Past exam of the astrophysics course of the University of Oxford / 2018 / 3 / d

For the Double-degenerate Type Ia supernova scenario, measure the Type Ia supernova rate per formed stellar mass in old galaxies, grouped by the time since their main star-formation episode. Correct the event counts for observing time and detection efficiency. In an ideal burst population, the normalized rate measures the delay-time distribution directly. If gravitational waves inspiral dominates the delay and the initial white-dwarf separations are uniformly distributed in separation, c predicts a slope $-3/4$.

More generally, if $dN/dA\propto A^{\beta }$ and $t_m\propto A^4$, the predicted rate scales as $t_m^{(\beta -3)/4}$. A distribution uniform in $\log A$ has $\beta =-1$ and gives $t_m^{-1}$. Thus a slope differing from $-3/4$ does not alone rule out mergers: the separation distribution must also be tested.

An extended star formation history gives a convolution, $R(t)=\int d\tau \mathrm{SFR}(t-\tau )\Psi (\tau )$, rather than the single-burst expression. The time since star formation stopped need not equal each progenitor's age. Stellar lifetimes before formation of the second white dwarf, common-envelope evolution, masses and mass ratios, metallicity, delay-dependent explosion efficiencies, galaxy-age estimates and survey selection all affect the comparison. Some CO-white-dwarf mergers need not make Type Ia supernovae, and other channels can contribute to the observed rate. Agreement in slope and normalization therefore supports the hypothesis under stated assumptions but is not a unique identification of the channel.