Develop perturbation theory for the preferred-frame thermogravity model

Develop a perturbative treatment of the preferred-frame thermogravity model, including predictions for the cosmic microwave background, weak lensing, and structure growth, to enable cosmological tests beyond the homogeneous background.

Background

The paper restricts its analysis to homogeneous background observables because local Lorentz invariance and diffeomorphism invariance are explicitly broken in the thermogravity framework. Consequently, different choices of the preferred frame beyond the background are physically distinct theories rather than gauge-equivalent descriptions.

A complete cosmological assessment therefore requires deriving perturbations in the preferred-frame theory rather than importing the standard gauge-invariant perturbation formalism of ΛCDM. The unresolved treatment must cover the cosmic microwave background, weak lensing, and structure growth, whose predictions may depend on the choice of preferred frame and associated perturbative dynamics.