Hubble's law#Hubble time

File:Three steps to the Hubble constant.jpg

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.{{cite journal |last=Slipher |first=V. M. |date=1913 |title=The Radial Velocity of the Andromeda Nebula |journal=Lowell Observatory Bulletin |volume=1 |issue=8 |pages=56–57 |bibcode=1913LowOB...2...56S }}{{Cite journal |last=Slipher |first=V. M. |date=1915 |title=Spectrographic Observations of Nebulae |journal=Popular Astronomy |volume=23 |pages=21–24 |bibcode=1915PA.....23...21S }}

{{Main|Friedmann–Lemaître–Robertson–Walker metric}}

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.{{Cite journal |last=Friedman |first=A. |date=1922 |title=Über die Krümmung des Raumes |journal=Zeitschrift für Physik |language=de |volume=10 |issue=1 |pages=377–386 |bibcode=1922ZPhy...10..377F |doi=10.1007/BF01332580 |s2cid=125190902}} Translated to English in {{Cite journal |last1=Friedmann |first1=A. |date=1999 |title=On the Curvature of Space |journal=General Relativity and Gravitation |volume=31 |issue=12 |pages=1991–2000 |bibcode=1999GReGr..31.1991F |doi=10.1023/A:1026751225741 |s2cid=122950995}} 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.

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."{{cite journal|last1=van den Bergh|first1=Sydney|title=The Curious Case of Lemaître's Equation No. 24|journal=Journal of the Royal Astronomical Society of Canada|volume=105|issue=4|page=151|arxiv=1106.1195|year=2011|bibcode=2011JRASC.105..151V}} It is now known that the alterations in the translated paper were carried out by Lemaître himself.{{cite book|last1=Block|first1=David|title='Georges Lemaitre and Stigler's law of eponymy' in Georges Lemaître: Life, Science and Legacy|date=2012|publisher=Springer|pages=89–96|edition=Holder and Mitton}}

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.

Edwin Hubble did most of his professional astronomical observing work at Mount Wilson Observatory,{{cite journal | last = Sandage | first = Allan | title = Edwin Hubble 1889–1953 | date = December 1989 | journal = Journal of the Royal Astronomical Society of Canada | volume = 83 | issue = 6 | pages = 351–362| bibcode = 1989JRASC..83..351S }} 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.

File:Hubble constant.JPG to Hubble's law.{{cite book |last=Keel |first=W. C. |date=2007 |title=The Road to Galaxy Formation |url=https://books.google.com/books?id=BUgJGypUYF0C&pg=PA7 |pages=7–8 |edition=2nd |publisher=Springer |isbn=978-3-540-72534-3 }} Various estimates for the Hubble constant exist.]]

The velocities and distances that appear in Hubble's law are not directly measured. The velocities are inferred from the redshift {{math|1=z = ∆λ/λ}} of radiation and distance is inferred from brightness. Hubble sought to correlate brightness with parameter {{mvar|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).

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.{{cite journal |last=Kirshner |first=R. P. |date=2003 |title=Hubble's diagram and cosmic expansion |journal=Proceedings of the National Academy of Sciences |volume=101 |issue=1 |pages=8–13 |bibcode=2004PNAS..101....8K |doi=10.1073/pnas.2536799100 |pmid=14695886 |pmc=314128 |doi-access=free }} A straight line of positive slope on this diagram is the visual depiction of Hubble's law.

{{main|Cosmological constant}}

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.{{cite web |title=What is a Cosmological Constant? |url=http://map.gsfc.nasa.gov/universe/uni_accel.html |publisher=Goddard Space Flight Center |access-date=2013-10-17 }} After Hubble's discovery that the universe was, in fact, expanding, Einstein called his faulty assumption that the universe is static his "greatest mistake". 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.{{cite book |last=Isaacson |first=W. |date=2007 |title=Einstein: His Life and Universe |url=https://archive.org/details/einsteinhislifeu0000isaa |url-access=registration |page=[https://archive.org/details/einsteinhislifeu0000isaa/page/n395 354] |publisher=Simon & Schuster |isbn=978-0-7432-6473-0 }}

The cosmological constant has regained attention in recent decades as a hypothetical explanation for dark energy.{{cite web |date=28 November 2007 |title=Einstein's Biggest Blunder? Dark Energy May Be Consistent With Cosmological Constant |url=https://www.sciencedaily.com/releases/2007/11/071127142128.htm |website=Science Daily |access-date=2013-06-02 }}

{{anchor|redshift}}

File:Velocity-redshift.JPG |volume=555 |pages=348–351 |arxiv=astro-ph/0011070 |bibcode=2001AIPC..555..348D |doi=10.1063/1.1363540 |citeseerx=10.1.1.254.1810 |s2cid=118876362 }}]]

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

$$v\; =\; H\_0\; \backslash ,\; D$$

where

• {{mvar|v}} is the recessional velocity, typically expressed in km/s.
• {{math|H₀}} is Hubble's constant and corresponds to the value of {{mvar|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 {{math|0}}. This value is the same throughout the universe for a given comoving time.
• {{mvar|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 {{math|1=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 {{mvar|D}} larger than the radius of the Hubble sphere {{math|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):

$$r\_\backslash text\{HS\}\; =\; \backslash frac\{c\}\{H\_0\}\; \backslash \; .$$

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), meaning that for any given galaxy, the recession velocity {{mvar|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 {{em|fixed}} distance {{mvar|D}} and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.{{cite web|title=Is the universe expanding faster than the speed of light?|url=http://curious.astro.cornell.edu/question.php?number=575|website=Ask an Astronomer at Cornell University|access-date=5 June 2015|archive-url=https://web.archive.org/web/20031123150109/http://curious.astro.cornell.edu/question.php?number=575|archive-date=23 November 2003}}

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.{{cite journal |last=Harrison |first=E. |date=1992 |title=The redshift-distance and velocity-distance laws |journal=The Astrophysical Journal |volume=403 |pages=28–31 | bibcode=1993ApJ...403...28H |doi=10.1086/172179|doi-access=free }}

The redshift {{mvar|z}} is often described as a redshift velocity, which is the recessional velocity that would produce the same redshift {{em|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.{{cite book |last=Madsen |first=M. S. |date=1995 |title=The Dynamic Cosmos |url=https://books.google.com/books?id=_2GeJxVvyFMC&pg=PA35 |page=35 |publisher=CRC Press |isbn=978-0-412-62300-4 }} In other words, to determine the redshift velocity {{math|v_rs}}, the relation:

$$v\_\backslash text\{rs\}\; \backslash equiv\; cz,$$

is used.{{cite book |last1=Dekel |first1=A. |last2=Ostriker |first2=J. P. |date=1999 |title=Formation of Structure in the Universe |url=https://books.google.com/books?id=yAroX6tx-l0C&pg=PA164 |page=164 |publisher=Cambridge University Press |isbn=978-0-521-58632-0 }}{{cite book |last=Padmanabhan |first=T. |date=1993 |title=Structure formation in the universe | url=https://books.google.com/books?id=AJlOVBRZJtIC&pg=PA58 |page=58 |publisher=Cambridge University Press |isbn=978-0-521-42486-8 }} That is, there is {{em|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{{cite book |last=Sartori |first=L. |date=1996 |title=Understanding Relativity |page=163, Appendix 5B |publisher=University of California Press |isbn=978-0-520-20029-6 }}

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

Here, {{math|λ_o}}, {{math|λ_e}} are the observed and emitted wavelengths respectively. The "redshift velocity" {{math|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.{{cite book|last=Sartori |first=L. |date=1996 |title=Understanding Relativity |pages=304–305 |publisher=University of California Press |isbn=978-0-520-20029-6 }}

Suppose {{math|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 {{math|D(t)}} between co-moving points increase proportionally to {{mvar|R}}. (The co-moving points are not moving relative to their local environments.) In other words:

$$\backslash frac\; \{D(t)\}\{D(t\_0)\}\; =\; \backslash frac\{R(t)\}\{R(t\_0)\},$$

where {{math|t₀}} is some reference time.Matts Roos, Introduction to Cosmology If light is emitted from a galaxy at time {{math|t_e}} and received by us at {{math|t₀}}, it is redshifted due to the expansion of the universe, and this redshift {{mvar|z}} is simply:

$$z\; =\; \backslash frac\; \{R(t\_0)\}\{R(t\_\backslash text\{e\})\}\; -\; 1.$$

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

$$v\_\backslash text\{r\}\; =\; d\_tD\; =\; \backslash frac\; \{d\_tR\}\{R\}\; D.$$

We now define the Hubble constant as

$$H\; \backslash equiv\; \backslash frac\{d\_tR\}\{R\},$$

and discover the Hubble law:

$$v\_\backslash text\{r\}\; =\; H\; D.$$

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 {{mvar|z}} approximately by making a Taylor series expansion:

$$z\; =\; \backslash frac\; \{R(t\_0)\}\{R(t\_e)\}\; -\; 1\; \backslash approx\; \backslash frac\; \{R(t\_0)\}\; \{R(t\_0)\backslash left(1+(t\_e-t\_0)H(t\_0)\backslash right)\}-1\; \backslash approx\; (t\_0-t\_e)H(t\_0),$$

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:

$$z\; \backslash approx\; (t\_0-t\_\backslash text\{e\})H(t\_0)\; \backslash approx\; \backslash frac\; \{D\}\{c\}\; H(t\_0),$$

or

$$cz\; \backslash approx\; D\; H(t\_0)\; =\; v\_r.$$

According to this approach, the relation {{math|1=cz = v_r}} is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See velocity-redshift figure.

Strictly speaking, neither {{mvar|v}} nor {{mvar|D}} in the formula are directly observable, because they are properties {{em|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 {{mvar|z}} much less than one), {{mvar|v}} and {{mvar|D}} will not have changed much, and {{mvar|v}} can be estimated using the formula {{math|1= v = zc}} where {{mvar|c}} is the speed of light. This gives the empirical relation found by Hubble.

For distant galaxies, {{mvar|v}} (or {{mvar|D}}) cannot be calculated from {{mvar|z}} without specifying a detailed model for how {{mvar|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: {{math|(1 + z)}} is the factor by which the universe has expanded while the photon was traveling towards the observer.

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,{{cite web |last=Scharping |first=Nathaniel |title=Gravitational Waves Show How Fast The Universe is Expanding |url=http://www.astronomy.com/news/2017/10/gravitational-waves-show-how-fast-the-universe-is-expanding |date=18 October 2017 |website=Astronomy |access-date=18 October 2017 }} 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.

The parameter {{mvar|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 {{math|H{{sub|0}}}} denoting the present-day value.

Another common source of confusion is that the accelerating universe does {{em|not}} imply that the Hubble parameter is actually increasing with time; since {{nowrap|$H(t)\; \backslash equiv\; \backslash dot\{a\}(t)/a(t)$,}} in most accelerating models $a$ increases relatively faster than {{nowrap|$\backslash dot\{a\}$,}} so {{mvar|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 {{nowrap|$q\; \backslash equiv\; -\; \backslash frac\; \{\backslash ddot\{a\}\backslash ,\; a\}\; \{\backslash dot\{a\}^2\}$,}} it follows that

$$\backslash frac\{dH\}\{dt\}\; =\; -H^2\; (1+q)$$

From this it is seen that the Hubble parameter is decreasing with time, unless {{math|q < -1}}; the latter can only occur if the universe contains phantom energy, regarded as theoretically somewhat improbable.

However, in the standard Lambda cold dark matter model (Lambda-CDM or ΛCDM model), {{mvar|q}} will tend to −1 from above in the distant future as the cosmological constant becomes increasingly dominant over matter; this implies that {{mvar|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.

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:

{{blockquote|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 {{em|every}} observer in an expanding universe will see objects receding from them.

Image:Friedmann universes.svg 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 ({{math|Ω_M}} for matter and {{math|Ω_Λ}} for dark energy).
A closed universe with {{math|Ω_M > 1}} and {{math|1= Ω_Λ = 0}} comes to an end in a Big Crunch and is considerably younger than its Hubble age.
An open universe with {{math|Ω_M ≤ 1}} and {{math|1= Ω_Λ = 0}} expands forever and has an age that is closer to its Hubble age. For the accelerating universe with nonzero {{math|Ω_Λ}} that we inhabit, the age of the universe is coincidentally very close to the Hubble age.]]

The value of the Hubble parameter changes over time, either increasing or decreasing depending on the value of the so-called deceleration parameter {{mvar|q}}, which is defined by

$$q\; =\; -\backslash left(1+\backslash frac\{\backslash dot\; H\}\{H^2\}\backslash right).$$

In a universe with a deceleration parameter equal to zero, it follows that {{math|1= H = 1/t}}, where {{mvar|t}} is the time since the Big Bang. A non-zero, time-dependent value of {{mvar|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 {{mvar|q}} was positive, indicating that the expansion is slowing down due to gravitational attraction. This would imply an age of the universe less than {{math|1/H}} (which is about 14 billion years). For instance, a value for {{mvar|q}} of 1/2 (once favoured by most theorists) would give the age of the universe as {{math|2/(3H)}}. The discovery in 1998 that {{mvar|q}} is apparently negative means that the universe could actually be older than {{math|1/H}}. However, estimates of the age of the universe are very close to {{math|1/H}}.

{{Main|Olbers' paradox}}

The expansion of space summarized by the Big Bang interpretation of Hubble's law is relevant to the old conundrum known as Olbers' paradox: If the universe were infinite in size, static, and filled with a uniform distribution of stars, then every line of sight in the sky would end on a star, and the sky would be as bright as the surface of a star. However, the night sky is largely dark.{{cite web |last1=Chase |first1=S. I. |last2=Baez |first2=J. C. |date=2004 |title=Olbers' Paradox |url=http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html |website=The Original Usenet Physics FAQ |access-date=2013-10-17}}{{cite book |last=Asimov |first=I. |date=1974 |chapter=The Black of Night |title=Asimov on Astronomy |publisher=Doubleday |isbn=978-0-385-04111-9 |chapter-url-access=registration |chapter-url=https://archive.org/details/asimovonastronom00isaa |url-access=registration |url=https://archive.org/details/asimovonastronom00isaa }}

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.

Instead of working with Hubble's constant, a common practice is to introduce the dimensionless Hubble constant, usually denoted by {{mvar|h}} and commonly referred to as "little h",{{cite journal |last=Croton |first=Darren J. |date=14 October 2013 |title=Damn You, Little h! (Or, Real-World Applications of the Hubble Constant Using Observed and Simulated Data) |journal=Publications of the Astronomical Society of Australia |volume=30 |url=https://www.cambridge.org/core/journals/publications-of-the-astronomical-society-of-australia/article/damn-you-little-h-or-real-world-applications-of-the-hubble-constant-using-observed-and-simulated-data/EB4B786F4500F897A589C3ED980C17F5 |doi=10.1017/pasa.2013.31 |arxiv=1308.4150 |bibcode=2013PASA...30...52C |s2cid=119257465 |access-date=8 December 2021}} then to write Hubble's constant {{math|H₀}} as {{math|h × 100 km⋅s⁻¹⋅Mpc⁻¹}}, all the relative uncertainty of the true value of {{math|H₀}} being then relegated to {{mvar|h}}.{{cite book |last=Peebles |first=P. J. E. |date=1993 |title=Principles of Physical Cosmology |publisher=Princeton University Press | isbn=978-0-691-07428-3}} The dimensionless Hubble constant is often used when giving distances that are calculated from redshift {{mvar|z}} using the formula {{math|1= d ≈ {{sfrac|c|H₀}} × z}}. Since {{math|H₀}} is not precisely known, the distance is expressed as:

$$cz/H\_0\backslash approx(2998\backslash times\; z)\backslash text\{\; Mpc\; \}h^\{-1\}$$

In other words, one calculates 2998 × {{mvar|z}} and one gives the units as Mpc {{math|h{{sup|-1}}}} or {{math|h{{sup|-1}}}} Mpc.

Occasionally a reference value other than 100 may be chosen, in which case a subscript is presented after {{mvar|h}} to avoid confusion; e.g. {{math|h{{sub|70}}}} denotes {{math|1= H{{sub|0}} = 70 h{{sub|70}}}} {{val||ul=km/s|upl=Mpc}}, which implies {{math|1= h{{sub|70}} = 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 {{math|H₀}} by {{val|1.75|e=-63}} (from definitions of parsec and Planck time), for example for {{math|1= H{{sub|0}} = 70}}, a Planck unit version of {{val|1.2|e=-61}} is obtained.

{{Main|Accelerating expansion of the universe}}

A value for {{mvar|q}} measured from standard candle observations of Type Ia supernovae, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the universe is currently "accelerating"{{cite journal |last=Perlmutter |first=S. |date=2003 |title=Supernovae, Dark Energy, and the Accelerating Universe |url=http://www.supernova.lbl.gov/PhysicsTodayArticle.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.supernova.lbl.gov/PhysicsTodayArticle.pdf |archive-date=2022-10-09 |url-status=live |journal=Physics Today |volume=56 |issue=4 |pages=53–60 |bibcode= 2003PhT....56d..53P |doi=10.1063/1.1580050 |citeseerx=10.1.1.77.7990 }} (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).

{{More citations needed section|date=March 2014}}

Start with the Friedmann equation:

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

where {{mvar|H}} is the Hubble parameter, {{mvar|a}} is the scale factor, {{mvar|G}} is the gravitational constant, {{mvar|k}} is the normalised spatial curvature of the universe and equal to −1, 0, or 1, and {{math|Λ}} is the cosmological constant.

If the universe is matter-dominated, then the mass density of the universe {{mvar|ρ}} can be taken to include just matter so

$$\backslash rho\; =\; \backslash rho\_m(a)\; =\; \backslash frac\{\backslash rho\_\{m\_\{0\}\}\}\{a^3\},$$

where {{math|ρ{{sub|m{{sub|0}}}}}} 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 for {{math|Ω{{sub|m}}}})

$$\backslash begin\{align\}$$

\rho_c &= \frac{3 H_0^2}{8 \pi G}; \\

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

\end{align}

therefore:

$$\backslash rho=\backslash frac\{\backslash rho\_c\; \backslash Omega\_m\}\{a^3\}.$$

Also, by definition,

$$\backslash begin\{align\}$$

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

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

\end{align}

where the subscript {{math|0}} refers to the values today, and {{math|1= a{{sub|0}} = 1}}. Substituting all of this into the Friedmann equation at the start of this section and replacing {{mvar|a}} with {{math|1= a = 1/(1+z)}} gives

$$H^2(z)=\; H\_0^2\; \backslash left(\; \backslash Omega\_m\; (1+z)^\{3\}\; +\; \backslash Omega\_k\; (1+z)^\{2\}\; +\; \backslash Omega\_\{\backslash Lambda\}\; \backslash right).$$

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:

$$\backslash rho\; =\; \backslash rho\_m\; (a)+\backslash rho\_\{de\}(a),$$

where {{mvar|ρ{{sub|de}}}} is the mass density of the dark energy. By definition, an equation of state in cosmology is {{math|1= P = wρc{{sup|2}}}}, and if this is substituted into the fluid equation, which describes how the mass density of the universe evolves with time, then

$$\backslash begin\{align\}$$

\dot{\rho}+3\frac{\dot{a}}{a}\left(\rho+\frac{P}{c^2}\right)=0;\\

\frac{d\rho}{\rho}=-3\frac{da}{a}(1+w).

\end{align}

If {{mvar|w}} is constant, then

$$\backslash ln\{\backslash rho\}=-3(1+w)\backslash ln\{a\};$$

implying:

$$\backslash rho=a^\{-3(1+w)\}.$$

Therefore, for dark energy with a constant equation of state {{mvar|w}}, {{nowrap|$\backslash rho\_\{de\}(a)=\; \backslash rho\_\{de0\}a^\{-3(1+w)\}$.}} If this is substituted into the Friedman equation in a similar way as before, but this time set {{math|1= k = 0}}, which assumes a spatially flat universe, then (see shape of the universe)

$$H^2(z)=\; H\_0^2\; \backslash left(\; \backslash Omega\_m\; (1+z)^\{3\}\; +\; \backslash Omega\_\{de\}(1+z)^\{3(1+w)\}\; \backslash right).$$

If the dark energy derives from a cosmological constant such as that introduced by Einstein, it can be shown that {{math|1= w = −1}}. The equation then reduces to the last equation in the matter-dominated universe section, with {{math|Ω{{sub|k}}}} set to zero. In that case the initial dark energy density {{math|ρ{{sub|de0}}}} is given by{{cite book|last1=Carroll|first1=Sean|title=Spacetime and Geometry: An Introduction to General Relativity|edition=illustrated|date=2004|publisher=Addison-Wesley|location=San Francisco|isbn=978-0-8053-8732-2|page=328|url=https://books.google.com/books?id=1SKFQgAACAAJ}}

$$\backslash begin\{align\}$$

\rho_{de0} &= \frac{\Lambda c^2}{8 \pi G} \,, \\

\Omega_{de} &=\Omega_{\Lambda}.

\end{align}

If dark energy does not have a constant equation-of-state {{mvar|w}}, then

$$\backslash rho\_\{de\}(a)=\; \backslash rho\_\{de0\}e^\{-3\backslash int\backslash frac\{da\}\{a\}\backslash left(1+w(a)\backslash right)\},$$

and to solve this, {{math|w(a)}} must be parametrized, for example if {{math|1= w(a) = w{{sub|0}} + w{{sub|a}}(1−a)}}, giving{{cite journal |last1=Heneka |first1=C. |last2=Amendola |first2=L. |date=2018 |title=General modified gravity with 21cm intensity mapping: simulations and forecast |journal=Journal of Cosmology and Astroparticle Physics |volume=2018 |issue=10 |page=004 |doi=10.1088/1475-7516/2018/10/004 |arxiv=1805.03629 |bibcode=2018JCAP...10..004H |s2cid=119224326 }}

$$H^2(z)=\; H\_0^2\; \backslash left(\; \backslash Omega\_m\; a^\{-3\}\; +\; \backslash Omega\_\{de\}a^\{-3\backslash left(1+w\_0\; +w\_a\; \backslash right)\}e^\{-3w\_a(1-a)\}\; \backslash right).$$

The Hubble constant {{math|H{{sub|0}}}} has units of inverse time; the Hubble time {{mvar|t{{sub|H}}}} is simply defined as the inverse of the Hubble constant,{{cite book |last1=Hawley |first1=John F. |title=Foundations of modern cosmology |last2=Holcomb |first2=Katherine A. |date=2005 |publisher=Oxford University Press |isbn=978-0-19-853096-1 |edition=2nd |location=Oxford [u.a.] |pages=304 |language=en-uk}} i.e.

$$t\_H\; \backslash equiv\; \backslash frac\{1\}\{H\_0\}\; =\; \backslash frac\{1\}\{67.8\; \backslash mathrm\{~(km/s)/Mpc\}\}\; =\; 4.55\backslash times\; 10^\{17\}\; \backslash mathrm\{~s\}\; =\; 14.4\; \backslash text\{\; billion\; years\}.$$

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,{{Cite book |last=Ridpath |first=Ian |title=A Dictionary of Astronomy |publisher=Oxford University Press |year=2012 |isbn=9780199609055 |edition=2nd |page=225 |language=en |doi=10.1093/acref/9780199609055.001.0001}} and it is different from the real age of the universe because the expansion is not linear; it depends on the energy content of the universe (see {{slink||Derivation of the Hubble parameter}}).

We currently appear to be approaching a period where the expansion of the universe is exponential due to the increasing dominance of vacuum energy. In this regime, the Hubble parameter is constant, and the universe grows by a factor E (mathematical constant) each Hubble time:

$$H\; \backslash equiv\; \backslash frac\{\backslash dot\; a\}\{a\}\; =\; \backslash textrm\{constant\}\; \backslash quad\; \backslash Longrightarrow\; \backslash quad\; a\; \backslash propto\; e^\{Ht\}\; =\; e^\{\backslash frac\{t\}\{t\_H\}\}$$

Likewise, the generally accepted value of 2.27 Es⁻¹ means that (at the current rate) the universe would grow by a factor of {{mvar|e}}{{sup|2.27}} in one exasecond.

Over long periods of time, the dynamics are complicated by general relativity, dark energy, inflation, etc., as explained above.

The Hubble length or Hubble distance is a unit of distance in cosmology, defined as {{math|cH{{sup|−1}}}} — 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.) Substituting {{math|1= D = cH{{sup|−1}}}} into the equation for Hubble's law, {{math|v {{=}} H₀D}} reveals that the Hubble distance specifies the distance from our location to those galaxies which are {{em|currently}} receding from us at the speed of light.

{{main|Hubble volume}}

The Hubble volume is sometimes defined as a volume of the universe with a comoving size of {{math|cH{{sup|−1}}}}. The exact definition varies: it is sometimes defined as the volume of a sphere with radius {{math|cH{{sup|−1}}}}, or alternatively, a cube of side {{math|cH{{sup|−1}}}}. Some cosmologists even use the term Hubble volume to refer to the volume of the observable universe, although this has a radius approximately three times larger.

File:Recent Hubble's Constant Values.png

The value of the Hubble constant, {{math|H{{sub|0}}}}, cannot be measured directly, but is derived from a combination of astronomical observations and model-dependent assumptions. Increasingly accurate observations and new models over many decades have led to two sets of highly precise values which do not agree. This difference is known as the "Hubble tension".

For the original 1929 estimate of the constant now bearing his name, Hubble used observations of Cepheid variable stars as "standard candles" to measure distance. The result he obtained was {{val|500|u=km/s|up=Mpc}}, much larger than the value astronomers currently calculate. Later observations by astronomer Walter Baade led him to realize that there were distinct "populations" for stars (Population I and Population II) in a galaxy. The same observations led him to discover that there are two types of Cepheid variable stars with different luminosities. Using this discovery, he recalculated Hubble constant and the size of the known universe, doubling the previous calculation made by Hubble in 1929.Baade, W. (1944) The resolution of Messier 32, NGC 205, and the central region of the Andromeda nebula. ApJ 100 137–146Baade, W. (1956) The period-luminosity relation of the Cepheids. PASP 68 5–16{{cite web|last=Allen|first=Nick|title=Section 2: The Great Debate and the Great Mistake: Shapley, Hubble, Baade|url=http://www.institute-of-brilliant-failures.com/section2.htm|website=The Cepheid Distance Scale: A History|access-date=19 November 2011|archive-url=https://web.archive.org/web/20071210105344/http://www.institute-of-brilliant-failures.com/section2.htm|archive-date=10 December 2007|url-status=dead}} He announced this finding to considerable astonishment at the 1952 meeting of the International Astronomical Union in Rome.

For most of the second half of the 20th century, the value of {{math|H{{sub|0}}}} was estimated to be between {{val|50|and|90|u=km/s|up=Mpc}}.

The value of the Hubble constant was the topic of a long and rather bitter controversy between Gérard de Vaucouleurs, who claimed the value was around 100, and Allan Sandage, who claimed the value was near 50. In one demonstration of vitriol shared between the parties, when Sandage and Gustav Andreas Tammann (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".{{Cite book |last=de Vaucouleurs |first=G. |title=The cosmic distance scale and the Hubble constant |publisher=Mount Stromlo and Siding Spring Observatories, Australian National University |year=1982}} In 1996, a debate moderated by John Bahcall between Sidney van den Bergh and Gustav Tammann 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. Incorporating the ΛCDM model, observations of high-redshift clusters at X-ray and microwave wavelengths using the Sunyaev–Zel'dovich effect, measurements of anisotropies in the cosmic microwave background radiation, and optical surveys all gave a value of around 50–70 km/s/Mpc for the constant.{{cite journal |title=Scaling the universe: Gravitational lenses and the Hubble constant |last=Myers |first=S. T. |date=1999 |journal=Proceedings of the National Academy of Sciences of the United States of America |volume=96 |issue=8 |pages=4236–4239 |doi=10.1073/pnas.96.8.4236 |doi-access=free |pmid=10200245 |pmc=33560|bibcode=1999PNAS...96.4236M }}

By the late 1990s, advances in ideas and technology allowed higher precision measurements.{{Cite journal |last=Turner |first=Michael S. |date=2022-09-26 |title=The Road to Precision Cosmology |url=https://www.annualreviews.org/content/journals/10.1146/annurev-nucl-111119-041046 |journal=Annual Review of Nuclear and Particle Science |language=en |volume=72 |pages=1–35 |doi=10.1146/annurev-nucl-111119-041046 |arxiv=2201.04741 |bibcode=2022ARNPS..72....1T |issn=0163-8998}}

However, two major categories of methods, each with high precision, fail to agree.

"Late universe" measurements using calibrated distance ladder techniques have converged on a value of approximately {{val|73|u=km/s|up=Mpc}}. Since 2000, "early universe" techniques based on measurements of the cosmic microwave background have become available, and these agree on a value near {{val|67.7|u=km/s|up=Mpc}}.{{Cite journal |last1=Freedman |first1=Wendy L. |last2=Madore |first2=Barry F. |date=2023-11-01 |title=Progress in direct measurements of the Hubble constant |url=https://doi.org/10.1088/1475-7516/2023/11/050 |journal=Journal of Cosmology and Astroparticle Physics |volume=2023 |issue=11 |article-number=050 |doi=10.1088/1475-7516/2023/11/050 |issn=1475-7516|arxiv=2309.05618 |bibcode=2023JCAP...11..050F }} (This accounts for the change in the expansion rate since the early universe, so is comparable to the first number.) Initially, this discrepancy was within the estimated measurement uncertainties and thus no cause for concern. However, as techniques have improved, the estimated measurement uncertainties have shrunk, but the discrepancies have not, to the point that the disagreement is now highly statistically significant. This discrepancy is called the Hubble tension.{{cite news |last=Mann |first=Adam |date=26 August 2019 |title=One Number Shows Something Is Fundamentally Wrong with Our Conception of the Universe – This fight has universal implications |work=Live Science |url=https://www.livescience.com/hubble-constant-discrepancy-explained.html |access-date=26 August 2019}}{{cite journal |last=di Valentino |first=Eleonora |display-authors=etal |date=2021 |title=In the realm of the Hubble tension—a review of solutions |journal=Classical and Quantum Gravity |volume=38 |issue=15 |article-number=153001 |doi=10.1088/1361-6382/ac086d |doi-access=free |arxiv=2103.01183 |bibcode=2021CQGra..38o3001D |s2cid=232092525}}

An example of an "early" measurement, the Planck mission published in 2018 gives a value for {{math|1= H{{sub|0}} =}} of {{val|67.4|0.5|u=km/s|up=Mpc}}. In the "late" camp is the higher value of {{val|74.03|1.42|u=km/s|up=Mpc}} determined by the Hubble Space Telescope{{Cite magazine|url=https://www.scientificamerican.com/article/best-yet-measurements-deepen-cosmological-crisis/|title=Best-Yet Measurements Deepen Cosmological Crisis|last=Ananthaswamy|first=Anil|date=22 March 2019|access-date=23 March 2019|magazine=Scientific American}}

and confirmed by the James Webb Space Telescope in 2023.{{Citation |last1=Riess |first1=Adam G. |title=Crowded No More: The Accuracy of the Hubble Constant Tested with High Resolution Observations of Cepheids by JWST |date=2023-07-28 |arxiv=2307.15806 |last2=Anand |first2=Gagandeep S. |last3=Yuan |first3=Wenlong |last4=Casertano |first4=Stefano |last5=Dolphin |first5=Andrew |last6=Macri |first6=Lucas M. |last7=Breuval |first7=Louise |last8=Scolnic |first8=Dan |last9=Perrin |first9=Marshall |journal=The Astrophysical Journal |volume=956 |issue=1 |article-number=L18 |doi=10.3847/2041-8213/acf769 |doi-access=free |bibcode=2023ApJ...956L..18R }}{{Cite web |date=2023-09-12 |title=Webb Confirms Accuracy of Universe's Expansion Rate Measured by Hubble, Deepens Mystery of Hubble Constant Tension – James Webb Space Telescope |url=https://blogs.nasa.gov/webb/2023/09/12/webb-confirms-accuracy-of-universes-expansion-rate-measured-by-hubble-deepens-mystery-of-hubble-constant-tension/ |access-date=2024-02-15 |website=blogs.nasa.gov |language=en-US}}

The "early" and "late" measurements disagree at the >5 σ level, beyond a plausible level of chance.{{Cite journal |last1=Riess |first1=Adam G. |last2=Yuan |first2=Wenlong |last3=Macri |first3=Lucas M. |last4=Scolnic |first4=Dan |last5=Brout |first5=Dillon |last6=Casertano |first6=Stefano |last7=Jones |first7=David O. |last8=Murakami |first8=Yukei |last9=Anand |first9=Gagandeep S. |last10=Breuval |first10=Louise |last11=Brink |first11=Thomas G. |last12=Filippenko |first12=Alexei V. |last13=Hoffmann |first13=Samantha |last14=Jha |first14=Saurabh W. |last15=Kenworthy |first15=W. D’arcy |date=July 2022 |title=A Comprehensive Measurement of the Local Value of the Hubble Constant with 1 km s−1 Mpc−1 Uncertainty from the Hubble Space Telescope and the SH0ES Team |journal=The Astrophysical Journal Letters |language=en |volume=934 |issue=1 |pages=L7 |doi=10.3847/2041-8213/ac5c5b |doi-access=free |bibcode=2022ApJ...934L...7R |issn=2041-8205|arxiv=2112.04510 }} The resolution to this disagreement is an ongoing area of active research.{{Cite journal|last1=Millea|first1=Marius|last2=Knox|first2=Lloyd|date=2019-08-10|title=Hubble constant hunter's guide|journal=Physical Review D |volume=101 |issue=4 |page=043533 |doi=10.1103/PhysRevD.101.043533 |language=en|arxiv=1908.03663}}

File:Measurements of the Hubble constant (H0) by different astronomical missions and groups until 2021.jpg

Since 2013 much effort has gone in to new measurements to check for possible systematic errors and improved reproducibility.

The "late universe" or distance ladder measurements typically employ three stages or "rungs". In the first rung distances to Cepheids are determined while trying to reduce luminosity errors from dust and correlations of metallicity with luminosity. The second rung uses

Type Ia supernova, explosions of almost constant amount of mass and thus very similar amounts of light; the primary source of systematic error is the limited number of objects that can be observed. The third rung of the distance ladder measures the red-shift of supernova to extract the Hubble flow and from that the constant. At this rung corrections due to motion other than expansion are applied.{{rp|2.1}}

As an example of the kind of work needed to reduce systematic errors, photometry on observations from the James Webb Space Telescope of extra-galactic Cepheids confirm the findings from the HST. The higher resolution avoided confusion from crowding of stars in the field of view but came to the same value for H₀.{{Cite journal |last1=Riess |first1=Adam G. |last2=Anand |first2=Gagandeep S. |last3=Yuan |first3=Wenlong |last4=Casertano |first4=Stefano |last5=Dolphin |first5=Andrew |last6=Macri |first6=Lucas M. |last7=Breuval |first7=Louise |last8=Scolnic |first8=Dan |last9=Perrin |first9=Marshall |last10=Anderson |first10=Richard I. |date=2023-10-01 |title=Crowded No More: The Accuracy of the Hubble Constant Tested with High-resolution Observations of Cepheids by JWST |journal=The Astrophysical Journal Letters |volume=956 |issue=1 |pages=L18 |doi=10.3847/2041-8213/acf769 |doi-access=free |issn=2041-8205|arxiv=2307.15806 |bibcode=2023ApJ...956L..18R }}{{Cite journal |last1=Verde |first1=Licia |last2=Schöneberg |first2=Nils |last3=Gil-Marín |first3=Héctor |date=2024-09-13 |title=A Tale of Many H0 |url=https://www.annualreviews.org/content/journals/10.1146/annurev-astro-052622-033813 |journal=Annual Review of Astronomy and Astrophysics |language=en |volume=62 |pages=287–331 |doi=10.1146/annurev-astro-052622-033813 |issn=0066-4146}}

The "early universe" or inverse distance ladder measures the observable consequences of spherical sound waves on primordial plasma density. These pressure waves – called baryon acoustic oscillations (BAO) – cease once the universe cooled enough for electrons to stay bound to nuclei, ending the plasma and allowing the photons trapped by interaction with the plasma to escape. The pressure waves then become very small perturbations in density imprinted on the cosmic microwave background and on the large scale density of galaxies across the sky. Detailed structure in high precision measurements of the CMB can be matched to physics models of the oscillations. These models depend upon the Hubble constant such that a match reveals a value for the constant. Similarly, the BAO affects the statistical distribution of matter, observed as distant galaxies across the sky. These two independent kinds of measurements produce similar values for the constant from the current models, giving strong evidence that systematic errors in the measurements themselves do not affect the result.{{rp|Sup. B}}

In addition to measurements based on calibrated distance ladder techniques or measurements of the CMB, other methods have been used to determine the Hubble constant.

One alternative method for constraining the Hubble constant involves transient events seen in multiple images of a strongly lensed object. A transient event, such as a supernova, is seen at different times in each of the lensed images, and if this time delay between each image can be measured, it can be used to constrain the Hubble constant. This method is commonly known as "time-delay cosmography", and was first proposed by Refsdal in 1964,{{cite journal |last1=Refsdal |first1=S. |title=On the Possibility of Determining Hubble's Parameter and the Masses of Galaxies from the Gravitational Lens Effect |journal=Monthly Notices of the Royal Astronomical Society |date=1 September 1964 |volume=128 |issue=4 |pages=307–310 |doi=10.1093/mnras/128.4.307|doi-access=free }} years before the first strongly lensed object was observed. The first strongly lensed supernova to be discovered was named SN Refsdal in his honor. While Refsdal suggested this could be done with supernovae, he also noted that extremely luminous and distant star-like objects could also be used. These objects were later named quasars, and to date (April 2025) the majority of time-delay cosmography measurements have been done with strongly lensed quasars. This is because current samples of lensed quasars vastly outnumber known lensed supernovae, of which <10 are known. This is expected to change dramatically in the next few years, with surveys such as LSST expected to discover ~10 lensed SNe in the first three years of observation.{{cite arXiv|eprint=2504.01068 |last1=Bronikowski |first1=M. |last2=Petrushevska |first2=T. |last3=Pierel |first3=J. D. R. |last4=Acebron |first4=A. |last5=Donevski |first5=D. |last6=Apostolova |first6=B. |last7=Blagorodnova |first7=N. |last8=Jankovič |first8=T. |title=Cluster-lensed supernova yields from the Vera C. Rubin Observatory and Nancy Grace Roman Space Telescope |date=2025 |class=astro-ph.GA }} For example time-delay constraints on H0, see the results from STRIDES and H0LiCOW in the table below.

In October 2018, scientists used information from gravitational wave events (especially those involving the merger of neutron stars, like GW170817), of determining the Hubble constant.{{cite web |last=Lerner |first=Louise |title=Gravitational waves could soon provide measure of universe's expansion |url=https://phys.org/news/2018-10-gravitational-universe-expansion.html |date=22 October 2018 |work=Phys.org |access-date=22 October 2018 }}{{cite journal |last1=Chen |first1=Hsin-Yu |last2=Fishbach |first2=Maya |last3=Holz |first3=Daniel E. |title=A two per cent Hubble constant measurement from standard sirens within five years |date=17 October 2018 |journal=Nature |volume=562 |issue=7728 |pages=545–547 |doi=10.1038/s41586-018-0606-0 |pmid=30333628 |bibcode=2018Natur.562..545C |arxiv=1712.06531 |s2cid=52987203 }}

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 stars, following the detection of the neutron star merger of GW170817, an event known as a dark siren.{{cite news |author=National Radio Astronomy Observatory |author-link=National Radio Astronomy Observatory |date=8 July 2019 |title=New method may resolve difficulty in measuring universe's expansion – Neutron star mergers can provide new 'cosmic ruler' |work=EurekAlert! |url=https://www.eurekalert.org/pub_releases/2019-07/nrao-nmm070819.php |access-date=8 July 2019}}{{cite news |last=Finley |first=Dave |title=New Method May Resolve Difficulty in Measuring Universe's Expansion |url=https://public.nrao.edu/news/new-method-measuring-universe-expansion/ |date=8 July 2019 |work=National Radio Astronomy Observatory |access-date=8 July 2019 }} Their measurement of the Hubble constant is {{val|73.3|+5.3|-5.0}} (km/s)/Mpc.{{cite journal |author=Hotokezaka, K. |display-authors=et al. |title=A Hubble constant measurement from superluminal motion of the jet in GW170817 |date=8 July 2019 |journal=Nature Astronomy |volume=3 |issue=10 |pages=940–944 |doi=10.1038/s41550-019-0820-1 |bibcode=2019NatAs...3..940H |arxiv=1806.10596 |s2cid=119547153 }}

Also in July 2019, astronomers reported another new method, using data from the Hubble Space Telescope and based on distances to red giant stars calculated using the tip of the red-giant branch (TRGB) distance indicator. Their measurement of the Hubble constant is {{val|69.8|+1.9|-1.9|u=km/s|up=Mpc}}.{{cite journal |last1=Freedman |first1=Wendy L. |author-link1=Wendy Freedman |last2=Madore |first2=Barry F. |last3=Hatt |first3=Dylan |last4=Hoyt |first4=Taylor J. |last5=Jang |first5=In-Sung |last6=Beaton |first6=Rachael L. |last7=Burns |first7=Christopher R. |last8=Lee |first8=Myung Gyoon |last9=Monson |first9=Andrew J. |last10=Neeley |first10=Jillian R. |last11=Phillips |first11=Mark M. |last12=Rich |first12=Jeffrey A. |last13=Seibert |first13=Mark |display-authors=6 |year=2019 |title=The Carnegie-Chicago Hubble Program. VIII. An Independent Determination of the Hubble Constant Based on the Tip of the Red Giant Branch |journal=The Astrophysical Journal |volume=882 |issue=1 |article-number=34 |arxiv=1907.05922 |bibcode=2019ApJ...882...34F |doi=10.3847/1538-4357/ab2f73 |s2cid=196623652 |doi-access=free }}

In February 2020, the Megamaser Cosmology Project published independent results based on astrophysical masers visible at cosmological distances and which do not require multi-step calibration. That work 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 predict that the Universe should be expanding more slowly than is currently observed.{{Cite journal|last=Castelvecchi|first=Davide|date=2020-07-15|title=Mystery over Universe's expansion deepens with fresh data|journal=Nature|language=en|volume=583|issue=7817|pages=500–501|doi=10.1038/d41586-020-02126-6|pmid=32669728|bibcode=2020Natur.583..500C|s2cid=220583383|doi-access=}}

In July 2023, an independent estimate of the Hubble constant was derived from a kilonova, the optical afterglow of a neutron star merger, using the expanding photosphere method.{{Cite journal |last1=Sneppen |first1=Albert |last2=Watson |first2=Darach |last3=Poznanski |first3=Dovi |last4=Just |first4=Oliver |last5=Bauswein |first5=Andreas |last6=Wojtak |first6=Radosław |date=2023-10-01 |title=Measuring the Hubble constant with kilonovae using the expanding photosphere method |url=https://www.aanda.org/articles/aa/abs/2023/10/aa46306-23/aa46306-23.html |journal=Astronomy & Astrophysics |language=en |volume=678 |article-number=A14 |doi=10.1051/0004-6361/202346306 |issn=0004-6361|arxiv=2306.12468 |bibcode=2023A&A...678A..14S }} Due to the blackbody nature of early kilonova spectra,{{Cite journal |last=Sneppen |first=Albert |date=2023-09-01 |title=On the Blackbody Spectrum of Kilonovae |journal=The Astrophysical Journal |volume=955 |issue=1 |article-number=44 |doi=10.3847/1538-4357/acf200 |doi-access=free |issn=0004-637X|arxiv=2306.05452 |bibcode=2023ApJ...955...44S }} such systems provide strongly constraining estimators of cosmic distance. Using the kilonova AT2017gfo (the aftermath of, once again, GW170817), these measurements indicate a local-estimate of the Hubble constant of {{val|67.0|+3.6|u=km/s|up=Mpc}}.{{Cite journal |last1=Sneppen |first1=Albert |last2=Watson |first2=Darach |last3=Bauswein |first3=Andreas |last4=Just |first4=Oliver |last5=Kotak |first5=Rubina |last6=Nakar |first6=Ehud |last7=Poznanski |first7=Dovi |last8=Sim |first8=Stuart |date=February 2023 |title=Spherical symmetry in the kilonova AT2017gfo/GW170817 |url=https://www.nature.com/articles/s41586-022-05616-x |journal=Nature |language=en |volume=614 |issue=7948 |pages=436–439 |doi=10.1038/s41586-022-05616-x |pmid=36792736 |arxiv=2302.06621 |bibcode=2023Natur.614..436S |s2cid=256846834 |issn=1476-4687}}

File:Hubbleconstants color.png

The cause of the Hubble tension is unknown,{{cite web |last1=Gresko |first1=Michael |title=The universe is expanding faster than it should be |url=https://www.nationalgeographic.com/science/article/the-universe-is-expanding-faster-than-it-should-be |archive-url=https://web.archive.org/web/20211217160427/https://www.nationalgeographic.com/science/article/the-universe-is-expanding-faster-than-it-should-be |url-status=dead |archive-date=December 17, 2021 |date=17 December 2021 |website=National Geographic |access-date=21 December 2021}} and there are many possible proposed solutions. The most conservative is that there is an unknown systematic error affecting either early-universe or late-universe observations. Although intuitively appealing, this explanation requires multiple unrelated effects regardless of whether early-universe or late-universe observations are incorrect, and there are no obvious candidates. Furthermore, any such systematic error would need to affect multiple different instruments, since both the early-universe and late-universe observations come from several different telescopes.

Alternatively, it could be that the observations are correct, but some unaccounted-for effect is causing the discrepancy. If the cosmological principle fails (see {{slink|Lambda-CDM model|Violations of the cosmological principle}}), then the existing interpretations of the Hubble constant and the Hubble tension have to be revised, which might resolve the Hubble tension.{{citation |last1=Abdalla |first1=Elcio |title=Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies |date=11 Mar 2022 |journal=Journal of High Energy Astrophysics |volume=34 |page=49 |arxiv=2203.06142 |bibcode=2022JHEAp..34...49A |doi=10.1016/j.jheap.2022.04.002 |s2cid=247411131 |last2=Abellán |first2=Guillermo Franco |last3=Aboubrahim |first3=Amin}} In particular, we would need to be located within a very large void, up to about a redshift of 0.5, for such an explanation to conflate with supernovae and baryon acoustic oscillation observations. Yet another possibility is that the uncertainties in the measurements could have been underestimated, but given the internal agreements this is neither likely, nor resolves the overall tension.

Finally, another possibility is new physics beyond the currently accepted cosmological model of the universe, the ΛCDM model.{{Cite journal |last=Vagnozzi |first=Sunny |date=2020-07-10 |title=New physics in light of the H₀ tension: An alternative view |url=https://link.aps.org/doi/10.1103/PhysRevD.102.023518 |journal=Physical Review D |volume=102 |issue=2 |article-number=023518 |doi=10.1103/PhysRevD.102.023518|arxiv=1907.07569 |bibcode=2020PhRvD.102b3518V |s2cid=197430820 }} There are very many theories in this category, for example, replacing general relativity with a modified theory of gravity could potentially resolve the tension,{{Cite journal |last1=Haslbauer |first1=M. |last2=Banik |first2=I. |last3=Kroupa |first3=P. |date=2020-12-21 |title=The KBC void and Hubble tension contradict LCDM on a Gpc scale – Milgromian dynamics as a possible solution |journal=Monthly Notices of the Royal Astronomical Society |volume=499 |issue=2 |pages=2845–2883 |arxiv=2009.11292 |bibcode=2020MNRAS.499.2845H |doi=10.1093/mnras/staa2348 |issn=0035-8711 |doi-access=free}}{{Cite journal |last1=Mazurenko |first1=S. |last2=Banik |first2=I. |last3=Kroupa |first3=P. |last4=Haslbauer |first4=M. |date=2024-01-21 |title=A simultaneous solution to the Hubble tension and observed bulk flow within 250/h Mpc |journal=Monthly Notices of the Royal Astronomical Society |volume=527 |issue=3 |pages=4388–4396 |arxiv=2311.17988 |bibcode=2024MNRAS.527.4388M |doi=10.1093/mnras/stad3357 |issn=0035-8711 |doi-access=free}} as can a dark energy component in the early universe,{{efn|In standard ΛCDM, dark energy only comes into play in the late universe – its effect in the early universe is too small to have an effect.}}{{Cite journal |last1=Poulin|first1=Vivian |last2=Smith|first2=Tristan L. |last3=Karwal|first3=Tanvi |last4=Kamionkowski|first4=Marc |date=2019-06-04 |title=Early Dark Energy can Resolve the Hubble Tension |journal=Physical Review Letters |volume=122 |issue=22 |article-number=221301 |doi=10.1103/PhysRevLett.122.221301 |pmid=31283280 |arxiv=1811.04083 |bibcode=2019PhRvL.122v1301P |s2cid=119233243 }} dark energy with a time-varying equation of state,{{efn|1=In standard ΛCDM, dark energy has a constant equation of state {{math|1= w = −1}}.}}{{cite journal|url=https://www.nature.com/articles/s41550-017-0216-z|title=Dynamical dark energy in light of the latest observations|journal=Nature Astronomy|date=2017|doi=10.1038/s41550-017-0216-z |last1=Zhao |first1=Gong-Bo |last2=Raveri |first2=Marco |last3=Pogosian |first3=Levon |last4=Wang |first4=Yuting |last5=Crittenden |first5=Robert G. |last6=Handley |first6=Will J. |last7=Percival |first7=Will J. |last8=Beutler |first8=Florian |last9=Brinkmann |first9=Jonathan |last10=Chuang |first10=Chia-Hsun |last11=Cuesta |first11=Antonio J. |last12=Eisenstein |first12=Daniel J. |last13=Kitaura |first13=Francisco-Shu |last14=Koyama |first14=Kazuya |last15=l'Huillier |first15=Benjamin |last16=Nichol |first16=Robert C. |last17=Pieri |first17=Matthew M. |last18=Rodriguez-Torres |first18=Sergio |last19=Ross |first19=Ashley J. |last20=Rossi |first20=Graziano |last21=Sánchez |first21=Ariel G. |last22=Shafieloo |first22=Arman |last23=Tinker |first23=Jeremy L. |last24=Tojeiro |first24=Rita |last25=Vazquez |first25=Jose A. |last26=Zhang |first26=Hanyu |volume=1 |issue=9 |pages=627–632 |arxiv=1701.08165 |bibcode=2017NatAs...1..627Z |s2cid=256705070 }} or dark matter that decays into dark radiation.{{cite journal|url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.92.061303|title=Reconciling Planck results with low redshift astronomical measurements|journal=Physical Review D|date=2015|doi=10.1103/PhysRevD.92.061303 |last1=Berezhiani |first1=Zurab |last2=Dolgov |first2=A. D. |last3=Tkachev |first3=I. I. |volume=92 |issue=6 |article-number=061303 |arxiv=1505.03644 |bibcode=2015PhRvD..92f1303B |s2cid=118169478 }} A problem faced by all these theories is that both early-universe and late-universe measurements rely on multiple independent lines of physics, and it is difficult to modify any of those lines while preserving their successes elsewhere. The scale of the challenge can be seen from how some authors have argued that new early-universe physics alone is not sufficient;{{cite web|url=https://astrobites.org/2021/05/17/template-post-5/|title=Solving the Hubble tension might require more than changing the early Universe|author=Laila Linke|publisher=Astrobites|date=17 May 2021}}{{Cite journal |last1=Vagnozzi|first1=Sunny |date=2023-08-30 |title=Seven Hints That Early-Time New Physics Alone Is Not Sufficient to Solve the Hubble Tension |journal=Universe |volume=9 |issue=9 |article-number=393 |doi=10.3390/universe9090393 |arxiv=2308.16628 |bibcode=2023Univ....9..393V |doi-access=free }} while other authors argue that new late-universe physics alone is also not sufficient.{{cite journal|title=Ruling Out New Physics at Low Redshift as a Solution to the H₀ Tension|author=Ryan E. Keeley and Arman Shafieloo|journal=Physical Review Letters |date=August 2023|volume=131 |issue=11 |article-number=111002 |doi=10.1103/PhysRevLett.131.111002 |pmid=37774270 |arxiv=2206.08440 |bibcode=2023PhRvL.131k1002K |s2cid=249848075 }} Nonetheless, astronomers are trying, with interest in the Hubble tension growing strongly since the mid 2010s.

• {{Annotated link|List of scientists whose names are used in physical constants}}
• S8 tension- a similar problem from another parameter of the ΛCDM model.
• {{Annotated link|Tests of general relativity}}

{{notelist}}

{{reflist}}

• {{Cite book

|last=Hubble |first=E. P.

|date=1937

|title=The Observational Approach to Cosmology

|publisher=Clarendon Press

|lccn=38011865

}}

• {{Cite book

|last=Kutner

|first=M.

|date=2003

|title=Astronomy: A Physical Perspective

|publisher=Cambridge University Press

|isbn=978-0-521-52927-3

|url-access=registration

|url=https://archive.org/details/astronomyphysica00kutn

}}

• {{Cite book

|last=Liddle |first=A. R.

|date=2003

|title=An Introduction to Modern Cosmology

|edition=2nd

|publisher=John Wiley & Sons

|isbn=978-0-470-84835-7

}}

• [http://map.gsfc.nasa.gov/universe/bb_tests_exp.html NASA's WMAP B ig Bang Expansion: the Hubble Constant]
• [http://www.ipac.caltech.edu/H0kp/H0KeyProj.html The Hubble Key Project]
• [http://cas.sdss.org/dr3/en/proj/advanced/hubble/ The Hubble Diagram Project]
• [https://www.forbes.com/sites/startswithabang/2019/05/03/cosmologys-biggest-conundrum-is-a-clue-not-a-controversy/ Coming to terms with different Hubble Constants] (Forbes; 3 May 2019)
• {{cite web|last=Merrifield|first=Michael|title=Hubble Constant|url=http://www.sixtysymbols.com/videos/hubble.htm|website=Sixty Symbols|publisher=Brady Haran for the University of Nottingham|date=2009}}

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