- Image 2 Caption: A cartoon
Hubble diagram
extending out to ∼ 300 Mpc.
- The Hubble diagram is plot of redshift velocity versus an direct observable kind of distance (usually a luminosity distance).
- The theoretical prediction for the plot
to 1st order
in small
cosmological redshift z
(i.e., when z << 1) is Hubble's law
### v

_{redshift}= H_{0}r_{observable},where v_redshift = zc is redshift velocity, r_observable is a direct observable kind of distance (usually a luminosity distance), and H_0 is Hubble constant. The subscripts can be dropped if you know what you mean.

By convention redshift velocity is measured in kilometers per second (km/s) and distance in megaparsec (Mpc).

The Hubble constant which is the slope of the best-fit line on the plot has units of (km/s)/Mpc.

- The theoretical prediction is from
expanding universe theory.
- We
**CANNOT**expect perfect agreement of observations with Hubble's law even if expanding universe theory is correct on average for the observable universe because of the following factors:- Observational errors in measuring z and the direct observable distance.
- The objects (e.g., galaxies
and quasars) in the actual
observable universe
have
peculiar velocities
(which are ordinary physical velocities)
superimposed on the
recession velocities
(which are growth rates of space
between us and points participating in the mean
expansion of the universe).
- Objects in the Local Group of galaxies
do
**NOT**participate in the expansion of the universe relative to us (who are in the Local Group) since the Local Group is a gravitationally bound system.You, me, the Milky Way, the Local Group, and all other gravitationally bound systems throughout the observable universe do

**NOT**expand. Only the space between gravitationally bound systems expands.

- Observational errors in measuring z and the direct observable distance.
- At first glance Hubble's law
suggests we are at the center of the
expanding universe
and it expands around us.
But actually, the expanding universe theory predicts a general scaling up of the observable universe. So every observer seems to see himself/herself at the center of expansion at first glance.

All distances between points participating in the mean expansion of the universe scale up with cosmic time (with zero time at the Big Bang) by the cosmic scale factor a(t).

The cosmic scale factor for our current epoch (conventionally cosmic time t_0) is conventionally set to 1 and given symbol a_0. Thus a(t=t_0) = a_0 = 1.

- The
expanding universe theory
is based on general relativity
(our best theory of gravity)
and certain very simplifying assumptions about the average behavior of the
observable universe.
Albert Einstein (1879--1955) failed to discover the expanding universe theory.

It was discovered independently by Alexander Alexandrovich Friedmann (1888--1925) and Georges Lemaitre (1894--1966) in the 1920s.

- Hubble's law is a simple
consequence of the
expanding universe theory.
Let r be a proper distance (i.e., a distance that can be measured at one instant in time with a ruler) between two points participating in the mean expansion of the universe. Now

### r = r

_{0}*a(t)where r_0 is the comoving distance (a time independent distance equal to the proper distance at the present cosmic time t_0), and a(t) is the cosmic scale factor. So all r's scale up and down with cosmic time as determined by a(t).

Taking the derivative of r with respect to time t (a calculus operation) gives

### v = dr/dt = r

_{0}*(da/dt) ,where v = dr/dt is the rate of change of r and da/dt is the rate of change of a(t). Note v = dr/dt is the recession velocity. It's

**NOT**an ordinary velocity: it's a rate of growth of space.Now r_0 = v/(da/dt) and substituting this into the second to last equation and rearranging gives

### v = [(da/dt)/a(t)]*r ,

The quantity [(da/dt)/a(t)] depends on cosmic time, but

**NOT**on position. We evaluate it for the present cosmic time and call it the Hubble constant### H

_{0}= [(da/dt)/a(t)]_{0},where subscript 0 means evaluated at the present cosmic time.

Now we have the general Hubble's law

### v = H

_{0}r .But there is a big problem. The recession velocity v and the proper distance are

**NOT**in general direct observables.However, it can be shown that to 1st order in small z, we obtain the ordinary Hubble's law given above.

- The
Hubble constant
is
**NOT**predicted by theory. It has to obtained from a Hubble diagram of low-z objects such as the Hubble diagram in the figure or by some other means.There are many contemporary determinations of the Hubble constant. The values are converging to H_0 = 70 (km/s)/Mpc to within a few percent.

Nowadays H_0 = 70 (km/s)/Mpc is taken as the fiducial value for many calculations. Formulae containing the Hubble constant can be written as 70 (km/s)/Mpc * h_70, where h_70 = H_0/[70 (km/s)/Mpc] is left as a variable.

- The observational discovery in 1929
that Hubble's law
was obeyed by extragalactic systems (allowing for the deviations due the factors discussed above)
was the first strong observational evidence for the
expanding universe theory.
The evidence when first presented did convince people that the expanding universe was probably true. Other evidence since that time has confirmed the expanding universe theory.

- A bit of the history of the
discovery of Hubble's law
is given in
georges_lemaitre_cartoon.html.