text:
\section{Introduction}

The realization that the rates of decline of the brightnesses of Type
Ia supernovae (SNe Ia) are correlated with their peak luminosities
(Phillips 1992) has led to renewed efforts to use them as cosmological
distance markers (Hamuy et al. 1995, 1996a, 1996b; Riess, Press and
Kirshner 1995, 1996). Ongoing searches for high redshift ($z\sim
0.5-1$) SNe Ia have employed phenomenological models for these
correlations to constrain the variation of luminosity distance $D_L(z)$
with redshift; the results have been interpreted to imply the existence
of a nonzero cosmological constant (Perlmutter et al.\ 1998 hereafter
P98, Riess et al.\ 1998 hereafter R98).  Moreover, the results appear
to rule out the simplest version of a flat cosmology, in which the
density parameter for ``ordinary'' matter (including as yet
unidentified nonbaryonic material) $\Omega_M=1$ and the density
parameter for the cosmological constant $\Omega_\Lambda=0$.

Although the logical possibility that $\Omega_\Lambda\sim 1$
today has long been recognized (Einstein 1917), it is anathema
to many theorists, since the associated vacuum energy density
must be $\rho_{vac}\sim 10^{-122}M_{\rm P}^4$, where $M_{\rm P}$
is the Planck mass. (Theoretical and conceptual problems
with a nonzero cosmological constant so small compared with
its ``natural'' scale have been reviewed by Weinberg 1989
and Carroll, Press \& Turner 1992.) On the other hand,
there is some evidence from large scale structure simulations
that $\Omega_\Lambda\neq 0$ in a flat Universe fits the
observations well (Cen 1998). Conceivably, what
we interpret as a cosmological ``constant'' might be an
evolving field (e.g., Caldwell, Dave \& Steinhardt 1998;
Garnavich et al.\ 1998; Perlmutter, Turner \& White 1999).
In any case, a convincing demonstration that the expansion
rate of the Universe is increasing would have a revolutionary
impact on our understanding of fundamental physics.

In view of the importance of the  potential discovery of a nonzero
cosmological constant, we have undertaken an independent study of the
published data in an effort to understand their implications better.
Our motivations are both phenomenological and astrophysical (and may
end up being related ultimately). On the phenomenological side, we note
that three different analysis methods are used to compute distances,
the multicolor light curve shape (MLCS) method (Riess, Press \&
Kirshner 1995, 1996), the M15 or template fitting (TF) method (Phillips
1992; Hamuy et al.\ 1995, 1996a, 1996b),
and the stretch factor (SF) method (Perlmutter et al.\ 1997).  None of
these methods is a perfect description of reality. As we will show,
they are not always in agreement,  and there seems to be no physical or
phenomenological reason to prefer one to the other.

On the astrophysical side, we note that there are processes such
as evolution of the supernovae sample
that can mimic the effects of cosmology at high redshifts and which
are extremely difficult to constrain convincingly
with the current data.
Therefore, it is useful to ask at what level the current data
are able to distinguish the effects of cosmology from these
other processes.
We find that allowing for the possibility of
a redshift-dependent shift in SNe Ia peak magnitudes
(such that the most distant observed SNe Ia are dimmed by $\approx 0.2$
to 0.3 mag) renders $\Omega_M=1$ and $\Omega_\Lambda=0$ acceptable, and that this
is true for a variety of phenomenological models for the evolution.
We also present results of simulations that show
that if SNe Ia luminosities evolve with redshift, but evolution
is neglected in analyzing the data, then, given enough data,
the analyses will settle on precisely determined, but incorrect,
values for $\Omega_M$ and $\Omega_\Lambda$, and that the
incorrectness of the model will not be detectable with a
standard $\chi^2$ goodness-of-fit test.  However, we find that
the Hubble constant, $H_0$, is virtually unaffected by evolution.

We believe that it is unjustifiable to try to determine
cosmological parameters $\Omega_M$ and $\Omega_\Lambda$ from data on ``standardized''
candles, such as the peak luminosities of SNe Ia, without
allowing for the possibility of
source evolution. Our attitude is that an uncertain amount of
evolution must be presumed to occur, as a default; and the sensitivity
of the results to the uncertainty must be studied.
Hopefully, one can demonstrate
from the data that source evolution is absent or negligible.
\footnote{The observers have employed supplementary measurements---such as
source spectra---to argue that there is no compelling evidence
{\it for} evolution in the SNe Ia samples. If the physical
connection between the additional data and the estimators
of peak luminosity can be understood quantitatively, then
the supplementary measurements can be incorporated usefully
into analyses of models with evolution. Without
such physical understanding, these additional data cannot be
marshalled to argue {\it against} evolution.}
If that turns out to be untrue, as recent
examination of the risetimes of the light curves
of the SNe Ia sample has preliminarily indicated
(Riess, Filippenko, Li, \& Schmidt 1999),
one might hope instead to constrain
the parameters in an evolutionary model along with the cosmological
parameters. Optimistically, one would anticipate that this might
be accomplished once enough data are acquired. We argue, from
simulations employing simple, phenomenological models, that such
optimism may be unrealistic. What is needed is a better
physical understanding of the SN Ia process and its evolution
with redshift, before cosmological parameters can be determined
reliably from SNe Ia catalogues.  Such an understanding is
currently being sought by theorists (see, e.g., von Hippel, Bothun,
\& Schommer 1997; H\"oflich, Wheeler,
\& Thielemann 1998; Dom\'inguez et al.\ 1999).

In using observations of SNe Ia to determine cosmological parameters,
the raw data are combined by any of the three methods mentioned above
to derive single parameter summaries -- the distance moduli to the
sources. When a single catalogue of data is subjected to different
types of analysis, each of which derives one quantity per source, the
results of the individual analyses need not agree with one another
entirely, and there is information contained in the degree to which the
answers derived by the different methods differ\footnote{A simple
example might involve computing mean values of data using similar, but
not identical, weighting functions. Each weighted mean would then be a
slightly different superposition of all of the moments of the data
computed with one fiducial weighting function.}.  In the example under
consideration, the different analysis methods might probe slightly
different physical aspects of the SN Ia mechanism, and their
relationships to reality and to one another may be different at high
and low redshift.  Indeed, one clue that evolutionary effects are
important would be a systematic drift with increasing redshift between
the distance moduli implied by the MLCS and TF methods for the SNe Ia
observed and analyzed by R98 where identical SNe Ia data are subjected
to two different analysis methods.

In order to set the scale of interest for our investigations of
potential systematic effects in these data we plot, in Figure~1, the
joint credible regions for $\Omega_M$ and $\Omega_\Lambda$ for the
largest available data set (P98), analyzed as published (Figure~1$a$)
and after introducing a systematic offset to all the high redshift
distance moduli ($z > 0.15$) of $-0.1$ magnitudes (Figure 1$b$).  We
see that a correlated systematic shift of this size would have a major
impact on the interpretation of the data.  A nonzero cosmological
constant would still be favored, but the statistical significance of
the result would be much reduced.

In Section \ref{sec:review} we review the published data
that we employ in our study, as well as the salient features
of the light curve fitting methods. In Section \ref{sec:fit}
we compare the different fitting methods on a supernova by
supernova basis where possible.  In Section \ref{sec:stat} we
explore whether the data have sufficient shape information to
distinguish effects of cosmology from other cosmological effects such
as evolution.  We summarize our conclusions in Section
\ref{sec:conc}.

\section{Measurements of $\Omega_M$ and $\Omega_\Lambda$ Using Type Ia Supernovae }
\label{sec:review}

The traditional measure of distance to a SN is its distance modulus,
$\mu\equiv m_{\rm bol}-M_{\rm bol}$, the difference between its
bolometric apparent magnitude, $m_{\rm bol}$, and its bolometric absolute
magnitude, $M_{\rm bol}$.
In the Friedman-Robertson-Walker (FRW) cosmology,
when the (relative) peculiar velocity of
the source is negligible, the distance modulus is determined
by the source's redshift, $z$, according to
\begin{eqnarray}
\mu &=& 5 \log \left[D_L(z;\Omega_M,\Omega_\Lambda,H_0)\over
      1\;\hbox{Mpc}\right] +25 \nonumber\\
  &\equiv& f(z; \Omega_M,\Omega_\Lambda,H_0).
\label{eq:mu}
\end{eqnarray}
Here the luminosity distance $D_L(z;\Omega_M,\Omega_\Lambda,H_0)
=cH_0^{-1}d_L(z;\Omega_M,\Omega_\Lambda)$, where $c$ is the velocity of
light, $H_0$ is Hubble's constant at the present epoch, and the
dimensionless luminosity distance from redshift $z$ is
\begin{equation}
d_L(z;\Omega_M,\Omega_\Lambda)=
(1+z)|\Omega_k|^{-1/2} \mathop{\rm sinn}\nolimits \{ |\Omega_k|^{1/2}
\int_0^z dz [ (1+z)^2(1+\Omega_M z) - z(2+z)\Omega_\Lambda ]^{-1/2}\},
\end{equation}
with $\Omega_k = 1-\Omega_M-\Omega_\Lambda$, and $\mathop{\rm sinn}\nolimits(x)=\sinh(x)$ for $\Omega_k \geq 0$
and $\sin(x)$ for $\Omega_k \leq 0$~(e.g., Carroll, Press \& Turner 1992).
In principle, one could infer the cosmological parameters
$H_0$, $\Omega_M$, and $\Omega_\Lambda$ from the distribution of measured distance moduli
of sources at a variety of redshifts.

Several factors complicate implementation of such an analysis.  In
reality, bolometric data are not available, and one must infer $\mu$
using magnitudes $m_X$ and $M_X$ in some bandpass, $X$.  The bandpass
maps to a different region of the spectrum as a function of redshift
$z$, so $\mu$ cannot be calculated simply by taking the difference
between band-limited magnitudes; a $K$-correction term must be added
whose value depends not only on the source's redshift, but also on its
spectrum.  In addition, extinction along the line of sight increases
the apparent magnitude by some amount $A_X$ not due to the cosmological
effects modelled in \ceqn(eq:mu).  Further, the absolute magnitude of
the source---bolometric or band-limited---is not directly measured, but
must instead be inferred from other source properties.  Finally, the
inevitable presence of statistical uncertainties and peculiar
velocities further complicates straightforward use of \ceqn(eq:mu).

Of these complications, the need to infer the absolute
magnitude indirectly is the most troublesome.  Ideally, one seeks a population
of ``standard candles'' such that all members of the population
have the same $M$ (for convenience we henceforth drop the subscript
$X$).  If such a population could be identified,
the parameters $\Omega_M$ and $\Omega_\Lambda$ could be inferred even if the actual
value of $M$ for the population
were unknown (the remaining parameter, $H_0$, would remain
undetermined).  Historically, all attempts to identify such a
population have failed.  Particularly worrisome is the possibility that
some classes of objects that appear to be approximately
standard candles locally (at low redshift, where they can be
studied in detail) have evolved significantly, so that their
younger counterparts at high redshifts have different absolute
magnitudes, thwarting their use as cosmological distance indicators.

SNe Ia were briefly considered promising candidates for standard candles,
but observers quickly discovered that SNe Ia are not all identically
bright (Branch, 1987; Barbon, Rosino, \& Iijima 1989; Phillips et al.\
1987, 1992; Filippenko, et al.\ 1992a, 1992b; Leibundgut, et al.\ 1993).
The intrinsic dispersion in the peak absolute magnitudes of SNe Ia,
determined from studies of nearby events, is approximately 0.3 - 0.5 mag
(Schmidt et al.\ 1998). However, there is an apparent empirical
correlation between the rate of decline of the light curve of a given
SN Ia and its luminosity at maximum brightness that was first
quantified by Phillips (1992).  Various techniques have been developed to
take advantage of this correlation to determine the absolute magnitudes
of individual supernovae using their light curves (Phillips 1992; Hamuy
et al.\ 1995, 1996a, 1996b; Riess, Press \& Kirshner 1995, 1996;
Perlmutter et al.\ 1997); the relationships used in these analyses have
come to be known generically as ``Phillips relations.''  When applied to
nearby SNe Ia, these methods reduce the dispersion of the distance
moduli about the low-$z$ FRW distance modulus vs.\ redshift relation to
$\approx 0.15$ (Hamuy et al. 1996a, Riess, Press \& Kirshner 1996).

The goal of the high redshift supernovae searches is to observe a large
sample of supernovae at relatively large $z$, and understand their properties
well enough to infer reliable distance moduli for them, allowing
accurate determination of cosmological parameters.  Two experimental
groups have recently announced and published results from their
independent programs to discover and study high redshift supernovae for
this purpose (Perlmutter 1997 and P98; R98).  The resulting two data
sets share many low redshift SNe discovered by previous surveys, but
include different high redshift SNe, and differ in their analysis
methods.  We take advantage of the similarities and differences among
the data and methods used to assess the consistency or inconsistency of
the assumptions underlying the analyses.

P98 have published data on 60 SNe Ia.  Of these, 18 were discovered and
measured in the Cal\'an-Tololo survey (all at low redshift; Hamuy et
al.\ 1996c), and this group discovered 42 new SNe Ia at redshifts
between 0.17 and 0.83.  The $\mu$ values are inferred using the SF
light curve fitting method and are typically uncertain to $\pm 0.2$
magnitudes (``$1\sigma$'').  The SF method (Perlmutter et al.\ 1997;
P98) is based on fitting a time-stretched version of a single standard
template to the observed light curves.  The stretch factor, $s$, is
then used to estimate the absolute magnitude of the SNe Ia via a linear
relationship that is determined jointly with the cosmological
parameters.  The quoted $\mu$ values include a correction for
extinction in the Galaxy based on the detailed model of Burstein \&
Heiles (1982).

R98 have published results based on 50 SNe Ia. Of these, 37, including
27 at low redshift ($z<0.15$) and ten at high redshift ($z>0.15$) have
well-sampled light curves in addition to spectroscopic information; the
quoted ``$1\sigma$'' uncertainties for $\mu$ for these SNe Ia are
typically smaller than $\pm 0.2$ mag at high $z$ for determinations by
either MLCS or TF light curve fitting method.  The data for 17 of the
SNe Ia at low redshift come from the Cal\'an-Tololo survey (Hamuy et
al. 1996c).  We focus our attention on these 37 best-observed SNe Ia,
which dominate the analysis in R98.  These authors extensively
tabulate their reduced data and provide detailed information
about their fitting techniques, thus facilitating independent analysis
of their conclusions.

R98 employ two different methods to estimate the distance modulus based
on information from the light curves.  The TF method~(Hamuy et al.
1996a) fits a set of light curve templates with different values of
$\Delta m_{15}$, the total decline in brightness from peak to 15 days
afterward, to observations of a particular SN Ia. By interpolating
between the values of $\chi^2$ for the fits to the various templates, a
minimum $\chi^2$ value of $\Delta m_{15}$ for the SN Ia is estimated.
The peak absolute magnitude is deduced from the independently
calibrated linear relationship between $M$ and $\Delta m_{15}$. The
MLCS method consists of fitting an observed light curve to a
superposition of a standard light curve and weighted additional
templates that parametrize the differences among SNe Ia (e.g., Riess,
Press \& Kirshner 1996); the outcome of the fits for a particular SN Ia
consists of the weights associated with the deviations from the
standard, which in turn determine the difference between its peak
absolute magnitude and the standard's. The fits are done for more than
one color, and reddening and extinction are inferred from color
dependences (Riess, Press \& Kirshner 1996, R98).  Originally the MLCS
method used a rather small training set to determine the requisite
templates (Riess, Press \& Kirshner 1996), but R98 now train on a
considerably larger set of nearby SNe Ia to find them.  Both the MLCS
and TF methods are calibrated on nearby SNe Ia in the Hubble flow
$(z < 0.15)$ and then applied to the SNe Ia discovered at high redshift.
The quoted $\mu$ values include a correction for local extinction
derived from the Burstein and Heiles model, and in addition the MLCS
method uses color dependence to estimate corrections for the extinction
and reddening due to absorbing material in the host galaxy.

Schematically, we can consider a lightcurve fitting method to
estimate the distance modulus for supernova number $i$ according to
the following model:
\begin{equation}
\mu_i = m_i - (M_0 + \Delta_i).
\eqn(mu-model)
\end{equation}
Here $m_i$ is the peak apparent magnitude for the SN, $M_0$ is
a fiducial absolute magnitude (a single constant for a particular
method), and $\Delta_i$ is a shift so that the peak absolute
magnitude for the SN is given by $(M_0+\Delta_i)$.  For the
purposes of this paper, we have ignored $K$-corrections and
extinction in \ceqn(mu-model)\ (one can consider them to have
been accounted for in the $m_i$ estimates and their uncertainties).
These corrections could potentially be important sources of
systematic error; the observing teams have gone to some
lengths to constrain the sizes of such errors.  Here we concentrate
on the possibility that systematic errors are introduced by the
lightcurve fitting algorithms entering the analyses via
the shifts $\Delta_i$.  We will seek information about such errors by
comparing the shifts {\em across} methods, rather than through
analysis of the internal consistency of a particular method.

The various fitting methods differ in how $m_i$ is interpolated from
the observed (incompletely sampled) lightcurve, in the choice for
$M_0$, and in how the shifts $\Delta_i$ are determined from the
(multicolor) lightcurve shapes.  The MLCS method provides $\Delta_i$
directly from fitting to a family of templates parameterized by
$\Delta_i$.  For the TF method, $\Delta_i = \beta_{\rm TF}(\Delta
m_{15} - 1.1)$, with $\beta_{\rm TF}$ a constant determined by fits.
For both the MLCS and TF methods, $M_0$ is inferred through the use of
SNe Ia that have Cepheid distances, and the various parameters
specifying the shift as a function of the lightcurve shape are set by
analyses of low redshift SNe.  For the SF method, $\Delta_i =
\alpha(s-1)$, where $s$ is the above-mentioned stretch factor and
$\alpha$ is a constant estimated jointly with cosmological parameters
in fits to the entire survey.  $M_0$ is simply set at an arbitrary
value; accordingly, no attempt is made to infer $H_0$ using SNe
analyzed with the SF method. In principle, each of the quantities on
the right hand side of \ceqn(mu-model)\ has uncertainty associated with
it, and the resulting errors in the estimates for these quantities are
correlated.  But only the combination given by \ceqn(mu-model)\ appears
in cosmological fits, so the lightcurve fitting results can be
summarized by the best-fit absolute magnitude $\hat\mu_i$ and the total
$\mu_i$ uncertainty $\sigma_i$ for each SN.  These quantities, and the
shifts $\Delta_i$, are the focus of our study.

Figure 2 shows histograms of the shifts deduced from the MLCS (R98),
TF (R98) and SF (P98) methods for the observed SNe Ia.  Since the choice
of $M_0$ can vary from method to method, we do not expect the histograms
to be aligned.  However, differences in histogram shape would indicate
that the various methods are correcting SNe in different and possibly
inconsistent ways.  While the three methods claim to reduce the
dispersion in the magnitude-redshift relationship at low $z$, it is
clear from the figure that they produce rather different distributions
of shifts.  Although the SF method has been applied to a different set of
SNe Ia than the other methods, this alone cannot explain the obvious
differences between the shapes of the histograms (we note that 14 SNe
are common to all three methods).  Most striking is that
the distribution is extremely narrow for the SF method, indicating that,
by this measure, the P98 SNe Ia sample consists almost entirely of
standard candles, or that for this sample of SNe Ia, the adopted
brightness-decline rate relation is not valid. This suggests to us that
these methods may be sensitive to different aspects of the SN Ia
phenomenon.  A consequence of this is that if the properties of SNe Ia
change with redshift, the relationships between the $\mu$ estimates
produced by the three methods could be $z$-dependent.  A search
for such a dependence could thus provide information about redshift
dependence of SNe Ia properties. In the following section we use
exploratory methods to search for evidence of this and other kinds of
dependences.

\section{Sleuthing}
\label{sec:fit}

Our approach in this section is driven by our belief that it is not
sufficient to settle for the consistency of the final cosmological
inferences of the MLCS, TF and SF analyses.  We should expect
consistency between them (statistically) on a supernova-by-supernova
basis where such a comparison is possible.

Since R98 use two different methods to compute distance moduli for
their sample of 37 SNe Ia, we can compare the results and search for
systematic differences between them.  Both the TF and MLCS techniques
are calibrated using the same set of nearby SNe Ia in the Hubble
flow and there is only one set of observational data for each SN Ia;
consequently, the uncertainties for the two methods are highly
correlated.  Another comparison set is the group of 14 supernovae from
the Cal\'an Tololo survey that are included in both R98 and P98.  Since
all fitting methods make use of the same published light curves for
this sample of 14 events, the inferred quantities for this sample will
also be highly correlated.

\subsection{Pointwise Consistency}

In Figure 3, we compare the distance moduli measured with the different
techniques on common samples of SNe Ia.  In Figure 3a, we show
$\mu_{\rm MLCS}$ vs $\mu_{\rm TF}$ for the 10 high redshift SNe Ia analyzed in
R98, and in Figure 3b, we show $\mu_{\rm MLCS}$ vs $\mu_{\rm SF}$ for the 14
common Cal\'an-Tololo SNe Ia analyzed by both SF and MLCS methods.  The
error bars for $\mu$ are derived from the uncertainties in the
individual distance moduli for each supernova (R98,P98), except that we
have removed the contribution associated with the intrinsic
dispersion of the SNe Ia sample, estimated to be $\sigma_{\rm int} = 0.10$
at low $z$, $\sigma_{\rm int} = 0.15$ at high $z$,
\footnote{The error due
to intrinsic dispersion in the SNe Ia sample is estimated to be
somewhat larger in P98; $\sigma_{int} = 0.17$.  For the purposes of
comparison we remove the smaller estimated value of the correlated
error.}
and we have removed the contribution associated with the
peculiar velocity of the SNe Ia ($\sigma_v = 300$ in P98, $\sigma_v =
200$ in R98).  Both the errors in the distance modulus from intrinsic
dispersion in the sample and from the peculiar velocity of the
SNe are completely correlated among the different methods.  We
have not removed the correlations due to $K$-corrections, photometry
and extinction (e.g., Schmidt et al.\ 1998), because there is
insufficient published information for us to do so properly;
consequently, we have overestimated the uncorrelated portion of the
distance modulus error somewhat.

{}From Figure 3 it is clear that the estimates for the distance moduli
from the different methods are strongly correlated, as they should be.
However, there is more dispersion  in these plots than we would expect
based on the quoted errors.  A fit of a straight line of slope 1 gives
a $\chi^2/\nu$ (with $\nu$ the number of degrees of freedom)
of 22.8/9 for Figure 3a and 21.2/13 for Figure 3b
indicating that there are errors associated with the analysis methods
that have not been accounted for.

We can pursue this type of comparison further with the R98 data where
all of the SNe Ia have been fully analyzed with two independent
methods.  In Figure 4 we compare the MLCS and TF estimates of various
quantities that are used in inferring the distance moduli of the SNe Ia
events.  For the 37 SNe 1a analyzed in R98, Figure 4$a$ shows the host
galaxy extinction, $A$, 4$b$ shows the correction to the absolute
magnitude, $\Delta$, and 4$c$ illustrates the peak apparent magnitude,
$m$, calculated with the MLCS and TF analysis methods.  (The individual
errors for the extinction and $\Delta$ are not published but can be
crudely estimated to be of order 0.1 magnitudes.)  Again, there is more
dispersion evident in these plots than might be expected from the
quoted or estimated errors except for the correlation plot of
$m_{\rm MLCS}$ versus $m_{\rm TF}$.  The peak apparent magnitudes inferred via
the two methods, which are the quantities most directly related to the
raw data,  are in excellent agreement.

\subsection{Redshift and Luminosity Dependence}

In Figure 5, we plot the difference $\Delta\mu \equiv \mu_{\rm MLCS} - \mu_{\rm TF}$
between the distance moduli determined
from MLCS and TF respectively, as a function of $z$.
The error bars for $\Delta\mu$ are derived from
the uncertainties in the individual distance moduli except that,
as described above, we have removed the contribution associated with
the intrinsic dispersion of the SNe Ia sample.
Formally, we use
$\sigma^2_{\Delta\mu}=\sigma^2_{\rm MLCS}+\sigma^2_{\rm TF}-2\sigma_{\rm corr}^2$
to calculate the error bars shown in Figure 5.
Although the data are somewhat scattered at both high and low $z$,
Figure 5 shows that the MLCS and TF methods agree rather well at
low $z$, apart from significant dispersion ($\sigma\sim 0.2$
mag) but there are hints of disagreement at large $z$, where
the dispersion, at least, appears larger, and the mean may also be shifted.

While it is possible that the appearance of Figure 5 at large $z$
merely reflects small number statistics, Figure 6 suggests that
the incompatibility between TF and MLCS could be systematic. In
Figure 6, we plot $\Delta\mu$ versus $M_{B}^{\rm AV}$, an estimated
absolute magnitude, defined by

\begin{equation}
M_B^{\rm AV} = (M_B^{\rm MLCS} + M_B^{\rm TF})/2 \label{eq:M}
\end{equation}
where
\begin{eqnarray}
M_B^{\rm MLCS} & = & m_B^{\rm MLCS} - \mu_{\rm MLCS} - A_B^{\rm MLCS} \nonumber \\
M_B^{\rm TF} & = & m_B^{\rm TF} - \mu_{\rm TF} - A_B^{\rm TF}
\end{eqnarray}
and $A_B^{\rm MLCS(TF)}$ is an estimate of the  extinction due to the host
galaxy in
the MLCS (TF) correction scheme.  For $z>0.15$, R98
provided all of the information necessary to calculate
$M_B^{\rm MLCS}$, $M_B^{\rm TF}$ and hence $M_B^{\rm AV}$, but at low $z$,
$M_B$ was only given for a subset of the SNe Ia in Hamuy et al.\ (1996a).~\footnote{The
zero point reference for $M_B$ may be somewhat different for the
high redshift and low redshift data which may account for the
fact that in Figure 6, the low redshift supernovae seem to be, on average,
less luminous by about 0.5 magnitudes.  Our conclusions are robust
against a shift in the zero point of the magnitude scale for the low
redshift supernovae.}

According to Figure 6, the difference between $\mu_{\rm MLCS}$ and
$\mu_{\rm TF}$ appears to be correlated with the estimated intrinsic
brightness, $M_B^{\rm AV}$, at high $z$, but not at low $z$. (Recall that
the error bars on $\Delta\mu$ are overestimates, as explained above.)
A similar correlation is evident if $\Delta\mu$ is plotted against
$\Delta_{\rm MLCS}(\Delta_{\rm TF})$, the difference in maximum absolute
magnitudes for the observed SNe Ia and the fiducial SNe Ia according to
the MLCS(TF) method. (R98 tabulates $\Delta_{\rm MLCS}$ and $\Delta_{\rm TF}$
for all SNe Ia in their sample.) Figure~6 suggests that, at high
$z$, one of the
analysis schemes, MLCS or TF, either under-corrects or over-corrects
for the luminosity variations in the SNe Ia sample. Since no such
systematic trend is evident at low $z$, we cannot know which method, if
either, yields the more accurate value of distance modulus.
It is relatively uncontroversial to say that the two methods are not
identical, either at low or high $z$, and hence must probe SNe Ia
physics in slightly different, and as yet ill-understood ways
(H\"oflich and Kholkov 1996, H\"oflich, Wheeler and Thielemann 1998).
Thus, the indications of $z$-dependence implied by Figures 5 and 6, while
still based on relatively few events, are not especially surprising.

A worrisome feature of Figures 5 and 6 is that imperfect corrections
for luminosity variations can alter the conclusions about $\Omega_M$ and
$\Omega_\Lambda$ that we draw from these data. To illustrate this point, we
computed $1\sigma$ confidence contours in ($\Omega_\Lambda,\Omega_M$)
space with separate fits to intrinsically bright and intrinsically dim
SNe Ia; the results are shown in Figure 7.~\footnote{In preparing
Figure 7, we have included a contribution to the uncertainty arising
from dispersion in galaxy redshift using the technique described in
R98.} The separation into `bright' and `dim' was somewhat arbitrary,
and we have verified that making different choices does not affect the
overall conclusion.~\footnote{ For the plots shown, we have chosen
$M_B^{AV}< -19.45$ as intrinsically bright and  $M_B^{AV}> -19.45$ as
intrinsically dim for MLCS and TF data.  For the SF data, we separated
intrinsically bright from intrinsically dim using $\alpha(s-1) >0 $ and
$\alpha(s-1) <0 $, respectively. Note that $\alpha(s-1)$ can be
calculated from the information in Tables 1 and 2 of P98.}
The $1\sigma$ confidence level contours for the combined data (all
$M_B^{AV}$) are also shown as dashed curves.  Figure 7 indicates a
systematic difference between the cosmology favored by intrinsically
bright versus intrinsically dim SNe Ia when the MLCS method is used;
the effect is much less pronounced for TF and seems to be of the
opposite sign for the SF method. The trend may be understood if the
MLCS (SF) method tends to overestimate (underestimate) the luminosities
of intrinsically bright SNe Ia at high redshift. Such a trend for the
MLCS data  is also consistent with Figure 6.

The set of plots in Figures 2 through 7 indicate that the analysis
methods disagree in their inferences of $\mu, A$, and $\Delta$ at a
level that is not covered by the quoted errors.  We can only
speculate on the sources of the discrepancies.  However until these
methods are understood more systematically, it will be difficult to
avoid assigning additional systematic errors to the measured distance
moduli with sizes that reflect the systematic differences between the
methods, and this will weaken the statistical
significance of the results substantially.

\subsection{Validity of Phillips Relations}

So far we have been investigating the light curve fitting methods as
possible sources of systematic error.  Potentially, there are other
effects that can mimic cosmology that are extremely difficult to
constrain with the present data.  The most pernicious, discussed at
some length by the observers themselves, is evolution of the SN Ia
population.  It is extremely difficult to put reliable quantitative
limits on evolution, and it cannot be excluded conclusively using the
currently available spectral and color information.  Furthermore, there
is already some evidence  in the current data that the high redshift
sample does not have the same properties as the low redshift sample.

The strongest evidence that the lightcurve corrections improve our
knowledge of the SN absolute magnitude would be a demonstration that
they reduce the dispersion
of the SN distance moduli about the best-fit cosmology.
(For example, Riess, Press \& Kirshner 1996 showed that MLCS reduces
the dispersion about Hubble's Law for low $z$ SNe Ia.) To test this,
we have
compared the dispersion between the data and the predictions of the
best-fit cosmology with and without the corrections, $\Delta_i$, inferred
from the light curve fitting.  \footnote{We redetermine the best fit
cosmology when we remove the corrections.}
We adopt the quantity
\begin{equation}
D^2 = {1\over N} \sum_i [\hat\mu_i - f(z_i;H_0,\Omega_m,\Omega_\lambda)]^2
\end{equation}
as a measure of the dispersion,
where $\hat\mu_i$ is the estimated distance modulus for SN $i$, $z_i$
is its redshift, the function $f(z)$ is defined by
\ceqn(eq:mu), and $N$ is the number of SNe Ia in the sample.  We
compute $D$ separately for high and low $z$; the results are
given in Table~1.

For both MLCS and TF, which are calibrated on low $z$ SNe Ia,
we see that the dispersion of the low redshift data is reduced
substantially by incorporating the corrections derived from the relation
between light curve shape and luminosity at maximum brightness.  At
high redshift, no such improvement is seen. The dispersion of the
high z data about the best fit cosmology is virtually unchanged by the
incorporation of either the MLCS or TF corrections.

For SF, there is little evidence from Table I that the corrections
reduce the dispersion in the data at all.  Recall that in the SF
parameterization, the relation between light curve width and luminosity
corrections is parameterized by $\Delta_{SF}= \alpha(s-1)$ where
$\alpha$ is inferred from a global fit to the data at all redshift.  As
was shown in Figure 2, the corrections $\Delta_{SF}$ are quite small so
it is not surprising that they do little to reduce the
dispersion in the data.  As stated before, the SF method finds little,
if any, correlation between light curve width and absolute luminosity
when averaged over all redshift.  What is startling is that the low
redshift sample used in P98 is almost identical to the ``peak
subsample'' of Hamuy et al. (Hamuy et al.\ 1996a).  As detailed in that
reference, that low redshift sample does show a significant correlation
between light curve width and peak luminosity.  If a strong correlation
is present in the low redshift sample and only a very weak correlation
is evident in the full sample, one is led to suspect that the correlation
is not present in the high
redshift sample; the large number of
high-redshift SNe leverage the joint fit.

\section{Accounting For Possible Evolution}
\label{sec:stat}

Both R98 and P98 assume implicitly that the same light curve fitting methods
may be applied at all redshifts sampled.  This assumption is
only valid if the light curve shape is correlated with peak luminosity
in the same way at both high and low redshift.  Given the evidence we
have presented that this may not be true, which indicates, at least
circumstantially, that the SNe Ia population evolves, we feel
it is necessary to explore whether the data published so far actually are able
to actually distinguish the effects of evolution from those of
cosmology.

Such effects fall under the rubric of ``systematic errors''---because
they are not ``random,'' their effects on one's final inferences are
difficult to account for in the conventional frequentist approach to
statistical inference.  However, both teams have adopted the Bayesian
approach for their final analyses (though not for all intermediate
stages of their analyses).  As noted by Jeffreys (1961), the Bayesian
approach is particularly apt for studying the effects of systematic
error because of its broader notion of uncertainty.  A Bayesian
probability density describes how probability is distributed among
the possible values of a parameter, rather than how values of the
parameter are distributed among some hypothetical population.  This
permits statistical calculations with quantities that are not ``random''
in the frequentist sense.  In particular, as Jeffreys noted,
systematic errors are treatable simply by introducing parameterized
models for the errors and marginalizing (integrating over) the
extra parameters to obtain one's final inferences.

This procedure, when followed blindly, has the potential to weaken
one's conclusions unjustifiably.  For example, one could simply
introduce a systematic dependence that is identical to the physical
dependence one is studying, but with a duplicated set of parameters.
This duplication would prevent useful constraints from being placed on
the parameters, since any measured effect could be ``blamed'' on the
duplicated systematic dependence.  Thus Jeffreys emphasized the need to
compare models with and without systematic error terms using the ratio
of the model probabilities, the {\em odds} favoring one model over
another.  The odds can be written as the product of the prior odds
(expressing information from other data, or possibly a subjective
comparison of the models) and a {\em Bayes factor} determined entirely
by the data, the models, and the sizes of the model parameter spaces.
If we know or strongly believe a systematic effect to be present
without consideration of the new data before us, then obviously the
systematic error model should be used; the prior odds would lead us to
this conclusion even if the Bayes factor is indecisive.  If we have no
strong prior evidence for a systematic error, one takes the prior odds
to be unity and relies on the data alone for determining if the effect
is present, taking the Bayes factor to be the odds.  An appealing
aspect of Bayesian model comparison is that the Bayes factor implements
an automatic ``Ockham's razor'' that penalizes models for the sizes of
their parameter spaces.  Thus model complexity is accounted for by the
Bayes factor.  Except in unusual cases, needlessly increasing a model's
complexity by simply duplicating terms prevents the Bayes factor
from favoring the more complicated model.  We
provide a brief review of Bayes factors in Appendix~A; standard
references reviewing their use are Kass and Raftery (1995) and
Wasserman (1997).

\subsection{Systematic Error in $H_0$}

To illustrate this approach, we show how it can be used to
quantitatively account for systematic error introduced by the uncertain
Cepheid distances used to infer $M_0$ in the MLCS and TF methods.  (The
SF analysis used a ``Hubble-constant-free'' parameterization and thus
could avoid explicit treatment of $M_0$ and the Hubble constant.)  We
write the true value of $M_0$ as $(\hat M_0+\delta)$, where $\hat M_0$
is the estimate used for calculating $\hat\mu_i$, and the new term,
$\delta$, represents the constant (but unknown) error introduced by
using Cepheid data to calculate $\hat M_0$.  We describe the likelihood
function for analyzing the SNe Ia data in some detail in Appendix B.
The final (approximate) likelihood is equivalent to what one would
find from modelling the tabulated $\hat\mu_i$ estimates according to,
\begin{eqnarray}
\hat\mu_i &=& f(z_i) + \delta + n_i\\
  &=& g(z_i) - \eta + \delta + n_i
\eqn(mu-delta)
\end{eqnarray}
where $f(z_i)$ is the cosmological distance modulus relation
defined in \ceqn(eq:mu), and $n_i$ is a random error term
whose probability distribution is a Gaussian with zero mean
and standard deviation $\sigma_i$.  In the second line, we
have separated out the $H_0$ dependence of $f(z_i)$ into
\begin{equation}
\eta\equiv 5\log\left(h \over c_2 \right) - 25,
\eqn(eta-def)
\end{equation}
where $H_0 = h\times 100\hbox{km}\,\hbox{s}^{-1}\,\hbox{Mpc}^{-1}$,
and $c_2$ is the speed of light in units of $10^2\hbox{km}\,\hbox{s}^{-1}$;
$g(z_i)$ contains the remaining $\Omega_M$- and $\Omega_\Lambda$-dependent part of $f(z_i)$.

It is clear from \ceqn(mu-delta)\ that $\eta$ (and thus $H_0$)
is degenerate with $\delta$; we cannot hope to learn about one
without independent knowledge of the other.  But $\delta$ is
constrained by our knowledge of the uncertainty of the Cepheid
distance scale.  In particular, R98 summarize the uncertainties
as introducing an error with a standard deviation of $d=0.21$ magnitudes
(corresponding to $\approx 10$\% uncertainty in $H_0$).
We account for this by introducing a prior distribution for $\delta$
that is a zero-mean Gaussian with standard deviation $d$.

Our model now has four parameters, $\delta$, $h$, $\Omega_M$, and $\Omega_\Lambda$.
The likelihood function for the data is the product of $N$ Gaussian
distributions specified by \ceqn(mu-delta)\ and is proportional to the
exponential of a familiar $\chi^2$ statistic.  The full joint posterior
distribution is the product of this and priors for the parameters,
including the informative prior for $\delta$.\footnote{The priors for
$\Omega_M$, and $\Omega_\Lambda$ we take to be flat over the region shown in our
plots, excluding the ``No Big Bang'' region; the prior for $h$
we take to be flat in the logarithm.}  We can summarize our
inferences for the cosmological parameters by integrating over
$\delta$; this can be done analytically and is described in
Appendix~B.  If we want to focus on the conclusions for $h$, we
numerically integrate over $\Omega_M$ and $\Omega_\Lambda$.  The result, for the MLCS
data, is the marginal distribution for $h$ shown as the rightmost solid
curve in Figure~8.  The best-fit value is $h=0.645$, and a 68.3\%
credible region has a half-width $\sigma_h = 0.063$.  This is
approximately equal to the ``total uncertainty'' on $H_0$ estimated by
R98 using standard ``rules of thumb'' for accounting for systematic
error; we have shown how this estimate could be justified by a formal
calculation.  For the TF data, the marginal posterior is plotted as the
leftmost solid curve in Figure~8, and $h= 0.627\pm 0.062$.

Of greater current interest are the implications for the density
parameters.  The marginal distribution for $\Omega_M$ and $\Omega_\Lambda$ is
found by integrating out $\delta$ and $h$.  This can be done
analytically (see Appendix~B).  Contours of the resulting
distributions, found using both the MLCS and TF data,
appear in Figure~9.  They are identical to
contours found using a model without $\delta$, and essentially
reproduce the results reported in R98 (minor differences result
from our omission of the ``snapshot'' SNe).

In this case, we know that $M_0$ has been estimated using the Cepheid
data, and that this estimate has systematic error.  Formally, the prior
odds favoring the model with $\delta$ over one with $\delta=0$ is thus
infinite.  The Bayes factor comparing these models is exactly equal to
one (this is because the SNe Ia data can tell us nothing about $\delta$; see
Appendix~A for discussion of this property of Bayes factors), so the
posterior odds is equal to the prior odds.  Since we know these errors
to be present, we take this $\delta$ model to be our ``default'' model
when calculating subsequent Bayes factors in this section.

We conclude our discussion of this model by summarizing the evidence in
the data for a nonzero cosmological constant, presuming the $\delta$
model to be true.  In R98 and P98, the marginal posterior probability
that $\Omega_\Lambda>0$ was presented as such a summary; this probability was
found to equal 99.6\% (2.9$\sigma$), 99.99\% (3.9$\sigma$), and 99.8\%
(3.1$\sigma$) in the MLCS, TF, and SF analyses, respectively,
apparently indicating strong evidence that $\Omega_\Lambda$ is nonzero.  But this
quantity is not a correct measure of the strength of the evidence that
$\Omega_\Lambda \ne 0$.  This probability would equal unity if negative values of
$\Omega_\Lambda$ were considered unreasonable a priori,
yet presumably even in this case one
would not consider the data to demand a nonzero cosmological constant
with absolute certainty.  The correct quantity to calculate is the odds
in favor of a model with $\Omega_\Lambda \ne 0$ versus a model with $\Omega_\Lambda=0$.
Considering such models to be equally probable a priori, this is given
by the Bayes factor comparing these models.
\footnote{These Bayes factor calculations can also be viewed as providing the
posterior probability that $\Omega_\Lambda=0$ by putting a prior probability of
0.5 on the $\Omega_\Lambda=0$ line; in the calculations reported in R98 and P98,
this line has zero prior probability (only finite intervals in
$(\Omega_M,\Omega_\Lambda)$ have nonzero prior probability in their analyses).}
We find $B=5.4$ using the MLCS data and $B=6.8$ using the SF data, each
indicating positive but not strong evidence for a nonzero cosmological
constant (presuming there is no evolution).  The TF data give $B=86$,
indicating strong evidence for a nonzero $\Omega_\Lambda$ (again, presuming no
evolution).  Without clear criteria identifying one method as superior
to the others, the data are equivocal about a nonzero cosmological
constant, even without accounting for the effects of possible
evolution.

Similarly, R98 report the number of standard deviations that the
$\Omega_M=1$, $\Omega_\Lambda=0$ point is away from the best-fit $(\Omega_M,\Omega_\Lambda)$ as a
measure of the evidence against the hypothesis that matter provides the
closure density; they state this hypothesis is ruled out at the
$7\sigma$ and $9\sigma$ levels using the MLCS and TF methods,
respectively.  Again, a proper assessment of the hypothesis that
$\Omega_M=1$ and $\Omega_\Lambda=0$ requires that one give this hypothesis a finite,
nonzero prior probability.  For the MLCS method, the Bayes factor
favoring a model with any $(\Omega_M,\Omega_\Lambda)$ over one with $\Omega_M=1$ and $\Omega_\Lambda=0$
is $B=2.3\times10^4$, so that with equal prior odds the probability for
the latter model is $p=1/(1+B)\approx 5\times10^{-5}$ ($\approx
4.5\sigma$).  For the TF method, we find $B=2.1\times10^7$, so that
$p\approx 5\times10^{-8}$ ($\approx 5.8\sigma$).  These are small
probabilities and indicate very strong evidence against the simpler
model, but they are much larger than the probabilities associated with
$7\sigma$ and $9\sigma$ significances ($\approx 2\times10^{-11}$ and
$3\times10^{-18}$, respectively, for two degrees of freedom).  The
incorrect summary statistics used in the previous analyses have
exaggerated the evidence for a nonzero cosmological constant,
irrespective of whether or not one considers the effects of evolution.

\subsection{Models With Evolution}

Without a detailed physical idea of the cause of evolution, we cannot
explore truly realistic models.  Instead we consider two illustrative
examples.  We first consider a model (Model I) that generalizes the
$\delta$ model just described by adding an additional offset,
$\epsilon$, for the high redshift SNe; we
apply this model only to data from R98.  As a model of physical
evolution, this is certainly too simple, but it is illustrative since
for the R98 sample, the observed high redshift SNe Ia are all in a
fairly narrow band in redshift near $z\sim 0.5$.  Essentially this
model merely permits differences in luminosities between $z
\lesssim 0.1$ and $z \sim 0.5$ as a consequence of evolution.  On a purely
phenomenological level, the model might be considered more realistic
because the low and high redshift SNe are not treated equally in the
R98 analysis: the MLCS and TF relations are calibrated using only low
redshift SNe.  Thus this model can be understood as allowing for a
systematic offset when extrapolating the methods beyond the training
set.  For this model, we use the same prior for $\delta$ as in our
default model (zero-mean Gaussian with standard deviation $d=0.21$ mag).
The prior for $\epsilon$ we also take to be a zero-mean
Gaussian but with a different width $e$.
The prior width, $e$, can be viewed as
a description of the scale of errors we
might expect from extrapolating low redshift properties to high
redshift.

Physically, we might expect evolution to lead to continuous variation
of SN Ia properties with redshift.  Also, the P98 analysis uses low and
high redshift SNe Ia together in calibrating the luminosity/decline
rate relation, so there is no clear separation of their data into low
and high $z$ subsamples.  Thus, Model~I is inappropriate for
phenomenological modeling of systematic effects from lightcurve fitting
of P98 data.  Consequently, we consider a second model (Model~II) which
assumes that the intrinsic luminosities of SNe Ia scale like a power of
$1 + z$ as a result of evolution.  This second model corresponds to
replacing \ceqn(mu-delta)\ with
\begin{equation}
\hat\mu_i
  = g(z_i) - \eta + \delta + \beta \ln (1+z) + n_i,
\eqn(mu-db)
\end{equation}
where $\delta$ again represents Cepheid uncertainty in $M_0$ (relevant
only when we apply this model to MLCS or TF data), and $\beta$
parameterizes the evolution.  We use a Gaussian prior for $\beta$ with
zero mean and standard deviation $b$.

For both models, we explore the dependence of the results on the prior
width ($e$ and $b$) to see how external constraints on evolution
(presently unknown) could affect the analysis.  We examine values that
allow evolutionary changes of up to a few tenths of a magnitude for
sources with $z\sim 1$.  These characteristic magnitude shifts are
comparable to the intrinsic dispersion seen in low redshift SNe Ia
(Schmidt et al.\ 1998), which may be taken as a rough indication of the
range of variation of peak magnitude with physical conditions in the
explosions.  Some current theoretical studies of possible sources of
$z$-dependent variations in SNe Ia luminosities also find magnitude
changes of this size to be reasonable (see, e.g., H\"oflich, Wheeler,
and Thielemann 1998; Dom\'inguez et al.\ 1999).

The new parameters in both models appear linearly in the model
equations and can thus be marginalized analytically.  Appendix~B
describes the calculations.  Reality could be and probably is far more
complicated than either model, but the sparsity of the present data do
not justify consideration of more sophisticated models.

\subsubsection{Model I}

Figure~10$a$ shows contours of the marginal density for $\Omega_M$ and $\Omega_\Lambda$
using Model~I to analyze the MLCS data and taking the prior width
for $\epsilon$ to be $e=0.1$ mag. Figure~10$b$ shows similar
results using the TF data.  It is clear from these figures that the
presence of a redshift-dependent shift of order $e$ greatly weakens our
ability to constrain $\Omega_M$ and $\Omega_\Lambda$ from the SNe Ia data.  The Bayes
factor for this model over the default $\delta$ model is $1.1$ for
both MLCS and TF; the data alone are indecisive about whether such a
redshift-dependent error is present.
\footnote{The $\chi^2$ values for the default fits are already
acceptable, so one might worry that the more complicated models are
``overfitting.'' But the maximum likelihoods for models I and II are
only slightly greater than those found with the default model.  Bayes
factors account for overfitting and it is not playing a role here.  The
operation of Bayes factors is discussed further in Appendix A.}
Presuming it is present, the Bayes factor favoring nonzero $\Omega_\Lambda$ over
$\Omega_\Lambda=0$ is reduced significantly from what is found using
the default model; it is $1.1$ for MLCS and $3.8$ for TF.

These results are sensitive to our knowledge of the evolution.
Figures~10$c$ and 10$d$ show the MLCS and TF results again, but
this time with $e=0.2$ mag; the credible regions have grown
even larger in size.  Now the Bayes
factor for Model~I over the default $\delta$ model is $1.3$ for MLCS
and 1.2 for TF; the data remain indecisive about the presence of
an evolutionary offset.  Presuming it is present, the
Bayes factor favoring nonzero $\Omega_\Lambda$ over $\Omega_\Lambda=0$ is $0.7$ for MLCS
(i.e., slightly favoring $\Omega_\Lambda=0$) and $1.2$ for TF.

As one might expect from these results, the most likely value of $\epsilon$
tends to be positive, making the more distant sources dimmer than the
nearer ones due to evolution rather than cosmology.  For example, in
Figures~10$c$ and 10$d$, when $\Omega_M=1$ and $\Omega_\Lambda=0$ (a point within the
95.4\% credible regions), $\epsilon=0.31\pm0.06$ for MLCS and
$\epsilon=0.17\pm0.06$ for TF.

The constraints placed on $H_0$ by the SNe Ia data arise mostly from
the low redshift objects, so one would not expect allowance for
evolution to drastically affect the $H_0$ inferences.  The analysis
bears this out.  The dashed curves in Figure~8 show the marginal
distributions for $h$ based on the MLCS and TF data using Model~I with
$e=0.2$; they differ little from the distributions found using the
reference model with no evolution.  Similar results are found using
Model~II.

\subsubsection{Model II}

Figure~11 shows results from analysis of the SF data using Model~II.
Figure~11$a$ shows contours of the marginal density for $\Omega_M$ and $\Omega_\Lambda$
for the $\beta=0$ case as a reference; these contours essentially
duplicate the results of Fit C in P98.  Figure~11$b$ shows similar
contours, but allowing for a nonzero $\beta$; the prior standard
deviation for $\beta$ was $b=0.25$.  Figure~11$c$ repeats the
analysis with $b=0.5$.  Again we find that the possibility
of evolution significantly weakens the constraints on the density
parameters, but if the amount of evolution can be bounded, useful
limits might result.  The Bayes factor for evolution vs.\ no
evolution is $1.0$ for $b=0.25$ and $1.1$ for $b=0.5$, so the
data alone are indecisive about the presence or absence of this type of
evolution.  We find similar results when using this model to analyze
the MLCS and TF data.

Figure~12 shows how these findings depend on the prior uncertainty
for $\beta$.  The solid curve shows how the Bayes factor favoring
a nonzero $\Omega_\Lambda$ over $\Omega_\Lambda=0$ depends on $b$; only for $b\lesssim0.1$
does the Bayes factor remain near the value of 6.8 found assuming
there is no evolution.  The dashed curve shows the Bayes factor
for Model~II versus the default model with no evolution; for
no value of $b$ in the range of the plot can the data clearly
distinguish evolution from cosmology.  This emphasizes the need
to use information independent of the $\mu$-$z$ relation to
constrain evolution.

\subsubsection{Flat Cosmologies}

So far we have assessed the evidence for nonzero $\Omega_\Lambda$ by
comparing models with $\Omega_\Lambda=0$ to models with arbitrary
$\Omega_\Lambda$, as was done in the R98 and P98 analyses.  However,
many cosmologists would consider flat models, with
$\Omega_\Lambda = 1 - \Omega_M$, to be of special relevance (e.g., because
of inflationary arguments).  We have thus analyzed models
constrained in this way, using our default model and models
I and II.

Figure~13 shows the marginal posterior distributions for $\Omega_M$ (and,
equivalently, for $\Omega_\Lambda = 1 - \Omega_M$) presuming a flat cosmology.  The
three panels show analyses of the distance moduli reported using the
MLCS and TF data with Model~I (top and middle,
respectively), and using
the SF data with Model~II (bottom).  The solid curves
show results based on the default model; the short-dashed curves show
results with a small amount of evolution allowed ($e = 0.1$ or
$b=0.25$), and the long-dashed curves show results with a larger amount
of evolution allowed ($e = 0.2$ or $b=0.5$).  As in the previous cases,
the Bayes factors comparing models with evolution to the default model
are all nearly equal to one.  Also, as was found before, accounting for
the possibility of evolution significantly weakens the evidence for
nonzero $\Omega_\Lambda$.  However, if one restricts attention to flat models, the
evidence for nonzero $\Omega_\Lambda$ is stronger than it is if one allows nonflat
cosmologies.  For the default model presuming no evolution, the Bayes
factors favoring nonzero $\Omega_\Lambda$ over a flat model with $\Omega_M=1$ are
$2.1\times10^4$ (MLCS), $2.5\times10^6$ (TF) and $5.0\times 10^3$ (SF),
much larger values than were found in the comparison using nonflat $\Omega_\Lambda$
models discussed above.  But these values fall dramatically if one
allows for evolution.  For models with a small amount of evolution
allowed, they decrease to 20, 48, and 14, respectively, indicating
positive but not compelling evidence for nonzero $\Omega_\Lambda$.  For models
with a larger amount of evolution allowed, they decrease further to
2.4, 2.5, and 2.3, indicating no significant evidence for nonzero
$\Omega_\Lambda$.

\subsection{Simulations}

Figure~14 elucidates why introducing the possibility of evolution
weakens our ability to constrain $\Omega_M$ and $\Omega_\Lambda$ so greatly.  The thick
solid curve shows $g(z)$ for the best-fit cosmology from fits to the SF
data presuming no evolution ($\Omega_M=0.75$, $\Omega_\Lambda=1.34$).  The dotted curve
shows the same function for the flat $\Omega_M=1$ ($\Omega_\Lambda=0$) cosmology (which
lies within the 68.3\% credible region in Figure~11$c$), and the dashed
curve shows the evolutionary component $\beta\ln(1+z)$ for the best-fit
value of $\beta$ given this cosmology ($\beta = 0.83$).  The thin solid
curve shows the sum of the dotted and dashed curve.  Over the range of
redshift covered by the data ($z\lesssim 1$) and even beyond,
$(\Omega_M,\Omega_\Lambda,\beta) = (0.75, 1.34, 0)$ and $(\Omega_M,\Omega_\Lambda,\beta) = (1,0,0.83)$
are indistinguishable if one allows evolution of this type, unless one
can determine $\mu$ to significantly better than the $\sim20$\%
accuracy currently obtained at $z\lesssim 1$.  However, we note that
the models differ substantially at larger redshift (by about 0.8 mag),
which offers some hope of discerning evolution. We caution, though,
that the best fit values of $(\Omega_M,\Omega_\Lambda)$ are likely to be different for
data extending to $z\approx 2$, either with or without evolution, so
the comparison is not truly apt, and moreover $(\Omega_M,\Omega_\Lambda)=(0.75,1.34)$
might be considered implausible intrinsically by many cosmologists.  To
amplify this point, we also compare $(\Omega_M,\Omega_\Lambda,\beta)=(1,0, 0.83)$ with
another cosmology, $(\Omega_M,\Omega_\Lambda)=(0.3,0.7)$, within the 68.3\% credible
region of the no-evolution analysis.  As is shown by the bold,
long-dashed curve in Figure~14, $\mu$ accuracies better than 10\% out
to $z \approx 2$ would be needed to distinguish these cosmologies from one
another out.  We have systematically explored a wide
range of cosmologies and found similar results:  simple power law
evolution can make widely disparate cosmologies appear remarkably
similar.  Put another way, the differences between cosmologies with
various $\Omega_M$ and $\Omega_\Lambda$ are well mimicked by power-law evolution to
redshifts beyond those currently accessible in supernova surveys.  We
emphasize that we did not choose the form of the evolution to produce
this degeneracy; this is a standard phenomenological model for
evolution.  We have found similar behavior with another simple model
for evolution consisting of a power law in lookback time.

To assess how well evolution can mimic cosmology, we analyzed simulated
data consisting of $\mu$ values with added Gaussian noise at redshifts
that themselves had added Gaussian noise.  The simulations were
designed to roughly mimic possible future data like that reported in
P98 (this simplified the analysis since $H_0$ need not be accounted for
explicitly as it would have to be for data like those reported in R98).
The redshifts of the first 16 SNe in our simulated data sets were
chosen to be similar to those of the 16 low-redshift SNe in P98
($z\lesssim 0.1$); the redshifts for the remaining simulated data were
chosen randomly from a uniform distribution over some specified
interval.  We added redshift errors with a standard deviation of 0.002,
and $\mu$ errors with standard deviations equal to those reported by
P98 for the 16 low-$z$ SNe, and equal to 0.25 magnitudes for the
high-$z$ SNe (a typical value for $\mu$ values reported in P98).

Figure~15 shows typical results.  Here we simulated data from a flat,
$\Omega_M=1$ ($\Omega_\Lambda=0$) cosmology with evolution described by Model~II with
$\beta=0.5$.  Figure~15$a$ shows the results of an analysis assuming no
evolution, with 38 simulated SNe redshifts in the interval $[0.3,1]$
(54 total simulated points).  This corresponds to a sample size equal
to that used in the P98 analysis and extending over a similar range of
redshift.  The cross indicates the best-fit parameter values, the dot
indicates the true values, and the contours bound credible regions of
various sizes.  One would reject the true model as being improbable if
evolution is ignored.  Figure~15$b$ shows a similar plot, with the
number of high-$z$ SNe increased so the total sample size is now 200,
with the high-$z$ points now spread over $[0.3,1.5]$.  The contours
have shrunk considerably, converging around a point well away from the
truth.  In both figures, the best-fit point has an excellent
$\chi^2/\nu$ ($53.6/52$ for the small data set, $201/198$ for the large
one).  Evolution mimics cosmology so well that standard ``goodness of
fit'' reasoning can lead one to conclude, mistakenly, that pure
cosmology (with no evolution) is an adequate description of data of
this quality even when substantial evolution is present.

Figure~15$c$ shows the results
of an analysis of the larger data set using a model that
includes evolution; the marginal posterior (with $\beta$
marginalized) is shown.  The credible regions now contain
the true model, but they are large even for a data set four
times the size of the currently published surveys, and
extending to significantly higher redshift.  The Bayes factor
is of order unity, showing that data of this quality is
not capable of distinguishing between models with and
without evolution. This is further testimony to the approximate
degeneracy between cosmology and evolution, at least at
$z\lesssim 1.5$.

The extent to which evolution corrupts the results depends both
on the true cosmology and on the amount of evolution allowed.
Independent constraints
on the amount of evolution could thus play an important role
in allowing useful contraints to be placed on the cosmology.
They would enter the analysis via the prior for $\beta$.
Comparison of Figures~11$a$ through 11$c$ shows how constraints on
the amount of evolution can affect one's final inferences.

\section{Conclusions}
\label{sec:conc}

Systematic uncertainty may enter the analysis of any data set as
a result of real physical effects that are not accounted
for explicitly.
As an example,  the use of observations
of distant galaxies
to measure the cosmological deceleration parameter had to
confront the systematic
errors introduced by the
fact that not only are galaxies not standard candles, but
their luminosities also evolve with time (e.g., Tinsley 1968;
Weinberg 1972; Ostriker \& Tremaine 1975; Tinsley
1977; Sandage 1988; Yoshii \& Takahara 1988; Bruzual 1990; Peebles 1993).
A principal goal of this paper
has been to present a study of the systematic error due to evolution
in attempts to determine $\Omega_M$ and $\Omega_\Lambda$ from
observations of SNe Ia.

One focus of this paper has been to see if
there are indications that the SNe Ia population has evolved
from $z\sim 0.5-1$ to $z\ll 1$. We have presented two arguments
that this might be so. The first is that a comparison of
the peak luminosities estimated for individual SNe Ia by
two different methods, MLCS and TF, are not entirely
consistent with one another at high redshifts, $z\sim 0.5$.
We have asserted that
the two methods very likely sample slightly different
aspects of the SN Ia mechanism, and should not be expected
to agree completely. If evolution were entirely absent,
though, the differences between them should not depend on
redshift, contrary to the admittedly sketchy evidence of
the data. A second hint that SNe Ia evolve with redshift
is that while the three luminosity estimators, SF as well as
MLCS and TF, reduce the dispersion of distance moduli
about best fit models at low redshift, they do not at
high redshift.

These studies were intended to give us impetus to pursue
the more fundamental point of this paper, namely that
evolution {\it must} be considered possible, even if
there are no ``smoking guns'' that seem to require it.
Ideally, one should attempt to constrain the parameters of
an evolutionary model at the same time as determining
the parameters of the cosmological model. As we stated
at the outset, changes in peak SN Ia magnitude of order 0.1
magnitudes out to $z\sim 1$ would alter the ranges of
acceptable cosmological models substantially.
The dispersion of SNe Ia peak magnitudes at low $z$ is approximately
0.3-0.5 mag (Schmidt et al. 1998), which might indicate a plausible
range of variation for diverse physical conditions. Using theoretical
models, H\"oflich, Wheeler \& Thielemann (1998) argue that a similar
range of variation of peak luminosities could arise as a consequence
of changes in composition which might be due to evolution.

To get an idea of how allowing for the possibility of
evolution would affect one's
ability to constrain cosmological parameters, we considered
two different models. In one, we assumed that there is
a constant magnitude shift between low and high redshift.
We also considered a model in which the
peak magnitudes of SNe Ia evolve continuously, with
$\delta m(z)=\beta\ln(1+z)$.
In applying these models, prior
assumptions about the amplitude of possible magnitude
changes of SNe Ia between low and high redshifts are
needed to evalute the systematic error that might be
introduced.  At present, little is known, so our
calculations allow a range of possibilities.  To
do this, we assume Gaussian prior probability distributions for the
(unknown) parameters of the evolutionary models.  These priors
express a preference for no evolution, but have adjustable standard
deviations that encapsulate prior notions about how large possible evolutionary
effects might be. The results presented in P98 and R98 correspond to setting
these standard deviations to zero; i.e., no
evolution at all. We adopt a more conservative viewpoint, and present
results for different choices of the ranges of magnitude evolution
that are allowed {\it a priori}. Significantly, when we permit peak magnitude
changes out to $z\sim 1$
comparable to (and even somewhat smaller than) the range observed
for low redshift SNe Ia (Schmidt et al. 1998), the implied systematic
uncertainty in $\Omega_M$
and $\Omega_\Lambda$ becomes so large that the data cannot constrain
these cosmological parameters usefully.
However, our ability to determine $H_0$ is virtually unaffected by evolution.

In order to assess the extent to which the data favor
models allowing evolution over ones without evolution, we computed
Bayes factors. The Bayes factor between classes of models
with and without luminosity evolution are equivalent
to the odds ratio between them if there is no {\it
a priori} reason to prefer one over the other. In
all cases, we found that the Bayes factors are
of order unity, which means that the data themselves
do not favor either model. If we accounted for
a prior prejudice that evolution does occur, the odds
would disfavor models in which the SNe Ia population
has the same properties at all redshifts.

The two models we have considered illustrate how well evolution can
mimic cosmology. The less realistic model merely allowed a shift
in the magnitudes
of high $z$ SNe Ia relative to low $z$ SNe Ia by a fixed (but uncertain) amount,
$\delta m$. Since the SNe Ia in the R98 sample were predominantly at
$z\approx 0.3-0.5$, the cosmological magnitude shift (relative to
Hubble's law or any other fiducial cosmology) varies little over the
entire redshift range they span. Clearly, for the high $z$ SNe Ia in
this sample, one only knows that there is a total magnitude shift
between $z\lesssim 0.1$ and $z\sim 0.5$, not how much is due to
cosmology and how much to evolution.  The characteristic magnitude
shifts due to evolution needed for a cosmological model with
$(\Omega_M,\Omega_\Lambda)=(1,0)$ are $\approx 0.2-0.3$.  Ultimately, a model in
which there is simply a constant magnitude difference between low and
high $z$ SNe Ia should fail to model data spanning a lage range of
redshifts.

More daunting is the success of models allowing a continuous
magnitude shift, $\delta m(z)=\beta\ln(1+z)$. While it
is unsurprising that such models would be approximately degenerate
with cosmology at low $z$, where the combined magnitude
shift, relative to Hubble flow, is
$[1.086(1-\Omega_M/2+\Omega_\Lambda)+ \beta]z$
(e.g., Weinberg 1972),
it is remarkable that a continuous magnitude shift with
this simple form cannot be discerned out to at least
$z\approx 1$. Our simulations show that even if there is
truly no evolution, so that reality corresponds to certain
values of $\Omega_M$ and $\Omega_\Lambda$ with $\beta=0$,
models with $\beta\neq 0$ and $(\Omega_M^{\rm eff},
\Omega_\Lambda^{\rm eff})\neq(\Omega_M,\Omega_\Lambda)$ yield
apparent magnitudes that are indistinguishable from the
truth within differences in distance modulus
$\approx 0.1$ mag.
Differences between select cosmological models may be larger
at higher $z$, but often remain within $\sim 0.1$ mag out to
$z\approx 2$.

We also use simulations to explore the converse situation where we
neglect evolution in the analysis of samples of evolving SNe Ia.
As an example, we simulated a set of 200 SNe Ia distance moduli,
including 184 high-$z$ SNe with redshifts uniformly
distributed over $0.3\leq z\leq 1.5$, in a cosmological model with
$(\Omega_M,\Omega_\Lambda,\beta)=(1.0, 0.0, 0.5)$.  We then analyzed
the data with evolution neglected entirely.  The result was that given
enough SNe Ia, the analysis would pick out a small range of ``allowed''
values of $(\Omega_M,\Omega_\Lambda)$, but centered around incorrect
values. The true cosmology was well outside the $3\sigma$ credible
region for these simulations, yet the (incorrect) best-fit model would
be judged excellent by a standard $\chi^2$ goodness-of-fit test.

What is needed to separate evolution from cosmology is both
detections of greater numbers of SNe Ia at high redshift
with detailed measurements of light curves and spectra, and,
equally important, a
better physical understanding of the SN Ia process. In particular, one
would like to be able to link the Phillips relations,
lightcurve risetimes and spectra, uniquely, to
internal conditions in the explosions themselves, to be able to
understand how they might evolve with redshift (see, e.g.,
von Hipple, Bothun, \& Schommer 1997; H\"oflich,
Wheeler, and Thielemann 1998; Dom\'inguez et al.\ 1999).  This would
allow construction of realistic, not phenomenological, models for
evolution, and one might hope to be able to constrain the parameters of
these models along with cosmological ones. The analogue in galactic
astronomy is the use of population synthesis models to study the
cosmological evolution of the luminosity function, which {\it might}
permit, given enough data, simultaneous fits for cosmological
parameters (Yoshii \& Takahara 1988, Bruzual 1990). Such
detailed physical modelling might lead to a detailed, quantitative
connection between the peak luminosities of SNe Ia and their spectra,
which would allow additional information to be useful quantitatively in
fitting for $\Omega_M$ and $\Omega_\Lambda$. R98 and P98 have argued,
using the spectral data, that there is no compelling evidence {\it for}
evolution, but that does not translate into a convincing argument {\it
against} evolution unless the salient features of the source spectra
can be connected unambiguously to peak luminosity.
In fact, since this paper was submitted,
Riess, Filippenko, Li \& Schmidt (1999) have claimed that the
rise times of low and high redshift SNe Ia are different even
though earlier studies found no comparably strong evidence for spectral
evolution.

In the end, what all cosmologists want to know is the probability that
the cosmological constant is nonzero.  The Bayes factor provides
straightforward mathematical machinery for doing this calculation,
whether or not evolution is included in the analysis.  When
the possibility of evolution
is not included in the analysis, and no prior assumptions are made
about the spatial geometry of the Universe, the Bayes factor for
$\Omega_\Lambda\neq 0$ compared to $\Omega_\Lambda=0$ is $B=5.4$ using
the MLCS method, $B=6.8$ using the SF method, and $B=86$ using the TF
method, which if one is not prejudiced either way, only favors nonzero
$\Omega_\Lambda$ equivocally. (There may be reasons to be prejudiced
one way or the other; see for example Turner 1999 for a theoretical
cosmologist's point of view.) When the possibility of evolution is
accounted for in the analysis, the
values of the analogous Bayes factors depend on one's prior
assumptions, but rather conservatively $B\lesssim 1$.  Thus, if we do
not discriminate among open, closed and flat cosmological models, the
data alone do not choose between $\Omega_\Lambda\neq 0$ and $\Omega_\Lambda=0$ once the
possibility of evolution is taken into account. However, if the
Universe is presumed to be flat spatially, then the case for $\Omega_\Lambda\neq
0$ is stronger. If evolution is presumed not to occur, we find Bayes
factors $B=2.1\times 10^4$ (MLCS), $2.5\times 10^6$ (TF) and $5.0\times
10^3$ (SF), decisive odds in favor of nonzero $\Omega_\Lambda$. Weak evolution
($e=0.1$ in Model I or $b=0.25$ in Model II) lowers these values to
$B=20$ (MLCS), 4.8 (TF) and 14 (SF), which still favors $\Omega_\Lambda\neq 0$
positively but not nearly as persuasively. If evolution is allowed
to be somewhat
more pronounced, but still at a plausible level ($e=0.2$ or $b=0.5$),
the Bayes factors fall to $B=2.4$ (MLCS), 2.5 (TF) and 2.3 (SF), which
is scant evidence for a non-vanishing cosmological constant. Once
again, the ability of the data to distinguish $\Omega_\Lambda\neq 0$ from $\Omega_\Lambda=0$
depends sensitively on prior assumptions about evolution of SNe Ia, and
underscores the importance of placing independent constraints on the
possible range of variation of their peak luminosities with redshift.

\acknowledgments

We have benefited from conversations and correspondence with Adam
Riess, Peter Garnavich and Bill Press.  We thank Saul Perlmutter
for providing a draft of P98 prior to publication.
We thank Margaret Geller, Saul Teukolsky, \'Eanna Flanagan,
David Chernoff, Ed Salpeter, and Sidney Drell for comments on
earlier drafts of this paper.
PSD gratefully acknowledges support from the John Simon Guggenheim
Memorial Foundation.  This work was supported by NASA grants NAG5-3097,
NAG5-2762, NAG5-3427, and NSF grant PHY-9310764.

redpajama_set_name: RedPajamaArXiv


text:
Q: iFrame not challenging for credentials I'm trying to create a phonegap app which simply holds either img tags or iframes, all of which come from a number of different servers on our network.
The current challenge I'm facing is, when you navigate to, for example, http://my-intranet-domain/image.png in the browser, you are given a login dialog to enter your active directory credentials; when you put this as <img src="http://my-intranet-domain/image.png" /> in the phonegap app (or even Chrome and Safari on a macbook), you simply get 401 Errors (no challenge for credentials).
How can I:

*

*Challenge for usernames and passwords in this situation

and, possibly:

*After you get the username and password, remember this so you don't have to ask again (there are a number of different servers these images/iframes may come from, but they're all on our domain -- if you have to authenticate, you should only authenticate once).

A: The webviews don't present a dialog by default. You may want to look into cordova-plugin-auth-dialog which enables this behavior for Cordova apps. It can persist credentials for the session and on iOS it can persist them forever.

redpajama_set_name: RedPajamaStackExchange


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text:
namespace Halley {
	class ScriptComment final : public ScriptNodeTypeBase<void> {
	public:
		String getId() const override { return "comment"; }
		String getName() const override { return "Comment"; }
		String getIconName(const ScriptGraphNode& node) const override { return "script_icons/comment.png"; }
		ScriptNodeClassification getClassification() const override { return ScriptNodeClassification::Comment; }

		Vector<SettingType> getSettingTypes() const override;
		std::pair<String, Vector<ColourOverride>> getNodeDescription(const ScriptGraphNode& node, const World* world, const ScriptGraph& graph) const override;
		gsl::span<const PinType> getPinConfiguration(const ScriptGraphNode& node) const override;
		String getLargeLabel(const ScriptGraphNode& node) const override;
	};
}

redpajama_set_name: RedPajamaGithub


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