3.1. Mass Constraints

Direct measurements of neutrino masses have given only upper limits (see also the chapter by Robertson and Wilkerson). A secure upper limit on the electron neutrino mass is roughly 15 eV. The Particle Data Group [23] notes that a more precise limit cannot be given since unexplained effects have resulted in significantly negative measurements of m(_e)² in tritium beta decay experiments. However, this problem is at least partially resolved, and the latest experimental upper limits on the electron neutrino mass are 2.8 eV from the Mainz [24] and 2.5 eV from the Troitsk [25] tritium beta decay experiments (both 95% C.L.). There is an upper limit on an effective Majorana neutrino mass of ~ 1 eV from neutrinoless double beta decay experiments [26]. The upper limits from accelerator experiments on the masses of the other neutrinos are m(_µ)< 0.17 MeV (90% CL) and m() < 18 MeV (95% CL) [23, 27], but since stable neutrinos with such large masses would certainly ``overclose the universe'' (i.e., contribute such a large cosmological density that the universe could never have attained its present age), cosmology implies a much lower upper limit on these neutrino masses.

Before going further, it will be necessary to discuss the thermal history of neutrinos in the standard hot big bang cosmology in order to derive the corresponding constraints on their mass. Left-handed neutrinos of mass 1 MeV remain in thermal equilibrium until the temperature drops to T _d, at which point their mean free path first exceeds the horizon size and they essentially cease interacting thereafter, except gravitationally [28]. Their mean free path is, in natural units ( = c = 1), ~ [ n_e^±]^-1 ~ [(G²_F T²) (T³)]^-1, where G_F 10^-5 GeV^-2 is the Fermi constant that measures the strength of the weak interactions. The horizon size is _h ~ (G )^-1/2 ~ M_Pl T^-2, where the Planck mass M_{P l} G^-1/2 = 1.22 x 10¹⁹ GeV. Thus _h / ~ (T / T _d)³, with the neutrino decoupling temperature

(1)

After T drops below 1/2 MeV, e⁺e^- annihilation ceases to be balanced by pair creation, and the entropy of the e⁺e^- pairs heats the photons. Above 1 MeV, the number density n _i of each left-handed neutrino species and its right-handed antiparticle is equal to that of the photons, n, times the factor 3/4 from Fermi versus Bose statistics. But then e⁺e^- annihilation increases the photon number density relative to that of the neutrinos by a factor of 11/4. ⁽¹⁾ As a result, the neutrino temperature T_{, 0} = (4/11)^1/3 T_{, 0}. Thus today, for each species,

(2)

where (T₀ / 2.7 K). With the cosmic background radiation temperature T₀ = 2.728 ± 0.004 K measured by the FIRAS instrument on the COBE satellite [29]., T_{, 0} = 1.947 K and n_{, 0} = 112 cm^-3.

Since the present cosmological matter density is

(3)

it follows that

(4)

where the sum runs over all neutrino species with M _i 1 MeV. (Heavier neutrinos will be discussed in the next paragraph.) Observational data imply that _m h² 0.1-0.3, since _m 0.3-0.5 and h 0.65 ± 0.1 [17]. Thus if all the dark matter were light neutrinos, the sum of their masses would be 9-28 eV.

In deriving eq. (4), we have been assuming that all the neutrino species are light enough to still be relativistic at decoupling, i.e. lighter than an MeV. The bound (4) shows that they must then be much lighter than that. In the alternative case that a neutrino species is nonrelativistic at decoupling, it has been shown [30, 31, 32, 33, 34] that its mass must then exceed several GeV, which is not true of the known neutrinos (_e, _µ, and ). (One might at first think that the Boltzmann factor would sufficiently suppress the number density of neutrinos weighing a few tens of MeV to allow compatibility with the present density of the universe. It is the fact that they ``freeze out'' of equilibrium well before the temperature drops to their mass that leads to the higher mass limit.) We have also been assuming that the neutrino chemical potential is negligible, i.e. that | n - n | << n. This is very plausible, since the net baryon number density (n_b - n) 10^-9 n, and big bang nucleosynthesis restricts the allowed parameters 35] (see also the chapter by Fuller).

¹ In the argument giving the 11/4 factor, the key ingredient is that the entropy in interacting particles in a comoving volume S_I is conserved during ordinary Hubble expansion, even during a process such as electron-positron annihilation, so long as it occurs in equilibrium. That is, S_I = g_I(T) N(T) = constant, where N = n V is the number of photons in a given comoving volume V, and g_I = (g_B + 7/8 g_F)_I is the effective number of helicity states in interacting particles (with the factor of 7/8 reflecting the difference in energy density for fermions versus bosons). Just above the temperature of electron-positron annihilation, g_I = g + 7/8 x g_e = 2 + 7/8 x 4 = 11/2; while below it, g_I = g = 2. Thus, as a result of the entropy of the electrons and positrons being dumped into the photon gas at annihilation, the photon number density is thereafter increased relative to that of the neutrinos by a factor of 11/4.) Back.