Issue 
A&A
Volume 649, May 2021



Article Number  A61  
Number of page(s)  6  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202039179  
Published online  12 May 2021 
TDCOSMO
V. Strategies for precise and accurate measurements of the Hubble constant with strong lensing
^{1}
Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA
email: sibirrer@stanford.edu
^{2}
Physics and Astronomy Department, University of California, Los Angeles, CA 90095, USA
Received:
13
August
2020
Accepted:
26
February
2021
Strong lensing time delays can measure the Hubble constant H_{0} independently of any other probe. Assuming commonly used forms for the radial mass density profile of the lenses, a 2% precision has been achieved with seven TimeDelay Cosmography (TDCOSMO) lenses, in tension with the H_{0} from the cosmic microwave background. However, without assumptions on the radial mass density profile – and relying exclusively on stellar kinematics to break the masssheet degeneracy – the precision drops to 8% with the current data obtained using the seven TDCOSMO lenses, which is insufficient to resolve the H_{0} tension. With the addition of external information from 33 Sloan Lens ACS (SLACS) lenses, the precision improves to 5% if the deflectors of TDCOSMO and SLACS lenses are drawn from the same population. We investigate the prospect of improving the precision of timedelay cosmography without relying on mass profile assumptions to break the masssheet degeneracy. Our forecasts are based on a previously published hierarchical framework. With existing samples and technology, 3.3% precision on H_{0} can be reached by adding spatially resolved kinematics of the seven TDCOSMO lenses. The precision improves to 2.5% with the further addition of kinematics for 50 nontimedelay lenses from SLACS and the Strong Lensing Legacy Survey. Expanding the samples to 40 timedelay and 200 nontimedelay lenses will improve the precision to 1.5% and 1.2%, respectively. Timedelay cosmography can reach sufficient precision to resolve the Hubble tension at 3–5σ, without assumptions on the radial mass profile of lens galaxies. By obtaining this precision with and without external datasets, we will test the consistency of the samples and enable further improvements based on even larger future samples of timedelay and nontimedelay lenses (e.g., from the Rubin, Euclid, and Roman Observatories).
Key words: gravitational lensing: strong / methods: observational / galaxies: kinematics and dynamics / distance scale / cosmological parameters
© ESO 2021
1. Introduction
Almost a century after it was first measured, the Hubble constant H_{0} still remains arguably the most debated number in cosmology. In the past few years, a discrepancy has emerged between a number of local measurements, and inferences from earlyUniverse probes such as the cosmic microwave background (CMB) and Big Bang nucleosynthesis, under the assumption of flat Λ cold dark matter (ΛCDM) cosmology (see, e.g., Verde et al. 2019, for a recent summary). If this discrepancy is real, and not due to unknown systematic uncertainties in multiple measurements, it implies that the standard ΛCDM model is not sufficient, and new physical ingredients beyond this model are required. From a theoretical standpoint, a number of possible solutions – for example, involving early dark energy – have been proposed (e.g., Knox & Millea 2020, and references therein), often requiring finetuning of free parameters in order to avoid violating other observational constraints. From an observational point of view, besides improving the precision of the measurements, significant attention has turned to the systematic investigation of unknown systematic uncertainties (e.g., Riess et al. 2019, 2020; Freedman et al. 2020).
Strong lensing time delays (hereafter timedelay cosmography, Treu & Marshall 2016, and references therein) provide a onestep measurement of H_{0} that is independent of any other probe, and are thus a powerful contribution to this debate. By assuming standard forms for the mass density profile of earlytype galaxies – consistent with Xray (e.g., Humphrey & Buote 2010) and stellar kinematic observations (e.g., Cappellari 2016, and references therein) – the H0LiCOW/COSMOGRAIL/STRIDES/SHARP (hereafter TDCOSMO^{1}) collaborations achieved 2% precision on H_{0} (Rusu et al. 2020; Wong et al. 2020; Chen et al. 2019; Shajib et al. 2020; Millon et al. 2020), in excellent agreement with the local distance ladder measurement by the SH0ES team (Riess et al. 2019) and more than 3σ statistical tension with earlyUniverse probes (e.g., Aiola et al. 2020). In summary, if the mass density profiles are well described by a powerlaw or a constant masstolight ratio plus a Navarro et al. (1997) dark matter halo^{2}, the tension is significant from the strong lensing measurements alone, corroborating other measurements, and new physics may be required.
Given the important implications of the above discrepancy, the TDCOSMO collaboration is performing a systematic investigation of possible systematic effects. Millon et al. (2020), Gilman et al. (2020) did not find any systematic uncertainty sufficient to resolve the discrepancy, if the two assumed forms of the mass density profile are valid. Attention therefore turned to relaxing the radial profile assumption. Birrer et al. (2020, hereafter TD4) addressed the issue in the most direct way, by choosing a parametrization of the radial mass density profile that is maximally degenerate with H_{0}, via the masssheet transform (MST, Falco et al. 1985). With this more flexible parametrization, H_{0} is only constrained by the measured time delays and stellar kinematics, increasing the uncertainty on H_{0} from 2% to 8% for the TDCOSMO sample of seven lenses, without changing the mean inferred value significantly.
TD4 introduce a hierarchical framework in which external datasets can be combined with the timedelay lenses to improve the precision. These latter authors achieved a 5% precision measurement on H_{0} by combining the TDCOSMO lenses with stellar kinematic measurements of a sample of lenses from the Sloan Lens ACS (SLACS) survey with no timedelay information (Bolton et al. 2008; Auger et al. 2009). The mean of the TDCOSMO+SLACS measurement is offset with respect to the TDCOSMOonly value, in the direction of the CMB value, although still statistically consistent given the uncertainties^{3}. The shift in the mean could be real or it could be due to an intrinsic difference between the deflectors in the TDCOSMO and SLACS samples, arising from selection effects. For example, the two samples are well matched in stellar velocity dispersion, but they differ in redshift; the TDCOSMO sample is source selected and composed mostly of quadruply imaged quasars, while SLACS is deflector selected and dominated by doubly imaged galaxies.
In this paper we outline a twopronged strategy to improve the precision of timedelay cosmography with flexible radial mass profile assumptions, as described in TD4. We use the formalism introduced by TD4 to forecast the precision of H_{0} attainable by improving the kinematic data of the TDCOSMO sample and by expanding and improving the kinematic data of external datasets when drawn from the same underlying deflector galaxy population. We show that, by pursuing both avenues on existing samples and with current technology, one can recover most of the precision achieved through previous stronger assumptions on the mass profile and at the same time test for internal consistency of the TDCOSMO and external datasets, thus verifying a key assumption of TD4. This dual strategy will also be beneficial in the longer term, when the sample size of both timedelay and nontimedelay lenses will expand by order of magnitudes, but the latter will always be a subset of the former due to the observational cost of measuring time delays. We stress that the point of this paper is not to discuss whether specific assumptions about the mass density profile of massive elliptical galaxies are valid or not, but rather to show that with sufficient data one can achieve 2–3% precision on H_{0} without making those assumptions. Of course, as a byproduct, following our proposed strategy, it is also possible to tell whether previous assumptions are sufficient to provide an accurate H_{0} measurement.
This paper is organized as follows. In Sect. 2 we summarize the hierarchical framework and its assumptions, referring the reader to TD4 for details. In Sect. 3 we describe the datasets used for the forecast. In Sect. 4 we present the forecasts. In Sect. 5 we draw our conclusions. Our conclusions are independent of the specific value of H_{0} chosen for the forecast. However, when necessary for visual clarity, we adopt a value of H_{0} = 70 km s^{−1} Mpc^{−1} and Ω_{m} = 0.3. A standard flat ΛCDM cosmology is assumed, with a uniform prior of H_{0} in [0, 150] km s^{−1} Mpc^{−1} and a tight prior on Ω_{m} based on relative distance measurements from type Ia supernovae with 𝒩(μ = 0.3, σ = 0.022) (i.e., comparable to Scolnic et al. 2018). The code used for the analysis presented in this work is available on GitHub^{4}.
2. Summary of the hierarchical framework
2.1. Background
The MST leaves the relative imaging observables unchanged but scales the predicted time delays, posing a fundamental limitation on the power of imaging data to constrain the radial mass profile of strong gravitational lenses, and in turn, H_{0}. In terms of the convergence field, the MST describes a rescaling of a given lens mass profile at angular coordinate θ, κ(θ), with a factor λ, while simultaneously adding a sheet of mass with constant convergence (1 − λ) as
The inferred H_{0} value from the measured time delays scales as
The stellar kinematics of the deflector galaxy – a lensingindependent mass tracer – can constrain the MST for a given family of mass profiles. The constraints on λ depend on the precision of the stellar velocity dispersion (σ^{P}) measurement as
Current uncertainties on the stellar velocity dispersion measurements of order 5%–10% imply that the joint analysis of timedelay and nontimedelay lens samples is required to constrain λ. Equation (3) does not include additional model uncertainties beyond λ in the prediction of the velocity dispersion. The kinematic modeling generally requires a 3D deprojected mass and stellar distribution model. The measured velocity dispersion is luminosity weighted, seeing integrated, and measured in projection along the line of sight. A key component in the interpretation of the velocity dispersion measurement, and thus the inference of λ, is the anisotropy distribution of the stellar orbits
where and are the radial and tangential velocity dispersions, respectively.
2.2. Implementation of the hierarchical framework
We adopt the framework introduced by TD4. Here we provide a brief summary for convenience referring to TD4 for details, including parametrization and adopted priors.
The TD4 framework drastically reduces the mass profile assumptions on individual lenses with respect to previous work, and quantifies any potential effect of the MST with the MST parameter λ applied to a powerlaw radial mass density profile that is maximally degenerate with H_{0}. This approach is similar to that of Birrer et al. (2016) who encoded the MST with a source size regularization in the inference.
Stellar velocity dispersion is assumed to be isotropic in the center and radial in the outer part, following the Osipkov (1979), Merritt (1985) form
where r_{eff} is the halflight radius of the deflector and a_{ani} is the anisotropy scaling factor.
To account for covariances in the parameters and priors on the MST and the stellar anisotropy, TD4 introduced a hierarchical framework to describe the MST parameter λ and the anisotropy parameter a_{ani} at the lens population level, assuming that the lenses are drawn from the same parent population.
The framework is validated on the TimeDelay Lens Modeling Challenge Rung3 mock lenses generated from hydrodynamical simulations (Ding et al. 2021). An external data set of gravitational lenses with kinematics and imaging constraints can be incorporated under the assumption that the deflectors are drawn from the same population as those of the timedelay lenses. Both unresolved and spatially resolved velocity dispersion measurements can be used in this framework. The spatially resolved measurements are particularly useful to constrain the anisotropy of the stellar orbits.
3. Future data sets
We envision parallel improvements in the quality of the data for the timedelay lenses in the TDCOSMO samples (Sect. 3.1) and external datasets composed of nontimedelay lenses (Sect. 3.2). We consider two cases: (i) improvements that can be made with existing^{5} facilities and samples (current scenario); and (ii) gains that can be made with future samples and/or facilities (future scenario). The scenarios are summarized in Table 1.
Observing scenarios and forecasted H_{0} precision.
A clarification is needed for the spatially resolved kinematics. There is an uncertainty floor on the calibration of stellar velocity dispersion due to systematic effects such as the match between stellar templates and target composite stellar populations and knowledge of the instrumental properties. To account for this floor, our stated precision is the overall uncertainty on the mean of the stellar velocity dispersion across the target, while the shape of the velocity dispersion profile is constrained taking into account the covariance between spatial bins.
3.1. Timedelay lenses
One limiting factor of the current TDCOSMO dataset is the precision of the unresolved stellar velocity dispersion measurements, which range between 5 and 10%. A first improvement is to bring all the uncertainties to 5%, which has been demonstrated to be feasible with groundbased spectrographs given sufficient data quality. This is the TDCOSMO5% scenario.
An additional improvement consists in spatially resolved stellar velocity dispersion of the TDCOSMO sample. Such data can be obtained from the ground in the optical in seeinglimited mode (e.g., with MUSE/VLT or KCWI/Keck; hereafter TDCOSMO+OIFU), or in the infrared with adaptive optics correction (e.g., with OSIRIS/Keck; hereafter TDCOSMO+AOIFU). JWST will enable a further improvement over groundbased spatially resolved kinematics owing to its superior stability and absence of emission and absorption from the Earth’s atmosphere (hereafter TDCOSMO+JWSTIFU). In the future scenario we expand the sample to 40 timedelay lenses and assume we can use 30 m class Extremely Large Telescopes (ELTs) with adaptive optics (hereafter TDCOSMO+ELTIFU). JWST data for this future sample would give a similar precision on H_{0}.
3.2. External datasets
There are currently three limiting factors to the external dataset used in TD4, namely (i) the precision of aperture velocity dispersion measurements; (ii) the absolute calibration and sample size of integral field data; and (iii) the overall sample size. In the current scenario, we consider two ways to overcome these limitations. The first is to take 50 lenses from the current SLACS and Strong Lensing Legacy Survey (SL2S) (Sonnenfeld et al. 2013, 2015) samples, selected for data quality and to match the TDCOSMO sample in velocity dispersion, with unresolved velocity dispersion measured with 5% precision (hereafter “+50”). The second involves taking 50 lenses with spatially resolved velocity dispersion measured from seeinglimited integral field data (within reach of current generation integral field spectrographs; hereafter “+50IFU”). In the future scenario, we add 200 nontimedelay lenses to the 40 timedelay lenses described above. We stress that only time delays constrain H_{0} and therefore the external datasets have to be used in combination with the timedelay lenses. Considering all the combinations, we are forecasting a total of 24 scenarios (see Table 1).
3.3. Limitations
We make a few simplifications relative to TD4 to facilitate the exploration of the information content of the data sets; such simplifications are used solely to compare their statistical properties, and are not used in estimating the final uncertainties. We assume that (i) λ and a_{ani} are singlevalued intrinsic distributions without scatter; (ii) lineofsight convergence (as a contribution to λ) has zero average with a known spread of 2%; and (iii) all lenses have the same Einstein radius and halflight radius and thus do not incorporate an additional parametrization to encompass a potential trend in λ as a function of projected distance from the center of the deflector. The first two assumptions do not affect our forecast precision significantly, considering that lineofsight effects are only a minor contribution to the statistical error budget. A more sophisticated representation of the third effect could in principle improve the precision of the measurement based on unresolved velocity dispersion, by providing a form of spatially resolved kinematics for the ensemble, given the range in Einstein and halflight radii for the real samples^{6}.
Our forecasts are robust to the details of the mock samples. For completeness and repeatability we specify that we adopted uniform priors on deflector and source redshift and typical values and measurement uncertainties for the Einstein radii, effective radii, and slope of the mass density profile prior to MST, as detailed in the Jupyter Notebook^{7}.
It is important to state some of the key simplifying assumptions of TD4, namely that (i) spherical case of the Jeans equation for kinematic modeling; and (ii) no rotational support, i.e., no bulk rotation of the lens. Most lenses are slow rotators (Barnabè et al. 2011) and therefore we expect the approximation to be valid to first order (see, Yıldırım et al. 2020, for forecasts based on nonspherical kinematics). The integral field data proposed in this paper would allow the hierarchical framework to be expanded to include departures from spherical symmetry and pure pressure support, extend the anisotropy model, and mitigate this potential source of systematic uncertainty.
4. Forecasts
In examining the performance of our proposed strategies it is worth using as a reference the 2% precision achieved under the assumption of powerlaw or composite radial mass profiles (Wong et al. 2020; Millon et al. 2020), and the 1% precision forecasted by Shajib et al. (2018) under the same assumption. This is the precision floor for our forecast with the current and future samples of timedelay measurements and we show that the MST can be controlled to get fairly close to this level.
In the TDCOSMOonly current scenario (Fig. 1), spatially resolved kinematics enables a precision of 3.5% for JWST. Ground based technology reaches approximately 4.7%, a substantial improvement over the 8.5% without IFU, limited fundamentally by the absolute precision that can be achieved on the stellar velocity dispersion owing to instrumental effects (AOIFU) and contamination from QSO light (OIFU).
Fig. 1. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of the seven TDCOSMO lenses (current scenario) as specified in Table 1 column δH_{0} (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 
In the TDCOSMO+external current scenario (Fig. 2), adding only unresolved velocity dispersion does not improve the precision very much because of the massanisotropy degeneracy (e.g., Courteau et al. 2014, and references therein), and our assumption that all the lenses have the same Einstein and effective radii. However, adding IFU data breaks that degeneracy and recovers almost the same level of precision as making assumptions on the radial mass profile (2.5–2.7% vs. 2%).
Fig. 2. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of the seven TDCOSMO lenses (current scenario) observed with aperture spectroscopy of 5% precision as well as in the case where external data sets are added, as specified in Table 1 (TDCOSMO5% row) (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 
We note that all the combinations that have some IFU data and at least unresolved velocity dispersion for the external dataset achieve a precision better than 3% (Table 1). In this mode, the MSTrelated uncertainty on H_{0} is 1.6%, subdominant in regard to timedelay measurements, the angular lens model component, and the lineofsight convergence of the TDCOSMO sample of seven lenses.
Similar considerations apply to the future scenarios illustrated in Figs. 3 and 4, except for the precision that reaches 1.2–1.5% by virtue of the larger samples. A significant component of the error budget at the 1% level arises from the uncertainty in the relative expansion history of the Universe (in our case the prior on Ω_{m}). It is encouraging that, thanks to the external datasets, we can reach a similar precision to that forecasted by Shajib et al. (2018). These latter authors broke the MST by assuming the mass profile is a power law. We break it with spatially resolved kinematics and external datasets.
Fig. 3. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of a future sample of 40 TDCOSMO lenses (future scenario) as specified in Table 1 in the row of TDCOSMO5% (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 
Fig. 4. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of a future sample of 40 TDCOSMO lenses (future scenario) observed with aperture spectroscopy of 5% precision and additional external data sets specified in Table 1 in the row of TDCOSMO5% (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 
5. Conclusions
We describe two strategies to measure H_{0} with 2.5–3.5% precision with gravitational time delays while accounting for the uncertainty introduced by the masssheet transformation. The first is based on current samples of 7 timedelay lenses and existing technology and the second is based on adding 50 nontimedelay lenses. The same strategies, applied to a future sample of 40 timedelay and 200 nontimedelay lenses can achieve 1.2–1.5% precision. The keys to achieving this precision are spatially resolved kinematics and the inclusion of datasets of nontimedelay lenses in a hierarchical framework.
These two strategies are not mutually exclusive and both should be pursued. The TDCOSMOonly approach has the advantage of not relying on the assumption that the timedelay and nontimedelay galaxies are drawn from the same parent population. With this additional assumption, the TDCOSMO+external approach allows for further improvement in precision. The precision of each approach is sufficient to test the mutual consistency among different samples while simultaneously fitting for H_{0}. If verified, potentially with the extension of the hierarchical framework, the consistency will enable the cosmological exploitation of larger samples of nontimedelay lenses that are expected to be discovered by future surveys (Oguri & Marshall 2010).
Following our proposed strategies, timedelay cosmography will, in the near future, have sufficient precision to distinguish the current ∼8% difference between early and late Universe measurements at the 3−5σ level, without relying on assumptions on the radial^{8} mass profile of lens galaxies^{9} to break the masssheet degeneracy.
The TD4 measurements are in statistical agreement with each other and with the earlier H0LiCOW/SHARP/STRIDES measurements based on radial mass profile assumptions. TD4 is also consistent, by construction, with the study by Shajib et al. (2021), because they share the same measurements for SLACS. Shajib et al. (2021) concluded that NFW+stars (using wider priors on mass and concentration than earlier H0LiCOW/SHARP/STRIDES measurements) is a sufficiently accurate description of the mass density profile of the SLACS lenses. However, small departures from those forms are allowed by the data, resulting in the uncertainties quoted by TD4.
See, e.g., Padmanabhan et al. (2004) for an example of derivation of ensemble radial profiles from integrated velocity dispersion measurements.
Acknowledgments
SB and TT thank the TDCOSMO team for useful discussions, and in particular Anowar Shajib for and Frederic Courbin for internal review and final read, respectively. TT acknowledges support from the National Science Foundation through grants NSFAST1906976 and NSFAST1906976 and, from NASA through grants HSTGO15320 and HSTGO15652, from the Moore Foundation through grant 8548, and from the Packard Foundation through a Packard Research Fellowship. This work uses the following opensource software packages: LENSTRONOMY (Birrer & Amara 2018), HIERARC (Birrer et al. 2020), ASTROPY (Astropy Collaboration 2013, 2018), EMCEE (ForemanMackey et al. 2013), CORNER
References
 Aiola, S., Calabrese, E., Maurin, L., et al. 2020, JCAP, 12, 047 [Google Scholar]
 Astropy Collaboration (Robitaille, T. P., et al.) 2013, A&A, 558, A33 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Astropy Collaboration (PriceWhelan, A. M., et al.) 2018, AJ, 156, [Google Scholar]
 Auger, M. W., Treu, T., Bolton, A. S., et al. 2009, ApJ, 705, 1099 [NASA ADS] [CrossRef] [Google Scholar]
 Barnabè, M., Czoske, O., Koopmans, L. V. E., Treu, T., & Bolton, A. S. 2011, MNRAS, 415, 2215 [NASA ADS] [CrossRef] [Google Scholar]
 Birrer, S., & Amara, A. 2018, Phys. Dark Univ., 22, 189 [NASA ADS] [CrossRef] [Google Scholar]
 Birrer, S., Amara, A., & Refregier, A. 2016, J. Cosmol. Astropart. Phys., 2016, 020 [Google Scholar]
 Birrer, S., Shajib, A. J., Galan, A., et al. 2020, A&A, 643, A165 [CrossRef] [EDP Sciences] [Google Scholar]
 Bolton, A. S., Burles, S., Koopmans, L. V. E., et al. 2008, ApJ, 682, 964 [NASA ADS] [CrossRef] [Google Scholar]
 Cappellari, M. 2016, ARA&A, 54, 597 [NASA ADS] [CrossRef] [Google Scholar]
 Chen, G. C. F., Fassnacht, C. D., Suyu, S. H., et al. 2019, MNRAS, 490, 1743 [Google Scholar]
 Courteau, S., Cappellari, M., de Jong, R. S., et al. 2014, Rev. Mod. Phys., 86, 47 [NASA ADS] [CrossRef] [Google Scholar]
 Ding, X., Treu, T., Birrer, S., et al. 2021, MNRAS, 503, 1096 [Google Scholar]
 Falco, E. E., Gorenstein, M. V., & Shapiro, I. I. 1985, ApJ, 289, L1 [Google Scholar]
 ForemanMackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306 [Google Scholar]
 Freedman, W. L., Madore, B. F., Hoyt, T., et al. 2020, ApJ, 891, 57 [Google Scholar]
 Gilman, D., Birrer, S., & Treu, T. 2020, A&A, 642, A194 [CrossRef] [EDP Sciences] [Google Scholar]
 Humphrey, P. J., & Buote, D. A. 2010, MNRAS, 403, 2143 [NASA ADS] [CrossRef] [Google Scholar]
 Knox, L., & Millea, M. 2020, Phys. Rev. D, 101, 043533 [Google Scholar]
 Merritt, D. 1985, AJ, 90, 1027 [NASA ADS] [CrossRef] [Google Scholar]
 Millon, M., Galan, A., Courbin, F., et al. 2020, A&A, 639, A101 [CrossRef] [EDP Sciences] [Google Scholar]
 Navarro, J. F., Frenk, C. S., & White, S. D. M. 1997, ApJ, 490, 493 [NASA ADS] [CrossRef] [Google Scholar]
 Oguri, M., & Marshall, P. J. 2010, MNRAS, 405, 2579 [NASA ADS] [Google Scholar]
 Osipkov, L. P. 1979, Pisma v Astronomicheskii Zhurnal, 5, 77 [Google Scholar]
 Padmanabhan, N., Seljak, U., Strauss, M. A., et al. 2004, New Astron., 9, 329 [Google Scholar]
 Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, ApJ, 876, 85 [Google Scholar]
 Riess, A. G., Yuan, W., Casertano, S., Macri, L. M., & Scolnic, D. 2020, ApJ, 896, L43 [CrossRef] [Google Scholar]
 Rusu, C. E., Wong, K. C., Bonvin, V., et al. 2020, MNRAS, 498, 1440 [Google Scholar]
 Scolnic, D. M., Jones, D. O., Rest, A., et al. 2018, ApJ, 859, 101 [Google Scholar]
 Shajib, A. J., Treu, T., & Agnello, A. 2018, MNRAS, 473, 210 [NASA ADS] [CrossRef] [Google Scholar]
 Shajib, A. J., Birrer, S., Treu, T., et al. 2020, MNRAS, 494, 6072 [Google Scholar]
 Shajib, A. J., Treu, T., Birrer, S., & Sonnenfeld, A. 2021, MNRAS, 503, 2380 [Google Scholar]
 Sonnenfeld, A., Gavazzi, R., Suyu, S. H., Treu, T., & Marshall, P. J. 2013, ApJ, 777, 97 [NASA ADS] [CrossRef] [Google Scholar]
 Sonnenfeld, A., Treu, T., Marshall, P. J., et al. 2015, ApJ, 800, 94 [NASA ADS] [CrossRef] [Google Scholar]
 Treu, T., & Marshall, P. J. 2016, A&ARv, 24, 11 [Google Scholar]
 Verde, L., Treu, T., & Riess, A. G. 2019, Nat. Astron., 3, 891 [Google Scholar]
 Wong, K. C., Suyu, S. H., Chen, G. C. F., et al. 2020, MNRAS, 498, 1420 [Google Scholar]
 Yıldırım, A., Suyu, S. H., & Halkola, A. 2020, MNRAS, 493, 4783 [CrossRef] [Google Scholar]
All Tables
All Figures
Fig. 1. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of the seven TDCOSMO lenses (current scenario) as specified in Table 1 column δH_{0} (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 

In the text 
Fig. 2. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of the seven TDCOSMO lenses (current scenario) observed with aperture spectroscopy of 5% precision as well as in the case where external data sets are added, as specified in Table 1 (TDCOSMO5% row) (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 

In the text 
Fig. 3. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of a future sample of 40 TDCOSMO lenses (future scenario) as specified in Table 1 in the row of TDCOSMO5% (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 

In the text 
Fig. 4. Forecast precision on H_{0}, the MST parameter λ, and the anisotropy parameter a_{ani} for different spectroscopic scenarios of a future sample of 40 TDCOSMO lenses (future scenario) observed with aperture spectroscopy of 5% precision and additional external data sets specified in Table 1 in the row of TDCOSMO5% (https://github.com/sibirrer/TDCOSMO_forecast/blob/master/forecast.ipynb). 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.