Modeling nonlinear scales for dynamical dark energy cosmologies with COLA

1. Problem Statement

Upcoming Stage-IV galaxy surveys (e.g., LSST, Euclid) will provide high-precision data on matter clustering at small, non-linear scales. However, modeling these scales for cosmologies beyond the standard $\Lambda$CDM model (specifically those with dynamical dark energy) is computationally prohibitive.

• The Bottleneck: Full $N$-body simulations are the gold standard for accuracy but are too expensive (tens of thousands of CPU hours per run) to generate the massive suites required for Bayesian parameter inference ($O(10^5 - 10^6)$ points).
• The Gap: While high-precision emulators exist for $\Lambda$CDM and simple extensions (like $w_0w_a$CDM), generating new $N$-body suites for every candidate dynamical dark energy model is unfeasible.
• Current Approximations: Existing workarounds often assume that the non-linear "boost" (the ratio of non-linear to linear power spectrum) derived from $\Lambda$CDM can be applied to extended cosmologies without modification. The authors argue this introduces uncontrolled systematics and biases in parameter constraints.

2. Methodology

The authors propose and validate a hybrid emulation framework that combines the computational efficiency of the COmoving Lagrangian Acceleration (COLA) method with the high accuracy of existing $\Lambda$CDM emulators.

A. The COLA Simulation Suite

• Algorithm: COLA combines Lagrangian Perturbation Theory (2LPT) for large scales with a Particle-Mesh (PM) scheme for small scales. It is 1–2 orders of magnitude faster than full $N$-body simulations.
• Setup:
  • Code: fml (Fast Multi-Level).
  • Box Size: $L = 1024 \, h^{-1}\text{Mpc}$ with $N_{part} = 1024^3$ particles.
  • Resolution: Force grid $N_{mesh} = 2048^3$; Power spectra calculated on a $1024^3$ grid.
  • Sampling: 700 cosmologies in the $w_0w_a$CDM parameter space using Latin Hypercube Sampling.
  • Technique: "Pairing-and-fixing" was used to suppress sample variance.
  • Cost: ~40 minutes per simulation on 128 cores (Total training cost: ~153,600 CPU-hours).

B. The Hybrid Emulator Architecture

Instead of training a neural network (NN) to predict the absolute power spectrum, the authors train it to predict the non-linear boost correction relative to a high-fidelity reference.

1. Definition: The boost is defined as $B(k, z) = P_{NL}(k, z) / P_L(k, z)$.
2. Hybrid Formula: The final predicted boost for a dynamical dark energy cosmology $\theta$ is:
   $$\tilde{B}_{COLA}(k, z | \theta) = B_{N-body}(k, z | \theta_p) \times \frac{B_{COLA}(k, z | \theta)}{B_{COLA}(k, z | \theta_p)}$$
   Where $\theta_p$ is the projection of $\theta$ onto the $\Lambda$CDM subspace ($w_0=-1, w_a=0$), and $B_{N-body}$ is the benchmark boost from EuclidEmulator2.
3. Machine Learning:
   • Input: Normalized cosmological parameters ($\Omega_m, \Omega_b, h, A_s, n_s, w_0, w_a$).
   • Target: Principal Components (PCA) of the COLA boost ratios.
   • Architecture: A fully connected neural network with 3 hidden layers (1024 neurons each) and a parametric activation function.
   • Goal: The NN learns the difference between COLA and the high-precision reference, correcting COLA's small-scale inaccuracies.

C. Validation Strategy

• Benchmark: EuclidEmulator2 (EE2).
• Comparison Method: A simulated cosmic shear analysis mimicking the LSST Year 1 (LSST-Y1) survey.
• Metrics:
  • Bias: $1\sigma$ shifts in parameters ($\Omega_m, S_8, w_0, w_a$).
  • Figure of Bias (FoB): Multivariate bias metric.
  • Figure of Merit (FoM): Precision of constraints.
  • Goodness-of-Fit: $\Delta\chi^2$ distributions comparing predictions against the benchmark.

3. Key Contributions

1. Extension to $w_0w_a$CDM: Successfully extended the COLA-based hybrid emulation method from $w$CDM to the more general and widely used $w_0w_a$CDM dynamical dark energy model.
2. Validation of the Hybrid Approach: Demonstrated that combining COLA with a $\Lambda$CDM reference emulator effectively mitigates COLA's intrinsic small-scale errors, achieving $\lesssim 2\%$ accuracy in the non-linear boost up to $k \approx 1 \, h/\text{Mpc}$.
3. Quantification of Approximation Errors: Provided a rigorous comparison showing that the common practice of using $\Lambda$CDM boosts for extended cosmologies (EE2 $\Lambda$CDM) introduces significant biases, whereas the COLA hybrid method does not.
4. LSST-Y1 Readiness: Validated the emulator's performance in a realistic cosmological inference setting, proving it is suitable for upcoming precision surveys.

4. Key Results

A. Emulator Accuracy

• Raw COLA: Shows ~10% error at $k=1 \, h/\text{Mpc}$ compared to $N$-body.
• Hybrid Emulator: Reduces this error significantly. At $k=1 \, h/\text{Mpc}$, 90% of test cosmologies have emulation errors within 2% when compared to EuclidEmulator2.
• Comparison to $\Lambda$CDM Projection: The standard approach of projecting to $\Lambda$CDM (EE2 $\Lambda$CDM) yields errors of ~7.5% at the same scale, an order of magnitude worse than the hybrid method.

B. Parameter Constraints (LSST-Y1 Simulation)

• Bias: The COLA emulator produces parameter constraints that are unbiased compared to the benchmark.
  • All 1D parameter biases are within $0.3\sigma$.
  • The 7D Figure of Bias (FoB) is < 0.35 for all tested fiducial cosmologies (including extreme values of $\Omega_m$ and $\sigma_8$).
• Precision (FoM):
  • The COLA emulator yields constraints slightly tighter than the benchmark (up to ~19% higher FoM at aggressive scales), likely due to COLA's specific noise characteristics, but remains statistically consistent.
  • In contrast, the EE2 $\Lambda$CDM approach artificially inflates the FoM by ~47–58% (over-optimistic constraints) and shifts central values by nearly $1\sigma$ for $S_8$ in non-$\Lambda$CDM fiducials.
• Goodness-of-Fit: The $\Delta\chi^2$ distribution for the COLA method is narrow and centered near zero, indicating excellent agreement with the benchmark. The EE2 $\Lambda$CDM approach shows a broad distribution, indicating poor fit quality for dynamical dark energy models.

5. Significance and Conclusion

This work establishes COLA-based hybrid emulators as a computationally efficient and physically robust alternative to full $N$-body simulations for extended cosmologies.

• Efficiency: It reduces the computational cost of generating training data by orders of magnitude while maintaining $N$-body level precision for parameter inference.
• Reliability: It avoids the systematic biases introduced by assuming $\Lambda$CDM non-linear corrections apply to dynamical dark energy models.
• Future Outlook: The methodology is flexible and can be applied to other beyond-$\Lambda$CDM scenarios, such as modified gravity or coupled dark energy. While baryonic feedback is not yet included, the framework is ready to incorporate such corrections in future iterations.

In summary, the paper provides a practical solution for the "non-linear modeling bottleneck" in the era of precision cosmology, enabling accurate constraints on dynamical dark energy using next-generation survey data.