Editing Lambda-CDM model

{{Short description|Mathematical model of the Big Bang}}
{{Redirect|Standard cosmological model|other uses|Standard model (disambiguation)}}
{{more citations needed|date=February 2024}}
{{Cosmology|comp/struct}}

The '''Lambda-CDM''', '''Lambda cold dark matter''', or '''ΛCDM''' model is a [[mathematical model]] of the [[Big Bang]] theory with three major components:
# a [[cosmological constant]], denoted by [[lambda]] (Λ), associated with [[dark energy]];
# the postulated [[cold dark matter]], denoted by CDM;
# ordinary [[matter]].

It is the current ''standard model'' of Big Bang cosmology,<ref name=DeruelleUzan>{{Cite book |last1=Deruelle |first1=Nathalie |url=https://academic.oup.com/book/43967 |title=Relativity in Modern Physics |last2=Uzan |first2=Jean-Philippe |date=2018-08-30 |publisher=Oxford University Press |isbn=978-0-19-878639-9 |editor-last=de Forcrand-Millard |editor-first=Patricia |edition=1 |language=en |doi=10.1093/oso/9780198786399.001.0001 |author-link=Nathalie Deruelle |author-link2=Jean-Philippe Uzan}}</ref> as it is the simplest model that provides a reasonably good account of:
* the existence and structure of the [[cosmic microwave background]];
* the [[Observable universe#Large-scale structure|large-scale structure]] in the distribution of galaxies;
* the observed [[abundance of the chemical elements|abundances]] of [[Big Bang nucleosynthesis|hydrogen (including deuterium), helium, and lithium]];
* 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 when disparate observed properties of the universe appeared mutually inconsistent, and there was no consensus on the makeup of the energy density of the universe.

The ΛCDM model has been successful in modeling a broad collection of astronomical observations over decades. Remaining issues challenge the assumptions of the ΛCDM model and have led to many alternative models.<ref name="Snowmass21"/>

== 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 |last=Navas |first=S. |last2=Amsler |first2=C. |last3=Gutsche |first3=T. |last4=Hanhart |first4=C. |last5=Hernández-Rey |first5=J. J. |last6=Lourenço |first6=C. |last7=Masoni |first7=A. |last8=Mikhasenko |first8=M. |last9=Mitchell |first9=R. E. |last10=Patrignani |first10=C. |last11=Schwanda |first11=C. |last12=Spanier |first12=S. |last13=Venanzoni |first13=G. |last14=Yuan |first14=C. Z. |last15=Agashe |first15=K. |date=2024-08-01 |title=Review of Particle Physics |url=https://link.aps.org/doi/10.1103/PhysRevD.110.030001 |journal=Physical Review D |language=en |volume=110 |issue=3 |doi=10.1103/PhysRevD.110.030001 |issn=2470-0010}}</ref>{{rp|25.1.1}}  These component densities become parameters extracted when the model is constrained to match astrophysical observations. The model aims to describe the observable universe from approximately 0.1&nbsp;s to the present.<ref name=DeruelleUzan/>{{rp|605}}

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}}

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 [[Metric tensor (general relativity)|metric]] that is well documented, both as the [[redshift]] of prominent spectral absorption or emission lines in the light from distant galaxies, and as the time dilation in the light decay of supernova luminosity curves. Both effects are attributed to a [[Doppler shift]] in electromagnetic radiation as it travels across expanding space. Although this expansion increases the distance between objects that are not under shared gravitational influence, it does not increase the size of the objects (e.g. galaxies) in space. Also, since it originates from ordinary general relativity, it, like general relativity, allows for distant galaxies to recede from each other at speeds greater than the speed of light; local expansion is less than the speed of light, but expansion summed across great distances can collectively exceed the speed of light.<ref name=DavisLineweaver>{{Cite journal |last1=Davis |first1=Tamara M. |last2=Lineweaver |first2=Charles H. |date=2004 |title=Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe |url=https://www.cambridge.org/core/product/identifier/S132335800000607X/type/journal_article |journal=Publications of the Astronomical Society of Australia |language=en |volume=21 |issue=1 |pages=97–109 |doi=10.1071/AS03040 |arxiv=astro-ph/0310808 |bibcode=2004PASA...21...97D |issn=1323-3580}}</ref>

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>

[[Dark matter]] is postulated in order to account for gravitational effects observed in very large-scale structures (the "non-keplerian" [[rotation curve]]s of galaxies;<ref>{{cite journal |author1= Persic, M.|display-authors=etal |title= The universal rotation curve of spiral galaxies — I. The dark matter connection  |journal=[[Monthly Notices of the Royal Astronomical Society]]  |date=1996  |volume=281   |issue=1 |pages=27–47 |doi= 10.1093/mnras/278.1.27 |doi-access=free |bibcode= 1996MNRAS.281...27P |arxiv=astro-ph/9506004}}</ref>                                                                                                                                   the [[gravitational lens]]ing of light by galaxy clusters; and the enhanced clustering of galaxies) that cannot be accounted for by the quantity of observed matter.<ref>{{Cite journal |last1=Bertone |first1=Gianfranco |last2=Hooper |first2=Dan |date=2018-10-15 |title=History of dark matter |url=https://link.aps.org/doi/10.1103/RevModPhys.90.045002 |journal=Reviews of Modern Physics |language=en |volume=90 |issue=4 |page=045002 |doi=10.1103/RevModPhys.90.045002 |issn=0034-6861|arxiv=1605.04909 |bibcode=2018RvMP...90d5002B }}</ref>
The ΛCDM model proposes specifically [[cold dark matter]], hypothesized as:
* Non-baryonic: Consists of matter other than protons and neutrons (and electrons, by convention, although electrons are not baryons)
* 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, 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
| first1 = Massimo
| last2 = Salucci
| first2 = Paolo
| date = 1992-09-01
| title = The baryon content of the Universe
| url = http://mnras.oxfordjournals.org/content/258/1/14P
| journal = Monthly Notices of the Royal Astronomical Society
| language = en
| volume = 258
| issue = 1
| pages = 14P–18P
| doi = 10.1093/mnras/258.1.14P
| doi-access = free
| issn = 0035-8711
| arxiv = astro-ph/0502178 |bibcode = 1992MNRAS.258P..14P | s2cid = 17945298
}}</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>&nbsp;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 10 000 K) for several hundred thousand years, a state that is detectable as a residual [[cosmic microwave background]], or CMB, a very low-energy radiation emanating from all parts of the sky. The "Big Bang" scenario, with cosmic inflation and standard particle physics, is the only cosmological model consistent with the observed continuing expansion of space, the observed distribution of [[Big Bang nucleosynthesis|lighter elements in the universe]] (hydrogen, helium, and lithium), and the spatial texture of minute irregularities ([[Anisotropy|anisotropies]]) in the CMB radiation. Cosmic inflation also addresses the "[[horizon problem]]" in the CMB; indeed, it seems likely that the universe is larger than the observable [[particle horizon]].<ref>{{Cite journal |last=Davis |first=Tamara M. |last2=Lineweaver |first2=Charles H. |date=January 2004 |title=Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe |url=https://www.cambridge.org/core/journals/publications-of-the-astronomical-society-of-australia/article/expanding-confusion-common-misconceptions-of-cosmological-horizons-and-the-superluminal-expansion-of-the-universe/EFEEEFD8D71E59F86DDA82FDF576EFD3 |journal=Publications of the Astronomical Society of Australia |language=en |volume=21 |issue=1 |pages=97–109 |doi=10.1071/AS03040 |issn=1323-3580}}</ref>

== 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}}

== Parameters ==
Multiple variants of the ΛCDM model are used with some differences in parameters.<ref name=PDG-2024/>{{rp|loc=25.1}} One such set is outlined in the table below. 
{| class="wikitable"
|+ Planck Collaboration Cosmological parameters
! &emsp;&emsp;
! Description<ref name="Planck-2013">The parameters used in the Planck series of papers are described in Table 1 of {{Cite journal |last=Ade |first=P. a. R. |last2=Aghanim |first2=N. |last3=Armitage-Caplan |first3=C. |last4=Arnaud |first4=M. |last5=Ashdown |first5=M. |last6=Atrio-Barandela |first6=F. |last7=Aumont |first7=J. |last8=Baccigalupi |first8=C. |last9=Banday |first9=A. J. |last10=Barreiro |first10=R. B. |last11=Bartlett |first11=J. G. |last12=Battaner |first12=E. |last13=Benabed |first13=K. |last14=Benoît |first14=A. |last15=Benoit-Lévy |first15=A. |date=2014-11-01 |title=Planck 2013 results. XVI. Cosmological parameters |url=https://www.aanda.org/articles/aa/full_html/2014/11/aa21591-13/aa21591-13.html |journal=Astronomy & Astrophysics |language=en |volume=571 |pages=A16 |doi=10.1051/0004-6361/201321591 |issn=0004-6361}}</ref>
! Symbol
! Value-2018<ref name="Planck 2018">
{{cite journal
 |author=Planck Collaboration
 |year=2020
 |title=Planck 2018 results. VI.&nbsp;Cosmological parameters
 |journal=Astronomy & Astrophysics
 |volume=641 |at=page&nbsp;A6 (see PDF page&nbsp;15, Table&nbsp;2: "Age/Gyr", last&nbsp;column)
 |doi=10.1051/0004-6361/201833910
 |arxiv=1807.06209 |bibcode=2020A&A...641A...6P
 |s2cid=119335614
}}</ref>
|-
! rowspan="6" {{vert header|&ensp;Independent parameters}}
| Baryon density today{{efn|name=physical density|The "physical baryon density parameter" Ω<sub>b</sub> {{var|h}}<sup>2</sup> is the "baryon density parameter" Ω<sub>b</sub> multiplied by the square of the reduced Hubble constant {{nowrap|1= {{var|h}} = {{var|H}}<sub>0</sub> / (100&nbsp;km⋅s<sup>−1</sup>⋅Mpc<sup>−1</sup>)}}.<ref>[https://web.archive.org/web/20120305082531/http://www.lsst.org/files/docs/sciencebook/SB_A.pdf Appendix A] of the [http://www.lsst.org/lsst/scibook LSST Science Book Version 2.0] {{Webarchive|url=https://web.archive.org/web/20130226112941/http://www.lsst.org/lsst/scibook |date=2013-02-26 }}</ref><ref>p. 7 of [https://web.archive.org/web/20140421213818/http://wfirst.gsfc.nasa.gov/science/fomswg/fomswg_technical.pdf Findings of the Joint Dark Energy Mission Figure of Merit Science Working Group]</ref> Likewise for the difference between "physical dark matter density parameter" and "dark matter density parameter".}}
| align="center" | Ω<sub>b</sub> {{var|h}}<sup>2</sup>
| {{val|0.0224|0.0001}}
|-
| Cold dark matter density today{{efn|name=physical density}}
| align="center" | Ω<sub>c</sub> {{var|h}}<sup>2</sup>
| {{val|0.120|0.001}}
|-
| 100 × approximation to r∗/DA (CosmoMC)
| align="center" | 100<math>\theta_{MC}</math>
| {{val|1.04089|0.00031}}
|-
| [[Reionization]] [[optical depth]]
| align="center" | {{var|τ}}
| {{val|0.054|0.007}}
|-
| Log power of the primordial curvature perturbations
| align="center" |<math>\ln(10^{10}A_s)</math>
| {{val|3.043| 0.014}}
|-
| Scalar spectrum power-law index 
| align="center" | {{var|n}}<sub>s</sub>
| {{val|0.965|0.004}}
|-
! rowspan="6" {{vert header|&ensp;&emsp;&emsp;Fixed parameters}}
| Total matter density today (inc. massive neutrinos
| align="center" | Ω<sub>m</sub> {{var|h}}<sup>2</sup>
| 0.1428 ± 0.0011
|-
| Equation of state of dark energy
| align="center" | {{var|w}}
| w<sub>0</sub> = −1
|-
| Tensor/scalar ratio
| align="center" | {{var|r}}
| r<sub>0.002</sub> <  0.06
|-
| Running of spectral index
| align="center" |<math>d n_\text{s} / d \ln k</math>
| 0
|-
| Sum of three neutrino masses
| align="center" |<math>\sum m_\nu</math>
| 0.06 [[electronvolt (mass)|eV/{{var|c}}{{sup|2}}]]
|-
| Effective number of relativistic degrees of freedom
| align="center" | ''N''<sub>eff</sub>
| {{val|2.99|0.17}}
|-
! rowspan="10" {{vert header|&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;&emsp;Calculated Values}}
| [[Hubble constant]]
| align="center" | {{var|H}}<sub>0</sub>
| {{val|67.4|0.5|u=km⋅s<sup>−1</sup>⋅[[parsec|Mpc]]<sup>−1</sup>}}
|-
| [[Age of the universe]]
| align="center" | {{var|t}}<sub>0</sub>
| {{val|13.787|0.020|e=9}} years<ref name="Planck 2018age">
{{cite journal
 |author=Planck Collaboration
 |year=2020
 |title=Planck 2018 results. VI.&nbsp;Cosmological parameters
 |journal=Astronomy & Astrophysics
 |volume=641 |at=page&nbsp;A6 (see PDF page&nbsp;15, Table&nbsp;2: "Age/Gyr", last&nbsp;column)
 |doi=10.1051/0004-6361/201833910
 |arxiv=1807.06209 |bibcode=2020A&A...641A...6P
 |s2cid=119335614
}}
</ref>
|-
| [[Dark energy]] density parameter{{efn|name=density| Density parameters are expressed relative to a critical density {{var|ρ}}{{sub|crit}}, which is the total density of matter/energy needed for the universe to be spatially flat: {{nowrap|1=Ω{{sub|{{var|x}}}} = {{var|ρ}}{{sub|{{var|x}}}} / {{var|ρ}}{{sub|crit}}}}.<ref name=Peacock-1998>{{Cite book |last=Peacock |first=J. A. |url=https://www.cambridge.org/core/product/identifier/9780511804533/type/book |title=Cosmological Physics |date=1998-12-28 |publisher=Cambridge University Press |isbn=978-0-521-41072-4 |edition=1 |doi=10.1017/cbo9780511804533}}</ref>{{rp|74}}}}
| align="center" | Ω<sub>Λ</sub>
| {{val|0.6847|0.0073}}
|-
| The present root-mean-square matter fluctuation,<br>averaged over a sphere of radius 8''h''<sup>−1</sup> Mpc
| align="center" | {{var|σ}}<sub>8</sub>
| {{val|0.811|0.006}}
|-
| Redshift of reionization (with uniform prior)
| align="center" | {{var|z}}<sub>re</sub>

| {{val|7.68|0.79}}
|}
{{Clear}}
The [[Planck (spacecraft)|Planck]] collaboration version of the ΛCDM model is based on six [[parameter]]s: baryon density parameter; dark matter density parameter; scalar spectral index; two parameters related to curvature fluctuation amplitude; and the probability that photons from the early universe will be scattered once on route (called reionization optical depth).<ref name="Planck-2013"/> Six is the smallest number of parameters needed to give an acceptable fit to the observations; other possible parameters are fixed at "natural" values, e.g. total density parameter = 1.00, dark energy equation of state = −1.

The parameter values, and uncertainties, are estimated using computer searches to locate the region of parameter space providing an acceptable match to cosmological observations. From these six parameters, the other model values, such as the [[Hubble's law|Hubble constant]] and the [[dark energy]] density, can be calculated.

{{notelist}}

== Historical development ==
The discovery of the [[cosmic microwave background]] (CMB) in 1964 confirmed a key prediction of the [[Big Bang]] cosmology. From that point on, it was generally accepted that the universe started in a hot, dense state and has been expanding over time. The rate of expansion depends on the types of matter and energy present in the universe, and in particular, whether the total density is above or below the so-called critical density.{{citation needed|date=February 2024}}

During the 1970s, most attention focused on pure-baryonic models, but there were serious challenges explaining the formation of galaxies, given the small anisotropies in the CMB (upper limits at that time). In the early 1980s, it was realized that this could be resolved if cold dark matter dominated over the baryons, and the theory of [[cosmic inflation]] motivated models with critical density.{{citation needed|date=February 2024}}

During the 1980s, most research focused on cold dark matter with critical density in matter, around 95% CDM and 5% baryons: these showed success at forming galaxies and clusters of galaxies, but problems remained; notably, the model required a Hubble constant lower than preferred by observations, and observations around 1988–1990 showed more large-scale galaxy clustering than predicted.{{citation needed|date=February 2024}}

These difficulties sharpened with the discovery of CMB anisotropy by the [[Cosmic Background Explorer]] in 1992, and several modified CDM models, including ΛCDM and mixed cold and hot dark matter, came under active consideration through the mid-1990s. The ΛCDM model then became the leading model following the observations of [[Accelerating universe|accelerating expansion]] in 1998, and was quickly supported by other observations: in 2000, the [[BOOMERanG experiment|BOOMERanG]] microwave background experiment measured the total (matter–energy) density to be close to 100% of critical, whereas in 2001 the [[2dF Galaxy Redshift Survey|2dFGRS]] galaxy redshift survey measured the matter density to be near 25%; the large difference between these values supports a positive Λ or [[dark energy]]. Much more precise spacecraft measurements of the microwave background from [[WMAP]] in 2003–2010 and ''[[Planck (spacecraft)|Planck]]'' in 2013–2015 have continued to support the model and pin down the parameter values, most of which are constrained below 1&nbsp;percent uncertainty.{{citation needed|date=February 2024}}

== Successes ==
Among all cosmological models, the ΛCDM model has been the most successful; it describes a wide range of astronomical observations with remarkable accuracy.<ref name="Snowmass21"/>{{rp|58|q=...the standard ΛCDM cosmological model provides a remarkable description of a wide range of astrophysical and cosmological probes}} The notable successes include:
* Accurate modeling the high-precision CMB angular distribution measure by the [[Planck (satellite)|Planck mission]]<ref name="Planck-2018-legacy">{{Cite journal |last1=Aghanim |first1=N. |last2=Akrami |first2=Y. |last3=Arroja |first3=F. |last4=Ashdown |first4=M. |last5=Aumont |first5=J. |last6=Baccigalupi |first6=C. |last7=Ballardini |first7=M. |last8=Banday |first8=A. J. |last9=Barreiro |first9=R. B. |last10=Bartolo |first10=N. |last11=Basak |first11=S. |last12=Battye |first12=R. |last13=Benabed |first13=K. |last14=Bernard |first14=J.-P. |last15=Bersanelli |first15=M. |date=2020-09-01 |title=Planck 2018 results - I. Overview and the cosmological legacy of Planck |url=https://www.aanda.org/articles/aa/full_html/2020/09/aa33880-18/aa33880-18.html |journal=Astronomy & Astrophysics |language=en |volume=641 |pages=A1 |doi=10.1051/0004-6361/201833880 |arxiv=1807.06205 |bibcode=2020A&A...641A...1P |issn=0004-6361}}</ref> and [[Atacama Cosmology Telescope]].<ref name="Aiola-ACT-2020">{{Cite journal |last1=Aiola |first1=Simone |last2=Calabrese |first2=Erminia |last3=Maurin |first3=Loïc |last4=Naess |first4=Sigurd |last5=Schmitt |first5=Benjamin L. |last6=Abitbol |first6=Maximilian H. |last7=Addison |first7=Graeme E. |last8=Ade |first8=Peter A. R. |last9=Alonso |first9=David |last10=Amiri |first10=Mandana |last11=Amodeo |first11=Stefania |last12=Angile |first12=Elio |last13=Austermann |first13=Jason E. |last14=Baildon |first14=Taylor |last15=Battaglia |first15=Nick |date=2020-12-01 |title=The Atacama Cosmology Telescope: DR4 maps and cosmological parameters |journal=Journal of Cosmology and Astroparticle Physics |volume=2020 |issue=12 |pages=047 |doi=10.1088/1475-7516/2020/12/047 |arxiv=2007.07288 |bibcode=2020JCAP...12..047A |issn=1475-7516}}</ref><ref name="Snowmass21"/>
* Accurate description of the linear [[Polarization (cosmology)|E-mode polarization]] of the CMB radiation due to fluctuations on the surface of last scattering events.<ref name="Dutcher-EMode-2021">{{Cite journal |last1=Dutcher |first1=D. |last2=Balkenhol |first2=L. |last3=Ade |first3=P. A. R. |last4=Ahmed |first4=Z. |last5=Anderes |first5=E. |last6=Anderson |first6=A. J. |last7=Archipley |first7=M. |last8=Avva |first8=J. S. |last9=Aylor |first9=K. |last10=Barry |first10=P. S. |last11=Basu Thakur |first11=R. |last12=Benabed |first12=K. |last13=Bender |first13=A. N. |last14=Benson |first14=B. A. |last15=Bianchini |first15=F. |date=2021-07-13 |title=Measurements of the E -mode polarization and temperature- E -mode correlation of the CMB from SPT-3G 2018 data |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.104.022003 |journal=Physical Review D |language=en |volume=104 |issue=2 |page=022003 |doi=10.1103/PhysRevD.104.022003 |arxiv=2101.01684 |bibcode=2021PhRvD.104b2003D |issn=2470-0010}}</ref><ref name="Snowmass21"/>
* 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 Bang nucleosynthesis]].<ref name="Cooke-2014">{{Cite journal |last=Cooke |first=Ryan J. |last2=Pettini |first2=Max |last3=Jorgenson |first3=Regina A. |last4=Murphy |first4=Michael T. |last5=Steidel |first5=Charles C. |date=2014-01-03 |title=PRECISION MEASURES OF THE PRIMORDIAL ABUNDANCE OF DEUTERIUM |journal=The Astrophysical Journal |volume=781 |issue=1 |pages=31 |doi=10.1088/0004-637x/781/1/31 |issn=0004-637X}}</ref> The observed abundance matches the one derived from the nucleosynthesis model with the value for baryon density derived from CMB measurements.<ref name="Turner"/>{{rp|4.1.2}}
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}}

== Challenges ==
Despite the widespread success of ΛCDM in matching observations of our universe, cosmologists believe that the model may be an approximation of a more fundamental model.<ref name="Snowmass21">{{cite journal|author1=Elcio Abdalla|author2=Guillermo Franco Abellán|author3=Amin Aboubrahim|display-authors=2|title=Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies|journal=Journal of High Energy Astrophysics |arxiv=2203.06142v1|date=11 Mar 2022|volume=34 |page=49 |doi=10.1016/j.jheap.2022.04.002 |bibcode=2022JHEAp..34...49A |s2cid=247411131 }}</ref><ref name="cern-courier">{{cite web|url=https://cerncourier.com/a/exploring-the-hubble-tension/|title=Exploring the Hubble tension|author=Matthew Chalmers|website=[[CERN Courier]]|date=2 July 2021|access-date=25 March 2022}}</ref><ref name="Turner">{{cite journal|author1=Michael Turner|title=The Road to Precision Cosmology|journal=Annual Review of Nuclear and Particle Science|volume=32|arxiv=2201.04741|date=12 Jan 2022|pages=1–35 |doi=10.1146/annurev-nucl-111119-041046|bibcode=2022ARNPS..72....1T |s2cid=245906450 }}</ref>

=== Lack of detection ===
Extensive searches for dark matter particles have so far shown no well-agreed detection, while dark energy may be almost impossible to detect in a laboratory, and its value is [[cosmological constant problem|extremely small]] compared to [[Vacuum energy|vacuum energy theoretical predictions]].{{citation needed|date=February 2024}}

=== 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]]) on a large enough scale: "the universe looks the same whoever and wherever you are."<ref>Andrew Liddle. ''An Introduction to Modern Cosmology (2nd ed.).'' London: Wiley, 2003.</ref> This [[cosmological principle]] allows a metric, [[Friedmann–Lemaître–Robertson–Walker metric]], to be derived and developed into a theory to compare to experiments. Without the principle, a metric would need to be extracted from astronomical data, which may not be possible.<ref>{{cite book|title=Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity|author=[[Steven Weinberg]]|isbn=978-0-471-92567-5|year=1972|publisher=John Wiley & Sons, Inc.}}</ref>{{rp|408}} The assumptions were carried over into the ΛCDM model.<ref name="Colin et al">{{cite journal|title=Evidence for anisotropy of cosmic acceleration|author1=Jacques Colin|author2=Roya Mohayaee|author3=Mohamed Rameez|author4=Subir Sarkar|journal=Astronomy and Astrophysics|volume=631|doi=10.1051/0004-6361/201936373|arxiv=1808.04597|date=20 November 2019|pages=L13|bibcode=2019A&A...631L..13C|s2cid=208175643|access-date=25 March 2022|url=https://www.aanda.org/articles/aa/full_html/2019/11/aa36373-19/aa36373-19.html}}</ref> However, some findings suggested violations of the cosmological principle.<ref name="Snowmass21"/><ref name="FLRW breakdown"/>

==== Violations of isotropy ====
Evidence from [[galaxy cluster]]s,<ref>{{cite web|url=https://www.scientificamerican.com/article/do-we-live-in-a-lopsided-universe1/|title=Do We Live in a Lopsided Universe?|author=Lee Billings|website=[[Scientific American]]|date=April 15, 2020|access-date=March 24, 2022}}</ref><ref>{{cite journal|url=https://www.aanda.org/articles/aa/full_html/2020/04/aa36602-19/aa36602-19.html|title=Probing cosmic isotropy with a new X-ray galaxy cluster sample through the LX-T scaling relation|author1=Migkas, K.|author2=Schellenberger, G.|author3=Reiprich, T. H.|author4=Pacaud, F.|author5=Ramos-Ceja, M. E.|author6=Lovisari, L.|journal=Astronomy & Astrophysics|volume=636|issue=April 2020|page=42|doi=10.1051/0004-6361/201936602|date=8 April 2020|arxiv=2004.03305|bibcode=2020A&A...636A..15M|s2cid=215238834|access-date=24 March 2022}}</ref> [[quasar]]s,<ref>{{cite journal|title=A Test of the Cosmological Principle with Quasars|author1=Nathan J. Secrest|author2=Sebastian von Hausegger|author3=Mohamed Rameez|author4=Roya Mohayaee|author5=Subir Sarkar|author6=Jacques Colin|journal=The Astrophysical Journal Letters|volume=908|issue=2|doi=10.3847/2041-8213/abdd40|arxiv=2009.14826|date=February 25, 2021|pages=L51|bibcode=2021ApJ...908L..51S|s2cid=222066749|doi-access=free }}</ref> and [[type Ia supernova]]e<ref>{{cite journal|url=https://iopscience.iop.org/article/10.1088/0004-637X/810/1/47|title=Probing the Isotropy of Cosmic Acceleration Traced By Type Ia Supernovae|author1=B. Javanmardi|author2=C. Porciani|author3=P. Kroupa|author4=J. Pflamm-Altenburg|journal=The Astrophysical Journal Letters|volume=810|issue=1|doi=10.1088/0004-637X/810/1/47|arxiv=1507.07560|date=August 27, 2015|page=47|bibcode=2015ApJ...810...47J|s2cid=54958680|access-date=March 24, 2022}}</ref> suggest that isotropy is violated on large scales.{{citation needed|date=February 2024}}

Data from the [[Planck Mission]] shows hemispheric bias in the [[cosmic microwave background]] in two respects: one with respect to average temperature (i.e. temperature fluctuations), the second with respect to larger variations in the degree of perturbations (i.e. densities). The [[European Space Agency]] (the governing body of the Planck Mission) has concluded that these anisotropies in the CMB are, in fact, statistically significant and can no longer be ignored.<ref name="Planck">{{cite web | url=http://sci.esa.int/planck/51551-simple-but-challenging-the-universe-according-to-planck/ | title=Simple but challenging: the Universe according to Planck | work=[[ESA Science & Technology]] | orig-date=March 21, 2013 |date= October 5, 2016 | access-date=October 29, 2016}}</ref>

Already in 1967, [[Dennis Sciama]] predicted that the cosmic microwave background has a significant dipole anisotropy.<ref name="sciama">{{cite journal|title=Peculiar Velocity of the Sun and the Cosmic Microwave Background|url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.18.1065|author=Dennis Sciama|journal=Physical Review Letters|volume=18|issue=24|doi=10.1103/PhysRevLett.18.1065|date=12 June 1967|pages=1065–1067|bibcode=1967PhRvL..18.1065S|access-date=25 March 2022|url-access=subscription}}</ref><ref>{{cite journal|title=On the expected anisotropy of radio source counts|url=https://academic.oup.com/mnras/article/206/2/377/1024995|author1=G. F. R. Ellis|author2=J. E. Baldwin|journal=Monthly Notices of the Royal Astronomical Society|volume=206|issue=2|doi=10.1093/mnras/206.2.377|date=1 January 1984|pages=377–381|access-date=25 March 2022|doi-access=free}}</ref> In recent years, the CMB dipole has been tested, and the results suggest our motion with respect to distant radio galaxies<ref>{{cite journal |last1=Siewert |first1=Thilo M. |last2=Schmidt-Rubart |first2=Matthias |last3=Schwarz |first3=Dominik J. |title=Cosmic radio dipole: Estimators and frequency dependence |journal=Astronomy & Astrophysics |year=2021 |volume=653 |pages=A9 |doi=10.1051/0004-6361/202039840 |arxiv=2010.08366|bibcode=2021A&A...653A...9S |s2cid=223953708 }}</ref> and quasars<ref>{{cite journal |last1=Secrest |first1=Nathan |last2=von Hausegger |first2=Sebastian |last3=Rameez |first3=Mohamed |last4=Mohayaee |first4=Roya |last5=Sarkar |first5=Subir |last6=Colin |first6=Jacques |title=A Test of the Cosmological Principle with Quasars |journal=The Astrophysical Journal |date=25 February 2021 |volume=908 |issue=2 |pages=L51 |doi=10.3847/2041-8213/abdd40 |arxiv=2009.14826 |bibcode=2021ApJ...908L..51S |s2cid=222066749 |issn=2041-8213 |doi-access=free }}</ref> differs from our motion with respect to the [[cosmic microwave background]]. The same conclusion has been reached in recent studies of the [[Hubble diagram]] of [[Type Ia supernovae]]<ref>{{cite journal |last1=Singal |first1=Ashok K. |title=Peculiar motion of Solar system from the Hubble diagram of supernovae Ia and its implications for cosmology |journal=Monthly Notices of the Royal Astronomical Society |year=2022 |volume=515 |issue=4 |pages=5969–5980 |doi=10.1093/mnras/stac1986 |doi-access=free |arxiv=2106.11968}}</ref> and [[quasars]].<ref>{{cite journal |last1=Singal |first1=Ashok K. |title=Solar system peculiar motion from the Hubble diagram of quasars and testing the cosmological principle |journal=Monthly Notices of the Royal Astronomical Society |year=2022 |volume=511 |issue=2 |pages=1819–1829 |doi=10.1093/mnras/stac144 |doi-access=free |arxiv=2107.09390}}</ref> This contradicts the cosmological principle.{{citation needed|date=February 2024}}

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|page= 131302 |doi=10.1103/PhysRevLett.117.131302|pmid=27715088|arxiv=1605.07178|bibcode = 2016PhRvL.117m1302S |s2cid=453412}}</ref>

==== 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}}

=== El Gordo galaxy cluster collision ===
{{main|El Gordo (galaxy cluster)}}

[[El Gordo (galaxy cluster)|El Gordo]] is a massive interacting galaxy cluster in the early Universe (<math>z = 0.87</math>). The extreme properties of [[El Gordo (galaxy cluster)|El Gordo]] in terms of its redshift, mass, and the collision velocity leads to strong (<math>6.16\sigma</math>) tension with the ΛCDM model.<ref name="Asencio">{{Cite journal|last1=Asencio|first1=E|last2=Banik|first2=I|last3=Kroupa|first3=P|date=2021-02-21|title=A massive blow for ΛCDM – the high redshift, mass, and collision velocity of the interacting galaxy cluster El Gordo contradicts concordance cosmology|journal=Monthly Notices of the Royal Astronomical Society|volume=500|issue=2|pages=5249–5267|doi=10.1093/mnras/staa3441|arxiv=2012.03950|bibcode=2021MNRAS.500.5249A|issn=0035-8711|doi-access=free}}</ref><ref name="Asencio_2023">{{Cite journal|last1=Asencio|first1=E|last2=Banik|first2=I|last3=Kroupa|first3=P|date=2023-09-10|title=A massive blow for ΛCDM – the high redshift, mass, and collision velocity of the interacting galaxy cluster El Gordo contradicts concordance cosmology|journal=The Astrophysical Journal|volume=954|issue=2|pages=162|doi=10.3847/1538-4357/ace62a|doi-access=free|arxiv=2308.00744|bibcode=2023ApJ...954..162A|issn=1538-4357}}</ref> The properties of [[El Gordo (galaxy cluster)|El Gordo]] are however consistent with cosmological simulations in the framework of [[MOND]] due to more rapid structure formation.<ref name="Katz">{{Cite journal|last1=Katz|first1=H|last2=McGaugh|first2=S|last3=Teuben|first3=P|last4=Angus|first4=G. W.|date=2013-07-20|title=Galaxy Cluster Bulk Flows and Collision Velocities in QUMOND|journal = The Astrophysical Journal|volume=772|issue=1|page=10|doi=10.1088/0004-637X/772/1/10|arxiv=1305.3651|bibcode=2013ApJ...772...10K|issn=1538-4357|doi-access=free}}</ref>

=== KBC void ===
{{main|KBC void}}

The [[KBC void]] is an immense, comparatively empty region of space containing the [[Milky Way]] approximately 2 billion light-years (600 megaparsecs, Mpc) in diameter.<ref name="kbc">{{Cite journal | last1 = Keenan | first1 = Ryan C. | last2 = Barger | first2 = Amy J. | last3 = Cowie | first3 = Lennox L. | title = Evidence for a ~300&nbsp;Mpc Scale Under-density in the Local Galaxy Distribution | journal = The Astrophysical Journal | volume = 775 | year = 2013 | issue = 1 | page = 62 | doi = 10.1088/0004-637X/775/1/62 |arxiv = 1304.2884 |bibcode = 2013ApJ...775...62K | s2cid = 118433293 }}</ref><ref name="siegel">{{cite web|url=https://www.forbes.com/sites/startswithabang/2017/06/07/were-way-below-average-astronomers-say-milky-way-resides-in-a-great-cosmic-void/#4d53c7cd6d05|title=We're Way Below Average! Astronomers Say Milky Way Resides In A Great Cosmic Void|last=Siegel|first=Ethan|work=[[Forbes]]|access-date=2017-06-09}}</ref><ref name="Snowmass21"/> Some authors have said the existence of the KBC void violates the assumption that the CMB reflects baryonic density fluctuations at <math>z = 1100</math> or Einstein's theory of [[general relativity]], either of which would violate the ΛCDM model,<ref name="Haslbauer">{{Cite journal|last1=Haslbauer|first1=M|last2=Banik|first2=I|last3=Kroupa|first3=P|date=2020-12-21|title=The KBC void and Hubble tension contradict LCDM on a Gpc scale – Milgromian dynamics as a possible solution|journal=Monthly Notices of the Royal Astronomical Society|volume=499|issue=2|pages=2845–2883|doi=10.1093/mnras/staa2348|arxiv=2009.11292|bibcode=2020MNRAS.499.2845H|issn=0035-8711|doi-access=free}}</ref> while other authors have claimed that supervoids as large as the KBC void are consistent with the ΛCDM model.<ref>{{Cite journal|last1=Sahlén|first1=Martin|last2=Zubeldía|first2=Íñigo|last3=Silk|first3=Joseph|date=2016|title=Cluster–Void Degeneracy Breaking: Dark Energy, Planck, and the Largest Cluster and Void|journal=The Astrophysical Journal Letters|volume=820|issue=1|pages=L7|doi=10.3847/2041-8205/820/1/L7|issn=2041-8205|arxiv=1511.04075|bibcode=2016ApJ...820L...7S|s2cid=119286482 |doi-access=free }}</ref>

=== 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
 |title=One Number Shows Something Is Fundamentally Wrong with Our Conception of the Universe – This fight has universal implications
 |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" is a name for another question mark for the ΛCDM model.<ref name="Snowmass21"/> The <math>S_8</math> parameter in the ΛCDM model quantifies the amplitude of matter fluctuations in the late universe and is defined as
<math display="block">S_8 \equiv \sigma_8\sqrt{\Omega_{\rm m}/0.3}</math>

Early- (e.g. from [[Cosmic microwave background|CMB]] data collected using the Planck observatory) and late-time (e.g. measuring [[weak gravitational lensing]] events) facilitate increasingly precise values of <math>S_8</math>. However, these two categories of measurement differ by more standard deviations than their uncertainties. This discrepancy is called the <math>S_8</math> tension. The name "tension" reflects that the disagreement is not merely between two data sets: the many sets of early- and late-time measurements agree well within their own categories, but there is an unexplained difference between values obtained from different points in the evolution of the universe. Such a tension indicates that the ΛCDM model may be incomplete or in need of correction.<ref name="Snowmass21"/>

Some values for <math>S_8</math> are {{val|0.832|0.013}} (2020 [[Planck (spacecraft)|Planck]]),<ref>{{cite journal |last1=Planck Collaboration |last2=Aghanim |first2=N. |last3=Akrami |first3=Y. |last4=Ashdown |first4=M. |last5=Aumont |first5=J. |last6=Baccigalupi |first6=C. |last7=Ballardini |first7=M. |last8=Banday |first8=A. J. |last9=Barreiro |first9=R. B. |last10=Bartolo |first10=N. |last11=Basak |first11=S. |last12=Battye |first12=R. |last13=Benabed |first13=K. |last14=Bernard |first14=J.-P. |last15=Bersanelli |first15=M. |date=September 2020 |title=Planck 2018 results: VI. Cosmological parameters (Corrigendum) |url=https://www.aanda.org/10.1051/0004-6361/201833910e |journal=Astronomy & Astrophysics |volume=652 |pages=C4 |doi=10.1051/0004-6361/201833910e |issn=0004-6361|hdl=10902/24951 |hdl-access=free }}</ref> {{val|0.766|0.020|0.014}} (2021 [https://kids.strw.leidenuniv.nl/ KIDS]),<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 web |last=Wood |first=Charlie |date=8 September 2020 |title=A New Cosmic Tension: The Universe Might Be Too Thin |url=https://www.quantamagazine.org/a-new-cosmic-tension-the-universe-might-be-too-thin-20200908/ |website=[[Quanta Magazine]]}}</ref> {{val|0.776|0.017}} (2022 [[Dark Energy Survey|DES]]),<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> {{val|0.790|0.018|0.014}} (2023 DES+KIDS),<ref>{{Cite journal |last1=Dark Energy Survey |last2=Kilo-Degree Survey Collaboration |last3=Abbott |first3=T.M.C. |last4=Aguena |first4=M. |last5=Alarcon |first5=A. |last6=Alves |first6=O. |last7=Amon |first7=A. |last8=Andrade-Oliveira |first8=F. |last9=Asgari |first9=M. |last10=Avila |first10=S. |last11=Bacon |first11=D. |last12=Bechtol |first12=K. |last13=Becker |first13=M. R. |last14=Bernstein |first14=G. M. |last15=Bertin |first15=E. |date=2023-10-20 |title=DES Y3 + KiDS-1000: Consistent cosmology combining cosmic shear surveys |url=https://astro.theoj.org/article/89164-des-y3-kids-1000-consistent-cosmology-combining-cosmic-shear-surveys |journal=The Open Journal of Astrophysics |volume=6 |page=36 |doi=10.21105/astro.2305.17173 |issn=2565-6120|arxiv=2305.17173 |bibcode=2023OJAp....6E..36D }}</ref> {{val|0.769|0.031|0.034}} – {{val|0.776|0.032|0.033}}<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><ref>{{Cite journal |last1=Dalal |first1=Roohi |last2=Li |first2=Xiangchong |last3=Nicola |first3=Andrina |last4=Zuntz |first4=Joe |last5=Strauss |first5=Michael A. |last6=Sugiyama |first6=Sunao |last7=Zhang |first7=Tianqing |last8=Rau |first8=Markus M. |last9=Mandelbaum |first9=Rachel |last10=Takada |first10=Masahiro |last11=More |first11=Surhud |last12=Miyatake |first12=Hironao |last13=Kannawadi |first13=Arun |last14=Shirasaki |first14=Masato |last15=Taniguchi |first15=Takanori |date=2023-12-11 |title=Hyper Suprime-Cam Year 3 results: Cosmology from cosmic shear power spectra |url=https://link.aps.org/doi/10.1103/PhysRevD.108.123519 |journal=Physical Review D |language=en |volume=108 |issue=12 |page=123519 |doi=10.1103/PhysRevD.108.123519 |issn=2470-0010|arxiv=2304.00701 |bibcode=2023PhRvD.108l3519D }}</ref><ref>{{Cite journal |last=Yoon |first=Mijin |date=2023-12-11 |title=Inconsistency Turns Up Again for Cosmological Observations |url=https://physics.aps.org/articles/v16/193 |journal=Physics |language=en |volume=16 |issue=12 |pages=193 |doi=10.1103/PhysRevD.108.123519|arxiv=2304.00701 |bibcode=2023PhRvD.108l3519D }}</ref><ref>{{Cite web |last=Kruesi |first=Liz |date=19 January 2024 |title=Clashing Cosmic Numbers Challenge Our Best Theory of the Universe |url=https://www.quantamagazine.org/clashing-cosmic-numbers-challenge-our-best-theory-of-the-universe-20240119 |website=[[Quanta Magazine]]}}</ref> (2023 [https://hsc.mtk.nao.ac.jp/ssp/ HSC-SSP]), {{val|0.86|0.01}} (2024 [[EROSITA]]).<ref>{{Cite journal |last1=Ghirardini |first1=V. |last2=Bulbul |first2=E. |last3=Artis |first3=E. |last4=Clerc |first4=N. |last5=Garrel |first5=C. |last6=Grandis |first6=S. |last7=Kluge |first7=M. |last8=Liu |first8=A. |last9=Bahar |first9=Y. E. |last10=Balzer |first10=F. |last11=Chiu |first11=I. |last12=Comparat |first12=J. |last13=Gruen |first13=D. |last14=Kleinebreil |first14=F. |last15=Krippendorf |first15=S. |date=February 2024 |title=The SRG/EROSITA all-sky survey |journal=Astronomy & Astrophysics |volume=689 |pages=A298 |doi=10.1051/0004-6361/202348852 |arxiv=2402.08458}}</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> Values have also obtained using [[Peculiar velocity|peculiar velocities]], {{val|0.637|0.054}} (2020)<ref>{{Cite journal |last1=Said |first1=Khaled |last2=Colless |first2=Matthew |last3=Magoulas |first3=Christina |last4=Lucey |first4=John R |last5=Hudson |first5=Michael J |date=2020-09-01 |title=Joint analysis of 6dFGS and SDSS peculiar velocities for the growth rate of cosmic structure and tests of gravity |url=https://academic.oup.com/mnras/article/497/1/1275/5870121 |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=497 |issue=1 |pages=1275–1293 |doi=10.1093/mnras/staa2032 |doi-access=free |issn=0035-8711|arxiv=2007.04993 }}</ref> and {{val|0.776|0.033}} (2020),<ref>{{Cite journal |last1=Boruah |first1=Supranta S |last2=Hudson |first2=Michael J |last3=Lavaux |first3=Guilhem |date=2020-09-21 |title=Cosmic flows in the nearby Universe: new peculiar velocities from SNe and cosmological constraints |url=https://academic.oup.com/mnras/article/498/2/2703/5894929 |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=498 |issue=2 |pages=2703–2718 |doi=10.1093/mnras/staa2485 |doi-access=free |issn=0035-8711|arxiv=1912.09383 }}</ref> among other methods.

=== Axis of evil ===
{{main|Axis of evil (cosmology)}}
{{#section:Axis of evil (cosmology)|lead}}

=== Cosmological lithium problem ===
{{main|Cosmological lithium problem}}

The actual observable amount of lithium in the universe is less than the calculated amount from the ΛCDM model by a factor of 3–4.<ref name=fields11>{{cite journal |last=Fields |first=B. D. |date=2011 |title=The primordial lithium problem |journal=[[Annual Review of Nuclear and Particle Science]] |volume=61 |issue=1 |pages=47–68 |doi=10.1146/annurev-nucl-102010-130445| doi-access=free |arxiv=1203.3551 |bibcode=2011ARNPS..61...47F}}</ref><ref name="Snowmass21"/>{{rp|141}} If every calculation is correct, then solutions beyond the existing ΛCDM model might be needed.<ref name="fields11" />

=== Shape of the universe ===
{{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"/>

=== Violations of the strong equivalence principle ===
{{main|Strong equivalence principle}}
The ΛCDM model assumes that the [[strong equivalence principle]] is true. However, in 2020 a group of astronomers analyzed data from the Spitzer Photometry and Accurate Rotation Curves (SPARC) sample, together with estimates of the large-scale external gravitational field from an all-sky galaxy catalog. They concluded that there was highly statistically significant evidence of violations of the strong equivalence principle in weak gravitational fields in the vicinity of rotationally supported galaxies.<ref>{{Cite journal|arxiv = 2009.11525|doi = 10.3847/1538-4357/abbb96|title = Testing the Strong Equivalence Principle: Detection of the External Field Effect in Rotationally Supported Galaxies|year = 2020|last1 = Chae|first1 = Kyu-Hyun|last2 = Lelli|first2 = Federico|last3 = Desmond|first3 = Harry|last4 = McGaugh|first4 = Stacy S.|last5 = Li|first5 = Pengfei|last6 = Schombert|first6 = James M.|journal = The Astrophysical Journal|volume = 904|issue = 1|page = 51|bibcode = 2020ApJ...904...51C|s2cid = 221879077 | doi-access=free }}</ref> They observed an effect inconsistent with [[tidal force|tidal effects]] in the ΛCDM model. These results have been challenged as failing to consider inaccuracies in the rotation curves and correlations between galaxy properties and clustering strength.<ref>{{Cite journal |last1=Paranjape |first1=Aseem |last2=Sheth |first2=Ravi K |date=2022-10-04 |title=The phenomenology of the external field effect in cold dark matter models |url=https://academic.oup.com/mnras/article/517/1/130/6713954 |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=517 |issue=1 |pages=130–139 |doi=10.1093/mnras/stac2689 |doi-access=free |issn=0035-8711|arxiv=2112.00026 }}</ref> and as inconsistent with similar analysis of other galaxies.<ref>{{Cite journal |last1=Freundlich |first1=Jonathan |last2=Famaey |first2=Benoit |last3=Oria |first3=Pierre-Antoine |last4=Bílek |first4=Michal |last5=Müller |first5=Oliver |last6=Ibata |first6=Rodrigo |date=2022-02-01 |title=Probing the radial acceleration relation and the strong equivalence principle with the Coma cluster ultra-diffuse galaxies |url=https://www.aanda.org/articles/aa/abs/2022/02/aa42060-21/aa42060-21.html |journal=Astronomy & Astrophysics |language=en |volume=658 |pages=A26 |doi=10.1051/0004-6361/202142060 |issn=0004-6361 |quote=We hence do not see any evidence for a violation of the strong equivalence principle in Coma cluster UDGs, contrarily to, for instance, Chae et al. (2020, 2021), for disc galaxies in the field. Our work extends that of Bílek et al. (2019b) and Haghi et al. (2019a), which is limited to DF44 and makes the result all the more compelling. We recall that the MOND predictions do not involve any free parameter.

|doi-access=free |arxiv=2109.04487 |bibcode=2022A&A...658A..26F }}</ref>

=== Cold dark matter discrepancies ===
{{main|Cold dark matter#Challenges}}
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).{{citation needed|date=January 2025}} Other proposals by theoretical astrophysicists of cosmological alternatives to Einstein's general relativity that attempt to account for dark energy or dark matter include [[f(R) gravity]], [[Scalar–tensor theory|scalar–tensor theories]] such as {{ill|galileon|ko}} theories (see [[Galilean invariance]]), [[brane cosmology|brane cosmologies]], the [[DGP model]], and [[massive gravity]] and its extensions such as [[bimetric theory|bimetric gravity]].{{citation needed|date=February 2024}}

==== Cuspy halo problem ====
{{main|Cuspy halo problem}}
The density distributions of dark matter halos in cold dark matter simulations (at least those that do not include the impact of baryonic feedback) are much more peaked than what is observed in galaxies by investigating their rotation curves.<ref>{{Cite journal |title=The cored distribution of dark matter in spiral galaxies|year=2004 |last1=Gentile |first1=G. |last2=Salucci |first2=P. |journal=Monthly Notices of the Royal Astronomical Society |volume=351 |issue=3 |pages=903–922 |doi=10.1111/j.1365-2966.2004.07836.x |doi-access=free |arxiv=astro-ph/0403154 |bibcode = 2004MNRAS.351..903G|s2cid=14308775 }}</ref>

==== Dwarf galaxy problem ====
{{main|Dwarf galaxy problem}} 
Cold dark matter simulations predict large numbers of small dark matter halos, more numerous than the number of small dwarf galaxies that are observed around galaxies like the [[Milky Way]].<ref name=Klypin>{{cite journal |last1=Klypin |first1=Anatoly |last2=Kravtsov |first2=Andrey V. |last3=Valenzuela |first3=Octavio |last4=Prada |first4=Francisco |year=1999 |title=Where are the missing galactic satellites? |journal=Astrophysical Journal |volume=522 |issue=1 |pages=82–92 |doi=10.1086/307643 |bibcode=1999ApJ...522...82K |arxiv=astro-ph/9901240|s2cid=12983798 }}</ref>

==== Satellite disk problem ====
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>

==== High-velocity galaxy problem ====
Galaxies in the [[NGC 3109]] association are moving away too rapidly to be consistent with expectations in the ΛCDM model.<ref>{{Cite journal|last1=Banik|first1=Indranil|last2=Zhao|first2=H|date=2018-01-21|title=A plane of high velocity galaxies across the Local Group|journal=Monthly Notices of the Royal Astronomical Society|volume=473|issue=3|pages=4033–4054|doi=10.1093/mnras/stx2596|arxiv=1701.06559|bibcode=2018MNRAS.473.4033B|issn=0035-8711|doi-access=free}}</ref> In this framework, [[NGC 3109]] is too massive and distant from the [[Local Group]] for it to have been flung out in a three-body interaction involving the [[Milky Way]] or [[Andromeda Galaxy]].<ref>{{Cite journal|last1=Banik|first1=Indranil|last2=Haslbauer|first2=Moritz|last3=Pawlowski|first3=Marcel S.|last4=Famaey|first4=Benoit|last5=Kroupa|first5=Pavel|date=2021-06-21|title=On the absence of backsplash analogues to NGC 3109 in the ΛCDM framework|journal=Monthly Notices of the Royal Astronomical Society|volume=503|issue=4|pages=6170–6186|doi=10.1093/mnras/stab751|arxiv=2105.04575|bibcode=2021MNRAS.503.6170B|issn=0035-8711|doi-access=free}}</ref>

==== Galaxy morphology problem ====
If galaxies grew hierarchically, then massive galaxies required many mergers. [[Galaxy merger|Major mergers]] inevitably create a classical [[Bulge (astronomy)|bulge]]. On the contrary, about 80% of observed galaxies give evidence of no such bulges, and giant pure-disc galaxies are commonplace.<ref name="kormendy2010">{{cite journal |last1=Kormendy |first1=J. |author1-link=John Kormendy |last2=Drory |first2=N. |last3=Bender |first3=R. |last4=Cornell |first4=M.E. |title=Bulgeless giant galaxies challenge our picture of galaxy formation by hierarchical clustering |year=2010 |journal=[[The Astrophysical Journal]] |volume=723 |issue=1 |pages=54–80 |doi=10.1088/0004-637X/723/1/54 |arxiv=1009.3015 |bibcode=2010ApJ...723...54K|s2cid=119303368 }}</ref> The tension can be quantified by comparing the observed distribution of galaxy shapes today with predictions from high-resolution hydrodynamical cosmological simulations in the ΛCDM framework, revealing a highly significant problem that is unlikely to be solved by improving the resolution of the simulations.<ref name="Haslbauer2022">{{cite journal |last1=Haslbauer|first1=M|last2=Banik|first2=I|last3=Kroupa|first3=P|last4=Wittenburg|first4=N|last5=Javanmardi|first5=B|title=The High Fraction of Thin Disk Galaxies Continues to Challenge ΛCDM Cosmology|date=2022-02-01|journal=[[The Astrophysical Journal]]|volume=925|issue=2|page=183|doi=10.3847/1538-4357/ac46ac|issn=1538-4357|arxiv=2202.01221|bibcode=2022ApJ...925..183H|doi-access=free}}</ref> The high bulgeless fraction was nearly constant for 8&nbsp;billion years.<ref name="sachdeva2016">{{cite journal |last1=Sachdeva |first1=S. |last2=Saha |first2=K. |title=Survival of pure disk galaxies over the last 8&nbsp;billion years |year=2016 |journal=The Astrophysical Journal Letters |volume=820 |issue=1 |pages=L4 |doi=10.3847/2041-8205/820/1/L4 |arxiv=1602.08942 |bibcode=2016ApJ...820L...4S|s2cid=14644377 |doi-access=free }}</ref>

==== Fast galaxy bar problem ====
If galaxies were embedded within massive halos of [[cold dark matter]], then the bars that often develop in their central regions would be slowed down by [[dynamical friction]] with the halo. This is in serious tension with the fact that observed galaxy bars are typically fast.<ref name="Roshan2021">{{Cite journal|last1=Mahmood|first1=R|last2=Ghafourian|first2=N|last3=Kashfi|first3=T|last4=Banik|first4=I|last5=Haslbauer|first5=M|last6=Cuomo|first6=V|last7=Famaey|first7=B|last8=Kroupa|first8=P|date=2021-11-01|title=Fast galaxy bars continue to challenge standard cosmology|journal=Monthly Notices of the Royal Astronomical Society|volume=508|issue=1|pages=926–939|doi=10.1093/mnras/stab2553|doi-access=free|arxiv=2106.10304|bibcode=2021MNRAS.508..926R|hdl=10023/24680|issn=0035-8711}}</ref>

==== Small scale crisis ====
Comparison of the model with observations may have some problems on sub-galaxy scales, possibly predicting [[Dwarf galaxy problem|too many dwarf galaxies]] and too much dark matter in the innermost regions of galaxies. This problem is called the "small scale crisis".<ref>{{Cite journal
 | title =Synopsis: Tackling the Small-Scale Crisis
 |journal = Physical Review D|volume = 95|issue = 12|page = 121302| last =Rini
 | first =Matteo
 |doi = 10.1103/PhysRevD.95.121302|year = 2017|arxiv = 1703.10559|bibcode = 2017PhRvD..95l1302N|s2cid = 54675159}}</ref>  These small scales are harder to resolve in computer simulations, so it is not yet clear whether the problem is the simulations, non-standard properties of dark matter, or a more radical error in the model.

==== High redshift galaxies ====
Observations from the [[James Webb Space Telescope]] have resulted in various galaxies confirmed by [[spectroscopy]] at high redshift, such as [[JADES-GS-z13-0]] at [[cosmological redshift]] of 13.2.<ref name="NASA-milestone">{{cite web|title = NASA's Webb Reaches New Milestone in Quest for Distant Galaxies|url = https://blogs.nasa.gov/webb/2022/12/09/nasas-webb-reaches-new-milestone-in-quest-for-distant-galaxies/|first = Thaddeus|last = Cesari|date = 9 December 2022|access-date = 9 December 2022}}</ref><ref name="Curtis-Lake2022">{{cite web|display-authors = etal|first1 = Emma|last1 = Curtis-Lake|title = Spectroscopy of four metal-poor galaxies beyond redshift ten|url = https://webbtelescope.org/files/live/sites/webb/files/home/webb-science/early-highlights/_documents/2022-061-jades/JADES_CurtisLake.pdf|date = December 2022| arxiv=2212.04568 }}</ref> Other candidate galaxies which have not been confirmed by spectroscopy include [[CEERS-93316]] at cosmological [[redshift]] of 16.4.

Existence of surprisingly massive galaxies in the early universe challenges the preferred models describing how dark matter halos drive galaxy formation. It remains to be seen whether a revision of the Lambda-CDM model with parameters given by Planck Collaboration is necessary to resolve this issue. The discrepancies could also be explained by particular properties (stellar masses or effective volume) of the candidate galaxies, yet unknown force or particle outside of the [[Standard Model]] through which dark matter interacts, more efficient baryonic matter accumulation by the dark matter halos, early dark energy models,<ref name="SmithEtAl-2022">{{cite journal|title=Hints of early dark energy in Planck, SPT, and ACT data: New physics or systematics?|author1=Smith, Tristian L.|author2=Lucca, Matteo|author3=Poulin, Vivian|author4=Abellan, Guillermo F.|author5=Balkenhol, Lennart|author6=Benabed, Karim|author7=Galli, Silvia|author8=Murgia, Riccardo|journal=Physical Review D|volume=106|issue=4|date=August 2022|page=043526 |doi=10.1103/PhysRevD.106.043526|arxiv=2202.09379|bibcode=2022PhRvD.106d3526S|s2cid=247011465 }}</ref> or the hypothesized long-sought [[Population III stars]].<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><ref name="SciAm2022">{{cite web|title=Astronomers Grapple with JWST's Discovery of Early Galaxies|url=https://www.scientificamerican.com/article/astronomers-grapple-with-jwsts-discovery-of-early-galaxies1/|last=O'Callaghan|first=Jonathan|website=[[Scientific American]] |date=6 December 2022|access-date=10 December 2022}}</ref><ref name="BehrooziEtAl">{{cite journal|title=The Universe at z > 10: predictions for JWST from the UNIVERSEMACHINE DR1|author1= Behroozi, Peter|author2=Conroy, Charlie|author3=Wechsler, Risa H.|author4=Hearin, Andrew|author5=Williams, Christina C.|author6=Moster, Benjamin P.|author7=Yung, L. Y. Aaron|author8=Somerville, Rachel S.|author9=Gottlöber, Stefan|author10=Yepes, Gustavo|author11=Endsley, Ryan|journal=Monthly Notices of the Royal Astronomical Society|volume=499|issue=4|pages=5702–5718|date=December 2020|doi=10.1093/mnras/staa3164|doi-access= free|arxiv=2007.04988|bibcode=2020MNRAS.499.5702B}}</ref><ref name="SpringelHernquist">{{cite journal|title=The history of star formation in a Λ cold dark matter universe|author1=Volker Springel|author2=Lars Hernquist|journal=Monthly Notices of the Royal Astronomical Society|volume=339|issue=2|pages=312–334|date=February 2003|doi=10.1046/j.1365-8711.2003.06207.x|doi-access=free |arxiv=astro-ph/0206395|bibcode=2003MNRAS.339..312S |s2cid=8715136 }}</ref>

=== Missing baryon problem ===
{{main|Missing baryon problem}}
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".

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.

=== Conventionalism ===
It has been argued that the ΛCDM model has adopted [[conventionalism|conventionalist stratagems]], rendering it [[falsifiability|unfalsifiable]] in the sense defined by [[Karl Popper]]. When faced with new data not in accord with a prevailing model, the conventionalist will find ways to adapt the theory rather than declare it false. Thus dark matter was added after the observations of anomalous galaxy rotation rates. [[Thomas Kuhn]] viewed the process differently, as "problem solving" within the existing paradigm.<ref>{{Cite journal | doi=10.1016/j.shpsb.2016.12.002| title=Cosmology and convention| journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics| volume=57| pages=41–52| year=2017| last1=Merritt| first1=David| arxiv=1703.02389| bibcode=2017SHPMP..57...41M| s2cid=119401938}}</ref>

== Extended models ==
<!-- Please do not update any numbers in this table without providing the source and updating them all. They should be updated with the next WMAP data release. -->
{| class="wikitable floatright"
|+ Extended model parameters<ref name=oldwmap>Table 8 on p. 39 of {{cite journal | author = Jarosik, N. et al. (WMAP Collaboration) | 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 | page = 14 | url = http://lambda.gsfc.nasa.gov/product/map/dr4/pub_papers/sevenyear/basic_results/wmap_7yr_basic_results.pdf |access-date=2010-12-04| bibcode = 2011ApJS..192...14J | year = 2011 | arxiv = 1001.4744 | doi = 10.1088/0067-0049/192/2/14 | hdl = 2152/43001 | s2cid = 46171526 }} (from NASA's [http://lambda.gsfc.nasa.gov/product/map/dr4/map_bibliography.cfm WMAP Documents] page)</ref>
! Description
! Symbol
! Value
|-
| Total density parameter
| align="center" |<math>\Omega_\text{tot}</math>
| {{val|0.9993|0.0019}}<ref>{{cite journal |last=Zyla | first= P.A. |display-authors=etal |collaboration=[[Particle Data Group]]|title=Cosmological Parameters|journal= Prog. Theor. Exp. Phys. | year=2020 | volume=083C01 | url = https://pdg.lbl.gov/2020/reviews/rpp2020-rev-cosmological-parameters.pdf}}</ref>
|-
| Equation of state of dark energy
| align="center" |<math>w</math>
| {{val|-0.980|0.053}}
|-
| Tensor-to-scalar ratio
| align="center" |<math>r</math>
| < 0.11, {{var|k}}<sub>0</sub> = 0.002 Mpc<sup>−1</sup> (<math>2\sigma</math>)
|-
| Running of the spectral index
| align="center" |<math>d n_s / d \ln k</math>
| {{val|-0.022|0.020}}, {{var|k}}<sub>0</sub> = 0.002&nbsp;Mpc<sup>−1</sup>
|-
| Sum of three neutrino masses
| align="center" |<math>\sum m_\nu</math>
| < 0.58 [[electronvolt (mass)|eV/{{var|c}}{{sup|2}}]] (<math>2\sigma</math>)
|-
| Physical neutrino density parameter
| align="center" |<math>\Omega_\nu h^2</math>
| < 0.0062
|}
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&nbsp;−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 below shows results for each of the possible "6+1" scenarios with one additional variable parameter; this indicates that, as of 2015, there is no convincing evidence that any additional parameter is different from its default value.

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.

== See also ==
* [[Bolshoi cosmological simulation]]
* [[Galaxy formation and evolution]]
* [[Illustris project]]
* [[List of cosmological computation software]]
* [[Millennium Run]]
* [[Weakly interacting massive particle]]s (WIMPs)
* The ΛCDM model is also known as the standard model of cosmology, but is not related to the [[Standard Model]] of particle physics.
* [[Inhomogeneous cosmology]]
{{clear}}

== References ==
{{reflist|30em}}

== Further reading ==
* {{cite arXiv |last1=Ostriker |first1=J. P. |last2=Steinhardt |first2=P. J. |date=1995 |title=Cosmic Concordance|eprint=astro-ph/9505066 }}
* {{cite book|last1=Ostriker|first1=Jeremiah P.|last2=Mitton|first2=Simon|title=Heart of Darkness: Unraveling the mysteries of the invisible universe|date=2013|publisher=[[Princeton University Press]]|location=Princeton, NJ|isbn=978-0-691-13430-7}}
* {{cite journal |last1=Rebolo |first1=R. |display-authors=etal  |year=2004 |title=Cosmological parameter estimation using Very Small Array data out to ℓ= 1500 |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=353 |issue=3 |pages=747–759 |arxiv=astro-ph/0402466 |bibcode = 2004MNRAS.353..747R |doi = 10.1111/j.1365-2966.2004.08102.x|doi-access=free |s2cid=13971059 }}

== External links ==
* [http://www.astro.ucla.edu/~wright/cosmolog.htm Cosmology tutorial/NedWright]
* [http://www.mpa-garching.mpg.de/galform/millennium-II/ Millennium Simulation]
* [http://lambda.gsfc.nasa.gov/product/map/dr3/parameters_summary.cfm WMAP estimated cosmological parameters/Latest Summary]

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