Testing the dark energy with gravitational lensing statistics
Abstract
We study the redshift distribution of two samples of earlytype gravitational lenses, extracted from a larger collection of 122 systems, to constrain the cosmological constant in the CDM model and the parameters of a set of alternative dark energy models (XCDM, DvaliGabadadzePorrati and Ricci dark energy models), in a spatially flat universe. The likelihood is maximized for when considering the sample excluding the Sloan Lens ACS systems (known to be biased toward large imageseparation lenses) and noevolution, and when limiting to gravitational lenses with image separation and noevolution. In both cases, results accounting for galaxy evolution are consistent within 1. The present test supports the accelerated expansion, by excluding the null hypothesis (i.e., ) at more than , regardless of the chosen sample and assumptions on the galaxy evolution. A comparison between competitive world models is performed by means of the Bayesian information criterion. This shows that the simplest cosmological constant model that has only one free parameter is still preferred by the available data on the redshift distribution of gravitational lenses. We perform an analysis of the possible systematic effects, finding that the systematic errors due to sample incompleteness, galaxy evolution and model uncertainties approximately equal the statistical errors, with presentday data. We find that the largest sources of systemic errors are the dynamical normalization and the highvelocity cutoff factor, followed by the faintend slope of the velocity dispersion function.
Subject headings:
cosmology: cosmological parameters — gravitational lensing: strong — methods: statistical1. Introduction
In the last 15 years, several complementary observational probes on cosmological scales have found strong evidence for an accelerating expansion of the universe: distance measurements of distant Type Ia supernovae (SNe Ia; Riess et al. (1998); Perlmutter et al. (1999)), the observations of the cosmic microwave background anisotropies (WMAP; Bennett et al. (2003)), and the baryon acoustic oscillations (BAOs) in the power spectrum of matter extracted from galaxy catalogs (Percival et al., 2007). By assuming General Relativity, a negative pressure component has been invoked as the most suitable mechanism for the observed acceleration, the simplest way of which is the cosmological constant , with a constant equationofstate (EoS) parameter . Dynamical dark energy models have also been proposed in the literature, such as the quintessence (Ratra & Peebles, 1988; Caldwell et al., 1998), phantom (Caldwell et al., 2002), quintom (Feng et al., 2005, 2006; Guo et al., 2005), DvaliGabadadzePorrati (DGP) (Dvali et al., 2000; Zhu & Alcaniz, 2005; Zhu & Sereno, 2008), and Ricci dark energy (RDE) models (Gao et al., 2009; Li et al., 2010).
The existence of a large number of theoretical models has triggered a variety of observational tests, based, for instance, on the angular sizeredshift data of compact radio sources (Alcaniz, 2002; Zhu & Fujimoto, 2002), the ageredshift relation (Alcaniz et al., 2003), the lookback time to galaxy clusters (Pires et al., 2006), Xray luminosities of galaxy clusters, and the Hubble parameter data (Gaztañaga et al., 2009; Stern et al., 2010; Cao, Zhu & Liang, 2011; Cao, Liang & Zhu, 2011). In this context, strong gravitational lensing plays an important role, providing cosmological tests, such as gravitational lensing statistics (Kochanek, 1996a; Zhu, 1998; Cooray & Huterer, 1999; Chiba & Yoshii, 1999; Chae et al., 2002; Sereno, 2005; Biesiada, Piórkowska, & Malec, 2010; Cao, Zhu & Zhao, 2011; Cao & Zhu, 2012; Cao et al., 2012), Einstein rings in galaxyquasar systems (Yamamoto & Futamase, 2001), clusters of galaxies acting as lenses on background high redshift galaxies (Sereno, 2002; Sereno & Longo, 2004), and time delay measurements (Schechter, 2004). Results from techniques based on gravitational lensing are complementary to other methods and can provide restrictive limits on the cosmological parameters. In this paper we focus on one interesting lensing statistic suggested by Kochanek (1992) and further discussed and developed in literature (e.g., Helbig & Kayser (1996), Ofek, Rix & Maoz (2003)).
Fukugita, Futamase, & Kasai (1990) (but see also Nemiroff (1989)) demonstrated that the gravitational lens expected redshift increases with a larger cosmological constant, . Kochanek (1992) derived the probability distribution of the redshift of the lens, as a function of the cosmological parameters, at fixed image separation and source redshift. His analysis of a small sample of four lenses strongly favored a null cosmological constant with respect to models with large in a flat cosmological model. Helbig & Kayser (1996) further discussed the method, assuming no evolution of the galaxy population and no detection of lensing galaxies beyond a given magnitude. Successively, Kochanek (1996a) extended his analysis to include gravitational lenses with not measured redshift, in order to account for the selection effects. He obtained the upper limit at the CL, assuming a flat cosmological model. Ofek, Rix & Maoz (2003) presented a new analysis of expected lens redshift distribution, taking into account number and mass evolution of the lens population, and applied this method to constrain both the cosmological and massevolution parameter spaces. They could obtain a strong upper limit on the cosmological constant ( at the CL), in a flat universe with no lens evolution.
Compared with other related methods based on the full lensing probability distribution, all of the uncertainties in the absolute value of the optical depth can be eliminated in the differential probability of Kochanek (1992), since it is determined by integrating the full lensing probability distribution over lens redshifts. Moreover, the sharp cutoff of this relative probability at high redshift makes the quantity a more powerful cosmological probe (Ofek, Rix & Maoz, 2003).
Following works investigating the evolving lens population have concluded that galaxy evolution is not strongly constrained by the redshift distribution test and does not significantly affect lensing statistics (Mitchell et al., 2005; Capelo & Natarajan, 2007; Oguri et al., 2012). However, the evolution of mass and number density can introduce large statistical errors and bias in the analysis of the lens redshift distribution. Therefore, it is mandatory to consider the mass and density evolution into the statistical analysis of the redshift distribution of gravitational lenses. Other limitations include systematic effects due to a sample of gravitational lenses which completeness might be not homogenous as a function of the lensed image separation and the lens redshift (Capelo & Natarajan, 2007).
Oguri et al. (2012) have presented a comprehensive statistical analysis of the sample of 19 lensed quasars found in the Sloan Digital Sky Survey (SDSS) Quasar Lens Survey (SQLS). This sample is used to determine both the cosmological constant and evolution of the massive lensing galaxies. When considering a noevolution case, a null cosmological constant is rejected at 6 level, providing an independent evidence for the accelerated expansion.
The purpose of this paper is to extend our previous statistical analysis based on the angular separation distribution of the lensed images (Cao, Zhu & Zhao, 2011) by using the redshift distribution test to obtain novel constraints on the parameters of spatially flat cosmological models. With this aim, we use a large sample of 122 gravitational lenses drawn from the Sloan Lens ACS (SLACS) Survey and other sky surveys.
The first aim is to obtain new constraints on the cosmological constant, by assessing both statistical and systematic uncertainties, mainly due to galaxy evolution and sample selection. Then, we also compare a number of alternative dark energy models with different numbers of parameters, in our analysis we apply, following Davis et al. (2007); Li et al. (2010), a model comparison statistic, i.e., the socalled Bayesian information criterion (BIC; (Schwarz, 1978)).
With respect to recent works (Oguri et al., 2012), we use a larger, not homogeneous sample and focus our attention on the determination of the cosmological constant and the comparison with other alternative dark energy models, as the current sample size does not allow a firm determination of the rate of mass and number evolution of massive galaxies.
The paper is organized as follows. In Section 2, the basics of gravitational lensing statistics is introduced, also allowing for number and mass evolution of the lens population. We conduct a literature survey for known systems, listing their basic parameters and defining two statistical samples to perform the redshift test. In Section 3, we introduce four cosmological models, and show the results of constraining cosmological parameters using the Markov Chain Monte Carlo method, with and without galaxy evolution. In Section 4, we assess the possible presence of selection effects and systematic biases in our galaxy sample. Finally, we present the main conclusions and discussion in Section 5.
2. The redshift test and the sample
Following Ofek, Rix & Maoz (2003), the differential optical depth per unit redshift is
(1) 
where is the comoving number density and is the strong lensing crosssection.
Earlytype galaxies are accurately described as singular isothermal spheres (SIS), and it is shown that radial mass distribution and ellipticity of the lens galaxy are unimportant in altering the cosmological constraints (Maoz & Rix, 1993; Kochanek, 1996b). The SIS density profile is
(2) 
where is the velocity dispersion and is the projected distance from the center. In Section 4, we will discuss the systematic uncertainties introduced by this assumption. The corresponding strong lensing crosssection is
(3) 
where , , and are the angular diameter distances between the observer and the lens, the lens and the source and the observer and the source, respectively. Under a FriedmanWalker metric with null space curvature, the angular diameter distance reads
(4) 
where is the Hubble constant and is the dimensionless expansion rate dependent on redshift and cosmological model parameters p. The two multiple images will form at an angular separation
(5) 
We use the empirically determined velocity dispersion distribution function (VDF) of earlytype galaxies: following previous works, our sample are limited to lensing earlytype galaxies. The VDF is generally modeled by a modified Schechter function as (Sheth et al., 2003)
(6) 
where , , , are the faintend slope, highvelocity cutoff, characteristic number density, and velocity dispersion, respectively. In the following analysis we use the results of Choi et al. (2007), who analyzed data from the SDSS Data Release 5 to derive the VDF of earlytype galaxies. The bestfit values of those VDF parameters are Mpc, where is in units of 100 km s Mpc, km s, , and .
We also allow for evolution of the quantities and , by adopting the following parameterization:
(7) 
where and are constant quantities.
In the following, we check the effect of redshift evolution by letting and be free parameters in Section 3, instead of adopting the evolution of predicted by the semianalytic model of Kang et al. (2005); Chae (2007). Moreover, since the main goal of this paper is to constrain cosmological parameters, we first consider and as free parameters, obtain their bestfit values and probability distribution function, and marginalize them to determine constraints on the relevant cosmological parameters of interest.
Straightforward calculations lead to the optical depth per unit redshift for a system with image separation and source redshift ,
(8) 
where the normalization,
(9) 
and
(10) 
For a given lens system, gives the relative probability of finding the lens at different redshift:
(11)  
Besides the probability of lensing by a deflector at , , the total optical depth for multiple imaging and the probability that a background source lensed by an SIS galaxy with image separation , (Cao, Zhu & Zhao, 2011; Cao & Zhu, 2012), can also be obtained by integrating the differential probability in Equation (8).
Moreover, it should be noted that the image separation is taken into account as a prior in the method applied in this paper, which makes it an advantage to use almost all the known lenses (Kochanek, 1992). We have compiled a list of 122 gravitational lenses from a variety of sources in literature. Their basic data (lens and source redshifts both and the largest image separations) are summarized in Table 1. As mentioned above, in order to build an homogeneous galaxy sample, we limit our analysis to galaxies with earlytype morphology.
The main source is given by the SLACS project (Bolton et al., 2008), providing 59 lenses in our list. These lenses have redshifts in the range from to 0.5, making the lower redshift part of our overall sample, with the lensed sources ranging from to 1.2 (Bolton et al., 2008). As a consequence of the initial spectroscopic selection method, all the SLACS gravitational lenses have known spectroscopic redshifts for both source and lens, giving the SLACS sample an immediate scientific advantage over stronglens candidate samples selected from imaging data. However, it is known that the SLACS sample is biased towards moderately largeseparation lenses (; (Arneson et al., 2012)), leading to biased estimates in the redshift test (Capelo & Natarajan, 2007).
A majority of the sample was observed as part of the CASTLES program ^{1}^{1}1http://cfawww.harvard.edu/castles/, but it also contains gravitational lenses found in the COSMOS survey^{2}^{2}2http://cosmos.astro.caltech.edu/(Faure et al., 2008) and the Extended Growth Strip (EGS; (Moustakas et al., 2007)), including six additional COSMOS and EGS systems discovered recently: COSMOS5921+0638, COSMOS0056+1226, COSMOS0245+1430, ”Cross”, and ”Dewdrop”. Finally, we also include five earlytype gravitational lenses from Lenses Structure and Dynamics survey (LSD, (Koopmans & Treu, 2002, 2003; Treu & Koopmans, 2004)), spanning the redshift range (Q00472808, HST15433+5352, MG2016, CY22013201 and CFRS03.1077).
Moreover, lenses dominated by a group or cluster potential will affect the constraint results (Keeton & Zabludoff, 2004; Oguri et al., 2005; Faure et al., 2011) and previous versions of the lens redshift test have certainly gone to some effort to exclude these systems. Therefore, systems known to be strongly affected by the presence of a group (e.g., B1359+154) or galaxy cluster (e.g., Q0957+561, SDSS1004+4112, and B2108+213) have been excluded from our sample. Moreover, we adopt the image separation criterion of to remove lenses that are influenced by complex environments such as clusters (Ofek, Rix & Maoz, 2003). On this basis we have removed Q00472828, and RXJ0921+4529 from the final sample. However, it should be noted that this technique is not biased by this criterion, because the redshift of a lens that has a large separation is very low compared with the source redshift(Ofek, Rix & Maoz, 2003).
The redshift distribution test requires a statistically complete and well characterized sample. As our list includes galaxies from a variety of surveys, using very different observational strategies and discovery spaces (the SDSS spectroscopic sample, Hubble Space Telescope (HST) field sky survey, etc.), it is mandatory to verify the completeness of our final sample of gravitational lenses and its usability for the redshift test. In our analysis, we will use the two following samples (see Table 2).
Sample A. Sixtythree lenses from the above list, excluding the whole SLACS sample. This sample is extracted by the same parent population as the primary sample investigated in Capelo & Natarajan (2007).
Sample B. Seventyone lenses with image separation larger than . This choice is motivated by the fact that the SLACS sample is biased toward moderately largeseparation lenses and large velocity dispersions and is less than 50% complete below (Arneson et al., 2012).
We will discuss the samples selection functions and the impact on our results in Section 4.
Lens Name  ()  Ref  Lens Name  ()  Ref  

J00290055  0.9313  0.227  1.92  1  J221929001743  1.0232  0.2888  1.472  4 
J00370942  0.6322  0.1955  2.943  1,2  J022511045433  1.1988  0.2380  3.54  4 
J0044+0113  0.1965  0.1196  1.58  1,3  J022610042011  1.232  0.4943  2.306  4 
J0109+1500  0.5248  0.2939  1.38  1  MG2016  3.263  1.004  3.12  5 
J02160813  0.5235  0.3317  2.303  1,2,3  HST15433+5352  2.092  0.497  1.18  5,6,7 
J0252+0039  0.9818  0.2803  2.08  1  CY22013201  3.900  0.320  0.830  5,6,7 
J03300020  1.0709  0.3507  2.2  1,3  CFRS03.1077  2.941  0.938  2.48  5,6,7 
J04050455  0.8098  0.0753  1.6  1  Q0142100  8  
J0728+3835  0.6877  0.2058  2.5  1  PMNJ01340931  9,10,11  
J0737+3216  0.5812  0.3223  2.065  2,3  B0218+357  12,13,14  
J0822+2652  0.5941  0.2414  2.34  1  CFRS03.1077  15  
J0841+3824  0.6567  0.1159  2.82  1  MG0414+0534  16,17  
J0912+0029  0.3239  0.1642  3.23  1,2  HE04351223  18  
J09350003  0.467  0.3475  1.74  1,3  HE05123329  19  
J0936+0913  0.588  0.1897  2.18  1  B0712+472  20  
J0946+1006  0.6085  0.2219  2.76  1  MG0751+2716  17  
J0955+0101  0.3159  0.1109  1.82  1  HS0818+1227  21  
J0956+5100  0.4699  0.2405  2.642  1,2  SBS0909+523  22  
J0959+4416  0.5315  0.2369  1.92  1,2  RXJ0911+0551  23  
J0959+0410  0.535  0.126  1.995  1  FBQ0951+2635  25  
J1016+3859  0.4394  0.1679  2.18  1  BRI09520115  26  
J1020+1122  0.553  0.2822  2.4  1  J100424.9+122922  27  
J1023+4230  0.696  0.1912  2.82  1  LBQS10090252  28  
J1029+0420  0.6154  0.1045  2.02  1  Q1017207  29  
J1032+5322  0.329  0.1334  2.06  1  FSC10214+4724  30  
J1103+5322  0.7353  0.1582  2.04  1,3  B1030+071  20  
J1106+5228  0.4069  0.0955  2.46  1,3  HE11041805  31  
J1112+0826  0.6295  0.273  2.98  1,3  PG1115+080  32  
J1134+6027  0.4742  0.1528  2.2  1  B1152+200  33  
J1142+1001  0.5039  0.2218  1.96  1,3  Q1208+101  34  
J11430144  0.4019  0.106  3.36  1  HST14113+5211  22  
J1153+4612  0.8751  0.1797  2.1  1  HST14176+5226  35  
J1204+0358  0.6307  0.1644  2.62  1,3  B1422+231  32  
J1205+4910  0.4808  0.215  2.44  1  SBS1520+530  8,36  
J1213+6708  0.6402  0.1229  2.84  1,3  MG1549+3047  37  
J1218+0830  0.7172  0.135  2.9  1,3  B1600+434  20  
J1250+0523  0.7953  0.2318  2.26  1,2  B1608+656  38  
J13300148  0.7115  0.0808  1.706  1,2  PMNJ16320033  39  
J1402+6321  0.4814  0.2046  2.775  1,2  FBQ1633+3134  40  
J1403+0006  0.473  0.1888  1.66  1  MG1654+1346  41  
J1416+5136  0.8111  0.2987  2.74  1,3  PKS1830211  42  
J1420+6019  0.5351  0.0629  2.097  1,2  PMNJ18383427  43  
J1430+4105  0.5753  0.285  3.04  1  B1933+507  44  
J1432+6317  0.6643  0.123  2.52  1,3  B2045+265  45  
J14360000  0.8049  0.2852  2.24  1,3  HE21492745  8, 46  
J1443+0304  0.4187  0.1338  1.62  1,3  Q2237+0305  47  
J14510239  0.5203  0.1254  2.08  1,3  SDSS02460825  1.68  0.724  1.2  48 
J1525+3327  0.7173  0.3583  2.62  1,3  B0850+054  3.93  0.59  0.68  49 
J15310105  0.7439  0.1596  3.42  1,3  SDSS0903+5028  3.605  0.388  3.0  50 
J1538+5817  0.5312  0.1428  2  1,3  HE11130641  1.235  0.75  0.88  51 
J1621+3931  0.6021  0.2449  2.58  1,3  Q11311231  0.658  0.295  3.8  52 
J16270053  0.5241  0.2076  2.42  1,2  SDSS1138+0314  2.44  0.45  1.34  53, 54 
J1630+4520  0.7933  0.2479  3.618  1,2  SDSS1155+6346  2.89  0.176  1.96  55 
J1636+4707  0.6745  0.2282  2.18  1  SDSS12260006  1.12  0.52  1.26  53, 54 
J22380754  0.7126  0.1371  2.54  1,3  WFI20334723  1.66  0.66  2.34  53, 55 
J2300+0022  0.4635  0.2285  2.494  1,2  HE00471756  1.66  0.41  1.54  56 
J2303+1422  0.517  0.1553  3.278  1,2  COSMOS5921+0638  3.15  0.551  1.6  57, 58 
J23210939  0.5324  0.0819  3.2  1,2  COSMOS0056+1226  0.81  0.361  2.4  57, 59 
J2341+0000  0.807  0.186  2.88  1,3  COSMOS0245+1430  0.779  0.417  3.08  57,59 
J021737051329  1.847  0.6458  2.536  4  ”Cross”  3.40  0.810  2.44  60 
J141137+565119  1.420  0.3218  1.848  4  ”Dewdrop”  0.982  0.580  1.52  60 
Subsample  Definition 

Sample A 
63 lenses from Table 1 excluding the SLACS sample 
Sample B  71 lenses with image separation larger than 
Sample C  51 lenses with image separation not larger than 
SLACS  59 lenses from the whole SLACS sample 
Full sample  122 lenses from Table 1 
3. Statistical analysis
Our statistical analysis is based on the maximum likelihood technique. For a sample of multiply imaged sources, the likelihood of the observed lens redshift given the statistical lensing model is defined by
(12) 
where is the particular differential probability given by Equation(11) normalized to one, and are the cosmological model parameters (e.g., , ), for the th multiply imaged source. Accordingly, the is defined as follows:
(13) 
and he bestfit model parameters are determined by minimizing . Our analysis is based on the publicly available package COSMOMC (Lewis & Bridle, 2002).
We consider four different cosmological models to be tested with the observed lens redshift distribution: the CDM model and three phenomenological models in which the vacuum energy is described as a dynamical quantity: the socalled XCDM model with the EoS a free parameter, the DGP model arising from the brane world theory, and the RDE models. These models are motivated by the well known finetuning and coincidence problems of the standard CDM model.
We note that the previous precision cosmological observational data have hinted that both the RDE and dark energy with EoS may have dubious stability problems (Feng & Li, 2009; Amani, 2011), and the DGP model has already been ruled out observationally (Fang et al., 2008; Durrer & Maartens, 2010; Maartens & Koyama, 2010), so it is indicated that these are just supposed to be a representative set, instead of viable candidates for dark energy.
It is well known that the statistics alone is not sufficient to provide an effective way to make a comparison between different models. In this paper, we use the BIC as a model selection criterion (Schwarz, 1978). The BIC is defined by
(14) 
where , , are the maximum likelihood, the number of parameters, and the number of data points, respectively. Under this selection criterion, a positive evidence against the model with the higher BIC is defined by a difference and a strong evidence is defined by .
A spatially flat universe is assumed throughout the paper, which is strongly supported a combined 5years Wilkinson Microwave Anisotropy Probe (WMAP5), BAOs, and SN data (Hinshaw et al., 2009). As mentioned above, we also consider the case of an evolving population of lensing galaxies in order to assess the accuracy of our results. We consider simultaneous constraints on the galaxy evolution and cosmological parameters. In order to derive the probability distribution function for the cosmological parameters of interest, we marginalize and and perform fits of different cosmological scenarios on both Samples A and B. Results are shown in Figure 1  6 and summarized in Table 5.
3.1. The standard cosmological model (Cdm)
In the simplest scenario, the dark energy is a cosmological constant, , i.e., a component with constant EoS . If spatial flatness of the FriedmannRobertsonWalker metric is assumed, the Hubble parameter according to the Friedmann equation is
(15) 
where and represent the density parameters of matter (both baryonic and nonbaryonic components) and cosmological constant, respectively. As , this model has only one independent parameter.
We consider constraints obtained both samples defined above. While considering Sample A, the likelihood is maximized, , for with no redshift evolution and with redshift evolution, see Figure 1. Hereafter, uncertainties denote the statistical 68.3% confidence limit for one parameter, determined by . Data are consistent with the noredshiftevolution case () at . Specifically, the measured values of the two parameters are and .
When using the Sample B, we find, in the noevolution scenario, , consistent with the result from Sample A. When allowing for galaxy evolution, we find , , and .
In both cases, our findings are very close to the ones obtained from the ESSENCE supernova survey data, in the flat case (Davis et al., 2007), and from the combined WMAP 5year, BAO, and SN Union data (Komatsu et al., 2009) with the bestfit parameter . Moreover, both samples used here exclude with large confidence (4 level) the null hypothesis of a vanishing , as obtained also by Oguri et al. (2012) in their statistical analysis on the SQLS data, providing independent evidence of the accelerated expansion.
While detailed analysis on the constraints of the redshift distribution test on the hierarchical models of galaxy evolution is beyond the scope of this work, we notice that these results obtained with both samples are in broad agreement with previous studies (Capelo & Natarajan, 2007; Oguri et al., 2012), in which no strong evidence for any evolution of the parameters and was found. Moreover, we find that the degenerate direction in the evolution parameters corresponds to a constant lensing probability, as already been noted by Oguri et al. (2012).
Previous studies on lensing statistics (Chae, 2003; Ofek, Rix & Maoz, 2003) considering a notevolving velocity dispersion function have obtained consistent results with the galaxy number counts (Im et al., 2002). Mitchell et al. (2005); Capelo & Natarajan (2007) assumed a nonevolving shape for the VDF and obtained results consistent with earlier results. To sum up, all previous results on redshift evolution from strong lensing statistics appear to be in agreement with noevolution of earlytype galaxies. As we only considered earlytype lensing galaxies, our results further confirm this conclusion.
3.2. Constraints on selected dark energy models
Now we focus on the constraints obtained on selected dark energy models. Here, we only use Sample A, as results obtained with Sample B are coherent at level.
3.2.1 Dark energy with constant equation of state (XCDM)
While deviating from the simple case , the EoS of dark energy can also give birth to an accelerated university expansion. In a zerocurvature universe, the Hubble parameter reads:
(16) 
Obviously, when flatness is assumed (), it is a twoparameter cosmological model, .
The bestfit values of the parameters are: ; with no redshift evolution and ; with redshift evolution, see Figure 3 and Figure 4 for the confidence limits in the plane. However, we note that the lower limits on the parameter are probably an artifact of the prior , which may be tested and constrained with future larger lens sample. Also in this cosmological scenario, the lens redshift data are consistent with no redshift evolution: when marginalizing over , we find and . The Einstein’s cosmological constant () is still consistent within 1. Meanwhile, compared to the cosmological constant model, this flat cosmology with constant EoS dark energy provides a lower , but a higher BIC: with no redshift evolution and with redshift evolution. However, we note that comparing with the cosmological constant model, the twoparameter XCDM model performs relatively well under the information criterion test.
3.2.2 DvaliGabadadzePorrati model (DGP)
In a brane world theory, the accelerated expansion of the universe can be explained by the gravity leaking out into the bulk at large scales with a crossover scale (Dvali et al., 2000). The Friedmann equation is
(17) 
where is the Planck mass and is the crossover scale at which the induced 4dimensional Ricci scalar dominates. In this model, the Hubble parameter expresses as
(18) 
and the DGP model has one one free parameter in a flat universe.
For the DGP model, the bestfit parameters are with no redshift evolution and with redshift evolution (See Figure 5). We find that the DGP model is somehow worse than the CDM model with this observational data. While its is larger than that of the CDM model by about 1.2, it gives with no redshift evolution and with redshift evolution.
3.2.3 Ricci dark energy (RDE) model
Following Gao et al. (2009); Li et al. (2010), the average radius of the Ricci scalar curvature might provide an infrared cutoff length scale. Accordingly, the DE energy density is
(19) 
and the Hubble parameter reads
(20) 
where is a positive constant to be determined. This is a twoparameter model with .
For the RDE model, the bestfit values of the parameters are with no redshift evolution, and with redshift evolution. In Figure 6, we plot the likelihood contours in the plane. Given the largest information criterion result: with no redshift evolution, and with redshift evolution, the RDE model performs the worst in all the cosmological models considered in this paper.
4. Biases and possible systematic effects
Thus far we have considered only statistical errors. Indeed, cosmological tests based on strong lensing have been somehow controversial since Kochanek (1996a), in particular for the possible biases associated with sample selection (Capelo & Natarajan, 2007). In this section, we discuss several possible sources of systematic errors, including: sample incompleteness, unknown survey selection function, uncertainties in the lensing galaxy properties and lens modeling, in order to verify their effect on the cosmological constraints.
First of all, one general concern is given by the fact that strong gravitational lenses are a biased sample of galaxies. Most of the previous works found no evidence for a biased sample of the lensing population with respect to massive earlytype galaxies (see, e.g., Treu et al. (2006)). On the other hand, Faure et al. (2011) found possible evidence for the stellar mass of lensing earlytype galaxies to evolve significantly with redshift. However, it is still not clear whether this supports a stronger lensing bias toward massive objects at high redshift or if it is a consequence of the possible higher proportion of massive and high stellar density galaxies at high redshift. This could be addressed in dedicated numerical simulations (van de Ven et al., 2009; Mandelbaum et al., 2009), as the available lens samples cannot allow yet to discriminate between the two alternatives.
We now estimate the systematic errors due to statistical sample incompleteness. As both our Samples (A, B) have been put together from different surveys, differences in the observing strategies (and selection functions) may cause systematical errors hard to estimate. In order to evaluate the effects due to a selection bias, we have rederived the best estimate on (with a noevolving lens population) by using the whole catalog of lenses and two additional, smaller samples (see Table 2): the first one includes only galaxies from the SLACS, and the second one the gravitational lenses with separations not larger than , including 59 and 51 systems, respectively. All of these three samples clearly suffer from strong selection effects or are very inhomogeneous and are therefore not suitable for deriving constraints by means of the redshift distribution test. However, they can shed light on the amplitude of the possible systematic errors due to sample inhomogeneity, incompleteness, and selection bias toward nonrepresentative systems. For instance, it is well known that the SLACS catalog is characterized by a selection function favoring moderately massive ellipticals (e.g., Arneson et al. (2012)); also the somehow extreme case of Sample C is not complete (below ) as the probability to detect a lensing galaxy is related to the image separation as very smallseparation lenses easily escape detection (both in presentday imaging and spectroscopic surveys). Therefore, we expect that the estimates of obtained from these samples allow us to establish an upper limit on the systematic errors due to a not welldefined lens sample.
Results are shown in Figure 7. When using the apparently incomplete sample of small separation lenses (Sample C) we find . This rather smaller value can be related to the fact that lowermass lenses, producing smallseparation images, will tend to be located at redshifts lower than expected in a large model. Hence, this determines a slightly lower value for the cosmological constant.
When adopting the whole SLACS sample, we find . Finally, by considering the whole, inhomogeneous list of 122 lenses, we obtain . When we compare these values with the best estimate obtained from Sample A, we note that systematic errors do not exceed on the cosmological constant.
We have assumed all the lenses to be isolated systems, with negligible lineofsight contamination. However, current studies find that proximate galaxies (Cohn & Kochanek, 2004) and environmental groups (Keeton & Zabludoff, 2004) can have various effects on the primary lens galaxies, even though the most significant effects only appear to be biasing the galaxy ellipticities and image multiplicities. Nevertheless, most of lenses in Sample B come from the SLACS survey where the role of environment has been assessed in Treu et al. (2009). Namely, it was found that for SLACS lenses the typical contribution from external mass distribution is no more than a few percent. Therefore, the environmental effects on observed image separations appear to be relatively small (certainly smaller than statistical errors arising from the current sample size of lenses).
In addition to the main lens galaxy, the contribution from lineofsight density fluctuations to the lens potential should also be taken into consideration. Based on the final lens sample from SQLS, Oguri et al. (2012) have investigated the lineofsight effect and found that its effect on the total lensing probability is rather small compared with the contribution of other systematic errors to the systematic error on (see Table 2 in Oguri et al. (2012)). Meanwhile, it is noted that this effect may also have a larger impact on lensing probabilities for the lens samples with larger images separations (Oguri et al., 2005; Faure et al., 2009).
Evolution of the VDF and model uncertainties can introduce additional systematic errors. Here we estimate these systematic errors on the constraint results of the flat CDM with the full sample ( lenses), in a similar way as done in Oguri et al. (2012). The analysis in Section 3 suggests that unconstrained redshift evolution of the velocity function is one of the most significant sources of systematic error. An additional source of uncertainty is the relation between velocity dispersions and image separations. This uncertainty is not only related to the difference between the velocity dispersion of the mass distribution and the observed stellar velocity dispersion (White & Davis, 1996), but also many complexities such as the velocity dispersion normalization factor for nonspherical galaxies (Oguri et al., 2012) and the detailed luminosity profiles of galaxies.
Hence, we introduce the parameter that combines the velocity dispersion and the stellar velocity dispersion
(21) 
The Einstein radius given by Equation 10 is then modified as (Kochanek, 1992; Ofek, Rix & Maoz, 2003):
(22) 
More specifically, we have kept as a free parameter to mimic the following effects of: (1) systematic errors in the difference between and ; (2) the influence of lineofsight mass contamination (Keeton, Kochanek, & Seljak, 1997); (3) softened isothermal sphere model which may change the typical image separations (Narayan & Bartelmann, 1996); and (4) the effect of secondary lenses on observed image separation. Recently, by combining stellar kinematics (central velocity dispersion measurements) with lensing geometry (Einstein radius determination from position of images), Cao et al. (2012) have tested a combined gravitational lens data set including 70 data points from SLACS and LSD, and obtained constraints on consistent with the previous results (Ofek, Rix & Maoz, 2003).
We consider the flat CDM model and obtain simultaneous constraints on and with the full sample ( lenses). The constraint result is shown in Figure 8, with the marginalized constraints and . It is obvious that the cosmological constraint on becomes much weaker, though the concordance CDM is still favored within error region. A vanishing cosmological constant is still ruled at 1.
The slightly larger value of indicates that larger is preferred by the lens statistics with used in this paper. This tendency is consistent with the previous result of Oguri et al. (2012) and may help explain why the observed image separation seems to be higher than model predictions (Capelo & Natarajan, 2007). However, we find that the case still consists with the data better to than 2. To be more specific, in order to take into account the measurement uncertainty and the approximations we are doing (no lens ellipticity accounted and no external shear), we estimate a fiducial error of on the values of image separations (Grillo et al., 2008), which is equivalent to a uncertainty on the parameter .
The velocity distribution function given by Equation 6 is another important source of systematic error on the final results. While adopting the bestfit values of the VDF measurement in the SDSS Data Release 5 by Choi et al. (2007) as our fiducial model, we investigate how the cosmological results are altered by introducing the uncertainties on and as listed in Table 3. With the results of measurements on the VDF, we vary the parameter of interest while fixing the other parameters at their bestfit values. For example, based on the sample, we vary the faintend slope by and find that this effect is quite negligible when compared to the present accuracy of the test (Mitchell et al., 2005).
The complete set of standard priors and allowances included in the analysis of the above systematics is summarized in Table 3. As shown in Figure 8 and 9 and Table 4, by comparing their contribution to the systematic error on for the flat CDM with the full sample, we find that the largest sources of systemic error are the dynamical normalization and the highvelocity cutoff , followed by the faintend slope of the VDF. This finding is consistent with the earlier results in Oguri et al. (2012). Meanwhile, this result is also compatible to the concordance cosmological model (, and ). Indeed, current samples of lenses do not allow to discriminate between an and a dynamical dark energy component.
Parameter  Allowance 

EoS of DE () 

DE density in XCDM ()  
DE density in CDM ()  
Matter density ()  
Evolution of ()  
Evolution of ()  
Normalization factor ()  
Faintend slope ()  
Highvelocity cutoff () 
Systematics  

5. Conclusion and discussion
Since the discovery of the accelerating expansion of the universe, in addition to the standard CDM cosmological model, a large number of theoretical scenarios have been proposed for the acceleration mechanism. Examples include the quintessence (Ratra & Peebles, 1988; Caldwell et al., 1998), phantom (Caldwell et al., 2002), quintom (Feng et al., 2005, 2006; Guo et al., 2005) and the Chaplygin gas (Kamenshchik et al., 2001; Bento et al., 2002; Alam et al., 2003; Zhu, 2004; Zhang & Zhu, 2006). All these acceleration mechanisms should be tested with various astronomical observations such as SNeIa, WMAP (Komatsu et al., 2009) and BAO (Percival et al., 2007). However, it is still important to use as many as different observational probes to set constraints on the cosmological parameters. In this work, we have followed this direction and used the redshift distribution of two welldefined samples of lensing, elliptical galaxies drawn from a large catalog of 122 gravitational lenses from a variety of surveys (see Table 1). The BIC is also applied in this analysis to assess various dark energy models and make a comparison.
Considering the two cases with and without the redshift evolution of the velocity function of galaxies, we have analyzed four dark energy models (the CDM, the XCDM with constant , the DGP and the RDE models) under a flat universe assumption. For each model, we have calculated the bestfit values of its parameters and found its value. Results are plotted in Figure 16.
The fit and information criteria results are summarized in Table 5. It is shown that for the zerocurvature CDM model, the likelihood is maximized, for with no redshift evolution and with redshift evolution, when using the Sample A (see Section 2). Consistent results (within ) are derived when the alternative Sample B is considered.
We have also derived simultaneous constraints on the redshift evolution of the parameters and of the velocity function. The constraints in the plane also indicate that the data are consistent with the noredshiftevolution case () at , with and . Both in the noevolution and galaxy evolution scenario, we rule out with high confidence a vanishing cosmological constant, with both the lens samples (at confidence larger than ), as recently found by Oguri et al. (2012) by adopting a smaller and independent lens sample. Therefore, the redshift test adds an independent evidence to the accelerating expansion.
The obtained likelihood distributions shown in Figure 1 are also in agreement with the results from analyzing data of WMAP 5year results with the BAO and SNe Union data, and the largescale structures in the SDSS luminous red galaxies (Spergel et al., 2003; Tegmark et al., 2004; Eisenstein et al., 2005), which implies that gravitational lensing statistics provides an independent and complementary support on the CDM model.
We give a graphical representation of the BIC test Figure 10. Following the CDM model, the DGP model is the only oneparameter model that reduces to the CDM but gives a worse fit, as we obtained close values for the matter density. The XCDM model gives a comparably good fit, but has one additional free parameter, that is in accordance with the bestfit CDM model (within 1 range for the EoS parameter). The other twoparameter model, RDE, provides a worst fit to the data, though the difference in BIC () indicates no clear evidence against it. Therefore, while still not firmly ruling out competing world models, the redshift distribution test clearly favors the cosmological constant model, a conclusion in accordance with previous works (Davis et al., 2007).
Model  Constraint result  BIC 

CDM ()  0  
()  0  
XCDM ()  ;  1.72 
()  ;  1.55 
DGP ()  1.20  
()  1.19  
RDE ()  1.78  
()  1.62  
In order to asses the reliability of our results and the related systematic errors, we have introduced three additional lens samples, characterized by different degrees of inhomogeneity and incompleteness, and rederived an estimate for the cosmological constant assuming a not evolving lens population. Two lens samples (the whole heterogeneous catalog of 122 systems and the SLACS sample) give results fully consistent with those discussed above, confirming us that systematic errors due to sample selection are not larger than statistical uncertainties.
Our model involves several uncertainties and assumptions that introduce additional systematic errors in our cosmological analysis (see Table 3). By comparing the contribution of each of these systematic errors to the systematic error on the flat CDM model (see Figure 8 and 9 and Table 4), we find that the largest sources of systematic error are the dynamical normalization and the highvelocity cutoff factor, followed by the faintend slope of the velocity dispersion function, which is consistent with the earlier results in Oguri et al. (2012). Moreover, the comparable systematic errors suggest the importance of careful studies of the systematics for robust cosmological constraints from lens statistics.
Finally, we note that four important effects should be mentioned. Evolution of the source population can also matter the technique applied in this paper, which might have a small secondorder effect on the statistics (Oguri et al., 2012). We have also neglected systematic uncertainties due to the effect of smallscale inhomogeneities on largescale observations. In fact, the inhomogeneous matter distribution can affects light propagation and the related cosmological distances (Sereno et al., 2001, 2002), although its effect on the total lensing statistics is small (Covone et al., 2005). Meanwhile, the lens redshift test applied in this paper may also lose the statistical power of absolute lensing probabilities. The last one is, multiple errors or biases in the method could easily be canceled out, which may lead the result to be a statistical fluke (Maoz, 2005).
Despite some of its inherent difficulties, the redshift distribution test, with either larger gravitational lensing samples from future widefield surveys such as PanSTARRS and Large Synoptic Survey Telescope by taking advantage of timedomain information (Oguri & Marshall, 2010) or a joint investigation with other cosmological observations, could be helpful for advancing such applications and provide more stringent constraints on the cosmological parameters.
We acknowledge fruitful discussions with M. Paolillo. We thank the anonymous referee for his/her valuable comments which helped us to improve the paper. GC thanks I. Skripnikova for enlightening discussions. 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, and the Excellent Doctoral Dissertation of Beijing Normal University Engagement Fund.
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