262
Superstring cosmology
In general, a thennaJ contribution Sr (from the gas of string modes in thennal
equilibrium) should be added to the action. It has the fonn
Sr = (/ dNX) / dtJGooF(A;,fJJGoo)
(9.70)
where F is the free energy and fJ = r- I in units where the Boltzmann constant
k8 = 1. The complete action is
Stot = So + Sr.
(9.71)
Varying this action with respect to <1>, Ai and G oo gives (exercise 3), after
combining field equations,
N
(ci»2 - L(~;)2 = e~ E
(9.72)
;=1
••
••
1 ~
Ai - Ai = ,e P;
(9.73)
N
(9.74)
;=1
The thennodynamics that has been employed here is as follows. The free energy
is
F = E - rs
(9.75)
where E is the energy and S is the entropy with
S = _ (aF) = fJ2 (aF) .
ar
(9.76)
v
afJ v
The pressure in the ith direction is
Pi = -(:~t.
(9.77)
As a consequence of (9.75) and (9.76),
E=F_r(aF) =F+fJ(aF) .
(9.78)
aT v
afJ v
With this functionaJ dependence of F,
aF
E = F+2aGoo
(9.79)