# Hubble's law

Hubble's law is the statement in astronomy that the redshift in light coming from distant galaxies is proportional to their distance. The law was first formulated by Edwin Hubble in 1929.

If one assumes that this redshift is caused by galaxies moving away from us then it leads to a picture of an expanding universe and, by extrapolating back in time, to the Big Bang theory. The redshift may also be explained by ambient dust and gas between here and the distant galaxy; a longer distance means more dust and gas exists between them, which would mean a larger redshift (a possible source of "dark matter"). Hubble compared the distances to nearby galaxies to their redshift, found a linear relationship. His estimate of the proportionality constant, known as Hubble's constant (and now also as "Hubble's parameter" since it turned out to be not a constant but a parameter that depends on time in a way suggesting accelerating expansion of the universe), was however off by a factor of about 10. Furthermore, if one takes Hubble's original observations and then uses the most accurate distances and velocities currently known, one ends up with a random scatter plot with no discernable relationship between redshift and distance. Nevertheless an approximately linear relationship between redshift and distance was confirmed by observations after Hubble. The law can be stated as follows:

[itex]v = H_0 D[itex]

where [itex]v[itex] is the redshift, typically expressed in km/s (velocity that the galaxy should move with, away from us, to produce this redshift through Doppler effect), H0 is Hubble's parameter (at the observer, as denoted by the subscript), and [itex]D[itex] is the current distance from the observer to the galaxy, measured in megaparsecs: Mpc.

One can derive Hubble's law mathematically if one assumes that the universe expands (or shrinks) and that the universe is homogeneous, meaning that all points within it are equal.

[A better definition of a 'homogeneous space' is one in which local (i.e. neighborhood of a point) properties of concern (say gravity, pressure, density, etc.) are constant throughout a region. The universe is notoriously inhomogeneous on a human scale, as also on stellar/galactic scales as neighborhoods within my house have physical properties very different from those within the interior of the sun much less neighborhoods within the Swartzchild radius of Our Galaxy's central black hole. On quantum scales, no regions anywhere are homogeneous. On cosmic scales, where galaxies are small, some regions are fairly homogeneous, but over all, the entire universe (as we know it from the background microwave radiation) is also notoriously inhomogeneous. But homogeneity is not the essential issue here.]

The mathematical derivation of an idealized Hubble's Law for a uniformly accelerating expanding universe is a fairly elementary theorem of geometry in 3-dimensional Cartesian/Newtonian coordinate space, which, considered as a metric space, is entirely homogeneous and anisotropic (properties do not vary with direction). Simply stated the theorem is this:

```  Any two points which are moving away from the origin, each
along straight lines and with speed proportional to distance
from the origin, will be moving away from each other with a
speed proportional to their distance apart.
```

For most of the second half of the 20th century the value of [itex]H_0[itex] was estimated to be between 50 and 90 km/s/Mpc. The value of the Hubble constant was the topic of a long and rather bitter controversy between Gérard de Vaucouleurs who claimed the value was 100 and Allan Sandage who claimed the value was 50. The Hubble Key Project significantly improved the determination of the value and in May 2001 published its final estimate of 72+/-8  km/s/Mpc. In 2003 the satellite WMAP further improved that determination to 71+/-4, using a completely independent method, based in the measurement of anisotropies in the cosmic microwave background radiation.

Hubble's constant is "constant" in the sense that it is believed to work for all velocities and distances right now. The value of H (usually called Hubble parameter to distinguish it from its value now, the Hubble constant) decreases over time however. If one assumes that all galaxies retain their speed relative to us and do not accelerate or decelerate, then we have D = vt and it follows that H = 1/t, where t is the time since the Big Bang. This allows to estimate the age of the universe from H.

Based on recent observations, it is now believed that galaxies accelerate away from us, which means that H > 1/t (but still decreases over time) and the estimate 1/H0 (between 11 and 20 billion years) for the age of the universe is too low.

• The distance D to nearby galaxies can be estimated for example by comparing their apparent luminosity to their postulated absolute luminosity.

If the galaxies are far away, one has to take as D the distance to the galaxy right now, not when the light from it was emitted. This distance is extremely hard to determine.

• The velocity v is defined to be the time rate of change of D.

For relatively nearby galaxies, the velocity v can be determined from the galaxy's redshift z using the formula vzc where c is the speed of light. However, only the velocity due to the expansion of the universe should be used: all galaxies move relative to each other independent of the expansion of the universe, and these relative velocities, called peculiar velocities are not accounted for by Hubble's law. For far away galaxies, v cannot easily be determined from the redshift z and can be larger than c.

• Systems that are gravitationally bound, such as galaxies or our planetary system, are not subject to Hubble's law and do not expand.

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