Epoch of reionization window. I. Mathematical formalism

The 21 cm line provides a powerful probe of astrophysics and cosmology at high redshifts, but unlocking the potential of this probe requires the robust mitigation of foreground contaminants that are typically several orders of magnitude brighter than the cosmological signal. Recent simulations and observations have shown that the smooth spectral structure of foregrounds combines with instrument chromaticity to contaminate a “wedge”-shaped region in cylindrical Fourier space. While previous efforts have explored the suppression of foregrounds within this wedge, as well as the avoidance of this highly contaminated region, all such efforts have neglected a rigorous examination of the error statistics associated with the wedge. Using a quadratic estimator formalism applied to the interferometric measurement equation, we provide a framework for such a rigorous analysis (incorporating a fully covariant treatment of errors). Additionally, we find that there are strong error correlations at high spatial wave numbers that have so far been neglected in sensitivity derivations. These error correlations substantially degrade the sensitivity of arrays relying on contributions from long baselines, compared to what one would estimate assuming uncorrelated errors.

Figure 1

A schematic of the EoR window in the cylindrical $k_{\perp }{k}_{\parallel }$ Fourier plane. At the lowest $k_{\perp }$, errors increase because of limits on an instrument’s field of view. High $k_{\perp }$ modes are probed by the longest baselines of an interferometer array, and the sensitivity drops to zero beyond $k_{\perp }$ scales corresponding to these baselines. Spectral resolution limits the sensitivity at large $k_{\parallel }$. The lowest $k_{\parallel }$ are in principle limited by cosmic variance, but in practice the larger concern is limited bandwidth and the foreground contamination, which intrinsically resides at low $k_{\parallel }$. As one moves towards higher $k_{\perp }$, however, the foregrounds leak out to higher $k_{\parallel }$ in a characteristic shape known as the “foreground wedge." The remaining parts of the Fourier plane are thermal-noise dominated, allowing (with a large collecting area or a long integration time) a clean measurement of the power spectrum in this “EoR window."

Figure 2

Angular Fourier coordinate $u$ probed by a variety of baseline lengths, plotted as a function of observing frequency. Taking a Fourier transform of the frequency spectrum of data from a single baseline essentially amounts to taking a Fourier transform along a solid line in this figure. This is a good approximation to taking a Fourier transform along the “true” frequency axis for short baselines.

Figure 3

Expected foreground bias [Eq. (27)] for the basic estimator defined in Sec. 4, where ${\mathbf{E}}_{\alpha }\propto {\mathbf{N}}^{-1}{\mathbf{C}}_{,\alpha }{\mathbf{N}}^{-1}$. For this basic estimator, the chromatic nature of the instrument results in foreground contamination in the form of a characteristic wedge at high $k_{\perp }$.

Figure 4

Sample window functions on the $k_{\perp }{k}_{\parallel }$ plane for the basic estimator defined by ${\mathbf{E}}_{\alpha }\propto {\mathbf{N}}^{-1}{\mathbf{C}}_{,\alpha }{\mathbf{N}}^{-1}$. The leftmost plot shows a set of window functions centered at $k_{\parallel }=0.25h{\mathrm{Mpc}}^{-1}$; the middle plot shows a set centered at $k_{\parallel }=0.66h{\mathrm{Mpc}}^{-1}$; the rightmost plot at $k_{\parallel }=3.28h{\mathrm{Mpc}}^{-1}$. Within each plot, the window functions are centered, from left to right, at $k_{\perp }=0.0042h{\mathrm{Mpc}}^{-1}$, $k_{\perp }=0.024h{\mathrm{Mpc}}^{-1}$, and $k_{\perp }=0.078h{\mathrm{Mpc}}^{-1}$. The black solid line in each plot indicates the rough extent of the foreground wedge [Eq. (13) with ${\theta }_0=\pi /2$]. For window functions centered at high $k_{\perp }$, there is substantial elongation in the $k_{\parallel }$ direction, causing higher $k_{\parallel }$ modes to pick up foreground power, resulting in a foreground-contaminated wedge.

Figure 5

Expected foreground error bars [given by ${({\mathbf{\Sigma }}_{\alpha \alpha })}^{1/2}$ for the $\alpha$th band power or $k_{\perp }{k}_{\parallel }$ cell] for the basic estimator defined by ${\mathbf{E}}_{\alpha }\propto {\mathbf{N}}^{-1}{\mathbf{C}}_{,\alpha }{\mathbf{N}}^{-1}$. The foreground wedge also shows up in the error bars of a power spectrum measurement.

Figure 6

Sections of the measurement error correlation matrix [Eq. (52)] for our basic ${\mathbf{E}}_{\alpha }\propto {\mathbf{N}}^{-1}{\mathbf{C}}_{,\alpha }{\mathbf{N}}^{-1}$ estimator. Each plot shows the measurement error correlation between all $k_{\parallel }$ bins for fixed $k_{\perp }$ (from left to right, $k_{\perp }=0.0042h{\mathrm{Mpc}}^{-1}$, $0.050h{\mathrm{Mpc}}^{-1}$, and $0.13h{\mathrm{Mpc}}^{-1}$). It is often assumed in the literature that errors in two $k_{\parallel }$ cells are uncorrelated if the cells are more than $\mathrm{\Delta }{k}_{\parallel }\sim 2\pi \frac{H_{0}E(z)}{c(1+z{)}^2}\frac{1}{B_{\text{band}}}$ apart. Overlaid near the bottom left corner of each plot is a semitransparent square of size $\mathrm{\Delta }{k}_{\parallel }\times \mathrm{\Delta }{k}_{\parallel }$. Comparing the sizes of these squares to the width of the off-diagonal correlations, one sees that such an assumption holds only at low $k_{\perp }$. At high $k_{\perp }$ the errors are highly correlated, reducing the number of independently measurable modes.

Figure 7

Effective number of independent cells $N_{\text{eff}}$ as a function of $N_c$, the number of Fourier cells included in an averaging of $k_{\parallel }$ modes (going from low to high $k_{\parallel }$). The approximately linear relationship seen for the low $k_{\perp }$ curve is indicative of uncorrelated modes. On the other hand, the high $k_{\perp }$ curve approaches linearity only when high $k_{\parallel }$ modes dominate the average. Its initially slow increase at low $k_{\parallel }$ shows that the modes are highly correlated. Thus, for power spectrum sensitivity calculations to be reliable, one must take error correlations into account or risk overestimating an instrument’s sensitivity.