1. Tunneling cosmological state and origin of SM Higgs inflation A.O.Barvinsky Theory Department, Lebedev Physics Institute, Moscow based on works with A.Yu.Kamenshchik C.Kiefer A.Starobinsky C.Steinwachs QUARKS - 2010

2. Introduction Lorentzian spacetime Euclidean spacetime No-boundary vs tunneling wavefunctions (hyperbolic nature of the Wheeler-DeWitt equation): Euclidean action of quasi-de Sitter instanton Tunneling ( - ) : probability maximum at the maximum of the potential No-boundary ( + ) : probability maximum at the mininmum of the potential vs infrared catastrophe no inflation inflatonother fields Problem of quantum initial conditions for inflationary cosmology

3. contradicts renormalization theory for ( - ) Beyond tree level: inflaton probability distribution: Both no-boundary (EQG path integral) and tunneling (WKB approximation) do not have a clear operator interpretation We suggest a unified framework for no-boundary and tunneling states as two different calculational prescriptions for the path integral of the microcanonical ensemble in quantum cosmology, the tunneling state being consistent with renormalization

4. Apply it to the Higgs inflation model with a strong non-minimal curvature coupling Higgs doublet CMB for GUT inflation : B. Spokoiny (1984); D.Salopek, J.Bond & J. Bardeen (1989); R. Fakir& W. Unruh (1990); A.Barvinsky & A. Kamenshchik (1994, 1998) F.Bezrukov & M.Shaposhnikov (2008-2009): Standard Model Higgs boson as an inflaton With the Higgs mass in the range 136 GeV < M H < 185 GeV the SM Higgs can drive inflation with the observable CMB spectral index n s ¸ 0.94 and a very low T/S ratio r' 0.0004. A.O.B & A.Kamenshchik, C.Kiefer, A.Starobinsky, C.Steinwachs (2008-2009): This model generates initial conditions for the inflationary background in the form of the sharp probability peak in the distribution function of an inflaton for the TUNNELING state of the above type. A.O.B, A.Kamenshchik, C.Kiefer, C.Steinwachs (Phys. Rev. D81 (2010) 043530, arXiv:0911.1408):

5. Plan Cosmological quantum states revisited microcanonical density matrix no-boundary vs tunneling states New status of the no-boundary state ; Hartle-Hawking state as a member of the microcanonical ensemble massless conformal fields vs heavy massive fields Tunneling state for heavy massive fields SM Higgs inflation RG improved effective action inflationary CMB parameters inflaton probability distribution peak – initial conditions for inflation Conclusions

6. 3-metric and matter fields-- conjugated momenta lapse and shift functions constraints Range of integration over Lorentzian Canonical (phase-space or ADM) path integral in Lorentzian theory: Cosmological quantum states revisited Microcanonical density matrix A.O.B., Phys.Rev.Lett. 99, 071301 (2007) Wheeler-DeWitt equations

7. EQG density matrix D.Page (1986) on S 3 £ S 1 Statistical sum: including as a limiting (vacuum) case S 4 (thermal) Lorentzian path integral = Euclidean Quantum Gravity (EQG) path integral with the imaginary lapse integration contour: Euclidean metricEuclidean action

8. minisuperspace backgroundquantum “matter” – cosmological perturbations Euclidean FRW metric 3-sphere of a unit size scale factorlapse quantum effective action of  on minisuperspace background Minisuperspace-quantum matter decomposition:

9. Semiclassical expansion and saddle points: No periodic solutions of effective equations with imaginary Euclidean lapse N (Lorentzian spacetime geometry). Saddle points exist for real N (Euclidean geometry): Deformation of the original contour of integration into the complex plane to pass through the saddle point with real N>0 or N<0 gauge equivalent N<0 gauge equivalent N>0

10. gauge (diffeomorphism) inequivalent!

11. New status of the no-boundary state Two cases: 1)massless conformally coupled quantum fields 2) heavy massive quantum fields

12. thermal part conformal anomaly and Casimir energy part instanton period in units of conformal time --- inverse temperature energies of field oscillators on a 3-sphere Free energy (bosonic case): coefficient of the Gauss-Bonnet term in the conformal anomaly Massless quantum fields conformally coupled to gravity cosmological constant

13. Hartle-Hawking state as a member of the microcanonical ensemble pinching a tubular spacetime density matrix representation of a pure Hartle-Hawking state – vacuum state of zero temperature T~1/  :

14.  ’ Transition to statistical sums thermal instantons Hartle-Hawking (vacuum) instanton

15. bounded range of the cosmological constant elimination of the vacuum no-boundary state: # of conformal fields new QG scale k- folded garland, k=1,2,3,… 1- fold, k=1 Saddle point solutions --- set of periodic ( thermal) garland-type instantons with oscillating scale factor ( S 1 X S 3 ) and vacuum Hartle-Hawking instantons ( S 4 ),.... S4S4

16. No-boundary state: heavy massive quantum fields Effective Planck mass (reduced) and cosmological constants Analytic continuation – Lorentzian signature dS geometry: Probability distribution on the ensemble of dS universes: S 4 instanton (vacuum): infrared catastrophe no inflation local inverse mass expansion

17. Tunneling state: heavy massive quantum fields Effective Planck mass (reduced) and cosmological constant Probability distribution of the ensemble of dS universes: S 4 (vacuum) instanton: no periodic solutions:

18. SM Higgs inflation inflaton-graviton sector of SM inflaton non-minimal curvature coupling Non-minimal coupling constant EW scale

19. Running coefficient functions: RG equations: running scale: anomalous scaling RG improved effective action Local gradient expansion: top quark mass

20. Overall Coleman-Weinberg potential: Anomalous scaling Anomalous scaling in terms of SU(2),U(1) and top-quark Yukawa constants Determines inflationary CMB parameters Determines the running of the ratio /  2 – CMB amplitude

21. end of inflation horizon crossing – formation of perturbation of wavelength k related to e-folding # Inflationary CMB parameters WMAP normalization at amplitude spectral index T/S ratio WMAP+BAO+SN at 2  CMB compatible range of the Higgs mass A.O.B, A.Kamenshchik, C.Kiefer,A.Starobinsky and C.Steinwachs (2008-2009): e-folding #

22. Einstein frame potential Probability maximum at the maximum of this potential! Inflaton probability distribution peak

25. Location of the probability peak – maximum of the Einstein frame potential: Quantum width of the peak: RG Quantum scale of inflation from quantum cosmology (A.B.& A.Kamenshchik, Phys.Lett. B332 (1994) 270) ! due to RG

26. Conclusions Effect of heavy SM sector and RG running --- small negative anomalous scaling: analogue of asymptotic freedom A complete cosmological scenario is obtained in SM Higgs inflation: i) formation of initial conditions for the inflationary background (a sharp probability peak in the inflaton field distribution) and ii) the ongoing generation of the WMAP compatible CMB perturbations on this background. in the Higgs mass range Path integral formulation of the tunneling cosmological state is suggested as a special calculational prescription for the microcanonical statistical sum in cosmology. Within the local gradient expansion it remains consistent with UV renormalization