General relativity has passed hundreds of tests, but two puzzles—dark matter, first noticed by Fritz Zwicky in 1933 and confirmed by Vera Rubin in the 1970s, and the accelerated expansion of the universe, discovered in 1998 by Adam Riess and colleagues—demand going beyond Einstein’s gravity. Modern sky surveys, such as the cosmic microwave background and large-scale structure, hint at a possible parity violation on cosmological scales. If left and right are not equivalent in the cosmos, gravitational waves should carry an imprint of this asymmetry.
The theoretical framework includes an action with an arbitrary function f(R) and a Chern–Simons term dynamically coupled to a scalar field. The field equations are derived in two variants: the metric one, where the connection is the Christoffel symbol, and the Palatini one, where metric and connection are independent. For a cosmological background (flat FRW universe), modified Friedmann equations are obtained, and at the level of linear tensor perturbations, a propagation equation for gravitational waves follows. The analysis is carried out in Fourier space with a switch to a circular polarization basis, which separates left and right modes. The key difference from standard GR is the appearance of terms with opposite signs for the two polarizations, responsible for birefringence.
In the Palatini formalism, both amplitude and velocity birefringence are found, while the metric version gives only an amplitude effect. When transitioning to a de Sitter background approximation (appropriate for the dark energy era), explicit expressions for the friction coefficient Ξ and squared frequency ω² show that the f(R) addition acts as a multiplicative factor 1/$f_R$. Comparing three f(R) models—Hu–Sawicki, exponential, and hyperbolic—with SNIa, BAO, and CMB data yields $f_{R0}$ values from 0,51 to 0,99. This means an enhancement of birefringence by 1–10% in the Palatini case and up to a doubling in the metric case. Constraints on the coupling parameter from LIGO/Virgo observations: from the gravitational wave speed of the neutron star merger in the binary system GW170817 we get |αϑ'_0 H0/(κ $f_{R0}$)| < $10^{-7}$, from phase distortions in the GWTC-3 catalog — < $10^{-14}$, from amplitude ones — < $10^{-5}$. Moreover, the dependence of the effect on redshift turns out to be polynomial, not linear in distance as assumed in phenomenological models.
Thus, f(R) gravity with a Chern–Simons term not only reproduces dynamical dark energy but also leaves a unique polarization signature on gravitational waves. The presence of velocity birefringence in the Palatini approach provides a fundamental way to distinguish between formalisms with data. Joint constraints accounting for the f(R) effect narrow the allowed parameter space of the theory and can explain why strong fields in the Solar System do not show such deviations—if a screening mechanism is at work.
The future of this line of research involves analyzing the full cosmological background without the simplifying assumption of constant curvature, which will require numerical integration accounting for scalar field evolution. Next-generation observatories—LISA, Einstein Telescope, and Cosmic Explorer—are expected to detect birefringence over a wide range of redshifts, up to z∼10, and thereby test the non-monotonic behavior predicted by the model.
The results will impact the interpretation of data from gravitational-wave detectors, early-universe cosmology, and the theory of modified gravity. In particular, they connect dark energy measurements with polarization tests of GR.
It is necessary to consider scenarios with a non-zero derivative of f(R), include the contribution of the scalar field to background dynamics, and investigate possible mechanisms for restoring GR in strong fields to avoid conflicts with observations near neutron stars and black holes.
The work touches on several unresolved problems: the nature of dark energy (whether it is a cosmological constant or a dynamical field), observational hints of a global parity violation in the cosmos, and the issue of quantizing gravity, for which string theory and loop gravity often lead to Chern–Simons-like additions in the low-energy limit.
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This paper examines f(R) gravity with a Chern–Simons term in both the metric and Palatini formalisms. We derive modified Friedmann equations and the propagation equation for gravitational waves with circular polarization, revealing birefringence. Using three specific f(R) models and data from SNIa, BAO, and CMB, we estimate the magnitude of the effect and obtain constraints from LIGO/Virgo observations.
Where its strength is
We show that f(R) models consistent with cosmological data can enhance both amplitude and velocity birefringence of gravitational waves to levels potentially detectable by future instruments. We obtain a polynomial dependence of the effect on redshift, which differs from phenomenological models.
Where it could go next
Worth trying
Why this may matter
If future gravitational-wave detectors indeed observe polarization asymmetry, it would provide direct evidence for the violation of fundamental symmetries in gravity and allow us to distinguish between competing extensions of general relativity. Our approach offers a concrete way to connect laboratory physics of gravitational waves with cosmological observations, potentially shedding light on the nature of dark energy.
Nearby works in our archive
The closest work in the archive sits at 0.69 by meaning — the usual level for our corpus.
Icelandic spar, known to Vikings as a ‘sunstone’, helped find the sun in cloudy weather. Today, its physical analogue in cosmology may help ‘see’ dark energy through the polarization of gravitational waves.