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.

Mean 21 cm brightness at band centre $16.2\,\mathrm{mK}$

Spin temperature $195\,\mathrm{K}$

First line-of-sight mode used $0.195\,h\,\mathrm{Mpc^{-1}}$

Model power over the HERA limit at

z=7.9

$0.0168$

Injected signal recovered $99.9\,\%$

Set the first-star history

Observed band centre

159

\nu_c/\mathrm{MHz}

Antenna spacing

14.6

b/\mathrm{m}

Starlight couples the spin

17

z_\alpha

X-rays heat the gas

12

z_h

Drag the right dish, or the two handles in the history chart.

The default record is ready.

History details, filter and instrument

Temperature added by heating

300

\Delta T_h/\mathrm{K}

Final coupling strength

20

x_{\alpha,\infty}

Coupling transition width

1.5

\Delta z_\alpha

Heating transition width

1.5

\Delta z_h

Reionization midpoint

7.68

z_{\rm re}

Delay buffer beyond the horizon

300

\tau_{\rm buf}/\mathrm{ns}

Order of the smooth fit

3

n

Random field seed

21

s

$x_\alpha=\frac{x_{\alpha,\infty}}{2}\left[1+\tanh\frac{z_\alpha-z}{\Delta z_\alpha}\right],\quad T_K=T_{\rm ad}+\frac{\Delta T_h}{2}\left[1+\tanh\frac{z_h-z}{\Delta z_h}\right]$

These steps are a teaching parameterization, not a calculation of how stars form. The unheated temperature comes from a frozen CAMB table.

$\Delta^2_{21}=\delta T_b^2\left(1+\mu^2\right)^2\Delta^2_m(k,z)$

The prediction keeps only density fluctuations; full models add spin-temperature and ionization terms. The export holds the full record, units and settings.

HERA Collaboration 2022, ApJ 925, 221

Parsons et al. 2012: the delay spectrum

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

CMBGasSpinBrightness

One baseline: delay spectrum

Simulated record, no noise; dashed: ideal expectation

ForegroundHydrogenAfter treatmentExpectation

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 limitModelThrough 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.

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 misconceptionThe HERA points are the measured hydrogen power.

Better mental modelThey 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 misconceptionThe predicted curve is what a real cosmic dawn would produce.

Better mental modelIt 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 misconceptionSubtracting a smooth fit removes only foregrounds.

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

Run the experiment

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.
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.
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.
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.

21 cm Cosmic Dawn and Radio Foregrounds

Find faint hydrogen under a bright radio sky

Core mathematical model

Spin temperature and brightness

Delay transform of one baseline

From delay to cosmological power

Common difficulties

An upper limit is not a detection

The model keeps density only

Filtering is not free

Run the experiment

Watch the sky change channel by channel

Drag the coupling and heating histories

Move the second antenna

Compare the two treatments

Related experiments and articles

Articles

From hydrogen brightness to a HERA upper limit