Gravity with free initial conditions: A solution to the cosmological constant problem testable by CMB B -mode polarization

Abstract

In standard general relativity the universe cannot be started with arbitrary initial conditions, because four of the ten components of the Einstein's field equations (EFE) are constraints on initial conditions. In the previous work it was proposed to extend the gravity theory to allow free initial conditions, with a motivation to solve the cosmological constant problem. This was done by setting four constraints on metric variations in the action principle, which is reasonable because the gravity's physical degrees of freedom are at most six. However, there are two problems about this theory; the three constraints in addition to the unimodular condition were introduced without clear physical meanings, and the flat Minkowski spacetime is unstable against perturbations. Here a new set of gravitational field equations is derived by replacing the three constraints with new ones requiring that geodesic paths remain geodesic against metric variations. The instability problem is then naturally solved. Implications for the cosmological constant $\Lambda$ are unchanged; the theory converges into EFE with nonzero $\Lambda$ by inflation, but $\Lambda$ varies on scales much larger than the present Hubble horizon. Then galaxies are formed only in small $\Lambda$ regions, and the cosmological constant problem is solved by the anthropic argument. Because of the increased degrees of freedom in metric dynamics, the theory predicts new non-oscillatory modes of metric anisotropy generated by quantum fluctuation during inflation, and CMB B-mode polarization would be observed differently from the standard predictions by general relativity.