⇦Chinese Physics Letters, 2016, Vol. 33, No. 7, Article code 079801 Cosmic Constraints to the $w$CDM Model from Strong Gravitational Lensing * Jie An(安洁)1, Bao-Rong Chang(常葆荣)1, Li-Xin Xu(徐立昕)1,2** Affiliations 1Institute of Theoretical Physics, School of Physics and Optoelectronic Technology, Dalian University of Technology, Dalian 116024 2State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190 Received 7 April 2016 *Supported by the National Natural Science Foundation of China under Grant No 11275035.
**Corresponding author. Email: lxxu@dlut.edu.cn Citation Text: An J, Chang B R and Xu L X 2016 Chin. Phys. Lett. 33 079801 Abstract We study the cosmic constraint to the $w$CDM (cold dark matter with a constant equation of state $w$) model via 118 strong gravitational lensing systems which are compiled from SLACS, BELLS, LSD and SL2S surveys, where the ratio between two angular diameter distances $D^{\rm obs}=D_{\rm A}(z_{\rm l},z_{\rm s})/D_{\rm A}(0,z_{\rm s})$ is taken as a cosmic observable. To obtain this ratio, we adopt two strong lensing models: one is the singular isothermal sphere model (SIS) and the other one is the power-law density profile (PLP) model. Via the Markov chain Monte Carlo method, the posterior distribution of the cosmological model parameters space is obtained. The results show that the cosmological model parameters are not sensitive to the parameterized forms of the power-law index $\gamma$. Furthermore, the PLP model gives a relatively tighter constraint to the cosmological parameters than that of the SIS model. The predicted value of ${\it \Omega}_{\rm m}=0.31^{+0.44}_{-0.24}$ by the SIS model is compatible with that obtained by Planck2015: ${\it \Omega}_{\rm m}=0.313\pm0.013$. However, the value of ${\it \Omega}_{\rm m}=0.15^{+0.13}_{-0.11}$ based on the PLP model is smaller and has $1.25\sigma$ tension with that obtained by Planck2015. DOI:10.1088/0256-307X/33/7/079801 PACS:98.80.-k, 98.80.Es © 2016 Chinese Physics Society Article Text When the light rays pass through astronomical objects (galaxies, cluster of galaxies), the cosmological gravitation field bends the paths traveled by light from distant source to us. These light paths respond to the distribution of mass. Therefore, the positions of the source and the image are related by the simple lens equation. The information of the invisible matter can be deduced by making a thorough inquiry into the relation between sources and images. If one can measure the redshifts of the source and lens, the velocity dispersion of the mass distribution, the separated image, then one might be able to infer the distribution of the mass in our Universe. Since the first discovery of the strong gravitational lensing in $Q0957+561$ by Walsh et al. in 1979,[1] it has become a powerful probe in the study of cosmology. Actually, in the last few years, the strong lensing systems have been used as a useful probe to determine the cosmological model parameter space.[2-20] Up to now, the data points have amounted to 118 as collected in Ref. [4]. These 118 strong lensing systems are compiled from four surveys: SLACS, BELLS, LSD and SL2S. The Sloan lens ACS survey (SLACS) and the BOSS emission-line lens survey (BELLS) are spectroscopic lens surveys in which candidates are selected respectively from Sloan digital sky survey (SDSS) data and baryon oscillation spectroscopic survey (BOSS). BOSS has been initiated by upgrading SDSS-I optical spectrograph.[21] The distribution of 118 strong lensing systems on $z_{\rm l}$–$z_{\rm s}$ space is shown in Fig. 1. These data points allow us to investigate the properties of dark energy.
cpl-33-7-079801-fig1.png
Fig. 1. Scatter plot of 118 strong lensing systems in terms of $z_{\rm l}$ and $z_{\rm s}$ from four surveys. One can see a fair coverage of redshifts in the combined sample.
Strong gravitational lensing occurs whenever the source, the lens and the observer are so well aligned that the observer–source direction lies inside the so-called Einstein ring of the lensing. To extract the geometric information from a strong lensing system, one should specify a lensing model connecting the source and the image. In fact, the geometric information can be derived from the ratio $$ D^{\rm obs}(z_{\rm l},z_{\rm s})=D_{\rm A}(z_{\rm l},z_{\rm s})/D_{\rm A}(0,z_{\rm s}),~~ \tag {1} $$ between two angular diameter distances $D_{\rm A}(z_{\rm l},z_{\rm s})$ (the angular diameter distance between lens and source) and $D_{\rm A}(0,z_{\rm s})$ (the angular diameter distance between the observer and the source) which has been adopted as an observable. In theory, this angular diameter distance reads $$\begin{alignat}{1} \!\!\!\!\!\!\!\!\!D_{\rm A}(z_1,z_2;p)=\frac{c}{H_0}\frac{1}{1+z_2}\int_{z_1}^{z_2}\frac{dz'}{E(z';p)},~ (z_2>z_1),~~ \tag {2} \end{alignat} $$ for a spatially flat cosmology, where $H_0$ is the Hubble constant, and $E(z;p)$ is the dimensionless expansion rate which depends on redshift $z$ and cosmological parameters $p$. For the $w$CDM model considered in this study, $p$ should be $$ p=\{H_0, {\it \Omega}_{\rm m},w\},~~ \tag {3} $$ and the dimensionless expansion rate reads $$ E^2(z;p)={\it \Omega}_{\rm m} (1+z)^3+{\it \Omega}_w(1+z)^{3(1+w)},~~ \tag {4} $$ where ${\it \Omega}_{\rm m}$ is dimensionless density parameter of matter, and ${\it \Omega}_w=1-{\it \Omega}_{\rm m}$ is dimensionless density parameter of dark energy for a spatially flat cosmology. It is clear that the Hubble parameter $H_0$ cannot be constrained, once the ratio $D^{\rm obs}(z_{\rm l},z_{\rm s})=D_{\rm A}(z_{\rm l},z_{\rm s})/D_{\rm A}(0,z_{\rm s})$ is adopted as the cosmic observable. Now we have only two cosmological model parameters ${\it \Omega}_{\rm m}$ and $w$ remaining to be determined by the strong lensing systems. On the other hand, once the power-law density profile is assumed, one would like to understand the degeneracy between ${\it \Omega}_{\rm m}$ and the power law index, and also the impact on the cosmological parameters due to the evolution of the power law index. This is the main motivation of this work. To have the ratio of $D^{\rm obs}(z_{\rm l},z_{\rm s})=D_{\rm A}(z_{\rm l},z_{\rm s})/D_{\rm A}(0,z_{\rm s})$ for a strong lensing system, one needs to specify a lensing model which relates the Einstein radius to the projected mass density. Here we will mainly focus on the singular isothermal sphere (SIS) or singular isothermal ellipsoid (SIE) profile and a generalized spherically symmetric power-law mass distribution model $\rho \sim r^{-\gamma}$. The spherically symmetric power-law model was proposed to consider the possible deviation from the isothermal profile and its evolution with redshift due to the structure formation theory.[22-24] The SIS is recovered when $\gamma=2$ is respected. In the following we give a brief review of these two models. Analyses of isothermal mass properties have been widely applied in statistical analysis of lenses. Under the first order approximation, the SIS model is a good choice to obtain mean parameters of galaxies in the statistical analysis. Theoretically the Einstein radius (square) in the SIS model is given by $$ \theta_{\rm E}^2=\frac{4G}{c^2}\frac{D_{\rm A}(z_{\rm l},z_{\rm s})}{D_{\rm A} (0,z_{\rm s})D_{\rm A}(0,z_{\rm l})}M_{\rm lens},~~ \tag {5} $$ where $M_{\rm lens}$ is the mass inside the Einstein radius, $c$ is the speed of light, and $G$ is the gravitational constant. The velocity dispersion $\sigma_{\rm SIS}$ (square) has the following formula $$ \sigma_{\rm SIS}^2=\frac{G}{\pi}\frac{M_{\rm lens}}{R_{\rm E}},~~ \tag {6} $$ where $R_{\rm E}=\theta_{\rm E}D_{\rm A}(0,z_{\rm l})$ is the physical Einstein radius. Thus the Einstein radius can be recast in terms of the velocity dispersion by eliminating $M_{\rm lens}$ and $R_{\rm E}$, $$ \theta_{\rm E}=4\pi\frac{D_{\rm A}(z_{\rm l},z_{\rm s})}{D_{\rm A}(0,z_{\rm s})}\frac{\sigma_{\rm SIS}^2}{c^2}.~~ \tag {7} $$ One introduces the relation between the stellar velocity dispersion $\sigma_0$ and the velocity dispersion $\sigma_{\rm SIS}$ in the form of $$ \sigma_{\rm SIS}=f_{\rm E}\sigma_0,~~ \tag {8} $$ where $f_{\rm E}$ is a free model parameter. Then if the observed values of the Einstein radius and the velocity dispersion are obtained, one has the observed ratio easily, $$ D^{\rm obs}_{\rm SIS} =\frac{ D_{\rm A}(z_{\rm l}, z_{\rm s})}{D_{\rm A}(0, z_{\rm s})} = \frac{c^{2}\theta_{\rm E}}{4\pi {f_{\rm E}}^2 {\sigma_0}^2}.~~ \tag {9} $$ According to Refs. [6,15], the value of $f_{\rm E}$ is in the range of $0.8^{1/2} < f_{\rm E} < 1.2^{1/2}$. To eliminate the effects and uncertainties caused by the free parameter $f_{\rm E}$, Wang et al. proposed to use the ratio $\mathcal{D}^{\rm obs}_{ij}=\theta_{E_{i}}\sigma_{0_{j}}^2/\theta_{E_{j}}\sigma_{0_{i}}^2$ as cosmic observations to constrain the cosmological model, where $i$ and $j$ denote the order numbers of the lensing systems used in Ref. [7]. In this way, the effects and uncertainties caused by the free parameter $f_{\rm E}$ are eliminated completely. However in this work, $f_{\rm E}$ together with the cosmological model parameters will be treated as a free model parameters. When we obtain the observable $D^{\rm obs}$ from the observed values of the Einstein radius $\theta_{\rm E}$ and the velocity dispersion $\sigma_0$, the uncertainties are also introduced into the error budget. Via the error propagation equation, one can derive the $1\sigma$ error for $D^{\rm obs}_{\rm SIS}$ via the equation $$\begin{align} \sigma^2_{D({\rm SIS})}=\,&(D^{\rm obs}_{\rm SIS})^2\Big[\Big(\frac{\sigma_{\theta_{\rm E}}}{\theta_{\rm E}}\Big)^2\\ &+4\Big(\frac{\sigma_{\sigma_0}}{\sigma_0}\Big)^2+4\Big(\frac{\sigma_{f_{\rm E}}}{f_{\rm E}}\Big)^2\Big],~~ \tag {10} \end{align} $$ where $\sigma_{\theta_{\rm E}}$, $\sigma_{\sigma_0}$ and $\sigma_{f_{\rm E}}$ are the uncertainties of $\theta_{\rm E}$, $\sigma_0$ and $f_{\rm E}$, respectively. Fortunately, $\sigma_0$ and $\sigma_{\sigma_0}$ can be obtained from observations. Following the SLACS team, we take 5% error for $\theta_{\rm E}$ and $f_{\rm E}$ due to the fractional uncertainty of the Einstein radius at the level of 5%. According to Ref. [25], it can be seen that choosing $\sigma_{\theta_{\rm E}}$ the level of 5% is reasonable. For a general power-law density profile model, the mass density distribution is given by $\rho\sim r^{-\gamma}$. Since lensing mass inside the Einstein radius is determined by $\theta_{\rm E}$, for a spherically symmetric lensing system, we can shift the coordinate origin to the center of symmetry and reduce light deflection to a one-dimensional problem, $$ M(\xi)=2\pi\int^{\xi}_{0}{\it \Sigma}(\xi')\xi'd\xi',~~ \tag {11} $$ where $M(\xi)$ is the mass enclosed within the radius $\xi$, and ${\it \Sigma}(\xi)$ is the projected mass density for lensing. For the case of an Einstein ring, the mass $M_{\rm lens}$ inside the Einstein radius is given by $$ M_{\rm lens}=\pi{\it \Sigma}_{\rm cr}R_{\rm E}^2,~~ \tag {12} $$ where the meaning of $R_{\rm E}$ has been noted above, and ${\it \Sigma}_{\rm cr}$ is the critical projected mass density, $$ {\it \Sigma}_{\rm cr}=\frac{c^{2}}{4\pi G}\frac{D_{\rm A}(0,z_{\rm s}) }{D_{\rm A}(0,z_{\rm l})D_{\rm A} (z_{\rm l},z_{\rm s})}.~~ \tag {13} $$ Thus the lens mass $M_{\rm lens}$ inside the Einstein radius reads $$ M_{\rm lens}=\frac{c^2}{4G}\frac{D_{\rm A}(0,z_{\rm s})D_{\rm A}(0,z_{\rm l})}{D_{\rm A}(z_{\rm l},z_{\rm s})} \theta^2_{\rm E}.~~ \tag {14} $$ After solving the spherical Jeans equation, one can assess the dynamical mass inside the aperture projected to lens plane and can scale it to the Einstein radius $$ M_{\rm dyn}= \frac{\pi}{G}\sigma^2_{\rm ap}D_{\rm A}(0,z_{\rm l})\theta_{\rm E}\Big(\frac{\theta_{\rm E}} {\theta_{\rm ap}}\Big)^{2-\gamma}f(\gamma),~~ \tag {15} $$ where $$\begin{alignat}{1} f(\gamma)=\,&-\frac{1}{\sqrt{\pi}}\frac{(5-2\gamma)(1-\gamma)}{3-\gamma}\\ &\cdot \frac{{\it \Gamma}(\gamma-1)}{{\it \Gamma}(\gamma-3/2)} \Big[\frac{{\it \Gamma}(\gamma/2-1/2)} {{\it \Gamma}(\gamma/2)}\Big]^2,~~ \tag {16} \end{alignat} $$ and $\sigma_{\rm ap}$ is the velocity dispersion inside the aperture. For a lens galaxy with a projected mass $M_{\rm lens}$ inside the Einstein radius $R_{\rm E}$, the luminosity weighted average line-of-sight velocity dispersion inside an aperture $R_{\rm ap}$ is given by $$ \sigma_{\rm ap}^2=\frac{G}{\pi}\frac{M_{\rm lens}}{R_{\rm E}}f(\gamma)\Big(\frac{R_{\rm ap}}{R_{\rm E}}\Big)^{2-\gamma}.~~ \tag {17} $$ In addition, the relationship between $\sigma_{\rm ap}$ and $\sigma_0$ is given by[26,27] $$ \sigma_0=\sigma_{\rm ap}\Big(\frac{\theta_{\rm eff}}{2\theta_{\rm ap}}\Big)^{0.04}.~~ \tag {18} $$ By setting $M_{\rm lens} =M_{\rm dyn}$, one obtains $$ \theta_{\rm E}=4\pi \frac{\sigma^2_{\rm ap}}{c^2}\frac{D_{\rm A}(z_{\rm l},z_{\rm s})} {D_{\rm A}(0,z_{\rm s})}\Big(\frac{\theta_{\rm E}}{\theta_{\rm ap}}\Big)^{2-\gamma}f(\gamma).~~ \tag {19} $$ Now we have the observable in the power-law density profile model, $$ D^{\rm obs}_{\rm PLP}=\frac{c^2\theta_{\rm E}}{4\pi \sigma^2_{\rm ap}}\Big(\frac{\theta_{\rm ap}} {\theta_{\rm E}}\Big)^{2-\gamma}f^{-1}(\gamma).~~ \tag {20} $$ In this model, the $1\sigma$ error of $D_{\rm PLP}^{\rm obs}$ coming from the uncertainties of $\sigma_{\rm ap}$ and $\theta_{\rm E}$ can be written as $$ \sigma^2_{\rm D(PLP)}=(D^{\rm obs}_{\rm PLP})^2 \Big[4\Big(\frac{\sigma_{\sigma_{\rm ap}}} {\sigma_{\rm ap}}\Big)^2+(1-\gamma)^2\Big(\frac{\sigma_{\theta_{\rm E}}}{\theta_{\rm E}}\Big)^2\Big],~~ \tag {21} $$ where we also take 5% error for $\theta_{\rm E}$ and $\sigma_{\rm ap}$.
cpl-33-7-079801-fig2.png
Fig. 2. The curves of $\gamma$ of two models in terms of redshift $z_{\rm l}$ from 0 to 1. Here we set $\gamma_0=2.0$ and $\gamma_{\rm a}=-0.1$. They are quite different on the redshifts of the lensing systems in the catalog.
It is suggested that the mass density power-law index $\gamma$ of massive elliptical galaxies evolves with respect to redshift. A linear relationship with $z_{\rm l}$ is assumed to be $\gamma(z_{\rm l})=2.12^{+0.03}_{-0.04}-0.25^{+0.10}_{-0.12}\times z_{\rm l}+0.17\pm 0.02 ({\rm scatter})$ by combining the lensing sample from SLACS, SL2S and LSD. Here we just quote the results in the case of the $w$CDM model obtained in Ref. [4]: $\gamma_0=2.06\pm0.09$, $\gamma_1=-0.09\pm0.16$, where 118 lensing systems were used under the assumption of a fixed best-fit value of ${\it \Omega}_{\rm m}$ obtained by Planck2013. Although, by this assumption, a relative tighter constraint to the equation of sate of dark energy could be obtained, the results and conclusion are limited to the special case. More importantly, one would like to see the power in the constraining cosmological model beyond the only equation of sate of dark energy, that is, ${\it \Omega}_{\rm m}$. Furthermore, we would like to know the possible degeneracy between ${\it \Omega}_{\rm m}$ and $\gamma$, and the possible dependence of cosmological model parameter space on the parameterized forms of $\gamma$. Therefore, in this work, we choose two parameterized forms of $\gamma$ as follows: $$\begin{align} {\rm Model~I:}~~\gamma(z_{\rm l}) =\,& \gamma_0+\gamma_{\rm a} z_{\rm l},~~ \tag {22} \end{align} $$ $$\begin{align} {\rm Model~I\!I:}~~\gamma(z_{\rm l}) =\,& \gamma_0+\gamma_{\rm a} \frac{z_{\rm l}}{1+z_{\rm l}},~~ \tag {23} \end{align} $$ to study the parameterized form dependence issue, with $\gamma_0$ and $\gamma_{\rm a}$ being free constant parameters. Model II is inspired by the Chevalier–Polarski–Linder (CPL) dark energy model already taken as a very natural Taylor expansion at present, i.e., $a=1$. The evolutions of $\gamma$ of the two models in terms of redshift $z_{\rm l}$ from 0 to 1, corresponding to the 118 data points (Fig. 1), are shown in Fig. 2. Here $\gamma_0=2.0$ and $\gamma_{\rm a}=-0.1$ are used. As a comparison to model I, model II is not degenerate even at quite small redshifts. They are easy to distinguish on the redshifts of the lensing systems in the catalog. This deviation on the parameterized forms of $\gamma$ is useful to verify our conjecture. To see the constraining power to the cosmological model and the possible degeneracy between ${\it \Omega}_{\rm m}$ and $\gamma$, we relax ${\it \Omega}_{\rm m}$ as a free cosmological parameter. To obtain the model parameter space, we perform a global fitting via the Markov chain Monte Carlo method which is based on the publicly available CosmoMC package.[28] The posterior likelihood $\mathcal{L}\sim\exp[-\chi^2(p)/2]$ is given by calculating $\chi^2(p)$ with $$ \chi^2(p)= \sum_{i=1}^{118}\frac{(D_{i}^{\rm th}(p)-D_{i}^{\rm obs})^2}{\sigma_{D,i}^2},~~ \tag {24} $$ where $p$ is the model parameter vector, $i$ denotes the $i$th strong lensing system, $\sigma_{D,i}^2$ is the corresponding $1\sigma$ variance of $D^{\rm obs}_{i}$. The theoretical values of $D_{i}^{\rm th}(p)$ are calculated by $$ D_{i}^{\rm th}(p)=\frac{D_{\rm A}(z_{\rm l},z_{\rm s};p)}{D_{\rm A}(0,z_{\rm s};p)} =\frac{\int_{z_{\rm l}}^{z_{\rm s}}\frac{dz'}{E(z';p)}}{\int_{0}^{z_{\rm s}}\frac{dz'}{E(z';p)}},~~ \tag {25} $$ where $z_{\rm l}$ and $z_{\rm s}$ are the redshifts at the lens and source of the lensing system, respectively. For the SIS (PLP) model, the observed value $D_{i}^{\rm obs}$ is derived from Eq. (9) (Eq. (20)) and its $1\sigma$ error $\sigma_{D,i}$ is calculated by Eq. (10) (Eq. (21)). The corresponding redshifts of the 118 lensing systems, the observed values of $\theta$, $\sigma$ and their $1\sigma$ error bars can be found in Table 1 of Ref. [4].
cpl-33-7-079801-fig3.png
Fig. 3. The 68.3% and 95.4% confidence regions for the $w$CDM model obtained from 118 strong lensing systems in the SIS model.
In the SIS model, we have three cosmological model parameters and one SIS model parameter, i.e., $$ p=\{H_0, {\it \Omega}_{\rm m}, w, f_{\rm E}\}.~~ \tag {26} $$ Due to the fact that the ratio $D_{\rm A}(z,z_{\rm s})/D_{\rm A}(0,z_{\rm s})$ is taken as observable, the Hubble parameter $H_0$ cannot be constrained, therefore $H_0$ is marginalized in our analysis in the SIS model and power-law density profile models. The constraint results are shown in Fig. 3. In $1\sigma$ region, one has ${\it \Omega}_{\rm m}=0.31^{+0.44}_{-0.24}$, which is compatible with that obtained by Planck2015: ${\it \Omega}_{\rm m}=0.313\pm0.013$.[29] The value of $w=-2.2^{+1.7}_{-2.4}$ in $1\sigma$ region implies that our universe is undergoing an accelerated expansion. It confirms the findings by SN Ia independently. The value of $f_{\rm E}=0.986^{+0.047}_{-0.041}$ centering around $f_{\rm E}\sim 1$ is also compatible with previous results.[6,15]
Table 1. The same as the SIS model but for the power-law density profile models: model I $\gamma(z_{\rm l})= \gamma_0+\gamma_{\rm a} z_{\rm l}$ and model II $\gamma(z_{\rm l})=\gamma_0+\gamma_{\rm a} z_{\rm l}/(1+z_{\rm l})$.
Parameter 95% limits model I 95% limits model II
${\it \Omega}_{\rm m}$ $0.15^{+0.13}_{-0.11} $ $0.16^{+0.14}_{-0.11} $
$w$ $-1.25^{+0.66}_{-0.86} $ $-1.23^{+0.66}_{-0.87} $
$\gamma_0$ $2.11^{+0.10}_{-0.11} $ $2.11^{+0.11}_{-0.12} $
$\gamma_{\rm a}$ $-0.10^{+0.19}_{-0.20} $ $-0.16^{+0.39}_{-0.40} $
cpl-33-7-079801-fig4.png
Fig. 4. The {68.3%} and {95.4%} confidence regions for the $w$CDM model obtained from 118 strong lensing systems for the power-law density profile models: model I $\gamma(z_{\rm l})=\gamma_0+\gamma_{\rm a} z_{\rm l}$ and model II $\gamma(z_{\rm l})=\gamma_0+\gamma_{\rm a} z_{\rm l}/(1+z_{\rm l})$.
cpl-33-7-079801-fig5.png
Fig. 5. The relative relation of two velocity dispersions in terms of $\gamma$ and $R_{\rm ap}$.
For the power-law density profile model, we also have three cosmological model parameters and two profile index parameters, i.e., $$ p=\{H_0, {\it \Omega}_{\rm m}, w, \gamma_0, \gamma_{\rm a}\}.~~ \tag {27} $$ As stated in the previous subsection, $H_0$ is marginalized. The constraint results are shown in Table 1 and Fig. 4. As shown in Fig. 4, we obtain almost the same distribution for the cosmological model space, i.e., ${\it \Omega}_{\rm m}=0.15^{+0.13}_{-0.11}$, $w=-1.25^{+0.66}_{-0.86}$ for model I and ${\it \Omega}_{\rm m}=0.16^{+0.14}_{-0.11}$ and $w=-1.23^{+0.66}_{-0.87}$ for model II, although the distribution for $\gamma$ parameters are different. This means that the cosmological model parameters ${\it \Omega}_{\rm m}$ and $w$ are insensitive to the parameterization form of $\gamma$. There is no significant difference for $\Delta\chi^2=0.7$ for two parameterized forms. Therefore, the cosmological parameter space does not depend on the parameterized forms of $\gamma$. In the contour plot for ${\it \Omega}_{\rm m}-\gamma$, there is no significant degeneracy between ${\it \Omega}_{\rm m}$ and $\gamma$. However, $\gamma_0$ and $\gamma_{\rm a}$ are anti-correlated for centering around $\gamma \sim 2$. As a comparison to the SIS model, the power-law density profile model can give a relatively tight constraint to the cosmological model parameters and favors small values of ${\it \Omega}_{\rm m}$. However, we should note that the values of ${\it \Omega}_{\rm m}=0.15^{+0.13}_{-0.11}$ have $1.25\sigma$ tension with that obtained by Planck2015 ${\it \Omega}_{\rm m}=0.313\pm0.013$.[29] The obvious difference on the best-fit values of ${\it \Omega}_{\rm m}$ between the SIS model and the PLP model can be interpreted from the velocity dispersions. From Eqs. (6) and (17), we can see that the velocity dispersion of SIS is a constant, and a power function for the PLP model. Here we set $\sigma_{\rm SIS}^2=1$, $\gamma$ from 1.9 to 2.0, and $R_{\rm a}p/R_{\rm ap}$ from 0.3 to 3.0, which is corresponding to the redshifts of the lensing systems in the catalog and our results. We plot the relative relation of two velocity dispersions in terms of $\gamma$ and $R_{\rm ap}$ in Fig. 5. Astronomy tell us that the deviation of rotational velocity of galaxies from theoretical prediction implies that there is dark matter among the galaxies. In Fig. 5, when $\gamma$ is fixed, we can see two different curves of rotational velocity, and the SIS model is more uniform and ideal than PLP, therefore the former prefers more dark matter. That is the reason why the best-fit values of ${\it \Omega}_{\rm m}$ of the SIS model are larger than PLP's. We thank an anonymous referees for helpful improvement of this study. References 0957 + 561 A, B: twin quasistellar objects or gravitational lens?Constraining Equation of State of Dark Matter: Including Weak Gravitational LensingStrong lensing systems as a probe of dark energy in the universeCOSMOLOGY WITH STRONG-LENSING SYSTEMSCosmic equation of state from strong gravitational lensing systemsConstraints on cosmological models from strong gravitational lensing systemsSTRONG GRAVITATIONAL LENSING AND ITS COSMIC CONSTRAINTSGRAVITATIONAL LENSING STATISTICS AS A PROBE OF DARK ENERGYGRAVITATIONAL LENSING STATISTICAL PROPERTIES IN GENERAL FRW COSMOLOGIES WITH DARK ENERGY COMPONENT(S): ANALYTIC RESULTSThe Cosmic Lens All-Sky Survey: statistical strong lensing, cosmological parameters, and global properties of galaxy populationsImproved Cosmological Constraints from Gravitational Lens StatisticsThe Sloan Lens ACS Survey. II. Stellar Populations and Internal Structure of Early‐Type Lens GalaxiesTHE STRUCTURE AND DYNAMICS OF MASSIVE EARLY-TYPE GALAXIES: ON HOMOLOGY, ISOTHERMALITY, AND ISOTROPY INSIDE ONE EFFECTIVE RADIUSIs There a Cosmological Constant?The redshift distribution of gravitational lenses revisited: constraints on galaxy mass evolutionLimits on the Evolution of Galaxies from the Statistics of Gravitational LensesConstraints on Scalar-Field Dark Energy from the Cosmic Lens All-Sky Survey Gravitational Lens StatisticsConstraints on a φCDM model from strong gravitational lensing and updated Hubble parameter measurementsSDSS-III: MASSIVE SPECTROSCOPIC SURVEYS OF THE DISTANT UNIVERSE, THE MILKY WAY, AND EXTRA-SOLAR PLANETARY SYSTEMSTHE BOSS EMISSION-LINE LENS SURVEY (BELLS). I. A LARGE SPECTROSCOPICALLY SELECTED SAMPLE OF LENS GALAXIES AT REDSHIFT ∼0.5THE SL2S GALAXY-SCALE LENS SAMPLE. II. COSMIC EVOLUTION OF DARK AND LUMINOUS MASS IN EARLY-TYPE GALAXIESTHE SL2S GALAXY-SCALE LENS SAMPLE. IV. THE DEPENDENCE OF THE TOTAL MASS DENSITY PROFILE OF EARLY-TYPE GALAXIES ON REDSHIFT, STELLAR MASS, AND SIZECosmological parameters from strong gravitational lensing and stellar dynamics in elliptical galaxiesMulticolour CCD surface photometry for E and S0 galaxies in 10 clustersSpectroscopy for E and S0 galaxies in nine clustersCosmological parameters from CMB and other data: A Monte Carlo approach
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