The HERA-19 Commissioning Array: Direction-dependent Effects

The cosmological signal of interest for 21 cm cosmology studies is effectively unpolarized, and we therefore use the pseudo-Stokes³³ I visibility to measure it (e.g., Moore et al. 2013); this is defined V^I = Vⁿⁿ + V^ee, which is the trace of ${ \mathcal V }$:

$$\begin{eqnarray}\begin{array}{rcl}{V}_{{ij}}^{I}(\nu ) & = & \mathrm{Tr}({{ \mathcal V }}_{{ij}})=\int \mathrm{Tr}({{ \mathcal J }}_{i}{ \mathcal C }{{ \mathcal J }}_{j}^{\dagger })\exp (-2\pi i\nu {\boldsymbol{b}}\cdot \hat{s}/c){\rm{d}}{\rm{\Omega }}\\ & = & \int ({{ \mathcal M }}_{00}I+{{ \mathcal M }}_{01}Q+{{ \mathcal M }}_{02}U+{{ \mathcal M }}_{03}V)\\ & & \times \exp (-2\pi i\nu {\boldsymbol{b}}\cdot \hat{s}/c){\rm{d}}{\rm{\Omega }},\end{array}\end{eqnarray} \tag{ 7 }$$

where I, Q, U. and V are the true Stokes sky and are functions of direction and frequency, and ${{ \mathcal M }}_{{ab}}(\hat{s},\nu )$ are the instrumental Mueller matrix elements:

$$\begin{eqnarray}&&{{ \mathcal M }}_{{ab}}(\hat{s},\nu )=\mathrm{Tr}({\sigma }_{a}{ \mathcal J }{\sigma }_{b}{{ \mathcal J }}^{\dagger })\end{eqnarray} \tag{ 8 }$$

and σ_i are the Pauli matrices (where the indices are reordered from the quantum mechanical convention to an order that gives the ordering of the Stokes vector as (I, Q, U, V); see, e.g., Shaw et al. 2015).

We simulated the HERA feed, faceted parabolic dish, and analog signal chain using CST³⁴ to generate the complex ${\boldsymbol{E}}$-field receptivity patterns, as described in Fagnoni & de Lera Acedo (2016; also see public HERA Memo #21³⁵ ), and then formed ${ \mathcal J }$ and ${ \mathcal M }$ as described above. Examples of ${{ \mathcal M }}_{{ij}}$ at 120 MHz and 160 MHz (our low and high bands of interest; see Section 3.2) are shown in Figure 1, projected in the R.A./decl. basis. Note that this basis has a singularity at the South Pole, leading to wide-field asymmetries in Q and U. Due to the large spread in dynamic ranges between ${{ \mathcal M }}_{00}$, other diagonal terms, and off-diagonal terms, we use separate color maps for each. All of the dynamic ranges are normalized to the peak of ${{ \mathcal M }}_{00}$, which is 1 at zenith. The off-diagonal terms are 2-8 orders of magnitude less than the diagonal terms.

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The key for these matrices are the mappings of Stokes parameters into pseudo-Stokes visibilities, following

$$\begin{eqnarray}{{ \mathcal M }}_{{ab}}(\hat{s},\nu )=\left(\begin{array}{cccc}I\to {V}^{I} & I\to {V}^{Q} & I\to {V}^{U} & I\to {V}^{V}\\ Q\to {V}^{I} & Q\to {V}^{Q} & Q\to {V}^{U} & Q\to {V}^{V}\\ U\to {V}^{I} & U\to {V}^{Q} & U\to {V}^{U} & U\to {V}^{V}\\ V\to {V}^{I} & V\to {V}^{Q} & V\to {V}^{U} & V\to {V}^{V}\end{array}\right),\end{eqnarray} \tag{ 9 }$$

where pseudo-Stokes visibilities are formed as

$$\begin{eqnarray}&&\left(\begin{array}{c}{V}^{I}\\ {V}^{Q}\\ {V}^{U}\\ {V}^{V}\end{array}\right)=\displaystyle \frac{1}{2}\left(\begin{array}{cccc}1 & 0 & 0 & 1\\ 1 & 0 & 0 & -1\\ 0 & 1 & 1 & 0\\ 0 & -i & i & 0\end{array}\right)\left(\begin{array}{c}{V}^{{nn}}\\ {V}^{{ne}}\\ {V}^{{en}}\\ {V}^{{ee}}\end{array}\right).\end{eqnarray} \tag{ 10 }$$

At low frequencies and the large scales probed by many low-frequency interferometers, Stokes I is extremely bright compared to the other Stokes parameters (Bernardi et al. 2009, 2010; Jelić et al. 2014, 2015; Asad et al. 2015; Kohn et al. 2016; Lenc et al. 2017; Moore et al. 2017). Moreover, only a few polarized point sources have been observed at frequencies below 300 MHz (Bernardi et al. 2013; Asad et al. 2016; Lenc et al. 2017). Farnes et al. (2014) showed evidence for systematic depolarization of steep-spectrum point sources toward low frequencies, causing low polarization fractions (≪1%) below 300 MHz.

The "linear polarization leakage beam" is shown for the two central frequencies of this analysis in Figure 2. This quantity is the magnitude of the spin-2 function ${{ \mathcal M }}_{{IQ}}+i{{ \mathcal M }}_{{IU}}$ and represents the amplitude of the direction-dependent leakage of Stokes Q and U into I.

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These factors make the first row of ${ \mathcal M }$, which represents $I\to {V}^{I},{V}^{Q},{V}^{U},{V}^{V}$, the most interesting for low-frequency polarized power spectra, as with limited calibration we can expect leakage from Stokes I into the other Stokes parameters to dominate over Stokes Q, U, and V emission alone.

We produced simulations ${ \mathcal V }$ using our fully polarized formalism for the HERA-19 commissioning array, described below, using an unpolarized model of the low-frequency sky from the Global Sky Model (GSM; de Oliveira-Costa et al. 2008; Price 2016; Zheng et al. 2017) at the appropriate R.A. range to match our observations. These simulations are based on the same source code as Martinot et al. (2018).

Forming power spectra from these visibilities allowed for a comparison of our data to a "leakage only" regime. We discuss the process for forming power spectra in Section 3.3, and the simulated power spectra are shown in comparison to those from data in Section 4.

In addition to the mixing of Stokes parameters due to the primary beam, it is possible to mix them in a direction-independent way. Calibration errors are capable of leaking signal between pseudo-Stokes visibilities independent of the sky (Thompson et al. 2008). Again focusing on the {V^I, V^Q, V^U, V^V} components of this leakage, we have:

• 1.  
  ${V}^{I}\to {V}^{Q}$ occurs through errors in calibrating the complex voltage gain factors for each dipole arm.
• 2.  
  ${V}^{I}\to {V}^{U}$ occurs through the sum of off-diagonal gain terms (D-terms; the receptivity of dipole arm "n" to an electric field vector aligned with arm "e" and vice versa).
• 3.  
  ${V}^{I}\to {V}^{V}$ occurs through the difference in D-terms between two feeds.

We detail how we obtain the direction-independent Jones terms in the next section.

To identify samples contaminated by RFI, a two-dimensional median filter in time and frequency was applied to the visibility data to smooth out high pixel-to-pixel variations, and remove significant outliers that were likely unphysical. The variance of the resulting data was computed, and points with a z-score greater than 6 (i.e., points where the value is more than 6σ away from the mean) were flagged as initial seeds for RFI extraction. A two-dimensional watershed algorithm was applied using these seeds as starting points, enlarging the regions of RFI-contamination to neighboring pixels with z-scores greater than 2, until all such pixels were flagged. Figure 4 shows the fractional RFI flag occupancy per time (displayed in LST) and frequency across the 8 days of observations. The majority of the band is relatively clear of RFI. Some clear features are: the FM radio band (below 110 MHz), ORBCOMM satellite communications (137 MHz), an ISS downlink (150 MHz), and VHF TV channels (above 170 MHz).³⁶ The galaxy, when transiting zenith at LST ≈ 17.75 hr, is so bright that it appears to degrade our ability to flag RFI.

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Four antennas were identified during the commissioning as having anomalous behavior. These are marked with red "X"s in Figure 3 and were omitted from further analysis. Before calibration, we manually flagged the edges of the band (below 110 MHz and above 190 MHz), where spectral behavior is dominated by the high and low pass filtering in the HERA signal chain (DeBoer et al. 2017).

HERA is designed to be calibrated using redundant calibration techniques (Dillon & Parsons 2016), but for this preliminary view of HERA commissioning data, we used image-based calibration. Future studies with deeper integrations targeting EoR detections will take advantage of redundancy to obtain more precise calibration solutions (DeBoer et al. 2017). We used the casa (McMullin et al. 2007) package for calibration, taking advantage of its CLEAN, gaincal and bandpass functions. We first converted from HERA's native miriad to a uvfits file format using pyuvdata (Hazelton et al. 2017); this could then be ingested by casa.

The brightest calibration sources near the decl. -30° stripe—for example, those used in previous PAPER analyses like Pictor A (Jacobs et al. 2013)—were not available for this observing window (10.5–23 hr R.A.), and the few long baselines in the array available made calibration using individual fainter point sources difficult. We therefore developed a calibration method using the Galactic Center (GC; taken to be at α, δ = 17^h 45^m 40^s, −29^d 0^m 28^s) as modeled by the GSM. Specifically, we selected four minutes of data centered at the transit of the GC to use for calibration. The visibilities were phased from drift-scan mode to a single phase-center chosen to be the LST at the start of the observations. Because this phasing imperfectly approximates the tracking telescope assumed by casa, the length of the observation was chosen to minimize the effects of beam-dependent gain variations as the GC transited. The calibration was done as a two-step process. First, we built an initial model with the GC as an unpolarized, flat-spectrum source with flux density scaled to a reference point of 1 Jy. This allowed us to solve for the large antenna-based delay terms using gaincal (gaintype='K'; typically tens of nanoseconds) and a complex bandpass using bandpass. A single solution was obtained for the 4 minute observation file for both calibration types, and for the bandpass solution a solution was obtained for each unflagged ∼100 kHz channel, resulting in a complex, frequency-dependent gain for each feed. With this first calibration in place, the second step was to interactively CLEAN the image to obtain a more accurate model of the GC extended structure. We still assumed an unpolarized source, but allowed multiple components within a two degree radius centered on the GC. This was followed by a second round of delay and bandpass calibration to the multi-component extended model, completely analogous to the first round of calibration. At this point, an overall frequency-dependent amplitude was required to scale the gains from the arbitrary 1 Jy normalization. For this we used our simulation of the GC from the GSM (converted to units of Jy) to determine a single, spectrally smooth function for all antennas to make the spectrum of the observations match the simulations.

Clearly, this is an incomplete calibration model. The assumption that the GC is unpolarized is probably adequate, due to the large optical depth toward the GC (Oppermann et al. 2012), resulting in near-complete depolarization in the plane of the Galaxy (Wolleben et al. 2006). Moreover, we expect significant beam depolarization due to the large solid angle of the synthesized beam (see Table 1). Other assumptions are less obviously correct. The GC structure is only partially modeled, and we have assumed the GSM provides an accurate calibration. We have also assumed that the direction-independent Jones matrix is diagonal. Although in principal casa is capable of solving for D-terms using the polcal task, we did not find that the solutions obtained using only an unpolarized GC model were stable or improved the image quality. The lack of polarized point sources as calibrators limits our interpretive power for addressing some aspects of polarization leakage, which we discuss in Section 4.

The calibration we have obtained serves to correct an initial large cable delay per antenna, which aligns all of the power spectra at zero delay, and sets the overall flux scale. The resulting complex antenna-based gains are shown in Figure 5. The gain amplitudes are clearly very similar in shape, with one outlier, and they cluster with 25% of each other. After removing the phase due to the delay term, the resulting phases show only small variations around their mean. The derived bandpasses are clearly spectrally smooth, and thus, even if there are errors, we expect that they will not add additional spectral structure to the power spectrum (see Section 3.3). These gains were applied to all 8 nights of observations. It was found that this produced smaller day-to-day calibration variability than calibrating each day separately to the GC. An estimate of the remaining variation is discussed in Section 4.3.

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Figure 6 shows the effect of calibration on the visibilities of three nominally redundantly spaced baselines. Shown in that figure are the phases of three Vⁿⁿ visibilities from 14.7 m baselines before and after calibration. There were no shared antennas between the visibilities shown. The qualitative agreement is obvious, providing a consistency check on the solutions, and showing our sky-based model achieves redundancy without assuming it. However, small-scale variations between baselines are still seen. This not unexpected; the antennas are likely to be non-identical, and we have evidence based on closure phase (which is insensitive to calibration errors) that redundant baselines do not see the sky identically (Carilli et al. 2018).

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In Figure 7, we show images formed from the simulated pseudo-Stokes visibilities (top panels) and our observations (bottom panels). These are multi-frequency synthesis images, where we used all unflagged frequencies on either side of the band edges, from 115 to 188 MHz. The primary beam has not been deconvolved. All images shown were produced using the same four-minute interval used for calibration. Note that at HERA's latitude the GC transits 2° north of zenith, while the HERA primary beam has a FWHM of ∼10° at 150 MHz (Neben et al. 2016). For the simulated visibilities, we flagged the same antennas and channels as in the data. As expected for a compact array, the Stokes I images capture only a low-resolution view of the GC. The simulated and observed visibilities form remarkably similar images in Stokes I, and Q and U clearly share features in common, due to leakage from I to Q and U through the primary beam (recall that the simulations are unpolarized). In Stokes V, the simulated map has a significantly smaller amplitude of features compared to the actual image generated from data. The presence of emission at the location of the Galactic Center not due to primary beam leakage is consistent with direction-independent gain errors at the few percent level in amplitude (for Stokes Q) and D-terms at ∼1% relative to the diagonal gains (for Stokes V) (Thompson et al. 2008). Note that the Stokes U image is broadly consistent with a large fraction of power coming from I leakage through the primary beam, though there is some additional power as well. We consider the implications for the power spectrum in Section 4.

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Power spectra were formed in a fashion similar to the method used in Pober et al. (2013) and Kohn et al. (2016). The actual implementation of the code is available as part of the GitHub HERA-Team repository hera_pspec.³⁷ We briefly review the method here.

Parsons et al. (2012b) defined the delay transform as the Fourier transform of a visibility for baseline ij and pseudo-Stokes parameter P along the frequency axis

$$\begin{eqnarray}&&{\tilde{V}}_{{ij}}^{P}(\tau ,t)=\int d\nu {\tilde{V}}_{{ij}}^{P}(\nu ,t){e}^{2\pi i\nu \tau }.\end{eqnarray} \tag{ 11 }$$

We selected two relatively RFI-free 20 MHz sub-bands (Figure 4); 115 to 135 MHz and 152 to 172 MHz, henceforth referred to as the "low band" and the "high band," in which to compute the power spectrum. An extremely conservative cut on RFI was used such that any integration that had RFI flagged in the 20 MHz sub-band was excluded from the analysis. This cut was performed separately for the two bands. In each case, approximately 35% of the data were retained after this cut. The bands were then multiplied by a Blackman–Harris window, centered on their central frequencies, before Fourier transforming, in order to minimize sidelobes. This windowing led to a noise-effective bandwidth of 10 MHz. We note that this bandwidth is appropriate for EoR analyses because the H i signal is, to a reasonable approximation, coeval over the corresponding redshift range (Furlanetto et al. 2006). However, this resolution (approximately 100 ns in delay as compared to 194 ns for the longest baseline in this study) does limit our ability to resolve certain features in the power spectrum. We also note that using a Blackman–Harris window will induce a correlation between adjacent τ modes; this should be kept in mind when interpreting plots, as all delay bins are plotted.

The power at each delay-mode and baseline can be represented in terms of their respective Fourier components k_∥ and k_⊥ (Parsons et al. 2012b; Thyagarajan et al. 2015a):

$$\begin{eqnarray}\begin{array}{rcl}P({k}_{\parallel },{k}_{\perp }) & \approx & | {\tilde{V}}_{{ij}}^{P}(\tau ){| }^{2}\displaystyle \frac{{X}^{2}Y}{{{\rm{\Omega }}}_{{pp}}{B}_{{pp}}}{\left(\displaystyle \frac{{c}^{2}}{2{k}_{B}{\nu }^{2}}\right)}^{2},\\ {k}_{\parallel } & = & \displaystyle \frac{2\pi {\nu }_{21\mathrm{cm}}H(z)}{c{\left(1+z\right)}^{2}}\tau ,\\ {k}_{\perp } & = & \displaystyle \frac{2\pi }{D(z)\lambda }b\end{array}\end{eqnarray} \tag{ 12 }$$

for: cosmological bandwidth B_pp and cosmological angular area of the beam Ω_pp, ${\nu }_{21\mathrm{cm}}\approx 1420\,\mathrm{MHz}$, baseline length b, wavelength of observation λ, Hubble parameter H(z), transverse comoving distance D(z), and redshift-dependent scalars X and Y (Parsons et al. 2012a). Note that the angular area of the beam refers to the diagonal components of the Mueller matrices shown in Figure 1. For further discussion of forming polarized power spectra in k-space, refer to Nunhokee et al. (2017).

To avoid a noise bias when forming the power spectrum, we cross-multiplied consecutive integrations (each having independent noise), rephasing the zenith angle of the latter to the former:

$$\begin{eqnarray}&&| {\tilde{V}}_{{ij}}^{P}(\tau ,t){| }^{2}\approx | {\tilde{V}}_{{ij}}^{P}(\tau ,t)\times {\tilde{V}}_{{ij}}^{P}(\tau ,t+{\rm{\Delta }}t){e}^{i{\theta }_{{ij},\mathrm{zen}}({\rm{\Delta }}t)}| ,\end{eqnarray} \tag{ 13 }$$

where ${\theta }_{{ij},\mathrm{zen}}({\rm{\Delta }}t)$ was the appropriate phasing for baseline ij and Δt = 10.7 s.

Pseudo-stokes power spectra were formed for each pair of integrations, for every baseline, according to Equation (13). Power spectra from baselines of identical lengths were then averaged together for all observation times over 8 days. The resulting "1D" power spectra for each baseline length were then concatenated to form a two-dimensional power spectrum (that is, arranged into the (k_⊥, k_∥) plane). Note that all averaging in this study was performed after forming power spectra, not by averaging visibilities; this incoherent averaging is non-optimal for achieving the greatest sensitivity. Future work will be able to test the features of the polarized beam and foregrounds to much greater depth.

Several features of the power spectra are readily apparent. The first is that foreground emission appears in a relatively narrow band near k_∥ = 0. Another is that the the shape of the power spectrum as a function of k_∥is both sharply peaked and relatively featureless. We note that in a similar study of 2D polarized power spectra in Kohn et al. (2016), PAPER measurements showed a comparably "filled" region of Fourier space out to the horizon delay (i.e., to directions corresponding to zenith angle ± 90°), with some supra-horizon leakage (e.g., Pober et al. 2013) into the EoR window itself. The power spectra in Figures 8 and 9 show similar behavior, though in this case part of the reason is the low resolution (∼100 ns; see Section 3.3) of the delay transform (Equation (11)) due to the small spectral bandwidth, and the small horizon delay associated with the short baselines of the array. Thus, we are not able to verify the prediction of Thyagarajan et al. (2015a) and Neben et al. (2016) that for dishes such as HERA, the region between a delay of about 50 ns (set by the width of the antenna primary beam) and the horizon should be free of foreground emission. Similarly, an "excess" of power near the horizon delay, as predicted by the same authors, is not observable, again due to the blurring of features in k_∥by the resolution. Along the ${k}_{\perp }$ direction in Stokes I, the amplitude declines as a function of ${k}_{\perp }$, as expected for diffuse Galactic emission with a power-law angular power spectrum (larger fluctuation power on large scales). This trend is also observed for the other Stokes parameters as well.

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A notable feature in the observational data is a peak in power above the noise level at a delay of ∼1000 ns, independent of the baseline length or the frequency band. There are ∼150 m coaxial cables connecting the HERA dishes to the correlator³⁸ and we have evidence that some of this power is due to a cable reflection at this stage of the signal chain producing an alias of the foreground signal; see HERA Memo #39³⁹ (Ewall-Wice 2017). However, this signal should appear at a delay corresponding to 1300 ns, or twice the propagation time in the coaxial cable. It appears that most of the signal present here is present at a smaller delay, and its origin is not understood.

In Figure 10, we show a direct comparison between the power spectra of the data and the simulations for the shortest baseline length for all Stokes parameters for both bands. Figure 11 shows a zoom of the same data close to the wedge. Recall that the simulations include only a Stokes I sky component and no simulated calibration errors, so the signal in the simulated pseudo-Stokes Q, U, and V power spectra is due solely to wide-field beam leakage (Figure 1). The Stokes I model is the diffuse emission of the GSM, which should be accurate at the scales probed by a 14.7 m baseline, as the model includes scales down to 1°. The power spectra of the simulated data have been computed in the same way as the data.

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In comparing to the foreground simulations, there are two things to notice. The first is the isolation, or the degree to which the foregrounds remain within the range of ${k}_{\parallel }$ defined by their intrinsic smoothness and by the mode-mixing of the interferometer, and which can be characterized by the width of the foregrounds in k_∥(or τ) space. The second is the dynamic range, defined here as the ratio between the k_∥ = 0 peak and the smallest value of the power spectrum. This smallest value may be limited either by noise (in the case of the data) or by the Blackman–Harris window (for the noise-free simulations). We note that the simulations provide a reasonable standard by which to judge both isolation and dynamic range: in the absence of any systematic effects, the width of the foregrounds in delay space cannot be narrower than that captured by the simulated foreground spectral structure and instrument mode-mixing, and the dynamic range is as expected from the window function. Thus, over a range of about 8 orders of magnitude in the Stokes I power spectrum, limited by the noise in the data, the isolation in the data agrees well with the simulation, arguing that the calibrated data (at the current noise level) do not have significant spectral structure beyond that intrinsically present. It is worth pointing out there is some evidence for broadening of the range in k_∥near the noise floor. For the the other pseudo-Stokes spectra, the isolation is worse: the power spectrum of the data is noticeably wider than the simulation.

Referring to Figure 11, calibrating the raw data to the simulation as described in Section 3.2 reproduces the total power in I very well in the high band, and leads to a total power ∼40% higher than the simulation in the low band. This discrepancy is consistent with the overall amplitude of the calibration derived from the GC disagreeing by ∼20% in the low band. It is not clear whether this is due to errors in the GSM or the primary beam model at low frequencies. In the analysis that follows, we have increased the amplitude of the simulated power spectrum in the low band to agree with the real data because we are primarily concerned with the relative power between different pseudo-Stokes parameters; this rescaling allows for a more even treatment of the two bands.

In the high-band analysis, the beam leakage modeled in the simulation is of roughly equal magnitude for both Q and U; however, the data show a markedly stronger pseudo-Q than U. Because there is no strong reason to believe the sky is highly anisotropic between Q and U over the R.A. included here, this difference argues for another interpretation. The amplitude difference between pseudo-Q and U is ∼10% in the visibilities (rather than the power spectra), which can be accounted for by a ∼5% relative miscalibration in the antenna-based gain amplitudes between the E and N polarized feeds of the antenna. The pseudo-Q visibilities are constructed by differencing the calibrated nn and ee visibilities, which can make relative amplitude differences between the two polarizations more pronounced. Thus, assuming the excess power in pseudo-Q can be attributed to this difference in gain amplitude between the two polarized feeds, we can use pseudo-U as a measurement of the excess polarized power on the sky not accounted for in the beam leakage from the simulations. This interpretation leads to a combined fractional polarization of ∼10%. This result is on the high end of the range of measurements of ∼1.6%–4.5% fractional polarization at 150 MHz on large scales from Jelić et al. (2015) and Lenc et al. (2016). We note that whatever the interpretation of the linear pseudo-Stokes spectra, they do not provide strong evidence for high rotation measure emission, which would be present at k_∥ > 0.1 for ${RM}\gt 10\,\mathrm{rad}\ {{\rm{m}}}^{-2}$ in the high band (Moore et al. 2017), although noise prevents probing levels as deep as those in Asad et al. (2018).

As with pseudo-Q and U, pseudo-V has excess power in the measured power spectra compared to the reference simulations. Measurements in the literature do not suggest a significant amount of large-scale circularly polarized emission at the frequencies measured, so this excess power is most likely due to miscalibration of the instrument. Relative phase errors in the gain solutions of the cross-polarized instrumental visibilities can lead pseudo-U to leak into pseudo-V. However, the measured amplitudes of the two power spectra would imply that this systematic error in phase angle must be of order π/2, or that these phase errors are nearly maximally rotated with respect to the correct value. A more plausible explanation for this excess power in pseudo-V is direction-independent leakage through the D-terms of the Jones matrix, which leak power from pseudo-I (Thompson et al. 2008). As discussed in Section 3.2, D-terms were entirely neglected for the calibration performed in this analysis. Assuming that these D-terms are the source of the leaked power from pseudo-I to pseudo-V, their amplitude relative to the diagonal elements of the Jones matrix would be ∼3%. This is similar to D-term levels from other low-frequency instruments such as MWA-32, which was found to have ∼2% D-terms (G. Bernardi 2019, private communication).

The low-band measurements tell a similar story, with the notable difference that the simulated pseudo-Q and U power spectra exhibit a greater discrepancy than the high-band measurements. This difference may be attributable to the general difficulty of accurately calibrating the low band due to model and beam uncertainties. Performing the same analysis as for the high band, we again find that the inferred polarization fraction from the pseudo-U visibilities is ∼10%, and the excess power in pseudo-Q can be accounted for by gain amplitude errors of ∼5%. The relative amplitude of the D-terms to account for the excess power in pseudo-V is closer to 5%, which is slightly higher than the high band but still plausible given the behavior of similar instruments. These results suggest that future precision calibration efforts should include analysis of D-terms in order to accurately model the instrumental effects on the measured visibilities.

One estimate of the system temperature of the observations was formed from the calibrated values of the autocorrelations. These were compared against the values obtained from the simulation. Over the R.A. range observed, which was heavily weighted toward the GC and much of the Galactic Plane, the system temperature estimated in this way was 1230 K for the high band and 4000 K for the low band, which was consistent with the simulated autocorrelations (DeBoer et al. 2017, also see the public HERA Memo #16⁴⁰ ).

The system temperature was converted to a noise level in the power spectrum according to the formalism in Parsons et al. (2012b), with the inclusion of a baseline-number dependence:

$$\begin{eqnarray}&&{P}_{\mathrm{Noise}}(k)\approx \displaystyle \frac{1}{2{\rm{\Delta }}t\sqrt{{N}_{\mathrm{LST}}}{N}_{\mathrm{days}}{N}_{\mathrm{bl}}}{X}^{2}{{YB}}_{\mathrm{NE}}{{\rm{\Omega }}}_{\mathrm{eff}}{T}_{\mathrm{sys}}^{2}.\end{eqnarray} \tag{ 14 }$$

In the above equation, Δt is integration time, N_LST is the number of LST hours used per day (12 hr), N_bl is the number of baselines (which differed per k_⊥ bin), X and Y are cosmological scalars defined in Parsons et al. (2012b), B_NE is the noise-equivalent bandwidth and Ω_eff is the effective beam area, as defined in Parsons et al. (2014). Figure 12 shows power as a function of baseline length for k_∥ = 0 h Mpc⁻¹ (solid lines) and the average over $| {k}_{\parallel }| \gt 1\,{\rm{h}}$ Mpc⁻¹ (dotted–dashed lines). Though not shown in Figures 8 and 9, the frequency sampling of the instrument is sensitive to delays up to 5000 ns, and this region defines our white noise level. To match the observed noise level of the power spectrum in Stokes I, the high band required an assumed system temperature of 2400 K, higher than that expected based on the sky and beam model from the simulations, or from the measured calibrated autocorrelations. The cause of this excess is not understood. Figure 12 also shows that both the noise and calibration of the instrument were stable at the 20% level in the visibilities (50% in the power spectrum) over the 8 days presented here.

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