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

Corollary: Explicit clock and crossover

Replacing the clock: compatibility, crossover, and cutoff › An inverse-square example · lines 1023–1049 · proof included or linked

Hawk05-C@1#cor:clock, Hawk05-C@1#eq:tP

Within the positive-cutoff interval,

M (t) t_{cut} - t_{0} A (M) = [M_{0}^{3} - 3 λ (t - t_{0})]^{1/3}, = \frac{M _{0} ^{3} - M _{cut} ^{3}}{3 λ}, = ξ c_{BH} (M_{0}^{2} - M^{2}) .

The formal crossover mass and time are

M_{P} t_{P} - t_{0} F (t_{P}) = M_{0} \frac{ξ}{1 + ξ}, = \frac{M _{0} ^{3}}{3 λ} [1 - (\frac{ξ}{1 + ξ})^{3/2}], = \frac{ξ}{1 + ξ} B (M_{0}) .

They belong to the retained model only if $M_{cut} \leq M_{P}$ . No value after $t_{cut}$ is predicted.

Proof. Integrate $M^{2} d M = - λ d t$ , then solve $A (M) = B (M)$ . The cutoff condition is Theorem thm:crossover.

Displayed formulas in this passage

unlabeled · line 1026
$M (t) t_{cut} - t_{0} A (M) = [M_{0}^{3} - 3 λ (t - t_{0})]^{1/3}, = \frac{M _{0} ^{3} - M _{cut} ^{3}}{3 λ}, = ξ c_{BH} (M_{0}^{2} - M^{2}) .$
Hawk05-C@1#eq:tP · line 1032
$M_{P} t_{P} - t_{0} F (t_{P}) = M_{0} \frac{ξ}{1 + ξ}, = \frac{M _{0} ^{3}}{3 λ} [1 - (\frac{ξ}{1 + ξ})^{3/2}], = \frac{ξ}{1 + ξ} B (M_{0}) .$