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{{Short description|Metric based on the exact solution of Einstein's field equations of general relativity}}
{{Physical cosmology |expansion}}
The '''Friedmann–Lemaître–Robertson–Walker metric''' ('''FLRW'''; {{IPAc-en||ˈ|f|r|iː|d|m|ə|n|_|l|ə|ˈ|m|ɛ|t|r|ə|...}}) is a [[Lorentzian metric|metric]] that describes a [[homogeneity (physics)#Translation invariance|homogeneous]], [[isotropic]], [[metric expansion of space|expanding]] (or otherwise, contracting) [[universe]] that is [[path-connected]], but not necessarily [[simply connected space|simply connected]].<ref>For an early reference, see Robertson (1935); Robertson ''assumes'' multiple connectedness in the positive curvature case and says that "we are still free to restore" simple connectedness.</ref><ref name="LaLu95">{{Cite journal |last1=Lachieze-Rey |first1=M. |last2=Luminet |first2=J.-P. |date=1995 |title=Cosmic Topology |journal=[[Physics Reports]] |volume=254 |issue=3 |pages=135–214 |arxiv=gr-qc/9605010 |bibcode=1995PhR...254..135L |doi=10.1016/0370-1573(94)00085-H |s2cid=119500217}}</ref><ref name="Ellis98">{{Cite conference |last1=Ellis |first1=G. F. R. |last2=van Elst |first2=H. |date=1999 |title=Cosmological models (Cargèse lectures 1998) |series=NATO Science Series C |volume=541 |pages=1–116 |arxiv=gr-qc/9812046 |bibcode=1999ASIC..541....1E |isbn=978-0792359463 |editor=Marc Lachièze-Rey |book-title=Theoretical and Observational Cosmology}}</ref> The general form of the metric follows from the geometric properties of homogeneity and isotropy. Depending on geographical or historical preferences, the set of the four scientists – [[Alexander Friedmann]], [[Georges Lemaître]], [[Howard P. Robertson]] and [[Arthur Geoffrey Walker]] – are variously grouped as '''Friedmann''', '''Friedmann–Robertson–Walker''' ('''FRW'''), '''Robertson–Walker''' ('''RW'''), or '''Friedmann–Lemaître''' ('''FL'''). When combined with Einstein's field equations the metric gives the [[Friedmann equation]] which has been developed into the ''Standard Model'' of modern [[Physical cosmology|cosmology]],<ref name="Goobar">{{Cite book |last1=Bergström |first1=Lars |url=https://books.google.com/books?id=CQYu_sutWAoC&pg=PA61 |title=Cosmology and particle astrophysics |last2=Goobar |first2=Ariel |date=2008 |publisher=Praxis Publ |isbn=978-3-540-32924-4 |edition=2. ed., reprinted |series=Springer Praxis books in astronomy and planetary science |___location=Chichester, UK |page=61}}</ref> and the further developed [[Lambda-CDM model]].
==
The metric is a consequence of assuming that the mass in the universe has constant density – homogeneity – and is the same in all directions – isotropy. Assuming isotropy alone is sufficient to reduce the possible motions of mass in the universe to radial velocity variations. The [[Copernican principle]], that our observation point in the universe is the equivalent to every other point, combined with isotropy ensures homogeneity. 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}} Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.<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|65}}
To measure distances in this space, that is to define a metric, we can compare the positions of two points in space moving along with their local radial velocity of mass. Such points can be thought of as ideal galaxies. Each galaxy can be given a clock to track local time, with the clocks synchronized by imagining the radial velocities run backwards until the clocks coincide in space. The [[equivalence principle]] applied to each galaxy means distance measurements can be made using [[special relativity#Invariant interval|special relativity]] locally. So a distance <math>d\tau</math> can be related to the local time {{mvar|t}} and the coordinates:
<math display="block">c^2d\tau^2 = c^2dt^2 - dx^2 - dy^2 -dz^2</math>
An isotropic, homogeneous mass distribution is highly symmetric. Rewriting the metric in spherical coordinates reduces four coordinates to three coordinates. The radial coordinate is written as a product of a [[Comoving and proper distances|comoving]] coordinate, {{mvar|r}}, and a time dependent scale factor {{mvar|R(t)}}. The resulting metric can be written in several forms. Two common ones are:
<math display="block">c^2d\tau^2 = c^2dt^2 - R^2(t)\left(dr^2+ S^2_k(r) d\psi^2\right)</math>
or
<math display="block">c^2 d\tau^2 = c^2 dt^2 - R^2(t)\left( \frac{dr^2}{1 - kr^2} + r^2 d\psi^2\right)</math>
where <math>\psi</math> is the angle between the two locations and
<math display="block">S_{-1}(r) = \sinh(r), S_0 = 1, S_1 = \sin(r).</math>
(The meaning of {{mvar|r}} in these equations is not the same). Other common variations use a dimensionless scale factor
<math display="block">a(t) = \frac{R(t)}{R_0}</math>
where time zero is now.<ref name=Peacock-1998/>{{rp|p=70}}
=== 2-dimensional analogy ===
The time dependent scale factor <math>R(t)</math>, which plays a critical role in cosmology, has an analog in the radius of a sphere. A sphere is a 2 dimensional surface embedded in a 3 dimensional space. The radius of a sphere lives in the third dimension: it is not part of the 2 dimensional surface. However, the value of this radius affects distances measured on the two dimensional surface. Similarly the cosmological scale factor is not a distance in our 3 dimensional space, but its value affects the measurement of distances.<ref>{{Cite book |last1=Tropp |first1=Eduard A. |url=https://www.cambridge.org/core/product/identifier/9780511608131/type/book |title=Alexander A Friedmann: The Man who Made the Universe Expand |last2=Frenkel |first2=Viktor Ya. |last3=Chernin |first3=Artur D. |date=1993-06-03 |publisher=Cambridge University Press |isbn=978-0-521-38470-4 |edition=1 |translator-last=Dron |translator-first=Alexander |doi=10.1017/cbo9780511608131 |translator-last2=Burov |translator-first2=Michael}}</ref>{{rp|p=147}}
== FLRW models ==
{{main|Friedmann equations}}
To apply the metric to cosmology and predict its time evolution requires Einstein's field equations together with a way of calculating the density, <math>\rho (t),</math> such as a [[equation of state (cosmology)|cosmological equation of state]].
This process allows an approximate analytic solution of [[Einstein field equations|Einstein's field equations]] <math>G_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}</math> giving the [[Friedmann equations]] when the [[energy–momentum tensor]] is similarly assumed to be isotropic and homogeneous.
Models based on the FLRW metric and obeying the Friedmann equations are called '''FRW models'''.<ref name=Peacock-1998/>{{rp|p=73}}
Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.<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|65}}
These models are the basis of the standard [[Big Bang]] cosmological model including the current [[Lambda-CDM model|ΛCDM]] model.<ref name=PDG-2024>{{Cite journal |last1=Navas |first1=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 |page=030001 |doi=10.1103/PhysRevD.110.030001 |issn=2470-0010|hdl=20.500.11850/695340 |hdl-access=free }}</ref>{{rp|loc=25.1.3}}
== General metric ==
The FLRW metric assume [[homogeneity (physics)#Translation invariance|homogeneity]] and [[isotropy]] of space.<ref>{{Cite book |last=Weinberg |first=Steven |title=Gravitation and cosmology: principles and applications of the general theory of relativity |date=1972 |publisher=Wiley |isbn=978-0-471-92567-5 |___location=New York}}</ref>{{rp|p=404}} It also assumes that the spatial component of the metric can be time-dependent. The generic metric that meets these conditions is
<math display="block">-c^2\mathrm{d}\tau^2 = -c^2\mathrm{d}t^2 + {a(t)}^2 \mathrm{d}\mathbf{\Sigma}^2,</math>
where <math>\mathbf{\Sigma}</math> ranges over a 3-dimensional space of uniform curvature, that is, [[elliptical space]], [[Euclidean space]], or [[hyperbolic space]]. It is normally written as a function of three spatial coordinates, but there are several conventions for doing so, detailed below. <math>\mathrm{d}\mathbf{\Sigma}</math> does not depend on <math>t</math> – all of the time dependence is in the function {{nowrap|1=<math>a(t)</math>,}} known as the "[[Scale factor (cosmology)|scale factor]]".
=== Reduced-circumference polar coordinates ===
In reduced-circumference polar coordinates the spatial metric has the form<ref>{{Cite book |last=Wald |first=Robert M. |title=General relativity |date=1984 |publisher=University of Chicago Press |isbn=978-0-226-87032-8 |___location=Chicago |page=116}}</ref><ref>{{Cite book |last=Carroll |first=Sean M. |title=Spacetime and geometry: an introduction to general relativity |date=2019 |publisher=Cambridge University Press |isbn=978-1-108-48839-6 |___location=New York |pages=329–333}}</ref>
<math display="block">\mathrm{d}\mathbf{\Sigma}^2 = \frac{\mathrm{d}r^2}{1-k r^2} + r^2 \mathrm{d}\mathbf{\Omega}^2, \quad \text{ where } \mathrm{d}\mathbf{\Omega}^2 = \mathrm{d}\theta^2 + \sin^2 \theta \, \mathrm{d}\phi^2.</math>
<math>k</math> is a constant representing the curvature of the space. There are two common unit conventions:
* <math>k</math> may be taken to have units of length<sup>−2</sup>, in which case <math>r</math> has units of length and <math>a(t)</math> is unitless. {{nowrap|1=<math>k</math> is}} then the [[Gaussian curvature]] of the space at the time when {{nowrap|1=<math>a(t) = 1</math>.}} {{nowrap|1=<math>r</math> is}} sometimes called the reduced [[circumference]] because it is equal to the measured circumference of a circle (at that value of {{nowrap|1=<math>r</math>),}} centered at the origin, divided by <math>2\pi</math> (like the <math>r</math> of [[Schwarzschild coordinates]]). Where appropriate, {{nowrap|1=<math>a(t)</math> is}} often chosen to equal 1 in the present cosmological era, so that <math>\mathrm{d}\mathbf{\Sigma}</math> measures [[comoving distance]].
* Alternatively, <math>k</math> may be taken to belong to the set {{nowrap|{{mset|−1, 0, +1}}}} (for negative, zero, and positive curvature respectively). Then {{nowrap|1=<math>r</math> is}} unitless and <math>a(t)</math> has units of length. When {{nowrap|1=<math>k = \pm1</math>,}} <math>a(t)</math> is the [[Radius of curvature (mathematics)|radius of curvature]] of the space and may also be written {{nowrap|1=<math>R(t)</math>.}}
A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature—circumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is [[elliptical geometry|elliptical]], i.e. a 3-sphere with opposite points identified.)
=== Hyperspherical coordinates ===
In ''hyperspherical'' or ''curvature-normalized'' coordinates the coordinate <math>r</math> is proportional to radial distance; this gives
<math display="block">\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}r^2 + S_k(r)^2 \, \mathrm{d}\mathbf{\Omega}^2</math>
where <math>\mathrm{d}\mathbf{\Omega}</math> is as before and
<math display="block">S_k(r) = \begin{cases}
\sqrt{k}^{\,-1} \sin (r \sqrt{k}), &k > 0 \\
r, &k = 0 \\
\sqrt{|k|}^{\,-1} \sinh
\end{cases}</math>
As before, there are two common unit conventions:
* <math>k</math> may be taken to have units of length<sup>−2</sup>, in which case <math>r</math> has units of length and <math>a(t)</math> is unitless. {{nowrap|1=<math>k</math> is}} then the [[Gaussian curvature]] of the space at the time when {{nowrap|1=<math>a(t) = 1</math>.}} Where appropriate, <math>a(t)</math> is often chosen to equal 1 in the present cosmological era, so that <math>\mathrm{d}\mathbf{\Sigma}</math> measures [[comoving distance]].
* Alternatively, as before, <math>k</math> may be taken to belong to the set {{mset|−1 ,0, +1}} (for negative, zero, and positive curvature respectively). Then <math>r</math> is unitless and <math>a(t)</math> has units of length. When {{nowrap|1=<math>k = \pm1</math>,}} <math>a(t)</math> is the [[radius of curvature]] of the space and may also be written {{nowrap|1=<math>R(t)</math>.}} Note that when {{nowrap|1=<math>k = +1</math>,}} {{nowrap|1=<math>r</math> is}} essentially a third angle along with <math>\theta</math> and {{nowrap|1=<math>\phi</math>.}} The letter <math>\chi</math> may be used instead {{nowrap|1=of <math>r</math>.}}
Though it is usually defined piecewise as above, <math>S</math> is an [[analytic function]] of both <math>k</math> and {{nowrap|1=<math>r</math>.}} It can also be written as a [[power series]]
<math display="block">S_k(r)
= \sum_{n=0}^\infty \frac{{\left(-1\right)}^n k^n r^{2n+1}}{(2n+1)!}
or as
<math display="block">S_k(r) = r \; \mathrm{sinc} \, (r \sqrt{k}), </math>
where <math>\mathrm{sinc}</math> is the unnormalized [[sinc function]] and <math>\sqrt{k}</math> is one of the imaginary, zero or real square roots {{nowrap|1=of <math>k</math>.}} These definitions are valid for {{nowrap|1=all <math>k</math>.}}
=== Cartesian coordinates ===
When <math>k = 0</math> one may write simply
<math display="block">\mathrm{d}\mathbf{\Sigma}^2 = \mathrm{d}x^2 + \mathrm{d}y^2 + \mathrm{d}z^2.</math>
This can be extended to <math>k \ne 0</math> by defining
<math display="block">\begin{align}
x &= r \cos \theta \,, \\
y &= r \sin \theta \cos \phi \,, \\
z &= r \sin \theta \sin \phi \,,
\end{align}</math>
where <math>r</math> is one of the radial coordinates defined above, but this is rare.
==Curvature==
=== Cartesian coordinates ===
In flat <math>(k=0)</math> FLRW space using Cartesian coordinates, the surviving components of the [[Ricci tensor]] are<ref>{{Cite book |last=Wald |first=Robert M. |title=General relativity |date=1984 |publisher=University of Chicago Press |isbn=978-0-226-87032-8 |___location=Chicago |page=97}}</ref>
<math display="block"> R_{tt} = - 3 \frac{\ddot{a}}{a}, \quad R_{xx}= R_{yy} = R_{zz} = c^{-2} \left(a \ddot{a} + 2 \dot{a}^2\right) </math>
and the Ricci scalar is
<math display="block"> R = 6 c^{-2} \left(\frac{\ddot{a}(t)}{a(t)} + \frac{\dot{a}^2(t)}{a^2(t)}\right).</math>
=== Spherical coordinates ===
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are<ref>{{cite web|url=http://icc.ub.edu/~liciaverde/Cosmology.pdf |title=Cosmology |page=23|archive-url=https://web.archive.org/web/20200111005125/http://icc.ub.edu/~liciaverde/Cosmology.pdf|archive-date=Jan 11, 2020}}</ref>{{Failed verification|date=February 2025|reason following equations do not match this source.}}
<math display="block">\begin{align}
R_{tt} &= - 3 \frac{\ddot{a}}{a}, \\[1ex]
R_{rr} &= \frac{c^{-2} \left(a\ddot{a} + 2\dot{a}^2\right) + 2k}{1 - kr^2} \\[1ex]
R_{\theta\theta} &= r^2 \left[c^{-2} \left(a\ddot{a} + 2\dot{a}^2\right) + 2k\right] \\[1ex]
R_{\phi\phi} &= r^2\sin^2(\theta) \left[c^{-2} \left(a\ddot{a} + 2\dot{a}^2\right) + 2k\right]
\end{align}</math>
and the Ricci scalar is
<math display="block"> R = \frac{6}{c^2} \left(\frac{\ddot{a}(t)}{a(t)} + \frac{\dot{a}^2(t)}{a^2(t)} + \frac{c^2k}{a^2(t)}\right).</math>
== Name and history ==
{{also|Friedmann equations#History}}
{{primary|section|date=February 2025}}
In 1922 and 1924 the Soviet mathematician [[Alexander Friedmann]]<ref>{{Cite journal |last=Friedmann |first=Alexander |author-link=Alexander Friedmann |date=1922 |title=Über die Krümmung des Raumes |journal=Zeitschrift für Physik A |volume=10 |issue=1 |pages=377–386 |bibcode=1922ZPhy...10..377F |doi=10.1007/BF01332580 |s2cid=125190902}}<!-- Фридман, Александр Александрович --></ref><ref>{{Cite journal |last=Friedmann |first=Alexander |author-link=Alexander Friedmann |date=1924 |title=Über die Möglichkeit einer Welt mit konstanter negativer Krümmung des Raumes |journal=Zeitschrift für Physik A |language=de |volume=21 |issue=1 |pages=326–332 |bibcode=1924ZPhy...21..326F |doi=10.1007/BF01328280 |s2cid=120551579}}<!-- Фридман, Александр Александрович --> English trans. in 'General Relativity and Gravitation' 1999 vol.31, 31–</ref> and in 1927, [[Georges Lemaître]], a Belgian priest, astronomer and periodic professor of physics at the [[Catholic University of Leuven (1834–1968)|Catholic University of Leuven]], arrived independently at results<ref>{{citation |last=Lemaître |first=Georges |author-link=Georges Lemaître |title=Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulæ |date=1931 |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=91 |issue=5 |pages=483–490 |bibcode=1931MNRAS..91..483L |doi=10.1093/mnras/91.5.483|doi-access=free }} ''translated from'' {{citation |last=Lemaître |first=Georges |author-link=Georges Lemaître |title=Un univers homogène de masse constante et de rayon croissant rendant compte de la vitesse radiale des nébuleuses extra-galactiques |date=1927 |journal=Annales de la Société Scientifique de Bruxelles |volume=A47 |pages=49–56 |bibcode=1927ASSB...47...49L}}</ref><ref>{{citation |last=Lemaître |first=Georges |author-link=Georges Lemaître |title=l'Univers en expansion |date=1933 |journal=Annales de la Société Scientifique de Bruxelles |volume=A53 |pages=51–85 |bibcode=1933ASSB...53...51L}}</ref> that relied on the metric.
[[Howard P. Robertson]] from the US and [[Arthur Geoffrey Walker]] from the UK explored the problem further during the 1930s.<ref>{{citation |last=Robertson |first=H. P. |author-link=Howard P. Robertson |date=1935 |title=Kinematics and world structure |journal=[[Astrophysical Journal]] |volume=82 |pages=284–301 |bibcode=1935ApJ....82..284R |doi=10.1086/143681}}</ref><ref>{{citation |last=Robertson |first=H. P. |author-link=Howard P. Robertson |date=1936 |title=Kinematics and world structure II |journal=[[Astrophysical Journal]] |volume=83 |pages=187–201 |bibcode=1936ApJ....83..187R |doi=10.1086/143716|doi-access=free }}</ref><ref>{{citation |last=Robertson |first=H. P. |author-link=Howard P. Robertson |date=1936 |title=Kinematics and world structure III |journal=[[Astrophysical Journal]] |volume=83 |pages=257–271 |bibcode=1936ApJ....83..257R |doi=10.1086/143726|doi-access=free }}</ref><ref>{{citation |last=Walker |first=A. G. |author-link=Arthur Geoffrey Walker |date=1937 |title=On Milne's theory of world-structure |journal=[[Proceedings of the London Mathematical Society]] |series=Series 2 |volume=42 |issue=1 |pages=90–127 |doi=10.1112/plms/s2-42.1.90|bibcode=1937PLMS...42...90W }}</ref> In 1935 Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic (as noted above, this is a geometric result and is not tied specifically to the equations of general relativity, which were always assumed by Friedmann and Lemaître).
This solution, often called the Robertson–Walker ''metric'' since they proved its generic properties, is different from the dynamical "Friedmann–Lemaître" ''models'', which are specific solutions for ''a''(''t'') that assume that the only contributions to stress–energy are cold matter ("dust"), radiation, and a cosmological constant.
== Current status ==
{{See also|Shape of the universe}}{{unsolved|physics|Is the universe homogeneous and isotropic at large enough scales, as claimed by the [[cosmological principle]]?<ref name="Snowmass21" /><ref>{{Cite web |last=Billings |first=Lee |date=April 15, 2020 |title=Do We Live in a Lopsided Universe? |url=https://www.scientificamerican.com/article/do-we-live-in-a-lopsided-universe1/ |access-date=March 24, 2022 |website=[[Scientific American]]}}</ref><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. |date=April 2020 |title=Probing cosmic isotropy with a new X-ray galaxy cluster sample through the L X – T scaling relation |url=https://www.aanda.org/10.1051/0004-6361/201936602 |journal=Astronomy & Astrophysics |volume=636 |issue=April 2020 |pages=A15 |arxiv=2004.03305 |bibcode=2020A&A...636A..15M |doi=10.1051/0004-6361/201936602 |issn=0004-6361 |s2cid=215238834 |access-date=24 March 2022}}</ref> Is the CMB dipole purely kinematic, or does it signal a possible breakdown of the FLRW metric?<ref name="Snowmass21" /> Is the Friedmann–Lemaître–Robertson–Walker metric valid in the late universe?<ref name="Snowmass21" /><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 |pages=184001 |doi=10.1088/1361-6382/ac1a81 |arxiv=2105.09790 |bibcode=2021CQGra..38r4001K |s2cid=234790314 |issn=0264-9381}}</ref>}}
The current standard model of cosmology, the [[Lambda-CDM model]], uses the FLRW metric. By combining the observation data from some experiments such as [[Wilkinson Microwave Anisotropy Probe|WMAP]] and [[Planck (spacecraft)|Planck]] with theoretical results of [[Ehlers–Geren–Sachs theorem]] and its generalization,<ref>See pp. 351ff. in {{citation |last1=Hawking |first1=Stephen W. |last2=Ellis |first2=George F. R. |author2-link=George Francis Rayner Ellis |title=The large scale structure of space-time |publisher=Cambridge University Press |isbn=978-0-521-09906-6 |date=1973|title-link=The large scale structure of space-time }}. The original work is Ehlers, J., Geren, P., Sachs, R.K.: Isotropic solutions of Einstein-Liouville equations. J. Math. Phys. 9, 1344 (1968). For the generalization, see {{citation |last1=Stoeger |first1=W. R. |last2=Maartens |first2=R |last3=Ellis |first3=George |author3-link=George Francis Rayner Ellis |title=Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem |journal=Astrophys. J. |volume=39 |date=2007 |pages=1–5 |doi=10.1086/175496 |bibcode=1995ApJ...443....1S|doi-access=free }}.</ref> astrophysicists now agree that the early universe is almost homogeneous and isotropic (when averaged over a very large scale) and thus nearly a FLRW spacetime. That being said, attempts to confirm the purely kinematic interpretation of the Cosmic Microwave Background (CMB) dipole through studies of radio galaxies <ref>See Siewert et al. for a recent summary of results {{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 J. |last2=Hausegger |first2=Sebastian von |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=2021-02-25 |volume=908 |issue=2 |pages=L51 |doi=10.3847/2041-8213/abdd40|arxiv=2009.14826 |bibcode=2021ApJ...908L..51S |s2cid=222066749 |doi-access=free }}</ref> show disagreement in the magnitude. Taken at face value, these observations are at odds with the Universe being described by the FLRW metric. Moreover, one can argue that there is a maximum value to the Hubble constant within an FLRW cosmology tolerated by current observations, <math>H_0</math> = {{val|71|1|u=km/s/Mpc}}, and depending on how local determinations converge, this may point to a breakdown of the FLRW metric in the late universe, necessitating an explanation beyond the FLRW metric.<ref>{{cite journal |last1=Krishnan |first1=Chethan |last2=Mohayaee |first2=Roya |last3=Ó Colgáin |first3=Eoin |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=2021-05-25|volume=38 |issue=18 |page=184001 |doi=10.1088/1361-6382/ac1a81 |arxiv=2105.09790 |bibcode=2021CQGra..38r4001K |s2cid=234790314 }}</ref><ref name="Snowmass21">{{Cite journal |last=Abdalla |first=Elcio |display-authors=etal |date=June 2022 |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 |language=en |volume=34 |pages=49–211 |arxiv=2203.06142v1 |bibcode=2022JHEAp..34...49A |doi=10.1016/j.jheap.2022.04.002 |s2cid=247411131}}</ref>
== References ==
{{reflist}}
== Further reading ==
* {{Cite book |last=North |first=John David |url=https://books.google.com/books?id=LK6pQgAACAAJ |title=The measure of the universe: a history of modern cosmology |date=1990 |publisher=Dover Publications |isbn=978-0-486-66517-7 |___location=New York}}
* {{Cite journal |last=Harrison |first=E. R. |date=1967 |title=Classification of Uniform Cosmological Models |journal=Monthly Notices of the Royal Astronomical Society |language=en |volume=137 |issue=1 |pages=69–79 |bibcode=1967MNRAS.137...69H |doi=10.1093/mnras/137.1.69 |issn=0035-8711 |doi-access=free}}
* {{Cite book |last=D'Inverno |first=Ray |url=https://archive.org/details/introducingeinst0000dinv |title=Introducing Einstein's relativity |date=1992 |publisher=Clarendon Press ; Oxford University Press |isbn=978-0-19-859686-8 |edition=Repr |___location=Oxford [England] : New York |url-access=registration}}. ''(See Chapter 23 for a particularly clear and concise introduction to the FLRW models.)''
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