Updates on Warped Brane Inflation

Transcription

1 Updates on Warped Brane Inflation Heng-Yu Chen Department of Physics UW-Madison March 31, 2009/ Department of Physics, Purdue University Based on , , , [hep-th] with [with Jinn-Ouk Gong, Ling-Yan Hung, Peter Ouyang and Gary Shiu].

2 It is good if early universe underwent a period of Inflation: Explain Flatness and Horizon problems in otherwise highly successful Big Bang cosmology, as well as predicts nearly scale-invariant density perturbations. Inflation is driven by dynamical Inflaton Field φ, whose potential V(φ) is highly UV sensitive. Sustained period of Inflation can be stopped by dimension six, Planck suppressed operator: V(φ) O 4 M P 2 φ 2 η < O 4 >. (1) V CMB Need a UV completion to estimate such correction to ensure η 1. Primodial gravitational wave/tensor-scalar ratio r is constrained by [Lyth] r CMB 0.01( φ/m p ) 2. (2) Future dectection limit r 0.01, need φ M p and also need to constrain φ 4 (φ/m p ) n, n > 0 using e.g. UV global symmetry. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 2 / 24

3 Non-Gaussianity: Single field, slow-roll Inflation yields negligible NG parameter f NL ε [Maldacena]. To have significant NG, can consider generalized single field inflation, e.g. DBI-inflation [Silverstein-Tong]: S = 1 2 d 4 x ( ) g Mp 2 R +2P(X,φ), X = 1 2 µφ µ φ. (3) The higher derivatives terms clearly requires UV completion. String theory can serve as UV completion of Inflation, but a viable model needs to satisfy compactification constraints. Exciting observational signatures in next decade e.g. WMAP/PLANCK or CMBPol, serving as tests for string theory. The cosmological data can help us constrain string landscape, or alternatively can also pursue PLANCK/CMBPol Inverse Problem. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 3 / 24

4 Consider a warped throat in string compactification, generated by localized fluxes backreacting on bulk compact Calabi-Yau [KKLMMT]. An explicit example: Warped deformed conifold [Klebanov-Strassler], generated by 1 (2π) 2 α A F 3 = M, 1 (2π) 2 α B H 3 = K, (4) where A and B denotes the two dual three cycles of the conifold. The deformation parameter ɛ, is stabilized by flux superpotential at exponentially small value [GKP]: z 2 1 +z2 2 +z2 3 +z2 4 = ɛ2, ɛ 2/3 /R UV = a 0 = e 2πK 3gsM. (5) The dilaton-axion τ = C 0 +i/g s is also fixed such that g s is small. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 4 / 24

5 Take a D3D3 pair, D3 is driven to the tip by fluxes, D3 is weakly attracted by warped Coulombic interaction ( V D3D3 ( y y D ) 0 = U ) D 0 (ρ,z) 16 y y 4, D 0 = 2T 3 a0 4, (6) where U(ρ,z) is universal volume modulus and y y is D3D3 seperation, usually identified with canonical inflaton φ. Landscape of brane inflation potentials arises from different moduli-stabilizations and other UV bulk effects: V(φ) = V D3D3 (φ) +V stab. (φ) +V bulk (φ). (7) D-brane inflation in a warped throat allows for detailed description of V(φ), and definite predictions for suitable input parameters. When y y a 0 α, D3D3 annihilate through open string tachyon condensation, reheating of universe can occur through production of closed string modes coupling to other fields. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 5 / 24

6 The moduli-stabilization effects are encoded in F-term scalar potential V F (z,ρ) = e K (z,ρ) ( D Σ W (z,ρ) 2 3 W (z,ρ) 2), (8) where ρ = σ +iχ is volume modulus and {z A } are brane moduli. The universal Kähler potential K (z, ρ) is given by [DeWolfe-Giddings]: K (z,ρ) = 3log(ρ + ρ γk(z A,z A )) = 3logU(ρ,z). (9) It would be interesting to investigate if K (z,ρ) is corrected in a warped background [Chen-Nakayama-Shiu, work in progress]. Other than complex structure moduli, still need to stabilize ρ and {z A } using non-perturbative superpotential [Ganor, BHK, BDKMMM]: W NP (z,ρ) = A(z)exp( aρ), a = 2π n (10) This can be generated by D7 (gaugino condensate) or Euclidean D3 (instanton) wrapping supersymmetric four cycles. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 6 / 24

7 Without z-dependence in W NP, σ is fixed at σ 0, expanding around it yields ) V(φ) = V(0) (1 + φ 2 /3Mp η 2/3. (11) This is a version of SUGRA η-problem and insufficient number of e-folds are generated [KKLMMT], need extra correction to inflaton potential! In the simplest KKLT scenario, the gravitino mass m 2 3/2 (σ 0) = e K W (σ 0 ) 2 in the uplifted vacuum is coupled to the potential barrier V AdS Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 7 / 24

8 The requirement of preventing runaway decompactification gives a generic bound on Hubble scale [Kallosh-Linde]: H m 3/2 (σ 0 ). (12) This implies either m 3/2 TeV (bad for LHC ) or low scale inflation (bad for PLANCK/CMBPol ). Improvement on η problem: Including φ-dependence from D3 D7 one-loop threshold correction in W NP [BDKMMM]: ( ) f (za ) 1/n Aexp( aρ) A(z A )exp( aρ), A(z A ) = A 0, f (0) where f (z a ) is the holomorphic four cycle embedding function in singular conifold. The factor A 0 depends on stabilized moduli. The same functional dependence on z A /φ of W NP (z,ρ) carries over to deformed conifold [Chen-Hung-Shiu]. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 8 / 24

9 W NP (z,ρ) partially breaks SO(4) isometry of deformed conifold, stabilizing broken isometry directions. Still exist residual light scalars, spectators during inflation, but coupled to φ at the end of inflation through V tachyon ( y y ) and give distinct signatures. Stabilizing σ instantaneously instead, using adiabatic approximation and introduce φ dependence: σ V(φ,σ) σ (φ) = 0, V(φ,σ 0) V(φ,σ (φ)) No quadratic correction, η(φ) can only be made small piece-wise by explicit fine-tuning, inflation near inflection point 2 φv(φ) = 0 at large radius [BDKM] Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 9 / 24

10 In remaining seminar, we shall consider following questions: Can there be other possible constructions of brane inflation? Can Hubble scale be greater than gravitino mass? Can we constrain the parameter space of warped brane inflation? Can the remaining light scalars give any interesting effects? Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 10 / 24

11 The supersymmetric D7 branes wrapping Σ 4 in deformed conifold allow for interesting inflationary and phenomenological model buildings, but few conditions need to be satisfied due to non-trivial background fluxes [MMMS, GMM]: The four cycle Σ 4 is holomorphic w.r.t. complex structure of CY three-fold. The generalized gauge field strength F = B 2 +2πα F 2 needs to satsify F (2,0) = F (0,2) = 0, J F = tanhθ ( ) e A J J e A F F. (13) 2 The field strength F also needs to satisfy consistency condition: df = H 3, (14) which is equivalent to cancellation of global anomaly [Freed-Witten]. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 11 / 24

12 Using these conditions to examine all known supersymmetric D7 embeddings for singular conifold, none except one remains supersymmetric in deformed conifold without F 2 [Chen-Ouyang-Shiu]. The special SUSY D7 embedding in deformed conifold is given by z 1 = µ [Kuperstein]. The other embeddings given as w n 1 1 w n 2 2 w n 3 3 w n 4 4 = µ 4i=1 n i with w 1 = (z 1 +iz 2 )/ 2 etc. [Ouyang, Karch-Katz, ACR] all require additional F 2. Consider next simplest case w 1 = µ, can only numerically solved F 2 in asymptotic limit with ansatz F = α(θ 1,2 )P + γ(θ 1,2 )Q [Klebanov-Tseytlin]: Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 12 / 24

13 Having established the existence of supersymmetric D7 in deformed conifold, we can construct inflationary models with multiple gaugino condensates [Chen-Hung-Shiu]. The superpotential becomes similar to the Racetrack type [Burgess et al]: W (z,ρ) = W 0 +A(z)e 2π n 1 ρ +B(z)e 2π n 2 ρ. (15) Can show from D7 action that the dependence on F cancel between DBI and CS terms, also confirmed by an one-loop open string scattering between D3/D7. Only consider simplified situation where two stacks of D7 branes are non-intersecting Kuperstein embeddings z 1 µ 1,2 = 0, this ensures no extra unsaturated zero modes. However we can still consider different relative positions between the mobile D3 and two stacks of D7 gaugino condensates. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 13 / 24

14 The simplest case r < µ 2 2/3 < µ 1 2/3, implies that inflationary trajectory is insensitive to additonal D7 brane stacks: More interesting case µ 2 2/3 < r < µ 1 2/3, a metastable local minimum can also be naturally generated to stabilize D3: Inflation occurs through accidental tuning of the microscopic parameters. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 14 / 24

15 Having Racetrack-like superpotential also allows us to avoid Kallosh-Linde bound. Assuming the perturbative superpotential W 0 can be tuned such that additional supersymmetric Minkowski vacuum exists, such that V F (σ Mink. ) = 0: The quantity e K W 2 is by definition vanishing in Minkowski vacuum, and even with uplifting, gravitino mass m 2 3/2 (σ 0) e K W (σ Mink. ) 2 0 remains very small. The remaining question is if a viable brane inflation model can be found? Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 15 / 24

16 The answer is Yes, We Can! Can easily obtain parameters allow for inflection point inflation: We can obtain around 260 e-folds and have H/m 3/2 (σ 0 ) 270. More favorable than modular inflation, where more complicated superpotential W is needed, and needs to invoke UV global symmetry. For warped brane inflation, only minimal superpotential is required, and no additional UV symmetry required. A existence proof for an inflationary model with H m 3/2, futher parameter scanning is required to achieve a model with m 3/2 Tev and consistent cosmological predictions, typically requires H Gev. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 16 / 24

17 The resultant potential V stab. (φ)+v D3D3 (φ) for the full deformed conifold takes the form: 2a 2 κ 2 A 0 2 g(τ) 2/n { e 2aσ (τ) U[τ,σ (τ)] U[τ,σ (τ)] ( 1 W ) } 0 e aσ (τ) +F(τ) 6 a A 0 [g(τ)] 1/n D(φ) + U[τ,σ (τ)] 2. (16) It generically contains large number of parameters: {n, A 0, W 0,s,ɛ,µ }, there are landscape of possible inflatonary trajectories. Explicit compactification constraints, such backreactions can loosely restrict their values, however can use observational data instead. The potential above allows for the precise identification of end of inflation point, hence the CMB scale for comparisons. However for this to be meaningful, it is important to include both V D3D3 (φ) +V stab. (φ), and the important bulk corrections V bulk (φ). Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 17 / 24

18 We can parametrize the bulk physics using AdS/CFT! [BDKKM] Consider a coupling between a conifold CFT operator O and a bulk moduli field X via Kähler potential: K = c d 4 θxx Mp O V = c F X 2 Mp O. (17) V is an irrelevant operator, but can still affect the Inflationary trajectory. From GKP, the perturbation due to bulk effects on D3 is encoded in δ(e 4A α) Φ ( ) J 1 J 2 R (r/r UV ). (18) The spectrum of {Φ ( ) } have been tabulated in [Ceresole et al], and we J 1 J 2 R care about the ones with the smallest (least irrelevant). The resultant perturbation in the inflaton potential is given by V(φ) = c V 0 (φ/φ UV ), (19) the leading corrections are = 3/2 (chiral) and = 2 (non-chiral). Other higher terms are also possible. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 18 / 24

19 It is important to examine how sensitive the predictions based on V D3D3 (φ) +V stab. (φ) to these corrections [Chen-Gong]. We used the data from SDSS+WMAP5 for the scalar power spectrum P R and spectral index n R as observational constraints: V P R = 24π 2 εmp 4 = (2.41 ±0.22) 10 9, n R = 1 6ε +2η = ± A toy scanning is presented here in α β plane for V D3D3 (φ) +V stab. (φ), where α = ɛ/µ, β = T 3 /6ɛ/M p : ÈË Ö Ö ÔÐ Ñ ÒØ ÐÓ ½¼ ÈÊ ÒÊ Æ «Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 19 / 24

20 However once the bulk perturbation V bulk (φ) is added to V D3D3 (φ) +V stab. (φ): V bulk (φ) = V F (0,σ F ) [ ( ) φ 3/2 ( φ c 3/2 +c 2 φ UV φ UV ) 2 ], (20) with natural values of c 3/2,c 2 O(1), the observationally consistent parameter sets becomes invalid. Due to the delicate nature of inflection point inflation, one needs to fine-tune c 3/2,c to preserve the earlier predictions. Similar to small field inflation, where some couplings are unnaturally small. The bulk effects are important, and should be included in systematic inflationary parameter scannings. Only subspaces α β and c 2 c 3/2 planes were scanned, full parameter space scanning including generic bulk effects is required. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 20 / 24

21 The open string tachyon potential near the tip is given by V Tachyon ( y y ) T 3 T 2 (ɛ 4/3 [τ 2 + τ 2 Ω 2 + Ω 3 ] a 2 0 α ). (21) Lyth Effect: The quantum fluctuations of the light residual isometry fields θ 0 (t) + δθ(t,x) gives spatial variation to φ e (t,x) [Lyth, Lyth-Riotto]. This gives spatial variation in total No. of e-folds, using δn-formalism [Lyth, Stewart-Sasaki], it gives extra contributions to power spectrum: P 2 e = 1 4ε e ϑ 2 e ϑ 2 c ϑ 2 e ( Hcmb 2πM p ) 2, c.f. P 2 cmb = 1 2ε CMB ( HCMB 2πM p It is possible P 2 e P 2 CMB, and we need P 2 CMB + P 2 e Can also obtain expression for non-gaussian parameter f NL : ) 2. f NL = 5 6 (η e 2ε e ) + 5 θ 2 c M p ɛe 3 θe 2. (22) θc 2 θe 2 Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 21 / 24

22 There are criteria for such effect to be significant: The initial value for θ 0 θ e needs to be tuned 1/(Γ 3 gs M) for inflationary trajectory to reach tachyon surface. Inflationary trajectory cannot incident tachyon surface orthogonally, occurs if Coulombic attraction V D3D3 ( y) dominates. Other than D3 at the tip, additional sources needed to uplift C.C. after D3D3 annihilation, e.g. D-term uplifting [BKQ] or distant D3. Typically unwarped, the distant sources can naturally dominate over V D3D3 ( y), allow decoupling its scale D 0 from V F ( y). Can engineer (V F (r,σ) V F (ɛ 2/3,σ F )) V Coulomb by tuning A 0, such that Coulombic attraction becomes subdominant. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 22 / 24

23 Here we apply our scenario to A Delicate Universe [BDKM] Consider D7 embedding z 1 µ = 0 [Kuperstein], preserving SO(3) SO(4). The angular stable trajectory for entire deformed conifold preserving SO(2) z 1 = ɛcosh τ 2 z 1 = r 3/2 2 (Large radius). (23) Slow-roll inflation continues near the tip, ε 1. The vev for heavy directions can be obtained, canonical normalized fields from the residual directions accidentally degenerate along stable trajectory. The Delicate Universe is protected from Lyth Effect. However, there are many other D7 embeddings and associated, such as w 1 = µ 1 or its generalizations. The systematics described here can potentially distinguish different brane embeddings! Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 23 / 24

24 In this seminar, we have discussed: The existence of supersymmetric D7 brane embeddings in warped deformed conifold. New class of brane inflation models, with multiple stacks of moduli-stabilizing D7s. In particular Kallosh-Linde bound can be avoided. Outline the systematics for constraining the parameter space of warped brane inflation including bulk corrections with observational data. Systematic study of the potentially dominating Lyth/multi-field effect at the end of warped brane inflation. Would be interesting to study other configurations, e.g intersecting D7s, as they can give distinct signatures. Heng-Yu Chen (UW-Madison) Updates on Warped Brane Inflation 24 / 24