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.

Chemical-potential distortion

\mu

$4.19\times10^{-5}$

Compton distortion

y

$3.43\times10^{-7}$

Last scattering peaks at redshift $1081$

Free electrons per hydrogen at

z=200

$3.39\times10^{-4}$

Release a burst of heat

When the heat is released

z_h=2\times10^{5}

\log_{10}(1+z_h)

How much energy

\Delta\rho/\rho_\gamma=3\times10^{-5}

\log_{10}(\Delta\rho/\rho_\gamma)

Drag the glowing release on the timeline: sideways for time, up or down for energy.

More numbers for this release

Temperature shift, absorbed into the measured temperature $\Delta T/T$: $2.63\times10^{-8}$
Redshift where the optical depth reaches one $\tau=1$: $1092$
Width of the visibility peak $\Delta z_{\rm FWHM}$: $203$
Share of the energy given to the photons $Q_\gamma/Q$: $100\,\%$
Share spent on ionization and excitation $Q_{\rm ion}/Q$: $0\,\%$

Without any release this model reaches unit optical depth at redshift 1092; Planck 2018 reports $z_*=1089.92\pm0.25$ . The frozen HyRec-2 file differs from the hydrogen-only curve by at most 3.9 percent below redshift 1500.

Course: how the universe became transparent

Recompute these numbers with the saved scripts

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 $x_e$

Sahathree-level, no releasewith this releaseHyRec-2 file, with helium

Where the light last scattered $g(z)$

no releasewith this release

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

chemical-potential shapeCompton shapedotted: COBE/FIRAS 95% limits

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 misconceptionHydrogen recombines when Saha says half the atoms are neutral.

Better mental modelSaha 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 misconceptionHeat released very early should be easy to detect.

Better mental modelFully 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 misconceptionThe smooth branching gives the exact distortion seen by an experiment.

Better mental modelBetween 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

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.
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.
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.
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.

Recombination, Visibility and CMB Spectral Distortions

What the gas and the light record

Free electrons per hydrogen $x_e$

Where the light last scattered $g(z)$

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

Follow a burst of heat through cosmic history

Core mathematical model

Saha equilibrium

The bottleneck in the effective three-level atom

Visibility of last scattering

Where the heat goes

Common difficulties

Saha is not the history

A temperature shift is invisible

The branching is approximate

Run the experiment

Start with the chemical-potential case

Drag the release earlier

Move it into the Compton era

Delay the last scattering

Related experiments and articles

Articles

From a blackbody to limits on hidden energy

What the gas and the light record

Free electrons per hydrogen xex_exe​

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

Spectrum left after refitting the temperature ΔIν=I0(μM+yY)\Delta I_\nu=I_0(\mu M+yY)ΔIν​=I0​(μM+yY)

Core mathematical model

Saha equilibrium

The bottleneck in the effective three-level atom

Visibility of last scattering

Where the heat goes

Common difficulties

Saha is not the history

A temperature shift is invisible

The branching is approximate

Run the experiment

Start with the chemical-potential case

Drag the release earlier

Move it into the Compton era

Delay the last scattering

Related experiments and articles

Articles

From a blackbody to limits on hidden energy

Free electrons per hydrogen $x_e$

Where the light last scattered $g(z)$

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