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| {{original research |reason=reliable source needed to validate statements and derivations |date=May 2013}}
| | I'm Berenice (25) from Oberlungwitz, Germany. <br>I'm learning Japanese literature at a local university and I'm just about to graduate.<br>I have a part time job in a college.<br><br>Take a look at my web page - [http://Coucou.co/index.php?do=/profile-156254/info/info/ Fifa 15 Coin Generator] |
| {{Cosmology|cTopic=Expanding universe}}
| |
| '''Hubble's law''' is the name for the observation in [[physical cosmology]] that: (1) objects observed in deep space (extragalactic space, ~10 [[megaparsec]]s or more) are found to have a [[Doppler shift]] interpretable as relative velocity away from the [[Earth]]; and (2) that this Doppler-shift-measured velocity, of various [[galaxy|galaxies]] receding from the Earth, is approximately [[Proportionality (mathematics)|proportional]] to their distance from the Earth for galaxies up to a few hundred megaparsecs away.<ref name="riess99" /><ref name="perlmutter99" /><!-- --> This is normally interpreted as a direct, physical observation of the expansion of the spatial volume of the observable universe.<ref name=Coles>
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| {{cite book
| |
| |editor-last=Coles |editor-first=P.
| |
| |year=2001
| |
| |title=Routledge Critical Dictionary of the New Cosmology
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| |url=http://books.google.com/?id=BgNGWVr5yhIC&pg=PA202
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| |page=202
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| |isbn=0-203-16457-1
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| |publisher=[[Routledge]]
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| }}</ref>
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| | |
| The motion of astronomical objects due solely to this expansion is known as the '''Hubble flow'''.<ref>
| |
| {{cite web
| |
| |title=Hubble Flow
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| |url=http://astronomy.swin.edu.au/cosmos/h/hubble+flow
| |
| |work=The Swinburne Astronomy Online Encyclopedia of Astronomy
| |
| |publisher=[[Swinburne University of Technology]]
| |
| |accessdate=2013-05-14
| |
| }}</ref> Hubble's law is considered the first observational basis for the [[metric expansion of space|expanding space paradigm]] and today serves as one of the pieces of evidence most often cited in support of the [[Big Bang]] model.
| |
| | |
| Although widely attributed to [[Edwin Hubble]], the law was first derived from the [[General Relativity]] equations by [[Georges Lemaître]] in a 1927 article where he proposed that the [[Metric expansion of space|Universe is expanding]] and suggested an estimated value of the rate of expansion, now called the '''Hubble constant'''.<ref>
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| {{cite journal
| |
| |last=Lemaître |first=G. |author-link=Georges Lemaître
| |
| |title=Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques
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| |year=1927
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| |journal=[[Annales de la Société Scientifique de Bruxelles A]]
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| |volume=47 |issue= |pages=49–56
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| |bibcode=1927ASSB...47...49L
| |
| }} Partially translated in {{Cite journal
| |
| |last=<!-- --> |first=<!-- -->
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| |title=<!-- -->
| |
| |year=1931
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| |journal=[[Monthly Notices of the Royal Astronomical Society]]
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| |volume=91 |issue= |pages=483–490
| |
| |bibcode=1931MNRAS..91..483L
| |
| }}</ref><ref>
| |
| {{cite journal
| |
| |last=van den Bergh |first=S.
| |
| |year=2011
| |
| |title=The Curious Case of Lemaitre's Equation No. 24
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| |journal=[[Journal of the Royal Astronomical Society of Canada]]
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| |volume=105 |issue=4 |pages=151
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| |arxiv=1106.1195
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| |bibcode=2011JRASC.105..151V
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| |doi=
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| }}</ref><ref>
| |
| {{cite book
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| |last=Block |first=D. L.
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| |year=2012
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| |chapter=Georges Lemaitre and Stiglers Law of Eponymy
| |
| |editor1-last=Holder |editor1-first=R. D.
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| |editor2-last=Mitton |editor2-first=S.
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| |title=Georges Lemaître: Life, Science and Legacy
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| |series=Astrophysics and Space Science Library
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| |volume=395 |pages=89–96
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| |arxiv=1106.3928
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| |bibcode=2012ASSL..395...89B
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| |doi=10.1007/978-3-642-32254-9_8
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| |isbn=978-3-642-32253-2
| |
| }}</ref><ref>
| |
| {{cite journal
| |
| |last=Reich |first=E. S.
| |
| |date=27 June 2011
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| |title=Edwin Hubble in translation trouble
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| |url=http://www.nature.com/news/2011/110627/full/news.2011.385.html
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| |journal=[[Nature News]]
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| |doi=10.1038/news.2011.385
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| }}</ref><ref>
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| {{cite journal
| |
| |last=Livio |first=M.
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| |year=2011
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| |title=Lost in translation: Mystery of the missing text solved
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| |journal=[[Nature (journal)|Nature]]
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| |volume=479 |issue=7372 |pages=171
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| |bibcode=2011Natur.479..171L
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| |doi=10.1038/479171a
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| }}</ref><ref>
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| {{cite journal
| |
| |last1=Livio |first1=M.
| |
| |last2=Riess |first2=A.
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| |year=2013
| |
| |title=Measuring the Hubble constant
| |
| |journal=[[Physics Today]]
| |
| |volume=66 |issue=10 |pages=41
| |
| |bibcode=2013PhT....66j..41L
| |
| |doi=10.1063/PT.3.2148
| |
| }}</ref> Two years later [[Edwin Hubble]] confirmed the existence of that law and determined a more accurate value for the constant that now bears his name.<ref>
| |
| {{cite journal
| |
| |last=Hubble |first=E.
| |
| |year=1929
| |
| |title=A relation between distance and radial velocity among extra-galactic nebulae
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| |url=http://www.pnas.org/cgi/reprint/15/3/168
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| |journal=[[Proceedings of the National Academy of Sciences]]
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| |volume=15 |issue=3 |pages=168
| |
| |bibcode=1929PNAS...15..168H
| |
| |doi=10.1073/pnas.15.3.168
| |
| }}</ref> The recession velocity of the objects was inferred from their [[redshifts]], many measured earlier by [[Vesto Slipher]] (1917) and related to velocity by him.<ref name=MS_Longair>
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| {{cite book
| |
| |last=Longair |first=M. S.
| |
| |year=2006
| |
| |title=The Cosmic Century
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| |url=http://books.google.com/?id=z0vlYHQZHJcC&pg=RA2-PA109
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| |page=109
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| |publisher=[[Cambridge University Press]]
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| |isbn=0-521-47436-1
| |
| }}</ref>
| |
| | |
| The law is often expressed by the equation {{nowrap|''v'' {{=}} ''H''<sub>0</sub>''D''}}, with ''H''<sub>0</sub> 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 velocity ''v'' (i.e. the [[derivative]] of proper distance with respect to cosmological time coordinate; see ''[[Comoving distance#Uses of the proper distance|Uses of the proper distance]]'' for some discussion of the subtleties of this definition of 'velocity'). The SI unit of ''H''<sub>0</sub> is s<sup>−1</sup> but it is most frequently quoted in ([[kilometre|km]]/[[second|s]])/[[Megaparsecs|Mpc]], thus giving the speed in km/s of a galaxy {{convert|1|Mpc|km|sigfig=3}} away. The reciprocal of ''H''<sub>0</sub> is the [[Hubble time]].
| |
| | |
| ==Observed values==
| |
| {| class="wikitable sortable" style="width:100%; font-size:96%;"
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| |-
| |
| ! Date published
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| ! Hubble constant <br/> (km/s)/Mpc
| |
| ! Observer
| |
| ! class="unsortable"| Citation
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| ! class="unsortable" | Remarks / methodology
| |
| <!-- Add entries in reverse chronological order (newest at the top). Do not remove older entries unless retracted by publisher. Add notes in remarks column as needed. -->
| |
| |-
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| | 2013-03-21
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| | {{val|67.80|0.77}}
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| | [[Planck (spacecraft)#2013 data release|Planck Mission]]
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| | <ref name="planck_overview">
| |
| {{cite arxiv
| |
| |last=Bucher |first=P. A. R.
| |
| |coauthors=''et al.'' ([[Planck Collaboration]])
| |
| |year=2013
| |
| |title=Planck 2013 results. I. Overview of products and scientific Results
| |
| |eprint=1303.5062
| |
| |class=astro-ph.CO
| |
| }}</ref><ref name="ESA-20130321">
| |
| {{cite web
| |
| |date=21 March 2013
| |
| |title=Planck reveals an almost perfect universe
| |
| |url=http://www.esa.int/Our_Activities/Space_Science/Planck/Planck_reveals_an_almost_perfect_Universe
| |
| |publisher=[[European Space Agency|ESA]]
| |
| |accessdate=2013-03-21
| |
| }}</ref><ref name="NASA-20130321">
| |
| {{cite web
| |
| |title=Planck Mission Brings Universe Into Sharp Focus
| |
| |url=http://www.jpl.nasa.gov/news/news.php?release=2013-109&rn=news.xml&rst=3739
| |
| |date=21 March 2013
| |
| |publisher=[[Jet Propulsion Laboratory|JPL]]
| |
| |accessdate=2013-03-21
| |
| }}</ref><ref name="NYT-20130321">
| |
| {{cite news
| |
| |last=Overbye |first=D.
| |
| |title=An infant universe, born before we knew
| |
| |url=http://www.nytimes.com/2013/03/22/science/space/planck-satellite-shows-image-of-infant-universe.html
| |
| |date=21 March 2013
| |
| |work=[[New York Times]]
| |
| |accessdate=2013-03-21
| |
| }}</ref><ref name="NBC-20130321">
| |
| {{cite web
| |
| |last=Boyle |first=A.
| |
| |date=21 March 2013
| |
| |title=Planck probe's cosmic 'baby picture' revises universe's vital statistics
| |
| |url=http://cosmiclog.nbcnews.com/_news/2013/03/21/17397298-planck-probes-cosmic-baby-picture-revises-universes-vital-statistics
| |
| |work=[[NBC News]]
| |
| |accessdate=2013-03-21
| |
| }}</ref>
| |
| | The [[European Space Agency|ESA]] [[Planck Surveyor]] was launched in May 2009. Over a four-year period, it performed a significantly more detailed investigation of cosmic microwave radiation than earlier investigations using [[HEMT]] [[radiometer]]s and [[bolometer]] technology to measure the [[CMB]] at a smaller scale than [[WMAP]]. On 21 March 2013, the European-led research team behind the Planck cosmology probe released the mission's data including a new CMB all-sky map and their determination of the Hubble constant.
| |
| |-
| |
| | 2012-12-20
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| | {{val|69.32|0.80}}
| |
| | [[Wilkinson Microwave Anisotropy Probe|WMAP]] (9-years)
| |
| | <ref>
| |
| {{cite journal
| |
| |last1=Bennett |first1=C. L.
| |
| |coauthors=''et al.''
| |
| |year=2013
| |
| |title=Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Final maps and results
| |
| |journal=[[The Astrophysical Journal Supplement Series]]
| |
| |volume=208 |issue=2 |pages=20
| |
| |arxiv=1212.5225
| |
| |bibcode=2013ApJS..208...20B
| |
| |doi=10.1088/0067-0049/208/2/20
| |
| }}</ref>
| |
| |
| |
| |-
| |
| | 2010
| |
| | {{val|70.4|+1.3|-1.4}}
| |
| | WMAP (7-years), combined with other measurements.
| |
| | <ref name="wmap7parameters">
| |
| {{cite journal
| |
| |last1=Jarosik |first1=N.
| |
| |coauthors=''et al.''
| |
| |year=2011
| |
| |title=Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Sky maps, systematic errors, and basic results
| |
| |journal=[[The Astrophysical Journal Supplement Series]]
| |
| |volume=192 |issue=2 |pages=14
| |
| |arxiv=1001.4744
| |
| |bibcode=2011ApJS..192...14J
| |
| |doi=10.1088/0067-0049/192/2/14
| |
| }}</ref>
| |
| | These values arise from fitting a combination of WMAP and other cosmological data to the simplest version of the ΛCDM model. If the data are fit with more general versions, ''H''<sub>0</sub> tends to be smaller and more uncertain: typically around {{val|67|4|u=(km/s)/Mpc}} although some models allow values near {{val|63|u=(km/s)/Mpc}}.<ref>Results for ''H''<sub>0</sub> and other cosmological parameters obtained by fitting a variety of models to several combinations of WMAP and other data are available at the [[NASA]]'s [http://lambda.gsfc.nasa.gov/product/map/current/parameters.cfm LAMBDA website].</ref>
| |
| |-
| |
| | 2010
| |
| | {{val|71.0|2.5}}
| |
| | WMAP only (7-years).
| |
| | <ref name="wmap7parameters"/>
| |
| |
| |
| |-
| |
| | 2009-02
| |
| | {{val|70.1|1.3}}
| |
| | WMAP (5-years). combined with other measurements.
| |
| | <ref name="WMAP2009">{{cite journal
| |
| |last=Hinshaw |first=G.
| |
| |coauthors=''et al.'' (WMAP Collaboration)
| |
| |year=2009
| |
| |title=Five-year Wilkinson Microwave Anisotropy Probe observations: Data processing, sky maps, and basic results
| |
| |journal=[[The Astrophysical Journal Supplement]]
| |
| |volume=180 |issue=2 |pages=225–245
| |
| |arxiv=0803.0732
| |
| |bibcode=2009ApJS..180..225H
| |
| |doi=10.1088/0067-0049/180/2/225
| |
| }}</ref>
| |
| |
| |
| |-
| |
| | 2009-02
| |
| | {{val|71.9|2.6|-2.7}}
| |
| | WMAP only (5-years)
| |
| | <ref name="WMAP2009"/>
| |
| |
| |
| |-
| |
| | 2006-08
| |
| | {{val|77.6|+14.9|-12.5}}
| |
| | [[Chandra X-ray Observatory]]
| |
| | <ref>
| |
| {{cite journal
| |
| |last1=Bonamente |first1=M.
| |
| |last2=Joy |first2=M. K.
| |
| |last3=Laroque |first3=S. J.
| |
| |last4=Carlstrom |first4=J. E.
| |
| |last5=Reese |first5=E. D.
| |
| |last6=Dawson |first6=K. S.
| |
| |year=2006
| |
| |title=Determination of the cosmic distance scale from Sunyaev–Zel'dovich effect and Chandra X‐ray measurements of high‐redshift galaxy clusters
| |
| |journal=[[The Astrophysical Journal]]
| |
| |volume=647 |issue= |pages=25
| |
| |arxiv=astro-ph/0512349
| |
| |bibcode=2006ApJ...647...25B
| |
| |doi=10.1086/505291
| |
| }}</ref>
| |
| |
| |
| |-
| |
| | 2007
| |
| | {{val|70.4|1.5|-1.6}}
| |
| | WMAP (3-years)
| |
| | <ref>
| |
| {{Cite journal
| |
| |last=Spergel |first=D. N.
| |
| |coauthors=''et al.'' (WMAP Collaboration)
| |
| |year=2007
| |
| |title=Three-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Implications for cosmology
| |
| |journal=[[The Astrophysical Journal Supplement Series]]
| |
| |volume=170 |issue=2 |pages=377–408
| |
| |arxiv=astro-ph/0603449
| |
| |bibcode=2007ApJS..170..377S
| |
| |doi=10.1086/513700
| |
| }}</ref>
| |
| |
| |
| |-
| |
| | 2001-05
| |
| | {{val|72|8}}
| |
| | [[Hubble Space Telescope]]
| |
| | <ref name=Freedman2001>
| |
| {{cite journal
| |
| |last=Freedman |first=W. L.
| |
| |coauthors=''et al.''
| |
| |year=2001
| |
| |title=Final results from the Hubble Space Telescope Key Project to measure the Hubble constant
| |
| |journal=[[The Astrophysical Journal]]
| |
| |volume=553 |issue=1 |pages=47–72
| |
| |arxiv=astro-ph/0012376
| |
| |bibcode=2001ApJ...553...47F
| |
| |doi=10.1086/320638
| |
| }}</ref>
| |
| | This project established the most precise optical determination, consistent with a measurement of ''H''<sub>0</sub> based upon [[Sunyaev-Zel'dovich effect]] observations of many [[galaxy cluster]]s having a similar accuracy.
| |
| |-
| |
| | prior to 1996
| |
| | 50–90 (est.)
| |
| |
| |
| | <ref name="Overbye">
| |
| {{cite book
| |
| |last=Overbye |first=D.
| |
| |year=1999
| |
| |chapter=Prologue
| |
| |title=Lonely Hearts of the Cosmos
| |
| |page=1''ff'' |edition=2nd
| |
| |publisher=[[HarperCollins]]
| |
| |isbn=978-0-316-64896-7
| |
| }}</ref>
| |
| |
| |
| |-
| |
| | 1958
| |
| | 75 (est.)
| |
| | [[Allan Sandage]]
| |
| | <ref>
| |
| {{cite journal
| |
| |last=Sandage |first=A. R.
| |
| |title=Current problems in the extragalactic distance scale
| |
| |year=1958
| |
| |journal=[[The Astrophysical Journal]]
| |
| |volume=127 |issue=3 |pages=513–526
| |
| |bibcode=1958ApJ...127..513S
| |
| |doi=10.1086/146483
| |
| }}</ref>
| |
| | This was the first good estimate of ''H''<sub>0</sub>, but it would be decades before a consensus was achieved.
| |
| |-
| |
| |}
| |
| | |
| ==Discovery==
| |
| A decade before Hubble made his observations, a number of [[physicists]] and [[mathematicians]] had established a consistent theory of the relationship between [[spacetime|space and time]] by using [[Einstein's field equations]] of [[general relativity]]. Applying the most [[Big Bang#Theoretical underpinnings|general principles]] to the nature of the [[universe]] yielded a [[Dynamics (mechanics)|dynamic]] solution that conflicted with the then-prevailing notion of a [[static universe]].
| |
| | |
| ===FLRW equations===
| |
| In 1922, [[Alexander Friedman]]n derived his [[Friedmann equations]] from [[Einstein field equations|Einstein's field equations]], showing that the universe might expand at a rate calculable by the equations.<ref>
| |
| {{Cite journal
| |
| |last=Friedman |first=A.
| |
| |year=1922
| |
| |title=Über die Krümmung des Raumes
| |
| |journal=[[Zeitschrift für Physik]]
| |
| |volume=10 |issue=1 |pages=377–386
| |
| |bibcode=1922ZPhy...10..377F
| |
| |doi=10.1007/BF01332580
| |
| }} Translated in {{Cite journal
| |
| |last1=<!-- --> |first1=<!-- -->
| |
| |year=1999
| |
| |title=<!-- -->
| |
| |journal=[[General Relativity and Gravitation]]
| |
| |volume=31 |issue=12 |pages=1991–2000
| |
| |bibcode=1999GReGr..31.1991F
| |
| |doi=10.1023/A:1026751225741
| |
| }}</ref> The parameter used by Friedmann is known today as the [[scale factor (universe)|scale factor]] which can be considered as a [[scale invariant]] form of the [[proportionality constant]] of Hubble's law. [[Georges Lemaître]] independently found a similar solution in 1927. The Friedmann equations are derived by inserting the [[Friedmann-Lemaître-Robertson-Walker metric|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 Theory|Steady State]] theories of cosmology.
| |
| | |
| ===Shape of the universe===
| |
| Before the advent of [[Big Bang|modern cosmology]], there was considerable talk about the [[Size of the universe|size]] and [[shape of the universe|shape]] of the [[universe]]. In 1920, the famous [[Shapley-Curtis debate]] took place between [[Harlow Shapley]] and [[Heber Doust Curtis|Heber D. Curtis]] over this issue. Shapley argued for a small universe the size of the [[Milky Way galaxy]] and Curtis argued that the universe was much larger. The issue was resolved in the coming decade with Hubble's improved observations.
| |
| | |
| ===Cepheid variable stars outside of the Milky Way===
| |
| Edwin Hubble did most of his professional astronomical observing work at [[Mount Wilson Observatory]], 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" took over.
| |
| | |
| ===Combining redshifts with distance measurements===
| |
| {{Disputed-section|date=December 2013}}
| |
| [[File:Hubble constant.JPG|thumb |250px |Fit of [[#Redshift velocity|redshift velocities]] to Hubble's law.<ref name=Keel>{{cite book
| |
| |last=Keel |first=W. C.
| |
| |year=2007
| |
| |title=The Road to Galaxy Formation
| |
| |url=http://books.google.com/?id=BUgJGypUYF0C&pg=PA7
| |
| |pages=7–8 |edition=2nd
| |
| |publisher=[[Springer (publisher)|Springer]]
| |
| |isbn=3-540-72534-2
| |
| }}</ref> Various estimates for the Hubble constant exist. The HST Key ''H''<sub>0</sub> Group fitted type Ia supernovae for redshifts between 0.01 and 0.1 to find that ''H''<sub>0</sub> = 71 ± 2 (statistical) ± 6 (systematic) km s<sup>−1</sup>Mpc<sup>−1</sup>,<ref name=Freedman2001/> while Sandage ''et al.'' find ''H''<sub>0</sub> = 62.3 ± 1.3 (statistical) ± 5 (systematic) km s<sup>−1</sup>Mpc<sup>−1</sup>.<ref name=Weinberg>
| |
| {{cite book
| |
| |last=Weinberg |first=S.
| |
| |year=2008
| |
| |title=Cosmology
| |
| |url=http://books.google.com/?id=nqQZdg020fsC&pg=PA28
| |
| |page=28
| |
| |publisher=[[Oxford University Press]]
| |
| |isbn=0-19-852682-2
| |
| }}</ref>]]
| |
| | |
| The parameters that appear in Hubble’s law: velocities and distances, are not directly measured. In reality we determine, say, a supernova brightness, which provides information about its distance, and the redshift ''z'' = ∆''λ''/''λ'' of its spectrum of radiation. Hubble correlated brightness and parameter ''z''.
| |
| | |
| Combining his measurements of galaxy distances with [[Vesto Slipher]] and [[Milton Humason]]'s measurements of the [[redshifts]] associated with the galaxies, Hubble discovered a rough proportionality between redshift of an object and its distance. Though there was considerable [[variance|scatter]] (now known to be caused by [[peculiar velocity|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.)
| |
| | |
| At the time of discovery and development of Hubble's law it was acceptable to explain redshift phenomenon as a [[Doppler shift]] in the context of special relativity, and use the Doppler formula to associate redshift ''z'' with velocity. Today the velocity-distance relationship of Hubble's law is viewed as a theoretical result with velocity to be connected with observed redshift not by the Doppler effect, but by a cosmological model relating recessional velocity to the expansion of the universe. Even for small ''z'' the velocity entering the Hubble law is no longer interpreted as a Doppler effect, although at small ''z'' the velocity-redshift relation for both interpretations is the same.
| |
| | |
| ====Hubble Diagram====
| |
| Hubble's law can be easily depicted in a "Hubble Diagram" in which the velocity (assumed approximately proportional to the redshift) of an object is plotted with respect to its distance from the observer.<ref>
| |
| {{cite journal
| |
| |last=Kirshner |first=R. P.
| |
| |year=2003
| |
| |title=Hubble's diagram and cosmic expansion
| |
| |journal=[[Proceedings of the National Academy of Sciences]]
| |
| |volume=101 |issue=1 |pages=8–13
| |
| |bibcode=2003PNAS..101....8K
| |
| |doi=10.1073/pnas.2536799100
| |
| }}</ref> A straight line of positive slope on this diagram is the visual depiction of Hubble's law.
| |
| | |
| ===Cosmological constant abandoned===
| |
| {{main|Cosmological constant}}
| |
| After Hubble's discovery was published, [[Albert Einstein]] abandoned his work on the [[cosmological constant]], which he had designed to modify his equations of general relativity, to allow them to produce a static solution which, in their simplest form, model either an expanding or contracting universe.<ref name=mapcc>
| |
| {{cite web
| |
| |title=What is a Cosmological Constant?
| |
| |url=http://map.gsfc.nasa.gov/universe/uni_accel.html
| |
| |publisher=[[Goddard Space Flight Center]]
| |
| |accessdate=2013-10-17
| |
| }}</ref> After Hubble's discovery that the Universe was, in fact, expanding, Einstein called his faulty assumption that the Universe is static his "biggest mistake".<ref name=mapcc /> On its own, general relativity could predict the expansion of the universe, which (through [[Tests of general relativity|observations]] such as the [[Gravitational lens|bending of light by large masses]], or the [[Perihelion precession of Mercury|precession of the orbit of Mercury]]) could be experimentally observed and compared to his theoretical calculations using particular solutions of the equations he had originally formulated.
| |
| | |
| In 1931, Einstein made a trip to Mount Wilson to thank Hubble for providing the observational basis for modern cosmology.<ref>
| |
| {{cite book
| |
| |last=Isaacson |first=W.
| |
| |year=2007
| |
| |title=Einstein: His Life and Universe
| |
| |url=http://books.google.com/books?id=cdxWNE7NY6QC&pg=PA354
| |
| |page=354
| |
| |publisher=[[Simon & Schuster]]
| |
| |isbn=0-7432-6473-8
| |
| }}</ref>
| |
| | |
| The cosmological constant has regained attention in recent decades as a hypothesis for [[dark energy]].<ref>
| |
| {{cite web
| |
| |date=28 November 2007
| |
| |title=Einstein's Biggest Blunder? Dark Energy May Be Consistent With Cosmological Constant
| |
| |url=http://www.sciencedaily.com/releases/2007/11/071127142128.htm
| |
| |publisher=[[Science Daily]]
| |
| |accessdate=2013-06-02
| |
| }}</ref>
| |
| | |
| {{anchor|redshift}}
| |
| | |
| ==Interpretation==
| |
| [[File:Velocity-redshift.JPG|thumb |250px |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. See Davis and Lineweaver.<ref name=D&L>
| |
| {{cite journal
| |
| |last1=Davis |first1=T. M.
| |
| |last2=Lineweaver |first2=C. H.
| |
| |year=2001
| |
| |title=Superluminal Recessional Velocities
| |
| |chapter=<!-- -->
| |
| |journal=[[AIP Conference Proceedings]]
| |
| |volume=555 |pages=348–351
| |
| |arxiv=astro-ph/0011070
| |
| |bibcode=2001AIPC..555..348D
| |
| |doi=10.1063/1.1363540
| |
| }}</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>v = H_0 \, D</math>
| |
| | |
| where
| |
| * <math>v</math> is the recessional velocity, typically expressed in km/s.
| |
| * ''H''<sub>0</sub> is Hubble's constant and corresponds to the value of <math>H</math> (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 ''0''. This value is the same throughout the universe for a given [[comoving time#Comoving coordinates|comoving time]].
| |
| * <math>D</math> 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 ''v'' = ''dD/dt'').
| |
| | |
| Hubble's law is considered a fundamental relation between recessional velocity and distance. However, the relation between recessional velocity and redshift depends on the cosmological model adopted, and is not established except for small redshifts.
| |
| | |
| For distances ''D'' larger than the radius of the [[Hubble sphere]] ''r''<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>r_{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 the expansion of the universe is accelerating (''see'' [[Accelerating universe]]), meaning that for any given galaxy, the recession velocity dD/dt is increasing over time as the galaxy moves to greater and greater distances; however, the Hubble parameter is actually thought to be decreasing with time, meaning that if we were to look at some ''fixed'' distance D and watch a series of different galaxies pass that distance, later galaxies would pass that distance at a smaller velocity than earlier ones.<ref>[http://curious.astro.cornell.edu/question.php?number=575 Is the universe expanding faster than the speed of light?] (see final paragraph)</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 unambiguous for experimental observation. The relation of redshift to recessional velocity is another matter. For an extensive discussion, see Harrison.<ref name=Harrison>
| |
| {{cite journal
| |
| |last=Harrison |first=E.
| |
| |year=1992
| |
| |title=The redshift-distance and velocity-distance laws
| |
| |journal=[[The Astrophysical Journal]]
| |
| |volume=403 |issue= |pages=28–31
| |
| |bibcode=1993ApJ...403...28H
| |
| |doi=10.1086/172179
| |
| }}</ref>
| |
| | |
| ====Redshift velocity====
| |
| The redshift ''z'' often is described as a ''redshift velocity'', which is the recessional velocity that would produce the same redshift ''if'' it were caused by a linear [[Doppler effect]] (which, however, is not the case, as the shift is caused in part by a [[Metric expansion of space|cosmological expansion of space]], and because the velocities involved are too large to use a non-relativistic formula for Doppler shift). This redshift velocity can easily exceed the speed of light.<ref name=Madsen>
| |
| {{cite book
| |
| |last=Madsen |first=M. S.
| |
| |year=1995
| |
| |title=The Dynamic Cosmos
| |
| |url=http://books.google.com/?id=_2GeJxVvyFMC&pg=PA35
| |
| |page=35
| |
| |publisher=[[CRC Press]]
| |
| |isbn=0-412-62300-5
| |
| }}</ref> In other words, to determine the redshift velocity ''v''<sub>rs</sub>, the relation:
| |
| | |
| :<math> v_{rs} \equiv cz \ ,</math>
| |
| | |
| is used.<ref name=Dekel>
| |
| {{cite book
| |
| |last1=Dekel |first1=A.
| |
| |last2=Ostriker |first2=J. P.
| |
| |year=1999
| |
| |title=Formation of Structure in the Universe
| |
| |url=http://books.google.com/?id=yAroX6tx-l0C&pg=PA164
| |
| |page=164
| |
| |publisher=[[Cambridge University Press]]
| |
| |isbn=0-521-58632-1
| |
| }}</ref><ref name=Padmanabhan>
| |
| {{cite book
| |
| |last=Padmanabhan |first=T.
| |
| |year=1993
| |
| |title=Structure formation in the universe
| |
| |url=http://books.google.com/?id=AJlOVBRZJtIC&pg=PA58
| |
| |page=58
| |
| |publisher=[[Cambridge University Press]]
| |
| |isbn=0-521-42486-0
| |
| }}</ref> 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 [[Relativistic Doppler effect|Fizeau-Doppler formula]]<ref name=Sartori>
| |
| {{cite book
| |
| |last=Sartori |first=L.
| |
| |year=1996
| |
| |title=Understanding Relativity
| |
| |page=163, Appendix 5B
| |
| |publisher=[[University of California Press]]
| |
| |isbn=0-520-20029-2
| |
| }}</ref>
| |
| | |
| :<math>z = \frac{\lambda_o}{\lambda_e}-1 = \sqrt{\frac{1+v/c}{1-v/c}}-1 \approx \frac{v}{c} \ .</math>
| |
| | |
| Here, ''λ''<sub>o</sub>, ''λ''<sub>e</sub> are the observed and emitted wavelengths respectively. The "redshift velocity" ''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. This discussion is based on Sartori.<ref name=L_Sartori>
| |
| {{cite book
| |
| |last=Sartori |first=L.
| |
| |year=1996
| |
| |title=Understanding Relativity
| |
| |pages=304–305
| |
| |publisher=[[University of California Press]]
| |
| |isbn=0-520-20029-2
| |
| }}</ref>
| |
| | |
| ====Recessional velocity====
| |
| {{Citation needed |reason=reliable source needed to validate derivation |date=May 2013}}
| |
| Suppose ''R(t)'' is called the ''[[Scale factor (Universe)|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 distances ''D(t)'' between co-moving points increase proportionally to ''R''. (The co-moving points are not moving relative to each other except as a result of the expansion of space.) In other words:
| |
| | |
| :<math>\frac {D(t)}{D(t_0)} = \frac {R(t)}{R(t_0)} \ , </math>
| |
| | |
| where ''t<sub>0</sub>'' is some reference time. If light is emitted from a galaxy at time ''t<sub>e</sub>'' and received by us at ''t<sub>0</sub>'', it is red shifted due to the expansion of space, and this redshift ''z'' is simply:
| |
| | |
| :<math>z = \frac {R(t_0)}{R(t_e)} - 1 \ . </math>
| |
| | |
| Suppose a galaxy is at distance ''D'', and this distance changes with time at a rate ''d<sub>t</sub>D ''. We call this rate of recession the "recession velocity" ''v<sub>r</sub>'':
| |
| | |
| :<math>v_r = d_tD = \frac {d_tR}{R} D \ . </math>
| |
| | |
| We now define the Hubble constant as
| |
| | |
| :<math>H \equiv \frac {d_tR}{R} \ , </math>
| |
| | |
| and discover the Hubble law:
| |
| | |
| :<math> v_r = H D \ . </math>
| |
| | |
| From this perspective, Hubble's law is a fundamental relation between (i) the recessional velocity contributed by the expansion of space and (ii) the distance to an object; the connection between redshift and distance is a crutch used to connect Hubble's law with observations. This law can be related to redshift ''z'' approximately by making a [[Taylor series]] expansion:
| |
| | |
| :<math> 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> z \approx (t_0-t_e)H(t_0) \approx \frac {D}{c} H(t_0) \ , </math> or <math> cz \approx D H(t_0) = v_r \ . </math>
| |
| According to this approach, the relation ''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 ''v'' nor ''D'' in the formula are directly observable, because they are properties ''now'' of a galaxy, whereas our observations refer to the galaxy in the past, at the time that the light we currently see left it.
| |
| | |
| For relatively nearby galaxies ([[redshift]] ''z'' much less than unity), ''v'' and ''D'' will not have changed much, and ''v'' can be estimated using the formula <math>v = zc</math> 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 travelling towards the observer.
| |
| | |
| ===Expansion velocity vs relative velocity===
| |
| In using Hubble's law to determine distances, only the velocity due to the expansion of the universe can be used. Since gravitationally interacting galaxies move relative to each other independent of the expansion of the universe, these relative velocities, called [[peculiar velocities]], need to be accounted for in the application of Hubble's law.
| |
| | |
| The [[Fingers of God|Finger of God]] effect is one result of this phenomenon. In [[Virial theorem|systems that are gravitationally bound]], such as galaxies or our planetary system, the expansion of space is a much weaker effect than the attractive force of gravity.
| |
| | |
| ===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:
| |
| | |
| :''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===
| |
| [[Image:Friedmann universes.svg|thumb|400px|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 (Ω<sub>M</sub> for [[matter]] and Ω<sub>Λ</sub> for [[dark energy]]). A "closed universe" with Ω<sub>M</sub> > 1 and Ω<sub>Λ</sub> = 0 comes to an end in a [[Big Crunch]] and is considerably younger than its Hubble age. An "open universe" with Ω<sub>M</sub> ≤ 1 and Ω<sub>Λ</sub> = 0 expands forever and has an age that is closer to its Hubble age. For the [[accelerating universe]] with nonzero 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 sign of the so-called [[deceleration parameter]] <math>q</math> which is defined by
| |
| | |
| :<math>q = -\left(1+\frac{\dot H}{H^2}\right).</math>
| |
| | |
| 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 <math>q</math> 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 ''q'' was positive, indicating that the expansion is slowing down due to gravitational attraction. This would imply an age of the universe less than 1/''H'' (which is about 14 billion years). For instance, a value for ''q'' of 1/2 (once favoured by most theorists) would give the age of the universe as 2/(3''H''). The discovery in 1998 that ''q'' is apparently negative means that the universe could actually be older than 1/''H''. However, estimates of the [[age of the universe]] are very close to 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]], [[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. 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 upon the [[Big Bang]] theory and in part upon the [[Hubble expansion]]. In a universe that exists for a finite amount of time, only the light of finitely many stars has had a chance to reach us yet, and the paradox is resolved. Additionally, in an expanding universe distant objects [[recessional velocity|recede]] from us, which causes the light emanating from them to be [[redshift]]ed and diminished in brightness.<ref>
| |
| {{cite web
| |
| |last=Chase |first=S. I.
| |
| |last2=Baez |first2=J. C.
| |
| |year=2004
| |
| |title=Olbers' Paradox
| |
| |url=http://math.ucr.edu/home/baez/physics/Relativity/GR/olbers.html
| |
| |work=The Original Usenet Physics FAQ
| |
| |accessdate=2013-10-17
| |
| }} See also {{cite book
| |
| |last=Asimov |first=I.
| |
| |year=1974
| |
| |chapter=The Black of Night
| |
| |title=Asimov on Astronomy
| |
| |publisher=[[Doubleday (publisher)|Doubleday]]
| |
| |isbn=0-385-04111-X
| |
| }}</ref>
| |
| | |
| ===Dimensionless Hubble parameter===
| |
| Instead of working with Hubble's constant, a common practice is to introduce the '''dimensionless Hubble parameter''', usually denoted by ''h'', and to write the Hubble's parameter ''H<sub>0</sub>'' as 100 ''h'' km [[second|s]] <sup>−1</sup> [[megaparsec|Mpc]]<sup>−1</sup>, all the uncertainty relative of the value of ''H<sub>0</sub>'' being then relegated on ''h''.<ref>
| |
| {{cite book
| |
| |last=Peebles |first=P. J. E.
| |
| |year=1993
| |
| |title=Principles of Physical Cosmology
| |
| |publisher=[[Princeton University Press]]
| |
| |isbn=
| |
| }}</ref>
| |
| | |
| ==Determining the Hubble constant==
| |
| [[File:Hubble-constant-vers2.png|thumb|right|Value of the Hubble Constant including measurement uncertainty above measurement method]]
| |
| The value of the Hubble constant is estimated by measuring the [[redshift]] of distant galaxies and then [[cosmic distance ladder|determining the distances to the same galaxies]] (by some other method than Hubble's law). Uncertainties in the physical assumptions used to determine these distances have caused varying estimates of the Hubble constant.
| |
| | |
| ===Earlier measurement and discussion approaches===
| |
| For most of the second half of the 20th century the value of <math>H_0</math> was estimated to be between 50 and {{nowrap|90 (km/s)/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.<ref name="Overbye"/> In 1996, a debate moderated by [[John Bahcall]] between [[Gustav Tammann]] and [[Sidney van den Bergh]] 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 [[Lambda-CDM model|ΛCDM model]] of the universe in the late 1990s. With the [[ΛCDM]] model observations of high-redshift clusters at X-ray and microwave wavelengths using the [[Sunyaev-Zel'dovich effect]], measurements of anisotropies in the [[cosmic microwave background radiation]], and optical surveys all gave a value of around 70 for the constant.{{Citation needed|date=July 2009}}
| |
| | |
| The consistency of the measurements from all these methods lends support to both the measured value of <math>H_0</math> and the [[Lambda-CDM model|ΛCDM model]].
| |
| | |
| ''See table of measurements above for many recent and older measurements.''
| |
| | |
| ===Acceleration of the expansion===
| |
| {{Main|Accelerating universe}}
| |
| A value for <math>q</math> 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.
| |
| |year=2003
| |
| |title=Supernovae, Dark Energy, and the Accelerating Universe
| |
| |url=http://www.supernova.lbl.gov/PhysicsTodayArticle.pdf
| |
| |journal=[[Physics Today]]
| |
| |volume=56 |issue=4 |pages=53–60
| |
| |arxiv=
| |
| |bibcode= 2003PhT....56d..53P
| |
| |doi=10.1063/1.1580050
| |
| }}</ref> (although the Hubble factor is still decreasing with time, as mentioned above in the [[Hubble's law#Interpretation|Interpretation]] section; see the articles on [[dark energy]] and the [[ΛCDM model]]).
| |
| | |
| ==Derivation of the Hubble parameter==
| |
| {{Citation needed |reason=reliable source needed to validate derivation |date=May 2013}}
| |
| Start with the [[Friedmann equations|Friedmann equation]]:
| |
| | |
| :<math>H^2 \equiv \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2}+ \frac{\Lambda c^2}{3},</math>
| |
| | |
| where <math>H</math> is the Hubble parameter, <math>a</math> is the [[scale factor (universe)|scale factor]], G is the [[gravitational constant]], <math>k</math> is the normalised spatial curvature of the universe and equal to −1, 0, or +1, and <math>\Lambda</math> is the [[cosmological constant]].
| |
| | |
| ===Matter-dominated universe (with a cosmological constant)===
| |
| {{Citation needed |reason=reliable source needed to validate derivation |date=May 2013}}
| |
| If the universe is [[Matter-dominated era|matter-dominated]], then the mass density of the universe <math>\rho</math> can just be taken to include matter so
| |
| | |
| :<math>\rho = \rho_m(a) = \frac{\rho_{m_{0}}}{a^3},</math>
| |
| | |
| where <math>\rho_{m_{0}}</math> is the density of matter today. We know for nonrelativistic 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>\Omega_m</math>)
| |
| | |
| :<math>\rho_c = \frac{3 H^2}{8 \pi G};</math>
| |
| | |
| :<math>\Omega_m \equiv \frac{\rho_{m_{0}}}{\rho_c} = \frac{8 \pi G}{3 H_0^2}\rho_{m_{0}};</math>
| |
| | |
| so <math>\rho=\rho_c \Omega_m /a^3.</math> Also, by definition,
| |
| | |
| :<math>\Omega_k \equiv \frac{-kc^2}{(a_0H_0)^2}</math>
| |
| | |
| and
| |
| | |
| :<math>\Omega_{\Lambda} \equiv \frac{\Lambda c^2}{3H_0^2},</math>
| |
| | |
| where the subscript nought refers to the values today, and <math>a_0=1</math>. Substituting all of this in into the Friedmann equation at the start of this section and replacing <math>a</math> with <math>a=1/(1+z)</math> gives
| |
| | |
| :<math>H^2(z)= H_0^2 \left( \Omega_M (1+z)^{3} + \Omega_k (1+z)^{2} + \Omega_{\Lambda} \right).</math>
| |
| | |
| ===Matter- and dark energy-dominated universe===
| |
| {{Citation needed |reason=reliable source needed to validate derivation |date=May 2013}}
| |
| If the universe is both [[Matter-Dominated Era|matter-dominated]] and [[dark energy]]- dominated, then the above equation for the Hubble parameter will also be a function of the [[equation of state (cosmology)|equation of state of dark energy]]. So now:
| |
| | |
| :<math>\rho = \rho_m (a)+\rho_{de}(a),</math>
| |
| | |
| where <math>\rho_{de}</math> is the mass density of the dark energy. By definition an equation of state in cosmology is <math>P=w\rho c^2</math>, and if we substitute this into the fluid equation, which describes how the mass density of the universe evolves with time,
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| :<math>\dot{\rho}+3\frac{\dot{a}}{a}\left(\rho+\frac{P}{c^2}\right)=0;</math>
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| :<math>\frac{d\rho}{\rho}=-3\frac{da}{a}\left(1+w\right).</math>
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| If w is constant,
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| :<math>\ln{\rho}=-3\left(1+w\right)\ln{a};</math>
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| :<math>\rho=a^{-3\left(1+w\right)}.</math>
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| Therefore for dark energy with a constant equation of state w, <math>\rho_{de}(a)= \rho_{de0}a^{-3\left(1+w\right)}</math>. If we substitute this into the Friedman equation in a similar way as before, but this time set <math>k=0</math> which is assuming we live in a spatially flat universe, (see [[Shape of the Universe]])
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| :<math>H^2(z)= H_0^2 \left( \Omega_M (1+z)^{3} + \Omega_{de}(1+z)^{3\left(1+w \right)} \right).</math>
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| If dark energy does not have a constant equation-of-state w, then
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| :<math>\rho_{de}(a)= \rho_{de0}e^{-3\int\frac{da}{a}\left(1+w(a)\right)},</math>
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| and to solve this we must parametrize <math>w(a)</math>, for example if <math>w(a)=w_0+w_a(1-a)</math>, giving
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| :<math>H^2(z)= H_0^2 \left( \Omega_M a^{-3} + \Omega_{de}a^{-3\left(1+w_0 +w_a \right)}e^{-3w_a(1-a)} \right).</math>
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| Other ingredients have been formulated recently.<ref>
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| {{cite journal
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| |last1=Tawfik |first1=A.
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| |last2=Harko |first2=T.
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| |year=2012
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| |title=Quark-hadron phase transitions in the viscous early universe
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| |journal=[[Physical Review D]]
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| |volume=85 |issue=8 |pages=084032
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| |arxiv=1108.5697
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| |bibcode=2012PhRvD..85h4032T
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| |doi=10.1103/PhysRevD.85.084032
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| }}</ref><ref>
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| {{cite journal
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| |last1=Tawfik |first1=A.
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| |year=2011
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| |title=The Hubble parameter in the early universe with viscous QCD matter and finite cosmological constant
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| |journal=[[Annalen der Physik]]
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| |volume=523 |issue=5 |pages=423
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| |arxiv=1102.2626
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| |bibcode=2011AnP...523..423T
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| |doi=10.1002/andp.201100038
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| }}</ref><ref>
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| {{cite journal
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| |last1=Tawfik |first1=A.
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| |last2=Wahba |first2=M.
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| |last3=Mansour |first3=H.
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| |last4=Harko |first4=T.
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| |year=2011
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| |title=Viscous quark-gluon plasma in the early universe
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| |journal=[[Annalen der Physik]]
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| |volume=523 |issue=3 |pages=194
| |
| |arxiv=1001.2814
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| |bibcode=2011AnP...523..194T
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| |doi=10.1002/andp.201000052
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| }}</ref> In certain era, where the high energy experiments seem to have a reliable access in analyzing the property of the matter dominating the background geometry, with this era we mean the quark-gluon plasma, the transport properties have been taken into consideration. Therefore, the evolution of the Hubble parameter and of other essential cosmological parameters, in such a background are found to be considerably (non-negligibly) different than their evolution in an ideal, gaseous, non-viscous background.
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| ==Units derived from the Hubble constant==
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| ===Hubble time===
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| {{confusing|date=April 2013}}
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| The Hubble constant ''H''<sub>0</sub> has units of inverse time, i.e. ''H''<sub>0</sub> ≈ {{val|2.3|e=-18|u=s<sup>−1</sup>}}. "Hubble time" is defined as 1/''H''<sub>0</sub>. The value of Hubble time in the [[lambda-CDM model|standard cosmological model]] is {{val|4.35|e=17|u=s}} or 13.8 billion years. {{Harv|Liddle|2003|p=57}} The phrase "expansion timescale" means "Hubble time".{{citation needed|date=October 2013}}
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| The Hubble unit is defined as ''hH''<sub>0</sub>, where ''h'' is around 1, and denotes the uncertainty in ''H''<sub>0</sub>. ''H''<sub>0</sub> is 100 km/s / Mpc = 1 dm/s/pc. The unit of time, then has as many seconds as there are decimetres in a parsec.
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| As mentioned above, ''H''<sub>0</sub> is the current value of Hubble parameter ''H''. In a model in which speeds are constant, ''H'' decreases with time. In the naive model where ''H'' is constant the Hubble time would be the time taken for the universe to increase in size by a factor of e (because the solution of ''dx''/''dt'' = ''xH''<sub>0</sub> is ''x'' = ''s''<sub>0</sub>exp(''H''<sub>0</sub>''t''), where ''s''<sub>0</sub> is the size of some feature at some arbitrary initial condition ''t'' = 0).
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| Over long periods of time the dynamics are complicated by [[general relativity]], [[dark energy]], [[Inflation (cosmology)|inflation]], etc., as explained above.
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| ===Hubble length===
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| The Hubble length or Hubble distance is a unit of distance in cosmology, defined as ''cH''<sub>0</sub><sup>-1</sup>—the speed of light multiplied by the Hubble time. It is equivalent to 4,228 million parsecs or 13.8 billion light years. (The numerical value of the Hubble length in light years is, by definition, equal to that of the Hubble time in years.) The Hubble distance would be the distance between the Earth and the galaxies which are ''currently'' receding from us at the speed of light, as can be seen by substituting {{nowrap|''D'' {{=}} ''c''/''H''<sub>0</sub>}} into the equation for Hubble's law, {{nowrap|''v'' {{=}} ''H''<sub>0</sub>''D''}}.
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| ===Hubble volume===
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| {{main|Hubble volume}}
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| The Hubble volume is sometimes defined as a volume of the universe with a [[comoving distance|comoving]] size of ''cH''<sub>0</sub>. The exact definition varies: it is sometimes defined as the volume of a sphere with radius ''cH''<sub>0</sub>, or alternatively, a cube of side ''cH''<sub>0</sub>. 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.
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| ==See also==
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| * [[Cosmology]]
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| * [[Dark energy]]
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| * [[Dark matter]]
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| * [[Tests of general relativity]]
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| ==Notes==
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| {{reflist|30em|refs=
| |
| | |
| <ref name="riess99">{{cite journal | author=Riess, A. et al. | title=Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant |date=September 1998 | journal=The Astronomical Journal | volume=116 | pages=1009–1038 | doi=10.1086/300499 | bibcode=1998AJ....116.1009R | issue=3|arxiv = astro-ph/9805201 }}</ref>
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| <ref name="perlmutter99">{{cite journal | author=Perlmutter, S. et al. | title=Measurements of Omega and Lambda from 42 High-Redshift Supernovae |date=June 1999 | journal=The Astrophysical Journal | volume=517 | issue=2 | pages=565–586 | doi=10.1086/307221 | bibcode=1999ApJ...517..565P|arxiv = astro-ph/9812133 }}</ref>
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| }}
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| ==References==
| |
| * {{Cite book
| |
| |last=Hubble |first=E. P.
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| |year=1937
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| |title=The Observational Approach to Cosmology
| |
| |publisher=[[Clarendon Press]]
| |
| |lccn=38011865
| |
| }}
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| * {{Cite book
| |
| |last=Kutner |first=M.
| |
| |year=2003
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| |title=Astronomy: A Physical Perspective
| |
| |publisher=[[Cambridge University Press]]
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| |isbn=0-521-52927-1
| |
| }}
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| * {{Cite book
| |
| |last=Liddle |first=A. R.
| |
| |year=2003
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| |title=An Introduction to Modern Cosmology
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| |edition=2nd
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| |publisher=[[John Wiley & Sons]]
| |
| |isbn=0-470-84835-9
| |
| }}
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| ==Further reading==
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| * {{Cite journal
| |
| |last1=Freedman | first1=W. L.
| |
| |last2=Madore | first2=B. F.
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| |year=2010
| |
| |title=The Hubble Constant
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| |journal=[[Annual Review of Astronomy and Astrophysics]]
| |
| |volume=48 |issue= |pages=673
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| |arxiv=1004.1856
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| |bibcode=2010ARA&A..48..673F
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| |doi=10.1146/annurev-astro-082708-101829
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| }}
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| ==External links==
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| * [http://map.gsfc.nasa.gov/universe/bb_tests_exp.html NASA's WAMP - Big Bang Expansion: the Hubble Constant]
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| * [http://www.ipac.caltech.edu/H0kp/H0KeyProj.html The Hubble Key Project]
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| * [http://cas.sdss.org/dr3/en/proj/advanced/hubble/ The Hubble Diagram Project]
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| * {{cite web|last=Merrifield|first=Michael|title=Hubble Constant|url=http://www.sixtysymbols.com/videos/hubble.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|year=2009}}
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| * [http://alemanow.narod.ru/hubbles.htm Hubble's quantum law.]
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| {{DEFAULTSORT:Hubble's Law}}
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| [[Category:Edwin Hubble]]
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| [[Category:Large-scale structure of the cosmos]]
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| [[Category:Physical cosmology]]
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