# Hubbles law

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.

## 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 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 approximately 400 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.

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:

${\displaystyle v=H_{0}D}$

where ${\displaystyle v}$ is the recessional velocity due to redshift, typically expressed in km/s. H0 is Hubble's constant and corresponds to the value of ${\displaystyle H}$ (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. ${\displaystyle D}$ is the proper distance that the light had traveled from the galaxy in the rest frame of the observer, measured in megaparsecs: Mpc.

For relatively nearby galaxies,..... the velocity v can be determined from the galaxy's redshift z using the formula ${\displaystyle v=zc}$ where c is the speed of light, or 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. 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 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 isotropic (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.

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

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

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. This allows to estimate the age of the universe from H. A non-zero, time-dependent value of ${\displaystyle q}$ simply requires integration of the Friedmann equations from now until a point when the comoving horizon size was zero.

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 billion 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

## Measuring the Hubble constant

For most of the second half of the 20th century the value of ${\displaystyle H_{0}}$ 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. 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. 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 (km/s)/Mpc, using a completely independent method, based in the measurement of anisotropies in the cosmic microwave background radiation.

A value for ${\displaystyle q}$ was measured from standard candle observations of Type Ia supernovae was determined in 1998 to be negative which implied, to the surprise of many astronomers, the universe is accelerating (see the article on dark energy).