131
average earlier observations of the light curve, which is a re-
sult of the rolling-search techniques frequently used to find and
follow-upSNe.Hence,whencomparinglow-ztohigh-zSNe,the
fitted light curve parameters are obtained from slightly different
parts ofthe lightcurve. The mismatchbetween template andthe
data light curve might thus be more pronounced in one sample
than the other. To quantify the effect, we have performed an ex-
tensive Monte Carlo simulation. A set of BVR light curve tem-
plates are obtainedfrom a quarticspline fitto data, includingthe
well-observed SNe 1990N, 1994D, 1998aq, 2001el, 2002bo,
2003du, 2004eo,and 2005cf(Strovink 2007).Thetemplates are
thenused to samplerandom realizations ofthe light curves with
cadence, S/N, and date of the first detection of the nearby and
distant SN sample. These simulated light curves are then fitted.
The difference in thestretch and color-correctedpeak magnitude
between the nearby and distant sample can be used to estimate
the systematic uncertainty. Forthe nine templates we obtain the
averagedifferencebetween nearbyanddistantSNe of0.02mag
with an rms scatter of 0.015. We adopt an associated systematic
uncertainty ofM ¼ 0:02 mag due to this.
Thereisanothersourceof uncertaintyarisingfromthediversity
ofSNe Ia lightcurves.Ifa certainclass of SNe is misrepresented
(e.g., if they are brighter than average for their typically fitted
stretchvalue)andifthefractionofsuchSNechangesasafunction
of redshift, it will lead to a systematic bias in the cosmological
parameters. Appendix D has addressed this issue by subdividing
the sample according to stretch and redshift. Ifa significant light
curve misrepresentationwere present,onewouldexpecttoseedif-
ferences in the fitted lightcurve–correctionparameters.No sta-
tisticallysignificantdifferences havebeenobserved,andweassign
no additionalcontribution to theuncertainties from suchaneffect.
The light curve model is based on a spectral template series.
IttherebyeliminatestheneedforaseparateK-correction(seex3.3).
The modelhasbeentrainedwithnearbySNe dataand hencewill
be affectedbysystematic errorsassociatedwith thattraining data.
These are largestforthe U band, which suffers from lowtraining
statistics anddifficultfluxcalibration.However,the validityofthe
model in the U bandhas beenverifiedwith the SNLSdata setto
better than 0.02 mag (Astier et al. 2006). Here we adopt their
assessment of the resulting systematic error of M ¼ 0:02.
5.4. Photometric Zero Points
With present methods, ground-based photometric zero-point
calibration is generally limited to an accuracy of k1% (Stubbs
&Tonry 2006). The largestcontributiontothe photometric error
of the peak magnitude arises from the color correction M
c.Thecolormeasurementis basedonthemeasurementofthe
relative flux in two (or more) bands and as a result some of the
uncertainties cancel. Nevertheless, since the colorofSNe at dif-
ferent redshifts are obtained from different spectral regions, the
uncertainty in the reference Vega spectrum limits the achiev-
able accuracyto c 0:01
0:015 mag (Stritzinger etal.2005;
Bohlin & Gilliland 2004).
Here we assume an uncertainty of M ¼ 0:03 for the pho-
tometricpeakmagnitudedue tozero-pointcalibration.Partofthis
uncertainty is common to all samples (as the same set ofcalibra-
tion stars is being used),while the otherpartissample-dependent
(e.g., tied to the calibration procedure), and we divide the error
equally among the two categories.
5.5. Malmquist Bias
Malmquist bias arises in flux-limited surveys, when SNe are
detected because they are overly bright. What matters for cos-
mology is whether the bias is different for the low-z and high-z
samples.Perlmutter et al.(1999), Knop et al. (2003), and Astier
etal.(2006)have evaluated theeffects of Malmquistbias forthe
SCPandSNLSSNsamplesas wellasthenearby SNsample and
found that they nearly cancel. Since an individual estimate of
Malmquistbias forallthe different samples is beyond the scope
of this work, we attribute a conservative systematic uncertainty
ofM ¼ 0:02(Astier et al. 2006)forallsamples,which is con-
sistent with previous estimates.
In addition, we investigated whether the significant redshift
dependence of the Hubble residuals observed for the Miknaitis
etal.(2007)sample(seex4.4),ifinterpretedasduetoMalmquist
bias,exceeds ourclaimeduncertainty.Asimulationwasperformed
in which we introduced a magnitude cutoff such that the result-
ing slope, d/dz, matches the observed slope of 0.6. The as-
sociated Malmquist bias with that sample is then 0.05 mag.
If this is compared to the average Malmquist bias obtained for
magnitude-limited searches, the extra bias is only 0.03 mag
larger—not much larger than the systematic uncertainty we
have adopted. While we do not treat the ESSENCE data sam-
ple differently from the others, we note that Wood-Vasey et al.
(2007) made their extinction prior redshift-dependent to ac-
count for the fact that at higher redshifts an increasingly larger
fraction of the reddened SNe was not detected. The linear color
correction employed in our analysis is independent of a prior
and therefore unaffected by a redshift-dependent reddening
distribution.
5.6. Gra
v
itational Lensing
Gravitationallensingdecreases themodeofthe brightness dis-
tribution and causes increased dispersion in the Hubble diagram
at high redshift (see Fig. 10). The effect has been discussed in
detailinthe literature(Sasaki1987;Linder1988; Bergstro¨m et al.
2000; Holz & Linder 2005). We treat lensing as a statistical
phenomenon only, although with the detailed optical and NIR
data available forthe GOODS field, the mass distribution in the
line of sight and hence the lensing (de)magnification may be
estimated for individual SNe (Jo¨nsson et al. 2006). What is
importantforthisworkisthat theyfindnoevidence forselection
effects (i.e., Malmquist bias) due to lensing of the high-redshift
SNe.
Considering both strong and weak lensing, Holz & Linder
(2005) found that lensing will add a dispersion of 0:093z mag,
which, if the statistics of SNe is large enough, can be approxi-
mated as an additional Gaussian error. Here we added the ad-
ditionaldispersionfrom gravitationallensinginquadraturetothe
‘‘constant’’ systematic dispersion and observational error. This
effectively deweights the high-redshift SNe. However, only for
the highest-redshiftSNeisthe additionaluncertaintycomparable
to that of the intrinsic dispersion.
Fluxmagnificationanddemagnificationeffectsdue toover-or
underdensities of matter near the line of sight cancel. But one
obtains a biasifmagnitudes instead offluxes areused.However,
the bias is 0:004z mag and therefore still much smaller than the
statistical errorontheluminosity distance obtained from the en-
semble of high-redshift SNe. While not yet relevant for this
analysis,future high-statistics samples will have to take this ef-
fect into account.
Anotherpotentialbiasisintroducedbythe3outlierrejection,
since the lensing PDF is asymmetric. Using the PDFs of Holz
&Linder (2005) we have checked that the bias is never larger
than0:006z mag.Wetaketheworst-casevalue of0.009mag(i.e.,
for a SNe at z 1:5) as a conservative systematic uncertainty
for gravitational lensing,since this is still an almost negligible
value.
IMPROVED COSMOLOGICAL CONSTRAINTS
765
No. 2, 2008
