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I am trying to compute the polarisation moments in the tight coupling limit which is an exercise from Dodelson's Modern cosmology where the evolution equation is given by

$$\dot{\Theta}_p+ik\mu\Theta_p=-\dot{\tau}[-\Theta_p+\frac{1}{2}(1-P_2(\mu))(\Theta_2+\Theta_{p2}+\Theta_{p0}],$$

where $\Theta_{p2}$ $\Theta_{p0}$ are the quadrupole and monopole of the polarization field which I want to express in terms of $\Theta_2$ and $P_l$ is the Legendre polynomial.

I know that in the tight coupling limit, $\dot{\tau}$ which is proportional to the visibility function is very large, the right-hand side of the equation must be very small to compensate. Hence I have:

$$ \Theta_p=\frac{1}{2}(1-P_2(\mu))(\Theta_2+\Theta_{p2}+\Theta_{p0}) \tag 1 $$

The hint is then to expand $\Theta_p$ in terms of Legendre polynomials, keeping only the monopole and the quadrupole but I don't know which expression to be used for the expansion. I.e isn't $(1)$ already being the Legendre expansion of $\Theta_p$?

I want to know which expression of $\Theta_p(\mu)$ should I use to find the coefficients in the expansion,

$$a_m=\frac{2m+1}{2}\int^1_{-1}P_m(\mu)\Theta_p(\mu),$$

so that I can then match coefficients with the expressions in $(1)$.

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