Riemann-Eddington theory: Incorporating matter, degravitating the   cosmological constant
  Here we show that, Eddington's pure affine gravity, when extended with Riemann curvature, leads to gravitational field equations that incorporate matter. This Riemanned Eddington gravity outfits a setup in which matter gravitates normally with Newton's constant but vacuum gravitates differently with an independent gravitational constant. This novel setup enables degravitation of the vacuum to observed level not by any fine-tuning but by a large hierarchy between its gravitational constant and its energy density. Remarkably, degravitation of the cosmological constant is local, causal and natural yet only empirical because the requisite degravitation condition is not predicted by the theory. 
Durmus A. Demir
gr-qc hep-ph hep-th