---
title: "21 cm Cosmic Dawn and Radio Foregrounds Interactive Lab | Sandbox Physics"
description: "How can faint hydrogen be found under a radio sky thousands of times brighter?"
canonical_url: "https://sandboxphysics.com/labs/cosmic-dawn-21cm-foregrounds/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/cosmic-dawn-21cm-foregrounds/"
last_modified: "2026-11-14"
---

# 21 cm Cosmic Dawn and Radio Foregrounds

**How can faint hydrogen be found under a radio sky thousands of times brighter?** Radio light from our Galaxy is thousands of times brighter than the 21 cm signal of early hydrogen, but its spectrum is smooth. Two antennas turn a smooth spectrum into short delays, so the bright emission stays inside a wedge set by the antenna spacing and faint hydrogen can be measured outside it. A response that changes with frequency leaks emission outward, and filters can remove signal too.

Drag when starlight couples and heats hydrogen, then watch the same neutral field become channel maps, a delay spectrum, a foreground wedge and a comparison with published HERA limits.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This is a teaching simulation, not a reionization or instrument pipeline. The hydrogen history uses smooth tanh steps for Lyman-alpha coupling and heating that the reader sets; they are not a source-population or radiative-transfer calculation. Unheated gas follows a frozen CAMB 1.6.0 (RECFAST) temperature table. Collisional coupling is omitted, which is adequate below redshift thirty, and the colour temperature equals the gas temperature. Reionization follows the Planck 2018 tanh history with midpoint 7.68. The brightness keeps the full optical-depth factor with Hubble-flow line widths. The map is one Gaussian random realization of linear matter fluctuations from the Eisenstein-Hu no-wiggle spectrum normalized to Planck 2018 sigma-8, with linear redshift-space distortion, drawn at band centre without light-cone evolution. Spin-temperature, coupling and ionized-bubble fluctuations are omitted, so the predicted power comes from density alone; full models add these terms and can differ substantially. The sky patch spans 0.32 radian; the uniform mean brightness is left out of the visibilities. Foregrounds are schematic: about 280 K diffuse synchrotron at 150 MHz with index minus 2.55, a bright band standing in for the Galactic plane and 48 power-law point sources out to the horizon. The beam is an ideal 14 metre Airy pattern; HERA dishes, mutual coupling, cable reflections, calibration, noise, the ionosphere and radio interference are absent. Delay spectra use a Blackman-Harris taper over 16 MHz in 128 channels and the HERA normalization. The HERA points are the published H1C IDR2 Field 1 bandpowers, never mixed with simulated data.

Coupling and heating history

Ideal reference, teaching history

CMB Gas Spin Brightness

One baseline: delay spectrum

Simulated record, no noise; dashed: ideal expectation

Foreground Hydrogen After treatment Expectation

Eight baselines: the foreground wedge

Simulated record, after the chosen treatment

HERA limits and this history

Real observation: HERA H1C Field 1; line: ideal reference

HERA upper limit Model Through HERA window

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

Below every limit means allowed. The model omits heating and ionization patchiness.

How to read the plots

History: the upper panel shows temperatures on a log scale and the lower panel the mean brightness; the shaded strip is the observed band.

Delay spectrum: the solid vertical line is the horizon; the shading marks modes the chosen treatment does not use.

Wedge: colour shows power on a log scale. Foregrounds fill the region under the dashed horizon line; longer baselines reach higher.

## Find faint hydrogen under a bright radio sky

**Background** Before the first stars, neutral hydrogen filled the Universe. Its 21 cm line reaches us redshifted to tens or hundreds of megahertz, where our own Galaxy shines thousands of times more brightly.

**Why it matters** The timing of the faint hydrogen signal records when the first stars coupled and heated the gas. Reading it needs a way to keep the bright Galaxy out of the measurement.

- **Focus question**: Why can a signal ten thousand times fainter than the sky still be measured, and what can spoil that?
- **One-sentence intuition**: Smooth-spectrum emission maps to short delays, at most the light-travel time across the antenna spacing. Hydrogen structure along the line of sight fills longer delays. Chromatic instruments and aggressive filters blur that separation.

### Core mathematical model

#### Spin temperature and brightness

$$
T_S^{-1}=\frac{T_\gamma^{-1}+x_\alpha T_K^{-1}}{1+x_\alpha},\qquad \delta T_b=\frac{T_S-T_\gamma}{1+z}\left(1-e^{-\tau}\right)
$$

Starlight coupling pulls the spin temperature from the CMB toward the gas temperature. Cold gas absorbs against the CMB; heated gas emits.

#### Delay transform of one baseline

$$
\widetilde V(\tau)=\int w(\nu)\,V(\nu)\,e^{2\pi i\nu\tau}\,d\nu,\qquad |\tau|\le\frac{|\mathbf b|}{c}\ \text{for smooth spectra}
$$

A source away from the zenith adds a fringe whose delay grows with the antenna spacing and its angle from the zenith. Smooth spectra keep foreground power inside the horizon delay, apart from taper and beam leakage.

#### From delay to cosmological power

$$
P(k)=\frac{X^2Y}{\Omega_{pp}B_{pp}}\,|\widetilde V|^2,\qquad k_\parallel=\frac{2\pi\tau}{Y},\qquad \Delta^2=\frac{k^3P}{2\pi^2}
$$

Two conversion factors turn angles and frequencies into comoving distances at the band redshift. HERA reports the dimensionless power in millikelvin squared.

### Common difficulties

#### An upper limit is not a detection

**Typical misconception** The HERA points are the measured hydrogen power.

**Better mental model** They are the estimate plus two standard deviations, with negative estimates set to zero. Residual systematics dominate, so they bound the signal from above.

#### The model keeps density only

**Typical misconception** The predicted curve is what a real cosmic dawn would produce.

**Better mental model** It keeps only density fluctuations. Fluctuations in heating, coupling and ionization are left out, and a full model must add them.

#### Filtering is not free

**Typical misconception** Subtracting a smooth fit removes only foregrounds.

**Better mental model** The fit also absorbs smooth parts of the signal. An injection test measures that loss and a correction must be applied.

### Run the experiment

1. 01

   #### Watch the sky change channel by channel

   Let the scan run and compare Hydrogen only with Whole sky.

   **What to observe:** Hydrogen structure changes from channel to channel, while the foreground barely changes and is thousands of times brighter.

2. 02

   #### Drag the coupling and heating histories

   Drag the two handles in the history chart or use the sliders.

   **What to observe:** Early coupling with late heating gives deep absorption; early heating turns the band into faint emission.

3. 03

   #### Move the second antenna

   Drag the right dish along the ground.

   **What to observe:** The fringes tighten and the horizon line in the delay spectrum moves out, so fewer clean modes remain.

4. 04

   #### Compare the two treatments

   Choose A smooth fit removes signal and read the recovery readout.

   **What to observe:** Avoidance keeps the injected signal; the fit uses more modes but loses part of the signal and leaves foreground residuals.

Physical conclusion

The 21 cm search works by keeping foregrounds in a predictable corner of delay space. Every instrument effect or filter that moves that boundary has to be measured, including how much signal it removes.

## Related experiments and articles

Hearing the first stars in 21 cm hydrogen · Reading path

### From hydrogen brightness to a HERA upper limit

Articles on this path that use this experiment, followed by the other experiments they suggest.

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. [Article **Reading how HERA kept foregrounds out of its first limits**](https://sandboxphysics.com/stories/cosmic-dawn-21cm/reading-hera-foreground-avoidance/)

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)