From: Greg Aldering (aldering@panisse.lbl.gov)
Date: Sun Apr 20 2003 - 13:01:10 PDT
Hi Rob,
Here is some text on the Malmquist bias. In the end it is a little
hand-wavey since we have little of the information needed to
do a real calculation.
I am copying it to deepnews, since otherwise people won't see it
until your next draft comes out.
There are some notes that the end which should be kept as
documentation. You could imbed them in the .tex file if you like.
- Greg
Malmquist bias -
As most of our supernovae are from flux-limited samples, they will
suffer Malmquist bias \cite{malmquist24, malmquist36}. This effect was
discussed extensively in \citet{p99} for the P99 dataset, and here we
update that discussion to include our new HST SNe~Ia. For the
measurement of the cosmological parameters, it is the difference
between the Malmquist bias of the low-redshift and high-redshift
samples which matters. In particular, the probability of $\Lambda > 0$
is enhanced only if the the low-redshift SNe suffer more Malmquist bias
than the high-redshift SNe, as this makes the high-redshift SNe~Ia seem
fainter.
The P99 high-redshift dataset was estimated to have little Malmquist
bias (0.01~mag) because the SN discovery magnitudes were decorrelated
with the measured peak magnitudes. However, for the new HST sample,
nine of the eleven SNe~Ia selected from full search samples were found
almost exactly at maximum light. This may reflect a spectroscopic flux
limit superimposed on the original search flux limit since only
spectroscopically confirmed SNe~Ia were considered, and of those,
generally the higher redshift SNe~Ia from a given search were chosen,
for HST for follow-up. In particular, the SNe~Ia selected for
follow-up from the fall 1997 search were all found at maximum light,
while all but SN~1998aw from the spring 1998 search were found at
maximum light. SN~2000fr was found well before maximum. Thus, the new
dataset is likely to suffer more Malmquist bias than the P99 dataset.
Further complicating the interpretation for the high-redshift SNe is
the fact that our new HST SNe are spread over a wide range in redshift,
such that a single brightness correction for Malmquist bias causes a
more complicated change in the fitted cosmological parameters. This is
unlike the situation in P99 in which most SNe were at $z\sim0.5$.
Following the calculation in P99 for a high-redshift flux-limited SN
sample we estimate that the maximum Malmquist bias for the ensemble of
HST SNe is $\sim0.03$~mag. However, we caution that it is SNe near the
flux-limit which are most strongly biased, and therefore, that a
subsample comprised of the highest redshift members drawn from a larger
flux-limited sample will be more biased. When combined with the P99
high-redshift SNe, the bias is likely to be $\sim0.02$~mag since both
samples have roughly the same statistical weight.
As for the low-redshift SNe~Ia, in P99 we established that since most
of the SNe~Ia from the \citet{hamuy96} flux-limited search were found
near maximum, that sample suffered about 0.04~mag of Malmquist bias.
On the other hand, the R98 SNe~Ia were discovered using a
galaxy-targeted technique, which therefore is not limited by the SN
flux, and may be more akin to a volume-limited sample \cite{li01}.
Thus, the addition of the R98 SNe~Ia could slightly reduce the overall
Malmquist bias of the low-redshift sample. If we were to assume no
Malmquist bias for the R99 SNe~Ia, and allowing for the fact that they
contribute only $sim1/3$ the statistical weight of the H96 SNe, we
estimate that the Malmquist bias in the current low-redshift sample
is roughly 0.03~mag.
Given that the new HST high-redshift SNe sample suffers more Malmquist
bias than the P99 sample, and that the enlarged low-redshift sample is
likely to have less Malmquist bias than the low-redshift sample used in
P99, the overall bias towards apparently fainter SNe~Ia at high
redshift should be less than in P99. In particular, the sign of the
bias is working to artificially decrease the statistically infered
$P(\Lambda>0)$. Thus, if anything, the Malmquist bias in the present
sample works to enhance confidence in the confirmation of an
accelerating universe presented in this paper. In addition, since the
intrinsic dispersion decreases from $\sim0.17$~mag to $\sim0.10$~mag
after extinction correction, the Malmquist bias in the extinction
corrected fits is almost halved.
We note that since Malmquist bias results in the selection of
overly-bright SNe at the limits of a flux-limit survey, and since the
flux-limit can be strongly correlated with redshift\footnote{they are
100\% correlated for a single field, but this correlation can be
diluted by combining fields of different depths}, this bias can result
in an apparent distortion of the shape of the Hubble diagram. This may
affect estimates of the dark energy equation of state. The selection
effects for the current high-redshift SNe are not sufficiently
well-defined, nor are the constraints on the dark energy equation of
state sufficiently strong, to warrant modeling of this effect with the
current datasets. However, for future work, much better control of the
selection criteria for SNe~Ia at both low- and high-redshift will be
required in order to properly estimate the impact of this small,
but nearly inescapable, bias.
\bibitem[Li, Filippenko, \& Riess(2001)]{li01} Li, W.,
Filippenko, A.~V., \& Riess, A.~G.\ 2001, \apj, 546, 719
\bibitem[Malmquist(1924)]{malmquist24} Malmquist, K. G.,
1924, Medd. Lund Astron. Obs. Ser. II, 32, 64
\bibitem[Malmquist(1936)]{malmquist36} Malmquist, K. G.,
1936, Stockholm Observatory Medd., no. 26
%Notes:
%
% IAU SCP z mR mR mI mI mB tile tile mR
% disc peak disc peak mag mag apsig lim
%----------------------------------------------------------------
%1997ek 97201 0.863 24.2 24.2 24.58
%1997eq 97198 0.538 22.7 22.65 23.23
%1997ez 97226 0.778 24.0 24.0 24.39
%1998be 9878 0.644 23.7 23.31 23.89 23.77 6.0 24.0
%1998as 98122 0.355 22.2 22.20 22.71 22.02 35.5 24.2
%1998aw 9855 0.440 24.9 22.64 23.29 24.98 5.1 25.0
%1998ax 98109 0.497 22.6 22.59 23.20 22.32 14.4 25.2
%1998ay 98104 0.638 23.3 23.28 23.91 22.91 9.3 23.6
%1998ba 9819 0.430 22.3 22.34 22.94 22.32 14.8 22.8
%1998bi 98142 0.740 24.0 23.5 24.00 23.65 6.5 23.9
%2000fr c00-008 0.543 23.8 22.4 23.14
%
%All but 2000fr and 1998aw have discovery and mB_peak very tightly
%correlated.
%
%For 1997 the correlation is almost 100% - residuals are less than 0.01 mag!
%
%For 1998, without 1998ax, 1998be, and 1998bi, the RMS is only 0.05 mag.
%These three SNe are the only ones not found almost exactly at peak.
%
%Plan: examine mean offset of 1998ax, 1998be, and 1998bi from best-fit
% compared to others. Also look at dmag vs z for those at peak.
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