Herein, we make use of a double-sided power spectral density defined on a whole range of frequencies, both negative and positive, that yields the following connection to the variance of an arbitrary observable ˆo(t):
∫ ∞ Var[oˆ(t)] ≡ ⟨ˆo2(t)⟩ − ⟨ˆo(t)⟩2 = dΩSo(Ω ), − ∞ 2π
It is worth noting that in the GW community, the sensitivity of GW detectors as well as the individual noise sousces are usually characterized by a single-sided power spectral density S+ (Ω ) o, that is simply defined on positive frequencies Ω ≥ 0. The connection between these two is straightforward: S+(Ω) = 2So (Ω ) o for Ω ≥ 0 and 0 otherwise.