Appendix A Derivation of Thermal Fluctuation of φ Our aim is to calculate the thermal fluctuation of φ that exists when the field is in thermal equilibrium with a thermal bath. This derivation has been worked through following Ref. [56]. As we are dealing with the thermal fluctuation of φ about φ = 0 , we will simply call δφ by φ . The solution to the perturbation equation where ¨ φ − ∇ $^{2}$φ + V ′′ φ = 0 (A.1) φ = 1 √$_{16}$$_{π}$ 3 ∫ 1 √ ω$_{k}$ ( e − iω$_{k}$t + i k · $^{x}$a − k + e iω$_{k}$t − i k · $^{x}$a + k ) d $^{3}$k (A.2) ω$_{k}$ = √ k $^{2}$+ ∣ m 2 φ, eff ∣ (A.3) is 114