From: Greg Aldering (aldering@panisse.lbl.gov)
Date: Sun Apr 13 2003 - 10:38:57 PDT
>> I realize there are few points to do this with; in Table 3 the 5
>> highest redshift SNe have large quoted errors in E(B-V), which I assume
>> has something to do with the assigned intrinsic uncertainty in U-B at
>> max.
>
>That's not it; the current table uses *no* intrinsic uncertainty in U-B.
>It has to do with two things. First, the higher redshift are dimmer, so
>the R-I colors at max tend to have higher uncertainties anyway. Second,
>R-I at different redshifts has different leverage on E(B-V). Considered
>to zeroth order, E(U-B)/E(B-V) is something like 0.75, so for supernovae
>where R predominantly measures U and I predominantly measures V, you'd
>expect 1.33 times the E(B-V) error for a given uncertainty in R-I. (The
>way I in practice work it out is figure out what E(B-V) I need to
>reproduce the color, figure out what E(B-V) I need to reproduce a
>slightly different color, and then scale dE(B-V) as dR-I times the
>difference in calculated E(B-V)'s divided by the offset in color I
>used-- i.e., just a numerical implemetnation of the whole derivative
>error propogation formula.)
Ah, this is a horse of a different color! When I saw the huge jump in
errors for the 5 highest-z SNe I had assumed this included the
intrinsic color error. To help others see this jump, below I list the
E(B-V) uncertainties for the HST SNe sorted by redshift:
z e(E(B-V))
(mag)
-----------------
0.355 0.035
0.430 0.045
0.440 0.030
0.497 0.053
0.538 0.038
0.543 0.036
0.638 0.091 <---
0.644 0.072 <---
0.740 0.063 <---
0.778 0.089 <---
0.863 0.096 <---
You can see that once a SN has z > 0.6 its uncertainty increases by a
factor of 2-3x relative to the lower z SNe. Try plotting this and you
see a strong discontinuity at z ~ 0.6. I had expected the measurement
errors to increase smoothly with redshift. When I saw the
discontinuity, I assumed it was due to the inclusion of intrinsic
color error. (In fact, in an earlier draft I could have sworn that e(E(B-V))
was > 0.08 for all of the z > 0.6 SNe, which seemed to reinforce this
conclusion.)
So, if Table 3 has the correct uncertainties, sans any intrinsic color
dispersion, then this casts the discussion of the intrinsic U-B
uncertainty in a different light. I had thought we were increasing our
error bars by 2-3x by including intrinsic U-B uncertainty for z > 0.6.
But now, as you can see, if we were to use an intrinsic U-B uncertainty
of 0.09 mag we would increase the E(B-V) error bars for the 5 highest
redshift SNe by only ~50%. If we use an intrinsic U-B uncertainty of
0.04 mag it would hardly matter.
Here are how the errors on A_B would change for different U-B intrinsic
color errors:
z e(A_B) w/ 0.09 w/ 0.04
mag mag mag
-----------------------------------
0.355 0.14
0.430 0.18
0.440 0.12
0.497 0.22
0.538 0.16
0.543 0.15
0.638 0.37 0.53 0.41
0.644 0.30 0.47 0.34
0.740 0.26 0.45 0.31
0.778 0.36 0.52 0.40
0.863 0.39 0.54 0.43
Roughly speaking, then, the errors are roughly 10% and 50% worse when using
intrinsic U-B errors of 0.04 or 0.09 mag respectively. (In terms of
telescope time this is like an efficiency decrease of 24% and 2.2x for
those 5 SNe, so it is a big waste if we have to use 0.09 mag.)
My conclusion from all this is that for z > 0.6 we have not measured
extinctions for individual galaxies which are meaningful relative to
the dmag ~ 0.25 size of the accelerating universe signal, as even with
no intrinsic color dispersion eA_B > 0.25 for all z > 0.6 SNe in this
sample. In the paper we can say we *have* measured the extinction well
for 6 high redshift SNe, and that these measurements demonstrate for
*individual SN host galaxies* the relatively low extinction measured
*statistically* in P99 for an ensemble of high-redshift SNe. And
moreover, 5 additional even higher redshift SNe as an ensemble also
indicate small extinction for the observed SNe~Ia. We can go on to say
that the well-measured extinctions made possible with HST allow a
meaningful measurement of the cosmological parameters without any
statistical argument (as in P99) or use of a biasing Bayesian prior (as
in R98).
We should still try to get an estimate of the intrinsic U-B color
dispersion, but it may be less important than had been assumed (or at
least I had assumed). A few of us are working on firming up the
intrinsic U-B color dispersion using empirical and modeling approaches.
We will also have to see whether intrinsic U-B color dispersion plays
much of a role in the OM-w plots.
Cheers,
Greg
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