Implications of the lens redshift distribution of strong lensing systems: cosmological parameters and the global properties of earlytype galaxies
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Abstract
In this paper, we assemble a welldefined sample of earlytype gravitational lenses extracted from a large collection of 158 systems, and use the redshift distribution of galacticscale lenses to test the standard cosmological model (\(\varLambda \)CDM) and the modified gravity theory (DGP). Two additional subsamples are also included to account for possible selection effect introduced by the detectability of lens galaxies. Our results show that independent measurement of the matter density parameter (\(\varOmega _m\)) could be expected from such strong lensing statistics. Based on future measurements of strong lensing systems from the forthcoming LSST survey, one can expect \(\varOmega _m\) to be estimated at the precision of \(\varDelta \varOmega _m\sim 0.006\), which provides a better constraint on \(\varOmega _m\) than Planck 2015 results. Moreover, use the lens redshift test is also used to constrain the characteristic velocity dispersion of the lensing galaxies, which is well consistent with that derived from the optical spectroscopic observations. A parameter \(f_E\) is adopted to quantify the relation between the lensingbased velocity dispersion and the corresponding stellar value. Finally, the accumulation of detectable galactic lenses from future LSST survey would lead to more stringent fits of \(\varDelta f_E\sim 10^{3}\), which encourages us to test the global properties of earlytype galaxies at much higher accuracy.
1 Introduction
The current accelerating expansion of the Universe, which is supported by the observations of Type Ia supernovae (SNIa) [1, 2] in combination with independent estimates of cosmic microwave background (CMB) [3] and large scale structure (LSS) [4], has become one of the fundamental challenges to standard models in particle physics and modern cosmology. As was pointed out in an increasing body of literature [5, 6], this cosmic acceleration can be attributed to an energy component with negative pressure (the so called dark energy), which dominates the universe at late times and causes the observed accelerating expansion. The other possibility is to contemplate modifications to the Friedman– Robertson–Walker models arising from extra dimensions, which has triggered many theoretical speculations in the socalled braneworld scenarios [7]. However, the potential of certain type of observational data, even though ever increasing, does not yet allow us to differentiate the two likely explanations for the observed cosmic acceleration. For instance, the SNIa data would not be sufficient to place stringent constraint on cosmological parameters, if taken alone separately from the other approaches [8]. Indeed, the power of modern cosmology lies in building up consistency rather than in single and precise experiments [9], which indicates that every alternative method of restricting cosmological parameters is desired. Following this direction, a number of combined analyses involving baryonic fraction the xray gas mass fraction of clusters [10], radio observations of the Sunyaev–Zeldovich effect together with Xray emission [11], and ultracompact structure in intermediateluminosity radio quasars [12, 13] have been performed in the literature, which were able to constrain the cosmological parameters consistent with the analysis of Type Ia supernovae. In this paper, we will assemble a large sample of strongly gravitationally lensing systems (SGL) [14, 15] to examine whether the lens redshift distribution test can be utilized – not only to optimize the parameters in \(\varLambda \)CDM model – but also to carry out comparative studies between competing cosmologies.
In the past decades, an cosmological examination of galacticscale SGL systems, based on the derived angular diameter distances between the source, the lens, and the observer, has been applied to test a diverse range of dynamical dark energy models. For example, the XCDM model in which dark energy is described by a hydrodynamic energymomentum tensor with a constant EoS coefficient, and the holographic dark energy model arising from the holographic principle of quantum gravity theory [16]. On the other hand, the first attempt to determine cosmological parameters from the redshift distribution of the lensing galaxies was presented in Ofek et al. [17], which investigated the viability of using lens redshift test to place additional constraints on dark energy models. The original (to our knowledge) formulations of this approach can be traced back to Kochanek [18], Helbig and Kayser [19], Kochanek [20]. The purpose of this paper is to extend our previous statistical analysis based on the angular separation distribution of the lensed images [21, 22], and show how the lens redshift distribution of the most recent and significantly improved observations of earlytype gravitational lenses (158 combined systems) can be used to provide accurate estimates of the cosmological constant in the \(\varLambda \)CDM model and the parameters of alternative cosmological models.
More importantly, in the framework of the concordance cosmological model (\(\varLambda \)CDM) and a singular isothermal ellipsoid (SIE) model for galactic potentials, strong lensing statistics have been most often used for a different purpose: to study the number density of lensing galaxies as a function of redshift, i.e., the velocity dispersion function (VDF) of potential lenses, since strong lensing probability is proportional to the comoving number density times \(\sigma ^4\) (the image separation is proportional to \(\sigma ^2\)) [23]. Meanwhile, considering the fact that earlytype galaxies dominate the lensing cross sections due to their larger central mass concentrations, gravitational lenses therefore provide a unique massselected sample to study the global properties of earlytype galaxies over a range of redshifts (up to \(z\sim 1\)). A pioneer work was made by Chae et al. [24], which investigated the VDF of earlytype galaxies based on the distribution of lensed image separations observed in the Cosmic Lens AllSky Survey (CLASS) and the PMNNVSS Extragalactic Lens Survey (PANELS). However, the statistical lens sample of lensed systems, which are required to be complete for image separations, is too small to provide accurate estimates. In this work, focusing a larger sample of 158 gravitational lenses drawn from the Sloan Lens ACS (SLACS) Survey and other sky surveys [14, 15], we will use the distribution of lens reshifts to provide independent constraints on the velocity dispersion function of earlytype galaxies (\(z\sim 1.0\)), especially the characteristic velocity dispersion (\(\sigma _*\)) in a solely lensingbased VDF.
This paper is organized as follows. In Sect. 2 we briefly describe the methodology and the lens redshift data from various surveys. In Sect. 3 we introduce two prevalent cosmologies and show the fitting results on the relevant cosmological parameters. In Sect. 4 we present the constraints on a model VDF of earlytype galaxies and discuss their implications. The final conclusions are summarized in Sect. 5.
2 Methodology and observations
3 Cosmological model and results
Summary of the cosmological constraints from the lens redshift distribution of current strong lensing observations
Cosmological model  Data  Cosmological fit 

\(\varLambda \)CDM (\(\nu _n=\nu _v=0\))  Current SGL (full sample)  \(\varOmega _m=0.315\pm 0.085\) 
(\(\nu _n=\nu _v=0\))  Current SGL (sample A)  \(\varOmega _m=0.291\pm 0.109\) 
(\(\nu _n=\nu _v=0\))  Current SGL (sample B)  \(\varOmega _m=0.355\pm 0.125\) 
(\(\nu _n\ne \nu _v\ne 0\))  Current SGL (full sample)  \(\varOmega _m=0.274\pm 0.076\) 
(\(\nu _n\ne \nu _v\ne 0\))  Current SGL (sample A)  \(\varOmega _m=0.254\pm 0.096\) 
(\(\nu _n\ne \nu _v\ne 0\))  Current SGL (sample B)  \(\varOmega _m=0.314\pm 0.116\) 
DGP (\(\nu _n=\nu _v=0\))  Current SGL (full sample)  \(\varOmega _m=0.243\pm 0.077\) 
(\(\nu _n=\nu _v=0\))  Current SGL (sample A)  \(\varOmega _m=0.238\pm 0.105\) 
(\(\nu _n=\nu _v=0\))  Current SGL (sample B)  \(\varOmega _m=0.263\pm 0.111\) 
(\(\nu _n\ne \nu _v\ne 0\))  Current SGL (full sample)  \(\varOmega _m=0.207\pm 0.067\) 
(\(\nu _n\ne \nu _v\ne 0\))  Current SGL (sample A)  \(\varOmega _m=0.204\pm 0.092\) 
(\(\nu _n\ne \nu _v\ne 0\))  Current SGL (sample B)  \(\varOmega _m=0.228\pm 0.101\) 
3.1 The standard cosmological model (\(\varLambda \)CDM)
By fitting the \(\varLambda \)CDM model to the current 158 strong lensing systems, we get \(\varOmega _m=0.315\pm 0.085\) in the case of no evolution model (\(\nu _n=\nu _v=0\)), which is well consistent with the results given by the recent data release of Planck observations [50]. One can clearly see that the currently compiled strong lensing data improves the constraints on model parameters significantly. Considering Sample A and Sample B, the likelihood is maximized at \(\varOmega _m=0.291\pm 0.109\) and \(\varOmega _m=0.355\pm 0.125\) with no redshift evolution. More importantly, we find that different galaxy evolution models will slightly affect the constraints on the model parameter: the evolution of the quantities \(n_{*}\) and \(\sigma _{*}\) will shift the matter density parameter to a lower value. For the three strong lensing samples defined in Sect. 2, the bestfitted values and the 1\(\sigma \) limits are \(\varOmega _m=0.274\pm 0.076\) (Full sample), \(\varOmega _m=0.254\pm 0.096\) (Sample A), and \(\varOmega _m=0.314\pm 0.116\) (Sample B) using the powerlaw evolution model after Kang et al. [38]. These results are shown in Fig. 2 and Table 1. We remark here that our results strongly suggest that larger and more accurate sample of the strong lensing data can become an important complementary probe to test the properties of dark energy. This conclusion is strengthened by the comparison of our cosmological fits from the redshift distribution of a larger sample and those from the absolute lensing probability for a smaller sample of optical and radio lenses (\(\varOmega _m=0.3^{+0.2}_{0.1}\)) [51].
Another important issue is the comparison of our cosmological results with earlier studies done using other alternative probes. We turn to the observational Hubble parameter data (OHD) to verify this point. The Hubble parameter H(z) at 31 different redshifts was obtained from the differential ages of passively evolving galaxies, while 10 more Hubble parameter data were determined recently from the radial BAO size method (see Qi et al. [52] for more details). With the latest OHD data comprising 41 data points, we obtain the bestfit values of the cosmological parameters in the flat \(\varLambda \)CDM model: \({\varOmega _m}=0.255\pm 0.030\) and \(H_0=70.4\pm 2.5 \; \mathrm {kms}^{1} \; \mathrm {Mpc}^{1}\) at 68.3% confidence level. For a good comparison, fits on the matter density parameter are also plotted in Fig. 2 (with Hubble constant marginalized). One may observe that the results obtained from the lens redshift test are well consistent with the OHD fits, although larger uncertainties may arise due to possible evolution of the quantities \(n_{*}\) and \(\sigma _{*}\). Such excellent consistency could also be clearly seen through the comparison with WMAP 5year data combined with BAO and SN Union data sets [53], in which the bestfit parameters are given as \(\varOmega _m=0.274\) and \(H_0=70.5 \mathrm {kms}^{1} \; \mathrm {Mpc}^{1}\) for the flat \(\varLambda \)CDM model. In contrast, recent CMB anisotropy measurements by Planck data favors a higher value of \(\varOmega _m\) and thus a larger matter density in the \(\varLambda \)CDM model. Based on the fullmission Planck observations of temperature and polarization anisotropies of the CMB radiation, Planck Collaboration (2015) gave the bestfit parameter: \({\varOmega _m}=0.308\pm 0.012\) and \(H_0=67.8\pm 0.9 \; \mathrm {kms}^{1} \; \mathrm {Mpc}^{1}\) [50]. Let us note that the matter density parameter inferred from CMB and OHD data are highly dependent on the value of the Hubble constant, considering the well known strong degeneracy between \(\varOmega _m\) and \(H_0\). Therefore independent measurement of \(\varOmega _m\) from strong lensing statistics could be expected and indeed is revealed here.
3.2 Dvali–Gabadadze–Porrati model (DGP)
Working on the DGP model, we obtain the fitting results from two cases of evolution models of lensing galaxies, which are displayed in Fig. 3 and Table 1. The marginalized 1\(\sigma \) constraints of the parameters are: \(\varOmega _m=0.243\pm 0.077\) with no redshift evolution and \(\varOmega _m=0.207\pm 0.067\) with redshift evolution. In both cases, the strong lensing statistics imposes a strong bound on \(\varOmega _m\), which is similar to what was obtained when the dark energy models are explored with the ratio of (angulardiameter) distances between lens and source and between observer and lens [14, 29]. Working on the two subsamples, the bestfit values of the parameters are: \(\varOmega _m=0.238\pm 0.105\) (with no redshift evolution), \(\varOmega _m=0.204\pm 0.092\) (with redshift evolution) for Sample A, and \(\varOmega _m=0.263\pm 0.111\) (with no redshift evolution), \(\varOmega _m=0.228\pm 0.101\) (with redshift evolution) for Sample B. More interestingly, we also note the DGP model, which has already been ruled out observationally considering the precision cosmological observational data [54, 55, 56], seems to be a representative set instead of viable candidates for dark energy. Such tendency is also strongly hinted by the fitting results derived from the lens redshift test and the latest Hubble parameter data (see Fig. 3).
Now it is worthwhile to make some comments on the results obtained above. Firstly, comparing to the previous analysis with a smaller sample [22, 29], our results strongly suggest that larger and more accurate sample of SGL data can become an important complementary probe to other standard ruler data. More importantly, the advantage of our method lies in the benefit of being independent of the Hubble constant. Consequently, \(H_0\) and its uncertainty do not influence the final cosmological results. Secondly, in the framework of two cosmologies classified into different categories, the null hypothesis of a dominant matter density (\(\varOmega _m\sim 1\)) is excluded at large confidence level (\(>4\sigma \)). Therefore, our results has provided independent evidence for the accelerated expansion of the Universe, which is the most unambiguous result of the current dataset. Thirdly, considering the general concern that strong gravitational lenses could be a biased sample of galaxies, we note that systematic errors due to sample incompleteness do not exceed \(\sim 0.1\) on the matter density parameter. Finally, although constraints on the hierarchical models of galaxy evolution is beyond the scope of this work, simple evolution of the velocity dispersion function does not significantly affect the lensing statistics and thus the derivation of cosmological information. This conclusion agrees very well with the previous studies on lensing statistics of earlytype galaxies [57, 58, 59].
3.3 Cosmology from future LSST observations
The lensing constraints on the cosmological parameters are already quite competitive compared with those from other methods. However, they still suffer from the small number of lenses in our statistical sample. The redshift distribution test, with larger gravitational lensing samples from future widefield surveys, could be helpful for advancing such applications. Following the recent analysis [60, 61], benefit from the improved depth, area and resolution, the next generation wide and deep sky surveys will increase the current galacticscale lens sample sizes by orders of magnitude in the near future. Recent analytical work has forecast the number of galacticscale lenses to be discovered in the forthcoming photometric surveys [62]. With a large increase to the known strong lens population, current work could be extended to a new regime: in the framework of lens redshift test, what kind of cosmological results one could obtain from \(\sim \) 10,000 discoverable lens population in the forthcoming Large Synoptic Survey Telescope (LSST) survey.
4 Constraints on lensing based characteristic velocity dispersion
Summary of the lensing based characteristic velocity dispersion and its corresponding ratio to the stellar velocity dispersion, based on the lens redshift distribution of current strong lensing observations
Galaxy evolution model  Data  Lensing based velocity dispersion (km/s)  Ratio 

\(\nu _n=\nu _v=0\)  Current SGL (full sample)  \(\sigma _{*,lens}=219.1\pm 5.5\)  \(f_E=1.010\pm 0.025\) 
\(\nu _n=\nu _v=0\)  Current SGL (sample A)  \(\sigma _{*,lens}=224.5\pm 11.3\)  \(f_E=1.034\pm 0.052\) 
\(\nu _n=\nu _v=0\)  Current SGL (sample B)  \(\sigma _{*,lens}=217.3\pm 6.3\)  \(f_E=1.001\pm 0.029\) 
\(\nu _n\ne \nu _v\ne 0\)  Current SGL (full sample)  \(\sigma _{*,lens}=221.6\pm 5.6\)  \(f_E=1.021\pm 0.026\) 
\(\nu _n\ne \nu _v\ne 0\)  Current SGL (sample A)  \(\sigma _{*,lens}=228.3\pm 11.7\)  \(f_E=1.052\pm 0.054\) 
\(\nu _n\ne \nu _v\ne 0\)  Current SGL (sample B)  \(\sigma _{*,lens}=219.4\pm 6.4\)  \(f_E=1.011\pm 0.029\) 
In this work, we consider constraining a model VDF of earlytype galaxies using the statistics of strong gravitational lensing. More specifically, the distribution of lens redshift is mainly applied to place limits on the characteristic velocity dispersion. Moreover, considering the strong degeneracy between the shape of the VDF (\(\alpha \), \(\beta \)) and the characteristic velocity dispersion (\(\sigma _*\)) [24], the focus of this work is: What would be the constrained value of \(\sigma _*\) if \(\alpha \) and \(\beta \) are fixed by a stellar VDF? We obtain a solely lensingbased VDF assuming no and passive evolution of earlytype galaxies, which will then be compared with the measured VDF in the local universe. Figure 6 shows the fits on \(\sigma _*\) for the case of fixing \(\alpha \) and \(\beta \) by the typespecific VDF [35]: \(\sigma _{*,lens}=219.1\pm 5.5\) km/s (with no redshift evolution), \(\sigma _{*,lens}=221.6\pm 5.6\) km/s (with redshift evolution) for the full sample, \(\sigma _{*,lens}=224.5\pm 11.3\) km/s (with no redshift evolution), \(\sigma _{*,lens}=228.3\pm 11.7\) km/s (with redshift evolution) for Sample A, and \(\sigma _{*,lens}=217.3\pm 6.3\) km/s (with no redshift evolution), \(\sigma _{*,lens}=219.4\pm 6.4\) km/s (with redshift evolution) for Sample B. Our results demonstrate the strong consistency between the lensingbased value of \(\sigma _{*,lens}\) and the corresponding stellar values for the adopted stellar VDF, which, to some extent agrees with the velocity dispersion profiles of a sample of 37 elliptical galaxies using a Jaffe stellar density profile and the SIS model for the total mass distribution [67].
Let us note here that the velocity dispersion \(\sigma _{*,lens}\) of the mass distribution and the observed stellar velocity dispersion \(\sigma _{*,stellar}\) need not be the same. We adopt a parameter \(f_E=\sigma _{*,lens}/\sigma _{*,stellar}\) that relates the velocity dispersion and the spectroscopically measured central stellar dispersion. Based on the three different strong lensing samples, we obtain the following bestfitting values and corresponding 68% confidence level uncertainties: \(f_E=1.010\pm 0.025\) (with no redshift evolution), \(f_E=1.021\pm 0.026\) (with redshift evolution) for the full sample, \(f_E=1.034\pm 0.052\) (with no redshift evolution), \(f_E=1.052\pm 0.054\) (with redshift evolution) for Sample A, and \(f_E=1.001\pm 0.029\) (with no redshift evolution), \(f_E=1.011\pm 0.029\) (with redshift evolution) for Sample B. It is apparent that for each case, the consistency between the velocity dispersion for our powerlaw lens model and the spectroscopically measured central stellar dispersion is supported within \(1\sigma \) C.L. However, the constrained results on \(f_E\) parameter are still particularly interesting. What would be an appropriate interpretation of such possible disagreement between \(\sigma _{*,lens}\) and \(\sigma _{*,stellar}\)? Note that the real earlytype galaxies can be divided into the luminous stellar component and the extended dark matter halo component. Based on the Xray properties of the first Xraycomplete optically selected sample of elliptical galaxies, White and Davis [68] discussed the kinetic temperature of the gas and the stars. The derived results and other independent results [69, 70] indicate that dark matter halos are dynamically hotter than the luminous stars, which strongly implies a greater velocity dispersion of dark matter than the visible stars. More recently, Treu and Koopmans [49] used a sample of five individual lens systems to determine the ratio of the SIE velocity dispersion to the stellar velocity dispersion, producing a mean value of \(f_E=1.15\pm 0.05\) from optical spectroscopic observation of the lensing galaxies. Therefore, our results presented in Fig. 7 and Table 2 robustly indicate the possible presence of dark matter, in the form of a mass component with velocity dispersion greater than stellar velocity dispersion.
5 Conclusion and discussion

Firstly of all, with the current catalog of 158 gravitational lenses, we evaluate the power of direct measurements of lens redshift distribution on constraining two popular cosmological models. For the concordance \(\varLambda \)CDM model, we have found \(\varOmega _m=0.315\pm 0.085\) with no redshift evolution and \(\varOmega _m=0.274\pm 0.076\) with redshift evolution. For the DGP braneworld scenario, the current strong lensing systems provide the constraints on the matter density parameter as \(\varOmega _m=0.243\pm 0.077\) with no redshift evolution and \(\varOmega _m=0.207\pm 0.067\) with redshift evolution. More importantly, the DGP model, which has already been ruled out observationally considering the precision cosmological observational data, seems to be a representative set instead of viable candidates for dark energy. Two additional subsamples are also included to account for possible selection effect introduced by the detectability of lens galaxies, which confirms that systematic errors due to sample selection are not larger than statistical uncertainties. Whereas, there are several sources of systematics we do not consider in this paper. For instance, although the average total powerlaw density slope of observed earlytype galaxies has been found to be close to isothermal within a few effective radii [71], the scatter of other galaxy structure parameters, especially those characterizing the stellar distribution in the lensing galaxies, could be an important source of systematic errors on the final results. An influential paper by Hernquist [72] suggested that a brand new Hernquist profile can provide a good approximation to the luminosity distribution of spherical galaxies. Such density profile, which resembles an elliptical galaxy or dark matter halo with \(r^{1}\) at small radii and \(r^{4}\) at large radii, has found widespread astrophysical applications in the literature [73, 74, 75]. Therefore, we perform a sensitivity analysis to investigate how the cosmological constraint on flat \(\varLambda \)CDM is altered by the luminosity density profile. In the framework of a general mass model for the totalmass density and luminosity density (Eq. (3)), the luminositydensity slope is varying as \(\delta =\)2.00, 2.09, and 2.20, while totalmass density parameter is fixed at its bestfit value (\(\gamma =2.09\)) from the totalmass and stellarvelocity dispersion measurements of a sample of SLACS lenses [30]. In general, one can see from Fig. 9 that the derived value of \(\varOmega _m\) is sensitive to the adopted luminosity density profiles, i.e., a steeper stellar density profile in the earlytype galaxies will shift the matter density parameter to a relatively lower value. This illustrates the importance of using auxiliary data to improve constraints on the luminosity density parameter, with future highquality integral field unit (IFU) data [76].

The advantage of our method lies in the benefit of being independent of the Hubble constant. Therefore independent measurement of \(\varOmega _m\) from strong lensing statistics could be expected and indeed is revealed here. More interestingly, one may also observe that simple VDF evolution does not significantly affect the lensing statistics and thus the derivation of cosmological information, if all galaxies are of early type. In the framework of two cosmologies classified into different categories, the null hypothesis of a vanishing dark energy density is excluded at large confidence level (\(>4\sigma \)). Therefore, our results has provided independent evidence for the accelerated expansion of the Universe, which is the most unambiguous result of the current dataset.

Moreover, we have quantified the ability of a future measurements of SGL from the forthcoming LSST survey, which may detect tens of thousands of lenses for the most optimistic scenario [62]. In the framework of the two cosmological models, one can expect \(\varOmega _m\) to be estimated with the precision of \(\varDelta \varOmega _m\sim 0.006\). Therefore, with about 10,000 discoverable SGL systems in forthcoming surveys, the lens redshift test places more stringent constraints on the matter density parameter, compared with the combined results from Planck temperature and lensing data (\(\varDelta \varOmega _m=0.012\)) [50]. Therefore, we have added some support to the argument that the lens redshift distribution, with more detectable galacticscale lenses from the forthcoming surveys, can eventually be used to carry out stringent tests on various cosmological models.

Finally, the currently available lens redshift distribution, which constitutes a promising new cosmic tracer, may also allow us to obtain stringent constraints on the global properties of earlytype galaxies. We use mainly the distribution of lens redshift to constrain the characteristic velocity dispersion (with fixed shape of the velocity function), and thus obtain a solely lensingbased VDF for \(z_l\sim 1.0\). Our results demonstrate the strong consistency between the lensingbased value of \(\sigma _{*,SIE}\) and the corresponding stellar value \(\sigma _{*,stellar}\) for the adopted stellar VDF in the local universe. Furthermore, a parameter \(f_E=\sigma _{*,SIE}/\sigma _{*,stellar}\) is adopted to quantify the relation between the two velocity dispersions, which is fit to \(f_E=1.010\pm 0.025\) (with no redshift evolution) and \(f_E=1.021\pm 0.026\) (with redshift evolution) from the full SGL sample. Therefore, our results agrees with the respective values of \(f_E\) derived in the previous studies, which robustly indicates the possible presence of dark matter halos in the earlytype galaxies, with velocity dispersion greater than stellar velocity dispersion. More importantly, this statistical lensing formalism, when applied to larger samples of strong lensing systems, can provide much more stringent constraints and one can expect \(f_E\) to be estimated with \(10^{3}\) precision.
Footnotes
 1.
In the framework of sphericallysymmetric distribution, the dimensionless surface mass density (convergence) of the lens galaxies can be written as \(\kappa (\theta )=\frac{3\gamma }{2}\left( \theta _E/\theta \right) ^{\gamma 1}\), where \(\theta \) is the angular radius projected to lens plane [28].
 2.
We have also performed a sensitivity analysis through Monte Carlo simulations, in which \(\nu _n\) and \(\nu _v\) were respectively characterized by Gaussian distributions with 10% uncertainty. The results showed the uncertainties of VDF evolution parameters have negligible effects on the final cosmological constraints.
 3.
Notes
Acknowledgements
This work was supported by National Key R&D Program of China no. 2017YFA0402600; the National Natural Science Foundation of China under Grants nos. 11705107, 11475108, 11503001, 11373014, 11073005 and 11690023; Beijing Talents Fund of Organization Department of Beijing Municipal Committee of the CPC; and the Opening Project of Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences. Y. Pan was supported by the Scientific and Technological Research Program of Chongqing Municipal Education Commission (Grant no. KJ1500414); and Chongqing Municipal Science and Technology Commission Fund (cstc2015jcyjA00044, and cstc2018jcyjAX0192).
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