Manuscript IV@1 · IV.tex · section · 351cb2d7

An explicit positive detector coefficient

Model 1: a fixed detector and its contrast › An explicit positive detector coefficient · lines 1237–1309

IV@1#eq:d1, IV@1#eq:mean1

Let $A_{0} = A (0)$ , $c_{0} = ∣ A_{0} ∣^{2} > 0$ . Choose a real compact bump $η_{0} = 1$ near zero and take scalar data

φ_{*} (x) = - \frac{1}{2} (A_{0})_{ij} x^{i} x^{j} η_{0} (x), ϖ_{*} = 0.

Let $ϕ_{*}$ be the free solution and $ψ_{*} = S_{t_{0}} (e) ϕ_{*}$ . Its zero correction data imply

\partial_{t}^{2} ψ_{*} (t_{0}, 0) = L_{e} ϕ_{*} (t_{0}, 0) = c_{0} .

Define finite constants on $0 \leq s \leq 1, ∣ x ∣ \leq 1$ :

M_{t} = sup ∣ \partial_{t}^{3} ψ_{*} (t_{0} + s, x) ∣, M_{x} = sup ∣ \nabla_{x} \partial_{t}^{2} ψ_{*} (t_{0} + s, x) ∣.

Set

δ = min {\frac{1}{2}, \frac{c _{0}}{4 ( 1 + M _{t} )}}, r_{0} = min {\frac{1}{2}, \frac{c _{0}}{4 ( 1 + M _{x} )}} .

Then

\partial_{t}^{2} ψ_{*} (t_{0} + s, x) \geq c_{0} /2, ψ_{*} (t_{0} + s, x) \geq c_{0} s^{2} /4

for $0 \leq s \leq δ, ∣ x ∣ \leq r_{0}$ .

Choose nonnegative unit-integral bumps $β_{t} \in C_{c}^{\infty} ((1/2, 1))$ and $β_{x} \in C_{c}^{\infty} (B (0, 1))$ , and define

f_{*} (t, x) = \frac{1}{δ r _{0} ^{3}} β_{t} ((t - t_{0}) / δ) β_{x} (x / r_{0}) .

Consequently,

d_{*} = ℓ_{e, f_{*}} (ϕ_{*}) = \int f_{*} ψ_{*} d^{4} x \geq d_{0} := \frac{c _{0} δ ^{2}}{16} > 0.

The scalar coherent label for real data is

z_{φ, ϖ} = \frac{ω}{2} φ + \frac{i}{2 ω} ϖ .

Direct expansion of (eq:ug) gives

⟨ Coh (z_{φ, ϖ}), Φ (u_{g}) Coh (z_{φ, ϖ}) ⟩ = 2 Re ⟨ u_{g}, z_{φ, ϖ} ⟩ = ℓ_{g} (φ, ϖ) .

Put $z_{*} = z_{φ_{*}, 0}$ , and choose

Ψ_{0} = Coh (0) \otimes Coh (z_{*}), Ψ_{*} = Coh (1/ 2) \otimes Coh (z_{*}) .

Their gravitational covariances agree; their $Q$ -means differ by one and their $Π_{0}$ -means by zero. Hence

⟨ Ψ_{*}, A^{(1)}_{f_{*}} (λ) Ψ_{*} ⟩ - ⟨ Ψ_{0}, A^{(1)}_{f_{*}} (λ) Ψ_{0} ⟩ = λ d_{*} .

Displayed formulas in this passage

IV@1#eq:d1 · line 1277
$d_{*} = ℓ_{e, f_{*}} (ϕ_{*}) = \int f_{*} ψ_{*} d^{4} x \geq d_{0} := \frac{c _{0} δ ^{2}}{16} > 0.$
IV@1#eq:mean1 · line 1303
$⟨ Ψ_{*}, A^{(1)}_{f_{*}} (λ) Ψ_{*} ⟩ - ⟨ Ψ_{0}, A^{(1)}_{f_{*}} (λ) Ψ_{0} ⟩ = λ d_{*} .$
unlabeled · line 1241
$φ_{*} (x) = - \frac{1}{2} (A_{0})_{ij} x^{i} x^{j} η_{0} (x), ϖ_{*} = 0.$
unlabeled · line 1247
$\partial_{t}^{2} ψ_{*} (t_{0}, 0) = L_{e} ϕ_{*} (t_{0}, 0) = c_{0} .$
unlabeled · line 1252
$M_{t} = sup ∣ \partial_{t}^{3} ψ_{*} (t_{0} + s, x) ∣, M_{x} = sup ∣ \nabla_{x} \partial_{t}^{2} ψ_{*} (t_{0} + s, x) ∣.$
unlabeled · line 1257
$δ = min {\frac{1}{2}, \frac{c _{0}}{4 ( 1 + M _{t} )}}, r_{0} = min {\frac{1}{2}, \frac{c _{0}}{4 ( 1 + M _{x} )}} .$
unlabeled · line 1263
$\partial_{t}^{2} ψ_{*} (t_{0} + s, x) \geq c_{0} /2, ψ_{*} (t_{0} + s, x) \geq c_{0} s^{2} /4$
unlabeled · line 1272
$f_{*} (t, x) = \frac{1}{δ r _{0} ^{3}} β_{t} ((t - t_{0}) / δ) β_{x} (x / r_{0}) .$
unlabeled · line 1285
$z_{φ, ϖ} = \frac{ω}{2} φ + \frac{i}{2 ω} ϖ .$
unlabeled · line 1291
$⟨ Coh (z_{φ, ϖ}), Φ (u_{g}) Coh (z_{φ, ϖ}) ⟩ = 2 Re ⟨ u_{g}, z_{φ, ϖ} ⟩ = ℓ_{g} (φ, ϖ) .$
unlabeled · line 1297
$Ψ_{0} = Coh (0) \otimes Coh (z_{*}), Ψ_{*} = Coh (1/ 2) \otimes Coh (z_{*}) .$