Realistic Inflation Models and Primordial Gravity Waves

Transcription

1 Journal of Physics: Conference Series Realistic Inflation Models and Primordial Gravity Waves To cite this article: Qaisar Shafi 2010 J. Phys.: Conf. Ser Related content - Low-scale supersymmetry from inflation Kazunori Nakayama and Fuminobu Takahashi - Gauge non-singlet (GNS) inflation in SUSY GUTs Jochen P Baumann - Right-Handed Sneutrinos as Thermal Dark Matter Frank Deppisch and Apostolos Pilaftsis View the article online for updates and enhancements. This content was downloaded from IP address on 15/12/2018 at 02:27

2 Realistic Inflation Models and Primordial Gravity Waves Qaisar Shafi Bartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA Abstract. We consider a variety of realistic inflationary potentials and discuss their predictions for the tensor to scalar ratio r, a canonical measure of gravity waves generated during inflation. For a Standard Model gauge singlet inflaton with a Higgs-like potential, for example, we find that r 0.02, which may be observed by the Planck satellite. We also identify supersymmetric hybrid inflation models where r can be as high as 0.03, again within the reach of Planck. The scalar spectral index in these models lies within the WMAP one sigma bounds. Introduction Successful primordial inflation should: Explain the observed flatness and high degree of isotropy; Provide origin of δt T ; Offer testable predictions for n s, r, dn s /dlnk; Recover Hot Big Bang Cosmology; Explain the observed baryon asymmetry; Offer plausible CDM candidate; Need physics beyond the Standard Model (SM)? Inflation is usually driven by some potential V(φ) and the relevant slow-roll parameters are defined as ( ) ǫ = m2 p V 2, ( ) 2 V η = m 2 V p V. 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. c 2010 Ltd 1

3 The tensor to scalar ratio r is related to the energy scale of inflation by V(φ 0 ) 1/4 = r 1/4 GeV. The amplitude of the curvature perturbation is given by ( 2 R = 1 V/m 4) p 24π 2 ǫ = (WMAP7 normalization). φ=φ 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 max[ǫ(φ e ), η(φ e ) ] = 1. Higgs Inflation Consider the following Higgs Potential: V (φ) = V 0 [1 ( φ M) 2 ] 2. (tree level) Here φ is a gauge singlet field. As discussed in Refs. [1, 2], slow roll inflation may occur from V Φ Below vev BV inflation Above vev AV inflation M Φ Figure 1. The Higgs potential. either above or below the VEV M. For shorthand, we henceforth denote these regimes as BV (below VEV) and AV (above VEV) solutions. As shown in Fig. 2, WMAP data favors BV inflation, whereas, the AV branch is found to lie outside of the 1-σ bounds in Ref. [2]. Consider next the following interaction of inflaton φ with some GUT symmetry breaking scalar boson Φ: L int = λ2 Φ 2 φ2 Φ 2. (1) Including radiative corrections (quantum smearing) we have ( m 2 M 2 ) [ ( ) ] φ 2 2 [ ( ) φ V(φ) 1 +Aφ 4 ln 1 ]+ AM4, (2) 4 M M 4 4 where V(φ = 0) V 0 = m2 M AM4 4 and A = Nλ4 Φ. Note that we can use Minkowski space 32π 2 CW corrections provided the propagating fields have masses H (Hubble constant). 2

4 0.15 QP Φ 2 limit 0.10 r 0.05 N 0 50 N 0 55 N Figure 2. r vs. n s for tree level Higgs inflation, shown together with the WMAP 1-σ (68% confidence level) bounds [3]. The blue dotted and blue dashed curves correspond to number of e-foldings N 0 = 50, N 0 = 60, respectively. Small (big) green circles correspond to the quadratic potential (QP) with N 0 = 50 (N 0 = 60). n s A A A CWP N 0 55 QP Φ 2 limit r n s A 0 CWP N 0 60 Figure 3. r vs. n s for the tree level Higgs potential (A = 0) and the radiatively corrected Higgs potential (A = , , ). Blue and red curves represent the predictions of Coleman-Weinberg Potential (CWP) with the number of e-foldings N 0 = 55 and N 0 = 60, respectively. Small (big) green circles correspond to quadratic potential (QP) with N 0 = 50 (N 0 = 60). The predictions of the radiatively corrected Higgs potential are shown in Fig. 3 and have been discussed in Ref. [4]. The parameter A here quantifies the radiative corrections. The CWP prediction represents the maximal smearing of the radiatively corrected Higgs inflation results. Note that r 0.02 if n s Thus, Planck will test Higgs inflation soon! 3

5 60 50 Α 1 1 MSSM Αi Α 2 1 Α Log 10 GeV Figure 4. The gauge coupling unification in MSSM. Susy Hybrid Inflation Some attractive features of supersymmetry (susy) are: Resolution of the gauge hierarchy problem Unification of the SM gauge couplings at M GUT GeV (see Fig. 4) Cold dark matter candidate (LSP), typically a neutralino. Other good reasons: Radiative electroweak breaking String theory requires susy Leading candidate is the MSSM (Minimal Supersymmetric Standard Model). Thesusy hybridmodel [5, 6, 7, 8] is an attractive scenario in which inflation can be associated with symmetry breaking G H Simplest inflation model is based on W = κs(φφ M 2 ). S = gauge singlet superfield, (Φ,Φ) belong to suitable representation of G. We need Φ,Φ pair in order to preserve susy while breaking G H at scale M TeV, susy breaking scale. R-symmetry ΦΦ ΦΦ, S e iα S, W e iα W 4

6 S M V Κ 2 M M 1 Figure 5. The supersymmetric hybrid potential. W is a unique renormalizable superpotential. Some examples of gauge groups: G = U(1) B L, (Supersymmetric superconductor) G = SU(5) U(1), (Φ = 10-plet of SU(5)), (Flipped SU(5)) 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 = SO(10), (Φ = 16) The susy hybrid tree level potential is V F = κ 2 (M 2 Φ 2 ) 2 +2κ 2 S 2 Φ 2, with susy vacua Φ = Φ = M, S = 0. Take into account radiative corrections (because during inflation V 0 and susy is broken by F S = κm 2 ): Mass splitting in Φ Φ One-loop radiative corrections m 2 ± = κ2 S 2 ±κ 2 M 2, m 2 F = κ2 S 2 V 1loop = 1 64π 2 Str[M 4 (S)(ln M2 (S) Q )] 5

7 In the inflationary valley (Φ = 0) ) V κ 2 M (1+ 4 κ2 NF(x) 8π 2 where x = S /M and F(x) = 1 4 Tree Level plus radiative corrections: ( (x 4 +1 ) ) ln (x4 1) x +2x 2 ln x x ln κ2 M 2 x 2 Q 3. 2 ns log 10 Κ log 10 r n s δt/t (M/M P ) attractive scenario (M M G ) n s 1 1 N The minimal Kähler potential can be expanded as The sugra scalar potential is given by where we have defined and z i {Φ,Φ,S,...}. V F = e K/m2 p D zi W W K = S 2 + Φ 2 + Φ 2. ( K 1 ij D z i WD z j W 3m 2 p W 2), z i +m 2 p K z i W; K ij 2 K z i z j Take into account sugra corrections, radiative corrections and soft susy breaking terms [8]:, ( ) M 4 V κ 2 M (1+ 4 x 4 m p 2 + κ2 N ( m3/2 8π 2 F(x)+a x) + κm ) ( m3/2 x) 2 κm (3) where a = 2 2 A cos[args +arg(2 A)], x = S /M and S m P. Note: No η problem with minimal (canonical) Kähler potential! 6

8 ns ns a log 10 Κ log 10 Κ log 10 r a n s r 10 4 within 2-σ bounds of WMAP data (see [10]) Now consider flipped SU(5) SU(5) U(1) X Chiral superfields are arranged as ( ) ( d c Q u c 10 1 = ν c, 5 3 = L ), 1 5 = e c Compared to standard SU(5), these multiplets correspond to the interchange u c d c, e c ν c Flipped SU(5) (Minimal) SU(5) (Minimal) Low scale susy Yes Yes Doublet-triplet splitting Yes Fine tuning Dimension 5 proton decay Eliminated! Challenging µ problem No Fine tuning Inflation Easy Difficult Dimension 6 proton decay τ p yrs τ p yrs Monopole problem No Yes Seesaw mechanism Automatic No Charge quantization No Yes Unification of gauge couplings Can be arranged Yes CDM Yes Yes Consider susy hybrid inflation in flipped SU(5), where Φ is a 10-plet 7

9 Allowing the soft mass squared to vary, the potential appears as ( ) 4 V κ 2 M (1+ 4 M x 4 mp 2 + κ2 NF(x)+a 8π 2 ( ) m3/2 x κm + ( ) ) 2 MS x κm If MS 2 < 0 the soft susy breaking mass squared term drives the spectral index toward red-tilted values The minimal model consistent with n s = leads to proton lifetime predictions of order years (see Ref. [11]) 16.5 n s 0.967, m 3 2 1TeV, 10, M S n s 0.967, m 3 2 1TeV, 10, M S 2 0 log 10 M GeV a 0 log 10 Κ a log 10 M S GeV log 10 M S GeV Figure 6. 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 (see Ref. [11]). The minimal susy hybrid inflation model yields r values 10 4 ; A more general analysis with a non-minimal Kähler potential can lead to larger r-values (see Fig. 7 and Ref. [12]) (Planck) r ns Figure 7. r vs n s for susy hybrid model (see Ref. [12]). 8

10 Summary The predictions of r (primordial gravity waves) for various models of inflation are as follows: Gauge Singlet Higgs Inflation: Susy Higgs (Hybrid) Inflation: r 10 4 (minimal), Results from PLANCK are eagerly awaited! Acknowledgments r 0.02 for n s 0.96 r 0.03 (non-minimal) I would like to thank Mansoor Ur Rehman for his help in preparing this talk. The work reported here is supported in part by the Department of Energy Grant No. DE-FG02-91ER References [1] R. Kallosh and A. D. Linde, JCAP 0704, 017 (2007) [arxiv: [hep-th]]. [2] M. U. Rehman, Q. Shafi and J. R. Wickman, Phys. Rev. D 78, (2008) [arxiv: [hep-ph]]. [3] E. Komatsu et al., arxiv: [astro-ph.co]. [4] M. U. Rehman and Q. Shafi, Phys. Rev. D 81, (2010) [arxiv: [astro-ph.co]]. [5] G. R. Dvali, Q. Shafi and R. K. Schaefer, Phys. Rev. Lett. 73, 1886 (1994) [arxiv:hep-ph/ ]. [6] E. J. Copeland, A. R. Liddle, D. H. Lyth, E. D. Stewart and D. Wands, Phys. Rev. D 49, 6410 (1994) [arxiv:astro-ph/ ]. [7] A. D. Linde and A. Riotto, Phys. Rev. D 56, 1841 (1997) [arxiv:hep-ph/ ]. [8] V. N. Senoguz and Q. Shafi, Phys. Rev. D 71, (2005) [arxiv:hep-ph/ ]. [9] M. ur Rehman, V. N. Senoguz and Q. Shafi, Phys. Rev. D 75, (2007) [arxiv:hep-ph/ ]. [10] M. U. Rehman, Q. Shafi and J. R. Wickman, Phys. Lett. B 683, 191 (2010) [arxiv: [hep-ph]]. [11] M. U. Rehman, Q. Shafi and J. R. Wickman, Phys. Lett. B 688, 75 (2010) [arxiv: [hep-ph]]. [12] Q. Shafi and J. R. Wickman, arxiv: [hep-ph]. 9