When fitting for h, we would like to use a weighting scheme which maximally downweights the CMB anisotropies while retaining as much information about the SZE signal as is possible. Naively, it seems that the way to do this would be to weight the data by the expected CMB variance at each baseline. However, the CMB is highly correlated, making it inappropriate as weights in the modelfitting. Instead, one must rotate the data matrix into a basis where the correlations have been removed. This method which we describe here, is related to signal-to-noise eigenmode studies used in CMB power spectrum analyses2.
First, we make a comparison to the basic case of determining σ, the rms of one-dimensional Gaussian noise with zero mean using a maximum likelihood estimator. The probability of obtaining the data set for a givenσis
P(σ)∝ N Y i=1 e−v2 i/(2σ2) σ , (5.1)
wherevirepresents each data point, andNis the total number of data points. In order to determine
theσ which best describes the data set, we would maximize this probability P with respect toσ, and of course we would find that the maximum likelihoodσis simply the standard deviation in the data.
Similarly, we assume the CMB fluctuations are Gaussian noise with zero mean. The strength of the CMB fluctuations is a function of angular scale as described by the power spectrum, so data points corresponding to differing angular scales are assigned their ownσi’s. If the CMB were not
correlated on different data points, we could write the product in Equation 5.1 as
P =e
−VTD−1V/2 √
detD , (5.2)
where V is the vector containing the individual visibility data pointsvi, D is the diagonal matrix
containing the correspondingσ2
i’s, and detDis its determinant. In a real observation, thermal noise
(which we assume to be independent on different baselines) would be added to theσ2
i’s. Theσ2i’s
then represent the noise due to CMB+thermal effects that corrupt each data point.
Now, we consider that the CMB signals on differing baselines of an interferometer are correlated since baselines of similar length see almost the same fluctuations. In place of D, we define a
covariance matrix C, which describes the correlation between different baseline pairs, and whose
elements are
Cij =hvivji, (5.3)
whereviis the observed visibility for one baseline pair, andvj is the visibility from another baseline
pair. Thus, the diagonal elements of C are identical to those of the matrix D above. In the case
of C we also add thermal noise along the diagonal. The covariance matrix contains much more
information than the simple diagonal matrix; however, it is not immediately obvious how to extract this new information, as there is no longer a single quantity like theσi’s with which we can weight the
data points in the SZE fitting. We must transform the data to a basis where there is an equivalent to theσi’s.
Each interferometer data point has both a real and an imaginary component, so in practice we must define separate covariance matrices for each. The CMB covariance matrix for the real component is always real and symmetric, and the covariance matrix for the imaginary component is always complex and hermitian. In either of these two cases, the covariance matrix can always be decomposed into an system of orthogonal eigenvectors,
C =AΛA−1, (5.4)
whereAis the matrix containing the eigenvectors along its columns, and Λis the diagonal matrix
containing the square of the eigenvalues, λi. Because we know the eigenvectors are orthogonal,
A−1=AT, and we can easily write
Table 5.2 Comparison of predicted errors inh−1/2 for no weighting and eigenmode weighting
Cluster β-FWHM σnowt σeigwt
A85 8.80 0.373 0.292 A399 12.54 0.423 0.379 A401 8.58 0.272 0.210 A478 3.77 0.251 0.183 A754 16.96 0.291 0.264 A1651 6.68 0.437 0.324 A2597 1.92 0.902 0.589
CMB error inh−1/2for sample 0.178 0.130
H0for sample with uncertainty due to CMB 67+25−16 67+17−12
Substituting Equation 5.5 into Equation 5.2 in place of D−1 yields in the exponent
VTC−1V =VTAΛ−1ATV. (5.6)
Hence the eigenvalue decomposition essentially tells us how to get to a new basis Y, whereY = ATV, in which the CMB covariance matrix has been diagonalized. The weights for different data
points are now independent, and they are given by the eigenvalues,λi, which describe the “variance”
corresponding to each data point in the new basis. We transform both the data and the model to this new basis and weight the data points by the independentλi’s, utilizing all of the information
contained in the covariance matrix. We call this the “eigenmode” weighting method.