There is a wide consensus on the correct dynamics of the background in loop quantum cosmology. In this article we make a systematic investigation of the duration of inflation by varying what we think to be the most important "unknowns" of the model: the way to set initial conditions, the amount of shear at the bounce and the shape of the inflaton potential.

The prediction of a phase of inflation whose number of e-folds is constrained is an important feature of loop quantum cosmology. This work aims at giving some elementary clarifications on the role of the different hypotheses leading to this conclusion. We show that the duration of inflation does not depend significantly on the modified background dynamics in the quantum regime.Loop quantum gravity (LQG) is a nonperturbative and background-independent quantization of general relativity (GR). It relies on the Sen-Ashtekar-Barbero variables, that is SU(2) valued connections and conjugate densitized triads. The quantization is obtained using holonomies of the connections and fluxes of the densitized triads. Loop quantum cosmology (LQC) is an effective theory based on a symmetry reduced version of LQG. In LQC, the big bang is believed to be replaced by a bounce due to repulsive quantum geometrical effects (see [1] for a review). For the flat homogeneous and isotropic background cosmology that we consider in this work, the effective LQC-modified Friedmann equation iswhere H ≡ (ȧ/a) is the Hubble parameter, ρ is the total energy density and ρ B is the critical density at the bounce (expected to be of the order of the Planck density). The dot refers to a coordinate time derivative. We assume that the dominating energy component in the early universe is a scalar field φ, with potential V = 1 2 m 2 φ 2 . The total energy density can be written as ρ = 1 2φ 2 + V . Based on cosmic microwave background (CMB) measurements and under most reasonable assumptions for the length of observable inflation (between horizon exit of the pivot scale and the end of the inflationary phase), one obtains m 10 −6 m Pl . The equation of motion for the scalar field isThere are different ways to statistically estimate the duration of inflation in this framework.At a fixed energy density, ρ 0 , one can first ask the following question: for a given number of e-folds N , what is the fraction of trajectories, i.e. solutions to Eq.(2), that lead to a phase of slow-roll inflation lasting more than N e-folds? It should be noticed that the set of trajectories can be parametrized by {a 0 , φ 0 }. As the energy density has been fixed, the initial time derivative of the scalar field,φ 0 , is determined in terms of ρ 0 and φ 0 . This also implies that φ 0 can only take values within a finite interval, ranging from −( √ 2ρ 0 /m) to ( √ 2ρ 0 /m). In a flat universe, the value of the scale factor has no physical meaning. The number of e-folds of inflation depends on φ 0 but not on a 0 : N = N (φ 0 ; m, ρ 0 ). So the fraction of trajectories that achieve a phase of inflation lasting more than N e-folds can be written as µ = (m∆φ 0 )/(2 √ 2ρ 0 ), where ∆φ 0 is the range of initial values of the scalar field that yields the required inflationary phase. It is then necessary to evaluate µ as a function of N . There are two cases in which this can be done analytically: (i) at low energy, ρ 0 m 2 , and (ii) at high energy ρ 0 m 2 . At low energy, the calculation of Gi...