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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 $\mathbf{k}$

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After thinking about it a bit more I think what Baumann was getting at is this:

The statement

$F_*$ and $G_*$ depend only on $|\mathbf k|$ even though the initial perturbations depend on $\mathbf k$

Physically means that

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 $|\mathbf k|$

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