Anisotropic Evolution of D-Dimensional FRW

Spacetime

Anisotropic Evolution of D-Dimensional FRW

SpacetimeChad A. MiddletonMesa State CollegeFebruary 19, 2009

Cosmologyis 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 hisGeneral Theory of Relativity

describes the curvature of spacetime

describes the matter & energy w/in spacetime

G 8GT

G

T

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. Isotropic2. Homogeneous

Maximally Symmetric Space

For a Homogeneous & Isotropic Universe…

… 3 possible Geometries

Recent data indicates

that the Universe

is flathttp://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*

ds2 dt 2 a2(t) dx 2 dy 2 dz2

Choose a perfect fluid stress-energy tensor

T diag (t), p(t), p(t), p(t)

* 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)

2

2

2

2

2

3

a

a

a

ap

a

a

Choose an “equation of state”

p w

For radiation:

For pressure-less matter:

For a vacuum:

w 0

w 1/3

w 1

Density as a function of the scale factor

(a) crit v m

a3 r

a4

Radiation dominated:

Matter dominated:

Vacuum energy dominated:

a(t) ~ t 2 / 3

a(t) ~ t1/ 2

a(t) ~ eHt

Data from Type Ia Supernovae, WMAP and SDSS implies…

The Universe is flat

The expansion of the Universe is ACCELERATING!

Seems to indicate a Vacuum Energy

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

The Cosmological Constant Problem

th ~ 2 10110 erg /cm3

From the zero-point energies of vacuum fluctuations…

obs ~ 2 10 10erg /cm3

Taking the ratio yields..

th

obs ~ 10120

Cosmological observations imply…

http://www.upscale.utoronto.ca/GeneralInterest/Harrison/BlackHoleThermo/VirtualPair.gif

The Ultraviolet Catastrophe

lim 0

uRJ () 8kT

4

Consider the energy density, u(λ), of an ideal blackbody...

lim 0

uPL () 8hc

5

1

ehc / kT 1

0

The resolution of the Ultraviolet Catastrophe

led to Quantum Mechanics

“Modern Physics” by Paul A. Tipler & Ralph A. Llewellyn, 5th edition, W.H. Freeman and Company

A Quantum Theory of Gravity?

In QFT, particles are treated as mathematical points.

Points of QFT 1D Strings

2 Types Closed & Open

Different Vibrational Modes Different particles

String Essentials…

http://eskesthai.blogspot.com/2006_02_01_archive.html

String Theory demands Extra Dimensions

Compactified Extra Dimensions

Non-Compactified Extra Dimensions

Two possible descriptions

http://www.damtp.cam.ac.uk/user/tong/string.html.jpg

http://www.columbia.edu/cu/record/23/18/11c.gif

Kaluza-Klein Compactification

R(5) R(4 ) 1

4F F

Consider a 5D theory, w/ the 5th dimension periodic…

F A A

A' A where

x 4 x 4 2R

•Kaluza, Theodor (1921) Akad. Wiss. Berlin. Math. Phys. 1921: 966–972•Klein, Oskar (1926) Zeitschrift für Physik, 37 (12): 895–906

http://images.iop.org/objects/physicsweb/world/13/11/9/pw1311091.gif

D-Dimensional FRW Cosmology

Choose a perfect fluid stress-energy tensor

T diag (t), p(t), p(t), p(t), pd (t),..., pd (t),

where is the higher dimensional pressure

pd (t)

ds2 dt2 a2(t) dx2 dy2 dz2 b2(t)nmdyndym

Choose the flat Robertson-Walker metric

D-dimensional FRW field equations

ab

ba

b

bd

b

bd

a

a

a

ap

ab

bad

b

bdd

b

bd

a

a

a

ap

ab

bad

b

bdd

a

a

d

3)2(2

1)1(3

2)1(2

12

3)1(2

13

2

2

2

2

2

2

2

2

2

2

2

2

An Incomplete History…

b(t) 1

an (t)

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]

˜ p p 1

3dn( pd )

ab

bat

)(

Choose a 4D and higher dimensional Equation of State

p wpd v

Remarkably, the equations decouple…

b(t) 1

an (t)1 0dn a(t)(dn 3)dt 1/ d

The FRW field equations become…

xxxadn

xxadnvwv

vwv

1

1)(3

02

)(30

)3(

)3(

where

x(t) a3bd a(3 dn )(t) 1 0dn a(t)(dn 3)dt

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?