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Lecture 4 on String Cosmology: Brane Inflation and Cosmic Superstrings

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Lecture 4 on String Cosmology: Brane Inflation and Cosmic Superstrings Henry Tye Cornell University and HKUST Asian Winter School, Japan, January 2012 BraneBrane W worldorld Brane world in Type IIB r " -30 10 m + W µ! G Inflationary Scenario Quantum Fluctuations V( ! ) Slow!Roll Region Damped Oscillations, Reheating ! How is inflation realized in brane world ? Brane inflation Dvali and H.T. hep-ph/9812483 Inflaton is an open string mode Inflaton potential comes from the closed string exchange D3-anti-D3 brane inflation D3 Relatively flat potential ? anti-D3 C.P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh, R. Zhang, hep-th/0105204 G. Dvali, Q. Shafi and S. Solganik, hep-th/ 0105203 Flux compactification where all moduli of the 6-dim. manifold are stabilized D7−branes D7−branes warped throat Figure 1: A schematic picture of the Calabi-Yau manifold is presented here. The large Giddings,circle given by Kachru,dashed line Polchinskirepresent the 3-cycle where NS-NS three form H3 is turned on. The smaller circle in the throat stands for the 3-cycle where the R-R three form F3 Kachru,is turned onKallosh,. Also shown Linde,are D7-br Trivedianes wraping 4-cycKKLTles. There mavacuumy exists a number of throats likande the manyone shown othershere. There is a mirror image of the entire picture due to the IIB/Z2 orientifold operation. From the zero mode we obtain the usual relation between the gravity strength in four dimensions and the fundamental mass scale of the higher dimensional theory 2 D 2 D 4 2A 2 M = M − d − y g e Ψ . (2.21) P s | mn| (0) ⇥ One may choose Ψ(0) = 1 as a convention, but in order to compare its magnitude to the excited modes magnitude we keep it as Ψ(0) which is of course a constant. For the excited mode we impose the following normalization condition D 2 D 4 2A 2 M − d − y g e Ψ (y) Ψ (y) = M δ (2.22) s | ab| (m) (m0) P mm0 ⇥ After this general discussions we would like to find the KK spectrum of the gravitons and other closed string modes in the KS background. We postpone the spectrum analysis until section 6 after some introduction of the KS background. 3. A Throat in the Calabi-Yau Manifold A KKLT vacuum involves a Calabi-Yau (CY) manifold with fluxes [1]. Consider F- theory compactified on an elliptic CY 4-fold X. The F-theory 4-fold is a useful way 7 The KKLMMT scenario D7−branes D7−branes D3-brane anti-D3-brane Figure 1: A schematic picture of the Calabi-Yau manifold is presented here. The large circle given by dashed line represent the 3-cycle where NS-NS three form H3 is turned on. The smaller circle in the throat stands for the 3-cycle where the R-R three form F3 is turnedKachru,on. Also sho Kallosh,wn are D7-br Linde,anes wrapi Maldacena,ng 4-cycles. There MacAllister,may exists a num bTrivedi,er of throats like the one shown here. There ishep-th/0308055a mirror image of the entire picture due to the IIB/Z2 orientifold operation. From the zero mode we obtain the usual relation between the gravity strength in four dimensions and the fundamental mass scale of the higher dimensional theory 2 D 2 D 4 2A 2 M = M − d − y g e Ψ . (2.21) P s | mn| (0) ⇥ One may choose Ψ(0) = 1 as a convention, but in order to compare its magnitude to the excited modes magnitude we keep it as Ψ(0) which is of course a constant. For the excited mode we impose the following normalization condition D 2 D 4 2A 2 M − d − y g e Ψ (y) Ψ (y) = M δ (2.22) s | ab| (m) (m0) P mm0 ⇥ After this general discussions we would like to find the KK spectrum of the gravitons and other closed string modes in the KS background. We postpone the spectrum analysis until section 6 after some introduction of the KS background. 3. A Throat in the Calabi-Yau Manifold A KKLT vacuum involves a Calabi-Yau (CY) manifold with fluxes [1]. Consider F- theory compactified on an elliptic CY 4-fold X. The F-theory 4-fold is a useful way 7 2THEAUTHOR What is eternal inflation ? a(t) eHt ⇠ H2 1/G⇤ ⇠ Consider a patch of size 1/H. Suppose T 3/H. ⇠ 3 3 9 The universe would have grown by a factor of e · = e . If inflation ends in one patch, there are still many other (causally disconnected) patches which continue to inflate. So inflation never ends. Ln g e− ⇠ The inflationary properβ(tiesg) ndependsln(g/gc) maximum dimension ∆max = 6, 7, and 7.8, and we take Nf =1, 2, 6oftheD3-branecoordinates⇠ to be dynamicalsensitiv fields. The coeffielcientsyc onhave rmsthe size Q, andpr the rescaledoper quantitiestiesc ˆ = of the throat LM LM c /Q are drawn from a distribution that has unit variance. We begin at x = x 0.9, LM M 0 8⌘ 14 = 0 ✓1 = ✓2 = φ1 = φ2 = =1.0 , with arbitrary radial velocityx ˙ 0,arbitraryangular10− >Gµ>10− ⌘˙ { and} compactification. velocity 0 in the direction, and all otherProbability angular velocities vanishing. with We would N now likee-folds: to understand how the observables depend on the input parameters Q, ∆max, Nf ,and ,andon ˙ M the initial data x0, x˙ 0, 0. 3 P (N ) N − e ⇠ e D3 Radial mode + 5 angular modes Figure 1: Examples of downward-spiralingN. Agarwal, trajectories R. for Bean, a particular L. McAllister, realization of theShiu poten-G. Xu,and 1103.2775 Underwood, tial. The black dotsin mark 60 and 120 e-folds before the end of inflation (7 of the 8 curves shown achieve Ne > 120); inflation occurs along an inflection point that is not necessarilyBean, parallel to Shandera, HT, Xu, the radial direction. Red curves have nonvanishing initial angular velocities ˙ 0, while blue curves have S˙ 2=0. x S3 Baumann, Klebanov, Maldacena, Steinhardt, 0 McAllister, Dymarsky, Seiberg, Murugan, . Burgess, Cline, Dasgupta, Firouzjahi, Stoica, . 4 Results for the Homogeneous Background As a first step, we study the evolution of the homogeneous background. In 4.1, we show that § for fixed initial conditions, the probability of Ne e-folds of inflation is a power law, and we show that the exponent is robust against changes in the input parameters ∆ , N ,and . In max f M 4.2 we present a simple analytic model that reproduces this power law. We study the e↵ect § of varying the initial conditions in 4.3, and we discuss DBI inflation in 4.4. § § 9 Outline Introduction Cosmic String Network Brane Inflation Microlensing Other SummaryPossibilities Some simple scenarios Features: • Mobile D3s D7 • warped throats D3 • anti-D3s in throats D3 D3 • Wrapped D7s • DBI D3 Henry Tye Brane Inflation : an Overview 7/32 Testing Brane Inflation • Compare power spectrum and its running • Tensor mode (B mode polarization) • Non-Gaussianity • Steps (from Gauge-gravity duality) AData blip inat CMBhigh k power (l) will spectrum improve ACBAR Kuo etc., astro-ph/0611198 A smallPow steper spectrum in the potential can generate such a blip Adam, Cresswell, Easther, astro-ph/0102236 As the D3 brane moves down the throat: The inflationarCascadey properties depends sensitively on the properties of the throat Klebanov-Strasslerand thr compactification.oat r = r SU((K + 1)M) × SU(KM) 0 l = 1 r = r SU((K − 1)M) × SU(KM) 1 l = 2 D3 SU((K − 1)M) × SU((K − 2)M) r = r2 ............. Radial............. mode + SU 5M angular× SU M modes (2 ) ( ) Shiu and Underwood, in Bean, Shandera, HT, Xu, S2 x S3 Baumann, Klebanov, Maldacena, Steinhardt, McAllister, Dymarsky, Seiberg, Murugan, . Burgess, Cline, Dasgupta, Firouzjahi, Stoica, . RG flow and Seiberg duality transition RG Flow and Seiberg Duality 8π2 T = g2 ˆ S b = 2 U T2 T (N (2 S 2 ( − ) U 1) M (N ) ) ) (2) M T1 ) + Cascade (N N SU U( S ) (1 T 1 ˆb = 0 ln(r/r0) ln(r2/r0) ln(r1/r0) 0 • The anomalous mass dimension has a correction that depends on which step the RG flow is at. This means that the coupling flows depend on which step the flow is at. • Using gauge/gravity duality, we see that the dilaton runs and it has a kink at the position where Seiberg duality transition takes place. • this leads to steps in the warp factor, which then leads to steps in the inflaton potential. φ˙2 h4(φ), it is most sensitive to the step in the warp factor. φ˙2 increasing by a factor ∼ of (1 + 2b)acrossastepdecreasesPR by a factor of (1 + 2b). The CMB data shows that around l 20, there is a dip in the power spectrum by about 20% in k-space, which gives ∼ 2b 0.2. Using (2.9), we have ∼ 3g M 1 s 0.2 . (5.11) 8π (p +1)3 ∼ ! " i Given that g M 33 to fit the spacing of the steps, we immediately get s ≈ pi =2. (5.12) With input of 4 quantities : the position l 20, the fractional height (size) ∆T/T 0.2 Predictions ∼ # and the width ∆l 5ofthe2ndstepaswellastheposition(atl 2) of the first step, p ∼ ∼ • Afterwe can fitting use ∆l thel and feature∆T/T at pl −~3 to20 get in aWMAP complete data, set of predictions: ∝ ∝ pl∆T/T ∆lp it predicts additional 2 2 0.7 1 • ∼ ∼ ∼ 3 20 0.2 5 steps : their positions, ∼ ∼ 4 170 0.08 40 their heights and their ∼ ∼ ∼ 5 1300 0.04 260 widths.
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  • Towards Uniform T-Duality Rules

    Towards Uniform T-Duality Rules

    Proceedings of Institute of Mathematics of NAS of Ukraine 2002, Vol. 43, Part 2, 659–662 Towards Uniform T-Duality Rules Alexei J. NURMAGAMBETOV Institute for Theoretical Physics, NSC “Kharkov Institute of Physics and Technology”, 1 Akademicheskaya Str., 61108, Kharkov, Ukraine Center for Theoretical Physics, Texas A&M University, College Station, TX 77843, USA E-mail: [email protected] In this contribution based on the talk given at the SUSY’01, Dubna, Russia, I discuss the reasons of appearing the different sets of fields entering into the T-duality transformations and a way to construct the uniform T-duality rules. 1 Introduction Discovery in the past decade of the new class of non-compact symmetries allowed to revise one of the main problems of Superstrings: The puzzle of having too much “fundamental” theories. The resolution was in the conjecture on the M-theory in framework of which it turned out possible to unify all the five different superstring theories. The useful tool for establishing the M-theory evidence are dualities connected to the new symmetries of perturbative or non-perturbative sectors of Superstrings. In this Contribution I would like to discuss some questions related to the so-called T-duality (see [1] for review). Namely, I will focus my attention on the following: Q1: What are the different ways of T-duality rules derivation? Q2: Do the T-duality rules coincide? Q3: How uniform T-duality rules could be derived? For the sake of simplicity I restrict myself in what follows to the pure classical frames, that means neglecting the dilaton field, which receives its corrections from quantum effects, and will consider backgrounds with single isometry direction.