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Hubble's law describes the observation in physical cosmology that the velocity at which various galaxies are receding from the Earth is proportional to their distance from us. The law was first formulated by Edwin Hubble in 1929 after nearly a decade of observations. The recession velocity of the objects was inferred from their redshifts, many measured much earlier by Vesto Slipher (1917) and related to velocity by him. It is considered the first observational basis for the expanding space paradigm and today serves as one of the pieces of evidence most often cited in support of the Big Bang model.

The law is often expressed by the equation v = H0D, with H0 the constant of proportionality (the Hubble constant) between the distance D to a galaxy and its velocity v. The SI unit of H0 is s-1 but it is most frequently quoted in (km/s)/Mpc, thus giving the speed in km/s of a galaxy one Megaparsec away. The reciprocal of H0 is the Hubble time.

The most recent observational determination of the proportionality constant obtained in 2009 by using the Hubble Space Telescope yielded a value of H0 =  .The results agree closely with an earlier measurement of H0 =  obtained in 2001 also by Hubble. In August 2006, a less-precise figure was obtained independently using data from NASA's Chandra X-ray Observatory: H0 = 77 (km/s)/Mpc or about 2.5×10−18 s−1 with an uncertainty of ± 15%. NASA summarizes existing data to indicate a constant of 70.8 ± 1.6 (km/s)/Mpc if space is assumed to be flat, or 70.8 ± 4.0 (km/s)/Mpc otherwise.

## Discovery

A 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 nature of the universe yielded a dynamic solution that conflicted with the then prevailing notion of a static universe.

### FLRW equations

In 1922, Alexander Friedmann derived his Friedmann equations from Einstein's field equations, showing that the universe might expand at a rate calculable by the equations. The parameter used by Friedmann 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. The Friedmann equations are derived by inserting the metric for a homogeneous and isotropic universe into Einstein's field equations 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.

### Shape of the universe

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

### Cepheid variable stars outside of the Milky Way

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. They continued to be called "nebulae" and it was only gradually that the term "galaxies" took over.

### Combining redshifts with distance measurements

The parameters that appear in Hubble’s law: velocities and distances, are not directly measured. In reality we determine, say, a supernova brightness, which provides information about its distance, and the redshift z = ∆λ/λ of its spectrum of radiation. Hubble correlated brightness and parameter z.

Combining his measurements of galaxy distances with Vesto Slipher's measurements of the redshifts associated with the galaxies, Hubble discovered a rough proportionality between redshift of an object and its distance. Though there was considerable scatter (now known to be caused by peculiar velocities), Hubble was able to plot a trend line from the 46 galaxies he studied and obtain a value for the Hubble constant of 500 km/s/Mpc (much higher than the currently accepted value due to errors in his distance calibrations). (See cosmic distance ladder for details.)

At the time of discovery and development of Hubble’s law it was acceptable to explain redshift phenomenon as a Doppler shift in the context of special relativity, and use the Doppler formula to associate redshift z with velocity. However, this approximate relation between velocity and redshift is accurate only for values of z somewhat less than 1. Today the velocity-distance relationship of Hubble's law is viewed as a theoretical result with velocity to be connected with observed redshift not by the Doppler effect, but by a cosmological model relating recessional velocity to the expansion of the universe. Even for small z the velocity entering the Hubble law is no longer interpreted as a Doppler effect, although at small z the velocity-redshift relation for both interpretations is the same.

In 1968, 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.

#### Hubble Diagram

Hubble's Law can be easily depicted in a "Hubble Diagram" in which the velocity (assumed approximately proportional to the redshift) of an object is plotted with respect to its distance from the observer. A straight line of positive slope on this diagram is the visual depiction of Hubble's Law.

### The cosmological constant abandoned

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 later termed this work his "greatest blunder" since the assumption of a static universe had prevented him from predicting the expanding universe. Einstein made a famous trip to Mount Wilson in 1931 to thank Hubble for providing the observational basis for modern cosmology. However, the cosmological constant has regained attention in recent decades as a hypothesis for dark energy.

## Interpretation

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

v = H_0 \, D

where
• v is the recessional velocity, 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 comoving time.
• D is the proper distance from the galaxy to the observer, measured in mega parsecs (Mpc), in the 3-space defined by given cosmological time. (Recession velocity is just v = dD/dt).

Hubble's law is considered a fundamental relation between recessional velocity and distance. However, the relation between recessional velocity and redshift depends on the cosmological model adopted, and is not established except for small redshifts.

For distances D larger than the radius of the Hubble sphere rHS objects recede at a rate faster than the speed of light:

r_{HS} = \frac{c}{H_0} \ .

Inasmuch as the Hubble "constant" is not constant at all, but varies with time in a manner dictated by the choice of cosmological model, the radius of the Hubble sphere may increase or decrease over various time intervals. The subscript '0' indicates the value of the Hubble constant today.

### Redshift velocity and recessional velocity

Redshift can be measured by determining the wavelength of a known transition, such as hydrogen α-lines for distant quasars, and finding the fractional shift compared to a stationary reference. Thus redshift is a quantity unambiguous for experimental observation. The relation of redshift to recessional velocity is another matter. For an extensive discussion, see Harrison.

#### Redshift velocity

The redshift z often is described as a redshift velocity, which is the recessional velocity that would produce the same redshift if it were caused by a linear Doppler effect (which, however, is not the case, as the shift is caused in part by a cosmological expansion of space, and because the velocities involved are too large to use a non-relativistic formula for Doppler shift). This redshift velocity can easily exceed the speed of light. In other words, to determine the redshift velocity vrs, the relation:

v_{rs} \equiv cz \ ,

is used. That is, there is no fundamental difference between redshift velocity and redshift: they are rigidly proportional, and not related by any theoretical reasoning. The motivation behind the "redshift velocity" terminology is that the redshift velocity agrees with the velocity from a low-velocity simplification of the so-called Fizeau-Doppler formula

z = \frac{\lambda_o}{\lambda_e}-1 = \sqrt{\frac{1+v/c}{1-v/c}}-1 \approx \frac{v}{c} \ .

Here, λo, λe are the observed and emitted wavelengths respectively. The "redshift velocity" vrs is not so simply related to real velocity at larger velocities, however, and this terminology leads to confusion if interpreted as a real velocity. Next, the connection between redshift or redshift velocity and recessional velocity is discussed. This discussion is based on Sartori.

#### Recessional velocity

Suppose R(t) is called the scale factor of the universe, and increases as the universe expands in a manner that depends upon the cosmological model selected. Its meaning is that all measured distances D(t) between co-moving points increase proportionally to R. (The co-moving points are not moving relative to each other except as a result of the expansion of space.) In other words:

\frac {D(t)}{D(t_0)} = \frac {R(t)}{R(t_0)} \ ,

where t0 is some reference time. If light is emitted from a galaxy at time te and received by us at t0, it is red shifted due to the expansion of space, and this redshift z is simply:

z = \frac {R(t_0)}{R(t_e)} - 1 \ .

Suppose a galaxy is at distance D, and this distance changes with time at a rate dtD . We call this rate of recession the "recession velocity" vr:

v_r = d_tD = \frac {d_tR}{R} D \ .

We now define the Hubble constant as

H \equiv \frac {d_tR}{R} \ ,

and discover the Hubble law:

v_r = H(t) D \ .

From this perspective, Hubble's law is a fundamental relation between (i) the recessional velocity contributed by the expansion of space and (ii) the distance to an object; the connection between redshift and distance is a crutch used to connect Hubble's law with observations. This law can be related to redshift z approximately by making a Taylor series expansion:

z = \frac {R(t_0)}{R(t_e)} - 1 \approx \frac {R(t_0)} {R(t_0)\left(1+(t_e-t_0)H(t_0)\right)}-1 \approx (t_0-t_e)H(t_0) \ ,

If the distance is not too large, all other complications of the model become small corrections and the time interval is simply the distance divided by the speed of light:

z \approx (t_0-t_e)H(t_0) \approx \frac {D}{c} H(t_0) \ , or cz \approx D H(t_0) = v_r \ .
According to this approach, the relation cz = vr is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See velocity-redshift figure.

### Observability of parameters

Strictly speaking, neither v nor D in the formula are directly observable, because they are properties now of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it.

For relatively nearby galaxies (redshift z much less than unity), v and D will not have changed much, and v can be estimated using the formula v = zc where c is the speed of light. This gives the empirical relation found by Hubble.

For distant galaxies, v (or D) cannot be calculated from z without specifying a detailed model for how H changes with time. The redshift is not even directly related to the recession velocity at the time the light set out, but it does have a simple interpretation: (1+z) is the factor by which the universe has expanded while the photon was travelling towards the observer.

### Expansion velocity vs relative 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 for in the application of 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 a much weaker effect than the attractive force of gravity.

### Idealized Hubble's Law

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.

In fact this applies to non-Cartesian spaces as long as they are locally homogeneous and isotropic; specifically to the negatively- and positively-curved spaces frequently considered as cosmological models (see shape of the universe).

An observation stemming from this theorem is that seeing objects recede from us on Earth is not an indication that Earth is near to a center from which the expansion is occurring, but rather that every observer in an expanding universe will see objects receding from them.

### The ‘ultimate fate’ and age of the universe

The value of the 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 = -\left(1+\frac{\dot H}{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.

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 (once favoured by most theorists) 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, estimates of the age of the universe are, by coincidence, very close to 1/H.

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 causes the light emanating from them to be redshifted and diminished in brightness. Although both effects contribute, the redshift is the less important of the two.

## Determining the Hubble constant

The value of the Hubble constant is estimated by measuring the redshift of distant galaxies and then determining the distances to the same galaxies (by some other method than Hubble's law). Uncertainties in the physical assumptions used to determine these distances have caused varying estimates of the Hubble constant. For most of the second half of the 20th century the value of H_0 was estimated to be between 50 and .

### Disputes over Hubble's constant

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 around 100 and Allan Sandage who claimed the value was near 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.

This difference was partially resolved with the introduction of the ΛCDM model of the universe in the late 1990s.

### The ΛCDM model

With the ΛCDM 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.

### Using Hubble space telescope data

The Hubble Key Project (led by Dr. Wendy L. Freedman, Carnegie Observatories) used Hubble space telescope to establish the most precise optical determination in May 2001 of , consistent with a measurement of H_0 based upon Sunyaev-Zel'dovich effect observations of many galaxy clusters having a similar accuracy.

### Using WMAP data

The most precise cosmic microwave background radiation determinations were , by WMAP in 2003, and , for measurements up to 2006. The five year release from WMAP in 2008 finds .[7086]

These values arise from fitting a combination of WMAP and other cosmological data to the simplest version of the ΛCDM model. If the data is fitted with more general versions, H_0 tends to be smaller and more uncertain: typically around although some models allow values near 63 (km/s)/Mpc.Results for H_0 and other cosmological parameters obtained by fitting a variety of models to several combinations of WMAP and other data are available at the NASA's LAMBDA website.

### Using Chandra X-ray Observatory data

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 , with an uncertainty of about 15%.The consistency of the measurements from all these methods lends support to both the measured value of H_0 and the ΛCDM model.

### Acceleration of the expansion

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 ΛCDM model).

## Derivation of the Hubble parameter

H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}+ \frac{\Lambda}{3},

where H is the Hubble parameter, a is the scale factor, G is the gravitational constant, k is the normalised spatial curvature of the universe and equal to −1, 0, or +1, and \Lambda is the cosmological constant.

### Matter-dominated universe (with a cosmological constant)

If the universe is matter-dominated, then the mass density of the universe \rho can just be taken to include matter so

\rho = \rho_m(a) = \frac{\rho_{m_{0}}}{a^3},

where \rho_{m_{0}} is the density of matter today. We know for nonrelativistic particles that their mass density decreases proportional to the inverse volume of the universe so the equation above must be true. We can also define (see density parameter for \Omega_m)

\rho_c = \frac{3 H^2}{8 \pi G};

\Omega_m \equiv \frac{\rho_{m_{0}}}{\rho_c} = \frac{8 \pi G}{3 H_0^2}\rho_{m_{0}};

so \rho=\rho_c \Omega_m /a^3. Also, by definition,

\Omega_k \equiv \frac{-kc^2}{(a_0H_0)^2}

and

\Omega_{\Lambda} \equiv \frac{\Lambda c^2}{3H_0^2},

where the subscript nought refers to the values today, and a_0=1. Substituting all of this in into the Friedman equation at the start of this section and replacing a with a=1/(1+z) gives

H^2(z)= H_0^2 \left( \Omega_M (1+z)^{3} + \Omega_k (1+z)^{2} + \Omega_{\Lambda} \right).

### Matter- and dark energy-dominated universe

If the universe is both matter-dominated and dark energy-dominated, then the above equation for the Hubble parameter will also be a function of the equation of state of dark energy. So now:

\rho = \rho_m (a)+\rho_{de}(a),

where \rho_{de} is the mass density of the dark energy. By definition an equation of state in cosmology is P=w\rho c^2, and if we substitute this into the fluid equation, which describes how the mass density of the universe evolves with time,

\dot{\rho}+3\frac{\dot{a}}{a}\left(\rho+\frac{P}{c^2}\right)=0;
\frac{d\rho}{\rho}=-3\frac{da}{a}\left(1+w\right).

If w is constant,

\ln{\rho}=-3\left(1+w\right)\ln{a};

\rho=a^{-3\left(1+w\right)}.

Therefore for dark energy with a constant equation of state w, \rho_{de}(a)= \rho_{de0}a^{-3\left(1+w\right)}. If we substitute this into the Friedman equation in a similar way as before, but this time set k=0 which is assuming we live in a spatially flat universe, (see Shape of the Universe)

H^2(z)= H_0^2 \left( \Omega_M (1+z)^{3} + \Omega_{de}(1+z)^{3\left(1+w \right)} \right).

If dark energy does not have a constant equation-of-state w, then

\rho_{de}(a)= \rho_{de0}e^{-3\int\frac{da}{a}\left(1+w(a)\right)},

and to solve this we must parametrize w(a), for example if w(a)=w_0+w_a(1-a), giving

H^2(z)= H_0^2 \left( \Omega_M a^{-3} + \Omega_{de}a^{-3\left(1+w_0 +w_a \right)}e^{-3w_a(1-a)} \right).

## Units derived from the Hubble constant

### Hubble time

The Hubble constant H_0 has units of inverse time, i.e. H_0 ~ 2.29×10−18 s−1. We can therefore define “Hubble time” as 1/H_0. The value of Hubble time in the standard cosmological model is 4.35×1017 s or 13.8 billion years. The phrase "expansion timescale" means "Hubble time".[7087]. If the value of H_0 were to stay constant, a naive interpretion of the Hubble time is that it is the time taken for the universe to increase in size by a factor of e (because the solution of dx/dt = xH_0 is x = s_0exp(H_0t), where s_0 is the size of some feature at some arbitrary initial condition t = 0). However, over long periods of time the dynamics are complicated by relativity, dark energy, inflation, etc., as explained above.

### Hubble length

The Hubble length is a unit of distance in cosmology, defined as c/H_0—the speed of light multiplied by the Hubble time. It is equivalent to 4228 million parsecs or 13.8 billion light years. (The numerical value of the Hubble length in light years is, by definition, equal to that of the Hubble time in years.)

### Hubble volume

The Hubble volume is sometimes defined as a volume of the universe with a comoving size of c/H_0. The exact definition varies: it is sometimes defined as the volume of a sphere with radius c/H_0, or alternatively, a cube of side c/H_0. Some cosmologists even use the term Hubble volume to refer to the volume of the observable universe, although this has a radius approximately three times larger.

## Notes

1. 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)
2. Friedman, A: Über die Krümmung des Raumes, Z. Phys. 10 (1922), 377-386. (English translation in: Gen. Rel. Grav. 31 (1999), 1991-2000.)
3. R. P. Kirshner, Hubble's Diagram and Cosmic Expansion, Online Article
4. It is argued that such motion is compatible with special relativity because every observer is the center of their own Hubble sphere, and the objects moving faster than the speed of light are therefore outside the reach of any inertial frame of reference. See
5. . A pdf file can be found here.
6. 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.
7. . Preprint available here.
8. ; available online at LAMBDA
9. Chandra independently determines Hubble constant in Spaceflight Now

## References

• Hubble, E.P.., The Observational Approach to Cosmology (Oxford, 1937)
• Eng, A.E.., A New Approach to Starlight Runs (Oswego, 1985)