---
title: "Recombination, Visibility and CMB Spectral Distortions Interactive Lab | Sandbox Physics"
description: "The background light is almost a blackbody. What history can its tiny departures still record?"
canonical_url: "https://sandboxphysics.com/labs/recombination-spectral-distortions/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/recombination-spectral-distortions/"
last_modified: "2026-11-14"
---

# Recombination, Visibility and CMB Spectral Distortions

**The background light is almost a blackbody. What history can its tiny departures still record?** The background light stays a near-perfect blackbody only if extra heat arrives very early. Heat released before redshift about two million is smoothed into a slightly hotter blackbody. Later heat leaves a chemical-potential or Compton-type distortion in the spectrum, and energy released around recombination can also delay the moment the universe became transparent.

Drag a burst of heat along cosmic history. Watch it turn into a hotter blackbody, a spectral distortion, or a delay in the moment the universe became transparent.

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

Model assumptions

This is an offline teaching model, not a fit to observed data. The background is flat, with matter, a cosmological constant, photons and massless neutrinos, using the same numbers as the HyRec-2 input file shipped with CLASS version 3.3.4. Only hydrogen is followed: helium recombination and the electrons it supplies are left out. Hydrogen starts in Saha equilibrium at redshift 1800 and then follows the effective three-level atom with the RECFAST fudge factor 1.14 (Seager, Sasselov and Scott 1999), with the gas temperature set equal to the radiation temperature. The energy release is a Gaussian in the logarithm of one plus redshift with width 0.15; its size is the total released energy as a fraction of the photon energy density. Energy is deposited on the spot: the split between ionization, excitation and heat follows Chen and Kamionkowski (2004). Heat becomes a temperature shift, a chemical-potential distortion or a Compton distortion through the smooth branching approximation of Chluba (2013), with the thermalization redshift scaling used in CLASS. A full Green’s-function treatment projected onto the COBE/FIRAS bands can differ from this approximation by up to a factor of two between redshift ten thousand and one hundred thousand. Below redshift about two hundred the gas no longer passes its heat to the photons efficiently, so the release cannot be placed after redshift 300. The residual spectrum shows only the distortion left after refitting the temperature; foregrounds, recombination lines, damping of small-scale sound waves and nonlinear effects are omitted. Gold points are a sampled extract of the frozen HyRec-2 output file distributed with CLASS version 3.3.4; it includes helium, and Sandbox Physics did not run CLASS or HyRec. The COBE/FIRAS 95 percent limits come from Fixsen and colleagues (1996). Photon paths, their scattering zigzag and the two-piece timeline scale on the stage are schematic.

## What the gas and the light record

Offline model; gold points are a frozen HyRec-2 file

### Free electrons per hydrogen $x_e$

Saha three-level, no release with this release HyRec-2 file, with helium

### Where the light last scattered $g(z)$

no release with this release

### Spectrum left after refitting the temperature $\Delta I_\nu=I_0(\mu M+yY)$

chemical-potential shape Compton shape dotted: COBE/FIRAS 95% limits

## Follow a burst of heat through cosmic history

**Background** For its first few hundred thousand years the universe was a hot plasma. Photons scattered off free electrons so often that the light kept a blackbody spectrum. When protons and electrons combined into hydrogen, the photons stopped scattering and the universe became transparent.

**Why it matters** Any extra heat added along the way could leave a mark on that light. Where the mark ends up depends on when the heat arrived.

- **One-sentence intuition**: Very early heat is fully thermalized and only raises the temperature. Between redshift about two million and fifty thousand it leaves a chemical-potential distortion; later it leaves a Compton distortion. Heat near or after recombination mostly changes the ionization history, which moves the last-scattering surface.

### Core mathematical model

#### Saha equilibrium

$$
\frac{x_e^2}{1-x_e}=\frac{1}{n_{\rm H}}\left(\frac{2\pi m_e k_BT}{h^2}\right)^{3/2}e^{-E_I/k_BT}
$$

Holds only while ionization and recombination are both fast. It predicts recombination too early, because real atoms get stuck in the excited level.

#### The bottleneck in the effective three-level atom

$$
-\frac{dx_e}{dt}=C\left[\alpha_B n_{\rm H}x_e^2-\beta_B(1-x_e)e^{-h\nu_\alpha/k_BT}\right],\quad C=\frac{1+K\Lambda_{2s}n_{1s}}{1+K(\Lambda_{2s}+\beta_B)n_{1s}}
$$

An atom in the second level reaches the ground state with a probability called the escape factor, by two-photon decay or by Lyman-alpha redshifting out of the line. Otherwise it is ionized again.

#### Visibility of last scattering

$$
g(z)=e^{-\tau}\frac{d\tau}{dz},\qquad \frac{d\tau}{dz}=\frac{c\,\sigma_T n_e}{(1+z)H}
$$

The probability density that a photon we see today last scattered at a given redshift. Its area is one when the optical depth to the start is large.

#### Where the heat goes

$$
\mu\approx1.401\int J_\mu J_{\rm bb}\,\frac{d(Q/\rho_\gamma)}{dz}\,dz,\qquad y\approx\frac14\int J_y\,\frac{d(Q/\rho_\gamma)}{dz}\,dz
$$

A smooth approximation (Chluba 2013). The blackbody visibility removes very early heat; the two branching functions split the rest between the chemical-potential and Compton shapes.

### Common difficulties

#### Saha is not the history

**Typical misconception** Hydrogen recombines when Saha says half the atoms are neutral.

**Better mental model** Saha falls below one percent ionized by redshift 1100, but the three-level atom still has about fourteen percent there. The bottleneck keeps electrons free longer.

#### A temperature shift is invisible

**Typical misconception** Heat released very early should be easy to detect.

**Better mental model** Fully thermalized heat makes a blackbody of a slightly different temperature. We only measure one temperature today, so that change cannot be separated from it.

#### The branching is approximate

**Typical misconception** The smooth branching gives the exact distortion seen by an experiment.

**Better mental model** Between redshift ten thousand and one hundred thousand the shape is neither pure chemical-potential nor pure Compton. A Green’s-function calculation projected onto real bands can differ by up to a factor of two.

### Run the experiment

1. 01

   #### Start with the chemical-potential case

   Keep the default release and read the spectrum residual.

   **What to observe:** Photons have had time to share the energy but not to make new photons. The residual dips at low frequency and turns positive above about 125 gigahertz.

2. 02

   #### Drag the release earlier

   Drag the glowing marker left of the blackbody-restored boundary.

   **What to observe:** The distortion fades and the photons turn white: the heat is erased into a slightly hotter blackbody.

3. 03

   #### Move it into the Compton era

   Choose Hot electrons boost photons.

   **What to observe:** The residual changes shape and its zero moves to about 218 gigahertz. Electrons hotter than the photons push photons to higher frequency.

4. 04

   #### Delay the last scattering

   Choose Delayed last scattering, then Dark-age ionization glow.

   **What to observe:** A tiny release near recombination moves the visibility peak to lower redshift. A later one leaves more free electrons at redshift 200 while the spectrum barely changes.

Physical conclusion

The spectrum records heat released between redshift about two million and recombination. The ionization history records energy released near and after recombination. Together they bound how much energy the early universe could have hidden.

## Related experiments and articles

How the universe became transparent · Reading path

### From a blackbody to limits on hidden energy

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

1. [Article **What can tiny departures from a perfect blackbody record?**](https://sandboxphysics.com/stories/cosmic-transparency/a-nearly-perfect-blackbody/)
2. [Article **Saha balance and the recombination bottleneck**](https://sandboxphysics.com/stories/cosmic-transparency/saha-and-the-recombination-bottleneck/)
3. [Article **Optical depth and the visibility function**](https://sandboxphysics.com/stories/cosmic-transparency/optical-depth-and-the-visibility-function/)
4. [Article **Thermalization, chemical-potential and Compton distortions**](https://sandboxphysics.com/stories/cosmic-transparency/thermalization-mu-and-y/)
5. [Article **Comparing Saha, a three-level atom and a frozen CLASS reference**](https://sandboxphysics.com/stories/cosmic-transparency/compare-recombination-codes/)
6. [Article **How to read the decaying dark matter limits of Lucca and colleagues**](https://sandboxphysics.com/stories/cosmic-transparency/reading-energy-injection-limits/)

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)