CSST WL preparation I: forecast the impact from non-Gaussian covariances and requirements on systematics control

ABSTRACT

The precise estimation of the statistical errors and accurate removal of the systematical errors are the two major challenges for the stage IV cosmic shear surveys. We explore their impact for the China Space Station Telescope (CSST) with survey area |${\sim} 17\,500\deg ^2$| up to redshift ∼4. We consider statistical error contributed from Gaussian covariance, connected non-Gaussian covariance, and super-sample covariance. We find the non-Gaussian covariances, which is dominated by the super-sample covariance, can largely reduce the signal-to-noise ratio of the two-point statistics for CSST, leading to an ∼1/3 loss in the figure of merit for the matter clustering properties (σ₈–Ω_m plane) and 1/6 in the dark energy equation of state (w₀–w_a plane). We further put requirements of systematics mitigation on intrinsic alignment of galaxies, baryonic feedback, shear multiplicative bias, and bias in the redshift distribution, for an unbiased cosmology. The 10⁻²–10⁻³ level requirements emphasize strong needs in related studies, to support future model selections and the associated priors for the nuisance parameters.

1 INTRODUCTION

Weak gravitational lensing (WL) represents a fundamental tool for investigating cosmology, gravity, dark matter, and dark energy (Refregier 2003; Mandelbaum 2018). The synergy between WL and cosmic microwave background (CMB) observations can yield even more robust results, as it possesses greater constraining power and effectively breaks parameter degeneracies (Planck Collaboration I 2020; Abbott et al. 2022). None the less, the tension observed between the CMB data at redshift z ∼ 1100 and the results obtained from late-time galaxy surveys at z < ∼1, possibly caused by unexplained systematic errors or new physics beyond the Lambda cold dark matter cosmological model, poses a significant challenge when employing their combined analysis (Hildebrandt et al. 2017; Hikage et al. 2019; Hamana et al. 2020; Planck Collaboration I 2020; Asgari et al. 2021; Heymans et al. 2021; Abbott et al. 2022; Amon et al. 2022; Secco et al. 2022). A broad range of investigations have been conducted to address the ‘S₈’ tension, encompassing diverse systematics (Yao et al. 2017, 2020; Fong et al. 2019; Kannawadi et al. 2019; Pujol et al. 2020; Wright et al. 2020b; Mead et al. 2021; Amon et al. 2022; Secco et al. 2022; Yamamoto et al. 2023), an array of statistical methods (Lin & Ishak 2017; Shan et al. 2018; Chang et al. 2019; Asgari et al. 2021; Harnois-Déraps et al. 2021; Joachimi et al. 2021; Leauthaud et al. 2022; Sánchez et al. 2022; Liu et al. 2023), and the possibility of new physics (Jedamzik, Pogosian & Zhao 2021). For further reading, we recommend consulting some recent review articles on this topic (Mandelbaum 2018; Perivolaropoulos & Skara 2021).

To comprehensively address the underlying causes of this tension, various cosmological probes are necessary owing to their distinct sensitivities to systematics and cosmology. Numerous recent observations are preparing to investigate an extended range of redshifts, sky patches, algorithms, and equipment properties. Prominent among these stage IV galaxy surveys are the Dark Energy Spectroscopic Instrument (DESI; DESI Collaboration 2016a, b), the Legacy Survey of Space and Time (LSST; LSST Science Collaboration 2009) at the Vera C. Rubin Observatory, Euclid (Laureijs et al. 2011), Roman Space Telescope (also known as WFIRST, Spergel et al. 2015), and the China Space Station Telescope (CSST; Gong et al. 2019).

In this work, we investigate how accurately CSST can constrain cosmology using its cosmic shear two-point statistics. We test the impact from several important sources of statistical errors and systematical errors. More specifically, we investigate the loss of constraining power in terms of signal-to-noise (S/N) ratio of the observables and figure of merit (FoM) of the constrained parameters, due to the non-Gaussian covariances (Takada & Hu 2013; Joachimi et al. 2021). We investigate potential biases in cosmological parameters by analysing different levels of residual bias in the mitigation of intrinsic alignment (IA; Catelan, Kamionkowski & Blandford 2001; Hirata & Seljak 2004; Bridle & King 2007; Yao et al. 2017, 2020), baryonic feedback (Mead et al. 2015, 2021; Schneider & Teyssier 2015; Chen et al. 2023a), shear multiplicative bias (Mandelbaum et al. 2018; Kannawadi et al. 2019; Giblin et al. 2021; Liu et al. 2021), and bias in the redshift distribution (van den Busch et al. 2020; Hildebrandt et al. 2021; Newman & Gruen 2022; Peng et al. 2022; Xu et al. 2022). These four types of systematics are considered most significant (⁠|${\sim} 10~{{\ \rm per\ cent}}$| level contamination in the observable, and ∼1σ level potential bias in the cosmology after mitigation, see Asgari et al. 2021; Abbott et al. 2022; Li et al. 2023) in the current stage III surveys, and there impacts could be more significant as the statistical constraining power further improves for stage IV. We therefore provide calibration requirements for CSST to assist future studies in mitigating those systematics.

This work is organized as follows. In Section 2, we briefly introduce the theoretical predictions to the observables, and the theories for different statistical errors and systematic errors. In Section 3, we describe the CSST data we expect. In Section 4, we forecast the cosmic shear measurements with tomography, and the impact of statistical errors and systematic errors on cosmological parameters. We summarize our findings in Section 5.

2 THEORY

This section provides a brief review of the cosmic shear two-point statistics theory, how different statistical errors and systematical errors affect the observable, and how we make the forecast with Fisher formalism. We assume Ω_k = 0 for spatial curvature, which renders the comoving radial distance and the comoving angular diameter distance identical.

2.1 Cosmic shear

We employ the lensing convergence autocorrelation in Fourier space, i.e. the lensing angular power spectrum (Asgari et al. 2021),

which is a weighted projection from the 3D non-linear matter power spectrum P_δ(k, z) to the 2D galaxy-lensing convergence angular power spectrum C^κκ(ℓ). In this work, the non-linear matter power spectrum is calculated by halofit (Takahashi et al. 2012; Mead et al. 2015, 2021). It also depends on the comoving distance χ, and the lensing efficiency as a function of the lens position (given the distribution of the source galaxies) q_s(χ), which is written as

where χ_s and χ_l denote the comoving distance to the source and the lens, respectively, while n_s(χ_s) = n_s(z)dz/dχ_s denotes the distribution of the source galaxies as a function of comoving distance. The ‘|$\rm s$|’ symbols for the source can be replaced by an index for different redshift bins, such as different tomographic bin index i or j. In this work, we consider a flat Universe with Ω_k = 0 as weak lensing is not sensitive to the spatial curvature.

The real space shear–shear autocorrelation function can be obtained through the Hankel transformation

where J_0/4(x) is the Bessel function of the first kind with order 0/4.

Therefore, by observing the correlation ξ_+/−,ij or the shear–shear angular power spectrum |$C^{GG}_{ij}=C^{\kappa \kappa }_{ij}$|⁠, we can derive the constraints on the cosmological parameters through equation (1), P_δ(k) and χ(z). To obtain a precise constraint on cosmology, many sources of statistical errors and systematic errors need to be considered.

2.2 Covariances

We consider three components of the covariance to account for the statistical error for cosmic shear:

namely, the Gaussian covariance, the connected non-Gaussian (cNG) covariance, and the super-sample covariance (SSC).

The Gaussian covariance is based on a common assumption that the fluctuation of the underlying matter field is Gaussian. It is calculated by

where |$\delta _{\ell _1,\ell _2}$| is the Kronecker delta function; C^GG is the shear–shear angular power spectrum; |$N^{\rm GG}=4\pi f_{\rm sky}\gamma _{\rm rms}^2/N_{\rm G}$| is the shot noise for C^GG, where f_sky is the fraction of sky of the overlapped area, N_G is total number of the galaxies for the source.

The cNG covariance (Takada & Jain 2004) provides impact from non-Gaussian distribution of the density field due to late-time non-linear evolution, so that the higher order perturbation enters the covariance. It is calculated by

where T_m is the matter trispectrum, calculated using a halo model formalism (Joachimi et al. 2021). We adopt the Navarro–Frenk–White halo profile (Navarro, Frenk & White 1996) along with a concentration–mass relation (Duffy et al. 2008), a halo mass function (Tinker et al. 2008), and a halo bias (Tinker et al. 2010).

The SSC (Takada & Hu 2013; Takahashi et al. 2019; Euclid Collaboration 2023c) account for the selection effect of limited observational window, in which the background overdensity can deviate from the ensemble average of the Universe. It is calculated by

where the derivative of |$\partial P_{\rm \delta }/\partial \delta _{\rm b}$| gives the response of the matter power spectrum to a change of the background density contrast δ_b, while |$\sigma ^2_{\rm b}$| denote the variance of the background matter fluctuations in the given footprint. Later, we will show the footprint of the CSST cosmic shear observations, which is used to calculate |$\sigma ^2_{\rm b}$|⁠:

The A_eff is the effective area for the corresponding spherical harmonic coefficient a_ℓm of a probe’s mask, with index μ or ν representing different masks, if applicable. And |$P_{\delta }^{\rm lin}$| is the linear matter power spectrum.

2.3 Systematics

In this work, we consider four major sources of systematics, which can potentially bias the observables at |${\sim} 10~{{\ \rm per\ cent}}$| level if unaddressed, in the current stage III surveys (Hikage et al. 2019; Asgari et al. 2021; Abbott et al. 2022). Other smaller systematics such as source clustering (Yu et al. 2015), beyond Born approximation (Fabbian, Calabrese & Carbone 2018), beyond Limber approximation (Fang et al. 2020), and potential CSST-based systematics (similar to Euclid Collaboration 2023b) are left for future studies.

2.3.1 Intrinsic alignment (IA)

Weak lensing uses the gravitationally lensed ‘optical’ shape γ^G of the source galaxy to probe the matter and gravity of the lens. The ‘dynamical’ shape of a galaxy before being lensed is affected by its local large-scale structures, causing the IA γ^I, which is therefore a source of systematics. Considering the shape noise γ^N, the overall observed shape reads

Therefore, in two-point statistics, the target <γ^Gγ^G> will be contaminated. In terms of the angular power spectrum,

Here, C^γγ is the observed angular power spectrum. C^GG is the target shear–shear power spectrum, which is identical to the convergence power spectrum as in equation (1). C^IG and C^GI are the shear-IA angular power spectra, which write

And C^II is the IA–IA angular power spectra, given by

The 3D matter-IA power spectrum |$P_{\rm \delta ,\gamma ^I}$| and the 3D IA power spectrum |$P_{\rm \gamma ^I}$| are based on IA physics. In this work, we use the most widely used non-linear alignment (NLA) model (Hikage et al. 2019; Asgari et al. 2021; Abbott et al. 2022):

which are both proportional to the non-linear matter power spectrum P_δ, suggesting that the IA is caused by the gravitational tidal field (Catelan, Kamionkowski & Blandford 2001; Hirata & Seljak 2004; Bridle & King 2007). It is also affected by ρ_m,0 = ρ_critΩ_m,0 (the mean matter density of the Universe at z = 0), the empirical amplitude C₁ = 5 × 10⁻¹⁴(h²M_sun/Mpc⁻³) taken from Brown et al. (2002), and D(z), the linear growth factor normalized to D(z = 0) = 1. The IA amplitude A_IA can be luminosity-dependent (Joachimi et al. 2011) or redshift-dependent (Chisari et al. 2016; Samuroff, Mandelbaum & Blazek 2021; Yao et al. 2020; Tonegawa & Okumura 2022).

The IA modelling in equations (14) and (15) can be replaced by more complicated models such as Krause, Eifler & Blazek (2016), Blazek et al. (2017), and Fortuna et al. (2021) for different galaxy samples (Samuroff, Mandelbaum & Blazek 2021; Yao et al. 2020; Zjupa, Schäfer & Hahn 2020). Its mitigation can alternatively be implemented with extra observables using self-calibration methods (Zhang 2010a, b; Yao et al. 2017, 2023b).

2.3.2 Baryonic feedback

The modelling of the matter power spectrum P(k) normally uses N-body simulations (Takahashi et al. 2012; Euclid Collaboration 2019), so that the corresponding density profiles are the dark-matter-only case. The existence of baryonic matter and their associated non-gravitational and powerful process, so-called baryonic feedback, can further change the clustering features in P(k), especially in the small scales (Jing et al. 2006; Schneider & Teyssier 2015; Mead et al. 2021). The precise modelling of the baryonic feedback is essential for future cosmological observations (Aricò et al. 2020; Martinelli et al. 2021; Chen et al. 2023a).

In this work, we examine the impact of residual baryonic feedback following the baryonic correction model (BCM; Schneider & Teyssier 2015):

which alters the dark-matter-only power spectrum P_DM with a correction term F, written as

Here, G(k|M_c, η_b, z) represents the suppression from gas dynamics, including active galactic nucleus feedback, supernovae feedback, etc. S(k|k_s) describes the increase of clustering at small scale, due to central galaxy stars. Their further expansion for this fitting formula reads:

with the associated values z_c = 2.3, |$B_0=0.105{\rm log}_{10}\left[\frac{M_c}{\rm Mpc/h}\right]-1.27$|⁠, and model parameters M_c = 10^14.08 [Mpc h⁻¹] (mass scale), η_b = 0.5 (relation to the escape radius), and k_s = 55 [h Mpc⁻¹] (star component scale).

We consider the unmodelled residual bias in the matter power spectrum in the form of

with an amplitude A_BCM to describe how precise we need to understand the true underlying baryonic physics.

2.3.3 Shear multiplicative bias

The galaxy shear measurement can suffer from low S/N of the dim galaxies and residuals from the point spread function deconvolution (Zhang et al. 2023). In the first order, the measured/observed shear can be described as a linear distortion from the true shear:

where the two-component column vector |$\boldsymbol{ \mathit{ \gamma}}$| represents the combination of γ_i (with i = 1, 2), while |$\boldsymbol{\mathit{c}}$| represents the corresponding shear additive bias. The 2 × 2 matrix |$\boldsymbol{\sf M}$| contains the shear multiplicative bias |$m_{ij}=\partial \gamma ^{\rm obs}_i/\partial \gamma ^{\rm true}_j$|⁠. Generally, for the ‘gold’ samples with good shear measurements, we have |m| ≪ 1 and |c| ≪ |γ|.

In this work, we consider the most common case of a homogeneous and isotropic multiplicative bias m₁₁ = m₂₂ = m, m₁₂ = m₂₁ = 0, and negligible additive bias c_i = 0, similar to the current stage III observations can achieve (Hikage et al. 2019; Asgari et al. 2021; Amon et al. 2022). We note that the non-vanishing additive bias can also be removed considering it mainly enters ξ₊ (equation 3) but not ξ₋ (equation 4), or applying cross-correlations. Similarly, a z-dependent m can be further limited with cross-correlations (Liu et al. 2021; Yao et al. 2023a). In this case, the weak lensing power spectrum is changed by

2.3.4 Bias in redshift distribution

As weak lensing requires a large amount of galaxies to suppress the intrinsic shape noise and subtract the cosmological lensing shear signals, photometric redshift (photo-z) is preferred over spectroscopic redshift (spec-z) for its low observational cost. However, the accuracy of photo-z does not satisfy the requirement for the current stage III and future stage IV weak lensing surveys, therefore careful calibration of the true redshift distribution is needed (Buchs et al. 2019; Alarcon et al. 2020; van den Busch et al. 2020; Wright et al. 2020a; Asgari et al. 2021; Abbott et al. 2022; Xu et al. 2023).

The main impact of redshift accuracy on cosmology is the mean value of the source redshift distribution. Assume that the mean value is biased by Δz, then the redshift distribution will change from n(z) to n(z − Δz), which changes the n_s(χ_s(z_s)) and therefore the lensing efficiency q_s through equation (2) and the theoretical estimation of C^κκ through equation (1).

In this work, we consider a systematic shift in all redshift bins with the same amount Δz, which can lead to a systematic shift in all cosmological parameters. We note that the bias in redshift distribution is very sample-dependent, therefore the shift is in principle redshift-dependent. However, the z-dependent shift is highly based on the selection function of each specific survey, which is hard to put in the theoretical forecast. Also, the z-dependent bias can be easily identified with cross-correlations (van den Busch et al. 2020; Xu et al. 2023) or simply remove a certain z-bin (Asgari et al. 2021; Li et al. 2023). So, for a concise demonstration, we use an identical z-bias in this work, which also strongly degenerates with the cosmology and is hard to detect.

2.4 Forecast

2.4.1 Fisher formalism

We use Fisher matrix (Huterer et al. 2006; Coe 2009; Kirk et al. 2012; Clerkin et al. 2015; Yao et al. 2017) to pass the statistical uncertainties of CSST observations to the cosmological parameters, to estimate the cosmological constraints and the potential bias from different residual systematics. The Fisher matrix is calculated as:

where in between () are all matrices. The vector |$\boldsymbol{C}^{\kappa \kappa }$| is the column data vector that contains all the i, j combinations and ℓ bins in equation (1), with total length of |$\mathscr {N}_{\rm data}=[(\mathscr {N}_{\rm tomo}+1)\mathscr {N}_{\rm tomo}]/2\times \mathscr {N}_{\ell }$|⁠, where |$\mathscr {N}_{\rm tomo}$| is the number of tomographic/redshift bins and |$\mathscr {N}_{\ell }$| is the number of angular bins. Its partial derivative with respect to the cosmological parameters |$\boldsymbol{p}$|⁠, |$\left(\frac{\partial \boldsymbol{C}^{\kappa \kappa }}{\partial \boldsymbol{p}}\right)$|⁠, is therefore a |$\mathscr {N}_{\rm data}\times \mathscr {N}_{\rm para}$| matrix, with |$\mathscr {N}_{\rm para}$| corresponds to the number of cosmological parameters. The inverse of the covariance (Cov⁻¹) uses the covariance matrices introduced in Section  2.2, and is, therefore, a |$\mathscr {N}_{\rm data}\times \mathscr {N}_{\rm data}$| matrix.

By using equation (25), we transform the covariance of the data vector to those of the cosmological parameters, with likelihood |$-2{\rm ln}\mathscr {L}=\boldsymbol{p}^T\left(F\right)\boldsymbol{p}$|⁠. The |$\mathscr {N}_{\rm para}\times \mathscr {N}_{\rm para}$| Fisher matrix F contains direct information of the variance on each parameter |$\sigma ^2_\alpha =\left(F^{-1}\right)_{\alpha \alpha }$| and the covariance between different parameters |$\sigma ^2_{\alpha \beta }=\left(F^{-1}\right)_{\alpha \beta }$|⁠. The constraining power in the two-parameter space can also be evaluated with FoM, defined as FoM_αβ = [det(F⁻¹)_αβ]^−1/2.

2.4.2 Biases due to residual systematics

To estimate how different residual biases can shift the best-fitting cosmology, we first use the introduce systematics in Section  2.3 to estimate a residual bias |$\Delta \boldsymbol{C}^{\kappa \kappa }$| in the data vector. Then, the corresponding shift in the cosmological parameters |$\boldsymbol{p}$| can be written as

considering first-order approximation to the likelihood (Huterer et al. 2006; Yao et al. 2017).

We note that in this work, we aim at getting a general requirement on the systematics, to guide future calibration works on different systematics. For some systematics such as IA and baryonic feedback, as the true model to describe the physics is still unknown, it is trivial to consider the marginalization over their nuisance parameters, due to different model’s parameters can have different degeneracies with the cosmological parameters. For any of the systematics mitigation methods, validation with simulation is also a crucial link, which can both estimate the potential residual bias and give priors to the nuisance parameters. Assuming the simulations can give strong enough priors, we no longer need to consider the constraining power loss due to marginalization. We therefore aim at getting the requirements from the residual systematics first, then check how it is affected by considering the marginalization of the nuisance parameters.

2.4.3 Signal-to-noise (S/N) definition

The conventional S/N definition uses amplitude fitting (Yao et al. 2023a). For a given measurement |$\boldsymbol{w}_{\rm data}$| and an assumed theoretical model |$\boldsymbol{w}_{\rm model}$|⁠, we fit an amplitude A to the likelihood:

so that a posterior of |$A^{+\sigma _A}_{-\sigma _A}$| can be obtained, where σ_A is the Gaussian standard deviation for the amplitude. Then, the corresponding S/N is A/σ_A.

Under the frame of Fisher formalism, we have |$\boldsymbol{w}_{\rm data} = \boldsymbol{w}_{\rm model}$| with a single free parameter A for the forecast. Similar to the procedures in Section  2.4.1, one can estimate the S/N is |$A/\sigma _A=\sqrt{\left(\boldsymbol{w}_{\rm model}\right)^T \left({\rm Cov}^{-1}\right) \left(\boldsymbol{w}_{\rm model}\right)}$|⁠.

3 SURVEY PROPERTIES

The CSST is a space-based project aiming at mapping the Universe with both photometric and slitless spectroscopic observations, covering 17 500 deg² of the sky. Its imaging survey contains seven photometric bands (NUV, u, g, r, i, z, y) with wavelength coverage from 255 to 1000 nm (Gong et al. 2019; Zhan 2021). The 5σ point sources detection limit in nominal AB magnitude is r ∼ 26 (Cao et al. 2018).

We present the CSST shear catalogue properties for this forecast work. For reliable shear and photo-z measurements, we consider a reduction in the galaxy number density from 28 gal arcmin⁻² (Gong et al. 2019) to 20 gal arcmin⁻² (Liu et al. 2023) with appropriate S/N cuts. For more detailed studies considering blending and masking (Chang et al. 2013), it will require a PhoSim (Peterson et al. 2015)-like simulator that highly mimics the CSST galaxies, which is under development. However, the blending problem is less significant comparing with LSST (Liu et al. 2023).

In Fig. 1, we show the expected redshift distribution of the CSST source galaxies. The photo-z distribution is obtained by applying the CSST observational limits to the COSMOS photo-z galaxies (Ilbert et al. 2009; Cao et al. 2018). We divide the galaxies into seven tomographic bins with (almost) equal numbers of galaxies, for higher total S/N following Moskowitz et al. (2023). We remove galaxies with photo-z z_p < 0.1 as they contribute little to the total cosmological signals due to low lensing efficiency as in equation (2). We assume the true-z follows a Gaussian probability distribution function (PDF) around the photo-z, namely

where we adapt the photo-z scatter σ_z = 0.05 and photo-z bias Δ_z = 0 for the fiducial analysis (Cao et al. 2018). The resulting true-z distributions are also shown as the solid curves in Fig. 1. A non-vanishing photo-z bias Δ_z is equivalent to a systematic shift in the overall redshift distribution with Δz, introduced in Section  2.3.4.

We consider galaxy shape noise γ_rms = 0.27 that is close to the other stage IV surveys (Yao et al. 2017). It will enter the shot noise term N^GG as shown in equation (6). For a more detailed estimation of how much additional statistical error can be introduced from the shear measurement, a more realistic imaging simulation is needed to highly mimic the CSST galaxy properties.

We also estimate how the sky coverage will change considering simple masking of the bright galaxies and bright stars. We use a healpix map (Górski et al. 2005; Zonca et al. 2019) with N_side = 4096 (∼0.74 arcmin² per pixel) and remove the regions within ±19.2 deg of the galactic latitude and the ecliptic latitude. The remaining region is |${\sim} 17\,572\deg ^2$|⁠, which approximates the CSST target sky. We further remove bright sources, which are likely to be low-z objects with low lensing efficiency and can contaminate their nearby galaxies. We remove pixels that contain any galaxy with a magnitude brighter than 18.5 (in B band from de Vaucouleurs et al. 1991), and any star with a magnitude brighter than 18 from GAIA DR3 (Gaia Collaboration 2016, 2023). The resulting footprint is shown in Fig. 2, with the sky coverage reduced to |${\sim} 14\,877\deg ^2$|⁠. The pixel size is significantly larger than the common size for bright object removal (Coupon et al. 2018); therefore, the resulting footprint is a conservative estimation.

4 RESULTS

4.1 Statistics

We first study the statistical constraining power from cosmic shear. Two different scale cuts are applied: a fiducial scale cut with 30 < ℓ < 3000, and a conservative scale cut with 30 < ℓ < 1000 in case the small-scale physics are not modelled correctly. We consider logarithmic angular binning with bin width Δℓ = 0.2ℓ, following other stage IV surveys (Yao et al. 2017).

Based on the survey properties in Section  3, the tomographic cosmic shear angular power spectra can be calculated, shown in Fig. 3. Under the assumption of Gaussian covariance, we find a significantly high S/N of cosmic shear signal, with total S/N of ∼512 and ∼317 for the fiducial scale cut and the conservative scale cut, respectively.

We then compare the impact from the non-Gaussian covariances, as introduced in equations (7) and (8). We find that the cNG covariance has a much smaller contribution to the total covariance, compared with the Gaussian covariance and the SSC. The comparison is shown in Fig. 4. We find that the existence of SSC does not significantly increase the errorbars in Fig. 3. However, it introduces a significantly strong correlation between different data points, shown as the non-diagonal terms in the Gaussian + SSC case in Fig. 4.

The SSC is more dominant compared with the Gaussian covariance at small scales, shown in each small cube in the right panel of Fig. 4. Each cube corresponds to the covariance between a certain |$C^{\rm GG}_{ij}(\ell _1)$| versus |$C^{\rm GG}_{mn}(\ell _2)$| pair. The diagonal feature in each cube mainly comes from the Gaussian covariance (ℓ₁ = ℓ₂) and the off-diagonal features come from the SSC term. It can be seen that in many small cubes, the diagonal feature fade away when it goes to a smaller scale (bottom-right corner). The finding of SSC being more dominant at small scales agrees with Takahashi et al. (2019).

When the total covariance including Gaussian + cNG + SSC is applied, S/N of the fiducial analysis will be reduced from ∼512 to ∼460, mainly due to the contribution from SSC. This result also agrees with Takahashi et al. (2019) that SSC is a dominant statistical error in the next stage cosmic shear studies. None the less, this reduced S/N is still much stronger than the current stage III observations can achieve (Abbott et al. 2022).

We further show the cosmological constraints considering the Gaussian covariance only as well as the full covariance in Fig. 5. Similar to the S/N results, when considering the full covariance, the constraining power suffers from a significant loss compared with the case of using Gaussian covariance only. All the cosmological parameters, especially in the σ₈ versus Ω_m plane for the large-scale structure studies, and w₀ versus w_a plane for the dark energy equation of state, experience enlargement in the contour due to the SSC. Some FoM values are also calculated in Fig. 5. Overall, we conclude the impact of SSC is non-negligible for CSST cosmic shear studies.

4.2 Impact of the systematics

We study how different systematics can bias the cosmological results, considering four different systematics we are most interested in, see Section  2.3. For a certain type of residual systematics, its cosmological impact is quantified by equation (26). The statistical constraints are identical to Fig. 5, considering the full covariance with contribution from Gaussian, cNG, and SSC. We alter the residual amplitude for each systematics so that its maximum resulting shift in all parameter spaces is at a similar level as the 2σ contour, which is easier to see. The assumed residuals are A_IA = 0.07 for IA, m = 0.07 for shear multiplicative bias, A_BCM = 0.4 for baryonic feedback, and Δz = 0.012 for mean redshift.

The shifts of the best-fitting cosmological parameters due to the residuals of different systematic effects is shown in Fig. 6. It is clear that different residual ΔC(ℓ) can bias the cosmology towards different directions with different amounts. We then raise requirements for the systematics control based on those biases. We define two different tolerances: a visual tolerance and a probability tolerance.

(1) The visual tolerance, or the 68 per cent contour tolerance, requires all the shifts to be within the boundary of the 1σ confidence contour for a certain type of systematics. This tolerance can be directly measured by changing the residual systematics in terms of {A_IA, m, A_BCM, Δz}, then observing its imprint in an updated bias shift figure, similar to Fig. 6. The actual measurements are shown in Fig. A1.

(2) The probability tolerance is 0.2 of the visual tolerance. This number comes from the length of a systematic shift that bias the cosmology right on to the edge of the 1 − σ contour in 2D space (i.e. the visual tolerance, and we refer to this length as σ_2D) correspond to ∼1.52 times the 1 − σ uncertainty in the projected 1D space (σ_1D) (Coe 2009). While in 1D PDF, we historically require the bias to be <0.31σ_1D for a 95 per cent overlap with the ideal 1D PDF (Massey et al. 2013). Therefore, the overall tolerance is |$0.31\sigma _{\rm 1D}\frac{\sigma _{\rm 2D}}{1.52\sigma _{\rm 1D}}\sim 0.2\sigma _{\rm 2D}$|⁠, which we also refer to as the 95 per cent probability tolerance. This definition is close to what has been used for Euclid (Laureijs et al. 2011; Massey et al. 2013), but we emphasize more on bias control for all the cosmological parameters rather than the dark energy parameters only.

We present the requirements for systematics control for the CSST cosmic shear studies in Table 1. For different angular scale cuts, we show the requirements on the residual systematics in terms of the visual tolerance (68 per cent contour) the probability tolerance (95 per cent prob.), and in which parameter space the tolerance is triggered (contaminated the most). The one with our fiducial cuts 30 < ℓ < 3000 requires a 95 per cent probability tolerance presented at the bottom of the table, which we will refer as the requirement for CSST systematics control Δ_req. Generally, the allowed residual systematics are at 10⁻²–10⁻³ level of the full contamination, which is a very strong requirement for future systematics-mitigation works.

We notice the requirement on m changes very little for the two different scale cuts. This is because this limitation mainly comes from the Ω_m–σ₈ plane, and when adding the information in 1000 < ℓ < 3000, the contour does not change much in the m-bias direction in Fig. 6. The main improvement comes from the direction that is perpendicular to the m-direction. Also, we note that our probability tolerance of |m| < 0.0026 is very close to the Euclid requirement of |m| < 0.002 (Laureijs et al. 2011), considering that we introduced the non-Gaussian covariances, which magnifies the Ω_m–σ₈ plane contour size by a factor of ∼1.5.

We also notice that when adding the small-scale information, IA also starts to have more impact in the dark energy equation of state, becoming equivalent to the Ω_m–σ₈ plane, see in Table 1. This also emphasizes the importance of considering IA at small scales and its possible deviation from the assumed tidal alignment model (Blazek, Vlah & Seljak 2015; Blazek et al. 2017; Fortuna et al. 2021; Kurita et al. 2021; Zjupa, Schäfer & Hahn 2020; Shi et al. 2021; Secco et al. 2022).

We see that the baryonic feedback can bias {h, n_s} more than the other cosmological parameters. This is because its effects mainly changes the small scale of the matter power spectrum as in equation (16). And this effect can be absorbed by the parameters that control the overall slope of the power spectrum, namely h and n_s. This is also why by expanding the angular scale from ℓ < 1000 to ℓ < 3000, the requirement on A_BCM becomes more strict comparing with the other nuisance parameters.

4.3 Marginalization over nuisance parameters

We further study the impact from marginalization over nuisance parameters, as they can steal some constraining power both due to increasing the parameter space and due to their degeneracy with the cosmological parameters. We test three cases for this purpose: (1) flat priors for all the nuisance parameters; (2) if our understanding for the systematics are accurate so that the requirement of probability tolerance Δ_req = 0.2σ_2D can be achieved with comparable statistical error, so that σ_prior = Δ_req; (3) if our a priori knowledge cannot reach the requirement in (2), but only σ_prior = 10Δ_req.

We note that the above case (3) uses comparable priors to what we can achieve for stage III data:

(A_IA): For IA, one can constrain the NLA model up to |${\sim} 10~{{\ \rm per\ cent}}$| precision with simulations (Shi et al. 2021; Hoffmann et al. 2022) or through self-calibration with extra observable (Yao et al. 2020, 2023b).

(A_BCM): For baryonic feedback, one can constrain the signal with |${\sim} 10~{{\ \rm per\ cent}}$| level precision using the small-scale data from cosmic shear (Chen et al. 2023a) or using the integrated thermal Sunyaev-Zel'dovich effect (tSZ, Pandey et al. 2023), while the significance could potentially be stronger with the next-stage observations (Chen, Zhang & Yang 2023b).

(m and Δ_z): the current priors for the multiplicative bias obtained from image simulations for stage III surveys are around 1 per cent level (Asgari et al. 2021; Abbott et al. 2022; Li et al. 2023). Most priors for redshift bias are also around 1 per cent level accuracy, based on different redshift inferences.

The impact of the priors are shown in Fig. 7. For case (1) with no assumed a priori knowledge, the nuisance parameters will take a large amount of the constraining power, leading to a significantly enlarged contour in blue. If the priors can achieve the level of the requirement/tolerance in Table 1, which is our case (2), the strong degeneracy between the cosmological parameters and the nuisance parameters can be efficiently broken, leading to the orange contours which is very close to the full covariance case in Figs 5 and 6. If we consider the very pessimistic case (3) that our prior understanding for the systematics for stage IV data will not improve comparing with stage III, we end up with the green contours, which is ∼2 times broader than the orange case (2) in the posteriors that we are interested in (Ω_m, σ₈, w₀, w_a). In this case, our requirements for the residual systematics in Table 1 can be relaxed by a factor of ∼2. This also states the importance in the systematics studies – we need to demonstrate that the mitigation methods are not only accurate, but also with high significance, so that the loss in the cosmological constraints can be well controlled.

5 CONCLUSIONS

In this work, we build a realistic set-up for the future CSST cosmic shear observation, and address the problem of how non-Gaussian covariances and residual systematics can change our cosmological analysis. We consider cNG covariance and SSC in terms of statistics, and residual systematics from IA, baryonic feedback, and measurements of shear multiplicative bias and mean redshift bias from n(z) reconstruction. We obtained very strong requirements on the residuals, from 10⁻² to 10⁻³ of the assumed contamination parameters, see in Table 1.

In terms of statistical errors, we demonstrated in Section 4.1 that the impact from cNG covariance is small, while SSC has an important effect that can enlarge the 2D confidence contours of some key cosmological parameters, as seen in Fig. 5. We, therefore, emphasize the importance of taking it into consideration in future data analysis. We also suggest careful investigation of the SSC with real data considering n(z) distribution and inhomogeneous galaxy distribution due to observational variation. The fact that SSC depends on the survey footprint also suggests that in order to maximize the scientific outcome for early-stage CSST data, the design of the survey strategy to minimize SSC is important.

In terms of systematical errors, we considered how different systematics can bias the cosmological results in different ways, shown in Fig. 6. We use the 2D parameter space which is mostly affected by a certain type of systematics to describe the tolerance level of the residual bias. This approach is different from the conventional analyses which normally focus on the bias in the dark energy equation of state (Massey et al. 2013). However, we still have comparable requirements in terms of shear multiplicative bias. The strong requirements for future CSST systematics control are beyond the current constraints on those effects (Schneider & Teyssier 2015; Yao et al. 2017, 2023b; Asgari et al. 2021; Abbott et al. 2022; Peng et al. 2022; Chen et al. 2023a; Xu et al. 2023 ). Therefore, we emphasize the importance of pushing new techniques, developing realistic simulations, and combining different approaches (Alarcon et al. 2020) to further constrain those systematics.

In this analysis, we further investigate the effects of different assumed priors for the nuisance parameters. In Fig. 7, we show that if the priors can reach the level of the requirement in Table 1, the constraining power loss due to the nuisance parameters are negligible. If the priors are weaken by ∼10 times, which is comparable to the priors for stage III surveys, the constraints on the main cosmological parameters will be weaken by a factor of ∼2. A flat prior case is also shown for your reference (even though it is not practical). Therefore, we emphasize that the systematics removal requires not only high accuracy, but also high significance. This means the high-fidelity simulations we use for systematics need to be large enough, and their combinations with observational methods (for example self-calibration) are also important.

The studies here primarily concern the cosmic-shear two-point correlation analyses. To fully utilize the CSST weak lensing data for cosmological constraints, other statistical tools beyond the two-point correlations are necessary (Shan et al. 2018; Martinet et al. 2021; Euclid Collaboration 2023a). The impact of the systematics on these alternative statistics deserve further careful investigations (Yuan et al. 2019; Harnois-Déraps et al. 2021; Zhang et al. 2022).

ACKNOWLEDGEMENTS

This work was supported by National Key R&D Program of China grant no. 2022YFF0503403. JY acknowledges the support from NSFC grant no. 12203084, the China Postdoctoral Science Foundation grant no. 2021T140451, and the Shanghai Post-doctoral Excellence Program grant no. 2021419. HS acknowledges the support from NSFC under grant no. 11973070, the Shanghai Committee of Science and Technology grant no. 19ZR1466600, and Key Research Program of Frontier Sciences, CAS, grant no. ZDBS-LY-7013. PZ acknowledges the support of NSFC grant no. 11621303, the National Key R&D Program of China grant no. 2020YFC22016. ZF acknowledges the support from NSFC grant nos 11933002 and U1931210. We acknowledge the support from the science research grants from the China Manned Space Project with grant nos CMS-CSST-2021-A01, CMS-CSST-2021-A02, CMS-CSST-2021-B01, and CMS-CSST-2021-B04. We acknowledge the usage of the following packages pyccl,¹healpy,²matplotlib,³astropy,⁴ and scipy⁵ for their accurate and fast performance and all their contributed authors. This work has made use of data from the European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC; https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.

DATA AVAILABILITY

The data used to produce the figures in this work are available through https://doi.org/10.5281/zenodo.7813033.

Footnotes

References

APPENDIX A: THE 68 PER CENT CONTOUR TOLERANCES

We show the 68 per cent contour tolerances for the systematics in Fig. A1. The maximum shifts are right on the edge of the contours, while the values of the parameters correspond to the 68 per cent contour tolerances with 30 < ℓ < 3000 in Table 1. We note the tips of the arrows in Fig. 6 are not very accurate, and are only for exhibition. Here, we use Fig. A1 to show the actual values.