Projection Effects of Large-scale Structures on Weak-lensing Peak Abundances

Photons are subject to the gravity of cosmic structures and are deflected when they propagate toward us. As a result, the observed images differ from their original ones. This phenomenon is referred to as the gravitational lensing effect. In the WL regime, the effect leads to small changes in size and shape of the images.

Theoretically, the WL effect can be described by the lensing potential ϕ. Its gradient gives rise to the deflection angle, and the second derivatives are related directly to the observational consequence of the lensing effect. Specifically, the convergence κ and the shear γ, characterizing the size and the shape changes, respectively, are given by Bartelmann & Schneider (2001):

$$\begin{eqnarray}&&\kappa =\displaystyle \frac{1}{2}{{\rm{\nabla }}}^{2}\phi ,\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{\gamma }_{1}=\displaystyle \frac{1}{2}\left(\displaystyle \frac{{\partial }^{2}\phi }{{\partial }^{2}{x}_{1}}-\displaystyle \frac{{\partial }^{2}\phi }{{\partial }^{2}{x}_{2}}\right),\quad {\gamma }_{2}=\displaystyle \frac{{\partial }^{2}\phi }{\partial {x}_{1}\partial {x}_{2}},\end{eqnarray} \tag{ 2 }$$

where ${\boldsymbol{x}}=({x}_{1},{x}_{2})$ is the two-dimensional angular vector. Under the Born approximation, the convergence κ is the projected density fluctuation weighted by the lensing kernel and given by

$$\begin{eqnarray}\kappa ({\boldsymbol{x}}) & = & \displaystyle \frac{3{H}_{0}^{2}{{\rm{\Omega }}}_{m}}{2}{\displaystyle \int }_{0}^{{\chi }_{{\rm{H}}}}d\chi ^{\prime} {\displaystyle \int }_{\chi ^{\prime} }^{{\chi }_{{\rm{H}}}}d\chi \\ & & \times \,\left[{p}_{{\rm{s}}}(\chi )\displaystyle \frac{{f}_{K}(\chi -\chi ^{\prime} ){f}_{K}(\chi ^{\prime} )}{{f}_{K}(\chi )a(\chi ^{\prime} )}\right]\delta [{f}_{K}(\chi ^{\prime} ){\boldsymbol{x}},\chi ^{\prime} ],\end{eqnarray} \tag{ 3 }$$

where χ is the comoving radial distance, ${\chi }_{{\rm{H}}}=\chi (z=\infty )$, a is the cosmic scale factor, f_K is the comoving angular diameter distance, δ is the 3D density fluctuation, and p_s is the source distribution function. The cosmological parameter H₀ is the Hubble constant.

We define the lensing window function as

$$\begin{eqnarray}&&w(\chi ^{\prime} )={\int }_{\chi ^{\prime} }^{{\chi }_{{\rm{H}}}}d\chi {p}_{{\rm{s}}}(\chi )\displaystyle \frac{{f}_{K}(\chi -\chi ^{\prime} )}{{f}_{K}(\chi )};\end{eqnarray} \tag{ 4 }$$

the corresponding power spectrum of κ is then

$$\begin{eqnarray}&&{C}_{{\ell }}=\displaystyle \frac{9{H}_{0}^{4}{{\rm{\Omega }}}_{m}^{2}}{4}{\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi ^{\prime} \displaystyle \frac{{w}^{2}(\chi ^{\prime} )}{{a}^{2}(\chi ^{\prime} )}{P}_{\delta }\left(\displaystyle \frac{{\ell }}{{f}_{K}(\chi ^{\prime} )},\chi ^{\prime} \right),\end{eqnarray} \tag{ 5 }$$

where P_δ is the power spectrum of 3D matter density perturbations.

Observationally, the brightness quadrupole moment tensor of a source galaxy can be measured, and from that, the source ellipticity can be extracted. The WL effect on observed images can then be described by the Jacobian matrix of the lensing equation, which reads

$$\begin{eqnarray}{\boldsymbol{A}} & = & \left(\begin{array}{cc}1-\kappa -{\gamma }_{1} & -{\gamma }_{2}\\ -{\gamma }_{2} & 1-\kappa +{\gamma }_{1}\end{array}\right)\\ & = & (1-\kappa )\left(\begin{array}{cc}1-{g}_{1} & -{g}_{2}\\ -{g}_{2} & 1+{g}_{1}\end{array}\right),\end{eqnarray} \tag{ 6 }$$

where g_i = γ_i/(1 − κ) is the reduced shear. Considering the intrinsic ellipticity of a source galaxy, the observed ellipticity written in the complex form (Seitz & Schneider 1997) is

$$\begin{eqnarray}{\boldsymbol{\epsilon }}=\left\{\begin{array}{ll}\displaystyle \frac{{{\boldsymbol{\epsilon }}}_{{\rm{s}}}+{\boldsymbol{g}}}{1+{{\boldsymbol{g}}}^{{\boldsymbol{* }}}{{\boldsymbol{\epsilon }}}_{{\rm{s}}}}; & \mathrm{for}\ | {\boldsymbol{g}}| \leqslant 1\\ \displaystyle \frac{1+{\boldsymbol{g}}{{\boldsymbol{\epsilon }}}_{{\rm{s}}}^{* }}{{{\boldsymbol{\epsilon }}}_{{\rm{s}}}^{* }+{{\boldsymbol{g}}}^{{\boldsymbol{* }}}}, & \mathrm{for}\ | {\boldsymbol{g}}| \gt 1\end{array}\right.,\end{eqnarray} \tag{ 7 }$$

where ${\boldsymbol{\epsilon }}$ and ${{\boldsymbol{\epsilon }}}_{{\rm{s}}}$ are the observed and intrinsic ellipticities of a source, respectively. The symbol * represents the complex conjugate operation. It is seen that the observed ellipticity is closely related to the WL shear. For κ ≪ 1, ${\boldsymbol{\epsilon }}\approx {{\boldsymbol{\epsilon }}}_{{\rm{s}}}+{\boldsymbol{\gamma }}$. Without considering the intrinsic alignments, the correlation analyses of ${\boldsymbol{\epsilon }}$ can thus give rise directly to an estimate of the WL shear correlation (e.g., Fu et al. 2008; Kilbinger et al. 2013; Fu et al. 2014; Becker et al. 2016; Jee et al. 2016; Hildebrandt et al. 2017).

Alternatively, because of the physical relation between the shear and the convergence as seen in Equation (1) and Equation (2), it is possible to reconstruct the convergence κ field from the observed ellipticities after a suitable smoothing (e.g., Kaiser 1993; Bartelmann 1995; Seitz & Schneider 1995; Squires & Kaiser 1996; Jullo et al. 2014). We note that κ is the weighted projection of the density fluctuations, and thus the structures can be better seen in the κ field than in the shear field. Compared to previous observations targeting individual clusters, the current survey cameras have a large field of view, typically ∼1° × 1°, and thus the boundary effects on the convergence reconstruction can be in good control. Cosmological studies using the reconstructed convergence fields have been carried out for different WL surveys (e.g., Shan et al. 2012; Van Waerbeke et al. 2013; Shan et al. 2014; J. Liu et al. 2015; X. K. Liu et al. 2015, 2016).

In this paper, we concentrate on WL peaks identified in convergence fields, and in particular we study the LSS projection effects on high-peak abundances.

Physically, the WL convergence field reflects the projected density distribution weighted by the lensing kernel. Peaks there should correspond to the projected mass concentrations. Studies show that for a high peak, its signal is primarily contributed by a single massive halo located in the line of sight (e.g., Yang et al. 2011; X. K. Liu et al. 2014; J. Liu & Haiman 2016).

In Figure 1, we zoom in to two high peaks from our ray-tracing simulations to be described in detail in Section 3. The horizontal axes are for the redshift of the lens planes, and the vertical axes show the relative contribution of each lens plane to the final peak signal ${{ \mathcal K }}_{\mathrm{peak}}$. The upper panels show the noiseless cases from ray-tracing simulations, and the lower panels are for the cases adding the shape noise from intrinsic ellipticities of source galaxies. The left ones are for a peak with the source galaxies at z_s = 0.71 and n_g = 10 arcmin⁻². The insert in the lower panel shows the zoom-in local image of the noisy peak. Here we apply a Gaussian smoothing with the window function

$$\begin{eqnarray}&&{W}_{{\theta }_{G}}({\boldsymbol{\theta }})=\displaystyle \frac{1}{\pi {\theta }_{G}^{2}}\exp \left(-\displaystyle \frac{| {\boldsymbol{\theta }}{| }^{2}}{{\theta }_{G}^{2}}\right).\end{eqnarray} \tag{ 8 }$$

We take θ_G = 2.0 arcmin in this paper. For this peak, ${{ \mathcal K }}_{\mathrm{peak}}\approx 0.0835$ for the noiseless case (upper), and ${{ \mathcal K }}_{\mathrm{peak}}\approx 0.114$ for the noisy case (lower). It is seen clearly that the lens plane at z ≈ 0.18 contributes dominantly to the peak signal. Further examination finds that a massive halo is located there. In the noiseless case, the LSS effect from other lens planes only accounts for less than 10% (negative) of the peak signal, as indicated by the black bar. In the noisy case, the dominant halo contributes ∼80% of the peak signal. The shape noise as indicated by the blue bar contributes ∼25%, and LSS from other planes contributes ∼−5%. The right panels are for the case with z_s = 2.05 and n_g = 20 arcmin⁻². Here the dominant halo is at z ≈ 0.7. In this case, the LSS effect increases to ∼−15% because of the increase in z_s. The shape noise contribution is ∼10%.

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These two examples show that for high peaks, the halo approach to modeling the WL peak abundances is a physically viable approach. The shape noise and the LSS projection effect can be regarded as perturbations to the signal from the dominant halo. For relatively shallow surveys, the shape noise is much larger than the LSS effect, and we expect the good performance of the F10 model, which takes into account the shape noise but without including the LSS effect. For deep surveys, however, the two perturbations are comparable, and the stochastic LSS effect cannot be neglected.

As a comparison, we show in Figure 2 a low peak with z_s = 2.05 and n_g = 20 arcmin⁻². The peak signal is ${{ \mathcal K }}_{\mathrm{peak}}\approx 0.0422$. It is seen that the signals are from the cumulative effect of the line-of-sight mass distribution, and no dominant halo contribution can be found. For these peaks, a modeling methodology other than the halo approach is needed.

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In this paper, we focus on high peaks, and we present our model for high-peak abundances including LSS projection effects. Similar to F10, we assume that the signal of a true high peak is mainly from a single massive halo. The shape noise contributes a random component to the reconstructed convergence field. For stochastic LSS, from Figure 1, we see that they add onto the final peak signal in a zigzag way, leading to a statistically random perturbation. Thus it is appropriate to also model the LSS projection effect as a random field. Therefore, in our model, the convergence field in a halo region can be written as

$$\begin{eqnarray}&&{ \mathcal K }={{ \mathcal K }}_{{\rm{H}}}+{{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N },\end{eqnarray} \tag{ 9 }$$

where ${{ \mathcal K }}_{{\rm{H}}}$ is the contribution from the halo convolved with a window function corresponding to the smoothing operation made in the convergence reconstruction. It is regarded as a known quantity given the density profile of the halo. The smoothed shape noise field ${ \mathcal N }$ is assumed to be Gaussian due to the central limit theorem (e.g., van Waerbeke 2000). We note that, in general, the overall convergence field from simulations shows non-Gaussianity due to the nonlinearity of structure formation. In our consideration here, however, ${{ \mathcal K }}_{\mathrm{LSS}}$ is the stochastic LSS contribution excluding the massive halo part, which is already explicitly split out as ${{ \mathcal K }}_{{\rm{H}}}$. It can be more Gaussian than the overall convergence field. Also, ${{ \mathcal K }}_{\mathrm{LSS}}$ is from small additive contributions from different lens planes in the high-peak case, and $| {{ \mathcal K }}_{\mathrm{LSS}}| \ll | {{ \mathcal K }}_{{\rm{H}}}|$. Therefore, as an approximation, we assume that ${{ \mathcal K }}_{\mathrm{LSS}}$ is also a Gaussian random field. Its validity will be extensively tested in Section 3 by comparing the model predictions for high-peak counts with the results from simulations.

A similar consideration was mentioned in Shirasaki et al. (2015) but without really calculating the LSS contribution. Also, they only concentrated on the influence of the random field on the central peak signals of halos. In our modeling here, we take into account specifically the stochastic LSS and calculate the total peak counts, including both the central ones from massive halos and the peaks from the random field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$ inside halo regions as well as outside halo regions. In other words, to apply our model for cosmological studies, we can simply use all of the high peaks identified from convergence maps without the need to go through additional analyses to locate true halo-associated peaks.

With the Gaussian assumptions for the two random fields, the total field ${ \mathcal K }$ in Equation (9) is also a Gaussian random field. More specifically, it is the Gaussian random field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$ modulated by the known halo contribution ${{ \mathcal K }}_{{\rm{H}}}$. Following the same procedures shown in F10, we can then calculate the number of peaks in a halo region. Two features need to be addressed. First, the original peak signal from the halo is affected by the existence of the two random fields, which not only generates scatters but also leads to a positive shift for the signal (F10; Shirasaki et al. 2015). Second, the height distribution of peaks generated purely by the stochastic part ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$ is modulated by the halo convergence profile ${{ \mathcal K }}_{{\rm{H}}}$. With the halo mass function, we can then compute statistically the number of peaks per unit area in regions occupied by massive halos. For peaks outside the halo regions, we can calculate the peak abundances simply from the Gaussian field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$.

In formulae, for high-peak abundances, we have (F10)

$$\begin{eqnarray}&&{n}_{\mathrm{peak}}(\nu )d\nu ={n}_{\mathrm{peak}}^{c}(\nu )d\nu +{n}_{\mathrm{peak}}^{n}(\nu )d\nu ,\end{eqnarray} \tag{ 10 }$$

where $\nu ={ \mathcal K }/{\sigma }_{0}$ with ${\sigma }_{0}^{2}={\sigma }_{\mathrm{LSS},0}^{2}+{\sigma }_{{\rm{N}},0}^{2}$ being the total variance of the field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$, and ${n}_{\mathrm{peak}}^{c}(\nu )$ and ${n}_{\mathrm{peak}}^{n}(\nu )$ are, respectively, the number density of peaks per unit ν centered at ν inside and outside halo regions.

We emphasize that our model is applicable for high peaks in which the signals of true peaks are dominated by single massive halos (see Figure 1). Thus we consider halos with mass M ≥ M_* as major contributors to the halo regions. Simulation analyses show that M_* ∼ 10¹⁴ h⁻¹ M_⊙ is a proper choice (Wei et al. 2018). The rest from smaller halos and the correlations between halos are included in the stochastic LSS part. Then, for ${n}_{\mathrm{peak}}^{c}(\nu )$, we have

$$\begin{eqnarray}{n}_{\mathrm{peak}}^{c}(\nu ) & = & \displaystyle \int {dz}\ \displaystyle \frac{{dV}(z)}{{dzd}{\rm{\Omega }}}{\displaystyle \int }_{{M}_{* }}^{\infty }{dM}\ n(M,z)\\ & & \times \,{\displaystyle \int }_{0}^{{\theta }_{\mathrm{vir}}}d\theta (2\pi \theta ){\hat{n}}_{\mathrm{peak}}^{c}(\nu ,M,z,\theta ),\end{eqnarray} \tag{ 11 }$$

where n(M, z) is the comoving halo mass function, ${\hat{n}}_{\mathrm{peak}}^{c}(\nu ,M,z,\theta )$ is the number density of peaks at θ, and θ_vir = R_vir(M, z)/D_A(z) is the angular virial radius of a halo with mass M at redshift z. Here R_vir and D_A are the virial radius of the halo and the angular diameter distance to the halo, respectively.

Considering the Gaussian random field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$ modulated by the halo term ${{ \mathcal K }}_{{\rm{H}}}$, following the calculations in F10, we have

$$\begin{eqnarray}&&{\hat{n}}_{\mathrm{peak}}^{c}(\nu ,M,z,\theta )=\exp \left[-\displaystyle \frac{{({{ \mathcal K }}_{{\rm{H}}}^{1})}^{2}+{({{ \mathcal K }}_{{\rm{H}}}^{2})}^{2}}{{\sigma }_{1}^{2}}\right]\\ &&\quad \times \,\left[\displaystyle \frac{1}{2\pi {\theta }_{* }^{2}}\displaystyle \frac{1}{{(2\pi )}^{1/2}}\right]\exp \left[-\displaystyle \frac{1}{2}{\left(\nu -\displaystyle \frac{{{ \mathcal K }}_{{\rm{H}}}}{{\sigma }_{0}}\right)}^{2}\right]\\ &&\quad \times \,{\displaystyle \int }_{0}^{\infty }{dx}\left\{\displaystyle \frac{1}{{[2\pi (1-{\gamma }^{2})]}^{1/2}}\right.\\ &&\quad \times \,\exp \left[-\displaystyle \frac{{[x+({{ \mathcal K }}_{{\rm{H}}}^{11}+{{ \mathcal K }}_{{\rm{H}}}^{22})/{\sigma }_{2}-\gamma (\nu -{{ \mathcal K }}_{{\rm{H}}}/{\sigma }_{0})]}^{2}}{2(1-{\gamma }^{2})}\right]\\ &&\quad \times \,F(x)\}\end{eqnarray} \tag{ 12 }$$

and

$$\begin{eqnarray}F(x) & = & \exp \left[-\displaystyle \frac{{({{ \mathcal K }}_{{\rm{H}}}^{11}-{{ \mathcal K }}_{{\rm{H}}}^{22})}^{2}}{{\sigma }_{2}^{2}}\right]\\ & & \times \,{\displaystyle \int }_{0}^{1/2}{de}8({x}^{2}e){x}^{2}(1-4{e}^{2})\exp (-4{x}^{2}{e}^{2})\\ & & \times \,{\displaystyle \int }_{0}^{\pi }\displaystyle \frac{d\psi }{\pi }\exp \left[-4{xe}\cos (2\psi )\displaystyle \frac{({{ \mathcal K }}_{{\rm{H}}}^{11}-{{ \mathcal K }}_{{\rm{H}}}^{22})}{{\sigma }_{2}}\right].\end{eqnarray} \tag{ 13 }$$

Here, ${\theta }_{* }^{2}=2{\sigma }_{1}^{2}/{\sigma }_{2}^{2}$, $\gamma ={\sigma }_{1}^{2}/({\sigma }_{0}{\sigma }_{2})$, ${{ \mathcal K }}_{{\rm{H}}}^{i}={\partial }_{i}{{ \mathcal K }}_{{\rm{H}}}$, and ${{ \mathcal K }}_{{\rm{H}}}^{{ij}}\,={\partial }_{{ij}}{{ \mathcal K }}_{{\rm{H}}}$. Different from that in F10, here the quantities ${\sigma }_{i}^{2}$ (i = 0, 1, 2) are, respectively, the moments of the total random field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$ and its first and second derivatives. Specifically, ${\sigma }_{i}^{2}={\sigma }_{\mathrm{LSS},i}^{2}+{\sigma }_{{\rm{N}},i}^{2}$.

For the number density of peaks contributed by those outside halo regions, ${n}_{\mathrm{peak}}^{n}(\nu )$, we have

$$\begin{eqnarray}&&{n}_{\mathrm{peak}}^{n}(\nu )=\displaystyle \frac{1}{d{\rm{\Omega }}}\\ &&\times \,\left\{{n}_{\mathrm{ran}}(\nu )\left[d{\rm{\Omega }}-\displaystyle \int {dz}\ \displaystyle \frac{{dV}(z)}{{dz}}{\displaystyle \int }_{{M}_{* }}^{\infty }{dM}\ n(M,z)(\pi {\theta }_{\mathrm{vir}}^{2})\right]\right\},\end{eqnarray} \tag{ 14 }$$

where n_ran(ν) is the number density of peaks from the random field ${{ \mathcal K }}_{\mathrm{LSS}}+{ \mathcal N }$ without halo modulations, and it can be calculated from Equation (12) by setting the halo-related quantities to zero.

From the above, we see that the stochastic LSS effects occur specifically in quantities of ${\sigma }_{i}^{2}={\sigma }_{\mathrm{LSS},i}^{2}+{\sigma }_{{\rm{N}},i}^{2}$. For the shape noise part, we have (e.g., van Waerbeke 2000)

$$\begin{eqnarray}&&{\sigma }_{{\rm{N}},i}^{2}={\int }_{0}^{\infty }\displaystyle \frac{{\ell }d{\ell }}{2\pi }{{\ell }}^{2i}{C}_{{\ell }}^{{\rm{N}}},\end{eqnarray} \tag{ 15 }$$

where ${C}_{{\ell }}^{{\rm{N}}}$ is a power spectrum of the smoothed noise field ${ \mathcal N }$. For the Gaussian smoothing, we have

$$\begin{eqnarray}&&{\sigma }_{{\rm{N}},0}^{2}=\displaystyle \frac{{\sigma }_{\epsilon }^{2}}{4\pi {n}_{g}{\theta }_{G}^{2}},\end{eqnarray} \tag{ 16 }$$

where σ is the rms amplitude of the intrinsic ellipticities. We further have ${\sigma }_{{\rm{N}},0}\,:{\sigma }_{{\rm{N}},1}\,:{\sigma }_{{\rm{N}},2}=1\,:\sqrt{2}/{\theta }_{G}\,:2\sqrt{2}/{\theta }_{G}^{2}$.

For σ_{LSS, i}, they are physical quantities and need to be computed in a cosmology-dependent way. In other words, σ_{LSS, i} also contributes to the cosmological information embedded in WL peak counts. Given a power spectrum for the LSS convergence field, ${C}_{{\ell }}^{\mathrm{LSS}}$, we have

$$\begin{eqnarray}&&{\sigma }_{\mathrm{LSS},i}^{2}={\int }_{0}^{\infty }\displaystyle \frac{{\ell }d{\ell }}{2\pi }{{\ell }}^{2i}{C}_{{\ell }}^{\mathrm{LSS}}.\end{eqnarray} \tag{ 17 }$$

To calculate ${C}_{{\ell }}^{\mathrm{LSS}}$, we adopt the following approach. From Equation (5), the WL power spectrum C_ℓ can be obtained from the integration of the weighted 3D nonlinear power spectrum P_δ. For P_δ, it can be computed using the simulation-calibrated halo model (Takahashi et al. 2012) and has been included in different numerical packages, such as CAMB (Lewis et al. 2000). In the language of the halo model, the overall P_δ consists of contributions from the one-halo term of all halos and the two-halo term considering the correlations between halos. In our model here, halos with M ≥ M_* have been separated out as ${{ \mathcal K }}_{{\rm{H}}}$. Thus, to compute the leftover stochastic LSS effects, we subtract the one-halo term from halos with M ≥ M_* from the overall power spectrum:

$$\begin{eqnarray}&&{P}_{\delta }^{\mathrm{LSS}}[k,\chi (z)]={P}_{\delta }[k,\chi (z)]-{{P}_{\delta }^{1{\rm{H}}}| }_{M\geqslant {M}_{* }}[k,\chi (z)].\end{eqnarray} \tag{ 18 }$$

It is seen that ${P}_{\delta }^{\mathrm{LSS}}$ contains the one-halo term from halos with M < M_* and the two-halo term between all the halos including the ones with M ≥ M_*. For the one-halo term ${{P}_{\delta }^{1{\rm{H}}}| }_{M\geqslant {M}_{* }}$, we have (e.g., Cooray & Sheth 2002)

$$\begin{eqnarray}&&{{P}_{\delta }^{1{\rm{H}}}| }_{M\geqslant {M}_{* }}[k,\chi (z)]=\displaystyle \frac{4\pi }{{\bar{\rho }}^{2}}{\int }_{{M}_{* }}^{\infty }{dM}\,{M}^{2}{W}^{2}(k,M)n(M,z),\end{eqnarray} \tag{ 19 }$$

where $\bar{\rho }$ is the mean matter density of the universe, and W(k, M) is the Hankel transformation of the spherically symmetric halo density profile ρ(r, M), given by

$$\begin{eqnarray}&&W(k,M)=\displaystyle \frac{1}{M}{\int }_{0}^{{R}_{\mathrm{vir}}}{dr}\,\displaystyle \frac{\sin ({kr})}{{kr}}4\pi {r}^{2}\rho (r,M).\end{eqnarray} \tag{ 20 }$$

From ${P}_{\delta }^{\mathrm{LSS}}$, we can obtain ${C}_{{\ell }}^{\mathrm{LSS}}$ by

$$\begin{eqnarray}&&{C}_{{\ell }}^{\mathrm{LSS}}=\displaystyle \frac{9{H}_{0}^{4}{{\rm{\Omega }}}_{m}^{2}}{4}{\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi ^{\prime} \displaystyle \frac{{w}^{2}(\chi ^{\prime} )}{{a}^{2}(\chi ^{\prime} )}{P}_{\delta }^{\mathrm{LSS}}\left(\displaystyle \frac{{\ell }}{{f}_{K}(\chi ^{\prime} )},\chi ^{\prime} \right),\end{eqnarray} \tag{ 21 }$$

and subsequently ${\sigma }_{\mathrm{LSS},i}^{2}$ by Equation (17).

For the calculations of the one-halo term in Equation (19), we take the Navarro–Frenk–White (NFW) halo density profile given by Navarro et al. (1996, 1997):

$$\begin{eqnarray}&&\rho (r)=\displaystyle \frac{{\rho }_{{\rm{s}}}}{(r/{r}_{{\rm{s}}}){(1+r/{r}_{{\rm{s}}})}^{2}},\end{eqnarray} \tag{ 22 }$$

where ρ_s and r_s are the characteristic density and scale of a halo, respectively. The scale r_s reflects the compactness of a halo and is often given through the concentration parameter c_th = R_th/r_s, with R_th being the radius inside which the average density of a halo is Δ_th times the cosmic density. Here we adopt the virial radius R_vir and use the concentration–mass relation from Duffy et al. (2008) with

$$\begin{eqnarray}&&{c}_{\mathrm{vir}}(M,z)=5.72{\left(\displaystyle \frac{M}{{10}^{14}{h}^{-1}{M}_{\odot }}\right)}^{-0.081}{(1+z)}^{-0.71}.\end{eqnarray} \tag{ 23 }$$

For the halo mass function n(M, z), we use the one given in Watson et al. (2013), an empirical fitting formula derived from N-body simulations.

In Figure 3, we show ${C}_{{\ell }}^{\mathrm{LSS}}$ together with the overall C_ℓ and the massive one-halo term ${C}_{{\ell }}^{1{\rm{H}}}$ under different source galaxy distributions p(z), for which we adopt the following form:

$$\begin{eqnarray}&&p(z)\propto {z}^{2}\exp \left[-{\left(1.414\displaystyle \frac{z}{{z}_{\mathrm{med}}}\right)}^{1.5}\right],\end{eqnarray} \tag{ 24 }$$

where z_med is the median redshift. In the plots, we also show the power spectra of aperture mass under the U filtering (blue), to be discussed in Section 3.4. Here we focus on the Gaussian filter case (red). We see that, on large scales, ${C}_{{\ell }}^{\mathrm{LSS}}$ is very close to C_ℓ. On small scales with ℓ ∼ 2000, the LSS random field ${C}_{{\ell }}^{\mathrm{LSS}}$ is smaller than the overall convergence field C_ℓ due to the exclusion of the one-halo term from halos with M ≥ 10^14.0 h⁻¹ M_⊙.

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With Equations (10)–(21), the number density of high peaks taking into account the LSS projection effects can be computed. In Figure 4, we show the results from this model (solid lines) and the ones from F10 without the LSS (red lines). For the model with LSS effects, we also show the contributions of peaks inside (${n}_{\mathrm{peak}}^{c}$, dashed-dotted line) and outside (${n}_{\mathrm{peak}}^{n}$, dashed) halo regions. It is seen that in the considered cases, peaks with ν₀ ≳ 4 are dominantly from halo regions. For higher z_med, such domination shifts a little more toward higher ν₀.

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We note that in our model calculation, we directly obtain peak counts at different $\nu ={ \mathcal K }/{\sigma }_{0}$. On the other hand, observationally, we can only estimate the shape noise part σ_N,0 by randomly rotating the observed galaxies. Thus to be consistent with observational analyses, we first make a binning in terms of ${\nu }_{0}={ \mathcal K }/{\sigma }_{{\rm{N}},0}$ and then convert it to the binning in ν using the ratio of σ_N,0/σ₀ for model calculations. The results shown are the peak counts versus ν₀. The corresponding ${ \mathcal K }$ values are also listed in the upper horizontal axis. It is seen clearly that with the increase in the median redshift of source galaxies, the LSS projection effects become increasingly important.

In the next section, we will compare our theoretical results with those from ray-tracing simulations to validate the model performance.

We carry out ray-tracing simulations up to z = 3.0 based on large sets of N-body simulations. The simulation setting is the same as that in X. K. Liu et al. (2015), but with the number of simulations doubled. The fiducial cosmology is the flat ΛCDM model with the parameters of Ω_m, dark energy density Ω_Λ, baryonic matter density Ω_b, Hubble constant h, the power index of initial matter density perturbation power spectrum n_s, and σ₈ set to be (Ω_m, Ω_Λ, Ω_b, h, n_s, σ₈) = (0.28, 0.72, 0.046, 0.7, 0.96, 0.82).

For each set of ray-tracing calculations, we use 12 independent N-body simulation boxes to fill up to the region of a comoving distance ∼4.5 h⁻¹ Gpc to z = 3.0, as illustrated in Figure 5. Among them, eight small boxes each with the size of 320 h⁻¹ Mpc are padded between z = 0.0 and z = 1.0. In the redshift range of 1.0 < z ≤ 3.0, we pad four boxes of size 600 h⁻¹ Mpc. The number of particles of N-body simulations for both small and large boxes is 640³, and the corresponding mass resolution is ∼9.7 × 10⁹ h⁻¹ M_⊙ and ∼6.4 ×10¹⁰ h⁻¹ M_⊙, respectively. For each of the boxes, we start at z = 50 and generate the initial conditions using 2LPTic⁶ based on the initial power spectrum from CAMB⁷ (Lewis et al. 2000). The simulations are run by GADGET-2⁸ (Springel 2005) with the force softening length being ∼20 h⁻¹ kpc.

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In the multiplane ray-tracing calculations, up to z = 3, we use 59 lens planes with the corresponding redshifts z_l being listed in Table 1. We run the ray-tracing WL simulations using the same code described in X. K. Liu et al. (2014) in which we deal with the crossing-boundary problem of halos following the procedures used in Hilbert et al. (2009). We then generate 4 × (35 × 35) convergence and shear maps at each lens plane, denoted as κ(z_lens) and γ_i(z_lens), respectively, from a set of 12 N-body simulations. Each of the 35 × 35 maps is pixelized into 1024 × 1024 grids with a pixel size of ∼0.205 arcmin. We perform in total 24 sets of N-body simulations, which give rise to 24 × 4 × (35 × 35) =1176 deg² of κ(z_lens) and γ_i(z_lens). From them, we construct the final κ (or γ) maps corresponding to different source galaxy distributions as follows:

$$\begin{eqnarray}&&{\kappa }_{\mathrm{mock}}({\boldsymbol{\theta }})=\displaystyle \sum _{\mathrm{lens}}p({z}_{\mathrm{lens}})\kappa ({\boldsymbol{\theta }};{z}_{\mathrm{lens}})({z}_{\mathrm{lens}+1}-{z}_{\mathrm{lens}}),\end{eqnarray} \tag{ 25 }$$

where p(z) is the normalized source galaxy redshift distribution given in Equation (24). For a given p(z), we obtain 24 × 4 = 96 maps, each with the size of 35 × 35.

Because we aim to test our WL high-peak model, we concentrate on convergence maps directly here. We include the shape noise by adding a Gaussian noise field to the pixels of each of 24 × 4 = 96 convergence maps ${\kappa }_{\mathrm{mock}}({\boldsymbol{\theta }})$ with the variance given by

$$\begin{eqnarray}&&{\sigma }_{\mathrm{pix}}^{2}=\displaystyle \frac{{\sigma }_{\epsilon }^{2}}{2{n}_{g}{\theta }_{\mathrm{pix}}^{2}},\end{eqnarray} \tag{ 26 }$$

where we take σ = 0.4 and the pixel size of maps θ_pix = 0.205 arcmin. We then apply a Gaussian smoothing given by Equation (8) with θ_G = 2.0 arcmin to obtain the final smoothed noisy convergence maps for peak analyses.

To analyze the LSS effects and test our model performance, we consider different survey parameters, including the median redshift z_med, the number density n_g of the source galaxies, and the survey area S. These are listed in Table 2.

From our simulations, for each set of (z_med, n_g), we generate 96 noiseless convergence maps each with the size of 35 × 35. For each map, we then add a Gaussian-shape noise field and apply smoothing as described above. To suppress the fluctuations caused by a particular realization of the noise field, we perform noise-adding 20 times for each map with different seeds. Therefore, we have 20 × 96 maps with the shape noise included for each set of (z_med, n_g).

We first compare peak counts between model predictions and the simulation results. We identify a peak in a pixelized convergence map from simulations if its convergence value is higher than those of its eight neighboring pixels. Because Fourier transformations are involved in ray-tracing calculations and in the smoothing operations, there can be boundary effects in each of the 35 × 35 convergence maps. To avoid such a problem, in our peak analyses, we exclude the outermost 20 pixels along each side of the map. The leftover area is ∼11.31 deg² for each map, and the total is ∼96 × 11.31 ≈ 1086 deg² for each set of noise field realizations.

For our theoretical model calculations, the quantity M_* corresponds to the lower mass limit of halos above which the halos dominate the WL high-peak signals. We have performed χ² tests with respect to the simulated peak counts to find suitable M_*. We note that for z_med = 0.7 and n_g = 10 arcmin⁻², the shape noise is much larger than that of the LSS effects, and the model of F10 works equally well. In that case, using F10, M_* = 10^13.9 h⁻¹ M_⊙ gives results that are in good agreement with those from simulations. In our current model with the LSS effects, for all of the cases including the one with z_med = 0.7 and n_g = 10 arcmin⁻², M_* =10^14.0 h⁻¹ M_⊙ is a proper choice. We comment that, physically, the suitable choice of M_* depends on the halo mass function used in the model calculations. The specific value of M_* may also have a mild cosmology dependence, which may need to be taken into account in the future for very high precision studies. In this paper, we do not include this subtle effect.

The peak count comparison results are shown in Figures 6 and 7 corresponding to the survey conditions listed in Table 2. The green symbols are the results averaged over the corresponding 20 × 96 maps and then scaled to the considered survey area. The error bars are the corresponding Poisson errors. The blue lines are the results from our model including the LSS effect, and the red lines are from F10 without the LSS effect. Again, the results shown are the peak counts versus ν₀ defined by the shape noise σ_N,0. The corresponding κ values are indicated in the upper horizontal axes. In the bottom part of each panel, we show the fractional differences of the two models with respect to the simulation results.

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Figure 6 shows the results for S10small with z_med = 0.7, n_g = 10 arcmin⁻², and S = 150 deg², similar to the current accomplished WL surveys. In this case, σ_LSS,0 = 0.0057 and σ_N,0 = 0.0178 for θ_G = 2.0 arcmin. Thus the contribution from LSS is much smaller than that from the shape noise, and its effect on WL peak counts is rather weak considering the relatively large error bars. This is evident in the lower part of the panel. Both the blue and red lines agree with the simulation results very well for high peaks with ν₀ ≥ 4 with the fractional differences less than 10%.

In the upper left panel of Figure 7, we show the results of S10. In this case, the number density and the redshift distribution of source galaxies are the same as those in Figure 6, but with a larger survey area with S = 1086 deg². Thus the statistical errors of WL peak counts are smaller by ∼2.7 times than that of S10small. We see again that both models work well, and the model including the LSS effect (blue) gives better results for the left two bins.

In the upper right panel of Figure 7, the results of S20 with z_med = 0.7, n_g = 20 arcmin⁻², and S = 1086 deg² are presented. In this case, the LSS effect is the same as that of S10, but the shape noise is lower with σ_N,0 = 0.0126. Thus the relative contribution of the LSS effect should be stronger than for S10. We see that for ν₀ > 5, although both the blue line and red line agree with the simulation results within 10%, the red line is systematically lower, showing that the LSS effect starts to be important. For ν₀ < 5, the red line deviates significantly from the simulation results, but our current model including the LSS effect can give excellent predictions out to ν₀ ∼ 4.

The results for M20 with z_med = 1.0, n_g = 20 arcmin⁻², and S = 1086 deg² are shown in the lower left panel of Figure 7. Here the LSS contribution increases to σ_LSS,0 = 0.0082. Comparing to the upper right panel, we see that the model prediction without the LSS effect (red line) significantly underestimates the peak counts over the whole considered range. Taking into account the LSS effect, our improved model works very well to ${ \mathcal K }\gtrsim 0.06$, corresponding to ν₀ ≳ 4.5.

With even higher z_med, the LSS projection effect gets larger. The lower right panel of Figure 7 shows the results of D20 with z_med = 1.4, n_g = 20 arcmin⁻², and S = 1086 deg². In this case, σ_LSS,0 = 0.0108 is comparable to that from the shape noise with σ_N,0 = 0.0126. Without including the LSS projection effect, the underestimation is at the level of 30%, much larger than the statistical errors. While including the LSS effect, the model predictions (blue lines) are in excellent agreement with the simulation results to ${ \mathcal K }\gtrsim 0.063$, or ν₀ ≳ 5.

Note that for the three cases with n_g = 20 arcmin⁻², the WL lensing signal from a halo increases with the increase of z_med. Thus we see a somewhat increased lower limit of ν₀, above which our high-peak halo model applies from z_med = 0.7 to z_med = 1.4.

To demonstrate explicitly the LSS effect on the cosmological constraints derived from WL high-peak counts and how our improved model performs, here we run Markov chain Monte Carlo (MCMC) fitting using WL mock data.

For S10, S20, M20, and D20 in Table 2, we generate the WL peak count mock data by averaging over the 20 × 96 maps. For S10small, we scale the peak counts obtained for S10 to S = 150 deg². The central data points in different bins {N_i} for different cases are the same as those shown in Figures 6 and 7. Note that for the upper end of the peaks, we only use bins with N_i ≳ 10.

We employ the χ² fitting to constrain cosmological parameters from WL mock data. The χ² is defined as

$$\begin{eqnarray}&&{\chi }^{2}={{\boldsymbol{\Delta }}}^{T}\widehat{{{\boldsymbol{C}}}^{-1}}{\boldsymbol{\Delta }},\end{eqnarray} \tag{ 27 }$$

where ${\boldsymbol{\Delta }}$ is ${\boldsymbol{\Delta }}\equiv {\boldsymbol{N}}-\widehat{{\boldsymbol{N}}}$, with ${\boldsymbol{N}}$ being the mock data vector consisting of WL peak counts of different bins and $\widehat{{\boldsymbol{N}}}$ being the model predictions for these bins. The quantity ${\boldsymbol{C}}$ is the covariance matrix for peak counts between different bins. We calculate it using the simulation at the fiducial cosmology. Specifically, for each case in Table 2, we first obtain the covariance for an area S₀ corresponding to an individual simulated convergence map by calculating the variance of peaks [${{\boldsymbol{C}}}^{0}$]_ij between the ith and jth bin from 20 × 96 maps. We then scale ${{\boldsymbol{C}}}^{0}$ to the mock survey area S considered in different cases by

$$\begin{eqnarray}&&{\boldsymbol{C}}=\displaystyle \frac{S}{{S}_{0}}{{\boldsymbol{C}}}_{0}.\end{eqnarray} \tag{ 28 }$$

Studies have shown that this scaling can lead to a slight underestimate of the covariance for large S (Kratochvil et al. 2010). This, however, should not affect our conclusions regarding the bias resulting from neglecting the LSS effect and the validity of our new model. It is also noted that the cosmology dependence of the covariance is not considered here.

From ${\boldsymbol{C}}$, we can calculate its inverse ${{\boldsymbol{C}}}^{-1}$ and further the $\widehat{{{\boldsymbol{C}}}^{-1}}$:

$$\begin{eqnarray}&&\widehat{{{\boldsymbol{C}}}^{-1}}=\displaystyle \frac{R-{N}_{\mathrm{bin}}-2}{R-1}{{\boldsymbol{C}}}^{-1},\end{eqnarray} \tag{ 29 }$$

where R = 20 × 96 and N_bin is the number of bins of WL peak counts used in deriving cosmological constraints.

In our analyses, we concentrate on the constraints on Ω_m and σ₈, and we set all of the other cosmological parameters fixed to be the input values of the simulations. We implement the MCMC technique to explore the posterior probabilities of (Ω_m, σ₈) (X. K. Liu et al. 2015, 2016).

The constraining results for S10small are shown in Figure 8, where the blue and red contours are the results using the model presented in this paper including the LSS effect and the model of F10 without the LSS effect, respectively. The black cross indicates the input values of the two parameters for WL simulations. Consistent with that shown in Figure 6, the two constraints overlap significantly and the two models perform equally well. In this case, the LSS effect is negligible, and the application of the F10 model is well justified without introducing notable biases in the parameter constraints.

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The results for S10, S20, M20, and D20 are presented in Figure 9. Because the survey area is larger than that of S10small, the statistical errors are reduced considerably, resulting in smaller contours. For S10 (upper left), the blue and red contours still have a large overlap. The WL simulation input values are at the edge of the 1σ red region. The blue constraints from our improved model including the LSS effect, on the other hand, give better results.

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For S20 (upper right), σ_N,0 = 0.0126, σ_LSS,0 = 0.0057, and the total σ₀ = 0.0138. The fractional contribution from LSS is σ_LSS,0/σ₀ ≈ 0.41. Thus the LSS effect is already apparent. The constraints obtained by using the model of F10 are biased by more than 2σ. For M20 (lower left) and D20 (lower right), the shape noise is the same as that of S20. But the LSS effect is stronger with σ_LSS,0 = 0.0082 and 0.0109, and the corresponding fractional contribution to σ₀ is ∼0.55 and ∼0.65 for M20 and D20, respectively. Without the LSS effect, the derived constraints are severely biased by more than 3σ for M20 and even larger for D20. On the other hand, in all of the cases, our new model incorporating the LSS effect works excellently with the input values being aligned with the degeneracy direction and well within the 1σ region, as shown in blue.

It is known that WL effects depend on Ω_m and σ₈ in a degenerate way, and the derived constraints of the two parameters are highly correlated, as seen from Figures 8 and 9. Such a correlation is often described by a relation Σ₈ = σ₈(Ω_m/0.27)^α. In Table 3, we list the values of α and Σ₈ for different cases. These values are derived from the principal component analysis of the MCMC samples (for details, please see Section 4.1 of Tereno et al. 2005 or the PCA method in getdist⁹ ). Because the F10 model works well for S10small, for this case we also list the values obtained from the constraints using F10. For the other cases, we only show the results derived from the blue regions in Figure 9. We see that for S10small, we have α ≈ 0.434 from F10 and α ≈ 0.452 from our improved model. The two results are very similar and consistent with the one we obtained from WL peak analyses using CS82 (X. K. Liu et al. 2015). For S20, M20, and D20, the α value decreases somewhat with the increase of z_med. We show their 1σ contours together with the derived degeneracy directions in Figure 10. This indicates the potential of tomographic WL peak analyses, which we will explore in detail in our future studies. We also note that the α values derived from WL high-peak abundances are systematically smaller than those from cosmic shear correlations (Kilbinger 2015), showing the complementary nature of the two types of statistical analyses.

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The previous analyses show the results with a set of fiducial parameters. In this subsection, we test the validity of the model for different cases. In Figure 11, we show the results with Gaussian smoothing of different smoothing scales with θ_G = 1.5 arcmin (upper) and 3 arcmin (lower) for S20 on the left and M20 on the right. We see the model performs equally well as that of θ_G = 2 arcmin.

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In our model, we consider massive halos with M ≥ M_* as the dominant sources of high peaks. Simulations show that M_* ∼ 10¹⁴ h⁻¹ M_⊙ is an appropriate choice. The very precise value can have dependencies on, for example, the halo mass function and cosmological models. In our fiducial analyses, we take M_* = 10¹⁴ h⁻¹ M_⊙. To test the M_* sensitivity of our model predictions, in Figure 12, we show the differences of the model predictions with M_* = 10^13.9 h⁻¹ M_⊙ and 10^14.1 h⁻¹ M_⊙ with respect to that of the fiducial results. The data points are the differences between the simulation results and the fiducial model predictions with M_* = 10¹⁴ h⁻¹ M_⊙. The large and small error bars are for the survey area of ∼150 deg² and ∼1086 deg², respectively. It is seen, as expected, that a different choice of M_* has no effect on the predicted abundance of very high peaks. For peaks around ν₀ ≈ 4 in the considered cases, they show some effects. For surveys of ∼150 deg² and z_med = 0.7, the differences arising from a 0.1 dex variation of M_* are within the statistical errors for ν₀ ≥ 4. For surveys of ∼1000 deg², or higher z_med, the dependence on M_* becomes significant. We will investigate these issues in more detail in our future studies.

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In Figure 12, we extend the horizontal axis to ν₀ = 3. We see that at ν₀ ∼ 3, there are some deviations between the model predictions and the simulation results: the higher the z_med, the larger the deviations. From Figure 4, we see that in the considered cases, for peaks of ν₀ ∼ 3, a significant fraction of them are from the field regions resulting from the combined effects of LSS and shape noise. Thus they are more sensitive to the LSS properties than high peaks that are mainly from halo regions. The Gaussian approximation of the LSS effects needs to be improved to better account for these relatively low peaks, particularly for higher z_med, where the LSS effects are comparable to or even larger than the shape noise effects. This is another important effort in our future studies.

It is noted that our analyses are done using the convergence fields from simulations directly. On the other hand, observations measure the shape ellipticities of galaxies, which directly give rise to an estimate of the reduced shear g_i = γ_i/(1 − κ). To perform peak analyses in the convergence fields, in general, we need to first reconstruct them from the shear estimates using the relation between γ and κ. To avoid the reconstructions that may introduce systematic errors, the aperture mass M_ap statistics has been proposed with (e.g., Schneider 1996; van Waerbeke 1998; Jarvis et al. 2004)

$$\begin{eqnarray}&&{M}_{\mathrm{ap}}({\boldsymbol{\theta }})=\int {d}^{2}{\boldsymbol{\theta }}^{\prime} Q(| {\boldsymbol{\theta }}-{\boldsymbol{\theta }}^{\prime} | ){g}_{t}({\boldsymbol{\theta }}^{\prime} ),\end{eqnarray} \tag{ 30 }$$

where g_t is the tangential component of g with respect to ${\boldsymbol{\theta }}-{\boldsymbol{\theta }}^{\prime}$. In the regime of κ ≪ 1 and ${\boldsymbol{g}}\approx {\boldsymbol{\gamma }}$, M_ap is equal to applying a U filter to the κ field with

$$\begin{eqnarray}&&Q(\theta )=-U(\theta )+\displaystyle \frac{2}{{\theta }^{2}}\int \theta ^{\prime} d\theta ^{\prime} U(\theta ^{\prime} ).\end{eqnarray} \tag{ 31 }$$

It is required that the U filter is compensated with $\int {d}^{2}{\boldsymbol{\theta }}U({\boldsymbol{\theta }})=0$. Here we present the peak analysis results for M_ap obtained by applying a U filter to the simulated convergence fields to show the applicability of our model. We choose a particular filter set with (van Waerbeke 1998; Jarvis et al. 2004)

$$\begin{eqnarray}&&U(\theta ,{\theta }_{U})=\displaystyle \frac{1}{\pi {\theta }_{U}^{2}}\left(1-\displaystyle \frac{{\theta }^{2}}{{\theta }_{U}^{2}}\right)\exp \left(-\displaystyle \frac{{\theta }^{2}}{{\theta }_{U}^{2}}\right).\end{eqnarray} \tag{ 32 }$$

This filter has smoothed behaviors both in real and Fourier spaces and can be handled better computationally than sharply truncated filters (van Waerbeke 1998).

In Figure 3, we showed the comparison of power spectra in Gaussian and U filters. The U filter can filter out the large-scale contributions more efficiently than can the Gaussian smoothing. For a visual comparison, we show in Figure 13 the zoom-in maps of the Gaussian and U filters of a same field. We see that high peaks correspond well in the two cases. On the other hand, large-scale patterns are more apparent in the Gaussian-smoothed map. In Figure 14, we show σ_LSS,0/σ_N,0 for the two filters for different source redshifts and different smoothing scales. While the LSS effect increases with the source redshift and the smoothing scale in both cases, it is more significant in the Gaussian filter case than with the U filter, consistent with the analyses shown in Figures 3 and 13. In Figure 15, we show the signal-to-noise ratio comparison of the corresponding peaks under the two filters. We see that, in general, the signal-to-noise ratio is lower in the U filter than in the Gaussian smoothing, which indicates that our peak model can be applicable to lower peaks in the U-filter case. Figure 16 presents the peak number distribution of M_ap under the U filter from simulations and also from our model prediction. The results demonstrate that our model works well too in this case, starting from ν₀ ≈ 2.

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We should note that the equivalence of obtaining M_ap from U filtering of the convergence fields with its original definition is approximate under the assumption of g ∼ γ. For high peaks, however, such an approximation is not accurate enough. Thus to model high aperture-mass peaks better corresponding to real observational analyses, we need to work on the true M_ap fields derived directly by applying a Q filter to the reduced shear field g_t, which is much more computationally complicated and intensive. Great effort has been devoted to building such a model, and we will present it in our forthcoming paper by C. Z. Pan et al. (2018, in preparation).