Tunneling cosmological state and origin of SM Higgs inflation

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Tunneling cosmological state andorigin 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
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Introduction
Problem of quantum initial conditions for
inflationary cosmology
No-boundary vs tunneling wavefunctions
(hyperbolic nature of the Wheeler-DeWitt
equation)
inflaton
other fields
Euclidean spacetime
Euclidean action of quasi-de Sitter instanton
Lorentzian spacetime
No-boundary ( ) probability maximum at the
mininmum of the potential
vs
infrared catastrophe no inflation
Tunneling ( - ) probability maximum at the
maximum of the potential
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Beyond tree level inflaton probability
distribution
contradicts renormalization theory for (- )
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
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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 lt MH lt 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, arXiv0911.1408)
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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

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Cosmological quantum states revisited
A.O.B., Phys.Rev.Lett. 99, 071301 (2007)
Microcanonical density matrix
Wheeler-DeWitt equations
Canonical (phase-space or ADM) path integral in
Lorentzian theory
3-metric and matter fields
-- conjugated momenta
constraints
lapse and shift functions
Range of integration over Lorentzian
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Lorentzian path integral Euclidean Quantum
Gravity (EQG) path integral with the imaginary
lapse integration contour
Euclidean metric
Euclidean action
EQG density matrix D.Page (1986)
Statistical sum
on S3 S1
(thermal)
including as a limiting (vacuum) case S4
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Minisuperspace-quantum matter decomposition
3-sphere of a unit size
Euclidean FRW metric
scale factor
lapse
minisuperspace background
quantum matter cosmological perturbations
quantum effective action of ? on
minisuperspace background
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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 Ngt0 or Nlt0
gauge equivalent Nlt0
gauge equivalent Ngt0
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gauge (diffeomorphism) inequivalent!
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New status of the no-boundary state

• Two cases
• massless conformally coupled quantum fields
• 2) heavy massive quantum fields

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Massless quantum fields conformally coupled to
gravity
thermal part
cosmological constant
conformal anomaly and Casimir energy part
coefficient of the Gauss-Bonnet
term in the conformal anomaly
Free energy (bosonic case)
energies of field oscillators on a 3-sphere
instanton period in units of conformal time ---
inverse temperature
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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 T1/?
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Transition to statistical sums
thermal instantons
? ?
Hartle-Hawking (vacuum) instanton
? ?
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Saddle point solutions --- set of periodic
(thermal) garland-type instantons with
oscillating scale factor ( S1 X S3 ) and vacuum
Hartle-Hawking instantons ( S4 )
, ....
1- fold, k1
k- folded garland, k1,2,3,
S4
bounded range of the cosmological constant
new QG scale
elimination of the vacuum no-boundary state
of conformal fields
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No-boundary state heavy massive quantum fields
local inverse mass expansion
Effective Planck mass (reduced) and
cosmological constants
Analytic continuation Lorentzian signature dS
geometry
S4 instanton (vacuum)
Probability distribution on the ensemble of
dS universes
infrared catastrophe no inflation
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Tunneling state heavy massive quantum fields
Effective Planck mass (reduced) and
cosmological constant
S4 (vacuum) instanton
Probability distribution of the ensemble of
dS universes
no periodic solutions
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SM Higgs inflation
non-minimal curvature coupling
inflaton
inflaton-graviton sector of SM
EW scale
Non-minimal coupling constant
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RG improved effective action
Local gradient expansion
running scale
Running coefficient functions
top quark mass
anomalous scaling
RG equations
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Anomalous scaling
Overall Coleman-Weinberg potential
Anomalous scaling in terms of SU(2),U(1) and
top-quark Yukawa constants
Determines the running of the ratio ?/?2 CMB
amplitude
Determines inflationary CMB parameters
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Inflationary CMB parameters
end of inflation
e-folding
horizon crossing formation of perturbation of
wavelength k related to e-folding
WMAP normalization at
amplitude
spectral index
WMAPBAOSN at 2?
T/S ratio
CMB compatible range of the Higgs mass
A.O.B, A.Kamenshchik, C.Kiefer,A.Starobinsky
and C.Steinwachs (2008-2009)
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Inflaton probability distribution peak
Einstein frame potential
Probability maximum at the maximum of this
potential!
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Location of the probability peak maximum of the
Einstein frame potential
RG
Quantum scale of inflation from quantum cosmology
(A.B. A.Kamenshchik, Phys.Lett. B332 (1994) 270)
! due to RG
Quantum width of the peak
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Conclusions
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
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
Effect of heavy SM sector and RG running ---
small negative anomalous scaling analogue of
asymptotic freedom