8 NEUTRINO DECOUPLING

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8.1 General 

 

The neutrino decoupling epoch refers to a period in the early Universe, approximately one second after the Big Bang, when neutrinos ceased to be in thermal equilibrium with the rest of the particles in the primordial plasma. During this epoch, the temperature of the Universe was around 10 billion Kelvin, and the energy density was dominated by photons, electrons, positrons, and neutrinos.

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We can distinguish following steps :

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• As the Universe expanded and cooled (10¹⁰ K → 10⁷K), the temperature dropped below the threshold for electron-positron pair production, and the number of electrons and positrons began to decrease. At the same time, the photons became less tightly coupled to the electrons and positrons, and started to free-stream through the primordial plasma.

• As the temperature continued to drop, the neutrinos began to decouple from the rest of the particles. This process occurred because the weak interaction rate between neutrinos and the other particles became smaller than the Hubble expansion rate of the Universe, meaning that the neutrinos no longer had enough time to interact with the other particles and maintain thermal equilibrium.

• At the time of neutrino decoupling, the temperature of the Universe was around 2 billion Kelvin. The number density of neutrinos was roughly equal to the number density of photons, and the energy density of neutrinos was around 1% of the total energy density of the Universe.

• After decoupling, the neutrinos continued to free-stream through the Universe, essentially behaving like a separate, non-interacting component of the primordial plasma. This has important implications for the large-scale structure of the Universe, as the neutrinos can affect the growth of cosmic structures through their gravitational influence.

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The neutrinos that decoupled during this epoch form what we now call the Cosmic Neutrino Background (CNB), the universe's background particle radiation composed of neutrinos. They are sometimes known as relic neutrinos.  The CνB is a relic of the Big Bang; while the cosmic microwave background radiation (CMB) dates from when the universe was 379,000 years old, the CνB decoupled (separated) from matter when the universe was just one second old. It is estimated that today, the CνB has a temperature of roughly 1.95 K. [1]

As neutrinos rarely interact with matter, these neutrinos still exist today. They have a very low energy, around 10−4 to 10−6 eV.[1][2] Even high energy neutrinos are notoriously difficult to detect, and the CνB has energies around 1010 times smaller, so the CνB may not be directly observed in detail for many years, if at all.[2][3] However, Big Bang cosmology makes many predictions about the CνB, and there is very strong indirect evidence that the CνB exists.[2][3]

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8.2 Derivation of decoupling time  [1]

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Neutrinos are scattered (interfering with free streaming) by their interactions with electrons and positrons, such as the reaction

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The approximate rate of these interactions is set by the number density of electrons and positrons, the averaged product of the cross section for interaction and the velocity of the particles. The number density η of the relativistic electrons and positrons depends on the cube of the temperature T, so that η α T³. The product of the cross section and velocity for weak interactions for temperatures (energies) below W/Z boson masses (~100 GeV) is given approximately by                          , where        is Fermi's constant (as is standard in particle physics calculations, factors of the speed of light  c are set equal to 1). Putting it all together, the rate of weak interactions Γ is

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This can be compared to the expansion rate which is given by the Hubble parameter H with

 

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where G is the gravitational constant and ρ  is the energy density of the universe. At this point in cosmic history, the energy density is dominated by radiation, so that ρ α T⁴.  As the rate of weak interaction depends more strongly on temperature, it will fall more quickly as the universe cools. Thus when the two rates are approximately equal (dropping terms of order unity, including an effective degeneracy term which counts the number of states of particles which are interacting) gives the approximate temperature at which neutrinos decouple:

 

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Solving for temperature gives

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While this is a very rough derivation, it illustrates the important physical phenomena which determined when neutrinos decoupled. [1]

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