Effects of Dynamical Effects of Dynamical Compactification on d-Compactification on d-

Dimensional Gauss-Bonnet FRW Dimensional Gauss-Bonnet FRW CosmologyCosmology

Brett BolenBrett BolenWestern Kentucky UniversityWestern Kentucky University

Keith Andrew, Chad A. MiddletonKeith Andrew, Chad A. Middleton

OutlineOutline

►Einstein Gauss-Bonnet Field Equations Einstein Gauss-Bonnet Field Equations for FRWfor FRW

►Dynamical Compactification of extra Dynamical Compactification of extra dimensionsdimensions

►Calculation of effects on HCalculation of effects on H00, q and , q and equation of stateequation of state

►Conclusion and Future workConclusion and Future work

► Einstein-Hilbert ActionEinstein-Hilbert Action

► Field equationsField equations

4 + d dimensional FRW4 + d dimensional FRW

Assume K=0 (flat) and that Assume K=0 (flat) and that mnmn is is maximally symmetric such that the maximally symmetric such that the Riemann Tensor for Riemann Tensor for mnmn has the form has the form

Dynamic CompactifactionDynamic 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.

Einstein Equations w/o GB Einstein Equations w/o GB termsterms

d – number of extra dimensions

n- order of compactifaction

Gauss Bonnet equationsGauss Bonnet equations

3

2

54

4

42

2

54

3

2

34

4

22

2

32

4

4

11

2

21

2

2

3332

a

aa

a

a

a

a

a

ap

a

aa

a

a

a

a

a

ap

a

a

a

a

d

Field Equations

Effective pressureEffective pressure

By using the conservation equation one finds By using the conservation equation one finds thatthat

As pointed out by Mohammedi , this is simply a statement that dE = −P dV

together with the assumption that a~1/btogether with the assumption that a~1/bnn one one findsfinds

Effective pressureEffective pressure

Using the conservation equation together with Using the conservation equation together with the assumption that a~1/bthe assumption that a~1/bnn one finds 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

Determination of constants with = 0

The pressure in the extra d-dimensions isThe pressure in the extra d-dimensions is

This equation may be solved pertubatively by This equation may be solved pertubatively by considering the GB term as smallconsidering the GB term as small

Where C is a constant depending upon n and dWhere C is a constant depending upon n and dA and B are constants of integration which depend A and B are constants of integration which depend

upon the initial conditions upon the initial conditions

Einstein equationsEinstein equations

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

Equation of stateEquation 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.

GB Modification of HGB Modification of H00 and q and q00

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

0.05 0.1 0.15 0.2t

-20

-15

-10

-5

5

10

15

20H

0.05 0.1 0.15 0.2t

-40

-20

20

40

q H 2

Conclusions and Future WorkConclusions and Future Work

►Case with Case with in paper at in paper at hep-th/0608127hep-th/0608127

►Measurement of w for GB termMeasurement of w for GB term►FutureFuture

Statement on energy conditionsStatement on energy conditions Semi-classical statesSemi-classical states