What is the least speculative is the reheating caused by the
annihilation of electrons and positrons which something that we may be
able to measure in the future is we can measure the temperature of the
neutrino background. About one second after the Big Bang the neutrinos
had decoupled from the electrons and photons, which means that the
average time between interactions would be longer than the lifetime of
the universe, so thermal equilibrium would not be maintained. However,
the temperatures of the two sectors still evolved in the same way due to
adiabatic cooling until about 15 seconds after the Big Bang when
temperature became too low for electrons and positrons to be created.
So, after 15 seconds after the Big Bang due to adiabatic cooling,
positrons and electrons would start to vanish due to annihilation
without their numbers getting replenished. The energy released from the
net annihilation partially went into the internal energy of photons, as
part of it would have been expended in the work done due to the
adiabatic expansion. But the entropy of such a process is conserved, so
we can calculate the change in the temperature due to eliminating the
electrons and positions. The entropy per unit volume of a electron gas
in the extremely relativistic limit at zero chemical potential and
temperature T s given by:
S_e = 28 pi^5 c k^4/(45 h^3) T^3
This can be obtained from the Fermi-Dirac distribution, the chemical
potential is zero because we're considering an electron positron gas in
equilibrium with photons and photons have a chemical potential of zero.
The entropy density of a positron gas is, of course also equal to S_e.
The entropy density of a photon gas is:
S_f = 32 pi^5 c k^4/(45 h^3) T^3
Before the electrons and positrons annihilated, the entropy density of
the electron, positron, photon sector was:
S = S_f + 2 S_e
The neutrino sector has decoupled, but it will continue to have a
temperature that's equal to the temperature in the electron sector
provided the above formula for the entropy is valid. Once the electrons
have annihilated with the positrons, the temperature of the neutrino
sector will track the temperature of a hypothetical relativistic
electron positron gas in equilibrium with photons. The real temperature
of the photon sector will be lower, but the entropy density will be
exactly the same as that of the hypothetical electron, positron, photon
gas. The real temperature T is thus related to the hypothetical
temperature Th without annihilation, according to:
S_f(T) = S_f(Th) + 2 S_e(Th)
It then follows that:
Th = (4/11)^(1/3) T
The temperature of the neutrino sector will continue to track Th, so the
temperature of the cosmic background neutrinos today will be about 1.9
K.
Saibal
On 25-01-2020 05:59, Alan Grayson wrote:
If the universe began as very hot, and given that the current
temperature of the CMBR is very cool, 2.7 deg K, in what models is a
re-heating phase postulated, and why? TIA, AG
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