I try to explain my issue on this forum since it seems to be appropriate.

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 $P_{k}$ (generated by CAMB code) 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 $R_{8}= 8.0/0.67$ : 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 ?



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