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−50 Large-Field Inflation - Naturalness and String Theory −55 −60 −65 Declination [deg.] BICEP2: B signal −50 −55 −60 Planck Collaboration: The Planck mission −65 Angular scale 90◦ 6000 18◦ 1◦ 0.2◦ 50 0.1◦ 0.07◦ 0 Right ascension [deg.] 5000 D![µK2] 4000 3000 2000 1000 0 2 10 50 500 1000 Multipole moment, ! 1500 2000 2500 • Why use string theory in cosmology? singularity at the beginning! — need quantum gravity to resolve inflation is UV-sensitive — needs a high-scale theory what are dark energy & dark matter? • if we try to use string theory for cosmology — better build on generic consequences of string theory! string vacuum structure seems to be a “landscape” of extremely many vacua — this can accommodate a tiny C.C. false-vacuum eternal inflation & tunneling set cosmological initial conditions — hint from large-scale CMB anomalies? • if we try to use string theory for cosmology — better build on generic consequences of string theory! string compactification produces moduli: - moduli stabilization & moduli phenomenology (reheating, baryogenesis ...) - spectrum of massive scalars and light axions: - axions driving large-field inflation in string theory - ---> monodromy horizon problem of the hot Big Bang time t why so many causally disconnected regions with -5 ΔT/T ~ 10 ? today: 13.7 billion years CMB: 370,000 years Big Bang: t=0 Inflation ... • inflation: period quasi-exponential expansion of the very early universe (solves horizon, flatness problems of hot big bang ...) [Guth ‘80] • driven by the vacuum energy of a slowly rolling light scalar field: e.o.m.: φ̈ + 3H φ̇ + V = 0 ! [Linde; Albrecht & Steinhardt ‘82] Inflation ... • slow-roll inflation: [Linde; Albrecht & Steinhardt ‘82] scale factor grows exponentially : a ∼ e Ht ⇒ Ḣ 1 !≡− 2 $ H 2 ! V V " ! 2 %1 , 2 φ̇ ! V if : |φ̈| ! |3H φ̇| V !˙ $ %1 η≡ !H V !! 2 ȧ with the Hubble parameter: H = 2 ! const. ∼ V a φE!+∆φ ! dφ Ne e-folds Ne in a ~ e : Ne = Hdt = √ 2" 2 φE Inflation ... • inflation generates metric perturbations: scalar (us) & tensor 2 PS ∼ ∼ • H ∼ ! k ! δρ ρ "2 nS −1 scalar spectral index: nS = 1 − 6! + 2η and 2 PT ∼ H ∼ V window to GUT scale & direct measurement of inflation scale • tensor-to-scalar ratio: PT r≡ = 16! PS shades of difficulty ... • • tensor-to-scalar ratio links levels of difficulty: PT r ≡ = 16! ≤ 0.003 PS 2) r << O(1/Ne models: ∆φ ! O(MP ) • 50 Ne "2 ! ∆φ MP "2 [Lyth ’97] Small-Field inflation ... needs control of leading dim-6 operators ➥ enumeration & fine-tuning reasonable ⇒ 2) r = O(1/Ne models: ∆φ ∼ O(MP ) • ! needs severe fine-tuning of all dim-6 operators, or accidental cancellations ⇒ r = O(1/Ne) models: ! ∆φ ∼ Ne MP " MP ⇒ Large-Field inflation ... needs suppression of all-order corrections ➥ symmetry is essential! why strings? • We need to understand generic dim ≥ 6 operators Op≥6 ⇒ ∆η ! "p−4 φ ∼ V (φ) MP ! "p−6 φ ∼ !1 MP ∀p ≥ 6 if φ > MP • requires UV-completion, e.g. string theory: need to know string and α‘-corrections, backreaction effects, ... • detailed information about moduli stabilization necessary! • string theory manifestation of the supergravity eta problem shift symmetry • effective theory of large-field inflation: 1 1 2 4−p p L = R + (∂µ φ) − µ φ 2 2 • the last term — the potential — spoils the shift symmetry … • However, if: • quantum GR only couples to Tµν : δV (n) ∼ V0 ! V0 MP4 "n V0 = µ 4−p p , V0 ! " !! n V0 MP2 " V0 φ ! 4 MP not δV (n) n φ ∼ cn n MP • shift symmetry while field fluctuation interactions: 4−p µ (φ! + δφ) ∼ p 4−p p µ φ! ! 1+ " n n δφ cn n φ! # die out with increasing field displacement … • if the inflaton potential breaks the shift symmetry weakly & smoothly (means: with falling derivatives) — it does not matter, whether the shift symmetry is periodically broken, or secularly • a shift symmetry itself does not guarantee smoothness of breaking — need UV theory as input, for all models! axion monodromy [Silverstein & AW ’08] [McAllister, Silverstein & AW ’08] [Kaloper & Sorbo ’08] [Dong, Horn, Silverstein & AW ’10] [Lawrence, Kaloper & Sorbo ’11] [Marchesano, Shiu & Uranga ’14] [Blumenhagen & Plauschinn ’14] [Hebecker, Kraus & Witkowski ’14] [McAllister, Silverstein, AW & Wrase ’14] axion inflation in string theory ... • shift symmetry dictates use of string theory axions for large-field inflation - periodic, e.g. b= ! B2 b → b + (2π) , Σ2 2 since Sstring 1 ⊃ 2πα! ! B2 - field range from kinetic terms f < MP : S∼ ! 10 √ 2 d x −g|H3 | ⊃ B2 = bω2 ⇒ φ = fb , ! √ 1 2 d x −g4 4 (∂µ b) L 4 MP f = 2 < MP L [Banks, Dine & Gorbatov ’03] however, maybe not strict: [Grimm; Blumenhagen & Plauschinn ’14] axion inflation in string theory ... • large field-range from assistance effects of many fields - N-flation ... • [Dimopoulos, Kachru, McGreevy & Wacker ’05] [Easther & McAllister ’05] or monodromy - generic presence from fluxes & branes ! [Silverstein & AW ’08] [McAllister, Silverstein & AW ’08] [Dong, Horn, Silverstein & AW ’10] - cos-potential for 2 axions can align/tune for large-field direction as well [Kim, Nilles & Peloso ’04] [Berg, Pajer & Sjörs ’09] [Ben-Dayan, Pedro & AW ‘14] [Tye & Wong ‘14] M N to thep+1 M analogous gaugeMinvariance under A → A + dΛ0 in electro milarly, there are other potential fields denoted Cp+1 sourced by p-dimensiona axion monodromy — the general story → A + ∂ Λ , the field C transforms as C → gauge transformation A M M 0 bjects (Dp-branes) [23]. a one-form, 1 EM et term action contains the gauge-invariant terms is the Maxwell action, written in terms of the field strength F • = dA. In electromagnetism, the action contains the gauge-invariant terms EM Stueckelberg gauge symmetry: m, known as a Stueckelberg term, can arise from spontaneous symmetry " # ! √ 4 4 vacuum expectation M N value2√ 2 " # the of a charged field. d x −g F F −4ρ (A + ∂ M C) , (2.1) N + 2. . . 2 MSNEM = d x −gM FM NM F − ρ (AM + ∂M C) + . . . , ring theory, one finds generalizations of these Maxwell and Stueckelberg gauge transformation B → B + dΛ1 accompanied by appropriate shifts of − Λ →field A +CC ∂→ Λ0C, the field C transform here under the A gauge transformation Athe → A + ∂ Λ , as C → nsformation 0 M ⇒ Mtransforms M M 0 though we will focus on specific examples in type IIB string theory below, − Λ . The first term is the Maxwell action, written in terms of the field streng 0 e Maxwell action,terms written the F = dA. nsidering the relevant arisingininterms D = 10of type IIAfield stringstrength theory. There he second term, known as a Stueckelberg term, can arise from spontaneous string theory contains analogous gauge symmetries for al fields C with odd p, and it is useful to define the following generalized p as a Stueckelberg term, can arise from spontaneous4 symmetry reaking, with ρ the vacuum expectation value of a charged field. hat respect all the gauge symmetries of the theory: NSNS and RR axions - e.g.transformation IIA:4 extended to a combined • m In expectation value of a charged field. type II string theory, one finds generalizations of these Maxwell and S H = dB , transformation δBand = dΛStueckelberg y,rms, onewith finds generalizations of Bthese the gauge → B Maxwell + dΛ1 accompanied 1 , by appropria F = Q , ⇒ 0 0 he C fields. Although we will focus on specific in −F type IIB string th p δC1 = Λ , sformation B → B + dΛ1 accompaniedexamples by appropriate shifts of 0 1 B , relevant terms arising in D t us start byF̃considering = 10 type IIA string th 2 = dC1 + F0the δC . will focus on specific examples in type IIB string theory below, 3 = −F 0 Λ1 ∧ B 1 e have potential fields C with odd p, and it is useful to define the following p F B ∧ B , (2.2) = dC + C ∧ H + F̃ 4 3 1 3 0 he relevant terms arising in D = 10 type IIA string theory. There 2 eld strengths that respect all the gauge symmetries the dimensionality theory: The effective action starting from aoftotal D= with odd p, and it is useful to define the following generalized 5 type IIB similar nteger. These are gauge-invariant, with the transformation B → B + dΛ • to 1 ! " # √ 1 1 H = dB , 10 2 all the gauge symmetries of the theory: d into −to breaking x open −G strings, |H|but+ |F̃ the type I string, in which closed strings are unstable 2 4 3 % % dimensional spacetime, thebydual 6-form flux Qin6Shiu ≡ #10 F4’14] = M F6th) [Marchesano, Uranga In F̃5 we do not find anorFequivalently working directly the&M ten-dimensional 1 ∧ B ∧ B term [Blumenhagen & Plauschinn ’14] models in §3, we will incorporate the analogue in type IIB string theory of these ad However, T-duality on a circle, including the duality between D7-branes and D8-br flux monodromy [Hebecker, Kraus & Witkowski ’14] requires thiswill coupling to be present upon dimensional reduction. This indeed fluxes, which yield interesting behavior in some cases, but for now will works focus [McAllister, Silverstein, AWwe & Wrase ’14] precisely [25]. Specifically, consider reducing typethis, IIB we theory leading contributions to the potential at large ten-dimensional field range. Given haveon anaac fluxes a potential for the axions: 9generate 9 ∼ 9 6 the x form direction, x = x + 2π), with the(along schematic & ' ! 9 C 0 = x Q 0 + C0 , √ 1 1 10 2 2 2 2 4 − ! 4 d x −G |H| + |Q B| + |Q B ∧ B| + γ g |Q B ∧ B| + . . . . 0 0 4 0 9 s C2 = x Q0 B + C2 , gs2 α • are fluctuations of the potential fields about the background. Substituting where C p Here in the last term and the ellipses we have allowed for corrections that could be produces periodically spaced set of multiple branches into (2.7), we find an effective F ∧ B ∧ B contribution to F̃ , and an effective F ∧ B 1 5 1 from the tree-level four-point and higher-point functions (γ4 being an order 1 numb in F̃ In the four-dimensional effective theory, 2there are many contributions of this 3 . large-field 2 of potentials: have also set to zero the contribution from |F̃6 | = |C3 ∧ H + Q0 B ∧ B ∧ B/6| , ha leading to axion potentials of the schematic form • mind situations where H flux is present in order to contribute to moduli stabilization, (n) n 2 a + Q(n−1) an−1 + · · · + Q(0) )2 (Q p0 minimizes f (χ, the . . . ) |F̃6 | term at zero. ! More generally,+there · · · ∼should f˜(χ, . . be . ) ainteresting for a %configu 1, L2n 5 Similar comments apply in the more generic cases with D > 10 [24]. /2 is a positive integer, and “χ, . . . ” r where we have denoted the axion field by a, n = p0 6 See e.g. equation 12.1.25 of [23]. However, we caution the reader that we follow the sign conve (i) as additional scalar fields, whose important effects we to the moduli fields χ, as well for given flux quanta Q potential is non-periodic – a given branch [25], not those of [23]. Q(i) change by brane-flux tunneling – 8Q(i) shift absorbed by a-shift – many 7 branches, full theory has periodicity flux monodromy • p-form axions get non-periodic potentials from coupling to branes or fluxes/field-strengths • produces periodically spaces set of multiple branches of large-field potentials: V (φ) ∼ µ φ + Λ cos(φ/f ) 4−p p 4 φ with the Lagrangian density 1 1 µ 1 2 2 2 We∗ present in Figure 1. In what follows we will focus on the axionm (∂φ) F φ R − − + F , L = (4) pl which we dub “natural chaotic (4) scale string 4-form models of [12, 21], 2 2 48 24 1 1 2 µ picture 1 monodromy flux 2 24D effective ∗ T mpl (∂φ) F(4)which φ F −[Kaloper − ’08]from + [Dubovsky, , (1) L = as inflation”, aR benchmark to discuss Lawrence & Roberts ’11] & theory Sorbo (4) 2 2 48 24 [Lawrence, Kaloper & Sorbo ’11] [Kaloper & Lawrence ’14] thewhere general F phenomenon of axion monodromy. We start tribu = dA is a four-form field strength, (4) (3) … see talk with the Lagrangian density 4D effective action: F field strength, axion ɸ : • 4 dA(3) is a gauge four-form fieldThe strength, with mome where F(4)a = three-form field. canonical A(3) peri gauge field. The canonical momentum A(3) pa three-form ∗ 1 1 µ 1 2 2 2 ∗ = F − µφ is quantized in, units of The thetheo el A123 − (∂φ) − F(4) + φ F(4) (1) L∗= mpl R(4) pA123 = F(4) − µφ is 2 2 2 quantized 48 in units 24 of the electric tributions ; using this, one may write the Hamiltoni charge e 2 ; using this, one may write the Hamiltonian as charge e = dA is a four-form field strength, with F 4D effective Hamiltonian: • where Natural Chaotic Inflation and UV Sensitivity Nemanja Kaloper (4) (3) Department of Physics, University of California, Davis, Davis, CA 95616 periodicity "2for n gauge field. The canonical momentum A(3) a three-form ! 1 1 1 ! " 1 1 1 2 2 2 2 2 2 2 ∗ ne ) + +in (∇φ) µφ H(p pA123 H = =F(4) −= is (p quantized units the ne of++ ) + µφelectric . +(2) φ(∇φ) φµφ 2 2 2 2 2 2 2 as charge e ; using this, one may write the Hamiltonian Albion Lawrence Martin Fisher School of Physics, Brandeis University, Waltham, MA 02454 If the recent measurement of B-mode polarization by BICEP2 is due to primordial gravitational waves, it implies that inflation was driven by energy densities at the GUT scale MGU T ∼ 2 × 1016 GeV . This favors single-field chaotic inflation models. These models require transplanckian excursions of the inflaton, forcing one to address the UV completion of the theory. We use a benchmark 4d effective field theory of axion-4-form inflation to argue that inflation driven by a quadratic potential (with small corrections) is well motivated in the context of high-scale string theory models; that it presents an interesting incitement for string model building; and the dynamics of the UV completion can have observable consequences. 22 . four 04.2912v1 [hep-th] 10 Apr 2014 2 ! " If φ If → φφ + f , we must have µf = e for consistency. 1 1 1 2 2 2 for n ≥ 1 ⇒ µf , n → n + 1 2= e µf → φ + f , we must have = e for consist ne + µφ . (2) H = (pφ ) + (∇φ) + The quantum2number 2n can be shifted by the nucleation 2 four-form, V( The quantum number n can be shifted by the nucle of If membranes, leading to the multivalued potential for φ 2 φ → φ + f , we must have µf = e for consistency. again, axion unwound into multiple • of membranes, leading to the multivalued potential shown in Figurenumber 1. Atntree level, if membrane nucleation If M The quantum can be shifted by the nucleation n=1 n=2 branches, n jumps by flux tunneling, shown in Figure 1. At tree level, if membrane nucle is suppressed, one has a model of chaotic inflation with a port of membranes, leading to the multivalued potential for φ periodicity by summing over branches n=0 shown in potential. Figure 1. AtThe tree fundamental level, quadratic of inflation the is suppressed, one has aif membrane modelperiodicity ofnucleation chaotic &w M If Muvn=3Not 1 2 2 is suppressed, one has a model of chaotic inflation with a portant if scalar implies that corrections to V = µ φ take the erat quadratic potential. The fundamental periodicity o 2 BRX-TH-676 INTRODUCTION The detection of B-mode polarization by the BICEP2 experiment [1, 2] gives tantalizing evidence that quantum gravity may be directly relevant for observational cosmology. If the future checks confirm that the BICEP2 Bmodes are generated by primordial gravitational waves, this will give support to the idea that these waves are induced by quantum fluctuations of the graviton. Furthermore, simple chaotic inflation models such as V = 12 m2 φ2 [3] are in excellent current agreement with the data. To generate the observed density fluctuations in the CMB and large scale structure, chaotic inflation models occur 16 < M4 at energy densities ∼ GeV )4 . Any GU T ∼ (2 × 10 models generating observable primordial gravitational radiation require transplanckian excursions of the inflaton in field space of order ∆φ ∼ 10mpl [4, 5] over the course of inflation.1 Therefore all such models are sensitive to the UV completion at the Planck scale, requiring a good 4 n quadratic potential. The fundamental periodicity of the picture taken from: [Kaloper, Lawrence ‘14] flux axion monodromy with moduli stabilization [Hebecker, Kraus & Witkowski ’14] [McAllister, Silverstein, AW & Wrase ’14] • type IIB string theory: ! d x 10 " $ # # 2 2 |dB| # # 2 2 + |F | + |F | + F̃ # # 1 3 5 gs2 with: • F̃5 = dC4 − B2 ∧ F3 + C2 ∧ H3 + F1 ∧ B2 ∧ B2 ɸ2 , ɸ3, ɸ4 terms … - generically flattening of the potential from adjusting moduli and/or flux rearranging its distribution on its cycle - ‘sloshing’, while preserving flux quantization [Dong, Horn, Silverstein & AW ’10] uct of three two-tori, (T ) (perhaps later we will generalize to higher .1 IIB example on product manifold and complex st aces). For simplicity take them to be rectangular tori, y ≡ y + L , y ≡ 1 1 1 2 flux axion monodromy with moduli stabilization justment 6 = L1 L2 , so the total internal volume V is L . [McAllister, Put 3-form fluxAW & Wrase ’14] Silverstein, 2 et us work (1) in type(2)IIB string theory,(1)including the |F ∧ B ∧ B| term (se 1 (3) (2) (3) (i) 2 (i) 2 2 2 2 F3 =•Q31simple dy1 ∧torus dy1 example: ∧ dy1 + Q32ds dy2= ∧ dy ∧ dy (26) L12(dy ) + L (dy ) 2 2 1 2 2 3 Consider a product of three two-tori, i=1 (T ) (perhaps later we will gen 2 3 For simplicity take them to be rectangular tori, y nus Riemann surfaces). ! 1 pt labels the three T ’s. Thatb is, (i) we have(i)Q31 units of flux on the product axion (i) 2B = 6 dy ∧ dy 1 2 +L L of=flux L1 Lon total internal volume V is L . Put 3-for 2 the les and Q32 units the product of the three y2 cycles. 2 . Denote 2 ,Lso i=1 d 1-form flux in the symmetric (1) configuration (2) (3) (1) (2) (3) F3 = Q31 dy1 ∧ dy1 ∧ dy1 + Q32 dy2 ∧ dy2 ∧ dy2 fluxes 3 ! Q1 (i) 2 F = dy (27) here the superscript1 labels the three T ’s. That is, we have Q31 units of flu 1 L 1 i=1 (i) (i the three y1 cycles and Q32 units of flux on the product of the three y2 L φ b 2 3 u = , = i) Include effective 4d action gives ɸ -potential: • quantized 1-form flux in the symmetric configuration F1 . L1 MP L2 ! $ " # ons 3 4 2 4 2 ! 3 b Q32 Q (i) 1 (i) s ! 2 ḃ 4 g 2 4 2 3 2 3 b L ∼ MP 4 + MP 1 Q(i)1 L (i) u + Q u + + µ φ ∼ φ̇ F = dy 1 31 1 3 (28) LB = L 22 dy1 ∧ dy2L2 u L 1 i=1 L i=1 6 2 2 " use Riemann • (i) surfaces: can fix Vol = L as well & get m ɸ hat is, Q1 = dy1 F1 . 8 3 ! 10 Planck Collaboration: Constraints on inflation Model ΛCDM + tensor Parameter ns r0.002 −2∆ ln Lmax Planck+WP 0.9624 ± 0.0075 < 0.12 0 Planck+WP+lensing 0.9653 ± 0.0069 < 0.13 0 Planck + WP+high-! 0.9600 ± 0.0071 < 0.11 0 0.00 Tensor-to-Scalar Ratio (r0.002) 0.05 0.10 0.15 0.20 0.25 Table 4. Constraints on the primordial perturbation parameters in the ΛCDM+r model from Planck c The constraints are given at the pivot scale k∗ = 0.002 Mpc−1 . Pla Pla Pla Na Po Lo Con vex Con cav e R2 V V V V N∗ 0.94 0.96 0.98 Primordial Tilt (ns ) 1.00 N∗ Fig. 1. Marginalized joint 68% and 95% CL regions for ns and r0.002 from Planck in combination wit the theoretical predictions of selected inflationary models. open questions ... • BICEP2 may provide evidence for primordial tensor modes with r = 0.05 … 0.16 — if so, only large-field inflation survives … • axion monodromy provides one avenue for large field inflation in string theory - technically natural & distinctive predictions ... • many powers … possible ; we need generalizations ... harder look at universality, generic distinctiveness from field theory models & potential backreaction issues in string models 2/3 ɸ 4 ɸ