Forecasting the Constraint on the Hu-Sawicki $f(R)$ Modified Gravity in the CSST $3 imes2$pt Photometric Survey

We forecast the constraint on the Hu-Sawicki $f(R)$ model from the photometric survey operated by the Chinese Space Station Survey Telescope (CSST). The simulated $3\times2$pt data of galaxy clustering, weak lensing, and galaxy-galaxy lensing measurements within 100 deg$^{2}$ are used in the analysis. The mock observational maps are constructed from a light cone, redshift sampling and noise. The angular power spectra are measured with pseudo-$C_\ell$ estimators and compared to theory in the same basis using validated weighting functions and an analytic covariance matrix that includes Gaussian, connected non-Gaussian, and super-sample terms. We model the theoretical spectra using two methods. The first one uses MGCAMB to compute the linear modified-gravity clustering power spectra, and the second one adopts the FREmu emulator with a baseline of nonlinear $Λ$CDM prescription. Parameter inference is performed with Cobaya, and the cosmological and modified-gravity parameters are sampled within the emulator training domain, which is jointly fitted with the systematic parameters. We find that the predictions from the two methods are in good agreement at the overlapping large scales, and the emulator method can correctly provide additional high-$\ell$ information. The $1σ$ upper bounds of $\log_{10}|f_{R0}|$ are found to be $<-5.42$ for cosmic shear only case and $<-5.29$ for the 100 deg$^2$ CSST $3\times2$pt probe. The full CSST photometric survey with 17,500 deg$^2$ survey area is expected to further improve the constraint precision by about one order of magnitude. Our results demonstrate that the CSST $3\times2$pt survey can deliver strict tests on $f(R)$ gravity.

💡 Research Summary

This paper presents a detailed forecast of how well the upcoming Chinese Space Station Survey Telescope (CSST) photometric survey can constrain the Hu‑Sawicki form of f(R) modified gravity. The authors focus on a 100 deg² mock data set that mimics the CSST 3 × 2pt analysis, i.e., the joint measurement of galaxy clustering, weak gravitational lensing (cosmic shear), and galaxy‑galaxy lensing. The mock catalogs are built from the large‑volume Jiutian N‑body simulation (6144³ particles in a 1 h⁻¹ Gpc box) using a semi‑analytic galaxy model calibrated to deep multi‑band observations. Photometric redshifts are assigned with a scatter of σ_z = 0.05(1+z), and galaxies are split into four tomographic bins covering 0 < z < 3, yielding an effective source density of about 26 arcmin⁻².

Angular power spectra for all auto‑ and cross‑correlations are measured with pseudo‑Cℓ estimators on flat‑sky patches, covering multipoles ℓ = 50–3000 in logarithmic bins. The authors construct an analytic covariance matrix that includes three contributions: a Gaussian term, a connected non‑Gaussian (cNG) term, and a super‑sample covariance (SSC) term. This comprehensive treatment captures both the shot‑noise and shape‑noise contributions as well as the coupling between large‑scale modes and the finite survey window, which is essential for realistic error budgeting in future CSST analyses.

The theoretical modeling of the modified‑gravity spectra is performed in two complementary ways. First, the linear matter power spectrum for a given f(R) parameter is computed with MGCAMB, which solves the modified Einstein–Boltzmann equations under the quasi‑static approximation. Second, the authors employ the FREmu emulator, which starts from a ΛCDM nonlinear prescription (e.g., HALOFIT or HMcode) and adds the f(R) correction calibrated on a suite of N‑body runs. FREmu is valid only within its training domain, so the authors restrict the analysis to scales where both methods overlap and verify that they agree on large scales. FREmu, however, provides additional high‑ℓ information (ℓ > 1000) that is not captured by the linear MGCAMB approach.

Parameter inference is carried out with the Cobaya framework, sampling the standard cosmological parameters (Ω_m, σ₈, h, n_s, …), the modified‑gravity parameter log₁₀|f_R0|, and a suite of nuisance parameters that model photometric‑redshift shifts (Δz_i, σ_z_i), shear calibration biases (m_i), galaxy bias (b_i), and intrinsic alignments (A_IA, η_IA). The likelihood incorporates the full 3 × 2pt data vector and the analytic covariance described above.

The main results are as follows. Using only the cosmic‑shear (κκ) spectra, the 1σ upper bound on the Hu‑Sawicki parameter is log₁₀|f_R0| < −5.42. When the full 3 × 2pt combination (galaxy clustering gg, shear κκ, and galaxy‑shear gκ) is employed, the constraint tightens to log₁₀|f_R0| < −5.29 (1σ). This improvement reflects the added statistical power of cross‑correlations and the ability of the FREmu emulator to exploit small‑scale information. The authors note that these limits are comparable to, or slightly stronger than, current Stage‑II surveys (e.g., KiDS‑1000, HSC‑Y1) when those analyses are limited by nonlinear modeling uncertainties.

Looking ahead to the full CSST photometric survey covering 17 500 deg², the authors extrapolate the statistical errors assuming they scale with the square root of the survey area. Under this assumption, the 1σ bound could improve by roughly an order of magnitude, reaching log₁₀|f_R0| ≈ −6. Such a constraint would be competitive with the most stringent existing limits from joint analyses of SPT clusters, DES weak lensing, and HST calibrations, which currently sit around log₁₀|f_R0| ≈ −5.3 (95% CL).

In summary, the paper demonstrates that the CSST 3 × 2pt photometric survey, even on a modest 100 deg² footprint, can deliver meaningful tests of f(R) gravity, and that the full survey will be capable of delivering constraints an order of magnitude tighter. The work underscores the importance of accurate covariance modeling, the synergy between linear Boltzmann solvers and nonlinear emulators, and the inclusion of systematic nuisance parameters in achieving robust modified‑gravity limits. The authors conclude that CSST will play a pivotal role in the next generation of cosmological tests of General Relativity.