Cosmology from the Chinese Space Station Optical Survey (CSS-OS)

Following Cao et al. (2018), we adopt a galaxy redshift distribution n(z) derived from the COSMOS catalog for the CSS-OS photometric survey (Capak et al. 2007; Ilbert et al. 2009). This catalog contains about 220,000 galaxies in 2 deg², and has a similar magnitude limit as the CSS-OS with i ≤ 25.2 for galaxy observation. Although the survey area of the CSS-OS is much larger, it can represent the redshift and magnitude distributions, and galaxy types observed by the CSS-OS. The CSS-OS redshift distribution derived from this catalog is shown by the dotted line in Figure 2. We can find that the redshift distribution has a peak around z = 0.6, and can extend to z ∼ 4.

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In order to extract more information from the weak lensing data, we divide the redshift range into different photo-z tomographic bins, and study the auto and cross power spectra of these bins. As shown in Figure 2, for instance, we divide the redshift range into four photo-z bins (gray vertical lines). The first three bins have equal intervals with Δ_z = 0.6, and the last bin occupies the rest of the redshift range of redshift distribution.¹⁴ Note that the fitting results of the cosmological parameters also depend on the number of photo-z bins (e.g., Huterer 2002). Under the assumptions of the systematics as discussed in Section 2.2, we find that although a greater number of bins may further improve the constraint results, the improvement is not very significant (averaging a factor of ∼1.3 and ∼1.5 on the standard deviations of cosmological parameters for the five and six photo-z bins cases, respectively). Considering the purpose of this work, a four-bin division is adequate for our study. A greater number of bins can be used in the real data analysis, which may help to improve the constraints on the cosmological parameters.

The real galaxy redshift distribution in the ith photoz bin can be expressed as (e.g., Ma et al. 2006)

$$\begin{eqnarray}&&{n}_{i}(z)={\int }_{{z}_{{\rm{p}},{\rm{l}}}^{i}}^{{z}_{{\rm{p}},{\rm{u}}}^{i}}{{dz}}_{{\rm{p}}}\ n(z)\ p({z}_{{\rm{p}}}| z),\end{eqnarray} \tag{ 1 }$$

where ${z}_{{\rm{p}},{\rm{l}}}^{i}$ and ${z}_{{\rm{p}},{\rm{u}}}^{i}$ are the lower and upper limits of the ith photo-z bin, n(z) is the total redshift distribution, and $p({z}_{{\rm{p}}}| z)$ is the photo-z distribution function given the real redshift z. We assume it takes the form of

$$\begin{eqnarray}&&p({z}_{{\rm{p}}}| z)=\displaystyle \frac{1}{\sqrt{2\pi }{\sigma }_{z}}\ \exp \left[-\displaystyle \frac{{\left(z-{z}_{{\rm{p}}}-{\rm{\Delta }}z\right)}^{2}}{2{\sigma }_{z}^{2}}\right],\end{eqnarray} \tag{ 2 }$$

where Δz and σ_z are the redshift bias and scatter, respectively, which vary as functions of redshift. In this work, we assume that they are constants in different photo-z bins, and treat them as free parameters when fitting the data. Then the n_i(z) in Equation (1) can be reduced to

$$\begin{eqnarray}&&{n}_{i}(z)=\displaystyle \frac{1}{2}n(z)\left[\mathrm{erf}({x}_{{\rm{u}}}^{i})-\mathrm{erf}({x}_{{\rm{l}}}^{i})\right],\end{eqnarray} \tag{ 3 }$$

where erf(x) is the error function, and

$$\begin{eqnarray}&&{x}_{{\rm{u}}/{\rm{l}}}^{i}=\left({z}_{{\rm{p}},\ {\rm{u}}/{\rm{l}}}^{i}-z+{\rm{\Delta }}z\right)/\sqrt{2}{\sigma }_{z}.\end{eqnarray} \tag{ 4 }$$

Note that we always have $n(z)={\sum }_{i}{n}_{i}(z)$, no matter what form n_i(z) takes.

In Figure 2, the solid blue, green, orange, and red lines show the n_i(z) for the four photo-z bins with Δz = 0 and σ_z = 0.05. In the COSMOS catalog, we find that there are about 208,000 galaxies with σ_z ≤ 0.05, which take about 95% of the whole sample (Cao et al. 2018). This number can be used, as a reference, to estimate the galaxy number density in the CSS-OS weak lensing survey. We will adopt the n_i(z) shown in Figure 2 and the estimated number density in the following WL discussion.

Considering intrinsic alignments and systematics, the measured shear power spectrum at a given multipole ℓ for the ith and jth tomographic bins can be estimated by (e.g., Huterer et al. 2006; Amara & Refregier 2008)

$$\begin{eqnarray}&&{\widetilde{C}}_{\gamma }^{{ij}}({\ell })=(1+{m}_{i})(1+{m}_{j}){C}_{\gamma }^{{ij}}({\ell })+{\delta }_{{ij}}\displaystyle \frac{{\sigma }_{\gamma }^{2}}{{\bar{n}}_{i}}+{N}_{\mathrm{add}}^{{ij}}({\ell }),\end{eqnarray} \tag{ 5 }$$

where ${C}_{\gamma }^{{ij}}({\ell })$ is named as the signal power spectrum, which is composed of three components (e.g., Hildebrandt et al. 2016; Joudaki et al. 2017; Troxel et al. 2017)

$$\begin{eqnarray}&&{C}_{\gamma }^{{ij}}({\ell })={P}_{\kappa }^{{ij}}({\ell })+{C}_{\mathrm{II}}^{{ij}}({\ell })+{C}_{\mathrm{GI}}^{{ij}}({\ell }).\end{eqnarray} \tag{ 6 }$$

Here ${P}_{\kappa }^{{ij}}({\ell })$ is the convergence power spectrum, which is the desired galaxy shear power spectrum for cosmological analysis. The ${C}_{\mathrm{II}}^{{ij}}({\ell })$ and ${C}_{\mathrm{IG}}^{{ij}}({\ell })$ are Intrinsic–Intrinsic (II), and Gravitational-Intrinsic (GI) power spectra, respectively. They are accounting for the intrinsic galaxy alignment effects, that "II" denotes the correlation of the intrinsic ellipticities between neighboring galaxies, and "GI" means the correlation between the intrinsic ellipticity of a foreground galaxy and the gravitational shear of a background galaxy (see, e.g., Joachimi et al. 2016). Assuming Limber and flat-sky approximations (Limber 1954), we have

$$\begin{eqnarray}&&{P}_{\kappa }^{{ij}}({\ell })={\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \ \displaystyle \frac{{q}_{i}(\chi ){q}_{j}(\chi )}{{r}^{2}(\chi )}{P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right),\end{eqnarray} \tag{ 7 }$$

where χ is the comoving radial distance, χ_H = χ(z = 5) denotes the horizon distance, and r(χ) is the comoving angular diameter distance. P_m is the matter power spectrum, which is calculated by the halo model (Cooray & Sheth 2002). q_i(χ) is the lensing weighting function in the ith tomographic bin

$$\begin{eqnarray}&&{q}_{i}(\chi )=\displaystyle \frac{3{{\rm{\Omega }}}_{{\rm{m}}}{H}_{0}^{2}}{2{c}^{2}}\displaystyle \frac{r(\chi )}{a(\chi )}{\int }_{\chi }^{{\chi }_{{\rm{H}}}}d\chi ^{\prime} \,{n}_{i}(\chi ^{\prime} )\displaystyle \frac{r(\chi ^{\prime} -\chi )}{r(\chi ^{\prime} )},\end{eqnarray} \tag{ 8 }$$

where H₀ = 100 h is the Hubble constant, c is the speed of light, and a = 1/(1 + z) is the scale factor. n_i(χ) is the normalized source galaxy distribution of the ith tomographic bin, and $\int {n}_{i}(\chi )d\chi =1$. The intrinsic–intrinsic and gravitational-intrinsic power spectra are given by

$$\begin{eqnarray}&&{C}_{\mathrm{II}}^{{ij}}({\ell })={\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \displaystyle \frac{{n}_{i}(\chi ){n}_{j}(\chi ){F}_{i}(\chi ){F}_{j}(\chi )}{{r}^{2}(\chi )}{P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right),\end{eqnarray} \tag{ 9 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{C}_{\mathrm{GI}}^{{ij}}({\ell }) & = & {\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \displaystyle \frac{{q}_{i}(\chi ){n}_{j}(\chi ){F}_{j}(\chi )}{{r}^{2}(\chi )}{P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right)\\ & & +\,{\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \displaystyle \frac{{q}_{j}(\chi ){n}_{i}(\chi ){F}_{i}(\chi )}{{r}^{2}(\chi )}{P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right).\end{array}\end{eqnarray} \tag{ 10 }$$

Here F_i(χ) is written as

$$\begin{eqnarray}&&{F}_{i}(\chi )=-{A}_{\mathrm{IA}}{C}_{1}{\rho }_{{\rm{c}}}\displaystyle \frac{{{\rm{\Omega }}}_{{\rm{m}}}}{D(\chi )}{\left(\displaystyle \frac{1+z}{1+{z}_{0}}\right)}^{{\eta }_{\mathrm{IA}}}{\left(\displaystyle \frac{{L}_{i}}{{L}_{0}}\right)}^{{\beta }_{\mathrm{IA}}},\end{eqnarray} \tag{ 11 }$$

where ${C}_{1}=5\times {10}^{-14}\ {h}^{-2}{M}_{\odot }^{-1}{\mathrm{Mpc}}^{3}$, ρ_c is the present critical density, D(χ) is the linear growth factor normalized to unity at z = 0, and z₀ = 0.6 and L₀ are pivot redshift and luminosity, respectively. A_IA, η_IA, and β_IA are free parameters in this model. For simplicity, we fix β_IA = 0 here, i.e., we do not consider luminosity dependence, because the change of the average luminosity can be ignored across different tomographic bins (Hildebrandt et al. 2016; Joudaki et al. 2017). The fiducial values of A_IA and η_IA are set to be −1 and 0, respectively, when producing mock data.

In the shot-noise term ${N}_{\gamma }^{\mathrm{shot}}$ of Equation (5), ${\sigma }_{\gamma }^{2}=0.04$ is the shear variance per component caused by intrinsic ellipticity and measurement error, and ${\bar{n}}_{i}$ is the average galaxy number density in a given tomographic bin per steradian. According to the mock CSS-OS catalog, we assume ∼100,000 galaxies with σ_z ≤0.05 per deg² (with ∼9.0% outlier fraction) can be observed by the CSS-OS. This corresponds to a total density ∼28 arcmin⁻², and ${\bar{n}}_{i}=7.9$, 11.5, 4.6, and 3.7 arcmin⁻² for the four redshift bins. Note that the number density estimated here can be smaller in the real observations, considering the complicated instrumental and astrophysical uncertainties. We need to perform realistic simulations to evaluate these effects in future work.

Besides, we also consider the systematic errors in the CSS-OS weak lensing measurements. In Equation (5), m_i accounts for the effect of the multiplicative error in the ith redshift bin, which is averaged over all directions and galaxies in that bin. We assume it varies independently in different tomographic bins, and treat it as a free parameter in a given photo-z bin in the fitting process (Troxel et al. 2017). We set m_i = 0 as a fiducial value in each bin when generating mock data.

The ${N}_{\mathrm{add}}^{{ij}}$ in Equation (5) is the additive error, which can be generated by the anisotropy of the PSF (Huterer et al. 2006). In principle, ${N}_{\mathrm{add}}^{{ij}}({\ell })$ could vary at different scales and in different redshift bins, and also can appear in the correlations between bins (Huterer et al. 2006; Amara & Refregier 2008; Amara et al. 2010). For simplicity, we adopt an average constant ${\bar{N}}_{\mathrm{add}}$ over all scales of all auto and cross shear power spectra for different tomographic bins (Zhan 2006). Based on the estimates of STEP (the Shear Testing Programme) and GREAT10 (the Gravitational Lensing Accuracy Testing 2010), the ${\bar{N}}_{\mathrm{add}}$ can be controlled within 10⁻¹⁰ with S/N ∼ 10 in the Stage IV weak lensing surveys (Heymans et al. 2006; Massey et al. 2007, 2013; Kitching et al. 2012). We find that the S/N of most CSS-OS galaxies is around 7 for the most important g, r, and i bands (Cao et al. 2018). Although the ${\bar{N}}_{\mathrm{add}}$ are expected to achieve ∼10⁻¹⁰ when adding up the flux of all seven CSST imaging bands, we would take a conservative estimate with ${\bar{N}}_{\mathrm{add}}={10}^{-9}$ as a moderate value in this work. We will discuss ${\bar{N}}_{\mathrm{add}}={10}^{-8}$ and 10⁻¹⁰ as pessimistic and optimistic cases.

The covariance matrix of the shear power spectra can be estimated by Hu & Jain (2004), Huterer et al. (2006), and Joachimi et al. (2008)

$$\begin{eqnarray}&&\begin{array}{l}\mathrm{Cov}\left[{\widetilde{C}}_{\gamma }^{{ij}}({\ell }),{\widetilde{C}}_{\gamma }^{{mn}}({\ell }^{\prime} )\right]\\ =\,\displaystyle \frac{{\delta }_{{\ell }{\ell }^{\prime} }}{{f}_{\mathrm{sky}}{\rm{\Delta }}l(2{\ell }+1)}\left[{\widetilde{C}}_{\gamma }^{{im}}({\ell }){\widetilde{C}}_{\gamma }^{{jn}}({\ell })+{\widetilde{C}}_{\gamma }^{{in}}({\ell }){\widetilde{C}}_{\gamma }^{{jm}}({\ell })\right],\end{array}\end{eqnarray} \tag{ 12 }$$

where ${\widetilde{C}}_{\gamma }^{{ij}}({\ell })$ is the observed shear power spectrum given by Equation (5), and f_sky is the sky coverage fraction of the survey. The CSS-OS can cover 17,500 deg², but after removing masked area (covering image defects, reflections, ghosts, etc.), we assume an effective area of ∼15,000 deg² can be used in the data analysis (Hildebrandt et al. 2016; Abbott et al. 2017; Troxel et al. 2017).

In Figure 3, we show the power spectra discussed above for the one photo-z bin case (i.e., no tomographic bins). We assume that the shot-noise and additive terms can be eliminated in the data analysis,¹⁵ and thus the mock data points of the ${C}_{\gamma }^{{ij}}$ can be derived as shown in blue circles with error bars. A random Gaussian distribution derived from the covariance matrix is added to each data point. The ${C}_{\gamma }^{{ij}}$ for the four photo-z bins are shown in Figure 4. To avoid the nonlinear effects, we only take account of the data at ℓ < 3000.

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We adopt the zCOSMOS catalog (DR3 release) to simulate the CSST spectroscopic survey result. The zCOSMOS redshift survey observes in the COSMOS field using the VIMOS spectrograph mounted at the Melipal Unit Telescope of the VLT (Lilly et al. 2007, 2009). It covers 1.7 deg² with a magnitude limit I_AB ≃ 22.5, which is close to survey depth of the CSS-OS. The spectral coverage is 5550–9450 Å with a spectral resolution R ∼ 600. There are about 20,000 sources in this catalog, and after selecting high-quality data suggested by the zCOSMOS team, we obtain about 16,600 sources (80% of the total) with reliable spectroscopic redshifts.

The redshift distribution of this mock CSST spectroscopic catalog is shown in Figure 5. We can see that it has a peak at z = 0.3 − 0.4, and can extend to z ∼ 2.5. Since there are not many galaxies at high redshifts, we only consider the sources at z < 1.5 in the galaxy clustering analysis (∼40 galaxies are out of this range). In order to study the evolution of the equation of state (EoS) of dark energy and other cosmological parameters, we divide the redshift range into five tomographic bins with Δ_z = 0.3 assuming no spec-z outliers for simplicity. The galaxy number fractions in the five spectroscopic bins are 0.17, 0.37, 0.35, 0.10, and 0.01, respectively. Note that a greater number of redshift bins can be used in the real data analysis, depending on the number density of observed galaxies.

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The redshift-space galaxy power spectrum in (k, μ) dimensions can be measured by the CSST spectroscopic survey, which can be expanded in Legendre polynomials (Ballinger et al. 1996; Taylor & Hamilton 1996)

$$\begin{eqnarray}&&{P}_{g}^{({\rm{s}})}(k,\mu )=\displaystyle \sum _{{\ell }}{P}_{{\ell }}^{g}(k){{ \mathcal L }}_{{\ell }}(\mu ),\end{eqnarray} \tag{ 13 }$$

where (s) denotes the redshift space, μ = k_∥/k is the cosine of the angle between the direction of wavenumber k and the line of sight, ${{ \mathcal L }}_{{\ell }}(\mu )$ is the Legendre polynomials that only the first few nonvanishing orders ℓ = (0, 2, 4) are considered in the linear regime, and ${P}_{{\ell }}^{g}(k)$ is the multipole moments of the power spectrum. Considering the Alcock–Paczynski effect (Alcock & Paczynski 1979), the galaxy multipole power spectra are given by

$$\begin{eqnarray}&&{P}_{{\ell }}^{g}(k)=\displaystyle \frac{2{\ell }+1}{2{\alpha }_{\perp }^{2}{\alpha }_{\parallel }}{\int }_{-1}^{1}d\mu \,{P}_{g}^{({\rm{s}})}(k^{\prime} ,\mu ^{\prime} ){{ \mathcal L }}_{{\ell }}(\mu ).\end{eqnarray} \tag{ 14 }$$

Here ${\alpha }_{\perp }={D}_{{\rm{A}}}(z)/{D}_{{\rm{A}}}^{\mathrm{fid}}(z)$ and ${\alpha }_{\parallel }={H}^{\mathrm{fid}}(z)/H(z)$ are the scaling factors in the transverse and radial directions, respectively, and the superscript "fid" means the quantities in the fiducial cosmology. The $k^{\prime} =\sqrt{{k}_{\parallel }^{{\prime} 2}+{k}_{\perp }^{{\prime} 2}}$ and $\mu ^{\prime} ={k}_{\parallel }^{{\prime} }/k^{\prime}$ are the apparent wavenumber and cosine of angle, where ${k}_{\parallel }^{{\prime} }={k}_{\parallel }/{\alpha }_{\parallel }$ and ${k}_{\perp }^{{\prime} }={k}_{\perp }/{\alpha }_{\perp }$. Assuming there is no peculiar velocity bias, the apparent redshift-space galaxy power spectrum can be written as

$$\begin{eqnarray}&&{P}_{g}^{({\rm{s}})}(k^{\prime} ,\mu ^{\prime} )={P}_{g}(k^{\prime} ){\left(1+\beta {\mu }^{{\prime} 2}\right)}^{2}\,{ \mathcal D }(k^{\prime} ,\mu ^{\prime} ),\end{eqnarray} \tag{ 15 }$$

where ${P}_{g}(k^{\prime} )={b}_{g}^{2}\,{P}_{{\rm{m}}}(k^{\prime} )$ is the apparent real-space galaxy power spectrum, b_g is the galaxy bias, P_m is the matter power spectrum, and β = f/b_g where $f=d\,\mathrm{ln}D(a)/d\,\mathrm{ln}\,a$ is the growth rate. ${ \mathcal D }(k^{\prime} ,\mu ^{\prime} )$ is the damping term at small scales, which is given by

$$\begin{eqnarray}&&{ \mathcal D }(k^{\prime} ,\mu ^{\prime} )=\exp \left[-{\left(k^{\prime} \mu ^{\prime} {\sigma }_{{\rm{D}}}\right)}^{2}\right].\end{eqnarray} \tag{ 16 }$$

Here ${\sigma }_{{\rm{D}}}^{2}={\sigma }_{v}^{2}+{\sigma }_{R}^{2}$, where σ_v = σ_v0/(1 + z) is the velocity dispersion (Scoccimarro 2004; Taruya et al. 2010), and we assume σ_v0 = 7 Mpc/h for the measurements of emission line galaxies in the CSS-OS (Blake et al. 2016; Joudaki et al. 2017). It will be set as a free parameter in the fitting process. The σ_R = c σ_z/H(z) is the smearing factor for the small scales below the spectral resolution of spectroscopic surveys, where H(z) is the Hubble parameter, and ${\sigma }_{z}=(1+z){\sigma }_{z}^{0}$ (Wang et al. 2009). Based on the instrumental design of the CSST, we set ${\sigma }_{z}^{0}=0.002$ as the accuracy of the spectral calibration. Note that this accuracy could be larger in the real observation, and we need to run simulations in the future for further confirmation.

After adding the shot-noise term and systematics, we obtain the total multipole power spectra of the ath spec-z bin

$$\begin{eqnarray}&&{\widetilde{P}}_{{\ell }}^{g,a}(k)={P}_{{\ell }}^{g,a}(k)+\displaystyle \frac{1}{{\bar{n}}_{g}^{a}}+{N}_{\mathrm{sys}}^{g,a},\end{eqnarray} \tag{ 17 }$$

where ${\bar{n}}_{g}^{a}$ is the galaxy number density in the ath spec-z bin. We take an average value ${\bar{n}}_{g}^{a}$ in a given redshift bin, and find that ${\bar{n}}_{g,\mathrm{ori}}^{a}=3.4\times {10}^{-2}$, 1.1 × 10⁻², 5.5 × 10⁻³, 1.2 × 10⁻³ (Mpc/h)⁻³, and 7.9 × 10⁻⁵ from the mock CSS-OS catalog for the five spec-z bins, respectively. Besides, we consider an effective redshift factor ${f}_{\mathrm{eff}}^{{z}_{s}}$ to account for the fraction of galaxies that can achieve the required accuracy of the CSS-OS spec-z calibration with ${\sigma }_{z}^{0}=0.002$ in the slitless grating observations (Wang et al. 2010). We assume the fraction decreases as the redshift increases, i.e., ${f}_{\mathrm{eff}}^{{z}_{{\rm{s}}}}={f}_{\mathrm{eff}}^{{z}_{{\rm{s}}},0}/(1+z)$, where we set ${f}_{\mathrm{eff}}^{{z}_{{\rm{s}}},0}=0.3$, 0.5, and 0.7, respectively, as the pessimistic, moderate, and optimistic cases. So the final number density in the ath spec-z bin should be ${\bar{n}}_{g}^{a}={f}_{\mathrm{eff}}^{{z}_{s},a}\,{\bar{n}}_{g,\mathrm{ori}}^{a}$. The systematics ${N}_{\mathrm{sys}}^{g}$ due to instrumentation effects of the CSST slitless gratings are assumed to be a constant in different tomographic bins, or it can be seen as the average value ${\bar{N}}_{\mathrm{sys}}^{g}$ over all redshift bins and scales. Combining the three assumptions of ${f}_{\mathrm{eff}}^{{z}_{{\rm{s}}},0}$, we jointly assume three values, i.e., ${\bar{N}}_{\mathrm{sys}}^{g}={10}^{5}$, 5 × 10⁴, and 10⁴ (Mpc/h)³, as pessimistic, moderate, and optimistic cases in the CSST galaxy clustering surveys.

The errors of the multipole power spectra in the ath bin are then given by

$$\begin{eqnarray}&&{\sigma }_{{P}_{{\ell }}^{g}}^{a}(k)=2\pi \sqrt{\displaystyle \frac{1}{{V}_{{\rm{S}}}^{a}\,{k}^{2}{\rm{\Delta }}k}}\ {\widetilde{P}}_{{\ell }}^{g,a}(k),\end{eqnarray} \tag{ 18 }$$

where ${V}_{{\rm{S}}}^{a}$ is the survey volume of the ath bin. We assume a 15,000 deg² sky coverage for the galaxy clustering spectroscopic survey, which is the same as the photometric imaging survey area used in the weak lensing, after masking the area with bad measurements. For a constant n_g in a given redshift bin, the corresponding effective survey volume for a multipole component of power spectrum can be defined as

$$\begin{eqnarray}&&{V}_{\mathrm{eff}}^{{\ell },a}(k)\equiv {\left[1+\displaystyle \frac{1}{{\bar{n}}_{g}^{a}{P}_{{\ell }}^{g,a}(k)}+\displaystyle \frac{{N}_{\mathrm{sys}}^{g,a}}{{P}_{{\ell }}^{g,a}(k)}\right]}^{-2}{V}_{{\rm{S}}}^{a}.\end{eqnarray} \tag{ 19 }$$

In Figure 6, we show the mock multipole power spectra P₀^g, P₂^g, and P₄^g. In order to avoid the nonlinear effect, we only consider the data at k < 0.2 h/Mpc. The errors of the mock data are generated by assuming ${f}_{\mathrm{eff}}^{{z}_{s},0}=0.5$ and ${\bar{N}}_{\mathrm{sys}}^{g}=5\,\times {10}^{4}$ (Mpc/h)³. In Figure 7, the V_eff of the P₀^g for each redshift bin are shown. The dashed, solid, and dotted curves are for the optimistic, moderate, and pessimistic cases.

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In addition to the redshift-space power spectrum, the CSS-OS three-dimensional (3D) galaxy clustering data also can be analyzed using other methods, such as topology (Park & Kim 2010), tomographic Alcock–Paczynski (Li et al. 2016), and three-point correlation function (Takada & Jain 2003), that can extract more cosmological information. We will discuss these methods in future work.

We first consider the two-dimensional (2D) angular galaxy power spectrum for the ath and bth spec-z bins, which can be derived from the 3D galaxy power spectrum by assuming Limber approximation (Limber 1954; Kaiser 1992; Hu & Jain 2004)

$$\begin{eqnarray}&&{C}_{g}^{{ab}}({\ell })={\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \ \displaystyle \frac{{b}_{g}^{2}(k,\chi ){\hat{n}}_{g}^{a}(\chi ){\hat{n}}_{g}^{b}(\chi )}{{r}^{2}(\chi )}{P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right),\end{eqnarray} \tag{ 20 }$$

where b_g(k, χ) is the galaxy bias, which depends on the scales and redshifts. For simplicity, we assume it is a constant at different scales and only varies as a function of redshift b_g(z) when producing mock data. ${\hat{n}}_{g}^{a}(\chi )$ is the normalized galaxy redshift distribution for the ath spec-z bin, that we have $\int {\hat{n}}_{g}^{a}(\chi )d\chi =1$. Incorporating the shot-noise term and systematics, the total angular galaxy power spectrum can be written as

$$\begin{eqnarray}&&{\widetilde{C}}_{g}^{{ab}}({\ell })={C}_{g}^{{ab}}({\ell })+\displaystyle \frac{{\delta }_{{ab}}}{{\bar{n}}_{g}^{{\prime} a}}+{N}_{\mathrm{sys}}^{{\prime} \,g,{ab}}({\ell }),\end{eqnarray} \tag{ 21 }$$

where ${\bar{n}}_{g}^{{\prime} a}$ is the average galaxy number density in the ath spec-z bin per steradian, and it is found to be 0.46, 1.0, 0.94, 0.29, and 0.02 arcmin⁻² for the five bins (∼2.7 arcmin⁻² totally), respectively. ${N}_{\mathrm{sys}}^{{\prime} \,g,{ab}}({\ell })$ is the systematic noise for the angular galaxy power spectrum, and we assume it is a constant for different scales and redshift bins. We find that the result is not sensitive to the ${N}_{\mathrm{sys}}^{{\prime} \,g}$ as long as it is less than 10⁻⁸, i.e., the shot-noise term (the second term) in Equation (21) is relatively large and dominant.

Note that the cross power spectra vanish in the spectroscopic surveys (e.g., the case we discuss here), because there is no overlapping region between spec-z bins. However, for the photometric surveys, both the auto and cross angular galaxy power spectra are important, and can be used to calibrate the galaxy bias and redshift distributions when cooperating with the weak lensing survey (Hu & Jain 2004; Zhan 2006). The CSS-OS also can probe the angular galaxy power spectrum in the photometric imaging survey, and we will discuss it in our future work. In Figure 8, we show the 2D angular galaxy power spectra for the five spec-z bins.

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Then we can calculate the angular galaxy–galaxy lensing power spectrum, i.e., cross-correlating the galaxy clustering and weak lensing surveys, of the ath galaxy spec-z bin and the ith photo-z bin. Considering the intrinsic alignment effect, it can be expressed as (Joudaki et al. 2017)

$$\begin{eqnarray}&&{C}_{g\gamma }^{{ai}}({\ell })={C}_{g\kappa }^{{ai}}({\ell })+{C}_{g{\rm{I}}}^{{ai}}({\ell }),\end{eqnarray} \tag{ 22 }$$

where the ${C}_{g\kappa }^{{ai}}$ and ${C}_{g{\rm{I}}}^{{ai}}$ are given by

$$\begin{eqnarray}&&{C}_{g\kappa }^{{ai}}({\ell })={\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \ \displaystyle \frac{{b}_{g}(k,\chi ){\hat{n}}_{g}^{a}(\chi ){q}_{i}(\chi )}{{r}^{2}(\chi )}{P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right),\end{eqnarray} \tag{ 23 }$$

and

$$\begin{eqnarray}\begin{array}{rcl}{C}_{g{\rm{I}}}^{{ai}}({\ell }) & = & {\int }_{0}^{{\chi }_{{\rm{H}}}}d\chi \ \displaystyle \frac{{b}_{g}(k,\chi ){\hat{n}}_{g}^{a}(\chi ){n}_{i}(\chi ){F}_{i}(\chi )}{{r}^{2}(\chi )}\\ & \times & {P}_{{\rm{m}}}\left(\displaystyle \frac{{\ell }+1/2}{r(\chi )},\chi \right).\end{array}\end{eqnarray} \tag{ 24 }$$

The covariance matrix for the galaxy–galaxy lensing power spectra is given by Hu & Jain (2004)

$$\begin{eqnarray}\begin{array}{rcl} & & \mathrm{Cov}\left[{C}_{g\gamma }^{{ai}}({\ell }),{C}_{g\gamma }^{{bj}}({\ell }^{\prime} )\right]\\ & = & \displaystyle \frac{{\delta }_{{\ell }{\ell }^{\prime} }}{{f}_{\mathrm{sky}}{\rm{\Delta }}l(2{\ell }+1)}\left[{\widetilde{C}}_{g}^{{ab}}({\ell }){C}_{\gamma }^{{ij}}({\ell })+{C}_{g\gamma }^{{aj}}({\ell }){C}_{g\gamma }^{{ib}}({\ell })\right].\end{array}\end{eqnarray} \tag{ 25 }$$

In Figure 9, we show the mock galaxy–galaxy lensing power spectra for the first photo-z and second spec-z bins. The green solid, light blue dashed–dotted, red dashed curves denote the total, galaxy–galaxy lensing, and galaxy intrinsic power spectra, respectively. For comparison, the corresponding convergence (dark blue dashed–dotted) and angular galaxy (orange dashed–dotted) power spectra are also shown. The data points with error bars are obtained by assuming ${\bar{N}}_{\mathrm{add}}={10}^{-9}$ and ${N}_{\mathrm{sys}}^{{\prime} \,g}={10}^{-8}$.

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We can define the coefficients of the cross power spectra of weak lensing and galaxy clustering between the photometric and spectroscopic redshift bins, which is given by

$$\begin{eqnarray}&&{r}_{{ai}}({\ell })=\displaystyle \frac{{C}_{g\kappa }^{{ai}}({\ell })}{\sqrt{{C}_{g}^{{aa}}({\ell }){P}_{\kappa }^{{ii}}({\ell })}}.\end{eqnarray} \tag{ 26 }$$

In Figure 10, we calculate the coefficients of the cross power spectra for the four photometric and five spectroscopic bins. For instance, in the top panel, we find that the ${z}_{{\rm{p}}}^{1}$ bin only has significant correlations with the ${z}_{{\rm{s}}}^{1}$ and ${z}_{{\rm{s}}}^{2}$ bins, because they cover a similar redshift range (see Figures 2 and 5).

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In Figure 11, the fitting results of Ω_m versus σ₈ and w₀ versus w_a are shown. We explore three cases, i.e., the moderate, optimistic, and pessimistic assumptions (in solid, dashed, and dotted curves), about the systematics of the CSST WL and galaxy clustering surveys. The top, middle, and bottom panels show the constraint results from the WL, galaxy clustering, and joint (WL+galaxy clustering+galaxy–galaxy lensing) surveys, respectively. The gray lines show the fiducial values of the parameters.

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For the CSST WL survey, we find that ${{\rm{\Omega }}}_{{\rm{m}}}={0.304}_{-0.008}^{+0.009}$, ${\sigma }_{8}={0.796}_{-0.010}^{+0.010}$, ${w}_{0}=-{1.003}_{-0.021}^{+0.025}$, and ${w}_{a}=-{0.013}_{-0.073}^{+0.078}$ in 1σ confidence level (C.L.) for the moderate systematic case with ${\bar{N}}_{\mathrm{add}}={10}^{-9}$. We also show the constraint results of Ω_m versus ${S}_{8}\equiv {\sigma }_{8}{({{\rm{\Omega }}}_{m}/0.3)}^{0.5}$ in Figure 12. We find that ${S}_{8}={0.801}_{-0.008}^{+0.009}$ in the moderate case (solid lines). This constraint result of the cosmological parameters is improved on average by a factor of ∼6 more than that from the Dark Energy Survey (DES) and Kilo Degree Survey (KiDS) (Hildebrandt et al. 2016; Troxel et al. 2017). The improvement can be even larger in the optimistic case (${\bar{N}}_{\mathrm{add}}={10}^{-10}$), and at least a factor of ∼3 in the pessimistic case (${\bar{N}}_{\mathrm{add}}={10}^{-8}$). This enhancement is due to several advantages of the CSST WL survey, e.g., the large survey area (17,500 deg²), excellent image quality (small and regular PSF),¹⁶ and accurate photo-z calibration, which can efficiently suppress both statistical and systematical errors.

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In the CSST galaxy clustering survey, we find that ${{\rm{\Omega }}}_{{\rm{m}}}\,={0.286}_{-0.012}^{+0.012}$, ${\sigma }_{8}={0.796}_{-0.019}^{+0.017}$, ${w}_{0}=-{1.005}_{-0.031}^{+0.029}$, and ${w}_{a}\,={0.021}_{-0.050}^{+0.061}$ in 1σ C.L. for the moderate systematic case (${f}_{\mathrm{eff}}^{{z}_{{\rm{s}}},0}=0.5$ and ${\bar{N}}_{\mathrm{sys}}^{g}=5\times {10}^{4}$ (Mpc/h)³). This leads to a factor of ∼5 improvement at least compared to the current galaxy clustering surveys, such as the SDSS-III Baryon Oscillation Spectroscopic Survey, WiggleZ, 2-degree Field Lensing Survey (2dFLenS), etc. (Blake et al. 2016; Wang et al. 2016; Zhao et al. 2016; Hinton et al. 2017). The constraint results could be better or worse with an average factor of 2–3 in the optimistic (${f}_{\mathrm{eff}}^{{z}_{{\rm{s}}},0}=0.7$ and ${\bar{N}}_{\mathrm{sys}}^{g}=1\times {10}^{4}$ (Mpc/h)³) and pessimistic (${f}_{\mathrm{eff}}^{{z}_{{\rm{s}}},0}=0.3$ and ${\bar{N}}_{\mathrm{sys}}^{g}=1\times {10}^{5}$ (Mpc/h)³) cases. The improvement is mainly caused by the CSST spectroscopic survey having a deep magnitude limit (see Table 1) and large effective survey volume (see Figure 7).

Since the CSST photometric and spectroscopic surveys are planned to perform simultaneously and cover the same sky area, their joint constraint would be convenient and powerful, which gives ${{\rm{\Omega }}}_{{\rm{m}}}={0.304}_{-0.010}^{+0.003}$, ${\sigma }_{8}={0.803}_{-0.008}^{+0.005}$, ${w}_{0}=-{0.996}_{-0.023}^{+0.010}$, and ${w}_{a}={0.044}_{-0.038}^{+0.034}$ in 1σ C.L. for the moderate systematic case. For the optimistic and pessimistic cases, the constraints can be tighter or looser by a factor of ∼1.3 and 1.7, respectively. Comparing to current similar joint fitting results, e.g., KiDS-450 + 2dFLenS (Joudaki et al. 2017), we find that the CSST joint survey can enhance the constraints of the cosmological parameters by one order of magnitude when assuming the moderate systematics.

Since the systematics have been included in the analysis, fitting biases with respect to the best fits appear in the constraint results according to the fiducial values of the cosmological parameters (gray dashed lines), especially for the moderate and pessimistic cases. By comparing the results of the three systematical assumptions, we find that well-controlled systematics not only can shrink the probability contours but also efficiently suppress the fitting biases of the cosmological parameters (see dashed contours in each survey). Besides, it seems that the fitting biases are relatively larger or more apparent in the joint constraint results (e.g., see the result of w_a). This means that better controlling of the systematics may be required in the CSS-OS joint fits. The detailed discussion of the systematical parameters can be found in the next section.

In Figure 13, we show the comparison of the results from the WL, galaxy clustering, and joint surveys. As can be seen, the CSST WL and galaxy clustering surveys have similar constraint strength on Ω_m. On the other hand, the WL is more powerful to constrain σ₈ than the galaxy clustering by a factor of ∼2, because the WL survey explores 2D power spectra integrating over large redshift range. On the other hand, the CSST galaxy clustering survey can provide comparable or even a bit more stringent fitting results on w₀ and w_a than the WL survey. The joint CSST surveys of WL+galaxy clustering+galaxy–galaxy lensing can further improve the fitting results, which give at least ∼2σ enhancement on the constraints of cosmological parameters, compared to the WL only or galaxy clustering only survey.

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The constraint results of all cosmological parameters with the mild assumption of the systematics for the WL, galaxy clustering, and joint surveys can be found in the Appendix.

In Figure 14, the 1D PDFs of the intrinsic alignment parameters A_IA and η_IA are shown for the WL (in blue curves) and joint (in green curves) surveys. We can find that there is no significant improvement on the constraint of A_IA for the joint constraints, while a factor of ∼1.3 tighter for η_IA compared to the WL survey. This indicates that the joint fitting can be helpful to extract the redshift evolution effect of intrinsic alignment.

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The 1D PDFs of the redshift calibration bias Δz_i, stretch factor s_zⁱ, and multiplicative error m_i, are shown in Figure 15. The top and bottom panels show the results from the WL and joint surveys, respectively. We can see that the best fits of the WL systematic parameters are close to their fiducial values (in 1σ C.L.), which means our fitting process can correctly extract the systematics as free parameters. This is also useful to reduce the effect of the systematics on the constraints of cosmological parameters, and can help suppressing the fitting biases (see Figure 11). In order to retain small fitting biases of the cosmological parameters (keeping the fiducial values in 1σ C.L.), as shown in the top panels of Figure 15, we need to control the systematical parameters in 1σ C.L. of their PDFs at least. This requires $| {\rm{\Delta }}{z}_{i}| \lt 0.02$, $| {\mathrm{log}}_{10}({s}_{z}^{i})| \lt 0.05$ (about 10%),¹⁷ and $| {m}_{i}| \lt 0.02$ for the CSST WL survey. When performing the joint fitting, the constraint results of the systematics can be improved by a factor of ∼1.5, which implies that it is helpful to include other surveys (here the galaxy clustering survey) to eliminate the WL systematics in the fitting process.

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In Figure 16, the 1D PDFs of galaxy bias b_g and velocity dispersion σ_v0 assuming the moderate systematics for the five spec-z bins are shown. For the galaxy clustering survey (top panels), we find that the dispersions of the best fits of b_g and σ_v0 from the fiducial values are within ±0.1, and the 1σ ranges 0.03–0.05 for b_g and 0.1–0.2 for σ_v0. The dispersions and uncertainties of b_g and σ_v0 can be further suppressed in the joint fitting (bottom panel), especially for σ_v0. We find that the dispersions of b_g and σ_v0 can be restricted within ±0.05, and the 1σ within 0.01–0.02 for b_g and 0.02–0.06 for σ_v0. (A factor of ∼2 and ∼4 improvement, respectively.) This indicates that the CSS-OS can provide accurate constraints on the galaxy bias and velocity dispersion.

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