Manuscript Hawk02-A@1 · Hawk02-A.tex · definition · 7fbae745

Definition: Recorded scalar emission law

Stage 1: the fixed-background comparison object › Geometry, field, state, and normalization · lines 372–399

Hawk02-A@1#def:source, Hawk02-A@1#eq:source

The comparison number density is

n_{ℓ} (ω; M) = \frac{Γ _{ℓ} ( ω ; M )}{e ^{8 π GM ω} - 1}, \frac{d ^{2} N _{ℓ m}}{d u d ω} = \frac{n _{ℓ} ( ω ; M )}{2 π} .

For the selected orthonormal packet functions $h_{i}$ define

b_{i} = \int_{0}^{\infty} h_{i} (ω) a_{ℓ m} (ω) d ω, K_{ij} = ⟨ b_{j}^{†} b_{i} ⟩ = \int_{0}^{\infty} h_{i} (ω) h_{j} (ω) n_{ℓ} (ω; M) d ω .

The comparison state is centered and gauge-invariant quasifree, so $⟨ b_{i} b_{j} ⟩ = 0$ .

Displayed formulas in this passage

Hawk02-A@1#eq:source · line 375
$n_{ℓ} (ω; M) = \frac{Γ _{ℓ} ( ω ; M )}{e ^{8 π GM ω} - 1}, \frac{d ^{2} N _{ℓ m}}{d u d ω} = \frac{n _{ℓ} ( ω ; M )}{2 π} .$
unlabeled · line 387
$b_{i} = \int_{0}^{\infty} h_{i} (ω) a_{ℓ m} (ω) d ω, K_{ij} = ⟨ b_{j}^{†} b_{i} ⟩ = \int_{0}^{\infty} h_{i} (ω) h_{j} (ω) n_{ℓ} (ω; M) d ω .$