On Friday, January 24, 2020 at 10:57:43 PM UTC-7, smitra wrote: > > 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 >
Thanks a lot! This merits further study before any comment is warranted. AG > > > > > > > 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 > > > > -- > > You received this message because you are subscribed to the Google > > Groups "Everything List" group. > > To unsubscribe from this group and stop receiving emails from it, send > > an email to [email protected] <javascript:>. > > To view this discussion on the web visit > > > https://groups.google.com/d/msgid/everything-list/e30841c5-3969-41b7-bba6-a1387310dbf7%40googlegroups.com > > > [1]. > > > > > > Links: > > ------ > > [1] > > > https://groups.google.com/d/msgid/everything-list/e30841c5-3969-41b7-bba6-a1387310dbf7%40googlegroups.com?utm_medium=email&utm_source=footer > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/52e66dbc-e32b-495d-b911-733827991ee9%40googlegroups.com.
