Hubble Law and the Expansion of the Universe

Redshifts

The Hubble Law, the relationship between the redshift of a distant galaxy and its distance from the Earth, had its foundations starting in 1914 with the observations of Vesto M. Slipher. V. M. Slipher noted that distant galaxies all showed redshifts, i.e., This was not true for nearby galaxies, as indeed, the Andromeda galaxy at a distance of ~2,300,000 light years shows a blueshift, it is moving towards us at a speed of ~ 100 km per second. Hmmmm. Is this consistent with Hubble's Law? Results The measurement of the redshifts to galaxies is straightforward and relatively easy to do. The major uncertainty in determining the Hubble relationship comes from problems in determining the distances to galaxies. We now consider this issue.

Distance Determinations

This is such an important problem that the Hubble Space Telescope made the determination of the distance scale one of its principal goals. The usual way in which distances are found is compare the apparent brightness of an object to its true brightness. This technique relies on the inverse square law and knowing the intrinsic brightness of the object observed. Objects for which the intrinisic brightnesses are known are referred to as standard candles. The distance follows naturally from the observed brightness of an object from the relation: Here, L is the intrinsic luminosity and D is the distance to the source.

The big problem with the method is that there is no single standard candle which can be used to measure distances to all objects in the Universe. Different standard candles are used to find distances in different regimes. Each successively brighter standard candle is used to calibrate the next one. A typical distance chain. The idea of successive calibrations is referred to as bootstrapping.

A problem with bootstrapping methods is that since each successive step is built on the previous step, the error in the method grows with each step and the method may become unreliable. For most extra-galactic things, people use Cepheid variable stars (light curve) to start the bootstrapping, i.e., Cepheid variables are considered to be primary distance indicators. Note that this assumes that we know the intrinsic luminosities of Cepheid variables. The pulsation periods of Cepheid variables are strongly correlated with their average luminosities in the sense that the longer their pulsation periods, the brighter they are on average. The Cepheid luminosities are calibrated in the first few steps of the above chain, i.e., distance to Sun (Astronomical Unit) ===> distances to nearby low mass stars via parallax ===> observations of star clusters which contain low mass stars yield distances to the clusters (===> can infer the distances and hence luminosities of the bright stars in the cluster and thankfully many clusters contain Cepheid variables).

Typical extra-galactic distance ladders are

Milky Way -----> 12 M l.y. ------> 80 M l.y.  ---------> 300 M l.y. --------->  5 B l.y.
         
         Cepheids     large HII regions,     Galaxy methods (types,        SN
                       brightest stars        n-th brightest galaxy in 
                                              a cluster, ... )

          ------------------------------------------------->

                      Tully-Fisher method

Interestingly, results based on the two chains yield different answers for
H(now) ===> different ages for the Universe. The range of answers is
H(now) ~ 15 - 30 km per second / Million light years !! or age ranges from
8 - 16 billion years.  The discrepancy needs to be removed. The 
Hubble Space Telescope 
offered the possibility for a definitive answer as 
one would be able to see Cepheid variables to greater distances in the 
Universe.

          -----------------> 50 M l.y.
                      Virgo cluster of galaxies
             Cepheids

Even by Virgo, the different methods gave distance estimates which differed by around a factor of 2. HST observed Cepheids in M100 (and see HST press release and animation, a galaxy in the nearby Virgo cluster of galaxies, and re-calibrated the Tully-Fisher method. The HST results suggest that However, there is still uncertainty in this result due to things like the peculiar velocities of galaxies (recall the local vs. global motions of galaxies).