Band Power Window Functions

our first-year papers, it has since been adopted into our analysis pipeline and will be used for all upcoming spectrum measurements. Also note that we could continue to differentiateDto be able to approximate the likelihood over successively larger areas. Since, when we are far from the maximum, the error in the step is predominantly due to the third derivative rather than the difference between DandF, we may be able to converge in fewer steps, though I have yet to investigate this in detail.

Incidentally, the errors in the band powers are easy to estimate when we have an (approximate) curvature matrix. To reasonably high accuracy for most experiments, the error onqB is simply that of the Gaussian approximation to the likelihood surface,FBB⁻¹ (see, e.g., Press et al., 1992). There are also higher accuracy approximations available for more detailed work (Bond et al., 2000), and one can always map out the likelihood surface by direct evaluation, but for the CBI these give very similar results to the errors (for further discussion, see Sievers et al., 2003).

hqBi=X

ℓ

φBℓCℓ (2.40)

where Cℓ is the true power spectrum and φBℓ are the window functions describing the response of qB to the true power spectrum. Unfortunately no such set of coefficients exists valid for allCℓ

because maximum likelihood is a non-linear method—the shift in the power spectrum from adding twice a signal is not exactly twice the shift from adding the original signal. We can, however, come up with such a set of coefficients if we restrict ourselves to the region around the maximum where the curvature is well described by F. In order to do this we need the new expected gradient of the likelihood when we add in the new signal. If Wℓ is the expected covariance from our new signal, then on average we have

∆∆^T →Wℓ+∆∆^T (2.41)

We can then use Equation 2.32 to estimate the new derivative

dlog (L) dqB = 1

2T r

Wℓ+∆∆^T −C

C⁻¹WBC⁻¹

(2.42)

If we are at the maximum, then the∆∆^T −C part of the gradient is equal to zero, and we are left with the expected gradient due to the new signal

dlog (L) dqB

= 1

2T r WℓC⁻¹WBC⁻¹

(2.43)

The expected shift in the band powers can then be calculated by doing a Newton-Raphson iteration

hdqBi= 1

2FBB^−1′T r WℓC⁻¹WB^′C⁻¹

dCℓ (2.44)

Now we have used no properties unique to the CMB to understand the response of the qB to Wℓ. This means that we could substituteany expected signal and see how theqBresponds. For instance, we can calculate the expected contribution to the power spectrum from a population of faint radio

point sources that are statistically isotropic. If the covariance describing the point sources is Siso, then their effect on the power spectrum is

hdqBi=1

2FBB^−1′T r WℓC⁻¹SisoC⁻¹

(2.45)

We can also estimate our sensitivity to a fractional uncertainty ofǫin our measured noise

hdqBi=1

2FBB^−1′T r WℓC⁻¹NC⁻¹

ǫ (2.46)

We use this algorithm in Mason et al. (2003) and Pearson et al. (2003) to measure the response of the CBI power spectrum toCℓas well as to errors in noise and source corrections.

It is worth a discussion of computational issues involved in measuring the filters, as they can easily far exceed the total computational effort required to measure the power spectrum itself. The best way to proceed depends on if one desires just a few filters (i.e. noise and source filters) or very many (for finely sampled window functions). If we desire many filters, then the fastest course of action is to calculate and store the set of matrices C⁻¹WBC⁻¹, and form the gradient vector by taking the trace of each of them multiplied by Wℓ. This requires an expensive initial step of order 2nBn³, which can easily be an order of magnitude more work than measuring the power spectrum. However each additional filter requires only ann² operation, so it is the most efficient way to calculate lots of filters. We can speed matters up considerably if we only require a few (< nB) filters. First, note that the trace remains unchanged if we write it as

C⁻¹SC⁻¹WB (2.47)

for some matrix S whose filter we desire. This is clearly true if

T r(A) =T r B⁻¹AB

(2.48)

It is indeed generally true (see any linear algebra text), but I shall prove it for the specific case of symmetric matrices. If we decompose B into its eigenvalues and eigenvectors, we have

T r VΛ⁻¹V^TAVΛV^T

(2.49)

V^TAV is just a rotation of A, and doesn’t affect the trace, since a rotation doesn’t change the eigenvalues. Similarly, the outer pair of V and V^T is also a rotation and doesn’t affect the trace.

So, if we rewrite the rotated A as A^∗, then the trace is now

T r Λ⁻¹A^∗Λ

(2.50)

We can carry out this multiplication element by element to get the ij^thelement of the product is A^∗_i,jΛ^Λ_jⁱ. This will in general change all elements except those for whichi=j - in other words, the matrix changesexcept for the elements along the diagonal. Clearly, this leaves the trace unchanged.

Now to get the filter from Equation 2.47, we need to calculate C⁻¹SC⁻¹, but then can take the trace of the set ofnB products quickly with only ordern²operations. So we have a choice between doing 2 matrix multiplications per filter, or 2nB matrix multiplications to get arbitrarily many filters.

Chapter 3

First CBI Results

We first used a simplified version of the formalism of Section 2 to analyze the first few months of CBI data, released in Padin et al. (2001a).