Manuscript En01-B@2 · En01-B-v2.tex · section · f226d1eb

One shared amplitude, not a free amplitude per setting

Residual force noise under independent calibration › One shared amplitude, not a free amplitude per setting · lines 1186–1214

En01-B@2#eq:scalar0, En01-B@2#eq:scalar1

Back to manuscript

Let $K_{x} (ρ) = [x, [x, ρ]]$ . The narrow residual model is

L_{0, s} (t; b) L_{1, s}^{(2)} (t; b) = L_{known, s} (t) - b f_{s} (t) K_{x}, = L_{known, s} (t) - [b f_{s} (t) + λ A_{r_{c}} g_{s} (t)] K_{x} .

Here $f_{s}, g_{s} \geq 0$ are declared functions of stage, time, and accessible history. The same amplitude $b$ is used across settings. Independent environmental characterization supplies

b \in I = [b_{-}, b_{+}], 0 \leq b_{-} \leq b_{+}, Δ b = b_{+} - b_{-} .

The reference's vibrational localization has a potential-dependent transfer function and can vanish during free motion Pino2018 (Sec. II). Thus proportionality to a persistent gravitational contribution is not inferred from the words “force noise”. It is an explicit model condition.

Displayed formulas in this passage

En01-B@2#eq:scalar0 · line 1190
$L_{0, s} (t; b) L_{1, s}^{(2)} (t; b) = L_{known, s} (t) - b f_{s} (t) K_{x}, = L_{known, s} (t) - [b f_{s} (t) + λ A_{r_{c}} g_{s} (t)] K_{x} .$
En01-B@2#eq:scalar1 · line 1190
$L_{0, s} (t; b) L_{1, s}^{(2)} (t; b) = L_{known, s} (t) - b f_{s} (t) K_{x}, = L_{known, s} (t) - [b f_{s} (t) + λ A_{r_{c}} g_{s} (t)] K_{x} .$
unlabeled · line 1203
$b \in I = [b_{-}, b_{+}], 0 \leq b_{-} \leq b_{+}, Δ b = b_{+} - b_{-} .$

One shared amplitude, not a free amplitude per setting

Displayed formulas in this passage

Nearby in this section