Coaxing cosmic 21 cm fluctuations from the polarized sky using $m$-mode analysis

In this paper we continue to develop the $m$-mode formalism, a technique for efficient and optimal analysis of wide-field transit radio telescopes, targeted at 21 cm cosmology. We extend this formalism to give an accurate treatment of the polarized sky, fully accounting for the effects of polarization leakage and cross polarization. We use the geometry of the measured set of visibilities to project down to pure temperature modes on the sky, serving as a significant compression, and an effective first filter of polarized contaminants. As in our previous work, we use the $m$-mode formalism with the Karhunen-Loève transform to give a highly efficient method for foreground cleaning, and demonstrate its success in cleaning realistic polarized skies observed with an instrument suffering from substantial off axis polarization leakage. We develop an optimal quadratic estimator in the $m$-mode formalism which can be efficiently calculated using a Monte Carlo technique. This is used to assess the implications of foreground removal for power spectrum constraints where we find that our method can clean foregrounds well below the foreground wedge, rendering only scales $k_{\parallel }<0.02h{\mathrm{Mpc}}^{-1}$ inaccessible. As this approach assumes perfect knowledge of the telescope, we perform a conservative test of how essential this is by simulating and analyzing data sets with deviations about our assumed telescope. Assuming no other techniques to mitigate bias are applied, we find we recover unbiased power spectra when the per-feed beamwidth to be measured to 0.1%, and amplifier gains to be known to 1% within each minute. Finally, as an example application, we extend our forecasts to a wideband 400–800 MHz cosmological observation and consider the implications for probing dark energy, finding a pathfinder-scale medium-sized cylinder telescope improves the Dark Energy Task Force figure of merit by around 70% over Planck and Stage II experiments alone.

Figure 1

A schematic of a cylinder telescope, consisting of two cylinders aligned north south on the ground. Each cylinder is of width $W$, and has $N_{\text{feeds}}$ regularly spaced a distance $D$ apart. In this paper we will only consider cylinders which are touching, making the total width of the array $2W$. The cylinders are assumed to be long enough that there are no optical differences between feeds at the edge and in the center of the array.

Figure 2

The primary beam of the cylinder telescope forms a long strip on the sky from north to south. This figure illustrates the transfer into an instrumental Stokes I ($XX+YY$ polarizations), from the total intensity on the sky (left panel), and from the polarized sky only (right panel). The red contour in the top panel marks the half power point of the beam.

Figure 3

Sensitivity of the array to temperature, derived from the square-root-inverse function of the diagonal elements of the Fisher matrix $({\mathcal{F}}_{(lm)(lm)}{)}^{-1/2}$. The plot above shows the ${\log }_{10}$ of the sensitivity in units of $\mu \mathrm{K}$. The sensitivity to the three remaining Stokes parameters is largely identical. The dashed black lines mark the $m$ corresponding to the separation between the cylinders, and the total width of the cylinders. As we would expect the sensitivity peaks in $m$ at the zero separation, and the single cylinder separation. It then falls off rapidly at the edge of the telescope.

Figure 4

The information about the sky does not spread throughout the space of visibilities but is contained in a subspace, a linear combination of the measured signals which does not span the whole visibility space. Directions orthogonal to this subspace are excited only by the instrumental noise, and contain no information about the sky. The left panel illustrates the geometry of the full visibility space, showing the sky subspace as a plane. In the right panel we show only the sky plane. Within this sky subspace, there are yet lower dimensional subspaces that the total intensity (labeled $T$) and polarized ($P$) signals get mapped to. However, they need not be orthogonal, an effect we must take into account. One way of treating this is to project onto the space orthogonal to polarization (labeled $T^{\prime }$), this eliminates polarized contamination at the expense of some sensitivity to total intensity. This is discussed in detail in the text.

Figure 5

The singular values of $\overline{\mathbf{B}}$ for the 400 MHz channel after removal of the polarized modes. Large singular values represent modes on the sky that are well measured. We see that at each $m$ there are less than 100 measured degrees of freedom from the sky, with the spectrum dropping off very steeply beyond this. This is a significant saving, before compression there are twice as many modes as there are unique baselines, including positive and negative $m$s. In our example there are 762 unique baselines (without autocorrelations), so there would be $\sim 1500$ modes.

Figure 6

Here we plot two KL modes (with $m=20$) as they would look on the sky: one of the most foregroundlike modes ($S/F=4\times {10}^{-13}$), and one of the most signal-like ($S/F=170$). The lower, square panels show the frequency direction of a cut through the sky indicated on the upper panel with a black line. Though they are derived in visibility space, when projected back to the sky, they appear as we would expect with the foreground mode having a smooth frequency spectrum, and the signal mode oscillating. Modes at either end of the spectrum, like the ones plotted, are easy to interpret, this is not generally true of the intermediate modes.

Figure 7

To remove foregrounds from our data (point $O$), we separate our space into two subspaces of foreground contaminated modes, and signal modes (denoted by $\mathbf{F}$ and $\mathbf{S}$). These spaces are not guaranteed to be orthogonal. Inverting with the pseudoinverse gives the linear combination of signal vectors with the same amplitude, however, the resulting vector $P$ is clearly contaminated by foregrounds (as the projection onto $\mathbf{F}$ is nonzero). The full inverse gives point $I$, which has the same projection onto $\mathbf{S}$, but contains no foregrounds, however, it is necessarily a combination of both $\mathbf{S}$ and $\mathbf{F}$.

Figure 8

This figure illustrates the process of foreground removal on simulations of the radio sky. The top row of panels shows sky maps of the individual components: unpolarized foregrounds, polarized foregrounds (showing Stokes $Q$ only) and the 21 cm signal. On the bottom row we show the total intensity maps we would make after foreground cleaning visibilities from our example telescope. Particularly, the lower middle panel is showing the leakage of polarized foregrounds into the total intensity. Again, the square panels show the frequency direction for the slice of the sky indicated with a black line. Both the polarized and unpolarized foregrounds become substantially suppressed, whereas the 21 cm signal is largely unaffected. In this example we have discarded modes with $S/F<10$. This leaves a clear correspondence between the original signal simulation and the foreground subtracted signal, while leaving the foreground residuals over ten times smaller in amplitude.

Figure 9

Forecast errors on the power spectrum as a fraction of its fiducial value for the 400–500 MHz band. The three panels show the predicted errors without foregrounds (left), with unpolarized foregrounds (center) and with polarized foregrounds (right). The dashed line indicated the predicted bound of the “foreground wedge,” showing that with perfect knowledge of our instrument foregrounds can be successfully cleaned well into this region.

Figure 10

Biasing of the power spectrum from complex gain perturbations with amplitude ${\sigma }_g=10\%$, 1% and 0.1%, again for observations of 400–500 MHz. The bias is given as a fraction of the statistical error. Regions where this ratio is less than one (shown in blue) indicate where the systematic errors are subdominant compared to the statistical errors. Again we indicate the foreground wedge with a dashed line, however in this case we note that most of the bias lies within this region.

Figure 11

The response of the primary beam to fractional changes in the $X$ and $Y$ dipole $E$-plane widths. Similar to Fig. 2 we illustrate the transfer from the total intensity and polarized sky, into an instrumental Stokes I combination, however, here we show the derivative with respect to changes in the $E$-plane width of the $X$ and $Y$ feeds. The first two plots show the change of the total intensity response with changes in the $E$-plane of the $X$ and $Y$ dipoles; the second two plots show the changes in the polarization response, again corresponding to changes in the $X$ and $Y$ feeds. For instance a 1% change in each dipoles width changes each response by 1% of the corresponding plot (to first order).

Figure 12

Power spectrum biasing for 10%, 1% and 0.1% shifts from the fiducial $E$-plane width. These biases are given in units of $\sigma$ for each band, values greater than one indicate where this systematic error dominates the statistical error. These are the biases of the minimum variance estimator (so as to avoid issues with the power spectrum deconvolution). Here we can see that unknown fluctuations in the beamwidth of more than 0.1% give rise to significant power spectrum biases.

Figure 13

This figure shows the fractional error on each power spectrum bin, for different frequency bands between 400 and 800 MHz. This clearly illustrates the increasing angular resolution as we move to higher frequencies. Again, the red dashed lines indicate the location of the foreground wedge.

Figure 14

Constraints on the expansion history as a function of redshift, shown relative to a fiducial $\mathrm{\Lambda }\mathrm{CDM}$ cosmology. The red line shows the mean expansion history predicted from the Planck constraints on $w_0,w_a$ [60] (combined with Union 2 supernovae data [61]). The grey lines show a selection of expansion histories randomly drawn from the $\text{Planck}+\text{Union}2$ posterior distribution to illustrating the trajectories which are consistent with those data sets. The black errorbars show what could be achieved with a medium sized cylinder experiment which allows us to place constraints directly on $D_M(z)$ and $H_0(z)$. As we can see, the best discrimination comes at low redshift from the 600–800 MHz bands (assuming a $w_0,w_a$ cosmology).

Figure 15

Constraints on the dark energy equation of state. We show the constraints for the example cylinder with Planck only (large, red), and with Planck and Stage II experiments (smaller, blue). The lighter and darker contours for each illustrate the $2\sigma$ and $1\sigma$ bounds respectively.

Figure 16

The top panel shows the polarization direction and fraction of the 600 MHz slice of a polarized simulation of the Galaxy. This clearly demonstrates the effect of Faraday depolarization towards the Galactic center. The lower panel shows the effective correlation length as measured across the sky, smoothed on 10° scales.