Realistic Inflation Models

Transcription

1 Realistic Inflation Models Qaisar Shafi Bartol Research Institute Department of Physics and Astronomy University of Delaware Fort Lauderdale, Florida December 19, /51

2 Outline Introduction Standard Model Inflation GUTs and Inflation Supersymmetry, Supergravity and Inflation Radiative Corrections and Precision Cosmology Conclusions 2 /51

3 Hot Big Bang Cosmology (SM+GR) But Hot Big Bang Cosmology cannot explain: High degree of CMB isotropy Origin of δt T Flatness n B /s (baryon asymmetry) Cold Dark Matter (CDM) WMAP Collaboration /51

4 Hot Big Bang Cosmology (SM+GR) In recent years ΛCDM has become the leading candidate for the description of the universe. WMAP Collaboration 2008 Where does ΛCDM come from? 4 /51

5 Inflationary Cosmology [Guth, Linde, Albrecht & Steinhardt, Starobinsky, Mukhanov, Hawking,... ] Successful Primordial Inflation should: Explain flatness, isotropy; Provide origin of δt T ; Offer testable predictions for n s, r, dn s /dln k; Recover Hot Big Bang Cosmology; Explain the observed baryon asymmetry; Offer plausible CDM candidate Physics Beyond the SM? 5 /51

6 Physics Beyond the SM Solar and atmospheric neutrino oscillation experiments require new physics beyond the SM Simplest Extensions 1 Just add 2-3 SM singlet ( right handed ) neutrinos and use seesaw mechanism to understand why m ν m charged matter 2 Gauge U(1) B L global symmetry of SM requires 3 RH-ν s to cancel anomalies Will discuss these two possible extensions of the SM in the context of inflation Gauge hierarchy problem (M W M P ) Electric charge quantization Strong CP problem (axions) 6 /51

7 Beyond the SM Supersymmetry (Gauge hierarchy problem, LSP dark matter) Grand Unification (Charge quantization, proton decay, gauge coupling unification) Technicolor (Complicated models) Large Extra Dimensions (Black holes at LHC?) Warped Extra Dimensions (TeV scale KK excitations) Universal Extra Dimensions (KK dark matter candidate) 7 /51

8 Slow-roll Inflation The slow-roll parameters may be defined as ( ) ǫ = m2 p V 2, ( ) 2 V η = m 2 V p V. The spectral index n s and the tensor to scalar ratio r are given by n s 1 dln 2 R d lnk, r 2 h, 2 R where 2 h and 2 R are the spectra of primordial gravity waves and curvature perturbation respectively. Assuming slow-roll approximation (i.e. (ǫ, η ) 1), the spectral index n s and the tensor to scalar ratio r are given by n s 1 6ǫ + 2η, r 16ǫ. 8 /51

9 Slow-roll Inflation The tensor to scalar ratio r can be related to the energy scale of inflation via V 1/4 = r 1/4 GeV. The amplitude of the curvature perturbation is given by ( 1 R = 2 V 3/2 3πm 3 V = p )φ=φ 5 (WMAP5 normalization) 0 The number of e-folds after the comoving scale l 0 = 2π/k 0 has crossed the horizon is given by N 0 = 1 φ0 ( V ) m 2 p φ e V dφ. Inflation ends when ǫ(φ e ) 1 or η(φ e ) 1. 9 /51

10 Standard Model Inflation? [Bezrukov, Magnin and Shaposhnikov; De Simone, Hertzberg and Wilczek.] [Barvinsky, Kamenshchik, Kiefer, Starobinsky and Steinwachs.] [Clark, Liu, Love and Veldhuis.] Consider the following action with non-minimal coupling: S J = dx 4 ) 2 g{ H 2 λ(h H v m2 P R + ξh H R} In the Einstein frame the potential turns out to be: V E (φ) λ G 4 4 φ4 ( 1+ ξ G2 φ 2 m 2 P ) 2 ; G(t) = exp[ t 0 dt γ(t )/(1 + γ(t ))] where H T = 1 2 (0,v + φ), t = log[ φ M t ] and γ(t) is the anomalous dimension of the Higgs field. For large φ values, V E (φ) gives rise to inflation. 10 /51

11 Standard Model Inflation? m h 134 GeV m h 128 GeV Mt GeV Ψ ΛG 4 ΛG Mt GeV Mt GeV Mt GeV t log Φ Mt t log Φ Mt λ(t) G(t) 4 vs. t, between pivot-scale t 0 = log[ φ 0 M t ] and inflation end scale t e = log[ φe M t ] The spectral index is ( ) n s 1 2 N e + ψ 0 ψ0 (λ G 4 ) 3 ; ψ (λ G 4 ) 0 ξ φ0 m P with 2 R λ ξ 2 N 2 e 72π 2 ( ξ N e 6 2 π R ) λ 10 4 (!) 11 /51

12 Standard Model Inflation? M t GeV M t GeV ns M t GeV m h The horizontal line is the classical tree level result which is independent of Higgs mass. All other curves include quantum corrections with different values of M t. 12 /51

13 Standard Model Inflation? M t GeV r M t GeV M t GeV n s r vs. n s. The black circle corresponds to the classical tree level result. All other curves include quantum corrections with different values of M t. 13 /51

14 Type I Seesaw with One Right Handed Neutrino [N. Okada, M. Rehman and Q. Shafi, 09] y Ν 0, 0.1 ns 0.97 y Ν y Ν 0.3 y Ν m h n s vs. m h in the simplified model (with one right handed neutrino) of type I seesaw for various values of neutrino Dirac coupling y ν, with M R = GeV and M t = GeV. 14 /51

15 Type I Seesaw with Three Right Handed Neutrinos MR GeV MR GeV ns 0.97 ns MR GeV MR GeV MR GeV MR GeV m h m h n s vs. m h for hierarchical (left panel) and inverted-hierarchical (right panel) neutrino mass spectrum for various values of M R and M t = GeV. 15 /51

16 Type I Seesaw with Three Right Handed Neutrinos M R GeV r M R GeV M R GeV n s r vs. n s for hierarchical neutrino mass spectrum for various values of M R and M t = GeV. The black circle corresponds to the classical tree level result. 16 /51

17 Challenges for SM inflation [Burgess, Lee and Trott; Barbon and Espinosa, 09] Consider g µν = η µν + hµν m P, so that the term ξ φ 2 R yields ξ m P φ 2 η µν 2 h µν + This suggests an effective cut-off scale Λ m P ξ m P. The energy scale of inflation is estimated to be E i V 1/4 0 = λ1/4 m P 4 1/4 ξ Λ = m P ξ. Thus it is not clear how reliable are the calculations. 17 /51

18 Non-Supersymmetric GUTs and Inflation [Q. Shafi and A. Vilenkin, 1984] Consider Coleman-Weinberg (CW) and Higgs Potentials for inflation in Non-Supersymmetric GUTs V (φ) = A φ 4 [ln( φ M V (φ) = V 0 [1 ) 1 4 ] + A M4 4 (CW Potential) ( φ M ) 2 ] 2 (Higgs Potential) Here φ is a gauge singlet field. Coleman Weinberg Potential V Φ V Φ Higgs Potential M Φ [Rehman, Shafi and Wickman, 08] [Kallosh and Linde, 07] M Φ 18 /51

19 Non-Supersymmetric GUTs and Inflation [V. N. Senoguz, Q. Shafi, 2007] 1 ns Log 10 V GeV Below vev inflation 19 /51

20 Non-Supersymmetric GUTs and Inflation ns Log 10 V Φ 0 Below vev inflation 20 /51

21 Predictions for CW and Higgs Potentials [M. Rehman, Q. Shafi and J. Wickman, 2008] r CW Potential BV CW Potential AV Higgs Potential BV Higgs Potential AV Φ Φ Φ 2 1 Φ n s 21 /51

22 Proton Decay and WMAP CW Potential BV CW Potential AV Higgs Potential BV Higgs Potential AV Φ 2 ns 0.94 Φ Φ 4 1 Φ log 10 V Φ GeV M X 2-4V 1/ GeV (CWP) V 1/4 0 (GeV) V (φ 0 ) 1/4 (GeV) n s r τ p years (proton lifetime) 22 /51

23 Why Supersymmetry? Resolution of the gauge hierarchy problem Unification of the SM gauge couplings at M GUT GeV Cold dark matter candidate (LSP) Other good reasons: Radiative electroweak breaking String theory requires susy Leading candidate is the MSSM (Minimal Supersymmetric Standard Model) 23 /51

24 Why Supersymmetry? Α 1 1 MSSM Αi Α 2 1 Α Log 10 GeV 24 /51

25 Susy Hybrid Inflation [Dvali, Shafi, Schaefer; Copeland, Liddle, Lyth, Stewart, Wands 94] [V. N. Senoguz, Q. Shafi 04; A. D. Linde, A. Riotto 97] Attractive scenario in which inflation can be associated with symmetry breaking G H SUSY hybrid inflation is defined by the superpotential inflaton (singlet) = S, R-symmetry W = κs (Φ Φ M 2 ) waterfall field = (Φ,Φ) G Φ Φ Φ Φ, S e iα S, W e iα W W is unique renormalizable superpotential Some examples of gauge groups: G = U(1) B L, (Supersymmetric superconductor) G = 3 c 2 L 2 R 1 B L, (Φ = (1, 1, 2, +1)) G = 4 c 2 L 2 R, (Φ = (4, 1, 2)) G = SU(5) U(1), (Φ = 10), G = SO(10), (Φ = 16) 25 /51

26 Susy Hybrid Inflation SUSY vacua Φ = Φ = M, S = 0 Potential V F = κ 2 (M 2 Φ 2 ) 2 + 2κ 2 S 2 Φ 2 S M V Κ 2 M M 1 26 /51

27 Susy Hybrid Inflation Mass splitting in Φ Φ m 2 ± = κ2 S 2 ± κ 2 M 2, m 2 F = κ2 S 2 One-loop radiative corrections V 1loop = 1 64π 2 Str[M 4 (S)(ln M2 (S) Q )] In the inflationary valley (Φ = 0) ) V κ 2 M (1 4 + κ2 N 8π F(x) 2 where x = S /M and ( (x F(x) = ) ) 1) ln (x4 x + 2x 2 ln x x ln κ2 M 2 x 2 Q /51

28 Susy Hybrid Inflation Tree Level plus radiative corrections: ns log 10 Κ log 10 r ns n s 1 ( m p ) ( ) 2 κ 2 N M F (x 8π 2 0 ) r 4(1 n s ) F (x 0 ) F (x 0 ) x N /51

29 Minimal Sugra Hybrid Inflation The minimal Kähler potential can be expanded as K = S 2 + Φ 2 + Φ 2 The SUGRA scalar potential is given by V F = e K/m2 p where we have defined ( K 1 ij D z i WD z j W 3m 2 p W 2) D zi W W z i + m 2 p K z i W; K ij and z i {Φ,Φ,S,...} 2 K z i z j 29 /51

30 Minimal Sugra Hybrid Inflation [V. N. Senoguz, Q. Shafi 04; A. D. Linde, A. Riotto 97] Take into account SUGRA corrections, radiative corrections and soft SUSY breaking terms, the potential is given by V κ 2 M 4 (1 + ( M m p ) 4 x κ2 N 8π 2 F(x) + a ( ) m3/2 x κm + ( ) ) 2 MS x κm where a = 2 2 A cos[arg S + arg(2 A)] and S m P. Note: No η problem with minimal Kähler potential 30 /51

31 Minimal Sugra Hybrid Inflation [M. Rehman, Q. Shafi, J. Wickman, 09] 1.02 a 1 ns a 0 a log 10 Κ ( ) 2 n s M mp x ( [ m P ) ( ) ] 2 2 M κ2 N 8π F (x 2 0 ) + 2 MS κ M 31 /51

32 Minimal Sugra Hybrid Inflation 17.0 log 10 M GeV a 1 a 0 a log 10 Κ 32 /51

33 Minimal Sugra Hybrid Inflation 17.0 a 1 log M GeV a 1 a n s 33 /51

34 Minimal Sugra Hybrid Inflation log 10 r a 1 a r 4 ( [ m P ) ( ) 2 4 M 2 M mp x κ2 N F (x 8π 2 0 ) + am 3/2 κm n s ( ) ] MS κm x0 34 /51

35 Minimal Sugra Hybrid Inflation 16.5 ns 0.967, m3 2 1TeV, 10, MS ns 0.967, m3 2 1TeV, 10, MS 2 0 log 10 M GeV a 1 a 1 a 0 log 10 Κ a 1 a 0 a log 10 MS GeV log 10 MS GeV log 10 (M/GeV) and log 10 (κ) vs. log 10 ( MS /GeV) in the flipped SU(5) model (N = 10), with n s fixed at the central value for r /51

36 Non-Minimal Sugra Hybrid Inflation [M. Bastero-Gil, S. F. King, Q. Shafi; M. Rehman, V. N. Senoguz, Q. Shafi 06] The Kähler potential including non-minimal terms can be expanded as K = S 2 + Φ 2 + Φ 2 S +κ 4 S S + κ 2 Φ 2 S 4m 2 Sφ + κ 2 Φ 2 S p m 2 p Sφ + κ 6 m 2 SS +. p 6m 4 p The potential is of the following form ( ( ) 2 ( ) 4 V κ 2 M 4 1 κ M S m p x 2 + γ M x 4 S m p ( ) ( ) ) m3/2 x 2 +a κm + MS x κm, where γ S = 1 7κ S 2 + 2κ 2 S. 2 + κ2 N 8π 2 F(x) 36 /51

37 Non-Minimal Sugra Hybrid Inflation n s s = 0 s = s = s = 0.01 s = ( ) 2 n S 1 2κ S + 6γ M S m p x 2 0 ( m p ) ( ) 2 κ 2 N M 8π F (x 2 0 ) 37 /51

38 MSSM µ Problem In MSSM, the term µh u H d is both supersymmetric and also gauge invariant, and we require that µ GeV in order to implement radiative EW breaking µ, in principle, could be of order M GUT /M P? In SUSY hybrid inflation models, the R-symmetry forbids the bare µ term. However, λh u H d S is allowed, λ a dimensionless coefficient. After SUSY breaking, S m 3/2 /κ, so that µ m 3/2 λ/κ. 38 /51

39 r Chaotic Inflation (Linde) 0.4 Driven by potentials m 2 φ 2 or λφ 4 Where does φ come from? One promising candidate is right handed sneutrino: m2 Φ 2 ΛΦ ns W N mñ2 V m 2 N 2 ; m GeV Komatsu et al, WMAP Collaboration 2008 Challenge: Does this survive SUGRA corrections? (since N > M P during inflation) Answer: With N > M P during inflation, SUGRA corrections overwhelm this scenario: W = mφ 2 large vev of W during inflation potential negative/no inflation 39 /51

40 Chaotic Inflation Shift symmetry imposed on the inflaton (removes the nasty expotential factor in the inflaton potential) K(Φ,Φ ) invariant for Φ Φ + icm, so K is a function of Φ + Φ, i.e., K(Φ + Φ ) However, with W = m Φ 2, chaotic inflation will not take place due to the large vev of W. To cure this introduce a field X with W = m X Φ. During inflation, X is assumed to be stablized at the origin, so that W vanishes. (Kawasaki, Yamaguchi, Yanagida) Why is W = m X Φ the only relevant term? 40 /51

41 Chaotic Inflation Gravitino Problem If Φ transforms under a shift symmetry, the following term is allowed: K = c(φ + Φ ), with c O(1) in Planck units The inflaton Φ decays into susy breaking sector as well as the MSSM sector, leading to thermal and non-thermal gravitino overproduction. One possible solution: Introduce a discrete symmetry which suppresses/eliminates the linear term. 41 /51

42 Chaotic Inflation If Φ is RH sneutrino, it is charged under the shift symmetry. It does not have a Majorana mass, so it is not clear how leptogenesis/ν oscillations work out. 42 /51

43 Radiative Corrections and Precision Cosmology Chaotic: potential is quadratic (V m 2 φ 2 ) or quartic (V λφ 4 ) Simplest (renormalizable) potential Predicts significant primordial gravity waves Hybrid: inflation driven by φ field, vacuum energy provided by second scalar field χ ( ) 2 V (φ,χ) = κ 2 M 2 χ2 4 + m 2 φ λ2 χ 2 φ 2 4 Compelling and robust Translates well to SUSY Non-SUSY model results in n s > 1 (unless in chaotic-like regime) SUSY case has n s < 1 43 /51

44 Tree Level Inflation Models vs. WMAP ΛΦ Φ 0 1 Φ 0 1 r m2 Φ 2 r Not Allowed Flat Potential Φ n s n s Komatsu et. al. (WMAP Collaboration), /51

45 Radiative Corrections in Chaotic Inflation Quadratic: V (φ) = 1 2 m2 φ 2 ± Aφ 4 ln φ2 µ 2 Quartic: V (φ) = λφ 4 ± Aφ 4 ln φ2 µ 2 Fermion-dominated corrections ( ) (say from right handed neutrinos) V Φ Φ 45 /51

46 Radiative Corrections in Chaotic Inflation Results [V. N. Senoguz and Q. Shafi, 2008] Tensor to scalar ratio r vs. the spectral index n s for the potential V = (1/2)m 2 φ 2 κφ 4 ln(φ/m P ) (solid curve) and for the potential V = (1/4!)λφ 4 κφ 4 ln(φ/m P ) (dashed curve). The points on the curves correspond to the tree level predictions for φ 2 and φ 4 potentials. 46 /51

47 Radiative Corrections in Non-Higgs Inflation At its tree level, Non-Higgs potential (V NHP ) can be written as V NHP (φ) = 1 2 m2 φ 2 + λ 4! φ4. Including one loop corrections it modifies as V RCNHP (φ) = 1 2 m2 φ 2 + λ 4! φ4 ± κφ 4 ln(φ/m P ). 47 /51

48 Radiative Corrections in Non-Higgs Inflation Q2P Κ 1 NHP r 0.10 RCQ4P Κ RCQ2P n s r vs. n s for quadratic (Q2P), radiatively-corrected quadratic (RCQ2P), radiatively-corrected quartic (RCQ4P), Non-Higgs (NHP) and radiatively corrected Non-Higgs (RCNHP) potentials. Here κ 1 = and κ 2 = and the small and big black circles correspond to Q2P with N 0 = 50 and N 0 = 60 respectively. 48 /51

49 Radiative Corrections & Precision Cosmology Radiatively-corrected inflation models; Chaotic Inflation Permits low values of r (e.g. 0 r 0.2 to agree with WMAP5 2σ bound) Hybrid Inflation Red spectrum (n s < 1) can be produced for Planckian or sub-planckian values of the inflaton Better agreement with WMAP5 Generically 2 solutions of n s for each A 49 /51

50 Conclusions One of the most important challenges in our field is to find a Standard Model of Inflationary Cosmology. A large class of realistic inflation models, based on grand unification and/or low scale supersymmetry are consistent with the data currently available from WMAP and other observations. Non-supersymmetric GUT inflation models typically predict an observable value for the tensor to scalar ratio r. Supersymmetric hybrid inflation, on the other hand, predicts a tiny r value ( 10 4 ), even though the symmetry breaking scale associated with inflation is comparable to M GUT. 50 /51

51 Conclusions Hopefully, the PLANCK Satellite will soon shed light on this fundamental parameter r. In addition, we expect the LHC to provide answers to the following long-standing and very basic questions: Is there a SM Higgs boson and what is its mass? Is there low scale supersymmetry? Is the LSP stable? Mass? To achieve truly significant progress in particle physics and cosmology we need both the LHC and the PLANCK Satellite. They hold the key to future developments in these fields. 51 /51