Journal ArticleDOI

# Measurements of Omega and Lambda from 42 High-Redshift Supernovae

01 Jun 1999-The Astrophysical Journal (IOP Publishing)-Vol. 517, Iss: 2, pp 565-586

Abstract: We report measurements of the mass density, Omega_M, and cosmological-constant energy density, Omega_Lambda, of the universe based on the analysis of 42 Type Ia supernovae discovered by the Supernova Cosmology Project. The magnitude-redshift data for these SNe, at redshifts between 0.18 and 0.83, are fit jointly with a set of SNe from the Calan/Tololo Supernova Survey, at redshifts below 0.1, to yield values for the cosmological parameters. All SN peak magnitudes are standardized using a SN Ia lightcurve width-luminosity relation. The measurement yields a joint probability distribution of the cosmological parameters that is approximated by the relation 0.8 Omega_M - 0.6 Omega_Lambda ~= -0.2 +/- 0.1 in the region of interest (Omega_M <~ 1.5). For a flat (Omega_M + Omega_Lambda = 1) cosmology we find Omega_M = 0.28{+0.09,-0.08} (1 sigma statistical) {+0.05,-0.04} (identified systematics). The data are strongly inconsistent with a Lambda = 0 flat cosmology, the simplest inflationary universe model. An open, Lambda = 0 cosmology also does not fit the data well: the data indicate that the cosmological constant is non-zero and positive, with a confidence of P(Lambda > 0) = 99%, including the identified systematic uncertainties. The best-fit age of the universe relative to the Hubble time is t_0 = 14.9{+1.4,-1.1} (0.63/h) Gyr for a flat cosmology. The size of our sample allows us to perform a variety of statistical tests to check for possible systematic errors and biases. We find no significant differences in either the host reddening distribution or Malmquist bias between the low-redshift Calan/Tololo sample and our high-redshift sample. The conclusions are robust whether or not a width-luminosity relation is used to standardize the SN peak magnitudes.
Topics: , Omega (56%), Supernova Legacy Survey (53%), Hubble's law (53%), Cosmological constant (53%)

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THE ASTROPHYSICAL JOURNAL, 517 :565È586, 1999 June 1
MEASUREMENTS OF ) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE
S. PERLMUTTER,1 G. ALDERING,G.GOLDHABER,1 R. A. KNOP,P.NUGENT,P.G.CASTRO,2 S. DEUSTUA,S.FABBRO,3
A. GOOBAR,4 D. E. GROOM,I.M.HOOK,5 A. G. KIM,1,6 M. Y. KIM,J.C.LEE,7 N. J. NUNES,2 R. PAIN,3
C. R. PENNYPACKER,8 AND R. QUIMBY
Institute for Nuclear and Particle Astrophysics, E. O. Lawrence Berkeley National Laboratory, Berkeley, CA 94720
C. LIDMAN
European Southern Observatory, La Silla, Chile
R. S. ELLIS,M.IRWIN, AND R. G. MCMAHON
Institute of Astronomy, Cambridge, England, UK
P. RUIZ-LAPUENTE
Department of Astronomy, University of Barcelona, Barcelona, Spain
N. WALTON
Isaac Newton Group, La Palma, Spain
B. SCHAEFER
Department of Astronomy, Yale University, New Haven, CT
B. J. BOYLE
Anglo-Australian Observatory, Sydney, Australia
A.VFILIPPENKO AND T. MATHESON
Department of Astronomy, University of California, Berkeley, CA
A. S. FRUCHTER AND N. PANAGIA9
Space Telescope Science Institute, Baltimore, MD
H. J. M. NEWBERG
Fermi National Laboratory, Batavia, IL
AND
W. J. COUCH
University of New South Wales, Sydney, Australia
(THE SUPERNOVA COSMOLOGY PROJECT)
Received 1998 September 8 ; accepted 1998 December 17
ABSTRACT
We report measurements of the mass density, and cosmological-constant energy density, of)
M
, )
"
,
the universe based on the analysis of 42 type Ia supernovae discovered by the Supernova Cosmology
Project. The magnitude-redshift data for these supernovae, at redshifts between 0.18 and 0.83, are Ðtted
jointly with a set of supernovae from the Supernova Survey, at redshifts below 0.1, to yieldCala
n/Tololo
values for the cosmological parameters. All supernova peak magnitudes are standardized using a SN Ia
light-curve width-luminosity relation. The measurement yields a joint probability distribution of the
cosmological parameters that is approximated by the relation in the region0.8)
M
[ 0.6)
"
B [0.2 ^ 0.1
of interest For a Ñat cosmology we Ðnd (1 p statistical)()
M
[ 1.5). ()
M
] )
"
\ 1) )
M
flat \ 0.28
~0.08
0.09
~0.04
0.05
(identiÐed systematics). The data are strongly inconsistent with a " \ 0 Ñat cosmology, the simplest
inÑationary universe model. An open, " \ 0 cosmology also does not Ðt the data well: the data indicate
that the cosmological constant is nonzero and positive, with a conÐdence of P("[0) \ 99%, including
the identiÐed systematic uncertainties. The best-Ðt age of the universe relative to the Hubble time is
Gyr for a Ñat cosmology. The size of our sample allows us to perform a variety oft
0
flat \ 14.9
~1.1
1.4(0.63/h)
statistical tests to check for possible systematic errors and biases. We Ðnd no signiÐcant di†erences in
either the host reddening distribution or Malmquist bias between the low-redshift sampleCala
n/Tololo
and our high-redshift sample. Excluding those few supernovae that are outliers in color excess or Ðt
residual does not signiÐcantly change the results. The conclusions are also robust whether or not a
width-luminosity relation is used to standardize the supernova peak magnitudes. We discuss and con-
strain, where possible, hypothetical alternatives to a cosmological constant.
Subject headings: cosmology: observations È distance scale È supernovae: general
1 Center for Particle Astrophysics, University of California, Berkeley, California.
2 Instituto Superior Lisbon, Portugal.Te
cnico,
3 LPNHE, CNRS-IN2P3, and University of Paris VI and VII, Paris, France.
4 Department of Physics, University of Stockholm, Stockholm, Sweden.
5 European Southern Observatory, Munich, Germany.
6 PCC, CNRS-IN2P3, and de France, Paris, France.Colle

ge
7 Institute of Astronomy, Cambridge, England, UK.
8 Space Sciences Laboratory, University of California, Berkeley, California.
9 Space Sciences Department, European Space Agency.
565

566 PERLMUTTER ET AL. Vol. 517
1. INTRODUCTION
Since the earliest studies of supernovae, it has been sug-
gested that these luminous events might be used as standard
candles for cosmological measurements (Baade 1938). At
closer distances they could be used to measure the Hubble
constant if an absolute distance scale or magnitude scale
could be established, while at higher redshifts they could
determine the deceleration parameter (Tammann 1979;
Colgate 1979). The Hubble constant measurement became
a realistic possibility in the 1980s, when the more homoge-
neous subclass of type Ia supernovae (SNe Ia) was identiÐed
(see Branch 1998). Attempts to measure the deceleration
parameter, however, were stymied for lack of high-redshift
supernovae. Even after an impressive multiyear e†ort by
et al. (1989), it was only possible toNÔrgaard-Nielsen
follow one SN Ia, at z \ 0.31, discovered 18 days past its
peak brightness.
The Supernova Cosmology Project was started in 1988 to
address this problem. The primary goal of the project is the
determination of the cosmological parameters of the uni-
verse using the magnitude-redshift relation of type Ia super-
novae. In particular, Goobar & Perlmutter (1995) showed
the possibility of separating the relative contributions of the
mass density, and the cosmological constant, ",to)
M
,
changes in the expansion rate by studying supernovae at a
range of redshifts. The Project developed techniques,
including instrumentation, analysis, and observing stra-
tegies, that make it possible to systematically study high-
redshift supernovae (Perlmutter et al. 1995a). As of 1998
March, more than 75 type Ia supernovae at redshifts
z \ 0.18È0.86 have been discovered and studied by the
Supernova Cosmology Project (Perlmutter et al. 1995b,
1996, 1997a, 1997b, 1997c, 1997d, 1998a).
A Ðrst presentation of analysis techniques, identiÐcation
of possible sources of statistical and systematic errors, and
Ðrst results based on seven of these supernovae at redshifts
z D 0.4 were given in Perlmutter et al. (1997e; hereafter
referred to as P97). These Ðrst results yielded a conÐdence
region that was suggestive of a Ñat, " \ 0 universe but with
a large range of uncertainty. Perlmutter et al. (1998b) added
a z \ 0.83 SN Ia to this sample, with observations from the
Hubble Space Telescope (HST ) and Keck 10 m telescope,
providing the Ðrst demonstration of the method of separat-
ing and " contributions. This analysis o†ered prelimi-)
M
nary evidence for a lowÈmass-density universe with a
best-Ðt value of assuming " \ 0. Indepen-)
M
\ 0.2 ^ 0.4,
dent work by Garnavich et al. (1998a), based on three
supernovae at z D 0.5 and one at z \ 0.97, also suggested a
low mass density, with best-Ðt for " \ 0.)
M
\[0.1 ^ 0.5
Perlmutter et al. 1997f presented a preliminary analysis
of 33 additional high-redshift supernovae, which gave a
conÐdence region indicating an accelerating universe and
barely including a low-mass " \ 0 cosmology. Recent inde-
pendent work of Riess et al. (1998), based on 10 high-
redshift supernovae added to the Garnavich et al. (1998a)
set, reached the same conclusion. Here we report on the
complete analysis of 42 supernovae from the Supernova
Cosmology Project, including the reanalysis of our pre-
viously reported supernovae with improved calibration
data and improved photometric and spectroscopic SN Ia
templates.
2. BASIC DATA AND PROCEDURES
The new supernovae in this sample of 42 were all dis-
covered while still brightening, using the Cerro Tololo
Inter-American Observatory (CTIO) 4 m telescope with the
20482 pixel prime-focus CCD camera or the 4 ] 20482 pixel
Big Throughput Camera.10 The supernovae were followed
with photometry over the peak of their light curves and
approximately 2È3 months further (D40È60 days rest
frame) using the CTIO 4 m, Wisconsin-Indiana-Yale-
NOAO (WIYN) 3.6 m, ESO 3.6 m, Isaac Newton Telescope
(INT) 2.5 m, and the William Herschel Telescope (WHT) 4.2
m telescopes. (SN 1997ap and other 1998 supernovae have
also been followed with HST photometry.) The supernova
redshifts and spectral identiÐcations were obtained using
the Keck I and II 10 m telescopes with the Low-Resolution
Imaging Spectrograph (Oke et al. 1995) and the ESO 3.6 m
telescope. The photometry coverage was most complete in
Kron-Cousins R-band, with Kron-Cousins I-band photo-
metry coverage ranging from two or three points near peak
to relatively complete coverage paralleling the R-band
observations.
Almost all of the new supernovae were observed spectro-
scopically. The conÐdence of the type Ia classiÐcations
based on these spectra taken together with the observed
light curves, ranged from ““ deÐnite ÏÏ (when Si II features
were visible) to ““ likely ÏÏ (when the features were consistent
with type Ia and inconsistent with most other types). The
lower conÐdence identiÐcations were primarily due to host-
galaxy contamination of the spectra. Fewer than 10% of the
original sample of supernova candidates from which these
SNe Ia were selected were conÐrmed to be nonÈtype Ia, i.e.,
being active galactic nuclei or belonging to another SN
subclass; almost all of these nonÈSNe Ia could also have
been identiÐed by their light curves and/or position far from
the SN Ia Hubble line. Whenever possible, the redshifts
were measured from the narrow host-galaxy lines rather
than the broader supernova lines. The light curves and
several spectra are shown in Perlmutter et al. (1997e, 1997f,
1998b); complete catalogs and detailed discussions of the
photometry and spectroscopy for these supernovae will be
presented in forthcoming papers.
The photometric reduction and the analyses of the light
curves followed the procedures described in P97. The super-
novae were observed with the Kron-Cousins Ðlter that best
matched the rest-frame B and V Ðlters at the supernovaÏs
redshift, and any remaining mismatch of wavelength cover-
age was corrected by calculating the expected photometric
di†erenceÈthe ““ cross-Ðlter K-correction ÏÏÈusing template
SN Ia spectra as in Kim, Goobar, & Perlmutter (1996). We
have now recalculated these K-corrections (see Nugent et
al. 1998) using improved template spectra, based on an
extensive database of low-redshift SN Ia spectra recently
made available from the survey (Phillips et al.Cala
n/Tololo
1999). Where available, IUE and HST spectra (Cappellaro,
Turatto, & Fernley 1995; Kirshner et al. 1993) were also
added to the SN Ia spectra, including those published pre-
viously (1972E, 1981B, 1986G, 1990N, 1991T, 1992A, and
1994D: in Kirshner & Oke 1975; Branch et al. 1993; Phil-
lips et al. 1987; Je†ery et al. 1992; Meikle et al. 1996; Patat
et al. 1996). In Nugent et al. (1998) we show that the K-
corrections can be calculated accurately for a given day on
the supernova light curve and for a given supernova light-
10 Big Throughput Camera information is provided by G. Bernstein &
J. A. Tyson, 1998, at http://www.astro.lsa.umich.edu/btc/user.html.

No. 2, 1999 ) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE 567
curve width from the color of the supernova on that day.
(Such a calculation of K-correction based on supernova
color will also automatically account for any modiÐcation
of the K-correction due to reddening of the supernova; see
Nugent et al. 1998. In the case of insigniÐcant reddening the
SN Ia template color curves can be used.) We Ðnd that these
calculations are robust to mis-estimations of the light-curve
width or day on the light curve, giving results correct to
within 0.01 mag for light-curveÈwidth errors of ^25% or
light-curve phase errors of ^5 days even at redshifts where
Ðlter matching is the worst. Given small additional uncer-
tainties in the colors of supernovae, we take an overall sys-
tematic uncertainty of 0.02 mag for the K-correction.
The improved K-corrections have been recalculated for
all the supernovae used in this paper, including those pre-
viously analyzed and published. Several of the low-redshift
supernovae from the survey have relativelyCala
n/Tololo
large changes (as much as 0.1 mag) at times in their K-
corrected light curves. (These and other low-redshift super-
novae with new K-corrections are used by several
independent groups in constructing SN Ia light-curve tem-
plates, so the templates must be updated accordingly.) The
K-corrections for several of the high-redshift supernovae
analyzed in P97 have also changed by small amounts at the
light-curve peak mag] and somewhat[*K(t \ 0) [ 0.02
larger amounts by 20 days past peak [*K(t \ 20) [ 0.1
mag]; this primarily a†ects the measurement of the rest-
frame light-curve width. These K-correction changes
balance out among the P97 supernovae, so the Ðnal results
for these supernovae do not change signiÐcantly. (As we
discuss below, however, the much larger current data set
does a†ect the interpretation of these results.)
As in P97, the peak magnitudes have been corrected for
the light-curve width-luminosity relation of SNe Ia:
m
B
corr \ m
B
] *
corr
(s) , (1)
where the correction term is a simple monotonic func-*
corr
tion of the ““ stretch factor,ÏÏ s, that stretches or contracts the
time axis of a template SN Ia light curve to best Ðt the
observed light curve for each supernova (see P97; Perlmut-
ter et al. 1995a; Kim et al. 1999; Goldhaber et al. 1999; and
see Phillips 1993; Riess, Press, & Kirshner 1995, 1996
[hereafter RPK96]). A similar relation corrects the V -band
light curve, with the same stretch factor in both bands. For
the supernovae discussed in this paper, the template must
be time-dilated by a factor 1 ] z before Ðtting to the
observed light curves to account for the cosmological
lengthening of the supernova timescale (Goldhaber et al.
1995; Leibundgut et al. 1996a; Riess et al. 1997a). P97 cal-
culated by translating from s to (both describ-*
corr
(s) *m
15
ing the timescale of the supernova event) and then using the
relation between and luminosity as determined by*m
15
Hamuy et al. (1995). The light curves of the Cala
n/Tololo
supernovae have since been published, and we have directly
Ðtted each light curve with the stretched template method
to determine its stretch factor s. In this paper, for the light-
curve width-luminosity relation, we therefore directly use
the functional form
*
corr
(s) \ a(s [ 1) (2)
and determine a simultaneously with our determination of
the cosmological parameters. With this functional form, the
supernova peak apparent magnitudes are thus all
““ corrected ÏÏ as they would appear if the supernovae had the
light-curve width of the template, s \ 1.
We use analysis procedures that are designed to be as
similar as possible for the low- and high-redshift data sets.
Occasionally, this requires not using all of the data avail-
able at low redshift, when the corresponding data are not
accessible at high redshift. For example, the low-redshift
supernova light curves can often be followed with photo-
metry for many months with high signal-to-noise ratios,
whereas the high-redshift supernova observations are gen-
erally only practical for approximately 60 rest-frame days
past maximum light. This period is also the phase of the
low-redshift SN Ia light curves that is Ðtted best by the
stretched-template method and that best predicts the lumi-
nosity of the supernova at maximum. We therefore Ðtted
only this period for the light curves of the low-redshift
supernovae. Similarly, at high redshift the rest-frame
B-band photometry is usually much more densely sampled
in time than the rest-frame V -band data, so we use the
stretch factor that best Ðts the rest-frame B-band data for
both low- and high-redshift supernovae, even though at
low-redshift the V -band photometry is equally well
sampled.
Each supernova peak magnitude was also corrected for
Galactic extinction, using the extinction law of Cardelli,A
R
,
Clayton, & Mathis (1989), Ðrst using the color excess,
at the supernovaÏs Galactic coordinates pro-E(B[V )
SFÔD
,
vided by Schlegel, Finkbeiner, & Davis (1998) and thenÈ
for comparisonÈusing the value provided byE(B[V )
BÔH
D. Burstein & C. Heiles (1998, private communication; see
also Burstein & Heiles 1982). Galactic extinction, wasA
R
,
calculated from E(B[V ) using a value of the total-to-selec-
tive extinction ratio, speciÐc to eachR
R
4 A
R
/E(B[V ),
supernova. These were calculated using the appropriate
redshifted supernova spectrum as it would appear through
an R-band Ðlter. These values of range from 2.56 atR
R
z \ 0 to 4.88 at z \ 0.83. The observed supernova colors
were similarly corrected for Galactic extinction. Any extinc-
tion in the supernovaÏs host galaxy or between galaxies was
not corrected for at this stage but will be analyzed separa-
tely in ° 4.
All the same corrections for width-luminosity relation,
K-corrections, and extinction (but using wereR
B
\ 4.14)
applied to the photometry of 18 low-redshift SNe Ia
(z ¹ 0.1) from the supernova survey (HamuyCala
n/Tololo
et al. 1996) that were discovered earlier than 5 days after
peak. The light curves of these 18 supernovae have all been
reÐtted since P97, using the more recently available photo-
metry (Hamuy et al. 1996) and our K-corrections.
Figures 1 and 2a show the Hubble diagram of e†ective
rest-frame B magnitude corrected for the width-luminosity
relation,
m
B
eff \ m
R
] *
corr
[ K
BR
[ A
R
, (3)
as a function of redshift for the 42 Supernova Cosmology
Project high-redshift supernovae, along with the 18
low-redshift supernovae. (Here is theCala
n/Tololo K
BR
cross-Ðlter K-correction from observed R band to rest-
frame B band.) Tables 1 and 2 give the corresponding IAU
names, redshifts, magnitudes, corrected magnitudes, and
their respective uncertainties. As in P97, the inner error bars
in Figures 1 and 2 represent the photometric uncertainty,
intrinsic dispersion of SN Ia magnitudes that remain after

568 PERLMUTTER ET AL. Vol. 517
FIG. 1.ÈHubble diagram for 42 high-redshift type Ia supernovae from the Supernova Cosmology Project and 18 low-redshift type Ia supernovae from the
Supernova Survey after correcting both sets for the SN Ia light-curve width-luminosity relation. The inner error bars show the uncertainty dueCala
n/Tololo
to measurement errors, while the outer error bars show the total uncertainty when the intrinsic luminosity dispersion, 0.17 mag, of light-curveÈwidth-
corrected type Ia supernovae is added in quadrature. The unÐlled circles indicate supernovae not included in Ðt C. The horizontal error bars represent the
assigned peculiar velocity uncertainty of 300 km s~1. The solid curves are the theoretical for a range of cosmological models with zero cosmologicalm
B
eff(z)
constant: on top, (1, 0) in middle, and (2, 0) on bottom. The dashed curves are for a range of Ñat cosmological models: on()
M
, )
"
) \ (0, 0) ()
M
, )
"
) \ (0, 1)
top, (0.5, 0.5) second from top, (1, 0) third from top, and (1.5, [0.5) on bottom.
applying the width-luminosity correction. For these plots,
the slope of the width-brightness relation was taken to be
a \ 0.6, the best-Ðt value of Ðt C discussed below. (Since
both the low- and high-redshift supernova light-curve
widths are clustered rather closely around s \ 1, as shown
in Fig. 4, the exact choice of a does not change the Hubble
diagram signiÐcantly.) The theoretical curves for a universe
with no cosmological constant are shown as solid lines for a
range of mass density, 1, 2. The dashed lines)
M
\ 0,
represent alternative Ñat cosmologies, for which the total
mass energy density (where)
M
] )
"
\ 1 )
"
4 "/3H
0
2).
The range of models shown are for (0.5,()
M
, )
"
) \ (0, 1),
0.5), (1, 0), which is covered by the matching solid line, and
(1.5, [0.5).
3. FITS TO )
M
AND )
"
The combined low- and high-redshift supernova data sets
of Figure 1 are Ðtted to the Friedman-Robertson-Walker
(FRW) magnitude-redshift relation, expressed as in P97:
m
B
eff 4 m
R
] a(s [ 1) [ K
BR
[ A
R
\ M
B
] 5 log D
L
(z; )
M
, )
"
) , (4)
where is the ““ Hubble-constantÈfree ÏÏ lumi-D
L
4 H
0
d
L
nosity distance and log is theM
B
4 M
B
[ 5 H
0
] 25
““ Hubble-constantÈfree ÏÏ B-band absolute magnitude at
maximum of a SN Ia with width s \ 1. (These quantities
are, respectively, calculated from theory or Ðtted from
apparent magnitudes and redshifts, both without any need
for The cosmological-parameter results are thus alsoH
0
.
completely independent of The details of the ÐttingH
0
.)
procedure as presented in P97 were followed, except that
both the low- and high-redshift supernovae were Ðtted
simultaneously, so that and a, the slope of the width-M
B
luminosity relation, could also be Ðtted in addition to the
cosmological parameters and For most of the)
M
)
"
.
analyses in this paper, and a are statistical ““ nuisance ÏÏM
B
parameters; we calculate two-dimensional conÐdence
regions and single-parameter uncertainties for the cosmo-
logical parameters by integrating over these parameters, i.e.,
da.P()
M
, )
"
) \ // P()
M
, )
"
, M
B
, a)dM
B
As in P97, the small correlations between the photo-
metric uncertainties of the high-redshift supernovae, due to
shared calibration data, have been accounted for by Ðtting
with a correlation matrix of uncertainties.11 The low-
redshift supernova photometry is more likely to be uncor-
related in its calibration, since these supernovae were not
discovered in batches. However, we take a 0.01 mag system-
atic uncertainty in the comparison of the low-redshift
B-band photometry and the high-redshift R-band photo-
metry. The stretch-factor uncertainty is propagated with a
Ðxed width-luminosity slope (taken from the low-redshift
11 The data are available at http://www-supernova.lbl.gov.

No. 2, 1999 ) AND " FROM 42 HIGH-REDSHIFT SUPERNOVAE 569
FIG. 2.È(a) Hubble diagram for 42 high-redshift type Ia supernovae from the Supernova Cosmology Project and 18 low-redshift type Ia supernovae from
the Supernova Survey, plotted on a linear redshift scale to display details at high redshift. The symbols and curves are as in Fig. 1.Cala
n/Tololo
(b) Magnitude residuals from the best-Ðt Ñat cosmology for the Ðt C supernova subset, 0.72). The dashed curves are for a range of Ñat()
M
, )
"
) \ (0.28,
cosmological models: on top, (0.5, 0.5) third from bottom, (0.75, 0.25) second from bottom, and (1, 0) is the solid curve on bottom. The()
M
, )
"
) \ (0, 1)
middle solid curve is for Note that this plot is practically identical to the magnitude residual plot for the best-Ðt unconstrained cosmology()
M
, )
"
) \ (0, 0).
of Ðt C, with (c) Uncertainty-normalized residuals from the best-Ðt Ñat cosmology for the Ðt C supernova subset,()
M
, )
"
) \ (0.73, 1.32). ()
M
, )
"
) \
(0.28, 0.72).
supernovae; cf. P97) and checked for consistency after the
Ðt.
We have compared the results of Bayesian and classical,
““ frequentist,ÏÏ Ðtting procedures. For the Bayesian Ðts, we
have assumed a ““ prior ÏÏ probability distribution that has
zero probability for but otherwise has uniform)
M
\ 0
probability in the four parameters a, and For)
M
, )
"
, M
B
.
the frequentist Ðts, we have followed the classical statistical
procedures described by Feldman & Cousins (1998) to
guarantee frequentist coverage of our conÐdence regions in
the physically allowed part of parameter space. Note that
throughout the previous cosmology literature, completely

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Journal ArticleDOI
Eiichiro Komatsu1, Kristine M. Smith2, Jo Dunkley3, Charles L. Bennett4  +17 moreInstitutions (10)
Abstract: The combination of seven-year data from WMAP and improved astrophysical data rigorously tests the standard cosmological model and places new constraints on its basic parameters and extensions. By combining the WMAP data with the latest distance measurements from the baryon acoustic oscillations (BAO) in the distribution of galaxies and the Hubble constant (H0) measurement, we determine the parameters of the simplest six-parameter ΛCDM model. The power-law index of the primordial power spectrum is ns = 0.968 ± 0.012 (68% CL) for this data combination, a measurement that excludes the Harrison–Zel’dovich–Peebles spectrum by 99.5% CL. The other parameters, including those beyond the minimal set, are also consistent with, and improved from, the five-year results. We find no convincing deviations from the minimal model. The seven-year temperature power spectrum gives a better determination of the third acoustic peak, which results in a better determination of the redshift of the matter-radiation equality epoch. Notable examples of improved parameters are the total mass of neutrinos, � mν < 0.58 eV (95% CL), and the effective number of neutrino species, Neff = 4.34 +0.86 −0.88 (68% CL), which benefit from better determinations of the third peak and H0. The limit on a constant dark energy equation of state parameter from WMAP+BAO+H0, without high-redshift Type Ia supernovae, is w =− 1.10 ± 0.14 (68% CL). We detect the effect of primordial helium on the temperature power spectrum and provide a new test of big bang nucleosynthesis by measuring Yp = 0.326 ± 0.075 (68% CL). We detect, and show on the map for the first time, the tangential and radial polarization patterns around hot and cold spots of temperature fluctuations, an important test of physical processes at z = 1090 and the dominance of adiabatic scalar fluctuations. The seven-year polarization data have significantly improved: we now detect the temperature–E-mode polarization cross power spectrum at 21σ , compared with 13σ from the five-year data. With the seven-year temperature–B-mode cross power spectrum, the limit on a rotation of the polarization plane due to potential parity-violating effects has improved by 38% to Δα =− 1. 1 ± 1. 4(statistical) ± 1. 5(systematic) (68% CL). We report significant detections of the Sunyaev–Zel’dovich (SZ) effect at the locations of known clusters of galaxies. The measured SZ signal agrees well with the expected signal from the X-ray data on a cluster-by-cluster basis. However, it is a factor of 0.5–0.7 times the predictions from “universal profile” of Arnaud et al., analytical models, and hydrodynamical simulations. We find, for the first time in the SZ effect, a significant difference between the cooling-flow and non-cooling-flow clusters (or relaxed and non-relaxed clusters), which can explain some of the discrepancy. This lower amplitude is consistent with the lower-than-theoretically expected SZ power spectrum recently measured by the South Pole Telescope Collaboration.

10,928 citations

Journal ArticleDOI
Peter A. R. Ade1, Nabila Aghanim2, Monique Arnaud3, M. Ashdown4  +334 moreInstitutions (82)
Abstract: This paper presents cosmological results based on full-mission Planck observations of temperature and polarization anisotropies of the cosmic microwave background (CMB) radiation. Our results are in very good agreement with the 2013 analysis of the Planck nominal-mission temperature data, but with increased precision. The temperature and polarization power spectra are consistent with the standard spatially-flat 6-parameter ΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations (denoted “base ΛCDM” in this paper). From the Planck temperature data combined with Planck lensing, for this cosmology we find a Hubble constant, H0 = (67.8 ± 0.9) km s-1Mpc-1, a matter density parameter Ωm = 0.308 ± 0.012, and a tilted scalar spectral index with ns = 0.968 ± 0.006, consistent with the 2013 analysis. Note that in this abstract we quote 68% confidence limits on measured parameters and 95% upper limits on other parameters. We present the first results of polarization measurements with the Low Frequency Instrument at large angular scales. Combined with the Planck temperature and lensing data, these measurements give a reionization optical depth of τ = 0.066 ± 0.016, corresponding to a reionization redshift of . These results are consistent with those from WMAP polarization measurements cleaned for dust emission using 353-GHz polarization maps from the High Frequency Instrument. We find no evidence for any departure from base ΛCDM in the neutrino sector of the theory; for example, combining Planck observations with other astrophysical data we find Neff = 3.15 ± 0.23 for the effective number of relativistic degrees of freedom, consistent with the value Neff = 3.046 of the Standard Model of particle physics. The sum of neutrino masses is constrained to ∑ mν < 0.23 eV. The spatial curvature of our Universe is found to be very close to zero, with | ΩK | < 0.005. Adding a tensor component as a single-parameter extension to base ΛCDM we find an upper limit on the tensor-to-scalar ratio of r0.002< 0.11, consistent with the Planck 2013 results and consistent with the B-mode polarization constraints from a joint analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP B-mode data to our analysis leads to a tighter constraint of r0.002 < 0.09 and disfavours inflationarymodels with a V(φ) ∝ φ2 potential. The addition of Planck polarization data leads to strong constraints on deviations from a purely adiabatic spectrum of fluctuations. We find no evidence for any contribution from isocurvature perturbations or from cosmic defects. Combining Planck data with other astrophysical data, including Type Ia supernovae, the equation of state of dark energy is constrained to w = −1.006 ± 0.045, consistent with the expected value for a cosmological constant. The standard big bang nucleosynthesis predictions for the helium and deuterium abundances for the best-fit Planck base ΛCDM cosmology are in excellent agreement with observations. We also constraints on annihilating dark matter and on possible deviations from the standard recombination history. In neither case do we find no evidence for new physics. The Planck results for base ΛCDM are in good agreement with baryon acoustic oscillation data and with the JLA sample of Type Ia supernovae. However, as in the 2013 analysis, the amplitude of the fluctuation spectrum is found to be higher than inferred from some analyses of rich cluster counts and weak gravitational lensing. We show that these tensions cannot easily be resolved with simple modifications of the base ΛCDM cosmology. Apart from these tensions, the base ΛCDM cosmology provides an excellent description of the Planck CMB observations and many other astrophysical data sets.

10,334 citations

Journal ArticleDOI
David N. Spergel1, Licia Verde1, Hiranya V. Peiris1, Eiichiro Komatsu1  +16 moreInstitutions (7)
Abstract: WMAP precision data enable accurate testing of cosmological models. We find that the emerging standard model of cosmology, a flat � -dominated universe seeded by a nearly scale-invariant adiabatic Gaussian fluctuations, fits the WMAP data. For the WMAP data only, the best-fit parameters are h ¼ 0:72 � 0:05, � bh 2 ¼ 0:024 � 0:001, � mh 2 ¼ 0:14 � 0:02, � ¼ 0:166 þ0:076 � 0:071 , ns ¼ 0:99 � 0:04, and � 8 ¼ 0:9 � 0:1. With parameters fixed only by WMAP data, we can fit finer scale cosmic microwave background (CMB) measure- ments and measurements of large-scale structure (galaxy surveys and the Lyforest). This simple model is also consistent with a host of other astronomical measurements: its inferred age of the universe is consistent with stellar ages, the baryon/photon ratio is consistent with measurements of the (D/H) ratio, and the inferred Hubble constant is consistent with local observations of the expansion rate. We then fit the model parameters to a combination of WMAP data with other finer scale CMB experiments (ACBAR and CBI), 2dFGRS measurements, and Lyforest data to find the model's best-fit cosmological parameters: h ¼ 0:71 þ0:04 � 0:03 , � bh 2 ¼ 0:0224 � 0:0009, � mh 2 ¼ 0:135 þ0:008 � 0:009 , � ¼ 0:17 � 0:06, ns(0.05 Mpc � 1 )=0 :93 � 0:03, and � 8 ¼ 0:84 � 0:04. WMAP's best determination of � ¼ 0:17 � 0:04 arises directly from the temperature- polarization (TE) data and not from this model fit, but they are consistent. These parameters imply that the age of the universe is 13:7 � 0:2 Gyr. With the Lyforest data, the model favors but does not require a slowly varying spectral index. The significance of this running index is sensitive to the uncertainties in the Ly� forest. By combining WMAP data with other astronomical data, we constrain the geometry of the universe, � tot ¼ 1:02 � 0:02, and the equation of state of the dark energy, w < � 0:78 (95% confidence limit assuming w �� 1). The combination of WMAP and 2dFGRS data constrains the energy density in stable neutrinos: � � h 2 < 0:0072 (95% confidence limit). For three degenerate neutrino species, this limit implies that their mass is less than 0.23 eV (95% confidence limit). The WMAP detection of early reionization rules out warm dark matter. Subject headings: cosmic microwave background — cosmological parameters — cosmology: observations — early universe On-line material: color figure

10,236 citations

Journal ArticleDOI
Peter A. R. Ade1, Nabila Aghanim2, C. Armitage-Caplan3, Monique Arnaud4  +324 moreInstitutions (70)
Abstract: This paper presents the first cosmological results based on Planck measurements of the cosmic microwave background (CMB) temperature and lensing-potential power spectra. We find that the Planck spectra at high multipoles (l ≳ 40) are extremely well described by the standard spatially-flat six-parameter ΛCDM cosmology with a power-law spectrum of adiabatic scalar perturbations. Within the context of this cosmology, the Planck data determine the cosmological parameters to high precision: the angular size of the sound horizon at recombination, the physical densities of baryons and cold dark matter, and the scalar spectral index are estimated to be θ∗ = (1.04147 ± 0.00062) × 10-2, Ωbh2 = 0.02205 ± 0.00028, Ωch2 = 0.1199 ± 0.0027, and ns = 0.9603 ± 0.0073, respectively(note that in this abstract we quote 68% errors on measured parameters and 95% upper limits on other parameters). For this cosmology, we find a low value of the Hubble constant, H0 = (67.3 ± 1.2) km s-1 Mpc-1, and a high value of the matter density parameter, Ωm = 0.315 ± 0.017. These values are in tension with recent direct measurements of H0 and the magnitude-redshift relation for Type Ia supernovae, but are in excellent agreement with geometrical constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent level precision using Planck CMB data alone. We use high-resolution CMB data together with Planck to provide greater control on extragalactic foreground components in an investigation of extensions to the six-parameter ΛCDM model. We present selected results from a large grid of cosmological models, using a range of additional astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured over the standard six-parameter ΛCDM cosmology. The deviation of the scalar spectral index from unity isinsensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find an upper limit of r0.002< 0.11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles beyond the three families of neutrinos in the standard model. Using BAO and CMB data, we find Neff = 3.30 ± 0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the sum of neutrino masses. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of Neff = 3.046. We find no evidence for dynamical dark energy; using BAO and CMB data, the dark energy equation of state parameter is constrained to be w = -1.13-0.10+0.13. We also use the Planck data to set limits on a possible variation of the fine-structure constant, dark matter annihilation and primordial magnetic fields. Despite the success of the six-parameter ΛCDM model in describing the Planck data at high multipoles, we note that this cosmology does not provide a good fit to the temperature power spectrum at low multipoles. The unusual shape of the spectrum in the multipole range 20 ≲ l ≲ 40 was seen previously in the WMAP data and is a real feature of the primordial CMB anisotropies. The poor fit to the spectrum at low multipoles is not of decisive significance, but is an “anomaly” in an otherwise self-consistent analysis of the Planck temperature data.

6,641 citations

Journal ArticleDOI
David N. Spergel1, Rachel Bean2, Rachel Bean1, Olivier Doré3  +24 moreInstitutions (10)
Abstract: A simple cosmological model with only six parameters (matter density, Omega_m h^2, baryon density, Omega_b h^2, Hubble Constant, H_0, amplitude of fluctuations, sigma_8, optical depth, tau, and a slope for the scalar perturbation spectrum, n_s) fits not only the three year WMAP temperature and polarization data, but also small scale CMB data, light element abundances, large-scale structure observations, and the supernova luminosity/distance relationship. Using WMAP data only, the best fit values for cosmological parameters for the power-law flat LCDM model are (Omega_m h^2, Omega_b h^2, h, n_s, tau, sigma_8) = 0.1277+0.0080-0.0079, 0.02229+-0.00073, 0.732+0.031-0.032, 0.958+-0.016, 0.089+-0.030, 0.761+0.049-0.048). The three year data dramatically shrink the allowed volume in this six dimensional parameter space. Assuming that the primordial fluctuations are adiabatic with a power law spectrum, the WMAP data_alone_ require dark matter, and favor a spectral index that is significantly less than the Harrison-Zel'dovich-Peebles scale-invariant spectrum (n_s=1, r=0). Models that suppress large-scale power through a running spectral index or a large-scale cut-off in the power spectrum are a better fit to the WMAP and small scale CMB data than the power-law LCDM model: however, the improvement in the fit to the WMAP data is only Delta chi^2 = 3 for 1 extra degree of freedom. The combination of WMAP and other astronomical data yields significant constraints on the geometry of the universe, the equation of state of the dark energy, the gravitational wave energy density, and neutrino properties. Consistent with the predictions of simple inflationary theories, we detect no significant deviations from Gaussianity in the CMB maps.

5,799 citations

##### References
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Journal ArticleDOI
Abstract: We present spectral and photometric observations of 10 Type Ia supernovae (SNe Ia) in the redshift range 0.16 " z " 0.62. The luminosity distances of these objects are determined by methods that employ relations between SN Ia luminosity and light curve shape. Combined with previous data from our High-z Supernova Search Team and recent results by Riess et al., this expanded set of 16 high-redshift supernovae and a set of 34 nearby supernovae are used to place constraints on the following cosmo- logical parameters: the Hubble constant the mass density the cosmological constant (i.e., the (H 0 ), () M ), vacuum energy density, the deceleration parameter and the dynamical age of the universe ) " ), (q 0 ), ) M \ 1) methods. We estimate the dynamical age of the universe to be 14.2 ^ 1.7 Gyr including systematic uncer- tainties in the current Cepheid distance scale. We estimate the likely e†ect of several sources of system- atic error, including progenitor and metallicity evolution, extinction, sample selection bias, local perturbations in the expansion rate, gravitational lensing, and sample contamination. Presently, none of these e†ects appear to reconcile the data with and ) " \ 0 q 0 " 0.

15,427 citations

Journal ArticleDOI
Abstract: We present a full-sky 100 μm map that is a reprocessed composite of the COBE/DIRBE and IRAS/ISSA maps, with the zodiacal foreground and confirmed point sources removed. Before using the ISSA maps, we remove the remaining artifacts from the IRAS scan pattern. Using the DIRBE 100 and 240 μm data, we have constructed a map of the dust temperature so that the 100 μm map may be converted to a map proportional to dust column density. The dust temperature varies from 17 to 21 K, which is modest but does modify the estimate of the dust column by a factor of 5. The result of these manipulations is a map with DIRBE quality calibration and IRAS resolution. A wealth of filamentary detail is apparent on many different scales at all Galactic latitudes. In high-latitude regions, the dust map correlates well with maps of H I emission, but deviations are coherent in the sky and are especially conspicuous in regions of saturation of H I emission toward denser clouds and of formation of H2 in molecular clouds. In contrast, high-velocity H I clouds are deficient in dust emission, as expected. To generate the full-sky dust maps, we must first remove zodiacal light contamination, as well as a possible cosmic infrared background (CIB). This is done via a regression analysis of the 100 μm DIRBE map against the Leiden-Dwingeloo map of H I emission, with corrections for the zodiacal light via a suitable expansion of the DIRBE 25 μm flux. This procedure removes virtually all traces of the zodiacal foreground. For the 100 μm map no significant CIB is detected. At longer wavelengths, where the zodiacal contamination is weaker, we detect the CIB at surprisingly high flux levels of 32 ± 13 nW m-2 sr-1 at 140 μm and of 17 ± 4 nW m-2 sr-1 at 240 μm (95% confidence). This integrated flux ~2 times that extrapolated from optical galaxies in the Hubble Deep Field. The primary use of these maps is likely to be as a new estimator of Galactic extinction. To calibrate our maps, we assume a standard reddening law and use the colors of elliptical galaxies to measure the reddening per unit flux density of 100 μm emission. We find consistent calibration using the B-R color distribution of a sample of the 106 brightest cluster ellipticals, as well as a sample of 384 ellipticals with B-V and Mg line strength measurements. For the latter sample, we use the correlation of intrinsic B-V versus Mg2 index to tighten the power of the test greatly. We demonstrate that the new maps are twice as accurate as the older Burstein-Heiles reddening estimates in regions of low and moderate reddening. The maps are expected to be significantly more accurate in regions of high reddening. These dust maps will also be useful for estimating millimeter emission that contaminates cosmic microwave background radiation experiments and for estimating soft X-ray absorption. We describe how to access our maps readily for general use.

15,382 citations

Journal ArticleDOI
Abstract: We present a full sky 100 micron map that is a reprocessed composite of the COBE/DIRBE and IRAS/ISSA maps, with the zodiacal foreground and confirmed point sources removed. Before using the ISSA maps, we remove the remaining artifacts from the IRAS scan pattern. Using the DIRBE 100 micron and 240 micron data, we have constructed a map of the dust temperature, so that the 100 micron map can be converted to a map proportional to dust column density. The result of these manipulations is a map with DIRBE-quality calibration and IRAS resolution. To generate the full sky dust maps, we must first remove zodiacal light contamination as well as a possible cosmic infrared background (CIB). This is done via a regression analysis of the 100 micron DIRBE map against the Leiden- Dwingeloo map of H_I emission, with corrections for the zodiacal light via a suitable expansion of the DIRBE 25 micron flux. For the 100 micron map, no significant CIB is detected. In the 140 micron and 240 micron maps, where the zodiacal contamination is weaker, we detect the CIB at surprisingly high flux levels of 32 \pm 13 nW/m^2/sr at 140 micron, and 17 \pm 4 nW/m^2/sr at 240 micron (95% confidence). This integrated flux is ~2 times that extrapolated from optical galaxies in the Hubble Deep Field. The primary use of these maps is likely to be as a new estimator of Galactic extinction. We demonstrate that the new maps are twice as accurate as the older Burstein-Heiles estimates in regions of low and moderate reddening. These dust maps will also be useful for estimating millimeter emission that contaminates CMBR experiments and for estimating soft X-ray absorption.

13,819 citations

Journal ArticleDOI
Abstract: We present observations of 10 type Ia supernovae (SNe Ia) between 0.16 0) and a current acceleration of the expansion (i.e., q_0 0, the spectroscopically confirmed SNe Ia are consistent with q_0 0 at the 3.0 sigma and 4.0 sigma confidence levels, for two fitting methods respectively. Fixing a minimal'' mass density, Omega_M=0.2, results in the weakest detection, Omega_Lambda>0 at the 3.0 sigma confidence level. For a flat-Universe prior (Omega_M+Omega_Lambda=1), the spectroscopically confirmed SNe Ia require Omega_Lambda >0 at 7 sigma and 9 sigma level for the two fitting methods. A Universe closed by ordinary matter (i.e., Omega_M=1) is ruled out at the 7 sigma to 8 sigma level. We estimate the size of systematic errors, including evolution, extinction, sample selection bias, local flows, gravitational lensing, and sample contamination. Presently, none of these effects reconciles the data with Omega_Lambda=0 and q_0 > 0.

13,423 citations

Journal ArticleDOI
Abstract: The parameterized extinction data of Fitzpatrick and Massa (1986, 1988) for the ultraviolet and various sources for the optical and near-infrared are used to derive a meaningful average extinction law over the 3.5 micron to 0.125 wavelength range which is applicable to both diffuse and dense regions of the interstellar medium. The law depends on only one parameter R(V) = A(V)/E(B-V). An analytic formula is given for the mean extinction law which can be used to calculate color excesses or to deredden observations. The validity of the law over a large wavelength interval suggests that the processes which modify the sizes and compositions of grains are stochastic in nature and very efficient.

10,948 citations

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