Temperature anisotropies are
described by a 2-dimensional random
a unit vector on the sphere. This means
we imagine that the
temperature at each point has been randomly
selected from an underlying probability
distribution, characteristic of
the mechanism generating the perturbations
It is convenient to expand the field in spherical
Observationally, one works with sky brightness
integrated over the experimental beam
Given these relations and a CMB map, it is now
straightforward to construct the likelihood function,
whose role is to relate the
observed sky temperatures, which we arrange in a
data vector with elements
to the model parameters,
represented by a parameter vector
As advertised, for Gaussian fluctuations
(with Gaussian noise) this is simply a multivariate
Many experiments report temperature
differences; and even if the starting
point is a true map, one may wish to subject it
to a linear transformation in order to define
bands in l-space over which power estimates
are to be given. Thus, it is useful to generalize
our approach to arbitrary homogeneous, linear
data combinations, represented by a transformation
Since the transformation is linear, the
new data vector retains a multivariate
Gaussian distribution (with zero mean), but
with a modified covariance matrix:
As a consequence, the transformed pixels,
may be treated
in the same manner as the originals, and
so we will hereafter use the term generalized
pixels to refer to the elements of a general
data vector which may be either real sky pixels or
some transformed version thereof.
The elements of the new theory covariance matrix
are (using the summation convention)
An example is helpful. Consider a
simple, single difference
whose variance is given by
This may be written in terms of multipoles as
Band-powers are defined via Eq. (9).
One reduces the set of Cl contained within the
window to a single number by adopting a spectral form.
The so-called flat band-power,
is established by
An important remark at this stage concerns the construction of Fig. 1. We see here that this figure is only valid for Gaussian perturbations, because it relies in Eq. (7), which assumes Gaussianity at the outset. If the sky fluctuations are non-Gaussian, then these estimates must all be re-evaluated based on the true nature of the sky fluctuations, i.e., the likelihood function in Eq. (7) must be redefined. The same comment applies to any experiment which has an important non-Gaussian noise component - the likelihood function must incorporate this aspect in order to properly yield the power estimate and associated error bars.
What is the raison d'être for these band powers? The likelihood function is clearly greatly simplified if we can find a transformation which diagonalizes (signal plus noise). This can be done for a given model, but because depends on the model parameters, there is in general no unique such transformation valid for all parameter values. The one exception is for an ideal experiment (no noise, or uniform, uncorrelated noise) with full-sky coverage - in this case the spherical harmonic transformation is guaranteed, by Eq. (2), to diagonalize for any and all values of the model parameters. This linear transformation is represented by a matrix , where i= l2 + l + m + 1 is a unidimensional index for the pair (l,m). It is the role of band-powers to approximately diagonalize the covariance matrix in more realistic situations, where sky coverage is always limited and noise is never uniform (and sometimes correlated), and in such a way as to concentrate the power estimates in as narrow bands as possible. Since this is not possible for arbitrary parameter values, in practice one adopts a fiducial model (particular values for the parameters) to define a transformation which compromises between the desires for narrow and independent bands (Bond 1995; Tegmark et al. 1997; Tegmark 1997; Bunn & White 1997).
Consider, then, a situation in which
the band temperatures (that is, generalized pixels which
are the elements of the general
are independent random variables
is diagonal) and that the experimental noise
is spatially uncorrelated and uniform:
Although not immediately relevant to our present
goals, it is all the same instructive to consider the distribution of
This is most easily done by noting that the quantity
All the above relations are exact for the adopted situation - Eq. (16) is the complete likelihood function for the band-power defined by the generalized pixels satisfying Eq. (12). Such a situation could be practically realized on the sky by observing well separated generalized pixels to the same noise level; for example, a set of double differences scattered about the sky, all with the same signal-to-noise. This is rarely the case, however, as scanning strategies must be concentrated within a relatively small area of sky (one makes maps!). This creates important off-diagonal elements in the theory covariance matrix , representing correlations between nearby pixels due to long wavelength perturbation modes. In addition, the noise level is quite often not uniform and sometimes even correlated, adding off-diagonal elements to the noise covariance matrix. Thus, the simple form proposed in Eq. (12) is never achieved in actual observations. Nevertheless, as mentioned, even in this case one could adopt a fiducial theoretical model and find a transformation which diagonalizes the full covariance matrix , thereby regaining one important simplifying property of the above ideal situation. The diagonal elements of the matrix are then its eigenvalues. Because of the correlations in the original matrix, we expect there to be fewer significant eigenvalues than generalized pixels; this will be relevant shortly. One could then work with a reduced matrix consisting of only the significant eigenvalues, an approach reminiscent of the signal-to-noise eigenmodes proposed by Bond (1995), and also known as the Karhunen-Loeve transform (Bunn & White 1997; Tegmark et al. 1997). There remain two technical difficulties: the covariance matrix does not remain diagonal as we move away from the adopted fiducial model by varying - only when this band-power corresponds to the fiducial model is the matrix really diagonal. The second complicating factor is that the eigenvalues are not identical, which greatly simplified the previous calculation.
All of this motivates us to examine the possibility
that a likelihood function of
the form (16) could be applied,
with appropriate redefinitions of
We therefore proceed by renaming
and treating them as parameters to be adjusted
to best fit the full likelihood function.
Thus, given an actual band-power estimate,
(i.e., an experimental result), we propose
as an ansatz for the band-power likelihood function,
with parameters and :
|Figure 2: Comparison to the Saskatoon Q-band 1995 4-point difference. The value of the likelihood is plotted as a function of the band-power, , in both linear (left) and logrithmic (right) scales. The solid (black) curve in each case gives the true likelihood function, while the dashed (red) curve corresponds to the proposed approximation based on two confidence intervals. The dot-dashed (blue) curve is the ansatz with and adjusted to the 68% confidence interval (see text). A "2-winged Gaussian'' with different positive-going and negative-going errors is shown as the three-dotted-dashed (green) curve. All curves have been normalized to unity at their peaks|
|Figure 3: Comparison to the Saskatoon Q-band 1995 10-point difference. The line-styles are the same as in the previous figure; here for the dot-dashed (blue) line|
Unfortunately, most of the time only the 68% confidence interval is reported along with an experimental result (we hope that in the future authors will in fact supply at least two confidence intervals). Is there any way to proceed in this case? For example, one could try to judiciously choose and then adjust with Eq. (19). The most obvious choice for would be , although from our previous discussion, we expect this to be an upper limit to the number of significant degrees-of-freedom (the significant eigenvalues of ), due to correlations between pixels. The comparisons we are about to make in the following section show that a smaller number of effective pixels (i.e., value for ) is in fact required for a good fit to the true likelihood function. One could try other games, such as setting (scan length)/(beam FWHM) for unidimensional scans. This also seems reasonable, and certainly this number is less than or equal to the actual number of pixels in the data set, but we have found that this does not always work satisfactorily. The availability of a second confidence interval permits both parameters, and , to be unambiguously determined and in such a way as to provide the best possible approximation with the proposed ansatz.
Bond et al. (2000) have recently examined the nature of the likelihood function and discussed two possible approximations. The form of the ansatz just presented is in fact identical to one of their proposed approximations, parameterized by x and G. These parameters are simply related to our and as follows: and .
Notice that the above development and motivation for the ansatz essentially follow for a single band-power. A set of uncorrelated power estimates is then easily treated by simple multiplication. However, the approximation as proposed does not simultaneously account for several correlated band-powers, and it's accuracy is therefore limited by the extent to which such inter-band correlations are important in a given data set. As a further remark along these lines, we have noted that flat-band estimates of any kind, be it from a complete likelihood analysis or not, do not always contain all relevant experimental information, (Douspis et al. 2000); any method based on their use is then fundamentally limited by nature of the lost information.
The only way to test the ansatz is, of course, by direct comparison to the full likelihood function calculated for a number of experiments. If it appears to work for a few such cases, then we may hope that it's general application is justified. We now turn to this issue.
Copyright The European Southern Observatory (ESO)