Hubble's law, also known as the Hubble–Lemaître law, is the observation in

physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...

that

galaxies A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...

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 In physics, a redshift is an increase in the wavelength, and corresponding decrease in the frequency and photon energy, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and simultaneous increase in f ...

, a shift of the

light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...

they emit toward the red end of the

visible spectrum The visible spectrum is the portion of the electromagnetic spectrum that is visual perception, visible to the human eye. Electromagnetic radiation in this range of wavelengths is called ''visible light'' or simply light. A typical human eye wil ...

. Hubble's law is considered the first observational basis for the

expansion of the universe The expansion of the universe is the increase in distance between any two given gravitationally unbound parts of the observable universe with time. It is an intrinsic expansion whereby the scale of space itself changes. The universe does not exp ...

, and today it serves as one of the pieces of evidence most often cited in support of the

Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...

model. The motion of astronomical objects due solely to this expansion is known as the Hubble flow. It is described by the equation , with ''H''₀ the constant of proportionality—the Hubble constant—between the "proper distance" ''D'' to a galaxy, which can change over time, unlike the

comoving distance In standard cosmology, comoving distance and proper distance are two closely related distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to where a distant object would be at a spec ...

, and its speed of separation ''v'', i.e. the

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

of proper distance with respect to the

cosmological time Cosmic time, or cosmological time, is the time coordinate commonly used in the Big Bang models of physical cosmology. Such time coordinate may be defined for a homogeneous, expanding universe so that the universe has the same density everywhere at ...

coordinate. (See ' for some 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 away, and its value is about . However, the SI unit of ''H''₀ is simply s⁻¹, and the SI unit for the reciprocal of ''H''₀ is simply the second. The reciprocal of ''H''₀ is known as the

Hubble time Hubble's law, also known as the Hubble–Lemaître law, 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 ...

. The Hubble constant can also be interpreted as the relative rate of expansion. In this form ''H''₀ = 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 Edwin Powell Hubble (November 20, 1889 – September 28, 1953) was an Americans, American astronomer. He played a crucial role in establishing the fields of extragalactic astronomy and observational cosmology. Hubble proved that many objects ...

, the notion of the universe expanding at a calculable rate was first derived from

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

equations in 1922 by

Alexander Friedmann Alexander Alexandrovich Friedmann (also spelled Friedman or Fridman ; russian: Алекса́ндр Алекса́ндрович Фри́дман) (June 16 .S. 4 1888 – September 16, 1925) was a Russian and Soviet physicist and mathematician ...

. Friedmann published a set of equations, now known as the

Friedmann equations The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedman ...

, showing that the universe might be expanding, and presenting the expansion speed if that were the case. Then

Georges Lemaître Georges Henri Joseph Édouard Lemaître ( ; ; 17 July 1894 – 20 June 1966) was a Belgian Catholic priest, theoretical physicist, mathematician, astronomer, and professor of physics at the Catholic University of Louvain. He was the first to th ...

, 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. Hubble inferred the recession velocity of the objects from their

s, many of which were earlier measured and related to velocity by

Vesto Slipher Vesto Melvin Slipher (; November 11, 1875 – November 8, 1969) was an American astronomer who performed the first measurements of radial velocities for galaxies. He was the first to discover that distant galaxies are redshifted, thus providing th ...

in 1917. Though the Hubble constant ''H''₀ is roughly constant in the velocity-distance space 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.

Discovery

A decade before Hubble made his observations, a number of

physicists A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate caus ...

and

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

s had established a consistent theory of an expanding universe by using

Einstein's field equations In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...

. Applying the most general principles to the nature of the

universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. Acc ...

yielded a

dynamic Dynamics (from Greek δυναμικός ''dynamikos'' "powerful", from δύναμις ''dynamis'' "power") or dynamic may refer to: Physics and engineering * Dynamics (mechanics) ** Aerodynamics, the study of the motion of air ** Analytical dynam ...

solution that conflicted with the then-prevalent notion of a

static universe In cosmology, a static universe (also referred to as stationary, infinite, static infinite or static eternal) is a cosmological model in which the universe is both spatially and temporally infinite, and space is neither expanding nor contract ...

Slipher's observations

In 1912,

Vesto M. Slipher Vesto Melvin Slipher (; November 11, 1875 – November 8, 1969) was an American astronomer who performed the first measurements of radial velocities for galaxies. He was the first to discover that distant galaxies are redshifts, redshifted, thus p ...

measured the first

Doppler shift The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...

of a "

spiral nebula Spiral galaxies form a class of galaxy originally described by Edwin Hubble in his 1936 work ''The Realm of the Nebulae''highly controversial whether or not these

nebulae A nebula ('cloud' or 'fog' in Latin; pl. nebulae, nebulæ or nebulas) is a distinct luminescent part of interstellar medium, which can consist of ionized, neutral or molecular hydrogen and also cosmic dust. Nebulae are often star-forming region ...

were "island universes" outside our Milky Way.

FLRW equations

In 1922,

derived his

from

, 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 In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a '' scale factor'' that is the same in all directions. The result of uniform scaling is similar ...

and can be considered as a

scale invariant In physics, mathematics and statistics, scale invariance is a feature of objects or laws that do not change if scales of length, energy, or other variables, are multiplied by a common factor, and thus represent a universality. The technical term ...

form of the

proportionality constant In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...

of Hubble's law.

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 Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...

and

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...

. This idea of an expanding spacetime would eventually lead to the

and

Steady State In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...

theories of cosmology.

Lemaître's equation

In 1927, two years before Hubble published his own article, the Belgian priest and astronomer

was the first to publish research deriving what is now known as Hubble's law. According to the Canadian astronomer

Sidney van den Bergh Sidney Van den Bergh, OC, FRS (born 20 May 1929 in Wassenaar) is a retired Dutch-Canadian astronomer. He showed an interest in science from an early age, learning to read with books on astronomy. In addition to being interested in astronomy. ...

, "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." It is now known that the alterations in the translated paper were carried out by Lemaître himself.

Shape of the universe

Before the advent of modern cosmology, there was considerable talk about the size and

shape of the universe The shape of the universe, in physical cosmology, is the local and global geometry of the universe. The local features of the geometry of the universe are primarily described by its curvature, whereas the topology of the universe describes gen ...

. In 1920, the

Shapley–Curtis debate The Great Debate, also called the Shapley–Curtis Debate, was held on 26 April 1920 at the National Museum of Natural History, Smithsonian Museum of Natural History, between the astronomers Harlow Shapley and Heber Curtis. It concerned the natu ...

took place between

Harlow Shapley Harlow Shapley (November 2, 1885 – October 20, 1972) was an American scientist, head of the Harvard College Observatory (1921–1952), and political activist during the latter New Deal and Fair Deal. Shapley used Cepheid variable stars to estim ...

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 The Mount Wilson Observatory (MWO) is an astronomical observatory in Los Angeles County, California, United States. The MWO is located on Mount Wilson, a peak in the San Gabriel Mountains near Pasadena, northeast of Los Angeles. The observat ...

, home to the world's most powerful telescope at the time. His observations of

Cepheid variable A Cepheid variable () is a type of star that pulsates radially, varying in both diameter and temperature and producing changes in brightness with a well-defined stable period and amplitude. A strong direct relationship between a Cepheid varia ...

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

''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 Milton La Salle Humason (August 19, 1891 – June 18, 1972) was an American astronomer. He was born in Dodge Center, Minnesota. Biography Humason dropped out of school and had no formal education past the age of 14. Because he loved the m ...

'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 Peculiar motion or peculiar velocity refers to the velocity of an object relative to a ''rest frame'' — usually a frame in which the average velocity of some objects is zero. Galactic astronomy In galactic astronomy, peculiar motion refers to ...

—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 The cosmic distance ladder (also known as the extragalactic distance scale) is the succession of methods by which astronomers determine the distances to celestial objects. A ''direct'' distance measurement of an astronomical object is possible o ...

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. Today, in the context of general relativity, velocity between distant objects depends on the choice of coordinates used, and therefore, the redshift can be equally described as a Doppler shift or a cosmological shift (or gravitational) due to the expanding space, or some combination of the two.

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.

Cosmological constant abandoned

After Hubble's discovery was published,

Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ...

abandoned his work on the

cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field equ ...

, 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 generated 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. After Hubble's discovery that the universe was, in fact, expanding, Einstein called his faulty assumption that the universe is static his "biggest mistake". On its own, general relativity could predict the expansion of the universe, which (through

observations Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...

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 made a trip to Mount Wilson Observatory to thank Hubble for providing the observational basis for modern cosmology. The cosmological constant has regained attention in recent decades as a hypothetical explanation for

dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...

Interpretation

The discovery of the linear relationship between redshift and distance, coupled with a supposed linear relation between

recessional velocity Recessional velocity is the rate at which an extragalactic astronomical object recedes (becomes more distant) from an observer as a result of the expansion of the universe. It can be measured by observing the wavelength shifts of spectral lines e ...

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. * ''H''₀ is Hubble's constant and corresponds to the value of

H

(often termed the Hubble parameter which is a value that is time dependent and which can be expressed in terms of the

) 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 In Big Bang, standard cosmology, comoving distance and proper distance are two closely related distance measures (cosmology), distance measures used by cosmologists to define distances between objects. ''Proper distance'' roughly corresponds to wh ...

. *

D

is the proper distance (which can change over time, unlike the

, which is constant) from the

galaxy A galaxy is a system of stars, stellar remnants, interstellar gas, dust, dark matter, bound together by gravity. The word is derived from the Greek ' (), literally 'milky', a reference to the Milky Way galaxy that contains the Solar System. ...

to the observer, measured in

mega Mega or MEGA may refer to: Science * mega-, a metric prefix denoting 106 * Mega (number), a certain very large integer in Steinhaus–Moser notation * "mega-" a prefix meaning "large" that is used in taxonomy * Gravity assist, for ''Moon-Eart ...

parsec The parsec (symbol: pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to or (au), i.e. . The parsec unit is obtained by the use of parallax and trigonometry, and ...

s (Mpc), in the 3-space defined by given

. (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 In cosmology, a Hubble volume (named for the astronomer Edwin Hubble) or Hubble sphere, subluminal sphere, causal sphere and sphere of causality is a spherical region of the observable universe surrounding an observer beyond which objects recede ...

''r''_HS , objects recede at a rate faster than the

speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

(''See'' Uses of the proper distance for a discussion of the significance of this): :

r_\text = \frac \ .

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. Current evidence suggests that the expansion of the universe is accelerating (''see''

Accelerating universe Observations show that the expansion of the universe is accelerating, such that the velocity at which a distant galaxy recedes from the observer is continuously increasing with time. The accelerated expansion of the universe was discovered during ...

), 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.

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'' 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 The Doppler effect or Doppler shift (or simply Doppler, when in context) is the change in frequency of a wave in relation to an observer who is moving relative to the wave source. It is named after the Austrian physicist Christian Doppler, who d ...

(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 ''v''_rs, the relation: :

v_\text \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-1 = \sqrt-1 \approx \frac.

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.

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 Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...

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: :

\frac  = \frac,

where ''t''₀ is some reference time. If light is emitted from a galaxy at time ''t''_e and received by us at ''t''₀, it is redshifted due to the expansion of space, and this redshift ''z'' is simply: :

z = \frac  - 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: :

v_\text = d_tD = \frac  D.

We now define the Hubble constant as :

H \equiv \frac,

and discover the Hubble law: :

v_\text = H 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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...

expansion: :

z = \frac  - 1 \approx \frac  -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_\text)H(t_0) \approx \frac  H(t_0),

or :

cz \approx D H(t_0) = v_r.

According to this approach, the relation 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: 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 The "finger of God" ( ''’etsba‘ ’Ĕlōhîm'') is a phrase used in the Bible. In Exodus 8:16–20 it is used during the plagues of Egypt by the Egyptian magicians. In Exodus 31:18 and Deuteronomy 9:10 it refers to the method by which the T ...

effect is one result of this phenomenon. 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.

Time-dependence of Hubble parameter

The parameter

H

is commonly called the "''Hubble constant''", but that is a misnomer since it is constant in space only at a fixed time; it varies with time in nearly all cosmological models, and all observations of far distant objects are also observations into the distant past, when the “constant” had a different value. The "''Hubble parameter''" is a more correct term, with

H_0

denoting the present-day value. Another common source of confusion is that the accelerating universe does ''not'' imply that the Hubble parameter is actually increasing with time; since

H(t) \equiv \dot(t)/a(t)

, in most accelerating models

a

increases relatively faster than

\dot

, so H decreases with time. (The recession velocity of one chosen galaxy does increase, but different galaxies passing a sphere of fixed radius cross the sphere more slowly at later times.) On defining the dimensionless

deceleration parameter The deceleration parameter ''q'' in cosmology is a dimensionless measure of the cosmic acceleration of the expansion of space in a Friedmann–Lemaître–Robertson–Walker universe. It is defined by: :q \ \stackrel\ -\frac where a is the sca ...

q \equiv - \frac

, it follows that :

\frac = -H^2 (1+q)

From this it is seen that the Hubble parameter is decreasing with time, unless

q < -1

; the latter can only occur if the universe contains

phantom energy Phantom energy is a hypothetical form of dark energy satisfying the equation of state with w < -1. It possesses negative kinetic energy, and predic ...

, regarded as theoretically somewhat improbable. However, in the standard Lambda cold dark matter model (Lambda-CDM or ΛCDM model),

q

will tend to −1 from above in the distant future as the cosmological constant becomes increasingly dominant over matter; this implies that

H

will approach from above to a constant value of ≈ 57 (km/s)/Mpc, and the scale factor of the universe will then grow exponentially in time.

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 Cartesian means of or relating to the French philosopher René Descartes—from his Latinized name ''Cartesius''. It may refer to: Mathematics *Cartesian closed category, a closed category in category theory *Cartesian coordinate system, modern ...

/Newtonian coordinate space, which, considered as a

metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...

, is entirely homogeneous and isotropic (properties do not vary with location or direction). Simply stated the theorem is this: 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

). 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.

Ultimate fate and age of the universe

The value of the Hubble parameter changes over time, either increasing or decreasing depending on the value of the so-called

q

, which is defined by :

q = -\left(1+\frac\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 Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...

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/(3''H''). The discovery in 1998 that ''q'' is apparently negative means that the universe could actually be older than 1/''H''. However, estimates of the

age of the universe In physical cosmology, the age of the universe is the time elapsed since the Big Bang. Astronomers have derived two different measurements of the age of the universe: a measurement based on direct observations of an early state of the universe, ...

are very close to 1/''H''.

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 Olbers's paradox, also known as the dark night sky paradox, is an argument in astrophysics and physical cosmology that says that the darkness of the night sky conflicts with the assumption of an infinite and eternal static universe. In the hypo ...

: If the universe were

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music * Infinite (group), a South Korean boy band *''Infinite'' (EP), debut EP of American m ...

in size,

static Static may refer to: Places *Static Nunatak, a nunatak in Antarctica United States * Static, Kentucky and Tennessee *Static Peak, a mountain in Wyoming **Static Peak Divide, a mountain pass near the peak Science and technology Physics *Static el ...

, and filled with a uniform distribution of

star A star is an astronomical object comprising a luminous spheroid of plasma (physics), plasma held together by its gravity. The List of nearest stars and brown dwarfs, nearest star to Earth is the Sun. Many other stars are visible to the naked ...

s, then every line of sight in the sky would end on a star, and the sky would be as

bright Bright may refer to: Common meanings *Bright, an adjective meaning giving off or reflecting illumination; see Brightness *Bright, an adjective meaning someone with intelligence People * Bright (surname) * Bright (given name) *Bright, the stage na ...

as the surface of a star. However, the night sky is largely dark. Since the 17th century, astronomers and other thinkers have proposed many possible ways to resolve this paradox, but the currently accepted resolution depends in part on the Big Bang theory, and in part on the Hubble expansion: In a universe that exists for a finite amount of time, only the light of a finite number of stars has had enough time to reach us, and the paradox is resolved. Additionally, in an expanding universe, distant objects recede from us, which causes the light emanated from them to be redshifted and diminished in brightness by the time we see it.

Dimensionless Hubble constant

Instead of working with Hubble's constant, a common practice is to introduce the dimensionless Hubble constant, usually denoted by ''h'' and commonly referred to as "little h", then to write Hubble's constant ''H''₀ as ''h'' × 100 km⋅ s⁻¹⋅ Mpc⁻¹, all the relative uncertainty of the true value of ''H''₀ being then relegated to ''h''. The dimensionless Hubble constant is often used when giving distances that are calculated from

''z'' using the formula . Since ''H''₀ is not precisely known, the distance is expressed as: :

cz/H_0\approx(2998\times z)\texth^

In other words, one calculates 2998×z and one gives the units as

\texth^

h^\text.

Occasionally a reference value other than 100 may be chosen, in which case a subscript is presented after ''h'' to avoid confusion; e.g. h₇₀ denotes

H_0 = 70 \, h_ \, \mathrm

, which implies

h_ = h / 0.7

. This should not be confused with the dimensionless value of Hubble's constant, usually expressed in terms of Planck units, obtained by multiplying ''H''₀ by 1.75 × 10⁻⁶³ (from definitions of parsec and ''t''_P), for example for ''H''₀ = 70, a Planck unit version of 1.2 × 10⁻⁶¹ is obtained.

Acceleration of the expansion

A value for

q

measured from

standard candle The cosmic distance ladder (also known as the extragalactic distance scale) is the succession of methods by which astronomers determine the distances to celestial objects. A ''direct'' distance measurement of an astronomical object is possible o ...

observations of

Type Ia supernova A Type Ia supernova (read: "type one-A") is a type of supernova that occurs in binary systems (two stars orbiting one another) in which one of the stars is a white dwarf. The other star can be anything from a giant star to an even smaller white ...

e, 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, as mentioned above in the Interpretation section; see the articles on

and the

ΛCDM model The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a Parameter#Modelization, parameterization of the Big Bang physical cosmology, cosmological model in which the universe contains three major components: first, a cosmological constant de ...

Derivation of the Hubble parameter

Start with the

Friedmann equation The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedma ...

: :

H^2 \equiv \left(\frac\right)^2 = \frac\rho - \frac+ \frac,

where

H

is the Hubble parameter,

a

is the

, ''G'' is the

gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...

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,

where

\rho_

is the density of matter today. From the Friedmann equation and thermodynamic principles we know for non-relativistic 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 The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann ...

for

\Omega_m

) :

\rho_c = \frac;

\Omega_m \equiv \frac = \frac\rho_;

therefore: :

\rho=\frac.

Also, by definition, :

\Omega_k \equiv \frac

\Omega_ \equiv \frac,

where the subscript nought refers to the values today, and

a_0=1

. Substituting all of this into the Friedmann 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)^ + \Omega_k (1+z)^ + \Omega_ \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_(a),

where

\rho_

is the mass density of the dark energy. By definition, an equation of state in cosmology is

P=w\rho c^2

, and if this is substituted into the fluid equation, which describes how the mass density of the universe evolves with time, then :

\dot+3\frac\left(\rho+\frac\right)=0;

\frac=-3\frac(1+w).

If ''w'' is constant, then :

\ln=-3(1+w)\ln;

implying: :

\rho=a^.

Therefore, for dark energy with a constant equation of state ''w'',

\rho_(a)= \rho_a^

. If this is substituted into the Friedman equation in a similar way as before, but this time set

k=0

, which assumes a spatially flat universe, then (see

) :

H^2(z)= H_0^2 \left( \Omega_m (1+z)^ + \Omega_(1+z)^ \right).

If the dark energy derives from a cosmological constant such as that introduced by Einstein, it can be shown that

w=-1

. The equation then reduces to the last equation in the matter-dominated universe section, with

\Omega_k

set to zero. In that case the initial dark energy density

\rho_

is given by :

\rho_ =  \frac

and

\Omega_=\Omega_.

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

\rho_(a)= \rho_e^,

and to solve this,

w(a)

must be parametrized, for example if

w(a)=w_0+w_a(1-a)

, giving :

H^2(z)= H_0^2 \left( \Omega_m a^ + \Omega_a^e^ \right).

Other ingredients have been formulated recently.

Units derived from the Hubble constant

Hubble time

The Hubble constant

H_0

has units of inverse time; the Hubble time ''t_H'' is simply defined as the inverse of the Hubble constant, i.e. :

t_H \equiv \frac = \frac = 4.55\times 10^ \mathrm  = 14.4 \text.

This is slightly different from the age of the universe which is approximately 13.8 billion years. The Hubble time is the age it would have had if the expansion had been linear, and it is different from the real age of the universe because the expansion is not linear; they are related by a dimensionless factor which depends on the mass-energy content of the universe, which is around 0.96 in the standard

. We currently appear to be approaching a period where the expansion of the universe is exponential due to the increasing dominance of

vacuum energy Vacuum energy is an underlying background energy that exists in space throughout the entire Universe. The vacuum energy is a special case of zero-point energy that relates to the quantum vacuum. The effects of vacuum energy can be experimental ...

. In this regime, the Hubble parameter is constant, and the universe grows by a factor ''e'' each Hubble time: :

H \equiv \frac = \textrm \quad \Longrightarrow \quad a \propto e^ = e^

Likewise, the generally accepted value of 2.27 Es⁻¹ means that (at the current rate) the universe would grow by a factor of

e^

in one

exasecond An order of magnitude of time is usually a decimal prefix or decimal order-of-magnitude quantity together with a base unit of time, like a microsecond or a million years. In some cases, the order of magnitude may be implied (usually 1), like a ...

. Over long periods of time, the dynamics are complicated by general relativity, dark energy,

inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...

, etc., as explained above.

Hubble length

The Hubble length or Hubble distance is a unit of distance in cosmology, defined as

cH_0^

— the speed of light multiplied by the Hubble time. It is equivalent to 4,420 million parsecs or 14.4 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.) The Hubble distance would be the distance between the Earth and the galaxies which are ''currently'' receding from us at the speed of light, as can be seen by substituting

D=cH_0^

into the equation for Hubble's law, .

Hubble volume

The Hubble volume is sometimes defined as a volume of the universe with a comoving size of

cH_0^.

The exact definition varies: it is sometimes defined as the volume of a sphere with radius

cH_0^,

or alternatively, a cube of side

cH_0^.

Some cosmologists even use the term Hubble volume to refer to the volume of the

observable universe The observable universe is a ball-shaped region of the universe comprising all matter that can be observed from Earth or its space-based telescopes and exploratory probes at the present time, because the electromagnetic radiation from these obj ...

, although this has a radius approximately three times larger.

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 them by some other method than Hubble's law. This approach forms part of the

for measuring distances to extragalactic objects. Uncertainties in the physical assumptions used to determine these distances have caused varying estimates of the Hubble constant.

Hubble tension

Multiple methods have been used to determine the Hubble constant. "Late universe" measurements using calibrated distance ladder techniques have converged on a value of approximately . Since 2000, "early universe" techniques based on measurements of the cosmic microwave background have become available, and these agree on a value near . (This is accounting for the change in the expansion rate since the early universe, so is comparable to the first number.) As techniques have improved, the estimated measurement uncertainties have shrunk, but the range of measured values has not, to the point that the disagreement is now highly

statistically significant In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...

. This discrepancy is called the Hubble tension. In December 2021, ''National Geographic'' reported that the cause of the Hubble tension discrepancy is not known. However, if the

cosmological principle In modern physical cosmology, the cosmological principle is the notion that the spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale, since the forces are expected to act uniformly throu ...

fails (see " Violations of the cosmological principle" in the "

Lambda-CDM model The ΛCDM (Lambda cold dark matter) or Lambda-CDM model is a parameterization of the Big Bang cosmological model in which the universe contains three major components: first, a cosmological constant denoted by Lambda (Greek Λ) associated with d ...

" article), then the existing interpretations of the Hubble constant and the Hubble tension have to be revised, which might resolve the Hubble tension. One possibility is that the Hubble tension is caused by the

KBC Void The KBC Void (or Local Hole) is an immense, comparatively empty region of space, named after astronomers Ryan Keenan, Amy Barger, and Lennox Cowie, who studied it in 2013. The existence of a local underdensity has been the subject of many pieces o ...

, as measuring galactic

supernovae A supernova is a powerful and luminous explosion of a star. It has the plural form supernovae or supernovas, and is abbreviated SN or SNe. This transient astronomical event occurs during the last evolutionary stages of a massive star or when a ...

inside a void is predicted by some authors to yield a larger local value for the Hubble constant than cosmological measures of the Hubble constant. However, other work has found no evidence for this in observations, finding the scale of the claimed underdensity to be incompatible with observations which extend beyond its radius. Important deficiencies were subsequently pointed out in this analysis, leaving open the possibility that the Hubble tension is indeed caused by outflow from the KBC Void. Another possibility is that the Hubble tension calls for new physics beyond the ΛCDM model. Moritz Haslbauer and collaborators have proposed

modified Newtonian dynamics Modified Newtonian dynamics (MOND) is a hypothesis that proposes a modification of Newton's law of universal gravitation to account for observed properties of galaxies. It is an alternative to the hypothesis of dark matter in terms of explaining ...

as a possible solution to the Hubble tension, while

Marc Kamionkowski Marc Kamionkowski (born 1965) is an American theoretical physicist and currently the William R. Kenan, Jr. Professor of Physics and Astronomy at Johns Hopkins University. His research interests include particle physics, dark matter, inflation (c ...

and collaborators have proposed an early dark energy model as a possible solution to the Hubble tension. Others have suggested that at least a portion of the tension could be a result of underestimation of uncertainties in data.

Earlier measurement and discussion approaches

The observations of astronomer

Walter Baade Wilhelm Heinrich Walter Baade (March 24, 1893 – June 25, 1960) was a German astronomer who worked in the United States from 1931 to 1959. Biography The son of a teacher, Baade finished school in 1912. He then studied maths, physics and astro ...

led him to define distinct "

populations Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...

" for stars (Population I and Population II). The same observations led him to discover that there are two types of Cepheid variable stars. Using this discovery he recalculated the size of the known universe, doubling the previous calculation made by

Hubble The Hubble Space Telescope (often referred to as HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the first space telescope, but it is one of the largest and most versa ...

in 1929. He announced this finding to considerable astonishment at the 1952 meeting of the

International Astronomical Union The International Astronomical Union (IAU; french: link=yes, Union astronomique internationale, UAI) is a nongovernmental organisation with the objective of advancing astronomy in all aspects, including promoting astronomical research, outreac ...

in Rome. For most of the second half of the 20th century, the value of

H_0

was estimated to be between 50 and . The value of the Hubble constant was the topic of a long and rather bitter controversy between

Gérard de Vaucouleurs Gérard Henri de Vaucouleurs (25 April 1918 – 7 October 1995) was a French astronomer. Life and career Born in Paris, he had an early interest in amateur astronomy and received his undergraduate degree in 1939 at the Sorbonne in that city. ...

, who claimed the value was around 100, and

Allan Sandage Allan Rex Sandage (June 18, 1926 – November 13, 2010) was an American astronomer. He was Staff Member Emeritus with the Carnegie Observatories in Pasadena, California. He determined the first reasonably accurate values for the Hubble consta ...

, who claimed the value was near 50. In one demonstration of vitriol shared between the parties, when Sandage and

Gustav Andreas Tammann Gustav Andreas Tammann (24 July 1932 – 6 January 2019) was a German astronomer and academic. He served as director of the Astronomical Institute of the University of Basel; as a member of the European Space Agency Space Telescope Advisory Team ...

(Sandage’s research colleague) formally acknowledged the shortcomings of confirming the systematic error of their method in 1975, Vaucouleurs responded “It is unfortunate that this sober warning was so soon forgotten and ignored by most astronomers and textbook writers”. In 1996, a debate moderated by

John Bahcall John Norris Bahcall (December 30, 1934 – August 17, 2005) was an American astrophysicist, best known for his contributions to the solar neutrino problem, the development of the Hubble Space Telescope and for his leadership and development of th ...

between Sidney van den Bergh and

Gustav Tammann Gustav Andreas Tammann (24 July 1932 – 6 January 2019) was a German astronomer and academic. He served as director of the Astronomical Institute of the University of Basel; as a member of the European Space Agency Space Telescope Advisory Tea ...

was held in similar fashion to the earlier Shapley–Curtis debate over these two competing values. This previously wide variance in estimates was partially resolved with the introduction of the ΛCDM model of the universe in the late 1990s. 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 In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all spac ...

radiation, and optical surveys all gave a value of around 70 for the constant.

21st century measurements

More recent measurements from the

Planck mission ''Planck'' was a space observatory operated by the European Space Agency (ESA) from 2009 to 2013, which mapped the anisotropies of the cosmic microwave background (CMB) at microwave and infrared frequencies, with high sensitivity and small angu ...

published in 2018 indicate a lower value of , although, even more recently, in March 2019, a higher value of has been determined using an improved procedure involving the Hubble Space Telescope. The two measurements disagree at the 4.4 ''σ'' level, beyond a plausible level of chance. The resolution to this disagreement is an ongoing area of active research. In October 2018, scientists presented a new third way (two earlier methods, one based on redshifts and another on the cosmic distance ladder, gave results that do not agree), using information from

gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...

events (especially those involving the merger of neutron stars, like

GW170817 GW 170817 was a gravitational wave (GW) signal observed by the LIGO and Virgo detectors on 17 August 2017, originating from the shell elliptical galaxy . The signal was produced by the last minutes of a binary pair of neutron stars' inspir ...

), of determining the Hubble constant. In July 2019, astronomers reported that a new method to determine the Hubble constant, and resolve the discrepancy of earlier methods, has been proposed based on the mergers of pairs of

neutron star A neutron star is the collapsed core of a massive supergiant star, which had a total mass of between 10 and 25 solar masses, possibly more if the star was especially metal-rich. Except for black holes and some hypothetical objects (e.g. white ...

s, following the detection of the neutron star merger of GW170817, an event known as a

dark siren A dark siren is a gravitational wave event without electromagnetic counterparts. They are significant as part of a method used by astronomers to establish the Hubble constant, first suggested by Bernard F. Schutz in 1986. Dark sirens are caused by ...

. Their measurement of the Hubble constant is (km/s)/Mpc. Also in July 2019, astronomers reported another new method, using data from the

Hubble Space Telescope The Hubble Space Telescope (often referred to as HST or Hubble) is a space telescope that was launched into low Earth orbit in 1990 and remains in operation. It was not the first space telescope, but it is one of the largest and most versa ...

and based on distances to red giant stars calculated using the

tip of the red-giant branch Tip of the red-giant branch (TRGB) is a primary distance indicator used in astronomy. It uses the luminosity of the brightest red-giant-branch stars in a galaxy as a standard candle to gauge the distance to that galaxy. It has been used in conjun ...

(TRGB) distance indicator. Their measurement of the Hubble constant is (km/s)/Mpc. In February 2020, the Megamaser Cosmology Project published independent results that confirmed the distance ladder results and differed from the early-universe results at a statistical significance level of 95%. In July 2020, measurements of the cosmic background radiation by the

Atacama Cosmology Telescope The Atacama Cosmology Telescope (ACT) is a cosmological millimeter-wave telescope located on Cerro Toco in the Atacama Desert in the north of Chile. ACT makes high-sensitivity, arcminute resolution, microwave-wavelength surveys of the sky in ord ...

predict that the Universe should be expanding more slowly than is currently observed. Hubbleconstants color

References

Bibliography

* * *

External links

NASA's WMAP - Big Bang Expansion: the Hubble Constant

The Hubble Diagram Project

Coming to terms with different Hubble Constants
(

Forbes ''Forbes'' () is an American business magazine owned by Integrated Whale Media Investments and the Forbes family. Published eight times a year, it features articles on finance, industry, investing, and marketing topics. ''Forbes'' also re ...

; 3 May 2019) * {{DEFAULTSORT:Hubble's Law

Law Law is a set of rules that are created and are enforceable by social or governmental institutions to regulate behavior,Robertson, ''Crimes against humanity'', 90. with its precise definition a matter of longstanding debate. It has been vario ...

Large-scale structure of the cosmos Physical cosmology Equations of astronomy

Discovery

Slipher's observations

FLRW equations

Lemaître's equation

Shape of the universe

Cepheid variable stars outside the Milky Way

Combining redshifts with distance measurements

Hubble diagram

Cosmological constant abandoned

Interpretation

Redshift velocity and recessional velocity

Redshift velocity

Recessional velocity

Observability of parameters

Expansion velocity vs. relative velocity

Time-dependence of Hubble parameter

Idealized Hubble's law

Ultimate fate and age of the universe

Olbers' paradox

Dimensionless Hubble constant

Acceleration of the expansion

Derivation of the Hubble parameter

Matter-dominated universe (with a cosmological constant)

Matter- and dark energy-dominated universe

Units derived from the Hubble constant

Hubble time

Hubble length

Hubble volume

Determining the Hubble constant

Hubble tension

Earlier measurement and discussion approaches

21st century measurements

See also

References

Bibliography

Further reading

External links