From: Robert A. Knop Jr. (robert.a.knop@vanderbilt.edu)
Date: Wed Mar 26 2003 - 14:27:23 PST
My current leaning is to do the fits by allowing the alpha used in error
analysis to vary right along with the alpha being tested. This is the
cleanest and "most obvious" way to do it.
For P99, we calculated errors using a pre-fixed version of alpha, which
then turned out to be larger than the most probably version of alpha.
The rationale was twofold. First, the fits tended to like to have
larger and larger alpha because it expanded error bars. Second, there
is a built-in bias: you're more likely to find high stretch supernovae
in greater extinction, which would tend to *flatten out* your stretch
distribution, thereby giving you a falsely low alpha; however, you'd
expect the stretch uncertainty to propogate into a magnitude uncertainty
via the true slope.
In my (current) opinion, the procedure is cleaner if you just let error
alpha track the real alpha-- that way, you're really propogating your
errors as directly as possible. This opinion, however, complete ignores
the aforementioned systematic effect. The question is, though, how does
that systematic effect compare in the high and low redshift supernovae?
Things get pretty complicated if you really try to fix an alpha to avoid
this systematic. Additionally, I'm not sure I'm even seeing that
systematic effect.
I've fit just the low redshift supernovae, trying to figure out what
they think is the best value of alpha. See
http://brahms.phy.vanderbilt.edu/~rknop/scp/hst/#whatisalpha
In particular, when I throw out the worst outlers, I get a very similar
alpha with and without extinction corrections. If the aforementioned
systematic effect were biasing alpha, this shouldn't be the case.
Note that this alpha I get is very similar to the alpha I get when I fit
the "primary set" including the high-redshift supernovae, allowing the
error alpha to track the testing alpha. (Unsprising: you'd expect the
low redshift supernovae, with their better stretch measurements, to
domiante this.) Since alpha isn't balooning out of control, and since I
get what seems to be a "reasonable" value of alpha, this approach
doesn't seem loony. And, given that it's the simplest one to explain
(i.e. we did what was obvious, rather than something non-obvious based
on a systematic bias that requires explanation), it's easier just to do
this.
To see how letting erroralpha slide affects the cosmology, compare line
B-4 on the hst web page to either line B-1, or to line A-1 (filled) or
line A-5 (stroked). It makes enough of a difference-- and it's not *so*
obvious which really is exactly the right way to do it-- that we have a
fairly sizeable "fit method" systematic along the direction of the major
axis of the confidence ellipse.
Comment if you have a strong opinion on the matter.
-Rob
-- --Prof. Robert Knop Department of Physics & Astronomy, Vanderbilt University robert.a.knop@vanderbilt.edu
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