Based on a relationship proposed by G t Hooft we can see how, in broad
terms, entropy evolves as a function of scale factor under different energy
conditions in the Universe. A form of this relationship is;
exp[S] = dim { Hilbert} = D^N
Where S is entropy, D are degrees of freedom and N is the number of
quanta.
1) Expansion under an adiabatic energy condition. ( All quanta
relativistic)
rho_E= 1/R^4
E= rho*V= (1/R*4)*R^3 = 1/R =1/a
S=a*E= a8(1/a)=1
S is constant.
This would be the condition of the Universe post inflation prior to CP
violation. There is no thermodynamic arrow of time under this
energy condition.
2) Expansion in a pure De Sitter space. ( Inflation and end
times assuming DE is the CC.
rho=1
E= rho*V= R^3=-a^3
S=a*E=a*(a^3)=a^4
3) Expansion in a Universe with non zero mass particles
rho=1/R3=1/a^3
E=a*V=(1/a^3)*a^3 =1
S=a*E= a*(1)=a
More generally we can write
S= k_1 + k_2*a+k_3a^4
Where k_1 is the log of the degree of freedom of massless states, k_2 is
the log of the degrees of freedom of massive states and k_3 is the log of
the degree of freedom of dark energy.