Define the radial transfer integral
$$
F_l(k)=\int_0^{\chi_e}d\chi\,
\frac{\chi_e-\chi}{\chi\chi_e}
\frac{D_i(\tau_0-\chi)}{a(\tau_0-\chi)}j_l(k\chi),
$$
where $\delta_m(\mathbf k,\tau)=D_i(\tau)\delta_m(\mathbf k,\tau_i)$ and $D_i(\tau_i)=1$. With
$$
\langle\delta_m(\mathbf k,\tau_i)\delta_m^*(\mathbf k',\tau_i)\rangle
=(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k')P_i(k),
$$
the angular and radial integrations give
$$
\boxed{C_l^\psi
=\frac{18}{\pi}\Omega_{m,0}^2H_0^4
\int_0^\infty\frac{dk}{k^2}P_i(k)F_l^2(k)}.
$$
This expression makes the lensing signal a weighted projection of the evolving matter power spectrum.


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 310 / 3 / d / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-310/3/d/solution