Modified gravity: Brief summary Dec. 21-23. 2015 GR100, EWHA Woman’s University Modified gravity: Brief summary *Hyeong-Chan Kim Korea National University of Transportation

There are diverse number of modified theories. Introduction Most modified theories which does not contain GR in a limit, dead already. (80~) There are diverse number of modified theories. Variances: Direct modifications: Cartan, Brans-Dicke, Rosen bimetric… Quantized gravity: loop quantum gravity, Noncommutative geometry, Horava-Lifschitz, Unification with other forces: Kaluza-Klein, Supergravity Do everything: M-theory ... Deals direct modifications only.

Motivations for modified gravity General relativity (GR) is quite successful. However, Quantization of gravity (Noncommutative geometry, String theory, Horava-Lifschitz, … ) Low energy effective theories of (would be) quantum gravity are represented by the modified gravity theories. Higher curvuture terms as a quantum gravitational corrections. (f(R ), Gauss-Bonnet, Lovelock, Born-Infeld… ) Spacetime singularities indicate that the theory is incomplete. (Einstein-Cartan, Eddington-ispired Born-Infeld) Add other aspects e.g., spin (Einstein-Cartan), additional DoF (Brans-Dicke, Kaluza-Klein, … ) There exist phenomena which cannot be explained in GR, e.g, Dark energy (DGP) and Dark matter (MOND).

General Relativity Levi-Civita Connection Metric Curvature = matter Distance between two points Gravity or Acceleration at a given coordinates system: Determine the motion of particles. Matter determines that the spacetime how to curve.

Modification 1) Add new DoF Modify the RHS so that include new DoF: Scalar-tensor theory, Brans-Dicke, Tensor-Vector-Scalar theory,… EoM: add stress-tensors for new fields GR appears when the new DoFs are killed. Spacetime singularity happens when the density diverges. (There may exist other singularities). Same difficulties in quantization and singularities as GR.

Scalar-tensor theories Dimensionally reduced effective theories of KK, string theories. Jordan frame: Conformal transformation: Einstein frame: Most general with 2nd order EoM: Horndeski’s theory

Brans-Dicke Theory of gravity (1961) Motivation: Mach’s principle (The reference frame comes from the distribution of matters in the universe.) In GR, geometry is determined by mass distribution. However, it is not unique up to boundary condition. Set the Newton constant be dynamical (1/G  ). : general relativity Present experimental limit: Deviation from GR is tiny if exists.

General Relativity Levi-Civita Connection Metric Curvature = matter Distance between two points Gravity or Acceleration at a given coordinates system: Determine the motion of particles. Matter determines that the spacetime how to curve.

Modification 2) Add higher curvatures Modify the LHS so that include higher curvature terms: f(R), conformal gravity, Gauss-Bonnet gravity, Lovelock gravity, etc. EoM: Add higher curvature terms. GR appears in the low curvature limit. Spacetime singularity happens when the density diverges. There exists other singularities.

Modification 2) f(R) theory Generally covariant, Lorentz invariant. Fourth order equation of motion GR +massless scalar field with non-metric coupling with matter. 3 massless graviton= 2 tensor+1scalar. Conformal transform: Present bound: | a2| < 4 x 10-9 m2 = 2.3x 10-22GeV-2 (fifth force measurement)

General Relativity Levi-Civita Connection Metric Curvature = matter Distance between two points Gravity or Acceleration at a given coordinates system: Determine the motion of particles. Matter determines that the spacetime how to curve.

Modification 3) Change theory structure EiBI, Bi-gravity and other Palatini-f(R) gravity. Metric Affine -Connection Curvature Affine connection cannot be obtained from metric straight forwardly. Matter EiBI, Bi-gravity and other Palatini-(connection) based modified gravity: is not the spacetime curvature but the connection curvature. 1) GR happens in the zero matter limit. 2) Spacetime singularity may or may not appear even if the energy density diverges.

| |<3 x 105 m5s-2/kg (Casanellas,Pani,Lopes,Cardoso,2011) Born-Infeld Generalization of Eddington gravity, inequivalent to GR: Action ( ) is dependent on the connection only. The matter field couples with the metric only. Small = GR limit; Large = Eddington limit. is a dimensionless constant related with the cosmological constant. For vacuum, it is the same as GR. Inside matters, it deviates from GR. | |<3 x 105 m5s-2/kg (Casanellas,Pani,Lopes,Cardoso,2011) EiBI gravity(Banados, Ferreira; 2010)

The equation of motions Define auxiliary metric: Metric Affine -Connection Curvature Matter Now, the variation of the action w.r.t. gives, where, The metric compatibility determines, Variation with respect to the metric,

General Relativity Levi-Civita Connection Metric Curvature = matter Distance between two points Gravity or Acceleration at a given coordinates system: Determine the motion of particles. Matter determines that the spacetime how to curve.

Modification 4) Einstein Cartan theory The connection has non-zero torsion. Ricci tensor is asymmetric. Spinning particles feels the torsion. contorsion tensor The autoparallels are not necessarily the longest (shortest) lines. On the other hand, minimizing proper length:

Modification 4) Einstein Cartan theory EC emerges as the minimal description of spacetime of local gauge theory of Poincare group. Non-renormalizable (spin interaction ~ 4-fermion interaction, torsion has vanishing canonical momentum) The theory is different from GR at very high spin-densities of matter. For electron, Cosmological singularity can be avoided only under unrealistic circumstances.

theories not categorized There are many other theories that does not belong to these categories. Gauge gravity theory Chern-Simon theory in 2+1 dim Theories breaking diffeomorphism invariance Breaks Lorentz invariance: Horava-Lifschitz, Einstein Ether,… Massive gravity Extension to Higher dimensions

Testing modified theories Self-consistency: tachyon, ghost, higher order poles. Completeness: A theory of gravity must be capable of analyzing the outcome of every experiment of interest. Classical tests: gravitational redshift Gravitational lensing Anomalous perihelion advances of the planets Agreement with Newtonian mechanics and special relativity The Einstein equivalence principle Parametric post-Newtonian formalism(weak field) Strong gravity and gravitational waves (BH horizon, speed of GW, etc.) Cosmological tests (Dark matter, galaxy rotation curve, Tully-Fisher relation), gravitational lensing due to galactic cluster, CMB, (Dark energy: Supernova brightness )

arXiv:1501.07274, Testing GR with present and future astrophysical observations

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Fourth order gravity (second order of Riemann) Metric theories Fourth order gravity (second order of Riemann) f(R): higher power of Ricci scalar Gauss-Bonnet: Second order contractions of Riemann with second order EoM. Lovelock: Higher order contractions of Riemann with second order EoM. Scalar-tensor theories (e.g., Brans-Dicke) Vector-tensor theories Bimetric theories (two metrics: Rosen(1940) ) Massive gravity (+graviton with mass, ghost problem) Eddington-inspired Born-Infeld gravity

Gauge theory of gravity Non-Metric theories Einstein-Cartan theory Include torsion tensor Spin tensor Metric-Affine gravitation theory Generalization of Einstein Cartan theory Gauge theory of gravity