Bimodular Gravity: Unimodularising Bimetric Scalar-Tensor Gravity

TL;DR

Bimodular gravity extends unimodular concepts to a disformal two-metric scalar–tensor framework, imposing a unimodular constraint on both the gravity and matter metrics. The authors develop two main formalisms: BUG, with dual fixed determinants leading to a constant relative volume and a conserved bimodular combination $Λ = λ_1 + ν λ_2$, and BHT/BDUG, with dual diffeomorphism-invariant constraints that keep $λ_1$ and $λ_2$ individually constant while allowing the relative volume $ν$ to be dynamical; these yield inequivalent cosmologies. In BUG, the biscalar dynamics are kinematically locked (∇_μ X = 0) and the background follows a first-order autonomous flow, producing a time-varying dark energy equation of state that still realizes de Sitter only on specific branches; in BHT/BDUG, full biscalar dynamics are allowed, enabling genuine dynamical dark energy and de Sitter solutions with rolling φ. The paper also provides a diffeomorphism-invariant completion that reproduces BUG on shell, preserving full covariance while maintaining the unique bimodular structure. Overall, bimodular gravity offers distinct, testable predictions for expansion history, dark energy behaviour, and perturbation propagation, with potential implications for late-time acceleration and quantum consistency.

Abstract

It is the object of the present paper to unimodularise a disformal bimetric scalar-tensor theory, thereby defining what we call bimodular gravity. We impose one unimodular constraint per metric via multipliers $λ_{1,2}$ and show that two natural implementations-a dual fixed-determinant (BUG) and a dual diffeomorphism-invariant (BHT/BDUG) formulation-are classically inequivalent. In BUG the relative volume element $ν=\sqrt{1-2BX}$ is fixed, enforcing a kinematic constraint on the biscalar and we derive the "bimodular cosmological constant" $Λ=λ_1+νλ_2$. In BHT/BDUG, $λ_{1,2}$ are individually constant but $ν$ (hence $BX$) remains dynamical. Recasting the theory in an Einstein-frame form, we derive the biscalar sound speed and identify a subluminal domain $1+B(V+λ_2)>0$. At the background level, BUG admits constant-roll solutions governed by first-order flow, whereas BHT supports solutions with time-dependent roll. These structural differences yield distinct, in-principle testable predictions for the expansion history, the dark-energy equation of state, and the propagation of biscalar perturbations. Finally, we present a diffeomorphism-invariant completion that correlates the two HT volume forms, reproducing the $Λ$ of BUG on shell whilst maintaining full covariance.