Simple 1D model to study the effect of different observation strategies on the bandpower statistics.

  B. Racine, V. Buza, J. Kovac

Edited on December 14th: Simplification of the expression of the variances, and generalization to the case where the weights used in the bandpower estimators are not the hitcount maps corresponding to the noise (and the numerical test of these formula).

In this posting we use a 1D model to test analytic formulas of the statistics of signal and non-uniform noise bandpowers.
We then use this simple model to predict the evolution of these statistics depending on different observation strategies, for example varying the field of view of our telescopes.

1. Inputs

1.1 Hitcount maps

We created different types of hitcount maps in map domain to mimick observations with different field of views (circles with radius from 8 to 18 degrees).
They are based on an observation strategy close to the BK one. BICEP/Keck currently observes the sky at a fixed elevation (centered on 57.5 degrees) for 2*50 minutes and scan in azimuth with a width of 64.2 degrees, of which we keep the central constant speed 56.4 degrees. We step by 0.25 degrees every 2*50 minutes. Between 2 steps, the field moves by 2*50/60*15 = 25 degrees. These steps cover 5 degrees in elevation.
Thus, the telescope roughly points on a rectangle of height 5° and width (56.4°)*cos(57.5°)=30.3°. Due to the sky rotation, there is an additional slightly tapered 25°*cos(57.5°)=13.4°.
Here we approximate this observed patch on a flat sky by a rectangle of width 38 degrees and height 5 degrees, on a full sky of 203°x203°. We then convolve with a circle to simulate the field of view of the instrument. We vary the radius of these circles between 8 and 18 degrees. These different hitmaps are shown in Figure 1.

Figure 1: Simple simulated hitmaps for different fields of view.

For the rest of the posting, we will use a toy model in 1D with signal and noise vectors. For the hitcount vectors, we will use the cumulative distributions of these hitmaps, remapped to \(N_{\rm modes}\)=10000 by interpolation, and normalized so that their sum is 1. These 1D hitcounts are shown in Figure 2, compared to the similarily derived hitcount vectors of the DC04, DC04b, DC04c and BK14 hitmaps.
We can note that while BICEP2/Keck is supposed to have a 20° field of view, and BICEP3 has 28°, they appear shallower than our simple model. These are not circular field of view though, more of a square that is rotated every 48 hours. In this figure it seems like our simple model doesn't exactly reproduce the BK or B3 hitcount distribution. This could maybe better approximated by a tapered circular FOV. Here we mostly care about the general dependence on the FOV though.

Figure 2: Hitcount vectors

1.2 Noise

We generate \(N_{\rm sim}\)=10000 vectors \(x_i\), with \(N_{\rm modes}\)=10000 random Gaussian numbers with \(\sigma_n=1\) and \(\mu_n=0\).

We use the hitcount vectors to generate "observed" noise vectors: \begin{align*} n_i=\frac{x_i}{\sqrt{h_i}} \end{align*}
In Figure 3, we show one example of such noise vector for each hitcount case.

Figure 3: "Observed" noise.

1.2 Signal

We generated \(N_{\rm sim}\)=10000 vectors \(s_i\), with \(N_{\rm modes}\)=10000 random Gaussian numbers with \(\sigma_s=0.1\) and \(\mu_s=0\).

2. Analysis

We then want to estimate a single bandpower from a given data vector \(d_i\): \begin{align*} {\rm\widehat{BP}} &= \frac{\sum_i (w_i*d_i)^2}{\sum_i w_i^2 } \end{align*}
We compute the \(N_{sim}\) bandpowers estimates and compute the mean and variance.
We also compute the effective number of degrees of freedom \(k\): \begin{align*} k &= 2 \frac{ \\< {\rm BP}>^2}{{\rm Var(BP)}} \end{align*}
This is related to the "Effective \(f_{\rm sky}\)" of a map, defined as : \begin{align*} f_{\rm sky,eff} &= \frac{k}{N_{\rm modes}} \end{align*}

In figure 4, we compare the MC numerical results (colored points) to a model (grey bars), derived in these notes.
We show the general case here the weights used are different from the hitcount mas, as well as the case where w=h: