xviii Units and conventions
Example 1Calculate the number density of photons in the background radiation
today.In usual units, the temperature of the background radiation isT 2 .73 K.
In dimensionless Planckian units, this temperature is equal to
T2 .73 K
1. 416 × 1032 K
1. 93 × 10 −^32.
The number density of photons in natural units is
nγ=
3 ζ( 3 )
2 π^2T^3
3 × 1. 202
2 π^2(
1. 93 × 10 −^32
) 3
1. 31 × 10 −^96.
To determine the number density of photons per cubic centimeter, we must multiply
thedimensionlessdensity obtained by the Planckian quantity with the correspond-
ing dimension cm−^3 , namelyl−Pl^3 :
nγ 1. 31 × 10 −^96 ×(
1. 616 × 10 −^33 cm)− 3
310 cm−^3.Example 2Determine the energy density of the universe1safter the big bang
and estimate the temperature at this time. The early universe is dominated by ultra-
relativistic matter, and in natural units the energy densityεis related to the timet
via
ε=3
32 πt^2.
Thetime1sexpressed in dimensionless units is
t1s
5. 391 × 10 −^44 s1. 86 × 1043 ;
hence the energy density at this time is equal to
ε=3
32 π(
1. 86 × 1043
) 2 ^8.^63 ×^10 −^89
Planckian units. To express the energy density in usual units, we have to multiply
this number by the Planckian density,εPl= 5. 157 × 1093 gcm−^3 .Thus we obtain
ε(
8. 63 × 10 −^89
)
εPl 4. 45 × 105 gcm−^3.To make a rough estimate of the temperature, we note that in natural unitsε∼T^4 ,
henceT∼ε^1 /^4 ∼
(
10 −^88
) 1 / 4
= 10 −^22 Planckian units. In usual units,T∼ 10 −^22 TPl 1010 K1 MeV.From this follows the useful relation between the temperature in the early Universe,
measured in MeV, and the time, measured in seconds:TMeV=O( 1 )tsec−^1 /^2.