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So I wanted know the order of magnitude of this phase space invariant (equation $3.31a$ )$R$ in our universe (FLRW metric):
$$R = \int_{0}^{\infty} \int_{0}^{\infty} \int_{0}^{\infty} \frac{\mathcal{N}(x,p,t)}{\mathcal{E}} dp_x dp_y dp_z$$
where $\mathcal{N}(x,p,t)$ is the phase space distribu...
