Lambda-CDM model

The ΛCDM model is based on three postulates on the structure of spacetime:{{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}}{{rp|227}}

1. The cosmological principle, that the universe is the same everywhere and in all directions, and that it is expanding,
2. 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,{{rp|175}}
3. 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 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.{{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}}{{rp|7}} For example, a model might include baryons, photons, neutrinos, and dark matter.{{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|hdl=11384/149923 |hdl-access=free }}{{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 s to the present.{{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 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.{{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 ionized space, where the electrons can scatter the light, altering the anisotropies. This effect is characterized by one additional parameter.{{rp|25.1.3}}

The ΛCDM model includes an expansion of the spatial 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.{{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}}

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, $p\; =\; -\; \backslash rho\; c^\{2\}$, 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, $\backslash Omega\_\{\backslash Lambda\}$, is estimated to be 0.669 ± 0.038 based on the 2018 Dark Energy Survey results using Type Ia supernovae{{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 }} or {{val|0.6847|0.0073}} based on the 2018 release of Planck satellite data, or more than 68.3% (2018 estimate) of the mass–energy density of the universe.{{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}}

Dark matter is postulated in order to account for gravitational effects observed in very large-scale structures (the "non-keplerian" rotation curves of galaxies;{{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}} the gravitational lensing of light by galaxy clusters; and the enhanced clustering of galaxies) that cannot be accounted for by the quantity of observed matter.{{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 }}

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%{{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 }} of the mass–energy density of the universe. The remaining 4.9% 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.

{{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

}}

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 10¹⁵ K. This was immediately (within 10⁻²⁹ seconds) followed by an exponential expansion of space by a scale multiplier of 10²⁷ 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 lighter elements in the universe (hydrogen, helium, and lithium), and the spatial texture of minute irregularities (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.{{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|arxiv=astro-ph/0310808 }}

The expansion of the universe is parameterized by a dimensionless scale factor $a\; =\; a(t)$ (with time $t$ counted from the birth of the universe), defined relative to the present time, so $a\_0\; =\; a(t\_0)\; =\; 1$; the usual convention in cosmology is that subscript 0 denotes present-day values, so $t\_0$ denotes the age of the universe. The scale factor is related to the observed redshift $z$ of the light emitted at time $t\_\backslash mathrm\{em\}$ by

$$a(t\_\backslash text\{em\})\; =\; \backslash frac\{1\}\{1\; +\; z\}\backslash ,.$$

The expansion rate is described by the time-dependent Hubble parameter, $H(t)$, defined as

$$H(t)\; \backslash equiv\; \backslash frac\{\backslash dot\; a\}\{a\},$$

where $\backslash dot\; a$ is the time-derivative of the scale factor. The first Friedmann equation gives the expansion rate in terms of the matter+radiation density {{nowrap|$\backslash rho$,}} the curvature {{nowrap|$k$,}} and the cosmological constant {{nowrap|$\backslash Lambda$,}}{{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}}

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

where, as usual $c$ is the speed of light and $G$ is the gravitational constant.

A critical density $\backslash rho\_\backslash mathrm\{crit\}$ is the present-day density, which gives zero curvature $k$, assuming the cosmological constant $\backslash Lambda$ 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 }}}}

$$\backslash rho\_\backslash mathrm\{crit\}\; =\; \backslash frac\{3\; H\_0^2\}\{8\; \backslash pi\; G\}\; =\; 1.878\backslash ;47(23)\; \backslash times\; 10^\{-26\}\; \backslash ;\; h^2\; \backslash ;\; \backslash mathrm\{kg\{\backslash cdot\}m^\{-3\}\},$$

where $h\; \backslash equiv\; H\_0\; /\; (100\; \backslash ;\; \backslash mathrm\{km\{\backslash cdot\}s^\{-1\}\{\backslash cdot\}Mpc^\{-1\}\})$ 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 $\backslash Omega\_x$ for various species is defined as the dimensionless ratio{{rp|p=74}}

$$\backslash Omega\_x\; \backslash equiv\; \backslash frac\{\backslash rho\_x(t=t\_0)\}\{\backslash rho\_\backslash mathrm\{crit\}\; \}\; =\; \backslash frac\{8\; \backslash pi\; G\backslash rho\_x(t=t\_0)\}\{3\; H\_0^2\}$$

where the subscript $x$ is one of $\backslash mathrm\; b$ for baryons, $\backslash mathrm\; c$ for cold dark matter, $\backslash mathrm\{rad\}$ for radiation (photons plus relativistic neutrinos), and $\backslash Lambda$ for dark energy.{{citation needed|date=February 2024}}

Since the densities of various species scale as different powers of $a$, e.g. $a^\{-3\}$ for matter etc.,

the Friedmann equation can be conveniently rewritten in terms of the various density parameters as

$$H(a)\; \backslash equiv\; \backslash frac\{\backslash dot\{a\}\}\{a\}\; =\; H\_0\; \backslash sqrt\{\; (\backslash Omega\_\{\backslash rm\; c\}\; +\; \backslash Omega\_\{\backslash rm\; b\})\; a^\{-3\}\; +\; \backslash Omega\_\backslash mathrm\{rad\}\; a^\{-4\}\; +\; \backslash Omega\_k\; a^\{-2\}\; +\; \backslash Omega\_\{\backslash Lambda\}\; a^\{-3(1+w)\}\; \}\; ,$$

where $w$ is the equation of state parameter of dark energy, and assuming negligible neutrino mass (significant neutrino mass requires a more complex equation). The various $\backslash Omega$ parameters add up to $1$ by construction. In the general case this is integrated by computer to give the expansion history $a(t)$ 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 $\backslash Omega\_k$ is zero and $w\; =\; -1$, so this simplifies to

$$H(a)\; =\; H\_0\; \backslash sqrt\{\; \backslash Omega\_\{\backslash rm\; m\}\; a^\{-3\}\; +\; \backslash Omega\_\backslash mathrm\{rad\}\; a^\{-4\}\; +\; \backslash Omega\_\backslash Lambda\; \}$$

Observations show that the radiation density is very small today, $\backslash Omega\_\backslash text\{rad\}\; \backslash sim\; 10^\{-4\}$; if this term is neglected

the above has an analytic solution{{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}}

$$a(t)\; =\; (\backslash Omega\_\{\backslash rm\; m\}\; /\; \backslash Omega\_\backslash Lambda)^\{1/3\}\; \backslash ,\; \backslash sinh^\{2/3\}\; (\; t\; /\; t\_\backslash Lambda)$$

where $t\_\backslash Lambda\; \backslash equiv\; 2\; /\; (3\; H\_0\; \backslash sqrt\{\backslash Omega\_\backslash Lambda\}\; )\; \backslash \; ;$

this is fairly accurate for $a\; >\; 0.01$ or $t\; >\; 10$ million years.

Solving for $a(t)\; =\; 1$ gives the present age of the universe $t\_0$ in terms of the other parameters.{{citation needed|date=February 2024}}

It follows that the transition from decelerating to accelerating expansion (the second derivative $\backslash ddot\{a\}$ crossing zero) occurred when

$$a\; =\; (\; \backslash Omega\_\{\backslash rm\; m\}\; /\; 2\; \backslash Omega\_\backslash Lambda\; )^\{1/3\}\; ,$$

which evaluates to $a\; \backslash sim\; 0.6$ or $z\; \backslash sim\; 0.66$ for the best-fit parameters estimated from the Planck spacecraft.{{citation needed|date=February 2024}}

Multiple variants of the ΛCDM model are used with some differences in parameters.{{rp|loc=25.1}} One such set is outlined in the table below.

{{Clear}}

The Planck collaboration version of the ΛCDM model is based on six parameters: 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). 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 constant and the dark energy density, can be calculated.

{{notelist}}

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 expansion in 1998, and was quickly supported by other observations: in 2000, the BOOMERanG microwave background experiment measured the total (matter–energy) density to be close to 100% of critical, whereas in 2001 the 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 in 2013–2015 have continued to support the model and pin down the parameter values, most of which are constrained below 1 percent uncertainty.{{citation needed|date=February 2024}}

Among all cosmological models, the ΛCDM model has been the most successful; it describes a wide range of astronomical observations with remarkable accuracy.{{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 mission{{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}} and Atacama Cosmology Telescope.{{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}}
• Accurate description of the linear E-mode polarization of the CMB radiation due to fluctuations on the surface of last scattering events.{{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}}
• Prediction of the observed B-mode polarization of the CMB light due to primordial gravitational waves.{{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}}
• Observations of H₂O 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.{{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}}
• Predictions of the primordial abundance of deuterium as a result of Big Bang nucleosynthesis.{{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|arxiv=1308.3240 }} The observed abundance matches the one derived from the nucleosynthesis model with the value for baryon density derived from CMB measurements.{{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 polarization of the CMB, discovered in 2002 by DASI,{{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}} has been successfully predicted by the model: in the 2015 Planck data release,{{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 }} 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}}

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.{{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 }}{{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}}{{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 }}

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 extremely small compared to vacuum energy theoretical predictions.{{citation needed|date=February 2024}}

{{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) on a large enough scale: "the universe looks the same whoever and wherever you are."Andrew Liddle. An Introduction to Modern Cosmology (2nd ed.). London: Wiley, 2003. 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.{{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.}}{{rp|408}} The assumptions were carried over into the ΛCDM model.{{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}} However, some findings suggested violations of the cosmological principle.

Evidence from galaxy clusters,{{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}}{{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}} quasars,{{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 }} and type Ia supernovae{{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}} 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.{{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}}

Already in 1967, Dennis Sciama predicted that the cosmic microwave background has a significant dipole anisotropy.{{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}}{{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}} In recent years, the CMB dipole has been tested, and the results suggest our motion with respect to distant radio galaxies{{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 }} and quasars{{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 }} 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{{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}} and quasars.{{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}} 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{{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}} and an anomalous parity asymmetry{{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}} that may have an origin in the CMB dipole.{{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}} Separately, the CMB dipole direction has emerged as a preferred direction in studies of alignments in quasar polarizations,{{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}} scaling relations in galaxy clusters,{{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 $L\_\{\backslash text\{X\}\}-T$ 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}}{{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}} strong lensing time delay,{{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}} Type Ia supernovae,{{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 }} and quasars and gamma-ray bursts as standard candles.{{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 }} 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.{{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}}

The homogeneity of the universe needed for the ΛCDM applies to very large volumes of space.

N-body simulations in ΛCDM show that the spatial distribution of galaxies is statistically homogeneous if averaged over scales 260/h Mpc or more.{{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 }}

Numerous claims of large-scale structures reported to be in conflict with the predicted scale of homogeneity for ΛCDM do not withstand statistical analysis.{{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}}{{rp|7.8}}

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

El Gordo is a massive interacting galaxy cluster in the early Universe ($z\; =\; 0.87$). The extreme properties of El Gordo in terms of its redshift, mass, and the collision velocity leads to strong ($6.16\backslash sigma$) tension with the ΛCDM model.{{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}}{{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}} The properties of El Gordo are however consistent with cosmological simulations in the framework of MOND due to more rapid structure formation.{{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}}

{{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.{{Cite journal | last1 = Keenan | first1 = Ryan C. | last2 = Barger | first2 = Amy J. | last3 = Cowie | first3 = Lennox L. | title = Evidence for a ~300 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 }}{{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}} Some authors have said the existence of the KBC void violates the assumption that the CMB reflects baryonic density fluctuations at $z\; =\; 1100$ or Einstein's theory of general relativity, either of which would violate the ΛCDM model,{{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}} while other authors have claimed that supervoids as large as the KBC void are consistent with the ΛCDM model.{{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 }}

{{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,{{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 }} widely acknowledged to be a major problem for the ΛCDM model.

{{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

}}

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, 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.

The "$S\_8$ tension" is a name for another question mark for the ΛCDM model. The $S\_8$ parameter in the ΛCDM model quantifies the amplitude of matter fluctuations in the late universe and is defined as

$$S\_8\; \backslash equiv\; \backslash sigma\_8\backslash sqrt\{\backslash Omega\_\{\backslash rm\; m\}/0.3\}$$

Early- (e.g. from CMB data collected using the Planck observatory) and late-time (e.g. measuring weak gravitational lensing events) facilitate increasingly precise values of $S\_8$. However, these two categories of measurement differ by more standard deviations than their uncertainties. This discrepancy is called the $S\_8$ 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.

Some values for $S\_8$ are {{val|0.832|0.013}} (2020 Planck),{{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 }} {{val|0.766|0.020|0.014}} (2021 [https://kids.strw.leidenuniv.nl/ KIDS]),{{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 }}{{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}} {{val|0.776|0.017}} (2022 DES),{{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 }} {{val|0.790|0.018|0.014}} (2023 DES+KIDS),{{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 }} {{val|0.769|0.031|0.034}} – {{val|0.776|0.032|0.033}}{{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 }}{{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 }}{{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 }}{{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}} (2023 [https://hsc.mtk.nao.ac.jp/ssp/ HSC-SSP]), {{val|0.86|0.01}} (2024 EROSITA).{{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}}{{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}} Values have also obtained using peculiar velocities, {{val|0.637|0.054}} (2020){{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 }} and {{val|0.776|0.033}} (2020),{{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 }} among other methods.

{{main|Axis of evil (cosmology)}}

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

{{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.{{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}}{{rp|141}} If every calculation is correct, then solutions beyond the existing ΛCDM model might be needed.

{{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.{{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}} 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.{{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}}

{{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.{{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 }} They observed an effect inconsistent with 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.{{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 }} and as inconsistent with similar analysis of other galaxies.{{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 }}

{{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.{{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 }}

Milgrom, McGaugh, and 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 theories such as {{ill|galileon|ko}} theories (see Galilean invariance), brane cosmologies, the DGP model, and massive gravity and its extensions such as bimetric gravity.{{citation needed|date=February 2024}}

{{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.{{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 }}

{{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.{{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 }}

Dwarf galaxies around the Milky Way and 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.{{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 }} However, latest research suggests this seemingly bizarre alignment is just a quirk which will dissolve over time.{{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 }}

Galaxies in the NGC 3109 association are moving away too rapidly to be consistent with expectations in the ΛCDM model.{{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}} 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.{{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}}

If galaxies grew hierarchically, then massive galaxies required many mergers. Major mergers inevitably create a classical bulge. On the contrary, about 80% of observed galaxies give evidence of no such bulges, and giant pure-disc galaxies are commonplace.{{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 }} 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.{{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}} The high bulgeless fraction was nearly constant for 8 billion years.{{cite journal |last1=Sachdeva |first1=S. |last2=Saha |first2=K. |title=Survival of pure disk galaxies over the last 8 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 }}

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.{{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}}

Comparison of the model with observations may have some problems on sub-galaxy scales, possibly predicting too many dwarf galaxies and too much dark matter in the innermost regions of galaxies. This problem is called the "small scale crisis".{{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}} 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.

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.{{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}}{{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 }} 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,{{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 }} or the hypothesized long-sought Population III stars.{{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 }}{{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}}{{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}}{{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 }}

{{main|Missing baryon problem}}

Massimo Persic and Paolo Salucci{{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}} 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 $M\_\{\backslash rm\; b\}/L$), weighted with the luminosity function $\backslash phi(L)$ over the previously mentioned classes of astrophysical objects:

$$\backslash rho\_\{\backslash rm\; b\}\; =\; \backslash sum\; \backslash int\; L\backslash phi(L)\; \backslash frac\{M\_\{\backslash rm\; b\}\}\{L\}\; \backslash ,\; dL.$$

The result was:

$$\backslash Omega\_\{\backslash rm\; b\}=\backslash Omega\_*+\backslash Omega\_\backslash text\{gas\}=2.2\backslash times10^\{-3\}+1.5\backslash times10^\{-3\}\backslash ;h^\{-1.3\}\backslash simeq0.003\; ,$$

where $h\backslash simeq\; 0.72$.

Note that this value is much lower than the prediction of standard cosmic nucleosynthesis $\backslash Omega\_\{\backslash rm\; b\}\backslash simeq0.0486$, 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–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 haloes, filling the space between galaxies.{{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}} Together with the amount of baryons inside galaxies and surrounding them, the total amount of baryons in the late time Universe is compatible with early Universe measurements.

It has been argued that the ΛCDM model has adopted conventionalist stratagems, rendering it 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.{{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}}

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 ($\backslash Omega\_\backslash text\{tot\}$ may be different from 1); or quintessence rather than a cosmological constant where the equation of state of dark energy is allowed to differ from −1. Cosmic inflation predicts tensor fluctuations (gravitational waves). Their amplitude is parameterized by the tensor-to-scalar ratio (denoted $r$), which is determined by the unknown energy scale of inflation. Other modifications allow hot dark matter in the form of neutrinos 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 $r$ should be between 0 and 0.3, and the latest results are within those limits.

• Bolshoi cosmological simulation
• Galaxy formation and evolution
• Illustris project
• List of cosmological computation software
• Millennium Run
• Weakly interacting massive particles (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}}

{{reflist|30em}}

• {{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 }}

• [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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