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Implementing F (T ) Gravity in Boltzmann Codes: A Framework for Power Spectrum Computation

Robert Rugg, Shambel Akalu, Amare Abebe

TL;DR

The paper tackles the challenge of nonlinear power-law $f(T)$ gravity in Boltzmann codes by introducing a second-order Taylor expansion around $n=0$ for the power-law form $F(T) = \alpha(-T)^n$, enabling the CLASS solver to compute CMB and matter power spectra. It demonstrates that this approximation remains accurate for small $n$ (with SN data suggesting $n \le 0.05$), by comparing to a full numerical solution obtained via SciPy's $fsolve$ and showing $\sim$1% differences at $n=0.1$. The work details the background cosmology in the teleparallel framework, derives the modified Friedmann equations, and formulates the Newtonian-gauge perturbation equations for inclusion in CLASS. Overall, the approach provides a practical route to constrain dynamical dark energy in $f(T)$ gravity using CMB and large-scale structure data, while highlighting the signatures in the CMB acoustic peaks, damping tail, and matter clustering when $n$ is nonzero.

Abstract

This work investigates the nonlinearity of the power-law model of F(T) gravity, highlighting the inability of the Boltzmann solver CLASS to handle nonlinear models. As a workaround, a second-order Taylor expansion is applied to the nonlinear field equations, under the assumption that the extra degree of freedom n, which quantifies deviations from the currently favored cosmological model (ΛCDM), remains sufficiently small to preserve the key properties of the ΛCDM model. The validity of the Taylor expansion is supported by supernova data indicating n \leq 0.05, for which the power spectrum can be accurately computed within CLASS with a negligible truncation error.

Implementing F (T ) Gravity in Boltzmann Codes: A Framework for Power Spectrum Computation

TL;DR

The paper tackles the challenge of nonlinear power-law gravity in Boltzmann codes by introducing a second-order Taylor expansion around for the power-law form , enabling the CLASS solver to compute CMB and matter power spectra. It demonstrates that this approximation remains accurate for small (with SN data suggesting ), by comparing to a full numerical solution obtained via SciPy's and showing 1% differences at . The work details the background cosmology in the teleparallel framework, derives the modified Friedmann equations, and formulates the Newtonian-gauge perturbation equations for inclusion in CLASS. Overall, the approach provides a practical route to constrain dynamical dark energy in gravity using CMB and large-scale structure data, while highlighting the signatures in the CMB acoustic peaks, damping tail, and matter clustering when is nonzero.

Abstract

This work investigates the nonlinearity of the power-law model of F(T) gravity, highlighting the inability of the Boltzmann solver CLASS to handle nonlinear models. As a workaround, a second-order Taylor expansion is applied to the nonlinear field equations, under the assumption that the extra degree of freedom n, which quantifies deviations from the currently favored cosmological model (ΛCDM), remains sufficiently small to preserve the key properties of the ΛCDM model. The validity of the Taylor expansion is supported by supernova data indicating n \leq 0.05, for which the power spectrum can be accurately computed within CLASS with a negligible truncation error.

Paper Structure

This paper contains 5 sections, 10 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background cosmology of f(T) gravity
  3. The power-law model
  4. Newtonian-gauge perturbations in F(T) gravity
  5. Conclusions

Figures (3)

  • Figure 1: (Top) The Hubble parameter plotted for various values of $n$ using both the second-order Taylor expansion method and scipy's fsolve as a function of $z$. (Bottom) Percentage difference between the second order Taylor expansion method and scipy's fsolve at specific values of $n$ as a function of $z$ for $H^2$.
  • Figure 2: (Top) The power spectrum for the $\Lambda$CDM and $F(T)$ power law case plotted against temperature mapping data form the Planck mission for different multipole moments $l$. (Bottom) The percentage difference between the $\Lambda$CDM and the $F(T)$ power-law model over different multipole moments $l$.
  • Figure 3: The matter power spectrum for the $\Lambda$CDM and the $F(T)$ power-law case plotted for different wavenumbers ($k$). The percentage difference between the $\Lambda$CDM and the $F(T)$ power-law model with $n = 0.1$ is indicated in the subplot within.