Cosmology, Inflation & Compact Extra Dimensions

Chad A. MiddletonMesa 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 SpaceLine element in

Euclidean space

is the line

element measuring distance

is invariant under rotations

€

limΔ →0

(Δs)2 = (Δx)2 + (Δy)2 + (Δz)2

€

ds2

€

ds2€

ds2 = dx 2 + dy 2 + dz2

In 1905, Einstein submitted his Special Theory of Relativity

Lorentz Transformations

€

x'= γ x − vt( )

t'= γ t −vx

c 2

⎛

⎝ ⎜

⎞

⎠ ⎟

Length depends on reference frame

€

ds2 = dx 2 + dy 2 + dz2

€

≠dx'2 +dy'2 +dz'2 = ds'2

4D Minkowski SpacetimeLine element in

Minkowski (Flat) spacetime

is the line element measuring ‘length’

is invariant under ‘rotations’

€

ds2 = −c 2dt 2 + dx 2 + dy 2 + dz2

€

ds2

€

ds2

€

ds2 = ds'2

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

€

GAC =

1

2βTA

C

€

GAC

€

TAC

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

€

v = H × rHubble’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 space @ 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

€

TB = 2.725K ±18μK 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*

€

ds2 = −dt 2 + a2(t) dx 2 + dy 2 + dz2[ ]

Choose a perfect fluid stress-energy tensor

€

Tμν = diag ρ (t), p(t), p(t), p(t)[ ]

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)

€

0 = ˙ ρ + 3˙ a

a(ρ + p)

ρ

2β= 3

˙ a 2

a2

p

2β= − 2

˙ ̇ a

a+

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟

Density as a function of the scale factor

€

(a) = ρ crit Ωv +Ωm

a3+

Ωr

a4

⎛

⎝ ⎜

⎞

⎠ ⎟

Radiation dominated:

Matter dominated:

Vacuum dominated:

€

a(t) ~ t 2 / 3

€

a(t) ~ t1/ 2

€

a(t) ~ eHt

Inflation Why is the Universe so spatially flat homogeneous & isotropicWhere did the temperature

anisotropies come from?

€

a(t) ~ eHt

The Einstein-Hilbert Gauss-Bonnet field equations are

€

GAC + εGA

C =1

2κTA

C

where

€

GAC = RA

C −1

2gA

C R

GAC =

1

2RBDEFRBDEF − 4RBDRBD + R2

( )δAC

− 2RBDEARBDEC + 2RRAC − 4RD

B RBADC − 4RB

C RAB

( )

Gauss-Bonnet FRW Cosmology

Choose a perfect fluid stress-energy tensor

€

Tμν = diag ρ (t), p(t), p(t), p(t), pd (t),..., pd (t),[ ]

where is the higher dimensional pressure

€

pd (t)

€

ds2 = −dt 2 + a2(t) dx 2 + dy 2 + dz2[ ] + b2(t)γ nmdy ndym

Choose the flat Robertson-Walker metric

Dynamical Compactification of the Extra Dimensions

Extra dimensions compactify as the 3 spatial dimensions expand

€

b(t) ~1

an (t)

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

€

0 = ˙ ρ + 3˙ a

a(ρ + ˜ p )

ρ

2κ= η1

˙ a 2

a2+ εξ1

˙ a 4

a4

˜ p

2κ= −

1

3η1 2

˙ ̇ a

a+

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟˙ a 2

a2−

1

3εξ1 4

˙ ̇ a

a−

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟˙ a 2

a2

pd

2κ=

1

dn(2η1 + 3η 2)

˙ ̇ a

a+ 2

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟− dn(η1 + 3η 2)

˙ a 2

a2

⎡

⎣ ⎢

⎤

⎦ ⎥

+ε1

dn3(ξ1 + ξ 3)

˙ ̇ a

a−

1

3dn

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟+ ξ1

˙ ̇ a

a

⎡

⎣ ⎢

⎤

⎦ ⎥˙ a 2

a2

€

0 = ˙ ρ + 3˙ a

a(ρ + ˜ p )

ρ

2κ=

1

3η1 3

˙ a 2

a2

⎡

⎣ ⎢

⎤

⎦ ⎥

˜ p

2κ=

1

3η1 − 2

˙ ̇ a

a+

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

€

(ε = 0)

€

0 = ˙ ρ + 3˙ a

a(ρ + p)

ρ

2β= 3

˙ a 2

a2

p

2β= − 2

˙ ̇ a

a+

˙ a 2

a2

⎛

⎝ ⎜

⎞

⎠ ⎟

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)…

€

a(t) ~ tm + o ε / t 2( )

large regime (early Universe)…

€

a(t) ~ eHt + o t 2 /ε( ) 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