Your answer is little more than an opinion. I was looking for a derivation likely involving integration by parts, but something that took me step-by-step from the first equation to the authors analytical solution (or a disproof, if the author was wrong).
Many apologies. You are correct. The graph for ##f(\eta)=\frac{1}{6(1+R(\eta))}\left( \frac {16}{15}+ \frac{R(\eta)^2}{(1+R(\eta))}\right)## looks like this (I forgot the factor of 1/6 in the original plot):
Not sure what you're asking in the second question. The bottom axis is conformal...
There are several sources for this equation. Some write it as an indefinite integral, others write it as $$k_D^{-2}(\eta)=\int_0^{\eta} \frac{1}{6(1+R)}\left( \frac {16}{15}+ \frac{R^2}{(1+R)}\right)\frac{1}{\dot\tau} d\eta$$I believe this is the more rigorous form. Both R and ##\dot\tau## are...
I'm reading through the lecture notes of Wayne Hu regarding the Damping Scale of the CMB. He give the following steps to calculating the damping scale, ##k_D##:$$k_D^{-2}=\int \frac{1}{6(1+R)}\left( \frac {16}{15}+ \frac{R^2}{(1+R)}\right)\frac{1}{\dot\tau} d\eta$$Limiting...
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.
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.
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...