1. Anisotropic Evolution of D-Dimensional FRW Spacetime
Chad A. Middleton
Mesa State College
February 19, 2009

2. Cosmology
is the scientific study of the large scale properties of the Universe as a whole.
addresses questions like:
Is the Universe (in)finite in spatial extent?
Is the Universe (in)finite in temporal extent?
What are the possible geometries of the Universe?
What is the ultimate fate of the Universe?

3. In 1915, Einstein completes his General Theory of Relativity
describes the curvature of spacetime
describes the matter & energy w/in spacetime

4. Matter tells space how to curve Space tells matter how to move
Space is a dynamical structure whose shape is determined by the presence of matter and energy.
Matter tells space how to curve
Space tells matter how to move
“Spacetime and Geometry” by Sean Carroll, 1st edition,
Pearson publishing

5. Cosmological Principle
On sufficiently large distance scales, the Universe is
1. Isotropic
2. Homogeneous
 Maximally Symmetric Space

6. For a Homogeneous & Isotropic Universe… … 3 possible Geometries
Recent data indicates that the Universe is flat

7. Friedmann-Robertson-Walker (FRW) Cosmology
Choose the flat Robertson-Walker metric*
Choose a perfect fluid stress-energy tensor
* the Robertson-Walker metric describes a spatially
homogeneous, isotropic Universe evolving in time

8. The FRW Equations are…
density () & pressure (p) determine the evolution of the scale factor (a)

9. Choose an “equation of state”
For radiation:
For pressure-less matter:
For a vacuum:

10. Density as a function of the scale factor
Radiation dominated:
Matter dominated:
Vacuum energy dominated:

11. Data from Type Ia Supernovae, WMAP and SDSS implies…
The expansion of the
Universe is ACCELERATING!
The Universe is flat
Seems to indicate a Vacuum Energy

12. The Cosmological Constant Problem
From the zero-point energies of vacuum fluctuations…
Cosmological observations imply…
Taking the ratio yields..

13. The Ultraviolet Catastrophe
Consider the energy density, u(λ), of an ideal blackbody...
“Modern Physics” by Paul A. Tipler & Ralph A. Llewellyn, 5th edition, W.H. Freeman and Company
The resolution of the Ultraviolet Catastrophe
led to Quantum Mechanics

14. A Quantum Theory of Gravity?
In QFT, particles are treated as mathematical points.

15. String Essentials… Points of QFT  1D Strings 2 Types  Closed & Open
Different Vibrational Modes
 Different particles

16. String Theory demands Extra Dimensions
 Two possible descriptions
Compactified
Extra Dimensions
Non-Compactified Extra Dimensions

17. Kaluza-Klein Compactification
Consider a 5D theory, w/ the 5th dimension periodic…
where
Kaluza, Theodor (1921) Akad. Wiss. Berlin. Math. Phys. 1921: 966–972
Klein, Oskar (1926) Zeitschrift für Physik, 37 (12): 895–906

18. D-Dimensional FRW Cosmology
Choose the flat Robertson-Walker metric
Choose a perfect fluid stress-energy tensor
where is the higher dimensional pressure

19. D-dimensional FRW field equations

20. An Incomplete History…
Paul & Mukherjee , “Higher-dimensional Cosmology with Gauss-Bonnet terms and the Cosmological-Constant Problem” Phys. Rev. D42, 2595 (1990)
Mohammedi, “Dynamical Compactification, Standard Cosmology, and the Accelerated Universe” Phys. Rev. D65, (2002)
Andrew, Bolen, and Middleton, “Solutions of Higher Dimensional Gauss-Bonnet FRW Cosmology”, Grav. And Gen. Rel., Vol. 39, Num. 12 (2007) pps
Ito, “Accelerating Universe from Modified Kasner Model in Extra Dimensions”, arXiv: v2 [hep-th]

21. Choose a 4D and higher dimensional Equation of State
Remarkably, the equations decouple…

22. The FRW field equations become…
where

23. Conclusions This research is a work in progress. To do:
Solve the field equations for special cases (v = w, n = 0, 3 - dn = 0, etc.)
Is there a realistic compactification scenario?
Does this scenario produce a solution for the time evolution of a(t) that agrees with known data?
What does this model say about an early inflationary epoch, if anything?
What does this model say about a late-time acceleration?