From HEP & inflation to the CMB and back

Transcription

1 ESF Exploratory Workshop, Porto March 8th, 008 From HEP & inflation to the CMB and back Jonathan Rocher ULB - Brussels V,ψ ψ 0

2 Introduction : Grand Unified Theories and Inflation Part A : Constraints on inflation from CMB. Constraints on F-term inflation. D-term inflation and constraints 3. Some ways out Part B : Initial conditions for hybrid inflation. Dynamics of inflation. Is there a fine tuning of the Initial Conditions?? Part C : Constraints from the spectral index n S. Naïve approach. Difficulties to improve the calculation of radiative corrections Conclusions and open questions

3 : - Superpotential F-term case: W F κ S Φ Φ M S : inflaton field V Φ, S Φ, Φ - : pair of Higgs fields κ : superpotential coupling M : energy scale of inflation and SSB Important property : The inflationary phase ends with the SSB. Motivations : Take into account coupling with other fields. Perfectly flat potential. The radiative corrections induce a tilt for slow roll. No additional symmetry nor extra fields. κ 0000 M 4 00 M M M/ The SUGRA corrections don t spoil the inflation [Copeland et al 994]. No fine tuning needed to generate the anisotropies? See later. 400 S

4 Models based on SUn, SO0, E 6. Most standard phenomenological ingredients : Proton life-time measurements Super-Kamiokande Motivates : Supersymmetry M GUT sufficiently high τ p e 0 π Z of R-parity unbroken at low energy Bonus dark matter P > yr Oscillations of solar and atmospheric neutrinos [Super-K, 998], Requires mass to neutrinos via See-saw Requires B-L in G GUT and broken at high energy Bonus leptogenesis To explain the CMB data, and solve the monopole problem and others, a phase of inflation hybrid

5 :? GUT SO0 4 C L R 3 C L R B-L 3 C L R B-L G SM Z G SM Z 3 C L R B-L G Z SM 4 C L R G SM Z 3 C L R B-L G SM Z G SM Z : Monopoles : Cosmic strings INFLATION Standard Model time Conclusions : [Jeannerot, J.R., Sakellariadou 003] 34 schemes compatibles with all hypothesis for SO0, String formation always after inflation Same conclusion for [E 6, SU8, SU9, SO4] : strings are generic and generically form at the end of hybrid inflation µ M infl

6 Are these strings consistent with CMB data? What is their influence on temperature anisotropies? What can we learn from their low influence? CS not yet observed! C l A CS C CS l -A CS C infl l WMAP 3 : A CS <% [Bevis et al. 007]

7 M S W F Φ Φ κ Superpotential : [Dvali, et al. 994] S U S Y Ex. The Higgs field responsible for SSB : N6 if it corresponds to B-L in SO0. Contributions to CMB quadrupole anisotropies : α9-0 Λ ln ln ln 3 4 eff z z z z S Ν M S V κ π κ κ tens scal infl Q Q Q T T T T T T δ δ δ strings with ~ h G T T Q π µ µ α δ [Landriau & Shellard 03] Solution to the horizon problem : N Q 60 Normalization to COBE/WMAP : δt/t Q ~ z f : -

8 Cosmic strings contribution % Κ M 0 5 GeV Κ Coupling to see-saw mechanism, gravitino overproduction imposes κ < 0 -. WMAP3 : A CS < % 95% CL. [Bevis et al. 007] Conclusions for SO0 : κ < 0-6 Reduced fine tuning M < 0 5 GeV Constraints on the SSB scale! For E 6, the limit on M and the conclusion unchanged. gives a window on G GUT! [J.R., Sakellariadou 005] This SSB related to neutrino masses via see-saw in this framework.

9 - Motivated by HEP : Can be embedded in SUSY/SUGRA GUTs [Jeannerot 97] No η-problem in SUGRA framework G x U -> HxU -> H -> SM Effective low energy description of Brane inflation [Dasgupta et al 0] Singlet inflaton field -> brane separation D3 SSB driven by Higgs fields -> Tachyon condensation Cosmic strings formed -> D-brane formed Can be embedded in weakly coupled string theory where anomalous Us generate FI term [Lyth & Riotto 99] D7 ξ Tr Q M p

10 - : Introduction of an additional U factor, with gauge coupling g FI term ξ 0. Charges under U : QS0, QΦ ± ± modified if SuperConformal origin of SUGRA. Superpotential : W D λ SΦ Φ [Binétruy & Dvali 96, Halyo 96, Binétruy et al 004] S U G R A Minimal SUGRA : K min S - V K K g ξ g g ξ S ln z f z 6π Λ D eff or including first order non-renorm terms 4 c S c S b S min M p We get A CS as a function of 3 parameters : g, λ, ξ.

11 Cosmic strings contribution % g0 3 g0 4 g0 g0 gx WMAP Λ Strings contribution % Ξ 0 5 GeV The WMAP3 constraint on A CS to the CMB data imposes g <. 0 λ < ξ < 0 5 Reduced fine tuning GeV Same constraint The SUGRA corrections imply a lower limit on λ. [J.R. & Sakellariadou 005a, 005b]

12 ...? Invoke a curvaton mechanism cost one more field more free parameter Making the CS unstable Introducing some additional chiral superfields [Urestilla et al 004 ; Binétruy et al. 004] Assuming additional symmetries [Binétruy et al. 004] : Consider - triplet under some gauge SU. The SSB is then SU x U -> U as in electroweak SSB Can we solve the fine tuning problem with non-flat kähler geometry [Seto & Yokoyama 005]? NO! [J.R. Sakellariadou 005]

13 - - K K Case where b0 : c S c S b S min M p Cosmic string contribution % g 0,c c 0,,,5 z λ S S Exp g ξ f f M p Cosmic string contribution % g 0,c c 0,,, Λ Conclusions : Constraints on ξ, g and λ ~unchanged. Fine tuning for λ still necessary Λ

14 Case where b 0 : Cosmic string contribution % 00 g 0,c c 0, b 0,0.5,, z λ S S b S Exp g ξ f f M p 4 0 Conclusion : the fine tuning is always necessary! J.R., Sakellariadou Λ Cosmic string contribution % b, c c 0, g 0 3,0,0 00 Cosmic string contribution % b, c c, g 0 3,0, Λ Λ

15 The coupling constant, the energy scale and the inflaton value «measured» from the data. What INITIAL CONDITIONS I.C. lead to enough efolds? Is there fine tuning on these I.C.?

16 :? Study dynamics of non SUSY hybrid model toy model for many realistic models [Linde 99]: Equations of motion : ' 4, ψ λ λ ψ ψ m M V, 3 with 0 with 0,, 3 0 with 0,, 3 ψ ψ ψ ψ ψ ψ ψ ψ ψ V M H V H V H P i i & & & && & && ψ 0 V,ψ Goal : assuming some random initial conditions for the fields and ψ, we study the trajectory and if the slow roll conditions are realized calculate Nt. Definition : successful inflation if Normalization to COBE N tot > 60

17 Known situation : fine tuning on initial values of and ψ to have a sufficient slow roll and N tot >60 [Mendes & Liddle 000, Tetradis 998]. Mendes & Liddle 000 Tetradis 998 Questions : Is this true in the rest of the parameter space? How to interpret/understand the isolated successful dots?

18 .5 λ, λ M3x0 - M pl, m0-6 M pl.0 A Our results :.5 B In agreement with Tetradis. Φ C show some isolated solutions to the horizon problem Most of the parameter space is successful if one allows for transplanckian values [Clesse & J.R. to appear] Ψ Mendes & Liddle Tetradis

19 Interpretation with some successful trajectories : B A ψ ψ Conclusions : A type trajectories are oscillation slow roll in direction ψ C B type trajectories are «miraculous» : the system climb up the valley and slide back down C type trajectories are mostly chaotic inflation in the ψ direction

20 Can we constrain more D-term inflation with the spectral index? How to improve predictions for n S for D-term inflation?

21 :? Power spectrum of primordial fluctuations ns P k k Observations WMAP 5 : n s 0.96 ± 0.05 Predictions? If slow roll, n 6ε η with Naïve approach : S M ε η M -loop potential given by perturbative formula [Coleman-Weinberg 73], pl V V ' and pl V '' V S V0 α i Φ m Φ log µ i 4 Veff m ± ± b f calculate V, V in S Q, and then n S Confrontation to the data

22 Problem : at critical point, m Φ - 0, > logm diverge Divergence of V /V for S~S crit. Artificial behaviour of n S close to the critical point! η g0 -, λ0 -, ξ4x0 5 GeV S crit S Q S GeV Divergence of V S close to S crit Inflection point in VS Need improved potential when S Q close to S crit S S Q S crit crit 0 0. Problem : in the allowed parameter space, S Q very close to S crit λ

23 Renormalization Renormalization group group methods methods : Renormalisation group can resum the subleading log. ex : Vλ 4 is replaced by Vλ 4 Perturb. R.G.E. µ π λ λ λ ln 6 3 µ π λ λ λ ln 6 3 Toy model : V massive λ 4 in the broken symmetry phase Eq of Callan-Symanzick applied to the physical quantities : V, Γ n Γ : 6 h h O m m π λ β 0,,,, ln Λ Λ Λ µ λ β β β λ λ β µ µ µ m V m m d dv Eq of Callan-Symanzick applied to proper vertices to get β-functions :

24 Γ 4 : β λ 3λ h 6π Ο h Γ 0 : β Λ h 4 m O h 3π CC required if m 0!! Solution for V RG : we introduce running parameters to account for a change in µ : dci t dt β c, with ct 0 c and t ln µ / µ j Theorem : then the RG improved potential is constructed by [Bando et al. 993] a perturbative solution V pert,c i to the CS equation at L-loop β functions at L-loop V RG c i V pert c i in order to resum the leading L0, sub-leading L, logs.

25 λ 4 ± ± Λ ± ± Λ ± 3 ln 64 4! 3 ln 64 4! 4 RG tree cw 4 tree µ λ λ π λ µ λ λ π λ m m m V m m V V m V h h [Coleman-Weinberg 73], with t cannot diverge thus µ cannot vanish! Conclusion : Don t forget the CC!! [ ] / , 3, 3 0 / t κλ λ m Λ Λ t m m t t κλ κλ λ λ

26 The improvement? µ cst running µ which can follow m eff > Log stay small in a bigger range. CW contains only h terms RG contains infinite powers of h. All the leading divergences in the Logs are resumed. For t0 or at order h, the perturbative potential MUST be recovered. How to choose V pert and µ? Restored symmetry case m >0 Here m eff m λ/ Case : µ and V pert V CW Case : µ m λ/ and V pert V CW V RG Case In the case, µ is IR regulated by m and don t vanish anymore. Case

27 Broken symmetry case m <0 Here m eff -m λ/. V diverges at crit m /λ. V tree Study of V all choices below give a nonsingular V. Case : µ -m λ/, V pert V tree or V CW. CW Case : µ m λ/, V pert V CW. Case 3 : µ m λ/, V pert V CW and β- loop. V RG Case V RG case & 3 with V CW [J.R., Greene in prep 008]

28 Cosmic strings are powerful objects to constrain models of inflation motivated by HEP. D-term inflation suffers from fine tuning on its coupling constant due to Cosmic Strings formation λ<0-5. Situation unchanged if consider non-minimal Kahler or SUGRA from SCFT. When embedded in SUSY GUTs, F-term inflation generically produces Cosmic Strings at the end. Similar fine-tuning imposed on its coupling. The dynamic shows NO fine-tuning on the Initial Conditions of hybrid inflation if transplanckian field values allowed. The fine-tuned successful points are due to fortuitous trajectories in field space. To study predictions of D-term inflation about spectral index, one needs to improve the calculation of radiative correction. The usual renormalization group methods don t work so far. This improvement is motivated for many HEP models of inflation containing an inflection point as well as other aspects of cosmology preheating,.

29 Phenomenology of SUSY GUTs and orthogonal constraints Constraints from neutrinos mass for F-term inflation within SUSY GUTs see-saw. What constraint on M from gauge unification? Constraints from proton decay on other SSBs? Phenomenology of GUT models with multiple SSB not well known. Nature of strings in SUSY GUTs Generic nature of strings : are the strings abelian or Z or both? More realistic SSB pattern will involve additionnal discrete symmetries. Formation of hybrid defects or Y-junctions? Consequence for cosmology? RG improved potential for theories with inflection point How to construct an improved potential whose second derivative is non singular? Application to MSSM inflation & some string theory models Application to preheating of hybrid inflation Hybrid inflation & initial conditions More of the parameter space to study. For what models of inflation is it safe to consider super-planckian values of the fields?

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