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fab13
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- TL;DR Summary
- 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.
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 advanceRegardsIn 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.