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We provide an analytical solution to the quantum dynamics of a flatFriedmann-Lema\^itre- Robertson-Walker model with a massless scalar field inthe presence of a small and positive cosmological constant, in the context ofLoop Quantum Cosmology. We use a perturbative treatment with respect to themodel without a cosmological constant, which is exactly solvable. Our solutionis approximate, but it is precisely valid at the high curvature regime wherequantum gravity corrections are important. We compute explicitly the evolutionof the expectation value of the volume. For semiclassical states characterizedby a Gaussian spectral profile, the introduction of a positive cosmologicalconstant displaces the bounce of the solvable model to lower volumes and tohigher values of the scalar field. These displacements are state dependent, andin particular, they depend on the peak of the Gaussian profile, which measuresthe momentum of the scalar field. Moreover, for those semiclassical states, thebounce remains symmetric, as in the vanishing cosmological constant case.However, we show that the behavior of the volume is more intricate for genericstates, leading in general to a non-symmetric bounce.

The Effect of a positive cosmological constant on the bounce of Loop Quantum Cosmology