[Paper Review] FRCAMB: An $f(R)$ Code for Anisotropies in the Microwave Background

Lixin Xu|arXiv (Cornell University)|Jun 10, 2015

Cosmology and Gravitation Theories3 citations

TL;DR

This paper introduces FRCAMB, a modified version of the CAMB code that self-consistently solves the background and perturbation equations for any f(R) gravity model, enabling accurate computation of CMB and matter power spectra. The key contribution is a black-box tool that outputs cosmological observables without assuming a ΛCDM background, yielding log(|fR0−1|) = −5.71−0.16+0.07 from joint cosmic observations.

ABSTRACT

An $f(R)$ gravity model is proposed to realize a late time accelerated expansion of our Universe. To test the viability of an $f(R)$ gravity model through cosmic observations, the background evolution and the Einstein-Boltzmann equation should be solved for studying the effects on the cosmic microwave background power spectrum and on the matter power spectrum. In the market, we already have the modified versions of {\bf CAMB} code, for instance {\bf EFTCAMB} and {\bf MGCAMB}. However, in these publicly available Einstein-Boltzmann codes, a specific background cosmology, for example the $Λ$CDM or $w$CDM, is assumed. This assumption would be non-proper for a specific $f(R)$ model where the background evolution may be different from a $Λ$CDM cosmology. Therefore the main task for this paper is to present a code to calculate the anisotropies in the microwave background for any $f(R)$ gravity model based on {\bf CAMB} code, i.e. {\bf FRCAMB}, where the background and perturbation evolutions are included consistently. As results, one can treat {\bf FRCAMB} as a blackbox to output the CMB power spectrum and matter power spectrum, once an $f(R)$ function, its first two derivative with respect to $R$, i.e. $f_R\equiv df/dR$, $f_{RR}\equiv d^2f/dR^2$ and the reasonable values of the model parameters are inputted properly. As by-products, one can also output the effective equation of state of $f(R)$ model, the evolution of the dimensionless energy densities and other interesting cosmological quantities.

Motivation & Objective

• To develop a self-consistent numerical tool for f(R) gravity models that does not assume a ΛCDM or wCDM background cosmology.
• To enable accurate computation of CMB and matter power spectra for any f(R) model by solving the full Einstein-Boltzmann system.
• To provide a flexible, user-friendly code that treats f(R), fR, and fRR as inputs and outputs cosmological observables including effective EoS and energy density evolution.
• To test the viability of f(R) gravity models using combined geometric and dynamical cosmic observations, including Planck, BAO, SNe, HST, and RSD data.
• To constrain the HS model in the n=1 case with global fitting, achieving tighter constraints on fR0 than previous methods.

Proposed method

• FRCAMB is built as a modified version of the CAMB code, extending it to handle arbitrary f(R) gravity models through consistent integration of background and perturbation equations.
• The code numerically solves the modified Friedmann equations derived from the f(R) action, allowing arbitrary f(R) functions and their first two derivatives as input.
• Perturbation evolution is computed by solving the full Einstein-Boltzmann system, including photon, baryon, and cold dark matter fluid equations with modified gravity corrections.
• The code outputs the CMB power spectrum (Cl), matter power spectrum (P(k)), effective equation of state, and dimensionless energy density evolution as functions of redshift.
• A global fitting procedure using CosmoMC is applied to the model parameter space, incorporating Planck 2013, WMAP9, BAO, SNe Ia, HST, and RSD data to constrain f(R) parameters.
• The Gelman-Rubin R−1 convergence criterion (R−1 ≈ 0.02) ensures reliable confidence intervals for derived parameters.

Experimental results

Research questions

• RQ1 Can a CMB and matter power spectrum code be developed for f(R) gravity that does not assume a ΛCDM background cosmology?
• RQ2 How does the consistency of background and perturbation evolution in f(R) gravity affect the CMB power spectrum and matter power spectrum?
• RQ3 What are the cosmological constraints on the HS model in the n=1 case when using combined geometric and dynamical observations?
• RQ4 How do the results from FRCAMB compare with those from MGCAMB and EFTCAMB in terms of fR0 constraints?
• RQ5 What is the impact of including redshift-space distortion (RSD) data on the parameter space of f(R) gravity models?

Key findings

• FRCAMB successfully computes the CMB and matter power spectra for any f(R) model by solving the background and perturbation equations consistently, without assuming a ΛCDM background.
• The code outputs the effective equation of state, energy density evolution, and other cosmological quantities, enabling comprehensive model testing.
• For the HS model with n=1, the global fit yields log(|fR0−1|) = −5.71−0.16+0.07, consistent with previous results from MGCAMB and EFTCAMB.
• The constraint on fR0 is robust and consistent with current observational bounds, confirming the viability of the HS model within the f(R) framework.
• The inclusion of RSD data improves the constraint on the growth rate, reducing uncertainty in the fσ8(z) evolution and tightening parameter constraints.
• The age of the universe is constrained to 13.752±0.037 Gyr, and σ8 is found to be 0.8214−0.0098+0.0100, consistent with Planck 2013 results.