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) :
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 ?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 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