The total energy density of the relativistic particles at the beginning of the universe
E = κ T 4 where
π 2 7 κ = g + g 30 b 8 f
and gb is the total number of internal degrees of freedom of all relativistic bosons, g f of all relativistic fermions. Since from the Фридман equation
k 8 π G H2 + = ε , a 2 3 introducing the conformal time η so that dt = a ( η ) d η ,
da 1 H = dt a and
da 1 H = , dη a 2 the dynamics of the universe becomes
G 8 π G a’ 2 + k a 2 = ε a4 3 where the prime denotes the derivative with respect to η . Differentiating with respect to η ,
G 8 π G 8 π G 2 a’ a’’ + 2 k a a’ = 4 ε a 3 a’ + ε ’ a 4 3 3 i.e.
G 8 π G 8 π G ⋅ dt 2 a’ a’’ + 2 k a a’ = 4 ε a 3 a’ + ε a 4 3 3 dη videlicet
G 8 π G 8 π G ⋅ 2 a’ a’’ + 2 k a a’ = 4 ε a 3 a’ + ε a a 4. 3 3
Since the conservation law is
⋅ ε = - 3 H ( ε + p ), the dynamics of the universe is described by
G 8 π G 8 π G 2 a’ a’’ + 2 k a a’ = 4 ε a 3 a’ - 3 H a a 4( ε + p ) 3 3 that is
G 8 π G 8 π G 2 a’ a’’ + 2 k a a’ = 4 ε a 3 a’ - 3 a’ a3( ε + p ) 3 3 namely
G 8 π G a’’ + k a = 2 ε a 3 - 4 π G a 3 ( ε + p ) 3 i.e.
G 4 π G a’’ + k a = a3 ( ε - 3 p ). 3
In the case of radiation, p = ε /3,
a’’ + k a = 0 and if k = 0
a’’ = 0, consequently
a’ = a c where a c is constant and
a = a c η if a ( 0 ) = 0 . When k = -1
a’’ = a,
a = A e η + B e -η and if a ( 0 ) = 0
A = - B therefore
a = C ( e η - e -η ) that is
a = D sinh η .
If k = + 1
a’’ = - a,
a = M cos η + N sin η where M and N are constant. Introducing two new constants R and α
a = R sin α cos η + R cos α sin η that is
a = R sin ( η + α ) and if a ( 0 ) = 0
a = R sin η .
Mukhanov presents the solution in the form
sinh η, k = - 1; a = a m η, k = 0; sin η, k += 1; similar to Sean Carroll’s one. When k = 0 in a flat radiation-dominated universe, a = am η and integrating dt = a d η
η 2 t = a m 2 therefore a ∝ t and hence
1 H = . 2 t t
1 8 π G Substituting H = into H 2 = ε , the energy density 2t 3
3 E = 32 π G t 2 and equating, according to Mukhanov,
3 1 t = . 32 π κ T 2
Converting from Planckian units
-1/2 1 tseconds = 1.39 κ 2 TMeV and, since in some case κ ≃ 3.537, some author writes the formula in the form
1 t . µ s ≈ E 2 GeV
The reactions responsible for the coupling of neutrinos to the relativistic plasma are
_ - + ν ν e + e ⇌ e + e , ± ν ± ν e + e ⇌ e + e , _ _ ± ν ± ν e + e ⇌ e + e with both charged W± -bosons and the neutral Z-boson intermediation. The neutrinos decouple from the electrons when the collision time
-1 tν ≈ ( σνe ne ) becomes of the magnitude order of the cosmological time ( tseconds ) and a calculation shows that this happens when
Tν 1.5 MeV. e ≈
The chemical decoupling of baryons ( protons and neutrons ), essential for nucleosynthesis, occurs later when T ≃ 0.8 MeV. The - and τ-neutrinos decouple earlier than the electron neutrinos and in conclusion all the three neutrino species decouple when T ≃ me i.e.
T ≈ 0.5 MeV before the electrons-positron pairs begin to annihilate. Neutrino decoupling takes place 1 s after BB when the temperature was 10 billion kelvins. The computation of the neutrino decoupling time may be executed equalizing the weak interaction rate with the expansion rate of the universe. The rate of weak interactions is n σ v ; since the number of relativistic electrons and positrons
n ∝ T 3 and
σ 2 2 v ≈ G F T where G F is Fermi’s constant,
σ 2 5 n v ≈ G F T .
The expansion rate
8 π H = G ρ 3 and
ρ ∝ T 4 therefore equalizing
2 5 4 GF T ∼ G T and
T ∼ 1 MeV.
After decoupling, the neutrinos propagate and preserve the Planckian spectrum. Their temperature decreases and the energy released in the electron-positron annihilation heats the radiation, consequently the temperature of radiation is higher than the neutrinos temperature. Today the primordial neutrinos should have a temperature of 1.95 K according to the cosmological standard model but it is difficult to detect them.
Alessandro Rossini Doctor in nuclear engineering
40 Rome Street Vetralla VT 15th December 2010
Bibliography
Viatcheslav Mukhanov Physical foundations of Cosmology Wikipedia The Free Encyclopedia Massimiliano Lattanzi, Riccardo Benini and Giovanni Montani A possible signature of cosmic neutrino decoupling in the nHz region of the spectrum of primordial gravitational waves ( Classical and Quantum Gravity 21 September 2010 )