THEORY OF DARK ENERGY AND DARK MATTER MARCO HERNANDEZ, TIAN MA, AND SHOUHONG WANG Dedicated to Professor Louis Nirenberg on the occasion of his 90th birthday with great admiration and respect Abstract. In (Ma and Wang, 2014b), a new set of gravitational field equa- tions are derived based only on 1) the Einstein principle of general relativity, and 2) the principle of interaction dynamics, due to the the presence of dark energy and dark matter. With the field equations, we show that gravity can display both attractive and repulsive behavior, and the dark matter and dark energy are just a property of gravity caused by the nonlinear interactions of the gravitational potential gμν and its dual field. The main objectives of this paper are two-fold. The first is to study the PID-induced cosmological model, and to show explicitly, as addressed in (Ma and Wang, 2014a), that 1) dark matter is due to the curvature of space, and 2) dark energy corresponds to the negative pressure generated by the dual gravitational potential in the field equations, and maintains the stability of geometry and large scale structure of the Universe. Second, for the gravitational field outside of a ball of centrally symmetric matter field, there exist precisely two physical parameters dictating the two-dimensional stable manifold of asymptotically flat space-time geom- etry, such that, as the distance to the center of the ball of the matter field increases, gravity behaves as Newtonian gravity, then additional attraction due to the curvature of space (dark matter effect), and repulsive (dark energy effect). This also clearly demonstrates that both dark matter and dark energy are just a property of gravity. We note that the dark matter property of the gravity and the approximate gravitational interaction formula are consistent with the MOND theory proposed by (Milgrom, 1983); see also (Milgrom, 2014) and the references therein. Contents 1. Introduction 2 2. Dark Energy and Dark Matter Phenomena 4 3. PID Cosmological Model and Dark Energy 5 4. PID Gravitational Interaction Formula 9 5. Asymptotic Repulsion of Gravity 10 6. Simplified Gravitational Interaction Formula 14 7. Nature of Dark Matter and Dark Energy 16 Date : March 1, 2015. 1991 Mathematics Subject Classification. 35Q75, 37N20, 83C, 83F,. Key words and phrases. dark matter, dark energy, principle of interaction dynamics (PID), gravitational field equations, gravitational interaction formula, MOND, asymptotic flatness, stable manifold, negative curvature. The authors are grateful to Micka¨ el D. Chekroun for his insightful discussions. The work was supported in part by the US National Science Foundation, and by the Chinese National Science Foundation. 1