HUBBLE’S LAW

Hubble’s law
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 a galaxy is from the Earth, the faster it moves away. A
galaxy’s recessional velocity is typically determined by measuring its redshift, a shift in the
frequency of light emitted by the galaxy.
The discovery of Hubble’s law is attributed to work published by Edwin Hubble in
1929,[2][3][4] but the notion of the universe expanding at a calculable rate was first derived
from general relativity equations in 1922 by Alexander Friedmann. The Friedmann
equations showed the universe might be expanding, and presented the expansion speed if
that were the case.[5] Before Hubble, astronomer Carl Wilhelm Wirtz had, in 1922[6] and
1924,[7] deduced with his own data that galaxies that appeared smaller and dimmer had
larger redshifts and thus that more distant galaxies recede faster from the observer. In 1927,
Georges Lemaître concluded that the universe might be expanding by noting the
proportionality of the recessional velocity of distant bodies to their respective distances. He
estimated a value for this ratio, which—after Hubble confirmed cosmic expansion and
determined a more precise value for it two years later—became known as the Hubble
constant.[8][9][10][11][12] 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]
Hubble’s law is considered the first observational basis for the expansion of the universe, and is one of the pieces of evidence most often cited in
support of the Big Bang model.[8][17] The motion of astronomical objects due solely to this expansion is known as the Hubble flow.[18] 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
cosmic time coordinate.[a] 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.[19][20]
The Hubble constant is most frequently quoted in km/s/Mpc, which gives the speed of a galaxy 1 megaparsec (3.09×1019 km) away as 70 km/s.
Simplifying the units of the generalized form reveals that H0 specifies a frequency (SI unit: s−1), leading the reciprocal of H0 to be known as the
Hubble time (14.4 billion years). The Hubble constant can also be stated as a relative rate of expansion. In this form H0 = 7%/Gyr, meaning that, at
the current rate of expansion, it takes one billion years for an unbound structure to grow by 7%.
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.
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 objects 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.[22][23]
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.[24] 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.
Discovery
Slipher’s observations
FLRW equations
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.”
[25] It is now
known that the alterations in the translated paper were
carried out by Lemaître himself.
[10]
[26]
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.
Three steps to the Hubble constant
[21]
Cepheid variable stars outside the Milky Way
Edwin Hubble did most of his professional astronomical observing work at
telescope at the time. His observations of
Mount Wilson Observatory,
[27] home to the world’s most powerful
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 velocities and distances that appear in Hubble’s law are not directly measured. The
velocities are inferred from the redshift z = ∆λ/λ of radiation and distance is inferred
from brightness. Hubble sought to correlate brightness with 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).
[29]
Hubble diagram
Fit of
estimates for the
redshift velocities to Hubble’s law.
[28] Various
Hubble constant exist.
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.
[30] 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, a
term he had inserted into his
equations of general relativity to coerce them into producing the static solution he previously considered the correct state of the universe. The
Einstein equations in their simplest form model either an expanding or contracting universe, so Einstein introduced the constant to counter
expansion or contraction and lead to a static and flat universe.
[31] After Hubble’s discovery that the universe was, in fact, expanding, Einstein called
his faulty assumption that the universe is static his “greatest mistake”.
[31] 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.
[32]
In 1931, Einstein went to Mount Wilson Observatory to thank Hubble for providing the observational basis for modern cosmology.
[33]
The cosmological constant has regained attention in recent decades as a hypothetical explanation 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:
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 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.
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
A variety of possible recessional velocity vs. redshift
functions including the simple linear relation v = cz; a
variety of possible shapes from theories related to general
relativity; and a curve that does not permit speeds faster
than light in accordance with special relativity. All curves
are linear at low redshifts.
Hubble sphere rHS, objects recede at a rate faster than the
[34]
speed of light (See
Uses of the proper
distance for a discussion of the significance of this):
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.
accelerating (see
[35]
[28] Current evidence suggests that the expansion of the universe is
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.
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 unambiguously acquired from observation. Care is required,
however, in translating these to recessional velocities: for small redshift values, a linear relation of redshift to recessional velocity applies, but more
generally the redshift-distance law is nonlinear, meaning the co-relation must be derived specifically for each given model and epoch.
Redshift velocity
[36]
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 velocities involved are too large to use a non-relativistic formula for Doppler shift).
This redshift velocity can easily exceed the speed of light.
[37] In other words, to determine the redshift velocity vrs , the relation:
is used.
[38]
[39] 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
[40]
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.
[41]
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 their local environments.) In other words:
where t0 is some reference time.
[42] If light is emitted from a galaxy at time te and received by us at t0, it is redshifted due to the expansion of the
universe, and this redshift z is simply:
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:
We now define the Hubble constant as
and discover the Hubble law:
From this perspective, Hubble’s law is a fundamental relation between (i) the recessional velocity associated with the expansion of the universe 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:
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:
or
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 one), 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 traveling towards the observer.
Expansion velocity vs. peculiar 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,
[43] these relative velocities, called peculiar velocities, need to be
accounted for in the application of Hubble’s law. Such peculiar velocities give rise to
redshift-space distortions.
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. “Hubble parameter” is a more correct term, with H0 denoting the present-day value.
, in most accelerating models
Another common source of confusion is that the accelerating universe does not imply that the Hubble parameter is actually increasing with time;
since
increases relatively faster than
, 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 , it follows that
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
energy, regarded as theoretically somewhat improbable.
However, in the standard
phantom
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/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.
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
parameter q, which is defined by
deceleration
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. However,
age of the universe are very close to 1/H.
estimates of the
Olbers’ paradox
The expansion of space summarized by the Big Bang interpretation of Hubble’s law is relevant to the old conundrum known as
the universe were
infinite in size,
static, and filled with a uniform distribution of
the sky would be as
bright as the surface of a star. However, the night sky is largely dark.
Olbers’ paradox: If
stars, then every line of sight in the sky would end on a star, and
[44]
[45]
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 existed 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.
[44]
[45]
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”,
[29] then to write Hubble’s
constant H0 as h × 100 km⋅s
−1⋅
Mpc−1 , all the relative
uncertainty of the true value of H0 being then relegated to h.
[46]
The dimensionless Hubble constant is often used when giving
distances that are calculated from redshift z using the formula
d ≈ c
H0
× z. Since H0 is not precisely known, the distance is
expressed as:
In other words, one calculates 2998 × z and one gives the units as
Mpc h-1 or h-1 Mpc.
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 H0 = 70 h70 (
km/s)/
Mpc, which implies
h70 = 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 H0 by 1.75 ×10−63 (from definitions of
parsec and
tP), for example for H0 = 70, a Planck unit version of
1.2 ×10−61 is obtained.
Acceleration of the expansion
A value for q measured from
The
age and
ultimate fate of the universe can be determined by measuring the
Hubble constant today and extrapolating with the observed value of the
deceleration parameter, uniquely characterized by values of density parameters
(ΩM for
matter and ΩΛ for dark energy).
A closed universe with ΩM > 1 and ΩΛ = 0 comes to an end in a
and is considerably younger than its Hubble age.
Big Crunch
An open universe with ΩM ≤ 1 and ΩΛ = 0 expands forever and has an age
that is closer to its Hubble age. For the accelerating universe with nonzero ΩΛ
that we inhabit, the age of the universe is coincidentally very close to the Hubble
age.
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”
[47] (although the Hubble factor is still decreasing
with time, as mentioned above in the
Interpretation section; see the articles on
dark energy and the ΛCDM model).
Derivation of the Hubble parameter
Start with the
Friedmann equation:
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 Λ is the cosmological constant.
Matter-dominated universe (with a cosmological constant)
If the universe is
matter-dominated, then the mass density of the universe ρ can be taken to include just matter so
where ρm0
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
parameter for Ωm)
density
therefore:
Also, by definition,
where the subscript 0 refers to the values today, and a0 = 1. Substituting all of this into the Friedmann equation at the start of this section and
replacing a with a = 1/(1+z) gives
Matter- and dark energy-dominated universe
equation of state of dark energy. So now:
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
where ρde is the mass density of the dark energy. By definition, an equation of state in cosmology is P = wρc2 , and if this is substituted into the
fluid equation, which describes how the mass density of the universe evolves with time, then
If w is constant, then
implying:
Therefore, for dark energy with a constant equation of state w, . 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
[48]
shape of the universe)
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 Ωk set to zero. In that case the initial dark energy density ρde0 is give