# Hubble's law

 Physical cosmology Age of the universeBig Bang Blueshift Comoving distanceCosmic microwave background Dark energy Dark matter FLRW metric Friedmann equations Galaxy formationHubble's lawInflation Large-scale structure Lambda-CDM modelMetric expansion of space Nucleosynthesis Observable universeRedshift Shape of the universe Structure formation Timeline of the Big Bang Timeline of cosmology Ultimate fate of the universeUniverse Related topics Astrophysics General relativity Particle physics Quantum gravity

Hubble's law is the statement in physical cosmology that the redshift in light coming from distant galaxies is proportional to their distance. The law was first formulated by Edwin Hubble and Milton Humason in 1929 after nearly a decade of observations. It is considered the first observational basis for the expanding space paradigm and today serves as one of the most often cited pieces of evidence in support of the Big Bang. The most recent calculation of the constant, using the satellite WMAP began in 2003, yielding a value of 71 ± 4 (km/s)/ Mpc. As of the 2006 data, that figure has been refined to 70 +2.4−3.2 (km/s)/Mpc.

## Discovery

In the decade before Hubble made his observations, a number of physicists and mathematicians had established a consistent theory of the relationship between space and time by using Einstein's field equation of general relativity. Applying the most general principles to the question of the nature of the universe yielded a dynamic solution that conflicted with the then prevailing notion of a static Universe.

However, a few scientists continued to pursue the dynamical universe and discovered that it could be characterized by a metric that came to be known after its discoverers, namely Friedmann, Lemaître, Robertson, and Walker. When this metric was applied to the Einstein equations, the so-called Friedmann equations emerged which characterized the expansion of the universe based on a parameter known today as the scale factor which can be considered a scale invariant form of the proportionality constant of Hubble's Law. This idea of an expanding spacetime would eventually lead to the Big Bang and to the Steady State theories.

Before the advent of modern cosmology, there was considerable talk as to what was the size and shape of the universe. In 1920 a famous debate took place between Harlow Shapley and Heber D. Curtis over this very issue with Shapley arguing for a small universe the size of our Milky Way galaxy and Curtis arguing that the universe was much larger. The issue would be resolved in the coming decade with Hubble's improved observations.

Edwin Hubble.

Edwin Hubble did most of his professional astronomical observing work at Mount Wilson observatory, at the time the world's most powerful telescope. His observations of Cepheid variable stars in spiral nebulae enabled him to calculate the distances to these objects. Surprisingly these objects were discovered to be at distances which placed them well outside the Milky Way. The nebulae were first described as "island universes" and it was only later that the moniker "galaxy" would be applied to them.

Combining his measurements of galaxy distances with Vesto Slipher's measurements of the redshifts associated with the galaxies, Hubble discovered a rough proportionality of the objects' distances with their redshifts. Though there was considerable scatter (now known to be due to peculiar velocities), Hubble was able to plot a trend line from the 46 galaxies he studied and obtained a value for the Hubble constant of 500 km/ s/ Mpc, which is much higher than the currently accepted value due to errors in his distance calibrations. Such errors in determining distance continue to plague modern astronomers. (See the article on cosmic distance ladder for more details.)

In 1958 the first good estimate of H0, 75 km/s/Mpc, was published (by Allan Sandage). But it would be decades before a consensus was achieved (see 'Measuring the Hubble constant' below).

After Hubble's discovery was published, Albert Einstein abandoned his work on the cosmological constant which he had designed to allow for a static solution to his equations. He would later term this work his "greatest blunder" since the belief in a static universe was what prevented him from predicting the expanding universe. Einstein would make a famous trip to Mount Wilson in 1931 to thank Hubble for providing the observational basis for modern cosmology.

## Interpretation

The discovery of the linear relationship between recessional velocity and distance yields a straightforward mathematical expression for Hubble's Law as follows:

v = H0D

where v is the recessional velocity due to redshift, typically expressed in km/s. H0 is Hubble's constant and corresponds to the value of H (often termed the Hubble parameter which is a value that is time dependent) in the Friedmann equations taken at the time of observation denoted by the subscript 0. This value is the same throughout the universe for a given conformal time. D is the proper distance that the light had traveled from the galaxy in the rest frame of the observer, measured in mega parsecs: Mpc.

For relatively nearby galaxies, the velocity v can be estimated from the galaxy's redshift z using the formula v = zc where c is the speed of light. For far away galaxies, v can be determined from the redshift z by using the relativistic Doppler effect. However, the best way to calculate the recessional velocity and its associated expansion rate of spacetime is by considering the conformal time associated with the photon traveling from the distant galaxy. In very distant objects, v can be larger than c. This is not a violation of the special relativity however because a metric expansion is not associated with any physical object's velocity.

In using Hubble's law to determine distances, only the velocity due to the expansion of the universe can be used. Since gravitationally interacting galaxies move relative to each other independent of the expansion of the universe, these relative velocities, called peculiar velocities, need to be accounted when applying Hubble's law. The Finger of God effect is one result of this phenomenon discovered in 1938 by Benjamin Kenneally. Systems that are gravitationally bound, such as galaxies or our planetary system, are not subject to Hubble's law and do not expand.

The mathematical derivation of an idealized Hubble's Law for a uniformly 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 isotropic (properties do not vary with location or 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.
The ultimate fate of the universe and the age of the universe can both be determined by measuring the Hubble constant today and extrapolating with the observed value of the deceleration parameter, uniquely characterized by values of density parameters (Ω). A so-called "closed universe" (Ω>1) comes to an end in a Big Crunch and is considerably younger than its Hubble age. An "open universe" (Ω≤1) expands forever and has an age that is closer its Hubble age. For the accelerating universe that we inhabit, the age of the universe is coincidentally very close to the Hubble age.

The value of Hubble parameter changes over time either increasing or decreasing depending on the sign of the so-called deceleration parameter q which is defined by:

$q = -H^{-2}\left( {{\; dH}\over {\; dt}} + H^2 \right)$

In a universe with a deceleration parameter equal to zero, it follows that H = 1/t, where t is the time since the Big Bang. A non-zero, time-dependent value of q simply requires integration of the Friedmann equations backwards from the present time to the time when the comoving horizon size was zero.

We may define the "Hubble age" (also known as the "Hubble time" or "Hubble period") of the universe as 1/H, or 977793 million years/[H/(km/s/Mpc)]. The Hubble age comes to 13968 million years for H=70 km/s/Mpc, or 13772 million years for H=71 km/s/Mpc. The distance to a galaxy being approximately zc/H for small redshifts z, and expressing c as 1 light-year per year, this distance can be expressed simply as z times 13772 million light-years.

It was long thought that q was positive, indicating that the expansion is slowing down due to gravitational attraction. This would imply an age of the universe less than 1/H (which is about 14,000 million years). For instance, a value for q of 1/2 (one theoretical possibility) would give the age of the universe as 2/(3H). The discovery in 1998 that q is apparently negative means that the universe could actually be older than 1/H. In fact, independent estimates of the age of the universe come out fairly close to 1/H

For most of the second half of the 20th century the value of H0 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 80 and Allan Sandage who claimed the value was 40. In 1996, a debate moderated by John Bahcall between Gustav Tammann and Sidney van den Bergh was held in similar fashion to the earlier Shapley-Curtis debate over these two competing values. This difference was partially resolved with the introduction of the Lambda-CDM model of the Universe in the late 1990s. With this model observations of high-redshift clusters at X-ray and microwave wavelengths using the Sunyaev-Zel'dovich effect, measurements of anisotropies in the cosmic microwave background radiation, and optical surveys all gave a value of around 70 for the constant. In particular the Hubble Key Project (led by Dr. Wendy L. Freedman, Carnegie Observatories) gave the most accurate optical determination in May 2001 with its final estimate of 72±8 (km/s)/Mpc, consistent with a measurement of H0 based upon Sunyaev-Zel'dovich effect observations of many galaxy clusters having a similar accuracy. The highest accuracy cosmic microwave background radiation determinations were 71±4 (km/s)/Mpc, by WMAP in 2003, and 70 (km/s)/Mpc, +2.4/-3.2 for measurements up to 2006. With 1 parsec approximated to $3.086\times 10^{16}$ meters, in the metric system H0 is about $2.3\times 10^{-18}$(m/s)/m (or Hertz). The consistency of the measurements from all three methods lends support to both the measured value of H0 and the Lambda-CDM model.