Manuscript Hawk05-C@1 · Hawk05-C.tex · proposition · ba0719f1

Proposition: Power allowance and peak-location allowance

Replacing the clock: compatibility, crossover, and cutoff › Clock and peak stability · lines 1155–1196 · proof included or linked

Hawk05-C@1#prop:stability

Back to manuscript

Suppose a supplied actual power on the same mass interval satisfies

(1 - ρ) \frac{λ}{M ^{2}} \leq P_{act} (M) \leq (1 + ρ) \frac{λ}{M ^{2}}, 0 \leq ρ < 1.

Then its time to any mass $M$ in that interval satisfies

\frac{M _{0} ^{3} - M ^{3}}{3 λ ( 1 + ρ )} \leq t_{act} (M) - t_{0} \leq \frac{M _{0} ^{3} - M ^{3}}{3 λ ( 1 - ρ )} .

Separately, suppose the inverse-square envelope has an interior crossover and a function $f (t)$ satisfies $sup_{t} ∣ f (t) - F (t) ∣ \leq E$ . Every attained global maximum $t$ of $f$ obeys

∣ t - t_{P} ∣ \leq \frac{2 E}{v}, v = \frac{2 c _{BH} λ min { ξ , 1 }}{M _{0}} .

Proof. Integrate the bounds on $1/ P_{act} (M)$ . For the second assertion, the envelope has slope $2 c_{BH} λ ξ / M$ on its early branch and negative slope of magnitude $2 c_{BH} λ / M$ on its late branch. Both magnitudes are at least $v$ , so

F (t_{P}) - F (t) \geq v ∣ t - t_{P} ∣.

At a maximizing $t$ , $f (t) \geq f (t_{P})$ , which implies $F (t_{P}) - F (t) \leq 2 E$ .

Displayed formulas in this passage

unlabeled · line 1158
$(1 - ρ) \frac{λ}{M ^{2}} \leq P_{act} (M) \leq (1 + ρ) \frac{λ}{M ^{2}}, 0 \leq ρ < 1.$
unlabeled · line 1165
$\frac{M _{0} ^{3} - M ^{3}}{3 λ ( 1 + ρ )} \leq t_{act} (M) - t_{0} \leq \frac{M _{0} ^{3} - M ^{3}}{3 λ ( 1 - ρ )} .$
unlabeled · line 1176
$∣ t - t_{P} ∣ \leq \frac{2 E}{v}, v = \frac{2 c _{BH} λ min { ξ , 1 }}{M _{0}} .$
unlabeled · line 1190
$F (t_{P}) - F (t) \geq v ∣ t - t_{P} ∣.$

Nearby in this section

section: Clock and peak stability