What do the line segments on the BICEP2 B-mode polarization map mean?

The first image of BICEP2 visuals shows the "BICEP2 B-mode Signal", described as follows:

Gravitational waves from inflation generate a faint but distinctive twisting pattern in the polarization of the CMB, known as a "curl" or B-mode pattern. For the density fluctuations that generate most of the polarization of the CMB, this part of the primordial pattern is exactly zero. Shown here is the actual B-mode pattern observed with the BICEP2 telescope, with the line segments showing the polarization from different spots on the sky. The red and blue shading shows the degree of clockwise and anti-clockwise twisting of this B-mode pattern.

I think I understand the red and blue, but I don't get what the line segments mean. Another web page explains a similar visual as follows:

Summing over all incident waves [at a given point], the E-fields are roughly equal in all directions, but not quite. There will be one direction that has a slightly greater magnitude of E than the other directions (see figure to the left).

We can represent polarization as a line with length proportional to the excess magnitude in that direction and at an angle such that it is aligned with the direction of largest E.

Aside from the fact that the latter description refers to E-mode, seemingly treating it as isomorphic to B-mode polarization (I'm talking through my hat a bit here)... I would expect from my understanding of this description that the length of a line segment should correlate to (the "absolute value" of) the intensity of red or blue at that point. But in the first visual referred to above, the correlation doesn't really seem to hold.

(BTW, I'm assuming that each line segment represents information about the point at the center of the line segment, not about a point at one end, unlike a vector field visualization would. That's because these are not vectors, since they don't really have direction in a 360-degree sense, but rather they have 180-degree rotational symmetry. This is consistent with the fact that each line segment's center seems to be on a grid point. Pardon me if I'm belaboring the obvious, but it took me a while to realize this.)

So... what am I misunderstanding here? What aspect of the polarization do the line segments actually show?

To summarize the question a different way: what is the difference between what the length of the line segments represents, and what the intensity of the red/blue color represents?

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The confusion probably comes from the fact that E-fields and E-modes are entirely different (but are etymologically similar).

A photon carries an intrinsic polarization. In classical E&M, we think of a single light wave as an oscillation of electric and magnetic fields. In vacuum, things are nice and simple, and the propagation direction, the electric field $\vec{E}$, and the magnetic field $\vec{B}$ are mutually orthogonal. See the animation from Wikipedia:

The convention is to say the wave's polarization vector points in the same direction as $\vec{E}$. The bars in the map you show are parallel to $\vec{E}$ - they show the direction of polarization. As an aside, note that they are not arrows, since there is no physical distinction between e.g. "up" and "down" polarization; the E-field oscillates back and forth.

Now when we look at a patch of the CMB, we are gathering many photons. One might expect their polarizations to be random and thus cancel in the aggregate, but this doesn't quite happen. Just as there are temperature fluctuations (some regions of the sky have slightly higher-energy CMB photons than others), there are polarization fluctuations. Theories of the early universe generally give predictions for the fluctuations in both temperature and polarization (and various other statistics).

Helmholtz tells us we can take maps like the one you have and decompose it into the sum of curl-free parts and divergence-free parts. The curl-free parts are called "E-modes" by analogy with classical E&M (again, but this time having nothing to do with photons), where electric fields have no curl (at least if nothing is varying in time). Similarly, the divergence-free parts are called "B-modes" because the magnetic field in classical E&M has no divergence (i.e. no monopoles).

An important prediction of standard inflationary models of the early universe is that most fluctuations (e.g. matter density) cause no B-mode polarization fluctuations (but they may cause E-mode polarization fluctuations). Primordial gravitational waves, on the other hand, induce both modes of polarization. Thus if you look at the CMB polarization and see B-modes, you have support for inflationary models incorporating gravitational waves. You can infer properties of the early universe within some model by noting how strong those B-mode components are.

So what do the line segments show? They represent the direction (modulo a sign) of the electric field at that point in the CMB. The line length, by the way, shows how strong the polarization is. That is, it is not as though the photons are 100% aligned with the same polarization; rather, the average polarization deviates statistically from 0. The length represents the magnitude of this deviation.

To be slightly more accurate, the line segments represent the result after subtracting off the E-mode signal, leaving only the B-modes. The colors show the curl of the vector field (well, vectors modulo that $180^\circ$ symmetry mentioned earlier). Technically, the curl is another vector field, but since we're only looking at a 2D surface, I imagine the colors are chosen to represent the magnitude of the radial (in-plane/out-of-plane) component of the curl. Regions are red if the line segments more or less spiral clockwise as you move from outside inward toward the region in question, whereas blue indicates counterclockwise spiraling. Colors are more intense where this effect is stronger, i.e. the lines are more coherently spiraling together.

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Thanks for this explanation. I really was confusing E-modes with E-fields. I still don't completely understand the relationship between the line lengths (magnitude of deviation) and the color (twisting). Is there any necessary relationship between the two? – LarsH Mar 17 '14 at 21:27
I added a bit more to address that. The color map looks like it was defined to be the magnitude of the "z"-component of the curl of the field shown. – Chris White Mar 17 '14 at 23:38
I'm not quite sure about your very last sentence. Colours are more intense where the polarization is "curlier", and bear little relation to the strength of the (B mode) polarization. – Emilio Pisanty Mar 18 '14 at 11:18
@EmilioPisanty Yeah, I admit I didn't look up their exact coloring scheme - I just figured they'd do something similar to what I would have done. – Chris White Mar 19 '14 at 5:32
In the original bicepkeck.org/#figures_detection_B_mode fig3 there is a color code on the right in muKelvin so it is temperature – anna v Mar 19 '14 at 11:39