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One-loop kernels in scale-dependent Horndeski theory

Ziyang Zheng, Hanqiong Jia, Bilal Tüdes, Anton Chudaykin, Martin Kunz, Luca Amendola

TL;DR

The paper addresses nonlinear cosmological perturbations in theories with scale-dependent growth, focusing on Horndeski gravity. It develops a perturbation-theory framework in which the second- and third-order kernels depend only on the linear growing mode and two Horndeski functions $h_1$ and $h_c$, solvable via the Wronskian method up to a time integral. By incorporating bias and redshift-space distortions, it constructs the one-loop galaxy power spectrum within scale-dependent gravity and provides a practical pipeline for forecasts and data analyses. The approach is sufficiently general to accommodate other scale-dependent scenarios and neutrino masses, offering a stable, analytic route to precision LSS modeling beyond $\Lambda$CDM.

Abstract

We investigate the nonlinear evolution of cosmological perturbations in theories with scale-dependent perturbation growth, first in general and then focusing on Horndeski gravity. Within the framework of standard perturbation theory, we derive the second- and third-order kernels and show that they are fully determined by two effective functions, \( h_1 \) and \( h_c \), which parametrize deviations from general relativity. Using the Wronskian method, we obtain solutions for the nonlinear growth functions and present explicit expressions for the resulting kernels, including bias and redshift space distortions, valid in the limit in which the $k$-dependent part is subdominant. We show that the kernels are entirely dependent on the linear growing mode: once this is calculated, the kernels are analytic up to a time integral. We also include redshift-space distortions (RSD) and scale-dependent bias. Our approach provides a physically motivated framework for evaluating the one-loop galaxy power spectrum in scale-dependent theories, suitable for the forecasts and actual data analysis.

One-loop kernels in scale-dependent Horndeski theory

TL;DR

The paper addresses nonlinear cosmological perturbations in theories with scale-dependent growth, focusing on Horndeski gravity. It develops a perturbation-theory framework in which the second- and third-order kernels depend only on the linear growing mode and two Horndeski functions and , solvable via the Wronskian method up to a time integral. By incorporating bias and redshift-space distortions, it constructs the one-loop galaxy power spectrum within scale-dependent gravity and provides a practical pipeline for forecasts and data analyses. The approach is sufficiently general to accommodate other scale-dependent scenarios and neutrino masses, offering a stable, analytic route to precision LSS modeling beyond CDM.

Abstract

We investigate the nonlinear evolution of cosmological perturbations in theories with scale-dependent perturbation growth, first in general and then focusing on Horndeski gravity. Within the framework of standard perturbation theory, we derive the second- and third-order kernels and show that they are fully determined by two effective functions, and , which parametrize deviations from general relativity. Using the Wronskian method, we obtain solutions for the nonlinear growth functions and present explicit expressions for the resulting kernels, including bias and redshift space distortions, valid in the limit in which the -dependent part is subdominant. We show that the kernels are entirely dependent on the linear growing mode: once this is calculated, the kernels are analytic up to a time integral. We also include redshift-space distortions (RSD) and scale-dependent bias. Our approach provides a physically motivated framework for evaluating the one-loop galaxy power spectrum in scale-dependent theories, suitable for the forecasts and actual data analysis.

Paper Structure

This paper contains 21 sections, 158 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Linear equations
  3. General kernels of standard perturbation theory
  4. Second-Order Kernels
  5. Third order kernels
  6. Kernels in Horndeski gravity
  7. Second-order kernels
  8. Third-order kernels
  9. Including bias and RSD
  10. Power spectrum at one loop
  11. Conclusion
  12. Relation with the $\alpha$-parametrization
  13. Numerical tests
  14. Detailed derivations
  15. Kernels in the $k$-independent and EdS limits
  16. ...and 6 more sections

Figures (2)

  • Figure 1: Maximum relative error of the $f_{kz}$ approximation for $z < 3$ in parameter space. The color indicates the magnitude of the maximum relative error between the exact numerical solution and our approximation, with fixed $h_{1}=1.0$, $k=0.2 h/\text{Mpc}$, $H_0 = 73.0 \, \text{km/s/Mpc}$ and $\Omega_{m0}=0.32$.
  • Figure 2: Left: Comparison of the total growth rate $f$ for various parameter choices, with fixed $H_0 = 73.0 \, \text{km/s/Mpc}$ and $\Omega_{m0} = 0.32$. The dashed vertical line indicates redshift $z = 3$. Right: The growth rate at the current epoch as a function of $k$, with fixed $H_0 = 73.0 \, \text{km/s/Mpc}$, $h_{10}=0.2$, $\alpha_{t0}=2$ and $\Omega_{m0}=0.32$.