__General Maths__A function is harmonic if it obeys the Laplace equation.

To know every value of the function of a line segment you need to know:

- that the function satisfies the Laplace equation
- the values of the function on the boundary

To know every value of the function of a circle you need to know:

- that the function satisfies the Laplace equation

For a circle there is no boundary and so the function must be a constant. This is also true for a sphere.

On the CMB, physical observables are Q and U polarisation. The rank 2 polarisation tensor is defined in terms of Q and U. We can write this in terms of 2 scalar fields P_{E} and P_{B}. (This is a projection of Q, U into P_{E}, P_{B})

How can we transform the scalar fields P_{E} and P_{B} such that the physical quantity P_{ab} doesn't change?

eg for P_{E}

Adding any constant to P_{E} will preserve the physical quantity P_{ab}. Therefore the transformation is gauge invariant.

Q(E,B) and U(E,B) have both E and B components. We know that physically in the universe we expect the E mode to be much bigger than the B mode, so let's say Q(E,B) and U(E,B) are made up of ~99% E mode and ~1% B mode. The projection defined above gives us a method of writing down E and B in terms of Q and U. Eg for the flat sky we write B(l) as the linear combination:

Since the E mode dominates in Q(E,B) and U(E,B), we would expect most linear combinations of Q(E,B) and U(E,B) to be mostly E mode. The special way we have written our B mode (according to how it was defined in the projection), is such that the signal from the dominating E mode gets cancelled out and leaves just the B mode component.

We defined the E and B mode on a sphere, therefore with no information about a boundary. If we cut a piece of the full map out, the new cut map has a boundary, and our projection is not gauge invariant on the cut map.

Specifically, at the boundary of the cut map the projection is not valid and the special linear combination of Q(E,B) and U(E,B) which defines the B mode no longer projects out the dominating E mode.

This effect produces noise on the boundary of the cut map which is the "leakage" of the E mode into the B mode for non periodic boundary conditions.

(Thanks Liam Fitzpatrick, Jared Kaplan)

__CMB Case__On the CMB, physical observables are Q and U polarisation. The rank 2 polarisation tensor is defined in terms of Q and U. We can write this in terms of 2 scalar fields P_{E} and P_{B}. (This is a projection of Q, U into P_{E}, P_{B})

How can we transform the scalar fields P_{E} and P_{B} such that the physical quantity P_{ab} doesn't change?

eg for P_{E}

Adding any constant to P_{E} will preserve the physical quantity P_{ab}. Therefore the transformation is gauge invariant.

Q(E,B) and U(E,B) have both E and B components. We know that physically in the universe we expect the E mode to be much bigger than the B mode, so let's say Q(E,B) and U(E,B) are made up of ~99% E mode and ~1% B mode. The projection defined above gives us a method of writing down E and B in terms of Q and U. Eg for the flat sky we write B(l) as the linear combination:

We defined the E and B mode on a sphere, therefore with no information about a boundary. If we cut a piece of the full map out, the new cut map has a boundary, and our projection is not gauge invariant on the cut map.

Specifically, at the boundary of the cut map the projection is not valid and the special linear combination of Q(E,B) and U(E,B) which defines the B mode no longer projects out the dominating E mode.

This effect produces noise on the boundary of the cut map which is the "leakage" of the E mode into the B mode for non periodic boundary conditions.

(Thanks Liam Fitzpatrick, Jared Kaplan)

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