1/2 [dR(t)/dt]**2 = (4 pi G /3) rho(t) R(t)**2 + (CC/3) R(t)**2 - k
Can we make some initial observations?
as time --> t = 0, R(t) ---> 0 (Universe is small) and the density of
matter and radiation ---> infinity (get large)
===> k can be ignored
===>The CC term may be ignored if the cosmological constant, CC, is
not much larger than its currently measured value. The inflation
theory effectively has a huge CC for a short time.
Conditions in the Early Universe
Today, we live in a matter-dominated Universe. To see this, compare
the energy contained in matter compared to the energy contained in radiation
(photons)
Matter: rho(matter) c**2 ~ 10**(-11) Joules per cubic meter
Radiation: energy in CMBR ~ 10**(-14) Joules per cubic meter
Comment: these are actually very small numbers. In the Sun, we have
a radiation energy of 10**13 Joules per cubic meter.
Has matter always dominated the radiation in the Universe?
density of matter energy ~ mass c**2 / volume ~ 1 / R(t)**3
density of radiation ~ photon energy / volume ~ [1/R(t)] x [1/R(t)**3]
~ 1 / R(t)**4
===> matter/radiation ~ R(t)
So as R(t) ---> 0, matter becomes less important. The cross-over occurs
near the Epoch of Recombination and so in the interesting part of the
evolution of the Universe, the Universe is radiation dominated.