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* the [[accelerating expansion of the universe]] observed in the light from distant [[Galaxy|galaxies]] and [[supernova]]e.
The model assumes that [[general relativity]] is the correct theory of gravity on cosmological scales. It emerged in the late 1990s as a '''concordance cosmology''', after a period
The ΛCDM model has been successful in modeling a broad collection of astronomical observations over decades. Remaining issues
== Overview ==
The ΛCDM model is based on three postulates on the structure of [[spacetime]]:<ref name="Longair-2009">{{Cite book|date=2008 |title=Galaxy Formation
|author=Malcolm S. Longair |url=http://link.springer.com/10.1007/978-3-540-73478-9 |series=Astronomy and Astrophysics Library |language=en |___location=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |doi=10.1007/978-3-540-73478-9 |isbn=978-3-540-73477-2}}</ref>{{rp|227}} # The [[cosmological principle]], that the universe is the same everywhere and in all directions, and that it is expanding,
# A postulate by [[Hermann Weyl]] that the lines of spacetime ([[geodesics]]) intersect at only one point, where time along each line can be synchronized; the behavior resembles an expanding [[perfect fluid]],<ref name="Longair-2009"/>{{rp|175}}
# [[general relativity]] that relates the geometry of spacetime to the distribution of matter and energy.
This combination greatly simplifies the equations of general relativity into a form called the [[Friedmann equations]]. These equations specify the evolution of the [[Scale factor (cosmology)|scale factor]] of the universe in terms of the pressure and density of a perfect fluid. The evolving density is composed of different kinds of energy and matter, each with its own role in affecting the scale factor.<ref>{{Cite book |last=White |first=Simon |title=Physics of the Early Universe: Proceedings of the Thirty Sixth Scottish Universities Summer School in Physics, Edinburgh, July 24 - August 11 1989 |date=1990 |publisher=Taylor & Francis Group |isbn=978-1-040-29413-0 |edition=1 |series=Scottish Graduate Series |___location=Milton |chapter=Physical Cosmology}}</ref>{{rp|7}} For example, a model might include [[baryons]], [[photons]], [[neutrinos]], and [[dark matter]].<ref name=PDG-2024>{{Cite journal |
The most accurate observations which are sensitive to the component densities are consequences of statistical inhomogeneity called "perturbations" in the early universe. Since the Friedmann equations assume homogeneity, additional theory must be added before comparison to experiments. [[Inflation (cosmology)|Inflation]] is a simple model producing perturbations by postulating an extremely rapid expansion early in the universe that separates quantum fluctuations before they can equilibrate. The perturbations are characterized by additional parameters also determined by matching observations.<ref name=PDG-2024/>{{rp|25.1.2}}
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Finally, the light which will become astronomical observations must pass through the universe. The latter part of that journey will pass through [[reionization|ionized space]], where the electrons can scatter the light, altering the anisotropies. This effect is characterized by one additional parameter.<ref name=PDG-2024/>{{rp|25.1.3}}
The ΛCDM model includes an expansion of the spatial [[
The letter Λ ([[lambda]]) represents the [[cosmological constant]], which is associated with a vacuum energy or [[dark energy]] in empty space that is used to explain the contemporary accelerating expansion of space against the attractive effects of gravity. A cosmological constant has negative pressure, <math> p = - \rho c^{2} </math>, which contributes to the [[stress–energy tensor]] that, according to the general theory of relativity, causes accelerating expansion. The fraction of the total energy density of our (flat or almost flat) universe that is dark energy, <math>\Omega_{\Lambda}</math>, is estimated to be 0.669 ± 0.038 based on the 2018 [[Dark Energy Survey]] results using [[Type Ia supernova]]e<ref>{{Cite journal |arxiv = 1811.02374|author=DES Collaboration |title = First Cosmology Results using Type Ia Supernovae from the Dark Energy Survey: Constraints on Cosmological Parameters|journal = The Astrophysical Journal|volume = 872|issue = 2|pages = L30|year = 2018|doi = 10.3847/2041-8213/ab04fa|s2cid = 84833144 |doi-access=free |bibcode=2019ApJ...872L..30A }}</ref> or {{val|0.6847|0.0073}} based on the 2018 release of [[Planck (spacecraft)|''Planck'' satellite]] data, or more than 68.3% (2018 estimate) of the mass–energy density of the universe.<ref>{{Cite journal |arxiv = 1807.06209|author=Planck Collaboration|title = Planck 2018 results. VI. Cosmological parameters|journal = Astronomy & Astrophysics|year = 2020|volume = 641|pages = A6|doi = 10.1051/0004-6361/201833910|bibcode = 2020A&A...641A...6P|s2cid = 119335614}}</ref>
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* Cold: Its velocity is far less than the speed of light at the epoch of radiation–matter equality (thus neutrinos are excluded, being non-baryonic but not cold)
* Dissipationless: Cannot cool by radiating photons
* Collisionless: Dark matter particles interact with each other and other particles only through gravity and possibly the [[weak force]]
Dark matter constitutes about 26.5%<ref name="PDG2019">{{cite journal |first1=M. |last1= Tanabashi |display-authors=etal |collaboration=[[Particle Data Group]] |url=http://pdg.lbl.gov/2019/reviews/rpp2019-rev-astrophysical-constants.pdf |title=Astrophysical Constants and Parameters |publisher=[[Particle Data Group]] |year=2019 |access-date=2020-03-08 |journal=Physical Review D |volume=98 |issue=3 |page=030001|doi= 10.1103/PhysRevD.98.030001|doi-access=free |bibcode= 2018PhRvD..98c0001T }}</ref> of the mass–energy density of the universe. The remaining 4.9%<ref name="PDG2019"/> comprises all ordinary matter observed as atoms, chemical elements, gas and [[Plasma (physics)|plasma]], the stuff of which visible planets, stars and galaxies are made. The great majority of ordinary matter in the universe is unseen, since visible stars and gas inside galaxies and clusters account for less than 10% of the ordinary matter contribution to the mass–energy density of the universe.<ref>
{{cite journal
| last1 = Persic
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}}</ref>
The model includes a single originating event, the "[[Big Bang]]", which was not an explosion but the abrupt appearance of expanding [[spacetime]] containing radiation at temperatures of around 10<sup>15</sup> K. This was immediately (within 10<sup>−29</sup> seconds) followed by an exponential expansion of space by a scale multiplier of 10<sup>27</sup> or more, known as [[cosmic inflation]]. The early universe remained hot (above 10
== Cosmic expansion history ==
The expansion of the universe is parameterized by a [[dimensionless]] [[scale factor (cosmology)|scale factor]] <math>a = a(t)</math> (with time <math>t</math> counted from the birth of the universe), defined relative to the present time, so <math>a_0 = a(t_0) = 1 </math>; the usual convention in cosmology is that subscript 0 denotes present-day values, so <math>t_0</math> denotes the age of the universe. The scale factor is related to the observed [[Redshift#Expansion of space|redshift]]<ref name="Dodelson"/> <math>z</math> of the light emitted at time <math>t_\mathrm{em}</math> by
<math display="block">a(t_\text{em}) = \frac{1}{1 + z}\,.</math>
The expansion rate is described by the time-dependent [[Hubble parameter]], <math>H(t)</math>, defined as
<math display="block">H(t) \equiv \frac{\dot a}{a},</math>
where <math>\dot a</math> is the time-derivative of the scale factor. The first [[Friedmann equations|Friedmann equation]] gives the expansion rate in terms of the matter+radiation density {{nowrap|<math>\rho</math>,}} the [[Curvature of the universe|curvature]] {{nowrap|<math>k</math>,}} and the [[cosmological constant]] {{nowrap|<math>\Lambda</math>,}}<ref name="Dodelson">{{cite book |last=Dodelson |first=Scott |title=Modern cosmology |date=2008 |publisher=[[Academic Press]] |___location=San Diego, CA |isbn=978-0-12-219141-1 |edition=4}}</ref>
<math display="block">H^2 = \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, as usual <math>c</math> is the speed of light and <math>G</math> is the [[gravitational constant]].
A critical density <math>\rho_\mathrm{crit}</math> is the present-day density, which gives zero curvature <math>k</math>, assuming the cosmological constant <math>\Lambda</math> is zero, regardless of its actual value. Substituting these conditions to the Friedmann equation gives{{refn|name=constants|{{cite web|url=http://pdg.lbl.gov/2015/reviews/rpp2014-rev-astrophysical-constants.pdf |title=The Review of Particle Physics. 2. Astrophysical constants and parameters |author=K.A. Olive |collaboration=Particle Data Group |website=Particle Data Group: Berkeley Lab |date=2015 |access-date=10 January 2016 |archive-url=https://web.archive.org/web/20151203100912/http://pdg.lbl.gov/2015/reviews/rpp2014-rev-astrophysical-constants.pdf |archive-date= 3 December 2015 }}}}
<math display="block">\rho_\mathrm{crit} = \frac{3 H_0^2}{8 \pi G} = 1.878\;47(23) \times 10^{-26} \; h^2 \; \mathrm{kg{\cdot}m^{-3}},</math>
where <math> h \equiv H_0 / (100 \; \mathrm{km{\cdot}s^{-1}{\cdot}Mpc^{-1}}) </math> is the reduced Hubble constant.
If the cosmological constant were actually zero, the critical density would also mark the dividing line between eventual recollapse of the universe to a [[Big Crunch]], or unlimited expansion. For the Lambda-CDM model with a positive cosmological constant (as observed), the universe is predicted to expand forever regardless of whether the total density is slightly above or below the critical density; though other outcomes are possible in extended models where the [[dark energy]] is not constant but actually time-dependent.{{citation needed|date=February 2024}}
The present-day '''density parameter''' <math>\Omega_x</math> for various species is defined as the dimensionless ratio<ref name=Peacock-1998/>{{rp|p=74}}
<math display="block">\Omega_x \equiv \frac{\rho_x(t=t_0)}{\rho_\mathrm{crit} } = \frac{8 \pi G\rho_x(t=t_0)}{3 H_0^2}</math>
where the subscript <math>x</math> is one of <math>\mathrm b</math> for [[baryon]]s, <math>\mathrm c</math> for [[cold dark matter]], <math>\mathrm{rad}</math> for [[radiation]] ([[photon]]s plus relativistic [[neutrino]]s), and <math>\Lambda</math> for [[dark energy]].{{citation needed|date=February 2024}}
Since the densities of various species scale as different powers of <math>a</math>, e.g. <math>a^{-3}</math> for matter etc.,
the [[Friedmann equation]] can be conveniently rewritten in terms of the various density parameters as
<math display="block">H(a) \equiv \frac{\dot{a}}{a} = H_0 \sqrt{ (\Omega_{\rm c} + \Omega_{\rm b}) a^{-3} + \Omega_\mathrm{rad} a^{-4} + \Omega_k a^{-2} + \Omega_{\Lambda} a^{-3(1+w)} } ,</math>
where <math>w</math> is the [[Equation of state (cosmology)|equation of state]] parameter of dark energy, and assuming negligible neutrino mass (significant neutrino mass requires a more complex equation). The various <math> \Omega </math> parameters add up to <math>1</math> by construction. In the general case this is integrated by computer to give the expansion history <math>a(t)</math> and also observable distance–redshift relations for any chosen values of the cosmological parameters, which can then be compared with observations such as [[supernovae]] and [[baryon acoustic oscillations]].{{citation needed|date=February 2024}}
In the minimal 6-parameter Lambda-CDM model, it is assumed that curvature <math>\Omega_k</math> is zero and <math> w = -1 </math>, so this simplifies to
<math display="block"> H(a) = H_0 \sqrt{ \Omega_{\rm m} a^{-3} + \Omega_\mathrm{rad} a^{-4} + \Omega_\Lambda } </math>
Observations show that the radiation density is very small today, <math> \Omega_\text{rad} \sim 10^{-4} </math>; if this term is neglected
the above has an analytic solution<ref>{{cite journal|last1=Frieman|first1=Joshua A.|last2=Turner|first2=Michael S.|last3=Huterer|first3=Dragan|title=Dark Energy and the Accelerating Universe|journal=Annual Review of Astronomy and Astrophysics|year=2008|volume=46|issue=1|pages=385–432|arxiv=0803.0982|doi=10.1146/annurev.astro.46.060407.145243|bibcode=2008ARA&A..46..385F|s2cid=15117520}}</ref>
<math display="block"> a(t) = (\Omega_{\rm m} / \Omega_\Lambda)^{1/3} \, \sinh^{2/3} ( t / t_\Lambda) </math>
where <math> t_\Lambda \equiv 2 / (3 H_0 \sqrt{\Omega_\Lambda} ) \ ; </math>
this is fairly accurate for <math>a > 0.01</math> or <math>t > 10</math> million years.
Solving for <math> a(t) = 1 </math> gives the present age of the universe <math> t_0 </math> in terms of the other parameters.{{citation needed|date=February 2024}}
It follows that the transition from decelerating to accelerating expansion (the second derivative <math> \ddot{a} </math> crossing zero) occurred when
<math display="block"> a = ( \Omega_{\rm m} / 2 \Omega_\Lambda )^{1/3} ,</math>
which evaluates to <math>a \sim 0.6</math> or <math>z \sim 0.66</math> for the best-fit parameters estimated from the [[Planck (spacecraft)|''Planck'' spacecraft]].{{citation needed|date=February 2024}}
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|+ Planck Collaboration Cosmological parameters
!   
! Description<ref name="Planck-2013">The parameters used in the Planck series of papers are described in Table 1 of {{Cite journal |
! Symbol
! Value-2018<ref name="Planck 2018">
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* Prediction of the observed [[Polarization (cosmology)|B-mode polarization]] of the CMB light due to primordial gravitational waves.<ref name="Ade-BModes-2021">{{Cite journal |last1=Ade |first1=P. A. R. |last2=Ahmed |first2=Z. |last3=Amiri |first3=M. |last4=Barkats |first4=D. |last5=Thakur |first5=R. Basu |last6=Bischoff |first6=C. A. |last7=Beck |first7=D. |last8=Bock |first8=J. J. |last9=Boenish |first9=H. |last10=Bullock |first10=E. |last11=Buza |first11=V. |last12=Cheshire |first12=J. R. |last13=Connors |first13=J. |last14=Cornelison |first14=J. |last15=Crumrine |first15=M. |date=2021-10-04 |title=Improved Constraints on Primordial Gravitational Waves using Planck , WMAP, and BICEP/ Keck Observations through the 2018 Observing Season |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.127.151301 |journal=Physical Review Letters |language=en |volume=127 |issue=15 |page=151301 |doi=10.1103/PhysRevLett.127.151301 |pmid=34678017 |arxiv=2110.00483 |bibcode=2021PhRvL.127o1301A |issn=0031-9007}}</ref><ref name="Snowmass21"/>
* Observations of H<sub>2</sub>O emission spectra from a galaxy 12.8 billion light years away consistent with molecules excited by cosmic background radiation much more energetic – 16-20K – than the CMB we observe now, 3K.<ref name="Riechers-2022">{{Cite journal |last1=Riechers |first1=Dominik A. |last2=Weiss |first2=Axel |last3=Walter |first3=Fabian |last4=Carilli |first4=Christopher L. |last5=Cox |first5=Pierre |last6=Decarli |first6=Roberto |last7=Neri |first7=Roberto |date=February 2022 |title=Microwave background temperature at a redshift of 6.34 from H2O absorption |journal=Nature |language=en |volume=602 |issue=7895 |pages=58–62 |doi=10.1038/s41586-021-04294-5 |issn=1476-4687 |pmc=8810383 |pmid=35110755}}</ref><ref name="Snowmass21"/>
* Predictions of the primordial abundance of [[deuterium]] as a result of [[Big
In addition to explaining many pre-2000 observations, the model has made a number of successful predictions: notably the existence of the [[baryon acoustic oscillation]] feature, discovered in 2005 in the predicted ___location; and the statistics of weak [[gravitational lensing]], first observed in 2000 by several teams. The [[Cosmic microwave background#Polarization|polarization]] of the CMB, discovered in 2002 by DASI,<ref>{{cite journal |last1=Kovac|first1=J. M.|last2=Leitch|first2=E. M.|last3=Pryke|first3=C.|author3-link=Clement Pryke|last4=Carlstrom|first4=J. E.|last5=Halverson|first5=N. W. |last6=Holzapfel |first6=W. L.|title=Detection of polarization in the cosmic microwave background using DASI |journal=Nature |year=2002|volume=420|issue=6917 |pages=772–787 |doi=10.1038/nature01269 |pmid=12490941 |arxiv=astro-ph/0209478|bibcode=2002Natur.420..772K|s2cid=4359884|url=https://cds.cern.ch/record/582473}}</ref> has been successfully predicted by the model: in the 2015 ''Planck'' data release,<ref>{{cite journal |title=Planck 2015 Results. XIII. Cosmological Parameters |arxiv=1502.01589 |author1=Planck Collaboration |year=2016 |doi=10.1051/0004-6361/201525830 |volume=594 |issue=13 |journal=Astronomy & Astrophysics |page=A13 |bibcode=2016A&A...594A..13P|s2cid=119262962 }}</ref> there are seven observed peaks in the temperature (TT) power spectrum, six peaks in the temperature–polarization (TE) cross spectrum, and five peaks in the polarization (EE) spectrum. The six free parameters can be well constrained by the TT spectrum alone, and then the TE and EE spectra can be predicted theoretically to few-percent precision with no further adjustments allowed.{{citation needed|date=February 2024}}
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=== Violations of the cosmological principle ===
{{main|Cosmological principle|Friedmann–Lemaître–Robertson–Walker metric}}
The ΛCDM model, like all models built on the Friedmann–Lemaître–Robertson–Walker metric, assume that the universe looks the same in all directions ([[isotropy]]) and from every ___location ([[homogeneity (physics)|homogeneity]])
==== Violations of isotropy ====
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The CMB dipole is hinted at through a number of other observations. First, even within the cosmic microwave background, there are curious directional alignments<ref>{{cite journal |last1=de Oliveira-Costa |first1=Angelica |last2=Tegmark |first2=Max |last3=Zaldarriaga |first3=Matias |last4=Hamilton |first4=Andrew |title=The significance of the largest scale CMB fluctuations in WMAP |journal=Physical Review D |date=25 March 2004 |volume=69 |issue=6 |page=063516 |doi=10.1103/PhysRevD.69.063516 |arxiv=astro-ph/0307282 |bibcode=2004PhRvD..69f3516D |s2cid=119463060 |issn=1550-7998}}</ref> and an anomalous parity asymmetry<ref>{{cite journal |last1=Land |first1=Kate |last2=Magueijo |first2=Joao |title=Is the Universe odd? |journal=Physical Review D |date=28 November 2005 |volume=72 |issue=10 |page=101302 |doi=10.1103/PhysRevD.72.101302 |arxiv=astro-ph/0507289 |bibcode=2005PhRvD..72j1302L |s2cid=119333704 |issn=1550-7998}}</ref> that may have an origin in the CMB dipole.<ref>{{cite journal |last1=Kim |first1=Jaiseung |last2=Naselsky |first2=Pavel |title=Anomalous parity asymmetry of the Wilkinson Microwave Anisotropy Probe power spectrum data at low multipoles |journal=The Astrophysical Journal |date=10 May 2010 |volume=714 |issue=2 |pages=L265–L267 |doi=10.1088/2041-8205/714/2/L265 |arxiv=1001.4613 |bibcode=2010ApJ...714L.265K |s2cid=24389919 |issn=2041-8205}}</ref> Separately, the CMB dipole direction has emerged as a preferred direction in studies of alignments in quasar polarizations,<ref>{{cite journal |last1=Hutsemekers |first1=D. |last2=Cabanac |first2=R. |last3=Lamy |first3=H. |last4=Sluse |first4=D. |title=Mapping extreme-scale alignments of quasar polarization vectors |journal=Astronomy & Astrophysics |date=October 2005 |volume=441 |issue=3 |pages=915–930 |doi=10.1051/0004-6361:20053337 |arxiv=astro-ph/0507274 |bibcode=2005A&A...441..915H |s2cid=14626666 |issn=0004-6361}}</ref> scaling relations in galaxy clusters,<ref>{{cite journal |last1=Migkas |first1=K. |last2=Schellenberger |first2=G. |last3=Reiprich |first3=T. H. |last4=Pacaud |first4=F. |last5=Ramos-Ceja |first5=M. E. |last6=Lovisari |first6=L. |title=Probing cosmic isotropy with a new X-ray galaxy cluster sample through the <math>L_{\text{X}}-T</math> scaling relation |journal=Astronomy & Astrophysics |date=April 2020 |volume=636 |pages=A15 |doi=10.1051/0004-6361/201936602 |arxiv=2004.03305 |bibcode=2020A&A...636A..15M |s2cid=215238834 |issn=0004-6361}}</ref><ref>{{cite journal |last1=Migkas |first1=K. |last2=Pacaud |first2=F. |last3=Schellenberger |first3=G. |last4=Erler |first4=J. |last5=Nguyen-Dang |first5=N. T. |last6=Reiprich |first6=T. H. |last7=Ramos-Ceja |first7=M. E. |last8=Lovisari |first8=L. |title=Cosmological implications of the anisotropy of ten galaxy cluster scaling relations |journal=Astronomy & Astrophysics |date=May 2021 |volume=649 |pages=A151 |doi=10.1051/0004-6361/202140296 |arxiv=2103.13904 |bibcode=2021A&A...649A.151M |s2cid=232352604 |issn=0004-6361}}</ref> [[strong lensing]] time delay,<ref name="FLRW breakdown">{{cite journal |last1=Krishnan |first1=Chethan |last2=Mohayaee |first2=Roya |last3=Colgáin |first3=Eoin Ó |last4=Sheikh-Jabbari |first4=M. M. |last5=Yin |first5=Lu |title=Does Hubble Tension Signal a Breakdown in FLRW Cosmology? |journal=Classical and Quantum Gravity |date=16 September 2021 |volume=38 |issue=18 |page=184001 |doi=10.1088/1361-6382/ac1a81 |arxiv=2105.09790 |bibcode=2021CQGra..38r4001K |s2cid=234790314 |issn=0264-9381}}</ref> Type Ia supernovae,<ref>{{cite journal |last1=Krishnan |first1=Chethan |last2=Mohayaee |first2=Roya |last3=Colgáin |first3=Eoin Ó |last4=Sheikh-Jabbari |first4=M. M. |last5=Yin |first5=Lu |title=Hints of FLRW breakdown from supernovae |journal=Physical Review D |year=2022 |volume=105 |issue=6 |page=063514 |doi=10.1103/PhysRevD.105.063514 |arxiv=2106.02532|bibcode=2022PhRvD.105f3514K |s2cid=235352881 }}</ref> and quasars and [[gamma-ray bursts]] as [[standard candles]].<ref>{{cite journal |last1=Luongo |first1=Orlando |last2=Muccino |first2=Marco |last3=Colgáin |first3=Eoin Ó |last4=Sheikh-Jabbari |first4=M. M. |last5=Yin |first5=Lu |title=Larger H0 values in the CMB dipole direction |journal=Physical Review D |year=2022 |volume=105 |issue=10 |page=103510 |doi=10.1103/PhysRevD.105.103510 |arxiv=2108.13228|bibcode=2022PhRvD.105j3510L |s2cid=248713777 }}</ref> The fact that all these independent observables, based on different physics, are tracking the CMB dipole direction suggests that the Universe is anisotropic in the direction of the CMB dipole.{{citation needed|date=February 2024}}
Nevertheless, some authors have stated that the universe around Earth is isotropic at high significance by studies of the combined cosmic microwave background temperature and polarization maps.<ref name=Saadeh>{{cite journal| vauthors = Saadeh D, Feeney SM, Pontzen A, Peiris HV, McEwen, JD|title=How Isotropic is the Universe?|journal=Physical Review Letters|date=2016|volume=117|number=13|
==== Violations of homogeneity ====
The homogeneity of the universe needed for the ΛCDM applies to very large volumes of space.
[[N-body simulation]]s in ΛCDM show that the spatial distribution of galaxies is statistically homogeneous if averaged over scales 260[[Parsec#Megaparsecs and gigaparsecs|/h Mpc]] or more.<ref name=Yadav>{{cite journal|last=Yadav|first=Jaswant |author2=J. S. Bagla |author3=Nishikanta Khandai|title=Fractal dimension as a measure of the scale of homogeneity|journal=Monthly Notices of the Royal Astronomical Society|date=25 February 2010|volume=405|issue=3|pages=2009–2015|doi=10.1111/j.1365-2966.2010.16612.x |doi-access=free |arxiv = 1001.0617 |bibcode = 2010MNRAS.405.2009Y |s2cid=118603499 }}</ref>
Numerous claims of large-scale structures reported to be in conflict with the predicted scale of homogeneity for ΛCDM do not withstand statistical analysis.<ref name=Nadathur>{{cite journal|last=Nadathur|first=Seshadri|title=Seeing patterns in noise: gigaparsec-scale 'structures' that do not violate homogeneity|journal=Monthly Notices of the Royal Astronomical Society|date=2013|volume=434|issue=1|pages=398–406|doi=10.1093/mnras/stt1028|doi-access=free |arxiv=1306.1700|bibcode =2013MNRAS.434..398N|s2cid=119220579}}</ref><ref name="Snowmass21"/>{{rp|7.8}}
=== Hubble tension ===
{{main|Hubble tension}}
Statistically significant differences remain in values of the Hubble constant derived by matching the ΛCDM model to data from the "early universe", like the cosmic background radiation, compared to values derived from stellar distance measurements, called the "late universe". While systematic error in the measurements remains a possibility, many different kinds of observations agree with one of these two values of the constant. This difference, called the [[Hubble tension]],<ref name="di Valentino 2021 153001">{{cite journal |last1=di Valentino |first1=Eleonora |last2=Mena |first2=Olga |last3=Pan |first3=Supriya |last4=Visnelli |first4=Luca |last5=Yang |first5=Weiqiang |last6=Melchiorri |first6=Alessandro|last7=Mota|first7=David F.|last8=Reiss|first8=Adam G. |last9=Silk|first9=Joseph|author-link9=Joseph Silk|display-authors=3 |date=2021 |title=In the realm of the Hubble tension—a review of solutions |journal=Classical and Quantum Gravity |volume=38 |issue=15 |page=153001 |doi=10.1088/1361-6382/ac086d |arxiv=2103.01183|bibcode=2021CQGra..38o3001D |s2cid=232092525 }}</ref> widely acknowledged to be a major problem for the ΛCDM model.<ref name="cern-courier"/><ref name="LS-20190826">
{{cite news
|last=Mann |first=Adam
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|url=https://www.livescience.com/hubble-constant-discrepancy-explained.html
|date=26 August 2019 |work=[[Live Science]] |access-date=26 August 2019
}}</ref><ref name="Snowmass21"/><ref name="Turner"/>
Dozens of proposals for modifications of ΛCDM or completely new models have been published to explain the Hubble tension. Among these models are many that modify the properties of [[dark energy]] or of [[dark matter]] over time, interactions between dark energy and dark matter, unified dark energy and matter, other forms of dark radiation like [[sterile neutrinos]], modifications to the properties of gravity, or the modification of the effects of [[inflation (cosmology)|inflation]], changes to the properties of elementary particles in the early universe, among others. None of these models can simultaneously explain the breadth of other cosmological data as well as ΛCDM.<ref name="di Valentino 2021 153001"/>
=== ''S''<sub>8</sub> tension ===
The "<math>S_8</math> tension"
<math display="block">S_8 \equiv \sigma_8\sqrt{\Omega_{\rm m}/0.3}</math>Early- (e.g. from [[Cosmic microwave background|CMB]] data) and late-time (e.g. measuring [[weak gravitational lensing]]) measurements facilitate increasingly precise values of <math>S_8</math>. Results from initial weak lensing measurements found a lower value of <math>S_8</math>, compared to the value estimated from Planck<ref>{{Cite journal |last1=Fu |first1=L. |last2=Kilbinger |first2=M. |last3=Erben |first3=T. |last4=Heymans |first4=C. |last5=Hildebrandt |first5=H. |last6=Hoekstra |first6=H. |last7=Kitching |first7=T. D. |last8=Mellier |first8=Y. |last9=Miller |first9=L. |last10=Semboloni |first10=E. |last11=Simon |first11=P. |last12=Van Waerbeke |first12=L. |last13=Coupon |first13=J. |last14=Harnois-Deraps |first14=J. |last15=Hudson |first15=M. J. |date=2014-05-26 |title=CFHTLenS: cosmological constraints from a combination of cosmic shear two-point and three-point correlations |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=441 |issue=3 |pages=2725–2743 |doi=10.1093/mnras/stu754 |doi-access=free |issn=0035-8711}}</ref><ref>{{Cite journal |last1=Abdalla |first1=Elcio |last2=Abellán |first2=Guillermo Franco |last3=Aboubrahim |first3=Amin |last4=Agnello |first4=Adriano |last5=Akarsu |first5=Özgür |last6=Akrami |first6=Yashar |last7=Alestas |first7=George |last8=Aloni |first8=Daniel |last9=Amendola |first9=Luca |last10=Anchordoqui |first10=Luis A. |last11=Anderson |first11=Richard I. |last12=Arendse |first12=Nikki |last13=Asgari |first13=Marika |last14=Ballardini |first14=Mario |last15=Barger |first15=Vernon |date=June 2022 |title=Cosmology intertwined: A review of the particle physics, astrophysics, and cosmology associated with the cosmological tensions and anomalies |url=https://linkinghub.elsevier.com/retrieve/pii/S2214404822000179 |journal=Journal of High Energy Astrophysics |language=en |volume=34 |pages=49–211 |doi=10.1016/j.jheap.2022.04.002 |arxiv=2203.06142 |bibcode=2022JHEAp..34...49A }}</ref>. In recent years much larger surveys have been carried out, some of the preliminarily results also showed evidence of the same tension<ref>{{Cite journal |last1=Heymans |first1=Catherine |last2=Tröster |first2=Tilman |last3=Asgari |first3=Marika |last4=Blake |first4=Chris |last5=Hildebrandt |first5=Hendrik |last6=Joachimi |first6=Benjamin |last7=Kuijken |first7=Konrad |last8=Lin |first8=Chieh-An |last9=Sánchez |first9=Ariel G. |last10=van den Busch |first10=Jan Luca |last11=Wright |first11=Angus H. |last12=Amon |first12=Alexandra |last13=Bilicki |first13=Maciej |last14=de Jong |first14=Jelte |last15=Crocce |first15=Martin |date=February 2021 |title=KiDS-1000 Cosmology: Multi-probe weak gravitational lensing and spectroscopic galaxy clustering constraints |url=https://www.aanda.org/10.1051/0004-6361/202039063 |journal=Astronomy & Astrophysics |volume=646 |pages=A140 |doi=10.1051/0004-6361/202039063 |issn=0004-6361|arxiv=2007.15632 |bibcode=2021A&A...646A.140H }}</ref><ref>{{Cite journal |last1=Abbott |first1=T. M. C. |last2=Aguena |first2=M. |last3=Alarcon |first3=A. |last4=Allam |first4=S. |last5=Alves |first5=O. |last6=Amon |first6=A. |last7=Andrade-Oliveira |first7=F. |last8=Annis |first8=J. |last9=Avila |first9=S. |last10=Bacon |first10=D. |last11=Baxter |first11=E. |last12=Bechtol |first12=K. |last13=Becker |first13=M. R. |last14=Bernstein |first14=G. M. |last15=Bhargava |first15=S. |date=2022-01-13 |title=Dark Energy Survey Year 3 results: Cosmological constraints from galaxy clustering and weak lensing |url=https://link.aps.org/doi/10.1103/PhysRevD.105.023520 |journal=Physical Review D |language=en |volume=105 |issue=2 |page=023520 |doi=10.1103/PhysRevD.105.023520 |issn=2470-0010|arxiv=2105.13549 |bibcode=2022PhRvD.105b3520A |hdl=11368/3013060 }}</ref><ref>{{Cite journal |last1=Li |first1=Xiangchong |last2=Zhang |first2=Tianqing |last3=Sugiyama |first3=Sunao |last4=Dalal |first4=Roohi |last5=Terasawa |first5=Ryo |last6=Rau |first6=Markus M. |last7=Mandelbaum |first7=Rachel |last8=Takada |first8=Masahiro |last9=More |first9=Surhud |last10=Strauss |first10=Michael A. |last11=Miyatake |first11=Hironao |last12=Shirasaki |first12=Masato |last13=Hamana |first13=Takashi |last14=Oguri |first14=Masamune |last15=Luo |first15=Wentao |date=2023-12-11 |title=Hyper Suprime-Cam Year 3 results: Cosmology from cosmic shear two-point correlation functions |url=https://link.aps.org/doi/10.1103/PhysRevD.108.123518 |journal=Physical Review D |language=en |volume=108 |issue=12 |page=123518 |doi=10.1103/PhysRevD.108.123518 |issn=2470-0010|arxiv=2304.00702 |bibcode=2023PhRvD.108l3518L }}</ref>. However, other projects found that with increasing precision there was no significant tension, finding consistency with the Planck results<ref>{{Citation |last1=Wright |first1=Angus H. |title=KiDS-Legacy: Cosmological constraints from cosmic shear with the complete Kilo-Degree Survey |date=2025 |arxiv=2503.19441 |last2=Stölzner |first2=Benjamin |last3=Asgari |first3=Marika |last4=Bilicki |first4=Maciej |last5=Giblin |first5=Benjamin |last6=Heymans |first6=Catherine |last7=Hildebrandt |first7=Hendrik |last8=Hoekstra |first8=Henk |last9=Joachimi |first9=Benjamin}}</ref><ref>{{Cite web |last=Kruesi |first=Liz |date=4 March 2024 |title=Fresh X-Rays Reveal a Universe as Clumpy as Cosmology Predicts |url=https://www.quantamagazine.org/fresh-x-rays-reveal-a-universe-as-clumpy-as-cosmology-predicts-20240304/ |website=[[Quanta Magazine]]}}</ref><ref>{{Cite web |title=eROSITA relaxes cosmological tension |url=https://www.mpg.de/21542664/erosita-confirms-standard-model-of-cosmology |access-date=2025-07-24 |website=www.mpg.de |language=en}}</ref>.
=== Axis of evil ===
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{{main|Shape of the universe}}
The ΛCDM model assumes that the [[shape of the universe]] is of zero curvature (is flat) and has an undetermined topology. In 2019, interpretation of Planck data suggested that the curvature of the universe might be positive (often called "closed"), which would contradict the ΛCDM model.<ref>{{cite journal|url=https://www.nature.com/articles/s41550-019-0906-9|title=Planck evidence for a closed Universe and a possible crisis for cosmology|author1=Eleonora Di Valentino|author2=Alessandro Melchiorri|author3=Joseph Silk|journal=Nature Astronomy|volume=4|doi=10.1038/s41550-019-0906-9|arxiv=1911.02087|date=4 November 2019|issue=2|pages=196–203|s2cid=207880880|access-date=24 March 2022}}</ref><ref name="Snowmass21"/> Some authors have suggested that the Planck data detecting a positive curvature could be evidence of a local inhomogeneity in the curvature of the universe rather than the universe actually being globally a 3-[[manifold]] of positive curvature.<ref>{{cite journal|url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.87.081301|title=What if Planck's Universe isn't flat?|author1=Philip Bull|author2=Marc Kamionkowski|journal=Physical Review D|volume=87|issue=3|date=15 April 2013|page=081301|doi=10.1103/PhysRevD.87.081301|arxiv=1302.1617|bibcode=2013PhRvD..87h1301B|s2cid=118437535|access-date=24 March 2022}}</ref><ref name="Snowmass21"/>
=== Cold dark matter discrepancies ===
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Several discrepancies between the predictions of [[cold dark matter]] in the ΛCDM model and observations of galaxies and their clustering have arisen. Some of these problems have proposed solutions, but it remains unclear whether they can be solved without abandoning the ΛCDM model.<ref>{{Cite journal |arxiv=1006.1647 |title=Local-Group tests of dark-matter Concordance Cosmology: Towards a new paradigm for structure formation |year=2010 |last1=Kroupa |first1=P. |last2=Famaey |first2=B. |last3=de Boer |first3=Klaas S. |last4=Dabringhausen |first4=Joerg |last5=Pawlowski |first5=Marcel |last6=Boily |first6=Christian |last7=Jerjen |first7=Helmut |last8=Forbes |first8=Duncan |last9=Hensler |first9=Gerhard |journal=Astronomy and Astrophysics |volume=523 |pages=32–54 |doi=10.1051/0004-6361/201014892 |bibcode=2010A&A...523A..32K|s2cid=11711780 }}</ref>
[[Mordehai Milgrom|Milgrom]], [[Stacy McGaugh|McGaugh]], and [[Pavel Kroupa|Kroupa]] have criticized the dark matter portions of the theory from the perspective of [[galaxy formation]] models and supporting the alternative [[modified Newtonian dynamics]] (MOND) theory, which requires a modification of the [[Einstein field equations]] and the [[Friedmann equations]] as seen in proposals such as [[modified gravity theory]] (MOG theory) or [[tensor–vector–scalar gravity]] theory (TeVeS theory).{{
==== Cuspy halo problem ====
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Dwarf galaxies around the [[Milky Way]] and [[Andromeda Galaxy|Andromeda]] galaxies are observed to be orbiting in thin, planar structures whereas the simulations predict that they should be distributed randomly about their parent galaxies.<ref name=Pawlowski>{{cite journal |first1=Marcel |last1=Pawlowski |display-authors=etal |title=Co-orbiting satellite galaxy structures are still in conflict with the distribution of primordial dwarf galaxies |journal=Monthly Notices of the Royal Astronomical Society |volume=442 |issue=3 |pages=2362–2380 |year=2014 |arxiv=1406.1799|doi=10.1093/mnras/stu1005 |doi-access=free |bibcode=2014MNRAS.442.2362P }}</ref> However, latest research suggests this seemingly bizarre alignment is just a quirk which will dissolve over time.<ref name="Sawala">{{cite journal |first1=Till |last1=Sawala |first2=Marius |last2=Cautun |first3=Carlos |last3=Frenk |display-authors=etal |title=The Milky Way's plane of satellites: consistent with ΛCDM|journal=Nature Astronomy |year=2022 |volume=7 |issue=4 |pages=481–491 |arxiv=2205.02860|doi=10.1038/s41550-022-01856-z |bibcode=2023NatAs...7..481S|s2cid=254920916 }}</ref>
Using some of the first data from the [[James Webb Space Telescope]], a team of astronomers selected candidate massive galaxies in the early universe.<ref>{{Cite journal |last1=Labbé |first1=Ivo |last2=van Dokkum |first2=Pieter |last3=Nelson |first3=Erica |last4=Bezanson |first4=Rachel |last5=Suess |first5=Katherine A. |last6=Leja |first6=Joel |last7=Brammer |first7=Gabriel |last8=Whitaker |first8=Katherine |last9=Mathews |first9=Elijah |last10=Stefanon |first10=Mauro |last11=Wang |first11=Bingjie |date=April 2023 |title=A population of red candidate massive galaxies ~600 Myr after the Big Bang |url=https://www.nature.com/articles/s41586-023-05786-2 |journal=Nature |language=en |volume=616 |issue=7956 |pages=266–269 |doi=10.1038/s41586-023-05786-2 |pmid=36812940 |arxiv=2207.12446 |bibcode=2023Natur.616..266L |issn=1476-4687}}</ref> The existence of such massive galaxies in the early universe would challenge standard cosmology. <ref name="Boylan-Kolchin">{{cite journal|title=Stress testing ΛCDM with high-redshift galaxy candidates|first=Michael|last=Boylan-Kolchin|journal=Nature Astronomy |year=2023 |volume=7 |issue=6 |pages=731–735 |doi=10.1038/s41550-023-01937-7 |pmid=37351007 |pmc=10281863 |arxiv=2208.01611|bibcode=2023NatAs...7..731B |s2cid=251252960 }}</ref> Follow up spectroscopy revealed that most of these objects have [[Active galactic nucleus|Active Galactic Nuclei]], which boosts the galaxies brightness and caused the masses to be overestimated. <ref>{{Cite web |date=2025-07-01 |title=JWST's early galaxies didn't break the Universe. They revealed it. |url=https://bigthink.com/starts-with-a-bang/jwst-break-universe-revealed/ |access-date=2025-07-24 |website=Big Think |language=en-US}}</ref><ref>{{Cite journal |last1=Kocevski |first1=Dale D. |last2=Finkelstein |first2=Steven L. |last3=Barro |first3=Guillermo |last4=Taylor |first4=Anthony J. |last5=Calabrò |first5=Antonello |last6=Laloux |first6=Brivael |last7=Buchner |first7=Johannes |last8=Trump |first8=Jonathan R. |last9=Leung |first9=Gene C. K. |last10=Yang |first10=Guang |last11=Dickinson |first11=Mark |last12=Pérez-González |first12=Pablo G. |last13=Pacucci |first13=Fabio |last14=Inayoshi |first14=Kohei |last15=Somerville |first15=Rachel S. |date=June 2025 |title=The Rise of Faint, Red Active Galactic Nuclei at z > 4: A Sample of Little Red Dots in the JWST Extragalactic Legacy Fields |journal=The Astrophysical Journal |language=en |volume=986 |issue=2 |pages=126 |doi=10.3847/1538-4357/adbc7d |arxiv=2404.03576 |bibcode=2025ApJ...986..126K |doi-access=free |issn=0004-637X}}</ref> The high redshift galaxies which have been spectroscopically confirmed, such as [[JADES-GS-z13-0]], are much less massive and are consistent with the predictions from LCDM simulations run before JWST<ref>{{Cite journal |last1=McCaffrey |first1=Joe |last2=Hardin |first2=Samantha |last3=Wise |first3=John H. |last4=Regan |first4=John A. |date=2023-09-27 |title=No Tension: JWST Galaxies at \(z > 10\) Consistent with Cosmological Simulations |url=http://localhost:58547/article/88302-no-tension-jwst-galaxies-at-z-10-consistent-with-cosmological-simulations,%20https://astro.theoj.org/article/88302-no-tension-jwst-galaxies-at-z-10-consistent-with-cosmological-simulations |journal=The Open Journal of Astrophysics |language=en |volume=6 |page=47 |doi=10.21105/astro.2304.13755 |arxiv=2304.13755 |bibcode=2023OJAp....6E..47M }}</ref>. As a population, the confirmed high redshift galaxies are brighter than expected from simulations, but not to the extent that they violate cosmological limits.<ref>{{Cite journal |last1=Xiao |first1=Mengyuan |last2=Oesch |first2=Pascal A. |last3=Elbaz |first3=David |last4=Bing |first4=Longji |last5=Nelson |first5=Erica J. |last6=Weibel |first6=Andrea |last7=Illingworth |first7=Garth D. |last8=van Dokkum |first8=Pieter |last9=Naidu |first9=Rohan P. |last10=Daddi |first10=Emanuele |last11=Bouwens |first11=Rychard J. |last12=Matthee |first12=Jorryt |last13=Wuyts |first13=Stijn |last14=Chisholm |first14=John |last15=Brammer |first15=Gabriel |date=November 2024 |title=Accelerated formation of ultra-massive galaxies in the first billion years |url=https://ui.adsabs.harvard.edu/abs/2024Natur.635..311X/abstract |journal=Nature |language=en |volume=635 |issue=8038 |pages=311–315 |doi=10.1038/s41586-024-08094-5 |pmid=39537883 |arxiv=2309.02492 |bibcode=2024Natur.635..311X |issn=0028-0836}}</ref><ref>{{Citation |last1=Yung |first1=L. Y. Aaron |title=$Λ$CDM is still not broken: empirical constraints on the star formation efficiency at $z \sim 12-30$ |date=2025 |url=https://arxiv.org/abs/2504.18618 |access-date=2025-07-24 |arxiv=2504.18618 |last2=Somerville |first2=Rachel S. |last3=Iyer |first3=Kartheik G.}}</ref> Theorists are studying many possible explanations, including modifying cosmology, more efficient star formation and different stellar populations.<ref>{{Cite journal |last1=Sun |first1=Guochao |last2=Faucher-Giguère |first2=Claude-André |last3=Hayward |first3=Christopher C. |last4=Shen |first4=Xuejian |last5=Wetzel |first5=Andrew |last6=Cochrane |first6=Rachel K. |date=2023-10-01 |title=Bursty Star Formation Naturally Explains the Abundance of Bright Galaxies at Cosmic Dawn |journal=The Astrophysical Journal Letters |volume=955 |issue=2 |pages=L35 |doi=10.3847/2041-8213/acf85a |arxiv=2307.15305 |bibcode=2023ApJ...955L..35S |doi-access=free |issn=2041-8205}}</ref><ref>{{Cite journal |last1=Dekel |first1=Avishai |last2=Sarkar |first2=Kartick C |last3=Birnboim |first3=Yuval |last4=Mandelker |first4=Nir |last5=Li |first5=Zhaozhou |date=2023-06-08 |title=Efficient formation of massive galaxies at cosmic dawn by feedback-free starbursts |url=https://academic.oup.com/mnras/article/523/3/3201/7179993 |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=523 |issue=3 |pages=3201–3218 |doi=10.1093/mnras/stad1557 |doi-access=free |issn=0035-8711}}</ref>
=== Missing baryon problem ===
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Massimo Persic and Paolo Salucci<ref>{{Cite journal|last1=Persic|first1=M.|last2=Salucci|first2=P.|date=1992-09-01|title=The baryon content of the Universe|journal=Monthly Notices of the Royal Astronomical Society|volume=258|issue=1|pages=14P–18P|doi=10.1093/mnras/258.1.14P|arxiv=astro-ph/0502178|bibcode=1992MNRAS.258P..14P |issn=0035-8711|doi-access=free}}</ref> first estimated the baryonic density today present in ellipticals, spirals, groups and clusters of galaxies.
They performed an integration of the baryonic mass-to-light ratio over luminosity (in the following <math display="inline"> M_{\rm b}/L </math>), weighted with the luminosity function <math display="inline">\phi(L)</math> over the previously mentioned classes of astrophysical objects:
<math display="block">\rho_{\rm b} = \sum \int L\phi(L) \frac{M_{\rm b}}{L} \, dL.</math>
The result was:
<math display="block"> \Omega_{\rm b}=\Omega_*+\Omega_\text{gas}=2.2\times10^{-3}+1.5\times10^{-3}\;h^{-1.3}\simeq0.003 ,</math>
where <math> h\simeq 0.72 </math>.
Note that this value is much lower than the prediction of standard cosmic nucleosynthesis <math> \Omega_{\rm b}\simeq0.0486 </math>, so that stars and gas in galaxies and in galaxy groups and clusters account for less than 10% of the primordially synthesized baryons. This issue is known as the problem of the "missing baryons".
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The missing baryon problem is claimed to be resolved. Using observations of the kinematic [[Sunyaev–Zeldovich effect|Sunyaev–Zel'dovich effect]] spanning more than 90% of the lifetime of the Universe, in 2021 astrophysicists found that approximately 50% of all baryonic matter is outside [[dark matter halo]]es, filling the space between galaxies.<ref>{{Cite journal|last1=Chaves-Montero|first1=Jonás|last2=Hernández-Monteagudo|first2=Carlos|last3=Angulo|first3=Raúl E|last4=Emberson|first4=J D|date=2021-03-25|title=Measuring the evolution of intergalactic gas from z = 0 to 5 using the kinematic Sunyaev–Zel'dovich effect|url=https://academic.oup.com/mnras/article/503/2/1798/6184230|journal=Monthly Notices of the Royal Astronomical Society|language=en|volume=503|issue=2|pages=1798–1814|doi=10.1093/mnras/staa3782|doi-access=free |arxiv=1911.10690 |issn=0035-8711}}</ref> Together with the amount of baryons inside galaxies and surrounding them, the total amount of baryons in the late time Universe is compatible with early Universe measurements.
===
It has been argued that the ΛCDM model
== Extended models ==
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Extended models allow one or more of the "fixed" parameters above to vary, in addition to the basic six; so these models join smoothly to the basic six-parameter model in the limit that the additional parameter(s) approach the default values. For example, possible extensions of the simplest ΛCDM model allow for spatial curvature (<math>\Omega_\text{tot}</math> may be different from 1); or [[quintessence (physics)|quintessence]] rather than a [[cosmological constant]] where the [[Equation of state (cosmology)|equation of state]] of dark energy is allowed to differ from −1. Cosmic inflation predicts tensor fluctuations ([[gravitational wave]]s). Their amplitude is parameterized by the tensor-to-scalar ratio (denoted <math>r</math>), which is determined by the unknown energy scale of inflation. Other modifications allow [[hot dark matter]] in the form of [[neutrino]]s more massive than the minimal value, or a running spectral index; the latter is generally not favoured by simple cosmic inflation models.
Allowing additional variable parameter(s) will generally ''increase'' the uncertainties in the standard six parameters quoted above, and may also shift the central values slightly. The table
Some researchers have suggested that there is a running spectral index, but no statistically significant study has revealed one. Theoretical expectations suggest that the tensor-to-scalar ratio <math>r</math> should be between 0 and 0.3, and the latest results are within those limits.
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* [http://lambda.gsfc.nasa.gov/product/map/dr3/parameters_summary.cfm WMAP estimated cosmological parameters/Latest Summary]
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