116 Thermal History of the Universe
[Equation (5.24)], also called thefirst law of thermodynamics, followed by assuming
that matter behaved as an expanding nonviscous fluid at constant pressure푝.
When the collisions resulted in a stable energy spectrum,thermal equilibriumwas
established and the photons had theblackbody spectrumderived in 1900 by Max
Planck.
Let the number of photons of energyℎ휈per unit volume and frequency interval be
푛훾(휈). Then the photon number density in the frequency interval(휈,휈+d휈)is
푛훾(휈)d휈=8 휋
푐^3
휈^2 d휈
eℎ휈∕kT− 1. (6.10)
At the end of the nineteenth century some 40 years was spent trying to find this for-
mula using trial and error. With the benefit of hindsight, the derivation is straightfor-
ward, based on classical thermodynamics as well as on quantum mechanics, unknown
at Planck’s time.
Note that Planck’s formula depends on only one parameter, the temperature푇. Thus
the energy spectrum of photons in thermal equilibrium is completely characterized
by its temperature푇. The distribution [Equation (6.10)] peaks at the frequency
휈max≃ 3. 32 × 1010 푇 (6.11)in units of Hertz or cycles per second, where푇is given in degrees Kelvin.
The total number of photons per unit volume, or thenumber density푁훾, is found
by integrating this spectrum over all frequencies:
푁훾=
∫
∞0푛훾(휈)d휈≃ 1. 2022
휋^2
(
kT
푐ℏ) 3
. (6.12)
Hereℏrepresents the reduced Planck’s constantℏ=ℎ∕ 2 휋. The temperature푇may
be converted into units of energy by the dimensional relation퐸=kT. The solution of
the integral in this equation can be given in terms of Riemann’s zeta-function;휁( 3 )≈
1 .2020.
Since each photon of frequency휈is a quantum of energyℎ휈(this is the interpreta-
tion Planck was led to, much to his own dismay, because it was in obvious conflict with
classical ideas of energy as a continuously distributed quantity), the total energy den-
sity of radiation is given by theStefan–Boltzmann lawafterJosef Stefan(1835–1893)
andLudwig Boltzmann(1844–1906),
휀r=
∫∞0ℎ휈푛훾(휈)d휈=휋2
15푘^4 푇^4
ℏ^3 푐^3
≡푎S푇^4 , (6.13)
where all the constants are lumped into Stefan’s constant
푎S=4723 eV m−^3 K−^4.The blackbody spectrum is shown in Figure 8.1.
If the expansion is adiabatic and the pressure푝is constant so that d(pV)=푝d푉,
we recover thesecond law of thermodynamics.
The second law of thermodynamics states in particular that entropy cannot
decrease in a closed system. The particles in a plasma possess maximum entropy when
thermal equilibrium has been established. The assumption that the Universe expands