Editing Hubble's law (section)

== Interpretation ==
[[File:Velocity-redshift.JPG|thumb|upright=1.4|A variety of possible recessional velocity vs. redshift functions including the simple linear relation {{math|1=''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.<ref name="D&L">{{cite journal |last1=Davis |first1=T. M. |last2=Lineweaver |first2=C. H. |date=2001 |title=Superluminal Recessional Velocities |journal=[[AIP Conference Proceedings]] |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 }}</ref>]]
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:

<math display="block">v = H_0 \, D</math>

where
* {{mvar|v}} is the recessional velocity, typically expressed in km/s.
* {{math|''H''<sub>0</sub>}} is Hubble's constant and corresponds to the value of {{mvar|H}} (often termed the '''Hubble parameter''' which is a value that is [[time-variant system|time dependent]] and which can be expressed in terms of the [[Scale factor (cosmology)|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#Comoving coordinates|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-|mega]] [[parsec]]s (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''<sub>HS</sub>}}, objects recede at a rate faster than the [[speed of light]] (''See'' [[Comoving distance#Uses of the proper distance|Uses of the proper distance]] for a discussion of the significance of this):

<math display="block">r_\text{HS} = \frac{c}{H_0} \ . </math>

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.<ref name=Keel/> 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.<ref>{{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}}</ref>

=== 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.<ref name="Harrison">{{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 }}</ref>

==== Redshift velocity ====
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.<ref name="Madsen">{{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 }}</ref> In other words, to determine the redshift velocity {{math|''v''<sub>rs</sub>}}, the relation:

<math display="block"> v_\text{rs} \equiv cz,</math>

is used.<ref name="Dekel">{{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 }}</ref><ref name="Padmanabhan">{{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 }}</ref> 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 [[Relativistic Doppler effect|Fizeau–Doppler formula]]<ref name="Sartori">{{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 }}</ref>

<math display="block">z = \frac{\lambda_\text{o}}{\lambda_\text{e}}-1 = \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}-1 \approx \frac{v}{c}.</math>

Here, {{math|''λ''<sub>o</sub>}}, {{math|''λ''<sub>e</sub>}} are the observed and emitted wavelengths respectively. The "redshift velocity" {{math|''v''<sub>rs</sub>}} 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.<ref name="L_Sartori">{{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 }}</ref>

==== Recessional velocity ====

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 [[Physical cosmology|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:

<math display="block">\frac {D(t)}{D(t_0)} = \frac{R(t)}{R(t_0)},</math>

where {{math|''t''<sub>0</sub>}} is some reference time.<ref>Matts Roos, ''Introduction to Cosmology''</ref> If light is emitted from a galaxy at time {{math|''t''<sub>e</sub>}} and received by us at {{math|''t''<sub>0</sub>}}, it is redshifted due to the expansion of the universe, and this redshift {{mvar|z}} is simply:

<math display="block">z = \frac {R(t_0)}{R(t_\text{e})} - 1. </math>

Suppose a galaxy is at distance {{mvar|D}}, and this distance changes with time at a rate {{mvar|d<sub>t</sub>D}}. We call this rate of recession the "recession velocity" {{math|''v''<sub>r</sub>}}:

<math display="block">v_\text{r} = d_tD = \frac {d_tR}{R} D. </math>

We now define the Hubble constant as

<math display="block">H \equiv \frac{d_tR}{R}, </math>

and discover the Hubble law:

<math display="block"> v_\text{r} = H D. </math>

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:

<math display="block"> z = \frac {R(t_0)}{R(t_e)} - 1 \approx \frac {R(t_0)} {R(t_0)\left(1+(t_e-t_0)H(t_0)\right)}-1 \approx (t_0-t_e)H(t_0), </math>

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:

<math display="block"> z \approx (t_0-t_\text{e})H(t_0) \approx \frac {D}{c} H(t_0), </math>

or

<math display="block"> cz \approx D H(t_0) = v_r. </math>

According to this approach, the relation {{math|1=''cz'' = ''v''<sub>r</sub>}} is an approximation valid at low redshifts, to be replaced by a relation at large redshifts that is model-dependent. See [[#redshift|velocity-redshift figure]].

=== Observability of parameters ===
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.

=== 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,<ref name="AM-20170818">{{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 (magazine)|Astronomy]] |access-date=18 October 2017 }}</ref> 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 {{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|<math> H(t) \equiv \dot{a}(t)/a(t) </math>,}} in most accelerating models <math>a</math> increases relatively faster than {{nowrap|<math>\dot{a}</math>,}} 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|<math display="inline"> q \equiv - \frac {\ddot{a}\, a} {\dot{a}^2} </math>,}} it follows that

<math display="block"> \frac{dH}{dt} = -H^2 (1+q) </math>

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-CDM model|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 ≈&nbsp;57&nbsp;(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 coordinate system|Cartesian]]/Newtonian coordinate space, which, considered as a [[metric space]], is entirely [[Cosmological principle|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.

=== Ultimate fate and age of the universe ===
[[Image:Friedmann universes.svg|thumb|upright=1.9|The [[age of the universe|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 ({{math|Ω<sub>M</sub>}} for [[matter]] and {{math|Ω<sub>Λ</sub>}} for dark energy).<br>A '''closed universe''' with {{math|Ω<sub>M</sub> > 1}} and {{math|1= Ω<sub>Λ</sub> = 0}} comes to an end in a [[Big Crunch]] and is considerably younger than its Hubble age.<br>An '''open universe''' with {{math|Ω<sub>M</sub> ≤ 1}} and {{math|1= Ω<sub>Λ</sub> = 0}} expands forever and has an age that is closer to its Hubble age. For the accelerating universe with nonzero {{math|Ω<sub>Λ</sub>}} 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

<math display="block">q = -\left(1+\frac{\dot H}{H^2}\right).</math>

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 [[integral|integration]] of the Friedmann equations backwards from the present time to the time when the [[particle horizon|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/(3''H'')}}. 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''}}.

=== Olbers' paradox ===
{{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 [[Infinity|infinite]] in size, [[static universe|static]], and filled with a uniform distribution of [[star]]s, then every line of sight in the sky would end on a star, and the sky would be as [[brightness|bright]] as the surface of a star. However, the night sky is largely dark.<ref name="Chase_etal_2004">{{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}}</ref><ref name="Asimov1974">{{cite book |last=Asimov |first=I. |date=1974 |chapter=The Black of Night |title=Asimov on Astronomy |publisher=[[Doubleday (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 }}</ref>

Since the 17th&nbsp;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.<ref name=Chase_etal_2004/><ref name=Asimov1974/>

=== Dimensionless Hubble constant ===
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",<ref name=damnh>{{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}}</ref> then to write Hubble's constant {{math|''H''<sub>0</sub>}} as {{math|''h'' × 100 km⋅[[second|s]]<sup>−1</sup>⋅[[Parsec|Mpc]]<sup>−1</sup>}}, all the relative uncertainty of the true value of {{math|''H''<sub>0</sub>}} being then relegated to {{mvar|h}}.<ref>{{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}}</ref> 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''|''H''<sub>0</sub>}} × ''z''}}. Since {{math|''H''<sub>0</sub>}} is not precisely known, the distance is expressed as:

<math display="block">cz/H_0\approx(2998\times z)\text{ Mpc }h^{-1}</math>

In other words, one calculates 2998 × {{mvar|z}} and one gives the units as Mpc&nbsp;{{math|''h''{{sup|-1}}}} or {{math|''h''{{sup|-1}}}}&nbsp;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}}}}&nbsp;{{val||ul=km/s|upl=Mpc}}, which implies {{math|1= ''h''{{sub|70}} = ''h'' / 0.7}}.

This should not be confused with the [[Dimensionless physical constant|dimensionless value]] of Hubble's constant, usually expressed in terms of [[Planck units]], obtained by multiplying {{math|''H''<sub>0</sub>}} by {{val|1.75|e=-63}} (from definitions of parsec and [[Planck time|{{math|''t''<sub>P</sub>}}]]), for example for {{math|1= ''H''{{sub|0}} = 70}}, a Planck unit version of {{val|1.2|e=-61}} is obtained.

=== Acceleration of the expansion ===
{{Main|Accelerating expansion of the universe}}
A value for {{mvar|q}} measured from [[standard candle]] observations of [[Type Ia supernova]]e, which was determined in 1998 to be negative, surprised many astronomers with the implication that the expansion of the universe is currently "accelerating"<ref>{{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 }}</ref> (although the Hubble factor is still decreasing with time, as mentioned above in the [[#Interpretation|Interpretation]] section; see the articles on [[dark energy]] and the ΛCDM model).