Issue 
A&A
Volume 538, February 2012



Article Number  A43  
Number of page(s)  7  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201015940  
Published online  30 January 2012 
Constraints on cosmological models from lens redshift data
Department of Astronomy, Beijing Normal University, 100875 Beijing, PR China
email: zhuzh@bnu.edu.cn
Received: 17 October 2010
Accepted: 24 May 2011
Context. Stronglensing has developed into an important astrophysical tool for probing cosmology and galaxies (their structures, formations, and evolutions). Now several hundreds of stronglens systems produced by massive galaxies have been discovered, which may form welldefined samples useful for statistical analyses.
Aims. We aim to collect relatively complete lensredshift data from various large systematic surveys of gravitationally lensed quasars and check the possibility of using these as a future supplement to other cosmological probes.
Methods. We use the distribution of gravitationallylensed image separations observed in the Cosmic Lens AllSky Survey (CLASS), the PMNNVSS Extragalactic Lens Survey (PANELS), the Sloan Digital Sky Survey (SDSS) and other surveys, considering a singular isothermal ellipsoid (SIE) model for galactic potentials as well as improved new measurements of the velocity dispersion function of galaxies based on the SDSS DR5 data and recent semianalytical modeling of galaxy formation to constrain two darkenergy models (ΛCDM and constant w) under a flat universe assumption.
Results. We find that the current lensredshift data yield a relatively weak constraint on the model parameter Ω_{Λ} = 0.85_{0.18}^{+0.11} for the simplest cosmological constant model. However, by combing the redshift data with the cosmic macrowave background data, we obtain Ω_{Λ} = 0.78_{0.03}^{+0.02}; by combing the redshift data with the baryonic acoustic oscillation peak and the comic macrowave background data, we obtain a more stringent result: Ω_{Λ} = 0.75_{0.02}^{+0.02}. We obtain w < −0.52 at 68.3% CL with the lensredshift data for the constant w of a dynamical darkenergy model, therefore, the current quintessence darkenergy model is still included at 1σ and a supernegative equation of state is not necessarily favored by this strong lensing data. We notice that the joint data can also provide a good fit with Ω_{x} = 0.80_{0.17}^{+0.17}, w = 1.12_{1.88}^{+0.57} and Ω_{x} = 0.71_{0.07}^{+0.07}, w = 0.78_{0.34}^{+0.22}, respectively, which agree with the flat Λ CDM model at 1σ.
Key words: gravitational lensing: strong / cosmological parameters / dark energy
© ESO, 2012
1. Introduction
The discovery of strong gravitational lensing in Q0957+561 (Walsh et al. 1979) opened up a vast possibility of using stronglens systems in the study of cosmology and astrophysics. Up to now, strong lensing has developed into an important astrophysical tool for probing both cosmology (Fukugita et al. 1992; Kochanek 1993; Zhu 2000a,b; Chae et al. 2002; Chae 2003; Chae et al. 2004; Mitchell et al. 2005; York et al. 2005; Zhu & Mauro 2008a; Zhu et al. 2008b) and galaxy structures, formations, and evolutions (Zhu & Wu 1997; Mao & Schneider 1998; Jin et al. 2000; Keeton 2001; Kochanek & White 2001; Chae & Mao 2003; Ofek et al. 2003; Rusin & Kochanek 2005; Chae 2005; Chae et al. 2006; Koopmans et al. 2006; Treu et al. 2006). Now several hundred strong lens systems produced by massive galaxies have been discovered, but only ~90 galacticscale strong lenses with known redshift of the source and the image separation can form welldefined samples that are useful for statistical analyses.
These welldefined strong lenses are particularly useful not only for constraining the statistical properties of galaxies such as optical region velocity dispersions (Chae 2005; Chae et al. 2006) and galaxy evolutions (Chae & Mao 2003; Ofek et al. 2003), but also for constraining cosmological parameters such as the present matter density Ω_{m} and the equation of state of dark energy w (Chae 2003; Mitchell et al. 2005). For example, the CLASS statistical sample (Browne et al. 2003; Chae 2003) containing 13 lenses that strictly satisfy the welldefined selection criteria was first extensively used by Chae et al. (2002) and Chae (2003), who found Ω_{m} ≈ 0.3 in a flat cosmology with nonevolving galaxy populations. Then this CLASS sample was reanalyzed with the velocity dispersion function (VDF) of earlytype galaxies derived from the SDSS Data Release 1 (DR1 Sheth et al. 2003) galaxies (Mitchell et al. 2005).
In our paper, we summarize a complete lensredshift sample from various imaging surveys including the Cosmic Lens ALLSky Survey (CLASS; Browne et al. 2003; Myers et al. 2003), the JVAS, the Sloan Digital Sky Survey (SDSS; Oguri et al. 2006), the PMNNVSS Extragalactic Lens Survey (PANELS; Winn et al. 2001) and the Snapshot survey, which accumulates 29 galacticscale lenses so far to form a welldefined radioselected lens sample. Newly measured J1620+1203 (Kayo et al. 2010) from SDSS is also included.
An important point is the measurement of the velocity dispersion function (VDF) of galaxies. Chae (2005) found that the Sheth et al. (2003) VDF of earlytype galaxies underestimated their abundance based on the Wilkinson Microwave Anisotropy Probe (WMAP) 1st year cosmology (Spergel et al. 2003). Then, with a new method of classifying galaxies (Park & Choi 2005), Choi et al. (2007) made a new VDF measurement of earlytype galaxies based on the much larger SDSS Data Release 5 (DR5; AdelmanMcCarthy et al. 2007). The Choi et al. (2007) VDF has a much higher comoving number density of earlytype galaxies and a different shape for the lower velocity part compared with the (Spergel et al. 2003) VDF. Recently, Chae (2007) have determined the VDF of the latetype population using the TullyFisher relation and a SIE galaxy model that matches that of Sheth et al. (2003) relatively well. These authors also considered the scatter of the TullyFisher relation. More recently, Chae (2010) introduced a correction term for highvelocity dispersions and used the Monte Carlo method to separately generate the earlytype and latetype galaxies and to derive a total VDF for the entire population of galaxies (see also in Bernardi et al. 2010). However, the simulated data points for the total population of galaxies are not fitted well by the VDF of the morphologicallytyped galaxy populations, which might be the true nature of the total VDF and may be caused by the errors in the adopted correlations between luminosity and velocity especially at low velocities (Chae 2010).
Moreover, strong lensing has also been extensively applied to constrain dark energy, one of the most important questions of the modern cosmology ever since the observations of type Ia supernovae (SNe Ia), which first indicated an accelerating expansion of the universe at the present stage (Riess et al. 1998). Up to now, diverse analyses have revealed that our universe is spatially flat and consists of 70 percent dark energy with negative pressure, the remaining 30 percent are dust matter (cold dark matter plus baryons) and negligible radiation. Among diverse darkenergy models, the most simple candidate for the uniformly distributed dark energy is considered to be in the form of vacuum energy density or cosmological constant (Λ). However, the cosmological constant is always entangled with (i) a finetuning problem (the present amount of dark energy is too small compared with the fundamental scale) and (ii) a coincidence problem (darkenergy density is comparable to the critical density today). Alternatively, there are other choices, for example, an Xmatter component, which is characterized by an equation of state p = wρ, where −1 ≤ w ≤ 0 (Zhu 1998; Peebles & Ratra 2003). The goal of this work is to use the lensredshift test combined with the revised VDF of alltype galaxies based on the SDSS DR5 data to constrain two cosmological models.
This paper is organized as follows. In Sect. 2 we briefly describe the analysis method including assumptions about the lens population. We then present the lensredshift data from various surveys in Sect. 3. We introduce two prevalent cosmologies and show the results of constraining cosmological parameters using the MCMC method in Sect. 4. Finally, we conclude and make a discussion in Sect. 5.
2. Lensredshift test
The statistical lensing model used in this paper incorporates the (differential) lensing probabilities of specific image multiplicities for the multiplyimaged sources using the SIE lens model. The primary assumption we make is that the distribution of earlytype galaxies in luminosity is given by the Schechter (1976) form (1)Considering a powerlaw relation between luminosity (L) and velocity dispersion (σ), i.e. , we can describe the distribution of galaxies in velocity dispersion in the form of the modified Schechter function φ(σ) (Sheth et al. 2003; Mitchell et al. 2005) (2)where \begin{lxirformule}$\phi_*$\end{lxirformule} is the integrated number density of galaxies, σ_{ ∗ } is the characteristic velocity dispersion, with the following relations: , β = γ, and .
Following Eq. (2), the particular differential probability that a source at redshift z_{s} be multiply imaged by a distribution of galaxies at redshift z_{l} with a image separation Δθ can be written as (3)Meanwhile, Δθ_{ ∗ } is the characteristic image separation given by (4)where D(z_{1},z_{2}) is the angulardiameter distance between redshifts z_{1} and z_{2}. On the hypothesis that galaxies are not biased toward oblate or prolate shape, we choose the dynamical normalization factor λ ≈ 1 for the singular isothermal ellipsoid (SIE) model (Chae 2003, 2005). Though isothermal mass model would, in general, be too simplified to accurately model individual lenses (Chae et al. 2002), it is accurate enough as firstorder approximations to the mean properties of galaxies for the analyses of statistical lensing (Kochanek 1993; Mao & Kochanek 1994; Rix et al. 1994; Kochanek 1996; King et al. 1997; Fassnacht & Cohen 1998; Rusin & Kochanek 2005).
Following Chae (2003), the likelihood ℒ of the observed image separations for N_{L} multiplyimaged sources reads (5)Here δp_{l} is the particular differential probability for the lth multiplyimaged source. (g = e, s) is the weight factor given to the earlytype or the latetype populations, which satisfies . If the lensing galaxy type is unknown, we use (g = e, s) (Chae 2003). Accordingly, a “χ^{2}” is defined as (6)Notice that the χ^{2} here is free of the dimensionless Hubble constant h, which makes it an individual cosmological probe besides SNeIa, CMB, BAO etc. The “bestfit” model parameters are determined by minimizing the χ^{2} (Eq. (6)) and confidence limits on the model parameters are obtained by using the usual Δχ^{2} ( ≡ ) static, where is the global minimum χ^{2} for the bestfit parameters. Our calculating method is based on the publicly available package CosmoMC (Lewis & Bridle 2002).
On the side of the measurement of VDF, the first direct measurement of the VDF of earlytype galaxies was the SDSS DR1 (Sheth et al. 2003): (7)then Choi et al. (2007) obtained a new VDF based on the much larger SDSS DR5: (8)Obviously, the revised DR5 VDF, which has been proved to provide an efficient way to constrain darkenergy models combined with gravitational lensing statistics (Zhu & Mauro 2008a), is quite different from the DR1 VDF in the characteristic velocity dispersion at 1σ. While earlytype galaxies dominate strong lensing, latetype galaxies cannot be neglected. However, for the latetype galaxy population, the direct measurement of the VDF is complicated by the significant rotations of the disks. Chae (2007) estimated all parameters of Eq. (2) for the latetype population with the TullyFisher relation and SIE galaxy model: (9)which matches relatively well that of Sheth et al. (2003) who determined using a TullyFisher relation taking into account the scatter of the TullyFisher relation.
Currently, it is found that simple evolutions do not significantly affect the lensing statistics if all galaxies are earlytype (Mao & Kochanek 1994; Rix et al. 1994; Mitchell et al. 2005; Capelo & Natarajan 2007). Many previous studies on lensing statistics without evolutions of the velocity function have also derived appealing results that agree with the galaxy number counts (Im et al. 2002) and the redshift distribution of lens galaxies (Chae 2003; Ofek et al. 2003). However, in this paper, we consider galaxy evolutions both for earlytype and latetype galaxies with a recent semianalytical model of galaxy formation (Kang et al. 2005; Chae et al. 2006). Specially, the evolutions of both the number density σ_{ ∗ } and the characteristic velocity dispersion φ_{ ∗ } are (10)with the bestfit parameters of Kang et al. (2005): (ν_{n},ν_{v}) = ( − 0.229, − 0.01) for earlytype and (ν_{n},ν_{v}) = (1.24, − 0.186) for latetype galaxies.
3. Lensredshift data
Large systematic surveys of gravitationally lensed quasars provide a large statistical lens sample appropriate for studying cosmology. In this section, we summarize a sample from CLASS, SDSS observations, and recent largescale observations of galaxies, which will be used as the input for the statistical lensing model described in Sect. 2. Two main sources of the lens redshift data are the Cosmic Lens AllSky Survey and the Sloan Digital Sky Survey Quasar Lens Search.
3.1. CLASS
As the largest completed galactic mass scale gravitational lens search project, the Cosmic Lens AllSky Survey, along with its predecessor project, the JodrellBankVLA Astrometric Survey (JVAS) has confirmed 22 multiplyimaged systems out of a total of 16521 radio sources (Browne et al. 2003). Out of the entire CLASS sample including the JVAS sources, a subsample of 8958 sources containing 13 multiplyimaged systems constitutes a statistical sample that satisfies welldefined observational selection criteria (Browne et al. 2003), which can be used for the statistical analysis of gravitational lensing in this paper. In this work we need galacticscale strong lens samples that satisfy welldefined observational selection criteria. With the welldefined selection criteria from Browne et al. (2003); Chae (2003), two out of the four multiplyimaged sources in the JVAS, 0414+054 and 1030+074, are excluded from the final CLASS statistical sample because 0414+054 has a spectral index that is too steep and 1030+074 has two images whose faintertobrighter image fluxdensity ratio is less than 0.1. Meanwhile, we stress that the measured lens redshifts, source redshifts, image separations, image multiplicities and the lensing galaxy types (if determined) as shown in Table 1 are all needed through the likelihood function defined in Sect. 2 to constrain cosmological parameters. With this criterion, the CLASS sources B0850+054, B0445+123, B0631+519, B1938+666 are clearly excluded because of their unknown source redshifts, while for B0739+366 and J2004.1349, redshifts for source and lens are unavailable^{1}. Moreover, the measured image separations of 1359+154, 1608+656 and 2114+022 are not used because the observed angular sizes are caused by multiple galaxies within their critical radii.
3.2. SDSS
The Sloan Digital Sky Survey Quasar Lens Search (SQLS; Oguri et al. 2006) is a photometric and spectroscopic survey that covers nearly a quarter of the entire sky (York et al. 2000) and therefore provides a large statistical lens sample apposite for cosmology study and research. We tried to find suitable lens samples from the optical quasar catalog of the Sloan Digital Sky Survey (SDSS; York et al. 2000). The first complete lens sample from Data Release 3 selected from 22 683 lowredshift (0.6 < z < 2.2) is provided in Inada et al. (2008). It consists of 11 lensed quasars satisfying the following welldefined selection criteria: 1) the image separation is between 1′′ and 20′′ with quasars brighter than i = 19.1; 2) the flux ratio of faint to bright images is greater than 10^{0.5} for double lenses. We applied an additional cut to select an appropriate subsample: the lensing galaxy should be fainter than the quasar components, because a lens galaxy that is too bright will strongly affect the colors of the quasars (Richards et al. 2002). Four lensed quasars, Q0957+561, SDSS J1004+4112, SDSS J1332+0347, and SDSS J1524+4409 are removed with this cut. As in the CLASS, two more quasars, SDSS J1001+5027 and SDSS J1021+4913 are excluded because they have no redshift available. However, we successfully added four lensredshift data including SDSS J1620+1203, one of the eight newly discovered and confirmed twoimage lensed quasars by SDSS Quasar Lens Search (Kayo et al. 2010).
In Table 1 we summarize 29 stronglylensed sources (redshifts both for sources and lenses as well as the largest image separations and galaxy types) from the CLASS (JVAS), the SDSS, the PANELS, and the Snapshot, which constitute a welldefined combined statistical sample for model constraints.
Summary of stronglylensed sources.
4. Darkenergy models and constraint results
Below we choose two popular darkenergy models and examine whether they are consistent with the lensredshift data listed above.

1.
Cosmologically constant model.

2.
Dark energy with a constant equation of state.
The two models are currently viable candidates to explain the observed cosmic acceleration. On the one hand, given the current status of cosmological observations, there is no strong reason to go beyond the simple, standard cosmological model with a cosmological constant. On the other hand, to allow for deviations from the simple w = −1 case, it is interesting to explore dark energy with a constant equation of state model to make a comparison. Unless stated otherwise, throughout the paper we calculate the bestfit values and vary the parameters within their 2σ uncertainties for either class of model. Next, we shall outline the basic equations describing the evolution of the cosmic expansion in both darkenergy models and calculate the bestfit parameters.
4.1. Constraint on the standard cosmological model (ΛCDM)
4.1.1. Constraint from the lensredshift data
In the simplest scenario, the dark energy is simply a cosmological constant, Λ, i.e., a component with constant equation of state w = p/ρ = −1. If flatness of the FRW metric is assumed, the Hubble parameter according to the Friedmann equation is (11)where Ω_{m} and Ω_{Λ} parameterize the density of matter and cosmological constant, respectively. Moreover, in the zerocurvature case (Ω = Ω_{m} + Ω_{Λ} = 1), this model has only one independent parameter θ = Ω_{Λ}.
We plot the likelihood distribution function for this model in Fig. 1. The bestfit value of the parameter is . Obviously, the lensredshift data only give a relatively weak constraint on the model parameter Ω_{Λ}, though the universally recognized value of Ω_{Λ} = 0.75 is still included at 68.3% CL (1σ). To make a comparison, it is necessary to refer to the previous results: the current bestfit value from cosmological observations is Ω_{Λ} = 0.73 ± 0.04 in the flat case (Davis et al. 2007), which is in relatively stringent accordance with our result. Moreover, Komatsu et al. (2009) gave the bestfit parameter Ω_{m} = 0.274 for the flat Λ CDM model from the WMAP 5year results with the BAO and SN Union data. We find that the constraint result from the lensredshift data is marginally consistent with the previous works above.
Fig. 1 Likelihood distribution function for the Λ CDM model constrained by the lensredshift data. 
4.1.2. Joint analysis with BAO and CMB
For the BAO data, the parameter is used, which is independent of cosmological models and for a flat universe can be expressed as (12)where z_{BAO} = 0.35 and is from the SDSS (Eisenstein et al. 2005).
For the CMB data, the shift parameter ℛ is used, which may provide an effective way to constrain the parameters of dark energy models owning to its large redshift distribution. Derived from the CMB data, ℛ takes the form (13)where z_{CMB} = 1090 (Komatsu et al. 2009) is the redshift of recombination and the 5year WMAP data give ℛ = 1.710 ± 0.019 (Komatsu et al. 2009).
Fig. 2 Likelihood distribution function for the Λ CDM model constrained by the lensredshift data combined with CMB. 
Fig. 3 Likelihood distribution function for the Λ CDM model constrained by the lensredshift data combined with CMB and BAO. 
In Figs. 2 and 3 we show the result by combining the lens redshift data with CMB and CMB+BAO, respectively. The bestfit parameter is and , a relatively satisfactory result consistent with that of Li et al. (2010). Compared with Fig. 1, apparently, Ω_{Λ} is more tightly constrained with the joint data sets.
4.2. Dark energy with a constant equation of state (w)
4.2.1. Constraint from the lensredshift data
If we allow for a deviation from the simple w = −1, a component with a arbitrary, constant value for the equation of state could be introduced. The accelerated expansion can be obtained when w < −1/3. In a zerocurvature universe, the Hubble parameter for this generic dark energy component with density Ω_{x} then becomes (14)Obviously, when flatness is assumed (Ω = Ω_{m} + Ω_{Λ} = 1), it is a twoparameter model with the model parameters θ = { Ω_{x}, w } .
Fig. 4 w model constrained by the lensredshift data: likelihood contours at 68.3% and 95.4% CL in the Ω_{x} − w plane. 
For the lensredshift data only, the bestfit values of the parameters are Ω_{x} = 0.78,w = −1.15. Figure 4 shows the confidence limits in the Ω_{x} − w plane. On the one hand, we have w < −0.52 at 68.3% CL, which is quite different from the result of Chae (2007); on the other hand, Einstein’s cosmological constant (w = −1) is still favored within 1σ error region. Therefore, it seems that the present lensredshift data do not necessarily favor the phantom DE model with w < −1 (Caldwell 2002, 2003). However, it is still interesting to see whether this remains so with future lager and better lensredshift data.
4.2.2. Joint analysis with BAO and CMB
In Figs. 5 and 6 we plot the likelihood contours with the joint data by combining the lensredshift data with CMB and CMB+BAO in the Ω_{x} − w plane. The bestfit parameters are and . Notice that both Ω_{x} and w are more stringently constrained with the joint observational data. Meanwhile, the currently preferred values of w in this model still include the cosmological constant case: w = −1.01 ± 0.15 (Davis et al. 2007). Therefore, when the equation of state does not depend on the redshift, the dark energy is consistent with a flat cosmological constant model within 1σ error region.
Fig. 5 w model constrained by the lensredshift data combined with CMB: likelihood contours at 68.3% and 95.4% CL in the Ω_{x} − w plane. 
Fig. 6 w model constrained by the lensredshift data combined with CMB and BAO: likelihood contours at 68.3% and 95.4% CL in the Ω_{x} − w plane. 
5. Conclusion and discussion
Recent new observations, such as SNeIa, Wilkinson Microwave Anisotropy Probe (WMAP) (Komatsu et al. 2009) and baryon acoustic oscillations (BAO) (Percival et al. 2009), the time drift of subtended angles (Zhang & Zhu 2009), the updated Gammaray bursts (GRB) (e.g., Gao et al. 2012; Liang et al. 2011) have provided many robust tools to study the dynamical behavior of the universe. However, it is still important to use other different probes to set bounds on cosmological parameters.
In this work, we have followed this direction and used the distribution of gravitationallylensed image separations observed in the Cosmic Lens AllSky Survey (CLASS), the PMNNVSS Extragalactic Lens Survey (PANELS), the Sloan Digital Sky Survey (SDSS) and other surveys to constrain cosmological models with the new measurements of the velocity dispersion function of galaxies based on the SDSS DR5 data and recent semianalytical modeling of galaxy formation. Two darkenergy models (ΛCDM and constant w) were considered under a flat universe assumption.
For the zerocurvature Λ CDM model, although the lensredshift data can not tightly constrain the model parameter , a stringent constraint can be obtained by combining the lensredshift data with the comic macrowave background data and the baryonic acoustic oscillation peak data . Furthermore, we consider a flat cosmology with a constant w dark energy. For the lensredshift data, we have w < −0.52 at 68.3% CL, a result different from that of Chae (2007) with w < −1.2 (68.3% CL), therefore, these strong lensing data do not necessarily favor a supernegative equation of state for dark energy. However, Einstein cosmological constant (w = −1) is still included within 1σ error region. Likewise, adding CMB and CMB+BAO does lead to further improvements in parameter constraints with and , respectively. Therefore, it indicates that the cosmological constant model is still the best one to explain these lensredshift data, a conclusion in accordance with the previous works (Davis et al. 2007) and the results from the WMAP and the largescale structures in the SDSS luminous red galaxies (Spergel et al. 2003; Tegmark et al. 2004; Eisenstein et al. 2005).
However, we also notice that firstly, the implementation of singular isothermal ellipsoid model (SIE) may be a source of systematic errors. For example, a lens ellipticity of 0.4 can lead to a difference of ΔΩ_{m} ≈ −0.05 compared with the spherical case due to the variation of magnification bias and cross section (Huterer et al. 2005). Secondly, for the source of the lens redshift data, in this paper we simply discarded lens systems that do not have measured lens or source redshifts. This could as well possibly bring biases and will be considered in our future work. Thirdly, though the lens redshift test applied in this paper is free from the magnification bias arising from the uncertain source counts, it may also lose the statistical power of absolute lensing probabilities (or “lensing rates”). Lastly, though we used the VDF of galaxies based on a much larger SDSS Data Release in the stronglensing statistics, the accuracy of measurements on relevant parameters of VDF may also make a difference. Hopefully, large new samples of strong lenses will be expected to be obtained in future widefield imaging surveys such as the Large Synoptic Survey Telescope (LSST; Ivezić et al. 2000) etc. Meanwhile, within a few decades, next generation observation tools such as the Square Kilometre Array (e.g., Blake et al. 2004) will also improve the precision of lensing statistics by several orders of magnitude. Therefore, the lensredshift test can play an important role in uncovering the physical processes of galaxy formation and universe evolution with much larger and better lensredshift data.
Summarizing, from the above discussion we may safely arrive at a conclusion that the results from the observational lensredshift data agree relatively well, and furthermore, the lensredshift test can be seen as a future supplement to other cosmological probes.
We discard lens systems that do not have measured lens or source redshifts, which may possibly cause biases. For many multiplyimaged sources without measured source redshifts, a possible strategy is to take z_{s} = 2, which is the mean source redshift for the multiplyimaged sources with measured source redshifts. However, in order to ensure the accuracy of constraint, we choose to abandon such a choice in our paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China under the Distinguished Young Scholar Grant 10825313 and Grant 11073005, the Ministry of Science and Technology national basic science Program (Project 973) under Grant No. 2012CB821804, the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University.
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All Tables
All Figures
Fig. 1 Likelihood distribution function for the Λ CDM model constrained by the lensredshift data. 

In the text 
Fig. 2 Likelihood distribution function for the Λ CDM model constrained by the lensredshift data combined with CMB. 

In the text 
Fig. 3 Likelihood distribution function for the Λ CDM model constrained by the lensredshift data combined with CMB and BAO. 

In the text 
Fig. 4 w model constrained by the lensredshift data: likelihood contours at 68.3% and 95.4% CL in the Ω_{x} − w plane. 

In the text 
Fig. 5 w model constrained by the lensredshift data combined with CMB: likelihood contours at 68.3% and 95.4% CL in the Ω_{x} − w plane. 

In the text 
Fig. 6 w model constrained by the lensredshift data combined with CMB and BAO: likelihood contours at 68.3% and 95.4% CL in the Ω_{x} − w plane. 

In the text 
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