Improved recipes for peculiar velocity power spectra using Evolution Mapping

Technical Summary: Improved Recipes for Peculiar Velocity Power Spectra using Evolution Mapping

Problem Statement
Current and upcoming galaxy surveys (e.g., DES, DESI, Euclid, LSST) require precise theoretical models of non-linear structure formation to extract cosmological information from redshift-space distortions (RSD). A critical component of RSD modelling is the velocity divergence field ($\theta = -\nabla \cdot \mathbf{v}$), specifically its auto-power spectrum $P_{\theta\theta}(k)$ and its cross-power spectrum with the density field $P_{\delta\theta}(k)$.

Existing fitting prescriptions, notably those by Bel et al. (2019, hereafter B19), suffer from three primary limitations:

1. Numerical Convergence: Previous calibrations relied on simulation resolutions where sampling artifacts (arising from estimating velocity fields from discrete particle distributions) rendered measurements unreliable at scales as large as $k \sim 0.2\text{--}0.3 \, h\text{Mpc}^{-1}$.
2. Spurious $h$-dependence: B19 parametrise non-linear evolution using $\sigma_8$ (RMS density fluctuation smoothed on $8 \, h^{-1}\text{Mpc}$). Because the smoothing scale depends on the Hubble parameter $h$, this introduces unphysical dependencies on $h$. Consequently, models with identical linear physics but different $h$ values yield divergent predictions, compromising accuracy when extrapolating across cosmological parameter space.
3. Functional Form: Existing models do not fully capture mode-coupling effects or the non-linear damping of Baryon Acoustic Oscillations (BAO) without additional free parameters.

Methodology
The authors present new fitting functions calibrated on a suite of gravity-only $N$-body simulations (Aletheia, AletheiaMass, and AletheiaEmu) using the Evolution Mapping framework. This framework posits that when shape parameters ($\Theta_s$) are fixed, the impact of evolution parameters ($\Theta_e$) and redshift can be absorbed into a single clustering amplitude parameter, $\sigma_{12}$ (RMS density fluctuation smoothed on 12 Mpc), which is independent of $h$.

Key methodological steps include:

• Convergence Strategy: The authors utilize the AletheiaMass suite, which features multi-resolution simulations (varying box sizes and particle counts). They employ a "patching" strategy where measurements from different volumes are stitched together. A convergence scale $k_{\text{crop}}$ is defined for each box by comparing high- and low-resolution runs; the final power spectrum uses the measurement from the largest volume satisfying $k < k_{\text{crop}}$. This yields converged measurements up to $k \simeq 0.56 \, \text{Mpc}^{-1}$.
• Velocity Estimation: To mitigate sampling artifacts, the authors use the MC-Voronoi method. This involves re-sampling the velocity field onto a uniform glass-like distribution of points (volume-weighted) based on the nearest $N$-body particle, followed by Cloud-in-Cell (CIC) assignment and deconvolution.
• Functional Forms:
  • $P_{\theta\theta}(k)$: The authors propose an ansatz combining a damped linear term (using a de-wiggled linear power spectrum $P_{dw}$ to account for non-linear BAO damping) and an additive Gaussian term in $\ln k$ to capture mode-coupling. The parameters are functions of $\sigma_{12}$.
  • $P_{\delta\theta}(k)$: This is modelled as the geometric mean of the non-linear density spectrum $P_{\delta\delta}(k)$ (predicted by the Aletheia emulator) and the new $P_{\theta\theta}(k)$, multiplied by an exponential damping term dependent on $\sigma_{12}$.
• Calibration: Parameters are fitted using a Bayesian framework (nautilus) against the AletheiaMass suite, covering a wide range of $\sigma_{12}$ (0.2 to 1.0) and redshifts ($z \approx -0.4$ to $4.2$).

Key Contributions

1. New Fitting Functions: The paper provides calibrated formulas for $P_{\theta\theta}(k)$ and $P_{\delta\theta}(k)$ that depend explicitly on $\sigma_{12}$ and use physical units (Mpc$^{-1}$), eliminating spurious $h$-dependencies.
2. Robust Convergence: By implementing a conservative patching strategy with multi-resolution simulations, the authors achieve percent-level accuracy up to $k \simeq 0.56 \, \text{Mpc}^{-1}$, significantly extending the reliable range compared to previous works.
3. Validation of Evolution Mapping: The study confirms that expressing velocity statistics in terms of $\sigma_{12}$ preserves the evolution mapping relation, ensuring that cosmologies with identical linear physics and $\sigma_{12}$ yield consistent non-linear predictions regardless of their growth history or $h$ value.

Results

• Accuracy: The new prescriptions achieve 1–2% accuracy for both $P_{\theta\theta}(k)$ and $P_{\delta\theta}(k)$ over a wide range of $\sigma_{12}$ and scales up to $k \simeq 0.56 \, \text{Mpc}^{-1}$ within the calibration set.
• Comparison with B19: The new models systematically outperform B19. B19 predictions deviate significantly at small scales ($k > 0.2 \, \text{Mpc}^{-1}$) due to resolution limits and show unphysical offsets for cosmologies with different $h$ values. The new models correctly predict velocity statistics across different Hubble parameters.
• Growth History Effects: Using the Aletheia suite (which isolates growth history effects by fixing shape parameters but varying evolution parameters), the authors quantify deviations from perfect evolution mapping. They find that for most cosmologies of practical interest, neglecting growth history differences introduces errors below 1%. Significant deviations ($>1\%$) only occur for extreme, observationally ruled-out cosmologies (e.g., Einstein-de Sitter or extreme $h$ values).
• Cross-Spectrum: The $P_{\delta\theta}(k)$ model benefits from the Aletheia emulator for $P_{\delta\delta}(k)$, which implicitly captures some growth history dependence, further reducing errors.

Significance
The paper claims that the choice of parametrisation and units is critical for the robustness of velocity statistics models. By demonstrating that $h$-dependent units (like $\sigma_8$) introduce systematic biases, the authors provide strong empirical support for using physical units and $\sigma_{12}$ as the primary scaling parameter. These fitting functions offer a robust, percent-level description of velocity power spectra, directly applicable to RSD modelling in current and future galaxy redshift surveys. The authors note that while their current fitting functions are sufficient for practical applications, future work may involve constructing a dedicated velocity emulator to explicitly model residual growth-history deviations and achieve higher precision across the full cosmological parameter space.