Tutorial: Convolution by integration
Convolution by integration is the default method of convolution in this codebase, and in some ways it is the most straightforward one. It uses backward convolution (TODO: refer to notebook), which answers the question of ‘if this event were to happen at this point, and it has a delay time of such, then when would it have to be born?’. By looking up the birth time of the system given the current lookback time and the delay time of the event in question, we can then calculate the star formation rate information at that time.
The results of population-synthesis simulations generally provide a series of (binned) events/systems/rates \(\rm{N}_{i}\) which carry, after a step of re-weighting, a normalized yield \(Y_{i}\). This normalized yield is to be read as ‘the number/rate/… of that event/systems/rate/… per formed solar mass in stars’. To predict actual numbers/rates/… we should then multiply those systems with a star formation (rate/mass).
Combining the normalized yield \(\rm{Y}_{i}\) with the star formation rate/mass \(SFR_{i}\) thus allows us to make predictions about observable populations/rates/… .
We do this as follows:
for all events \(i\) we calculate the relevant star formation \(SFR_{i}\) based on the target lookback time L and the delay times \(t_{delay, i}\) associated with those events:
We then multiply this by the normalized yield \(Y_{i}\) to get the actual yield, \(Y^{*}_{i}\) of that event/….