Introduction
149
The dimension less measure of the total energy density is then defined by
00= Po
(5.15)
Pc
and. similarly. for a relic species X whose current energy density is Px.o. we
define
Ox = px.o
(5.16)
Pc
Thus.
L:0x.o=Oo
(5.17)
x
and the energy density for anyone relic species must be less than the total energy
density. so
Ox.o < 00
(5.18)
where data from the latest microwave anisotropy probe (WMAP) [I] give
00 = 1.02 ± 0.02.
(5.19)
The current energy density of a species X is given in terms of the current
abundance by
px.O = flx.omx = Yx.osomx
(5.20)
so provided we can calculate the current abundance Yx.o. and the current entropy
density so. a bound on the mass m x of any relic species may be obtained:
Pc 00
< - - .
(5.21)
mx
soYx.o
A stronger bound may be obtained by replacing 00 by Om where the latter derives
from the total matter content in the universe. The WMAP analysis gives
Omh2 = 0.135~:::
(5.22)
with h given by (5.14).
In the next section. we shall apply the foregoing considerations to neutrinos,
in order to see what can be inferred about their masses. In section 5.3 we shall
attempt a similar analysis for 'axions'. hypothetical particles that are required to
exist if the 'strong CP problem' of the standard SU(3) x SU(2) x U(I> model of
particle physics is solved by the 'Peccei-Quinn' mechanism. currently the only
known solution of this problem. Axions must be very light. like the neutrinos.
If they have survived until the present, their mass too is strongly constrained by
various astrophysical and cosmological data.