I need to apply a correction on $\sigma_{8}$ between linear and non-linear regime to keep it fixed (I make change the values of cosmological parameters at each iteration). I have to compute $\sigma_{8}$ from the $P_{k}$ and found the following relation (I put also the text for clarify the context) :

Part of this Klein Onderzoek is aimed at finding an estimate of the cosmological parameter \(\sigma_{8}\) from peculiar verlocity data only. \(\sigma_{8}\) is defined as the r.m.s. density variation when smoothed with a tophat-filter of radius of \(8 \mathrm{h}^{-1} \mathrm{Mpc} .[9]\) The definition of \(\sigma_{8}\) in formula-form is given by:

\[

\sigma_{8}^{2}=\frac{1}{2 \pi^{2}} \int W_{s}^{2} k^{2} P(k) d k

\]

where \(W_{s}\) is tophat filter function in Fourier space:

\[

W_{s}=\frac{3 j_{1}\left(k R_{8}\right)}{k R_{8}}

\]

where \(j_{1}\) is the first-order spherical Bessel function. The parameter \(\sigma_{8}\) is mainly sensitive to the power spectrum in a certain range of \(k\) -values. For large \(k,\) the filter function will become negligible and the integral will go to zero. For small \(k,\) the factor \(k^{2}\) in combination with the power spectrum factor \(k^{-3}\) will make sure that the integral is negligible. In other words, \(\sigma_{8}\) is mostly determined by the power spectrum within the approximate range \(0.1 \leq k \leq 2 .\) since \(\sigma_{8}\) is only sensitive to a certain range of \(k,\) any difference in the values of the Hubble uncertaintenty, the baryonic matter density and the total matter density will influence the found estimate.

**Question 1)**What numerical value have I got to take for $R_{8}$ in my code : for the instant, I put $R_{8}= 8.0/0.67=11.94\,$Mpc : is this correct ?

**Question 2)**The other issue is, for each correction on $A_{s}$, that I find with this expression a value roughly around : $\sigma_{8} = 0.8411 ........$ instead of standard (fiducial) value $\sigma_{8} = 0.8155 ........$ : there is a 4 percent of difference between both values : is the expression above right ?

Could anyone tell me a good way to compute $\sigma_{8}$ from $P_{k}$ generated by CAMB-1.0.12 ?

Regards