Renormalization-Free Galaxy Bias in Unified Lagrangian Perturbation Theory

TL;DR

This work develops a renormalization-free galaxy bias framework within Unified Lagrangian Perturbation Theory (ULPT), deriving a field-level expansion for biased tracers built solely from Galileon-type operators that mirror the intrinsic nonlinear dark-matter structure. The model yields analytic one-loop power spectra for galaxy auto- and galaxy–matter cross-correlations and evaluates them with fast FFT-based methods, requiring only four bias parameters (b1, b2^u, b3^u, N_ε) for power spectra and three for correlation functions. Validation against the Dark Emulator shows sub-percent accuracy up to k ~ 0.3 h Mpc^{-1} for typical biases, and up to k ~ 0.2 h Mpc^{-1} for strongly biased tracers, with configuration-space statistics agreeing down to tens of Mpc scales. The study also confirms a theoretical relation b_{K^2}^E = −3/4 b_2^E and demonstrates robustness across 100 cosmologies, highlighting ULPT as a physically consistent, efficient framework for nonlinear galaxy bias with broad applicability to redshift-space distortions and reconstruction; the open-source ULPTKit implements the numerical pipeline used herein.

Abstract

We present a renormalization-free framework for modeling galaxy bias based on Unified Lagrangian Perturbation Theory (ULPT). In this approach, the biased density fluctuation is built solely from Galileon-type operators associated with the intrinsic nonlinear growth of dark matter. This ensures the bias expansion is well defined at the field level, automatically satisfies statistical conditions of vanishing ensemble and volume averages, and removes the need for ad hoc renormalization. We derive analytic one-loop expressions for the galaxy-galaxy and galaxy-matter power spectra and implement an efficient numerical algorithm using \texttt{FFTLog} and \texttt{FAST-PT}, enabling rapid and accurate evaluation. The model requires only a minimal set of bias parameters: three parameters are sufficient to describe correlation functions in configuration space, while four parameters are needed for power spectra in Fourier space. To test accuracy, we jointly fit halo auto- and cross-spectra from the \textit{Dark Emulator}, covering nine redshift-mass combinations with 100 cosmologies each. A single set of bias parameters reproduces both spectra within $\sim1\%$ up to $k \simeq 0.3\,h\,\mathrm{Mpc}^{-1}$ for typical linear bias $b_1 \sim 0.8$-2, and to $k \simeq 0.2\,h\,\mathrm{Mpc}^{-1}$ for $b_1 \sim 3$. The same parameters also match two-point correlation functions down to $r \simeq 15\,h^{-1}\mathrm{Mpc}$. Moreover, ULPT predicts the relation $b_{K^2}^{\mathrm{E}} = -\tfrac{3}{4} b_2^{\mathrm{E}}$, validated against $N$-body results. These results demonstrate that ULPT provides a physically consistent and efficient model for nonlinear galaxy bias, with applications to redshift-space distortions, bispectra, and reconstruction. The numerical implementation is released as the open-source Python package https://github.com/naonori/ulptkit.

Figures (11)

• Figure 1: Nonlinear bias correction terms $P_X$ for $X = \{ b_1 b_2,\, b_2 b_2,\, b_1 b_3 \}$ at redshift $z = 0$, normalized by the no-wiggle linear matter power spectrum $P_{\rm nw}(k)$. All correction terms smoothly approach zero in the large-scale limit $k \to 0$, as expected for physically consistent nonlinear contributions. This infrared behavior highlights the renormalization-free nature of the ULPT bias expansion.
• Figure 2: Comparison between the ULPT predictions and Dark Emulator outputs for the halo--halo auto power spectrum $P_{\rm hh}(k)$ and the halo--matter cross power spectrum $P_{\rm hm}(k)$, shown across all redshift and halo mass bins. In each panel, the upper sub-panel displays the ratios $P_{\rm hh}/(b_1^2 P_{\rm nw})$ and $P_{\rm hm}/(b_1 P_{\rm nw})$, where $P_{\rm nw}$ denotes the no-wiggle linear matter power spectrum. The lower sub-panel shows the relative deviation between the ULPT and emulator predictions, defined as $\Delta_{P}[\%] = 100 \times (P_{\rm ULPT} - P_{\rm Emu})/P_{\rm Emu}$. Magenta and blue lines denote $P_{\rm hh}$ and $P_{\rm hm}$, respectively; solid and dashed curves indicate ULPT fits, while points represent emulator data. In nearly all cases, ULPT achieves better than 1% accuracy up to $k = 0.3\, h\,\mathrm{Mpc}^{-1}$, except for the highest redshift and mass bin ($z = 1.0$, $\log_{10}(M/M_\odot) = 13.5$), where sub-percent agreement extends up to $k = 0.2\, h\,\mathrm{Mpc}^{-1}$.
• Figure 3: Same as Fig. \ref{['fig:Phh_Phm']}, but for the hybrid analysis at $z = 0$, in which the matter power spectrum $P_{\rm m}$ from the emulator is substituted for the ULPT prediction. The residuals for both $P_{\rm hh}(k)$ and $P_{\rm hm}(k)$ remain within 1% over the entire fitting range, with $P_{\rm hh}(k)$ achieving sub-0.5% accuracy across most scales. The reduced chi-squared values are significantly improved, indicating that the dominant source of discrepancy between the ULPT and emulator predictions at $z = 0$ originates from the nonlinear matter power spectrum.
• Figure 4: Minimum reduced chi-squared values obtained from joint fits to $P_{\rm hh}(k)$ and $P_{\rm hm}(k)$, plotted as a function of the maximum wavenumber $k_{\rm max}$. Each line corresponds to a different combination of redshift and halo mass. A lower $\chi^2_{\rm min}/\mathrm{dof}$ indicates better agreement between ULPT predictions and Dark Emulator outputs.
• Figure 5: Comparison between the ULPT predictions and Dark Emulator outputs for the halo--halo auto correlation function $\xi_{\rm hh}(r)$ and the halo--matter cross correlation function $\xi_{\rm hm}(r)$, shown for all redshift and halo mass bins. In each panel, the upper sub-panel displays the correlation functions themselves, while the lower sub-panel shows the relative deviation between the ULPT and emulator predictions, defined as $\Delta_{\xi}[\%] = 100 \times (\xi_{\rm ULPT} - \xi_{\rm Emu}) / \xi_{\rm Emu}$. Magenta and blue colors denote $\xi_{\rm hh}$ and $\xi_{\rm hm}$, respectively; solid and dashed lines indicate the ULPT predictions, while points represent emulator data. ULPT reproduces both correlation functions to within 1% accuracy over the range $15 \leq r \lesssim 45\, h^{-1} \mathrm{Mpc}$, with only modest deviations (2--4%) at larger separations.