In 1917, two years after completing the theory of general relativity (GR), Einstein began to explore the implications of his theory for cosmology.

He started by introducing a key assumption, the cosmological principle: The universe is homogeneous and isotropic. In other words: the universe has no ``special'' places and no ``special'' directions.

The cosmological principle does not apply locally, since the sun, the Milky Way, and nearby galaxies all mark ``special'' directions.

In 1917, no empirical evidence supported the cosmological principle because ``maps'' of the universe didn't extend very far.

Today deep maps of the universe provide excellent evidence that the cosmological principle applies on very large scales. Einstein made a brilliant guess.


Einstein also introduced another assumption (almost without realizing that it was an assumption): The universe is static (not expanding or contracting).

He found that he must add another term to the equations of GR --- a ``repulsive force,'' exerted by empty space, that balances the attraction of gravity --- to make such a universe possible.

Einstein called this term the ``cosmological constant''; today physicists usually call it vacuum energy, energy of empty space.

In the 1920's, Friedmann and Lemaitre showed that without a cosmological constant, GR implies that a homogeneous universe must expand.


1910-1920: Slipher measures Doppler shifts of bright spiral nebulae. He finds that they are moving fast, up to ~ 2000 km/s, nearly all away from the Milky Way

1923: Hubble finds Cepheids in Andromeda => spiral nebulae are spiral galaxies

1923-1929: Hubble estimates distances to ~ 20 galaxies, using Cepheid variables and brightest stars. He also develops his classification scheme for galaxies.

1929: Hubble announces a new discovery -- a galaxy's recession velocity (and hence its Doppler shift) is proportional to its distance. Implication: the universe is expanding, in a systematic way.

Hubble's 1929 data were rather marginal. Hubble's law is now very well established, through later efforts of Hubble and many astronomers since.



In symbols:

v = Hd.

v = recession velocity, measured from Doppler shift [km/s]
d = distance [Mpc]
H = Hubble's constant [km/s/Mpc]

In words: A galaxy's recession velocity, which can be measured from its Doppler shift, is proportional to its distance. The constant of proportionality is Hubble's constant H.



(1) What is the recession velocity of a galaxy 10 Mpc away, assuming H=70 km/s/Mpc?

The velocity is v = Hd = (70 km/s/Mpc) x (10 Mpc) = 700 km/s.

(2) By measuring a galaxy's Doppler shift, I determine that it is receding from the earth at 7000 km/s. How distant is the galaxy, assuming H=70 km/s/Mpc?

The distance is d = v/H = (7000 km/s) / (70 km/s/Mpc) = 100 Mpc.


Since distant galaxies are moving away in every direction, it seems as though we're in the center of the universe.

We're not. Observers in other galaxies would also see distant galaxies (including our own) receding from them, also obeying Hubble's law.

There is no ``center'' of the expansion. The Hubble flow is an expansion of space not an expansion into space. Galaxies are carried along by the expansion, like corks in a current.

A 2-dimensional analogy

Represent galaxies by points on a spherical balloon.

As the balloon inflates (expansion of space), galaxies are carried apart.

The expansion has no center; every galaxy sees others moving away, with velocity proportional to distance.

Note: In the real universe, galaxies move apart from each other, but the galaxies themselves do not expand because gravity holds them together.


Hubble's law is not a perfect description. In addition to moving with the expansion of space, a galaxy can have a peculiar velocity, caused by the gravitational pull of nearby galaxies.

A more accurate formula is

v = Hd + vpec.

Typical peculiar velocities are 100--1000 km/s. When the distance is large, Hd is always much bigger than vpec.

Andromeda is close (small d), and is actually moving towards us.

In terms of the balloon analogy, peculiar velocities correspond to small motions of galaxies along the balloon's surface.

Hubble's law also breaks down when the implied velocity v approaches the speed of light. We'll need a more sophisticated interpretation of redshift for these cases (lecture 24).

Bottom line: v=Hd is very accurate for v between 5000 km/s and 50,000 km/s, pretty accurate for v between 1000 km/s and 100,000 km/s.


Since a receding galaxy's spectral lines are Doppler shifted to the red, a galaxy's recession velocity is often called its redshift.

To discover the v=Hd relation, Hubble needed galaxies with measured redshifts and distances.

A galaxy's distance is hard to measure directly, but its redshift is relatively easy to measure from its spectrum.

[Note, however, that the fractional change in the wavelength of light is only v/c, so except for the most distant galaxies, the redshift does not significantly change a galaxy's color.]

Once we know Hubble's law, we can use it to infer a galaxy's distance from its redshift:

d = v/H.

By measuring galaxy spectra, we can map the distribution of galaxies in 3 dimensions. The largest such maps today have about 20,000 galaxies.

These maps show that galaxies and clusters lie in enormous sheets and filaments, separated by gigantic voids.

Shortcomings of the redshift approach:


The cosmological principle: the universe is homogeneous and isotropic.

General relativity, combined with the cosmological principle, predicts an expanding universe.

Doppler shifts show that distant galaxies are receding.

Hubble's law: v=Hd. Space is expanding.

The Hubble constant H is an important number for cosmology, but it's difficult to measure. The best current estimate is about 70 km/s/Mpc.

Peculiar velocities => the expansion is not perfectly smooth: v = Hd + vpec.

Knowing Hubble's law, we can determine a galaxy's distance from its redshift: d=v/H.

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Updated: 1997 February 23 [dhw]