Formula to Compute $\sigma_{8}$ for correction in non-linear regime

Question

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 $R_{8}= 8.0/0.67$ : 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?

Answer 1

Firstly, the variance of density perturbations can be calculated using the matter power spectrum $P(k)$. To do so, we coarse-grain our density field with a suitable window function $W_s(k;R)$, where $R$ represents the size of our window. (see Galaxy Formation and Evolution by H. Mo, F. Bosch, S. White) Thus:

$$‎\delta_s(\textbf{x};R) \equiv \int \delta(\textbf{x}^{\prime})W_s(|\textbf{x}^{\prime}-\textbf{x}|;R) d^3\textbf{x}^{\prime}$$

Note that there are several options for the window function. An easy and indeed natural option would be a Top-hat in real space, which gives Bessel's function in Fourier space: $$‎W(x;R) = \begin{array}{lr}‎ ‎ \frac{3}{4\pi R^3} & \mbox{($x \le R$)}\\‎ ‎ 0 & \mbox{($x > R$)} \nonumber‎ ‎\end{array}‎$$

Then, $$‎S \equiv \sigma^2(R) = \langle\delta_s^2(\textbf{x};R)\rangle‎ ‎= \int‎\Delta^2(k)\ ‎W_s^2 (k;R) \ d(\ln k)$$ where $\Delta^2(k) \equiv k^3P(k)/2\pi^2$ is the dimensionless power spectrum. In addition, the power spectrum changes with time (or redshift). So, with the help of a transfer function (see Eisenstein & Hu for an example), one can build $P(k,z)$ as well. Subsequently, one can calculate $\sigma^2(R,z)$.

Secondly, $\sigma_8^2$ is the variance for a window size of $8 h^{-1}Mpc$ (i.e. $\sigma^2(R=8 h^{-1})$) at $z=0$. This size is about the size when structure formation's regime, changed from linear to non-linear. Well, it appears that in standard texts people use a real-space Top-Hat filter to calculate this, but one shall use any other appropriate option.

As an answer to your first question, it depends on the normalization you have used for physical parameters in your code, but if you are working with CAMB, putting $R_8$ as $8 h^{-1}$ is OK.

About the second question, I think one peculiarity is the numerical integration of the interpolated $P(k)$ (noting that CAMB returns an interpolated object) from zero to infinity (so feel free to sample as many points as you wish, for a precise result). Of course, the integrand is neglectable in low and high $k$ but taking these ranges into account as well might play a role. Plus, I have recently experienced a "bug" in extrapolating the non-linear $P(k)$ for $k>100$ in CAMB. It seems that CAMB uses a power-law function to extrapolate the tail of $P(k)$ which is reasonable but doesn't work properly. It is worthwhile to mention that comparing the obtained $P(k)$ with observations might give a clue as well.