Tunneling cosmological state and origin of SM Higgs inflation

Tunneling cosmological state andorigin of SM Higgs inflation

A.O.Barvinsky

Theory Department, Lebedev Physics Institute, Moscow

based on works withA.Yu.KamenshchikC.KieferA.StarobinskyC.Steinwachs

QUARKS - 2010

Introduction

Lorentzian spacetime

Euclideanspacetime

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 catastropheno inflation

inflaton other fields

Problem of quantum initial conditions for inflationary cosmology

contradicts renormalization theory for (- )

Beyond tree level: inflaton probability distribution:

Both no-boundary (EQG path integral) and tunneling (WKB approximation) do nothave a clear operator interpretation

We suggest a unified framework for no-boundary and tunneling states as twodifferent calculational prescriptions for the path integral of the microcanonicalensemble in quantum cosmology, the tunneling state being consistent withrenormalization

Apply it to the Higgs inflation model with a strong non-minimal curvaturecoupling

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 < MH < 185 GeV

the SM Higgs can drive inflation with the observable CMB spectral index ns¸ 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):

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

3-metric and matter fields -- conjugated momenta

lapse and shiftfunctions

constraints

Range of integration over Lorentzian

Canonical (phase-space or ADM) path integral in Lorentzian theory:

Cosmological quantum states revisited

Microcanonicaldensity matrix

A.O.B., Phys.Rev.Lett. 99, 071301 (2007)

Wheeler-DeWitt equations

EQG density matrixD.Page (1986)

on S3£ S1

Statistical sum:

including as a limiting (vacuum) case S4

(thermal)

Lorentzian path integral =Euclidean Quantum Gravity (EQG) path integral with the imaginary lapse integration contour:

Euclidean metric Euclidean action

minisuperspace background quantum “matter” – cosmological perturbations

Euclidean FRW metric

3-sphere of a unit size

scale factorlapse

quantum effective actionof on minisuperspacebackground

Minisuperspace-quantum matter decomposition:

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<0gauge equivalent N>0

gauge (diffeomorphism) inequivalent!

New status of the no-boundary state

Two cases:

1) massless conformally coupled quantum fields

2) heavy massive quantum fields

thermal part

conformal anomaly and Casimir energy part

instanton period in units of conformal time --- inverse temperature

energies of fieldoscillators 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

Hartle-Hawking state as a member of the microcanonical ensemble

pinching a tubularspacetime

density matrix representation of a pure Hartle-Hawking state – vacuum state of zero temperature T~1/:

’

’

Transition to statistical sums

thermal instantons

Hartle-Hawking(vacuum) instanton

bounded range of the cosmologicalconstant

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 ( S1 X S3 ) and vacuum Hartle-Hawking instantons ( S4 )

, ....

S4

No-boundary state: heavy massive quantum fields

Effective Planck mass (reduced) and cosmologicalconstants

Analytic continuation – Lorentziansignature dS geometry:

Probability distributionon the ensemble of dSuniverses:

S4 instanton (vacuum):

infrared catastropheno inflation

local inverse mass expansion

Tunneling state: heavy massive quantum fields

Effective Planck mass (reduced) and cosmologicalconstant

Probability distributionof the ensemble of dSuniverses:

S4 (vacuum) instanton:

no periodic solutions:

SM Higgs inflation

inflaton-gravitonsector of SM

inflaton non-minimal curvature coupling

Non-minimal coupling constant

EW scale

Running coefficient functions:

RG equations:

running scale:

anomalous scaling

RG improved effective action

Local gradient expansion:

top quark mass

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

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 rangeof the Higgs mass

A.O.B, A.Kamenshchik, C.Kiefer,A.Starobinsky and C.Steinwachs (2008-2009):

e-folding #

Einstein frame potential

Probability maximum at the maximum of this potential!

Inflaton probability distribution peak

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

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