A CLIPPING METHOD TO MITIGATE THE IMPACT OF CATASTROPHIC PHOTOMETRIC REDSHIFT ERRORS ON WEAK LENSING TOMOGRAPHY

Gravitational shear can be simply related to the lensing convergence: the weighted mass distribution integrated along the line of sight (see, e.g., Bartelmann & Schneider 2001 for a thorough review and references therein). Photometric redshift information on source galaxies allows us to subdivide galaxies into redshift bins, enabling more cosmological information to be extracted, which is referred to as lensing tomography (e.g., Hu 1999; Huterer 2002; Takada & Bridle 2007). In the context of cosmological gravitational lensing, assuming the flat-sky approximation and the Limber approximation, the lensing power spectrum of the ith and jth tomographic bins can be expressed as

$$\begin{equation} P^{\kappa }_{ij}(\ell) = \int _{0}^{\infty } dz \frac{W_i(z)W_j(z)}{\chi ^2(z)H(z)} P_\delta \!\left(k=\frac{l}{\chi }; z\right), \end{equation} \tag{ 1 }$$

where H(z) is the Hubble expansion rate, χ is the comoving angular diameter distance up to redshift z, and P_δ(k, z) is the three-dimensional matter power spectrum at scale k and at redshift z. The lensing weight function W_(i)(χ) in the ith tomographic redshift bin, defined to lie between the redshifts z_i and z_i+1, is given by

$$\begin{equation} W_i(z) = \frac{3}{2}\Omega _{\rm m0} H^2_0 g_i(z)(1+z), \end{equation} \tag{ 2 }$$

and

$$\begin{eqnarray*} g_i(z)&=& \left\lbrace \begin{array}{@{}ll} {\displaystyle \chi (z) \int _{{\rm max}(z,z_i)}^{z_{i+1}}\!\!dz^{\prime } \frac{n(z^{\prime })}{\bar{n}_i} \left[1-\frac{\chi (z)}{\chi (z^{\prime })}\right]}, & z<z_{i+1}\nonumber \\ 0,& z>z_{i+1}, \end{array} \right. \end{eqnarray*}$$

where n(z) is the redshift distribution of galaxies, and $\bar{n}_i$ is the average number density of galaxies residing in the ith tomographic bin (or the redshift range z = [z_i, z_i+1]): $\bar{n}_i=\int _{z_i}^{z_{i+1}}dz^{\prime } n(z^{\prime })$.

In practice, the power spectrum measured from a galaxy survey has shot noise contamination arising from the finite sampling of galaxy images. Hence, the measured power spectrum becomes

$$\begin{equation} C_{ij}^\kappa (l)=P^\kappa _{ij}(l)+\frac{\sigma _\epsilon ^2}{\bar{n}_i} \delta ^K_{ij}, \end{equation} \tag{ 3 }$$

where σ is the rms intrinsic ellipticities per component and δ^K_ij is the Kronecker delta symbols; δ^K_ij = 1 when i = j, otherwise δ^K_ij = 0.

Note that the distribution n(z) appearing in Equation (2) denotes the underlying true redshift distribution of galaxies used in lensing analysis. However, the distribution needs to be inferred from photo-z information of individual galaxies available from multi-color imaging data sets. This generally causes biases in the lensing power spectrum in the presence of photo-z errors. As long as tomographic redshift bins are broad enough, O(10⁷) galaxies are available in each bin for a Subaru-type survey with ∼1000 deg² sky coverage. Therefore, the statistical errors of photo-z are not problematic: the lensing power spectrum is primarily sensitive to the mean redshift of source galaxies. Instead, a precise knowledge of the mean redshift in each tomographic bin is required not to have any significant biases in best-fit parameters compared to the statistical errors, as studied by Huterer et al. (2006).

To assess the required photo-z accuracies for lensing tomography, an important fact we should keep in mind is that the lensing measurement for planned wide-field surveys is not shot noise limited. Hence a significant fraction of galaxies with ill-defined photo-zs can be discarded, without severely degrading parameter accuracies (Jain et al. 2007). With these considerations in mind, we will address in the following how to define an adequate subsample of galaxies for a given multi-color data set.

Figure 1 gives a quick summary of the impact of redshift uncertainty on the lensing power spectrum for the no-tomography case, i.e., a single redshift bin. Here, for simplicity, we assumed the redshift distribution given by the analytical form, n(z) ∝ z²exp [ − (z/z₀)²] with z₀ = 1, corresponding to the mean redshift $\langle z\rangle = 2z_0/\sqrt{\pi }\simeq 1.12$. (Note that the following results are all computed using simulated galaxy catalogs that have different redshift distributions). The plot shows that a 5% change in the mean redshift causes a 10% level change in the power-spectrum amplitude, and the amount of the change varies with multipoles due to the projection of the nonlinear matter power spectrum. This change can be compared with the effect of the dark energy equation of state and the statistical errors at each multipole bin expected for the power-spectrum measurement. Clearly such a bias in the mean redshift is problematic for planned surveys.

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To make a mock catalog we need to properly take into account the redshift distribution of galaxies, which varies with the range of magnitudes considered. For example, fainter galaxies are preferentially at higher redshifts. This is the so-called magnitude–redshift relation. We use the magnitude–redshift relation estimated from the COSMOS catalog of galaxies selected by the Subaru i-band magnitudes (Ilbert et al. 2009). The COSMOS catalog provides the most accurate current photometric redshifts because the photo-z are estimated from 30 broad, intermediate, and narrowbands covering from UV, optical to mid-infrared. Also the photo-z estimates are calibrated by the spectroscopic subsample. Ilbert et al. (2009) studied subsamples of photo-z galaxies for different limiting magnitudes and showed that the resulting redshift distributions are well fitted by the polynomial form:

$$\begin{equation} n(z) = A \frac{z^{a b}+z^{a}}{z^{b}+c}, \end{equation} \tag{ 4 }$$

where A, a, b, and c are the fitting parameters. The best-fit parameters for different magnitudes in the range i = [22, 25] are given in Table 2 in Ilbert et al. (2009). The COSMOS data is deep enough in the i band, and can be safely considered as a magnitude limit sample for i < 25. The COSMOS catalog also includes information on the angular number counts of galaxies for a given magnitude range as well as on the estimated galaxy SED type for each galaxy. To model a hypothetically deeper survey such as we are interested in, we extrapolate the fitting parameters to obtain the redshift distribution for fainter galaxies down to i = 25.8.

We thus generate a mock i-band photometric catalog of galaxies such that the resulting catalog satisfies the magnitude and redshift relation for different ranges of i-band magnitudes. Figure 2 shows the magnitude–redshift distributions for the simulated catalog containing about 10⁵ galaxies.

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For a given simulated galaxy labeled with some i-band magnitude and redshift z, we need to model the SED from which the apparent magnitudes can be computed for a given set of filters. We use the publicly available library GISSEL98 (Bruzual & Charlot 1993, 2003; Bolzonella et al. 2000) to model the synthetic galaxy spectrum. The galaxy SEDs are generated to represent SEDs from early- to late-types (elliptical, S0, Sa, Sb, Sc, Sd, Im, and starburst). To model these populations—composite stellar populations (CSPs)—the single stellar population (SSP) is convolved with a model star formation history:

$$\begin{equation} f_{\rm CSP}(t) = \int _0^t \psi (t-\tau)f_{\rm SSP}(\tau) d\tau. \end{equation} \tag{ 5 }$$

Note that the SSP is modeled in Bruzual & Charlot (1993), with the initial stellar mass function given in Miller & Scalo (1979). The function ψ(t) is the star formation rate (SFR) at galaxy age t. We assumed ψ(t) ∝ exp(−t/τ) with τ = 1, 2, 3, 5, 15, and 30 Gyr for elliptical, S0, Sa, Sb, Sc, and Sd galaxies, respectively. For a starburst galaxy, star formation occurs instantaneously, while ψ = constant for an irregular (Im) galaxy. The metallicity evolves self-consistently with galaxy age, and we checked that different models of metallicity have little effect on the photo-z estimates (also see Bolzonella et al. 2000). We randomly chose the age of each simulated galaxy from 221 different ages in the range of t = [0, 20] Gyr, where the age interval is determined according to GISSEL98.

Figure 3 demonstrates simulated SEDs for starburst, elliptical, irregular, and spiral galaxies at six different ages. The dust extinction is modeled following Calzetti et al. (2000) with A_V in the range [0, 2.0].

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To make a mock galaxy catalog containing various galaxy populations, we used the publicly available code HyperZ. In doing this, we need an appropriate mixture of different galaxy SED types. We employed the composition (SB, E, S, Im) = (0.52, 0.035, 0.40, 0.045) over the entire redshift range, which was chosen so as to match the composition of best-fit galaxy SED types found in the COSMOS catalog with i < 25.⁹ For this purpose, the command make_catalog in HyperZ was slightly modified in such a way so that the resulting catalog satisfied the magnitude–redshift relations for each magnitude range in Section 3.1 and the assumed composition of galaxy SED types, because the unmodified make_catalog generates a catalog in which redshift, reference magnitude, age, galaxy SED type, and the amount of dust extinction are randomly assigned to each galaxy.

Once the SED is specified for each simulated galaxy at redshift z, it is straightforward to compute the apparent magnitudes for a given set of filters taking into account the redshifted spectrum at observed wavelengths. The photo-z estimate is sensitive to the details of observational parameters: the wavelength coverage, the transmission curve of a given filter, the exposure time, the limiting magnitude, and so on. We consider the parameters that match those of the planned Subaru HSC survey: our default filter set is g'r'i'z'y' (hereafter the prime superscripts are sometimes omitted), and the 5σ limiting magnitudes (2'' aperture) are set to g = 26.5, r = 26.4, i = 25.8, z = 24.9, and y = 23.7, respectively, assuming an exposure time of 15 minutes for each passband, 3 days from new moon, and 1.2 air mass at the Subaru Telescope site.¹⁰

We also study how adding other bands, u-band data and NIR data, into the optical data above can improve photo-z accuracies. Having a wider wavelength coverage helps break degeneracies in photo-z estimates; more specifically, it helps discriminate the Lyman break and 4000 angstrom break from the multi-color data. We consider here the u-band data that can be delivered from the Canada–France–Hawaii Telescope (CFHT), and also the J, H, K_s(hereafter K) bands of the planned VIKING (VISTA Kilo-Degree Infrared Galaxy) survey. The 5σ limiting magnitudes are u = 25.0, J = 22.1, H = 21.5, and K = 21.2, respectively. The set of filters and the depths we consider in this paper are summarized in Table 1.

A detailed study of the optimal filter parameters, for example, the number of filters and filter resolution, in terms of minimizing the outlier fraction or photo-z scatters, are found in Jouvel et al. (2010).

Finally, we include statistical errors in the apparent magnitudes. Assuming the sky noise limit, we simply model this magnitude error as Gaussian fluctuations with width given by

$$\begin{equation} \Delta m \simeq 2.5\log \left(1+\frac{1}{\mbox{SN}} \right), \end{equation} \tag{ 6 }$$

where SN is the signal-to-noise ratio for a given galaxy; SN is computed once the galaxy's apparent magnitude and depth in the filter are given. The magnitude error is computed as follows. First, the sky noise is added to the observed flux of a galaxy, causing a deviation from the true flux as f^obs = f₀ + Δf = f₀(1 + 1/SN). Then the magnitude error above is computed as Δm = −2.5log(1 + 1/SN) because m₀ + Δm = −2.5log f₀(1 + 1/SN) + constant. Strictly speaking, even for a Gaussian sky noise, the magnitude error does not obey a Gaussian distribution due to the log-mapping. However, the Gaussian approximation on Δm holds for galaxies with sufficiently high SN values, which we will assume for the following results.

Note that a galaxy which has its apparent magnitude near the limiting magnitude may be excluded from or included in the sample in the presence of the magnitude error. While our simulated galaxies are all i-band selected, some galaxies may have apparent magnitudes below the detection limit in other passbands. We use such an upper limit on the flux in the photo-z estimate, which improves the photo-zs to some extent. In doing this, the flux for such an undetected galaxy in a given filter is set to the magnitude corresponding to the halved limiting flux f_lim/2. Also, we note that the systematic offset of photometry may cause an additional uncertainty in magnitude measurement, which in this paper is ignored. Such a zero-point magnitude offset can be calibrated, for example, by using a spectroscopic redshift subsample (Ilbert et al. 2009).

According to the procedures described above we made a mock photometric catalog containing about 10⁵ galaxies down to magnitude i = 25.8. Although we tried to make a realistic mock catalog based on the COSMOS catalog, some of our treatments may be still oversimplified: for example, we assumed an SSP for galaxy SEDs. The simplified assumptions may make our results somewhat optimistic. A more accurate method to overcome these obstacles would be to use the real data, including spectra, for a representative subsample of imaging galaxies. However, such spectroscopic data, especially for faint galaxies of interest, are still limited; this will be our future work.

By combining multi-passband magnitudes of a given imaging galaxy, its redshift can be estimated without spectroscopic observation—the so-called photo-z. There are various techniques that have been developed: the template fitting method (Sawicki et al. 1997; Bolzonella et al. 2000), the template method combined with prior information (magnitude prior and so on; Benítez 2000; Mobasher et al. 2004), the method including a self-calibration based on a training spectroscopic set (Collister & Lahav 2004, and references therein).

In this paper, we use the publicly available code, Le Phare,¹¹ which is a template fitting method. The photo-z for each galaxy is estimated based on the χ² fitting:

$$\begin{equation} \chi ^2 = \sum _{i}^{N_{f}} \frac{ \left[ f_i^{\rm obs} - \alpha f(T,z,E) \right]^2 }{\sigma _i^2}, \end{equation} \tag{ 7 }$$

where f^obs_i is the observed flux in the ith filter, f(T, z, E) is the model flux which is given as a function of galaxy SED type (T), redshift (z), and the amount of dust extinction (E), and σ_i is the magnitude error. Note that galaxy SED type is modeled according to the method described in Section 3.2. The summation in Equation (7) runs over the number of filters considered (N_f). The extra factor parameter α, which is the same in all the filters, is introduced in Equation (7) because the photo-z is estimated only from colors, the relative amplitudes of fluxes in different filters, not from the absolute fluxes. Therefore, there are (N_f−1) constraints given the data of N_f filters. The best-fit redshift parameter, i.e., the best-fit photo-z, is obtained by minimizing the χ² value with varying other model parameters.

If the location of spectral features such as the Lyman break and the 4000 Å break is captured given the wavelength coverage of observed filters and the magnitude depths taken, the redshift is robustly estimated. On the other hand, a misidentification of the spectral features causes a degeneracy in redshift estimation, often yielding multiple solutions at low and high redshifts. Hence the photo-z method based on broadband photometry generally yields a large fraction of outliers, where the best-fit photo-z can be far from the true redshift. To quantify the photo-z accuracy for each galaxy we will use the following two quantities: (1) the goodness-of-fit parameter for the template fitting, and (2) the width of likelihood function of redshift estimation.

The goodness of fit for the template fitting of a given galaxy may be defined as

$$\begin{equation} \chi ^2_\nu \equiv \frac{\chi ^2_{\rm min}(z_{\rm bf})}{N_{f}-1}, \end{equation} \tag{ 8 }$$

where χ²_min is the minimum χ² value for the best-fit model and redshift, z_bf is the best-fit redshift, and N_f − 1 is the number of colors available. Note that the quantity above, χ²_ν, is not the same as the reduced χ², which is defined as the number of constraints minus the number of model parameters. The number of model parameters are equal for all the galaxies, so χ²_ν gives a measure of the goodness of fit.

We also use the width of likelihood function of redshift estimation for each galaxy defined as

$$\begin{equation} {\rm Var}(z) \equiv \int _0^\infty \!dz (z - z_{\rm bf})^2 p(z) \frac{1}{(1+z_{\rm bf})}, \end{equation} \tag{ 9 }$$

where p(z) is the likelihood function given as p(z) ∝ exp [ − χ²(z)/2], which is normalized to satisfy ∫dzp(z) = 1. We compute the likelihood function p(z) by fixing other model parameters to their best-fit values. The normalization factor (1 + z_bf) is introduced based on the fact that the photo-z accuracy scales with (1 + z). Compared to σ(z), the local standard deviation of photo-z estimation, which is obtained from Δχ² ⩽ 1, the quantity Var(z) is sensitive to the outlier probability with |z − z_bf| ≫ 1 due to the weight (z − z_bf)². Hence for galaxies whose likelihood function has multiple peaks, i.e., multiple redshift solutions, the quantity Var(z) tends to be larger. Similar quantities to Var(z) are also used in previous works (Mobasher et al. 2004; Wolf 2009) where the primary purpose was to improve the photo-z performance for a majority of galaxies. In this paper, we use the figure-of-merit quantity Var(z) mainly for identifying photo-z outliers.

Note that in the following results we use z_bf to construct the redshift distribution of galaxies. An alternative method, which may be less sensitive to photo-z outliers, is summing the photo-z posterior likelihood function p(z) over all the sampled galaxies to obtain the overall redshift distribution (Wittman 2009; Cunha et al. 2009). Calibrating the mean redshift distribution is another important issue that needs to be carefully studied (Ma & Bernstein 2008; Bordoloi et al. 2009), but is beyond the scope of this paper.

Figure 4 demonstrates examples of the photo-z fitting for two simulated galaxies. It can be seen that a galaxy which has a ill-defined photo-z estimate, i.e., a wider likelihood function of redshift estimation, tends to have a larger value of Var(z). In particular, even if the likelihood function locally has a narrow peak around the best-fit redshift, therefore even if the photo-z error looks apparently small, the value Var(z) becomes larger if the likelihood has multiple peaks (i.e., the case of multiple redshift solutions), as demonstrated in the upper panel. On the other hand, a galaxy with reliable photo-z estimate has a small value of Var(z).

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While the quantity Var(z) is empirically defined as an indicator of photo-z outliers, Figure 5 gives a quantitative study of how Var(z) can characterize the photo-z likelihood function. The figure shows the distributions of simulated galaxies in the Var(z)–Δz plane, where Δz is the difference between true and photometric redshifts defined as Δz ≡ (z_bf − z_true)/(1 + z_true) for each galaxy. According to the properties of their photo-z likelihood functions, we divide galaxies into two subsamples: one (thick black line) is defined from galaxies (about 25% of all the galaxies) that have a single peak and therefore a reliable photo-z estimate, while the other (thin gray line) is from galaxies of multiple peaks, respectively. Note that the second- and higher-order peaks are defined as the local maximum values of the likelihood if their heights are higher than 10% of the first peak height. One can find from the figure that the quantity Var(z) nicely separates galaxies that tend to have greater photo-z biases and multiple solutions of photo-zs, i.e., degenerate photo-z estimates; most of the galaxies where Var(z) ≳ 0.1 have multiple peaks.

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We will use the Fisher matrix formalism to estimate accuracies of estimating parameters given the lensing power spectrum. The Fisher matrix is given by

$$\begin{equation} F_{\alpha \beta }=\sum _{\ell =\ell _{\rm min}}^{\ell _{\rm max}} \sum _{i,j,m,n} \frac{\partial C^{\kappa }_{ij}(\ell)}{\partial p_\alpha } {\bf C}^{-1}\!\!\left[ C^\kappa _{ij}(\ell),C^\kappa _{mn}(\ell) \right] \frac{\partial C^{\kappa }_{mn}(\ell)}{\partial p_\beta }, \end{equation} \tag{ 10 }$$

where p_α denotes a set of cosmological parameters, the matrix C denotes the covariance matrix, and C⁻¹ denotes the inverse matrix. In this paper, we simply use the covariance matrix given by the first term in Equation (9) in Takada & Jain (2009), assuming the Gaussian errors on power-spectrum measurements. The Gaussian error assumption is adequate for our purpose because the impact of non-Gaussian errors on parameter estimation is not significant as long as a multi-parameter fitting is considered as shown in Takada & Jain (2009). The marginalized 1σ error on the αth parameter p_α is given by σ²(p_α) = (F⁻¹)_αα, where F⁻¹ is the inverse of the Fisher matrix. Throughout this paper, we set l_min = 5 and l_max = 3000 for the minimum and maximum multipoles as in the summation above. Note that all the parameter forecasts given below are for the lensing tomography combined with the expected Planck information, which is obtained by simply adding the two Fisher matrices of lensing and cosmic microwave background (CMB): ${\boldsymbol{F}}_{\rm WL+CMB}={\boldsymbol{F}}_{\rm WL}+{\boldsymbol{F}}_{\rm CMB}$.

As we explained for Equation (1), the lensing power spectrum is sensitive to the underlying true redshift distribution of galaxies, n(z). For a multi-color imaging survey, however, the distribution n(z) needs to be estimated from the available photo-z information. In this procedure, the photo-z errors affect weak lensing experiments. The most dangerous effect is a systematic bias in parameter estimations: if the inferred redshift distribution has a bias in the mean redshift compared to the true one, the redshift bias may cause significant biases in cosmological parameters. In order to quantify the biases in cosmological parameters caused by photo-z errors, we use the following method based on the Fisher matrix formalism in Huterer & Takada (2005) (see also Appendix B of Joachimi & Schneider 2009 for the detailed derivation):

$$\begin{eqnarray} \delta p_\alpha &=& \sum _{\beta } \big[{\boldsymbol{F}}_{\rm WL+CMB}^{-1}\big]_{\alpha \beta } \sum _{\ell }\sum _{i,j,m,n} \frac{\partial C^\kappa _{ij}(\ell)}{\partial p_\beta } \nonumber \\ && \times {\bf C}^{-1}\big[C^{\kappa }_{ij}(\ell),C^\kappa _{mn}(\ell ^{\prime })\big] \left[ C^{\kappa }_{mn}(\ell ^{\prime })-C^{\kappa,{\rm photo-z}}_{mn}(\ell ^{\prime }) \right],\nonumber \\ \end{eqnarray} \tag{ 11 }$$

where ${\boldsymbol{F}}^{-1}_{\rm WL+CMB}$ is the inverse of the Fisher matrix, and δp_α denotes a bias in the αth parameter, the difference between the best-fit and true values. In the equation above, the spectrum C^κ_mn(l) is the underlying true power spectrum, while C^κ,photo-z_mn(l) is the spectrum estimated from the redshift distribution inferred based on the photo-z information. In the presence of the photo-z errors, generally C^κ_ij ≠ C^κ,photo-z_ij, thereby causing a bias in parameter estimation. We can compute both spectra, C^κ and C^κ,photo-z, from a simulated galaxy catalog for a hypothetical lensing survey. Note that in Equation (11), for simplicity, we have not considered any other nuisance parameters that model other systematic effects such as shape measurement errors (e.g., Huterer et al. 2006) and the inability to make precise model predictions arising from nonlinear clustering and baryonic physics (Huterer & Takada 2005; Rudd et al. 2008; Zentner et al. 2008).

To compute the parameter forecasts, we need to specify a fiducial cosmological model and survey parameters. Our fiducial cosmological model is based on the WMAP 5-year results (Komatsu et al. 2009): the density parameters for dark energy, CDM, and baryon are Ω_de(=0.74), Ω_cdmh²(=0.1078), and Ω_bh²(=0.0196) (note that we assume a flat universe); the primordial power-spectrum parameters are the spectral tilt, n_s(=1), and the normalization parameter of primordial curvature perturbations, A_s ≡ δ²_ζ(=2.3 × 10⁻⁹) (the values in the parentheses denote the fiducial model); and the dark energy equation of state parameter w₀(= −1). We used the publicly available code CAMB developed in Lewis & Challinor (2006) (also see Seljak & Zaldarriaga 1996) to compute the transfer function, and use the fitting formula in Smith et al. (2003) to compute the nonlinear mass power spectrum from which the lensing power spectrum is computed over the relevant range of angular scales.

Our fiducial survey roughly resembles the planned Subaru Hyper-Suprime Cam (HSC) Survey (Miyazaki et al. 2006). We adopt the set of filters (grizy) and the depths in each filter as given in Section 3. We will also study how the results change when the hypothetical Subaru survey is combined with other surveys that deliver the u-band data or/and the NIR data, which especially help to improve the photo-z accuracies. The survey area is throughout assumed to be Ω_s = 2000 deg². The redshift distribution of galaxies is computed for an assumed subsample of galaxies based on the photo-z information.

We may be able to construct a suitable subsample of galaxies in the sense that the impact of photo-z errors are minimized in order not to have more than 100% biases in cosmological parameters compared to the statistical errors. Hence a selection of adequate galaxies is important for weak lensing: this may be attained, for example, by discarding galaxies with ill-defined photo-zs. However, an important fact to keep in mind is that by discarding more galaxies, the statistical accuracy of parameter estimation is degraded due to the increased shot noise contamination in the power-spectrum measurement. Thus, there would be a trade-off point in defining a suitable galaxy subsample in terms of the parameter bias versus the statistical error for a given survey.

Throughout this paper, we work on i-band-selected galaxies assuming that the i-band data are used for the lensing shape measurement as was often done in previous lensing works. For our simulated galaxies, given the limiting magnitude i = 25.8 at 5σ significance, a sufficient number of photometric galaxies are available: the number density for the total galaxies is 80 arcmin⁻². However, not all the galaxies are usable for lensing measurements. First, in order to obtain a reliable shape measurement, galaxies used in the lensing analysis need to be well resolved, requiring them to have sufficient S/N values (say more than 10σ). Second, galaxies with ill-defined photo-z estimates are not useful because including them may cause a significant bias in parameter estimation.

With the above considerations in mind, we will use the following object selections or their combinations to make parameter forecasts.

• 1.  
  The restrictive range of i-band magnitudes (22.5 ⩽ i ⩽ 25). The range is a typical one used in the weak lensing analysis (e.g., Okabe et al. 2009). The faint-end magnitude cut may be imposed such that the selected galaxies have sufficiently high S/N values: i = 25 corresponds to S/N ≃ 10 in our simulations. The bright-end magnitude cut is not important, but usually imposed in practice to avoid galaxies with saturated pixels.
• 2.  
  The photo-z selection. We select only galaxies that have reasonably good photo-z estimates by imposing a threshold on the goodness of fit of photo-z estimation, χ²_ν ⩽ 2 (see Equation (8)). The clipping threshold χ²_ν = 2 is not a unique choice; we selected the value as one working example.
• 3.  
  The restrictive range of photo-zs (0.2 ⩽ z_bf ⩽ 1.5). The spectral features of galaxies in this range of redshifts can be captured relatively well by the wavelength coverage of optical filters. The lower redshift cut is introduced, because there is a strong degeneracy between galaxies at such low redshifts as z ≲ 0.2 and those at higher redshifts, especially in cases where that the u-band data are not available or shallower than the optical data considered in this paper.
• 4.  
  Clipping of photo-z outliers. By discarding galaxies with Var(z) (see Equation (9)) greater than a given threshold, which turn out to be mostly photo-z outliers, we define a subsample from the remaining galaxies that have relatively reliable photo-zs. In the following, we study the performance of this clipping method by varying the clipping threshold values of Var(z).

Table 2 gives the fraction of remaining galaxies compared to the original sample, where galaxies in each subsample are selected with object selection criteria described above for a given set of filters. The column denoted by "χ²_ν ⩽ 2" shows the fraction of galaxies selected when imposing the threshold on the goodness of fit for each galaxy. It is clear that this clipping discards only a small fraction of galaxies for all the cases of filter combinations. The second and third columns show the fractions when further imposing the restricted ranges of photometric redshifts and magnitudes, respectively.

Figure 6 shows the results using different galaxy catalogs with various combinations of the measured passbands (see Table 1), grizy, ugrizy, grizy+JHK, and ugrizy+JHK from the left- to right-column panels, respectively. Each panel in the top row shows the photo-z performance for the simulated objects with i' < 25.8 (>5σ), selected by imposing the condition that the goodness-of-fit χ²_ν < 2 (see Equation (8)). It is clear that, with broadband photometry alone, the photo-z accuracy is limited: a significant contamination of outliers is unavoidable, even when including NIR- and/or u-band data.

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The middle- and lower-row panels show the results obtained by further discarding photo-z outliers based on the clipping method with a given threshold on the quantity Var(z) (Equation (9)). The thresholds are chosen such that 40% or 70% of the objects in each top-row panel, which tend to have ill-defined photo-zs, are discarded, respectively. We found that this clipping method based on the photo-z likelihood function of each galaxy can efficiently discard galaxies with ill-defined photo-zs, especially when combined with the NIR and u-band data.

To be more explicit, the upper panel shows the fraction of galaxies included in the subsample after discarding galaxies with Var(z) greater than a given threshold denoted on the vertical axis. The solid and dotted curves show the results without and with the JHK data in addition to the optical data, grizy. Note that including the CFHT-type u-band data of the assumed depth, as given in Table 1, has little effect on the two results. For a given particular value of Var(z), the dotted curve has more remaining galaxies than the solid curve, implying that these NIR data help improve the photo-z accuracies on an individual galaxy basis, i.e., indicating that the shape of the photo-z likelihood function is narrowed by adding the NIR data for most of the galaxies.

Figure 7 shows a similar result to the previous figure, but for brighter galaxy samples, selected with i < 25 or equivalently with S/N values greater than 10σ in the i band. These brighter galaxies are more suitable for the accurate shape measurement as discussed in Section 4.3. We find that the photo-z accuracy is improved compared to Figure 6.

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We are now in a position to use the photo-z galaxy catalogs, constructed in the preceding sections, to make the trade-off analysis on the number of galaxies within a sample versus how "clean" the tomographic redshift intervals are. Then we study the impact of photo-z errors on parameter estimation assuming the hypothetical lensing tomography experiment.

Figure 8 shows the redshift distributions of each tomographic bin made by subdividing the photo-z galaxies into three intervals of photometric redshifts, z_p < 0.8, and 0.8 < z_p < 1.5, and z_p > 1.5. The redshift intervals are chosen such that each redshift interval contains similar number densities for the original mock catalog of galaxies. Note that the redshift binning is fixed in the following analysis for simplicity, which helps us to compare the results of different galaxy catalogs. Also note that three redshift bins are a minimal choice of lensing tomography for constraining the dark energy equation of state parameter "w" to a reasonable accuracy by efficiently breaking parameter degeneracies in the lensing power spectrum (e.g., Takada & Jain 2004).

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The upper plot in Figure 8 shows the tomographic redshift distributions constructed from different photo-z catalogs in Figure 6, where different panels correspond to the different sets of filters and the different clipping thresholds. The shaded regions in each panel show the photometric redshift distributions of galaxies, which have sharp cutoffs in the distributions due to the sharp redshift binning, while the solid line histograms show the underlying true redshift distributions. As can be found from the top-row panels, if all the galaxies are used, the resulting redshift distributions have significant overlaps between different redshift bins due to a significant contamination of photo-z outliers for any combinations of filters. On the other hand, the middle- and bottom-row panels show that, when 40% or 70% of photo-z outliers are discarded by imposing the corresponding thresholds on Var(z), respectively, the overlaps can be increasingly reduced. In particular, when the optical data are combined with the NIR data such as those expected from the VIKING survey, the resulting subsamples have almost no overlap, if about 70% of galaxies are discarded. We should note that such a clean redshift binning can greatly reduce a possible contamination of the intrinsic ellipticity alignments arising from the physically close pairs of galaxies in the similar redshifts (Takada & White 2004).

The lower plot shows similar results for higher S/N samples with 22.5 < i < 25 whose scatter plots are seen in Figure 7. Again, by using the clipping method with a wider coverage of wavelengths, a clean subsample with almost no overlap between redshift bins can be obtained.

As has been stressed, the weak lensing power spectra are, to the zeroth-order approximation, sensitive to the mean redshifts of each redshift bin, and less to the statistical errors of photo-zs or the detailed shape of redshift distribution. The level of systematic photo-z errors in each tomographic bin is quantified in Figure 9. The central values and error bars in this plot show the bias in mean redshift and the statistical error of the mean redshift, σ(〈Δz〉), where Δz is defined before (see Figure 5 caption) and the average 〈 ⋅ ⋅ ⋅ 〉 denotes the average over all the galaxies in the tomographic redshift bin.¹² Note that, for illustrative purposes, the error bars are scaled for a survey area of 1 arcmin², and therefore the corresponding errors for our fiducial survey area of 2000 deg² are much smaller than plotted, by a factor of $\sqrt{2000\times 60^2}\simeq 2700$.

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The middle- and bottom-row panels are the results of subsamples obtained by further imposing the condition 0.2 ⩽ z_p ⩽ 1.5 or 22.5 ⩽ i ⩽ 25, respectively. It is clear that, without clipping ill-defined photo-z galaxies, the bias and errors are significant. Note that the redshift bias for the no tomography case sometimes becomes smaller than in some tomographic bins (especially the highest redshift bins), because photo-z outliers at low- and high-redshifts cancel out to some extent in the no-tomography case. As can be seen from the middle-row panels, when the restricted redshift range of 0.2 < z < 1.5 is considered, a subsample with very accurate photo-zs is obtained, because spectral features of galaxies, especially the Lyman and 4000 Å breaks, are well captured by the sets of filters in this redshift range.

We next propagate the errors of tomographic redshifts into the dark energy parameter w, expected from the lensing power-spectrum measurements, based on the Fisher matrix formalism described in Section 4.2. To do this we address the following issues.

• 1.  
  The trade-off of dark energy constraint: the statistical accuracy of w, σ(w), versus the offset of the best-fit value from the true value, δw.
• 2.  
  The dark energy trade-off against different subsamples of galaxies.

These can be studied using the simulated galaxy catalogs. The statistical error of w can be reduced by including a greater number of galaxies in the sample for a fixed range of working multipoles (5 ⩽ ℓ ⩽ 3000), because the shot noise is more suppressed. On the other hand, a bias in the best-fit w due to the photo-z errors can be reduced by discarding photo-z outliers, leaving a smaller number of galaxies in the subsample. Hence a trade-off point in σ(w) versus δw may be found by compromising these competing effects.

Figure 10 shows the marginalized error σ(w) and the amount of bias |δw| as a function of number densities of galaxies included in the corresponding galaxy subsamples. Again, note that we considered the lensing tomography with three tomographic bins for a sky coverage of 2000 deg². The smaller number densities in the horizontal axis correspond to subsamples of galaxies where more galaxies with ill-defined photo-zs are discarded by imposing more stringent thresholds on Var(z), i.e., smaller threshold values of Var(z), in the clipping method. The different curves are for different combinations of filters. The error σ(w) is computed from the underlying true redshift distribution, i.e., for the case with perfect photo-zs, therefore specified by the number density in the horizontal axis. When δw ⩾ σ(w), the best-fit value of w can be different from the true value by more than the 1σ error; even if the true model has the cosmological constant (w = −1), the result of w ≠ −1 may be falsely inferred. Hence, a minimal requirement on photo-z accuracies can be assessed from the condition δw ⩽ σ(w).

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First, the plot shows that, as the subsample is restricted to galaxies with more accurate photo-zs, i.e., the smaller number densities, the bias in w is reduced to some extent. On the other hand, the error σ(w) is only slightly degraded because the constraint comes mainly from the sample variance limited regime for a given range of working multipoles (5 ⩽ l ⩽ 3000).

It is also shown that the bias can be reduced by adding the NIR and/or u-band data. However, the broadband data alone may not be sufficient to reduce the bias. The optimal range of redshifts needs to be considered, and a brighter subsample whose galaxies have higher S/N values in each filter is preferred to sufficiently reduce the bias, as implied from the middle and right panels. Depending on the available set of filters, the compromise point can be obtained around the number densities $\bar{n}_g=[10,30]$ arcmin⁻², i.e., more than 60% of ill-defined photo-z galaxies need to be discarded. It is also worth noting that combining the lensing constraints with other dark energy probes such as the baryon acoustic oscillation experiment may allow us to further calibrate photo-z errors by breaking parameter degeneracies.

We have so far paid special attention to how to eliminate photo-z outliers in order to obtain a subsample of galaxies suitable for tomographic lensing measurements. However, due to the limitation of photo-z accuracies, there may remain a residual bias in the tomographic redshift bins, even if photo-z outliers are completely removed. To study this, Figure 11 shows the results obtained by artificially discarding photo-z outliers according to the clipping criteria

$$\begin{equation} \log \frac{1+z_p}{1+z_s}>\pm t, \end{equation} \tag{ 12 }$$

with t = 0.1, 0.3 and 0.5, respectively. The figure shows that a bias in w cannot be fully eliminated even if the outliers are completely discarded. This implies that there remains a residual bias in the mean redshift for each tomographic bin due to asymmetric photo-z errors around z_p = z_s, therefore the residual biases would need to be calibrated, e.g., by using a spectroscopic training subsample (e.g., Ma & Bernstein 2008).

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An alternative way to identify photo-z outliers is using angular cross-correlations of galaxies between different photo-z bins (Newman 2008; Erben et al. 2009; Zhang et al. 2009; Schulz 2009). As implied in Figure 8, photo-z errors cause overlaps of galaxies between different redshift bins. Therefore, photo-z errors may cause non-vanishing cross-correlations of galaxies between different photo-z bins, if the galaxies indeed have similar true redshifts and therefore are physically correlated with each other. In other words, the cross-correlations can, albeit statistical, be used to monitor the contamination of photo-z outliers. In this subsection, we use our simulated photo-z catalogs to estimate the expected S/N values for measuring the cross-correlations assuming the same survey parameters we have considered.

Assuming the Limber approximation, the angular power spectra of galaxies in the ith and jth photo-z bins are given as

$$\begin{equation} C^{gg}_{ij}(\ell) = \int _0^\infty \!\!dz \frac{dz}{d\chi } \frac{b_i b_j}{\chi ^2} \frac{n_i(z)n_j(z)}{\bar{n}_i \bar{n}_j} P_\delta \!\!\left(\frac{\ell }{\chi }, z \right) + \frac{\delta _{ij}^K}{\bar{n}_i}, \end{equation} \tag{ 13 }$$

where n_i(z) is the underlying true redshift distribution for the ith photo-z bin and $\bar{n}_i$ is its mean number density. In the following, we consider a sharp redshift binning in photo-z space, however, the underlying true distributions generally have overlaps due to photo-z errors. We simply assume that the galaxy distribution in the ith bin is related to the matter distribution via constant bias parameter b_i, which is taken to b_i = 1 for all the photo-z bins for simplicity. To make this assumption reasonable, we restrict the following analysis to a range of low multipoles l < 500. Note that the cross power spectra are not affected by shot noise.

The strength of cross-correlations or redshift leakages can be quantified by the cross-correlation coefficients at each multipole:

$$\begin{equation} \mu _{ij}(\ell) = \frac{C_{ij}^{gg}(\ell)}{\sqrt{C_{ii}^{gg}(\ell)C_{jj}^{gg}(\ell)}}. \end{equation} \tag{ 14 }$$

The coefficient μ_ij ≃ 1 implies significant cross-correlations between the ith and jth bins compared to their auto-spectra, while μ_ij = 0 means no cross-correlation or no leakage of photo-z outliers into different bins. The total S/N values expected for measuring the cross-correlations can be estimated as

$$\begin{equation} \left(\frac{\mbox{S}}{\mbox{N}}\right)^2_{ij} \equiv \sum _{\ell =\ell _{\rm min}}^{\ell _{\rm max}} f_{\rm sky}(2\ell +1) \frac{C_{ij}^2}{ C_{ij}^2 + C_{ii} C_{jj} }. \end{equation} \tag{ 15 }$$

Here we simply assume the Gaussian covariances to model the statistical errors in measuring cross power spectra from a survey. Note that the error covariance includes the shot noise contamination via the auto spectra C_ii and C_jj. For the minimum and maximum multipoles used in the summation, we adopt ℓ_min = 5 and ℓ_max = 500, respectively.

Figure 12 studies the angular cross-correlations for our fiducial survey parameters with 2000 deg² coverage. Here we consider the photo-z galaxy catalog selected with ugrizy and χ²_ν < 2, and adopt 35 redshift bins over 0 < z_p < 3.5 with the bin width dz = 0.1 corresponding to a typical photo-z error on individual galaxy basis. The upper-left triangle in each panel shows the correlation coefficients of cross power spectra between the two different photo-z bins, μ_ij. The coefficients have large values around the diagonal terms, i.e., z_pi ≃ z_pj, because the photo-z errors cause significant overlaps between neighboring redshift bins. Compared with the results in Figure 6, one can find that photo-z outliers cause some isolated regions with significant correlation coefficients.

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The lower-right triangle shows the expected S/N, for measuring cross-correlations. Sufficiently high S/N values, say greater than 10, can be expected for redshift bins that have high correlation coefficients. Thus, monitoring the cross-correlations between different photo-z bins may allow us to further identify photo-z outliers, in a statistical sense. However, the genuine ability of the cross-correlation method to eliminate outliers or calibrate photo-z errors needs to be more carefully studied.

Finally, we remark on a more quantitative work done by Schulz (2009), which studied, based on mock simulations, the use of cross-correlations of photometric galaxies with an overlapping spectroscopic sample to calibrate the redshift distribution of the photometric galaxies without using the photo-z information. While using cross-correlations to identify photo-z outliers is not the main purpose of this paper, the promising result shown is that the redshift distribution can be reconstructed well by using a sufficiently large spectroscopic sample. However, one of the limiting factors found is that the reconstruction requires a sufficiently fine binning of spectroscopic redshifts, which tends to make the cross-correlation measurements noisy. Therefore, it would be interesting to study how the method can be further refined by combining the cross-correlation method and the photo-z information, and the combined method could relax requirements on the size of the spectroscopic calibration sample. We also note that, as pointed out in Bernstein & Huterer (2010), the cross-correlation method is affected by the lensing magnification bias, which may cause an apparent correlation between foreground and background galaxies even if there are no photo-z errors to cause redshift overlaps. This effect also needs to be included, and mock simulations would be useful for such a study of the cross-correlation method.