University of Oslo & Caltech Four-dimensional General Relativity from higher-dimensional Gauss-Bonnet gravity Ingunn Kathrine Wehus University of Oslo & Caltech Cosmo08, Madison August 26th, 2008 Based on Carroll and Wehus 2008, in preparation

Four-dimensional General Relativity from higher-dimensional Gauss-Bonnet gravity Natural generalization of GR in higher dimensions Special case of Lovelock gravity Compactification of the extra dimensions Give GR coupled to a scalar field in 4 dimensions Scalar field may be stabilized Gives pure GR, but with modified Planck mass May be relevant to the hierarchy problem

Gauss-Bonnet The Gauss-Bonnet theorem in differential geometry connects the geometry (curvature) of a surface to its topology (Euler characteristic) In 2 dimensions the Ricci scalar R is a topological invariant. The action gives trivial field equations. However, in higher dimensions (D > 2) the same Lagrangian gives GR Similarily, in 4 dimensions the Gauss-Bonnet scalar B is a topological invariant The action gives trivial field equations for D=4. For D > 4 it gives Gauss-Bonnet gravity.

Lovelock gravity In D = 2n (+1 if odd) dimensions the most general action yielding second order field equations for a dynamical spacetime metric is given by the Lovelock Lagrangian For example, the lowest order cases are

Gauss-Bonnet gravity The combined GR and Gauss-Bonnet action in D dimensions, gives the equations of motion where For a maximally symmetric spacetime this simplifies to For D > 4, there are two vacuum solutions, and

Compactification We now Kaluza-Klein compactify the extra dimensions, to obtain an effective theory for our ordinary four dimensional spacetime We then find

Dimensional Reduction Putting the previous expressions into the action and integrating over the extra dimensions gives where

From Jordan frame to Einstein frame The Weyl transformation, brings us to Einstein frame, where

Stabilization by matter Can various forms of matter help stabilize the scalar field? Some available components: Cosmological constant Magnetic field wrapped around extra dimensions Scalar fields Casimir energy due to the compact space Adding all these, one obtains the combined potential

Pure Gauss-Bonnet We first consider the special case of pure Gauss-Bonnet, which, after dimensional reduction, gives the following effective action In n = 2 extra dimensions, this action is already in Einstein frame

Scales and hierarchies Adding matter gives the following potential for the scalar field Choosing for instance For b stabilized, the field equations reduce to where

Summary Gauss-Bonnet gravity is a natural extension of General Relativity when considering higher dimensions Higher-dimensional Gauss-Bonnet gravity compactified to four dimensions reduces to General Relativity, coupled to a scalar field The details of this scalar-tensor theory differ from the GR case After stabilizing the scalar field, the four dimensional theory of gravity is pure General Relativity, but with a modified Planck mass