The perturbed universe in the deformed algebra approach of Loop Quantum
  Cosmology
  Loop quantum cosmology is a tentative approach to model the universe down to
the Planck era where quantum gravity settings are needed. The quantization of
the universe as a dynamical space-time is inspired by Loop Quantum Gravity
ideas. In addition, loop quantum cosmology could bridge contact with
astronomical observations, and thus potentially investigate quantum cosmology
modellings in the light of observations. To do so however, modelling both the
background evolution and its perturbations is needed. The latter describe
cosmic inhomogeneities that are the main cosmological observables. In this
context, we present the so-called deformed algebra approach implementing the
quantum corrections to the perturbed universe at an effective level by taking
great care of gauge issues. We particularly highlight that in this framework,
the algebra of hypersurface deformation receives quantum corrections, and we
discuss their meaning. The primordial power spectra of scalar and tensor
inhomogeneities are then presented, assuming initial conditions are set in the
contracting phase preceding the quantum bounce and the well-known expanding
phase of the cosmic history. These spectra are subsequently propagated to
angular power spectra of the anisotropies of the cosmic microwave background.
It is then shown that regardless of the choice for the initial conditions
inside the effective approach for the background evolution (except that they
are set in the contracting phase), the predicted angular power spectra of the
polarized B-modes exceed the upper bound currently set by observations. The
exclusion of this specific version of loop quantum cosmology establishes the
falsifiability of the approach, though one shall not conclude here that either
loop quantum cosmology or loop quantum gravity is excluded.
