I computing the CMB Damping Tail

[Quarkly]

TL;DR Summary

Trying to perform the actual calculations of D. Hu and M. White that will reproduce their work on analysing the CMB Damping Tail.

The formula for the Damping Tail in the CMB BAO analysis generally has the form:
$$\mathcal{D} (k)=\int_{0}^{\eta_0}\dot\tau e^{-\left[\frac {k}{k_D(\eta)} \right]^2}d\eta $$I can’t make sense of this. $\dot\tau$ is the Thompson (Differential) Opacity; the product of electron density, cross section and scale factor:$$\dot\tau(\eta)=n_e\sigma_Ta$$At η = 0 (t = 0), the electron density is infinite (how are there even electrons at this energy level?) and the scale factor is zero. How are we supposed to perform this integration when the Thompson Opacity is so obviously indeterminate at the lower limit of integration?

 

[kimbyd]

Summary: Trying to perform the actual calculations of D. Hu and M. White that will reproduce their work on analysing the CMB Damping Tail.

The formula for the Damping Tail in the CMB BAO analysis generally has the form:
$$\mathcal{D} (k)=\int_{0}^{\eta_0}\dot\tau e^{-\left[\frac {k}{k_D(\eta)} \right]^2}d\eta $$I can’t make sense of this. $\dot\tau$ is the Thompson (Differential) Opacity; the product of electron density, cross section and scale factor:$$\dot\tau(\eta)=n_e\sigma_Ta$$At η = 0 (t = 0), the electron density is infinite (how are there even electrons at this energy level?) and the scale factor is zero. How are we supposed to perform this integration when the Thompson Opacity is so obviously indeterminate at the lower limit of integration?

Many integrals that have a divergence right at the edge of the integral can be integrated without issue. Have you attempted to evaluate the integral yourself?

 

[Quarkly]

Yes. Neither me nor Mathematica can evaluate it. If you graph it, you'll see that it goes up as a power of 2 as it approaches $\eta=0$. Analytically, the density rises as a power of 3 as you move to the start of time and the scale shrinks by a factor of -1.

 

[kimbyd]

Yes. Neither me nor Mathematica can evaluate it. If you graph it, you'll see that it goes up as a power of 2 as it approaches $\eta=0$. Analytically, the density rises as a power of 3 as you move to the start of time and the scale shrinks by a factor of -1.

That's odd. Thinking of the situation physically, it makes no sense. Only a small range in scale factor should contribute to the optical depth calculations (the period when the universe is transitioning from opaque to transparent). I haven't done this calculation myself, but this reference might help:
https://arxiv.org/abs/astro-ph/9609079
They suggest an approximation that is slightly different from the equation you posted above (it has an extra factor of $e^{-\tau}$).

 

[Quarkly]

Thank you. I'm bouncing back and forth between the Hu paper (#12) and the Jungman paper (# 6). They don't seem to agree. I'm still wrestling with the Hu calculation, which seems like it's possible, but I just don't understand how Jungman's calculation is supposed to work.