Fullshape power spectrum for the Symmetron modified gravity model

This paper develops and validates a fullshape galaxy power spectrum pipeline for the Symmetron modified gravity model using perturbation theory and MCMC analysis, demonstrating its viability and similarity to the Hu-Sawicki F6 model through comparisons with EZMock simulations.

The Gist

The Big Picture: Is Gravity Broken?

Imagine the universe is a giant, expanding balloon. For decades, scientists have used a standard rulebook called General Relativity (Einstein's theory of gravity) to predict how the balloon expands and how the dots on it (galaxies) clump together. This rulebook works perfectly for our solar system.

However, when we look at the entire universe, something is weird. The universe isn't just expanding; it's speeding up. To explain this, standard physics invented a mysterious, invisible glue called Dark Energy. But Dark Energy is a bit of a "patch job"—we don't really know what it is.

This paper asks a different question: What if the rulebook itself is slightly wrong? What if gravity behaves differently on the massive scale of the universe compared to the small scale of a planet? This is called Modified Gravity (MG).

The Two "New Rules" Being Tested

The authors are testing two specific "new rulebooks" for gravity:

1. The Hu-Sawicki Model (The "F6" Model): Think of this as a gravity rule that gets a little stronger in empty space but acts normal in crowded places (like our solar system). It's like a rubber band that stretches easily in a vacuum but snaps tight when you pull it hard.
2. The Symmetron Model: This is the star of the show. Imagine gravity has a "secret mode." In dense areas (like near Earth), it hides its extra powers and acts exactly like Einstein's gravity. But in the vast, empty voids between galaxies, it "wakes up" and starts pulling things together differently. It's like a spy who acts normal at a party but changes their behavior when they are alone in the woods.

The Challenge: The "Fullshape" Puzzle

To test these theories, scientists look at the Power Spectrum. Imagine taking a photo of the entire universe and looking at the pattern of galaxy clusters.

• Standard Analysis: Usually, scientists just look at specific "peaks" in the pattern (like measuring the distance between two specific trees).
• Fullshape Analysis: This paper tries to analyze the entire photo, from the big gaps to the tiny clumps. It's like trying to solve a puzzle by looking at every single piece's shape and color, not just the corners.

The problem? Calculating the "Fullshape" for these new gravity models is incredibly slow. It's like trying to solve a Rubik's cube while someone is constantly changing the colors of the stickers. It takes too long to run the computer simulations needed to compare theories with real data.

The Solution: The "Shortcut" (fkPT)

The authors developed a clever shortcut called fkPT.

• The Analogy: Imagine you are baking a cake. The "full recipe" requires you to measure the temperature of the oven every single second and adjust the heat, which takes forever. The "shortcut" says, "Okay, we know the oven generally stays at 350°F, but let's just tweak the rising time of the dough based on the specific flour we used."
• In Physics: They realized that while the complex math changes, the way the galaxies grow (the "growth rate") follows a predictable pattern. They created a simple formula (a "parametrization") that approximates the complex math with less than 1% error. This turns a calculation that takes hours into one that takes seconds.

The "Screening" Mechanism

A major worry with new gravity theories is: "If gravity is different, why don't we feel it on Earth?"

The answer is the Screening Mechanism.

• The Analogy: Think of the Symmetron model as a chameleon. In a dense forest (high density, like our solar system), it blends in perfectly with the trees (acts like normal gravity). But in an open desert (low density, like deep space), it reveals its true, colorful self (modified gravity).
• The paper confirms that the Symmetron model successfully "hides" its extra forces in our solar system while still changing how galaxies clump together in the deep universe.

The Results: A Perfect Match

The authors ran their new "shortcut" method against computer simulations (called EZMocks, which are fake universes generated by computers to test theories).

1. Similarity: They found that the Symmetron model and the Hu-Sawicki model produce almost identical patterns of galaxy clustering. They are like two different brands of sneakers that look and feel exactly the same when you run.
2. Validation: When they fed their model into a "search engine" (an MCMC sampler) using data from a fake universe that wasn't modified (it was standard Einstein gravity), the model correctly said, "You know what? This looks just like standard gravity."
3. Readiness: This proves their "shortcut" method works. They are now ready to apply this to real data from the DESI survey (a massive project mapping millions of galaxies).

The Bottom Line

This paper is a tool-building project. The authors haven't discovered a new universe yet, but they have built a fast, accurate, and reliable calculator that allows scientists to test if gravity works differently on the cosmic scale.

They showed that:

• The Symmetron model is a viable candidate for explaining the universe's acceleration.
• Their new math shortcut makes testing these theories fast enough to use with real-world data.
• The universe might be playing a game of hide-and-seek with gravity, and we finally have a better way to find it.

Technical Summary

1. Problem Statement

The standard $\Lambda$CDM cosmological model, while successful, faces theoretical challenges regarding the nature of dark energy (e.g., the coincidence problem and the fine-tuning of the cosmological constant). Recent observational data from surveys like DESI and DES suggest that dynamical dark energy or Modified Gravity (MG) models may provide a better fit than $\Lambda$CDM.

To distinguish between these models, cosmologists rely on large-scale structure (LSS) observables, specifically the galaxy power spectrum (PS) and Redshift Space Distortions (RSD). However, modeling the LSS in MG theories is computationally expensive and theoretically complex due to:

1. Scale-dependent growth: Unlike General Relativity (GR), MG models introduce a scale-dependent growth rate $f(k, z)$ due to an extra scalar degree of freedom.
2. Non-linear screening: Mechanisms like the Symmetron screening are required to recover GR in high-density regions (solar system tests) but introduce complex non-linearities in the perturbation theory.
3. Computational cost: Calculating the full one-loop power spectrum for MG models requires solving differential equations for kernels at every step of a Markov Chain Monte Carlo (MCMC) analysis, which is often too slow for parameter inference with real data.

The authors aim to develop a robust pipeline to compute the fullshape galaxy power spectrum for the Symmetron model, validate it against simulations, and compare its viability with the widely studied Hu-Sawicki $f(R)$ model.

2. Methodology

The authors employ a comprehensive theoretical framework combining Perturbation Theory (PT), Effective Field Theory (EFT), and specific approximations to handle MG complexities.

A. Theoretical Framework

• Models: The study focuses on the Symmetron model (a scalar-tensor theory with a symmetry-breaking potential) and the Hu-Sawicki $f(R)$ model ($n=1$) as a benchmark.
• Linear Growth: They derive the scale-dependent growth rate $f(k, z)$ by solving the modified continuity and Euler equations. The growth is governed by a function $\mu(k, a)$ which modifies the Poisson equation.
• Non-linear Contributions:
  • They construct the full one-loop power spectrum using Standard Perturbation Theory (SPT) up to 1-loop order.
  • They incorporate Infrared (IR) Resummation to correctly model Baryon Acoustic Oscillations (BAO) damping.
  • They include EFT counterterms and shot noise to account for non-perturbative effects and galaxy bias.
• The $f_k$-PT Approximation: To reduce computational time for MCMC sampling, they utilize the $f_k$-Perturbation Theory ($f_k$PT) approximation.
  • Instead of calculating full, scale-dependent kernels ($F_n, G_n$) numerically at every step, they approximate the kernels using their large-scale limits ($k \to 0$) but retain the full scale-dependence in the linear growth rate $f(k, z)$.
  • Missing kinematic terms are compensated by renormalizing EFT counterterms.

B. Parametrization

• The authors propose an analytical parametrization for the scale-dependent growth rate:
  $$f(k) = c_1 \tanh^2(c_2 k) + c_3$$
  This formula approximates the numerical solution of the growth differential equation with a deviation of less than 0.6% for $k \leq 0.2 \, h/\text{Mpc}$, significantly speeding up calculations.

C. Validation and Data

• Code Implementation: The theory is implemented in the FOLPS-nu code (adapted for MG) and the fkpt code.
• Validation: Since high-precision non-linear simulations for the Symmetron model are currently insufficient for 1-loop RSD testing, the authors validate their pipeline using GR EZMocks (1000 realizations).
• Goal: They test if the Symmetron pipeline can successfully recover the GR limit (where the coupling parameter $\beta_0 \to 0$) when fitted to GR data.

3. Key Contributions

1. First Fullshape Analysis for Symmetron: This work provides the first comprehensive calculation of the fullshape power spectrum (including 1-loop PT, RSD, and EFT terms) specifically for the Symmetron model.
2. $f_k$PT Adaptation: They successfully adapt the $f_k$PT approximation to the Symmetron model, demonstrating that it is a viable, fast alternative to full numerical kernel calculations, with errors in multipoles remaining below 1%.
3. Growth Rate Parametrization: The introduction of the $\tanh^2$ parametrization for $f(k)$ offers a computationally efficient tool for future MCMC analyses of MG models.
4. Correction to IR-Resummation: The authors correct a previous implementation in the literature (Ref. [43]) by using the full scale-dependent growth rate $f(k)$ in the IR-resummation scheme rather than the constant $f_0$. They show this correction significantly impacts the power spectrum amplitude, particularly in the quadrupole.
5. Comparative Analysis: A direct comparison between Symmetron and Hu-Sawicki $f(R)$ (specifically the F6 variant) is established, showing their similarities in growth behavior and power spectrum structure.

4. Key Results

• Growth Rate Similarity: The Symmetron model exhibits a growth rate very similar to the Hu-Sawicki F6 model in the wavenumber interval $0.01 \leq k \leq 0.1$ and for redshifts where the model is viable. The deviation from $\Lambda$CDM is of the same order as F6, making it a viable candidate (unlike F4/F5 which are ruled out by observations).
• Parametrization Accuracy: The proposed analytical formula for $f(k)$ reproduces the numerical solution with < 0.6% deviation, validating its use in parameter inference.
• Power Spectrum Multipoles:
  • The monopole ($\ell=0$) and quadrupole ($\ell=2$) for Symmetron and F6 are nearly identical and distinct from $\Lambda$CDM.
  • The monopole is generally lower than $\Lambda$CDM, while the quadrupole is higher.
  • The use of the scale-dependent $f(k)$ in IR-resummation increases the power spectrum amplitude compared to using a constant $f_0$.
• MCMC Validation:
  • When fitting the Symmetron model to GR EZMocks data, the pipeline successfully recovers the GR limit ($\beta_0 \to 0$).
  • The recovered cosmological parameters ($\Omega_m, h, A_s, n_s$) and bias parameters are consistent with the input simulation values within $1\sigma$ confidence intervals.
  • The best-fit coupling parameter was constrained to $|\beta_0| < 3.8 \times 10^{-7}$, effectively recovering GR.

5. Significance and Future Outlook

• Readiness for Real Data: The study concludes that the pipeline is robust and ready to infer cosmological parameters from real data (e.g., DESI DR1/DR2). The ability to recover GR limits from MG pipelines is a crucial validation step.
• Computational Efficiency: By validating the $f_k$PT approximation and the growth parametrization, the authors have removed the primary computational bottleneck (solving differential equations for kernels) that previously hindered MG model testing with large datasets.
• Model Discrimination: The results highlight that Symmetron and F6 models are difficult to distinguish from each other using current fullshape analyses but are distinguishable from $\Lambda$CDM. Future work will focus on applying this pipeline to real DESI data to constrain the Symmetron parameters directly.

In summary, this paper bridges the gap between complex Modified Gravity theories and practical observational cosmology by providing a fast, accurate, and validated method to compute the fullshape power spectrum for the Symmetron model, paving the way for stringent tests of gravity with upcoming survey data.