{'fraction_non_alphanumeric': 0.037589190830423565, 'fraction_numerical': 0.020525277060877485, 'mean_word_length': 4.461117559277069, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Detailed cluster lensing profiles at large radii and the impact on cluster weak lensing studies

Using a large set of ray-tracing in N-body simulations, we examine lensing profiles around massive dark haloes in detail, with a particular emphasis on the profile at around the virial radii. We compare radial convergence profiles, which are measured accurately in the ray-tracing simulations by stacking many dark haloes, with our simple analytic model predictions. Our analytic models consist of a main halo, which is modelled by the Navarro-Frenk-White (NFW) density profile with three different forms of the truncation, plus the correlated matter (2-halo term) around the main halo. We find that the smoothly truncated NFW profile best reproduces the simulated lensing profiles, out to more than 10 times the virial radius. We then use this analytic model to investigate potential biases in cluster weak lensing studies in which a single, untruncated NFW component is usually assumed in interpreting observed signals. We find that cluster masses, inferred by fitting reduced tangential shear profiles with the NFW profile, tend to be underestimated by ~5-10% if fitting is performed out to ~10'-30'. In contrast, the concentration parameter is overestimated typically by ~20% for the same fitting range. We also investigate biases in computing the signal-to-noise ratio of weak lensing mass peaks, finding them to be<4% for significant mass peaks. In the Appendices, we provide useful formulae for the smoothly truncated NFW profile.


INTRODUCTION
Cosmological applications of clusters of galaxies rely on the inference of cluster masses. A challenge lies in the fact that clusters are dominated by dark matter, which accounts for ∼ 80% of the mass of the universe. While the distribution of dark matter in clusters can accurately be predicted using N -body simulations, it is difficult to relate the dark matter distribution with observable characteristics of clusters, such as gas temperatures, optical/X-ray luminosities, and member galaxy distributions, because it involves many complicated physical processes associated with gas cooling/heating and star formations. Indeed, recent cosmological constraints from X-ray analysis of clusters has mainly been limited by various astrophysical uncertainties (e.g., Vikhlinin et al. 2009;Mantz et al. 2010).
Weak gravitational lensing provides a direct probe of dark matter distributions in clusters. This method makes use of coherent tangential distortion of distant background galaxies induced by deep potential wells of clusters. For massive clusters, we can detect weak lensing signals out to virial radii, which allows direct measurements of virial masses (e.g., Dahle 2006;Broadhurst et al. 2008;Mahdavi et al. 2008;Okabe & Umetsu 2008;Oguri et al. 2009;Umetsu et al. 2009Umetsu et al. , 2011Medezinski et al. 2010;Okabe et al. 2010). Even for less massive clusters, one can measure an average mass of a given cluster sample by adding up lensing signals for clusters in a given sample. This stacked lensing technique has been applied to the Sloan Digital Sky Survey data to obtain constraints on cosmological parameters (Mandelbaum & Seljak 2007;Johnston et al. 2007;Sheldon et al. 2009;Rozo et al. 2010), and will offer a promising mass-calibration method in future wide-field optical imaging surveys (Oguri & Takada 2011;Rozo, Wu, & Schmidt 2011). In addition, stacked lensing signals much beyond virial radii probe clustering of massive haloes, which contains complementary information on halo masses (Oguri & Takada 2011).
Weak lensing also provides a fascinating way to identify massive haloes in a way unbiased to the physical state of the baryon, by searching for peaks in the mass map reconstructed from lensing shear measurements (Schneider 1996 . This idea has been applied to real data to demonstrate that massive clusters can indeed be identified (Miyazaki et al. 2002(Miyazaki et al. , 2007Gavazzi & Soucail 2007;Schirmer et al. 2007;Hamana et al. 2009;Kubo et al. 2009). Such shear-selected cluster catalogs not only offer a unique opportunity to study the relation between mass and light in a statistical manner (e.g., Geller et al. 2010), but also may provide an alternative way to constrain cosmological parameters through number counts of peaks (e.g., Kratochvil, Haiman, & May 2010).
The interpretation of weak lensing data is usually made by comparing observed signals with analytical model predictions. For the analytic calculations of lensing properties, it is customary to adopt a density profile proposed by Navarro, Frenk, & White (1997, hereafter NFW). The NFW profile has widely been used to extract information on cluster masses from weak lensing data. However, results of such analysis can be biased if the assumption on the NFW profile is not accurate. For instance, lensing signals are determined by all matter distributions along the line-of-sight, which cause the scatter and bias in mass estimates (Hoekstra 2003;Dodelson 2004;de Putter & White 2005;Marian, Smith, & Bernstein 2010;Mandelbaum et al. 2010;Becker & Kravtsov 2011;Hoekstra et al. 2011), which can modify lensing signals particularly at large radii. Indeed, earlier work using N -body simulations has found the significant contribution of correlated matter around haloes (e.g., Mandelbaum et al. 2005;Hayashi & White 2008;Tavio et al. 2008;Cacciato et al. 2009;Hilbert & White 2010;Masaki 2011). If the true density profile deviates from the NFW profile, it can also cause a systematic bias. Understanding such bias is clearly important for attempts to use clusters as a cosmological probe.
In this paper, we investigate cluster lensing profiles in details using a large set of ray-tracing in N -body simulations. We pay particular attention to lensing profiles around virial radii where the origin of lensing signals should change from main haloes to the correlated matter around clusters. Thanks to the large number of ray-tracing realisations, we can study lensing profiles out to very large radii, typically several tens time virial radii of clusters. We consider an analytic model that better fit the simulated signals, which is then used to explore potential biases of several weak lensing studies originating from the assumption of the NFW profile.
The structure of this paper is as follows. In Section 2 we present analytic models adopted in the paper. Detailed comparisons with ray-tracing simulations are made in Section 3. Using the analytic model calibrated by the ray-tracing, we study potential biases in various lens studies in Section 4. We summarise the results in Section 5. In Appendices A and B, we provide formulae which should be useful for various calculations with the smoothly truncated NFW profile.

Main Halo
In this paper, we study the radial profiles of the convergence and tangential shear around clusters. At small scales the profiles are dominated by the signals from the dark haloes associated with the clusters (the so-called 1-halo term). The current most popular model of the dark halo density profile is the profile proposed by NFW, which is defined by the following form (hereafter the NFW profile): ρNFW(r) = ρs (r/rs)(1 + r/rs) 2 . (1) The density parameter ρs is related to the virial mass Mvir defined such that the average density within the virial radius becomes equal to the nonlinear overdensity ∆vir, which we compute using the spherical collapse model (see, e.g., Nakamura & Suto 1997), times the mean matter density of the universe. Specifically, ρs is described as where cvir is the so-called concentration parameter defined by and m nfw (cvir) defined by The concentration parameter is known to be correlated with the halo mass and redshift. When necessary, we adopt the following relation: cvir(Mvir, z) = 7.26 Mvir which was derived from N -body simulations assuming best-fit cosmological parameters in the Wilkinson Microwave Anisotropy Probe (WMAP) third year results (Macciò, Dutton, & van den Bosch 2008), with the additional redshift dependence based on the simulation result of Duffy et al. (2008). A well-known advantage of the NFW profile in lensing studies is that there are analytic expressions for the radial profiles of the deflection angle, convergence, and shear (Bartelmann 1996;Wright & Brainerd 2000).
However, the NFW profile is not well-defined in the sense that the enclosed mass diverges logarithmically. Thus Takada & Jain (2003a,b) considered lensing by the NFW profile truncated at the virial radius (hereafter the TJ profile): with Θ(x) being the Heaviside step function. The lensing properties of this profile can also be computed analytically (see Takada & Jain 2003a,b). One potential problem of the TJ profile is that the shear and convergence profiles are not differentiable at the truncation radius, which causes the divergence in the flexion profile. Baltz, Marshall, & Oguri (2009) proposed a different form of the truncation (hereafter the BMO profile): ρBMO(r) = ρs (r/rs)(1 + r/rs) 2 where rt is the truncation radius. In the paper we mainly use the following dimensionless truncation radius τv ≡ rt rvir .
While both n = 1 and 2 have been considered in Baltz et al. (2009), in this paper we consider only n = 2 for simplicity. In the case, at large radii the density profile behaves as ρBMO(r) ∝ r −7 , and therefore the enclosed mass converges quickly. The three-dimensional enclosed mass of the profile is where τ ≡ rt/rs = τvcvir and we have fixed n = 2 in equation. (7). Again, the advantage of the choice of this specific form of the truncation is that lensing properties can be computed analytically (see Baltz et al. 2009).
In equation (7), we adopted the same density parameter ρs as used in the NFW profile, with Mvir as an input parameter of this model. This virial mass Mvir, strictly speaking, is different from a true virial mass defined from the requirement of the nonlinear overdensity. However, one advantage of this choice of the normalisation is that the "virial mass" of this model, Mvir, becomes rather close to the total mass Mtot. From the definition above, we can show that these two masses are related by (again for n = 2) Typically we have Mtot/Mvir ∼ 1 − 1.3 and M (< rvir)/Mvir ∼ 0.9 in the parameter range we are interested in.

Correlated matter
At the large scale, the correlated matter around the cluster (the so-called 2-halo term) contributes to the lensing profile (e.g., Johnston et al. 2007). Using the Limber's approximation, we compute the 2-halo term contribution for a cluster with mass M and at redshift z as (e.g., Oguri & Takada 2011) where k ℓ ≡ ℓ/{(1 + z)DA(z)}, DA(z) is the angular diameter distance, Σcrit is the (physical) critical mass density for lensing, Pm(k) is the linear matter power spectrum, and b h (M ) is the halo bias. We can compute the corresponding radial profile of the tangential shear γT(θ) simply by replacing the zero-th order Bessel function J0(x) in equation (13) to the second order Bessel function J2(x). The power spectrum Pm(k) is computed using the approximation of the transfer function presented by Eisenstein & Hu (1998). We adopt a model of the halo bias b h (M ) by Tinker et al. (2010), which has been calibrated using a large set of N -body simulations. The model has been presented as a function of several different mean overdensities ∆. We adopt a model with ∆ = 800, a larger overdensity than the virial overdensity of ∆ ≈ 300, so that the details of the truncation near the virial radius do not affect the conversion of the masses very much. For a given set of Mvir and cvir, we compute the mass for ∆ = 800 assuming the NFW profile (see, e.g., Hu & Kravtsov 2003).

Simulated data
In order to study the detailed radial profiles of lensing signals around massive haloes, we employ a large set of ray-tracing simulations presented by Sato et al. (2009). The ray-tracing simulations are based on 200 × 2 realisations of N -body simulations with the box sizes of 240 (particle mass mp = 5.4×10 10 h −1 M⊙) and 480h −1 Mpc (mp = 4.4×10 11 h −1 M⊙).
The number of particles in each N -body run is 256 3 . Outputs of the small and large box simulations at different redshifts are then placed to construct a light cone of the 5 • ×5 • region. We use the standard multiple lens plane algorithm to simulate lensing by intervening matter. We trace 2048 2 light rays backward from the observer, resulting in the angular grid size of 0.15 ′ . Thus our ray-tracing simulations take account of all density fluctuations along the line-of-sight, not just the matter distributions around massive haloes. While lens planes have been constructed out to z = 3, in this paper we consider only one source redshift of zs = 0.997, which is a typical mean source redshift of weak lensing analysis. Thanks to the large number of N -body runs, 1000 independent ray-tracing realisations can finally be generated by randomly shifting the simulation boxes. The total effective area of 25000deg 2 thus allows us to compute the average radial profiles quite accurately. Interested readers are referred to Sato et al. (2009) for more details of N -body simulations and ray-tracing technique. The assumed cosmological parameters are the matter density ΩM = 0.238, baryon density Ω b = 0.042, cosmological constant ΩΛ = 0.762, spectral index ns = 0.958, the normalisation of the matter fluctuation σ8 = 0.76, and the Hubble parameter h = 0.732, which we adopt in our analytic calculations too.
In the N -body simulations we identify dark haloes using the friend-of-friend (FOF) algorithm with the linking length of b = 0.2 to derive the total mass MFOF of each halo. We compute average convergence and tangential shear profiles in each mass and redshift bin. We consider 5 mass bins centred at MFOF = 10 13.75 , 10 14 , 10 14.25 , 10 14.5 , and 10 14.75 h −1 M⊙, with the bin size of ∆(log MFOF) = 0.1, and 7 redshift bins centred at z = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, and 0.7, with the bin size of ∆z = 0.04. The number of haloes used in each bin ranges from ∼ 20 to ∼ 20000. When stacking the profiles, we carefully choose a centre of each halo by identifying a peak in the convergence map in order to avoid any off-centring effect which essentially smears out central profiles (see Oguri & Takada 2011). The average profiles are computed as a function of θ/θvir, where θvir is the virial radius computed assuming Mvir = MFOF. We derive profiles in the radius range of 10 −2 < θ/θvir < 50, with the bin size of log(θ/θvir) = 0.1.

Fitting method
We fit the average convergence profile in each bin using the models described in Sec. 2. We basically fit the average profiles in the simulations with the sum of the 1-halo and 2-halo components, κ(θ) = κ 1h (θ) + κ 2h (θ), with three different models of the 1-halo components as described in Sec. 2.2. The standard χ 2 method is employed for fitting, with the error of convergence in each radial bin derived from the scatter in the simulations. While the average profiles are derived for given ranges of MFOF, it is not clear whether MFOF coincides with the virial mass Mvir used in the analytic models. In fact, previous work using N -body simulations has found that there are systematic bias and considerable scatter between MFOF and Mvir (e.g., White 2001;Tinker et al. 2008). Therefore, we perform fitting with leaving the virial mass Mvir as a free parameter to take account of such systematic effect. Unless otherwise stated, we regard the concentration parameter cvir as a free parameter, although our main conclusions are unchanged even if we fix the concentration parameter to the value obtained from equation (5). Thus there are 2 fitting parameters (Mvir, cvir) for the NFW and TJ profiles, and 3 fitting parameters (Mvir, cvir, τv) for the BMO profile.
It has been known that very central density profiles of dark haloes in N -body simulations are not reliable because of several numerical effects such as the two-body relaxation and the finite time step size. Based on the detailed analysis of Fukushige & Makino (2001), we estimate that density profiles are reliable down to ∼ 0.08rvir for our N -body simulations in which N ∼ 10 4 particles are included in each halo analysed in the present paper. Indeed, our analysis results also indicate that convergence and tangential shear profiles near the very centre (θ/θvir ∼ 0.01) tend to be significantly smaller than our analytic model calculations (see below). Thus, in all mass and redshift bins fitting is performed in the radial bins θ/θvir 0.079 to make sure that our fitting results are not affected by numerical artifacts.

Result
First we check the best-fit values of the truncation radius τv (eq. [8]) in the BMO profile. Figure 1 show the best-fit values for different redshift and mass bins. We find the bestfit values to be ∼ 2−3, showing no strong dependence on the mass and redshift. We consider two cases, the case that the concentration parameter is fixed to the value computed in equation (5) and the case that the concentration parameter is also fitted to the data. We find that the results of both cases agree reasonably well with each other, which implies that the best-fit values of τv is not sensitive to how the concentration parameter is treated. The median values are τv = 3.0 when the concentration parameter is fixed, and τv = 2.6 when the concentration parameter is fitted to the data.
To illustrate how well the different main halo models can reproduce the average profiles in ray-tracing simulations, in Figure 2 we show the comparisons of convergence and tangential shear profiles for a representative case. As clearly shown in the Figure, the BMO profile reproduce profiles in the ray-tracing simulations quite successfully for a wide range in radii. In contrast, the NFW and TJ profiles are less successful in fitting the profile around the virial radius, where the transition between 1-halo and 2-halo terms occurs. The NFW profile tends to overpredict the convergence profile, whereas the TJ profile clearly underpredicts the profile. For comparison, we fit the convergence profile using the 1-halo term only for the NFW and TJ profile. We find that the NFW profile can fit the total average profile quite well out to θ/θvir ∼ 2, beyond which the NFW profile clearly underpredicts the profile. The situation is worse for the TJ profile for which we can see the significant discrepancy already at θ/θvir ∼ 1.
The Figure also indicates that the analytic and simulated convergence profiles agree well with each other down to very small radii of θ/θvir ∼ 0.05, whereas the tangential shear profiles show large deviations already at θ/θvir ∼ 0.1. One reason for this is that shear signals are non-local. Since the tangential shear signal at certain radius reflects all the mass distributions at smaller radii, numerical effects appear at larger radii in the tangential shear profiles than in the convergence profiles. Figure 3 show residuals of the convergence profile, i.e., the fractional difference between profiles in simulations and best-fit analytic models, averaged over all redshift and mass bins. We find that the BMO profile can fit the convergence profiles in simulations quite well (< 5%) for a wide range of radii from the core of main haloes to more than 10 times the virial radii. We find the NFW profile generally overpredicts the convergence profile by ∼ 20 − 30% at θ/θvir ∼ 2. On the other hand, the TJ profile grossly underpredicts the convergence profile at θ/θvir ∼ 1. Thus we confirm the result shown in Figure 2 that the BMO profile can best describe convergence profiles in ray-tracing simulations. Again, we check residuals for the case that we include only 1-halo term in the analytic model, and find that the NFW profile can fit the convergence profiles quite well out to θ/θvir ∼ 1, but begins to underpredicts the profile quickly beyond the radius.  Figure 2 for a representative example) averaged over all redshift and mass bins, plotted as a function of the normalised radius θ/θ vir . Specifically we define the residual as (κ sim − κ)/κ, where κ sim is the average convergence profile from ray-tracing simulations and κ is the convergence profile of the best-fit analytic model. Lines are same as Figure 2. Upper: Residuals for fitting when only 1-halo term is considered (see also top panels of Figure 2). Again, the result for the BMO is similar to the NFW result. Lower: Residuals when both 1-halo and 2-halo terms are included in the analytic model (see also bottom panels of Figure 2). The thin solid line indicates the result when the truncation radius in the BMO profile is fixed to τv = 2.6, the median value among fitting results for all redshift and mass bins (see Figure 1).

IMPACT OF INACCURATE PROFILES ON CLUSTER WEAK LENSING STUDIES
In most weak lensing studies, the NFW profile has been adopted in comparing with observed lensing signals, without including any contribution from the 2-halo term. However, the difference between assumed and true profiles can induce systematic biases in the interpretation of results. In this section, we investigate such systematic biases, assuming the analytic model calibrated by ray-tracing simulations (see Sec. 3) as a true cluster lensing profile.

Shear profile fitting
The most popular method to measure cluster masses from weak lensing data is to fit the tangential shear profile with analytic model predictions, for which the untruncated NFW profile is usually adopted as the analytic model. Here we estimate how cluster masses derived by such shear profile fitting can be biased due to the deviation of the true cluster lensing profile from the one computed from the NFW profile. In actual weak lensing analysis of observed data, we measure reduced shear profiles around clusters, which are defined by g(θ) ≡ γT(θ)/ [1 − κ(θ)]. For a given reduced shear profile, we usually derive the best-fit mass by minimizing the following χ 2 : where g(θi) is observed reduced shear profile at the radius θ = θi and gNFW(θi; Mvir, cvir) is the corresponding analytic model prediction assuming the NFW profile with no 2-halo term. We compute g(θi) assuming the BMO profile which has been shown to best reproduce lensing profiles in ray- tracing simulations (Sec. 3). Assuming the uniform source galaxy density, we adopt the error σi ∝ 1/θi. The χ 2 is computed in the range θin < θ < θout with an interval of ∆(log θ) = 0.05. We fix the innermost radius to θin = 1 ′ , and see how the best-fit mass Mvir differs from the input mass as a function of θout. In fitting, we vary both Mvir and cvir to search for the best-fit parameter values. Figure 4 shows the results, the biases as a function of θout. We find that adopting a single NFW profile tends to underestimate virial masses for θout ∼ 10 ′ − 30 ′ , a typical outermost radii adopted in the real analysis of cluster weak lensing data. The exact amount of the biases appears to be sensitive to τv. For our canonical value of τv = 2.6, which was the median value among fitting results for many redshift and mass bins in ray-tracing simulations (see Figure 1) virial masses are underestimated by ∼ 5%. On the other hand, if we adopt a slightly smaller value of τv = 2, virial masses can be underestimated by more than 10%. If we choose very large values of θout, the bias becomes smaller because of counteracting effects of the overpredciton of signals near the virial radius and the underprediction of the signals at very large radii where the 2-halo term contributions dominates. Our result suggests that we can reduce such mass estimation bias by restricting the fitting range small, θout θvir.
Recently, Becker & Kravtsov (2011) studied biases in weak lensing cluster mass estimates in details using a cosmological N -body simulations, and found that weak lens-ing masses are biased low by ≈ 5 − 10%. They adopted θout = 15 ′ , 20 ′ , 25 ′ , and found a larger amount of the bias with increasing θout. Thus our results are consistent with their results, both qualitatively and quantitatively, and moreover provide a physical explanation for their finding.
In Figure 5, we show biases in concentration parameter estimates. We find that the concentration parameter is generally overestimated, typically by ∼ 20%. Clearly this is related to the mass estimation bias studied above. In lensing analysis with the NFW profile, there is a well-known degeneracy between mass and concentration parameter, and hence the underestimate of the mass has to be compensated by the increase of the concentration parameter in order to recover lensing signals near the cluster centre.
A caveat here is that the input Mvir in the BMO profile differs from the true virial mass of the profile, which may make the interpretation of this result somewhat difficult. In fact we can avoid this problem by considering larger nonlinear overdensity ∆, such as ∆ ≈ 1600 used by Becker & Kravtsov (2011), because the enclosed masses for the NFW and BMO profiles adopting the same ρs should become more similar for larger values of ∆. We confirm that our results shown in Figures 4 and 5 similarly hold for different nonlinear overdensities.

Weak lensing mass peak
Next we discuss how the uncertainty of lensing profiles can affect the interpretation and statistics of weak lensing mass peaks. The mass peak is usually characterised by the signalto-noise ratio: where κ peak is the filtered peak amplitude computed by with Q(θ) being the filter function. The noise σnoise is related to the intrinsic ellipticity and the number density of source galaxies, for which we assume σe = 0.35 and n gal = 30 arcmin −2 . In this paper, we consider two filter functions. One is the Gaussian filter (Kaiser & Squires 1993), and the other is a filter proposed by Hennawi & Spergel (2005, hereafter HS05), where x ≡ θ/θin and xout ≡ θout/θin. We adopt θin = 1 ′ for the Gaussian filter, and θin = 0.5 ′ and θout = 10 ′ for the HS05 filter, which are typical values adopted in the literature.
We study the bias in ν assuming the true profile to be the BMO profile. Figure 6 shows how values of ν are biased by adopting the NFW profile with no 2-halo term. We find that the bias is 4% for peaks with ν ∼ 3 and smaller for Figure 6. Contours of the biases in the signal-to-noise ratio ν defined in equation (15) in the mass-redshift plane. The bias is defined by (ν NFW − ν)/ν, where ν NFW is the signal-to-noise ratio computed using the NFW profile with no 2-halo term, and ν is the true signal-to-noise ratio computed from the BMO profile with τv = 2. Note that the bias is even smaller for the canonical truncation radius of τv = 2.6. Dotted lines show contours for ν = 3, 5, 7, and 9, from bottom to top. The top panel is the result for the Gaussian filter (eq. [17]), whereas the bottom panel is for the HS05 filter (eq. [18]).
higher ν. The bias is not so large because ν is mainly determined by the lensing profiles well inside the virial radius where the difference of the profiles are rather small. Note that the result shown in Figure 6 is for the truncation radius of τv = 2, which is smaller than our canonical value of τv = 2.6. The bias is even smaller if we adopt τv = 2.6.
In some previous work, the TJ profile has been used to compute ν instead of the NFW profile (e.g., Hamana et al. 2004). In Figure 7 we show the bias for the TJ profile. We confirm that the bias has a different sign from the case for the NFW profile, with a similar size of 4% for most of the parameter range we are interested in.
Therefore, in either case the effect of the outer profile on the calculation of the peak height ν is not very large, although it can be important for high-precision analytic predictions. We note that the result presented here is an averaged effect. In practice, there is a considerable scatter on the amount of correlated matter around clusters, which can affect the statistics of weak lensing mass peaks in various ways. The detailed study of statistical properties of weak lensing mass peaks is currently underway, which will be reported in a forthcoming paper (Hamana & Oguri, in preparation).

SUMMARY AND DISCUSSIONS
We have studied detailed lensing profiles and their impact on various cluster lensing studies. First, we compare our simple analytic model predictions, which consist of three different 1-halo terms and a 2-halo term, with a large set of ray-tracing in N -body simulations, with a particular emphasis on lensing profiles at around virial radii of massive haloes. Our ray-tracing solves the full light propagation from the source plane to the observer, and thus take accounts of all the matter fluctuations along the line-of-sight. We find that the BMO profile, which is essentially a smoothly truncated NFW profile, with the truncation radius of τv ∼ 2 − 3 with additional contribution from the 2-halo term best reproduces the lensing profiles in ray-tracing simulations. In contrast, the untruncated NFW profile tends to overpredict the convergence profiles by 20 − 30% at twice the virial radii when the 2-halo term is added. The TJ (hard-truncation) profile largely underpredicts the lensing signals at around the virial radii.
Next, assuming the BMO profile with the 2-halo term as the "true" cluster lensing profile, we have investigated biases in cluster weak lens studies coming from the adoption of the NFW profile as an analytic model. We find that, if observed reduced shear profiles are fitted by a single NFW compo-nent, as has commonly been done in cluster weak lensing analysis, we can underestimate cluster masses up to 5−10%. The bias is most significant when the outer boundary for fitting is set to ∼ 10 ′ − 30 ′ depending on the mass and redshift of the cluster, or equivalently several times the virial radius. We can reduce the bias by restricting the fitting range small with the outer boundary equal or smaller than the virial radius. On the other hand, the concentration parameter tends to be biased high (up to ∼ 20% or even higher) to compensate the underestimate of the mass. These results based on the analytic model appear to be consistent with recent numerical results by Becker & Kravtsov (2011).
In the reduced shear fitting above, we have considered only intrinsic shapes of galaxies as a source of the error, i.e., ignored the contribution of the large-scale structure (cosmic shear) to the error (e.g., Hoekstra 2003;Dodelson 2004;Hoekstra et al. 2011). The error from the large-scale structure becomes increasingly important at larger radii, and thus decreases the relative weight of bins at large radii, implying that it reduces the bias in the mass and concentration parameter estimates. The exact amount of the reduction of the bias, however, depends on the source galaxy number density assumed. In real cluster weak lensing analysis, colour cuts have often been applied in order to minimise the dilution effect by cluster member galaxies, which significantly decreases source galaxy number densities used for the analysis. Because of this, the contribution of the large-scale structure to the total error budget tends to be subdominant in real data analysis (e.g., Oguri et al. 2010), but the large-scale structure should still contribute significantly at very large radii, e.g., θ 20 ′ .
We have also examined the effect of the outer lensing profile on the signal-to-noise ratio ν of weak lensing mass peaks. We have considered two different filters, and found that ν can be biased by 4% for peaks with ν 3 by adopting a single NFW component in calculating ν. Although this level of the bias is not so large, it can be important in detailed comparisons of observed mass peaks with theory in the future.
While in this paper we have focused on biases due to modeling uncertainty of the outer lensing profiles, there are other effects that could also bias the weak lensing analysis. Among others, the halo triaxiality could have the most significant impact. Indeed, the large triaxiality of cluster-scale haloes suggests that the lensing signal is a strong function of the viewing angle (e.g., Oguri et al. 2005). We expect that the effect is almost averaged out by proper statistical analysis (e.g., Corless & King 2009), but the exact amount of the residual bias due to the triaxiality should be sensitive to how the cluster sample is selected.
We note that the analytic model considered in the paper is subject to a potential improvement. For instance, the 2-halo term considered in the paper is accurate in the largescale limit. Near the cluster we may have to consider various effects such as non-linearity and stochasticity of the halo bias for more accurate theoretical prediction. In addition, for more robust separation of signals between the main halo and correlated matter, it is important to compare the three-dimensional matter distribution around clusters with projected lensing signals, which is left for future work.