On the Physical Nature of the Scalar Mode Mass in the Jordan frame of a Metric $f(R)$ gravity

TL;DR

This paper investigates the scalar mode mass in $f(R)$ gravity within the Jordan frame, showing it must be much larger than the Hubble scale, challenging previous assumptions about its role in cosmology.

Contribution

It derives constraints on the scalar degree of freedom's mass in $f(R)$ gravity from cosmological observations, revealing a super-Hubble mass hierarchy.

Findings

Scalar mode mass exceeds the Hubble scale by several orders of magnitude.

Constraints from $ ext{Lambda CDM}$ deceleration parameter and Newton's constant variation.

Proper scalar mass definition involves adiabatic separation, leading to a super-Hubble scale.

Abstract

We analyze the Taylor expansion of metric $f(R)$ gravity in the Jordan frame around the General Relativity limit. By relating the scalar--tensor representation to the original $f(R)$ formulation, we derive constraints on the expansion parameters from the observed value of the present-day $\Lambda$CDM deceleration parameter and from cosmological bounds on the variation of Newton's constant. We show that these requirements imply that the scalar degree of freedom must have a mass exceeding the Hubble scale by several orders of magnitude. This result challenges the common assumption that the scalar mode can drive cosmological dynamics with a mass of order $H_0$. We provide a dynamical interpretation of this hierarchy by emphasizing that a proper definition of the scalar mass, in a field-theoretical sense, requires an adiabatic separation between background evolution and perturbations, which naturally leads to a super-Hubble mass scale.