Probing nonlinear structure formation beyond $\Lambda$CDM with the LSS bootstrap: a joint power spectrum and bispectrum analysis

Technical Summary: Probing nonlinear structure formation beyond $\Lambda$CDM with the LSS bootstrap

Problem and Motivation
Current and upcoming Stage-IV galaxy surveys (e.g., Euclid, DESI) are generating high-precision data on galaxy clustering, necessitating rigorous tests of the standard $\Lambda$CDM paradigm. While numerous extensions to $\Lambda$CDM exist (e.g., modified gravity, evolving dark energy), analyzing each individually is computationally expensive and lacks a unifying theoretical motivation, particularly given the absence of a compelling argument to privilege one specific model. Recent observational tensions, such as the DESI indication of $w < -1$, suggest the need for model-agnostic frameworks. The Large Scale Structure (LSS) bootstrap offers such a framework, utilizing fundamental symmetries (Equivalence Principle, rotational invariance) to constrain the analytic structure of perturbation theory kernels without assuming a specific Lagrangian. This work aims to determine how effectively galaxy clustering measurements can constrain departures from $\Lambda$CDM using this symmetry-driven approach, specifically focusing on the linear growth rate and the quadratic perturbation-theory kernel.

Methodology
The authors employ a joint analysis of the one-loop galaxy power spectrum and the tree-level bispectrum multipoles within the Effective Field Theory of Large Scale Structure (EFTofLSS). The analysis is conducted using the PyBird code, which has been extended to implement the LSS bootstrap parametrization.

• Theoretical Framework: The bootstrap approach constrains the convolution kernels ($F_n, G_n, K_n$) based on symmetry arguments rather than solving specific equations of motion. The authors introduce two fractional deviation parameters to quantify departures from $\Lambda$CDM:
  • $\epsilon_f$: Deviations in the linear growth rate $f(a)$.
  • $\epsilon_{d\gamma}$: Deviations in the quadratic perturbative parameter $d^{(2)}_\gamma(a)$, which governs nonlinear structure formation.
    Third-order bootstrap coefficients are fixed to their $\Lambda$CDM predictions, as the chosen observables (one-loop power spectrum and tree-level bispectrum) are primarily sensitive to second-order terms.
• Datasets:
  1. BOSS DR12: Observational data from the Luminous Red Galaxy (LRG) sample at $z=0.32$ (LOWZ) and $z=0.57$ (CMASS). The analysis includes power spectrum monopole, quadrupole, and hexadecapole, alongside the bispectrum monopole. Covariance is estimated using 2048 Patchy mock catalogs.
  2. PT Challenge Simulations: Mock galaxy catalogs generated from ten large-volume ($566 \, (h^{-1}\text{Gpc})^3$) periodic boxes with a flat $\Lambda$CDM cosmology. This dataset allows for null tests of the bootstrap parameters and evaluation of constraints in a volume $\sim 100$ times larger than BOSS.
• Analysis Pipeline: Markov Chain Monte Carlo (MCMC) methods (via MontePython) are used to explore the posterior distribution of cosmological parameters ($\omega_b, \omega_{cdm}, h, \ln A_s$), bias parameters ($b_1, b_2, b_4$), and the bootstrap parameters ($\epsilon_f, \epsilon_{d\gamma}$). The analysis accounts for redshift-space distortions and the Alcock-Paczynski effect. Scale cuts ($k_{max}$) are carefully chosen to ensure the validity of the tree-level bispectrum modeling while maximizing information extraction.

Key Contributions

• First MCMC Constraints: This work presents the first Markov Chain Monte Carlo-derived constraints on the LSS bootstrap parameters $\epsilon_f$ and $\epsilon_{d\gamma}$.
• Joint Power Spectrum and Bispectrum Analysis: The study demonstrates the necessity of combining the power spectrum with the bispectrum monopole (and quadrupole for large volumes) to break degeneracies and tighten constraints on model-independent parameters.
• Extension of PyBird: The authors successfully adapted the PyBird code to incorporate the bootstrap parametrization and the time-dependence of the kernels, enabling a state-of-the-art analysis of both observational and simulated data.
• Degeneracy Analysis: The paper investigates parameter degeneracies, particularly between the primordial amplitude $A_s$, linear bias $b_1$, and $\epsilon_f$. It shows that while the bispectrum provides marginal improvements in breaking the $A_s$-$b_1$-$\epsilon_f$ degeneracy, it significantly reduces projection effects on $\epsilon_{d\gamma}$ by breaking its degeneracy with quadratic bias parameters ($b_2, b_4$).

Results

• BOSS Analysis:
  • When $A_s$ is left free, strong degeneracies prevent precise constraints.
  • Fixing $A_s$ to the Planck best-fit value yields $\sim 7\%$ constraints on $\epsilon_f$ and $\sim 57\%$ constraints on $\epsilon_{d\gamma}$ when combining the power spectrum with the bispectrum monopole.
  • The inclusion of the bispectrum reduces the uncertainty on $\epsilon_f$ by $\sim 40\%$ and on $\epsilon_{d\gamma}$ by $\sim 47\%$ compared to power-spectrum-only analyses.
• PT Challenge Simulations:
  • Leveraging the $\sim 100\times$ larger volume, the analysis achieves $\sim 1\%$ precision on $\epsilon_f$ and $\sim 25\%$ on $\epsilon_{d\gamma}$.
  • The inclusion of the bispectrum quadrupole is crucial for obtaining unbiased constraints on $\epsilon_{d\gamma}$, reducing its uncertainty to the $\sim 0.25$ level.
  • The results confirm that the bootstrap parameters can be tightly constrained even without fixing $A_s$ for $\epsilon_{d\gamma}$, though fixing $A_s$ is necessary for robust inference of $\epsilon_f$ and bias parameters in the absence of external CMB data.

Significance and Claims
The authors claim that the LSS bootstrap offers a competitive, model-agnostic method for probing physics beyond $\Lambda$CDM using existing and future galaxy surveys. The results underscore the complementary roles of $\epsilon_f$ and $\epsilon_{d\gamma}$: $\epsilon_f$ primarily captures modifications to the background expansion (e.g., evolving dark energy), while $\epsilon_{d\gamma}$ captures modifications to nonlinear structure formation (e.g., modified gravity). By disentangling these effects, the framework clarifies the theoretical requirements for viable extensions of $\Lambda$CDM. The paper concludes that while current constraints are promising, the ultimate precision is limited by the reliability of the theoretical framework (specifically the modeling of redshift-space distortions at small scales). Future improvements will likely require including the bispectrum at the one-loop order to extend the analysis to higher $k_{max}$ values.