Big Bang Nucleosynthesis 139instructive to follow the arguments about how they affect the value of푌 4. Let us rewrite
the decoupling condition [Equation (6.58)] for neutrons
훤wi
퐻=퐴푇d^3 , (6.97)where퐴is the proportionality constant left out of Equation (5.64) and푇dis the decou-
pling temperature. An increase in the neutron mean life implies a decrease in the reac-
tion rate훤wiand therefore a decrease in퐴. At temperature푇dthe ratio of the reaction
rate to the expansion rate is unity; thus
푇d=퐴−^1 ∕^3. (6.98)Hence a longer neutron mean life implies a higher decoupling temperature and an
earlier decoupling time. As we have already seen, an earlier start of helium production
leads to an increase in푌 4.
The expansion rate퐻of the Universe is, according to Equations (6.43) and (6.45),
proportional to
√
푔∗, which in turn depends on the number of neutrino families퐹휈.
In Equations (6.46) we had set퐹휈=3. Thus, if there were more than three neutrino
families,퐻would increase and퐴would decrease with the same consequences as
in the previous example. Similarly, if the number of neutrinos were very different
from the number of anti-neutrinos, contrary to the assumptions in standard Big Bang
cosmology,퐻would also increase.
Light Element Abundance Observations. The value of훺bℎ^2 (or휂) is obtained in
direct measurements of the relic abundances of^4 He,^3 He,^2 HorD,and^7 Li from the
time when the Universe was only a few minutes old. Although the^4 He mass ratio푌 4
is about 0.25, the^3 He and^2 H mass ratios are less than 10−^4 and the^7 Li mass ratio as
small as a few times 10−^10 , they all agree remarkably well on a common value for휂.
If the observed abundances are indeed of cosmological origin, they must not be
affected significantly by later stellar processes. The helium isotopes^3 He and^4 He can-
not be destroyed easily but they are continuously produced in stellar interiors. Some
recent helium is blown off from supernova progenitors, but that fraction can be cor-
rected for by observing the total abundance in hydrogen clouds of different ages and
extrapolating to time zero. The remainder is then primordial helium emanating from
BBN. On the other hand, the deuterium abundance can only decrease; it is easily
burned to^3 He in later stellar events. Measurements of the deuterium abundance are
quite uncertain bacause of systematic errors. The case of^7 Li is complicated because
some fraction is due to later galactic cosmic ray spallation products.
The^4 He abundance is easiest to observe, but it is also least sensitive to훺bℎ^2 ,its
dependence is logarithmic, so only very precise measurements are relevant. The best
‘laboratories’ for measuring the^4 He abundance are a class of low-luminosity dwarf
galaxies called blue compact dwarf (BCD) galaxies, which undergo an intense burst of
star formation in a very compact region. The BCDs are among the most metal-deficient
gas-rich galaxies known (astronomers call all elements heavier than heliummetals).
Since their gas has not been processed through many generations of stars, it should
approximate well the pristine primordial gas.