Anisotropic Evolution of DDimensional FRW Spacetime

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Anisotropic Evolution of D-Dimensional FRW
Spacetime

• Chad A. Middleton
• Mesa State College
• February 19, 2009

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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 hisGeneral Theory of
Relativity

• describes the curvature of spacetime
• describes the matter energy w/in
  spacetime

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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
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Cosmological Principle

• On sufficiently large distance scales, the
  Universe is
• 1. Isotropic
• 2. Homogeneous
• ? Maximally Symmetric Space

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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
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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
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The FRW Equations are

• density (?) pressure (p) determine the
  evolution of the scale factor (a)

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Choose an equation of state

• For radiation
• For pressure-less matter
• For a vacuum

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Density as a function of the scale factor

• Radiation dominated
• Matter dominated
• Vacuum energy dominated

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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/Fig
ures/figure3.jpeg
http//map.gsfc.nasa.gov/media/060916/060916/320.j
pg
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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/Har
rison/BlackHoleThermo/VirtualPair.gif
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The Ultraviolet Catastrophe
Consider the energy density, u(?), 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
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A Quantum Theory of Gravity?
In QFT, particles are treated as mathematical
points.
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String Essentials

• Points of QFT
• ? 1D Strings
• 2 Types
• ? Closed Open
• Different Vibrational Modes
• ? Different particles

http//eskesthai.blogspot.com/2006_02_01_archive.h
tml
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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.j
pg
http//www.columbia.edu/cu/record/23/18/11c.gif
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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 966972
• Klein, Oskar (1926) Zeitschrift für Physik, 37
  (12) 895906

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D-Dimensional FRW Cosmology

• Choose the flat Robertson-Walker metric

• Choose a perfect fluid stress-energy tensor

where is the higher dimensional pressure
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D-dimensional FRW field equations
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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

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Choose a 4D and higher dimensional Equation of
State
Remarkably, the equations decouple
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The FRW field equations become
where
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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?