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Simulation budgeting for hybrid effective field theories

This paper forecasts the simulation requirements for training hybrid effective field theory (HEFT) emulators, demonstrating that fewer than 225 N-body simulations are sufficient to achieve 1–2% accuracy across a broad cosmological parameter space, thereby providing practical guidance for efficient emulator design in future analyses.

Original authors: Alexa Bartlett, Joseph DeRose, Martin White

Published 2026-04-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Alexa Bartlett, Joseph DeRose, Martin White

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for the entire planet, but instead of just looking at clouds, you are trying to understand the entire history of the universe's growth. You want to know how galaxies formed, how dark energy is stretching space, and how invisible neutrinos are moving around.

To do this, cosmologists use two main tools:

Math Formulas (Theory): Great for simple, smooth parts of the universe, but they break down when things get messy and clumpy.
Supercomputer Simulations (N-body): These are like massive video games where you drop billions of "particles" of dark matter into a box and watch them crash into each other over billions of years. They are incredibly accurate but extremely expensive to run.

This paper is about budgeting. The authors are asking: "How many of these expensive computer simulations do we actually need to run to build a perfect 'weather forecast' tool for the universe?"

Here is the breakdown of their findings using simple analogies.

1. The Problem: The "Supercomputer" is Too Expensive

Imagine you want to build a map of every possible shape a mountain range could take.

The Old Way: You hire a team of hikers to climb every single possible mountain to measure it. This would take forever and cost a fortune.
The New Way (HEFT): You use a mix of math and a few key measurements. You climb a few specific, representative mountains (simulations) and use a smart computer program (an Emulator) to guess what the other mountains look like based on those few samples.

The problem is: If you only climb 5 mountains, your guesses for the 6th one might be way off. If you climb 1,000, your guesses are perfect, but you've spent your whole life budget on hiking. The authors wanted to find the "Goldilocks" number: Just enough simulations to be accurate, but not so many that we go broke.

2. The Goal: How Accurate Do We Need to Be?

The authors set a target: They want their "guessing machine" (the emulator) to be 1% accurate for the most likely scenarios, and 2% accurate for the weird, unlikely ones.

Why 1%? Because the new telescopes coming online (like the Roman Space Telescope and the Vera Rubin Observatory) are so powerful that they will measure the universe with incredible precision. If our "guessing machine" is off by even a tiny bit, we might think we discovered a new law of physics when we actually just made a math error.

They also had to account for "messy" real-world factors:

Intrinsic Alignments: Galaxies aren't just floating randomly; they tend to line up like leaves in a stream due to local gravity. This messes up our measurements.
Baryons (Normal Matter): Stars and gas explode and push things around. This "feedback" changes how galaxies cluster. It's like trying to predict traffic patterns when you don't know if a parade is happening.

3. The Solution: The "Hybrid" Approach

The authors propose a Hybrid Effective Field Theory (HEFT). Think of this as a Chef's Recipe:

The Math (LPT): This is the basic recipe (flour, sugar, eggs). It works perfectly for the smooth batter.
The Simulations: This is the actual baking. We need to bake a few cakes to see exactly how the heat affects the rise.
The Emulator: This is the AI chef. It tastes the few cakes we baked and learns the rules of how the ingredients interact. Then, it can instantly tell you what a cake made with any combination of ingredients would taste like, without you having to bake it.

4. The Big Discovery: You Don't Need as Many as You Think

The authors ran a massive test using a "fake" simulation (a stand-in model) to see how many real simulations they would need.

The "Wide" Search (Tier 1): If you want to explore the entire universe of possibilities (including weird dark energy and heavy neutrinos), you need about 225 simulations.
The "Focused" Search (Tier 2): If you only care about the most likely scenarios (the "high-likelihood" zone), you only need about 80 simulations.

The Analogy:
Imagine you are trying to find the best pizza in a city.

The Old Fear: "We need to taste every pizza in the city (10,000 slices) to be sure!"
The Paper's Finding: "Actually, if you taste 80 representative slices from different neighborhoods, you can build a model that predicts the taste of any pizza in the city with 99% accuracy."

5. Practical Advice for the Future

The paper gives a "shopping list" for future cosmologists:

Don't over-engineer: You don't need the highest-resolution simulations (which are the most expensive) for everything. You can use "coarse" simulations for the big picture and save the expensive ones for the tricky parts.
Start late: You don't need to simulate the universe from the Big Bang. You can start the simulation later (like at redshift 12) and still get perfect results, saving massive amounts of computer time.
The "Steel Sample": If you pick a very specific, high-quality group of galaxies (a "steel sample") that are easy to measure, you might only need 100 simulations to get perfect results for major experiments like DESI or the Roman Telescope.

Summary

This paper is a cost-benefit analysis for the universe. It tells us that we don't need to spend millions of dollars on supercomputers to understand the cosmos. By using smart math, a few well-chosen simulations, and a clever "guessing" algorithm, we can achieve the precision needed for the next generation of telescopes with a budget that is actually manageable.

The Bottom Line: We can build a perfect map of the universe without climbing every single mountain. We just need to climb the right ones.

1. Problem Statement

Future cosmological surveys (e.g., DESI, LSST/Rubin, Euclid, Roman, CMB-S4) will provide high-precision measurements of large-scale structure (LSS). To extract cosmological parameters from these data, particularly on quasi-nonlinear scales ( $k \lesssim 1 \, h \text{ Mpc}^{-1}$ ), accurate theoretical models are required.

The Challenge: Pure analytical perturbation theory (PT) fails at small scales due to nonlinear growth, while running full $N$ -body simulations for every cosmology in a Markov Chain Monte Carlo (MCMC) analysis is computationally prohibitive.
The Solution: Hybrid Effective Field Theory (HEFT) combines $N$ -body simulations with analytical bias expansions. However, HEFT requires training "emulators" (surrogate models) on a grid of simulations.
The Gap: There is no clear consensus on the simulation budget (number of simulations, volume, resolution, and initial conditions) required to train HEFT emulators to meet the accuracy goals of next-generation surveys, especially for complex cosmologies involving massive neutrinos and dynamical dark energy ( $w_0wa\text{CDM} + m_\nu$ ).

2. Methodology

The authors developed a framework to forecast the computational cost of training HEFT emulators by defining accuracy requirements and simulating the training process using a surrogate model.

A. Accuracy Requirements

The authors established a target accuracy of 1% in the high-likelihood region and 2% over a broader parameter space for $k < 1 \, h \text{ Mpc}^{-1}$ and $z < 3$ . This goal was derived by analyzing:

Statistical Errors: Forecasted from future surveys (DESI, Rubin, SO).
Systematic Uncertainties: Intrinsic alignments (IA), baryonic feedback, photometric redshift errors, and bias modeling.
Bias Expansion Order: They demonstrated that while linear bias is insufficient, quadratic bias (plus a single cubic operator $:\delta^3:$ ) achieves $\sim 1\%$ accuracy up to $k \approx 0.6 \, h \text{ Mpc}^{-1}$ .

B. Simulation Specifications

To meet the 1% accuracy goal, the paper recommends specific simulation parameters:

Volume: $L_{\text{box}} = 1 \, h^{-1} \text{ Gpc}$ to minimize sample variance and missing power from super-box modes.
Resolution: Particle mass $\sim 10^{10} \, h^{-1} M_\odot$ (approx. $10^9$ particles). Crucially, the authors argue that resolving dark matter halos is unnecessary for HEFT; only the matter power spectrum needs to be resolved. This allows for coarser force resolution and larger time steps.
Initial Conditions: Generated at $z \approx 12$ using 3rd-order Lagrangian Perturbation Theory (LPT) to minimize discretization errors.
Variance Reduction: Use of Zeldovich Control Variates (ZCV) to suppress sample variance at low $k$ , allowing smaller box sizes or fewer realizations.

C. Emulator Training Strategy

Since running thousands of $N$ -body simulations for the study itself was too costly, the authors used a surrogate model:

Proxy: They combined HMcode2020 (for nonlinear clustering) with Lagrangian PT (for linear/one-loop terms) to mimic the behavior of true $N$ -body simulations.
Training Data: They generated training sets for an 8-dimensional parameter space ( $\{A_s, n_s, h, w_0, w_a, \omega_b, \omega_{cdm}, \sum m_\nu\}$ ) using Latin Hypercube Sampling.
Emulator Architecture: Used Principal Component Analysis (PCA) for dimensionality reduction and Polynomial Chaos Expansion (PCE) for interpolation. They emulated the log-ratio of the model to the 1-loop PT prediction to reduce dynamic range.

3. Key Contributions

Quantified Simulation Budgets: The paper provides concrete numbers for the number of simulations required to train HEFT emulators for various survey configurations and cosmological models.
Cost-Effective Simulation Design: It demonstrates that HEFT does not require expensive, high-resolution halo-resolving simulations. Coarse-resolution, large-volume simulations with optimized time-stepping can achieve the necessary accuracy, drastically reducing computational costs.
Systematic Error Budgeting: The authors rigorously break down how systematics (IA, baryons, bias) dictate the required emulator precision, showing that for many use cases, the emulator error can be made sub-dominant to other uncertainties.
Open Source Tools: The code used for the analysis and the implementation of ZCV for galaxy-matter cross-spectra are made publicly available.

4. Key Results

The authors forecasted the number of training simulations ( $N_{\text{train}}$ ) required for different use cases to keep the emulator error below a fraction of the statistical error:

Use Case	Parameter Space	Tier 1 (Broad Space)	Tier 2 (High Likelihood)	Notes
DES Y6 GGL + DESI BGS	$w_0wa\text{CDM} + m_\nu$	~600	~110	3D clustering available; error driven by shape noise.
SO CMB Lensing + DESI LRG	$w_0wa\text{CDM} + m_\nu$	~1500	~275	Driven by high- $k$ galaxy auto-spectrum ( $C_{gg}$ ) precision.
Roman (RST) Cosmic Shear	$w_0wa\text{CDM} + m_\nu$	~1500	~100	High source density but small sky area; high- $k$ modeling critical.
"Steel" Sample	$w_0wa\text{CDM} + m_\nu$	~700	~110	Optimized sample with low systematics; 3D clustering available.

Tier 1: Covers a broad parameter space; requires ~225 simulations to meet 2% accuracy goals.
Tier 2: Covers a narrower, high-likelihood space; requires as few as 80 simulations to meet 1% accuracy goals.
Scaling: The number of required simulations scales with the dimensionality of the parameter space. For an 8-parameter model, ~225 simulations are sufficient for a broad search, while restricted spaces need fewer.
Cost Implications: If using cheap Particle Mesh (PM) simulations with global time steps, the total cost is manageable ( $O(10^3)$ node-hours). If high-resolution halo-resolving simulations were used, the cost would be prohibitive ( $O(10^5-6)$ node-hours).

5. Significance

This work provides a practical roadmap for the cosmological community to design efficient simulation campaigns for the next decade of surveys.

Feasibility: It proves that achieving percent-level precision for complex cosmologies (including massive neutrinos and evolving dark energy) is computationally feasible without requiring exascale resources, provided the simulation design is optimized for HEFT (i.e., avoiding unnecessary halo resolution).
Efficiency: By identifying that ~100–200 simulations are often sufficient for high-precision constraints, the paper encourages the development of emulators that can be trained on modest computational budgets.
Strategic Guidance: It advises that for many analyses (especially those with 3D clustering), the "bottleneck" is not the emulator accuracy but the modeling of systematics (baryons, IA), suggesting that resources should be allocated to understanding these systematics rather than blindly increasing simulation resolution.

In summary, the paper successfully bridges the gap between theoretical requirements and computational reality, enabling the efficient deployment of HEFT emulators for future precision cosmology.