112 Thermal History of the Universe
6.1 Planck Time
Recall that in quantum mechanics it is always possible to associate the mass of a
particle푀with a wave having theCompton wavelength
휆=ℏ
Mc. (6.1)
In other words, for a particle of mass푀, quantum effects become important at dis-
tances of the order of휆. On the other hand, gravitational effects are important at dis-
tances of the order of the Schwarzschild radius. Equating the two distances, we find
the scale at which quantum effects and gravitational effects are of equal importance.
This defines thePlanck mass
푀P=√
ℏ푐∕퐺= 1. 221 × 1019 GeV푐−^2. (6.2)
From this we can derive the Planck energy푀P푐^2 and thePlanck time푡P=휆P∕푐= 5. 31 × 10 −^44 s. (6.3)Later on we shall make frequent use of quantities at the Planck scale. The reason for
associating these scales with Planck’s name is that he was the first to notice that the
combination of fundamental constants
휆P=√
ℏ퐺∕푐^3 = 1. 62 × 10 −^35 m (6.4)yielded a natural length scale. The particle symmetry at Planck time is characterized
by all fields except the inflaton field being exactly massless. Only when this symme-
try is spontaneously broken in the transition to a lower temperature phase do some
particles become massive.
Unfortunately, there is as yet no theory including quantum mechanics and gravita-
tion. Thus we have no description of the Universe before the Planck time, nor of the
Big Bang, because of a lack of theoretical tools.
There is strong evidence that this cosmic inflation happened when the energy scale
of the Universe was about three orders of magnitude lower than the Planck scale. We
shall postpone the discussion of inflation to Chapter 7.
6.2 The Primordial Hot Plasma
The primeval Universe may have developed through phases when some symmetry was
exact, followed by other phases when that symmetry was broken. The early cosmol-
ogy would then be described by a sequence ofphase transitions. Symmetry breaking
may occur through afirst-order phase transition, in which the field tunnels through a
potential barrier, or through asecond-order phase transition, in which the field evolves
smoothly from one state to another, following the curve of the potential.
An important bookkeeping parameter at all times is the temperature,푇.Whenwe
follow the history of the Universe as a function of푇, we are following a trajectory in