246 The cosmic microwave background
always be true for some sufficiently small range of parameter values. Then the
parameter space curvature matrix (also known as the Fisher information matrix)
is specified by
αij=∑
l∂ClT
∂si∂CTl
∂sj1
(ClT)^2. (7.37)
The variance in the determination of the parametersi from a set ofClTwith
variancesCTl after marginalizing over all other parameters is given by the
diagonal elementiof the matrixα−^1.
Estimates of this kind were first made by Jungmanet al (1996) and
subsequently refined by Zaldarriagaet al(1997) and Bondet al(1997), among
others. The basic result is that a map with pixels of a few arcminutes in size and a
signal-to-noise ratio of around one per pixel can determine,bh^2 ,mh^2 ,h^2 ,
Q,n,andzrat the few percent levelsimultaneously, up to the one degeneracy
mentioned earlier (see the table in Bondet al1997). Significant constraints
will also be placed onrandNν. This prospect has been the primary reason
that the microwave background has generated such excitement. Note that,h,
b,andare the classical cosmological parameters. Decades of painstaking
astronomical observations have been devoted to determining the values of these
parameters. The microwave background offers a completely independent method
of determining them with comparable or significantly greater accuracy, and
with fewer astrophysical systematic effects to worry about. The microwave
background is also the only source of precise information about the spectrum and
character of the primordial perturbations from which we arose. Of course, these
exciting possibilities hold only if the universe is accurately represented by a model
in the assumed model space. The model space is, however, quite broad. Model-
independent constraints which the microwave background provides are discussed
in section 7.6.
The estimates of parameter variances based on the curvature matrix would
be exact if the power spectrum always varied linearly with each parameter. This,
of course, is not true in general. Given a set of power spectrum data, we want
to know two pieces of information about the cosmological parameters: (1) What
parameter values provide the best-fit model? (2) What are the error bars on these
parameters, or more precisely, what is the region of parameter space which defines
a given confidence level? The first question can be answered easily using standard
methods of searching parameter space; generally such a search requires evaluating
the power spectrum for fewer than 100 different models. This shows that the
parameter space is generally without complicated structure or many false minima.
The second question is more difficult. Anything beyond the curvature matrix
analysis requires looking around in parameter space near the best-fit model. A
specific Monte Carlo technique employing a Metropolis algorithm has recently
been advocated (Christensen and Meyer 2000); such techniques will certainly
prove more flexible and efficient than recent brute-force grid searches (Tegmark
and Zaldarriaga 2000). As upcoming data-sets contain more information and