Decoupling (cosmology)

Photon decoupling occurred during the epoch known as the recombination. During this time, electrons combined with protons to form hydrogen atoms, resulting in a sudden drop in free electron density. Decoupling occurred abruptly when the rate of Compton scattering of photons ${\displaystyle \Gamma }$ was approximately equal to the rate of expansion of the universe ${\displaystyle H}$, or alternatively when the mean free path of the photons ${\displaystyle \lambda }$ was approximately equal to the horizon size of the universe ${\displaystyle H^{-1}}$. After this photons were able to stream freely, producing the cosmic microwave background as we know it, and the universe became transparent.[1]

The interaction rate of the photons is given by

${\displaystyle \Gamma ={\frac {c}{\lambda }}=n_{e}\sigma _{e}c}$

where ${\displaystyle n_{e}}$ is the electron number density, ${\displaystyle \sigma _{e}}$ is the electron cross sectional area, and ${\displaystyle c}$ is the speed of light.

In the matter-dominated era (when recombination takes place),

${\displaystyle H\varpropto a^{-{3/2}}}$

where ${\displaystyle a}$ is the cosmic scale factor. ${\displaystyle \Gamma }$ also decreases as a more complicated function of ${\displaystyle a}$, at a faster rate than ${\displaystyle H}$.[2] By working out the precise dependence of ${\displaystyle H}$ and ${\displaystyle \Gamma }$ on the scale factor and equating ${\displaystyle \Gamma =H}$, it is possible to show that photon decoupling occurred approximately 380,000 years after the Big Bang, at a redshift of ${\displaystyle z=1100}$ [3] when the universe was at a temperature around 3000 K.

Another example is the neutrino decoupling which occurred within one second of the Big Bang.[4] Analogous to the decoupling of photons, neutrinos decoupled when the rate of weak interactions between neutrinos and other forms of matter dropped below the rate of expansion of the universe, which produced a cosmic neutrino background of freely streaming neutrinos. An important consequence of neutrino decoupling is that the temperature of this neutrino background is lower than the temperature of the cosmic microwave background.

Decoupling may also have occurred for the dark matter candidate, WIMPs. These are known as "cold relics", meaning they decoupled after they became non-relativistic (by comparison, photons and neutrinos decoupled while still relativistic and are known as "hot relics"). By calculating the hypothetical time and temperature of decoupling for non-relativistic WIMPs of a particular mass, it is possible to find their density.[5] Comparing this to the measured density parameter of cold dark matter today of 0.222 ${\displaystyle \pm }$ 0.0026 [6] it is possible to rule out WIMPs of certain masses as reasonable dark matter candidates.[7]

• Recombination
• Chronology of the universe
• Wouthuysen–Field coupling