Figure 5 indicates that the next step
in the compression process is extracting bandpowers
from the map. What is a bandpower and how can it be extracted from the map?
To answer these questions, we must construct a new likelihood function, one
in which the estimated are the data. No theory predicts an individual , but all predict the distribution from which the individual
temperatures are drawn. For example, if the theory predicts Gaussian fluctuations,
then is distributed as a Gaussian with mean zero and covariance
equal to the sum of the noise covariance matrix
and the covariance due to the finite sample
of the cosmic signal . Inverting Equation (1)
and using Equation (2) for the ensemble average
where depends on the theoretical parameters through (see Equation (3)). Here , the window function, is proportional to the Legendre polynomial and a beam and pixel smearing factor . For example, a Gaussian beam of width , dictates that the observed map is actually a smoothed picture of true signal, insensitive to structure on scales smaller than . If the pixel scale is much smaller than the beam scale, . Techniques for handling asymmetric beams have also recently been developed [Wu et al, 2001,Wandelt & Gorski, 2001,Souradeep & Ratra, 2001]. Using bandpowers corresponds to assuming that is constant over a finite range, or band, of , equal to for . Plate 1 gives a sense of the width and number of bands probed by existing experiments.
For Gaussian theories, then, the likelihood function
where and is the number of pixels in the map. As before, is Gaussian in the anisotropies , but in this case are not the parameters to be determined; the theoretical parameters are the , upon which the covariance matrix depends. Therefore, the likelihood function is not Gaussian in the parameters, and there is no simple, analytic way to find the point in parameter space (which is multi-dimensional depending on the number of bands being fit) at which is a maximum. An alternative is to evaluate numerically at many points in a grid in parameter space. The maximum of the on this grid then determines the best fit values of the parameters. Confidence levels on say can be determined by finding the region within which , say, for limits.
This possibility is no longer viable due to the sheer volume of data. Consider the Boomerang experiment with . A single evaluation of involves computation of the inverse and determinant of the matrix , both of which scale as . While this single evaluation might be possible with a powerful computer, a single evaluation does not suffice. The parameter space consists of bandpowers equally spaced from up to . A blindly placed grid on this space would require at least ten evaluations in each dimension, so the time required to adequately evaluate the bandpowers would scale as . No computer can do this. The situation is rapidly getting worse (better) since Planck will have of order pixels and be sensitive to of order a bands.
It is clear that a ``smart'' sampling of the likelihood
in parameter space is necessary. The numerical problem, searching for
the local maximum
of a function, is well-posed, and a number of search algorithms might be used.
tends to be sufficiently structureless that these techniques
suffice. [Bond et al, 1998] proposed the Newton-Raphson
method which has become widely used. One expands the derivative of the
log of the likelihood function - which vanishes at the true maximum of - around a trial point in parameter space, . Keeping terms second order in leads to
where the curvature matrix is the second derivative of with respect to and . Note the subtle distinction between the curvature matrix and the
Fisher matrix in Equation (29), . In general, the curvature matrix
depends on the data, on the . In practice, though, analysts typically use the inverse of
the Fisher matrix in Equation (32). In that
case, the estimator becomes
quadratic in the data . The Fisher matrix is equal to
In the spirit of the Newton-Raphson method, Equation (33) is used iteratively but often converges after just a handful of iterations. The usual approximation is then to take the covariance between the bands as the inverse of the Fisher matrix evaluated at the convergent point . Indeed, [Tegmark, 1997b] derived the identical estimator by considering all unbiased quadratic estimators, and identifying this one as the one with the smallest variance.
Although the estimator in Equation (33) represents a improvement over brute force coverage of the parameter space - converging in just several iterations - it still requires operations which scale as . One means of speeding up the calculations is to transform the data from the pixel basis to the so-called signal-to-noise basis, based on an initial guess as to the signal, and throwing out those modes which have low signal-to-noise [Bond, 1995,Bunn & Sugiyama, 1995]. The drawback is that this procedure still requires at least one operation and potentially many as the guess at the signal improves by iteration. Methods to truly avoid this prohibitive scaling [Oh et al, 1999,Wandelt & Hansen, 2001] have been devised for experiments with particular scan strategies, but the general problem remains open. A potentially promising approach involves extracting the real space correlation functions as an intermediate step between the map and the bandpowers [Szapudi et al, 2001]. Another involves consistently analyzing coarsely pixelized maps with finely pixelized sub-maps [Dore et al, 2001].