Halo Mass-Concentration Relation at High-Mass End

The concentration-mass (c-M) relation encodes the key information of the assembly history of the dark matter halos, however its behavior at the high mass end has not been measured precisely in observations yet. In this paper, we report the measurement of halo c-M relation with galaxy-galaxy lensing method, using shear catalog of the Dark Energy Camera Legacy Survey (DECaLS) Data Release 8, which covers a sky area of 9500 deg^2. The foreground lenses are selected from redMaPPer, LOWZ, and CMASS catalogs, with halo mass range from 10^{13} to 10^{15} M_sun and redshift range from z=0.08 to z=0.65. We find that the concentration decreases with the halo mass from 10^{13} to 10^{14} M_sun, but shows a trend of upturn after the pivot point of ~10^{14} M_sun. We fit the measured c-M relation with the concentration model c(M)=C_0 (M/(10^{12} M_sun/h)^{-\gamma} [1+(M/M_0)^{0.4}], and get the values (C_0, \gamma, log(M_0) = (5.119_{-0.185}^{0.183}, 0.205_{-0.010}^{0.010}, 14.083_{-0.133}^{0.130}), and (4.875_{-0.208}^{0.209}, 0.221_{-0.010}^{0.010}, 13.750_{-0.141}^{0.142}) for halos with 0.08<=z<0.35 and 0.35<=z<0.65, respectively. We also show that the model including an upturn is favored over a simple power-law model. Our measurement provides important information for the recent argument of massive cluster formation process.


INTRODUCTION
As fundamental building blocks of the dark matter universe, halos are believed to follow a self-similar structure distribution. The most widely used density profile of dark matter halos are proposed by Navarro et al. (1995Navarro et al. ( , 1996Navarro et al. ( , 1997 (hereafter NFW), where the slope of density γ equals to 1 at inner part and grows to 3 at outer part. In such a model, one can define the concentration of a halo to be the ratio between the virial radius (r vir ) and the characteristic radius (r s ), where the density slope γ(r s ) = 2. Given the mass and concentration, the density distribution of a halo is determined.
The halo concentration is confirmed to vary with its redshift and mass both in observations and N-body simulations (e.g., McClintock et al. 2019;Shan et al. 2012Shan et al. , 2017Johnston et al. 2007;Cui et al. 2018). N-body simulations show that the relation between concentration and mass (called c-M relation hereafter) can be used to probe the formation and evolution history of the halo (e.g., Zhao et al. 2003a,b;Bullock et al. 2001), and the dependence of concentration on the halo mass and redshift can be described with a power-law function, c = α (M/M pivot ) β (1 + z) γ (e.g., Duffy et al. 2008;Bullock et al. 2001;Neto et al. 2007;Eke et al. 2001;Cui et al. 2018). Recently, however, it is argued that the power-law may not extend to the high mass end. Some N-body simulations (Klypin et al. 2011(Klypin et al. , 2016Prada et al. 2012;Ishiyama et al. 2020) predict the flattening and upturn of the c-M relation for massive halos, and the location of the upturn varies with redshift, which may due to the different formation procedure of massive halos at higher redshift.
On observational side, gravitational lensing is the only way to detect the profile and mass of the halo directly, without any assumption about its hydrodynamic state, symmetry, or profile. This makes it a unique way to get tight constraint of the structure and evolution of halos (Abbott et al. 2018(Abbott et al. , 2020Kwan et al. 2017). Several previous works have been undertaken to constrain the c-M relation with a variety of data sets, such as the Dark Energy Survey (DES) Science Verfication data (Melchior et al. 2017), DES Year 1 data (McClintock et al. 2019), DES Year 3 (Varga et al. 2021), the Cluster Lensing and Supernova Survey with Hubble data (CLASH, Merten et al. 2015;Sereno et al. 2015). Some more works aim to measure the X-ray concentration (Pointecouteau et al. 2005;Vikhlinin et al. 2006;Gastaldello et al. 2007;Sato et al. 2000;Buote et al. 2007;Comerford & Natarajan 2007). However, they usually describe the c-M relation with the power-law function, where the concentration decreases with the halo mass in a wide mass range.
In this paper, we perform the measurement of halo c-M relation with galaxy-galaxy lensing method, using shear catalog of the Dark Energy Camera Legacy Survey (DECaLS) Data Release 8, which covers a sky area of 9 500 deg 2 . The foreground lenses are selected from redMaPPer, LOWZ, and CMASS catalogs, with the range of halo mass 10 13 −10 15 M and redshift range 0.08 − 0.65. Throughout the paper, we take the M 200m and c 200m as the mass and concentration of the halo, which means the mean density in the halo is 200 times of the matter background density at the same redshift. We make use of different c-M models to fit the data and try to investigate whether an upturn exist.
The set of cosmological parameters used is obtained from Planck Collaboration et al. (2020), the Hubble constant H 0 = 67.4 km s −1 Mpc −1 , the baryon density parameter Ω b h 2 = 0.0224, cold dark matter density parameter Ω cdm h 2 = 0.120, matter fluctuation amplitude σ 8 = 0.811, the power index of primordial power spectrum n s = 0.965, and matter density parameter Ω m = 0.315. The structure of this paper is listed as follows. In Sec. 2, the source and lens catalogs are described. In Sec. 3, the lensing signal, lensing model, and systematics are shown. In Sec. 4, we discuss the c-M relation measurement and fitting. In Sec. 5 and Sec. 6, the discussion and conclusion are shown, respectively. The source galaxies used in our measurement are obtained from the Data Release 8 (DR8) of DECaLS survey, which is part of the Dark Energy Spectroscopic Instrument (DESI) Legacy Imaging Survey (Blum et al. 2016;Dey et al. 2019). The sky coverage of DECaLS DR8 is ∼ 9 500 deg 2 in grz bands.
In the DECaLS DR8 catalog, the sources from the Tractor catalog (Lang et al. 2014) are divided into five kinds of morphologies, including point sources, round exponential galaxies with a variable radius, DeVaucouleurs, exponential, and the composite model. The sources above 6σ detection limit in any stack are kept as candidates. The ellipticity of galaxy is estimated by a joint fitting on the optical images in three bands (g, r, and z). Then, we model the multiplicative and additive biases by cross-matching the DECaLS sources with external shear measurements (Phriksee et al. 2020;Yao et al. 2020;Zu et al. 2021), including Canada-France-Hawaii Telescope Stripe 82 (Moraes et al. 2014), Dark Energy Survey (Dark Energy Survey Collaboration et al. 2016, and Kilo-Degree Survey (Hildebrandt et al. 2017) sources.
We use the photometric redshift derived by Zou et al. (2019) with the k-nearest-neighbour (kNN) algorithm. The redshift of a target galaxy is derived with its k-nearest-neighbour in Spectral Energy Distribution (SED) space whose spectroscopic redshift is known. The photometric redshift (photo-z) is obtained from 5 photometric bands: three optical bands (g, r, and z), and two infrared bands (W 1 and W 2) from the Wide-Field Infrared Survey Explorer. We only take samples with r < 23 mag, resulting in a spectroscopic sample of about 2.2 million galaxies. The characteristics of the final catalog include redshift bias of ∆z norm = 2.4 × 10 −4 , accuracy of σ ∆znorm = 0.017, and outlier rate of about 5.1%. More details are discussed in Zou et al. (2019).

Lens catalogs
We use the redMaPPer cluster catalog v6.3 1 as the lens catalog, which is obtained from the Sloan Digital Sky Survey (SDSS) DR8 (Rykoff et al. 2014). The catalog includes 26 111 clusters with richness λ > 19. In order to get the evolution of c-M relation with redshift and mass, the redMaPPer cluster sample is separated into multiple redshift and richness bins, as shown in Fig. 1 and Tab. 1. The outliers with λ > 100 are removed to get the main characters of the whole sample without bias from a few outliers. In this step, 375 most massive clusters are removed, which account for 1.4% of the full sample. Then, the sample is separated into two redshift parts, i.e., low-z sample (0.08 ≤ z < 0.35, hereafter z1 sample) and high-z sample (0.35 ≤ z < 0.65, hereafter z2 sample). The number ratio of clusters in these two redshift samples are 40.3% and 59.7%, respectively. There are relatively more clusters in the high-z sample, this can partly compensate the weakness of remote cluster signal. The cluster richness are used as the mass proxy to separate the massive and less massive clusters. Each redshift sample is further divided into 5 richness bins (named λ4 − λ8 bins), with similar cluster number in each bin. This similarity of cluster number can control unnecessary bias when we measure and compare the weak lensing signal of each bin. Using the galaxy-galaxy lensing measurement, we can further obtain the density profile of clusters with different halo mass at different redshifts. Besides redMaPPer catalog, we also utilize the LOWZ and CMASS catalogs from SDSS-III BOSS DR10 (Ahn et al. 2014) to constrain the c-M relation at the low mass region. The redshift of LOWZ and CMASS halos are constraint within 0.08−0.35 and 0.35−0.65, respectively, to match with the redshift bins of redMaPPer halos. For these two catalogs, we take the stellar mass as the halo mass proxy and separate each catalog into 3 mass bins (named λ1 − λ3 bins), as shown in Fig. 1 and Tab. 1. To minimize the effect of outliers, we remove halos with stellar mass or redshift out of the listed range.  The gravitational well of the foreground halo produces a tangential shear of the source around the foreground halo, which stretches and aligns the source images along the tangential direction. Thus, the projected mass density of lens, Σ, is related to the azimuthally averaged tangential shear at projected radius R. The relation is The ∆Σ(R) is widely used to show the weak lensing signal, the reduced shear (g), which is defined as g = γ/(1 − κ), where κ is the dimensionless surface mass density and defined as κ = Σ(R)/Σ crit . In this paper, we also use ∆Σ(R) to show the detected signal.
The critical surface mass density, Σ crit , is defined as where the D l , D s , and D ls are the angular diameter distance to the lens, to source, and between the lens and source, respectively, and the c here is the constant of light velocity in vacuum. The Σ crit shows how the geometry of the lens-source system modulates the induced shear signal.
To obtain ∆Σ, we stack lens-source pairs in 10 logarithmic co-moving radius of 0.3 − 5 Mpc in this work. Only sources with z s > z l + 0.1 are considered, to avoid the mis-classification caused by the redshift uncertainty. With this set of parameters, as listed in Tab. 2, ∆Σ(R) is estimated for a given set of lenses using where γ ls t is the tangential shear, w ls = w n Σ −2 crit , and w n is the weight factor introduced to account for intrinsic scatter in ellipticity and the error of shape measurement (Miller et al. 2007(Miller et al. , 2013. The w n used in this work is defined as w n = 1/(σ 2 + σ 2 e ). The σ is the intrinsic ellipticity dispersion derived from the whole galaxy catalogue, and taken as 0.27 (Giblin et al. 2021). σ e is the error of the ellipticity measurement defined in Hoekstra et al. (2002).
The lensing signal is recalibrated as and where m is the multiplicative error as described in Sec. 2.1. We use the software SW OT 2 (Coupon et al. 2012) to detect the stacked signal. It is a fast tree-code to compute the two-point correlations, histograms, and galaxygalaxy lensing signal from large datasets. The theta projection is taken to measure the signal. The software can be parallelized for a maximum computational efficiency. We estimate the statistical error with a Jackknife resampling of 64 sub-regions with equal area, and remove one sub-sample at a time for each Jackknife realisation. Refer to Tab. 2 for SW OT parameter setting.

The lensing model
We fit the observation with a comprehensive model, including the central halo, the miscentered halo, and nearby halo term with offset.
Firstly, we use the NFW profile to estimate the contribution from the central halo.
In addition, we take into account the miscentering effect, which comes from the inaccurate determination of halo center, and can reduce the central signal greatly (Johnston et al. 2007). For a cluster miscentered by the distance R mis , the surface mass density is Σ mis = 2π 0 dθ Σ( R 2 + R 2 mis + 2RR mis cosθ)/2π. We assume a Gamma profile for the miscentering of the stacked signal (McClintock et al. 2019). The miscentering effect is characterized by two parameters, f mis and r mis , representing the fraction of offset halos and the offset distance, respectively. Finally, we use the two halo term to indicate the signal from nearby halos, which dominates at the cluster outskirt. The contribution from the two halo term is estimated from the non-linear scaling of the matter power spectrum as a function of redshift with the Halof it model, using the CAM B package 3 . More details about the model refer to the Cluster toolkit package 4 (Smith et al. 2003;Eisenstein & Hu 1998;Takahashi et al. 2012). Thus, the whole model is,

Boost factor
The boost effect (Sheldon et al. 2004;Mandelbaum et al. 2006) comes from the membership dilution biases when some foreground or member galaxies are misclassified as background sources. If the fraction of misclassified member galaxies for the cluster is f cl , the boost factor, B(R) = (1 − f cl ) −1 , is used to correct the diluted signal. In this work, we use the boost factor model (Eq. 7) referring to McClintock et al. (2019), which is constructed from the NFW profile and characterised with B 0 and scale radius R s . We use the typical values of B 0 = 0.1 and R s = 1.0 Mpc/h.

Reduced shear error
The reduced shear error comes from the fact that the measured signal is the reduced shear (g), instead of the shear (γ). This can be corrected by multiplying the model with Here, the Σ(R) includes the contribution of the central halo, nearby halos, and miscentering effect.

Photo-z bias
Besides the corrections mentioned above, we also consider the systematic uncertainties of photo-z (δ).
The redshift of source is the photometric redshift estimated by the kNN algorithm (see Sec. 2.1), and local linear regression is used as described in Zou et al. (2019). For each galaxy, our kNN photo-z algorithm provides a Gaussian estimation of the photo-z error. We further apply this Gaussian scatter for each galaxy to obtain the probability distribution function (PDF) of redshift, i.e., p photo (z s ). For each galaxy, we estimate the effective critical surface density with the redshift PDF as where i and j indicate the source and lens in a lenssource pair. For the case with z s < z l or the offset > 0.3 Mpc, the value of Σ −1 crit i,j = 0. We only consider the signal from the lens-source pair when their projected distance is smaller than 0.3 Mpc and when the background source has a higher redshift than the foreground lens. Furthermore, we compare the kNN photo-z catalog of the photometric redshift (PDF: p kNN photo (z s )) with a large reliable photo-z catalog (PDF: p true photo (z s )) obtained from UDS HSC+SPLASH (Mehta et al. 2018), ECDF-S (Cardamone et al. 2010), CFHTLS Deep + WIRDS (Bielby et al. 2010), and COSMOS (Laigle et al. 2016). As a result, the bias of photo-z is given as 4. RESULT The final parameters are estimated by the Markov Chain Monte Carlo (MCMC) fitting method using the EM CEE package 5 (Foreman-Mackey et al. 2013). We use 50 chains with the original length of 30 000 steps each, and discard the first 10 000 steps, similar with the implementation in Yang et al. (2020). The first part of chains are discarded to avoid the effect from first guess of parameters. The relation between the mass and concentration, obtained from the stacked signal of halos in redMaPPer, CMASS, and LOWZ catalogs, are shown in Fig. 3. All these three lens samples are divided into two redshift bins. And in each redshift bin, there are 5 mass bins for redMaPPer clusters, and 3 mass bins for CMASS, LOWZ galaxies, respectively. The criteria are shown in Tab. 1.

c-M relation measurement
For the performance of signal, only signal with the error smaller than 100% are used. We measure the signal in 15 logarithmic bins in the radius of 0.1 − 7 Mpc, and only fit the 10 bins in 0.3 − 5 Mpc, to avoid the contribution of central galaxy and nearby halos. We use maximum likelihood fitting method to calculate best parameters in the fitting. There are 4 parameters to estimate, i.e., halo mass (M 200m ), concentration (c 200m ), offset distance (r mis ), ratio of halo with offset (f mis ).
First, we fit the model with MCMC method. Gaussian priors are assumed for halo mass and concentration, whose central value and full width at half maximum (FWHM) are listed in Tab. 3. The r mis and f mis are assumed flat priors within 0−1 Mpc and 0−1, respectively, with initial starting value both set as 0.2. Second, the four parameters are fitted with MCMC method again, and all the four priors are assumed to be a Gaussian distribution, with the central value and the FWHM as the best fitted results in the previous step. Third, the r mis and f mis are fixed to the best fitted value obtained from the second step. And the mass and concentration of halo are fitted with MCMC method, with priors listed in Tab. 3. Fourth, the mass and concentration are assigned gaussian priors, with the central value and FWHM estimated from the third step. Finally, the halo mass is fixed to the value obtained in the 4th step, and the concentration is fitted with MCMC method, with the gaussian prior from the 4th step.
In Fig. 2, we show the MCMC fitting result of the z1λ6 sample step by step as an example. The fitting result for step 1, 2, 3, 4, 5 described in the last paragraph are shown in sequence as the panel 1 − 5 in the figure, and the final fitting are shown in the last panel. The corresponding χ 2 /ν is labeled out for each fitting.

c-M relation fitting
We fit the relation between the concentration and mass of halos with the model with upturn (called K16 model hereafter, shown in Equ. 12, referring to Equ. 24 in Klypin et al. 2016), and the power-law model (called PL model hereafter, shown in Equ. 13), To remove the effect of redshift, we fit the data with two redshift bins separately. In this step, we firstly estimate parameters with the maximum likelihood method. Then, the best fitting parameters are obtained using the MCMC fitting method. We assume parameters (C 0 , γ, M 0 ) as gaussian priors, whose central values are obtained from the maximum likelihood fitting, and FWHM as [2, 0.13, 1], respectively. In terms of the MCMC fitting setup, we take 50 chains with the length of 30 000 steps and only last 20 000 steps are taken into account. In Fig. 3

Comparison with previous observations
There are several works dedicated to measure the halo mass and concentration. In this section, we make a brief introduction of some previous works, followed by the comparison with our work. A summary of these works are shown in Tab. 5. Mandelbaum et al. (2008) performed the statistical weak lensing analysis around the halos of 170 640 isolated galaxies, 38 236 groups and 13 823 maxBCG clusters. It includes the halo of galaxy size to cluster size, with the mass of 10 12 −10 15 M /h. The estimated NFW concentration parameter c 200b decreases from ∼ 10 to 4 with halo mass. They fit the c-M relation with powerlaw function, and find the slope is in agreement with prediction of theory. However, the value of measured concentration is slightly smaller than the theoretical prediction and some other measurements. Millenium simulations predict the concentration becomes constant at higher mass range than the highest mass bin in their observation, and they only fit observation with power law relation in the work. Using WebPlotDigitizer 6 (Rohatgi 2020), we extract the data points in their Fig. 5. We discard the two leftmost and the rightmost data points because of the limited display. From these data points, we estimate the median uncertainties of the halo mass and concentration to be ∼ 8.7 × 10 12 M /h and ∼ 3.00, respectively. Covone et al. (2014) obtain the stacked shear profile of ∼ 2 000 optical-selected galaxy clusters, wth the shear catalog from the CFHTLenS. The whole sample is divided into six richness bins to obtain the stacked shear In the model fitting of the measurements, they take into account the theoretical ΛCDM models of the hierarchical structure growth. The best fit slope is consistent with Duffy et al. (2008), and the normalization differs within 1σ error.
In the work of Umetsu et al. (2014), they made a joint shear-and-magnification weak-lensing analysis. A sample of 16 X-ray-regular and 4 high-magnification galaxy clusters selected from CLASH are taken into account. The redshift spans a range between 0.19 and 0.69. The result also agree with the ΛCDM prediction, especially the model of Meneghetti & Rasia (2013). However, they didn't fit the measurements with the model with upturn. And the limited number of stacked clusters and the limited coverage of halo mass makes it difficult to distinguish the power-law model and the model with upturn.
With 19 X-ray selected galaxy clusters from CLASH, Merten et al. (2015) derived a new constraint of the c-M relation. The redshift of this sample covers 0.19 to 0.89. The estimation of c-M relation agree with the theoretical estimation at 90% confidence level. Using the combination of lensing reconstruction techniques of weak and strong lensing, they made a tight constraint of the mass and concentration, and the uncertainty is 0.8 × 10 14 M /h and 1.0, respectively. The observation matches well with the full sample of Meneghetti & Rasia (2013), and is similar with Bhattacharya et al. (2013). However, there is also no fitting of the measurement with the model with upturn, and the limited sample impedes an accurate constraint of the c-M relation.
In Shan et al. (2017), they constrain the c-M relation with the redMaPPER clusters and LOWZ, CMASS galaxies. The halo mass is 5×10 12 -2×10 14 M /h. They only consider halos with the redshift of 0.2 − 0.4 and 0.4 − 0.6, including 7425 and 27 358 halos respectively. With the large sample and accurate shear measurement from the CFHT Stripe 82 Survey, they constraint the uncertainty of halo mass within 0.06 × 10 14 M /h, 0.05 × 10 14 M /h, and the uncertainty of the concentration within 0.91, 0.86 for halos in two redshift bins. The measurements in two redshift bins are fitted with power-law model, and matches well with the simulation predictions (e.g., Duffy et al. 2008, Klypin et al. 2016). The comparison with power-law model and the model with upturn is not discussed in their work.
In this work, we focus on the stacked weak lensing signal of galaxy clusters in two redshift bins ( 0.08 ≤ z < 0.35 and 0.35 ≤ z < 0.65), and only consider the halo with the mass in the range of 5 × 10 12 -5 × 10 14 M /h. The sample size is 301 496 and 747 376 for the low redshift and high redshift bin. Our uncertainties for halo mass is 3.08 × 10 12 M /h and 3.94 × 10 12 M /h in low and high redshift bins, respectively, while the uncertainty of concentration is 0.093 and 0.108.
Compared with previous observations, this work has the largest sample size, with more than 1 million halos taken into account. In addition, the least massive halo considered in our work (∼ 5 × 10 12 M /h) is lower than most of previous observation measurement, except the Mandelbaum et al. (2008) (∼ 5 × 10 11 M /h) and Shan et al. (2017) (∼ 5 × 10 12 M /h). However, we do not consider the very massive halos (> 5 × 10 14 M /h) as in Mandelbaum et al. (2008), andUmetsu et al. (2014). Furthermore, we have a quite wide redshift coverage (0.08 − 0.65), which covers the whole redshift area considered in the previous mentioned works, except for some high redshift halos in Umetsu et al. (2014) and Merten et al. (2015). What is more import, we make a much Step 1 Step 2  Step 3 Step 4 Step 5  (2087) 3 . 4 3 . 6 3 . 8 4 . 0 C Step 5 10 0 R(Mpc) 10 1 ΔΣ χ 2 /ν = 1.033 CΔ=Δ3.704 0.092 −0.094 Figure 2. The fitting steps of λ6z1 as an example. The 1st to 5th panels corresponding to the step 1 − 5 in the Sec. 4.1. The last panel shows the fitting result. In the top-right sub-panel of first two panels in first two rows, as well as the plot in the last row, the data and models are overlaid. The contribution from the central model, the nearby halo plus the mis-centering effect, and the total model are shown in green dotted curve, magenta dash-dotted curve and blue solid curve, respectively. In addition, the value of reduced χ 2 is overlaid. In panels of the first row, best fitted parameters and errors are added to the top of the second column of sub-panels. The best fitted c is also shown in the last panel. In each sub-panel with histogram, the median value and 1σ error are shown in solid cyan and dashed black vertical lines, respectively. tight constraint of the halo mass and the concentration, 2 − 10 times better than previous works.

Comparison with previous cosmological simulations
In this section, we compare our measurement in the high redshift bin (0.35 ≤ z < 0.65) with simulations at the redshift of 0.5, as shown in Fig. 4. For the simulation in Zhao et al. (2009), we obtain the c-M relation with the halo evolution web-calculator 7 , with the power spec-7 http://www.shao.ac.cn/dhzhao/mandc.html trum type set as BBKS 1986 power spectrum (Bardeen et al. 1986

Correlation between parameters
To quantify the correlation between the four free parameters, M 200 , c, r mis , and f mis , we calculate the Pearson correlation coefficient (PCC) for each subsample with Equ. 14. The MCMC output of step 2, as described in Sec. 4.1, are used. As shown in Fig. 5, the absolute PCC value is < 0.5 for all sub-samples, except the one between the r mis and f mis . The correlation between these two parameters is expected for both of them contribute the miscentering effect.  LOWZ, and CMASS catalogs. Using the model of the central halo, nearby halo and miscentering effect, we model the weak lensing signal and get the value of halo mass and concentration for sub-samples in multiple redshift and mass bins. We obtain the c-M relation for the halos with mass range from 10 13 to 10 15 M and redshift range from z = 0.08 to z = 0.65. Compared with power-law model, our fitting of the c-M relation prefers the K16 model (Klypin et al. 2016), which includes a trend of upturn after the pivot point of ∼ 10 14 M . This is the first measurement of the c-M relation with DECaLS DR8 data. Our measurement shows the halo concentration with similar redshift decreases with the mass increases, except for the upturn at massive end, which happens at lower mass for high redshift halo. For halos with similar mass, the halo at large redshift has larger concentration.
In this paper, we measure the c-M relation to detect the possible upturn at high-mass end. Until now, there is still no consensus on the existence nor the reason of the upturn in the c-M relation. Some works claim it comes from the unrelaxed dynamical state of dark matter halo (Ludlow et al. 2012). Some others take the selection effect as the main contribution (Meneghetti & Rasia 2013). Some more works explain the high concentration of massive clusters as a consequence of the alignment of the major axis of the ellipsoidal halo and the line-of-sight (e.g., Corless et al. 2009;Sereno et al. 2013;Limousin et al. 2013;Sereno et al. 2018), with as-suming the shape of dark matter halo as triaxial instead of spherical.
In addition, there is a possible degeneracy between f mis and c. An over-estimated factor of miscentered halo is likely to over-estimate the concentration. This would happen more frequently at high redshift. Thus, to get more accurate constraint of halo concentration, we need a better centering strategy, such as the massive galaxies near the X-ray centroids (George et al. 2012). With the miscentering effect limited, the concentration would be measured more accurate, as well as the c-M relation.
Furthermore, the upturn of the c-M relation can also be explained by the major merger of massive clusters, when the subhalo moves to the central part radially and increase the concentration of the resulting halo (Klypin et al. 2011;Prada et al. 2012;Klypin et al. 2016). In our measurement, the K16 model (Klypin et al. 2016) with an upturn at massive regime shows a comparatively good fitting, suggesting that a trend of upturn exists. This measurement is important for the structure formation at high-mass end. We expect the next generation weak lensing surveys, such as Euclid (