Cosmology, Inflation & Compact Extra Dimensions Chad A. Middleton Mesa State College March 1, 2007 Keith Andrew and Brett Bolen, Western Kentucky University

Outline…  Einstein’s General Relativity  Distance  Big Bang Cosmology  FRW Cosmology  Inflation  D-dimensional General Relativity  Gauss-Bonnet FRW Cosmology  Dynamical Compactification  Inflationary-like expansion

3D Euclidean Space Line element in Euclidean space is the line element measuring distance is invariant under rotations

In 1905, Einstein submitted his Special Theory of Relativity  Lorentz Transformations  Length depends on reference frame

4D Minkowski Spacetime Line element in Minkowski (Flat) spacetime is the line element measuring ‘length’ is invariant under ‘rotations’

In 1915, Einstein gives the world his General Theory of Relativity is the Einstein tensor describing the curvature of space is the stress- energy tensor describing the matter

In 1929, Edwin Hubble discovers that the Universe is expanding! Hubble’s Law

Did the Universe begin with a “ Big Bang ”??  is not an explosion that happened at one point in space Big Bang - a time of infinite density, infinite temperature, and infinite spacetime curvature The “Big Bang”...  occurred at every place in one moment in time

In 1965, observational evidence for the Big Bang!! Arno Penzias & Robert Wilson Bell Lab Physicists calibrating the Bell Labs microwave antenna designed for satellite communications Awarded the 1978 Nobel Prize in physics for discovery of the Cosmic Microwave Background Radiation

COBE image of the Cosmic Microwave Background Radiation Light from when the Universe was 380,000 years old… Map of  K anisotropies

Spectrum of the Cosmic Microwave Background Radiation The excellent agreement with Planck’s law is the best fit ever measured! John Mather & George Smoot Awarded the 2006 Nobel Prize in physics “for their discovery of the blackbody form and anisotropy discovery of the CMB”

Antenna-fed television “snow”

On large-distance scales… the Universe is Homogeneous & Isotropic

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

Friedmann-Robertson-Walker (FRW) Cosmology  Choose the flat Robertson-Walker metric *  Choose a perfect fluid stress-energy tensor consisting of 3 noninteracting components - pressureless matter, radiation, & vacuum * 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)

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

Inflation Why is the Universe so spatially flat homogeneous & isotropic Where did the temperature anisotropies come from?

The Einstein-Hilbert Gauss-Bonnet field equations are where

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

Dynamical Compactification of the Extra Dimensions  Extra dimensions compactify as the 3 spatial dimensions expand

The D-dimensional FRW equations and the Conservation Equation are…

Upon redefinition of constants  4D Cosmology for arbitrary n & d !!  The D-dim FRW equations are …  Compare to 4-dim FRW equations …

Solution for the Scale Factor  small  regime (late Universe)…  large  regime (early Universe)…  Inflationary-like expansion!  4D-like expansion

Conclusions  D-dimensional General Relativity is identical to 4D GR when Dynamical Compactification is employed  Addition of a Gauss-Bonnet term yields Inflationary-like expansion in the early Universe

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

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

String Theory admits a variety of non-perturbative excitations of extended objects  D(irichlet)-branes  Closed strings  Spin-2 gravitons  Open strings  Spin-1, -1/2 SM particles