Towards Matter Inflation in Heterotic Compactifications

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1 Towards in Heterotic Compactifications Max-Planck-Institut für Physik (Werner-Heisenberg-Institut) Cornell University September 2, 2011 Based on work with S. Antusch, K. Dutta, J. Erdmenger arxiv & work in progress

2 Inflation η-problem F-term Hybrid Inflation Inflationary paradigm Inflation = Period of exponential expansion in very early universe. Guth 81; Linde 82; Albrecht, Steinhardt 82 Inflation is a successful paradigm which solves the flatness & horizon problem (T 2.7K) provides a seed for structure formation ( δt T 10 5 ) WMAP 7 year full sky temperature map Towards in Heterotic Compactifications

3 Inflation η-problem F-term Hybrid Inflation Slow-roll inflation Standard realization: slowly rolling scalar field φ ds 2 dt 2 + a(t) 2 d x 2, a(t) e Ht, H V (φ) must satisfy ɛ MP 2 V 2 V 2 1 & η V M2 P V m2 φ H 2 1 V (φ) 3M 2 P Towards in Heterotic Compactifications

4 Inflation η-problem F-term Hybrid Inflation η-problem Slow-roll inflation sensitive to Planck-scale physics: δv = c O 4 φ 2 M 2 P & O 4 V η c e.g. F-term inflation in supergravity: ) V F = e K/M2 P (K i j W 2 D i W D j W 3 MP 2, D i W = W i + K iw M 2 P with K = Φ 2 + X & only W X 0 Copeland, Liddle, Lyth, Stewart, Wands 94; Dine, Randall, Thomas 95 ( ) V F = W X φ 2 MP η 1 Solution: fine-tune against dots or impose symmetry Towards in Heterotic Compactifications

5 Inflation η-problem F-term Hybrid Inflation Example: F-term Hybrid Inflation Minimal W & K: W = κ Φ ( H 2 M 2), K = Φ 2 + H 2 Tree-level: φ 2 -term in V F cancels accidentally Copeland, Liddle, Lyth, Stewart, Wands 94 1-loop: slope from Coleman-Weinberg potential Pro: works with field values M P Con: implicit fine tuning of e.g. δk = c Φ 4 MP 2 Towards in Heterotic Compactifications

6 Superpotential Kähler Potential Superpotential Requirements: 1 Solve η-problem by special form of K during inflation W should fulfill Stewart 95 W W Φ 0, W X 0 2 Inflation ends via hybrid mechanism Linde 93; Dvali, Shafi, Schaefer 94 at φ = φ cr a tachyonic direction appears Minimal form of W : Arkani-Hamed, Cheng, Creminelli, Randall 03; Antusch, Dutta, Kostka 09 During inflation: X H 0 W = κ X (H 2 M 2 ) + λ f (Φ) H 2 Towards in Heterotic Compactifications

7 Superpotential Kähler Potential Kähler Potential Usual choice: shift symmetry e.g. Kawasaki, Yamaguchi, Yanagida 00; Arkani-Hamed, Cheng, Creminelli, Randall 03 Φ Φ + iα Alternative: Heisenberg symmetry Binetruy, Gaillard 87; Ellwanger, Schmidt 87; Gaillard, Murayama, Olive 95; Gaillard, Lyth, Murayama 98 T T + iα Φ Φ + β T T + β Φ β 2 Invariant combination: ρ T + T Φ 2 e.g. K = 3 ln ρ + k(ρ) X Towards in Heterotic Compactifications

8 Superpotential Kähler Potential Why? Why is it interesting to have the inflaton in the matter sector? Direct link between particle physics & inflation Hybrid phase transition and GUT breaking? Typically H M M GUT Inflaton in visible sector, e.g. right-handed sneutrino? Relate inflation to leptogenesis Extra constraints on inflaton potential from particle physics Minimally coupled SM Higgs excluded by EWSB vs. CMB Towards in Heterotic Compactifications

9 Superpotential Kähler Potential Matter Fields as Inflatons Heisenberg symmetry & structure of W Gauge non-singlet matter field as inflaton Antusch, Bastero-Gil, Baumann, Dutta, King, Kostka 10 W = κ X (HH c M 2 ) + λ Λ ΦΦc HH c K = 3 ln ρ + X 2 (1 βρ γ X 2 ) + H 2 + H c 2 ρ T + T Φ 2 Φ c 2 D-flat trajectory: Φ, Φ c 0, H H c X 0 Example(s): sneutrino inflation Φ = ν R, Φ c = ν c R in SU(4) c SU(2) L SU(2) R Pati-Salam model SO(10) GUT model Towards in Heterotic Compactifications

10 Superpotential Kähler Potential 1-Loop Corrections So far: inflaton potential flat at tree-level Slope provided at 1-loop by Coleman-Weinberg potential V 1-loop (φ) = 1 ( ( ) M(φ) [M(φ) 64π 2 STr 4 2 ln Q 2 3 )] 2 Contributions from 2 sectors: Waterfall sector ms 2 λ2 φ 4 ± κ 2 M 2, m 2 Λ 2 f λ2 Gauge sector Inflaton singlet under unbroken gauge group m A gφ Direct SUGRA gaugino masses (G = K + ln W 2 ) Λ 2 φ 4 mλ,ab 2 e G G i (G 1 ) i j f ab f ab Φ j ek W X X + O(W, X, W i X ) gauge sector contribution negligible if fab 0 X Towards in Heterotic Compactifications

11 Superpotential Kähler Potential 2-Loop Corrections If W κ X (HH c M 2 ) + mh 2 HHc & φ gauge non-singlet δm 2 g 4 W X 2 H 2 since H 2 W (4π) 4 mh 2 X 2 MP 2 Dvali 95 Here however φ charged only under massive gauge bosons extra factors of m A universal suppression of δm2 κ by 2 H 2 (4π) 4 Examples for contributing 2-loop diagrams H c, Hc H c, Hc A µ A µ A µ A µ A µ A µ φ φ φ φ φ φ δν + H H c, H c Towards in Heterotic Compactifications

12 Superpotential Kähler Potential Some Generalization Generalization of previous model: Antusch, Dutta, Erdmenger, Halter 11 W = a(t i ) X ( b(t i ) HH c Σ 2 ) + c(t i ) f (Φ 3,α ) HH c +... ( 3 2 ) K = ln ρ i + X 2 ( 1 + d(ρ 3 ) γ X 2) + i=1 ( 3 i=1 ρ q i,h i ) i=1 H 2 + ρ q i,x i ( 3 i=1 ρ q i,h c i with ρ i T i + T i α Φ i,α 2 and 0 q i,α < 1 ) H c During inflation: X H H c Φ 1,α Φ 2,α 0 Towards in Heterotic Compactifications

13 Superpotential Kähler Potential Some Comments f (Φ 3,α ) = gauge invariant (D-flat) product of Φ s e.g. N N of SU(N) or of E 6 etc. Need K γ X 4 to ensure m X H Kawasaki, Yamaguchi, Yanagida 00 Geometric interpretation: Covi, Gomez-Reino, Gross, Louis, Palma, Scrucca 08 If X 0, W X 0 R X X X X < 0 necessary for de Sitter vacua Non-canonical kinetic terms (T + T Φ 2 ) q Moduli-dependent superpotential couplings e at W X Σ 2 with Σ generated dynamically Σ carries moduli dependence Towards in Heterotic Compactifications

14 Superpotential Kähler Potential Moduli-dependent D-term VEVs A D-term example: D a = ξ + α Q a,α K ψ α ψ α = ξ + α Q a,α ( 3 i=1 ρ q i,α i ) ψ α 2 Simplest solution to D a = 0: ψ 2 ξ Q a ( 3 i=1 ρ q i i ) with Q a ξ < 0 More general solution to D a = 0: ψ n β β 0 with β β n β Q a,β ξ < 0 Relative size fixed e.g. by further D-term equations Towards in Heterotic Compactifications

15 Superpotential Kähler Potential Moduli-dependent F-term VEVs An F-term example: W e at X χφψ + Y φ ψ (1 + e bt χφ ψ ) consider φ, φ, ψ, ψ 0, X, Y 0 χ ebt φ ψ Parametrize moduli dependence of Σ Σ 3 i=1 ρ p i i e b i T i Towards in Heterotic Compactifications

16 Superpotential Kähler Potential Moduli Stabilization Moduli stabilized for suitable choice of a(t i ), Σ and d(ρ 3 ) e.g. a(t i ) Σ 2 M 2 e a 1T 1 +a 2 T 2, d(ρ 3 ) βρ 3 M 4 e a 1T 1 +a 2 T 2 2 V (T 1 + T 1 ) n 1 (T2 + T 2 ) n 2 ρ n 3 3 (1 + d(ρ 3)) Re T 1,2 O(1) and ρ 3 β 1 with masses H # V V! # Re T i! "! #! $! ρ 3 Towards in Heterotic Compactifications

17 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Heterotic Orbifolds Heterotic string theory = theory of closed strings Contains gauge group SO(32) or E 8 E 8 Orbifolds are toy models of Calabi-Yau compactifications: obtained as T 6 /Z N Strings on oribfolds can be untwisted closed in T 6 and T 6 /Z N twisted closed only in T 6 /Z N Corresponds to fields living in 10D bulk full orbifold untwisted 4D brane fixed point twisted 6D brane fixed torus twisted Towards in Heterotic Compactifications

18 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Heterotic Orbifolds MSSM-like models exist, e.g. heterotic mini-landscape based on T 6 /Z 6 or T 6 /(Z 2 Z 2 ) Buchmüller, Hamaguchi, Lebedev, Ratz 05-06; Lebedev, Nilles, Raby, Ramos-Sanchez, Ratz, Vaudrevange, Wingerter 06-08; Blaszczyk, Groot-Nibbelink, Ratz, Rühle, Trapletti, Vaudrevange 09 viable models contain anomalous U(1) a Identification of field content could be T i 3 universal untwisted Kähler moduli Φ i α associated untwisted matter fields X twisted sector field Technical challenge: twisted matter field VEVs (partial) blow-up of orbifold singularities how to compute reliably? Towards in Heterotic Compactifications

19 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Heisenberg Symmetry Heisenberg symmetry at tree-level for untwisted matter fields T i R 2 i + ib i Φ i,α 0 T i R 2 i + ib i + Φ i,α 2 10D picture g 0 limit: for λ a M harmonic Ellwanger, Schmidt 87 A a M Aa M + λa M B MN B MN A a [M λa N] Limit g 0 limit W 0 Worldsheet picture Accidental symmetry for A a M 1 Cveti c, Molera, Ovrut 89 Towards in Heterotic Compactifications

20 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Target Space Modular Invariance Low-energy effective action constraint by modular invariance T at + ib ict + d, ab cd = 1, a, b, c, d Z Kähler potential transforms as ( ) T + T K = ln(t + T ) ln (ict + d)( ic T + d) Invariance of supergravity action superpotential also transforms W (ict + d) 1 W One such symmetry for each T i Towards in Heterotic Compactifications

21 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Target Space Modular Invariance Matter fields also transform Φ α 3 (ic i T i + d i ) q i,α Φ α i=1 with modular weights q i,α determined by action of Z N on T 6 Generic superpotential term with matter fields ( 3 ) η(t i ) 2σ i α Φ nα α i=1 σ i = 1 + α n αq i,α & η(t ) = e πt 12 ( n 1 e 2πnT ) e πt 12 all matter couplings in W are 1 + e at +... or e at +... Towards in Heterotic Compactifications

22 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Dilaton Stabilization Dilaton S additional modulus Non-perturbative corrections to K Kähler stabilization (g 2 S + S) Shenker 90; Banks, Dine 94; Casas 96; Binetruy, Gaillard, Wu 96 & 97; Gaillard, Lyth, Murayama 98 K np (A 0 + A 1 g )e B g from worldsheet instanton corrections Anomalous U(1) a FI-parameter ξ (S + S) 1 moduli dependence of VEVs? Gaillard, Lyth, Murayama 98 D-term driven VEVs ψ 2 (S + S) 1 destabilizing F-term induced VEVs χ (S + S) p stabilizing Towards in Heterotic Compactifications

23 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Kähler Moduli Stabilization T 1 & T 2 stabilized if Copeland, Liddle, Lyth, Stewart, Wands 94 W X η(t 1 ) p 1 η(t 2 ) p 2 e a 1T 1 +a 2 T 2 Ideally: T 3 & Φ 3,α enter V only through ρ 3 R 2 3 Avoid superpotential stabilization W T3 W Φ3,α 0 Alternatives: α -corrections? Candelas, De La Ossa, Green, Parkes 91 ln V ln(v + ξ), ξ χ = 2 (h 1,1 h 2,1 ) Moduli-dependent threshold corrections to K X X? Antoniadis, Gava, Narain, Taylor 92 Φ α = 0 ln η(t ) 4 (T + T ) ln(t + T ) π 6 (T + T )+O(e 2πT ) Towards in Heterotic Compactifications

24 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Moduli Stabilization with Threshold Corrections In principle: can stabilize T 1, T 2, ρ 3 and l 1/(S + S) But: requires some tuning of parameters!'!& V!'!(! V!'!(!'!&! " # $% $& ρ3 l V V Re T1 Im T1 Towards in Heterotic Compactifications

25 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Slope for Inflaton Various sources for inflaton slope Loop corrections from waterfall & gauge sector Gaugino condensate W A(T i )e cs Corrections from W 0, W i X 0 & X 0 Corrections from D 0 Threshold corrections not only depending on ρ 3 α -corrections Requires systematic study to check if η 1 (in progress) Additionally: Corrections from complex structure stabilization? What if fluxes are turned on? Towards in Heterotic Compactifications

26 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Moduli Stabilization after Inflation Moduli stabilizing mechanism changes During inflation: moduli stabilization tied to W X 0 After waterfall phase transition: W X 0 Need extra terms e.g. gaugino condensate W A(T i )e cs Possible issues 1 Cosmological moduli problem? 2 Overshooting problem? 3 Reheating through moduli decays? Preliminary results Simulate field evolution for toy model with only T, Φ, H, X To evade 1 & 2 seems to require fine tuning Typically m T m Φ,H,X Reheating through moduli decays Towards in Heterotic Compactifications

27 Heterotic Orbifolds & Heisenberg Symmetry Superpotential & Target Space Modular Invariance Moduli Stabilization & Inflaton Slope Conclusion Matter inflation is phenomenologically interesting needs certain structure of K & W to work seems suitable to embed in heterotic compactifications stabilizes moduli differently during & after inflation Open issues Explicit realization in heterotic orbifolds? Better understanding of moduli stabilization other ways to stabilize dilaton? Reheating & moduli stabilization after inflation? Towards in Heterotic Compactifications