Computing sig8 for correction in non-linear regime

[fab13]

TL;DR

In the context of the sudy of C_l's stability for Fisher formalism, I need to apply, with CAMB code, a correction on 𝜎8 between linear and non-linear regime to keep it fixed (I make change the values of cosmological parameters at each iteration).

I need to apply, with CAMB code, 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 $R_{8}= 8.0/0.67=11.94$ : 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 ? Thanks in advanceRegards

 

[kimbyd]

You should be able to use CAMB to output $\sigma_8$ directly. In the Python docs, for instance, look at get_matter_transfer_data():
https://camb.readthedocs.io/en/latest/results.html

 

[fab13]

Thanks, isn't there a way to do it with a semi-analytical method since I am using mostly the Fortran90 source files ?