Galaxy-galaxy lensing in the VOICE deep survey

As discussed in F18, the photo-z estimation in the VOICE study is the peak value of the probability density function, and the photo-z data was derived using the $\verb|BPZ|$ software. We checked the comparison of the measurements of photo-z with the corresponding spec-z (Vaccari et al., 2010, 2016) for the matched 23638 galaxies in Fig. 1, and It shows good agreements to spec-z for corresponding photo-z on the whole. F18 presents the median value of the difference between photo-z and spec-z: $\delta z=(Z_{phot}-Z_{spec})/(1+Z_{spec})=0.008$, with median absolute deviation $\sigma=0.06$. For the photometric catalogue and shear catalogue, we consider the uncertainties of sources from $\verb|BPZ|$ are good enough to support the photo-z used for estimating galaxy-galaxy lensing signals.

Galaxy masses are derived usind a standard spectral energy distribution (SED) fitting software, Le-Phare¹¹1http://www.cfht.hawaii.edu/~arnouts/lephare.html(Arnouts et al., 1999; Ilbert et al., 2006). Since we want to ultimately study the halo properties of the lens sample, and relate these to their stellar mass properties, we have used the photometric galaxy multi-band (optical plus NIR) catalogue to estimate the stellar masses. Here Le-Phare is fed with the full 9-band photometry from the VOICE galaxy catalogue to produce, as output, the best stellar population parameters, including the age, metallicity, star formation rate and the stellar mass. The stellar population synthesis (SPS) models (Conroy & Wechsler, 2009) we have adopted to match the multi-band photometry are stellar templates from Bruzual & Charlot (2003) with a Chabrier (2003) initial mass function (IMF) and an exponential decaying star formation history.

For the Le-Phare run, we have used a broad set of models with different metallicities ($0.005\leq Z$/${Z_{\odot}}\leq 2.5$) and ages ($age\leq age_{\rm max}$), with the maximum age, $age_{\rm max}$, set by the age of the Universe at the redshift of the galaxy, with a maximum value at z = 0 of $13.5\,\rm Gyr$. We also considered internal extinction using the Calzetti et al. (1994) models. Finally, to reduce the degeneracies between the redshift and galaxy colours, we have fixed the galaxy redshift to the VOICE catalogue photo-z.

As mentioned in Sect. 3, the lens sample is based on the photometric galaxy catalogue, regardless any shear has been measured for them. Among these, we have selected galaxies in the range of $0.1<z_{l}(\verb|BPZ|)<0.35$, for which all $gri$-bands magnitudes are available (hereafter ’Full Lens sample’, or FLS in short). This catalogue is made of 46188 galaxies, containing positions and photo-z for each of them. The distribution of photo-z of this FLS, shown in Fig. 2, has a median of 0.29 presents that the galaxies are dominated FLS in the redshift bin from 0.30 to 0.35. The choice of this specific redshift range for the FLS is made to maximise the number of foreground galaxies to guarantee a galaxy density minimising the statistical errors over the two-dimensional lensing signal which is represented in Sect. 4.2.

The mean luminosity of the FLS is $M_{r}\sim-18.06$ with a scatter of $\sigma(M_{r})\sim 1.61$ mag, while the averaged logarithmic stellar mass is $\log M_{*}/M_{\odot}=8.56$, with a scatter of $\sigma(\log M_{*}/M_{\odot})=0.96$. This rather low mean value and large scatter imply that a large portion of the lens systems have low mass. Indeed, the stellar masses are distributed in the range $10^{6}-10^{12}M_{\odot}$, such as they cover a very large mass range going from dwarf galaxies to giant ellipticals. As we seek to obtain mean dark halo properties of the lens sample, which are a strong function of the stellar mass (e.g. Moster et al. 2013), averaging the mass profiles in this wide range of masses is poorly meaningful. Hence, we have decided to bin galaxies in stellar mass.

Furthermore, to study the halo properties as a function of galaxy types (e.g. Mandelbaum et al. 2006; Hudson et al. 2015), we roughly separate the passive red galaxies from some active bluer systems, we further bin them in colour.

The distribution of galaxy colours as a function of the $r-$band rest-frame magnitude is shown in Fig. 3. Here we can clearly identify a red sequence at the rest frame $[g-i]_{rest}>0.9$, for $M_{r}<-19$. Hence we have separate FLS into the Red and Blue lenses as the ones above and below the dashed line in Fig. 3, respectively. We have tried to use other classification of red and blue galaxies like $[u-g]_{rest}$, $[g-r]_{rest}$ (Bell et al., 2003), $[u-r]_{rest}$ (Baldry et al., 2004), but these other methods hardly catch the obvious bimodal distribution in the colour magnitude diagram for the classification of FLS as seen in Fig. 3. Fig. 2 shows the distributions of photometric redshifts of the Red and Blue lenses with that the means are 0.24 and 0.26, respectively.

In Fig. 4 we use the colour separation above to display the mass distribution of the two colour classes. This shows a clear bimodal distribution in $\log M_{*}$ for the Red and Blue Lens. The Red Lens sample contains 5822 galaxies and the Blue Lens sample 40366 with median $\log M_{*}/M_{\odot}$ of $9.88$ and $8.37$, respectively. Looking at the distributions of the stellar masses in Fig. 4, we can see that the most massive bin, such as $\log M_{*}/M_{\odot}>10.5$ is mainly occupied by the Red lenses, while in the bin $9.5<\log M_{*}/M_{\odot}<10.5$ there is a mix of Blue and Red lenses, although the former are dominant in absolute number. In the lower mass range, 7.0¡$\log M_{*}/M_{\odot}$¡9.5, the Blue lenses reach their peak at $\log M_{*}/M_{\odot}\sim 8.5$, while they start to become incomplete at lower masses. The Red lenses, though, are too little in the same mass bin, $8.5<\log M_{*}/M_{\odot}<9.5$, to produce a significant lensing signal. Hence to obtain a significant colour/mass separation of the FLS, we have defined the following samples:

1. 1.

   Blue Lens$\verb|-|$1 and $\verb|-|$2: $10^{7.0}<M_{*}/M_{\odot}<10^{9.5}$, $10^{9.5}<M_{*}/M_{\odot}<10^{10.5}$, respectively;

2. 2.

   Red Lens$\verb|-|$1 and $\verb|-|$2: $10^{9.5}<M_{*}/M_{\odot}<10^{10.5}$, $M_{*}/M_{\odot}>10^{10.5}$), respectively.

In Table 1 we summarise the galaxy number, the median of photo-z and logarithmic stellar mass for the four lens sub-samples. Hereafter, we will make use of the median values of these parameters to characterise the four lens samples.

As background galaxy sample (the sources), we select galaxies from the shear catalogue of VOICE$\verb|-|$CDFS with photo-z in the range of $0.3<z_{s}(\verb|BPZ|)<1.5$. The distribution of photo-zs is shown in Fig. 2.

The lens-source pairs are then selected using the condition that $\Delta z_{p}=z_{s}-z_{l}>0.2$. This criterion has been adopted to take into account the errors on the photometric redshifts and avoid confusion between background and foreground objects if too close in redshift. According to F18, the typical photo-z errors are $\sim 0.06$ for $z<0.83$ and $\sim 0.1$ for $z>0.83$, hence the use of $\Delta z_{p}>0.2$ allows us to separate foreground from background with $\sim 2\sigma$ significance.

Galaxy-galaxy lensing signal estimator is based on the measurement of tangential shear $\gamma_{t}$ from background sources, around the foreground lenses. As mentioned before, galaxy shapes have been measured in F18 using $\verb|LensFit|$, where galaxy ellipticities $\gamma_{1}$ and $\gamma_{2}$ have been derived from $\verb|OmegaCAM|$ images with an accuracy to $\leq$1$\%$. These measurements are used to estimate the tangential component $\gamma_{t}$ and cross component $\gamma_{\times}$ of shear signals of background sources around a lens galaxy for the $j$-th lens-source pair according to the equations

where $\phi_{j}$ is the angle between the separation vector of the $j$-th lens-source pair with the horizontal axis in Cartesian coordinate system centred on each object of lens. In weak lensing, the weak distortion of the intrinsic shape due to the warped space-time caused by the lenses of each independent background source is too small to be detected. Hence, in order to detect the shear signals, we need to average over large numbers of lens-source pairs to finally measure, in particular, the tangential component of the shear. This allows us to derive the signal around a lens sample in angular bins $\theta$ (Mandelbaum et al., 2005a; Luo et al., 2018),

where $\overline{\mathcal{R}}$ means the responsivity of source galaxies derived by the equations (5) to (7) in Jarvis et al. (2003), and here $w^{\prime}_{j}$ is the weight come from $\verb|LensFit|$ for the $j$-th lens-source pair.

This quantity is used next to derive a proxy of the mass density as a function of the angular distance from the common centre of the lens sample adopted to measure it.

The Excess Surface Density (ESD; $\Delta\Sigma$) is defined as the discrepancy between $\overline{\Sigma}(\leq R)$ with $\overline{\Sigma}(R)$ that are the averaged projected surface mass densities inside of radius $R$ and at radius $R$,

There is a connection between the ESD with the tangential shear from background sources. Indeed the $\Delta\Sigma$ can be written as (Hudson et al., 2015):

where the critical surface density $\Sigma_{\rm crit}$ is defined as

where $D_{l}$, $D_{s}$, and $D_{ls}$ are the angular diameter distance of the lens, background source, and that between two the objects, respectively.

In this equation, pairs are weighted by the $\Sigma^{-2}_{{\rm crit,}j}(z_{l},z_{s})$, such that we write the weights $w_{j}$, for the $j$-th lens-source pair, as

Hence, Eq. (4) states that we can estimate the ESDs directly from the mean tangential shear signal of sources around the lenses in the different bins of projected separation $R$.

However, we need to correct the shear for possible biases in the shear measurements by $\verb|LensFit|$. The shear calibration (Liu et al., 2018), brings a multiplicative, $m$, and an additive, $c$, bias into the shear estimation, that allow us to convert the observed shear into a ’true’ signals as

where $a$ presents two components ($a$=1, 2) of galaxy ellipticities. This calibration can be applied to our averaged ESD measurement as above to obtain a corrected mean ESD measurement. This is a function of lens redshift and writes as follows:

where $c_{1}$ and $c_{2}$ are the additive biases and $m_{1}$ and $m_{2}$ are the multiplicative biases, obtained as discussed in $F18$. The estimated values of $c_{1}$ and $c_{2}$ are $\sim$8$\times 10^{-4}$ and $\sim$3$\times 10^{-5}$ for $\gamma_{1}$ and $\gamma_{2}$, respectively (see $F18$). Being these values $<<1$, the produce produce almost no effect on the shear measurements. On the other hand, the multiplicative biases $m_{1}$ and $m_{2}$ are quite uniformly distributed in the range (-0.494,0.678) and (-0.362, 0.696), respectively, hence they have to taken into account.

Finally, to derive an unbiased ESD estimator, we need to subtract the tangential shear measurements around random points that ought not to have net lensing signal. This writes as:

According to the random test in Sect. 6.1, however, the ESD measurements from tangential shear signals of the random sample, $\Delta\Sigma^{\rm rand}(R)$, is generally consistent with zero. Here we would consider the noise, $\Delta\Sigma^{\rm rand}(R)$ of 100 times the number of random points corresponding to our samples, is subtracted in our final ESD measurements.

Despite we have payed attention to avoid the overlap between the lens and source pairs by considering the $z_{s}-z_{l}$ separation in photo-z larger than 0.2 (see Sect. 3.3), there can be still a fraction of background sources which are physically connected to the lenses, causing a scale-dependent bias of lensing signal due to galaxy clustering (Sheldon et al., 2004). To correct for the effect of this correlation between lens with background sources, we apply a multiplicative boost factor $B(R)$. This is defined as the ratio between the weighted number of background galaxies per unit area around the lens and the ones around random points:

where $i,j$ and $k,l$ denote the sources found around the real lens and the random points, respectively, and $w_{i,j}$ or $w_{k,l}$ are the weight for the pair between each background source with one lens or a random point (Sheldon et al., 2004).

In Fig. 5 we show the $B(R)$ in radial bins from $\sim$0.03 to 1.2 $h^{-1}$Mpc for the different selected samples. In particular, we see that the $B(R)$ is close to one at all radii only for the FLS and the low mass Blue lens sample, while for all other samples it become significantly larger than one for $R<0.05$ $h^{-1}$Mpc, with the Red massive sample showing the largest factor at all radii, from 1.1 at $R>0.05$ $h^{-1}$Mpc to 1.6 for $R\sim 0.03$ $h^{-1}$Mpc. The similarity of the $B(R)$ between the FLS and the low mass Blue lenses comes mainly from the fact that the FLS is numerically dominated by the Blue Lens$\verb|-|$1 sample, which, because of the large statistics and the more sparse distribution in space (i.e. low mass blue galaxies tend to be less clustered than red massive galaxies, (Zehavi et al., 2005)), has a lower chance to have an intrinsic excess of concentration. On the other end of the colour-mass selection, Red massive galaxies are known to cluster more (Zehavi et al., 2005) as they are, for example, the dominant population in cluster of galaxies.

The $B(R)$ of Blue Lens$\verb|-|$1 tend to 1 which is similar for FLS that presents the selection of lens-source pairs $\Delta z_{p}>0.2$ can clearly separate foreground lens and background sources out for the low stellar mass galaxies. The Blue Lens$\verb|-|$2 and Red Lens$\verb|-|$1 have a little bigger $B(R)$ tells there are increasing correlation between lens and sources. Especially for high stellar mass galaxies such as the Red Lens$\verb|-|$2, the boost factor reflects there is an obvious correlation effect we discussed before for lens-sources pairs. This allows us to compute an ’unbiased’ $\Delta\Sigma$ by multiplying the $\Delta\Sigma$ measured as in Sect. 4.2 by the boost factor $B(R)$.

To estimate the statistic errors on $\Delta\Sigma(R)$, we apply a bootstrap method to the covariance matrix of the $\Delta\Sigma(R)$. The dimensionless covariance matrix can be simply calculated by

where $V_{i.j}=\langle(\Delta\Sigma(R_{i})-\langle\Delta\Sigma(R_{i})\rangle)\cdot(\Delta\Sigma(R_{j})-\langle\Delta\Sigma(R_{j})\rangle)\rangle$, and it makes sense by changing corresponding index for $V_{i,i}$ and $V_{j,j}$. If $\Delta\Sigma(R)$ is subtracted by the random signal $\Delta\Sigma^{rand}(R)$, the ESD measurements will gain smaller covariance (Singh & Mandelbaum, 2016).

We calculate the covariance matrix from the correlations between different $\Delta\Sigma(R)$, which are re-sampled by bootstrapping for $10^{4}$ times, around the foreground galaxies from FLS objects. In our cases, the Hartlap correction (Hartlap et al., 2007) $(N_{s}-N_{points}-2)/(N_{s}-1)$ is negligible for covariance matrix cause it is very close to unity, where $N_{s}=10^{4}$ is the number of simulations and $N_{points}=9$ is the data points. In Fig. 6, the off-diagonal terms shows there are not highly correlation of $\Delta\Sigma(R)$ between different radial bins, but on the contrary, there are strong self-correlation of $\Delta\Sigma(R)$ that are reflected in the diagonal of the covariance matrix. This means that ESDs from different radial bins are mutually independent in the galaxy-galaxy lensing signal. Furthermore, we can calculate the reduced $\chi^{2}$ to qualify the goodness of the fit between model and data, using the equation

where $C^{-1}$ is the inverse covariance matrix, and degree of freedom $d.o.f=N_{points}-N_{para}-1$, where $N_{para}$ are the number of model parameters. Table 2 also shows the $\chi^{2}_{d.o.f}$ for the model results correspond to lens samples.

The measurements of $\Delta\Sigma(R)$ can be derived from the galaxy-matter cross-correlation $\xi_{gm}(r)$ so that the galaxy-galaxy lensing can give an effective estimator of the dark matter halo profile and galaxy environment in the area around the lens. The $\xi_{gm}(r)$ is the line-of-sight projection of galaxy-matter cross-correlation function, defined as (Luo et al., 2018):

which relates the surface mass density to a corresponding lens galaxy. The ESD would be calculated by the Eq. (3):

and

where $\overline{\rho}$ is the averaged background density of the universe.

Since Eq. (4) connects the observations with $\Delta\Sigma(R)$, it provides the method to estimate the distribution of whole underlying dark matter for foreground environments in observed region by fitting observation to the halo model.

In the following we will consider the total mass, contributing to the $\Delta\Sigma(R)$, made of two main terms: the one-halo term and two-halo term. The first term includes all the mass contained in stars, both from the central and satellite galaxies, and the dark matter main halo. The second term is the projected two-halo term that correlates the matter in other individual halos with the main host halo. In general, the contribution from one-halo term is dominated in the scales smaller than the virial radius of host halo, and the two-halo term is forced to have an effect to the scales larger than the virial radius. According to this definition the ESD can be written as:

which does not contain the contribution from the averaged background density of the universe, which does not give any contribution to the ESD, by definition.

Given the halo model defined above, the measured ESDs is used to constrain the halo properties of the corresponding lens sample. Specifically we want to constrain the three free parameters of the model, such as the virial halo mass, $M_{h}$, and the concentration, $c$, of the one-halo term, and the satellite fraction, $f_{sat}$.

To best fit the observed EDS, we use the $\verb|emcee|$ (Foreman-Mackey et al., 2013) python pipeline, which makes use of a standard Markov Chain Montecarlo (MCMC) procedure (Luo et al., 2022) to explore the likelihood function in the multidimensional parameter space. The maximum likelihood function is a Gaussian where the covariance matrix is estimated by bootstrap sampling. In our analysis, we have adopted a flat prior distribution with the host halo mass $log({M_{h}}/M_{\odot})$ in the range [9.5, 14.0], the concentration $c$ in the range [0.1, 20.0], and the satellite fraction, $f_{sat}$ in the range [0.0, 1.0]. We set a rather broad range for the parameters space for reducing the prior effects as far as possible.

The assessment of systematics is a crucial part of the galaxy-galaxy lensing analyses, as systematics impact the reliability of the results.

The first test is related to the B-mode signal. This represents the cross components of the galaxy-galaxy lensing signals, $\gamma_{\times}$, along a direction tilted by $45^{\circ}$ with respect to the tangential component, $\gamma_{t}$. The $\gamma_{t}$ produces itself a ESD cross component, $\Delta\Sigma_{\times}$, tilted with respect to the tangential components, the $\Delta\Sigma$. By definition, the B-mode signal is zero for an unbiased shear signal. Thus, any deviation from zero can indicate the presence of systematics in the ESD, which is diluted by the presence of a off-axis shear. Fig. 7 shows the B-mode signals $\Delta\Sigma_{\times}$ for the VOICE FLS. This is generally consistent with zero for all scales as expected for lensing though the most inner $\Delta\Sigma_{\times}$ point is somehow deviated from zero. The B-mode tests for the Red/Blue Lens sub-samples is discussed in details in Appendix. A. These also show basically no systematics, althought the error bars become, in some cases, rather large. Since all $\Delta\Sigma_{\times}$ are almost all consistent with zero, we conclude that the systematics, if any, are reasonably confined within the statistical errors.

We generate lens samples of random points as 100 times of the numbers correspond to the FLS, and Red/Blue Lens sub-samples, respectively. Fig. 8 shows the ESD signals $\Delta\Sigma^{\rm rand}$ of background sources, measured around the random points from FLS. This is, again, fairly consistent with zero overall, with marginal evidence of a positive signal in the first bin. The results for the Red/Blue Lens sub-samples are discussed in details in Appendix. A. They show a similar pattern, with random signal generally consistent with zero.

A final note of caution is needed about the innermost bin at $\sim 27$kpc h^-1, corresponding to $\sim 9^{\prime\prime}$ in angular scale. In both tests above, we have stressed a marginal systematic deviation of the $\Delta\Sigma_{\times}$ and $\Delta\Sigma^{\rm rand}$ from zero. There are two possible explanation that can mitigate the impact of this source of systematic. On the one hand, the lens sample dominated by faint galaxies with low stellar mass is different with Sifón et al. 2018 which finds an additive bias as a bright lens influences the shapes of background sources in small scale. So, the effect is negligible for these faint galaxies of our lens samples in this small scale. On the other hand, the $\Delta\Sigma_{\times}$ and $\Delta\Sigma^{\rm rand}$ represent the systematics which should tend to zero, but it is reasonable that both are deviated from zero if the counts of lens-source pairs decrease. The fact is that the area of innermost bin is smaller than outer bins. Therefore, there are less counts of sources around lens in the smaller scale that cause the bigger deviation from zero for the innermost ESD measurements. However, the $\Delta\Sigma_{\times}$ and $\Delta\Sigma^{\rm rand}$ for all radial bins are within 2$\sigma$ represent there are acceptable systematics so that we would keep the innermost ESD measurements.

In this section we present the halo model constraints of the VOICE lens samples using the MCMC procedure introduced in Sect. 5.2 to fit the ESD signals produced by the source sample.

The results of $\Delta\Sigma$ of the FLS are shown as blue points with error bars in Fig. 9. The error bars of $\Delta\Sigma$ are calculated from the standard errors based on the bootstrap via re-sampling $10^{4}$ times. In the same figure, the blue solid curve is the best fit line to the ESDs of the FLS, given by the total model, as the sum of all contribution of the different mass components, defined by the free parameters. In details: 1) the orange dash-dotted line represents the contribution from stellar mass of foreground galaxies, which is defined by the mean stellar mass derived from the stellar population analysis; 2) the green dashed line is the NFW model defined by the best fit parameters $c$ and $M_{h}$; 3) the red dashed line represents the satellite galaxies, defined by the other free parameter $f_{\rm sat}$; 4) the black dotted line describes the contribution from two-halo term. These contributions all are described in Sect. 5.1.

From Fig. 9, it is clear that the satellite component and the two-halo term dominate the large scale, while the stellar mass and mostly the dark halo dominate on small scales. Overall, the total fit is reasonably good with a reduced $\chi^{2}\approx$ 1.995 (d.o.f=5, see Table 2).

The marginalised posterior distributions of the three parameters obtained for the FLS sample is shown in Fig. 10. The three contours correspond, from the innermost to the outermost one, to $16\%$, $50\%$, $84\%$ confidence levels. For the FLS, the median of host halo mass is $10^{11.42}\ M_{\odot}$, which, compared to the stellar mass $M_{*}$, implies a $M_{h}/M_{*}=10^{2.93}$. It is evident, from both $M_{h}$ and $M_{*}$, that the sample is dominated by low mass systems, as also discussed in Sect. 3.2.

To explore the stellar to dark matter relation, we proceed to best fit also the other samples split by mass and colours, as defined in Sect. 3.2. In Fig. 11 we show the best fit models with contributions of different components for the $\Delta\Sigma$ from Red/Blue Lens sub-samples in the different mass bins adopted. We found the corrected $\Delta\Sigma$ of Blue Lens-1 dropped at the large scale due to the subtraction. The value of the blue low mass bin signal ($\Delta\Sigma=$0.18) is too small for the subtraction which means it is sensitive to $\Delta\Sigma^{rand}$ at the large scale for low mass lenses. So we measure the reduced chi-square $\chi^{2}_{d.o.f}$ with or without the last data point (d.o.f = 5 or 4) after subtracting the outermost data point in Table 2. And the outermost data point does not change the halo mass significantly, and we think the model fitting is generally consistent with data points cause the $\chi^{2}_{d.o.f=4}$ is 1.644 and p-value is 0.160.

Here we can appreciate the variance of the $\Delta\Sigma$ amplitude as a function of the sample mass. In particular the central peak of the most massive (Red) sample (bottom-right panel) is one order of magnitude larger than the one of the least (blue) massive sample. Looking at the typical systematics from the cross and random samples (see Appendix. A), it is evident that these are negligible with respect to the signal of the massive samples ($\log M_{*}/M_{\odot}>9.5$), while they migh affect the low mass sample ($\log M_{*}/M_{\odot}<9.5$). Another evident feature is that the stellar component seems to be more centrally concentrated for the massive systems, with respect to the less massive ones, while the satellite fraction decreases with the stellar mass (see also Table 2).

Overall, the total model allows us to fit rather well the ESD of all sample with reduced $\chi^{2}$ smaller than the FLS (see Table 2). The halo masses $M_{h}$ of the four samples have positive relation with the stellar masses $M_{*}$ that is physically reasonable. On the other hand, the concentrations $c$ does not show a clear (anti) correlation with the virial mass, as predicted from simulations (e.g. Neto et al. 2007).

To conclude this section, we compare the Stellar-to-Halo Mass Relations (SHMR) of our results with literature. Fig. 12 shows the comparisons between the SHMR results from Red/Blue Lens sub-samples in the similar redshift ranges with seven different curves which are the results from three models and galaxy-galaxy lensing (GGL) analyses of three survey: abundance matching (AM; Girelli et al. 2020, Rodríguez-Puebla et al. 2017), empirical modelling (EM; Behroozi et al. 2019), and the conditional luminosity function or halo occupation distribution (CLF/HOD; Yang et al. 2012); CFHTLenS (Hudson et al., 2015), KiDS+GAMA (Dvornik et al., 2020) and HSC (Wang et al., 2021). As we can see in Fig. 12, the SHMRs of Blue Lens$\verb|-|$2 and Red Lens$\verb|-|1\&2$ have good agreements with the results from other studies, but it is situated below these curves for Blue Lens$\verb|-|$1 that means the stellar mass is lower than these predictions under the certain low halo mass.