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Type Ia Supernovae and q 0
1
Bruno Leibundgut
European Southern Observatory, Karl­Schwarzschild­Strasse 2,
D­85748 Garching, Germany
Abstract. Type Ia Supernovae hold great promise to measure the cos­
mic deceleration. The diversity observed among these explosions, how­
ever, complicates their ability to measure cosmological parameters con­
siderably. The comparison of near and distant objects depends critically
on the global properties of the samples. Systematic errors are introduced
in cases where the samples are not compatible. Our discussion of system­
atic uncertainties emphasizes the dependence on the zero­point defined
by the nearby SNe Ia sample and the influence of the application of light
curve corrections to the luminosity. Other error sources are in principle
controllable. We illustrate the effects in an analysis of four SNe Ia at
z ? 0:4.
1. Introduction
It is more than 70 years since it was realized that the Universe is expanding,
yet we still do not know its exact dynamic state. The long debate on the
present­day value of the expansion rate, Hubble's constant, appears to be coming
to an end and most astronomers agree on similar values. The coming years
will see a discussion on the global change of the expansion rate and the final
fate of the Universe. Supernovae have been at the forefront of this research
for the last decade and will also play an important role in the near future by
measuring the deceleration q 0 and separating the contributions of
matter\Omega M
and a cosmological
constant\Omega \Lambda .
'Supernovae Type Ia [are] the best standard candles known so far' (Branch
& Tammann 1992). Even though this statement has been challenged in recent
years it holds pretty much true. The magnitude­redshift relation of Type Ia
Supernovae (SNe Ia) has always displayed a rather small scatter (Kowal 1968,
Tammann & Leibundgut 1990) and almost all SNe Ia exhibit very similar B, V ,
and I light curves (Leibundgut et al. 1991, Hamuy et al. 1996). A debate on
SN Ia uniformity has started with the discovery of a few under-- and overluminous
events (SN 1991bg: Filippenko et al. 1992a, Leibundgut et al. 1993, Turatto et
al. 1996; SN 1992K: Hamuy et al. 1994; SN 1991T: Filippenko et al. 1992b,
Phillips et al. 1992; SN 1992bc and SN 1992bo: Maza et al. 1994) which show
1 To appear in `Supernovae and Cosmology' eds. L. Labhardt, B. Binggeli, R. Buser, Basel:
University of Basel
1

many, but not all, signatures of what normally is called a SN Ia. There are
important differences in the appearance of these objects, but it seems that a
clear separation into subclasses is not possible, if they are indeed distinct from
the majority of SNe Ia. Even highly selected samples (Vaughan et al. 1995,
Sandage et al. 1996, Saha et al. 1997) show internal scatter in absolute peak
luminosity.
It is the maximum luminosity of SNe Ia which is normally used when they
are employed as standard candles. A limited and small range of peak brightness
is crucial. The total scatter of nearby SNe Ia in the Hubble flow ranges from
about 0.25 magnitudes in B (Hamuy et al. 1996, Saha et al. 1997) to 0.19
magnitudes in I (Hamuy et al. 1996). The scatter can further be reduced
by empirical corrections derived from distance­independent parameters which
correlate with the peak luminosity. The light curve shape is the most frequently
used method (Phillips 1993, Hamuy et al. 1996, Riess et al. 1996). Another
option is to apply appropriate selection criteria to limit the range of absolute
magnitudes of known SNe Ia in which case the pure standard candle approach
may still be valid (Vaughan et al. 1995, Sandage et al. 1996, Saha et al. 1997).
Defining subsets of objects is a valid procedure as long as the selection does
not result in a degeneracy. None of the light curve correlations found for SNe Ia
have so far been explained by theory. The use of SNe Ia in cosmology is one of
the main incentives to better understand SN Ia physics.
The current knowledge of the explosion physics and the radiation trans­
port in the highly non­thermal envelopes is far from being complete (Eastman
& Pinto 1993, Pinto & Eastman 1997, Eastman 1997, Pinto 1997, H¨oflich et
al. 1993, 1995, 1996, 1997). Detailed observations obtained for many nearby
objects indicate that some of the fundamental assumptions adopted a few years
back have been in error. Especially the assumption of a thermal radiation field
in SNe Ia is wrong and a more exact treatment is necessary. The emerging ra­
diation resembles a black body in the optical fairly well results, however, from
redistribution of the flux due to the many lines of iron and iron­like elements
blocking the UV radiation. A very clear sign of the non­thermal nature of the
radiation emitted by SNe Ia is the time of maximum light in different filter pass­
bands. For a simple, expanding ball of gas in thermal equilibrium the maximum
at longer wavelengths should be reached at later phases as the gas cools adiabat­
ically. This is not true for most SNe Ia where the I light curve peaks before the
B maximum is reached (Vacca & Leibundgut 1996, Contardo et al. 1998), a sign
of the wavelength dependence of the ejecta opacity. A detailed radiation model
explaining this observation is still lacking. The spectral individuality of SNe Ia
at maximum light has been recognized as due to small abundance differences in
the outer envelope which are of little relevance for the explosion physics itself
(e.g. Jeffery et al. 1992, Mazzali et al. 1993). It may be sensitive to some of the
precursor history, but not necessarily represent a true signature of fundamental
differences among the observed explosions.
The light curve tracks the temporal evolution of the energy release. Most
of the energy generated in the explosion by burning matter to nuclear statistical
equilibrium goes into unbinding the white dwarf. The rest is stored in radioactive
material synthesized in the explosion (Clayton 1964, Colgate & McKee 1969).
The conversion of the fl­rays from the radioactive decays to lower energy photons
2

and their escape from the ejecta defines the light curve (Arnett 1982, Leibundgut
& Pinto 1992). Since the release is highly time dependent due to the decreasing
column density and rapidly changing opacities in the envelope, it has been very
difficult to calculate light curves of SNe Ia. A technical complication is the
conversion of the total released energy into the filter passbands of observed light
curves requiring the calculation of the complete spectrum to obtain enough
wavelength points. The observers have had little input into the bolometric light
curves of SNe Ia (Leibundgut & Pinto 1992). Attempts have been made by
Suntzeff (1996) for SN 1992A with superb and Leibundgut (1996) for SN 1990N
with rudimentary data. Only recently have data sets, like the one for SN 1994D,
become available for a detailed determination of a bolometric flux evolution
(Vacca & Leibundgut 1996). Interestingly, the variations among SNe Ia persist
in the bolometric light curves and a variety of shapes is found. Most strikingly,
the second maximum observed in the infrared appears as an inflection of variable
strength in the bolometric light curves (Contardo et al. 1998). This points
to important fundamental differences in the energy release among individual
SNe Ia. The light curve correction procedures will have to be tested on the
bolometric light curves which are a much more physical quantity than the filter
light curves. It should be pointed out that the combination of these effects
indicates a more complicated picture than what has been captured in simple
scaling relations. They may be adequate to render SNe Ia suitable objects to
measure cosmological distances, but do not suffice to describe the physics of the
explosion.
Apart from the general difficulty of demonstrating the small luminosity
scatter of SNe Ia, their use also requires a careful error analysis. Section 2 will
outline how cosmology is done with standard candles and basically summarizes
work presented in fundamental papers of several decades ago. In x3 we apply
the theory to observed SNe Ia. A discussion of the sources of systematic errors
and how they are treated is given. The analysis of the first set of 4 SN Ia serves
as an illustrative example. We conclude (x4) with an assessment of how suitable
SNe Ia are for cosmology.
2. The use of standard candles in cosmology
The principle of standard candles is probably the simplest and most often used
method to measure cosmological parameters. The combination of the distance
modulus and the Hubble law at small redshifts provides a direct way to measure
the Hubble constant, H 0 . The dimming of a standard candle as a function of
redshift z is described by
m = 5 log z + 5 log c
H 0
+ M + 25:
Given the fixed absolute magnitude M of a known standard candle any
measurement of the apparent magnitude m of an object at redshift z provides
the value of Hubble's constant (in units of km s \Gamma1 Mpc \Gamma1 ). This is typically
shown in a Hubble diagram, log(cz) vs: m.
For cosmologically significant distances, where the effects of the matter and
energy contents of the Universe become significant, the luminosity distance is
defined by the integration over the line element along the line of sight.
3

All early papers on this subject used the series expansion
m = 5 log z + 1:086(1 \Gamma q 0 )z + 5 log c
H 0
+M + 25
(Heckmann 1942, Robertson 1955, Hoyle & Sandage 1956, Sandage 1961). q 0
here is the deceleration of the expansion. The integral of the line element can
be solved analytically only in some specific cases (e.g. negligible cosmological
constant: Mattig 1958; special cases including a cosmological constant: Mat­
tig 1968). The earliest publications (McVittie 1938, Heckmann 1942) already
warned of the dangers involved in the expansion of the exact equation, and
Mattig (1958) showed that for models without a cosmological constant a second
order term makes significant contributions.
A modern derivation of the relations for an expanding universe with a cos­
mological constant is given in Carroll et al. (1992). Using the Robertson­Walker
metric the luminosity distance in an expanding universe, allowing for a cosmo­
logical constant \Lambda, is
DL = (1 + z)c
H 0 jŸj 1=2
S
ae
jŸj 1=2
Z z
0
[Ÿ(1 + z 0 ) 2
+\Omega M (1 + z 0 ) 3
+\Omega \Lambda ] \Gamma1=2 dz 0
oe
:
Here\Omega M = 8úG
3H 2
0
ae M stands for the matter content, which depends only on
the mean matter density of the universe ae M ,
and\Omega \Lambda = \Lambda
3H 2
0
describes the con­
tribution of a cosmological constant to the expansion factor. Ÿ is the curvature
term and obeys
Ÿ = 1
\Gamma\Omega M
\Gamma\Omega \Lambda :
S(ü) takes the form
S(ü) =
8
!
:
sin(ü) Ÿ ! 0
ü for Ÿ = 0
sinh(ü) Ÿ ? 0:
The cosmic deceleration in these models is defined as q 0
=\Omega M
2
\Gamma\Omega \Lambda .
The dimming of standard candles in different cosmological models is nor­
mally displayed as a set of lines in the Hubble diagram (Sandage 1961, Lei­
bundgut & Spyromilio 1997, Perlmutter et al. 1997). It is, however, more
instructive to plot a diagram of the magnitude differences between the various
world models (Fig. 1, cf. also Schmidt et al. 1998, Garnavich et al. 1997).
The differences between the various models become more apparent in this dia­
gram. A standard candle in an open
universe(\Omega M = 0, \Lambda = 0) would appear
0.17 magnitudes fainter at a redshift of 0.3 than in a flat
universe(\Omega M = 0:5,
\Lambda = 0). This difference increases to 0.33 mag at z = 0:6 and 0.54 mag for z = 1:0
(Fig. 1). These are small values considering how difficult the observations are
and the corrections which are needed to obtain a significant measurement.
The present­day cosmic deceleration q 0 combines all energy sources con­
tributing to the change of the expansion rate of the universe. It thus represents
a fundamental parameter for the description of the Universe we live in. For
models without the cosmological constant the fate of the Universe is encapsu­
lated in q 0 . There is, however, a degeneracy in q 0 when a cosmological constant
4

0 .2 .4 .6 .8 1 1.2
­.5
0
.5
1
redshift
dm (0.6,0.6)
(1,0)
(0,1)
(0,0)
Garnavich et al. (1997)
Perlmutter et al. (1997)
Figure 1. Magnitude differences of a standard candle for various cos­
mological models. The lines for various combinations
of(\Omega M
;\Omega \Lambda ) are
shown. They correspond to q 0 = 0:5 (solid line), q 0 = 0 (dotted),
q 0 = \Gamma1 (dashed), and q 0 = \Gamma0:3 (grey).
is included (cf. Fig. 1). The separation
of\Omega M
and\Omega \Lambda requires the observation
of standard candles over a large range of redshifts (Goobar & Perlmutter 1995).
It is important to realize that the value of the Hubble constant is not re­
quired for the determination of q 0 . What is important is the apparent magni­
tude difference of a standard candle measured at two different redshifts. Distant
supernovae have to be compared to a set of nearby supernovae where q 0 , or
equivalently the curvature Ÿ, is negligible.
3. The cosmological deceleration as measured by SNe Ia
3.1. Systematics
Even though the scatter in peak luminosity is small, SNe Ia are not perfect
standard candles. Various effects can play an important role when they are ap­
plied to measure distances. Most importantly, the samples which are compared
for the derivation of the cosmological parameters have to be compatible and
commensurate.
Even a perfect standard candle suffers from extinction along the line be­
tween the object and the observer. The differences in the absorption along
different lines of sight, least known in the host galaxy, can modify the observed
luminosities. Supernovae must be checked carefully for reddening. SNe Ia dis­
play a range of intrinsic colors and can not be corrected for absorption based on
5

color alone. Rest­frame color still is a good indicator of unreddened objects as
there is a well­defined blue cutoff in the color distribution, but it can only be
measured through light curves in at least two filters. The concentration on SNe
in elliptical galaxies could partially remedy this problem (Branch & Tammann
1992), but a suitable sample of nearby SNe Ia in elliptical galaxies would have to
be established. There also appears to be a luminosity difference among SNe Ia
between star­forming and elliptical galaxies which may be a worry (Hamuy et
al. 1996, Schmidt et al. 1998).
Some correction methods treat reddening implicitly (Riess et al. 1996) with
the assumption that the color correlates with the peak luminosity. Light curves
in at least two filters are mandatory for this application.
Type Ia supernovae have to be recognized as such which is reliably done
through spectroscopy (Oke & Searle 1974, Filippenko 1997). Peculiar super­
novae have been distinguished by their spectral appearance (Filippenko et al.
1992a, 1992b, Phillips et al. 1992, Leibundgut et al. 1993, Branch et al. 1993).
Spectroscopic observations of supernovae are an indispensable requisite for any
supernova research. Contamination of the sample by supernovae of other types
is unacceptable and dangerous. Peculiar SNe Ia have to be detected as well. The
High­z Supernova Search (Schmidt 1997) thus has set out from the beginning to
obtain spectroscopy of all supernovae discovered in its search.
Any global differences in the composition and global properties of the com­
parison samples introduce systematic errors in the result. To illustrate this
consider the average luminosity of two samples which are used to derive cos­
mological parameters. If, due to Malmquist bias, the more distant sample has
a systematically higher luminosity than the nearby sample, and we would just
apply the standard candle paradigm, the mean distance difference would be un­
derestimated. The implications are the derivation of too large a value for the
Hubble constant or an overestimation of q 0 .
There are two more uncertainties when using SNe Ia for the determination
of q 0 . One is evolution and the other distortions due to gravitational lensing
effects. Evolution would be a clear violation of the standard candle paradigm.
Since we are looking at explosions, which occurred at a time when the universe
was only half its current age, we have to make sure that they are the same as
the ones we observe nearby. This would be a bold assumption if unchecked.
Depending on the progenitor stars of SNe Ia there could be substantial evolu­
tionary effects. If SNe Ia result from sub­Chandrasekhar­mass white dwarfs,
the mass dependence could introduce luminosity differences which would skew
the luminosity function (Ruiz­Lapuente et al. 1995, Canal et al. 1996). Since
more massive white dwarfs are formed first, it is conceivable that these distant
SNe Ia are from an intrinsically more massive population and thus also more
luminous, which would have an effect comparable to the Malmquist bias de­
scribed above. Even with Chandrasekhar­mass progenitors the luminosity could
be substantially changed due to abundance differences in the parent popula­
tions. The smaller metallicities could mean that the line opacities governing
the radiation escape in SNe Ia are significantly changed which would result in
changed luminosities in the observed filters. The only way to check any of the
above conjectures lies in carefully observing the distant objects and asserting
their similarity with the nearby sample. If the distant sample looks the same as
6

the nearby one in all aspects we can measure, we have a good chance that they
have the same average luminosity as well. It is not sufficient to exclude strange­
looking or non­conforming objects as they may be the sign of real differences
invalidating the standard candle approach altogether.
Gravitational lensing always affects light traveling over cosmological dis­
tances. Amplifications introduced by the lensing have been investigated in com­
bination with SNe Ia (Wambsganss et al. 1997). Gravitational lensing, unlike
absorption, is achromatic. It can not be inferred from the observed light alone.
Luckily, at the redshifts of the supernova searches (z ! 1 so far) the systematic
amplification (or dimming) of standard candles for current models of the mass
distribution is small (\Deltaq 0 ! 0:04 for all z ! 1; Wambsganss et al. 1997).
3.2. Observing distant supernovae
There are currently two large observational programs which try to determine
the deceleration of the universe using SNe Ia. They both make use of SNe Ia
as standard candles or modified versions thereof. The High­z Supernova Search
Team (Schmidt 1997, Schmidt et al. 1998, Garnavich et al. 1997, Leibundgut et
al. 1996, Leibundgut & Spyromilio 1997) and the Berkeley Cosmology Project
(Perlmutter et al. 1995, 1997) have discovered a large number of supernovae over
the last three years. In fact, these two supernova searches now contribute the
majority of all discoveries per year. Figure 2 shows a summary of all SNe Ia found
in these two searches up to June 1997. It should be compared to a similar figure
from 1996 (Leibundgut & Spyromilio 1997) to appreciate the rapid progress.
Both groups have mastered the art of detecting these extremely faint objects
very efficiently. The numbers reported are actually lower limits as both groups
now only announce bona fide supernovae at high redshifts. Not even all types
of SNe are reported any longer as the highest interest is in distant SNe Ia.
This should be taken as a word of caution when interpreting the numbers for
supernova rates. The discovery of SNe Ia dominates the distribution. Only very
few Type II supernovae have been announced at redshifts larger than 0.3 and a
single SN Ic has been discovered in the High­z SN Search.
The total number of SNe discovered in the two searches is 107 objects, 70 of
which are confirmed SNe Ia at z ? 0:25 (29 High­z SN Search and 41 Berkeley
Cosmology Project).
The emphasis has shifted from actually finding the supernovae to organiz­
ing the critical follow up observations. Spectroscopy is the prime classification
resource and also serves as a discriminator against unusual objects. Photometry
is required to establish the essential light curve parameters, if such a correction
should be applied.
These observations are challenging for any telescope. As an example con­
sider a SN Ia at z=0.5: the observed peak magnitude reaches 21.9 in the R band
(SN 1995K; Schmidt et al. 1998), but the crucial light curve parts for the decline
rate determination are around R = 23 and, if some of the light curve tail should
be observed, magnitude levels with R ? 25 have to be considered. It can not be
emphasized enough how important it is to achieve accurate photometry at these
magnitudes as only small deviations can result in substantial, in terms of the
effect to be measured, over­corrections. This is even more so, should systematic
effects creep in, which skew the correction unilaterally through, e.g., contami­
7

0 .2 .4 .6 .8 1
0
5
10
15
20
redshift
Supernova
Discoveries unclassified
Type Ic
Type II
Type Ia
High­z Team
0 .2 .4 .6 .8 1
redshift
unclassified
Type II
Type Ia
Berkeley
Figure 2. Supernova discoveries reported by the distant SN searches
in IAU Circular until July 1997. The top line indicates the total number
of SNe from both searches combined.
nation by the galaxy background. It is dangerous to ``blindly'' apply light curve
correction procedures without the assurance that the objects are indeed SNe Ia.
All observations obtained so far indicate that the distant supernovae look
very similar to the local ones (Leibundgut et al. 1996, Leibundgut & Spyromilio
1997, Perlmutter et al. 1997, Schmidt et al. 1998, Garnavich et al. 1997). Spec­
tra, light curves, and rest­frame colors of distant supernovae appear identical
to the bulk of the local population, and evolutionary effects are not apparent
(Schmidt et al. 1998).
Technical issues in the analysis are the accuracy of the photometry, the
calculation of K­corrections, any application of luminosity corrections, and un­
certainties in the zero­point defined by the nearby sample. While the photometry
and the K­corrections appear to be under control, the correction for the light
curve shape can become problematic. The maximum magnitude of a SN Ia is
measured fairly easily as the techniques employed in these searches by design
discover SNe close to maximum (Perlmutter et al. 1997, Schmidt et al. 1998).
Typical photometric errors of individual light curve points are about 0.05 to 0.15
magnitudes near the peak, but increase to 0.3 magnitudes a couple of weeks from
maximum. The error in the determination of the filter maximum is around 0.1
to 0.2 magnitudes. The uncertainties in the K­corrections are negligible (!0.05
magnitudes; Kim et al. 1996, Schmidt et al. 1998). Light curve corrections in­
troduce two uncertainties. They are defined by photometry points which are less
well measured than near maximum and the correction parameters themselves
carry uncertainties (Hamuy et al. 1996). It has yet to be shown unambiguously
8

that the light curve corrections also decrease the scatter in the distant sample.
Contrary to the simple determination of the maximum magnitude these methods
make use of the whole light curve shape. This is a very valuable asset when the
photometry errors are not significant, but can become problematic when applied
to points with large errors. In addition, the corrections change the zero­point of
the sample, a very contentious issue in the Hubble constant discussion (Hamuy
et al. 1996, Saha et al. 1997). It will be very important to resolve this ambiguity
also for the determination of q 0 .
The statistical uncertainty in the zero­point of the nearby SN Ia sample is
typically about 0.03 (Hamuy et al. 1996). The contribution to the uncertainty
in q 0 scales with 1=z and for a sample with a mean redshift of 0.5 contributes
about 0.06 to the total uncertainty in q 0 .
3.3. The first four High­z supernovae
SN 1995K is the first SN Ia discovered in the High­z SN Search (Leibundgut et
al. 1996, Schmidt et al. 1998). Three more supernovae have become available
recently (Garnavich et al. 1997). We will use this small sample of four SNe to
exemplify the analysis of distant SNe Ia and describe the systematic influence
of the basic assumptions.
The spectrum and the light curves of SN 1995K show no signs of any pecu­
liarity for a SN Ia at z = 0:478 (Leibundgut et al. 1996, Leibundgut & Spyromilio
1997, Schmidt et al. 1998). Once the time dilation has been removed the light
curve is comparable to regular nearby SNe Ia. K­corrections are nearly constant
throughout the observed phase due to the near match of the R and I filters
to rest­frame B and V , respectively, at z ú 0:5. The rest­frame color derived
for this object is (B \Gamma V ) ú \Gamma0:1, consistent with unreddened nearby objects
(Vaughan et al. 1995, Tammann & Sandage 1995). The peak magnitude in
rest­frame B and V is 22.9 and 23.0, respectively. It can thus be assumed that
it is a good candidate for the measurement of deceleration.
SN 1997ce and SN 1997cj are spectroscopically confirmed SNe Ia at red­
shifts of 0.44 and 0.50, respectively (Garnavich et al. 1997). Spectra of these
two supernovae show the characteristic Si II (SN 1997ce) and Ca II (SN 1997cj)
absorption. They both have a well sampled light curve from ground­based and
HST observations in rest­frame B and V without apparent peculiarities. The
color at maximum for both objects is also consistent with (B \Gamma V ) ú 0:0 indi­
cating that absorption is negligible. We find peak magnitudes of 22.9 (B) and
23.0 (V ) for SN 1997ce and 23.3 (B and V ) for SN 1997cj from simultaneous
fits to light curve templates.
The most distant supernova in the sample, SN 1997ck, is at a redshift of
z = 0:97 as deduced from a spectrum which shows a single, narrow emission line,
identified as [O II], from the galaxy. The time dilation in the light curve indicates
a redshift of 1:1\Sigma0:2 for the best fitting light curve of a nearby SN Ia (SN 1991T).
At this redshift the V band is shifted beyond 1¯m and no data could be obtained.
Hence we have no indication whether dust absorption is affecting this object.
The following analysis of the B light curve assumes negligible absorption.
Figure 1 indicates the position of these supernovae in comparison with
nearby SNe Ia (Hamuy et al. 1996). B and V observations have been com­
bined for the points of the High­z sample. This diagram has the advantage
9

.4 .5 .6 .7 .8 .9 1
­.2
0
.2
.4
.6
.8
1
redshift
dm
(1,0)
(0,1)
(0,0)
Figure 3. Same as Fig. 1 but after application of the light curve
correction. The arrow indicates the shift due to the zero­point change
(0.14 magnitudes). The faint dots indicate the former position of the
data( shifted in redshift for clarity). The errors have increased slightly
due to the additional uncertainty of the light curve fitting.
that both filter measurements can be combined in the same graph. This also
means that the observations in two filters, apart from their necessity for accurate
photometry and the check for absorption, increases the number of observations
in the data set thus constraining the uncertainty. The figure also displays the
supernovae reported by the Berkeley group (Perlmutter et al. 1997).
The diagram in Figure 1 assumes a perfect standard candle, i.e. no light
curve shape correction has been applied. It uses the zero­point given in equations
4 and 5 of Hamuy et al. (1996). There is a large overlap between the two data
sets, but a clear lack of bright supernovae in the High­z sample. We find values
for\Omega M
and\Omega \Lambda by ü 2 fitting. Only for the uncorrected B data do we find an
acceptable solution (ü 2 minimum), if the cosmological constant is ignored. The
best value in this case
is\Omega M = 0:1 \Sigma 0:6. The best solution is found
for\Omega M = 0:6
and\Omega \Lambda = 0:6 (corresponding to q 0 = \Gamma0:3) with a very large uncertainty in both
parameters. For the V filter data we find no solution
with\Omega \Lambda = 0
and\Omega M – 0.
The best solution lies
near\Omega M = 0:5
and\Omega \Lambda = 0:7 (q 0 = \Gamma0:45). The original
result for the Berkeley set was close
to\Omega M =
0:9,\Omega \Lambda = 0:1 (Perlmutter et al.
1997).
The picture changes, when we apply the correction for the light curve shape.
The model lines shift by 0.14 magnitudes due to the change of the zero­point
provided by the local sample (Hamuy et al. 1996; equations 7 and 8). At the
same time we have to correct the distant supernovae for their light curves. With
10

the exception of SN 1997ck, which seems to have been similar to SN 1991T in
light curve shape, all SNe are best approximated by the template light curve of
nearby SNe Ia (cf. Garnavich et al. 1997). Thus, any light curve corrections for
these objects are small (! 0:1 mag). The correction of the B maximum bright­
ness of SN 1997ck is 0.13 magnitudes assuming it was identical to SN 1991T.
The global effect is illustrated in Figure 3. The arrow indicates the shift
of the data points due to the change of the zero­point of the lines. After the
correction all points scatter around the line for an open
universe(\Omega M =
0;\Omega \Lambda =
0). A \Lambda­dominated universe is now favored with best fits
of(\Omega M
;\Omega \Lambda ) = (0:3; 0:6)
for the B data
and(\Omega M
;\Omega \Lambda ) = (0:3; 0:9) for V . These values correspond to
q 0 = \Gamma0:45 and q 0 = \Gamma0:75, respectively. The value
for\Omega M has decreased in the
B solution, because SN 1997ck, being identified as overluminous, has undergone
a larger correction than the other supernovae. The photometric uncertainty in
the measurement of this high­z supernova, however, reduces its contribution in
the ü 2 ­fit. Thus the signal is produced mainly by the 3 supernovae near z ú 0:45
and not by SN 1997ck, as is obvious from the fits to the V data. Note that any
reddening of SN 1997ck would make this supernova even brighter. We have to
emphasize that the number of objects is still small and it is certainly premature
to draw definite conclusions.
Since we do not have access to the individual photometry of the Berkeley
supernovae we can not perform the same analysis on their data. It should be
pointed out, however, that two of their SNe lie outside the observed range of
decline rates and have been excluded in their analysis. The correction of the
remaining objects nearly cancels the offset of the zero­point shift as they arrive
at almost the same result for the uncorrected and the corrected sample. This
is surprising considering the effect of the zero­point alone (Fig. 3). A direct
comparison of the two data samples is not possible, but we can compare SNe
which have similar redshifts. SN 1992bi had almost the same redshift (z =
0:458) as SN 1995K and undergoes a large correction, but was excluded from
the subsequent analysis, because it displayed a slower decline rate than observed
for any local SN Ia (Perlmutter et al. 1997). More interesting might be the
comparison of SN 1997ce with SN 1994al (z = 0:420) and SN 1994G (z = 0:425).
SN 1994G has a spectrum identifying the object as a genuine SN Ia, while the
galaxy spectrum of SN 1994al is described as consistent with an elliptical or S0
galaxy, which would identify this object also as a SN Ia. The observed rest­frame
B magnitudes at peak are 22.8 (SN 1994al) and 22.3 (SN 1994G). These have to
be compared to the peak magnitude of SN 1997ce which is 22.9. Thus, we have a
very good agreement between SN 1994al and SN 1997ce, but SN 1994G deviating
by about 0.5 magnitudes. Even after corrections for the light curve shape the
close agreement between SN 1994al and SN 1997ce remains, and SN 1994G still
is an outlier (Table 1 of Perlmutter et al. 1997). It is puzzling to find that our
result changes so dramatically (almost entirely due to the shift of the zero­point
of the local sample), while the analysis of Perlmutter et al. (1997) did not result
in a similar signal when they applied their light curve correction method.
11

4. Conclusions
The regular behavior of SNe Ia has already provided proof of another prediction
of models of an expanding universe. Time dilation contributes significantly to
the filter light curves and the spectral evolution of distant supernovae (Wilson
1939, Tammann 1979). The light curves of SN 1995K are dilated exactly by the
factor (1+z) predicted for an expanding universe (Leibundgut et al. 1996). The
Berkeley data set has been investigated for time dilation with similar conclusions
(Goldhaber et al. 1997). In cases like SN 1997ck, where there is the detection of
a single line in the galaxy spectrum, the time dilation can be used for consistency
checks, although one has to assume that the object indeed is a SN Ia. Retarded
spectral evolution, consistent with time dilation, has been observed in SN 1996bj
at a redshift of 0.574 (Riess et al. 1997). Nevertheless, there remain proposals of
static universe models which conform with the observed time dilation (Narlikar
& Arp 1997, Segal 1997).
With sufficient control the difficulties of SNe Ia to measure the current
state of the deceleration can be overcome. Light curves in two filters and spec­
troscopy near maximum light can identify extinction or peculiar SNe Ia, which
may prevent the correct application of the standard candle paradigm. Only a
very careful characterization of all measurable aspects of the sample of distant
supernovae and its comparison with the nearby set can lead to a successful mea­
surement of q 0 . The influence of uncertainties introduced by the local sample
should not be underestimated. They are sizeable contributions to the error with
which q 0 can be determined.
The largest systematic uncertainty is the application of luminosity correc­
tions. This is a rather intricate problem which is discussed in connection with
SNe Ia based determinations of H 0 , but applies equally to q 0 . Our analysis of
the small SN sample available shows that the light curve corrections produce a
different result than a simple comparison of peak magnitudes. For SNe Ia with
light curve shapes deviating only slightly from the average this effect is entirely
in the offset of the zero­point of the local supernova sample. For SNe Ia which are
overluminous as indicated by their light curves, the standard candle paradigm
only produces an upper limit for q 0 . The existing data at this point clearly favor
an open universe and even a contribution of a cosmological constant, although
the errors are still considerable.
As discussed the difference between an empty and a flat universe is 0.54
magnitudes at z = 1 (Fig. 1). This means that the influence of the zero­point
decreases and a measurement at larger redshifts is less sensitive to these sys­
tematics. The difficulty will be to obtain the accurate photometry to reliably
measure the light curve shape and apply the corrections. The High­z SN Search
has already discovered and followed supernovae at these redshifts. We confirm
in our analysis the result presented in Garnavich et al. (1997). The distant ob­
jects are essential to
measure\Omega M
and\Omega \Lambda independently (Goobar & Perlmutter
1995).
Due to Malmquist bias one is more likely to overestimate q 0 than to find
too small a value. On the other hand, any residual photometry errors which
would tend to widen the light curves would lead to the adoption of too bright a
luminosity for the supernovae.
12

Type Ia Supernovae may be the closest we know to a cosmological stan­
dard candle, yet the magnitude differences for different dynamical states of the
universe are very small and a tight control of the samples is indispensable. In
principle all the required information can be gathered by the observations. The
knowledge of the progenitor systems and the exact radiation physics are piv­
otal ingredients still missing from our picture of these explosions (Branch et al.
1995). With such an understanding the inference of the standard candle char­
acter and possible evolutionary scenarios can be tested. Until then we have to
rely on the control provided by careful observations and analysis.
Acknowledgments. I am grateful to Gustav Tammann for directing my
early research into supernovae and the never failing support he has given me
since I first walked into his office. The data discussed in this paper have been
collected by the High­z SN team. I am thankful to the team members, especially
Brian Schmidt and Peter Garnavich, on the work of the photometry reported in
this paper.
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