---
title: "Reading HERA's 21 cm upper limits"
description: "Read the markers, errors and windows in HERA's power spectrum, recompute two published table rows, and connect an upper limit to early-universe physics."
canonical_url: "https://sandboxphysics.com/stories/reionization/reading-hera-upper-limits/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/reionization/reading-hera-upper-limits/"
last_modified: "2026-11-12"
---

[Starlight and intergalactic hydrogen](https://sandboxphysics.com/stories/reionization/)

# Reading HERA's 21 cm upper limits

Sandbox Physics · 2026-11-12

A HERA upper limit constrains the strength of 21 cm brightness fluctuations over specified spatial scales. Reading it starts with the quantity estimated by each point, followed by its uncertainty and instrumental response. Locating individual ionized regions would also require the spatial phases that a power statistic discards.

Open the [published version of First Results from HERA Phase II](https://doi.org/10.3847/1538-4357/ae2d54), Astrophysical Journal 998:33, dated February 10, 2026. We will use Figure 16 on page 32 and Appendix F, Table 5 on page 41 of this 44-page paper. An [open institutional copy](https://api.repository.cam.ac.uk/server/api/core/bitstreams/d98f6b1c-b4da-41a6-ad94-e4343baacde5/content) provides the full text. The earlier [arXiv:2511.21289v1](https://arxiv.org/abs/2511.21289v1), posted November 26, 2025, contains the same Figure 16 identity but has different pagination. Use the journal page numbers given here.

## Identify the points, errors, noise and earlier limits

Figure 16 has eight panels for eight frequency bands. Lower observing frequency corresponds to higher redshift, so the panels sample different epochs. These results use a roughly two-week subset of Phase II observations. They are not successive photographs of one expanding bubble.

Begin with the lower panel labelled 176 MHz and redshift about seven. The horizontal axis is wavenumber $k$; larger wavenumber means a shorter spatial wavelength. The vertical quantity is

$$
\Delta^2(k)=\frac{k^3P(k)}{2\pi^2},
$$

where $P(k)$ is the three-dimensional power spectrum of brightness fluctuations. This form describes fluctuation strength per logarithmic interval of wavenumber. Its units are $\mathrm{mK^2}$. It is neither a local gas temperature nor an ionized volume fraction.

Pink markers show estimated power. Their vertical bars show the quoted $2\sigma$ errors, where $\sigma$ is the estimated thermal-noise error. Downward arrows indicate an estimate consistent with zero within these $2\sigma$ errors. The thick black dashed curve is labelled as a $1\sigma$ noise level. Purple steps in two panels show earlier Phase I upper limits. The estimate, twice its error, a noise prediction and an upper limit have different roles. Compare quantities of the same kind at compatible redshifts and scales. [Journal Figure 16 and §5.2](https://doi.org/10.3847/1538-4357/ae2d54)

The horizontal width of each error-bar top cap has a separate meaning: it shows the full width at half maximum of the window function. An estimated bandpower combines a range of nearby spatial scales. Its window gives the weights of that combination. A point therefore has a nominal horizontal position and a range of scales to which it responds.

## How a power estimate becomes negative

The squared amplitude of a mode in a noiseless map is nonnegative. A radio measurement must also deal with the positive contribution from receiver noise.

Imagine two observations of the same sky signal, $s+n_1$ and $s+n_2$, where the two noise contributions are independent and have zero mean. Cross-multiplying the observations retains their common signal. The noise cross terms vanish in an ensemble average, but a finite observation still fluctuates above and below that average. A cross-power estimate can consequently be negative. HERA forms cross-spectra between interleaved time subsets; §3.9.4 gives the actual procedure.

Empty pink markers identify negative estimates in Figure 16. A logarithmic vertical axis cannot place them at negative coordinates, so read the marker convention together with the errors and signed values in the table. A negative estimate is a possible outcome of a noisy measurement of a nonnegative underlying power.

## Recompute two rows of Table 5

The four repeated columns are wavenumber, estimated power, a one-sigma thermal error and a two-sigma upper limit. Here are two transcribed rows. All three power columns are in $\mathrm{mK^2}$.

**Table 7.1** Two published HERA rows, retaining the printed values.

| Redshift | Wavenumber $[h\,\mathrm{Mpc^{-1}}]$ | Estimate | One-sigma error | Published limit |
| --- | --- | --- | --- | --- |
| $16.78$ | $0.55$ | $252112$ | $439236$ | $1130584$ |
| $7.05$ | $0.70$ | $-494$ | $887$ | $1775$ |

These rows implement the relation

$$
U=\max\!\left(\widehat{\Delta^2},0\right)+2\sigma.
$$

The hat denotes an estimate from the data; $U$ is the tabulated upper limit. Retain a positive estimate, replace a negative estimate by zero for this operation, then add twice the error. The first row gives

$$
252112+2\times439236=1130584\;\mathrm{mK^2}.
$$

For the second, twice the printed error is $1774\,\mathrm{mK^2}$, compared with the published $1775\,\mathrm{mK^2}$. This final-digit difference is compatible with independently rounded columns; retain the published value. Adding the negative estimate directly would give $1280\,\mathrm{mK^2}$ and would use a different treatment of negative estimates.

This limit concerns fluctuation power for a particular band, scale window and analysis. It does not report a gas temperature of 1775 mK or a bubble diameter. Taking a square root gives a strength scale of about $42.1\,\mathrm{mK}$ for the spatial fluctuations described by that window.

## Convert the horizontal units before comparing scales

The symbol $h$ means $H_0/(100\,\mathrm{km\,s^{-1}\,Mpc^{-1}})$, the dimensionless Hubble parameter. It is not Planck's constant. For an example choice $h=0.674$,

$$
0.70\,h\,\mathrm{Mpc^{-1}}
=0.4718\,\mathrm{Mpc^{-1}},\qquad
\lambda=\frac{2\pi}{k}\simeq13.3\,\mathrm{cMpc}.
$$

The wavelength is a comoving distance, with the overall expansion factor divided out. Both wavenumber notations describe the same physical scale.

A periodic cube of side 20 cMpc has discrete modes, starting at $2\pi/L\simeq0.314\,\mathrm{Mpc^{-1}}$. Four finite mode-family averages in such a cube differ from the instrument-weighted bands in Figure 16. Comparing theory with data requires consistent redshift, units and statistical definitions, followed by the observational window response. Nearby horizontal coordinates are the beginning of that comparison.

## Follow the low-wavenumber excess back to the instrument

Return to the high points on the left of several panels. Smaller wavenumber describes larger spatial scales. Section 5.2 identifies mutual coupling as the main difficulty for this Phase II subset. An antenna reradiates some of the electromagnetic field that it receives; neighboring antennas receive that contribution too. Additional propagation paths introduce frequency structure and leak bright foreground emission into modes used to search for the faint cosmological signal.

This explains how an upgraded instrument can cover a wider frequency range yet yield weaker limits on some scales. Thermal sensitivity and foreground contamination must be considered separately. The paper both extends the redshift coverage of the limits and identifies coupling as a key problem for further analysis. The current result covers this two-week subset; eventual limits from the full Phase II observations also depend on subsequent data and foreground treatment.

## From an upper limit to an astrophysical statement

A model that predicts very strong fluctuations in an observed window can be challenged by a low upper limit. Inferring stellar or gas properties also requires its heating, ionization and spin-temperature history, together with the observational response and statistical comparison. Section 6 begins that larger analysis. The two-row arithmetic above is preparation for reading it, not a rerun of the inference.

The earlier chapter assumed a spin temperature far above the microwave background to convert neutral fraction into brightness. Cold gas can instead produce absorption and alter the fluctuation amplitude. Power strength alone therefore cannot determine the ionized volume fraction or locate each ionized region. [Furlanetto, Oh and Briggs on the temperature dependence of the signal](https://arxiv.org/abs/astro-ph/0608032v2)

Even a clear future power-spectrum detection would leave spatial information to recover. The [two-wave example in the preceding chapter](https://sandboxphysics.com/stories/reionization/reading-a-neutral-hydrogen-map/) already has identical powers with different relative phases. Imaging must retain more of this information through finite angular and frequency resolution and foreground treatment. Power first asks how much fluctuation occurs on each scale; imaging asks where it occurs.

## Three questions to carry to another result

1. A printed row has an estimate of $-20\,\mathrm{mK^2}$ and error $30\,\mathrm{mK^2}$. What upper limit follows from the relation above? What would direct addition of the negative estimate give?
2. For $h=0.674$, convert $k=0.5\,h\,\mathrm{Mpc^{-1}}$ into a value without $h$ and find the corresponding comoving wavelength.
3. Two maps have the same powers in all measured windows, with different dark-region shapes. Can these powers select the spatial map? Which information matters beyond them?

The first answer is $60\,\mathrm{mK^2}$; direct addition gives 40 and follows a different rule. The second is $0.337\,\mathrm{Mpc^{-1}}$, or a wavelength of about 18.6 cMpc. The third cannot be decided from the given powers alone. Spatial phases and modes lost through measurement or averaging also matter. This does not require every pair of maps to have equal power; it recognizes that the supplied statistic has not retained the full spatial field.

For the next paper, note its version and observing subset, distinguish the estimate from its error and limit, check the scale units and window, then follow the authors' connection to a physical model. Each step has a place in the original figure or table where you can check it yourself.

## Sources

1. [published version of First Results from HERA Phase II](https://doi.org/10.3847/1538-4357/ae2d54)
2. [open institutional copy](https://api.repository.cam.ac.uk/server/api/core/bitstreams/d98f6b1c-b4da-41a6-ad94-e4343baacde5/content)
3. [arXiv:2511.21289v1](https://arxiv.org/abs/2511.21289v1)
4. [Furlanetto, Oh and Briggs on the temperature dependence of the signal](https://arxiv.org/abs/astro-ph/0608032v2)

## Keep reading

Starlight and intergalactic hydrogen · Reading path

### Photons, ionized volume and a radio signal

1. [Article **How starlight changes intergalactic hydrogen**](https://sandboxphysics.com/stories/reionization/starlight-and-hydrogen/)
2. [Article **What does half ionized mean?**](https://sandboxphysics.com/stories/reionization/ionized-volume-and-local-fraction/)
3. [Article **Supplying photons to an expanding volume**](https://sandboxphysics.com/stories/reionization/photons-in-an-expanding-box/)
4. [Article **Where the photons go when ionized bubbles meet**](https://sandboxphysics.com/stories/reionization/when-ionized-bubbles-overlap/)
5. [Article **Reading ionized bubbles through neutral hydrogen**](https://sandboxphysics.com/stories/reionization/reading-a-neutral-hydrogen-map/)
6. [Article **Recompute a fixed-source ionization history**](https://sandboxphysics.com/stories/reionization/recompute-a-fixed-source-history/)
7. [You are here **Reading HERA's 21 cm upper limits**](https://sandboxphysics.com/stories/reionization/reading-hera-upper-limits/)