First Cosmological Results using Type Ia Supernovae from the Dark Energy Survey: Measurement of the Hubble Constant
E. Macaulay, R. C. Nichol, D. Bacon, D. Brout, T. M. Davis, B. Zhang, B. A. Bassett, D. Scolnic, A. Möller, C. B. D'Andrea, S. R. Hinton, R. Kessler, A. G. Kim, J. Lasker, C. Lidman, M. Sako, M. Smith, M. Sullivan, T. M. C. Abbott, S. Allam, J. Annis, J. Asorey, S. Avila, K. Bechtol, D. Brooks, P. Brown, D. L. Burke, J. Calcino, A. Carnero Rosell, D. Carollo, M. Carrasco Kind, J. Carretero, F. J. Castander, T. Collett, M. Crocce, C. E. Cunha, L. N. da Costa, C. Davis, J. De Vicente, H. T. Diehl, P. Doel, A. Drlica-Wagner, T. F. Eifler, J. Estrada, A. E. Evrard, A. V. Filippenko, D. A. Finley, B. Flaugher, R. J. Foley, P. Fosalba, J. Frieman, L. Galbany, J. García-Bellido, E. Gaztanaga, K. Glazebrook, S. González-Gaitán, D. Gruen, R. A. Gruendl, J. Gschwend, G. Gutierrez, W. G. Hartley, D. L. Hollowood, K. Honscheid, J. K. Hoormann, B. Hoyle, D. Huterer, B. Jain, D. J. James, T. Jeltema, E. Kasai, E. Krause, K. Kuehn, N. Kuropatkin, O. Lahav, G. F. Lewis, T. S. Li, M. Lima, H. Lin, M. A. G. Maia, J. L. Marshall, P. Martini, R. Miquel, P. Nugent, A. Palmese, Y.-C. Pan, A. A. Plazas, A. K. Romer, A. Roodman, E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, S. Serrano, I. Sevilla-Noarbe, R. Sharp, M. Soares-Santos, F. Sobreira, N. E. Sommer, E. Suchyta, E. Swann, et al. (9 additional authors not shown)
MMNRAS , 1–14 (2019) Preprint 29 May 2019 Compiled using MNRAS L A TEX style file v3.0
First Cosmological Results using Type Ia Supernovae fromthe Dark Energy Survey: Measurement of the Hubbleconstant
E. Macaulay (cid:63) , R. C. Nichol, D. Bacon, D. Brout, T. M. Davis, B. Zhang, , B. A. Bassett, , D. Scolnic, A. M¨oller, , C. B. D’Andrea, S. R. Hinton, R. Kessler, , A. G. Kim, J. Lasker, , C. Lidman, M. Sako, M. Smith, M. Sullivan, T. M. C. Abbott, S. Allam, J. Annis, J. Asorey, S. Avila, K. Bechtol, D. Brooks, P. Brown, D. L. Burke, , J. Calcino, A. Carnero Rosell, , D. Carollo, M. Carrasco Kind, , J. Carretero, F. J. Castander, , T. Collett, M. Crocce, , C. E. Cunha, L. N. da Costa, , C. Davis, J. De Vicente, H. T. Diehl, P. Doel, A. Drlica-Wagner, , T. F. Eifler, , J. Estrada, A. E. Evrard, , A. V. Filippenko, , D. A. Finley, B. Flaugher, R. J. Foley, P. Fosalba, , J. Frieman, , L. Galbany, J. Garc´ıa-Bellido, E. Gaztanaga, , K. Glazebrook, S. Gonz´alez-Gait´an, D. Gruen, , R. A. Gruendl, , J. Gschwend, , G. Gutierrez, W. G. Hartley, , D. L. Hollowood, K. Honscheid, , J. K. Hoormann, B. Hoyle, , D. Huterer, B. Jain, D. J. James, T. Jeltema, E. Kasai, , E. Krause, K. Kuehn, N. Kuropatkin, O. Lahav, G. F. Lewis, T. S. Li, , M. Lima, , H. Lin, M. A. G. Maia, , J. L. Marshall, P. Martini, , R. Miquel, , P. Nugent, A. Palmese, Y.-C. Pan, , A. A. Plazas, A. K. Romer, A. Roodman, , E. Sanchez, V. Scarpine, R. Schindler, M. Schubnell, S. Serrano, , I. Sevilla-Noarbe, R. Sharp, M. Soares-Santos, F. Sobreira, , N. E. Sommer, , E. Suchyta, E. Swann, M. E. C. Swanson, G. Tarle, D. Thomas, R. C. Thomas, B. E. Tucker, , S. A. Uddin, V. Vikram, A. R. Walker, and P. Wiseman (DES Collaboration) Author affiliations are shown in Appendix B
Accepted 2019 April 03. Received 2019 April 02; in original form 2018 November 08
ABSTRACT
We present an improved measurement of the Hubble constant ( H ) using the ‘inversedistance ladder’ method, which adds the information from 207 Type Ia supernovae(SNe Ia) from the Dark Energy Survey (DES) at redshift . < z < . to existingdistance measurements of 122 low redshift ( z < . ) SNe Ia (Low- z ) and measure-ments of Baryon Acoustic Oscillations (BAOs). Whereas traditional measurementsof H with SNe Ia use a distance ladder of parallax and Cepheid variable stars, theinverse distance ladder relies on absolute distance measurements from the BAOs tocalibrate the intrinsic magnitude of the SNe Ia. We find H = . ± . km s − Mpc − (statistical and systematic uncertainties, 68% confidence). Our measurement makesminimal assumptions about the underlying cosmological model, and our analysis wasblinded to reduce confirmation bias. We examine possible systematic uncertainties andall are below the statistical uncertainties. Our H value is consistent with estimatesderived from the Cosmic Microwave Background assuming a Λ CDM universe (PlanckCollaboration et al. 2018).
Key words: cosmology: observations – cosmology: cosmological parameters – cos-mology: distance scale (cid:63) email: [email protected] © a r X i v : . [ a s t r o - ph . C O ] M a y E. Macaulay et al.
The precise value of the Hubble constant ( H ) has againbecome one of the most debated topics in cosmology (seeFreedman 2017). This debate has been fuelled by the ap-parent disagreement between local, direct measurements of H , primarily Riess et al. (2016) who find H = . ± . km s − Mpc − , and estimates derived from the Cosmic Mi-crowave Background (CMB) which give H = . ± . kms − Mpc − (Planck Collaboration et al. 2018), assuming a Λ CDM universe. This discrepancy has increased to . σ with new parallax measurements to Cepheid variable starsby Riess et al. (2018) giving H = . ± . km s − Mpc − .This tension between the local measurements of theHubble constant and the Planck+ Λ CDM expectation maybe due to unknown systematic uncertainties in the variousobservations, flaws in the theoretical assumptions, and/orunder-estimation of the uncertainties on the measurementsof H (e.g., see discussion in Zhang et al. 2017b).Sample or cosmic variance has been proposed as apotential systematic effect for direct measurements of H .Cepheid variable stars can only be observed in the nearestgalaxies, and the number of such galaxies that also havea well-determined SN Ia with which to calibrate the SNIa luminosity zeropoint is small. Thus these measurementsonly probe a small cosmological volume with a low num-ber of galaxies for cross-calibration. However, Wu & Huterer(2017) used N -body simulations to evaluate the sample vari-ance, and found that it contributes a dispersion of 0.31 kms − Mpc − to the local measurements of H , which is toosmall to account for the discrepancy with Planck.The discrepancy in H may alternatively be due tophysics beyond the Λ CDM model (Bernal et al. 2016; DiValentino et al. 2016; Riess et al. 2016; Di Valentino et al.2018). A negative curvature ( Ω k <
0) could account for thediscrepancy, which would have implications for models ofcosmic inflation (de Putter et al. 2014; Grandis et al. 2016;Farooq et al. 2017). Modifications to gravity could cause alarger acceleration than expected in Λ CDM (Pourtsidou &Tram 2016; Di Valentino et al. 2017; Zhao et al. 2017). Alter-natively, an additional relativistic species at the CMB epochcould account for the tension (Moresco et al. 2012; Vagnozziet al. 2017). We note however that explanations for the H tension involving modified gravity, or an extra relativisticspecies, would increase tensions in measurements of σ (theamplitude of the matter power spectrum) and expectationsfrom Planck+ Λ CDM (e.g., Macaulay et al. 2013; Yang &Xu 2014; Joudaki et al. 2017; Costa et al. 2017).This tension has motivated the development of new,independent ways to measure H . For example, Birrer et al.(2019) find H = . + . − . km s − Mpc − from measurementsof time delays from strongly lensed quasars, and Guidorziet al. (2017) find H = . + . − . km s − Mpc − by using thegravitational wave event GW170817 as a standard siren.In this paper, we present a new measurement of H us-ing spectroscopically-confirmed SNe Ia from the Dark En-ergy Survey (see Flaugher et al. 2015; DES Collaborationet al. 2018, for details). Since SNe Ia are relative, not ab-solute, distance indicators, their intrinsic magnitude mustbe calibrated using an absolute distance measurement. Thisis the motivation behind the conventional distance ladderapproach of calibrating local SNe Ia using Cepheid variable stars and parallax. The approach we take is to calibratethe intrinsic magnitude of SNe Ia against the absolute dis-tance measurements from the Baryon Acoustic Oscillations(BAOs) at z > . (assuming the sound horizon from theCMB). We then use the calibrated SN Ia distances to tracethe expansion history of the Universe back to z = to de-termine H .We note that although BAO measurements alone couldderive a value for H (e.g., see Figure 1), it would rely on theassumption of a cosmological model to extrapolate the BAOmeasurements to z = (see e.g., DES Collaboration 2018).By using calibrated SNe Ia across a range of redshifts, wecan determine H more directly without assuming a specificcosmological model such as Λ CDM. H was first measured using this ‘inverse distance lad-der’ technique by Aubourg et al. (2015), who found H = . ± . km s − Mpc − with SNe Ia from the Joint LightCurve Analysis (JLA; Betoule et al. 2014) and BAO mea-surements from the Baryon Oscillation Spectroscopic Sur-vey (BOSS) Data Release Eleven (DR11). This result wasupdated with BOSS DR12 in Alam et al. (2017), finding H = . ± . km s − Mpc − .In this paper, we use 207 new, spectroscopically-confirmed SNe Ia from the DES Supernova Program (DES-SN3YR) to measure H with this inverse distance laddertechnique. While this sample contains fewer supernovae thanJLA, the DES-SN3YR sample has the key advantage ofspanning the entire redshift range of the available galaxyBAO measurements (e.g., z eff = . to . ) in a singlesurvey. This is not true of other inverse distance ladder mea-surements which rely on the JLA sample, because differentSN surveys must be combined in order to cover this red-shift range (e.g., the SDSS SN sample at z (cid:39) . , SNLS at z (cid:39) . ).We describe the data and method used for our analysisin Section 2, and our results in Section 3. We conclude inSection 4. We use a similar methodology as Aubourg et al. (2015),using BAO distance measurements to calibrate the SNe Ia.This breaks the well-known degeneracy between the SNe Iapeak absolute magnitude and H . While BAO data alonecan constrain H , these measurements typically assume aspecific cosmological model (e.g., a cosmological constant asin DES Collaboration 2018), since the BAO measurementsdo not have sufficient redshift coverage to determine H ( z ) on their own (Figure 1 illustrates this point). By combiningBAO and SNe Ia, we can relax the assumption of a specificcosmological model when determining H .However, we do still require some model for the redshift-distance relationship to extrapolate these data to z = .For this work, we adopt a cosmographical approach for theredshift-distance relationship, which is a smooth Taylor ex-pansion about redshift, that makes minimal assumptionsabout the underlying cosmological model (Muthukrishna &Parkinson 2016; Zhang et al. 2017a; Feeney et al. 2019). Weuse Equations 6, 7 and 8 in Zhang et al. (2017a) to deter-mine the luminosity distance D L ( z ) and Hubble parameter MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H H ( z ) as a function of redshift. D L ( z ) is given by D L ( z ) = z + C z + C z + C z + C z + ..., (1)where C = ( − q ) , (2) C = − ( − q − q + j ) , (3) C = ( − q − q + j + q j + s ) , (4)and C = (− + q + q + q + q + j − j − q j − q j − q s − s − l ) . (5) H ( z ) is given by H ( z ) = H [ + ( + q ) z + (− q + j ) z + ( q + q − q j − j − s ) z + (− q − q − q + q j + q j + q s + j − j + s + l ) z ] + ... (6)We find that including the lerk ( l ) parameter ( z ) in ourcosmographic model increases the Bayesian Information Cri-terion from 40.4 to 44.1, which indicates that including thisadditional parameter is not warranted by the data. The fit-ting parameters in this Taylor expansion are then H (Hub-ble constant), q (deceleration), j (jerk), and s (snap).This is consistent with G´omez-Valent & Amendola (2018)who found that fourth-order polynomials and above mademinimal improvement to the Bayesian and Akaike informa-tion criteria when fitting a larger set of data including bothSNe Ia and other cosmological data-sets. They also notedthat such higher-order polynomials had minimal effect onthe value of H they determined. We assume uniform priorson these cosmographical parameters and therefore, our re-sults are relatively insensitive to the details of the assumedunderlying late-time redshift-distance relationship.Throughout, we must assume the validity of the cos-mic distance-duality relation; that the luminosity distance D L ( z ) is related to the angular diameter distance D A ( z ) by D L ( z ) = D A ( z )( + z ) . This well-known relationship in cos-mology is applicable to general metric theories of gravity inwhich photons are conserved and travel on null geodesics(e.g., see Bassett & Kunz 2004).To determine H , we perform a combined analysis ofSNe Ia and BAO with a Gaussian prior on r s (the soundhorizon at recombination) based on CMB data. All thesedata are required and complementary, and assumed to be in-dependent. The individual likelihood functions are assumedto be Gaussian and given by ln L x = ∆ Tx C − ∆ x , (7)where x above is either SNe Ia or BAO data. C is the datacovariance matrix for either data-set, and ∆ x is the differencevector between the data-sets and their corresponding valuesin the cosmographical model. These likelihood functions are then combined to give ln L( Θ , M B , r s ) = ln L SN ( Θ , M B ) + ln L BAO ( Θ , r s ) + ln L r s ( r s ) , (8)where Θ = [ H , q , j , s ] is a common set of cosmographicparameters and M B is the SNe Ia absolute magnitude atpeak (see Section 2.2). The scale of the BAO is a well-established cosmological stan-dard ruler (e.g., Blake & Glazebrook 2003; Seo & Eisenstein2003; Eisenstein et al. 2005; Blake et al. 2011; Busca et al.2013; Anderson et al. 2014; Alam et al. 2017). With thephysical scale set by the sound horizon at recombination( r s ), BAOs provide absolute distance measurements over arange of redshifts. In order to measure the BAO signal fromgalaxy redshift surveys, a fiducial cosmology must be as-sumed in order to convert the observed angles and redshiftsinto distances.We emphasise that this does not imply that a BAOmeasurement is limited to a consistency test of that assumedfiducial cosmology, since most BAO analyses typically fit for α BAO , a dimensionless parameter measuring the ratio of theobserved BAO scale to the scale expected in the fiducialcosmology.An isotropic BAO analysis, where the BAO signal ismeasured from pairs of galaxies averaged over all angles,is sensitive to the volume averaged distance (e.g., D V , seeAubourg et al. 2015). We can relate the expected D V ( Θ , z ) to the observed α BAO and the fiducial BAO values by D V ( Θ , z ) r s = α BAO D fid V ( z ) r fid s , (9)where D V ( Θ , z ) = [ zD H ( Θ , z ) D M ( Θ , z )] / . (10) D H ( Θ , z ) is the Hubble distance, given by D H ( Θ , z ) = cH ( Θ , z ) , (11)and D M ( Θ , z ) is the comoving angular diameter distance.In our likelihood analysis, we use the observed D V ( z eff = . ) = ± ( r s / r fid s ) Mpc (68% confidence) taken fromCarter et al. (2018), based on a re-analysis of the 6-degreeField Galaxy Survey (Beutler et al. 2011) and Sloan DigitalSky Survey Main Galaxy Sample (Ross et al. 2015).At higher redshift, we use the ‘Consensus’ BOSS DR12data-set from Alam et al. (2017) which provides a two-dimensional description of the clustering, dividing the sepa-ration between pairs of galaxies into components across andalong the line-of-sight, now summarising the clustering withtwo parameters of α BAO ⊥ and α BAO | | , respectively. Their ex-pected values can now be related to their observed valuesby D M ( Θ , z ) r s = α BAO ⊥ D fid M ( z ) r fid s , (12)and D H ( Θ , z ) r s = α BAO | | D fid H ( z ) r fid s . (13) MNRAS000
0) could account for thediscrepancy, which would have implications for models ofcosmic inflation (de Putter et al. 2014; Grandis et al. 2016;Farooq et al. 2017). Modifications to gravity could cause alarger acceleration than expected in Λ CDM (Pourtsidou &Tram 2016; Di Valentino et al. 2017; Zhao et al. 2017). Alter-natively, an additional relativistic species at the CMB epochcould account for the tension (Moresco et al. 2012; Vagnozziet al. 2017). We note however that explanations for the H tension involving modified gravity, or an extra relativisticspecies, would increase tensions in measurements of σ (theamplitude of the matter power spectrum) and expectationsfrom Planck+ Λ CDM (e.g., Macaulay et al. 2013; Yang &Xu 2014; Joudaki et al. 2017; Costa et al. 2017).This tension has motivated the development of new,independent ways to measure H . For example, Birrer et al.(2019) find H = . + . − . km s − Mpc − from measurementsof time delays from strongly lensed quasars, and Guidorziet al. (2017) find H = . + . − . km s − Mpc − by using thegravitational wave event GW170817 as a standard siren.In this paper, we present a new measurement of H us-ing spectroscopically-confirmed SNe Ia from the Dark En-ergy Survey (see Flaugher et al. 2015; DES Collaborationet al. 2018, for details). Since SNe Ia are relative, not ab-solute, distance indicators, their intrinsic magnitude mustbe calibrated using an absolute distance measurement. Thisis the motivation behind the conventional distance ladderapproach of calibrating local SNe Ia using Cepheid variable stars and parallax. The approach we take is to calibratethe intrinsic magnitude of SNe Ia against the absolute dis-tance measurements from the Baryon Acoustic Oscillations(BAOs) at z > . (assuming the sound horizon from theCMB). We then use the calibrated SN Ia distances to tracethe expansion history of the Universe back to z = to de-termine H .We note that although BAO measurements alone couldderive a value for H (e.g., see Figure 1), it would rely on theassumption of a cosmological model to extrapolate the BAOmeasurements to z = (see e.g., DES Collaboration 2018).By using calibrated SNe Ia across a range of redshifts, wecan determine H more directly without assuming a specificcosmological model such as Λ CDM. H was first measured using this ‘inverse distance lad-der’ technique by Aubourg et al. (2015), who found H = . ± . km s − Mpc − with SNe Ia from the Joint LightCurve Analysis (JLA; Betoule et al. 2014) and BAO mea-surements from the Baryon Oscillation Spectroscopic Sur-vey (BOSS) Data Release Eleven (DR11). This result wasupdated with BOSS DR12 in Alam et al. (2017), finding H = . ± . km s − Mpc − .In this paper, we use 207 new, spectroscopically-confirmed SNe Ia from the DES Supernova Program (DES-SN3YR) to measure H with this inverse distance laddertechnique. While this sample contains fewer supernovae thanJLA, the DES-SN3YR sample has the key advantage ofspanning the entire redshift range of the available galaxyBAO measurements (e.g., z eff = . to . ) in a singlesurvey. This is not true of other inverse distance ladder mea-surements which rely on the JLA sample, because differentSN surveys must be combined in order to cover this red-shift range (e.g., the SDSS SN sample at z (cid:39) . , SNLS at z (cid:39) . ).We describe the data and method used for our analysisin Section 2, and our results in Section 3. We conclude inSection 4. We use a similar methodology as Aubourg et al. (2015),using BAO distance measurements to calibrate the SNe Ia.This breaks the well-known degeneracy between the SNe Iapeak absolute magnitude and H . While BAO data alonecan constrain H , these measurements typically assume aspecific cosmological model (e.g., a cosmological constant asin DES Collaboration 2018), since the BAO measurementsdo not have sufficient redshift coverage to determine H ( z ) on their own (Figure 1 illustrates this point). By combiningBAO and SNe Ia, we can relax the assumption of a specificcosmological model when determining H .However, we do still require some model for the redshift-distance relationship to extrapolate these data to z = .For this work, we adopt a cosmographical approach for theredshift-distance relationship, which is a smooth Taylor ex-pansion about redshift, that makes minimal assumptionsabout the underlying cosmological model (Muthukrishna &Parkinson 2016; Zhang et al. 2017a; Feeney et al. 2019). Weuse Equations 6, 7 and 8 in Zhang et al. (2017a) to deter-mine the luminosity distance D L ( z ) and Hubble parameter MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H H ( z ) as a function of redshift. D L ( z ) is given by D L ( z ) = z + C z + C z + C z + C z + ..., (1)where C = ( − q ) , (2) C = − ( − q − q + j ) , (3) C = ( − q − q + j + q j + s ) , (4)and C = (− + q + q + q + q + j − j − q j − q j − q s − s − l ) . (5) H ( z ) is given by H ( z ) = H [ + ( + q ) z + (− q + j ) z + ( q + q − q j − j − s ) z + (− q − q − q + q j + q j + q s + j − j + s + l ) z ] + ... (6)We find that including the lerk ( l ) parameter ( z ) in ourcosmographic model increases the Bayesian Information Cri-terion from 40.4 to 44.1, which indicates that including thisadditional parameter is not warranted by the data. The fit-ting parameters in this Taylor expansion are then H (Hub-ble constant), q (deceleration), j (jerk), and s (snap).This is consistent with G´omez-Valent & Amendola (2018)who found that fourth-order polynomials and above mademinimal improvement to the Bayesian and Akaike informa-tion criteria when fitting a larger set of data including bothSNe Ia and other cosmological data-sets. They also notedthat such higher-order polynomials had minimal effect onthe value of H they determined. We assume uniform priorson these cosmographical parameters and therefore, our re-sults are relatively insensitive to the details of the assumedunderlying late-time redshift-distance relationship.Throughout, we must assume the validity of the cos-mic distance-duality relation; that the luminosity distance D L ( z ) is related to the angular diameter distance D A ( z ) by D L ( z ) = D A ( z )( + z ) . This well-known relationship in cos-mology is applicable to general metric theories of gravity inwhich photons are conserved and travel on null geodesics(e.g., see Bassett & Kunz 2004).To determine H , we perform a combined analysis ofSNe Ia and BAO with a Gaussian prior on r s (the soundhorizon at recombination) based on CMB data. All thesedata are required and complementary, and assumed to be in-dependent. The individual likelihood functions are assumedto be Gaussian and given by ln L x = ∆ Tx C − ∆ x , (7)where x above is either SNe Ia or BAO data. C is the datacovariance matrix for either data-set, and ∆ x is the differencevector between the data-sets and their corresponding valuesin the cosmographical model. These likelihood functions are then combined to give ln L( Θ , M B , r s ) = ln L SN ( Θ , M B ) + ln L BAO ( Θ , r s ) + ln L r s ( r s ) , (8)where Θ = [ H , q , j , s ] is a common set of cosmographicparameters and M B is the SNe Ia absolute magnitude atpeak (see Section 2.2). The scale of the BAO is a well-established cosmological stan-dard ruler (e.g., Blake & Glazebrook 2003; Seo & Eisenstein2003; Eisenstein et al. 2005; Blake et al. 2011; Busca et al.2013; Anderson et al. 2014; Alam et al. 2017). With thephysical scale set by the sound horizon at recombination( r s ), BAOs provide absolute distance measurements over arange of redshifts. In order to measure the BAO signal fromgalaxy redshift surveys, a fiducial cosmology must be as-sumed in order to convert the observed angles and redshiftsinto distances.We emphasise that this does not imply that a BAOmeasurement is limited to a consistency test of that assumedfiducial cosmology, since most BAO analyses typically fit for α BAO , a dimensionless parameter measuring the ratio of theobserved BAO scale to the scale expected in the fiducialcosmology.An isotropic BAO analysis, where the BAO signal ismeasured from pairs of galaxies averaged over all angles,is sensitive to the volume averaged distance (e.g., D V , seeAubourg et al. 2015). We can relate the expected D V ( Θ , z ) to the observed α BAO and the fiducial BAO values by D V ( Θ , z ) r s = α BAO D fid V ( z ) r fid s , (9)where D V ( Θ , z ) = [ zD H ( Θ , z ) D M ( Θ , z )] / . (10) D H ( Θ , z ) is the Hubble distance, given by D H ( Θ , z ) = cH ( Θ , z ) , (11)and D M ( Θ , z ) is the comoving angular diameter distance.In our likelihood analysis, we use the observed D V ( z eff = . ) = ± ( r s / r fid s ) Mpc (68% confidence) taken fromCarter et al. (2018), based on a re-analysis of the 6-degreeField Galaxy Survey (Beutler et al. 2011) and Sloan DigitalSky Survey Main Galaxy Sample (Ross et al. 2015).At higher redshift, we use the ‘Consensus’ BOSS DR12data-set from Alam et al. (2017) which provides a two-dimensional description of the clustering, dividing the sepa-ration between pairs of galaxies into components across andalong the line-of-sight, now summarising the clustering withtwo parameters of α BAO ⊥ and α BAO | | , respectively. Their ex-pected values can now be related to their observed valuesby D M ( Θ , z ) r s = α BAO ⊥ D fid M ( z ) r fid s , (12)and D H ( Θ , z ) r s = α BAO | | D fid H ( z ) r fid s . (13) MNRAS000 , 1–14 (2019)
E. Macaulay et al.
The BOSS DR12 data-set consists of measurements of D M ( z ) and H ( z ) at three effective redshifts of z eff = [ . , . , . ] (6 measurements in total). The covariance matrix for thesesix measurements includes the correlation between D M ( z ) and H ( z ) at each z eff , and the correlation between these sixmeasurements in different redshift bins.In Equation 7, ln L BAO ( Θ , r s ) is then the combined like-lihood of the two BAO data-sets from Carter et al. (2018)and BOSS DR12. This likelihood requires knowledge of thesound horizon at recombination ( r s ), which depends on thesound speed at these earlier epochs and thus the baryondensity ( ω b ) and the total matter density ( ω cb ) in the earlyuniverse (see Equation 16 of Aubourg et al. 2015).In our analysis, we adopt a Gaussian prior on r s of . ± . Mpc (68% confidence) taken from the Planck2018 analysis (
TT,TE,EE+lowE result in Table 2). By usingthe value of r s derived from only the TT,TE,EE+lowE
Planckdata, we minimise our sensitivity to physics of the late-timeuniverse. Of these effects, CMB lensing is the most signifi-cant, although Feeney et al. (2019) note that even includingCMB lensing changes their value of H by less than the sta-tistical uncertainty on this measurement.We explore the sensitivity of our results to this priorin Appendix 4, but note that the Planck measurement of r s comes from their measurement of the baryon and to-tal matter densities, which in turn are only related to theheights of the acoustic peaks in the CMB power spectrumand not on their angular locations. Therefore, any dependen-cies introduced because of this Planck prior are based onlyon our correct understanding of plasma physics in the pre-recombination epoch, rather than assumptions about curva-ture and late-time dark energy, which are negligible in theearly Universe for many cosmological models. Type Ia supernovae are cosmological standard candles (e.g.,Riess et al. 1998; Perlmutter et al. 1999; Hicken et al. 2009;Kessler et al. 2009; Sullivan et al. 2010). In this analysis, weuse the DES-SN3YR sample of 207 new, spectroscopically-confirmed SNe Ia from the first three years of DES, which aresupplemented by 122 SNe Ia from the CfA3 (Hicken et al.2009), CfA4 (Hicken et al. 2012) and CSP Low- z sample(Hamuy et al. 2006; Contreras et al. 2010; Folatelli et al.2010; Stritzinger et al. 2011; Krisciunas et al. 2017) ( z < . ) described in part in Scolnic et al. (2018).The details of the DES-SN3YR sample are provided ina series of papers as part of the overall DES-SN 3 year cos-mology paper by DES Collaboration et al. (2019). D’Andreaet al. (2018) describes the spectroscopic follow-up obser-vations, Brout et al. (2019a) outlines the supernova scenemodel photometry, Lasker et al. (2019) details the DES pho-tometric corrections, Kessler et al. (2019) presents the sur-vey simulations and selection function. Brout et al. (2019b)presents validations of the sample, systematic uncertainties,light-curve fits, and distance measurements.Our analysis uses the distances and covariance matricesderived in Brout et al. (2019b) with the ‘BBC’ (BEAMSwith Bias Correction) method (Kessler & Scolnic 2017). Thedistances have been binned in 18 redshift bins (originally 20bins, but with 2 empty bins).The distance modulus, µ , for these supernovae is given by µ ( Θ , z ) = + ( D L ( Θ , z )) , (14)where D L ( Θ , z ) is the luminosity distance. We relate µ to theobserved SALT2 (Guy et al. 2007) light curve parameters by µ = m B − M B + α X − β C + ∆ m host + ∆ B , (15)where m B is the observed B -band peak magnitude, M B is theabsolute magnitude of the SNe Ia, X is the stretch parame-ter of the light curve, and C is the colour parameter. α and β are free parameters which are fitted for when calculatingthe distances. Also, ∆ m host is a correction applied for hostgalaxy masses of M stellar > M (cid:12) (see also Sullivan et al.2010; Kelly et al. 2010; Lampeitl et al. 2010b). The stellarmass measurements for the host galaxies were obtained fromfits to the DES SV galaxy photometry with the galaxy evolu-tion modelling code ZPEG (Le Borgne & Rocca-Volmerange2002) (see Smith et al. 2019, in prep. for details). ∆ B is theexpected µ -correction due to the survey selection functionfor both the DES-SN3YR sample as discussed in detail inBrout et al. (2019b). We use emcee (Foreman-Mackey et al. 2013) as our Markovchain Monte Carlo sampler to determine the joint likelihoodof our parameters ( H , q , j , s , r s , M B ) in Equation 8.These joint likelihoods are shown in Figure 2 along with themarginalised likelihood functions for all the fitted parame-ters.We blinded our analysis throughout the analysis to re-duce confirmation bias (e.g., Croft & Dailey 2015). Thisblinding has been achieved by preparing and testing all ourcodes and plots using either simulated DES-SN3YR samples(see Appendix A for details) or the existing JLA sample fromBetoule et al. (2014). We only replaced these testing datafiles with the genuine DES-SN3YR sample before submissionto the DES collaboration for internal collaboration review.Before unblinding, we reviewed the results with unknownrandom offsets added to the chains, so that the shape of thelikelihoods and the uncertainties could be assessed withoutinfluence from the maximum likelihood values of the chains.After unblinding, some minor updates to the SN-data co-variance matrix were introduced, including the use of twodifferent intrinsic dispersion values for the DES and Low- z samples. Updating our results after unblinding did not sig-nificantly change our results or conclusions.In Table 1 we summarise our H measurements withdifferent supernovae surveys. We note that the χ / DoF in-creases for all of the combined fits with SNe Ia data. Thisis consistent with the higher H values in the combined fitsthan the BAO-only case, and suggests that the best-fit cos-mographic model favours increased expansion at lower red-shifts than the BAO surveys.We note that our cosmographical method can reproducethe H measurement from Alam et al. (2017). Using the samedata-set (namely, JLA supernovae, BOSS DR12 BAOs, andthe older BAO measurements from Beutler et al. (2011) andRoss et al. (2015)), we find H = . ± . km s − Mpc − . Wedo not use the BAO measurements from Beutler et al. (2011) MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H H Data [ km s − Mpc − ] DoF χ DoF p -valueBAO Only 65.7 ± ± z ± ± z . ± .
20 1.16 0.28
Table 1.
A summary of our H measurements. We fit all the datasets with 6 free parameters (except for the BAO-only case, with5 free parameters, since we do not fit for M B ). and Ross et al. (2015) elsewhere, since they are supersededby the combined analysis of Carter et al. (2018).We find a larger uncertainty than Alam et al. (2017),which may be due to the more general model we are using forthe redshift-distance relation. Instead of the cosmographicmodel used here, Alam et al. (2017) use a ‘PolyCDM’ cos-mological model, which is the Λ CDM model, plus two ad-ditional arbitrary density components, Ω and Ω , whichscale with linear and quadratic order with redshift, in orderto allow for deviations from Λ CDM.We find that our best-fit parameters are consistent withprevious analyses. For example, we find M B = − . ± . mag (68% confidence) which is consistent with the valuefrom Aubourg et al. (2015) of M B = − . ± . mag.Moreover, Figure 2 illustrates that, as expected, SNe Ia aloneare unable to constrain this parameter, given the strong cor-relation with H (also seen in the figure).The cosmographical parameters are constrained by bothBAO and SNe Ia. The deceleration parameter shows a sig-nificant, negative value of q = − . ± . (68% confidence),even after marginalising over other parameters. This value isconsistent with other q measurements in the literature (e.g.,Lampeitl et al. (2010a) found q = − . ± . from just theSDSS SN sample) but less negative than expected for a Λ –dominated cosmology (e.g., q = − . from Capozzielloet al. 2008). Other cosmographical parameters ( j , s ) arebest constrained by the SNe Ia data (green contours), andconsistent with zero and expectations from Λ CDM given ouruncertainties.We find H = . ± . km s − Mpc − (68% confidence),with a reduced χ of 1.16. This measurement is consistentwith the Planck value of H = . ± . km s − Mpc − , butinconsistent at 2.5 σ to the recent Riess et al. (2018) localdistance ladder measurement.We make no further comment on this tension as thereis already significant literature (e.g., see Freedman 2017)on the possible systematic uncertainties involved in all mea-surements and/or interesting new physics that could be re-sponsible. However, we emphasise the independence of thisresult to that of Riess et al. (2018): although both mea-surements rely on SNe Ia, the Cepheid-calibrated distanceladder method of Riess et al. (2018) is very different to theBAO-calibrated inverse distance ladder method used here.Our measurement is in excellent agreement with previous in-verse distance ladder measurements, e.g., Alam et al. (2017)who used the JLA sample, as well more recent measurementsusing the Pantheon SN sample (Feeney et al. 2019; Lemoset al. 2019). In Figure 1, we provide an illustration of the inverse dis-tance ladder method and the importance of both BAO andSNe Ia data for deriving the best constraint on H . The SNeIa data is the DES-SN3YR sample (18 redshift bins) and theSN and BAO uncertainties are the square roots of the cor-responding diagonal elements of their covariance matrices.We also show our best-fit cosmographical model (with as-sociated 68% confidence band in red) as well as the best-fitBAO-only cosmographical model and uncertainty band (inblue). Although our H value is in excellent agreement with PlanckCollaboration et al. (2018), we emphasise that the use of an r s prior from Planck does not imply that our measured valueof H will inevitably agree with the value of H derived fromPlanck cosmological parameters assuming a Λ CDM cosmol-ogy. The value of r s is informed by only the baryon and mat-ter densities at z = ; there are many viable cosmologicalmodels which are consistent with only these two quantities(or, in other words, this value of r s ) that have wildly differentvalues of the Hubble constant at z = .Indeed, our BAO-only value of H = . ± . km s − Mpc − is lower than the Planck-derived value. The BAOdata only directly constrain the expansion down to z ∼ . .Extrapolating to z = thus leads to larger uncertaintiesfrom the BAO-only than from the BAO+SNe Ia becausethe SNe Ia probe down to z . . The consistency betweenour measurement and the derived Planck value is instead areflection of the consistency between the cosmology tracedby the SN and BAO data and the model used to derive thePlanck value. We now consider the possible systematic uncertainties thatmay affect our result. As in Brout et al. (2019b), we considercontributions to the total uncertainty from many contribu-tions of systematic uncertainty. To quantify the effect of eachsystematic uncertainty, we first repeat our analysis with onlythe statistical uncertainties included in the supernova datacovariance matrix, to find the statistical-only uncertainty on H , σ stat . We then repeat this analysis for each of the sourcesof systematic uncertainty, including each contribution of sys-tematic uncertainty in the supernova covariance matrix, tofind the combined uncertainty due to statistics and the par-ticular systematic, σ stat + sys . We then define the systematiconly uncertainty as σ sys = (cid:113) σ + sys − σ . (16)We consider systematic uncertainties due to DES andLow- z calibration, SALT fitting, Supercal calibration (Scol-nic et al. 2015), the intrinsic scatter model, colour param-eter parent populations, volume limits, peculiar velocities,flux uncertainty, spectroscopic efficiency, the use of refer-ence cosmology in validation simulations, the Low- z sample3 σ outlier cut, the parent population uncertainty, PS1 Co-herent Shift, and the use of two different values of the intrin-sic dispersion for the DES and Low- z samples (Brout et al. MNRAS000
A summary of our H measurements. We fit all the datasets with 6 free parameters (except for the BAO-only case, with5 free parameters, since we do not fit for M B ). and Ross et al. (2015) elsewhere, since they are supersededby the combined analysis of Carter et al. (2018).We find a larger uncertainty than Alam et al. (2017),which may be due to the more general model we are using forthe redshift-distance relation. Instead of the cosmographicmodel used here, Alam et al. (2017) use a ‘PolyCDM’ cos-mological model, which is the Λ CDM model, plus two ad-ditional arbitrary density components, Ω and Ω , whichscale with linear and quadratic order with redshift, in orderto allow for deviations from Λ CDM.We find that our best-fit parameters are consistent withprevious analyses. For example, we find M B = − . ± . mag (68% confidence) which is consistent with the valuefrom Aubourg et al. (2015) of M B = − . ± . mag.Moreover, Figure 2 illustrates that, as expected, SNe Ia aloneare unable to constrain this parameter, given the strong cor-relation with H (also seen in the figure).The cosmographical parameters are constrained by bothBAO and SNe Ia. The deceleration parameter shows a sig-nificant, negative value of q = − . ± . (68% confidence),even after marginalising over other parameters. This value isconsistent with other q measurements in the literature (e.g.,Lampeitl et al. (2010a) found q = − . ± . from just theSDSS SN sample) but less negative than expected for a Λ –dominated cosmology (e.g., q = − . from Capozzielloet al. 2008). Other cosmographical parameters ( j , s ) arebest constrained by the SNe Ia data (green contours), andconsistent with zero and expectations from Λ CDM given ouruncertainties.We find H = . ± . km s − Mpc − (68% confidence),with a reduced χ of 1.16. This measurement is consistentwith the Planck value of H = . ± . km s − Mpc − , butinconsistent at 2.5 σ to the recent Riess et al. (2018) localdistance ladder measurement.We make no further comment on this tension as thereis already significant literature (e.g., see Freedman 2017)on the possible systematic uncertainties involved in all mea-surements and/or interesting new physics that could be re-sponsible. However, we emphasise the independence of thisresult to that of Riess et al. (2018): although both mea-surements rely on SNe Ia, the Cepheid-calibrated distanceladder method of Riess et al. (2018) is very different to theBAO-calibrated inverse distance ladder method used here.Our measurement is in excellent agreement with previous in-verse distance ladder measurements, e.g., Alam et al. (2017)who used the JLA sample, as well more recent measurementsusing the Pantheon SN sample (Feeney et al. 2019; Lemoset al. 2019). In Figure 1, we provide an illustration of the inverse dis-tance ladder method and the importance of both BAO andSNe Ia data for deriving the best constraint on H . The SNeIa data is the DES-SN3YR sample (18 redshift bins) and theSN and BAO uncertainties are the square roots of the cor-responding diagonal elements of their covariance matrices.We also show our best-fit cosmographical model (with as-sociated 68% confidence band in red) as well as the best-fitBAO-only cosmographical model and uncertainty band (inblue). Although our H value is in excellent agreement with PlanckCollaboration et al. (2018), we emphasise that the use of an r s prior from Planck does not imply that our measured valueof H will inevitably agree with the value of H derived fromPlanck cosmological parameters assuming a Λ CDM cosmol-ogy. The value of r s is informed by only the baryon and mat-ter densities at z = ; there are many viable cosmologicalmodels which are consistent with only these two quantities(or, in other words, this value of r s ) that have wildly differentvalues of the Hubble constant at z = .Indeed, our BAO-only value of H = . ± . km s − Mpc − is lower than the Planck-derived value. The BAOdata only directly constrain the expansion down to z ∼ . .Extrapolating to z = thus leads to larger uncertaintiesfrom the BAO-only than from the BAO+SNe Ia becausethe SNe Ia probe down to z . . The consistency betweenour measurement and the derived Planck value is instead areflection of the consistency between the cosmology tracedby the SN and BAO data and the model used to derive thePlanck value. We now consider the possible systematic uncertainties thatmay affect our result. As in Brout et al. (2019b), we considercontributions to the total uncertainty from many contribu-tions of systematic uncertainty. To quantify the effect of eachsystematic uncertainty, we first repeat our analysis with onlythe statistical uncertainties included in the supernova datacovariance matrix, to find the statistical-only uncertainty on H , σ stat . We then repeat this analysis for each of the sourcesof systematic uncertainty, including each contribution of sys-tematic uncertainty in the supernova covariance matrix, tofind the combined uncertainty due to statistics and the par-ticular systematic, σ stat + sys . We then define the systematiconly uncertainty as σ sys = (cid:113) σ + sys − σ . (16)We consider systematic uncertainties due to DES andLow- z calibration, SALT fitting, Supercal calibration (Scol-nic et al. 2015), the intrinsic scatter model, colour param-eter parent populations, volume limits, peculiar velocities,flux uncertainty, spectroscopic efficiency, the use of refer-ence cosmology in validation simulations, the Low- z sample3 σ outlier cut, the parent population uncertainty, PS1 Co-herent Shift, and the use of two different values of the intrin-sic dispersion for the DES and Low- z samples (Brout et al. MNRAS000 , 1–14 (2019)
E. Macaulay et al. . . . . . Redshift , z c l n ( + z ) D − M [ k m s − M p c − ] BOSS6dF + SDSSBAO Fit DESLow − zBAO + SN Fit Figure 1.
Here we illustrate the inverse distance ladder method. The white data points are the BAO distance measurements, and theblack data points are the SNe Ia data. The DES-SN3YR sample comprises the higher redshift DES SNe (illustrated with hexagonalpoints), and the Low- z SNe (illustrated with triangular points). The red line shows our best-fit cosmographical model, and the shadedregion is the 68% confidence region. The blue dashed line and shaded region illustrates the equivalent constraints from just the BAOdata, without any supernovae. The blue, BAO-only region is very large at z > . because we fit only for r s , and not the absolute distancescale at the CMB. H value, and also the fractional contributionof the systematic compared to the statistical uncertainty.We find that the total systematic uncertainty from eachof these contributions is 72% of the statistical uncertainty inour measurement; variations below the statistical error arestill significant when comparing analysis of the same data.We note that the various individual systematic uncertaintieswill not necessarily add in quadrature to the total system-atic uncertainty, since each systematic introduces a differentweighting of the redshift bins.In Figure 3, we illustrate the effect of the Low- z sample3 σ outlier cut, the parent population uncertainty, a shift inthe absolute wavelength calibration of DES (PS1 CoherentShift), and the 2 σ int systematic uncertainties. In the upper panel, we illustrate the effect on the observed distances, µ ,relative to the statistical-only distances. In the lower panel,we illustrate the corresponding shift in H (again, relativeto the statistical-only result). One uncertainty in our analysis is our assumed prior on thesound horizon, based on the Planck CMB measurement. Themeasurement of r s is very similar between Planck 2013, 2015and 2018, and changing between the measurements has anegligible effect on our results. Moreover, Addison et al.(2018) and Lemos et al. (2019) showed that the H ten-sion was still present using non-Planck CMB data for r s (e.g., WMAP+SPT) or measurements of the primordial deu-terium abundance from damped Lyman- α systems to con- MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H Figure 2.
The parameter constraints on our model from supernovae (green, dot-dashed lines), BAO (blue, dashed lines), and both datasets combined (red, solid lines). We can see that the supernovae constraints on H and M B are degenerate, and r s is entirely unconstrained,although the cosmographic parameters of q , j and s (which affect the shape of the Hubble diagram) are well constrained. We can alsosee that the BAO-only constraint on H is correlated with all of the cosmographic parameters. The power of the combined fit is drivenby the degree of orthogonality of the individual constraints, particularly in the H - M B plane.. strain the baryon density, and thus sound horizon, indepen-dent of any measurement of the CMB power spectrum.That said, the existence of ‘dark radiation’, or an addi-tional relativistic species in the pre-recombination era, couldaffect the sound horizon. The most obvious candidate wouldbe massive neutrinos, but constraints on such particles are becoming increasingly tight from a combination of the CMBwith measurements of the large-scale structure. For exam-ple, Y`eche et al. (2017) provide a constraint on the sum ofthe neutrino masses of (cid:205) m µ < . eV (95% confidence) withjust data from the Lyman- α forest (LyAF). This constraintimproves to (cid:205) m µ < . eV (95% confidence) when com- MNRAS000
The parameter constraints on our model from supernovae (green, dot-dashed lines), BAO (blue, dashed lines), and both datasets combined (red, solid lines). We can see that the supernovae constraints on H and M B are degenerate, and r s is entirely unconstrained,although the cosmographic parameters of q , j and s (which affect the shape of the Hubble diagram) are well constrained. We can alsosee that the BAO-only constraint on H is correlated with all of the cosmographic parameters. The power of the combined fit is drivenby the degree of orthogonality of the individual constraints, particularly in the H - M B plane.. strain the baryon density, and thus sound horizon, indepen-dent of any measurement of the CMB power spectrum.That said, the existence of ‘dark radiation’, or an addi-tional relativistic species in the pre-recombination era, couldaffect the sound horizon. The most obvious candidate wouldbe massive neutrinos, but constraints on such particles are becoming increasingly tight from a combination of the CMBwith measurements of the large-scale structure. For exam-ple, Y`eche et al. (2017) provide a constraint on the sum ofthe neutrino masses of (cid:205) m µ < . eV (95% confidence) withjust data from the Lyman- α forest (LyAF). This constraintimproves to (cid:205) m µ < . eV (95% confidence) when com- MNRAS000 , 1–14 (2019)
E. Macaulay et al. . . . . . Redshift , z − . − . . . . . . ∆ H ( z ) [ k m s − M p c − ] − . − . − . . . . . . . ∆ µ Low − z 3 σ CutSys . Parent PS1 Coherent Shift2 σ int Figure 3.
Illustrating the effect of systematic uncertainties. In the upper panel we plot the change in distance modulus, ∆ µ . Thesupernova data are shown with points, and the best-fit models are shown with lines. For each systematic, we plot the residual comparedto the best-fit model and data with statistical only uncertainties. In the lower panel, we plot the inferred H ( z ) residual, again subtractingthe statistical-only result. The light-grey vertical lines are the 68% confidence error bars on the data (which we have plotted for only thePS1 Coherent Shift systematic, for clarity). The systematics are described in Section 4.1. bined with the CMB power spectrum, although adding inthe recent DES Y1 clustering analysis (DES Collaboration2018) relaxes these constraints to (cid:205) m µ < . eV.Verde et al. (2017) calculated the effect on r s due toall remaining observationally allowed contributions of darkradiation, and found r s = ± Mpc. We therefore testour sensitivity to the possibility of early dark radiation byrepeating our analysis with this wider prior on r s . Our re-sults are shown in Figure 4, and we find H = . ± . km s − Mpc − (compared to H = . ± . km s − Mpc − based on our original prior on r s ). As expected, this widerprior increases the uncertainty on H , making it more con-sistent with the value from Riess et al. (2018). We do notinclude this uncertainty in our uncertainty budget, since theevidence for early dark radiation is not well established, butprovide the value for comparison.As a final test of the sound horizon, we remove any prioron the sound horizon and fit for r s as a free parameter. Even without any prior on the value of r s , we are able to placesome constraints on r s (and, correspondingly, H ) with theminimal assumption that r s is the same for each of the BAOmeasurements.While this assumption would be insufficient in the caseof a single (volume averaged) BAO measurement, havingmultiple BAO measurements at different redshift ranges will– in principle – determine r s (modulo any uncertainty inthe cosmographic parameters). Moreover, in the case of 2DBAO measurements, the consistency of r s in the parallel andperpendicular measurements further constrains r s with theAlcock-Paczynski effect. In other words, the value of r s is –in principle – over-determined, up-to the uncertainties in thecosmographic parameters (which are themselves constrainedby the BAO measurements, and also independently by theSN data).With no prior on r s , we find r s = . ± . Mpc, whichis close to the Planck value, although with a much greater
MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H Description H shift σ syst σ syst / σ stat Total Stat. 0.000 1.048 1.00Total Sys. 0.162 0.760 0.72ALL Calibration -0.078 0.375 0.36DES Cal. -0.016 0.276 0.26Low- z Cal -0.026 0.254 0.24SALT 0.053 0.217 0.21ALL Other 0.004 0.661 0.63Intrinsic Scatter 0.129 0.330 0.31 z + . c , x Parent Pop. -0.031 0.249 0.24Low- z Vol. Lim. -0.081 0.124 0.12Flux Err. -0.004 0.179 0.17Spec. Eff -0.091 0.125 0.12Ref. Cosmo. -0.065 0.134 0.13Low- z σ Cut 0.498 0.193 0.18Sys. Parent 0.370 0.222 0.21PS1 Coherent Shift 0.064 0.246 0.232 σ int -0.068 0.231 0.22 Table 2.
The contributions of systematic uncertainties on our H measurement. For each term we quantify the shift in the valueof H , the value of the systematic uncertainty, and the fractionof the uncertainty compared to the statistical uncertainty. Theindividual uncertainties will not necessarily sum in quadratureto the total uncertainty, due to different redshift weightings andcorrelations in each term. uncertainty, which reduces our sensitivity to H , leading to avalue of H = . ± . km s − Mpc − . This test illustrates theimportance of knowing the absolute scale of the sound hori-zon, while also indicating that future BAO measurementsfrom galaxy redshift surveys (e.g., Euclid and DESI) shouldhelp constrain this parameter independent of the CMB andthus remove any reliance on early universe plasma physics.One possible concern is the assumption of a fiducial cos-mology in converting the galaxy angular positions and red-shifts observed in a galaxy redshift survey (such as BOSS)into the power spectrum of galaxy clustering where the BAOsignal is determined. As stated in Section 2.1, this issue isaddressed by assuming a scaling law (Equation 13) which‘dilates’ the distance being tested to the fiducial cosmologyused to calculate the galaxy power spectrum. The applicabil-ity of such scaling was first studied in detail by Padmanab-han & White (2008) who showed, using N -body simulations,a systematic uncertainty of only (cid:39) on α BAO over a widerange of α BAO values, or more importantly, over a wide rangeof alternative cosmologies.It is also worth stressing that the BAO signal is es-timated in a series of narrow redshift bins, thus allowingfor uncertainties between the assumed and fiducial cosmol-ogy to be minimised across any single redshift shell (assum-ing cosmologies with a smooth redshift-distance relationshipclose to Λ CDM). The effects of such redshift binning has re-cently been examined by Zhu et al. (2016) using mock galaxycatalogues that closely mimic BOSS, and they found theirBAO analysis and measurements remained unbiased evenwhen the assumed fiducial cosmology differed from the true(simulation) cosmology. They also confirmed an uncertaintyof just % on α BAO over a range of different assumptions(fiducial cosmologies, pivot redshifts, redshift-space distor- tion parameters, and galaxy bias models) which could befurther improved to the sub-percent level with future opti-mal redshift weighting schemes (e.g., Zhu et al. 2015). Forreference, Alam et al. (2017) assumed a 0.3% systematic un-certainty on α BAO from their fitting methodologies.We note that the redshifts at which the BAO measure-ments are made are approximate, because they are weightedaverages of the redshifts of all of the pairs of galaxies thatgo into generating the correlation function. The weightingcan depend on the choice of using average redshifts or aver-age distances, which adds some uncertainty to the redshiftof the distance anchor. However, since the slope in Figure 1is shallow, any uncertainty in the centre of the redshift binwould only add a small uncertainty to the measurement of H ( < − Mpc − ).Therefore, the systematic uncertainties on the BAOmeasurements should be subdominant at present, andshould not affect our conclusions given the larger statisti-cal uncertainties on our H results. In this section, we compare our results to earlier inversedistance ladder measurements by Aubourg et al. (2015)and Alam et al. (2017), using the JLA supernova sam-ple. Aubourg et al. (2015) found H = . ± . km s − Mpc − with BAOs from BOSS DR11, which changed onlymarginally to H = . ± . km s − Mpc − in Alam et al.(2017) with BOSS DR12. We note that these values are con-sistent with our value of H = . ± . Mpc − using theDES-SN3YR sample, and H = . ± . km s − Mpc − using JLA.We emphasize a number of differences in the methods tothese papers. Aubourg et al. (2015) and Alam et al. (2017)assumed a model for the redshift-distance relation based on Λ CDM, with the addition of two additional, arbitrary den-sity components, which scaled to linear and quadratic orderwith the scale factor, a , in order to allow for the possibilityof physics beyond Λ CDM. In order to constrain the BAOscale, r s , these papers applied priors on matter and baryondensities derived from the CMB, and used a fitting func-tion to calculate r s as a function of these parameters (seeAubourg et al. 2015).In contrast, our method is more physics-agnostic. Thecosmographic model we assume means that we do not haveto assume a Friedmann model for the redshift-distance rela-tion. The prior on the scale of r s also means that we are lesssensitive to the (albeit, well understood) plasma physics ofthe CMB.As for data, we additionally use the BAO measurementfrom Carter et al. (2018). As illustrated in Figure 1, webelieve that the inclusion of this BAO measurement leadsto our marginally lower value of H with the JLA sample:67.1, vs. 67.3 km s − Mpc − in Aubourg et al. (2015) andAlam et al. (2017).Although our value of H has a larger uncertainty thanin Alam et al. (2017) (1.3 vs. 1.0 km s − Mpc − ), this is dueto our more physics-agnostic model, as opposed to the data.Comparing both sets of SNe directly with our cosmographicmodel, we find the same uncertainty with DES-SN3YR as MNRAS000
The contributions of systematic uncertainties on our H measurement. For each term we quantify the shift in the valueof H , the value of the systematic uncertainty, and the fractionof the uncertainty compared to the statistical uncertainty. Theindividual uncertainties will not necessarily sum in quadratureto the total uncertainty, due to different redshift weightings andcorrelations in each term. uncertainty, which reduces our sensitivity to H , leading to avalue of H = . ± . km s − Mpc − . This test illustrates theimportance of knowing the absolute scale of the sound hori-zon, while also indicating that future BAO measurementsfrom galaxy redshift surveys (e.g., Euclid and DESI) shouldhelp constrain this parameter independent of the CMB andthus remove any reliance on early universe plasma physics.One possible concern is the assumption of a fiducial cos-mology in converting the galaxy angular positions and red-shifts observed in a galaxy redshift survey (such as BOSS)into the power spectrum of galaxy clustering where the BAOsignal is determined. As stated in Section 2.1, this issue isaddressed by assuming a scaling law (Equation 13) which‘dilates’ the distance being tested to the fiducial cosmologyused to calculate the galaxy power spectrum. The applicabil-ity of such scaling was first studied in detail by Padmanab-han & White (2008) who showed, using N -body simulations,a systematic uncertainty of only (cid:39) on α BAO over a widerange of α BAO values, or more importantly, over a wide rangeof alternative cosmologies.It is also worth stressing that the BAO signal is es-timated in a series of narrow redshift bins, thus allowingfor uncertainties between the assumed and fiducial cosmol-ogy to be minimised across any single redshift shell (assum-ing cosmologies with a smooth redshift-distance relationshipclose to Λ CDM). The effects of such redshift binning has re-cently been examined by Zhu et al. (2016) using mock galaxycatalogues that closely mimic BOSS, and they found theirBAO analysis and measurements remained unbiased evenwhen the assumed fiducial cosmology differed from the true(simulation) cosmology. They also confirmed an uncertaintyof just % on α BAO over a range of different assumptions(fiducial cosmologies, pivot redshifts, redshift-space distor- tion parameters, and galaxy bias models) which could befurther improved to the sub-percent level with future opti-mal redshift weighting schemes (e.g., Zhu et al. 2015). Forreference, Alam et al. (2017) assumed a 0.3% systematic un-certainty on α BAO from their fitting methodologies.We note that the redshifts at which the BAO measure-ments are made are approximate, because they are weightedaverages of the redshifts of all of the pairs of galaxies thatgo into generating the correlation function. The weightingcan depend on the choice of using average redshifts or aver-age distances, which adds some uncertainty to the redshiftof the distance anchor. However, since the slope in Figure 1is shallow, any uncertainty in the centre of the redshift binwould only add a small uncertainty to the measurement of H ( < − Mpc − ).Therefore, the systematic uncertainties on the BAOmeasurements should be subdominant at present, andshould not affect our conclusions given the larger statisti-cal uncertainties on our H results. In this section, we compare our results to earlier inversedistance ladder measurements by Aubourg et al. (2015)and Alam et al. (2017), using the JLA supernova sam-ple. Aubourg et al. (2015) found H = . ± . km s − Mpc − with BAOs from BOSS DR11, which changed onlymarginally to H = . ± . km s − Mpc − in Alam et al.(2017) with BOSS DR12. We note that these values are con-sistent with our value of H = . ± . Mpc − using theDES-SN3YR sample, and H = . ± . km s − Mpc − using JLA.We emphasize a number of differences in the methods tothese papers. Aubourg et al. (2015) and Alam et al. (2017)assumed a model for the redshift-distance relation based on Λ CDM, with the addition of two additional, arbitrary den-sity components, which scaled to linear and quadratic orderwith the scale factor, a , in order to allow for the possibilityof physics beyond Λ CDM. In order to constrain the BAOscale, r s , these papers applied priors on matter and baryondensities derived from the CMB, and used a fitting func-tion to calculate r s as a function of these parameters (seeAubourg et al. 2015).In contrast, our method is more physics-agnostic. Thecosmographic model we assume means that we do not haveto assume a Friedmann model for the redshift-distance rela-tion. The prior on the scale of r s also means that we are lesssensitive to the (albeit, well understood) plasma physics ofthe CMB.As for data, we additionally use the BAO measurementfrom Carter et al. (2018). As illustrated in Figure 1, webelieve that the inclusion of this BAO measurement leadsto our marginally lower value of H with the JLA sample:67.1, vs. 67.3 km s − Mpc − in Aubourg et al. (2015) andAlam et al. (2017).Although our value of H has a larger uncertainty thanin Alam et al. (2017) (1.3 vs. 1.0 km s − Mpc − ), this is dueto our more physics-agnostic model, as opposed to the data.Comparing both sets of SNe directly with our cosmographicmodel, we find the same uncertainty with DES-SN3YR as MNRAS000 , 1–14 (2019) E. Macaulay et al.
Figure 4.
Testing the sensitivity of our parameters to different priors and data cuts. The red contours show our original combinedconstraints from Figure 2. In cyan we show the constraints using just the Low- z sample, and in blue we show the effect of varying ourprior on the sound horizon, to allow for the possibility of early dark radiation. with JLA (1.3 km s − Mpc − ), even though DES-SN3YRcomprises 329 SNe, compared to 740 in JLA.Moreover, we also include several additional sources ofsystematic uncertainty which were not included by Betouleet al. (2014). For example, we allow for a systematic uncer-tainty due to a redshift-shift caused by large-scale inhomo-geneities (Calcino & Davis 2017). We also note in Figure 4that the uncertainty in the underlying parent population of SNe Ia is one of the dominant sources of systematic uncer-tainty (e.g., Guy et al. 2010; Conley et al. 2011).This smaller uncertainty on H is driven in part by theBBC analysis method (Kessler & Scolnic 2017), and alsoby the calibration consistency of DECam in the DES SNsample (Flaugher et al. 2015; Brout et al. 2019b). We note,however, that since DES-SN3YR and JLA share some SNe in MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H the Low- z sample, they cannot be combined, as the samplesare not entirely independent. We thank the anonymous referee for comments and sug-gestions which have considerably improved this paper. Wethank An˘ze Slosar for help and advice with the method andlikelihood code, and Paul Carter and Florian Beutler forhelp with the BAO data.E.M., R.C.N., D.B. acknowledge funding from STFCgrant ST/N000668/1. TC acknowledges the University ofPortsmouth for a Dennis Sciama Fellowship. D.S. is sup-ported by NASA through Hubble Fellowship grant HST-HF2-51383.001 awarded by the Space Telescope Science In-stitute, which is operated by the Association of Universitiesfor Research in Astronomy, Inc., for NASA, under contractNAS 5- 26555. We acknowledge funding from ERC Grant615929.A.V.F. is grateful for financial assistance from NSFgrant AST-1211916, the Christopher R. Redlich Fund, theTABASGO Foundation, and the Miller Institute for BasicResearch in Science (U.C. Berkeley).The UCSC team is supported in part by NASAgrants 14-WPS14-0048, NNG16PJ34G, NNG17PX03C,NSF grants AST-1518052 and AST-1815935, the Gordon &Betty Moore Foundation, the Heising-Simons Foundation,and by fellowships from the Alfred P. Sloan Foundation andthe David and Lucile Packard Foundation to R.J.F.This paper makes use of observations taken usingthe Anglo-Australian Telescope under programs ATACA/2013B/12 and NOAO 2013B-0317; the Gemini Obser-vatory under programs NOAO 2013A-0373/GS-2013B-Q-45, NOAO 2015B-0197/GS-2015B-Q-7, and GS-2015B-Q-8; the Gran Telescopio Canarias under programs GTC77-13B, GTC70-14B, and GTC101-15B; the Keck Obser-vatory under programs U063-2013B, U021-2014B, U048-2015B, U038-2016A; the Magellan Observatory under pro-grams CN2015B-89; the MMT under 2014c-SAO-4, 2015a-SAO-12, 2015c-SAO-21; the South African Large Telescopeunder programs 2013-1-RSA OTH-023, 2013-2-RSA OTH-018, 2014-1-RSA OTH-016, 2014-2-SCI-070, 2015-1-SCI-063, and 2015-2-SCI-061; and the Very Large Telescope un-der programs ESO 093.A-0749(A), 094.A-0310(B), 095.A-0316(A), 096.A-0536(A), 095.D-0797(A).Funding for the DES Projects has been provided bythe U.S. Department of Energy, the U.S. National Sci-ence Foundation, the Ministry of Science and Education ofSpain, the Science and Technology Facilities Council of theUnited Kingdom, the Higher Education Funding Council forEngland, the National Center for Supercomputing Applica-tions at the University of Illinois at Urbana-Champaign, theKavli Institute of Cosmological Physics at the Universityof Chicago, the Center for Cosmology and Astro-ParticlePhysics at the Ohio State University, the Mitchell Institutefor Fundamental Physics and Astronomy at Texas A&MUniversity, Financiadora de Estudos e Projetos, Funda¸c˜aoCarlos Chagas Filho de Amparo `a Pesquisa do Estado do Riode Janeiro, Conselho Nacional de Desenvolvimento Cient´ı-fico e Tecnol´ogico and the Minist´erio da Ciˆencia, Tecnologia e Inova¸c˜ao, the Deutsche Forschungsgemeinschaft and theCollaborating Institutions in the Dark Energy Survey.The Collaborating Institutions are Argonne NationalLaboratory, the University of California at Santa Cruz,the University of Cambridge, Centro de InvestigacionesEnerg´eticas, Medioambientales y Tecnol´ogicas-Madrid, theUniversity of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Ei-dgen¨ossische Technische Hochschule (ETH) Z¨urich, FermiNational Accelerator Laboratory, the University of Illi-nois at Urbana-Champaign, the Institut de Ci`encies del’Espai (IEEC/CSIC), the Institut de F´ısica d’Altes Ener-gies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit¨at M¨unchen and the associated Ex-cellence Cluster Universe, the University of Michigan, theNational Optical Astronomy Observatory, the University ofNottingham, The Ohio State University, the University ofPennsylvania, the University of Portsmouth, SLAC NationalAccelerator Laboratory, Stanford University, the Universityof Sussex, Texas A&M University, and the OzDES Member-ship Consortium.Based in part on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Obser-vatory, which is operated by the Association of Universi-ties for Research in Astronomy (AURA) under a cooperativeagreement with the National Science Foundation.The DES data management system is supported bythe National Science Foundation under Grant NumbersAST-1138766 and AST-1536171. The DES participants fromSpanish institutions are partially supported by MINECOunder grants AYA2015-71825, ESP2015-66861, FPA2015-68048, SEV-2016-0588, SEV-2016-0597, and MDM-2015-0509, some of which include ERDF funds from the Euro-pean Union. IFAE is partially funded by the CERCA pro-gram of the Generalitat de Catalunya. Research leading tothese results has received funding from the European Re-search Council under the European Union’s Seventh Frame-work Program (FP7/2007-2013) including ERC grant agree-ments 240672, 291329, and 306478. We acknowledge sup-port from the Australian Research Council Centre of Excel-lence for All-sky Astrophysics (CAASTRO), through projectnumber CE110001020, and the Brazilian Instituto Nacionalde Ciˆencia e Tecnologia (INCT) e-Universe (CNPq grant465376/2014-2).This manuscript has been authored by Fermi ResearchAlliance, LLC under Contract No. DE-AC02-07CH11359with the U.S. Department of Energy, Office of Science, Of-fice of High Energy Physics. The United States Governmentretains and the publisher, by accepting the article for pub-lication, acknowledges that the United States Governmentretains a non-exclusive, paid-up, irrevocable, world-wide li-cense to publish or reproduce the published form of thismanuscript, or allow others to do so, for United States Gov-ernment purposes.
REFERENCES
Addison G. E., Watts D. J., Bennett C. L., Halpern M., HinshawG., Weiland J. L., 2018, ApJ, 853, 119Alam S., et al., 2017, MNRAS, 470, 2617Anderson L., et al., 2014, MNRAS, 441, 24MNRAS000
Addison G. E., Watts D. J., Bennett C. L., Halpern M., HinshawG., Weiland J. L., 2018, ApJ, 853, 119Alam S., et al., 2017, MNRAS, 470, 2617Anderson L., et al., 2014, MNRAS, 441, 24MNRAS000 , 1–14 (2019) E. Macaulay et al.
Aubourg ´E., et al., 2015, Phys. Rev. D, 92, 123516Bassett B. A., Kunz M., 2004, ApJ, 607, 661Bernal J. L., Verde L., Riess A. G., 2016, J. Cosmology Astropart.Phys., 10, 019Betoule M., et al., 2014, A&A, 568, A22Beutler F., et al., 2011, MNRAS, 416, 3017Birrer S., et al., 2019, MNRAS, 484, 4726Blake C., Glazebrook K., 2003, ApJ, 594, 665Blake C., et al., 2011, MNRAS, 418, 1707Brout D., et al., 2019a, ApJ, 874, 106Brout D., et al., 2019b, ApJ, 874, 150Busca N. G., et al., 2013, A&A, 552, A96Calcino J., Davis T., 2017, Journal of Cosmology and Astro-Particle Physics, 2017, 038Capozziello S., Cardone V. F., Salzano V., 2008, Phys. Rev. D,78, 063504Carter P., Beutler F., Percival W. J., Blake C., Koda J., RossA. J., 2018, MNRAS, 481, 2371Conley A., et al., 2011, ApJS, 192, 1Contreras C., et al., 2010, AJ, 139, 519Costa A. A., Xu X.-D., Wang B., Abdalla E., 2017, J. CosmologyAstropart. Phys., 1, 028Croft R. A. C., Dailey M., 2015, Quarterly Physics Review No 1,pp 1–14D’Andrea C. B., et al., 2018, arXiv e-prints, p. arXiv:1811.09565DES Collaboration 2018, MNRAS, 480, 3879DES Collaboration et al., 2018, ApJS, 239, 18DES Collaboration et al., 2019, ApJ, 872, L30Di Valentino E., Melchiorri A., Silk J., 2016, Physics Letters B,761, 242Di Valentino E., Melchiorri A., Mena O., 2017, Phys. Rev. D, 96,043503Di Valentino E., Linder E. V., Melchiorri A., 2018, Phys. Rev. D,97, 043528Eisenstein D. J., et al., 2005, ApJ, 633, 560Farooq O., Ranjeet Madiyar F., Crandall S., Ratra B., 2017, ApJ,835, 26Feeney S. M., Peiris H. V., Williamson A. R., Nissanke S. M.,Mortlock D. J., Alsing J., Scolnic D., 2019, Phys. Rev. Lett.,122, 061105Flaugher B., et al., 2015, AJ, 150, 150Folatelli G., et al., 2010, AJ, 139, 120Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013,PASP, 125, 306Freedman W. L., 2017, Nature Astronomy, 1, 0169G´omez-Valent A., Amendola L., 2018, J. Cosmology Astropart.Phys., 4, 051Grandis S., Rapetti D., Saro A., Mohr J. J., Dietrich J. P., 2016,MNRAS, 463, 1416Guidorzi C., et al., 2017, ApJ, 851, L36Guy J., et al., 2007, A&A, 466, 11Guy J., et al., 2010, A&A, 523, A7Hamuy M., et al., 2006, PASP, 118, 2Hicken M., et al., 2009, ApJ, 700, 331Hicken M., et al., 2012, The Astrophysical Journal SupplementSeries, 200, 12Joudaki S., et al., 2017, MNRAS, 471, 1259Kelly P. L., Hicken M., Burke D. L., Mandel K. S., Kirshner R. P.,2010, ApJ, 715, 743Kessler R., Scolnic D., 2017, ApJ, 836, 56Kessler R., et al., 2009, ApJS, 185, 32Kessler R., et al., 2019, MNRAS, p. 472Krisciunas K., et al., 2017, AJ, 154, 211Lampeitl H., et al., 2010a, MNRAS, 401, 2331Lampeitl H., et al., 2010b, ApJ, 722, 566Lasker J., et al., 2019, MNRAS, 485, 5329Le Borgne D., Rocca-Volmerange B., 2002, A&A, 386, 446 Lemos P., Lee E., Efstathiou G., Gratton S., 2019, MNRAS, 483,4803Macaulay E., Wehus I. K., Eriksen H. K., 2013, Physical ReviewLetters, 111, 161301Moresco M., Verde L., Pozzetti L., Jimenez R., Cimatti A., 2012,J. Cosmology Astropart. Phys., 7, 053Muthukrishna D., Parkinson D., 2016, J. Cosmology Astropart.Phys., 11, 052Padmanabhan N., White M., 2008, Phys. Rev. D, 77, 123540Perlmutter S., et al., 1999, ApJ, 517, 565Planck Collaboration et al., 2018, ArXiv e-prints, 1807.06209,Pourtsidou A., Tram T., 2016, Phys. Rev. D, 94, 043518Riess A. G., et al., 1998, AJ, 116, 1009Riess A. G., et al., 2016, ApJ, 826, 56Riess A. G., et al., 2018, ApJ, 855, 136Ross A. J., Samushia L., Howlett C., Percival W. J., Burden A.,Manera M., 2015, MNRAS, 449, 835Scolnic D., et al., 2015, ApJ, 815, 117Scolnic D. M., et al., 2018, ApJ, 859, 101Seo H.-J., Eisenstein D. J., 2003, ApJ, 598, 720Stritzinger M. D., et al., 2011, AJ, 142, 156Sullivan M., et al., 2010, MNRAS, 406, 782Vagnozzi S., Giusarma E., Mena O., Freese K., Gerbino M., HoS., Lattanzi M., 2017, Phys. Rev. D, 96, 123503Verde L., Bellini E., Pigozzo C., Heavens A. F., Jimenez R., 2017,J. Cosmology Astropart. Phys., 4, 023Wu H.-Y., Huterer D., 2017, MNRAS, 471, 4946Yang W., Xu L., 2014, Phys. Rev. D, 89, 083517Y`eche C., Palanque-Delabrouille N., Baur J., du Mas des Bour-boux H., 2017, J. Cosmology Astropart. Phys., 6, 047Zhang M.-J., Li H., Xia J.-Q., 2017a, European Physical JournalC, 77, 434Zhang B. R., Childress M. J., Davis T. M., Karpenka N. V., Lid-man C., Schmidt B. P., Smith M., 2017b, MNRAS, 471, 2254Zhao G.-B., et al., 2017, Nature Astronomy, 1, 627Zhu F., Padmanabhan N., White M., 2015, MNRAS, 451, 236Zhu F., Padmanabhan N., White M., Ross A. J., Zhao G., 2016,MNRAS, 461, 2867de Putter R., Linder E. V., Mishra A., 2014, Phys. Rev. D, 89,103502
APPENDIX A: TESTS OF THEPARAMETER-FITTING CODE
In this section, we test our method with artificially gener-ated SN and BAO distances, to ensure we can recover theinput parameters used to generate the distances. To gener-ate the artificial distances, we use our cosmographic modelwith values chosen at H = . km s − Mpc − , q = − . , j = − . , s = − . , M B = − . , and r s = Mpc.To generate the artificial supernova distances, we calcu-lated fiducial distance moduli µ fid ( z ) at the redshift values ofthe genuine supernova data. To generate artificial distancemoduli with a realistic dispersion, we drew realisations froma correlated Gaussian distribution centred on µ fid with co-variance given by the genuine data covariance matrix.We took a similar approach to generating artificial dis-tances for the BOSS DR12 BAO measurements for this cos-mographic model. We first calculated a vector of fiducial H fid ( z ) and D fid m ( z ) at the three effective redshift bins of theBAO measurements. To generate realisations of the BAOdata, we then drew samples from a correlated Gaussian dis-tribution centred on this observation vector with covarianceof the genuine DR12 data covariance matrix. MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H In total, we generated one hundred artificial realisationsof DES-SN3YR and BOSS DR12 data-sets. These simula-tions are shown in Figure A1 along with the average of theserealisations (contours). These simulations confirm we canaccurately recover the input parameters, and the uncertain-ties on the simulated parameter measurements are consistentwith our genuine uncertainties.
APPENDIX B: AUTHOR AFFILIATIONS Institute of Cosmology and Gravitation, University ofPortsmouth, Portsmouth, PO1 3FX, UK Department of Physics and Astronomy, University ofPennsylvania, Philadelphia, PA 19104, USA School of Mathematics and Physics, University of Queens-land, Brisbane, QLD 4072, Australia ARC Centre of Excellence for All-sky Astrophysics(CAASTRO) The Research School of Astronomy and Astrophysics,Australian National University, ACT 2601, Australia African Institute for Mathematical Sciences, 6 MelroseRoad, Muizenberg, 7945, South Africa South African Astronomical Observatory, P.O.Box 9,Observatory 7935, South Africa Kavli Institute for Cosmological Physics, University ofChicago, Chicago, IL 60637, USA Department of Astronomy and Astrophysics, Universityof Chicago, Chicago, IL 60637, USA Lawrence Berkeley National Laboratory, 1 CyclotronRoad, Berkeley, CA 94720, USA School of Physics and Astronomy, University ofSouthampton, Southampton, SO17 1BJ, UK Cerro Tololo Inter-American Observatory, NationalOptical Astronomy Observatory, Casilla 603, La Serena,Chile Fermi National Accelerator Laboratory, P. O. Box 500,Batavia, IL 60510, USA Korea Astronomy and Space Science Institute, Yuseong-gu, Daejeon, 305-348, Korea LSST, 933 North Cherry Avenue, Tucson, AZ 85721,USA Department of Physics & Astronomy, University CollegeLondon, Gower Street, London, WC1E 6BT, UK George P. and Cynthia Woods Mitchell Institute forFundamental Physics and Astronomy, and Department ofPhysics and Astronomy, Texas A&M University, CollegeStation, TX 77843, USA Kavli Institute for Particle Astrophysics & Cosmology,P. O. Box 2450, Stanford University, Stanford, CA 94305,USA SLAC National Accelerator Laboratory, Menlo Park, CA94025, USA Centro de Investigaciones Energ´eticas, Medioambientalesy Tecnol´ogicas (CIEMAT), Madrid, Spain Laborat´orio Interinstitucional de e-Astronomia - LIneA,Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400,Brazil INAF, Astrophysical Observatory of Turin, I-10025 PinoTorinese, Italy Department of Astronomy, University of Illinois atUrbana-Champaign, 1002 W. Green Street, Urbana, IL 61801, USA National Center for Supercomputing Applications, 1205West Clark St., Urbana, IL 61801, USA Institut de F´ısica d’Altes Energies (IFAE), The BarcelonaInstitute of Science and Technology, Campus UAB, 08193Bellaterra (Barcelona) Spain Institut d’Estudis Espacials de Catalunya (IEEC), 08034Barcelona, Spain Institute of Space Sciences (ICE, CSIC), Campus UAB,Carrer de Can Magrans, s/n, 08193 Barcelona, Spain Observat´orio Nacional, Rua Gal. Jos´e Cristino 77, Riode Janeiro, RJ - 20921-400, Brazil Department of Astronomy/Steward Observatory, 933North Cherry Avenue, Tucson, AZ 85721-0065, USA Jet Propulsion Laboratory, California Institute of Tech-nology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA Department of Astronomy, University of Michigan, AnnArbor, MI 48109, USA Department of Physics, University of Michigan, AnnArbor, MI 48109, USA Department of Astronomy, University of California,Berkeley, CA 94720-3411, USA Miller Senior Fellow, Miller Institute for Basic Researchin Science, University of California, Berkeley, CA 94720,USA Santa Cruz Institute for Particle Physics, Santa Cruz,CA 95064, USA PITT PACC, Department of Physics and Astronomy,University of Pittsburgh, Pittsburgh, PA 15260, USA Instituto de Fisica Teorica UAM/CSIC, UniversidadAutonoma de Madrid, 28049 Madrid, Spain Centre for Astrophysics & Supercomputing, SwinburneUniversity of Technology, Victoria 3122, Australia CENTRA, Instituto Superior T´ecnico, Universidade deLisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal Department of Physics, ETH Zurich, Wolfgang-Pauli-Strasse 16, CH-8093 Zurich, Switzerland Center for Cosmology and Astro-Particle Physics, TheOhio State University, Columbus, OH 43210, USA Department of Physics, The Ohio State University,Columbus, OH 43210, USA Max Planck Institute for Extraterrestrial Physics,Giessenbachstrasse, 85748 Garching, Germany Universit¨ats-Sternwarte, Fakult¨at f¨ur Physik, Ludwig-Maximilians Universit¨at M¨unchen, Scheinerstr. 1, 81679M¨unchen, Germany Harvard-Smithsonian Center for Astrophysics, Cam-bridge, MA 02138, USA Department of Physics, University of Namibia, 340Mandume Ndemufayo Avenue, Pionierspark, Windhoek,Namibia Australian Astronomical Optics, Macquarie University,North Ryde, NSW 2113, Australia Sydney Institute for Astronomy, School of Physics, A28,The University of Sydney, NSW 2006, Australia Departamento de F´ısica Matem´atica, Instituto de F´ısica,Universidade de S˜ao Paulo, CP 66318, S˜ao Paulo, SP,05314-970, Brazil Department of Astronomy, The Ohio State University,Columbus, OH 43210, USA Instituci´o Catalana de Recerca i Estudis Avan¸cats,E-08010 Barcelona, Spain
MNRAS000
MNRAS000 , 1–14 (2019) E. Macaulay et al. L i k e li h oo d
10 5 0 5 10 s r s H q j .
20 19 .
15 19 .
10 19 .
05 19 .
00 18 . M B s . . . . . . . r s
66 68 70 72 74 H . . . . . . q j Figure A1.
The best-fit parameters for 100 mock realisations (see text for details). The black dashed lines show the input parametervalues for our mock realisations, while the light blue contours show the 68% confidence region for each of the hundred realisations. Thedark blue points show the maximum likelihood values for each realisation. The magenta ellipses are the one and two standard deviationsof these best-fit points, centred on the magenta crosses, at the averages of the individual maximum likelihood values. Division of Theoretical Astronomy, National Astronom-ical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo181-8588, Japan Institute of Astronomy and Astrophysics, AcademiaSinica, Taipei 10617, Taiwan Department of Physics and Astronomy, Pevensey Build-ing, University of Sussex, Brighton, BN1 9QH, UK Brandeis University, Physics Department, 415 SouthStreet, Waltham MA 02453 Instituto de F´ısica Gleb Wataghin, Universidade Estad-ual de Campinas, 13083-859, Campinas, SP, Brazil Computer Science and Mathematics Division, Oak RidgeNational Laboratory, Oak Ridge, TN 37831 Observatories of the Carnegie Institution for Science, 813
MNRAS , 1–14 (2019) irst Cosmological Results using SNe Ia from DES: Measurement of H Santa Barbara St., Pasadena, CA 91101, USA Argonne National Laboratory, 9700 South Cass Avenue,Lemont, IL 60439, USA
This paper has been typeset from a TEX/L A TEX file prepared bythe author.MNRAS000