Effects of Dynamical Compactification on d-Dimensional Gauss-Bonnet FRW Cosmology

1. Effects of Dynamical Compactification on d-Dimensional Gauss-Bonnet FRW Cosmology Brett Bolen Western Kentucky University Keith Andrew, Chad A. Middleton

2. Outline • Einstein Gauss-Bonnet Field Equations for FRW • Dynamical Compactification of extra dimensions • Calculation of effects on H0, q and equation of state • Conclusion and Future work

3. Einstein-Hilbert Action • Field equations

4. 4 + d dimensional FRW Assume K=0 (flat) and that gmn is maximally symmetric such that the Riemann Tensor for gmn has the form

5. Dynamic Compactifaction We make the assumption that the extra dimensions compactify as the 3 spatial dimensions expand as where n > 0 in order to insure that the scale factor of the compact manifold is both dynamical and compactifies as a function of time.

6. Einstein Equations w/o GB terms d – number of extra dimensions n- order of compactifaction

7. Gauss Bonnet equations

8. Field Equations

9. Effective pressure By using the conservation equation one finds that As pointed out by Mohammedi , this is simply a statement that dE = −P dV together with the assumption that a~1/bn one finds

10. Effective pressure Using the conservation equation together with the assumption that a~1/bn one finds where we have defined an “effective” pressure which an observer constrained to exist only upon the “usual” 3 spatial dimensions would see as

11. Determination of constants with l= 0 The pressure in the extra d-dimensions is This equation may be solved pertubatively by considering the GB term as small Where C is a constant depending upon n and d A and B are constants of integration which depend upon the initial conditions

12. Einstein equations The other 2 Einstein equations are used to obtain equations for r and p

13. Equation of state • Note, in the limit where n → 0, w = 1/3 which is the relationship one would expect for a radiation dominated universe. • Geometrical terms in the compactifacation are playing the same role as matter. • Thus, by demanding that w have a physical value; one may use this relationship to restrict the choices of n and d. For instance if d = 7, then n must be less then 1/2 if w is demanded to have a physically reasonable value of between 1 and −2.

14. GB Modification of H0 and q0 Note that in the large time limit (t → 1) these terms tend to their zeroth-order values. Plots of H and q H2 for d=7 and various n values

15. Conclusions and Future Work • Case with l in paper at hep-th/0608127 • Measurement of w for GB term • Future • Statement on energy conditions • Semi-classical states