= Solution
For the growing adiabatic mode in an epoch of constant $w$, the superhorizon potential is constant. The <Friedmann equation> gives $4\pi Ga^2(\bar\rho+\bar P)=\tfrac32(1+w)\mathcal H^2$, so the given relation implies
$$
\mathcal R=-\frac{5+3w}{3(1+w)}\phi.
$$
Thus $\phi_{\rm rad}=-2\mathcal R/3$ and $\phi_{\rm mat}=-3\mathcal R/5$. Photon continuity, neglecting its gradient term on <superhorizon scales>, gives $(\Theta_0-\phi)'=0$. The radiation-era initial condition fixes $\Theta_0-\phi=\mathcal R$. Through the matter-radiation transition it follows that $\Theta_{0*}=\mathcal R+\phi_{\rm mat}=2\mathcal R/5$. Since $\psi=\phi$, \b[the matched large-scale emission perturbation is]
$$
\boxed{(\Theta_0+\psi)_*=\frac{2\mathcal R}5-\frac{3\mathcal R}5=-\frac{\mathcal R}5.}
$$
Keeping the radiation value of $\Theta_0$ unchanged across the transition would give the wrong coefficient. This is the <Sachs-Wolfe radiation-to-matter matching> for the paper's sign convention for <comoving curvature perturbation>.
Neglect the <Integrated Sachs-Wolfe effect>, the superhorizon-suppressed Doppler term, observer monopole, and any observer kinematic dipole. Put $\mathbf x_0=0$ by <statistical homogeneity>, and define
$$
\langle\mathcal R(\mathbf k)\mathcal R^*(\mathbf k')\rangle
=(2\pi)^3\delta^{(3)}(\mathbf k-\mathbf k')\frac{2\pi^2}{k^3}\mathcal P_{\mathcal R}(k).
$$
The plane-wave expansion on $\mathbf x_*=-\chi_*\mathbf e$ gives angular multipoles
$$
a_{\ell m}=-\frac{4\pi}5(-i)^\ell\int\frac{d^3k}{(2\pi)^3}
\mathcal R(\mathbf k)j_\ell(k\chi_*)Y_{\ell m}^*(\hat{\mathbf k}).
$$
<Statistical isotropy> and orthonormal <spherical harmonics> then imply $\langle a_{\ell m}a_{\ell'm'}^*\rangle=\delta_{\ell\ell'}\delta_{mm'}C_\ell$. Radial integration supplies $(4\pi)^2(2\pi^2)/(2\pi)^3=4\pi$, giving \b[the large-angle angular power spectrum]
$$
\boxed{C_\ell\simeq\frac{4\pi}{25}\int d\ln k\,\mathcal P_{\mathcal R}(k)j_\ell^2(k\chi_*).}
$$
This assumes adiabatic growing modes, <matter domination> at emission, negligible anisotropic stress, and wavenumbers that are outside the <Hubble radius> then. It treats recombination as instantaneous, omits late potential evolution and rescattering, and omits lensing at this linear order. The formal primordial contribution is defined for $\ell>0$; the observed dipole has a large additional kinematic contribution, so the clean large-angle primordial comparison is usually $\ell\ge2$. A scale-invariant spectrum also yields the Sachs-Wolfe plateau $\ell(\ell+1)C_\ell=2\pi\mathcal P_{\mathcal R}/25$ for $\ell\ge1$.
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