Loop quantum cosmology (a symmetry-reduced quantum model of the Universe inspired by loop quantum gravity) extends the inflationary paradigm to the Planck era: the big bang singularity is replaced by a quantum bounce naturally followed by inflation. Testing for these models requires to compute the amount of cosmological perturbations produced in this quantum background and subsequently derives their footprints on the cosmic microwave background. This proceedings proposes a very brief summary of

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... this road from quantum gravity to cosmological observables, focusing on the case of a loop-quantum model of flat FLRW spaces with cosmological perturbations. The background evolution, passing through a bounce, is first presented. The case of cosmic inhomogeneities in a perturbative treatment about a quantum background are then discussed (with an emphasis on two specific theoretical constructions), and there respective predictions in terms of primordial power spectra for the tensor modes are outlined. A brief overview of loop quantum cosmology Julien Grain Introduction-Loop quantum cosmology (LQC) [1] is a tentative approach to built a quantum model of the Universe inspired by the ideas of loop quantum gravity (LQG) -a non-perturbative quantization of general relativity-[2], thus allowing for extending the inflationary paradigm to the Planck era where quantum gravity effects are no more negligible. This approach relies on minisuperspace ideas: the entire phase space of general relativity is reduced thanks to the isotropy and homogeneity symmetries, meaning that only FLRW metric are considered. This reduced phase space is then quantized using the loop representation. At a quantum level, the big bang singularity issue is solved for the transition to the null volume is forbidden, leading to the replacement of the big bang singularity by a quantum big bounce. This is understood at a more effective level by a regularization of the curvature. Classically, the curvature can be computed from holonomies evaluated around a plaquette and then, by shrinking the area of the plaquette to zero. (The FLRW space can be paved by identical plaquettes thanks to isotropy and homogeneity.) However, the kinematical Hilbert space in LQG is given by eigenstates of geometric (area and volume) operators which have discrete spectra. This leads to a minimal area gap and the area of the plaquette around which holonomies are evaluated cannot be shrinked to zero anymore, thus preventing the curvature to diverge. This is captured in the effective, modified Friedmann equation (for a flat FLRW metric with holonomy correction only): ȧ a 2 = 8πG 3 ρ 1 − ρ ρ c , with ρ c a maximal energy density precisely set by the minimal area gap (ρ c is of the order of the Planck energy density, though its explicit value depends on the value of the Barbero-Immirzi parameter, γ). In the effective picture, the bounce occurs when the Hubble parameter, H := (ȧ/a), vanishes, that is when the energy density of the content of the universe reaches ρ c . The classical equation is recovered for ρ ρ c . The cosmic history would therefore be a phase of classical contraction (ρ ρ c and ρ increases), then a quantum bounce (ρ ρ c ) making contact with a classical expansion (ρ ρ c and ρ decreases). (The efficiency of this approach has been shown thanks to both numerical and analytical arguments [3].) Obviously, LQC is not free of any hypotheses (though following the spirit of LQG, it attempts to make use of few of them), and its program is not fully accomplished yet. The robustness of any bouncing cosmologies should be tested against the evolution of anisotropies through the bounce, and now a rather wide amount of studies is devoted to this question (see [4] and references therein). Similarly, its robustness against of full derivation of the quantum cosmological history starting from LQG is still needed, and recent progresses have been made in e.g. spin foams or group field theory [5] . In any case, the last years of research in this field have paved a road going from quantum gravity to cosmological observables. Here we focus on a paradigmatic (and most advanced) example of such a program. We consider the case of LQC for flat FLRW spaces incorporating cosmological perturbations. The matter content is a massive scalar field allowing for having a phase of inflation shortly after the bounce. Finally, only holonomy corrections are considered here (see e.g. [6] for inverse volume corrections). Background-The effective equation of motions for the background are obtained by regularizing the curvature operator. The gravitational phase space is composed of the extrinsic curvature, k, and p := a 2 while the phase space for matter is the field configuration, ϕ, and its conjugate momentum,