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Cosmic Expansion

When we look far enough out beyond the Milky Way, we see that the radiation from galaxies is shifted to lower frequencies. The farther away a galaxy is, the larger this redshift, with the radiation of the most distant galaxies shifted to the red by roughly a factor of 5. This redshift has a straightforward interpretation: the galaxies are moving away from us and away from each other, and the redshift is a Doppler shift of the radiation. The farther away a galaxy is, the faster away it moves. The most distant galaxies that we see are moving away at close to the speed of light.

Measuring the redshift of a galaxy is easy. The emission lines of elements such as hydrogen have well defined frequencies in the optical and ultraviolet frequency bands, so by comparing the frequencies of the lines emitted by a galaxy to their laboratory frequencies, we can derive a redshift. The standard way of expressing the redshift of a galaxy in astronomy is through the variable z, which is defined by the equation

νobs = νemit/( 1 + z ),

where νobs is the observed frequency of an emission line, and νemit is the frequency of an emission line at the galaxy.

The other part of the problem, determining the distance to a galaxy, is a bear of a problem. The reason is that there is only one way to associate a physical measure of distance (metric units, for instance) to an extrasolar object, and that is to measure the object's parallax. This can only be done for nearby stars. To measure the distance to a galaxy, we must use a standard candle, a class of sources of fixed luminosity, that can be calibrated through parallax measurements. We are forced to calibrate some standard candles against other standard candles, which introduces a source of error in our standard candle distance ladder.

The two most trusted standard candles now in use to associate a distance with a redshift are the Cepheid variable stars, which are calibrated through their parallax, and the type 1a supernovae, which are calibrated against the Cepheid variable stars. With these two standard candles we are able to associate a redshift with a distance to a redshift of about z = 1.2.

When standard candles are applied to galaxies, one finds that their redshifts have a random component and a distance-dependent component; the first is interpreted as a random motion, and the second, which is called the Hubble flow, is interpreted as a motion away from us. As one goes out in distance, the contribution of the Hubble flow to the redshift becomes dominant.

One finds that for redshifts much less than unity, the distance of an object is proportional to its cosmological (Hubble flow) redshift. This proportionality is expressed through the variable H0, which is dubbed the Hubble constant. The name is something of an oxymoron, because the value of the Hubble constant changes as the universe expands, but the timescale for it to change in our day is around 10 billion years, so it is as constant as you can wish in a human life. The Hubble constant is a velocity divided by distance. At a redshift that is much less than unity, the velocity is proportional to cz; for low redshift, distance is therefore related to redshift by

d= c z/H0

The current value of the Hubble constant that is derived from type 1a supernovae studies is 60±10 km s-1 Mpc-1 (the unit Mpc is megaparsec). Objects at a redshift of 0.1 are therefore at 500 Mpc; in contrast, the neighboring Andromeda galaxy (M31) is at 0.73 Mpc.

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