anisotropic evolution of d-dimensional frw spacetime chad a. middleton mesa state college february...
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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?