4

trophysics lying behind the observations, leaving a small
residual error once we have carried out as good a fit as
possible to the data. The systematic imposes an upper
limit on the number of supernovae useful for reducing
the statistical error in the magnitude through Poisson
statistics. One example of such a systematic is nonstan-
dard host galaxy dust extinction. To model the slow
variation of astrophysical systematics we adopted a floor
to the magnitude error within a bin of width Az = 0.1
of dm = 0.02 (1.7/zma,) (1 + z)/2.7. Despite the error
growing with redshift, we see from Fig. 3 that the long
baseline of a deep survey provides crucial leverage.
Indeed this conclusion might be made even stronger.
Despite an increased magnitude error for short redshift
baselines, our adopted systematic might be said to be
overly generous to shallow surveys (e.g. it gives an error
of 0.02 at z = 0.5 for a survey reaching zma = 0.9),
since the level of the residual systematic will depend
on how elaborately the survey is designed. Without a
long redshift baseline, broad wavelength coverage into the
near infrared, spectral observations, a rapid observing ca-
dence, small point spread function, etc. this number can
be large. SNAP is specifically designed to achieve 0.02
mag. For a typical ground based survey, a more realistic
estimate might be 0.05 mag.
For the time variation w' in Fig. 4 the discrepancy due
to ignoring systematics is also strong. For any reason-
able prior on QM, systematics have an extreme effect for
shallow surveys: a factor ~ 5 degradation of our esti-
mate o(w') at zma$. = 0.5. Compare this to a mere 12%
(40%) degradation for zma. = 1.7 when the QM prior is
0.03 (0.01); this clearly shows the vast utility of including
supernovae at z > 1.5.
V. HERESIES COMPOUND
We have seen that low redshift sensitivity to the form
of the dark energy depends on idealized conditions: 1)
reduction of the parameter space by fixing the cosmo-
logical model (i.e. the matter density Qm), 2) reduction
of the parameter space by restricting the dark energy
model (i.e. ad hoc adoption of constant w, ignoring w'),
3) reducing errors by increasing statistics without limit
(i.e. no systematics floor from unknown uncertainties).
This perfect knowledge of cosmology, physics, and astro-
physics is unrealistic and misleading.
Compounding approximations takes us further from
reality. Here we take the three oversimplifications two at
a time to show the distortions they cause. The conclu-
sion in each case will be that realistic analysis of probing
dark energy leads inexorably to the necessity for the ob-
servations to extend beyond z > 1.5.
For clarity and conciseness, we demonstrate this in
simple illustrations. Fig. 5 shows the effects of correct-
ing the first two oversimplifications. When both QM and
the dark energy model (e.g. constant w) are not overas-

0.15
0.1
0
C
0
0.05
0

0

0.5

1

1.5

2

z
FIG. 5: Degeneracies due to the dark energy model,
e.g. equation of state value wo or evolution w', and to the
cosmological model, e.g. value of m, cannot be resolved
at low redshifts. In this differential magnitude-redshift
diagram the three parameters to be determined are var-
ied two at a time. Only at z ~ 1.7 do these very different
physics models exceed 0.02 mag discrimination; SNAP
will be able to distinguish them.
sumed, then degeneracies can lead to complete inability
to discriminate very different cases using only data from
a survey out to z < 1. A deep survey gains both by the
divergence of the curves and the longer redshift observa-
tion baseline. The curves in Fig. 5 would be distinguish-
able by SNAP, which will attain a precision, including
systematics, below 0.02 mag.
The effect of the second and third heresies is to mistake
the uppermost, more realistic curve on Fig. 3 for the
lowest one. Ignoring both time variation and systematics
would misestimate the errors by a factor 12.5 at zma,, =
0.5 but only 2 at zma = 1.7.
Finally, consider the first and third together: the ide-
alized case vs. realistic knowledge of the cosmology in the
form of flatness, a prior on QM of 0.03, and systematic
error. Fig. 6 illustrates several important properties:
1. w': A shallow survey is incapable of appreciably
limiting w', even for perfect assumptions; a medium
survey fails under any realistic conditions.
2. Depth: While there appears to be only moderate
difference between the results of a zma  = 0.9 and
1.7 survey under the ideal case, for the realistic
case the la constraints on wo, w' degrade by a full
sigma. Depth plus long redshift baselines immunize

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