Measuring the Hubble constant is one of the most important tasks in modern cosmology especially since not only it sets the age, the size, and the critical density of the Universe but also the recent direct 퐻 0 measurements from Type Ia supernovae (SNe), calibrated by the traditional Cepheid distance ladder (SH0ES collaboration; Riess et al. 2019), show a 4.4휎 tension with the Planck results under the assumption of ΛCDM model (e.g., Komatsu et al. 2011;Hinshaw et al. 2013;Planck Collaboration et al. 2018;Anderson et al. 2014;Kazin et al. 2014;Ross et al. 2015). However, a recent measurement of 퐻 0 from Collett et al. 2013), the TDCOSMOfoot_0 collaboration (Millon et al. 2020) has shown that one can provide robust constraints on both the angular diameter distance to the lens (퐷 d ; Jee et al. 2015) and the time-delay distance which is a ratio of the angular diameter distances in the system:
where 푧 d is the redshift of the lens, 퐷 s is the distance to the background source, and 퐷 ds is the distance between the lens and the source. These distances are used to determine cosmological parameters, primarily 퐻 0 (e.g., Suyu et al. 2014;Bonvin et al. 2016;Birrer et al. 2019;Chen et al. 2019;Rusu et al. 2019;Wong et al. 2020;Jee et al. 2019;Taubenberger et al. 2019;Shajib et al. 2020).
A blind analysis done by Wong et al. (2020) with this technique as part of the H0LiCOW program (Suyu et al. 2017), in collaboration with the COSMOGRAIL (e.g., Courbin et al. 2018) and SHARP (Chen et al. 2019, Fassnacht et al. in prep) programs, combined the data from six gravitational lens systemsfoot_1 , and inferred 퐻 0 = 73.3 +1.7 -1.8 km s -1 Mpc -1 , a value that was 3.8휎 away from the Planck results. The above work marginalized over two different kinds of mass profiles for the lensing galaxies in order to better estimate the uncertainties. The first description consists of a NFW dark matter halo (Navarro et al. 1996) plus a constant mass-to-light ratio stellar distribution (the "composite model"). The second description models the three dimensional total mass density distribution, i.e., luminous plus dark matter, of the galaxy as a power law (Barkana 1998), i.e., 휌(푟) ∝ 푟 -훾 (the power-law model). Millon et al. (2020) later combined six lenses from Wong et al. (2020) with one additional lens analyzed by Shajib et al. (2020) in the STRIDES program (Treu et al. 2018), and showed that even if we separate these two descriptions of the mass distribution of the lensing galaxy, the 퐻 0 measurements are consistent well within 1%. An independent check by Chen et al. (2019) using ground-based high-resolution adaptive optics (AO) imaging data from SHARPfoot_2 with three strongly lensed quasar also shows consistent results with Wong et al. (2020) and is 3.5휎 away from Planck results.
Given the growing statistical tension between 퐻 0 measurements, efforts by the TDCOSMO collaboration have gone into studying potential systematic uncertainties (Millon et al. 2020;Gilman et al. 2020). A crucial potential source of uncertainty is the assumptions on the radial density profile. Birrer et al. (2020) introduced a flexible parametrization on the mass model that is maximally degenerate with 퐻 0 through the mass-sheet trasformation (so-called internal MST; see also Schneider & Sluse 2013;Xu et al. 2016;Kochanek 2020Kochanek , 2021;;Chen et al. 2020), as a way to express departures from the standard assumptions in previous work (Blum et al. 2020;Shajib et al. 2021). With this parametrization, the main factor determining the precision of the cosmological inference is the stellar kinematics in the lensing galaxy (see discussion by Treu & Koopmans 2002;Koopmans et al. 2003;Jee et al. 2016;Birrer et al. 2016;Chen et al. 2020). With the MST parametrization, the uncertainty on 퐻 0 based on the 7 lens sample of Millon et al. (2020) goes from ∼2% to ∼ 8%, in a standard ΛCDM cosmology.
To further constrain the 퐻 0 value contributed from the MST and anisotropy parameters, Birrer et al. (2020) developed a hierarchical Bayesian framework by including external datasets, assuming they are drawn from the same population. When assuming that the TDCOSMO lenses and the SLACS samples are drawn from the single stellar-orbit anisotropy distribution (Bolton et al. 2004(Bolton et al. , 2006;;Auger et al. 2010), Birrer et al. (2020) inferred 73.3 ± 5.8 km s -1 Mpc -1 . Assuming that TDCOSMO and SLACS are also drawn from the same population in terms of both anisotropy and mass density profile, the inference on 퐻 0 shifted to 67.4 +4.1 -3.2 km s -1 Mpc -1 , which statistically agree with both Planck and SH0ES results. Increasing the number of the time-delay lens systems and using different external datasets are crucial to assess whether the difference between SLACS and TDCOSMO is real or a statistical fluctuation (Birrer & Treu 2021).
To expand the sample of analyzed AO-observed time-delay lenses (Chen et al. 2019) we study the J 0924+0219 lens system, which has AO imaging and archival HST imaging. In this work, we take into account both the internal and external MST and forecast the time delays. Since the velocity dispersion of the lensing galaxy is not yet measured, we predict the time delay based on the imaging data with the expected precision of the kinematic data. In Section 2, we describe the basic information on J 0924+0219 and describe the data acquisition and analysis; in Section 3, we describe the models we used for fitting the imaging; in Section 4, we make a time-delay prediction based on imaging data under the assumption of a flat ΛCDM model with fixed Ω m = 0.3. The conclusion is in Section 5.
The J 0924+0219 system (J2000: 09 h 24 m 55.87, 02 • 19 ′ 24. ′′ 9) is a quadruply-lensed quasar discovered by Inada et al. (2003). The main lensing galaxy is at a redshift of 푧 d = 0.394 ± 0.001 (Eigenbrod et al. 2006), and the source redshift is 푧 s = 1.524 ± 0.001 (Inada et al. 2003). The analysis in this paper is based on new Keck AO and archival HST observations of J 0924+0219. We describe the data acquisition and analysis in Section 2.1 and Section 2.2. We show the data from three HST bands and one Keck AO K ′ -band in Figure 1.
We use optical and near-infrared imaging of the system obtained from the HST archive. The archival data include NICMOS images through the F160W filter (total exposure time: 5311.52 seconds) taken with HST on November 23, 2003 and ACS/WFC images though the F814W filter (total exposure time: 2296 seconds) and F555W filter (total exposure time: 2188 seconds) taken with HST on November 18, 2003 (PID:9744, PI: C. Kochanek). We process the data using AstroDrizzle with standard settings, which removes the geometric distortions, corrects for sky background variations, and flags cosmic-rays. The final drizzled HST images with a scale of 0.05 ′′ per pixel are presented in Figure 1.
The AO imaging was obtained at K ′ -band with the Near-infrared Camera 2 (NIRC2), as part of the SHARP AO effort (Fassnacht et al., in prep). The target was observed with the narrow camera setup, which provides a roughly 10×10 ′′ field of view and a pixel scale of 10 milliarcsec (mas). There are three exposures of 300 seconds on December 30, 2011, seven exposures of 300 seconds on May 16, 2012, and four exposures of 300 seconds on May 18, 2012. The total exposure time was 4200 seconds. We follow our previous work (Chen et al. 2016(Chen et al. , 2019) ) and use the SHARP python-based pipeline, which performs a flat-field correction, sky subtraction, correction of the optical distortion in the images, and a coadditon of the exposures. For the distortion correction step, the images are resampled to produce final pixel scales of 10 mas pix -1 for the narrow camera. The narrow camera pixels samples well the AO PSF, which has typical FWHM values of 60-90 mas. To improve the modeling efficiency for the narrow camera data, we perform a 2×2 binning of the images produced by the pipeline to obtain images that have a 20 mas pix -1 scale. The final HST images are presented in Figure 1.
We describe the PSF models in Section 3.1, lens modeling in Section 3.2, kinematics modeling in Section 3.3, and time-delay prediction model in Section 3.4.
For the F160W band HST imaging, we use T (Krist & Hook 1997) to generate the PSFs with different spectral index, 휂푣 , of a power-law from -0.4 to -2.5 and different focusesfoot_4 from 0 to 10. Given the F160W band HST imaging, we find that the best-fit to the imaging is the PSF with focus equal to 0 and spectral index equal to -1.3. We use this T PSF as the initial guess and then apply the PSF-correction method of Chen et al. (2016) while modeling the F160W HST imaging. For the F814W and F555W bands, that were observed with the ACS with a larger field of view, we use one of the nearby bright stars as the initial guess of the PSF and apply the PSF-correction until the residuals stabilized. For the AO imaging, we follow the criteria described in Section 4.4.3 of Chen et al. ( 2016) and perform 9 iterative steps to create the final PSF and make sure the size of the PSF for convolution is large enough (1.18 ′′ × 1.18 ′′ ) such that the results are stable. The full width half maximum (FWHM) of the AO PSF is ∼ 75 mas. We show the reconstructed AO PSF in Figure 2 and the comparison of AO and HST PSF in Figure 3. Eigenbrod et al. (2006) first modeled this system with HST imaging and suspected that the second set of bluer arcs in F814W band (see Figure 1) inside and outside the area delimited by the red arcs in F160W band could be either a second source in different redshift or a star forming region in the source galaxy. We examine the possibility of a second source plane existing at a lower redshift than the source (푧 = 1.52) due to bluer color and find that the scenario is very unlikely, as the macro model determined by the red arc cannot reproduce a reasonable source for the blue arcs given a possible range of the source redshift from 푧 = 0.5 to 푧 < 1.52. In contrast, we do find that a star forming region can be reconstructed at the same source redshift. Faure et al. (2011) modeled the lens with high-resolution H and Ks imaging obtained using the ESO VLT with adaptive optics and the laser guide star system. They identified a luminous object, located ∼ 0.3 ′′ to the north of the lens galaxy, but showed that it cannot be responsible for the anomalous flux ratios. Many studies (e.g., Metcalf & Madau 2001;Bradač et al. 2002;Dalal & Kochanek 2002;Pooley et al. 2012;Schechter et al. 2014;Glikman et al. 2018;Badole et al. 2020) have shown that the macro model cannot explain the flux ratio, which suggested the presence of microlensing or dark matter substructures. Thus, to avoid possible biases caused by flux ratios, we only use the lensed quasar positions and the extended arc to constrain the mass model, which is also the standard procedure for 퐻 0 measurements in TDCOSMO collaboration. Gilman et al. (2020) also show that the presence of substructures do not bias 퐻 0 above the percent level. We use , a strong lens modeling code to model three HST bands and one Keck AO band simultaneously (Suyu & Halkola 2010;Suyu et al. 2012). We describe the models in the following for fitting the high resolution imaging data. We show the imaging, models, normalized residuals, and reconstructed sources in Figure 4. Note that since the source in F555W band has more clumpy star forming region, the reconstructed source is less regular with small-scale structures and more noisefoot_5 . In addition, the noiseoverfitting problem is due to the fact that the outer region of the source plane is under-regularized, but this effect will not underestimate the uncertainty because the uncertainty will be dominated by the timedelay and velocity dispersion measurements. Besides, we model the imaging with different source resolutions and marginalize over them to control the systematics.
• Power-law mass model+shear+Sérsic light model: we first choose the softened power-law elliptical mass distributions (SPEMD; Barkana 1998) density profile with the softening length close to zero -the main parameters include radial slope (훾), Einstein radius (휃 E ), Position angle (휃 푞 ) and the axis ratio of the elliptical isodensity contour (푞) -to simultaneously model the extended arcs seen in the three HST bands and one AO band, and reconstruct the source structure on a pixelated grid (Suyu et al. 2006). The power-law model is motivated by many studies which have shown that a power-law model provides a good description of the lensing galaxies and dynamical studies for galaxy-galaxy lensing (e.g., Koopmans et al. 2006Koopmans et al. , 2009;;Suyu et al. 2009;Auger et al. 2010;Barnabè et al. 2011;Sonnenfeld et al. 2013;Cappellari et al. 2015;Shajib et al. 2021). In the modeling, we found that two concentric Sérsic profiles are sufficient to describe the lensing light distribution of the HST F555W and HST F160W bands, while three concentric Sérsic profiles are needed for the HST F814W band and Keck AO band. Except for the parameters that describe the lens light center (휃 1,Light and 휃 2,Light ), which are linked together for the light profiles, the light parameters (position angles, ellipticities, and Sérsic index) are free. We list all parameters in Table 1 and Table 2, and show the important marginalised mass model parameters in Figure 5.
• Composite mass model+shear+chameleon light profile: we follow Chen et al. ( 2019) and test a composite (baryonic + dark matter) model. For the dark matter component we adopt the standard NFW profile (Navarro et al. 1996) with the following parameters: halo normalization (NFW 휅 s ), halo scale radius (NFW 푟 s ), halo minor-to-major axis ratio (NFW 푞), and associated position angle (NFW 휃 푞 ). This is motivated by Dutton & Treu (2014), who find that non-contracted NFW profiles are a good representation for the dark matter halos of massive elliptical galaxies (See also Shajib et al. 2021). The baryonic component is modeled by multiplying the lens surface brightness distribution by a constant M/L ratio parameter. For computational efficiency, we model the surface brightness with chameleon profile. The chameleon profile is the difference of two isothermal profiles and is a good approximation to a Sérsic profile over the range of interest (see details in Dutton et al. 2011). We link the baryonic matter to the chameleon light profiles of the F160W bands because it probes the rest-frame near-infrared and thus should be the best tracer of stellar mass (See also Chen et al. 2019;Wong et al. 2017). Since the degeneracy between the wings of the AO PSF and lens light could bias the inferred baryonic component, we do not use AO lens light to infer the baryonic distribution (Chen et al. 2019). However, when combining with HST imaging, the well-known HST PSF can provide the information of baryonic distribution (Chen et al. 2019). Future AO imaging with AO PSF reconstructed from telemetry data can break the degeneracy and directly infer the baryonic matter without the need of HST imaging (Chen et al. 2021). We set a Gaussian prior of 푟 푠 = 15.0± 2.0 ′′ based on the results of Gavazzi et al. (2007) for lenses in the SLACS sample, which encompasses the redshift of J 0924+0219. We list all parameters in Table 3 and Table 4, and show the important marginalised parameters in Figure 6.
To predict the time delays under the presence of the MST, velocity dispersion information is required to constrain the normalization of the 3D de-projected mass model. We follow Sonnenfeld et al. (2012) and calculate the three-dimensional radial velocity dispersion by numerically integrating the solutions of the spherical Jeans equation (Binney & Tremaine 1987)
where 푀 (푟) follows either the power-law mass or composite model.
For the stellar component, we assume a Hernquist profile (Hernquist 1990),
where 퐼 0 is the normalization term and the scale radius can be related to the effective radius by 푎 = 0.551푟 eff . To compare with the data, the seeing-convolved luminosity-weighted line-of-sight velocity dispersion can be expressed as
where 푅 is the projected radius, 퐼 (푅) is the light distribution, P is the PSF convolution kernel (Mamon & Łokas 2005), and A is the aperture. The streaming motions (e.g. rotation) are assumed to be zero. The luminosity-weighted line-of-sight velocity dispersion is given by
The predicted velocity dispersion can be simplified and wellapproximated (Birrer et al. 2016(Birrer et al. , 2020;;Chen et al. 2020) as where 퐽 contains the angular-dependent information including the parameters describing the 3D deprojected mass distribution, 휂 lens , the surface-brightness distribution in the lensing galaxy, 휂 light , and the stellar orbital anisotropy distribution, 훽 ani . 휅 ext and 휆 int represents the external MST and internal MST, respectively. We assume the anisotropy component has the form of an anisotropy radius, 푟 ani , in the Osipkov-Merritt (OM) formulation (Osipkov 1979;Merritt 1985),
where 푟 ani = 0 is pure radial orbits and 푟 ani → ∞ is isotropic with equal radial and tangential velocity dispersions. In our models, we use a scaled version of the anisotropy parameter,
where
The predicted time delay can be expressed as,
where 푐 is the speed of light and 휃, 훽, and 휙(휃) are the image coordinates, the source coordinates, and the Fermat potential (Blandford & Narayan 1986) without the presence of internal or external MST respectively. In the case of single aperture velocity dispersion, we can replace the MST terms (휆 int and 휅 ext ) with Equation ( 6) and the predicted time delays will directly relate to the velocity dis-persion via
The MST-related terms (i.e., 휅 ext and 휆 int ) canceled out in Equation (9). Thus, the uncertainty of the predicted time delays do not depend on the uncertainty of the mass along the line of sight or transformed mass profile via MST, and only rely on the precision of the velocity dispersion measurement, the redshift of the lens, and the angular diameter distance to the lens (See also similar discussion in Koopmans 2006). In other words, once the time delay and velocity dispersion are measured, the value of 퐷 d can be determined (Chen et al. 2020). When further including environmental information (which provide an estimation of 휅 ext ) and 퐷 s /퐷 ds information which comes from either external datasets or assumption of a cosmological model, one can further determine 휆 int (Birrer et al. 2020;Chen et al. 2020) and use it to further constrain 퐻 0 with 퐷 Δt from the population point of view (Birrer et al. 2020). Note that Birrer et al. (2020) use both 퐷 d and 퐷 Δt information to constrain 퐻 0 .
Due to the lack of velocity dispersion measurement, we express the observed velocity dispersion as 휎 ob 푣 = 휎푣 × 280 km s -1 , which is created by assuming a flat ΛCDM with fixed Ω m = 0.3, 퐻 0 = 70 km s -1 Mpc -1 , and 휆 int = 1 (i.e., no internal mass-sheet transformation) in the power-law model (Chen et al. 2020). We fold in an expected 5% uncertainty of the velocity dispersion measurement and present time delay predictions under the assumption of the ΛCDM model with fixed Ω m = 0.3. For the velocity dispersion calculation, we assume the seeing is 1.0 ′′ and the aperture size is 1 ′′ × 1 ′′ . We show the predicted time delays in Figure 7 with various source resolutions. When we marginalized over different source resolutions of the power-law model, the power-law model predicts Δ푡 BA ℎ 휎-2 푣 = 6.75
+0.78 -0.68 days, Δ푡 CA ℎ 휎-2 푣 = 10.2 +1.2 -1.0 days, and Δ푡 DA ℎ 휎-2 푣 = 7.31 +0.86 -0.74 days. When we marginalized over different source resolutions of the composite model, the composite model predicts Δ푡 BA ℎ 휎-2 푣 = 6.99 +0.81 -0.71 days, Δ푡 CA ℎ 휎-2 푣 = 11.6 +1.4 -1.2 days, and Δ푡 DA ℎ 휎-2 푣 = 8.10 +0.96 -0.82 days. When we marginalized powerlaw and composite model, we obtain Δ푡 BA ℎ 휎-2 푣 = 6.89 +0.78 -0.74 days, Δ푡 CA ℎ 휎-2 푣 = 10.7 +1.6 -1.2 days, and Δ푡 DA ℎ 휎-2 푣 = 7.70 +0.97 -0.91 days.
Given the expected short time delay of this system, it will be challenging to measure the time delays within 10% uncertainty.
In this work, we use the high resolution Keck AO imaging data, collected by the SHARP team, and deep HST WFC3 images through the F160W filter, HST ACS/WFC images though F555W filter and F814W filter to simultaneously constrain the mass distribution of J 0924+0219 lens system. When assuming a ΛCDM model with fixed Ω m = 0.3, we find that the power-law model predicts Δ푡 BA ℎ 휎-2 푣 = 6.75
+0.78 -0.68 days, Δ푡 CA ℎ 휎-2 푣 = 10.2 +1.2 -1.0 days, and Δ푡 DA ℎ 휎-2 푣 = 7.31 +0.86 -0.74 days; the composite model (i.e., a NFW dark matter halo (Navarro et al. 1996) plus a constant mass-tolight ratio stellar distribution) predicts Δ푡 BA ℎ 휎-2 푣 = 6.99 +0.81 -0.71 days, Δ푡 CA ℎ 휎-2 푣 = 11.6 +1.4 -1.2 days, and Δ푡 DA ℎ 휎-2 푣 = 8.10 +0.96 -0.82 days. When we marginalized over the power-law and composite model, we obtain Δ푡 BA ℎ 휎-2 푣 = 6.89 +0.78 -0.74 days, Δ푡 CA ℎ 휎-2 푣 = 10.7 +1.6 -1.2 days, and Δ푡 DA ℎ 휎-2 푣 = 7.70 +0.97 -0.91 days. Future measurements of time delays with 10% uncertainty and velocity dispersion with 5% uncertainty would yield a 퐻 0 constraint of ∼ 15% precision.
It is important to note that our analysis is truly blind since the time delays and velocity dispersion are not yet measured. Once the velocity dispersion measurement and time delays are measured, the derived posteriors can be used to constrain the 퐻 0 . As part of the TDCOSMO effort, we are getting everything for this lens to have a high-quality 퐻 0 measurement under the assumptions of standard NFW profile and fixed M/L ratio. These assumptions are in general supported by Shajib et al. (2021) and are currently the standard in the TDCOSMO collaboration. Future work with including varying mass-to-light (M/L) ratio, allowing contracted/expanded NFW profile, and adapting axisymmetric Jeans equations are worth examining the systematics when spatially-resolved kinematics data are obtained.
http://www.tdcosmo.org/
Except the first lens, B1608+656, which was not done blindly, the subsequent five lenses in H0LiCOW are analyzed blindly with respect to the cosmological quantities of interest.
The Keck AO imaging data are part of the Strong-lensing High Angular Resolution Programme (SHARP;Fassnacht et al. in preparation)
MNRAS 000,1-11 (2015)
The flux per unit frequency interval is 퐹 휈 = 퐶휈 휂푣 , where 휂푣 is the power-law index and C is a constant; focus is related to the breathing of the secondary mirror, which is between 0 ∼ 10.MNRAS 000,1-11 (2015)
See also the same effect inWong et al. (2017).MNRAS 000,1-11 (2015)