# Express polarization moments in terms of temperature quadrupoles

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 $$(1)$$.