# American Institute of Mathematical Sciences

February  2014, 34(2): 335-366. doi: 10.3934/dcds.2014.34.335

## Gravitational Field Equations and Theory of Dark Matter and Dark Energy

 1 Department of Mathematics, Sichuan University, Chengdu 2 Department of Mathematics, Indiana University, Bloomington, IN 47405

Received  January 2013 Revised  May 2013 Published  August 2013

The main objective of this article is to derive new gravitational field equations and to establish a unified theory for dark energy and dark matter. The gravitational field equations with a scalar potential $\varphi$ function are derived using the Einstein-Hilbert functional, and the scalar potential $\varphi$ is a natural outcome of the divergence-free constraint of the variational elements. Gravitation is now described by the Riemannian metric $g_{\mu\nu}$, the scalar potential $\varphi$ and their interactions, unified by the new field equations. From quantum field theoretic point of view, the vector field $\Phi_\mu=D_\mu \varphi$, the gradient of the scalar function $\varphi$, is a spin-1 massless bosonic particle field. The field equations induce a natural duality between the graviton (spin-2 massless bosonic particle) and this spin-1 massless bosonic particle. Both particles can be considered as gravitational force carriers, and as they are massless, the induced forces are long-range forces. The (nonlinear) interaction between these bosonic particle fields leads to a unified theory for dark energy and dark matter. Also, associated with the scalar potential $\varphi$ is the scalar potential energy density $\frac{c^4}{8\pi G} \Phi=\frac{c^4}{8\pi G} g^{\mu\nu}D_\mu D_\nu \varphi$, which represents a new type of energy caused by the non-uniform distribution of matter in the universe. The negative part of this potential energy density produces attraction, and the positive part produces repelling force. This potential energy density is conserved with mean zero: $\int_M \Phi dM=0$. The sum of this potential energy density $\frac{c^4}{8\pi G} \Phi$ and the coupling energy between the energy-momentum tensor $T_{\mu\nu}$ and the scalar potential field $\varphi$ gives rise to a unified theory for dark matter and dark energy: The negative part of this sum represents the dark matter, which produces attraction, and the positive part represents the dark energy, which drives the acceleration of expanding galaxies. In addition, the scalar curvature of space-time obeys $R=\frac{8\pi G}{c^4} T + \Phi$. Furthermore, the proposed field equations resolve a few difficulties encountered by the classical Einstein field equations.
Citation: Tian Ma, Shouhong Wang. Gravitational Field Equations and Theory of Dark Matter and Dark Energy. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 335-366. doi: 10.3934/dcds.2014.34.335
##### References:
 [1] H. A. Atwater, "Introduction to General Relativity,", International Series of Monographs in Natural Philosophy, (1974). [2] G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints,, Physics Reports, 405 (2005), 279. doi: 10.1016/j.physrep.2004.08.031. [3] C. H. Brans and R. H. Dicke, Mach's principle and a relativistic theory of gravitation,, Physical Review (2), 124 (1961), 925. doi: 10.1103/PhysRev.124.925. [4] H. A. Buchdahl, Non-linear Lagrangians and cosmological theory,, Monthly Notices of the Royal Astronomical Society, 150 (1970), 1. [5] R. Caldwell, R. Dave and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state,, Phys. Rev. Lett., 80 (1998), 1582. doi: 10.1103/PhysRevLett.80.1582. [6] R. Caldwell and E. V. Linder, The limits of quintessence,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.141301. [7] S. Capozziello and M. De Laurentis, Extended theories of gravity,, Phys. Rept., 509 (2011), 167. doi: 10.1016/j.physrep.2011.09.003. [8] Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion,, C. R. Acad. Sci. (Paris), 174 (1922), 593. [9] B. Chow, P. Lu and L. Ni, "Hamilton's Ricci Flow,", Graduate Studies in Mathematics, (2006). [10] D. Clowe, et al., A direct empirical proof of the existence of dark matter,, Astrophys. J., 648 (2006). doi: 10.1086/508162. [11] T. Damour and G. Esposito-Farse, Tensor-multi-scalar theories of gravitation,, Class. Quantum Grav., 9 (1992), 2093. doi: 10.1088/0264-9381/9/9/015. [12] Joshua A. Frieman, Michael S. Turner and Dragan Huterer, Dark energy and the accelerating universe,, Annu. Rev. Astro. Astrophys., 46 (2008), 385. [13] M. L. Kutner, "Astronomy: A Physical Perspective,", Second edition, (2003). doi: 10.1017/CBO9780511802195. [14] L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics, Vol. 2. The Classical Theory of Fields,", Fourth edition, (1975). [15] T. Ma, "Manifold Topology,", (in Chinese) Science Press, (2010). [16] _______, "Theory and Methods of Partial Differential Equations,", (in Chinese) Science Press, (2011). [17] T. Ma and S. Wang, "Bifurcation Theory and Applications,", World Scientific Series on Nonlinear Science, (2005). doi: 10.1142/9789812701152. [18] _______, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,", Mathematical Surveys and Monographs, (2005). [19] _______, "Phase Transition Dynamics,", Springer-Verlag, (2013). [20] _______, Unified field equations coupling four forces and principle of interaction dynamics,, , (2012). [21] _______, Unified field theory and principle of representation invariance,, , (2012). [22] R. Massey, J. Rhodes, R. Ellis, N. Scoville, A. Leauthaud, et al., Dark matter maps reveal cosmic scaffolding,, Nature, 445 (2007), 286. [23] P. Peebles and B. Ratra, The cosmological constant and dark energy,, Rev. Mod. Phys., 75 (2003), 559. doi: 10.1103/RevModPhys.75.559. [24] S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae,, Astrophys. J., 517 (1999), 565. [25] Nikodem J. Popławski, Cosmology with torsion: An alternative to cosmic inflation,, Phys. Lett. B, 694 (2010), 181. doi: 10.1016/j.physletb.2010.09.056. [26] B. Ratra and P. Peebles, Cosmological consequences of a rolling homogeneous scalar field,, Phys. Rev. D, 37 (1988), 3406. doi: 10.1103/PhysRevD.37.3406. [27] A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant,, Astron. J., 116 (1998), 1009. doi: 10.1086/300499. [28] V. Rubin, W. K. Ford, Jr., Rotation of the Andromeda nebula from a spectroscopic survey of emission regions,, Astrophysical Journal, 159 (1970), 379. doi: 10.1086/150317. [29] C. Wetterich, Cosmology and the fate of dilatation symmetry,, Nucl. Phys. B, 302 (1988), 668. doi: 10.1016/0550-3213(88)90193-9. [30] C. M. Will, "Theory and Experiment in Gravitational Physics,", Second edition, (1993). [31] I. Zlatev, L.-M. Wang and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant,, Phys. Rev. Lett., 82 (1999), 896. doi: 10.1103/PhysRevLett.82.896. [32] F. Zwicky, On the masses of nebulae and of clusters of nebulae,, Astrophysical Journal, 86 (1937), 217.

show all references

##### References:
 [1] H. A. Atwater, "Introduction to General Relativity,", International Series of Monographs in Natural Philosophy, (1974). [2] G. Bertone, D. Hooper and J. Silk, Particle dark matter: Evidence, candidates and constraints,, Physics Reports, 405 (2005), 279. doi: 10.1016/j.physrep.2004.08.031. [3] C. H. Brans and R. H. Dicke, Mach's principle and a relativistic theory of gravitation,, Physical Review (2), 124 (1961), 925. doi: 10.1103/PhysRev.124.925. [4] H. A. Buchdahl, Non-linear Lagrangians and cosmological theory,, Monthly Notices of the Royal Astronomical Society, 150 (1970), 1. [5] R. Caldwell, R. Dave and P. J. Steinhardt, Cosmological imprint of an energy component with general equation of state,, Phys. Rev. Lett., 80 (1998), 1582. doi: 10.1103/PhysRevLett.80.1582. [6] R. Caldwell and E. V. Linder, The limits of quintessence,, Phys. Rev. Lett., 95 (2005). doi: 10.1103/PhysRevLett.95.141301. [7] S. Capozziello and M. De Laurentis, Extended theories of gravity,, Phys. Rept., 509 (2011), 167. doi: 10.1016/j.physrep.2011.09.003. [8] Élie Cartan, Sur une généralisation de la notion de courbure de Riemann et les espaces à torsion,, C. R. Acad. Sci. (Paris), 174 (1922), 593. [9] B. Chow, P. Lu and L. Ni, "Hamilton's Ricci Flow,", Graduate Studies in Mathematics, (2006). [10] D. Clowe, et al., A direct empirical proof of the existence of dark matter,, Astrophys. J., 648 (2006). doi: 10.1086/508162. [11] T. Damour and G. Esposito-Farse, Tensor-multi-scalar theories of gravitation,, Class. Quantum Grav., 9 (1992), 2093. doi: 10.1088/0264-9381/9/9/015. [12] Joshua A. Frieman, Michael S. Turner and Dragan Huterer, Dark energy and the accelerating universe,, Annu. Rev. Astro. Astrophys., 46 (2008), 385. [13] M. L. Kutner, "Astronomy: A Physical Perspective,", Second edition, (2003). doi: 10.1017/CBO9780511802195. [14] L. D. Landau and E. M. Lifshitz, "Course of Theoretical Physics, Vol. 2. The Classical Theory of Fields,", Fourth edition, (1975). [15] T. Ma, "Manifold Topology,", (in Chinese) Science Press, (2010). [16] _______, "Theory and Methods of Partial Differential Equations,", (in Chinese) Science Press, (2011). [17] T. Ma and S. Wang, "Bifurcation Theory and Applications,", World Scientific Series on Nonlinear Science, (2005). doi: 10.1142/9789812701152. [18] _______, "Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics,", Mathematical Surveys and Monographs, (2005). [19] _______, "Phase Transition Dynamics,", Springer-Verlag, (2013). [20] _______, Unified field equations coupling four forces and principle of interaction dynamics,, , (2012). [21] _______, Unified field theory and principle of representation invariance,, , (2012). [22] R. Massey, J. Rhodes, R. Ellis, N. Scoville, A. Leauthaud, et al., Dark matter maps reveal cosmic scaffolding,, Nature, 445 (2007), 286. [23] P. Peebles and B. Ratra, The cosmological constant and dark energy,, Rev. Mod. Phys., 75 (2003), 559. doi: 10.1103/RevModPhys.75.559. [24] S. Perlmutter, et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae,, Astrophys. J., 517 (1999), 565. [25] Nikodem J. Popławski, Cosmology with torsion: An alternative to cosmic inflation,, Phys. Lett. B, 694 (2010), 181. doi: 10.1016/j.physletb.2010.09.056. [26] B. Ratra and P. Peebles, Cosmological consequences of a rolling homogeneous scalar field,, Phys. Rev. D, 37 (1988), 3406. doi: 10.1103/PhysRevD.37.3406. [27] A. G. Riess, et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant,, Astron. J., 116 (1998), 1009. doi: 10.1086/300499. [28] V. Rubin, W. K. Ford, Jr., Rotation of the Andromeda nebula from a spectroscopic survey of emission regions,, Astrophysical Journal, 159 (1970), 379. doi: 10.1086/150317. [29] C. Wetterich, Cosmology and the fate of dilatation symmetry,, Nucl. Phys. B, 302 (1988), 668. doi: 10.1016/0550-3213(88)90193-9. [30] C. M. Will, "Theory and Experiment in Gravitational Physics,", Second edition, (1993). [31] I. Zlatev, L.-M. Wang and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant,, Phys. Rev. Lett., 82 (1999), 896. doi: 10.1103/PhysRevLett.82.896. [32] F. Zwicky, On the masses of nebulae and of clusters of nebulae,, Astrophysical Journal, 86 (1937), 217.
 [1] Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119 [2] Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306 [3] Claude Elbaz. Gravitational and electromagnetic properties of almost standing fields. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 835-848. doi: 10.3934/dcdsb.2012.17.835 [4] Angelo Alberti, Claudio Vidal. Singularities in the gravitational attraction problem due to massive bodies. Discrete & Continuous Dynamical Systems - A, 2010, 26 (3) : 805-822. doi: 10.3934/dcds.2010.26.805 [5] P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1199-1212. doi: 10.3934/dcdss.2011.4.1199 [6] David Usero. Dark solitary waves in nonlocal nonlinear Schrödinger systems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1327-1340. doi: 10.3934/dcdss.2011.4.1327 [7] François Gay-Balma, Darryl D. Holm, Tudor S. Ratiu. Variational principles for spin systems and the Kirchhoff rod. Journal of Geometric Mechanics, 2009, 1 (4) : 417-444. doi: 10.3934/jgm.2009.1.417 [8] Carlos J. Garcia-Cervera, Xiao-Ping Wang. Spin-polarized transport: Existence of weak solutions. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 87-100. doi: 10.3934/dcdsb.2007.7.87 [9] Leif Arkeryd. A kinetic equation for spin polarized Fermi systems. Kinetic & Related Models, 2014, 7 (1) : 1-8. doi: 10.3934/krm.2014.7.1 [10] Marco Cicalese, Matthias Ruf. Discrete spin systems on random lattices at the bulk scaling. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 101-117. doi: 10.3934/dcdss.2017006 [11] Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803 [12] Annalisa Cesaroni, Matteo Novaga. Volume constrained minimizers of the fractional perimeter with a potential energy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 715-727. doi: 10.3934/dcdss.2017036 [13] Weizhu Bao, Chunmei Su. Uniform error estimates of a finite difference method for the Klein-Gordon-Schrödinger system in the nonrelativistic and massless limit regimes. Kinetic & Related Models, 2018, 11 (4) : 1037-1062. doi: 10.3934/krm.2018040 [14] Miaomiao Niu, Zhongwei Tang. Least energy solutions of nonlinear Schrödinger equations involving the half Laplacian and potential wells. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1215-1231. doi: 10.3934/cpaa.2016.15.1215 [15] Toshiyuki Suzuki. Scattering theory for semilinear Schrödinger equations with an inverse-square potential via energy methods. Evolution Equations & Control Theory, 2019, 8 (2) : 447-471. doi: 10.3934/eect.2019022 [16] Pierre Carcaud, Pierre-Henri Chavanis, Mohammed Lemou, Florian Méhats. Evaporation law in kinetic gravitational systems described by simplified Landau models. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 907-934. doi: 10.3934/dcdsb.2010.14.907 [17] Xavier Cabré, Eleonora Cinti. Energy estimates and 1-D symmetry for nonlinear equations involving the half-Laplacian. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1179-1206. doi: 10.3934/dcds.2010.28.1179 [18] Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 1-29. doi: 10.3934/dcds.2019230 [19] Gisèle Ruiz Goldstein, Jerome A. Goldstein, Fabiana Travessini De Cezaro. Equipartition of energy for nonautonomous wave equations. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 75-85. doi: 10.3934/dcdss.2017004 [20] Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. On a system of semirelativistic equations in the energy space. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1343-1355. doi: 10.3934/cpaa.2015.14.1343

2018 Impact Factor: 1.143