Poisson bivectors and Poisson brackets on affine derived stacks

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Let Spec(A) be an affine derived stack. We give two proofs of the existence of a canonical map from the moduli space of shifted Poisson structures (in the sense of Pantev-Toën-Vaquié-Vezzosi, see http://arxiv.org/abs/1111.3209) on Spec(A) to the moduli space of homotopy (shifted) Poisson algebra structures on A. The first makes use of a more general description of the Poisson operad and of its cofibrant models, while the second in more computational and involves an explicit resolution of the Poisson operad.