# A revised test of cosmic curvature at high redshifts: the distance sum rule

## Abstract

Ultra-compact structure in radio quasars, with milliarcsecond angular sizes measured by very-long-baseline interferometry (VLBI), provides an important source of angular diameter distances that can be observed up to higher redshifts. In this paper, with the latest catalog of galactic-scale strong gravitational lensing systems and the VLBI observation of milliarcsecond compact structure in intermediate-luminosity quasars, we place constraints on the curvature of the universe through the well-known distance sum rule, without assuming any fiducial cosmological model. Assuming power-law density profiles for the total mass density of lensing galaxies (), we find that, although the zero cosmic curvature is still included within confidence level, a closed universe is seemed to be more favored in our analysis. In addition, in the framework of a more general lens model which allows the luminosity density profile to be different from that of the total-mass density profile , a weaker constraint on the curvature ( at 68% confidence level)is obtained indicates that a more general lens model does have a significant impact on the measurement of cosmic curvature. Finally, based on the mock samples of strong gravitational lenses and quasars with the current measurement accuracy, we find that with about 16000 strong lensing events (observed by the forthcoming LSST survey) combined with the distance information provided by 500 compact uv-coverage, one can constrain the cosmic curvature with an accuracy of , which is comparable to the precision of Planck 2015 results.

Department of Astronomy, Beijing Normal University, Beijing 100875, China

Department of Astrophysics and Cosmology, Institute of Physics, University of Silesia, 75 Pułku Piechoty 1, 41-500 Chorzów, Poland.

Electronic address: caoshuo@bnu.edu.cn

## 1. Introduction

Estimating the curvature of the Universe, which is described by the parameter (open (), flat () or closed ()), is a robust way to test the important assumption that the Universe, which will significantly influence our knowledge of the cosmic evolution, the nature of dark energy, the cosmic inflation and the ultimate fate of the Universe. Therefore, the measurement of the cosmic curvature has always been one of the most fundamental issues in modern cosmology. Fortunately, the increasing precision and breadth of astronomical observations could provide much tighter constraints on , most of which indicate that a flat Universe () in the framework of the standard CDM model is favored at very high confidence level. The latest constraint, , was derived by the newest Planck 2015 results of Cosmic Microwave Background (CMB) observations (Ade et al., 2016). However, we emphasize here this seemingly convincible evidence of zero cosmic curvature is strongly dependent on the standard CDM model (Ade et al., 2016). More importantly, such constraint on is obtained at just one high redshift (), which makes it rewarding to explore the behavior of cosmic geometry with observations spanning a much greater redshift range (Räsänen et al., 2015; Cai et al., 2016). Therefore, the measurement of through the model-independent method at lower redshifts has been extensively investigated in the literature (Clarkson et al., 2008; Li et al., 2016b; Wei & Wu, 2017; Räsänen et al., 2015; Xia et al., 2017). By combining observations of the expansion rate and cosmological distances, Clarkson et al. (2008) proposed a method to measure the spatial curvature of the Universe or even test the Friedmann-Lemaître-Robertson-Walker (FLRW) metric in a model-independent way, which has been fully implemented with updated observational data including supernovae Ia (SN Ia) (standard candles) and Hubble parameter (cosmic chronometers) (Li et al., 2016b; Wei & Wu, 2017). It was found by the authors that the cosmic curvature is well consistent with the flat case of . More recently, based on the distance sum rule, Räsänen et al. (2015) proposed that a joint analysis with the strong gravitational lensing (SGL) and SN Ia data may provide another method to measure the curvature in a model-independent way. As one of the successful predictions of General Relativity, strong gravitational lensing has provided a very useful astrophysical tool which can be used as a cosmic distance indicator independently of redshift (see Biesiada et al. (2011); Cao & Liang (2011); Cao & Zhu (2011a); Cao et al. (2012a)). The feasibility of this method inferred from standard rulers and standard candles was further discussed by Xia et al. (2017), who found that the constrained curvature parameter is close to zero from the newly-complied SGL sample including 118 galactic-scale systems (Cao et al., 2015a).

We remark that the measurements of in the previous works concentrated on the luminosity distance using SN Ia as standard candles. However, as was extensively discussed in the literature (Wei & Wu, 2017; Li et al., 2016b), the involvement of SN Ia data also leads to a large uncertainty to the determination of , due to several nuisance parameters characterizing the SN light-curves. More importantly, the limited redshift distribution of SN Ia, i.e., the available observation of this standard ruler covers relatively lower redshift range (), will lead to the fact that only a limited number of SGL systems can be used to perform constraints on and the high-redshift systems can not be calibrated by SN Ia (Xia et al., 2017). Finally, the application of SN Ia data in the distance sum rule is heavily dependent on the so-called Etherington duality principle connecting luminosity and angular diameter distances, the breakdown of which would indicate some mechanism of nonconservation of the number of photons on the path from the source to observer. Earlier discussions of this issue can be found in Holanda et al. (2011); Cao & Zhu (2011b); Cao & Liang (2011); Cao & Zhu (2014).

In this context, it is clear that the collection of observational data concerning angular diameter distance measurements at high redshifts does play a crucial role. Following this direction, in this paper, in order to alleviate the shortcoming of SN Ia in the distance sum rule, we turn to the VLBI observations of milliarcsecond compact structure in intermediate-luminosity quasars covering the redshift range , which are combined with the latest catalog of strong gravitational lensing systems to measure the cosmic curvature. As is well known, in order to measure the angular diameter distance, we always turn to objects of known comoving size acting as standard rulers, such as the sound horizon at the epoch of last scattering observed by baryonic acoustic oscillations (BAO). The standard yardstick estimated from total and correlated flux densities measured with radio interferometers was first noted by Gurvits (1994), who considered a sample comprising 337 active galactic nuclei (AGN). Note that the final sample, which does not show jet-like structure for any system, contains a wide class of extragalactic objects, including quasars, radio galaxies, BL Lac objects (blazars), etc. The ratio of these two quantities (total flux density and correlated flux density), i.e., the visibility modulus defines a characteristic angular size

(1) |

where is the interferometer baseline, measured in wavelengths. let us note here, due to the complication of the luminosity distribution of AGN, this estimated characteristic angular size and the true source size need not be the same. However, this quantity is still accurate enough to represent the mean properties of source sizes relevant to statistical analysis (Gurvits, 1994). Such type of measurements has been extensively used by several other authors to test the accelerating expansion of the universe (Gurvits et al., 1998; Jackson & Dodgson, 1997; Vishwakarma, 2001; Lima & Alcaniz, 2002; Zhu & Fujimoto, 2002; Chen & Ratra, 2003). On the other hand, such measurement of the apparent angular size of milliarcsecond structure is heavily dependent on the locations of the radio telescopes, the time of observations, and the length of the interferometer baseline. There were suggestions from Gurvits (1994) that an averaging a sample of sources could significantly alleviate the above possible effects, which, from a statistical point of view, is equivalent to two kinds of averaging: the direct averaging over the set of sources and the averaging over the different position angles of the interferometer baselines (which is especially important for elongated structure). Finally, we encounter a critical issue that still needs to be addressed in a systematic way: the linear sizes of compact radio sources might not be constant. Let us consider the case in which sources with extreme spectral indices and low luminosities are excluded from the full sample, which might alleviate the dependence of on the source luminosity and redshift (Gurvits et al., 1998; Vishwakarma, 2001). Such treatment has been extensively discussed in several follow-up papers (Lima & Alcaniz, 2002; Zhu & Fujimoto, 2002; Chen & Ratra, 2003), in which the full data set was distributed into certain number of redshift bins with about the same number of sources per bin. However, the typical value of the characteristic linear size remained one of the major uncertainties in their analysis. Confronted with such theoretical and observational puzzles, more recently, 181 milliarcsecond ultra-compact radio quasars observed by a 2.29 GHz VLBI all-sky survey were divided into three sub-samples according to their optical counterparts and luminosity: low-luminosity QSO, intermediate-luminosity QSO and high-luminosity QSO (Cao et al., 2017a). The final results showed that the statistical linear sizes of intermediate-luminosity QSO show negligible dependence on redshifts and intrinsic luminosity, which thus represent a new type of standard ruler in cosmology (Cao et al., 2017c; Qi et al., 2017; Xu et al., 2017; Zheng et al., 2017). Compared with the commonly used distance indicator SN Ia (), the remarkable advantage of this QSO sample lies in the fact that it can be observed up to very high redshift (). Therefore, the angular diameter distance provided by QSO could act as a bridge for the so-called “redshift desert”, which makes it rewarding to extend our exploration of cosmic curvature to the deep universe.

This paper is organized as follows: In section 2 we briefly describe the method, and then introduce the data used in our work. We discuss our results in section 3 and show in section 4 how large the source catalogue needs to be in order to achieve stringent constraint on the cosmic curvature comparable to Planck 2015 results. Finally, conclusions and discussions are presented in section 5.

## 2. Methodology and data

We perform our calculations in the context of a homogeneous and isotropic universe, the geometry of which can be described by the FLRW metric

(2) |

where represents the spatial curvature. The unperturbed evolution of FLRW metric is described by the Friedman equation, i.e., the Hubble parameter defined as a function about the scale factor and its derivative respect to time, (its present value is the Hubble constant ).

For a specific strong lensing system with the intervening galaxy acting as a lens (at redshift ), the multiple image separation of the source (at redshift ) depends only on the ratio of angular-diameter distances between lens and source () and between observer and source (), as long as one has a reliable model for the mass distribution within the lens (Grillo et al., 2008; Biesiada et al., 2011; Cao et al., 2012a, 2015a). In this context, given the relation between angular diameter distance and comoving distance, the dimensionless distance between the lensing galaxy at and the source at is given by

(3) | |||||

where . For convenience, we denote , and . According to the distance sum rule, these three dimensionless distances satisfy the following relation (Räsänen et al., 2015; Xia et al., 2017)

(4) |

Therefore, if the distance information of , and are obtained from the observations, the value of could be directly derived without the involvement of any specific cosmological model. In this paper, the distance ratio of can be extracted form the observations of SGL, while the information of other two distances, and , could be obtained through the angular size measurements of compact structure in radio quasars.

Moreover, it is well known that an upper limit on the comoving angular diameter distance, , is related to an a lower limit on the cosmic curvature, . Based on the precise observation of CMB, (Vonlanthen et al., 2010; Audren et al., 2013; Audren, 2014) as well as a recent measurement of Hubble constant with 3% uncertainty (Efstathiou, 2014), a prior of the curvature will be applied in our statistical analysis

(5) |

This lower limit on has also been extensively used in the previous works involving SN Ia as standard candles to investigate the cosmic curvature (Räsänen et al., 2015; Xia et al., 2017).

### 2.1. Strong gravitational lensing data

Since the first discovery of strong gravitational lensing system Q0957+561 (Walsh et al., 1979), the increasing size of detectable sample has made SGL a serious technique in the exploring of extragalactic astronomy (galactic structure studies) (Ofek et al., 2003; Cao et al., 2016) and in cosmology (Biesiada et al., 2011; Cao & Liang, 2011; Cao & Zhu, 2011a; Cao et al., 2012a, 2015a). Especially, strong lensing system with the intervening galaxy acting as a lens between the observer and source usually produces multiple images of the background source (quasar, supernova or galaxy). Moreover, because most of the cosmic stellar mass of the universe is located in early-type galaxies (or elliptical galaxies), in our analysis we will consider only SGL systems with early-type galaxies acting as the intervening lenses. Although the properties of early-type galaxies (their formation and evolution) are still not fully understood, in statistical gravitational lensing studies, the singular isothermal sphere (SIS) model and the singular isothermal ellipsoid (SIE) model are commonly used to describe the mass distribution of lensing galaxies acting as lenses.

The Einstein radius in a SIE lens is expressed as

(6) |

where is the speed of light, is the central velocity dispersion of the lensing galaxy and is a phenomenological coefficient with 20% uncertainty, i.e., (Cao et al., 2012b; Ofek et al., 2003). For , the SIE model reduces to the standard SIS model. However, for the rigid assumption of the SIS model, a assumption of the deviation from the isothermal profile and its evolution with redshift is more reasonable. Therefore, in this paper we will also generalize the SIS model to a spherically symmetric power-law mass distribution (Cao et al., 2015a). Finally, based on the power-law density profiles, we allow the luminosity density distribution to vary with that of the total-mass density. Therefore, in order to fully consider the effect of lens model in measuring cosmic curvature, the ratio of dimensionless distance between lens and source and that between observer and lens will be derived as

(7) |

in the framework of the above three mass distribution models.

A total sample of 118 galactic scale SGL systems from the Sloan Lens ACS Survey (SLACS), BOSS emission-line lens survey (BELLS), Lens Structure and Dynamics (LSD) and Strong Lensing Legacy Survey (SL2S) assembled by Cao et al. (2015a) are appropriate for this work, and we base our analysis on the methodology described above.

### 2.2. Radio quasar data

Currently, the possibility of using compact radio sources in the study of cosmological parameters and the physical properties of AGN is very attractive in the literature (Jackson & Dodgson, 1997; Vishwakarma, 2001; Lima & Alcaniz, 2002; Zhu & Fujimoto, 2002; Chen & Ratra, 2003). More interestingly, the advantage of radio quasars, compared with other standard rulers including baryon acoustic oscillations (BAO) (Percival et al., 2010), galaxy clusters (Bonamente et al., 2006), and strong lensing systems (Cao et al., 2012a, 2015a), lies in the fact that is that quasars are observed at much higher redshifts. Our procedure follows the phenomenological model originally proposed in Gurvits (1994) and later investigated in Gurvits et al. (1998), which quantifies the luminosity and redshift dependence of the linear sizes of quasars as

(8) |

where and are two constant parameters quantifying the ”angular size-redshift” and ”angular size-luminosity” relations. Therefore, constraints on and will help us to differentiate between sub-samples and determine standard rulers, i.e., the compact radio sources fulfiling the criteria of could act as standard rulers. In the same work, considering the VLBI visibility data for 337 active galactic nuclei (AGN), Gurvits (1994) provided a rough estimation of the dependence of the apparent angular sizes on the luminosity and redshifts: , and , under the assumption of a homogeneous and isotropic universe without cosmological constant. A regression analysis of the restricted sample with spectral index () and total luminosity (), gave a value of and (Gurvits et al., 1998). Following this direction, a recent attempt to determine the intrinsic linear sizes of the compact structure in 112 radio quasars, in the framework of the concordance CDM cosmology, was presented in Cao et al. (2015b), which found a substantial evolution of linear sizes with luminosity () and negligible cosmological evolution of the linear size (). More recently, the use of intermediate-luminosity quasars ( W/Hz W/Hz) as potential cosmological tracers was studied in detail in Cao et al. (2017a), which found that the intrinsic metric linear size shows negligible dependence on redshift and the source luminosity (, ). Following the same sample selection criteria, Cao et al. (2017b) presented observations of a sample of 120 high- quasars in the redshift range of , in which the linear size of this standard ruler was also calibrated to pc through a cosmology-independent technique (Cao et al., 2017b). See Table 1 of Cao et al. (2017b) for details of the quasar data and reference to the source papers.

Therefore, once the characteristic length is determined, the radio quasars in this sub-sample could act as a cosmological standard ruler, whose angular size is expressed as

(9) |

According to the relation between comoving distance and angular diameter distance, we can obtain the dimensionless distances and as

(10) |

### 2.3. The detailed procedures of our analysis

The difficulty of applying observational data (SGL and QSO) to Eq. (4) lies in the fact that there is not one-to-one correspondence between the redshift of SGL data providing distance ratio and the redshift of QSO data providing the information of distances. Fortunately, a model-independent method Gaussian processes (GP) (Seikel et al., 2012a), which has been widely used in various studies (Seikel et al., 2012b; Yang et al., 2015; Li et al., 2016a; Qi et al., 2016; Zhang & Xia, 2016), can be employed to reconstruct a function from data straightforwardly, without any parametric assumption. Therefore, we could use GP to reconstruct the profile of from the quasar sample, which make it possible to calibrate the distance up to the redshifts . Considering the redshift range of lenses and sources in these 118 SGL systems (, ), after the subtraction of those SGL systems with , there are 106 SGL systems (20 samples from BELLS, 57 samples from SLACS, and 6 samples from SL2S) left in our sample.

Fig. 1 shows the redshift coverage of different standard ruler data. Compared with the previous works concentrated on the luminosity distance using SN Ia as standard rulers (Xia et al., 2017), one can clearly see that the inclusion of quasars could result in a fair coverage of redshifts, which enables QSO an excellent complement to other observational probes at lower-redshift SN Ia data.

## 3. Results and discussions

In our analysis, the value of will be determined by applying the maximum likelihood method by using the emcee Python module (Foreman-Mackey et al., 2013) based a Markov chain Monte Carlo Markov chain (MCMC) code. Moreover, in order to fully consider the effect of lens model in measuring cosmic curvature, in our analysis two lens models will be used to characterize the total mass-profile and light-profile shapes of elliptical galaxies: 1) the total mass distribution is described by a generalized singular isothermal sphere (SIS) model with a spherically symmetric power-law mass distribution (); 2) the luminosity density distribution varies with that of the total-mass density (, ) (Cao et al., 2016).

The cosmic curvature is determined by using a minimization method

(11) |

where with the corresponding uncertainty from both SGL and QSO observations: . In our fits mass density power-law index p={} are taken as free parameters fitted together with cosmic curvature . Note that for the SGL data, uncertainties of the velocity dispersion are derived from the data set, while for all the lenses the fractional uncertainty of the Einstein radius is taken at the level of 5%. For the QSO data, it will contribute to the uncertainty of and due to distance reconstruction with GP.

### 3.1. Extended SIE lens

In the first model, we assume that the shape of the the total mass density follows the extended SIS model, i.e., a spherically symmetric power-law mass distribution , where is the spherical radius from the center of the lening galaxy. In combination of the spherical Jeans equation, the observational value of the angular-diameter distance ratio is expressed as (Cao et al., 2015a)

(12) |

where

(13) |

Note that is the velocity dispersion obtained inside an aperture radius (more precisely, luminosity-averaged line-of-sight velocity dispersion). More recently, the analysis performed by Ruff et al. (2011) has revealed the mass density power-law index of massive elliptical galaxies evolves with redshift, . The authors found the following results: . Such linear relation will be also used in our analysis, in which and are treated as free parameters together with cosmological ones. Similar works aimed at establishing the evolution of mass density profile with different samples of lens galaxies have also been attempted in the literature (Koopmans et al., 2006; Bolton et al., 2012).

Applying the -minimization procedure to 106 SGL systems, we obtain the one-dimensional and two-dimensional marginalized distribution with , uncertainties for the parameters , and , which are shown in Fig. 2. These contours show that at the level, the optimized parameter values are , and . Surprisingly, the one-dimensional marginalized distribution of implies that, although the zero cosmic curvature is still included within confidence level, a closed universe seems to be more favored in our analysis, which is also in well agreement with the results of previous works (Räsänen et al., 2015; Xia et al., 2017), but it is different from the flat case of supported by other independent and precise experiments (Ade et al., 2016). On the other hand, compared with the previous works concentrating on the luminosity distance using SN Ia as standard candles (Räsänen et al., 2015; Xia et al., 2017), constraint on the cosmic curvature is not greatly improved in our analysis, although a larger strong lensing sample comprising 106 available SGL systems can be used due to the wider redshift range of the QSO data. This may happen due to the larger error of QSO data. However, in theory, the maximum observable redshift of SN Ia can only reach about 1.7, while the one of QSO can reach about 6. Therefore, the observation of QSO is more likely to explore the behavior of cosmic geometry, and the more detailed discussion of this will be found in section 4.

Let us stress that we have not only constrained the cosmic curvature, but also considered the evolution of slope factor in the mass density profile of lensing galaxies. Different from a smaller combined lens sample combined with cosmology-independent method (Räsänen et al., 2015; Xia et al., 2017) as well as the full lens sample combined with various cosmologies (Cao et al., 2015a), our results suggest that the total density profile of early-type galaxies has become slightly shallower over cosmic time (), which is still consistent with the standard SIS model (, ) within 1.

### 3.2. Generalized SIE model

In the second case we consider the more complex SGL model, which allows the luminosity density profile to be different from the total-mass density profile , i.e., , . Moreover, considering the anisotropic distribution of three-dimensional velocity dispersion , the observational angular-diameter distance ratio will be revised as

(14) | |||||

where , and . Following the analysis of Cao et al. (2015a) based on the well-studied sample of nearby elliptical galaxies, a Gaussian distribution will be applied to characterize the anisotropy of velocity dispersion.

In Fig. 3, we show the 1-D and 2-D marginalized distributions with and contours for three model parameters, the 68% confidence level uncertainties of which are: , and . From the fitted range of and parameters, one can clearly note that the luminosity density profile of elliptical galaxies is different from that of total mass density profile (). In this context, the study of the mass density distribution in early-type galaxies, which contain most of the cosmic stellar mass of the universe and dominate the top end of the galaxy mass function, is a critical issue that still needs to be addressed in a statistical way. More in details, as was extensively discussed in the literature (Cao et al., 2016), our results have provided possible predictions on the density distribution of dark matter, which is differently spatially distributed than stars.

Moreover, in the framework of the generalized SIE model, the constraint on the curvature here is weaker than that obtained in the extended SIE model: at 68% C.L., which indicates that a more general lens model in SGL system does have a significant impact on the measurement of cosmic curvature. As much as we have learned from the current lens systems, several sources of uncertainty still remain, including the need to properly model the mass distribution within the lens. In order to provide a detailed quantitative assessment of what kind of strong lensing and quasar data are necessary to really provide stringent constraint on cosmic curvature, we will here produce mock samples of strong gravitational lenses and quasars based on the current measurement accuracy.

## 4. Monte Carlo Simulations with Mock Sample

Fortunately, ongoing and future imaging surveys will provide us the opportunity to discover up to three orders of magnitude more galactic-scale strong lenses than are currently known. For instance, according to the forecast and prediction of Collett (2015), the forthcoming Large Synoptic Survey Telescope (LSST) survey may potentially discover 120000 lenses for the most optimistic scenario. Therefore, the goal of our following work is to answer the question: “What precision can be achieved to determine cosmic curvature with such a significant increase of the number of strong lensing systems?” On the other hand, fractional uncertainty of the angular size of compact structure in radio quasars will be significantly reduced by both current and future VLBI surveys based on better uv-coverage, one can expect to derive distance information from this standard ruler approach on future VLBI data.

For our simulation we have to choose a fiducial cosmological model. For consistency between the quasar and strong lensing data, we choose the cosmological parameters of the fiducial flat CDM cosmology as: , , . For the quasar data to be observed by future VLBI surveys, the mock “” data is generated with fixed linear size of ILQSOs equal to pc. We have simulated 500 intermediate-luminosity quasars in the redshift range , for which the fractional uncertainty of the angular size “” was taken at a level of 3%. The redshift distribution of the sources as observed on Earth follows the luminosity function obtained from a combination of SDSS and 2dF (2SLAQ) surveys, the bright and faint end slopes of which agrees very well with those in the bolometric luminosity function. See Cao et al. (2017c) for more details of the specific procedure of QSO simulation. Moreover, in order to derive the distance information of and , we select SGL data with redshift obeying the criteria , whose angular diameter distance can be calibrated by QSO.

For the strong lensing data, we use the realistic simulated observations of 120000 events with elliptical galaxies acting as lenses in the forthcoming LSST survey (Collett, 2015). The simulation programs available on the github.com/tcollett/LensPop. Considering the recent result that only the medium-mass elliptical galaxies is consistent with the singular isothermal sphere case within 1 (Cao et al., 2016), we restrict the velocity dispersions of lensing galaxies to the intermediate range: and obtain 16000 strong lensing systems meeting the redshift criterion in compliance with QSO data used in parallel. Fig. 4 shows the redshift coverage of the simulated QSO and SGL sample. Following the analysis of Cao et al. (2016), the fraction uncertainty of the Einstein radius and the observed velocity dispersion are respectively taken at the level of 1% and 10%. The posterior probability density for is shown in Fig. 5. We find that with about 16000 strong lensing events combined with the distance information provided by 500 compact radio quasars, one can constrain the cosmic curvature with an accuracy of , which is comparable to the precision of Planck 2015 results.

## 5. Conclusions

Ultra-compact structure in radio quasars, with milliarcsecond angular sizes measured by very-long-baseline interferometry (VLBI), has provided an important source of angular diameter distances that can be observed up to very high redshifts. On the other hand, the distance sum rule method is becoming an important astrophysical tool for probing the curvature of the universe without assuming any fiducial cosmological model. In this paper, with the aim to alleviate the shortcoming of SN Ia in the distance sum rule, we turn to the VLBI observations of milliarcsecond compact structure in intermediate-luminosity quasars covering the redshift range , which are combined with the latest catalog of strong gravitational lensing systems to measure the cosmic curvature. Moreover, in order to fully consider the effect of lens model in measuring cosmic curvature, in our analysis two lens models are used to characterize the total mass-profile and light-profile shapes of elliptical galaxies.

When the shape of the total mass density follows the extended SIS model, i.e., a spherically symmetric power-law mass distribution , although the zero cosmic curvature is still included within confidence level, a closed universe seems to be more favored in our analysis, which is different from the flat case of supported by other independent and precise experiments (Ade et al., 2016). Compared with the previous works concentrating on the luminosity distance using SN Ia as standard candles (Räsänen et al., 2015; Xia et al., 2017), constraint on the cosmic curvature is not greatly improved in our analysis. Moreover, considering the evolution of slope factor in the mass density profile, our results suggest that the total density profile of early-type galaxies has become slightly shallower over cosmic time (), which is still consistent with the standard SIS model within 1. In the second case we consider the more complex SGL model, which allows the luminosity density profile to be different from the total-mass density profile , i.e., and . From the fitted range of and parameters, one can clearly note that the luminosity density profile of elliptical galaxies is different from that of total mass density profile (), which has provided possible predictions on the different density distribution of dark matter. Moreover, in the framework of the generalized SIE model, the constraint on the curvature here is weaker than that obtained in the extended SIE model: at 68% C.L., which indicates that a more general lens model in SGL system does have a significant impact on the measurement of cosmic curvature.

Given the limitations of the current sample, we have also investigated a detailed quantitative assessment of what kind of strong lensing and quasar data are necessary to really provide stringent constraint on cosmic curvature. On the one hand, ongoing and future imaging surveys will provide us the opportunity to discover up to three orders of magnitude more galactic-scale strong lenses than are currently known. On the other hand, fractional uncertainty of the angular size of compact structure in radio quasars will be significantly reduced by both current and future VLBI surveys based on better uv-coverage, one can expect to derive distance information from this standard ruler approach on future VLBI data. Based on the mock samples of strong gravitational lenses and quasars with the current measurement accuracy, we find that with about 16000 strong lensing events (observed by the forthcoming LSST survey) combined with the distance information provided by 500 compact uv-coverage), one can constrain the cosmic curvature with an accuracy of , which is comparable to the precision of Planck 2015 results. This encourages us to consider the possibility of testing the cosmic curvature at much higher accuracy with future surveys of strong lensing systems (Collett, 2015) and high-quality VLBI observations of radio quasars (Pushkarev & Kovalev, 2015).

## Acknowledgments

We would like to thank Zheng-Xiang Li and Jun-Qing Xia for their helpful discussions. This work was supported by National Key R&D Program of China No. 2017YFA0402600; the Ministry of Science and Technology National Basic Science Program (Project 973) under Grants No. 2014CB845806; the National Natural Science Foundation of China under Grants Nos. 11503001, 11373014, and 11690023; the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University; and the Opening Project of Key Laboratory of Computational Astrophysics, National Astronomical Observatories, Chinese Academy of Sciences. Jing-Zhao Qi is supported by China Postdoctoral Science Foundation under Grant No. 2017M620661.

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