How do we compare the dark energy "reach" of different methods and
different experiments? We cannot quantify the
probative power of dark energy methods in a strictly model-independent way,
since we do not know which aspects of the expansion history are
most important to measure. Nevertheless, some useful
figures of merit (FoMs) have been proposed to facilitate comparison of
methods and experimental designs. Examples include
the volume of the uncertainty ellipsoid for the dark
energy parameters or the thickness of the ellipsoid in its narrowest
direction
[Huterer
& Turner 2001].
In the Fisher matrix approach (Section 11.2),
these correspond to the inverse square root of the determinant and the
largest eigenvalue of the Fisher matrix, respectively. A special case of
the volume FoM is the inverse area of the Fisher-matrix-projected
ellipse in the *w*_{0} - *w*_{a} plane,

(37) |

where *F*^{w0
wa} is the Fisher matrix projected onto
the *w*_{0} - *w*_{a} plane. This choice
was adopted by the Dark Energy Task Force as a metric for comparing
methods and surveys and is shown in relative terms for Stage IV
space-based experiments in
Fig. 16. The DETF
FoM provides a simple yet useful metric for comparison, as it takes into
account the power of experiments to measure the temporal variation of
*w*. For generalizations, see
[Albrecht
& Bernstein 2007].

**11.2. Fisher information matrix**

The Fisher information matrix formalism allows a quick and easy way to estimate errors on cosmological parameters, given errors in observable quantities, and is particularly useful in experimental design. The Fisher matrix is defined as the (negative) Hessian of the log-likelihood function ,

(38) |

The second equality follows by assuming that
is Gaussian in the
observables; here
*µ* is the vector of mean values of the observables,
**C** is their covariance matrix, and _{,i}
denotes a derivative with respect to *i*th model parameter
*p*_{i}. The parameter vector
includes both
cosmological and any other model parameters needed to characterize the
observations. This expression often simplifies — for example, for
*N* observable quantities with mean values
**O**_{} and
a covariance matrix **C**
that does not depend on the cosmological parameters, the Fisher matrix
becomes *F*_{ij} =
_{, }
(**O**_{}
/
*p*_{i})
**C**_{}^{-1}(**O**_{}
/
*p*_{j}).

By the Cramer-Rao inequality, a model parameter
*p*_{i} cannot be measured to a precision better
than 1 / (*F*_{ii})^{1/2} when all other
parameters are fixed, or a precision
(*F*_{ii}^{-1})^{1/2} when all other
parameters are marginalized over. In practice, the Fisher matrix is a
good approximation to the uncertainties as long as the likelihood can be
approximated by a Gaussian, which is generally the case near the peak of
the likelihood and therefore in cases when the parameters are measured
with small errors. Conversely, if the errors are large, then the
likelihood is typically non-Gaussian, and the constraint region is no
longer elliptical but characteristically banana-shaped, as in
Fig. 8. In this case, the Fisher
matrix typically underestimates the true parameter errors and
degeneracies, and one should employ a Monte Carlo approach to error
estimation.