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Cosmology II - The nature of the universe
What kind of universe do we live in (positive, Negative, or zero curvature)
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Example1: space with positive curvature: the surface of a sphere
You would have flunkedSophomore geometry
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Interesting historical aside
Karl Friedrich Gauss thought of this, and sent out surveyors to test if the geometry in
Hanover really was Euclidean
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The case of zero curvature: Euclidean space
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Final case: that of negative curvature: the surface of a saddle
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Within the context of General Relativity, all three cases of curvature (positive, negative, zero) are
theoretical possibilities.
All three possibilities give universes which expand with time
The question is: what kind of universe do we live in?
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Point to emphasize
For all three types of curvature, the solutions of the equation have a(t)
increasing. The equations of General Relativity could have predicted Hubble’s
Law
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How to measure the curvature (or geometry) of our universe
• Measure angular size of a “rigid rod” as a function of distance (z).
• Measure the brightness of a “standard candle” as a function of distance (z).
• Unfortunately, real astronomical objects don’t want to stay constant over cosmological times.
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Result of solutions of equations for Friedmann Universe
Evolution of universeAnd its curvature dependOn the mean density
(1) Density less than critical density: negative curvature and indefinite expansion
(2) Density greater than critical density, positive curvature and future contraction
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Value of the critical density
• 10-26 kg/cubic meter ~ 10 hydrogen atoms/cubic meter
• Typical density in the interstellar medium is 1 million - 10 million hydrogen atoms/cubic meter
• Space could be very empty and still have a density greater than critical
Say it with equations!
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The age of the universe
In a Friedmann universe, the age depends on what sort of a(t) we have
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A consistency check for cosmological theories: are our estimates for the age
of the universe consistent with independent measurements of the
age of objects?
• Age of the Earth
• Age of globular cluster stars
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Connection with textbook: Skip (for now) discussionof cosmological constant and Dark Energy.
Will return to later, since they are an importantPart of modern cosmology
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The “Big Bang”
• Friedmann equation predicts a=0 in remote past
• This happened 14 Gyr ago if Omega=0• Happened (2/3)*14 Gyr ago if Omega =1• At that time, universe infinitely compressed• From that instant on, there was expansion of
universe, density drops, temperature drops, like aftermath of explosion
• Big Bang
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The Big Bang
The Big Bang was not like an explosion, in that it didn’t “explode
into nothing”. At the time of the BB, the universe was probably infinite in extent; the scale has
gotten bigger with time. Even if it was finite (K>0), it was unbounded
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A Reality Check
• All of this sounds pretty weird (and it’s about to get weirder), but it isn’t “made up”
• We have Hubble’s Law: the universe IS expanding
• We have the equations of General Relativity, exhaustively tested in physics experiments
• More to come
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The Big Bang from the inside out; start at t=0 and see what happens
• First few seconds: really weird stuff• First three minutes: whole universe hot and dense as
center of Sun. Nuclear reactions everywhere• 700,000 years after BB: universe cools to point where
hydrogen atoms combine from protons and electrons, making universe transparent
• Few hundred million years after BB: first ghostly protogalaxies
• One billion years after BB: birth of the quasars• 5 billion years after BB: galaxies as they are today