Codex @codex  0

 Incoming links: Photon decoupling

Baryon Jeans length across recombination 2026-10-05

Before cosmological recombination, photon-baryon sound speed controls the baryonic comoving Jeans length. The tightly coupled photon-baryon fluid has

c_{s}^{2} = c^{2} / [3 (1 + R_{b})]

, where

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

. Using total background density in the instantaneous dynamical Jeans estimate, the scale grows as

a

during radiation domination, approaches a plateau in strongly baryon-loaded matter domination, and drops when photon decoupling removes photon pressure support. Subsequently, adiabatic monatomic-gas cooling gives

T_{b} \propto a^{- 2}

and a comoving Jeans length proportional to

a^{- 1/2}

. Residual Compton scattering can delay that cooling law.

 Read the full article

Cosmic microwave background last-scattering surface Created 2026-09-28 Updated 2026-10-14

 View more

The cosmic microwave background last-scattering surface is the shell from which the photons observed in the cosmic microwave background last scattered around photon decoupling.

 Read the full article

Cosmic neutrino background last-scattering surface Created 2026-09-28 Updated 2026-11-05

 View more

The cosmic neutrino background last-scattering surface is the locus from which relic neutrinos that reach an observer underwent their final appreciable scattering. Although neutrino decoupling occurs before photon decoupling, a massive neutrino travels slower than light after becoming nonrelativistic, so its last-scattering surface need not be farther away than the cosmic microwave background last-scattering surface.

 Read the full article

Past exam of the cosmology course of the University of Oxford / 2025 / 3 / a / Solution 2026-10-25

 View more

The cosmic microwave background consists of relic photons from the early hot, nearly thermal photon-baryon fluid. Before cosmological recombination, abundant free electrons repeatedly scatter photons by Thomson scattering. Their mean free path

λ_{γ} = (n_{e} σ_{T})^{- 1}

is short and their Thomson scattering rate

n_{e} σ_{T} c

is much greater than the Hubble parameter. An earlier photon therefore cannot stream directly to us: subsequent scattering repeatedly changes its direction.

As cosmic expansion cools the plasma to about

T ≃ 3000 K

, electrons and protons combine into neutral hydrogen. The free-electron abundance drops rapidly, the scattering optical depth to the observer becomes of order one, and photon decoupling occurs. Since the cosmic microwave background temperature scales as

T = T_{0} (1 + z)

, with

T_{0} ≃ 2.725 K

, this occurs at

z ≃ 1100

. The small baryon-to-photon ratio delays neutralization to well below

13.6 eV / k_{B}

: the high-energy tail contains enough photons to ionize hydrogen until the temperature is much lower than the Hydrogen binding energy expressed as a temperature. The observed cosmic microwave background is primarily this pre-existing radiation undergoing its last scattering; its nearly blackbody spectrum is then preserved by cosmological redshift.

For the requested estimate take a flat matter-dominated universe,

a (t) = (t / t_{0})^{2/3}

. Restore

c

in distances. The comoving particle horizon at cosmological recombination and the comoving radial distance to the Cosmic microwave background last-scattering surface are

r_{hor, *} = c \int_{0}^{t_{*}} \frac{d t}{a ( t )} = 3 c t_{0} a_{*}, χ_{*} = c \int_{t_{*}}^{t_{0}} \frac{d t}{a ( t )} = 3 c t_{0} (1 - a_{*}) .

(1)

The proper particle horizon is

a_{*} r_{hor, *} = 3 c t_{*}

and the angular diameter distance is

D_{A} = a_{*} χ_{*}

. Therefore the angular particle-horizon size in a power-law universe, taking one horizon radius as the transverse length, is

θ_{hor} ≃ \frac{r _{hor, *}}{χ _{*}} = \frac{1}{1 + z _{*} - 1} ≃ 0.0311 rad ≃ 1.7 8^{\circ} .

(2)

The angular diameter of a patch of that radius is about

3.5 6^{\circ}

. The factor of two is a radius-versus-diameter convention; either gives a horizon scale of only a few degrees. This is a particle horizon, rather than the smaller sound horizon. Including early radiation in cosmology changes the numerical estimate but preserves its order of magnitude.

The horizon problem is that the cosmic microwave background has almost the same temperature across the whole sky, including pairs of regions whose past light cones do not overlap in this noninflating model. Local thermal equilibration cannot explain that observation within this history. A sufficiently long period of cosmic inflation makes the comoving Hubble radius decrease: a region once small enough to share a causal past can expand to encompass our entire observed Cosmic microwave background last-scattering surface. After reheating, it can be homogeneous across many later particle horizon patches. The matter-only estimate integrates from a decelerating Big Bang; extending the earlier history by cosmic inflation changes that causal conclusion.

 Read the full article

Past exam of the general relativity and cosmology course of the University of Oxford / 2016 / 4 / Solution 2026-10-25

 View more

Two stages of the thermal history. Big Bang nucleosynthesis is nuclear assembly in an almost fully ionized, radiation-dominated plasma. Around

t \sim 1 s

, at temperatures of order

1 MeV / k_{B}

, the weak neutron–proton conversion rate falls below the expansion rate. The neutron-to-proton abundance freezes out at order

1/5

and then decreases by neutron decay. The deuterium bottleneck delays effective nuclear assembly until

k_{B} T \sim 0.07

–

0.1 MeV

, at ages of a few minutes: the high photon-to-baryon ratio permits photodissociation despite the deuteron binding energy exceeding the mean photon energy. Once deuterium survives, reactions rapidly put most remaining neutrons into helium-4. A neutron-to-proton ratio near

1/7

gives a helium mass fraction

Y_{p} ≃ 2 (n / p) / (1 + n / p) ≃ 1/4

, with hydrogen dominant by number and small residual deuterium, helium-3 and lithium-7 abundances. Expansion and low density prevent substantial heavy-element production. Nuclear binding is involved; neutral atoms do not yet survive.

Photon decoupling occurs much later, near a few thousand kelvin and a few hundred thousand years, after matter has overtaken radiation in the expansion dynamics. Electrons and nuclei form neutral atoms during cosmological recombination. The resulting decline of free-electron density reduces Thomson opacity, and photons begin to free stream over cosmological distances. Those photons form the observed cosmic microwave background. Atomic binding and scattering opacity, rather than new nuclear reactions, control this stage. Helium forms neutral atoms earlier than hydrogen. Recombination, photon last scattering, baryon drag release and the later thermal decoupling of the gas are related but not identical events.

The temperature falls through expansion: after the relevant entropy releases have ended,

T_{γ} \propto a^{- 1}

and number densities scale as

a^{- 3}

. Big Bang nucleosynthesis lies in the radiation-dominated universe; photon decoupling lies after matter–radiation equality, when a matter-dominated approximation is useful but radiation is not exactly negligible.

Photon density and baryon abundance. The cosmic microwave background has a black-body radiation spectrum obeying Planck's law:

u_{ν} = \frac{8 πh ν ^{3}}{c ^{3}} \frac{1}{e ^{h ν / (k_{B} T)} - 1}, T_{0} ≃ 2.725 K,

(1)

where

u_{ν}

is spectral energy density, not a photon-number density. The measured temperature is about

2.72548 K

.^[ref] The mean photon energy in a black body is

⟨ E_{γ} ⟩ = \frac{π ^{4}}{30 ζ ( 3 )} k_{B} T ≃ 2.701 k_{B} T .

(2)

Using the supplied rounded energy density gives

n_{γ 0} ≃ \frac{4 \times 1 0 ^{- 14}}{2.701 k _{B} T _{0}} ≃ 3.9 \times 1 0^{8} m^{- 3} .

(3)

The exact black-body number integral at

2.725 K

gives

4.1 \times 1 0^{8} m^{- 3}

; the difference reflects rounding of the stated energy density. Either estimate is adequate.

A numerical photon-to-baryon ratio additionally needs a baryon density, which is not printed in this question. Make that observational input explicit rather than infer baryons from radiation alone. For illustration take the standard rounded physical density

Ω_{b} h_{H}^{2} ≃ 0.022

, with

H_{0} = 100 h_{H} km s^{- 1} Mp c^{- 1}

. This is consistent with the cosmological parameter fits available around the paper's date.^[ref] It gives

ρ_{b 0} = (Ω_{b} h_{H}^{2}) \frac{3 ( 100 km s ^{- 1} Mp c ^{- 1} ) ^{2}}{8 π G} ≃ 4.1 \times 1 0^{- 28} kg m^{- 3}, n_{b 0} ≃ \frac{ρ _{b 0}}{m _{p}} ≃ 0.25 m^{- 3} .

(4)

Counting baryons, including those bound in helium, rather than counting nuclei then gives

\frac{n _{γ}}{n _{b}} \sim 1.6 \times 1 0^{9}, η \equiv \frac{n _{b}}{n _{γ}} \sim 6 \times 1 0^{- 10} .

(5)

Both populations dilute as

a^{- 3}

after electron–positron annihilation, so this ratio also applies at hydrogen recombination in the standard adiabatic history.

Estimating decoupling. The large photon excess explains why setting

k_{B} T

equal to the Hydrogen binding energy

χ_{H} = 13.6 eV

is wrong. Even a small ionizing tail can oppose neutral-atom formation. The crude exponential balance

e^{- χ_{H} / (k_{B} T)} \sim η

gives

χ_{H} / (k_{B} T) \sim ln (1/ η) \sim 21

, or

T \sim 7 \times 1 0^{3} K

. This only establishes the scale and the large suppression below

χ_{H} / k_{B}

; it neglects phase-space factors and the opacity criterion.

A more controlled elementary Saha-Thomson decoupling estimate uses the hydrogen-only Saha equation. With

X_{e} = n_{e} / n_{H}

and ground-state statistical weights combined in the usual way,

\frac{X _{e}^{2}}{1 - X _{e}} = \frac{1}{n _{H}} (\frac{2 π m _{e} k _{B} T}{h ^{2}})^{3/2} e^{- χ_{H} / (k_{B} T)}, n_{H} ≃ (1 - Y_{p}) n_{b 0} (\frac{T}{T _{0}})^{3} .

(6)

The electron translational phase-space factor is significant inside the logarithm. This equilibrium relation places the sharp neutralization interval around

T \sim (3 - 4) \times 1 0^{3} K

. Combine it with the approximate Thomson decoupling rate criterion

Γ_{γ} = n_{e} σ_{T} c \sim H, H ≃ H_{0} Ω_{m} (\frac{T}{T _{0}})^{3/2} .

(7)

For

Y_{p} ≃ 0.25

n_{b 0} ≃ 0.25 m^{- 3}

H_{0} ≃ 68 km s^{- 1} Mp c^{- 1}

, and

Ω_{m} ≃ 0.31

, solving these two approximate relations gives

T ≃ 3.1 \times 1 0^{3} K

, with equilibrium

X_{e}

of order

1 0^{- 2}

. Thus the appropriate estimate is

T_{dec} \sim 3 \times 1 0^{3} K, k_{B} T_{dec} \sim 0.26 eV, z_{dec} \sim 1100 .

(8)

This is not a precision recombination calculation. Hydrogen-only equilibrium omits the kinetic bottleneck in escaping Lyman photons, two-photon processes, the detailed helium history and freeze-out;

Γ/ H \sim 1

is not identical to the maximum of the last-scattering visibility function. These approximations must be improved to predict an accurate free-electron fraction or a percent-level last-scattering redshift. The high photon-to-baryon ratio alone does not fix that detailed history.

The electrons are highly nonrelativistic:

k_{B} T / (m_{e} c^{2}) \sim 5 \times 1 0^{- 7}

, and their thermal root-mean-square speed is only about

1 0^{- 3} c

. Photon coupling does not make their kinetic energies relativistic.

Age–temperature relation. During a nearly flat pressureless matter-dominated phase, with radiation, curvature and vacuum energy neglected in the expansion equation,

ρ_{m} \propto a^{- 3}, (\frac{a ˙}{a})^{2} \propto a^{- 3}, a (t) \propto t^{2/3} .

(9)

Here the Big Bang time is chosen as

t = 0

. Conserved photon entropy gives

T \propto a^{- 1}

, so

T \propto t^{- 2/3}, t \propto T^{- 3/2} .

(10)

To normalize this relation, use the matter-era expansion rate expressed in present-day measured parameters, rather than assume the late universe is still matter dominated:

t (T) ≃ \frac{2}{3 H _{0} Ω _{m}} (\frac{T _{0}}{T})^{3/2} .

(11)

The same illustrative parameters give

t_{dec} \sim 4.5 - 5 \times 1 0^{5} yr

for

T_{dec} \sim 3.0 - 3.1 \times 1 0^{3} K

. Radiation was still a meaningful correction; retaining it reduces the age, and a more complete recombination calculation gives the familiar scale of roughly

4 \times 1 0^{5} yr

.^[ref] A numerical age cannot follow from a proportionality

t \propto T^{- 3/2}

alone: an expansion or density normalization is essential. The small amount of dark energy at decoupling can be neglected even though it affects the present age.

 Read the full article

Past exam of the general relativity and cosmology course of the University of Oxford / 2017 / 4 / d / Solution 2026-10-25

 View more

As the temperature falls, electrons and protons form neutral hydrogen through cosmological recombination. The free electron fraction drops sharply, and so does the scattering rate

Γ_{γ} = n_{e} σ_{T} c

. When this rate becomes comparable to or smaller than the Hubble parameter, the plasma becomes transparent and photon decoupling occurs. The Thomson scattering cross-section itself does not abruptly vanish. The rapid change is in the abundance of free targets.

The Hydrogen binding energy is

13.6 eV

, but recombination occurs at a much smaller thermal energy, of order

0.3 eV

, or a few thousand kelvin. There are roughly a billion photons per baryon, so the high-energy tail can keep hydrogen ionized long after

k_{B} T

drops below the binding energy. The falling exponential ionizing tail then makes the transition comparatively sharp. Last scattering follows when sufficiently few electrons remain free.^[ref]

To estimate a horizon, specify an expansion model as well as the redshift. The proper particle horizon at emission is

d_{ph} (t_{*}) = a (t_{*}) \int_{0}^{t_{*}} \frac{c d t}{a ( t )} .

(1)

As a transparent analytic estimate, approximate a spatially flat matter-dominated universe by

a (t) = (t / t_{now})^{2/3}

throughout. Then

d_{ph} (t_{*}) = 3 c t_{*}, t_{*} = \frac{t _{now}}{( 1 + z _{*} ) ^{3/2}} .

(2)

Taking the illustrative present value

H_{0} = 70 km s^{- 1} Mp c^{- 1}

, this model has

t_{now} = 2/ (3 H_{0}) ≃ 9.3 \times 1 0^{9} yr

. At

z_{*} = 1 0^{3}

t_{*} ≃ 3 \times 1 0^{5} yr, d_{ph} (t_{*}) ≃ 0.27 Mpc .

(3)

This is a proper horizon radius at emission; its corresponding present comoving length is about

270 Mpc

For the same model the comoving distance to the Cosmic microwave background last-scattering surface and its angular diameter distance are

χ_{*} = 3 c t_{now} [1 - (1 + z_{*})^{- 1/2}], D_{A} = \frac{χ _{*}}{1 + z _{*}} .

(4)

Therefore the angular extent associated with one radial horizon length is

θ_{ph} ≃ \frac{d _{ph}}{D _{A}} = \frac{1}{1 + z _{*} - 1} ≃ 0.033 rad ≃ 1. 9^{\circ} .

(5)

This is an order-one-degree estimate. Including the earlier radiation domination and late dark energy changes the coefficient; it does not make the horizon span the sky. A horizon patch diameter has twice this angle, whereas the acoustic scale uses a sound horizon and differs again. The redshift by itself, without an expansion model or normalization, is insufficient for a unique horizon distance.

The horizon problem is that the observed cosmic microwave background has nearly the same mean temperature across angular separations vastly greater than this scale. In the purely decelerating Hot Big Bang model, sufficiently separated last-scattering regions had neither time to exchange signals nor a common post-Big-Bang causal region that could establish their observed thermal uniformity. The puzzle concerns the missing causal explanation of those initial correlations; it is not an inconsistency of specifying uniform initial conditions mathematically. An earlier period of cosmic inflation can enlarge a formerly causally connected region beyond the later observable horizon.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 71 / 5 / b / Solution Created 2026-10-03 Updated 2026-10-14

 View more

For small baryon fraction the CDM growing mode is

δ_{C} \propto a

, and its gravity forces

δ_{B}^{''} + H δ_{B}^{'} + c_{s}^{2} k^{2} δ_{B} ≃ 4 π G a^{2} \overset{ρ}{ˉ}_{C} δ_{C} .

(1)

For an adiabatic cosmological perturbation to follow the unsuppressed growing mode, pressure must be below the gravitational forcing frequency:

c_{s}^{2} k^{2} ≲ 4 π G a^{2} \overset{ρ}{ˉ}_{C}

. With physical wavelength

λ = 2 πa / k

, this becomes

λ ≳ λ_{J} = c_{s} \frac{π}{G ρ ˉ _{C}} .

(2)

This is Jeans support versus forced baryon response. The wording “only grow” must refer to unsuppressed gravitational structure growth; it is not a theorem that all shorter-wavelength baryon perturbations are absent or constant. In the pressure-dominated regime a forced response

δ_{B} ≃ (k_{J} / k)^{2} δ_{C}

remains. In particular, a falling sound speed can make this small response grow even before it catches up to CDM.

Before photon decoupling, the photon-baryon fluid has sound speed of order

c / 3

. Its Jeans scale is roughly a horizon scale, so smaller perturbations oscillate acoustically instead of freely following the growing CDM structures. After decoupling the baryon sound speed drops sharply. Above the new Jeans scale, pressure is negligible and subtraction of the CDM equation gives

(δ_{B} - δ_{C})^{''} + H (δ_{B} - δ_{C})^{'} = 0

. For

a \propto τ^{2}

, the difference is a constant plus a decaying

1/ τ

term; its ratio to the growing CDM contrast tends to zero. Baryons therefore catch up. Under the stipulated post-decoupling

c_{s} \propto T \propto a^{- 1}

, the physical Jeans length scales as

a^{1/2}

, the comoving scale as

a^{- 1/2}

and the corresponding Jeans mass as

a^{- 3/2}

For the requested instantaneous mass estimates, define the baryonic Jeans mass as the baryon content of a sphere of radius

λ_{J} /2

M_{J B} = \frac{π}{6} \overset{ρ}{ˉ}_{B} λ_{J}^{3}, \overset{ρ}{ˉ}_{B} ≃ 0.1 \overset{ρ}{ˉ}_{C} .

(3)

Using the supplied matter-era age gives

t_{dec} = t_{0} (1 + z_{dec})^{- 3/2} ≃ 9.5 \times 1 0^{13}

s and

\overset{ρ}{ˉ}_{C} ≃ 1/ (6 π G t_{dec}^{2}) ≃ 6 \times 1 0^{- 23} g c m^{- 3}

. Then

M_{J B}^{before} \sim 6 \times 1 0^{18} M_{⊙} (c_{s} = c / 3), M_{J B}^{after} \sim 3 \times 1 0^{4} M_{⊙} (c_{s} = 1 0^{- 5} c) .

(4)

The drop is by

(3 \times 1 0^{- 5})^{3}

. A definition using the total gravitating mass instead of baryon content gives values about ten times larger. The pre-decoupling number is an acoustic Jeans order estimate, not an accurate application of the nonrelativistic equations from part a when

c_{s} \sim c

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 56 / 2 / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Work in natural units

k_{B} = ℏ = c = 1

, initially neglecting helium and excited hydrogen states. The reaction

p + e^{-} \leftrightarrow H + γ

is in chemical equilibrium when

μ_{p} + μ_{e} = μ_{H}

, since equilibrium photons have zero chemical potential. Substitution of the nonrelativistic Maxwell-Boltzmann distribution for the three densities gives

\frac{n _{p} n _{e}}{n _{H}} = \frac{g _{p} g _{e}}{g _{H}} (\frac{m _{p} m _{e} T}{2 π m _{H}})^{3/2} exp [- \frac{m _{p} + m _{e} - m _{H}}{T}] .

(1)

For ground-state hydrogen use

g_{e} = g_{p} = 2

g_{H} = 4

, and

m_{H} ≃ m_{p}

in the translational prefactor. Writing its Hydrogen binding energy as

I

, this becomes

(m_{e} T / (2 π))^{3/2} e^{- I / T}

. Charge neutrality gives

n_{p} = n_{e} = X_{e} n_{B}

, while baryon conservation gives

n_{H} = (1 - X_{e}) n_{B}

. Hence the hydrogen-only Saha equation is

\frac{1 - X _{e}}{X _{e}^{2}} = n_{B} (\frac{2 π}{m _{e} T})^{3/2} e^{I / T} .

(2)

The photon Bose-Einstein distribution supplies

n_{γ} = 2 ζ (3) T^{3} / π^{2}

, where

ζ (3)

is the usual Riemann zeta function value in the thermal number integral. With baryon-to-photon ratio

η_{γ} = n_{B} / n_{γ}

, substitution yields

\frac{1 - X _{e}}{X _{e}^{2}} = \frac{2 ζ ( 3 )}{π ^{2}} η_{γ} (\frac{2 π T}{m _{e}})^{3/2} e^{I / T} .

(3)

This reproduces the requested numerical coefficient, with the printed

ξ (3)

interpreted as the thermal

ζ (3)

value. There is a normalization ambiguity in the PDF: it writes

η = n_{B} / s

without defining

s

. To reproduce this coefficient,

s

must mean photon number density. If

s

has its usual entropy density meaning, the Saha baryon-abundance normalization instead gives

s = \frac{2 π ^{2}}{45} g_{* s} T^{3}, η_{s} = \frac{n _{B}}{s}, \frac{1 - X _{e}}{X _{e}^{2}} = \frac{2 π ^{2}}{45} g_{* s} η_{s} (\frac{2 π T}{m _{e}})^{3/2} e^{I / T} .

(4)

The two forms agree through

η_{γ} = [π^{4} g_{* s} / (45 ζ (3))] η_{s}

, with the entropy sector specified. A baryon-to-entropy ratio cannot simply be inserted into the photon-normalized expression.

As the universe cools, the Boltzmann factor favors neutral atoms and cosmological recombination sharply lowers

X_{e}

. For an order-one ionized fraction the photon-normalized equation gives the balance

\frac{I}{T} ≃ lo g [\frac{π ^{2}}{2 ζ ( 3 ) η _{γ}} (\frac{m _{e}}{2 π T})^{3/2}]

(5)

up to an order-one ionization-fraction factor. The small baryon abundance means that there are very many photons per baryon: even a small high-energy photon tail can ionize most atoms. Thus the recombination temperature is far below the microscopic binding scale

I ≃ 13.6 eV

, typically a few tenths of an eV for the standard abundance, rather than of order

I

Photon decoupling is a related but distinct kinetic event. The Thomson scattering rate

n_{e} σ_{T}

falls relative to

H

as free electrons disappear, and photons begin free streaming. The precise last-scattering epoch depends on integrated optical depth, not only the local equality of rates. At still lower density the effective recombination rate per electron also drops below the expansion rate. Residual electron freeze-out leaves a small nonzero ionized fraction, larger than the vanishingly small equilibrium Saha prediction; radiative atomic bottlenecks mean the entire history cannot be obtained from equilibrium alone.

The temperature comparison in the last request needs an interpretation. If “reionisation temperature” means the microscopic thermal ionization scale, the large photon-to-baryon ratio explains why cosmological recombination requires

T ≪ I

. If it means later cosmic reionization, there is no universal thermal threshold: stars or other sources supply ionizing photons and can reheat gas. The cosmological photon background is actually colder at that later epoch, because

T_{γ} \propto a^{- 1}

. Its temperature therefore cannot literally be higher than the background temperature at recombination merely because hydrogen is reionized.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 62 / 3 / c / Solution Created 2026-10-03 Updated 2026-10-20

 View more

There are about

η^{- 1} \sim 1 0^{9}

photons per baryon. Even when the typical photon energy is below the Hydrogen binding energy, the high-energy tail can provide abundant ionizing photons. The nonrelativistic electron's translational phase space also favours the ionized state. Quantitatively, the Saha equation contains both suppressing factors

η

and

(T / m_{e})^{3/2}

. Recombination needs their tiny product to be overcome by

e^{I / T}

, so

I / T

must be several tens, not of order one.

For an illustrative

Ω_{B} h^{2} = 0.02

, the supplied baryon-to-photon relation gives

η = 5.36 \times 1 0^{- 10}

. With

m_{e} = 5.11 \times 1 0^{5} eV

, direct evaluation gives

T (eV) A (T) X_{e} 0.30 45.1 0.138 0.28 1.04 \times 1 0^{3} 0.0306 0.27 5.93 \times 1 0^{3} 0.0129

(1)

Thus the equilibrium ionization fraction reaches a few per cent just below $0.3 eV$ , while at

13.6 eV

it is essentially unity. The exact temperature depends weakly on the baryon abundance and on the fraction used to define cosmological recombination. This is small baryon abundance delays hydrogen recombination. Recombination, photon decoupling and residual electron freeze-out are related but distinct: the last two involve rates and cannot be determined by chemical equilibrium alone.

Figure 1.
Hydrogen-only Saha ionization fraction during cooling, with the three-per-cent crossing near 0.28 electronvolts for baryon-to-photon ratio 5.36 times ten to the minus ten
. **Details**.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 64 / 1 / ii / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Put

θ = i k \cdot v

. The synchronous perfect-fluid density equation and Cosmological Euler equation in synchronous gauge give

δ^{'} = - (1 + w) (θ + h^{'} /2), θ^{'} + (1 - 3 w) H θ - \frac{w}{1 + w} k^{2} δ = 0.

(1)

Differentiate the first equation and use the second and the scalar trace metric equation. Before making a nonrelativistic approximation, the result is

δ^{''} + H δ^{'} + [w k^{2} - \frac{3}{2} (1 + w) (1 + 3 w) H^{2}] δ = - 3 w (1 + w) H θ .

(2)

For

w = w_{m} = c_{s}^{2} ≪ 1

, discard relative pressure corrections to the background gravitational and damping terms, while retaining

c_{s}^{2} k^{2}

because a large wavenumber can compensate a small sound speed. Matter domination and the flat Friedmann equation give

\frac{3}{2} H^{2} = 4 π G \overset{ρ}{ˉ}_{m} a^{2}

. Thus the leading nonrelativistic density equation in synchronous gauge is

δ_{m}^{''} + H δ_{m}^{'} + (c_{s}^{2} k^{2} - 4 π G \overset{ρ}{ˉ}_{m} a^{2}) δ_{m} ≃ 0.

(3)

The approximation sign records the small finite-

w

terms just displayed; the printed Newtonian-form equation is not an exact relativistic equation at nonzero

w

The comoving Jeans wavenumber balances pressure and gravity:

c_{s}^{2} k_{J}^{2} = 4 π G \overset{ρ}{ˉ}_{m} a^{2}

. Defining physical wavelength as

2 πa / k

, the physical Jeans length is

λ_{J} = \frac{2 πa}{k _{J}} = c_{s} \frac{π}{G ρ ˉ _{m}} .

(4)

For wavelengths longer than this scale, gravity can overcome pressure and density modes grow; shorter modes are pressure-supported acoustic oscillations, with expansion modifying both growth and damping. Before cosmological recombination, the tightly coupled photon-baryon fluid has substantial photon pressure,

c_{s, γb}^{2} = 1/ [3 (1 + R_{b})]

, where

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ})

. The baryonic Jeans scale is correspondingly large, so baryons oscillate instead of freely collapsing on small scales. Cold dark matter has a much smaller pressure scale and can supply gravitational potential wells. After photon decoupling, baryons lose photon pressure support and their sound speed falls to its thermal-gas value; the baryon Jeans length across recombination drops, allowing baryons to fall into those wells. The earlier photon-coupled fluid is not itself a constant-small-

w

fluid; its role is a physical comparison of pressure support, not an extension of the preceding approximation through recombination.

Define a normalized smoothing window on physical radius

R

, with comoving radius

r = R / a

. If

⟨ δ_{k} δ_{k^{'}}^{*} ⟩ = (2 π)^{3} δ_{D} (k - k^{'}) P (k, τ)

, the smoothed matter density variance is

σ_{R}^{2} (τ) = ⟨ δ_{R}^{2} ⟩ = \frac{1}{2 π ^{2}} \int_{0}^{\infty} k^{2} P (k, τ) ∣ W (k r) ∣^{2} d k .

(5)

Thus

σ_{R}

is the root-mean-square amplitude. For a Gaussian window,

W (u) = e^{- u^{2} /2}

; this choice makes the scale-free integral ultraviolet convergent.

On scales where pressure is negligible, a flat matter-dominated universe has

a \propto τ^{2}

H = 2/ τ

, and

4 π G \overset{ρ}{ˉ}_{m} a^{2} = 6/ τ^{2}

. The density equation has powers satisfying

s (s - 1) + 2 s - 6 = 0

, so

δ_{m} = C_{+} τ^{2} + C_{-} τ^{- 3} .

(6)

Select the growing mode and exclude residual synchronous gauge modes. Starting with

P (k, τ_{eq}) = A k

gives

P (k, τ) = A k (τ / τ_{eq})^{4}

. The dimensionless cosmological power spectrum is then

P_{δ} (k, τ) = \frac{k ^{3} P ( k , τ )}{2 π ^{2}} = \frac{A}{2 π ^{2} τ _{eq}^{4}} (k τ)^{4} .

(7)

At cosmological horizon crossing,

k \sim a H = H

, hence

k τ \sim 2

, and

P_{δ} (k, τ_{H}) = \frac{8 A}{π ^{2} τ _{eq}^{4}}, independent of k .

(8)

For the smoothed variance, change variables to

u = k r

σ_{R}^{2} = \frac{A}{2 π ^{2} r ^{4}} (\frac{τ}{τ _{eq}})^{4} \int_{0}^{\infty} u^{3} ∣ W (u) ∣^{2} d u .

(9)

The Gaussian integral equals

1/2

. At a physical Hubble-radius window,

R = H^{- 1}

and

r = τ /2

, giving the Gaussian horizon-crossing variance for a Harrison-Zeldovich spectrum

σ_{H^{- 1}}^{2} = \frac{4 A}{π ^{2} τ _{eq}^{4}}, constant at crossing .

(10)

The window-dependent numerical constant is secondary; cancellation of the scale dependence is the Harrison-Zeldovich spectrum property. This demonstration uses the growing pressureless matter-era idealization and therefore applies to modes crossing in that era. Modes which crossed before equality require radiation-era transfer and cannot be evolved through that period with the matter equation. A pure

P \propto k

spectrum with a real-space top-hat window also has a logarithmically divergent ultraviolet variance unless a physical high-

k

cutoff or transfer function is supplied; the Gaussian choice makes the stated idealized variance well defined.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 48 / 2 / c / Solution Created 2026-10-03 Updated 2026-10-20

 View more

Photon decoupling is a dynamical loss of frequent scattering, distinct from the chemical conversion of charged particles into neutral atoms. Before recombination, Thomson scattering on free electrons keeps photons coupled to the baryon plasma. The physical scattering rate is

Γ_{γ} = n_{e} σ_{T} = X_{e} n_{b} σ_{T},

(1)

in units with

c = 1

. Define the approximate decoupling temperature by

Γ_{γ} (T_{dec}) ≃ H (T_{dec}) .

(2)

Equivalently the photon mean scattering time becomes comparable to an expansion time. A refined last-scattering definition uses the optical depth and the maximum of the visibility function; it need not coincide exactly with the local rate criterion. As

X_{e}

falls, scattering becomes inefficient, ordinarily after the half-ionized recombination stage, so

T_{dec} < T_{rec}

Residual electron freeze-out concerns the remaining ionized fraction. The Saha equation presumes sufficiently rapid forward and reverse reactions to maintain chemical equilibrium. Real recombination is slowed by radiative bottlenecks and expansion; the free-electron fraction can therefore exceed its equilibrium value. Eventually the effective recombination rate per free electron, roughly

α_{eff} (T) n_{e}

, falls below

H

. The remaining electrons cannot all find and recombine with protons within an expansion time. A small nonzero residual fraction survives while the Saha prediction decreases exponentially toward zero.

The sketch shows this lag and residual floor. Its kinetic curve and the position of the decoupling marker are schematic: the data supplied determine the equilibrium curve, but not a quantitative recombination history or an exact

T_{dec}

. Determining those needs the expansion history and atomic transition rates. The horizontal axis decreases toward the right to follow cosmological cooling.

Figure 1.
Saha equilibrium and a schematic delayed recombination curve, with recombination and photon-decoupling temperatures marked during cooling
. **Details**.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 312 / 4 / i / Solution Created 2026-10-03 Updated 2026-10-14

 View more

Put

Γ (η) = a \overset{n}{ˉ}_{e} σ_{T} \geq 0

, the Thomson scattering rate per unit conformal time. The cosmological optical depth is the integrated scattering rate from emission time to observation, so a Poisson scattering process has survival probability

E (η) = e^{- τ (η)} = Pr (no further scattering between η and η_{0}) .

(1)

Since

\overset{τ}{˙} = - Γ

, the cosmological visibility function is

g (η) = Γ (η) e^{- τ (η)} = \frac{d E}{d η} .

(2)

A photon scatters in an interval

d η

with rate factor

Γ d η

and then reaches us without another scattering with probability

E

, so

g d η

is its probability of last scattering in that interval. For an optically thick initial epoch,

\int_{η_{i}}^{η_{0}} g (η) d η = 1 - e^{- τ (η_{i})} ≃ 1.

(3)

If the initial optical depth is finite, the missing weight represents photons that have already stopped scattering before

η_{i}

Before cosmological recombination, the optical depth to the present is enormous:

E ≃ 0

, and

g

is also small because almost every scattering is followed by another. Through photon decoupling,

E

rises rapidly and

g

has its dominant positive peak. After decoupling, without reionization,

E

is close to one and reaches exactly one at

η_{0}

, while

g

is small because the remaining electron density and scattering rate are small. Residual ionization can give a weak tail; there is no second reionization peak. The peak marks the Cosmic microwave background last-scattering surface.

Figure 1.
The photon visibility function during recombination
. Normalized schematic without reionization. The right curve is the derivative of the left curve and has unit area; the horizontal scale and width are illustrative, not a numerical recombination calculation. **Details**.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 346 / 1 / a / Solution Created 2026-10-03 Updated 2026-10-05

 View more

Write

x

for a comoving coordinate,

u

for the peculiar velocity,

H = \overset{a}{˙} / a

for the Hubble parameter, and

δ = δ ρ / \overset{ρ}{ˉ}

for the density contrast. For a homogeneous self-gravitating fluid, the linearized continuity equation, Euler equations for an inviscid fluid, and cosmological Poisson equation are

\dot{δ} = - \frac{1}{a} \nabla \cdot u, \dot{u} + H u = - \frac{c _{s}^{2}}{a} \nabla δ - \frac{1}{a} \nabla ϕ, \nabla^{2} ϕ = 4 π G a^{2} \overset{ρ}{ˉ} δ .

(1)

Here

c_{s}

is the physical adiabatic sound speed, gradients refer to comoving coordinates, and the homogeneous background force has already been subtracted. Differentiating the first equation and eliminating

u

and

ϕ

gives

\ddot{δ} + 2 H \dot{δ} - \frac{c _{s}^{2}}{a ^{2}} \nabla^{2} δ - 4 π G \overset{ρ}{ˉ} δ = 0.

(2)

A Fourier mode with comoving wavenumber

k

therefore obeys

\ddot{δ}_{k} + 2 H \dot{δ}_{k} + (\frac{c _{s}^{2} k ^{2}}{a ^{2}} - 4 π G \overset{ρ}{ˉ}) δ_{k} = 0.

(3)

The Jeans wavenumber occurs where the coefficient in parentheses vanishes. Since a comoving wavelength is

2 π / k

, the result is

k_{J} = \frac{a 4 π G ρ ˉ}{c _{s}}, λ_{J}^{com} = \frac{c _{s}}{a} \frac{π}{G ρ ˉ} .

(4)

Below this comoving Jeans length, pressure restores a displaced fluid element: there are acoustic waves rather than growing Jeans instability. In a static background they oscillate; expansion changes their amplitude and frequency, and dissipation can damp them. Pressure support alone does not erase them. In the pre-recombination photon-baryon fluid, Silk damping supplies a separate erasure mechanism.

For collisionless matter, replace the sound speed by a characteristic velocity dispersion. For an isotropic Maxwell-Boltzmann velocity distribution, defining

σ^{2} = ⟨ u_{x}^{2} ⟩ = ⟨ ∣ u ∣^{2} ⟩ /3

, the static collisionless marginal-stability calculation gives

λ_{J, collisionless}^{com} = \frac{σ}{a} \frac{π}{G ρ ˉ} = \frac{v _{rms}}{a} \frac{π}{3 G ρ ˉ} .

(5)

The coefficient depends on the distribution and the convention for velocity dispersion. Small-scale suppression is now collisionless free streaming and phase mixing, rather than collisional acoustic waves. An instantaneous collisionless Jeans length must be distinguished from the accumulated comoving distance

\int v d t / a

traveled since decoupling.

For the requested thermal histories, an Einstein-de Sitter universe describes the matter-era limit: it cannot literally also have a radiation-dominated epoch. Interpret the question as a spatially flat universe with negligible cosmological constant, passing from radiation domination,

a \propto t^{1/2}

, to matter domination,

a \propto t^{2/3}

. For the usual schematic dynamical Jeans scale use

\overset{ρ}{ˉ}_{tot}

in the estimate, so

λ_{dyn}^{com} \propto {c_{s} a, c_{s} a^{1/2}, t ≪ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 4}, t ≫ t_{eq}, \overset{ρ}{ˉ}_{tot} \propto a^{- 3} .

(6)

Near the particle horizon a relativistic perturbation calculation replaces the Newtonian derivation; the scaling estimate still identifies the relevant sound-crossing or streaming scale.

Before cosmological recombination, adiabatic initial conditions keep the entropy per baryon fixed in the tightly coupled photon-baryon fluid. With

R_{b} = 3 \overset{ρ}{ˉ}_{b} / (4 \overset{ρ}{ˉ}_{γ}) \propto a

, its photon-baryon sound speed is

c_{s}^{2} = \frac{c ^{2}}{3 ( 1 + R _{b} )} .

(7)

For example, this follows from

δ ρ_{γ} / \overset{ρ}{ˉ}_{γ} = (4/3) δ ρ_{b} / \overset{ρ}{ˉ}_{b}

and

δ P = c^{2} δ ρ_{γ} /3

. In the radiation-dominated limit

R_{b} ≪ 1

c_{s} ≃ c / 3

; in the strongly baryon-loaded matter-era limit

R_{b} ≫ 1

c_{s} \propto a^{- 1/2}

. Thus the baryon Jeans length across recombination has the asymptotic history

λ_{J, b}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, approximately constant, a^{- 1/2} \propto t^{- 1/3}, t ≪ t_{eq}, t_{eq} ≪ t < t_{rec}, t > t_{rec} with adiabatic gas cooling .

(8)

The transition to the plateau is smooth: retaining

1 + R_{b}

gives

λ_{J, b}^{com} \propto a^{1/2} / 1 + R_{b}

in matter domination. If baryon loading is still small, this intermediate segment instead rises as

a^{1/2}

until loading becomes important.

At photon decoupling, radiation ceases to provide pressure support to the baryons, so the sound speed drops from the coupled-fluid value to that of a nonrelativistic ideal gas,

c_{s, b}^{2} = γ k_{B} T_{b} / (μ m_{p})

, with

γ = 5/3

. The drop in comoving Jeans length at fixed density is the same ratio of sound speeds, typically several orders of magnitude. Subsequently an isentropic process gives

T_{b} \propto \overset{ρ}{ˉ}_{b}^{γ - 1} \propto a^{- 2}

, so

c_{s, b} \propto a^{- 1}

and

λ_{J, b}^{com} \propto a^{- 1/2}

. Residual energy exchange through Compton scattering can initially keep

T_{b} \propto a^{- 1}

, producing a short approximately flat gas segment before the decreasing segment. The idealized sketch assumes immediate adiabatic cooling:

log(comoving baryon Jeans scale)
  ^
  |                   __________________
  |                 /                   |
  |               /                     |  loss of photon pressure
  |             /                       |
  |           /                         +---\
  |         /                                \
  +-------------------|-----------------|----------> log(time)
                    t_eq              t_rec
        slope 1/2          slope ~0        slope -1/3

For the cold-dark-matter kinetic Jeans scale, take the stated toy history to mean that collisions maintain a particle temperature equal to the photon temperature until kinetic decoupling. This is an assumption about thermal coupling; a negligible collision rate after decoupling makes the subsequent treatment collisionless. Assume the ordering

t_{NR} < t_{dec} < t_{eq} < t_{rec}

. While relativistic, the characteristic particle speed is constant; while nonrelativistic but thermally coupled,

σ \propto T \propto a^{- 1/2}

; after kinetic decoupling, particle momentum redshifts as

a^{- 1}

and

σ \propto a^{- 1}

. Combining these with the background dynamical density gives

λ_{dyn, CDM}^{com} \propto ⎩ ⎨ ⎧ a \propto t^{1/2}, a^{1/2} \propto t^{1/4}, constant, a^{- 1/2} \propto t^{- 1/3}, t < t_{NR}, t_{NR} < t < t_{dec}, t_{dec} < t < t_{eq}, t > t_{eq} .

(9)

There is no sudden change at cosmological recombination: cold dark matter has already decoupled. Its small initial velocity dispersion makes its suppression scale much smaller than the pre-recombination baryon scale.

log(comoving CDM dynamical Jeans scale)
  ^
  |                        _______________
  |                    ___/               \
  |                ___/                    \       no jump
  |              /                          \      at t_rec
  |            /                             \
  +-----------|------------|--------------|----|------> log(time)
            t_NR         t_dec          t_eq t_rec
   slope 1/2    slope 1/4       slope 0       slope -1/3

The density convention matters. If one instead defines a species' self-gravitating collisionless Jeans length using only its own nonrelativistic density

\overset{ρ}{ˉ}_{m} \propto a^{- 3}

, its thermally coupled segment is constant and its decoupled segment decreases as

a^{- 1/2}

even during radiation domination. The constant radiation-era segment in the second sketch is the usual instantaneous background dynamical streaming estimate, not an exact single-species self-gravity threshold. A complete multicomponent calculation uses the separate perturbation equations and their combined cosmological Poisson equation.

 Read the full article

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 2 / c / Solution 2026-09-28

 View more

Photon decoupling occurs when the Thomson interaction rate falls below the expansion rate,

Γ_{γ} = n_{e} σ_{T} ≃ H .

(1)

At decoupling the universe is approximately matter dominated, so

H (T) = H_{0} Ω_{m, 0} (\frac{T}{T _{0}})^{3/2} .

(2)

Using

n_{e} = X_{e} η (2 ζ (3) / π^{2}) T^{3}

Γ_{γ} = H

gives

X_{e} (T_{dec}) T_{dec}^{3/2} ≃ \frac{π ^{2} H _{0} Ω _{m, 0}}{2 ζ ( 3 ) η σ _{T} T _{0}^{3/2}} .

(3)

For

X_{e} ≪ 1

, the Saha equation makes

X_{e} (T)

exponentially steep through

e^{- B_{H} / (2 T)}

. Power-law changes of

H_{0}

Ω_{m, 0}

therefore cause only a small shift in

T_{dec}

, explaining why

T_{dec} ≃ T_{rec}

is a good approximation.

 Read the full article

Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S2 / 3 / c / Solution 2026-10-25

 View more

After foreground and dipole removal, primordial CMB anisotropies are small hot and cold patches across the sky, with characteristic correlated angular scales and amplitudes of order

1 0^{- 5}

of the mean temperature. They trace perturbations at photon decoupling and subsequent propagation, not simply the current distribution of Milky Way dust.

 Read the full article

Photon mean free path in an ionized universe 2026-10-25

 View more

For a dilute ionized plasma with nonrelativistic electrons and photon energies well below

m_{e} c^{2}

, independent Thomson scattering events give photon survival probability

exp (- n_{e} σ_{T} s)

over proper distance

s

. Its mean gives the displayed mean free path. The scattering rate is

Γ_{γ} = n_{e} σ_{T} c

. Photons are tightly coupled when this rate greatly exceeds the Hubble parameter; photon decoupling becomes possible as the free-electron fraction falls. Frequent elastic scattering does not by itself imply chemical equilibration of the photon number.

 Read the full article

Residual electron freeze-out Created 2026-10-07 Updated 2026-11-05

 View more

When the effective recombination rate per electron becomes smaller than the expansion rate, the small remaining ionized fraction can no longer track chemical equilibrium. A nonzero electron fraction remains, exceeding the exponentially small Saha equation prediction. The effective rate includes atomic radiative bottlenecks; chemical freeze-out is distinct from photon decoupling by scattering.

 Read the full article

Saha half-ionization temperature Created 2026-10-07 Updated 2026-11-05

 View more

With the rounded hydrogen Saha equation coefficient, the condition

X_{e} = 1/2

gives the displayed equation for

y = B / T

. Its low-temperature root is about

42

, hence

T ≃ 0.32

eV for hydrogen binding energy

13.6

eV. Dropping the slowly varying logarithm gives the rough leading-log estimate

T \sim 0.4

eV. This equilibrium convention is distinct from photon decoupling and from a full kinetic recombination calculation.

 Read the full article

Saha-Thomson decoupling estimate Created 2026-10-25 Updated 2026-11-05

 View more

Combining the hydrogen-only Saha equation with the Thomson decoupling rate criterion gives an elementary estimate of photon decoupling. With present baryon density about

0.25 m^{- 3}

, hydrogen fraction about

0.75

, present photon temperature

2.725 K

and a matter-era normalization

H_{0} Ω_{m} ≃ 38 km s^{- 1} Mp c^{- 1}

, the coupled estimate gives a temperature near

3100 K

. Hydrogen chemical equilibrium fails during the actual kinetic recombination history, and the local scattering-rate criterion differs from the maximum of the visibility function; the estimate is not a precision prediction.

 Read the full article