Just to remind, $C_\ell$ is the variance of random variables $a_{\ell m}$ following a centered Gaussian PDF (in spherical harmonics of Legendre) :

$$C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)$$

  1. Second observable : $$ \sigma_{D, 2}^{2}=\dfrac{2 \sum_{\ell_{\min }}^{\ell_{\max }}(2 \ell+1)}{\left(f_{s k y} N_{p}^{2}\right)} $$ so : $$ \sigma_{o, 2}^{2}=\dfrac{\sigma_{D, 2}^{2}}{\left(\sum_{\ell_{\min }}^{\ell_{\max }}(2 \ell+1) C_{\ell}\right)^{2}} $$

  2. First observable : $$ \sigma_{D, 1}^{2}=\sum_{\ell_{\min }}^{\ell_{\max }} \dfrac{2}{(2 \ell+1)\left(f_{s k y} N_{p}^{2}\right)} $$ so : $$ \sigma_{o, 1}^{2}=\dfrac{\sigma_{D, 1}^{2}}{\left(\sum_{\ell_{\min }}^{\ell_{\max }} C_{\ell}\right)^{2}} $$

  3. Goal : I would like to prove than $\sigma_{o, 1}^{2}<\sigma_{o, 2}^{2}$ but I have difficulties to derive this inequality.

UPDATE : from the preliminary results of a colleague, it may show the contrary, i.e that the inequality is rather : $\sigma_{o, 2}^{2}<\sigma_{o, 1}^{2}$ But I have to double check, there should be an error since numerically, I find that $\sigma_{o, 1}^{2}<\sigma_{o, 2}^{2}$. Any help is welcome



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