CSST Cosmological Emulator III: Hybrid Lagrangian Bias Expansion Emulation of Galaxy Clustering

TL;DR

The paper tackles modeling galaxy clustering in the CSST era by developing a hybrid Lagrangian bias expansion emulator that merges a Lagrangian bias expansion with accurate N-body dynamics. Using the Kun simulation suite, it emulates eight cosmological parameters (including neutrino mass and dynamical dark energy $w_0w_a$) and employs Zel'dovich variance control to measure Lagrangian basis spectra with 1% precision up to $k\leq 1\,h\mathrm{Mpc}^{-1}$ and $0\le z\le 3$. The authors validate the emulator through leave-one-out tests and joint halo auto and halo–matter cross-spectrum fits across 46 cosmologies, achieving unbiased results up to $k\approx 0.57-0.7\,h\mathrm{Mpc}^{-1}$ depending on the counterterms and redshift. The work demonstrates the viability of precise, fast theoretical predictions for CSST galaxy clustering and provides a framework for extending to redshift-space and broader biased-tracer analyses, with public code to enable community adoption. Overall, this hybrid HEFT emulator offers a scalable path to extracting robust cosmological information from upcoming large-scale structure surveys.

Abstract

Galaxy clustering is an important probe in the upcoming China Space Station Telescope (CSST) survey to understand the structure growth and reveal the nature of the dark sector. However, it is a long-term challenge to model this biased tracer and connect the observable to the underlying physics. In this work, we present a hybrid Lagrangian bias expansion emulator, combining the Lagrangian bias expansion and the accurate dynamical evolution from $N$-body simulation, to predict the power spectrum of the biased tracer in real space. We employ the Kun simulation suite to construct the emulator, emulating across the space of 8 cosmological parameters including dynamic dark energy $w_0$, $w_a$, and total neutrino mass $\sum m_ν$. The sample variance due to the finite simulation box is further reduced using the Zel'dovich variance control, and it enables the precise measurement of the Lagrangian basis spectra up to the quadratic order. The emulation of basis spectra realizes 1% level accuracy, covering wavelength $ k \leq 1 \,{\rm Mpc}^{-1}h$ and redshift $0\leq z\leq 3$ up to the quadratic order field. To validate the emulator, we perform a joint fit to the halo auto power spectrum and the halo-matter cross power spectrum measured from 46 independent simulations. Depending on the choice of counterterm, the joint fit is unbiased up to $k_{\rm max}\simeq 0.7\,{\rm Mpc}^{-1}h$ within $1\sim 2$ percent accuracy, for all the redshift and halo mass samples. As part of the CSST cosmological emulator series, this emulator is expected to provide accurate theoretical predictions for the galaxy power spectrum in upcoming CSST survey.

Figures (14)

• Figure 1: Lagrangian basis fields $\mathcal{O}_i({\bf x}, z)$ in Eulerian space at redshift $z=1$, with physical size of the region $200\times 200\,h^{-2}{\rm Mpc}^2$ and projected depth $5\,h^{-1}{\rm Mpc}$. They are advected from the initial redshift $z_{\rm ini}=127$. The constant Lagrangian field $\mathcal{O}_i({\bf q})=1$ is the matter overdensity field $\mathcal{O}_i({\bf x},z)=\delta_m({\bf x},z)$ in Eulerian space. Notice that all the Lagrangian basis fields are subtracted the zero-lag before assigned to the targeted redshift, e.g., $\delta^2 \rightarrow \delta^2-\sigma^2_L$.
• Figure 2: The Lagrangian basis spectra measured from simulations (solid lines) are compared to the 1-loop theoretical calculations (dashed lines), using samples randomly selected from the 83 cosmologies employed in the emulator construction. The four selected redshift bins, $z=2.0$, $1.5$, $1.0$, and $0.0$, are indicated by different colors. The Zel'dovich variance control has been applied to the simulation spectra, and the absolute values of all power spectra are taken for visualization purposes. The simulation results are consistent with the 1-loop theory on large scales, particularly for the leading and quadratic order basis spectra. However, significant deviations appear at low redshift and small scales. The $\nabla^2\delta$ basis field is sensitive to the small-scale modes, and the $\delta^3$ field is not fully captured by the 1-loop perturbation theory. Specifically, the $P_{1\delta^3}(k)$ spectra measured from simulations are positive over the range $k \gtrsim 0.1\, \mathrm{Mpc}^{-1}h$ across all four redshift bins shown. In contrast, the 1-loop prediction remains negative throughout the entire $k$-range at $z=0.0$, despite exhibiting a similar trend in the figure.
• Figure 3: The sample variance fluctuations derived from Zel'dovich variance control are shown for a random selection of the 83 cosmologies used in the emulator construction. The power spectrum $P_{ij}^{ZZ}$ is computed from the corresponding Zel'dovich simulations with $3072^3$ particles, while the ensemble average ${\langle} P_{ij}^{ZZ}{\rangle}$ is obtained from the analytical theory. Line colors range from red to blue, indicating redshifts from $z=3.0$ to $z=0.0$. Across all redshifts and basis spectra, the extracted linear order noise remains unbiased because of the exact agreement between the Zel'dovich realizations and the precise analytical prediction.
• Figure 4: Leave-one-out 68th percentile fractional deviation of the emulator. Line colors range from red to blue, corresponding to redshifts from $z=3.0$ to $z=0.0$. The gray shaded regions indicate the $1\%$ deviation, where most of the quadratic order basis spectra fall below this threshold. For the basis spectra $P_{\delta^2s^2}$, the apparent divergence beyond the plot range is due to the denominator approaching zero, and does not imply a large absolute fractional error.
• Figure 5: In the field-level modeling of halo overdensity field, the fitting performance is quantified by the power spectrum of residual field $\hat{\varepsilon}({\bf k})\equiv \delta_h({\bf k}) - \hat{\delta}^{\rm HEFT}({\bf k})$. Here, $\delta_h({\bf k})$ is the true halo overdensity measured from the simulation and $\hat{\delta}^{\rm HEFT}({\bf k})$ is the Lagrangian bias expansion model. They are normalized by the mean number density $\bar{n}$, where $\bar{n}P_{\rm err}=1$ corresponds to the Poisson expectation. The results are shown for the Planck cosmology simulation at $z=1$, with 5 colors indicating the 5 halo mass bins within $11<\log(M)<14$. The dotted lines indicate the results with 4 bias parameters $\{b_1, b_2, b_s, b_\nabla\}$, while the solid lines indicate the results with 5 bias parameters $\{b_1, b_2, b_s, b_\nabla, b_3\}$. We seek the optimal bias parameters by fitting the large-scale $k$ modes up to $|{\bf k}|_{\rm max}=0.2\, {\rm Mpc}^{-1}h$, and as shown, the bias expansion models are consistent with the true halo distribution at $k\lesssim 1\,{\rm Mpc}^{-1}h$ in the field level. Further, the inclusion of $\delta^3$ field reduces the super-Poisson stochasticity arising from the small-scale clustering, and this reduction is significant for the low mass samples.