, negative curvature, then dS vacua possible”
SLIDE 10
Minimal dS vacua [Underwood et al 08]
V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4 V = μAcurvature ½ + ANSNS ½3 ¶ ¿ ¡2¡(nO6¡nD6)A6¿ ¡3+ @X
p
½3¡pARR
p
1 A ¿ ¡4
Acurvature » ¡ Z R6 Acurvature » ¡ Z R6
Negative Curvature:
Acurvature » ¡ Z R6 > 0 Acurvature » ¡ Z R6 > 0
Minimal dS vacua:
“If IIA, fluxes, O6/D6, negative curvature, then dS vacua possible”
4ac b2 = (const) X
p
ARR
p [Acurvature½2¡p + ANSNS½¡p]
4ac b2 = (const) X
p
ARR
p [Acurvature½2¡p + ANSNS½¡p] Can minimize
SLIDE 11
Summary
“If IIA, fluxes, O6/D6, internal curvature, then dS vacua possible”
Minimal Ingredients for 10d dS (in IIA): Examples…?
- “Twisted 3‐tori”: Can classify all possibilities. All fail – additional
moduli lead to runaway directions.
- “Twisted 6‐tori”: simple constructions all fail
- “Compact Hyperbolic Manifolds”: Seem to work!”
Lift to 10D? Solving 10D equations tricky…
[Caviezel et al, 08] [Flauger et al, 08] [Underwood et al, 08] [Underwood et al, 08] [Underwood et al, in progress]
SLIDE 12
Summary
“If IIA, fluxes, O6/D6, internal curvature, then dS vacua possible”
Minimal Ingredients for 10d dS (in IIA): Examples…?
- “Twisted 3‐tori”: Can classify all possibilities. All fail – additional
moduli lead to runaway directions.
- “Twisted 6‐tori”: simple constructions all fail
- “Compact Hyperbolic Manifolds”: Seem to work!”
Lift to 10D? Solving 10D equations tricky…
[Caviezel et al, 08] [Flauger et al, 08] [Underwood et al, 08] [Underwood et al, 08] [Underwood et al, in progress]
Q: Is nature telling us that dS vacua are inherently quantum, or do classical top‐down constructions exist? A: Minimal (necessary) ingredients known, but are they sufficient?
SLIDE 13
But with tradeoffs…
Example: Compact Hyperbolic Manifolds
Consider 6D internal space to be product of 3‐dimensional compact hyperbolic spaces: O6‐plane maps Meta‐stable solutions exist:
Curvature NSNS flux O6‐plane 0‐form flux 6‐form flux