Recipe to generate performance based S4 simulations with arbitrary sky distribution (in progress)

B. Racine & V. Buza

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,
scale the noise to a given S4 frequency and distribute this noise using a hitcount map following a given scan strategy.
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.

1. Inputs

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. Figure 1: This figure shows the effective sky fraction for the three BK15 bands: 95 GHz in red, 150 GHz in green, and 220 GHz in blue.

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 re-convolved by the beam, The definition that makes more physical sense for comparison between experiments includes the beam roll-off (i.e. not from the re-convolved bandpowers) as it represents how deep our data are at a given $$\ell$$. Here we show both for completeness. Figure 2: This figure shows the Survey Weight per detector year the three BK15 bands: 95 GHz in red, 150 GHz in green, and 220 GHz in blue.

2. Automatic Noise Fitting

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 $$\ell=30$$. Figure 3: This figure shows the input noise bandpowers: 95 GHz in red, 150 GHz in green, and 220 GHz in blue, as well as the 1/f noise fit in dashed black.

3. Making full-sky and cut-sky noise maps

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. Figure 4: This figure shows the noise maps genereated from the fitted noise spectra shown in the above figure, on a full sky, and after applying the inverse variance mask from BK15. For T, the scale goes from -50 to 50 $$\mu K$$, while Q and U go from -5 to 5 $$\mu K$$.

4. Power Spectrum Estimation and Noise Variance

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). Figure 5: This figure shows the output noise spectra: 95 GHz in red, 150 GHz in green, and 220 GHz in blue, as well as the mean of the 499 realizations in black, the input bandpowers as yellow dots, and the input fitted spectrum in dashed white. The input noise bandpowers shown here are "beam re-convolved".

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. Figure 5b: This figure shows the input bandpowers (circles), the mean of the spectra (solid lines), as well as the mean after they went through bandpower window function integration (dashed): 95 GHz in red, 150 GHz in green, and 220 GHz in blue. Note that these are $$\mathcal{D}_\ell$$s, not $$C_\ell$$s. The input noise bandpowers stds shown here are "beam re-convolved".

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: $\begin{equation} sf = \frac{\sum_i{m_i^2}}{\sum_i{m_i}}. \end{equation}$ Figure 6: This figure shows the noise standard deviation: 95 GHz in red, 150 GHz in green, and 220 GHz in blue. The full line shows the standard deviation of the 499 output spectra. The circles show the input noise std, and the dashed line shows the standard variation of the 499 "windowed spectra", i.e. after the 499 output spectra are weighted by the bandpower window functions. Note that these are $$\mathcal{D}_\ell$$s, not $$C_\ell$$s. The input noise bandpowers shown here are "beam re-convolved".

6. Reobtaining the survey weight per det-yr

We can then compute the output $$f_{sky}$$ and $$SW_{det-year}$$ using equations (2) and (3). Figure 7: This figure compares the input and output effective sky fraction: 95 GHz in red, 150 GHz in green, and 220 GHz in blue. The full line shows the standard deviation of the 499 output spectra. The circles show the input noise std, and the dashed line shows the standard variation of the 499 "windowed spectra", i.e. after the 499 output spectra are weighted by the bandpower window functions. Figure 8: This figure compares the input and output Survey Weight per detector year: 95 GHz in red, 150 GHz in green, and 220 GHz in blue. The full line shows the standard deviation of the 499 output spectra. The circles show the input noise std, and the dashed line shows the standard variation of the 499 "windowed spectra", i.e. after the 499 output spectra are weighted by the bandpower window functions. The input SWdy shown here are "beam re-convolved".