Biased tracers, Hybrid Effective Field Theory and Modified Gravity

This paper extends the Hybrid Effective Field Theory (HEFT) framework to modified gravity cosmologies, specifically $f(R)$ gravity, by detailing the perturbative biased expansion required to compute loop-corrected power spectra and proposing a strategy to adapt existing $\Lambda$CDM emulators for these beyond-standard-model scenarios.

The Gist

Imagine the universe as a giant, expanding loaf of raisin bread. As the bread rises, the raisins (which represent galaxies) move apart. But they don't just drift apart evenly; they clump together in some places and leave empty spaces in others. This clumping is called "structure formation."

For decades, scientists have tried to write a mathematical recipe to predict exactly how these raisins will clump. This paper is about improving that recipe, specifically for two reasons:

1. The "Biased" Raisins: Not all raisins are the same. Some are big, some are small, and they don't distribute themselves exactly like the dough (dark matter) around them. We need a way to model how these specific "biased" raisins cluster.
2. The "Modified" Dough: Most recipes assume the dough follows standard rules (General Relativity). But what if the dough follows slightly different, stranger rules (Modified Gravity)? This paper tests if our recipe still works when the rules of gravity change.

Here is a breakdown of the paper's journey using simple analogies:

1. The Two Ways to Bake (Eulerian vs. Lagrangian)

Scientists have two main ways to track the raisins:

• The "Fixed Grid" Method (Eulerian): Imagine a camera taking a picture of a specific spot in the kitchen. You watch the raisins flow through that spot. This is good for seeing the flow, but it gets messy when the dough gets too squished (non-linear).
• The "Follow the Raisin" Method (Lagrangian): Imagine you put a tiny GPS tracker on a specific raisin at the very beginning. You follow that raisin as it moves from its starting spot to its final spot. This paper uses this method because it handles the "squishing" of the dough much better.

2. The Hybrid Trick (HEFT)

The authors introduce a clever shortcut called Hybrid Effective Field Theory (HEFT).

• The Problem: Calculating the exact movement of every single raisin using pure math is incredibly hard and slow. Calculating it using super-computer simulations is accurate but requires massive computing power.
• The Solution: HEFT is like a "hybrid car." It uses the simple, fast math for the easy parts (where the dough is smooth) and borrows data from the heavy-duty computer simulations for the messy, squished parts. This gives you the speed of math with the accuracy of a simulation.

3. The Challenge: Modified Gravity

Most of these "hybrid cars" have been built and tested only for our current universe (called $\Lambda$CDM), where gravity works the way Einstein originally described.

• The Twist: The authors wanted to see if this hybrid method works if gravity is different. They specifically looked at $f(R)$ gravity, a theory where gravity gets stronger or weaker depending on the scale (like a chameleon changing colors).
• The Difficulty: In this modified gravity, the "growth" of the universe isn't uniform. It depends on the size of the clump. This breaks the simple math shortcuts scientists usually use.

4. What They Did

The team built a new, more flexible engine for their "hybrid car" to handle these weird gravity rules.

• Re-calculating the Map: They derived new mathematical maps (called "growth functions") that account for how gravity changes based on scale.
• Testing the Engine: They ran their new math against super-computer simulations (the "gold standard").
  • Result 1 (Standard Universe): When they tested it on our normal universe, the math worked perfectly, matching the simulations almost exactly.
  • Result 2 (Modified Gravity): When they tested it on the $f(R)$ gravity model, they found that the old, simple math shortcuts (called the "Einstein-de Sitter approximation") failed. They were like using a flat map to navigate a mountainous terrain—the old map just didn't show the hills and valleys correctly. Their new, complex math was required to get the right answer.

5. The Conclusion

The paper concludes that:

• The HEFT framework (the hybrid method) is robust and can be extended to work with these strange, modified gravity theories.
• However, you cannot use the old, simplified math shortcuts when dealing with modified gravity. You must use their new, more complex calculations that account for the changing rules of gravity.
• They have provided the necessary tools and "ingredients" for other scientists to update their galaxy survey models (like those from the DESI or Euclid missions) to test if our universe follows standard gravity or these modified rules.

In short: The authors took a powerful tool for mapping the universe, upgraded its engine to handle "weird gravity," and proved that while the old shortcuts work for our normal universe, they break down in these new scenarios. They have now handed the keys to the rest of the scientific community so they can drive this new vehicle to explore the cosmos.

Technical Summary

Technical Summary: Biased Tracers, Hybrid Effective Field Theory and Modified Gravity

Problem Statement
The analysis of Stage-IV galaxy surveys (e.g., DESI, LSST, Euclid, Roman) requires precise theoretical modeling of the power spectrum of biased tracers, such as galaxies, to constrain cosmological parameters. While perturbative methods and cosmological simulations are standard tools, they present a trade-off: perturbation theory offers computational efficiency but struggles with nonlinear scales, while simulations provide physical completeness but require expensive calibration and mapping to observables. The Hybrid Effective Field Theory (HEFT) framework has emerged as a solution, combining Lagrangian Perturbation Theory (LPT) expansions with outputs from dark-matter-only simulations to model the nonlinear regime efficiently. However, existing HEFT implementations and emulators (e.g., bacco, Aemulus) have been primarily developed within the $\Lambda$CDM paradigm or mild extensions (e.g., $w_0w_a$CDM). A significant gap exists in applying HEFT to modified gravity (MG) theories, such as $f(R)$ gravity, where structure growth is scale-dependent and screening mechanisms (e.g., chameleon) complicate the computation of LPT growth functions and loop integrals.

Methodology
The authors develop a framework to compute loop-corrected biased power spectra within the HEFT formalism for modified gravity cosmologies, specifically focusing on $f(R)$ gravity. The methodology proceeds through several stages:

1. Lagrangian Perturbation Theory (LPT) in General Relativity: The authors review the Eulerian and Lagrangian formulations of perturbation theory. In the Lagrangian picture, the evolution of dark matter is described by a displacement field $\Psi(q, \tau)$ mapping initial Lagrangian coordinates $q$ to final Eulerian positions $x$. The matter power spectrum is derived from the Jacobian of this mapping, utilizing the cumulant theorem to resum infrared (IR) displacements non-perturbatively.
2. Extension to Modified Gravity: The authors revisit the equations of motion for scalar-tensor theories (specifically the Hu-Sawicki $f(R)$ model). They derive the necessary growth factors up to third order, $D^{(1)}(k)$, $D^{(2)}(k_1, k_2)$, and $D^{(3),\text{symm.}}(k_1, k_2, k_3)$, which are now scale-dependent due to the modified Poisson equation and the presence of a scalar field. This involves solving coupled continuity, Euler, and Klein-Gordon equations iteratively.
3. Numerical Implementation of Generalized Kernels: The authors implement a numerical code to solve the generalized growth functions on a grid of momenta. They compute the Lagrangian kernels $L^{(n)}$ and the associated loop integrals ($Q_n, R_n$ functions) required for the one-loop matter power spectrum. They explicitly test the validity of the Einstein–de Sitter (EdS) approximation, which assumes scale-independent growth, against these full scale-dependent solutions.
4. Biased Tracer Expansion: The framework is extended to biased tracers using the local Lagrangian bias scheme. The tracer density is expanded in terms of the initial density field $\delta_L$, tidal shear $s^2$, and higher-order operators. The authors compute the relevant Lagrangian correlators involving both displacement fields and initial density fields, adapting the scalar functions ($Q_I, R_I$) for the scale-dependent MG context.
5. Validation: The implementation is validated against existing codes, specifically velocileptors (for EdS comparisons) and mgpt (for full MG comparisons), ensuring consistency in the calculation of loop integrals and power spectra.

Key Contributions

• Generalized LPT Framework for MG: The paper provides a detailed derivation and numerical implementation of LPT growth functions and kernels for $f(R)$ gravity, accounting for scale-dependent growth and chameleon screening effects up to third order.
• Quantification of EdS Approximation Failure: The authors demonstrate that the Einstein–de Sitter approximation, while highly accurate for $\Lambda$CDM (sub-percent errors), fails significantly in modified gravity scenarios. In $f(R)$ models, the EdS approximation deviates from exact solutions, particularly in the mildly nonlinear regime ($k \sim 0.1\text{--}0.4 \, h\text{Mpc}^{-1}$) and in squeezed triangle configurations where mode coupling is strongest.
• HEFT Extension Strategy: The work outlines the necessary ingredients to extend existing HEFT-based emulators (like bacco and Aemulus) to beyond-$\Lambda$CDM cosmologies. This involves replacing the standard EdS kernels with the generalized, scale-dependent kernels derived in this work.
• Validation of Loop Integrals: The authors verify their numerical integration of the generalized loop integrals ($Q_n, R_n$) against the mgpt code, showing excellent agreement across $\Lambda$CDM and $f(R)$ models.

Results

• Scale-Dependent Growth: The computed second- and third-order Lagrangian growth factors exhibit strong scale dependence in $f(R)$ gravity, with the most significant deviations occurring in squeezed configurations.
• Loop Integral Deviations: Comparisons of the $Q_n$ and $R_n$ functions show that while the EdS approximation reproduces $\Lambda$CDM results with high fidelity, it introduces visible errors in $f(R)$ gravity. These errors are most pronounced where loop corrections peak, indicating that using EdS kernels in MG analyses would lead to biased theoretical predictions.
• Matter Power Spectrum: The inclusion of full-kernel loop corrections in LPT yields a matter power spectrum that differs from the EdS prediction in $f(R)$ models. The exponential suppression (Zel'dovich contribution) is more accentuated in $f(R)$ due to larger displacement variances ($\sigma_L^2$).
• Biased Tracer Consistency: The generalized formalism successfully computes the biased tracer power spectrum, confirming that the standard local Lagrangian bias expansion remains applicable provided the underlying growth functions and loop integrals are treated with the correct scale dependence.

Significance
The paper establishes the theoretical and numerical foundation required to apply the Hybrid Effective Field Theory framework to modified gravity cosmologies. By explicitly quantifying the limitations of the Einstein–de Sitter approximation in scale-dependent growth scenarios, the authors argue that accurate modeling of future Stage-IV survey data in non-$\Lambda$CDM theories requires the use of generalized, scale-dependent LPT kernels. This work enables the extension of high-precision emulators to test theories of gravity beyond General Relativity, bridging the gap between perturbative efficiency and the complex nonlinear physics of modified gravity.