From: Greg Aldering (aldering@panisse.lbl.gov)
Date: Fri May 02 2003 - 09:39:08 PDT
First, so Rob doesn't miss it:
IT LOOKS LIKE THE 0.01 mag IS MISSING - I THOUGHT THIS WAS STATED.
WE HAVE 0.02 mag AT HIGH-Z and 0.03 mag AT LOW-Z. THE DIFFERENCE IS
0.01 mag, AND SHOULD BE STATED IN THE LAST PARAGRAPH OF 5.5!
Now to continue ...
Since the Malmquist bias section has drawn some questions, let me try
to address them and then we can see whether there is a better way to
say what little can be said.
First, realize that the standard Malmquist bias correction is applied
to the mean luminosity of an ensemble of standard candles. It is not
applied differentially based on the distance of the object, although
that is how the bias acts - differentially. There is a standard formula
for ordinary standard candles that relates the amount of bias to the
power-law slope of the counts of objects versus flux and the disperion
in the standard candle. Intuitively you can see that if the standard
candle has no dispersion, there can be no Malmquist bias. Likewise, the
bias is less if there are few objects in the parent population near
your flux limit because this implies that you have found all the
objects. If you find all the objects there can be no Malmquist
bias. In P99 I determined that the SN sample had a number-vs-flux
distribution consistent with that of the galaxy population as a whole,
and so used that as the power-law slope.
Next, it is important to realize how SNe are different than other
standard candles. Since SNe brighten and fade, and occur at random
epochs, it is possible that the discovery criterion is completely
uncorrelated with the peak brightness of the SN. In this case there is
no Malmquist bias because you have not selected against underluminous
(for a given stretch) objects. In P99 I used the formula from Jeff
Willick's paper to include this de-correlation suppression of Malmquist
bias. In this paper I did not use it, since most of the SNe were
discovered at maximum light, and hence their selection is strongly
correlated with their peak brightnesses.
Now, I will be the first to admit that the above approach is
approximate. What would be needed to do it right? Well, we would need
to know the flux-limit for each search field and determine which SNe
could have been found, using a full Monte Carlo of SNe brightening and
fading. (For P99 the flux limits were known - they have not been
determined for the current searches.) We would have account for the
fact that different scanners searched with different thresholds (which
we do not know because these are not saved). Next we would have to
account for the selection of candidates for spectroscopic follow-up,
which is again subjective. Next we have to simulate the spectroscopic
follow-up to determine whether we tend to get the best spectra of
overluminous SNe, and know how the host brightness and SN epoch
interact when doing spectral classification. Since we rarely get
spectra of all the candidates (this was especially true of the
March/April 1998 search), we have to not only know the cut-off for
candidates, but all the details of why one or another SN was chosen to
be observed over another. Finally, we have to account for the fact
that we wanted to send the highest redshift SNe to HST, but that we
also had a fixed number per region of sky that we could send.
Roll that all up into a Monte Carlo, and you're done! So, I hope you
can see such a job is impossible for the actual conditions of the SNe
we are reporting. This is much different than running a nice clean SNAP
simulation and then looking at Malmquist bias. I tried to get some
estimate in the face of all this. I tried not to worry the reader too
much, assuring them that the net effect was small, i.e. 0.01 mag.
OH, IT LOOKS LIKE THE 0.01 mag IS MISSING - I THOUGHT THIS WAS STATED
WE HAVE 0.02 mag AT HIGH-Z and 0.03 mag AT LOW-Z. THE DIFFERENCE IS
0.01 mag, AND SHOULD BE STATED IN THE LAST PARAGRAPH OF 5.5!
As a practical matter, I am not wedded to the paragraph starting
with "Since Malmquist bias ...". I wrote that isolation and didn't know
what tone the other sections would take. If there is a reason to
keep that paragraph, then a simple fix is to say "... to warrant
more detailed modeling of this effect than presented here." Next,
one could eliminate the clause ", nor are the constraints on the dark
energy equation of state suffciently strong," (although that statement
is still true).
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