$CPT$ Violating Electrodynamics and Chern-Simons Modified Gravity
Mingzhe Li, Yi-Fu Cai, Xiulian Wang, Xinmin Zhang

TL;DR
This paper investigates the inconsistencies in Chern-Simons modified electrodynamics coupled to gravity and proposes a consistent model by adding a Chern-Simons term to gravity, analyzing its effects on cosmic microwave background and gravitational waves.
Contribution
It introduces a consistent framework for Chern-Simons modified gravity coupled with electrodynamics, addressing divergence issues and exploring cosmological phenomenologies.
Findings
Symmetric but non-divergence-free energy-momentum tensor in the original model
Modified gravity sector with Chern-Simons term ensures consistency
Potential observable effects on cosmic microwave background and gravitational waves
Abstract
The electrodynamics with a Chern-Simons term violates Lorentz and symmetries with a non-vanishing . For a fixed vector , in this paper we point out that the energy-momentum tensor of this theory coupled to the gravity minimally is symmetric but not divergence free, which consequently makes the gravitational field equation inconsistent. To preserve the consistency, we introduce a Chern-Simons term in the gravity sector with the coefficient determined by the Lorentz and violating term in the electromagnetic field. Further we study the phenomenologies of the model on the cosmic microwave background radiation and the relic gravitational waves.
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