Low-redshift measurement of the sound horizon through gravitational time-delays
AAstronomy & Astrophysics manuscript no. main c (cid:13)
ESO 2020February 20, 2020
Low-redshift measurement of the sound horizonthrough gravitational time-delays
Nikki Arendse (cid:63) , Adriano Agnello , and Radosław J. Wojtak DARK, Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, 2100 Copenhagen, DenmarkReceived 28 May 2019 / Accepted 4 October 2019
ABSTRACT
Context.
The matter sound horizon can be infered from the cosmic microwave background within the Standard Model. Independentdirect measurements of the sound horizon are then a probe of possible deviations from the Standard Model.
Aims.
We aim at measuring the sound horizon r s from low-redshift indicators, which are completely independent of CMB inference. Methods.
We used the measured product H ( z ) r s from baryon acoustic oscillations (BAO) together with supernovae Ia to constrain H ( z ) / H and time-delay lenses analysed by the H0LiCOW collaboration to anchor cosmological distances ( ∝ H − ). Additionally,we investigated the influence of adding a sample of quasars with higher redshift with standardisable UV-Xray luminosity distances.We adopted polynomial expansions in H ( z ) or in comoving distances so that our inference was completely independent of anycosmological model on which the expansion history might be based. Our measurements are independent of Cepheids and systematicsfrom peculiar motions to within percent-level accuracy. Results.
The inferred sound horizon r s varies between (133 ±
8) Mpc and (138 ±
5) Mpc across di ff erent models. The discrepancywith CMB measurements is robust against model choice. Statistical uncertainties are comparable to systematics. Conclusions.
The combination of time-delay lenses, supernovae, and BAO yields a distance ladder that is independent of cosmology(and of Cepheid calibration) and a measurement of r s that is independent of the CMB. These cosmographic measurements are then acompetitive test of the Standard Model, regardless of the hypotheses on which the cosmology is based. Key words.
Gravitational lensing: strong – cosmological parameters – distance scale – early Universe
1. Introduction
The sound horizon is a fundamental scale that is set by thephysics of the early Universe and is imprinted on the cluster-ing of dark and luminous matter of the Universe. The mostprecise measurements of the sound horizon are obtained fromobservations of the acoustic peaks in the power spectrum ofthe cosmic microwave background (CMB) radiation, althoughthe inference partially depends on the underlying cosmologicalmodel. In particular, the recent
Planck satellite mission yieldeda sound horizon scale (at the end of the baryonic drag epoch)of r s = . ± .
26 Mpc. This was based on the spatially flatsix-parameter Λ CDM model, which provides a satisfactory fit toall measured properties of the CMB (Planck Collaboration et al.2018), and on the Standard Model of particle physics.The sound horizon remains fixed in the comoving coordi-nates since the last scattering epoch and its signature can beobserved at low redshifts as an enhanced clustering of galax-ies. This feature is referred to as baryon acoustic oscillations(BAO). When we assume that the sound horizon is calibrated bythe CMB, BAO observations can be used to measure distancesand the Hubble parameter at the corresponding redshifts. The re-sulting BAO constraints can then be extrapolated to z = , for in-stance, using type Ia supernovae (SNe), in order to determine thepresent-day expansion rate H . However, this inverse distanceladder procedure depends on the choice of cosmological modeland on the strong assumption that the current standard cosmo- (cid:63) email: [email protected] ; [email protected] ;[email protected] logical model provides an accurate and su ffi cient description ofthe Universe at the lowest and highest redshifts. The robust-ness of the standard cosmological model has recently been ques-tioned on the grounds of a strong and unexplained discrepancybetween the local H measured from SNe with distances cali-brated by Cepheids and its CMB-based counterpart (currently a4 . σ di ff erence; Riess et al. 2019). The inverse distance laddercalibrated on the CMB should therefore be taken with caution.Recently, Macaulay et al. (2019) performed an inverse-distance-ladder measurement of H adopting the baseline r s from Planck ,and therefore their inferred H agrees with CMB predictions, asexpected.Observations of BAO alone only constrain a combination ofthe sound horizon and a distance or the expansion rate at thecorresponding redshift, that is, r s / D ( z ) and r s H ( z ) . Using SNe,we can propagate BAO observables to redshift z = r s H that are fully independent of the CMB(L’Huillier & Shafieloo 2017; Shafieloo et al. 2018). The ex-trapolation to low redshifts can be performed using various cos-mographic techniques, so that the final measurement is essen-tially independent of cosmological model. Furthermore, com-bining BAO constraints with a low-redshift absolute calibrationof distances or the expansion history, we can break the intrin-sic degeneracy of the BAO between r s and H ( z ) and thus candetermine the sound horizon scale. The resulting measurementis based solely on low-redshift observations, and it is therefore Distances are defined more precisely below.Article number, page 1 of 6 a r X i v : . [ a s t r o - ph . C O ] F e b & A proofs: manuscript no. main an alternative based on the local Universe to the sound horizoninferred from the CMB.Several di ff erent calibrations of distances or the expansionhistory have been used to obtain independent low-redshift mea-surements of the sound horizon. The main results include the cal-ibration of H ( z ) estimated from cosmic chronometers (Heavenset al. 2014; Verde et al. 2017), the local measurement of H fromSNe with distances calibrated with Cepheids (Bernal et al. 2016),angular diameter distances to lens galaxies (Jee et al. 2016; Wo-jtak & Agnello 2019), and adopting the Hubble constant fromtime-delay measurements (Aylor et al. 2019), although the lastmeasurement is based on cosmology-dependent modelling (Bir-rer et al. 2019). Currently, the sound horizon is most preciselyconstrained by a combination of BAO measurements from the Baryon Oscillations Spectroscopic Survey (BOSS; Alam et al.2017), with a calibration from the
Supernovae and H for theEquation of State of dark energy project (SH0ES; Riess et al.2019). A significantly higher local value of the Hubble con-stant than its CMB-inferred counterpart implies a substantiallysmaller sound horizon scale than its analogue inferred from theCMB under the assumption of the standard Λ CDM model (Ayloret al. 2019). The discrepancy in H and r s may indicate a genericproblem of distance scale at lowest and highest redshifts withinthe flat Λ CDM cosmological model (Bernal et al. 2016).Here, we present a self-consistent inference of H and r s from BAO, SNe Ia, and time-delay likelihoods released bythe H0LiCOW collaboration (Suyu et al. 2010, 2014; Wonget al. 2017; Suyu et al. 2017; Birrer et al. 2019). We exam-ine flat- Λ CDM models as a benchmark and di ff erent classes ofcosmology-free models. Our approach allows us to determinethe local sound horizon scale in a model-independent manner.A similar method was employed by Taubenberger et al. (2019),who used SNe to extrapolate constraints from time-delays to red-shift z = , and thus to obtain a direct measurement of the Hub-ble constant that depends rather weakly on the adopted cosmol-ogy.This paper is organised as follows. The datasets, models, andinference are outlined in Section 2. Results are given in Section3, and their implications are discussed in Section 4.Throughout this work, comoving distances, luminosity dis-tances, and angular diameter distances are denoted by D M , D L , and D A , respectively. We also adopt the distance duality relations D M ( z < z ) = D L ( z < z ) / (1 + z ) , D A ( z < z ) = D M ( z < z ) / (1 + z ) , which should hold in all generality and whose va-lidity with current datasets has been tested (Wojtak & Agnello2019).
2. Datasets, models, and inference
We used a combination of di ff erent low-redshift probes to setdi ff erent distance measurements and di ff erent models for the ex-pansion history. All models inferred the following set of param-eters: H , r s , M (normalisation of the SN distance moduli), andcoe ffi cients parametrising the expansion history or distance asa function of redshift. Curvature Ω k is left as a free parameterin some models. The sample of high-redshift quasars introducestwo additional free parameters: the normalisation M and the in-trinsic scatter σ int of the quasar distance moduli. The first model, for homogeneity with previous literature,adopted a polynomial expansion of H ( z ) in z : H ( z ) = H (1 + B z + B z + B z ) + O ( z ) , (1) where the coe ffi cients are related to the standard kinematical pa-rameters, that is, the deceleration q , jerk j , and snap s , in thefollowing way (Xu & Wang 2011; Weinberg 2008; Visser 2004): B = + q B =
12 ( j − q ) B =
16 (3 q + q − j (3 + q ) − s ) . Model distances were computed through direct integration of1 / H ( z ) . This is preferred over a corresponding expansion in dis-tances (as chosen e.g. by Macaulay et al. 2019) in order to ensuresub-percent accuracy in the model distances (Arendse et al., inprep.).In our second chosen model family, H ( z ) was expanded as apolynomial in x = log(1 + z ) : H ( x ) = H (1 + C x + C x + C x ) + O ( x ) . (2)Plugging the Taylor expansion of z = x − ffi cients C i and the kinematical parameters: C = ln(10) (1 + q ) C = ln (10)2 ( − q + q + j + C = ln (10)6 (3 q + q (1 − j ) − s + . Here, distances were also computed through direct numerical in-tegration of 1 / H ( z ).In our third model choice, comoving distances were com-puted through expansion in y = z / (1 + z ) , and H ( z ) was obtainedthrough a general relation (Li et al. 2019), H ( z , Ω k ) = c ∂ D M ( z ) /∂ z (cid:115) + H Ω k c D M ( z ) . (3)When a polynomial expansion D M ( y ) = c H (cid:16) y + D y / + O ( y ) (cid:17) (4)is adopted, then the second-order coe ffi cient D is related to thedeceleration parameter q through q = − D . (5)Adopting multiple families of parametrisations, for H ( z )and / or for model distances allowed us to quantify the system-atics due to di ff erent ways of extrapolating the given distancemeasurements down to z = . This is equivalent to anothercommon choice of adopting di ff erent cosmologies to extend theCDM model, but with the important di ff erence that our chosenparametrisations are completely agnostic about what the under-lying cosmological model should be.Lastly, for the sake of comparison with widely adopted mod-els, we also adopted a Λ CDM model class, with a uniformprior Ω k = [ − . , .
0] on curvature, and with the constraint that Ω Λ + Ω m + Ω k = . A discrepancy in flat- Λ CDM ( Ω k =
0) be-tween CMB measurements and our low-redshift measurementswould then indicate that more general model families are re-quired, that is, possible departures from concordance cosmology,or that the Standard Model needs to be extended.
Article number, page 2 of 6ikki Arendse et al.: Low-redshift measurement of the sound horizon through gravitational time-delays
Our measurement relies on the complementarity of di ff erent cos-mological probes. BAO observations constrain r s H ( z ) at severaldi ff erent redshifts and independently of the CMB. Standard can-dles play the role of the inverse distance ladder, by means ofwhich the BAO constraints can be extrapolated to redshift z = . Finally, gravitational lensing time-delays place constraints on H , thus breaking the degeneracy between H and r s in the in-verse distance ladder of BAO and standardisable candles.In our study, we used pre-reconstruction (independent of cos-mological model) consensus measurements of the BAO from theBaryon Oscillations Spectroscopic Survey (Alam et al. 2017).For the relative luminosity distances, we employed binned dis-tance moduli of SNe Ia from the Pantheon sample (Scolnic et al.2018). We excluded possible changes due to the choice of SNsample by re-running our inference on distance moduli from JLA(Betoule et al. 2014), and with the current quality of data, there isno appreciable change in the results. Finally, we used constraintson time-delays of four strongly lensed quasars observed by theH0LiCOW collaboration (see Suyu et al. 2017; Birrer et al. 2019,and references therein). Results from a fifth lens have recentlybeen communicated by H0LiCOW (Rusu et al. 2019). We cur-rently use only results that have been reviewed, validated, andreleased.As an option that provides more precise distance indicators athigh redshifts, we used distance moduli estimated from a relationbetween UV and X-ray luminosity quasars, which was provedto be an alternative standard candle at high redshift (Risaliti &Lusso 2018). Risaliti & Lusso (2018) reported that quasar dis-tances at high redshift show a deviation from Λ CDM; however,the lack of any corroborative pieces of evidence does not al-low us to conclude if this deviation is a genuine cosmologicalanomaly H Expansion of H in zNo quasars, flatNo quasars, curvature All, flatAll, curvature7080 H Expansion of H in log(1+z)2 3 4Order7080 H Expansion of D in z/(1+z)
Fig. 1.
Inferred Hubble constant H (in km / s / Mpc) vs. the chosen modelfamily and expansion truncation. The fiducial values from each expan-sion model (displayed as squares) are chosen by considering the changein BIC score and in ln L m . a . p . vs. the change in degrees of freedom.The upper dashed line corresponds to the local measurement value of H = . − Mpc − with a Cepheid calibration, and the lowerdashed line corresponds to the Planck value of H = . − Mpc − .The shaded grey regions show the error bars. anomaly or an unaccounted-for systematic e ff ect. For this rea-son, we dismissed the quasar data at redshifts z > .
8, which isthe highest redshift of lensed quasars in our sample.
The best-fit parameters and credibility ranges of the di ff erentexpansion models were obtained by sampling the posterior us-ing a ffi ne-invariant Monte Carlo Markov chains (Goodman &Weare 2010), and in particular with the python module emcee (Foreman-Mackey et al. 2013). For the BAO and SN data set,the uncertainties are given by a covariance matrix C . The likeli-hood is obtained by L = p (data | model) ∝ e − χ / ,χ = r † C − r (6)where r corresponds to the di ff erence between the value pre-dicted by the expansion and the observed data.The high-redshift quasar sample contains significant intrin-sic scatter, σ int , which has to be modelled as an additional freeparameter. The total uncertainty on each quasar data point is thesum of σ i , the uncertainty of that data point, and σ int . This leadsto the following formula of the likelihood: L quasars = N (cid:88) i = e − r i / σ i + σ ) (cid:113) ( σ i + σ )2 π . (7)The likelihoods of the lensed quasars HE0435, RXJ1131, andB1608 of the H0LiCOW collaboration were given as skewedlog-normal distributions of their time-delay distances D ∆ t = (1 + z l ) D A , l D A , s / D A , ls . For the lensed quasar J1206, both the angu-lar diameter distance and the time delay distance were available H [km/s/Mpc] r s [ M p c ] r s [Mpc]PlanckThis work Fig. 2.
Inference on cosmological parameters, including the Hubbleconstant H and sound horizon r s , for the baseline case of flat- Λ CDMmodels using time-delay lenses, SN Ia, and BAO as late-time indicators.The outermost credibility contour contains 95% of the marginalisedposterior probability, and the innermost contour contains 68%.Article number, page 3 of 6 & A proofs: manuscript no. main in the form of a sample drawn from the model posterior distri-bution. A Gaussian kernel density estimator (KDE) was used tointerpolate a smooth distribution between the posterior points.The final log-likelihood that was sampled by emcee is a sumof the separate likelihoods of the SN, BAO, lensed quasars, andhigh-redshift quasars,ln ( L total ) = ln( L SN ) + ln( L BAO ) + ln( L lenses ) + ln( L quasars ) . (8)We note that the high-redshift quasar likelihood is optional in ourstudy. For all cosmographic models used in our work, parameterinference is carried out with or without quasar data, and bothresults are consistently reported.A uniform prior was used for all the free parameters, exceptwhen the high-redshift quasar sample was used. In that case, theintrinsic scatter σ int was also constrained to be larger than zero.This choice of priors does not seem to bias the inference accord-ing to current data and tests on flat- Λ CDM mocks.To choose the right order of expansion for each model, theBayesian information criterion (BIC) indicator was used,BIC = ln( N ) k − L m . a . p . ) , (9)where N is the number of data points, k is the number of freeparameters and L m . a . p . is the maximum a posteriori likelihood(i.e. evaluated where the posterior is maximised). The BIC scoreexpresses how well a model describes the data, with a lowerscore corresponding to a better agreement. It also introduces apenalty term for added complexity in a model. Table 1 displaysthe number of free parameters, maximum a posteriori likelihood,and the BIC score for four increasing orders of expansion. Theexpansion order with five free parameters provides the lowestBIC score. When more complexity is added to the model, theBIC value continuously increases, which supports the conclu-sion that higher expansion orders will be ruled out as well. Whenthe high-redshift quasar sample was added to the data collection,it changed the preferred order of expansion of model 3 from thirdto second order. Model 1 first second third fourthparameter order order order orderFree parameters 4 5 6 7ln L m . a . p . -60.8 -55.8 -55.3 -55.2BIC score 137.2 131.1 134.1 137.8 Model 2 first second third fourthparameter order order order orderFree parameters 4 5 6 7ln L m . a . p . -67.1 -56.8 -55.7 -55.0BIC score 149.8 133.2 134.9 137.4 Model 3 second third fourth fifthparameter order order order orderFree parameters 4 5 6 7ln L m . a . p . -61.0 -56.1 -56.0 -54.5BIC score 137.6 131.7 135.5 136.3 Table 1.
Overview of the number of free parameters, maximum a pos-teriori likelihood, and BIC score for di ff erent expansion orders for cos-mographic models 1, 2, and 3. These numbers were calculated usingthe four lenses, SN, and BAO points and assuming a flat Universe. Forexpansion in H (models 1 and 2) the second order is preferred, and forexpansion in distance (model 3) the third order is preferred. This corre-sponds to five free parameters in each of the models. r s [ M p c ] H [km/s/Mpc] . . . . K r s [Mpc] . . . . K Model 1Model 2Model 3CDM
Fig. 3.
Inference on the Hubble constant H and sound horizon r s fordi ff erent models (at fiducial truncation order for models 1-3), with free Ω k , using time-delay lenses, SN Ia, and BAO. While the inferred pa-rameters can change among models and among truncation choices, therelative discrepancy with CMB measurements remains the same. Thecredibility contours contain 95% of the marginalised posterior proba-bility. The grey point corresponds to the Planck value of H and r s andto a flat Universe.
3. Results and discussion
The inferred values from our inference are given in Tables 2 and3. For the sake of compactness, we report only the inferred val-ues for each model that correspond to the lowest BIC scores (andto a ∆ BIC > H as inferred bydi ff erent expansion orders. Plots of marginalised posteriors onselected cosmological parameters are given in Figures 2 and 3.The inferred values of the Hubble constant from Table 1,both its maximum a posteriori and uncertainty, vary between(73 . ± .
7) km s − Mpc − and (76 . ± .
0) km s − Mpc − . Theyare in full agreement with current results form the H0LiCOWand SH0ES collaborations, even despite the choice of generaland agnostic models in our method. This indicates that the dis-crepancy between Cepheid-calibrated H and that inferred fromCMB measurements is not due to (known and unknown) system-atics in the very low redshift range. The inferred sound horizon r s varies between (133 ±
8) Mpc and (138 ±
5) Mpc. The largestdiscrepancy with the value from CMB and Standard Model pre-dictions (147 . ± .
26 Mpc) is more significant for models thatare agnostic to the underlying cosmology.The systematic uncertainties, due to di ff erent model choices,are still within the range allowed by statistical uncertainties.However, they may become dominant in future measurementsaiming at percent-level precision. Adding UV-Xray standardis-able quasars generally raises the inferred value of H (and cor-respondingly lowers the inferred r s ), even though the normalisa-tion of their Hubble diagram is treated as a nuisance parameter.The addition of the quasar sample also results in lower valuesof Ω k . This suggests that the behaviour of distance modulus with Article number, page 4 of 6ikki Arendse et al.: Low-redshift measurement of the sound horizon through gravitational time-delays flat ( Ω k = Λ CDM) r s (Mpc) 135 . ± .
22 138 . ± .
97 137 . ± .
970 138 . ± . H r s (km s − ) 10091 . ± .
54 10095 . ± .
23 10069 . ± .
82 10046 . ± . H (km s − Mpc − ) 74 . ± .
92 73 . ± .
65 73 . ± .
67 72 . ± . q − . ± . − . ± . − . ± .
18 —ln L m . a . p . − . − . − . − . τ (Planck Λ CDM) 3.1 (2.0 σ ) 2.3 (1.6 σ ) 2.3 (1.7 σ ) 2.5 (1.8 σ )free Ω k parameter model 1 (second order) model 2 (second order) model 3 (third order) model 4 ( Λ CDM) r s (Mpc) 133 . ± .
57 137 . ± .
80 136 . ± .
05 139 . ± . H r s (km s − ) 10069 . ± .
97 10079 . ± .
20 10052 . ± .
32 10073 . ± . H (km s − Mpc − ) 75 . ± .
07 73 . ± .
86 73 . ± .
06 72 . ± . Ω k . ± .
23 0 . ± .
21 0 . ± . − . ± . q − . ± . − . ± . − . ± .
23 —ln L m . a . p . − . − . − . − . τ (Planck Λ CDM) 2.3 (1.6 σ ) 1.6 (1.3 σ ) 1.5 (1.2 σ ) 2.3 (1.6 σ ) Table 2.
Inference on the cosmological parameters from BAO + SNe + lenses in our four model classes, with or without imposed flatness. We listthe posterior mean and 68% uncertainties of the main parameters, the maximum a posteriori likelihood, the BIC score, and the odds τ that ourmeasurements of H and r s are consistent with those from the Planck observations, as derived for the standard flat- Λ CDM cosmological model. flat ( Ω k = Λ CDM) r s (Mpc) 132 . ± .
05 135 . ± .
84 131 . ± .
45 138 . ± . H r s (km s − ) 10124 . ± .
40 10111 . ± .
68 10186 . ± .
68 9999 . ± . H (km s − Mpc − ) 76 . ± .
90 74 . ± .
67 77 . ± .
52 72 . ± . q − . ± . − . ± . − . ± .
11 —ln L m . a . p . − . − . − . − . τ (Planck Λ CDM) 4.9 (2.7 σ ) 3.5 (2.2 σ ) 7.8 (3.5 σ ) 2.5 (1.7 σ )free Ω k parameter model 1 (second order) model 2 (second order) model 3 (second order) model 4 ( Λ CDM) r s (Mpc) 134 . ± .
00 140 . ± .
15 139 . ± .
40 143 . ± . H r s (km s − ) 10132 . ± .
61 10150 . ± .
94 10223 . ± .
08 10140 . ± . H (km s − Mpc − ) 75 . ± .
16 72 . ± .
89 73 . ± .
18 70 . ± . Ω k − . ± . − . ± . − . ± . − . ± . q − . ± . − . ± . − . ± .
17 —ln L m . a . p . − . − . − . − . τ (Planck Λ CDM) 2.4 (1.7 σ ) 0.6 (0.6 σ ) 2.6 (1.8 σ ) 1.9 (1.4 σ ) Table 3.
Same as for Table 1, but including UV-Xray quasars as standardisable distance indicators. redshift has su ffi cient constraining power on auxiliary cosmolog-ical parameters that in turn are degenerate with H in the time-delay lensing standardisation. For all cosmographic models, theintrinsic scatter in quasar distance moduli found in our analysisis 1.45 mag, which is fully consistent with the estimate reportedin Risaliti & Lusso (2018).We quantified the tension with CMB measurements throughthe two-dimensional inference on H and r s . Following Verdeet al. (2013), we estimated the odds that both measurements areconsistent by computing the following ratio: τ = (cid:82) (cid:82) ˆ p CMB ˆ p local d H d r s (cid:82) (cid:82) p CMB p local d H d r s , (10) where p is the marginalised probability distribution for r s and H from the CMB (Planck Collaboration et al. 2018) or ourstudy (in both cases approximated by Gaussians), while ˆ p is adistribution shifted to a fixed arbitrary point so that both mea-surements have the same posterior probability means. A moreintuitive scale representing the discrepancy between two mea-surements is a number-of-sigma tension, which can be derivedfrom the odds ratio. This is done by calculating the probabil-ity enclosed by a contour such that 1 /τ = e − r . The numberof sigma tension can then be calculated from the probability bymeans of the error function. We list the logarithm of the oddsand the number of sigma tension in Tables 2 and 3. Article number, page 5 of 6 & A proofs: manuscript no. main
The tension with Planck measurements from CMB is ap-proximately at a 2 σ level. While the uncertainties from somemodel families are larger, the corresponding H ( r s ) optimal val-ues are also higher (lower), and the tension remains the same.The curvature Ω k slightly alleviates the tension through larger H uncertainties, but the current data do not yield any evidenceof a departure from flatness.
4. Conclusions and outlook
Current data enable a ≈
3% determination of key cosmologicalparameters, in particular, the Hubble constant H and the soundhorizon r s , resulting in a ≈ σ Gaussian tension with predic-tions from CMB measurements and the Standard Model. Whilethis tension is robust against the choice of model family and istherefore independent of the underlying cosmology, the system-atics due to di ff erent model choices are currently comparableto the statistical uncertainties and may dominate percent-levelmeasurements of H . A simple estimate based on recent SH0ESmeasurements (Riess et al. 2019) and very recent five-lens mea-surements by H0LiCOW (Rusu et al. 2019) indicates a ≈ σ tension with CMB measurements within a flat- Λ CDM model.Our study also demonstrated the potential of constrainingthe curvature of the Universe solely based on low-redshift ob-servations and in a cosmology-independent manner. The currentprecision of 0 .
20 is insu ffi cient to test possible minimum depar-tures from flatness, mainly due to the accuracy in H from asmall sample of well-studied lenses. Samples of lenses with suit-able ancillary data are already being assembled (see e.g. Shajibet al. 2019). Future measurements of gravitational time-delaysfrom the Large Synoptic Survey Telescope can reach percent-level precision (Liao et al. 2015), making this method a highlycompetitive probe (Denissenya et al. 2018). Acknowledgements.
The authors were supported by a grant from VILLUMFONDEN (project number 16599). This project is partially funded by the Dan-ish Council for independent research under the project “Fundamentals of DarkMatter Structures”, DFF–6108-00470.We are grateful for the public release of the time-delay distance likelihoods bythe H0LiCOW collaboration, and interesting conversations with S. H. Suyu, F.Courbin and T. Treu.We thank Guido Risaliti for sharing distance moduli measured from high-redshiftquasars.We also thank the anonymous referee for constructive comments which helpedimprove our work.
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