Chern-Simons Modified General Relativity

TL;DR

This work develops a complete, coordinate-invariant formalism for CMB polarization on the full sky by expanding the polarization tensor in tensor spherical harmonics ${Y_{(lm)ab}^{ m G}}$ and ${Y_{(lm)ab}^{ m C}}$, yielding a clean E/B-like (G/C) decomposition. It derives exact expressions for temperature and polarization multipole moments for both scalar and tensor perturbations, and establishes parity constraints that distinguish scalar-only (no ${ m C}$-type) contributions from tensor-induced polarization. The paper also connects multipole moments to two-point correlation functions, provides map-simulation and estimator frameworks, and discusses noise, cosmic variance, and line-of-sight approaches, demonstrating practical pathways to constrain gravitational-parity violation and primordial gravitational waves. Finally, it discusses experimental prospects (e.g., MAP/COBRAS-SAMBA) and foregrounds, highlighting the significance of full-sky polarization analyses for cosmology and fundamental physics.

Abstract

Chern-Simons modified gravity is an effective extension of general relativity that captures leading-order, gravitational parity violation. Such an effective theory is motivated by anomaly cancelation in particle physics and string theory. In this review, we begin by providing a pedagogical derivation of the three distinct ways such an extension arises: (1) in particle physics, (2) from string theory and (3) geometrically. We then review many exact and approximate, vacuum solutions of the modified theory, and discuss possible matter couplings. Following this, we review the myriad astrophysical, solar system, gravitational wave and cosmological probes that bound Chern-Simons modified gravity, including discussions of cosmic baryon asymmetry and inflation. The review closes with a discussion of possible future directions in which to test and study gravitational parity violation.

Figures (1)

• Figure 1: The basis tensors $M_{(j)}^{ab}(k)$ traditionally used for polarization are discontinuous in the orientation of the polarization in the directions on the sky both parallel and anti-parallel to ${\bf k}$. The discontinuity is either as depicted in the left panel or as depicted in the right panel. Note that we have just switched the sign of the polarization between the two panels.