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Could loop quantum gravity corrections leave a
footprint in the primordial tensor spectrum ?
A. Barrau, Laboratoire de Physique Subatomique et
de Cosmologie, Grenoble, France J. Grain,
Laboratoire Astroparticules et Cosmologie, Paris,
France
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A few words about LQG
 Can we construct a quantum theory of spacetime
based only on the experimentally well confirmed
principles of general relativity and quantum
mechanics ?  L. Smolin, hep-th/0408048

• Four basic principles
• Any theory which is to have general relativity as
  a low energy limit must be background
  independant.
• Duality and diffeomorphism invariance may be
  consistently combined in a quantum theory.
• General relativity and all related theories can
  be formulated as gauge theories.
• Further, general relativity and related theories
  can be put in a special form in which they are
  constrained topological field theories.

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Basic physical picture
The Hilbert space of the theory H has an
orthonormal basis Ggt labeled by the embeddings
of the spin networks in the manifold S.
Basic results (nearly) randomly selected

• The area, volume and length operators have a
  discrete, finite spectra.
• The Wheeler deWitt equation is precisely
  recovered and can be solved exactly.
• The horizon entropy is completely explained in
  terms of the statistical mechanics of the state
  associated with the degrees of freedom on the
  horizon.
• Singularities are eliminated.
• The hawking radiation is recovered.
• Ultraviolet divergences of QFT are not present.
• There exist an exact physical state solution to
  the
  quatum constraint equation for any sign of ?.
• Corrections to the energy-momentum relations.
• Loop quantum cosmology is on the way.

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Holonomy corrections, basic picture
Bojowald Hossain, Phys. Rev. D (2007) 023508
Which translates, in a cosmological framework, in
Redifining the field
Which should be compared (pure general
relativity) to
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The n-1/2 case
With
Which leads to a Schrödinger equation
Which can be solved by a linear combination of
Bessel functions, leading to the spectrum
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A more detailed computation of the spectral index
leads to
For a de-Sitter inflation
and
for a more realistic slow roll inflation
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In the general case (dS)
The potential is therefore
with t-H?
IF ?2lt0
H1016 GeV
H103 GeV
? Blue spectrum
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One can easily define a transition time
So that modes with k/H gtgt (V(tt))1/2 are
significantly affected by LQG corrections.
IF ?2gt0
The potential becomes ?
Keeping in mind that
? Red spetrum
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The maximum of the potential and its location in
time can be displayed with H 1016 GeV
Combining all the constraints, the effect should
be noticeable if
Which translates in a wide sarameter space ?
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To do
WKB solutions
Power spectrum in slow-roll inflation
Background modifications ?
Hamiltonian constraints
Scalar perturbations
And mesurements !
LQG corrections could be probed by the next
generation of cosmology experiments