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Sorry if this is a technical question. I am studying the origin of CMB anisotropies and the tightly coupled limit of the Boltzmann equations. We have a fluid composed of ionized electrons and photons.

In this limit, I read that, in the Legendre expansion of $\Theta = \delta T / T$, in the Fourier space :

$$\Theta_{\ell}(k,t) = \int_{-1}^{1} \dfrac{d\mu}{2}P_{\ell} (\mu)\ \Theta(k,\mu,t) $$

only the monopole and dipole are non-zero. Why ?

I understand why very small scales are damped (essentially because the wavelength is much smaller than the damping scale that relates to mean free path, density and so on). But large scales are beyond horizon, so should have very little anisotropy ? And scales between sub-horizon scales and small damped scales should exist. As I see the things, $\Theta_{\ell}$ should be non zero only at medium scales, but here it's only $\ell =0,1$ that are non zero.

I think that I am confusing the wavelenght of perturbation with the order of multipole in Legendre expansion. For me, large scale = $\Theta_1 \to \Theta_{\ell\sim 10}$, etc, a bit like in the power spectrum.

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I will answer the question by first explaining the need for the tightly coupled limit.

(1) The equations describing the evolution of CMB temperature fluctuations are derived from $\frac{df }{dt} = 0$. Here $f$ is the distribution function of photons. If I know the distribution function, then I can find out the number density. I have set the right hand side to zero, so I am ignoring changes in number density of photons because of Thomson scattering. That is electrons can collide with photons of a given energy and change its energy, that would change the distribution function of photons and in the above equation the right hand side would be non-zero.

(2) Finding the number density of photons (or any other quantity you may be interested in ) via a distribution function is called the Kinetic approach. Note that you will also need to refer to a distribution function of electrons when setting the right hand side to non zero.

(3) An alternative to the Kinetic approach is the hydrodynamic approach. In this approach you have a single fluid. A single fluid element is made up of photons that are closely tracing the movements of electrons, and the electrons themselves closely trace the movements of protons (because of coulomb interaction). Naturally this approach will work as long as the collision between photons and electrons remains very high. But around about the time of last scattering the collision rate between photons and electrons drops significantly, and you can no longer study the movements of photons-electrons together instead you have to look at each one separately.

(4) At a fundamental level a fluid is characterized by its density, flow velocity and shear. In the Kinetic approach these correspond to the monopole, dipole and quadrupole moments. How about higher multipole moments? Well, I can not associate a meaning so clearly to them because the hydrodynamic approach only has the density, velocity and shear.

(5) "In this limit, I read that, in the Legendre expansion of $\Theta = \delta T / T$, in the Fourier space :

$\Theta_ℓ(k,t)=\int_1^{−1}\frac{d\mu}{2}P_ℓ(\mu) \Theta(k,\mu,t)$

only the monopole and dipole are non-zero. Why ?"

The quadrupole and higher moments are also non-zero, but they are much smaller then the magnitudes of monopole and dipole and can be safely ignored as long as the hydrodynamic limit holds, i.e. the interaction rate of photons with electrons remains high.

(6) "I think that I am confusing the wavelength of perturbation with the order of multipole in Legendre expansion".

Yes you are. A given $k$ roughly corresponds to a given $l$ (ell) only because of the integral:

$ \Theta_l(k,\eta_0) = \int_0^{\eta_0} S(k, \eta) j_{l}(k (\eta_0 - \eta)) d\eta$

The peak of the Bessel function is roughly at $ l \approx k(\eta_0 - \eta) $ and $\Theta_l(k,\eta_0)$ gets most of its contribution close to that peak. This way we can roughly associate a given $k$ with a given $l$. But you should not make this association when trying to understand the physics of CMB fluctuations about anything not related to the integral. For Cls the association works, for anything else it will lead you astray.

(7) I did not answer why successive higher order moments keep getting smaller. Leave that for another post.

(8) The differential equations describing the CMB temperature fluctuations are numerically unstable when the scattering rate between photons and electrons is large. This is fixed by ignoring the moments higher then the dipole with a theoretical justification that a tightly coupled fluid approximation is good enough.

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