Can we do transfer functions better?

[fjebaker]

I'm really not happy with how the transfer functions are currently defined and used. Since the integration domain is over $g$ when calculating the line profiles anyway, the conversion to $g^\ast$ is really inconvenient, as it introduces a number of numerical issues when calculating the transfer functions themselves. Consider the definition

$$ f := \frac{g}{\pi r} \sqrt{g^\ast (1 - g^\ast)} \left\lvert \frac{\partial(\alpha \beta}{\partial(r, g^\ast)} \right\rvert, $$

which is then used to find the flux through

$$ \text{d}F = g^2 I \frac{\pi r}{ \sqrt{g^\ast (1 - g^\ast)} } f \text{d}r \text{d} g^\ast. $$

This is itself used in the integral

$$ F = \int_{r=r0}^{r=r\text{max}}\int_{g^\ast=0}^{g^\ast=1} \text{d}F. $$

It's pretty obvious that most of the terms cancel. Care when integrating comes from the regions where $g^\ast \rightarrow { 0,1 }$, but that is because by the transfer functions go to zero at those points, but $\text{d}F \rightarrow \infty$ numerically, even though there is an analytic value. The prescription is then to have a buffer $h > 0$, where special edge integrations are performed instead.

The transfer functions could far more conveniently be defined

$$ f := g\left\lvert \frac{\partial(\alpha \beta}{\partial(r, g)} \right\rvert, $$

such that

$$ \text{d}F = g^2 I f \text{d}r \text{d} g, $$

and the integration is then performed directly over $g$ instead of $g^\ast$, which also then elides the need for a $g \rightarrow g^\ast$ Jacobian.

Possible gripes can easily be fixed in this formalism too:

• With $g^\ast$ we can integrate over the full energy. So too with $g$, since we need to rebin anyway when extracting the line profiles, so we just integrate over the limits of the bins.
• How would one know when to use which transfer function? Easy, simply set $f=0$ everywhere except $g\text{min} \leq g \leq g\text{max}$. Infact, one can even simplify the total integral this way too, by adding the transfer functions all together for different radi, and then integrating over $g$.

It seems the transfer functions could be so much easier to use when defined this way.

[fjebaker]

Something I failed to account for here is that the Jacobian has the behaviour that

$$ \lim{g\rightarrow g\text{min}, g_\text{max}}\left\lvert \frac{\partial(\alpha, \beta)}{\partial(r, g)} \right\rvert \rightarrow \infty, $$

which is supressed by including the $\sqrt{g^\ast ( 1 - g^\ast)}$ term. Reparameterising $\text{d} r$ and $\text{d} \theta$ on the disc would solve this problem, since when $\text{d} \alpha / \text{d} r$ (respectively $\beta$) diverges, the orthogonal derivative w.r.t. $\theta$ on the disc would go to zero, keeping the Jacobian finite.

Of course the issue with this is then having to store more information (angles in addition to redshift values), and sampling representative $\theta$ such that integrating over $\text{d} \theta$ still converges correctly.

[fjebaker]

Closing, since this is not really that pertinent anymore.