Hubble's law
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Hubble's law is a statement in physical cosmology which states 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[1] 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. In August, 2006, a less accurate figure was obtained independently using data from NASA's orbital Chandra X-ray Observatory: 77 (km/s)/Mpc with an uncertainty of ± 15%.[2]
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[edit] Discovery
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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 equations 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.
In 1922, Alexander Friedmann derived his Friedmann equations from the Einstein field equations, showing that the universe might expand at a rate calculable by the equations.[3] The parameter used by Friedman is known today as the scale factor which can be considered as a scale invariant form of the proportionality constant of Hubble's Law. Georges Lemaître independently found a similar solution in 1927. From the Friedmann equations, the Friedmann-Lemaître-Robertson-Walker metric was derived for a fluid with a given density and pressure. This idea of an expanding spacetime would eventually lead to the Big Bang and Steady State theories of cosmology.
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 issue with Shapley arguing for a small universe the size of the 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, the world's most powerful telescope at the time. 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 -- a frequent problem even for modern astronomers. (See cosmic distance ladder for 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.
[edit] 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 megaparsecs: 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. In systems that are gravitationally bound, such as galaxies or our planetary system, the expansion of space is (more than) annihilated by the attractive force of gravity.
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 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:
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
[edit] Olbers' paradox
The expansion of space summarized by the Big Bang interpretation of Hubble's Law is relevant to the old conundrum known as Olbers' paradox: if the Universe were infinite, static, and filled with a uniform distribution of stars (notice that this also requires an infinite number of stars), then every line of sight in the sky would end on a star, and the sky would be as bright as the surface of a star. However, the night sky is largely dark. Since the 1600s, astronomers and other thinkers have proposed many possible ways to resolve this paradox, but the currently accepted resolution depends in part upon the Big Bang theory and in part upon the Hubble expansion. In a universe that exists for a finite amount of time, only the light of finitely many stars has had a chance to reach us yet, and the paradox is resolved. Additionally, in an expanding universe distant objects recede from us which cause the light emanating from them to be redshifted and diminished in brightness. Both effects contribute (the redshift being the more important of the two; remember the original paradox was couched in terms of a static universe). The darkness of the night sky, then, provides a kind of confirmation for the Hubble expansion of the universe. [4]
[edit] Measuring the Hubble constant
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 
meters, in the metric system H0 is about 
(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.
A value for q measured from standard candle observations of Type Ia supernovae, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the universe is currently "accelerating" (although the Hubble factor is still decreasing with time; see the articles on dark energy and the Lambda-CDM model).
In August 2006, using NASA's Chandra X-ray Observatory, a team from NASA's Marshall Space Flight Center (MSFC) found the Hubble constant to be 77 (km/s)/Mpc, with an uncertainty of about 15%. [5]
[edit] Notes
- ^ Hubble, Edwin, "A Relation between Distance and Radial Velocity among Extra-Galactic Nebulae" (1929) Proceedings of the National Academy of Sciences of the United States of America, Volume 15, Issue 3, pp. 168-173 (Full article, PDF)
- ^ Chandra Confirms the Hubble Constant (2006-08-08). Retrieved on March 7, 2007.
- ^ Friedman, A: Über die Krümmung des Raumes, Z. Phys. 10 (1922), 377-386. (English translation in: Gen. Rel. Grav. 31 (1999), 1991-2000.)
- ^ S. I. Chase, Olbers' Paradox, entry in the Physics FAQ; see also I. Asimov, "The Black of Night", in Asimov on Astronomy (Doubleday, 1974), ISBN 0-385-04111-X.
- ^ Chandra independently determines Hubble constant in Spaceflight Now
[edit] See also
- Age of the universe
- Shape of the universe
- Hyperbolic geometry (the space is curved due to expansion)
- Euclidean geometry (the space is "flat" due to a balance)
- Elliptic geometry (the space is curved due to gravity)
[edit] References
- Kutner, Marc (2003). Astronomy: A Physical Perspective. Cambridge University Press. ISBN 0-521-52927-1.
- Hubble, E.P.., The Observational Approach to Cosmology (Oxford, 1937)
[edit] External links
- History of Hubble's constant by John Huchra
- The Hubble Key Project
- Final Results from the Hubble Space Telescope Key Project to Measure the Hubble Constant. Freedman et. al. The Astrophysical Journal, Volume 553, Issue 1, pp. 47-72.
- The Hubble Diagram Project
