Hubble's law, also known as the Hubble–Lemaître law,^[1] is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving away from Earth. The velocity of the galaxies has been determined by their redshift, a shift of the light they emit toward the red end of the visible spectrum. The discovery of Hubble's law is attributed to Edwin Hubble's work published in 1929.^[2]

Hubble's law is considered the first observational basis for the expansion of the universe, and today it serves as one of the pieces of evidence most often cited in support of the Big Bang model.^[3][4] The motion of astronomical objects due solely to this expansion is known as the Hubble flow.^[5] It is described by the equation v = H₀D, with H₀ the constant of proportionality—the Hubble constant—between the "proper distance" D to a galaxy, which can change over time, unlike the comoving distance, and its speed of separation v, i.e. the derivative of proper distance with respect to the cosmological time coordinate. (See Comoving and proper distances § Uses of the proper distance for discussion of the subtleties of this definition of velocity.)

The Hubble constant is most frequently quoted in (km/s)/Mpc, thus giving the speed in km/s of a galaxy 1 megaparsec (3.09×10¹⁹ km) away, and its value is about 70 (km/s)/Mpc. However, crossing out units reveals that H₀ is a unit of frequency (SI unit: s⁻¹) and the reciprocal of H₀ is known as the Hubble time. The Hubble constant can also be interpreted as the relative rate of expansion. In this form H₀ = 7%/Gyr, meaning that at the current rate of expansion it takes a billion years for an unbound structure to grow by 7%.

Although widely attributed to Edwin Hubble,^[6][7][8] the notion of the universe expanding at a calculable rate was first derived from general relativity equations in 1922 by Alexander Friedmann. Friedmann published a set of equations, now known as the Friedmann equations, showing that the universe might be expanding, and presenting the expansion speed if that were the case.^[9] Then Georges Lemaître, in a 1927 article, independently derived that the universe might be expanding, observed the proportionality between recessional velocity of, and distance to, distant bodies, and suggested an estimated value for the proportionality constant; this constant, when Edwin Hubble confirmed the existence of cosmic expansion and determined a more accurate value for it two years later, came to be known by his name as the Hubble constant.^{[3][10][11][12][2]} Hubble inferred the recession velocity of the objects from their redshifts, many of which were earlier measured and related to velocity by Vesto Slipher in 1917.^[13][14][15] Combining Slipher's velocities with Henrietta Swan Leavitt's intergalactic distance calculations and methodology allowed Hubble to better calculate an expansion rate for the universe.^[16]

Though the Hubble constant H₀ is constant at any given moment in time, the Hubble parameter H, of which the Hubble constant is the current value, varies with time, so the term constant is sometimes thought of as somewhat of a misnomer.^[17][18]

Discovery

A decade before Hubble made his observations, a number of physicists and mathematicians had established a consistent theory of an expanding universe by using Einstein 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-prevalent notion of a static universe.

Slipher's observations

In 1912, Vesto M. Slipher measured the first Doppler shift of a "spiral nebula" (the obsolete term for spiral galaxies) and soon discovered that almost all such nebulae were receding from Earth. He did not grasp the cosmological implications of this fact, and indeed at the time it was highly controversial whether or not these nebulae were "island universes" outside the Milky Way galaxy.^[20][21]

FLRW equations

In 1922, Alexander Friedmann derived his Friedmann equations from Einstein field equations, showing that the universe might expand at a rate calculable by the equations.^[22] The parameter used by Friedmann is known today as the scale factor and can be considered as a scale invariant form of the proportionality constant of Hubble's law. Georges Lemaître independently found a similar solution in his 1927 paper discussed in the following section. 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.

Lemaître's equation

In 1927, two years before Hubble published his own article, the Belgian priest and astronomer Georges Lemaître was the first to publish research deriving what is now known as Hubble's law. According to the Canadian astronomer Sidney van den Bergh, "the 1927 discovery of the expansion of the universe by Lemaître was published in French in a low-impact journal. In the 1931 high-impact English translation of this article, a critical equation was changed by omitting reference to what is now known as the Hubble constant."^[23] It is now known that the alterations in the translated paper were carried out by Lemaître himself.^[11][24]

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 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 the Milky Way

Edwin Hubble did most of his professional astronomical observing work at Mount Wilson Observatory,^[25] home to 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 replaced it.

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 and Milton Humason'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—the 'Hubble flow' is used to refer to the region of space far enough out that the recession velocity is larger than local 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). ^{[citation needed]}

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.^[29] A straight line of positive slope on this diagram is the visual depiction of Hubble's law.

Cosmological constant abandoned

After Hubble's discovery was published, Albert Einstein abandoned his work on the cosmological constant, which he had designed to modify his equations of general relativity to allow them to produce a static solution, which he thought was the correct state of the universe. The Einstein equations in their simplest form model either an expanding or contracting universe, so Einstein's cosmological constant was artificially created to counter the expansion or contraction to get a perfect static and flat universe.^[30] After Hubble's discovery that the universe was, in fact, expanding, Einstein called his faulty assumption that the universe is static his "biggest mistake".^[30] On its own, general relativity could predict the expansion of the universe, which (through observations such as the bending of light by large masses, or the precession of the orbit of Mercury) could be experimentally observed and compared to his theoretical calculations using particular solutions of the equations he had originally formulated.

In 1931, Einstein went to Mount Wilson Observatory to thank Hubble for providing the observational basis for modern cosmology.^[31]

The cosmological constant has regained attention in recent decades as a hypothetical explanation for dark energy.^[32]

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:

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

where

• ${\displaystyle v}$ is the recessional velocity, 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 and which can be expressed in terms of the scale factor) 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.
• ${\displaystyle D}$ is the proper distance (which can change over time, unlike the comoving distance, which is constant) 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 r_HS , objects recede at a rate faster than the speed of light (See Uses of the proper distance for a discussion of the significance of this):

$${\displaystyle r_{\text{HS}}={\frac {c}{H_{0}}}\ .}$$

Since the Hubble "constant" is a constant only in space, not in time, 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.^[26] Current evidence suggests that the expansion of the universe is accelerating (see Accelerating universe), meaning that for any given galaxy, the recession velocity dD/dt is increasing over time as the galaxy moves to greater and greater distances; however, the Hubble parameter is actually thought to be decreasing with time, meaning that if we were to look at some fixed distance D and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.^[34]

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.^[35]

Redshift velocity

The redshift z is often 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.^[36] In other words, to determine the redshift velocity v_rs, the relation:

$${\displaystyle v_{\text{rs}}\equiv cz,}$$

is used.^[37][38] 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.^[39]

$${\displaystyle z={\frac {\lambda _{\text{o}}}{\lambda _{\text{e}}}}-1={\sqrt {\frac {1+{\frac {v}{c}}}{1-{\frac {v}{c}}}}}-1\approx {\frac {v}{c}}.}$$

Here, λ_o, λ_e are the observed and emitted wavelengths respectively. The "redshift velocity" v_rs 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.^[40]

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 proper 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:

$${\displaystyle {\frac {D(t)}{D(t_{0})}}={\frac {R(t)}{R(t_{0})}},}$$

where t₀ is some reference time.^[41] If light is emitted from a galaxy at time t_e and received by us at t₀, it is redshifted due to the expansion of space, and this redshift z is simply:

$${\displaystyle z={\frac {R(t_{0})}{R(t_{\text{e}})}}-1.}$$

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

$${\displaystyle v_{\text{r}}=d_{t}D={\frac {d_{t}R}{R}}D.}$$

We now define the Hubble constant as

$${\displaystyle H\equiv {\frac {d_{t}R}{R}},}$$

and discover the Hubble law: