Lecture 34: Geometry of the Universe

Key Ideas:

Cosmological Principle:

• The Universe is Homogeneous and Isotropic on Large Scales.
• No special places or directions.

General Relativity predicts an expanding universe.

Cosmology

Cosmology is the study of the entire Universe:

• Physics of the Universe.
• Distribution of objects on all scales.
• Motions of objects in the Universe.
• Evolution of the Universe.
• Age, Origin, and Fate of the Universe.

The Universe in 1917

1917: Einstein explored the cosmological implications of General Relativity.

Observational State in 1917:

• Kapteyn model of the Milky Way was favored by some (but not all) astronomers.
• No agreement on the "spiral nebulae."
• First good calibrations of the P-L relation for Cepheids and RR Lyrae variables.

The Cosmological Principle

"The Universe is Homogeneous and Isotropic on the Largest Scales."

Critical assumption underlying Cosmology.

Homogeneous:

• No special places in the Universe.

Isotropic:

• No special directions

Homogeneity

When viewed on the largest scales:

• The average density of matter is about the same in all places in the Universe.
• The Universe is fairly smooth on large scales.

Does not apply locally:

• We see planets, stars, galaxies in regions nearby in space.
• The Universe is locally rather "lumpy".

Isotropy

When viewed on the largest scales:

• The Universe looks the same to all observers.
• The Universe looks the same in all directions as viewed by a particular observer.

Does not apply locally:

• We see different numbers of local objects in different directions.

The Dynamic Universe

Einstein applied the Cosmological Principle to General Relativity and got a surprise:

• The spacetime of the Universe could not be static and unchanging.
• The Universe must either expand or contract!

Astronomers assured him that no such general motion was observed.

The Cosmological Constant

To make the Universe static, he was required to add a new term in his equations:

The Cosmological Constant, Lambda:

• "Repulsive" gravitational force term
• Arises from empty space.
• Balances the effects of gravity.
• Halts the motion of spacetime.

Cosmic Expansion

1917: Vesto Slipher, Lowell Observatory

Measured the radial velocities of the brightest "spiral nebulae".

Results:

• 21 out of 25 spirals show a systematic redshift.
• Systematic motion away from us.
• Some velocities are large: >2000 km/second.

Einstein Modifies the Theory to Fit the Data

By introducing an ad hoc Cosmological Constant, Einstein missed being able to predict the expansion of the Universe.

1920s:

• Friedmann & Lemaitre showed that without a cosmological constant, GR predicts that the Universe is expanding.
• Hubble firmly established cosmic expansion observationally.

State of the Art

Einstein's guess about the homogeneity and isotropy of the Universe was brilliant and far ahead of the scanty empirical data of his time.

Modern observations bear out large-scale homogeneity and isotropy:

• Large-scale galaxy surveys.
• Maps of the cosmic background radiation.

Modern Cosmological Constant

In current cosmological theory, Lambda reappears in a somewhat altered form:

• Introduced as the "vacuum energy" of space.
• Quantum ground-state of empty space.
• Assumed to be either very small or zero.
• Recent evidence of accelerating universe

Distinction:

• Its use today is arguably better physically motivated than when Einstein used it in 1917.