B. Racine & V. Buza

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In this posting we present an end-to-end prescriptive recipe for making new
S4 simulations that are directly scaled from BK performance based noise
spectra.

We start from BK inputs:

- Survey weight per detector year, per bandpower,
- number of detector year,
- Equivalent \(f_{sky}\) per bandpower,

To test this, we use these BK inputs and the BK inverse variance to recover BK noise spectra and statistics -- through this process we will empirically derive a ell-dependent scaling factor that accounts for effects such as filtering that are not captured in these simple simulations.

We start by taking achieved noise BK15 bandpowers (\(N_\ell\)'s) and number of detector-years corresponding to these noise levels, and compute the effective sky fraction and survey weight per detector-year at each of the three BK frequencies.

As explained in Tegmark (1997) and Colin's posting, we can start from the sample variance of our \(N_\ell\): \[\begin{equation} \sigma(N_\ell) = \sqrt{\frac{2}{\Sigma_{\ell \in bin} (2 \ell + 1) f_{sky}}} N_\ell. \end{equation}\] We can then estimate the equivalent \(f_{sky}\) using the mean and standard deviation of the bandpowers of the 499 BK15 noise simulations: \[ \begin{equation} f_{sky} = \frac{2}{\Sigma_{\ell \in bin} (2 \ell + 1)} \left( \frac{N_\ell}{\sigma(N_\ell)} \right)^2=\frac{1}{\ell \Delta \ell}\left( \frac{N_\ell}{\sigma(N_\ell)} \right)^2 \end{equation} \] Where we used the fact that \(\Sigma_{\ell-\Delta \ell/2}^{\ell+\Delta \ell/2}(2\ell ' +1)=(\Delta\ell+1)(2\ell+1) ≈ \ell\Delta\ell\). Here \(\ell\) is the center of the bin, and \(\Delta\ell\) is it's width.

We then compute the survey weight per detector-year. It is simply the ratio:
\[\begin{equation}
SW_{det-year} = \frac{2f_{sky}}{n_{det}\times N_\ell}
\end{equation}
\]
where \(n_{det}\) is the number of detector-years. For BK15 [95 GHz, 150 GHz, 220 GHz] this is [1152, 5835, 1024] det-yrs.

The factor of 2 here comes from the fact that we report the survey weight for total Q and U polarization, whereas we only used the BB bandpowers here.

Remark: The noise bandpowers have had a suppression factor applied, that includes filtering and beam deconvolution. In the current exercise, we plan to share the \(SW_{det-year}\) that has been

To generated full sky noise map, we need a \(\ell\)-by-\(\ell\) noise spectrum. We fit a 1/f model: \[\begin{equation} N_\ell^{fit} = N_0 \left(1+\left(\frac{l}{l_{knee}}\right)^{\alpha}\right), \end{equation} \] to the beam "re-convolved" noise bandpowers. We set to 0 the power below \(\ell=30\).

We generate full sky noise \(N_{side}=512\) maps, using \(\texttt{synfast}\) from Healpix.

We then divide by the square root of the BK15 peak-normalized inverse variance map, \(m_i = \frac{V^{-1}}{max(V^{-1})} \) to recover the noise in-homogeneity of BK. This inverse variance map is generated by accumulating sign-flip noise from the timeline into the map.

We use Healpix' \(\texttt{anafast}\) to compute the power spectrum, masking the data with the same inverse variance "mask".

We rescale these spectra, dividing by the \(f_{sky-hit}\) factor:
\[\begin{equation}
f_{sky-hit} = \frac{\sum_i{m_i}}{N_{pix}}
\end{equation}
\]
where \(N_{pix}\) is the number of pixel in the full map, here 12*512*512, and \(m_i\) is the peak-normalized inverse-variance map.

The resulting spectra are distributed around the mean that lies on top of the input spectra (see figure 5).

We then integrate all these output \(\ell\)-by-\(\ell\) spectra in the measured bandpower window functions. In figure 5b we show the mean of these output bandpowers and in figure 6 the std.

While the binned mean lies on top of the input spectrum, for the standard deviation, we also need to apply an additional factor 1/sf: \[\begin{equation} sf = \frac{\sum_i{m_i^2}}{\sum_i{m_i}}. \end{equation} \]

We can then compute the output \(f_{sky}\) and \(SW_{det-year}\) using equations (2) and (3).