# Transfer function from initial Curvature Perturbations to CMB Temperature Anisotropies has no direction dependence?

In Chapter 7 of Baumann's Cosmology book when discussing how inhomogeneities in the primordial plamsa leads to correlations in CMB temperature anisotropies, he makes a simplification that I'm having trouble wrapping my head around.

The final result is that the CMB angular power spectrum $$C_\ell$$ can be related to the (dimensionless) power spectrum of primordial curvature perturbations $$\Delta_{\mathcal R}^2$$ with a transfer function function $$\Theta_\ell (k)$$: $$C_\ell = 4\pi \int d \ln k\ \Theta^2_\ell(k) \Delta^2_{\mathcal R}(k)$$ What confuses me is how we get to a transfer function which does not depend on the direction of the wave vector $$\mathbf k$$ and instead just the magnitude $$k = |\mathbf k|$$. A sketch of how Baumann gets here:

In what follows lets assume recombination happens instantaneously and we will work in the Newtonian gauge $$ds^2 = a^2(\eta) \left( -(1 + 2 \Psi)d\eta^2 + (1 - 2 \Phi)\delta_{ij}dx^i dx^j \right)$$

Baumann starts with the line-of-sight solution for a temperature anisotropy in a direction $$\hat {\mathbf n}$$ which corresponds to a position $$\mathbf{x} _* =\chi_* \hat{\mathbf{n} }$$ on the surface of last scattering. Let recombination occur at conformal time $$\eta_*$$ $$\Theta(\hat {\mathbf n}) \equiv \frac{\delta T} {\overline{T} } (\hat{\mathbf{n} } ) = \underbrace{\frac{1}{4}\delta_\gamma(\mathbf{x} _*,\eta_*)+ \Psi(\mathbf{x} _*,\eta_*)}_{\rm Sachs-Wolfe} +\underbrace{\hat{\mathbf{n} } \cdot \mathbf{v} _e(\mathbf{x} _*, \eta_*)}_{\rm Doppler}$$ Where in the above we will neglect the integrated Sachs-Wolfe term for this derivation

• $$\delta_\gamma$$ are density perturbations in the photon fluid
• $$\mathbf{v}_e=\nabla v_e$$ is the velocity of the electron that last scatters the CMB photon we observe. The equality follows from a Scalar-Vector-Tensor decomposition of $$\mathbf{v} _e$$ and considering only the scalar part.

We can then rewrite $$\Theta(\hat{\mathbf{n} } )$$ as a superposition of Fourier modes: $$\Theta(\hat{\mathbf{n} } ) = \int \frac{d^3k} {(2\pi)^3} e^{i \mathbf{k} \cdot (\chi_* \hat{\mathbf{n} } )}\left[ F(\mathbf{k} ,\eta_*) +i (\hat{\mathbf{k} } \cdot \hat{\mathbf{n} } )G(\mathbf{k} ,\eta_*) \right]$$ Where $$F \equiv \frac{1}{4}\delta_\gamma + \Psi\qquad G \equiv v_e$$ Then Baumann says

Since the evolution is linear, it is convinient to factor out the inital curvature perturbation $$\mathcal R_i(\mathbf{k} ) = \mathcal R(0, \mathbf{k} )$$ and write $$\Theta(\hat{\mathbf{n} } ) = \int \frac{d^3k}{(2\pi)^3} e^{i\mathbf{k} \cdot(\chi_* \hat{\mathbf{n} }) } \left[ F_*({\color{red} k}) + i (\hat{\mathbf{k} } \cdot \hat{\mathbf{n} }) G_*({\color{red} k})\right]\mathcal R_i(\mathbf{k} )$$ where $$F_*(k) = F(\mathbf{k} , \eta_*) / \mathcal R_i(\mathbf{k} )$$ and $$G_*(k) = G(\mathbf{k} ,\eta_*) / \mathcal R_i(\mathbf{k} )$$ are the transfer functions for the Sachs-Wolfe and Doppler terms. Note that these transfer functions only depend on the magnitude of the wavevector, $$k\equiv |\mathbf{k} |$$, while the initial perturbations are a function of its direction

This bolded statement, followed by some extra Legendre polynomial voodoo witchcraft, is what leads to the transfer function $$\Theta_\ell^2$$ to depend only on $$k$$ and not $$\mathbf{k}$$.

Should it be obvious that the bolded statement is true? I'm not super convinced that $$F(\mathbf{k} ,\eta_*)/\mathcal R_i(\mathbf{k} )$$ or $$G(\mathbf{k} ,\eta_*) / \mathcal R_i(\mathbf{k} )$$ should lead to a quantity that is independent of the direction of $$\mathbf{k}$$

$$F_*$$ and $$G_*$$ depend only on $$|\mathbf k|$$ even though the initial perturbations depend on $$\mathbf k$$
Even if the the initial perturbations have $$\mathbf k$$ dependence, the evolution of these perturbations via the transfer function does not depend on the direction of $$\mathbf k$$.
We know that initial perturbations satisfy statistical isotropy and for this to be maintained, which I think it should for linear evolution, the transfer function should not depend on any direction $$\mathbf k$$, just the magnitude $$|\mathbf k|$$