Issue 
A&A
Volume 628, August 2019



Article Number  L7  
Number of page(s)  5  
Section  Letters to the Editor  
DOI  https://doi.org/10.1051/00046361/201935980  
Published online  20 August 2019 
Letter to the Editor
The Hubble constant determined through an inverse distance ladder including quasar time delays and Type Ia supernovae
^{1}
MaxPlanckInstitut für Astrophysik, KarlSchwarzschildStr. 1, 85748 Garching, Germany
email: tauben@mpagarching.mpg.de
^{2}
PhysikDepartment, Technische Universität München, JamesFranckStr. 1, 85748 Garching, Germany
^{3}
Institute of Astronomy and Astrophysics, Academia Sinica, 11F of ASMAB, No.1, Section 4, Roosevelt Rd., Taipei 10617, Taiwan
^{4}
Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), Univ. of Tokyo, Kashiwa 2778583, Japan
^{5}
Department of Physics and Astronomy, University of California, Los Angeles, CA 900951547, USA
^{6}
Laboratoire d’Astrophysique, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland
^{7}
National Astronomical Observatory of Japan, 2211 Osawa, Mitaka, Tokyo 1818588, Japan
^{8}
Subaru Telescope, National Astronomical Observatory of Japan, 650 N Aohoku Pl, Hilo 96720, Japan
^{9}
Department of Physics, University of California, Davis, 1 Shields Avenue, Davis, CA 95616, USA
Received:
29
May
2019
Accepted:
20
July
2019
Context. The precise determination of the presentday expansion rate of the Universe, expressed through the Hubble constant H_{0}, is one of the most pressing challenges in modern cosmology. Assuming flat ΛCDM, H_{0} inference at high redshift using cosmic microwave background data from Planck disagrees at the 4.4σ level with measurements based on the local distance ladder made up of parallaxes, Cepheids, and Type Ia supernovae (SNe Ia), often referred to as Hubble tension. Independent cosmologicalmodelinsensitive ways to infer H_{0} are of critical importance.
Aims. We apply an inverse distance ladder approach, combining stronglensing timedelay distance measurements with SN Ia data. By themselves, SNe Ia are merely good indicators of relative distance, but by anchoring them to strong gravitational lenses we can obtain an H_{0} measurement that is relatively insensitive to other cosmological parameters.
Methods. A cosmological parameter estimate was performed for different cosmological background models, both for stronglensing data alone and for the combined lensing + SNe Ia data sets.
Results. The cosmologicalmodel dependence of stronglensing H_{0} measurements is significantly mitigated through the inverse distance ladder. In combination with SN Ia data, the inferred H_{0} consistently lies around 73–74 km s^{−1} Mpc^{−1}, regardless of the assumed cosmological background model. Our results agree closely with those from the local distance ladder, but there is a > 2σ tension with Planck results, and a ∼1.5σ discrepancy with results from an inverse distance ladder including Planck, baryon acoustic oscillations, and SNe Ia. Future stronglensing distance measurements will reduce the uncertainties in H_{0} from our inverse distance ladder.
Key words: gravitational lensing: strong / cosmological parameters / distance scale
© S. Taubenberger et al. 2019
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Open Access funding provided by Max Planck Society.
1. Introduction
Ever since Georges Lemaître and Edwin Hubble discovered that our Universe is expanding (Lemaître 1927, 1931; Hubble 1929), astronomers have sought to measure the Hubble constant H_{0} that characterises the presentday expansion rate. For decades there was the “factor of 2 controversy” in the H_{0} measurements, culminating in the Hubble Space Telescope Key Project that pinned down H_{0} to 72 ± 8 km s^{−1} Mpc^{−1} using the Cepheids distance ladder with several secondary distance indicators including Type Ia Supernovae (SNe Ia) (Freedman et al. 2001; Freedman 2017). In recent years, another controversy on H_{0} has emerged, particularly between the measurements based on the cosmic microwave background (CMB; H_{0} = 67.36 ± 0.54 km s^{−1} Mpc^{−1} for flat ΛCDM; Planck Collaboration VI 2018) and the local distance ladder (SH0ES programme; H_{0} = 74.03 ± 1.42 km s^{−1} Mpc^{−1}; Riess et al. 2019). The value of H_{0} inferred from the CMB depends on the background cosmology, and the 4.4σ tension between the Planck and SH0ES measurements refers to a standard flat ΛCDM cosmology with a spatially flat Universe consisting of cold dark matter and a dark energy that is described by the cosmological constant Λ.
This tension, if not resolved by systematic effects (e.g., Rigault et al. 2015, 2018; Jones et al. 2018; Roman et al. 2018), indicates new physics beyond flat ΛCDM (e.g., Di Valentino et al. 2018; Mörtsell & Dhawan 2018; Adhikari & Huterer 2019; Agrawal et al. 2019; Kreisch et al. 2019; Pandey et al. 2019; Poulin et al. 2019; Vattis et al. 2019). Independent measurements of H_{0} are particularly important in order to verify this tension, given the potential of discovering new physics. Methods based on Type IIP supernova expanding photospheres (Schmidt et al. 1994; Gall et al. 2016), water masers (Gao et al. 2016; Braatz et al. 2019), or standard sirens (Schutz 1986; The LIGO Scientific Collaboration et al. 2017) have recently provided independent H_{0} measurements. While they currently have uncertainties that are consistent with both the Planck and the SH0ES measurements, future measurements with larger samples of Type IIP supernovae, water masers, and standard sirens could reduce their uncertainties to help shed light on the H_{0} tension.
Gravitationally lensed quasars are another competitive and independent cosmological probe, particularly in measuring H_{0}. When a quasar is strongly lensed by a foreground galaxy, multiple timedelayed images of the quasar appear around the lens. By measuring the time delays between the multiple quasar images and modelling the mass distributions of both the lens galaxy and other structures along the line of sight, strong lensing provides a measurement of the timedelay distance (D_{Δt}), which is a combination of the angular diameter distances between the observer and the lens (D_{d}), the observer and the quasar (D_{s}), and the lens and the quasar (D_{ds}): D_{Δt} = (1 + z_{d})D_{d}D_{s}/D_{ds} (Refsdal 1964; Suyu et al. 2010; Treu & Marshall 2016). While D_{Δt} is inversely proportional and mostly (but not only) sensitive to H_{0}, the inference of H_{0} from D_{Δt} measurements depends on the cosmological model. In addition, stellar velocity dispersions of the foreground lens galaxies, if available, provide a determination of D_{d}, which can further constrain cosmological models (Paraficz & Hjorth 2009; Jee et al. 2015, 2016, 2019).
The H0LiCOW project (Suyu et al. 2017), in Collaboration with the COSMOGRAIL programme (Courbin et al. 2018), has assembled a sample of lensed quasar systems with exquisitely measured timedelay distances (Bonvin et al. 2017; Wong et al. 2017; Birrer et al. 2019). Through a blind analysis, Birrer et al. (2019) reported km s^{−1} Mpc^{−1} (3% uncertainty, including systematics) from the data of four lensed quasars, in flat ΛCDM. However, as in all cosmological experiments that measure distances outside the scope of the linear Hubble relation D = cz/H_{0}, the inference of H_{0} from D_{Δt} depends on the assumed background cosmology. While much focus in the community is on H_{0} in flat ΛCDM, a cosmologicalmodelindependent inference of H_{0} is valuable.
The inverse distance ladder (Aubourg et al. 2015; Cuesta et al. 2015) provides a way to infer H_{0} which is more modelindependent. The idea is to anchor the relative distances from SNe Ia with an absolute distance measurement. Supernova distances on their own are not absolute distances because of the unknown intrinsic luminosity of SNe (e.g. Leibundgut et al. 2017). Nonetheless, SNe map out the expansion history of the Universe very precisely, and by anchoring their distance scale with absolute distance measurements, cosmologicalmodelinsensitive absolute distance determinations become feasible. By anchoring the SN distance scale using distances measured from baryon acoustic oscillations (BAO), Macaulay et al. (2019) measured an H_{0} from the Dark Energy Survey consistent with that provided by the Planck Collaboration VI (2018) and that does not depend much on cosmological models, although the inference of H_{0} is strongly affected by the assumptions of the size of the sound horizon (Aylor et al. 2019). Recently, Jee et al. (2019) and Wojtak & Agnello (2019) anchored the SN distance scale using D_{d} measured from strongly lensed quasars, resulting in H_{0} values with ∼10% uncertainty, limited by the precision of the D_{d} measurements. With current data, lensed quasars yield tighter constraints on D_{Δt} than D_{d}. In this paper, we explore the use of D_{Δt} as anchor.
This Letter is organised as follows. In Sect. 2 we summarise the D_{Δt} measurements from the four H0LiCOW lenses, and in Sect. 3 we use these distances to infer H_{0} through the inverse distance ladder. We discuss the results in Sect. 4, and provide an outlook in Sect. 5. Throughout the paper, our parameter constraints correspond to the median values of the parameter probability distributions, with 68% credibility intervals given by the 16th and 84th percentiles.
2. Lensing timedelay distances
We use the posterior probability distribution of D_{Δt}, P(D_{Δt}), for the four lensed quasar systems that have been measured by the H0LiCOW Collaboration (listed in Table 1). For three systems (B1608+656, RX J1131−1231, and HE 0435−1223; Suyu et al. 2010, 2014; Sluse et al. 2017; Rusu et al. 2017; Wong et al. 2017; Tihhonova et al. 2018), P(D_{Δt}) is nicely described by the analytic fit
Lens redshifts (z_{d}) and source redshifts (z_{s}) of the strongly lensed quasars from H0LiCOW included in this study.
where x = D_{Δt}/(1 Mpc), and the fitted parameter values (λ_{D}, σ_{D}, μ_{D}) are listed in Table 3 of Bonvin et al. (2017). For the fourth lens system (SDSS 1206+4332), we use the Markov chain Monte Carlo (MCMC) results for D_{Δt} from Birrer et al. (2019)^{1}, and obtain P(D_{Δt}) through a kernel density estimator.
3. Inverse distance ladder: supernovae anchored with strongly lensed quasars
To determine the joint likelihood of cosmological parameters for different experiments and cosmological models, we employ the MontePython v3.1 MCMC sampling package (Audren et al. 2013; Brinckmann & Lesgourgues 2018), which is interfaced with the Boltzmann code CLASS (Lesgourgues 2011) for CMB calculations. As MCMC algorithm, we select MontePython’s MetropolisHastings sampler. For every combination of cosmological probes and assumed cosmological background model, we start with a relatively short MCMC chain (∼50 000 steps) with dynamic updates of the covariance matrix and jumping factor (known as the superupdate strategy in MontePython; Brinckmann & Lesgourgues 2018), which ensures an efficient sampling and an acceptance rate close to the optimal 25%. This is followed by a fully Markovian chain of 500 000 steps, where the covariance matrix and jumping factor are kept fixed at the previously determined values. Our long chains have acceptance rates between 15% and 30% and show good convergence.
The sampling includes the H_{0} and Ω_{CDM} parameters^{2} and, for cosmological models other than flat ΛCDM, combinations of Ω_{k}, w_{0}, and w_{a}. The priors employed for these cosmological parameters are summarised in Table 2. They can have an impact on the inferred parameter posteriors, so we make sure that they are either physically motivated or sufficiently conservative. In those runs where stronglensing data are combined with SN Ia data, four additional nuisance parameters (M_{B}, α, β, and Δ_{M}) are added. They represent the absolute Bband magnitude, the coefficients of light curve stretch (X_{1}) and colour (C) corrections, and the hostgalaxy mass step, respectively, in a SALT2 framework (Guy et al. 2007; Mosher et al. 2014; Betoule et al. 2014):
Priors on cosmological parameters (all uniform) as employed in the MontePython MCMC sampling.
The (luminosity) distance modulus, μ = 5log_{10}(D_{L}/1 Mpc)+25, is thereby calculated as the difference between the apparent peak magnitude in the rest frame B band (m_{B}), and the stretch and colourcorrected absolute Bband magnitude. The empirical mass stepcorrection Δ_{M} is only added for SN host galaxies with stellar masses ≥10^{10} M_{⊙} (Sullivan et al. 2010).
We first concentrate on the cosmological parameter inference using H0LiCOW D_{Δt} data of strongly lensed quasars alone. Four different background cosmologies are investigated: flat ΛCDM; flat wCDM with a redshiftindependent dark energy equationofstate parameter w, which can differ from −1 (corresponding to Λ); flat w_{0}w_{a}CDM with a redshiftdependent dark energy equationofstate parameter ; and nonflat ΛCDM, which covers the possibilities of a negatively or positively curved Universe. The resulting cosmological parameters are shown in Table 3. The energy densities of matter (Ω_{m}), a cosmological constant (Ω_{Λ}), or a more generic form of dark energy (Ω_{DE}) are not tightly constrained by the lensed quasars alone, but the effect of different background cosmologies is very weak for these parameters. For nonΛCDM models, w deviates from −1 by more than 1σ, while the curvature in the nonflat ΛCDM case is consistent with zero. The Hubble constant shows a moderately strong dependence on the background cosmology, in particular on the dark energy equation of state, being 72.9 in both the flat and nonflat ΛCDM cases, but > 80 in the flat wCDM and w_{0}w_{a}CDM cosmologies. This can be explained by the lensedquasar systems spanning redshifts between ∼0.3 (for the most nearby lens) and ∼1.8 (for the most distant quasar; see Table 1), and the necessary extrapolation to obtain the presentday expansion rate of the Universe being cosmologicalmodeldependent, for example due to degeneracy between H_{0} and w (Fig. 1).
Cosmological parameters extracted with MontePython MCMC sampling.
Fig. 1. Contour plots with 68% and 95% confidence regions for H_{0}, Ω_{m}, and w in a flat wCDM cosmology (lefthand side), and for H_{0}, Ω_{m}, and Ω_{k} in a nonflat ΛCDM cosmology (righthand side). Contours based on quasar time delays and SNe Ia (JLA compilation) alone are shown in blue and green, respectively, while those using the inverse distance ladder with both probes combined are overplotted in red. 
The dependence of H_{0} on the assumed background cosmology can be mitigated by combining the quasar D_{Δt} measurements with SN data from the joint light curve analysis (JLA), which is a compilation of 740 spectroscopically confirmed lowz, SDSSII, and SNLS SNe Ia (Betoule et al. 2014). In this inverse distance ladder approach, the SN Ia data are anchored near their highz end by the lensed quasars, and allow for a much improved measurement of the presentday expansion rate compared to the quasar time delays alone. We employ the same priors on cosmological parameters as before (Table 2). In addition to the cosmological models investigated in the lensesonly case, we now also include the more flexible nonflat wCDM and nonflat w_{0}w_{a}CDM models, which did not converge in the lensesonly MCMC chains. The results are again summarised in Table 3. The posteriors for Ω_{m} and Ω_{Λ} (or Ω_{DE}) have tightened up significantly compared to the lensesonly case, which is a merit of SNe Ia being able to map out the relative expansion history of the Universe very well. Similarly, the dark energy equationofstate parameter w in the nonΛCDM models is now better constrained, and very close to −1 in all models. The inferred curvature in the nonflat geometries is slightly larger than before, but still consistent with zero. The Hubble constant, finally, shows only a ∼2% variation with the assumed background cosmology, lying between 72.7 and 74.1 km s^{−1} Mpc^{−1} in all cases. The increased median and uncertainty observed for nonΛCDM cosmologies obtained from lensed quasars alone is no longer an issue when combined with SN Ia data.
For two selected background cosmologies (flat wCDM and nonflat ΛCDM), the full posterior distributions for the sampled cosmological parameters are shown in Fig. 1. The improved constraints on H_{0} and w (in flat wCDM) and on Ω_{m} (in nonflat ΛCDM) when including SN Ia data are evident. Figure 2 compares the marginalised 1D H_{0} posteriors obtained from lensed quasars alone with those obtained from a combination of lensing and SN Ia data for different cosmological models. Using the inverse distance ladder, the peaktopeak scatter in the median H_{0} values of these four models is impressively reduced from 11% to 1%.
Fig. 2. H_{0} posteriors for different cosmologies using H0LiCOW timedelay distance measurements of four strongly lensed quasars only (left), and using the combination of the lensing measurements with the JLA SN Ia data set (right). 
4. Discussion
We now investigate how our H_{0} results compare to those from other cosmological probes: the SH0ES (Riess et al. 2019) and Planck (Planck Collaboration VI 2018) experiments, and a Planck + BAO + SNe Ia inverse distance ladder (Aubourg et al. 2015).
The local distance ladder underlying the SH0ES determination of H_{0} is anchored to parallaxes at z = 0. It is therefore almost completely insensitive to the cosmological background model. As shown in Fig. 3, our inverse distance ladder measurements of H_{0} for different cosmologies all agree very nicely with the SH0ES results of H_{0} = 74.03 ± 1.42 km s^{−1} Mpc^{−1}.
Fig. 3. Comparison between the quasar timedelay + SNe Ia inverse distance ladder with other cosmological probes: CMB data from Planck (Planck Collaboration VI 2018; TT,TE,EE + lowE + lensing), a Planck + BAO + SNe Ia inverse distance ladder from Aubourg et al. (2015), and Cepheid + SN Ia data from the SH0ES project (Riess et al. 2019). The mean and 68% confidence intervals for H_{0} for different background cosmologies are shown for Planck and the two inverse distance ladders. The orangeshaded region reflects the 68% confidence interval for the SH0ES distance ladder, which is anchored locally and is thus insensitive to the cosmological background model. 
Inference of H_{0} based on CMB, on the contrary, takes place at z > 1000, and involves extrapolation to z = 0 by assuming a cosmological model. Hence, while the Planck results for H_{0} are very precise once a flat ΛCDM cosmology is assumed, they can vary widely if this assumption is dropped. The Planck Collaboration VI (2018) provides H_{0} values only for flat ΛCDM (H_{0} = 67.36 ± 0.54 km s^{−1} Mpc^{−1}) and nonflat ΛCDM ( km s^{−1} Mpc^{−1}) cosmologies, and already these differ significantly (see Fig. 3). With cosmologies that do not assume a cosmological constant, no meaningful constraints on H_{0} can be obtained from the CMB alone. Our inverse distance ladder results show a tension of just above 2σ with Planck for both flat ΛCDM and nonflat ΛCDM, which is lower than the tension between Planck and SH0ES owing to our larger error bars compared to SH0ES.
Finally, the Planck + BAO + SNe Ia inverse distance ladder of Aubourg et al. (2015) is anchored to BAO absolute distances at redshifts 0.1–0.6. CMB data are used to set the soundhorizon scale, which BAO distances are inversely proportional to. The inferred values of H_{0} for different cosmological models are in good agreement with each other, clustering between 67 and 68 km s^{−1} Mpc^{−1}, with uncertainties between 1.0% and 1.5%. The discrepancy with our lensingbased inverse distance ladder is between 1.3 and 1.9σ, which is not huge, but systematic. A possible origin of this discrepancy could be the adopted soundhorizon scale from Planck, which is the only earlyUniverse property that enters into the Aubourg et al. (2015) measurement.
5. Outlook
On their own SNe Ia are poor probes of the absolute distance scale of the Universe (and hence H_{0}). In our inverse distance ladder experiment, where an anchor is provided at high redshift by timedelay distances of strongly lensed quasars, their main role is to extrapolate these absolute distance measurements back to redshift zero. This allows us to constrain H_{0} in a way which is (1) rather insensitive to the assumed cosmological background model and (2) independent of Cepheids and the CMB. The Hubble constant derived from this approach (72.7–74.1 km s^{−1} Mpc^{−1}) is consistent with that obtained with the local distance ladder, but deviates at the ∼1.5–2σ level from results based on Planck CMB measurements. The origin of this discrepancy can only be speculated about, but the sound horizon determined by Planck certainly warrants further investigation.
The small (∼2%) dependence of H_{0} on the assumed cosmological model in our inverse distance ladder implies that the precision of the H_{0} inference of 3%–4% is currently limited by the D_{Δt} data for lensed quasars. So far the number of D_{Δt} measurements is small: merely four strongly lensed quasars have been published by the H0LiCOW Collaboration thus far. However, more are soon to come (Rusu et al. 2019a; Chen et al. 2019), and systematic searches through various surveys^{3} are yielding new lensed quasar systems. Some of these are now being monitored by the COSMOGRAIL Collaboration for timedelay measurements (Courbin et al. 2018; Millon et al. in prep.). With the upcoming Large Synoptic Survey Telescope (LSST) and Euclid surveys, many more D_{Δt} measurements are expected, both for strongly lensed quasars and for SNe. Accordingly, the statistical uncertainty on H_{0} from D_{Δt} measurements will shrink substantially in the upcoming years (Shajib et al. 2018), rendering the inverse distance ladder approach that couples timedelay distances with SN Ia data one of the most promising methods for solving the current Hubbletension puzzle.
The chain of the joint constraint on D_{Δt} and D_{d} is released on the H0LiCOW website (http://www.h0licow.org), and we focus on D_{Δt}, marginalising over D_{d}.
These surveys include the Dark Energy Survey (DES) (particularly STRIDES; Treu et al. 2018), Gaia, the HyperSuprime Cam (HSC) survey, the KiloDegree Survey (KiDS), the Panoramic Survey Telescope and Rapid Response System (PanSTARRS) and the Asteroid Terrestrialimpact Last Alert System (VSTATLAS) (e.g., Agnello et al. 2018; KroneMartins et al. 2018; Lemon et al. 2018; Spiniello et al. 2018; Rusu et al. 2019b).
Acknowledgments
We thank the H0LiCOW team for the public release of D_{Δt} likelihoods. SHS thanks the Max Planck Society for the support through the Max Planck Research Group. VB and FC acknowledge support from the Swiss National Science Foundation (SNSF). This project has received funding from the European Research Council (ERC) under the EU’s Horizon 2020 research and innovation programme (grant agreements No. 771776 and No. 787866). We thank Thejs Brinckmann for the helpful hints on how to run MontePython, Andreas Weiss for computing support, and the anonymous referee for constructive comments.
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All Tables
Lens redshifts (z_{d}) and source redshifts (z_{s}) of the strongly lensed quasars from H0LiCOW included in this study.
Priors on cosmological parameters (all uniform) as employed in the MontePython MCMC sampling.
All Figures
Fig. 1. Contour plots with 68% and 95% confidence regions for H_{0}, Ω_{m}, and w in a flat wCDM cosmology (lefthand side), and for H_{0}, Ω_{m}, and Ω_{k} in a nonflat ΛCDM cosmology (righthand side). Contours based on quasar time delays and SNe Ia (JLA compilation) alone are shown in blue and green, respectively, while those using the inverse distance ladder with both probes combined are overplotted in red. 

In the text 
Fig. 2. H_{0} posteriors for different cosmologies using H0LiCOW timedelay distance measurements of four strongly lensed quasars only (left), and using the combination of the lensing measurements with the JLA SN Ia data set (right). 

In the text 
Fig. 3. Comparison between the quasar timedelay + SNe Ia inverse distance ladder with other cosmological probes: CMB data from Planck (Planck Collaboration VI 2018; TT,TE,EE + lowE + lensing), a Planck + BAO + SNe Ia inverse distance ladder from Aubourg et al. (2015), and Cepheid + SN Ia data from the SH0ES project (Riess et al. 2019). The mean and 68% confidence intervals for H_{0} for different background cosmologies are shown for Planck and the two inverse distance ladders. The orangeshaded region reflects the 68% confidence interval for the SH0ES distance ladder, which is anchored locally and is thus insensitive to the cosmological background model. 

In the text 
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