Computing sig8 for correction in non-linear regime

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Dournac Fabien
Posts: 12
Joined: May 18 2019
Affiliation: IRAP
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Computing sig8 for correction in non-linear regime

Post by Dournac Fabien » December 25 2019

Hello,

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
Last edited by Dournac Fabien on January 02 2020, edited 1 time in total.

Antony Lewis
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Joined: September 23 2004
Affiliation: University of Sussex
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Re: Computing sig8 for correction in non-linear regime

Post by Antony Lewis » December 27 2019

You can get sigma8 directly from CAMB rather than calculating it externally.

Dournac Fabien
Posts: 12
Joined: May 18 2019
Affiliation: IRAP
Contact:

Re: Computing sig8 for correction in non-linear regime

Post by Dournac Fabien » December 28 2019

Thanks, I didn't find any documentation about how to compute it. Have I got to use python layer to get sigma8 ?

Or it may be an option to set into params.ini ?

Let me know please how to perform this computation.

Regards.

Antony Lewis
Posts: 1534
Joined: September 23 2004
Affiliation: University of Sussex
Contact:

Re: Computing sig8 for correction in non-linear regime

Post by Antony Lewis » December 28 2019

It's usually printed when you run the program for any run including the initial power spectrum. Otherwise using python interface is easiest.

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