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We prove explicit upper and lower bounds for the Poisson hierarchy, theaveraged $L^1$-moment spectra$\{\dfrac{\mathcal{A}_k\left(B_R^M\right)}{\text{vol}\left(S_R^M\right)}\}_{k=1}^\infty$,and the torsional rigidity $\mathcal{A}_1(B^M_R)$ of a geodesic ball $B^M_R$ ina Riemannian manifold $M^n$ which satisfies that the mean curvatures of thegeodesic spheres $S^M_r$ included in it, (up to the boundary $S^M_R$), arecontrolled by the radial mean curvature of the geodesic spheres$S^\omega_r(o_\omega)$ with same radius centered at the center $o_\omega$ of arotationally symmetric model space $M^n_\omega$. As a consecuence, we prove afirst Dirichlet eigenvalue $\lambda_1(B^M_R)$ comparison theorem and show thatequality with the bound $\lambda_1(B^\omega_R(o_\omega))$, (where$B^\omega_r(o_\omega)$ is the geodesic $r$-ball in $M^n_\omega$), characterizesthe $L^1$-moment spectrum $\{\mathcal{A}_k(B^M_R)\}_{k=1}^\infty$ as thesequence $\{\mathcal{A}_k(B^\omega_R)\}_{k=1}^\infty$ and vice-versa.

First Dirichlet eigenvalue and exit time moment spectra comparisons