This has been bugging me a bit since the BICEP announcement, but if there are any resources that answer my question in a simple way, they've been buried in a slew of over-technical or over-popularized articles; Wikipedia isn't much help either.
It is clear to me that when the CMB is described as having "E modes" and "B modes", some reference is being made to electric and magnetic fields. What is the precise nature of this reference? I suspect it is simply an appeal to the fact that the polarization can be split into a curl-free component, which is the gradient of something, and a divergence-free component, which is the curl of something else, and these are formally analogous to electric and magnetic fields. Calling them that way certainly brings to bear all our intuition from electrostatics and magnetostatics into how such modes can look. Is this suspicion correct? Or is there some actual electric or magnetic field involved (as, for example, in TE modes in a waveguide)?
Secondly, how exactly does one split the polarization field into these components? It's not quite the sort of 3D vector field to which Helmholtz's theorem applies:
It is a vector field over a sphere instead of all space. This sphere can be seen as the celestial sphere, or equivalently as the surface of last scattering.
It does not automatically have a magnitude on top of its direction, though I suspect one can happily use some measure of the degree of polarization for this. (Is that correct? If so, exactly what measure is used?)
How exactly is the polarization field defined, over what space, and exactly what mathematical machinery is used to split it into E modes and B modes? Are there analogues to the scalar and vector potential? If so, what do they physically represent?
Answer
Planck, BICEP, et al are all detecting electromagnetic radiation, but the "E-modes" and "B-modes" refer to polarization characteristics of this radiation, not the actual electric and magnetic fields. As you surmised, the names derive from an analogy to the decomposition of a vector field into curl-less (here "E" for electric or "G" for gradient) and divergence-less ("B" for magnetic or "C" for curl) components, as follows...
The first step is the measurement of the standard Stokes parameters $Q$ and $U$. In general, the polarization of monochromatic light is completely described via four Stokes parameters, which form a (non-orthonormal) vector space when the various waves are incoherent. For light propagating in the $z$ direction, with electric field:
$$ E_x = a_x(t) \cos(\omega_0 t - \theta_x (t)) \, \, , \quad E_y = a_y(t) \cos(\omega_0 t - \theta_y (t)) $$
the Stokes parameters are:
- $ I =
- $ Q =
- $ U = < 2 a_x a_y \cos(\theta_x - \theta_y) > $ , polarization at $\pm 45$ degrees
- $ V = < 2 a_x a_y \sin(\theta_x - \theta_y) > $ , left- or right-hand circular polarization
In cosmology, no circular polarization is expected, so $V$ is not considered. In addition, normalization of $Q$ and $U$ is traditionally with respect to the mean temperature $T_0$ instead of intensity $I$.
The definitions of $Q$ and $U$ imply that they transform under a rotation $\alpha$ around the $z$-axis according to: $$ Q' = Q \cos (2 \alpha) + U \sin (2 \alpha) $$ $$ U' = -Q \sin (2 \alpha) + U \cos (2 \alpha) $$
These parameters transform, not like a vector, but like a two-dimensional, second rank symmetric trace-free (STF) polarization tensor $\mathcal{P}_{ab}$. In spherical polar coordinates $(\theta, \phi)$, the metric tensor $g$ and polarization tensor are:
$$ g_{ab} = \left( \begin{array}{cc} 1 & 0 \\ 0 & \sin^2 \theta \end{array} \right) $$ $$ \mathcal{P}_{ab}(\mathbf{\hat{n}}) =\frac{1}{2} \left( \begin{array}{cc} Q(\mathbf{\hat{n}}) & -U(\mathbf{\hat{n}}) \sin \theta \\ -U(\mathbf{\hat{n}}) \sin \theta & -Q(\mathbf{\hat{n}})\sin^2 \theta \end{array} \right) $$
As advertised, this matrix is symmetric and trace-free (recall the trace is $g^{ab} \mathcal{P}_{ab}$).
Now, just as a scalar function can be expanded in terms of spherical harmonics $Y_{lm}(\mathbf{\hat{n}})$, the polarization tensor (with its two independent parameters $Q$ and $U$) can be expanded in terms of two sets of orthonormal tensor harmonics:
$$ \frac{\mathcal{P}_{ab}(\mathbf{\hat{n}})}{T_0} = \sum_{l=2}^{\infty} \sum_{m=-l}^{l} \left[ a_{(lm)}^G Y_{(lm)ab}^G(\mathbf{\hat{n}}) + a_{(lm)}^C Y_{(lm)ab}^C(\mathbf{\hat{n}}) \right]$$
where it turns out that:
$$ Y_{(lm)ab}^G = N_l \left( Y_{(lm):ab} - \frac{1}{2} g_{ab} {Y_{(lm):c}}^c\right) $$ $$ Y_{(lm)ab}^C = \frac{N_l}{2} \left( Y_{(lm):ac} {\epsilon^c}_b + Y_{(lm):bc} {\epsilon^c}_a \right)$$
where $\epsilon_{ab}$ is the completely antisymmetric tensor, "$:$" denotes covariant differentiation on the 2-sphere, and
$$ N_l = \sqrt{\frac{2(l-2)!}{(l+2)!}} $$
The "G" ("E") basis tensors are "like" gradients, and the "C" ("B") like curls.
It appears that cosmological perturbations are either scalar (e.g. energy density perturbations) or tensor (gravitational waves). Crucially, scalar perturbations produce only E-mode (G-type) polarization, so evidence of a cosmological B-mode is (Nobel-worthy) evidence of gravitational waves. (Note, however, that Milky-Way "dust" polarization (the "foreground" to cosmologists) can produce B-modes, so it must be well-understood and subtracted to obtain the cosmological signal.)
An excellent reference is Kamionkowski. See also Hu.
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