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

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?

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

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

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

For a Homogeneous & Isotropic Universe… … 3 possible Geometries Recent data indicates that the Universe is flat http://en.citizendium.org/images/thumb/1/1e/Omega/ratio/and/cosmological/morphology-990006b.jpg

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

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

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

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

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 http://nedwww.ipac.caltech.edu/level5/Carroll2/Figures/figure3.jpeg http://map.gsfc.nasa.gov/media/060916/060916/320.jpg

The Cosmological Constant Problem From the zero-point energies of vacuum fluctuations… Cosmological observations imply… Taking the ratio yields.. http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BlackHoleThermo/VirtualPair.gif

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

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

String Essentials… Points of QFT  1D Strings 2 Types  Closed & Open Different Vibrational Modes  Different particles http://eskesthai.blogspot.com/2006_02_01_archive.html

String Theory demands Extra Dimensions  Two possible descriptions Compactified Extra Dimensions Non-Compactified Extra Dimensions http://www.damtp.cam.ac.uk/user/tong/string.html.jpg http://www.columbia.edu/cu/record/23/18/11c.gif

Kaluza-Klein Compactification Consider a 5D theory, w/ the 5th dimension periodic… http://images.iop.org/objects/physicsweb/world/13/11/9/pw1311091.gif where Kaluza, Theodor (1921) Akad. Wiss. Berlin. Math. Phys. 1921: 966–972 Klein, Oskar (1926) Zeitschrift für Physik, 37 (12): 895–906

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

D-dimensional FRW field equations

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, 104018 (2002) Andrew, Bolen, and Middleton, “Solutions of Higher Dimensional Gauss-Bonnet FRW Cosmology”, Grav. And Gen. Rel., Vol. 39, Num. 12 (2007) pps. 2061-2071 Ito, “Accelerating Universe from Modified Kasner Model in Extra Dimensions”, arXiv: 0812.4326v2 [hep-th]

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

The FRW field equations become… where

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?