STATISTICAL AND SYSTEMATIC ERRORS IN THE MEASUREMENT OF WEAK-LENSING MINKOWSKI FUNCTIONALS: APPLICATION TO THE CANADA–FRANCE–HAWAII LENSING SURVEY

By using the Limber approximation (Limber 1954; Kaiser 1992) and Equation (2), one can calculate the convergence power spectrum as

$$\begin{eqnarray} P_{\kappa }(\ell) &=& \int _{0}^{\chi _s} {\rm d}\chi \frac{W(\chi)^2}{r(\chi)^2} P_{\delta }\left(k=\frac{\ell }{r(\chi)},z(\chi)\right), \end{eqnarray} \tag{ 4 }$$

where P_δ(k) is the three-dimensional matter power spectrum. The most direct measurement of weak-lensing two-point statistics is the two-point shear correlation function (2PCF) ξ_±. Theoretically, the 2PCFs are obtained by the Hankel transforms of a convergence power spectrum P_κ as

$$\begin{eqnarray} &&\xi _{\pm }(\theta) = \frac{1}{2\pi }\int _{0}^{\infty }{\rm d}\ell \, \ell \, P_{\kappa }(\ell) \,J_{0,4}(\ell \theta), \end{eqnarray} \tag{ 5 }$$

where J₀ and J₄ are the first-kind Bessel functions of the order of 0 and 4.

Schneider et al. (2002) show that the 2PCFs are estimated in an unbiased way by averaging over pairs of galaxies. In practice, the estimator $\hat{\xi }_{\pm }$ is calculated by

$$\begin{eqnarray} &&\hat{\xi }_{\pm }(\theta) = \frac{\sum _{ij}w_{i}w_{j}(\epsilon _{t}({\boldsymbol {\theta }}_{i})\epsilon _{t}({\boldsymbol {\theta }}_{j}) +\epsilon _{\times }({\boldsymbol {\theta }}_{i})\epsilon _{\times }({\boldsymbol {\theta }}_{j}))}{\sum _{ij}w_{i}w_{j}}, \end{eqnarray} \tag{ 6 }$$

where w_i is weight related to shape measurement and $\epsilon _{t,\times }({\boldsymbol {\theta }}_{i})$ is the tangential and cross component of ith source galaxy's ellipticity. The summation is taken over all galaxy pairs (i, j) with angular separation $|{\boldsymbol {\theta }}_{i}-{\boldsymbol {\theta }}_{j}| \in [\theta -\Delta \theta, \theta +\Delta \theta ]$.

MFs are morphological statistics for some smoothed random fields characterized by a certain threshold. In general, for a given D-dimensional smoothed field $\mathbb {S}^{D}$, one can calculate D + 1 MFs V_i. One can thus define 2+1 MFs on $\mathbb {S}^2$: V₀, V₁ and V₂. The MFs are defined, for a threshold of ν, as

$$\begin{eqnarray} V_{0}(\nu) &\equiv & \frac{1}{4\pi }\int _{Q_{\nu }}\, {\rm d}S, \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} V_{1}(\nu) &\equiv & \frac{1}{4\pi } \int _{\partial Q_{\nu }}\, \frac{1}{4} {\rm d}\ell, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} V_{2}(\nu) &\equiv & \frac{1}{4\pi } \int _{\partial Q_{\nu }}\, \frac{1}{2\pi }K{\rm d}\ell, \end{eqnarray} \tag{ 9 }$$

where Q_ν and ∂Q_ν represent the excursion set and its boundaries for a smoothed field $u({\boldsymbol {\theta }})$. They are given by

$$\begin{eqnarray} &&Q_{\nu } = \lbrace {\boldsymbol {\theta }}\, |\, u({\boldsymbol {\theta }}) > \nu \rbrace, \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} &&\partial Q_{\nu } =\lbrace {\boldsymbol {\theta }} \, |\, u({\boldsymbol {\theta }}) = \nu \rbrace. \end{eqnarray} \tag{ 11 }$$

For a given threshold, V₀, V₁, and V₂ describe the fraction of sky area, the total boundary length of contours, and the integral of the geodesic curvature K along the contours, respectively.

We summarize how to estimate lensing MFs from observed cosmic shear. Let us first define the weak-lensing mass map, i.e., the smoothed lensing convergence field:

$$\begin{eqnarray} &&{\cal K} ({\boldsymbol {\theta }}) = \int {\rm d}^2 \phi \ \kappa ({\boldsymbol {\theta }}-{\boldsymbol {\phi }}) U({\boldsymbol {\phi }}), \end{eqnarray} \tag{ 12 }$$

where U is the filter function to be specified below. We can calculate the same quantity by smoothing the shear field γ as

$$\begin{eqnarray} &&{\cal K} ({\boldsymbol {\theta }}) = \int {\rm d}^2 \phi \ \gamma _{t}({\boldsymbol {\phi }}:{\boldsymbol {\theta }}) Q_{t}({\boldsymbol {\phi }}), \end{eqnarray} \tag{ 13 }$$

where γ_t is the tangential component of the shear at position ${\boldsymbol {\phi }}$ relative to the point ${\boldsymbol {\theta }}$. The filter function for the shear field Q_t is related to U by

$$\begin{eqnarray} &&Q_{t}(\theta) = \int _{0}^{\theta } {\rm d}\theta ^{\prime } \ \theta ^{\prime } U(\theta ^{\prime }) - U(\theta). \end{eqnarray} \tag{ 14 }$$

We consider Q_t to be defined with a finite extent. In this case, one finds

$$\begin{eqnarray} &&U(\theta) = 2\int _{\theta }^{\theta _{o}} {\rm d}\theta ^{\prime } \ \frac{Q_{t}(\theta ^{\prime })}{\theta ^{\prime }} - Q_{t}(\theta), \end{eqnarray} \tag{ 15 }$$

where θ_o is the outer boundary of the filter function.

In the following, we consider the truncated Gaussian filter (for U) as

$$\begin{eqnarray} U(\theta) &=& \frac{1}{\pi \theta _{G}^{2}} \exp \left(-\frac{\theta ^2}{\theta _{G}^2} \right) -\frac{1}{\pi \theta _{o}^2}\left(1-\exp \left(-\frac{\theta _{o}^2}{\theta _{G}^2} \right) \right),\quad\quad \end{eqnarray} \tag{ 16 }$$

$$\begin{eqnarray} Q_{t}(\theta) &=& \frac{1}{\pi \theta ^{2}}\left[ 1-\left(1+\frac{\theta ^2}{\theta _{G}^2}\right)\exp \left(-\frac{\theta ^2}{\theta _{G}^2}\right)\right], \end{eqnarray} \tag{ 17 }$$

for θ ⩽ θ_o and U = Q_t = 0 elsewhere. Throughout the present paper, we adopt θ_G = 1 arcmin and θ_o = 15 arcmin. Note that this choice of θ_G is considered to be an optimal smoothing scale for the detection of massive galaxy clusters using weak lensing for z_source = 1.0 (Hamana et al. 2004).

It is important to use appropriately the weight associated with shape measurement when making smoothed convergence maps. In practice, we estimate ${\cal K}$ by generalizing Equation (13):

$$\begin{equation} {\cal K}({\boldsymbol {\theta }}_{i}) = \frac{\sum _{j}Q_{t}({\boldsymbol {\phi }}_{j})w_{j} \epsilon _{t}({\boldsymbol {\phi }}_{j}:{\boldsymbol {\theta }}_{i})}{\sum _{j}Q_{t}({\boldsymbol {\phi }}_{j})w_{j}}, \end{equation} \tag{ 18 }$$

where the summation in Equation (18) is taken over all the source galaxies that are located within θ_o from ith pixel. The weak-lensing convergence field ${\cal K}$ is then computed from the galaxy ellipticity data on regular grids with a grid spacing of 0.15 arcmin. In making the convergence map, we discard the pixels when the denominator in Equation (18) is equal to zero. The boundaries are defined by masking a pixel if the number of sources within θ_o from the pixel is less than $5\sqrt{15\pi \theta _{o}^2}$. This critical value effectively sets the signal-to-noise ratio of the number of sources inside a circle with radius of θ_o to be less than 5, on the assumption that the distribution of sources is approximated by a Poisson distribution. We repeat the above procedure for all the pixels. Note that the details of the procedure do not affect the final results significantly, as long as we impose the same conditions on all of the pixels, because our analysis is based on the comparison of two maps that have the same configuration of source positions.

We follow Lim & Simon (2012) to calculate the MFs from pixelated ${\cal K}$ maps. We convert a weak-lensing field ${\cal K}$ to $x = ({\cal K} - \langle {\cal K} \rangle)/\sigma _{0}$ where σ₀ is the standard deviation of ${\cal K}$. We set Δx = 0.2 from x = −5 to x = 5 for binning the threshold value.

We first run a number of cosmological N-body simulations to generate a three-dimensional matter density field. We use the parallel Tree-Particle Mesh code Gadget2 (Springel 2005). The simulations are run with 512³ dark matter particles in a volume of 480 or 960 h⁻¹Mpc on a side. We generate the initial conditions using a parallel code developed by Nishimichi et al. (2009) and Valageas & Nishimichi (2011), which employs the second-order Lagrangian perturbation theory (e.g., Crocce et al. 2006). The initial redshift is set to z_init = 50, where we compute the linear matter transfer function using CAMB (Lewis et al. 2000). Our fiducial cosmology adopts the following parameters: matter density Ω_m0 = 0.279, dark energy density Ω_Λ0 = 0.721, the amplitude of curvature fluctuations A_s = 2.41 × 10⁻⁹ at the pivot scale k = 0.002Mpc⁻¹, the parameter of the equation of state of dark energy w₀ = −1, Hubble parameter h = 0.700, and the scalar spectral index n_s = 0.972. These parameters are consistent with the WMAP nine-year results (Hinshaw et al. 2013). To investigate the degeneracy of the cosmological parameters, we also run the same set of simulations but with slightly different Ω_m0, w₀, and A_s. We summarize the simulation parameters in Table 1.

For ray-tracing simulations of gravitational lensing, we generate light-cone outputs using multiple simulation boxes in the following manner. Our small- and large-volume simulations are placed to cover the past light-cone of a hypothetical observer with an angular extent 10° × 10°, from z = 0 to 3, similarly to the methods in White & Hu (2000), Hamana & Mellier (2001), and Sato et al. (2009). Details of the configuration are found in the last reference. The angular grid size of our maps is 10°/4096 ∼ 0.15 arcmin. We use outputs from independent realizations when generating the light-cone outputs. We also randomly shift the simulation boxes in order to avoid the same structure appearing multiple times along a line-of-sight. In total, we generate 40 independent shear maps from four N-body simulations for each cosmological model.

Our purpose is to study observational effects on weak-lensing morphological statistics. To this end, we generate realistic mock weak-lensing catalogs by combining ray-tracing simulations and the CFHTLenS data (Van Waerbeke et al. 2013). The main advantage of our mock catalogs is that the observed positions on the sky of the source galaxies are directly used. Because of this, we can keep all the characteristics of the survey geometry the same as in CFHTLenS.

We locate the source galaxies in the pixel unit of our lensing map and then calculate the reduced shear signal g = γ/(1 − κ) at the galaxy positions. Ray-tracing is done up to the redshift of the galaxy as described in Section 4.1. In this step, a galaxy's redshift is set to be at the peak of the posterior PDF obtained from BPZ. This could cause systematic effects on morphological statistics originating from the inaccuracy of the photometric redshift estimation. We discuss the impact of the redshift distribution of sources on lensing morphological statistics later in Section 5.

We next consider the intrinsic ellipticity that is known to be a major error source in cosmic shear measurement. For each galaxy, we randomize the orientation of the observed ellipticity, while keeping its amplitude. The randomized ellipticity is then assigned as the intrinsic ellipticity _int at each galaxy's position. Seitz & Schneider (1997) show that the observed source ellipticity sheared by weak-lensing effect can be expressed as a function of _int and the reduced shear signal g. The final "observed" ellipticity is

$$\begin{eqnarray} &&{\boldsymbol {\epsilon }}_{\rm mock} = \frac{{\boldsymbol {\epsilon }}_{\rm int}+{\boldsymbol {g}}}{1+{\boldsymbol {g}}^{*}{\boldsymbol {\epsilon }}_{\rm int}}, \end{eqnarray} \tag{ 19 }$$

where ${\boldsymbol {\epsilon }}_{\rm mock}$ is represented as a complex ellipticity.

Finally, we incorporate calibration correction in the shear measurement. We assign the weight associated with the shape measurement of lensfit and the shear calibration correction following Heymans et al. (2012). The two factors determine the potential additive shear bias c and multiplicative bias m. We then apply the shear calibration correction to ${\boldsymbol {\epsilon }}_{\rm mock}$ by using bias factors m and c as

$$\begin{eqnarray} &&{\boldsymbol {\epsilon }}_{\rm mock} \rightarrow (1+m){\boldsymbol {\epsilon }}_{\rm mock}+{\boldsymbol {c}}. \end{eqnarray} \tag{ 20 }$$

In this step, we assume that there is no correlation between ${\boldsymbol {\epsilon }}$ and $m, {\boldsymbol {c}}$. We have explicitly calculated the correlation between ${\boldsymbol {\epsilon }}$ and $m, {\boldsymbol {c}}$ at the source galaxy positions using the CFHTLenS data set, and found that there is indeed no significant correlation between the quantities.

Through the above procedures, we have successfully included the following observational effects in our analysis: (1) nonlinear relation between the observed ellipticities and cosmic shear, (2) non-Gaussian distribution of the intrinsic ellipticities, (3) the masked survey area of CFHTLenS and the inhomogeneous angular distribution of the source galaxies, (4) imperfect shape measurements, and (5) the redshift distribution of the source galaxies. We note that all or many of these effects are often ignored in previous works on lensing morphological statistics.

We perform a Fisher analysis to produce a forecast for parameter constraints on Ω_m0, A_s, and w₀ with future weak-lensing surveys.

For a multivariate Gaussian likelihood, the Fisher matrix F_ij can be written as

$$\begin{eqnarray} &&F_{ij} = \frac{1}{2} {\rm Tr} [ A_{i} A_{j} + C^{-1} M_{ij} ], \end{eqnarray} \tag{ 21 }$$

where A_i = C⁻¹∂C/∂p_i, M_ij = 2(∂μ/∂p_i)(∂μ/∂p_j), C is the data covariance matrix, μ represents the assumed model, and ${\boldsymbol {p}} = (\Omega _{\rm m0}, A_s, w_0)$ are the main parameters. In the present study, we consider only the second term in Equation (21). Because C is expected to scale proportionally inverse to the survey area, the second term will be dominant for a large area survey (see, e.g., Eifler et al. 2009). We calculate the model template by averaging the MFs over 40 convergence maps with appropriate noises for each CFHTLenS field. Figure 2 shows the cosmological dependence of our model MFs thus calculated. We see clearly the behavior of the MFs as a function of p.

Zoom In Zoom Out Reset image size

We calculate the model 2PCFs using the fitting formula of nonlinear matter power spectrum of Takahashi et al. (2012), on the assumption that the source redshift distribution is well approximated by the sum of the posterior PDF with 0.2 < z_p < 1.3 given in Kilbinger et al. (2013).

To calculate the matrix M_ij, we approximate the first derivatives of the 2PCF and MFs with respect to cosmological parameter p_i as

$$\begin{eqnarray} &&\frac{\partial \mu }{\partial p_{i}} = \frac{\mu (p^{(0)}_{i}+\Delta p_{i}) - \mu (p^{(0)}_{i} - \Delta p_{i})}{2\Delta p_{i}}, \end{eqnarray} \tag{ 22 }$$

where ${\boldsymbol {p}}^{(0)}=(0.279, 2.41 \times 10^{-9}, -1.0)$ gives our fiducial model parameters and we set $\Delta {\boldsymbol {p}}=(0.025, 0.1 \times 10^{-9},0.2)$.

We construct the data vector D from a set of binned MFs and 2PCFs,

$$\begin{eqnarray} {D_{i}} &=& \lbrace V_{0}(x_{1}),...,V_{0}(x_{10}),V_{1}(x_{1}),...,V_{1}(x_{10}),\nonumber\\ && V_{2}(x_{1}),...,V_{2}(x_{10}),\nonumber \\ &&\xi _{+}(\theta _{1}),...,\xi _{+}(\theta _{10}), \xi _{-}(\theta _{1}),...,\xi _{-}(\theta _{10})\rbrace, \end{eqnarray} \tag{ 23 }$$

where $x_{i}=({\cal K}_{i} - \langle {\cal K} \rangle)/\sigma _{0}$ is the binned normalized lensing field. For the Fisher analysis, we use 10 bins in the range of x_i = [ − 3, 3].⁶ In this range of x, Equation (22) gives smooth estimates for M_ij. For the 2PCFs, we use 10 bins logarithmically spaced in the range of θ_i = [0.9, 300] arcmin. A data vector has 50 elements, 3 × 10 MFs, and 2 × 10 2PCFs.

We therefore need a 50 × 50 data covariance matrix for the Fisher analysis. To this end, we first use 1000 shear maps made by Sato et al. (2009). The maps have almost the same design as our simulations, but are generated for slightly different cosmological parameters (consistent with WMAP three-year results Spergel et al. 2007). The actual parameter differences are small, and also the dependence of the covariance matrix on cosmological parameters is expected to be weak. Each map covers a 5° × 5° sky and the source redshift is set to be z_source = 1. We model the intrinsic ellipticities by adding random ellipticities drawn from a 2D Gaussian to the simulated shear data. We set the rms of the intrinsic ellipticities to be 0.38 and the number of source galaxies is set to be 10 arcmin⁻². These are reasonable choices for our study here. In making the smoothed lensing map from the simulation outputs, we set the weight related to shape measurement to be unity. From the 1000 shear maps with appropriate noises, we can estimate the variances of the 2PCFs and MFs. We also estimate the statistical errors by randomly rotating the observed orientation of the ellipticities. Using these randomized catalogs, the data covariance matrices in each CFHTLenS field are obtained by the sum of sampling variance and the statistical error as

$$\begin{eqnarray} &&C_{\rm each} = C_{\rm cosmic} \left(\frac{A_{\rm each} \ {\rm deg}^2}{25 \ {\rm deg}^2}\right)^{-1} + C_{\rm stat}, \end{eqnarray} \tag{ 24 }$$

where C_cosmic is sampling variance, C_stat represents the statistical error in each CFHTLenS field, and A_each corresponds to the effective survey area. In the following, we assume that the four CFHTLenS fields are independent of each other statistically. The total inverse covariance matrix for the whole CFHTLenS data is the sum of $C_{\rm each}^{-1}$ over the W1, W2, W3, and W4 fields. We produce a forecast for future lensing surveys by simply scaling the data covariances by the survey area, assuming that the statistical error is identical to that in CFHTLenS. When calculating the inverse covariance, we include a debiasing correction, the so-called Anderson–Hartlap factor α = (n_real − n_bin − 2)/(n_real − 1) with n_rea = 1000 being the number of realization of simulation sets and n_bin = 50 being the number of total bins in our data vector (Hartlap et al. 2007).

We expect that Equation (24) provides a good approximation to the full covariance, but the accuracy needs to be addressed here. In the case of shear correlation functions, the covariance matrix consists of three components: a sampling variance, the statistical noise, and a third term coupling the two (Schneider et al. 2002). Because the MFs do not have the additivity of, e.g., V_i(ν₁ + ν₂) = V_i(ν₁) + V_i(ν₂), it is not clear if the MF covariance can be expressed similarly as the sum of the three contributions. We thus resort to estimating the MF covariance in a direct manner by using the large set of mock catalogs generated by the procedure shown in Section 4.2. Note that, in the procedure, we perform two randomization processes for a fixed cosmological model. One is to generate multiple realizations of the large-scale structure (by N-body simulations) and the other is to randomize intrinsic ellipticities of the source galaxies. We perform each process separately, technically by fixing a random seed of the other process, to evaluate each term in Equation (24). In principle, one can perform both of the processes simultaneously and derive the full covariance. However, this would require a huge number of mock catalogs. In Appendix B, we have done a simple but explicit check to validate that Equation (24) indeed provides a reasonably good approximation. The details of our test and the result are shown there.

We present a forecast for upcoming surveys such as HSC and LSST. We first derive constraints on the cosmological parameters for a 154 deg² area survey, for which we have the full covariance matrix obtained in the previous sections. We then consider two wide surveys with an area coverage of 1400 deg² (HSC) and 20,000 deg² (LSST). We simply scale the covariance matrix by a factor of 154/1400 or 154/20000 for them.

Let us begin with quantifying the statistical error associated with the real data. We have performed a Fisher analysis including the sampling variance and the statistical error. When we include the statistical error, the cosmological constraints are degraded by a factor of ∼2 for the CHFTLens survey. In Figure 3, the red error circle corresponds to the 1σ cosmological constraints including the sampling variance and the statistical error, while the blue one is obtained from the Fisher analysis without the statistical error.

Zoom In Zoom Out Reset image size

We are now able to present a forecast for future lensing surveys covering larger sky areas on the assumption that the data covariance is the same as that of CFHTLenS. Figure 4 shows the derived parameter constraints. The blue error circles show the 1σ constraints from the shear 2PCFs whereas the red circles are obtained from the lensing MFs. It is promising that, with Subaru HSC, we can constrain the dark energy equation of state w₀ with an error of Δw₀ ∼ 0.25 by the lensing MFs alone. Table 2 summarizes the expected constraints by future surveys. Combining the 2PCFs and the MFs can improve the constraints by a factor of ∼2 by breaking the degeneracy between the three parameters.

Zoom In Zoom Out Reset image size

It should be noted that this conclusion might seem slightly different from that of Kratochvil et al. (2012), who argue that adding the power spectrum does not effectively improve the constraints when all three MFs are already used. Our result suggests that combining the 2PCFs and the MFs improves cosmological parameter constraints appreciably. A precise account for the difference is not given by our study only, but there are many factors that can affect the parameter constraint. First of all, we characterize the amplitude of the matter power spectrum by the amplitude of curvature perturbations A_s at the cosmological recombination epoch whereas Kratochvil et al. (2012) adopt σ₈ at the present epoch as a parameter. The latter is the so-called derived parameter and has an internal degeneracy with Ω_m and w₀. Our result suggests that including the 2PCFs in the analysis can better constrain A_s, which in turn yields tighter constraints on the other parameters. Furthermore, our analysis includes observational effects such as survey mask regions and the source distribution directly. Altogether, these differences make it difficult to compare our results with those of previous works that mostly adopt idealized configurations.

In this section, we examine the effect of known systematics on measurement of the MFs. We follow Huterer et al. (2006) to estimate the bias in the cosmological parameter due to some possible systematics

$$\begin{eqnarray} &&\delta p_{\alpha } = F^{-1}_{\alpha \beta }\sum _{i,j} C^{-1}_{ij}\left(D^{\rm test}_{i}-D^{\rm fid}_{i}\right) \frac{\partial D^{\rm fid}_{j}}{\partial p_{\beta }} \end{eqnarray} \tag{ 25 }$$

where δp_α is the bias in the αth cosmological parameter, F_αβ is a Fisher matrix, D is the data vector, and C is the data covariance. The data vector ${\boldsymbol {D}}^{\rm fid}$ represents the theoretical template for our fiducial model. and ${\boldsymbol {D}}^{\rm test}$ is the test data vector that includes a known systematics effect. In this section, we use the data vector D consisting of the lensing MFs only. For ${\boldsymbol {D}}^{\rm fid}$, we use the average MFs over 40 mock catalogs from our fiducial cosmological model described in 4.2. The mock samples serve as a reference, for which we have assumed that (1) the source galaxy redshift is well approximated by the peak of the posterior PDF of photometric redshift, and (2) the observed shear is perfectly calibrated by a functional form shown in Heymans et al. (2012). We test these assumptions and quantify the net effects in a direct manner by generating and using another set of the mock catalogs for ${\boldsymbol {D}}^{\rm test}$ using the same N-body realizations as for our fiducial case.

Redshift Distribution

It is important to quantify the effect of the source redshift distribution and of the error in photometric redshifts on the lensing MFs, or indeed on any lensing statistics. We perform ray-tracing simulations by shooting rays to the farthest lens plane at z = 3, weighting the lensing kernel using a redshift distribution function of the sources.

Specifically, we follow the same manner as in Sato et al. (2009) to simulate the weak-lensing effect but the lensing kernel is slightly different from their simulation because of the wider source redshift distribution. In Sato et al. (2009), they assume z_p = 1 so that the lensing kernel can be calculated by the simple expression, i.e., r(χ_s − χ_l)r(χ_l)/r(χ_s), where χ_s and χ_l are the comoving distances of sources and of the lens, respectively. In the present study, we consider source redshift distribution p(χ). Then the lensing kernel for the lensing objects at χ_l should be replaced with $\int _{\chi _{l}}^{\chi _{H}}{\rm d}\chi _{s}p(\chi _s)r(\chi _{s}-\chi _{l})r(\chi _{l})/r(\chi _{s})$.

The source positions on the sky and all the other characteristics are kept the same as in our original mock catalogs, which themselves are derived from CHFTLenS. For the redshift distribution, we adopt the sum of the posterior PDF of photometric redshift for the galaxies with 0.2 < z_p < 1.3. Figure 5 compares the integrated redshift distribution with the histograms of the source redshifts. The latter is used in our fiducial simulations. The test data vector ${\boldsymbol {D}}^{\rm test}$ is calculated by averaging the MFs over the new 40 catalogs with the posterior weight described above.

Zoom In Zoom Out Reset image size

The main difference caused by the different redshift distributions is the amplitude of the standard deviation of ${\cal K}$. We see in Figure 6 that the net difference is as large as those found for cosmological models differing by Δw₀ = 0.2; this can obviously be a significant source of error in cosmological parameter constraints with upcoming future surveys. We estimate the resulting bias in the derived w₀ by using Equation (25). The uncertainties in the photometric redshifts can indeed induce a Δw₀ ∼ 0.1 bias in w₀. The exact values are summarized in Table 3.

Zoom In Zoom Out Reset image size

Table 3. Bias of Cosmological Parameter Estimation Due to Possible Systematics

	Ωm0	As × 109	w0
Redshift distribution	0.00707	−0.0254	−0.122
Calibration correction	−0.0224	0.110	−0.234

Download table as:  ASCIITypeset image

We have also studied the effect of source redshift clustering on the lensing MFs. The results are presented in Appendix A. Briefly, the source redshift clustering is found to be a minor effect, but we note that it could cause non-negligible bias in "precision" cosmology with, for example, the LSST lensing survey.

Shear Calibration Correction

We use multiple data sets. As a probe of large-scale structure, we use data from the nine-year WMAP data release (Bennett et al. 2013; Hinshaw et al. 2013). We use the output of Monte Carlo Markov Chains (MCMC) derived from the likelihood analysis with the CMB temperature and polarization power- and cross-spectrum in the WMAP9 data. Note that the MCMC we use here does not include other external data sets, such as small-scale CMB measurements, galaxy redshift surveys, and Hubble constant. We calculate the likelihood in our parameter space ${\boldsymbol {p}} = (\Omega _{\rm m0}, A_s, w_0)$ after marginalizing over the following three parameters: the reionization optical depth τ, scalar spectral index n_s and Hubble parameter H₀.

As a probe of matter distribution at low redshifts, we use the lensing MFs and the 2PCF calculated from the CFHTLenS data. We construct the data vector and covariances in the same manner as in Section 5.1, but we assume no covariances between the MFs and the 2PCF. To validate the assumption, we have actually calculated the parameter constraints by the Fisher analysis with/without covariances between the MFs and the 2PCF. We have found that the approximation does not affect the final results significantly for the current data set. The error in Ω_m0 increases only by ΔΩ_m0 = 5 × 10⁻⁴.

We sample the posterior of the cosmological parameters from the lensing 2PCF data set using the Population Monte Carlo (PMC) using the publicly available code ${\tt COSMO}\_{\tt PMC}$ (Kilbinger et al. 2011). Details of the PMC are found in Wraith et al. (2009). The method incorporates the cosmological dependence of the shear covariance in the manner described in Eifler et al. (2009). The same model parameters are adopted as in Kilbinger et al. (2013), with the smallest and largest angular bins being 0.9 and 300 arcmin. We consider the following set of cosmological parameters: ${\boldsymbol {p}}=({\Omega _{\rm m0}, \Omega _{\rm b0}, \sigma _{8}, H_{0}, n_s, w_0})$, where Ω_b0 is the baryon density and σ₈ normalizes the matter power spectrum. To compare the result derived from CMB and that from the lensing MFs, we calculate the value of A_s at each sample point in parameter space by using the following relation:

$$\begin{eqnarray} A_{s} &=& A_{s, \rm fid}\left(\frac{\sigma _{8}}{\sigma _{8, \rm fid}}\right)^2 \frac{SD_{+}^2|_{\rm fid}}{SD_{+}^2}, \end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray} S &=& \int _{0}^{\infty } \frac{{\rm d}^3 k}{(2\pi)^3}k^{n_s}T(k)^2|W_8(k)|^2, \end{eqnarray} \tag{ 27 }$$

where D₊ is the linear growth factor of matter density, T(k) is the transfer function, and W₈(k) is the top-hat function with a scale of 8 Mpc/h in the Fourier space. For the fiducial parameter set, we use the same parameters as the WMAP9 best-fit values.

In our PMC run, we perform 30 iterations to find a suitable importance function compared to the posterior. Also, 100,000 sample points are generated for each iteration. To obtain a large sample set, we combine the PMC samples with the five highest value of perplexity p, which is the conventional diagnostic that indicates the quality and effectiveness of the sampling. Our PMC run achieves p > 0.7 for the final samples; this criterion is the same as that adopted in the analysis in Kilbinger et al. (2013).

In the next sections, we study the following three cases: (1) likelihood analysis with the lensing MFs alone, (2) combined analysis with the lensing MFs and the 2PCF, and (3) combined analysis with the lensing MFs and CMB anisotropies. In the last analysis, we treat the lensing MFs data and the CMB data as being independent of each other.

In our maximum likelihood analysis, we assume that the data vector D is well approximated by the multivariate Gaussian distribution with covariance C. This assumption is reasonable for the case of joint analysis of the CMB and lensing power spectrum (Sato et al. 2010). In this case, the χ² statistics (log-likelihood) is given by

$$\begin{eqnarray} &&\chi ^2 = (D_{i}-\mu _{i}({\boldsymbol {p}}))C^{-1}(D_{j}-\mu _{j}({\boldsymbol {p}})) \end{eqnarray} \tag{ 28 }$$

where ${\boldsymbol {\mu }}({\boldsymbol {p}})$ is the theoretical prediction as a function of cosmological parameters. The theoretical prediction is computed in a three-dimensional parameter space. In sampling the likelihood function, we consider the limited parameter region as follows: Ω_m0 ∈ [0, 1], A_s × 10⁹ ∈ [0.1, 8.0] and w₀ ∈ [ − 6.5, 0.5]. The sampling number in each parameter is set to 100.

To estimate the MFs components in ${\boldsymbol {\mu }}$, we assume that the lensing MFs depend linearly on the cosmological parameters with the first derivatives calculated from Equation (22). We consider two components of the contribution of the data covariance in our likelihood analysis; one is the statistical error and sampling variance, which are estimated as in Section 5.1 while the other originates from the possible systematics as studied in Section 5.3. We denote the latter contribution as ${\boldsymbol {C}}^{\rm sys}$. We estimate ${\boldsymbol {C}}^{\rm sys}$ in a simple and direct manner using the differences of the MFs, as shown in Figure 6:

$$\begin{eqnarray} &&C^{\rm sys}_{ij} = \left[\left(D^{\rm zdist}_{i}-D^{\rm fid}_{i}\right)^2 + \left(D^{\rm scc}_{i}-D^{\rm fid}_{i}\right)^2\right]\delta ^{2D}_{ij}, \end{eqnarray} \tag{ 29 }$$

where ${\boldsymbol {D}}^{\rm fid}$ is the template MFs for our fiducial mock catalogs, ${\boldsymbol {D}}^{\rm zdist}$ is the average MFs over 40 catalogs reflecting the different source redshift distribution as shown in Figure 5, and ${\boldsymbol {D}}^{\rm scc}$ is estimated by averaging the MFs over 40 catalogs without shear calibration correction. The total covariance is the sum of the above two contributions.

We would like to examine the ability of the lensing MFs to break degeneracies between cosmological parameters. We first consider the concordance ΛCDM model, i.e., w₀ = −1. Figure 7 shows the marginalized constraints on Ω₀ and σ₈ in the two-parameter plane. Clearly, the lensing MFs can break the well-known degeneracy between Ω₀ and σ₈ that is apparent in the analysis using only the 2PCF. Interestingly, the marginalized constraints on each parameter can be improved by a factor of five to eight by adding the MFs. The final constraints in the case of ΛCDM model are summarized in Table 4.

Zoom In Zoom Out Reset image size

Table 4. Cosmological Parameter Constraints Obtained From the Maximum Likelihood Analysis

	Ωm0	σ8
2PCF alone	$0.396 \,{\pm}\, {}^{0.177}_{0.185}$	$0.695\,{\pm}\, {}^{0.203}_{0.202}$
MFs alone	0.295  ±  0.020	$0.855\,{\pm}\, {}^{0.060}_{0.060}$
MFs + 2PCF	0.282  ±  0.022	$0.782\,{\pm}\, {}^{0.042}_{0.042}$

Notes. The error bar shows the 68% confidence level. The concordance ΛCDM model is assumed for this pot.

Download table as:  ASCIITypeset image

Next, we explore models with a variant of dark energy. The equation of state parameter w₀ serves as an additional parameter here. The left panel in Figure 8 shows the marginalized constraints in the 2D plane by the lensing MFs and 2PCF. The red circle represents the result from the lensing MFs alone, whereas the green circle is the estimate derived from combining the lensing MFs and 2PCF. Interestingly, with the lensing MFs alone, the data set favors a low w₀.⁷ We have checked that our theoretical MF template can recover correctly the input cosmological parameters for the 40 mock data with a similar confidence level expected from the Fisher analysis. We thus argue that the trend of favoring low w₀ is likely attributed to the possible systematics as studied in Section 5.3, or to imperfect modeling of the dependence of the lensing MFs on the cosmological parameters.

Zoom In Zoom Out Reset image size

The right panel of Figure 8 shows the 68% and 95% confidence regions obtained from our joint analysis with the WMAP9 CMB data. The red region represents the results from lensing MFs alone. The blue region is the result obtained from CMB, and the green one represents constraints by combining the both. These figures show clearly that the lensing MFs are useful to improve the cosmological constraints by breaking the parameter degeneracies. The marginalized constraints for the three parameters are summarized in Table 5. It is worth making further effort to reduce the systematic errors in measurement of the lensing MFs.

Table 5. We Summarize the Parameter Constraints Obtained from the Maximum Likelihood Analysis

	Ωm0	As × 109	w0
MFs alone	0.205  ±  0.060	$2.18\,{\pm}\, {}^{0.60}_{0.60}$	−2.2  ±  0.8
MFs + 2PCF	$0.256\,{\pm}\, {}^{0.054}_{0.046}$	$1.92\,{\pm}\, {}^{0.65}_{0.65}$	$-1.60\,{\pm}\, {}^{0.76}_{0.57}$
MFs + CMB	$0.290\,{\pm}\, {}^{0.016}_{0.028}$	2.39  ±  0.07	−0.90  ±  0.11

Note. The error bar indicates the 68% confidence level.

Download table as:  ASCIITypeset image