Cosmology II - The nature of the universe

What kind of universe do we live in (positive, Negative, or zero curvature)

Example1: space with positive curvature: the surface of a sphere

You would have flunkedSophomore geometry

Interesting historical aside

Karl Friedrich Gauss thought of this, and sent out surveyors to test if the geometry in

Hanover really was Euclidean

The case of zero curvature: Euclidean space

Final case: that of negative curvature: the surface of a saddle

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?

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

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.

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

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!

The age of the universe

In a Friedmann universe, the age depends on what sort of a(t) we have

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

Connection with textbook: Skip (for now) discussionof cosmological constant and Dark Energy.

Will return to later, since they are an importantPart of modern cosmology

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

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

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

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