Download Cosmology and Geometry

The Shape and Fate of the Universe
•
•
•
•
•
Assumptions in cosmology
Olber’s paradox
Expansion of the Universe
Curvature of the Universe
Fate of the Universe
Assumptions in Cosmology
Copernican principle:
– We do not occupy a special place.
– There are no special places.
– The universe is homogeneous if viewed at
sufficently large scales.
– The laws of physics are the same everywhere.
Implications of the Copernican Principle
• The average density of matter and energy is
the same throughout the Universe.
• The same Hubble expansion law is seen for
all observers anywhere in the Universe.
• The curvature of the Universe is the same
everywhere.
How can we test the Copernican principle?
• Does the Universe look the same in all
directions? (Isotropy)
• Are the spectral lines from atoms the same in
distant galaxies?
• Do the same laws of gravity apply in other
galaxies?
Why is the night sky dark?
(Olber’s Paradox 1826)
• Or what is the temperature of the sky?
• Assume universe is static, infinite, and full
of stars like the Sun.
• Then every ray extending out from the
Earth will eventually intersect a star.
• So, the brightness of the sky at that point
will be determined by the surface brightness
of that star.
• But surface brightness is independent of
distance, so the whole sky should be as
bright as the Sun.
F d
2
d
2
F
   d0

Why is the night sky dark?
(Olber’s Paradox 1826)
• One of the assumptions (static, infinite,
and full of stars like the Sun) must be
incorrect.
• Thus, to have a dark night sky, the
Universe must be some combination of
• dynamic
• finite in time
• finite in extent
Hubble expansion v = H0d
Expansion of the Universe
Hubble expansion v = H0d
Time =
distance/velocity
= d/H0d
= 1/H0
= 1/(71 km/s/Mpc)
= 13.8 Gyr
We can only see out to distances ~ c/H0
Two dimensional geometry
• Only two directions: up/down and left/right
: north/south and east/west
• All motion of particles, light confined to two
dimensions
• Examples: black board, piece of paper, surface of
sphere, surface of donut, surface of saddle
Geometry
• How are the diameter and circumference of
a circle related?
• What is the sum of all of the angles in a
triangle?
Geometry in flat space
• Circumference = 2  radius
 = 3.1415926… 2 = 6.28…
• The sum of the angles in a triangle is 180º
Geometry in positively curved space
• Circumference < 2  radius
• The sum of the angles in a triangle > 180º
Geometry in negatively curved space
• Circumference > 2  radius
• The sum of the angles in a triangle < 180º
Curvature of the Universe
The curvature of the Universe is determined by:
• the density of matter and energy
– higher density produces positive curvature
• the expansion of the Universe
– more rapid expansion produces negative curvature
Fate of the Universe
Critical Density
At what density is the Universe balanced between expanding forever
versus contracting? Consider a small mass at the surface of a sphere:
1 2 GMm
E  0  K  U  mv 
2
R
4 3
v  H0R
M  R 
3
1 2 2 G4 3
H0 R 
R 
2
R3
3
2
 27
-3
C 
H 0  9.7 10 kg  m
8G
Curvature of the Universe
The curvature of the Universe is determined
by the density parameter 0

0 
C
0 < 1  negative curvature
0 > 1  positive curvature
Fate of the Universe
Age of the Universe
Imagine that the circumference of a circle
with a radius of 1000 Mpc was measured to
be 5500 Mpc. This would imply that
1. The Universe is flat.
2. The Universe is positively curved and will
eventually contract in a big crunch.
3. The Universe is negatively curved and will
expand forever.