# I computing the CMB Damping Tail

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• Quarkly
In summary, the CMB Damping Tail appears to be a function of energy and scale factor, and can be approximated with a power of 2. However, the calculation for it is difficult and there is disagreement between the results of two different papers.
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?

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Quarkly said:
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?

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.

Quarkly said:
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}##).

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.

## 1. What is the CMB damping tail?

The CMB damping tail refers to a specific feature in the cosmic microwave background (CMB) radiation. It is a gradual decrease in the temperature fluctuations of the CMB at small angular scales, caused by the diffusion of photons from hotter to colder regions of the early universe.

## 2. Why is it important to study the CMB damping tail?

The CMB damping tail provides valuable information about the early universe and its evolution. By studying the shape and amplitude of the tail, scientists can better understand the physical processes that occurred during the epoch of recombination, when the universe transitioned from a plasma to a neutral gas.

## 3. How is the CMB damping tail calculated?

The CMB damping tail is calculated using a mathematical model called the blackbody spectrum. This model takes into account the temperature and density of the early universe, as well as the properties of photons, to predict the shape and amplitude of the CMB damping tail.

## 4. What can we learn from the CMB damping tail?

Studying the CMB damping tail can provide insights into various cosmological parameters, such as the age, geometry, and composition of the universe. It can also help us understand the formation and evolution of large-scale structures, such as galaxies and galaxy clusters.

## 5. How does the CMB damping tail support the Big Bang theory?

The CMB damping tail is a key piece of evidence that supports the Big Bang theory. Its existence and characteristics are predicted by the theory and have been confirmed by observations. The CMB damping tail also provides important constraints on alternative cosmological models.

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