MODERN COSMOLOGY

Lepton number violation and neutrinos as HDM candidates 197

and

H∼

T^2

MP

(5.30)

so that
Tνd∼MP−^1 /^3 G−F^2 /^3 ∼1MeV, (5.31)

whereGFis the Fermi constant,Tdenotes the temperature,MPis the Planck
mass. Since this decoupling temperatureTνdis higher than the electron mass, then
the relic neutrinos are slightly colder than the relic photons which are ‘heated’ by
the energy released in the electron–positron annihilation. The neutrino number
density turns out to be linked to the number density of relic photonsnγby the
relation:
nν= 223 gνnγ, (5.32)

wheregν=2 or 4 according to the Majorana or Dirac nature of the neutrino,
respectively.
Then one readily obtains theνcontribution to
M:

ν= 0. 01 ×mν(eV)h− 02

gν

2

(

T 0

2. 7

) 3

. (5.33)

Imposing
νh^20 to be less than one (which comes from the lower bound on the

lifetime of the universe), one obtains the famous upper bound of 200(gν)−^1 eV
on the sum of the masses of the light and stable neutrinos:

∑

i

mνi≤ 200 (gν)−^1 eV. (5.34)

Clearly from equation (5.33) one easily sees that it is enough to have one
neutrino with a mass in the 1–20 eV range to obtain
νin the 0.1–1 range of
interest for the DM problem.

5.4.4 HDM and structure formation

Hence massive neutrinos with mass in the eV range are very natural candidates
to contribute to an
Mlarger than 0.1. The actual problem for neutrinos as
viable DM candidates concerns their role in the process of large-scale structure
formation. The crucial feature of HDM is the erasure of small fluctuations by
free-streaming: neutrinos stream relativistically for quite a long time until their
temperature drops toT ∼mν. Therefore a neutrino fluctuation in order to
be preserved must be larger than the distancedνtravelled by neutrinos during
such an interval. The mass contained in that space volume is of the order of the
supercluster masses:

MJ,ν∼dν^3 mνnν(T=mν)∼ 1015 M , (5.35)