In the early Universe, the three flavour active left-handed neutrinos ν_{α} and their CP conjugated states were kept in thermodynamical equilibrium with e.m. plasma by CC and NC weak interactions with charged leptons and quarks. In this regime the neutrino distributions are just fixed by thermodynamics to the Fermi-Dirac type

By using such distributions we get at the equilibrium

As the temperature decreases, CC and NC weak interactions rate drop down faster than H(T) and hence become not efficient enough in mantaining neutrinos at equilibrium. At this point one says that neutrinos decouple from the e.m. plasma (baryons, leptons, photon). To describe neutrino decoupling in details, one should use the set of Boltzmann equations governing the evolution of their distribution function. Nevertheless in the following we will assume a simplified approach that provides quite precise results.

In order to simplify the Boltzmann equations ruling neutrino distribution functions one can consider the leading processes contributing to the collisional term, namely the scattering over electrons/positrons and pair conversions,

When the temperature is at most few MeV, we can consider the weak reaction rates as due to a 4-fermions interaction term (W and Z not propagating). In this case the corresponding cross section times velocity for charged and neutral current interactions is of the order of

Therefore the weak interaction rate for neutrinos becomes

n_{e} denotes the electron/positron density, which is ≈ T^{3} as long as electrons are still relativistic. In that epoch the Universe was RD, hence the Hubble parameter was given by

The temperature for which the weak interaction rate, dropping down because of the expansion, reaches the value of H is called the Decoupling Temperature, hereafter denoted by T_{νD}.

From the above definition the decoupling temperature reads

The decoupling temperature is weakly dependent on g_{*} . For temperature of few MeV the relativistic d.o.f. give

One can get a better prediction for the decoupling temperature via the kinetic equation. The Boltzmann equations written in terms of the variables x=m a and y=p a, with m an arbitrary mass scale (usually taken ≈ m_{e}) read

During RD epoch H ≈ x^{-2}, in this case it is easy to find that the solutions are function of the combination y/x^{3}, or y T^{3}. In particular one gets a decoupling temperature momentum dependent, namely T_{νeD}= 2.7 y^{-1/3} and T_{ν(μ,τ)D}= 4.5 y^{-1/3}. By using the average momentum, <y> ≈ 3, one gets T_{νeD}= 1.87 MeV and T_{ν(μ,τ)D}= 3.12 MeV.

With *instantaneous decoupling limit* we denote the simplifying assumption of a neutrinos decoupling occurring instantaneously at T_{νD}. In reality the decoupling takes place over an extended range of time. In particular more energetic neutrinos will be kept in equilibrium with e.m. plasma longer than low energy neutrinos. This means that even neutrinos to some extent get profit of the entropy release due to e^{±} annihilations, and hence some thermal distortions will be imprinted in the neutrino distribution with respect to a standard Fermi-Dirac function. However these effects change the neutrino energy density at the percent level only, and thus will be neglected in the following.

After decoupling, neutrinos propagate freely, and their distribution only feels the effect of redshift of physical momentum. In the instantaneous decoupling approximation, we can assume the distribution at T_{νD} to be exactly a Fermi-Dirac one

with a_{D }the scale factor at T_{νD}. From that moment on the distribution function satisfying the Liouville equation for ultrarelativistic particles is

which is the initial distribution with a temperature redshifted T = T_{νD }a_{D}/a . Note that this temperature scaling is equivalent to the conservation of the neutrino entropy per comoving volume, namely

The neutrino to photon temperature ratio after neutrino decoupling can be easily obtained by using the conservation of entropy for the e.m. plasma. As long as electron/positron pairs are relativistic, the photon temperature simply scales as a^{-1} since the number of degrees of freedom g_{s} is constant. When the temperature drops below a value ≈ m_{e}, e.m. , the process cannot be efficiently compensated by the inverse pair production. This implies that electron/positron distributions start to be suppressed by the term e^{-me/T}, and practically e^{±} disappear, apart from a tiny relic electron density related by electric neutrality to the baryon number parameter η_{B}. In practice, the entropy release due to e^{± }annihilations reheat photons, whose temperature is then not decreasing as a^{-1 }for a while. Due to the smallness of weak reactions at this temperature, neutrinos practically get no profit of such reheating, nevertheless even though to a very small extent, such phenomenon leaves a footprint on neutrino distribution functions.

The entropy density of coupled e^{±} and γ has the following expression

where

The previous expression strongly simplifies for a temperature much larger than m_{e}, in fact we get

whereas for T << m_{e }we simply have g_{s} = 2. Since the entropy for the e.m. plasma is conserved inside a coomoving volume, namely

we get

At neutrino decoupling T/T_{ν }=1 and g_{s} = 11/2, whereas for T << m_{e }we simply have g_{s} = 2, hence the previous expression becomes

Using the known value of the CMB temperature today, T_{0}= 2.725 K, neutrinos at present have a temperature T_{ν0}= 1.945 K, or equivalently 1.7 10^{-4} eV in natural units.

As we have seen after the e^{+} - e^{- }annihilations the density number of photons and neutrinos read

with g_{γ} = 2, and

with g_{ν} = 6. By using the ratio T/T_{ν }determined in the framework of instantaneous neutrino decoupling, we get n_{ν} = 9/11 n_{γ}. This ratio allows to evaluate for the present value of photon number density, namely n_{γ0} = 410 cm^{-3}, the corresponding neutrino number density, n_{ν0} = 336 cm^{-3}. Absolutely larger than the present aboundance of baryons, namely n_{Bo} ≈ 10^{-7 }cm^{-3}. Hence photons and neutrinos are the most abundant species in the present Universe. In particular since neutrinos at the present are non relativistic (their energy is ≈ 10^{-4} eV << m_{νi}) we can write

Since Ω_{ν} ≤ Ω_{DM }≈ 0.2, we get for h ≈ 0.73, ∑_{i} m_{νi }≤ 10 eV.

If we consider negligible the neutrino chemical potentials, the number density of cosmic neutrinos for each flavour is given by

which leads to an extremely large flux with respect to other astrophysical neutrino sources, including solar neutrinos. By using the ratio between neutrino and photon temperatures, previously obtained, we can compute the total amount of radiation after e^{±} annihilations which results

Such result holds if:

- there are only three light neutrino species and no other relativistic particles;
- neutrino distributions are standard Fermi-Dirac functions with zero chemical potentials;
- we trust the instantaneous decoupling limit.

A convenient way to parameterize any deviation from these assumptions consists in defining the* effective number of neutrino species* N_{eff}

Let us consider the neutrino distribution functions with with x=μ, τ

The Boltzmann Equations

can be casted in the following form

where x = a(t) m_{e} and y= a(t) |p|

In addition one must satisfies the covariant energy conservation

where the total energy density reads

whereas the total pressure reads

The change in the electromagnetic plasma equation of state can be evaluated by first considering the corrections induced on the e^{±} and photon masses. They can be obtained perturbatively by computing the loop corrections to the self-energy of these particles. For the electron/positron mass, up to order α ≡ e^{2}/4 π we find the additional finite temperature contribution

where

The renormalized photon mass in the electromagnetic plasma is instead given, up to order α, by

The previous corrections modify the corresponding dispersion relations as

This in turn affects the total pressure and the energy density of the electromagnetic plasma

Expanding P with respect to δm_{e}^{2} and δm_{γ}^{2}, one obtains the first order correction

and from the previous relation one gets ρ^{int} as well.

The presence of the additional contributions P^{int} and ρ^{int} modify the evolution equation for z≡ T a(t), which now reads

where

and

with

The functions K’(ω) and J’(ω) stand for the first derivativeof K(ω) and J(ω) with respect to their argument. Note that neutrinos affect the final value of z through the terms ,which are not vanishing only if neutrinos have a non thermal behaviour.

In case one neglects the finite temperature QED corrections, the functions G_{1}(x/z) and G_{2}(x/z) vanish, and one recovers a very well known expression reported in literature. Notice that the presence of G_{2}(x/z) in the denominator of the r.h.s. of dz/dx makes, at least inprinciple, not correct to simply sum the neutrino contribution with the QED one.

The initial conditions are fixed by the thermodynamical equilibrium which is satisfied for T > 10 MeV

In particular we have

Asymptotically, as x → ∞, the prediction of the instantaneous decoupling approximation gives

z_{eq} → (11/4)^{1/3}

In the Figure we show the departure of z ≡ T/T_{ν} from z_{eq }for different values of the asymmetry parameters ξ_{α}

The asymptotic distribution functions are not perfectly thermodynamical distributions (out of equilibrium effects).

The effective number of neutrino species is used to parametrize the energy density stored in relativistic species, ρ_{R}, through the relation

where ρ_{γ} is the energy density of photons, whose value today isknown from the measurement of the Cosmic Microwave Background (CMB) temperature.

The previous exprtession can be also written as

Finally, from the previous definitions it is straightforward to get the following expression

In the Table are reported the values assumed by the several factors when different effects are considered.

We now consider the possibility that, at the stage of neutrino decoupling, there are extra relativistic degrees of freedom, provided by some X field excitations, which are assumed to have a thermal distribution with some temperature T_{X}. Their contribution ρ_{X} to the total energy density of the Universe can be parametrized in terms of an additional contribution in the effective number of neutrinos, as previously defined

where

and we have defined z_{X}= T_{X} a(t). The parameter g_{X} depends on the spin (g_{X}=1 for a real scalar, g_{X}=7/4 for a Weyl spinor, etc.) as well as on the additional internal degrees of freedom of the X particles. Notice that if the X excitations have decoupled between μ^{±} and e^{±} annihilation phases we simply have N_{X}= 4/7 g_{X }. For an earlier decoupling we have instead N_{X}< 4/7 g_{X}.

The presence of ρ_{X}, apart from introducing a new contribution to , slightly affects the results previously obtained, namely the relative change of neutrino energy density induced by incomplete neutrino decoupling, as well as the asymptotic photon temperature z^{fin}. In fact since their energy density increases the expansion rate of the Universe, we expect to decrease with growing N_{X}, and the ratio to become closer to unity, since neutrino decouplingprocess starts at earlier times.

If we denote with and the new values for these parameters asfunctions of N_{X} we therefore have

As it is seems more and more clear from neutrino experiments on solar and atmospheric neutrinos, it is unlikely that neutrino mass differences may be greater than ≈ 0.1 eV, unless we enlarge the standard scenario introducing sterile neutrino states. At the same time, it is also quite well established from Tritium decay data that ν_{e} mass is bound to be smaller or, at most, of the order of 1 eV. It is therefore clear that in this scenario all neutrino masses are completely negligible as far as their decoupling is concerned. However if their values is as large as ≈ 1 eV, they start to be relevant as the temperature approaches the range relevant for CMB, and the presence of a finite mass modifies of course the neutrino contribution to . It is interesting to consider how the effects of incomplete decoupling and QED thermal effects just studied would now affect the neutrino energy density. This is conveniently parametrized by the (time dependent) quantity

Since for m_{α} ≈ 1 eV active neutrino are fully elativistic for temperatures of the order of MeV, so we can still take at the e^{±} annihilation phase the results previously obtained, it is easy to see that is given by

where we have

*1*. Rèsumé of standard cosmology in FRWL

*2*. Thermodynamics of the expanding universe

*5*. Baryogenesis

*6*. Dark Matter

*7*. Primordial Nucleosynthesis: theory and experimental data

*8*. Theory of classical cosmological perturbations

*9*. Theory of Quantum Cosmological Perturbations

*10*. A Brief Introduction to Cosmic Microwave Background Anisotropy Formation

*11*. Cosmic Rays - I

*12*. Cosmic Rays - II

Progetto "Campus Virtuale" dell'Università degli Studi di Napoli Federico II, realizzato con il cofinanziamento dell'Unione europea. Asse V - Società dell'informazione - Obiettivo Operativo 5.1 e-Government ed e-Inclusion