skip to main content
DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

This content will become publicly available on April 29, 2020

Title: Steve: A Hierarchical Bayesian Model for Supernova Cosmology

Abstract

Here, we present a new Bayesian hierarchical model (BHM) named Steve for performing Type Ia supernova (SN Ia) cosmology fits. This advances previous works by including an improved treatment of Malmquist bias, accounting for additional sources of systematic uncertainty, and increasing numerical efficiency. Given light-curve fit parameters, redshifts, and host-galaxy masses, we fit Steve simultaneously for parameters describing cosmology, SN Ia populations, and systematic uncertainties. Selection effects are characterized using Monte Carlo simulations. We demonstrate its implementation by fitting realizations of SN Ia data sets where the SN Ia model closely follows that used in Steve. Next, we validate on more realistic SNANA simulations of SN Ia samples from the Dark Energy Survey and low-redshift surveys (DES Collaboration et al. 2018). These simulated data sets contain more than 60,000 SNe Ia, which we use to evaluate biases in the recovery of cosmological parameters, specifically the equation of state of dark energy, w. This is the most rigorous test of a BHM method applied to SN Ia cosmology fitting and reveals small w biases that depend on the simulated SN Ia properties, in particular the intrinsic SN Ia scatter model. This w bias is less than 0.03 on average, less thanmore » half the statistical uncertainty on w. These simulation test results are a concern for BHM cosmology fitting applications on large upcoming surveys; therefore, future development will focus on minimizing the sensitivity of Steve to the SN Ia intrinsic scatter model.« less

Authors:
ORCiD logo; ; ; ORCiD logo; ORCiD logo; ORCiD logo; ; ORCiD logo; ; ; ; ; ; ; ; ORCiD logo; ; ORCiD logo; ; more »; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ORCiD logo; ORCiD logo; ; ; ; ORCiD logo; ; ; ; ; ; ; ; ; ; ; ORCiD logo; ; ; ; ; ; ; ; ; ORCiD logo; ; ; ORCiD logo; ; ; « less
Publication Date:
Research Org.:
SLAC National Accelerator Lab., Menlo Park, CA (United States); Fermi National Accelerator Lab. (FNAL), Batavia, IL (United States); Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
Sponsoring Org.:
USDOE Office of Science (SC), High Energy Physics (HEP) (SC-25); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)
Contributing Org.:
DES Collaboration; DES
OSTI Identifier:
1526927
Alternate Identifier(s):
OSTI ID: 1529360; OSTI ID: 1531222; OSTI ID: 1582334
Report Number(s):
arXiv:1811.02381; FERMILAB-PUB-19-050-A-AD-AE-CD
Journal ID: ISSN 1538-4357
Grant/Contract Number:  
AC02-76SF00515; AC02-07CH11359; AC05-00OR22725; AC02-05CH11231
Resource Type:
Accepted Manuscript
Journal Name:
The Astrophysical Journal (Online)
Additional Journal Information:
Journal Name: The Astrophysical Journal (Online); Journal Volume: 876; Journal Issue: 1; Journal ID: ISSN 1538-4357
Publisher:
Institute of Physics (IOP)
Country of Publication:
United States
Language:
English
Subject:
79 ASTRONOMY AND ASTROPHYSICS; cosmology: supernovae

Citation Formats

Hinton, S. R., Davis, T. M., Kim, A. G., Brout, D., D’Andrea, C. B., Kessler, R., Lasker, J., Lidman, C., Macaulay, E., Möller, A., Sako, M., Scolnic, D., Smith, M., Wolf, R. C., Childress, M., Morganson, E., Allam, S., Annis, J., Avila, S., Bertin, E., Brooks, D., Burke, D. L., Rosell, A. Carnero, Kind, M. Carrasco, Carretero, J., Cunha, C. E., Costa, L. N. da, Davis, C., Vicente, J. De, DePoy, D. L., Doel, P., Eifler, T. F., Flaugher, B., Fosalba, P., Frieman, J., García-Bellido, J., Gaztanaga, E., Gerdes, D. W., Gruendl, R. A., Gschwend, J., Gutierrez, G., Hartley, W. G., Hollowood, D. L., Honscheid, K., Krause, E., Kuehn, K., Kuropatkin, N., Lahav, O., Lima, M., Maia, M. A. G., March, M., Marshall, J. L., Menanteau, F., Miquel, R., Ogando, R. L. C., Plazas, A. A., Sanchez, E., Scarpine, V., Schindler, R., Schubnell, M., Serrano, S., Sevilla-Noarbe, I., Soares-Santos, M., Sobreira, F., Suchyta, E., Tarle, G., Thomas, D., Vikram, V., and Zhang, Y. Steve: A Hierarchical Bayesian Model for Supernova Cosmology. United States: N. p., 2019. Web. doi:10.3847/1538-4357/ab13a3.
Hinton, S. R., Davis, T. M., Kim, A. G., Brout, D., D’Andrea, C. B., Kessler, R., Lasker, J., Lidman, C., Macaulay, E., Möller, A., Sako, M., Scolnic, D., Smith, M., Wolf, R. C., Childress, M., Morganson, E., Allam, S., Annis, J., Avila, S., Bertin, E., Brooks, D., Burke, D. L., Rosell, A. Carnero, Kind, M. Carrasco, Carretero, J., Cunha, C. E., Costa, L. N. da, Davis, C., Vicente, J. De, DePoy, D. L., Doel, P., Eifler, T. F., Flaugher, B., Fosalba, P., Frieman, J., García-Bellido, J., Gaztanaga, E., Gerdes, D. W., Gruendl, R. A., Gschwend, J., Gutierrez, G., Hartley, W. G., Hollowood, D. L., Honscheid, K., Krause, E., Kuehn, K., Kuropatkin, N., Lahav, O., Lima, M., Maia, M. A. G., March, M., Marshall, J. L., Menanteau, F., Miquel, R., Ogando, R. L. C., Plazas, A. A., Sanchez, E., Scarpine, V., Schindler, R., Schubnell, M., Serrano, S., Sevilla-Noarbe, I., Soares-Santos, M., Sobreira, F., Suchyta, E., Tarle, G., Thomas, D., Vikram, V., & Zhang, Y. Steve: A Hierarchical Bayesian Model for Supernova Cosmology. United States. doi:10.3847/1538-4357/ab13a3.
Hinton, S. R., Davis, T. M., Kim, A. G., Brout, D., D’Andrea, C. B., Kessler, R., Lasker, J., Lidman, C., Macaulay, E., Möller, A., Sako, M., Scolnic, D., Smith, M., Wolf, R. C., Childress, M., Morganson, E., Allam, S., Annis, J., Avila, S., Bertin, E., Brooks, D., Burke, D. L., Rosell, A. Carnero, Kind, M. Carrasco, Carretero, J., Cunha, C. E., Costa, L. N. da, Davis, C., Vicente, J. De, DePoy, D. L., Doel, P., Eifler, T. F., Flaugher, B., Fosalba, P., Frieman, J., García-Bellido, J., Gaztanaga, E., Gerdes, D. W., Gruendl, R. A., Gschwend, J., Gutierrez, G., Hartley, W. G., Hollowood, D. L., Honscheid, K., Krause, E., Kuehn, K., Kuropatkin, N., Lahav, O., Lima, M., Maia, M. A. G., March, M., Marshall, J. L., Menanteau, F., Miquel, R., Ogando, R. L. C., Plazas, A. A., Sanchez, E., Scarpine, V., Schindler, R., Schubnell, M., Serrano, S., Sevilla-Noarbe, I., Soares-Santos, M., Sobreira, F., Suchyta, E., Tarle, G., Thomas, D., Vikram, V., and Zhang, Y. Mon . "Steve: A Hierarchical Bayesian Model for Supernova Cosmology". United States. doi:10.3847/1538-4357/ab13a3.
@article{osti_1526927,
title = {Steve: A Hierarchical Bayesian Model for Supernova Cosmology},
author = {Hinton, S. R. and Davis, T. M. and Kim, A. G. and Brout, D. and D’Andrea, C. B. and Kessler, R. and Lasker, J. and Lidman, C. and Macaulay, E. and Möller, A. and Sako, M. and Scolnic, D. and Smith, M. and Wolf, R. C. and Childress, M. and Morganson, E. and Allam, S. and Annis, J. and Avila, S. and Bertin, E. and Brooks, D. and Burke, D. L. and Rosell, A. Carnero and Kind, M. Carrasco and Carretero, J. and Cunha, C. E. and Costa, L. N. da and Davis, C. and Vicente, J. De and DePoy, D. L. and Doel, P. and Eifler, T. F. and Flaugher, B. and Fosalba, P. and Frieman, J. and García-Bellido, J. and Gaztanaga, E. and Gerdes, D. W. and Gruendl, R. A. and Gschwend, J. and Gutierrez, G. and Hartley, W. G. and Hollowood, D. L. and Honscheid, K. and Krause, E. and Kuehn, K. and Kuropatkin, N. and Lahav, O. and Lima, M. and Maia, M. A. G. and March, M. and Marshall, J. L. and Menanteau, F. and Miquel, R. and Ogando, R. L. C. and Plazas, A. A. and Sanchez, E. and Scarpine, V. and Schindler, R. and Schubnell, M. and Serrano, S. and Sevilla-Noarbe, I. and Soares-Santos, M. and Sobreira, F. and Suchyta, E. and Tarle, G. and Thomas, D. and Vikram, V. and Zhang, Y.},
abstractNote = {Here, we present a new Bayesian hierarchical model (BHM) named Steve for performing Type Ia supernova (SN Ia) cosmology fits. This advances previous works by including an improved treatment of Malmquist bias, accounting for additional sources of systematic uncertainty, and increasing numerical efficiency. Given light-curve fit parameters, redshifts, and host-galaxy masses, we fit Steve simultaneously for parameters describing cosmology, SN Ia populations, and systematic uncertainties. Selection effects are characterized using Monte Carlo simulations. We demonstrate its implementation by fitting realizations of SN Ia data sets where the SN Ia model closely follows that used in Steve. Next, we validate on more realistic SNANA simulations of SN Ia samples from the Dark Energy Survey and low-redshift surveys (DES Collaboration et al. 2018). These simulated data sets contain more than 60,000 SNe Ia, which we use to evaluate biases in the recovery of cosmological parameters, specifically the equation of state of dark energy, w. This is the most rigorous test of a BHM method applied to SN Ia cosmology fitting and reveals small w biases that depend on the simulated SN Ia properties, in particular the intrinsic SN Ia scatter model. This w bias is less than 0.03 on average, less than half the statistical uncertainty on w. These simulation test results are a concern for BHM cosmology fitting applications on large upcoming surveys; therefore, future development will focus on minimizing the sensitivity of Steve to the SN Ia intrinsic scatter model.},
doi = {10.3847/1538-4357/ab13a3},
journal = {The Astrophysical Journal (Online)},
number = 1,
volume = 876,
place = {United States},
year = {2019},
month = {4}
}

Journal Article:
Free Publicly Available Full Text
This content will become publicly available on April 29, 2020
Publisher's Version of Record

Save / Share:

Works referenced in this record:

First Cosmology Results using Type Ia Supernovae from the Dark Energy Survey: Constraints on Cosmological Parameters
journal, February 2019


SPECTRA AND HUBBLE SPACE TELESCOPE LIGHT CURVES OF SIX TYPE Ia SUPERNOVAE AT 0.511 < z < 1.12 AND THE UNION2 COMPILATION
journal, May 2010


The Supernova Legacy Survey: measurement of $\Omega_{\mathsf{M}}$, $\Omega_\mathsf{\Lambda}$ and w from the first year data set
journal, January 2006


The ESO/VLT 3rd year Type Ia supernova data set from the supernova legacy survey
journal, August 2009


Supernova Simulations and Strategies for the dark Energy Survey
journal, June 2012


Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples
journal, August 2014


First Cosmology Results Using SNe Ia from the Dark Energy Survey: Analysis, Systematic Uncertainties, and Validation
journal, April 2019


Stan : A Probabilistic Programming Language
journal, January 2017

  • Carpenter, Bob; Gelman, Andrew; Hoffman, Matthew D.
  • Journal of Statistical Software, Vol. 76, Issue 1
  • DOI: 10.18637/jss.v076.i01

The reddening law of type Ia supernovae: separating intrinsic variability from dust using equivalent widths
journal, April 2011


Supernova Constraints and Systematic Uncertainties from the First Three Years of the Supernova Legacy Survey
journal, December 2010

  • Conley, A.; Guy, J.; Sullivan, M.
  • The Astrophysical Journal Supplement Series, Vol. 192, Issue 1
  • DOI: 10.1088/0067-0049/192/1/1

THE CARNEGIE SUPERNOVA PROJECT: FIRST PHOTOMETRY DATA RELEASE OF LOW-REDSHIFT TYPE Ia SUPERNOVAE
journal, January 2010


SPECTROSCOPIC PROPERTIES OF STAR-FORMING HOST GALAXIES AND TYPE Ia SUPERNOVA HUBBLE RESIDUALS IN A NEARLY UNBIASED SAMPLE
journal, December 2011


A Measurement of the Rate of Type Ia Supernovae at Redshift z ≈ 0.1 from the First Season of the SDSS‐II Supernova Survey
journal, July 2008

  • Dilday, Benjamin; Kessler, Richard; Frieman, Joshua A.
  • The Astrophysical Journal, Vol. 682, Issue 1
  • DOI: 10.1086/587733

THE CARNEGIE SUPERNOVA PROJECT: ANALYSIS OF THE FIRST SAMPLE OF LOW-REDSHIFT TYPE-Ia SUPERNOVAE
journal, December 2009

  • Folatelli, Gastón; Phillips, M. M.; Burns, Christopher R.
  • The Astronomical Journal, Vol. 139, Issue 1
  • DOI: 10.1088/0004-6256/139/1/120

THE CARNEGIE SUPERNOVA PROJECT: FIRST NEAR-INFRARED HUBBLE DIAGRAM TO z ∼ 0.7
journal, September 2009


The Sloan Digital sky Survey-Ii Supernova Survey: Technical Summary
journal, December 2007


TYPE-Ia SUPERNOVA RATES TO REDSHIFT 2.4 FROM CLASH: THE CLUSTER LENSING AND SUPERNOVA SURVEY WITH HUBBLE
journal, February 2014


IMPROVED CONSTRAINTS ON TYPE Ia SUPERNOVA HOST GALAXY PROPERTIES USING MULTI-WAVELENGTH PHOTOMETRY AND THEIR CORRELATIONS WITH SUPERNOVA PROPERTIES
journal, October 2011


Host Galaxy Identification for Supernova Surveys
journal, November 2016


SALT2: using distant supernovae to improve the use of type Ia supernovae as distance indicators
journal, February 2007


The Supernova Legacy Survey 3-year sample: Type Ia supernovae photometric distances and cosmological constraints
journal, November 2010


CfA3: 185 TYPE Ia SUPERNOVA LIGHT CURVES FROM THE CfA
journal, July 2009


CfA4: LIGHT CURVES FOR 94 TYPE Ia SUPERNOVAE
journal, May 2012

  • Hicken, Malcolm; Challis, Peter; Kirshner, Robert P.
  • The Astrophysical Journal Supplement Series, Vol. 200, Issue 2
  • DOI: 10.1088/0067-0049/200/2/12

IMPROVED DARK ENERGY CONSTRAINTS FROM ∼100 NEW CfA SUPERNOVA TYPE Ia LIGHT CURVES
journal, July 2009


ChainConsumer
journal, August 2016


Photometric Supernova Cosmology with Beams and Sdss-Ii
journal, May 2012


Dark energy two decades after: observables, probes, consistency tests
journal, December 2017


LSST: From Science Drivers to Reference Design and Anticipated Data Products
journal, March 2019

  • Ivezić, Željko; Kahn, Steven M.; Tyson, J. Anthony
  • The Astrophysical Journal, Vol. 873, Issue 2
  • DOI: 10.3847/1538-4357/ab042c

SN Ia host galaxy properties from Sloan Digital Sky Survey-II spectroscopy
journal, September 2013

  • Johansson, Jonas; Thomas, Daniel; Pforr, Janine
  • Monthly Notices of the Royal Astronomical Society, Vol. 435, Issue 2
  • DOI: 10.1093/mnras/stt1408

HUBBLE RESIDUALS OF NEARBY TYPE Ia SUPERNOVAE ARE CORRELATED WITH HOST GALAXY MASSES
journal, May 2010


First-Year Sloan Digital sky Survey-Ii Supernova Results: Hubble Diagram and Cosmological Parameters
journal, October 2009

  • Kessler, Richard; Becker, Andrew C.; Cinabro, David
  • The Astrophysical Journal Supplement Series, Vol. 185, Issue 1
  • DOI: 10.1088/0067-0049/185/1/32

SNANA: A Public Software Package for Supernova Analysis
journal, September 2009

  • Kessler, Richard; Bernstein, Joseph P.; Cinabro, David
  • Publications of the Astronomical Society of the Pacific, Vol. 121, Issue 883
  • DOI: 10.1086/605984

First cosmology results using Type Ia supernova from the Dark Energy Survey: simulations to correct supernova distance biases
journal, February 2019

  • Kessler, R.; Brout, D.; D’Andrea, C. B.
  • Monthly Notices of the Royal Astronomical Society, Vol. 485, Issue 1
  • DOI: 10.1093/mnras/stz463

TESTING MODELS OF INTRINSIC BRIGHTNESS VARIATIONS IN TYPE Ia SUPERNOVAE AND THEIR IMPACT ON MEASURING COSMOLOGICAL PARAMETERS
journal, January 2013


Correcting Type Ia Supernova Distances for Selection Biases and Contamination in Photometrically Identified Samples
journal, February 2017


Improved Cosmological Constraints from New, Old, and Combined Supernova Data Sets
journal, October 2008

  • Kowalski, M.; Rubin, D.; Aldering, G.
  • The Astrophysical Journal, Vol. 686, Issue 2
  • DOI: 10.1086/589937

Bayesian estimation applied to multiple species
journal, May 2007


THE EFFECT OF HOST GALAXIES ON TYPE Ia SUPERNOVAE IN THE SDSS-II SUPERNOVA SURVEY
journal, September 2010


Cosmological parameters from CMB and other data: A Monte Carlo approach
journal, November 2002


Application of Bayesian graphs to SN Ia data analysis and compression
journal, August 2016

  • Ma, Cong; Corasaniti, Pier-Stefano; Bassett, Bruce A.
  • Monthly Notices of the Royal Astronomical Society, Vol. 463, Issue 2
  • DOI: 10.1093/mnras/stw2069

TYPE Ia SUPERNOVA LIGHT CURVE INFERENCE: HIERARCHICAL MODELS IN THE OPTICAL AND NEAR-INFRARED
journal, April 2011

  • Mandel, Kaisey S.; Narayan, Gautham; Kirshner, Robert P.
  • The Astrophysical Journal, Vol. 731, Issue 2
  • DOI: 10.1088/0004-637X/731/2/120

The Type Ia Supernova Color–Magnitude Relation and Host Galaxy Dust: A Simple Hierarchical Bayesian Model
journal, June 2017

  • Mandel, Kaisey S.; Scolnic, Daniel M.; Shariff, Hikmatali
  • The Astrophysical Journal, Vol. 842, Issue 2
  • DOI: 10.3847/1538-4357/aa6038

TYPE Ia SUPERNOVA LIGHT-CURVE INFERENCE: HIERARCHICAL BAYESIAN ANALYSIS IN THE NEAR-INFRARED
journal, September 2009

  • Mandel, Kaisey S.; Wood-Vasey, W. Michael; Friedman, Andrew S.
  • The Astrophysical Journal, Vol. 704, Issue 1
  • DOI: 10.1088/0004-637X/704/1/629

Comparison of cosmological parameter inference methods applied to supernovae light curves fitted with salt-ii
journal, December 2013

  • March, M. C.; Karpenka, N. V.; Feroz, F.
  • Monthly Notices of the Royal Astronomical Society, Vol. 437, Issue 4
  • DOI: 10.1093/mnras/stt2114

Improved constraints on cosmological parameters from Type Ia supernova data: Improved constraints from SNIa
journal, November 2011


Measurements of Ω and Λ from 42 High‐Redshift Supernovae
journal, June 1999

  • Perlmutter, S.; Aldering, G.; Goldhaber, G.
  • The Astrophysical Journal, Vol. 517, Issue 2
  • DOI: 10.1086/307221

Real-Time Analysis and Selection Biases in the Supernova Legacy Survey
journal, July 2010


EVOLUTION IN THE VOLUMETRIC TYPE Ia SUPERNOVA RATE FROM THE SUPERNOVA LEGACY SURVEY
journal, July 2012


The absolute magnitudes of Type IA supernovae
journal, August 1993

  • Phillips, M. M.
  • The Astrophysical Journal, Vol. 413
  • DOI: 10.1086/186970

The Reddening-Free Decline Rate Versus Luminosity Relationship for Type [CLC]Ia[/CLC] Supernovae
journal, October 1999

  • Phillips, M. M.; Lira, Paulina; Suntzeff, Nicholas B.
  • The Astronomical Journal, Vol. 118, Issue 4
  • DOI: 10.1086/301032

COSMOLOGICAL CONSTRAINTS FROM MEASUREMENTS OF TYPE Ia SUPERNOVAE DISCOVERED DURING THE FIRST 1.5 yr OF THE Pan-STARRS1 SURVEY
journal, October 2014


Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant
journal, September 1998

  • Riess, Adam G.; Filippenko, Alexei V.; Challis, Peter
  • The Astronomical Journal, Vol. 116, Issue 3
  • DOI: 10.1086/300499

Evidence of environmental dependencies of Type Ia supernovae from the Nearby Supernova Factory indicated by local H α
journal, December 2013


zBEAMS: a unified solution for supernova cosmology with redshift uncertainties
journal, October 2017

  • Roberts, Ethan; Lochner, Michelle; Fonseca, José
  • Journal of Cosmology and Astroparticle Physics, Vol. 2017, Issue 10
  • DOI: 10.1088/1475-7516/2017/10/036

TYPE Ia SUPERNOVA RATE MEASUREMENTS TO REDSHIFT 2.5 FROM CANDELS: SEARCHING FOR PROMPT EXPLOSIONS IN THE EARLY UNIVERSE
journal, June 2014

  • Rodney, Steven A.; Riess, Adam G.; Strolger, Louis-Gregory
  • The Astronomical Journal, Vol. 148, Issue 1
  • DOI: 10.1088/0004-6256/148/1/13

Unity: Confronting Supernova Cosmology’S Statistical and Systematic Uncertainties in a Unified Bayesian Framework
journal, November 2015


The Data Release of the Sloan Digital Sky Survey-II Supernova Survey
journal, May 2018

  • Sako, Masao; Bassett, Bruce; Becker, Andrew C.
  • Publications of the Astronomical Society of the Pacific, Vol. 130, Issue 988
  • DOI: 10.1088/1538-3873/aab4e0

Measuring type ia Supernova Populations of Stretch and Color and Predicting Distance Biases
journal, May 2016


BAHAMAS: NEW ANALYSIS OF TYPE Ia SUPERNOVAE REVEALS INCONSISTENCIES WITH STANDARD COSMOLOGY
journal, August 2016

  • Shariff, Hikmatali; Jiao, Xiyun; Trotta, Roberto
  • The Astrophysical Journal, Vol. 827, Issue 1
  • DOI: 10.3847/0004-637X/827/1/1

THE CARNEGIE SUPERNOVA PROJECT: SECOND PHOTOMETRY DATA RELEASE OF LOW-REDSHIFT TYPE Ia SUPERNOVAE
journal, October 2011

  • Stritzinger, Maximilian D.; Phillips, M. M.; Boldt, Luis N.
  • The Astronomical Journal, Vol. 142, Issue 5
  • DOI: 10.1088/0004-6256/142/5/156

The dependence of Type Ia Supernovae luminosities on their host galaxies: SN Ia host galaxies
journal, May 2010


The Influence of Host Galaxies in Type Ia Supernova Cosmology
journal, October 2017

  • Uddin, Syed A.; Mould, Jeremy; Lidman, Chris
  • The Astrophysical Journal, Vol. 848, Issue 1
  • DOI: 10.3847/1538-4357/aa8df7

LIKELIHOOD-FREE COSMOLOGICAL INFERENCE WITH TYPE Ia SUPERNOVAE: APPROXIMATE BAYESIAN COMPUTATION FOR A COMPLETE TREATMENT OF UNCERTAINTY
journal, January 2013


Observational Constraints on the Nature of Dark Energy: First Cosmological Results from the ESSENCE Supernova Survey
journal, September 2007

  • Wood‐Vasey, W. M.; Miknaitis, G.; Stubbs, C. W.
  • The Astrophysical Journal, Vol. 666, Issue 2
  • DOI: 10.1086/518642