Statistics in astrophysical systems 175Whether or not superfluidityis a part oftheunderstandinginvolves speculation too
uncertain even for the average astrophysicist, and we will not be drawn into this
intriguing question!
1 5.2.1 After the bigbang
The nowadays orthodox view of big bang cosmology is based on three principal
pieces ofevidence: (1) theobservationby US astronomer EdwinHubbleandothers
that theuniverseis expanding, (2) theexistence ofthe cosmicbackgroundradiation,
and( 3 )the observation that 2 5 % of the mass of the universe is helium.
Theideaisthat, since theuniverseis now expanding, we canimagine running
time backwards towards a beginning, where the universe was effectivelyat a point
and had infinite temperature. Forward from that (almost unimaginable) big bang, we
haveasystem whichisgraduallyexpandingandcooling,withthe ‘normal’laws of
physics applyingafter about timet= 10 −^34 seconds. At that stage, the temperature
was aboutT= 1027 K.
Atthe presentday,t= 15 × 109 years, the temperaturehas cooledto that ofthe
uniform cosmic background radiation. Recent measurements from the COBE satellite
have confirmedthat our universe todayisfilledwithrather (but not exactly) spatially
uniform radiation. The spectraldensityoftheradiationfollows verypreciselythe
black-body radiation spectrum (see (9.13) and Fig. 9.7) corresponding to a thermal
equilibrium temperature of 2.72 6 K. This is seen as a logical legacy of the original
hotbig bangandwas predictedlongbeforeits eventualmeasurement. Itis a very
beautiful example of the photon gas treatment of Chapter 9.
The origins of the 25% helium involves many of the ideas introduced in our dis-
cussion of chemical reactions in Chapter 13, and inparticular concerns Boltzmann
factors of the type exp(−E/kkkBT). Consider first what happens up to aboutt= 10 −^3
seconds,bywhichtime the temperatureThas cooledto a mere 1 012 K. Theuniverse
will contain a lot ofphotons(γrays) of typical energyseveral timeskkkBT(about
1 00 MeV), using the black-body spectrum. This means that there will also be created
particles andantiparticles ofvarious sorts (thoughttobe quarksandother exotic par-
ticles, the ‘quark soup’). As the system cools, so does the energyscale of the photons
and hence also the upper mass of creatable particles (usingE=mc^2 ∼kkkBT).
Byt= 10 −^2 seconds, the temperaturehasfallen to around 1011 K, andthe energy
scale to 10 MeV. This is now too cold for anything but the particles with which we are
familiar: photons, electrons, neutrinos, neutrons andprotons. Nucleosynthesis (man-
ufacture ofheavier nucleons)has notyet started,since ampleγphotons are available
to break up any transiently formed deuterons. Thermal equilibrium prevails, hence the
interest to statisticalphysics. Because ofthis equilibrium, the neutron/proton ratiois
determinedsimply bythe temperature. We wouldpredict (correctly)that,inthermal
equilibrium, the number of neutrons divided by the number of protons is
NNNn/NNNp=exp(−mc^2 /kkkBT) ( 15 .1)