---
title: "Reading how HERA kept foregrounds out of its first limits"
description: "Read four sections of the HERA Phase I paper: the wedge and buffer, leakage past the horizon, instrument spectral structure and signal-loss corrections."
canonical_url: "https://sandboxphysics.com/stories/cosmic-dawn-21cm/reading-hera-foreground-avoidance/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/cosmic-dawn-21cm/reading-hera-foreground-avoidance/"
last_modified: "2026-11-12"
---

[Hearing the first stars in 21 cm hydrogen](https://sandboxphysics.com/stories/cosmic-dawn-21cm/)

# Reading how HERA kept foregrounds out of its first limits

Sandbox Physics · 2026-11-12

HERA's Phase I limits were obtained without subtracting any foreground. The team kept only modes beyond the horizon plus a $200\,\mathrm{ns}$ buffer, chose three stretches of sky with fainter foregrounds, suppressed instrument systematics by about two orders of magnitude, and corrected the power for a few percent of measured signal loss. This guide reads the four places in the paper where those choices are made and shows what each figure does and does not establish.

It assumes the delay spectrum and the wedge, from [how one baseline separates foregrounds from hydrogen](https://sandboxphysics.com/stories/cosmic-dawn-21cm/delay-spectrum-and-foreground-wedge/), and bandpowers, windows and injection tests, from [turning delay power into a bandpower](https://sandboxphysics.com/stories/cosmic-dawn-21cm/power-spectrum-window-and-signal-loss/). For how the final limits constrain reionization, read the companion guide [reading HERA's upper limits](https://sandboxphysics.com/stories/reionization/reading-hera-upper-limits/).

## Which version, and how claims are labelled

The paper is HERA Collaboration, "First Results from HERA Phase I: Upper Limits on the Epoch of Reionization 21 cm Power Spectrum", published in 2022 as ApJ 925, 221. This guide reads the first arXiv version, 2108.02263v1, in full. It was not compared line by line with the journal text, so section, equation and figure numbers below refer to v1. [HERA Collaboration 2022, arXiv v1](https://arxiv.org/abs/2108.02263v1)

Each point below carries one of four labels.

- Published result: measured or stated in the paper itself.
- Work in preparation: cited by the paper as "in prep." when v1 appeared. Check whether it has since been published before relying on its details.
- Model prediction: the Lab's simulation, not HERA data.
- Open question: something the paper itself leaves unexplained.

## Section 4.3 and Figures 14 and 15: the wedge in real data

Start with the statement that no foreground subtraction is performed. Everything that follows is avoidance. [HERA Collaboration 2022, §4.3](https://arxiv.org/abs/2108.02263v1)

Published result. Because HERA's dishes point straight up, the array cannot steer away from bright sources. Instead the team chose three fields in local sidereal time, 1.25 to 2.7 h, 4.5 to 6.5 h and 8.5 to 10.75 h, which avoid the radio galaxy Fornax A and the Galactic anticentre. Field 1 has the least foreground power and gives the best limits.

Published result. Figures 14 and 15 show the cylindrical power $P(k_\perp,k_\parallel)$ for each band and field. The dashed line is the horizon, the solid line the horizon plus $200\,\mathrm{ns}$, which is $\Delta k_\parallel=0.11\,h\,\mathrm{Mpc^{-1}}$ at $z=7.9$ and $0.10$ at $z=10.4$. The lower panels divide by the expected noise; for $k_\parallel>0.2\,h\,\mathrm{Mpc^{-1}}$ the root-mean-square of that ratio is one to within about 3 %, so the high-$k_\parallel$ data are mostly noise. Negative bandpowers appear as white cells.

Published result, with an open question. Power appears beyond the horizon, mostly at low $k_\perp$. The paper attributes part of it to the taper's footprint, says that this cannot explain all of it, and cites a speculation that residual cross-coupling between antennas is responsible. In Band 2, Field 2 there is also a slight excess beyond the buffer whose cause the paper calls unclear, with low-level unflagged radio interference as one possibility.

When you look at these figures, check three things: where the dashed and solid lines sit, whether the bright region stays below the solid line at each $k_\perp$, and where the noise ratio in the lower panels departs from one.

## Figure 16: how much dynamic range was reached

Published result. Figure 16 averages the power in shells of $|k|$ without the buffer cut and divides by the power at $k=0$. The pipeline reaches about $10^{9}$ between the foreground peak and the noise floor at high $k$. [HERA Collaboration 2022, §4.3, Fig. 16](https://arxiv.org/abs/2108.02263v1) The paper calls this necessary but not sufficient for a detection.

## Where beam chromaticity hides in this paper

The v1 text never names beam chromaticity. You have to recognise the mechanism under other names.

The general mechanism is in the methods literature. Any frequency structure in the instrument response, as well as in the source, widens the delay kernel and pushes foreground power above the horizon. [Parsons et al. 2012, §2.1](https://arxiv.org/abs/1204.4749v1) A dish's sidelobes change with frequency, so bright sources far from the pointing centre are the worst offenders. [Furlanetto, Oh and Briggs 2006, §9.3](https://arxiv.org/abs/astro-ph/0608032v2)

Published result. In HERA's paper the same physics appears as leakage. Section 4.3 notes that radio interference and residual instrumental gains leak foreground modes to higher $k_\parallel$, which is one reason for choosing quieter fields. Section 3.8.1 describes polarized emission from a pulsar, PSR J0742-2822, whose Faraday rotation makes it oscillate with frequency. It is visible in the linear polarizations but not appreciably in pseudo-Stokes I, and the paper notes that asymmetry between the two feeds' beams could leak it into Stokes I at a suppressed level. [HERA Collaboration 2022, §3.8.1, Fig. 9](https://arxiv.org/abs/2108.02263v1)

Model prediction. The Lab's chromatic-beam preset puts an ideal $14\,\mathrm m$ Airy beam whose width scales as $1/\nu$ on a $43.8\,\mathrm m$ baseline with a $200\,\mathrm{ns}$ buffer. In the modes the analysis uses, the leaked foreground is $1.5\times10^{3}$ times the hydrogen power. Freezing the beam at its band-centre shape lowers that to $0.20$. This shows the size of the mechanism in a schematic sky; it is not a statement about HERA's beam.

Open question. The paper does not separate how much of HERA's supra-horizon leakage comes from beam chromaticity rather than from cross-coupling or gains.

## Section 3.8: suppressing the instrument's own echoes

Published result. HERA Phase I has reflections in its $150\,\mathrm m$ coaxial cables, with sub-reflections spanning $100$ to $1200\,\mathrm{ns}$, and a cross-coupling systematic that sits at high delay, is nearly constant in time and is roughly symmetric in delay. The team fitted $25$ reflection terms between $150$ and $1500\,\mathrm{ns}$, and filtered the cross-coupling on baselines with an east-west projection of at least $14\,\mathrm m$. Together these steps suppress instrumental systematics by about two orders of magnitude for $0.1<k<0.8\,h\,\mathrm{Mpc^{-1}}$. [HERA Collaboration 2022, §3.8](https://arxiv.org/abs/2108.02263v1)

Read this section as the reason the window is usable at all. Without it these echoes would sit right where the hydrogen is measured. Read it also as a source of signal loss: a filter that removes a time-steady component at high delay can remove a little sky signal too, and the paper corrects for that in Section 3.11.

Work in preparation. The paper states that this step is vetted in a validation pipeline described by Aguirre et al., in preparation at the time.

## Sections 3.10 and 3.11: how much signal the pipeline removed

Published result. Averaging nearly redundant baselines coherently can make the signal partly cancel. HERA measures this with a metric $\Delta\chi$ that compares coherent and incoherent averages of foreground power at zero delay; the average loss is about 1 %. [HERA Collaboration 2022, §3.10, Fig. 10](https://arxiv.org/abs/2108.02263v1)

Published result. Table 2 lists four losses, given here for Band 1 with Band 2 in brackets: absolute calibration 9 % (8 %), cross-coupling filtering 3 % (1 %), time averaging 1 % (1 %) and redundant averaging 1 % (1 %). All are treated as independent of scale and of the reionization model, some by argument and some by test, and all are corrected in the final power spectra. [HERA Collaboration 2022, §3.11, Table 2](https://arxiv.org/abs/2108.02263v1)

Work in preparation. Section 5 summarizes a near end-to-end simulation by Aguirre et al. that recovered a realistic model for $k\ge0.192\,h\,\mathrm{Mpc^{-1}}$, and a semi-blinded run of an independent pipeline that agreed for $k$ above about $0.2$. [HERA Collaboration 2022, §5](https://arxiv.org/abs/2108.02263v1)

Model prediction. The Lab shows why the choice of method matters. Wedge avoidance returns $99.9\,\%$ of an injected signal; subtracting a third-order polynomial returns $82\,\%$.

## Section 4.4 and Figures 17 and 18: from avoidance to limits

Published result. When the cylindrical power is averaged into shells of $|k|$, every cell with $k_\parallel$ below the horizon plus buffer gets zero weight. Figure 17 shows the result with $2\sigma$ error bars and negative bandpowers set to zero. The best limits come from Field 1: $(30.76\,\mathrm{mK})^2$ at $k=0.192\,h\,\mathrm{Mpc^{-1}}$, $z=7.9$, and $(95.74\,\mathrm{mK})^2$ at $k=0.256$, $z=10.4$. [HERA Collaboration 2022, §4.4](https://arxiv.org/abs/2108.02263v1)

Published result. Figure 18 shows the Band 2, Field 1 windows. Their 16th to 84th percentile half-width is $0.031\,h\,\mathrm{Mpc^{-1}}$, and the lowest window has no response below $0.128$ because of the buffer. [HERA Collaboration 2022, Fig. 18](https://arxiv.org/abs/2108.02263v1)

A detail to read carefully. The text says the bandpowers are sampled at "$\Delta k=0.64\,h\,\mathrm{Mpc^{-1}}$, twice the native spacing". The public files and the paper's own Tables 3 and 4 use a grid of $0.064$, and the public notebook gives the spacing as $0.064$. This guide reports the printed value and does not change it. The method guide in this series uses the files directly.

A second detail. Equation 12 writes the normalization as $X^2Y/(\Omega_{pp}B)$, the usual form in which $X$ converts angle and $Y$ converts frequency to distance. Equations 13 and 14 then define $X$ through $c(1+z)^2/(\nu_{21}H)$, the line-of-sight factor, and $Y$ as the transverse comoving distance. Checking units resolves it: only angle squared times frequency gives $h^{-3}\mathrm{Mpc^3}$ from $\mathrm{sr\,Hz}$, and hera\_pspec computes the factor as the transverse conversion squared times the line-of-sight conversion. [hera\_pspec conversions.py](https://github.com/HERA-Team/hera_pspec/blob/30c77f02567e87a2db36cdc2f92adef4552a43f1/hera_pspec/conversions.py)

## Checklist for reading the paper yourself

- Confirm that the limits use avoidance, not subtraction, and find the $200\,\mathrm{ns}$ buffer in Figures 14 and 15.
- For each figure, ask whether it shows a measurement, a model or a test.
- Separate what the paper measured from what it cites as in preparation.
- Look for spectral structure from the instrument under the names leakage, gains, reflections and cross-coupling, even where beam chromaticity is not named.
- Read Table 2 as corrections already applied, and check that the limit you quote is the corrected one.
- When comparing a model, use the windows in Figure 18 or the public files, and the same rule $U=\max(\hat\Delta^2,0)+2\sigma$.

## Try it in the Lab

[21 cm cosmic dawn and radio foregrounds](https://sandboxphysics.com/labs/cosmic-dawn-21cm-foregrounds/): Use the chromatic-beam preset, then freeze the beam and compare the leakage.

## Sources

1. [HERA Collaboration 2022, arXiv v1](https://arxiv.org/abs/2108.02263v1)
2. [Parsons et al. 2012, §2.1](https://arxiv.org/abs/1204.4749v1)
3. [Furlanetto, Oh and Briggs 2006, §9.3](https://arxiv.org/abs/astro-ph/0608032v2)
4. [hera\_pspec conversions.py](https://github.com/HERA-Team/hera_pspec/blob/30c77f02567e87a2db36cdc2f92adef4552a43f1/hera_pspec/conversions.py)

## Keep reading

Hearing the first stars in 21 cm hydrogen · Reading path

### From hydrogen brightness to a HERA upper limit

1. [Article **Finding faint early hydrogen under a bright radio sky**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/finding-hydrogen-under-radio-foregrounds/)
2. [Article **Why early hydrogen glows or casts a shadow at 21 cm**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/spin-temperature-and-21cm-brightness/)
3. [Article **How one baseline separates smooth foregrounds from hydrogen**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/delay-spectrum-and-foreground-wedge/)
4. [Article **Turning delay power into a bandpower without losing the signal**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/power-spectrum-window-and-signal-loss/)
5. [Article **Recompute HERA's upper limits and compare a model through its window**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/recompute-hera-window-comparison/)
6. [You are here **Reading how HERA kept foregrounds out of its first limits**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/reading-hera-foreground-avoidance/)