A Model-independent Method to Determine H 0 Using Time-delay Lensing, Quasars, and Type Ia Supernovae

Absolute distances from strong lensing can anchor Type Ia Supernovae (SNe Ia) at cosmological distances giving a model-independent inference of the Hubble constant (H 0). Future observations could provide strong lensing time-delay distances with source redshifts up to z ≃ 4, which are much higher than the maximum redshift of SNe Ia observed so far. In order to make full use of time-delay distances measured at higher redshifts, we use quasars as a complementary cosmic probe to measure cosmological distances at redshifts beyond those of SNe Ia and provide a model-independent method to determine H 0. In this work, we demonstrate a model-independent, joint constraint of SNe Ia, quasars, and time-delay distances from strong lensed quasars. We first generate mock data sets of SNe Ia, quasar, and time-delay distances based on a fiducial cosmological model. Then, we calibrate the quasar parameters model independently using Gaussian process (GP) regression with mock SNe Ia data. Finally, we determine the value of H 0 model-independently using GP regression from mock quasars and time-delay distances from strong lensing systems. As a comparison, we also show the H 0 results obtained from mock SNe Ia in combination with time-delay lensing systems whose redshifts overlap with SNe Ia. Our results show that quasars at higher redshifts show great potential to extend the redshift coverage of SNe Ia and thus enable the full use of strong lens time-delay distance measurements from ongoing cosmic surveys and improve the accuracy of the estimation of H 0 from 2.1% to 1.3% when the uncertainties of the time-delay distances are 5% of the distance values.


INTRODUCTION
The simplest flat ΛCDM model explains a large range of current observations including cosmic microwave background radiation (CMB), Big Bang nucleosynthesis and baryon acoustic oscillation (BAO) measurements (Schlegel et al. 2009;Ade et al. 2014Ade et al. , 2016;;Aghanim et al. 2020;Alam et al. 2021).However, there are significant tensions between different data sets when ΛCDM is used to estimate some key cosmological parameters.One of the major issues is the discrepancy between the value of the Hubble constant measured by the multiple local-universe probes (Riess et al. 2018(Riess et al. , 2019;;Reid et al. 2019;Riess et al. 2022) and that inferred by early-universe probes under the assumption of ΛCDM cosmology (Aghanim et al. 2020).This tension has reached the 4σ to 6σ level (Di Valentino et al. 2021).
The tension either could be due to unknown systematic errors in the observations or could reveal new physics beyond ΛCDM.A model-independent method to determine H 0 from observations in the redshift gap between local-universe probes and early-universe probes is necessary to better assess the H 0 tension.Quasars are luminous persistent sources in the Universe which can be observed up to redshifts of z ≃ 7.5 (Mortlock et al. 2011).The magnifying effect of strong gravitational lensing can be used to observe quasars at even higher redshifts.With future surveys, the redshift of SNe Ia from Nancy Grace Roman Space Telescope (ROMAN) SN could reach z ∼ 2 with larger uncertainties (Hounsell et al. 2023).On the other hand, future surveys will provide us with more strong lensing system measurements with higher redshift (Oguri & Marshall 2010).There-fore, lensed quasars act as a potential cosmic probe at higher redshifts to shrink the redshift gap between the farthest observed SN Ia and CMB observations.
Recently, a feasible method to determine H 0 independent of the cosmological model that used strong lensed quasars and Type Ia supernovae (SNe Ia) with Gaussian Process (GP) regression has been presented in Liao et al. (2019Liao et al. ( , 2020)).Strong gravitational lensing of a variable source measures the time-delay distance D ∆t of the system and measuring the stellar velocity dispersion of the lens also yields a constraint on the angular diameter distance to the lens D d .One can anchor SNe Ia with these absolute distances and obtain an excellent constraint on the shape of the distance-redshift relation (Collett et al. 2019).In Liao et al. (2019Liao et al. ( , 2020)), the authors applied GP regression to SNe Ia data to get a model-independent relative distance-redshift relation and anchored the distance-redshift relation with D ∆t and D d from strong gravitational lensing to give the constraints on H 0 .
However, observations of strong lens systems summarized in Schmidt et al. (2023) show that the redshift of lensed quasars could reach as high as z ≃ 3.8, which is far beyond the highest redshift of observed SNe Ia so far.Therefore, looking for observations at higher redshifts is necessary to make full use of time-delay lensing systems.Recently Du et al. (2023) used gamma-ray burst (GRB) distances and H0LiCOW lenses with redshifts < 1.8 to infer H 0 .
Moreover, the linear relation between the log of the UV and X-ray luminosities allows quasars to be potentially used as standard candles at higher redshifts if well calibrated (Risaliti & Lusso 2015;Lusso & Risaliti 2017;Risaliti & Lusso 2019;Lusso et al. 2020;Khadka & Ratra 2021, 2020;Li et al. 2021).Thus, the combination of time-delay observations in strong lensed quasars and the linear relation between the log of the ultraviolet (UV) and X-ray luminosities of quasars can help us to determine H 0 model-independently.
In our work, we use GP regression to reconstruct the expansion history of the Universe model-independently.We first generate SNe Ia, quasar, and strong lens data set based on a fiducial cosmological model.Then, we calibrate the mock quasar data set by using GP regression to model-independently reconstruct the expansion history of the Universe from the mock SNe Ia data set following the previous work by (Li et al. 2021).Using the calibrated quasar data set, we further reconstruct the expansion history up to redshift of z ≃ 7.5 with GP.Then following Liao et al. (2019Liao et al. ( , 2020) ) we determine H 0 cosmological-model-independently using simu-lated strong lensed quasars with source redshifts up to 4 and calibrated unlensed quasars.
This paper is organized as follows: in Section 2, we describe the data sets we used in detail.The quasar calibration with GP regression from the latest SNe Ia observations, as well as the determination of H 0 from strong lens systems and calibrated quasars are shown in Section 3. We discuss our conclusions in Section 4.

DATA
Since there is a lack of necessary time-delay measurements for strong lensing systems, we are going to simulate the time-delay measurements based on a fiducial cosmological model.Moreover, to make the results convictive, we use simulated SNe Ia data as well as simulated quasar samples instead of the real data from observations.
In this section, we briefly describe the method of generating the mock data based on a fiducial cosmological model.Throughout our work, a flat-ΛCDM model with Ω m = 0.3 and H 0 = 70 km s −1 Mpc −1 is used as the fiducial cosmological model.We should emphasize here that following our previous works, we could have selected any cosmological model (as the fiducial model) for our analysis since we are performing a model-independent analysis for reconstructing the expansion history.We have chosen the standard flat-ΛCDM model as our fiducial model since the focus of this paper is on the high precision determination of H 0 using high redshift quasars.

Type Ia Supernovae
SNe Ia, which helped discover cosmic acceleration, are powerful standard candles that enable precise measurements of the expansion of the Universe.A most recent Pantheon+ sample was reported in (Scolnic et al. 2022) which consists of 1701 light curves of 1550 distinct SNe Ia ranging in redshift from z=0.001 to 2.26.This larger SNe Ia sample is a significant increase compared to the original Pantheon sample, especially at lower redshifts.
In this work, we generate a mock SNe Ia data set based on the Pantheon+ sample assuming a fiducial cosmological model.First, we obtain the luminosity distances of SNe Ia with and then the distance modulus can be calculated with The mock SNe Ia data, µ mock , are then generated from µ fid by adding noise as a random variable with a mean of zero and a variance characterized by the Pantheon+ covariance matrix.We use this mock SNe Ia data set along with the mock quasar data set to simultaneously calibrate the mock quasar data set and reconstruct the Universe's expansion history.

Quasar sample
Quasars act as standard candles based on the loglinear relation between the UVt and the X-ray luminosities log(L X ) = γ log(L UV ) + β 1 .This allows quasars to work as cosmic probes at higher redshifts to shrink the redshift gap between SNe Ia and the CMB if well calibrated since they can be observed up to the redshifts of z ≃ 7.5.So far, the largest quasar sample with both Xray and UV observations consists of ∼ 12, 000 objects.However, after applying several filtering steps to reduced the systematic effects, 2421 quasars with spectroscopic redshifts and X-ray observations from either Chandra or XMM-Newton in the redshift range of 0.009 < z < 7.54 were left in the final cleaned sample (Lusso et al. 2020).
In this work, we generate a mock quasar data set based on the quasar catalog described above assuming a fiducial cosmological model.First, we take the values of log(F UV ) fid from the actual measurements.Then, we calculate log(F X ) fid using log where β 2 = γ log(4π) − log(4π) + β 1 , F UV and F X are the fluxes measured at fixed rest-frame wavelengths, and D fid L is the luminosity distance relation of the fiducial cosmology.γ and β 2 are quasar parameters that need to be calibrated.These calibration parameters are degenerate with the cosmological parameters, or modelindependent distances we want to fit or reconstruct.Since β 2 is degenerate with H 0 , quasars can only measure relative distances, just like SNe Ia.Thus, we absorb H 0 into the parameter β = β 2 − (2γ − 2) log(H 0 ).This is to absorb multiple degenerate parameters which characterize the relative anchoring between the data and the expansion history into one parameter.Here, we use fiducial values for γ = 0.6430 and β = 7.88, which are the best-fit values from Li et al. (2021).The final sample of fluxes (log(F UV ) mock , log(F X ) mock ) are calculated from the fiducial values (log(F UV ) fid , log(F X ) fid ) by adding Gaussian random noise with a standard deviation (σ log(FUV) , σ log(FX) ) from the actual data set.

Strong lens time-delay distance data
A typical strongly lensed system, as used for timedelay cosmography, consists of a source quasar at cosmological distances, which is lensed by a foreground elliptical galaxy, and forms multiple images of the quasar and the arcs of the host galaxy.With years of observations of the light curves, one can measure the time delay between any two images, which, following the Fermat principle, arises from the different geometries and Shapiro time delays along the multiple paths.The time delay thus depends on both the geometry of the Universe and the gravitational field of the lens galaxy.The time delays ∆t can be used to measure a time-delay distance where ∆ϕ is the Fermat potential difference between the two images which is a function of lens mass profile parameters ξ lens , determined by high-resolution imaging of the host arcs.D ∆t is the time-delay distance which is a combination of three angular diameter distances D d , D s , and D ds where the subscripts d and s denotes the deflector (lens) and the source, respectively.Time-delay distance measurements can be used as a onerung distance ladder and are independent of the Cepheid distance ladder and early Universe physics.The angular diameter distance to the deflector lens itself, D d , can be obtained independently of the time-delay distance of the strong lens system.These D d measurements provide additional constraints on the expansion history beyond the D ∆t measurements.
With the increasing number of the wide-field imaging surveys, e.g., Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST), Euclid, and ROMAN, the number of the known strong lens systems is growing rapidly (Spergel et al. 2015;Akeson et al. 2019;Oguri & Marshall 2010;Collett 2015).New images of billions of galaxies are expected to be observed, of which ∼ 100, 000 are strong lens systems (Collett 2015).
In this work, we simulate angular diameter distances and time-delay distances for each strong lens system following the method described below: 1.In the fiducial cosmological model, the angular diameter distance can be calculated via with the redshift to the deflector z d and the redshift to the source z s , we can calculate D d and D s , respectively.Random noise following a normal distribution is added to the angular diameter distance.We analyze two cases, one where the noise being added is at the level of 5% and another at the level of 10%.The value depends on specific systems, observational conditions, algorithm and most importantly systematical errors.We adopted 5% for the best case and 10% for the worse case.
In a spatially flat universe, the distance between the lens and the source (D ds ) is calculated via 2. Then, the time-delay distance can be obtained with Equation (5).
3. We consider two cases, where the uncertainties on the simulated distances (both D d and D ∆t ) are taken to be 5% and 10%.
In this work, we use nine mock strongly lensed quasars based on those in Ertl et al. (2023) which is a subset of 30 quadruply imaged quasars in Schmidt et al. (2023).The sample will probably be analyzed by the TDCOSMO team in the next stage (Treu et al. 2022).The redshifts of strong lens systems are summarized in Table 1 which is part of Table A.3 from Ertl et al. (2023).These systems have higher source redshifts than H0LiCOW and might be well analyzed by the TDCOSMO team in the near future.The boldface shows the lensing systems whose redshifts in the redshift range of SNe Ia (z < 2.261).As can be seen from Table 1, only three strong lens systems are left in the redshift coverage of SNe Ia.
Following the simulation method described above, we obtain the simulated angular diameter distance to the lens and the time-delay distance, which are shown in the last two columns of Table 1.Throughout our work, we consider 5% and 10% of distance values as uncertainties for the mock time-delay distance D ∆t and angular diameter distance to the deflector D d .
We show the redshift distribution for the data set used in our analysis in Figure 1.

Quasar calibration
In this subsection, we briefly describe the method we used to simultaneously calibrate the quasar sample and reconstruct the expansion history using the SNe Ia data set and GP regression.
In order to calibrate the quasar parameters in a modelindependent way, we use cosmological distances from another cosmic probe -SNe Ia.Since the absolute brightness of SNe Ia is degenerate with H 0 , only the dimensionless, unanchored luminosity distances (D L H 0 ) can be constrained.We rewrite Eq. (3) as where s i = ln(1 + z i )/ ln(1 + z max ) and z max = 2.261 is the maximum redshift of the SNe Ia sample.σ f and ℓ are two hyperparameters that are marginalized over.φ is just a random function drawn from the distribution defined by the covariance function of Equation ( 9) and we take this function as φ(z) = ln H mf (z)/H(z) , i.e. the logarithm of the ratio between the reconstructed expansion history, H(z), and a mean function, H mf (z), which we choose to be the best-fit ΛCDM model from the Pan-theon+ data set.The mean function plays an important role in GP regression and the final reconstruction results are not quite independent of the mean function, however, it has a modest effect on the final reconstruction results because the values of hyperparameters help to trace the deviations from the mean function (Shafieloo et al. 2012;Shafieloo et al. 2013;Aghamousa et al. 2017).
Moreover, the true model should be very close to the flat ΛCDM model so it is reasonable to choose the best-fit flat ΛCDM model from Pantheon+ as a mean function.This choice allows us to perform a test of whether the data need some additional flexibility to fit the data beyond the ΛCDM model (Keeley et al. 2021).For the details of the reconstruction with GP, we refer the readers to (Rasmussen & Williams 2006;Holsclaw et al. 2010a,b;Holsclaw et al. 2011;Shafieloo et al. 2012;Shafieloo et al. 2013;Aghamousa et al. 2017;Keeley et al. 2021;Li et al. 2021;Hwang et al. 2023).
With the measurements of F UV from the quasar sample and D L H 0 from SNe Ia, we obtain log(F X ) SN following Equation ( 8).This allows us to compare the quasar data set and the SNe Ia data set with (10) where s 2 i = σ 2 log(FX) + γ 2 σ 2 log(FUV) + δ 2 .The intrinsic dispersion δ of the L X − L UV relation models various unknown physical properties that scatter the observed log(L X )-log(L UV ) trend by more than the measurement uncertainty (Risaliti & Lusso 2019;Lusso et al. 2020).
We then calculate the posterior distribution of the quasar parameters: the slope γ, the intercept β and the intrinsic dispersion parameter δ.We should note that the Hubble constant H 0 is absorbed into the parameter β.This is to absorb multiple degenerate parameters that characterize the relative anchoring between the data and the expansion history into one parameter.Based on the method described above, we use a Python package named emcee (Foreman-Mackey et al. 2013) to do the Markov Chain Monte Carlo analysis and flat priors are used for each parameter.
To make sure that our calibrated results give reasonable information about cosmology, we calculate log(D L H 0 ) versus z relation from the quasar fluxes with the calibrated quasar parameters through The results are shown in Figure 3.The blue points represent the unanchored distances from the calibrated quasar sample, which we use to anchor time-delay distances of strong lensing in later work.In Figure 3 we also show the log(D L H 0 ) obtained from the posterior of SNe Ia calculated with GP.Moreover, we check the consistency between the calibrated quasar sample and the unanchored luminosity distance from SNe Ia by estimating the normalized residual of log(F X ) SN which is calculated via where log(F X ) SN is obtained with Equation (8) and log(F X ) QSO is the quasar measurements.Figure 5.The posterior on H0 obtained with simulated time-delay lensing distance (D∆t) as well as lensing distance (D d ).
The left plot denotes the results with 5% uncertainties of the simulated distance values and the right plot denotes the results with 10% uncertainties when doing simulation.
we can see that the distribution of the normalized residual is a Gaussian distribution, which indicates that the log(F X ) data from quasar measurements is consistent with that derived from SNe Ia using calibrated quasar parameters.
Finding internal consistency between the calibrated quasar and SNe Ia data sets would show that quasars can be used as standard candles at higher redshift and are therefore powerful probes of cosmology.We also need to emphasize here that there are deviations from the standard ΛCDM model for quasars at higher redshift as standard candles.It is not clear so far that whether the deviations are due to new physics beyond the ΛCDM model or the evolution of calibration parametrizations.Hopefully, future surveys will provide us more copious and precise data for quasars which could help us solve this puzzle.

H 0 determination
In order to determine H 0 with our technique, we have to take the unanchored reconstructions of the expansion history from SNe Ia and quasars, which only measure relative distances, and anchor them with the strong lens data set, which does measure absolute distances.We first generate 1000 posterior samples of the H 0independent quantity D L H 0 from quasars with GP regression mentioned above and convert these unanchored luminosity distances to unanchored angular diameter distances D A H 0 .Then we evaluate the values of each of the 1000 D A H 0 curves at the lens and the source redshifts of the simulated strong lens systems to calculate 1000 values of H 0 D ∆t using where D ∆t is the time-delay distance.Comparing 1000 H 0 D ∆t and H 0 D d curves with simulated D ∆t andD d at the lens and source redshifts of the lensing systems, we calculate the likelihood with 14) and In the end, we marginalize over the realizations to form the posterior distribution of H 0 .
The posterior on H 0 in a flat ΛCDM model obtained with nine simulated strong lens systems in combination with quasars and SNe Ia are shown in solid lines in Figure 5.The left plot shows the H 0 estimation results taking 5% of the distance values as uncertainties while the right plot shows the H 0 estimation results taking 10% of the distance values as uncertainties.In Table 2 we summarize the numerical results.First, we consider the combination of D ∆t and D d from the nine simulated lensing systems.The method described above yields H 0 = 69.8±0.9 km s −1 Mpc −1 when taking 5% of the distance values as uncertainties and H 0 = 70.1 ± 1.7 km s −1 Mpc −1 when taking 10% of the distance values as uncertainties.In addition, we give the results from D

5.8%
As a comparison, we also use SNe Ia as standard candles following the same H 0 determination method described above to determine H 0 model-independently.However, only three strong lens systems are left in the redshift range of SNe Ia.The constraints are shown in dashed lines in Figure 5 and the best-fit values together with the 1σ uncertainties are summarized in Table 2.We obtain H 0 = 70.8± 1.5 km s −1 Mpc −1 taking 5% of the distance values as uncertainties and H 0 = 72.0+2.7 −3.2 km s −1 Mpc −1 taking 10% of the distance values as uncertainties with three time-delay lensing systems in combination with SNe Ia.We see that by using quasars and not just SNe Ia as standard candles, more time-delay lensing systems can be included because quasars are measured out to higher redshift, thus yielding a tighter constraint on H 0 .The precision of estimating H 0 can be improved from 2.1% to 1.3% with adding quasars as standard candles when the uncertainties of the time-delay distances are 5%.

CONCLUSION AND DISCUSSIONS
In this work, we develop a model-independent method to measure the Hubble constant with time-delay distances from strong lens systems combined with quasar and SNe Ia standard candles.
We first generate mock data sets of SNe Ia, quasars, and strong lenses based on a fiducial cosmological model.Then we apply our GP regression technique on the mock SNe Ia and quasar data sets to simultaneously calibrate the mock quasar data set and reconstruct the Universe's expansion history.This reconstruction extends out to a redshift of 7.5, further than any other reconstruction to date.We also demonstrate how to test the reliability of our calibrated results, namely by calculating the normalized residuals of the log F X with respect to the mock SNe Ia data set.If the normalized residuals follow a Gaussian distribution, then the calibration results are reliable.
Since both the quasar and SNe Ia data sets are unanchored, we then anchor the reconstruction of expansion history from those data sets with the time-delay distances from the mock strong lens data set.Previous model-independent reconstructions of the expansion history have only used SNe Ia.Since SNe Ia only extend to a redshift of 2.26, the model-independent reconstructions can only use the three strong lens systems with source redshifts less than 2.26.Using quasar as standard candles extends the redshift coverage over our modelindependent reconstruction, and thus we can use nine strong lens systems in our analysis.This yields a 1.3% precision on H 0 in an optimistic case (5% precision for strong lens distances) for a future strong lens data set and 2.4% precision in a less optimistic case (10% precision).
Fortunately, we will obtain more well-measured timedelay strong lens systems with the onset of cosmic surveys such as Roman, LSST, and Euclid.In addition to lensed quasars, strongly lensed transients, such as SN, are coming soon (Liao et al. 2022).With future timedelay distance measurements together with a larger and more precise quasar data set, one can obtain H 0 more precisely with the method described in this work and understand the H 0 tension better.

Figure 1 .Figure 2 .
Figure 1.The left plot shows the redshift distribution of the Quasar sample from Lusso et al. (2020) and SNe Ia from Pantheon+ sample (Scolnic et al. 2022) while the right plot shows the redshift distribution of the sources and deflector for the strong lens time-delay data set from Ertl et al. (2023).The redshift distribution of quasars at higher redshift are also displayed in the inner plot to make it more clear.

Figure 3 .
Figure 3. log(DLH0)-redshift relation for the mock quasars.The errorbars of log(DLH0) are obtained through error propagation and the black solid lines show log(DLH0) obtained from mock SNe Ia data and the dashed red line denotes flat ΛCDM model with H0 = 70km s −1 Mpc −1 and Ωm = 0.3 as comparison.

Figure 4 .
Figure 4. Residuals of the mock log(FX) values with respect to the predicted log(FX) values derived from the GP reconstructions of the mock SNe Ia compilation, normalized to the calibrated errors (observational and intrinsic).The right plot shows the histogram for ∆ log(FX) and the purple line shows the best Gaussian fit with µ = −0.04 and σ = 0.97.
d and D ∆t separately to quantify the contribution of D d in Figure 5.The constraints when the angular diameter distances to the lens deflector (D d ) are considered separately from the time-delay distances (D ∆t ) are largely equivalent.We can also see from Figure 5 that the constraining power of the combination of D d and D ∆t will improve a lot compared to the results from D d and D ∆t separately.

Table 1 .
The deflector and source redshifts and the simulated angular diameter distance and time-delay distance for our strong lens systems.We denote lensing systems whose redshifts are overlap with the redshifts of SNe Ia (z < 2.261) in boldface.

Table 2 .
The best-fit values for H0 and the corresponding 1σ uncertainties as well as the precision of the estimation when the precision of the measurement of the distances in the strong lens data set is 5% and 10%.For comparison, we include the case where we are limited to using the strong lens systems with source redshifts within the SNe Ia redshift range.