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The Friedmann鈥揕ema卯tre鈥揜obertson鈥揥alker (FLRW; /fri藧dmn lmtr .../) metric is a metric based on the exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected. [1] [2] [3] The general form of the metric follows from the geometric properties of homogeneity and isotropy; Einstein's field equations are only needed to derive the scale factor of the universe as a function of time. Depending on geographical or historical preferences, the set of the four scientists 鈥 Alexander Friedmann, Georges Lema卯tre, Howard P. Robertson and Arthur Geoffrey Walker 鈥 are customarily grouped as Friedmann or Friedmann鈥揜obertson鈥揥alker (FRW) or Robertson鈥揥alker (RW) or Friedmann鈥揕ema卯tre (FL). This model is sometimes called the Standard Model of modern cosmology, [4] although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 1930s.

General metric

The FLRW metric starts with the assumption of homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. The generic metric which meets these conditions is

where 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. does not depend on t 鈥 all of the time dependence is in the function a(t), known as the " scale factor".

Reduced-circumference polar coordinates

In reduced-circumference polar coordinates the spatial metric has the form [5] [6]

k is a constant representing the curvature of the space. There are two common unit conventions:

  • k may be taken to have units of length鈭2, in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature of the space at the time when a(t) = 1. r is sometimes called the reduced circumference because it is equal to the measured circumference of a circle (at that value of r), centered at the origin, divided by 2 (like the r of Schwarzschild coordinates). Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that measures comoving distance.
  • Alternatively, k may be taken to belong to the set {鈭1 ,0, +1} (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When k = 卤1, a(t) is the radius of curvature of the space, and may also be written R(t).

A disadvantage of reduced circumference coordinates is that they cover only half of the 3-sphere in the case of positive curvature鈥攃ircumferences beyond that point begin to decrease, leading to degeneracy. (This is not a problem if space is elliptical, i.e. a 3-sphere with opposite points identified.)

Hyperspherical coordinates

In hyperspherical or curvature-normalized coordinates the coordinate r is proportional to radial distance; this gives

where is as before and

As before, there are two common unit conventions:

  • k may be taken to have units of length鈭2, in which case r has units of length and a(t) is unitless. k is then the Gaussian curvature[ clarification needed] of the space at the time when a(t) = 1. Where appropriate, a(t) is often chosen to equal 1 in the present cosmological era, so that measures comoving distance.
  • Alternatively, as before, k may be taken to belong to the set {鈭1 ,0, +1} (for negative, zero, and positive curvature respectively). Then r is unitless and a(t) has units of length. When k = 卤1, a(t) is the radius of curvature of the space, and may also be written R(t). Note that when k = +1, r is essentially a third angle along with and . The letter may be used instead of r.

Though it is usually defined piecewise as above, S is an analytic function of both k and r. It can also be written as a power series

or as

where sinc is the unnormalized sinc function and is one of the imaginary, zero or real square roots of k. These definitions are valid for all k.

Cartesian coordinates

When k = 0 one may write simply

This can be extended to k 鈮 0 by defining

, and

where r is one of the radial coordinates defined above, but this is rare.


Cartesian coordinates

In flat FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are [7]

and the Ricci scalar is

Spherical coordinates

In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are [8]

and the Ricci scalar is


Einstein's field equations are not used in deriving the general form for the metric: it follows from the geometric properties of homogeneity and isotropy. However, determining the time evolution of does require Einstein's field equations together with a way of calculating the density, such as a cosmological equation of state.

This metric has an analytic solution to Einstein's field equations giving the Friedmann equations when the energy鈥搈omentum tensor is similarly assumed to be isotropic and homogeneous. The resulting equations are: [9]

These equations are the basis of the standard Big Bang cosmological model including the current 螞CDM model. [10] Because the FLRW model assumes homogeneity, some popular accounts mistakenly assert that the Big Bang model cannot account for the observed lumpiness of the universe. In a strictly FLRW model, there are no clusters of galaxies or stars, since these are objects much denser than a typical part of the universe. Nonetheless, the FLRW model is used as a first approximation for the evolution of the real, lumpy universe because it is simple to calculate, and models which calculate the lumpiness in the universe are added onto the FLRW models as extensions. Most cosmologists agree that the observable universe is well approximated by an almost FLRW model, i.e., a model which follows the FLRW metric apart from primordial density fluctuations. As of 2003, the theoretical implications of the various extensions to the FLRW model appear to be well understood, and the goal is to make these consistent with observations from COBE and WMAP.

If the spacetime is multiply connected, then each event will be represented by more than one tuple of coordinates.[ citation needed]


The pair of equations given above is equivalent to the following pair of equations

with , the spatial curvature index, serving as a constant of integration for the first equation.

The first equation can be derived also from thermodynamical considerations and is equivalent to the first law of thermodynamics, assuming the expansion of the universe is an adiabatic process (which is implicitly assumed in the derivation of the Friedmann鈥揕ema卯tre鈥揜obertson鈥揥alker metric).

The second equation states that both the energy density and the pressure cause the expansion rate of the universe to decrease, i.e., both cause a deceleration in the expansion of the universe. This is a consequence of gravitation, with pressure playing a similar role to that of energy (or mass) density, according to the principles of general relativity. The cosmological constant, on the other hand, causes an acceleration in the expansion of the universe.

Cosmological constant

The cosmological constant term can be omitted if we make the following replacements

Therefore, the cosmological constant can be interpreted as arising from a form of energy which has negative pressure, equal in magnitude to its (positive) energy density:

which is an equation of state of vacuum with dark energy.

An attempt to generalize this to

would not have general invariance without further modification.

In fact, in order to get a term which causes an acceleration of the universe expansion, it is enough to have a scalar field which satisfies

Such a field is sometimes called quintessence.

Newtonian interpretation

This is due to McCrea and Milne, [11] although sometimes incorrectly ascribed to Friedmann. The Friedmann equations are equivalent to this pair of equations:

The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarily a) is the amount which leaves through the sides due to the expansion of the universe plus the mass equivalent of the work done by pressure against the material being expelled. This is the conservation of mass鈥揺nergy ( first law of thermodynamics) contained within a part of the universe.

The second equation says that the kinetic energy (seen from the origin) of a particle of unit mass moving with the expansion plus its (negative) gravitational potential energy (relative to the mass contained in the sphere of matter closer to the origin) is equal to a constant related to the curvature of the universe. In other words, the energy (relative to the origin) of a co-moving particle in free-fall is conserved. General relativity merely adds a connection between the spatial curvature of the universe and the energy of such a particle: positive total energy implies negative curvature and negative total energy implies positive curvature.

The cosmological constant term is assumed to be treated as dark energy and thus merged into the density and pressure terms.

During the Planck epoch, one cannot neglect quantum effects. So they may cause a deviation from the Friedmann equations.

Name and history

The Soviet mathematician Alexander Friedmann first derived the main results of the FLRW model in 1922 and 1924. [12] [13] Although the prestigious physics journal Zeitschrift f眉r Physik published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein, who, on behalf of Zeitschrift f眉r Physik, acted as the scientific referee of Friedmann's work. Eventually Einstein acknowledged the correctness of Friedmann's calculations, but failed to appreciate the physical significance of Friedmann's predictions.

Friedmann died in 1925. In 1927, Georges Lema卯tre, a Belgian priest, astronomer and periodic professor of physics at the Catholic University of Leuven, arrived independently at results similar to those of Friedmann and published them in the Annales de la Soci茅t茅 Scientifique de Bruxelles (Annals of the Scientific Society of Brussels). [14] [15] In the face of the observational evidence for the expansion of the universe obtained by Edwin Hubble in the late 1920s, Lema卯tre's results were noticed in particular by Arthur Eddington, and in 1930鈥31 Lema卯tre's paper was translated into English and published in the Monthly Notices of the Royal Astronomical Society.

Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s. [16] [17] [18] [19] 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 Lema卯tre).

This solution, often called the Robertson鈥揥alker metric since they proved its generic properties, is different from the dynamical "Friedmann鈥揕ema卯tre" models, which are specific solutions for a(t) which assume that the only contributions to stress鈥揺nergy are cold matter ("dust"), radiation, and a cosmological constant.

Einstein's radius of the universe

Einstein's radius of the universe is the radius of curvature of space of Einstein's universe, a long-abandoned static model that was supposed to represent our universe in idealized form. Putting

in the Friedmann equation, the radius of curvature of space of this universe (Einstein's radius) is[ citation needed]

where is the speed of light, is the Newtonian constant of gravitation, and is the density of space of this universe. The numerical value of Einstein's radius is of the order of 1010 light years, or 10 billion light years, though modern telescopes can detect distant objects 13 billion light years away in various directions.

Current status

Unsolved problem in physics:

Is the universe homogeneous and isotropic at large enough scales, as claimed by the cosmological principle and assumed by all models that use the Friedmann鈥揕ema卯tre鈥揜obertson鈥揥alker metric, including the current version of 螞CDM, or is the universe inhomogeneous or anisotropic? [20] [21] [22] Is the CMB dipole purely kinematic, or does it signal a possible breakdown of the FLRW metric? [20] Even if the cosmological principle is correct, is the Friedmann鈥揕ema卯tre鈥揜obertson鈥揥alker metric valid in the late universe? [20] [23]

The current standard model of cosmology, the Lambda-CDM model, uses the FLRW metric. By combining the observation data from some experiments such as WMAP and Planck with theoretical results of Ehlers鈥揋eren鈥揝achs theorem and its generalization, [24] 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 [25] and quasars [26] 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, 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. [27] [20]


  1. ^ 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.
  2. ^ M. Lachieze-Rey; J.-P. Luminet (1995), "Cosmic Topology", Physics Reports, 254 (3): 135鈥214, arXiv: gr-qc/9605010, Bibcode: 1995PhR...254..135L, doi: 10.1016/0370-1573(94)00085-H, S2CID  119500217
  3. ^ G. F. R. Ellis; H. van Elst (1999). "Cosmological models (Carg猫se lectures 1998)". In Marc Lachi猫ze-Rey (ed.). Theoretical and Observational Cosmology. NATO Science Series C. Vol. 541. pp. 1鈥116. arXiv: gr-qc/9812046. Bibcode: 1999ASIC..541....1E. ISBN  978-0792359463.
  4. ^ L. Bergstr枚m, A. Goobar (2006), Cosmology and Particle Astrophysics (2nd ed.), Sprint, p. 61, ISBN  978-3-540-32924-4
  5. ^ Wald, Robert. General Relativity. p. 116.
  6. ^ Carroll, Sean. Spacetime and Geometry: An Introduction to General Relativity. pp. 329鈥333.
  7. ^ Wald, Robert. General Relativity. p. 97.
  8. ^ "Cosmology" (PDF). p. 23.
  9. ^ P. Ojeda and H. Rosu (2006), "Supersymmetry of FRW barotropic cosmologies", International Journal of Theoretical Physics, 45 (6): 1191鈥1196, arXiv: gr-qc/0510004, Bibcode: 2006IJTP...45.1152R, doi: 10.1007/s10773-006-9123-2, S2CID  119496918
  10. ^ Their solutions can be found in Rosu, Haret C.; Mancas, S. C.; Chen, Pisin (2015-05-05). "Barotropic FRW cosmologies with Chiellini damping in comoving time". Modern Physics Letters A. 30 (20): 1550100. arXiv: 1502.07033. Bibcode: 2015MPLA...3050100R. doi: 10.1142/S021773231550100x. ISSN  0217-7323. S2CID  51948117.
  11. ^ McCrea, W. H.; Milne, E. A. (1934). "Newtonian universes and the curvature of space". Quarterly Journal of Mathematics. 5: 73鈥80. Bibcode: 1934QJMat...5...73M. doi: 10.1093/qmath/os-5.1.73.
  12. ^ Friedmann, Alexander (1922), "脺ber die Kr眉mmung des Raumes", Zeitschrift f眉r Physik A, 10 (1): 377鈥386, Bibcode: 1922ZPhy...10..377F, doi: 10.1007/BF01332580, S2CID  125190902
  13. ^ Friedmann, Alexander (1924), "脺ber die M枚glichkeit einer Welt mit konstanter negativer Kr眉mmung des Raumes", Zeitschrift f眉r Physik A, 21 (1): 326鈥332, Bibcode: 1924ZPhy...21..326F, doi: 10.1007/BF01328280, S2CID  120551579 English trans. in 'General Relativity and Gravitation' 1999 vol.31, 31鈥
  14. ^ Lema卯tre, Georges (1931), "Expansion of the universe, A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebul忙", Monthly Notices of the Royal Astronomical Society, 91 (5): 483鈥490, Bibcode: 1931MNRAS..91..483L, doi: 10.1093/mnras/91.5.483 translated from Lema卯tre, Georges (1927), "Un univers homog猫ne de masse constante et de rayon croissant rendant compte de la vitesse radiale des n茅buleuses extra-galactiques", Annales de la Soci茅t茅 Scientifique de Bruxelles, A47: 49鈥56, Bibcode: 1927ASSB...47...49L
  15. ^ Lema卯tre, Georges (1933), "l'Univers en expansion", Annales de la Soci茅t茅 Scientifique de Bruxelles, A53: 51鈥85, Bibcode: 1933ASSB...53...51L
  16. ^ Robertson, H. P. (1935), "Kinematics and world structure", Astrophysical Journal, 82: 284鈥301, Bibcode: 1935ApJ....82..284R, doi: 10.1086/143681
  17. ^ Robertson, H. P. (1936), "Kinematics and world structure II", Astrophysical Journal, 83: 187鈥201, Bibcode: 1936ApJ....83..187R, doi: 10.1086/143716
  18. ^ Robertson, H. P. (1936), "Kinematics and world structure III", Astrophysical Journal, 83: 257鈥271, Bibcode: 1936ApJ....83..257R, doi: 10.1086/143726
  19. ^ Walker, A. G. (1937), "On Milne's theory of world-structure", Proceedings of the London Mathematical Society, Series 2, 42 (1): 90鈥127, Bibcode: 1937PLMS...42...90W, doi: 10.1112/plms/s2-42.1.90
  20. ^ a b c d Elcio Abdalla; Guillermo Franco Abell谩n; et al. (11 Mar 2022), "Cosmology Intertwined: A Review of the Particle Physics, Astrophysics, and Cosmology Associated with the Cosmological Tensions and Anomalies", Journal of High Energy Astrophysics, 34: 49, arXiv: 2203.06142v1, Bibcode: 2022JHEAp..34...49A, doi: 10.1016/j.jheap.2022.04.002, S2CID  247411131
  21. ^ Lee Billings (April 15, 2020). "Do We Live in a Lopsided Universe?". Scientific American. Retrieved March 24, 2022.
  22. ^ Migkas, K.; Schellenberger, G.; Reiprich, T. H.; Pacaud, F.; Ramos-Ceja, M. E.; Lovisari, L. (8 April 2020). "Probing cosmic isotropy with a new X-ray galaxy cluster sample through the LX-T scaling relation". Astronomy & Astrophysics. 636 (April 2020): 42. arXiv: 2004.03305. Bibcode: 2020A&A...636A..15M. doi: 10.1051/0004-6361/201936602. S2CID  215238834. Retrieved 24 March 2022.
  23. ^ Krishnan, Chethan; Mohayaee, Roya; Colg谩in, Eoin 脫; Sheikh-Jabbari, M. M.; Yin, Lu (16 September 2021). "Does Hubble Tension Signal a Breakdown in FLRW Cosmology?". Classical and Quantum Gravity. 38 (18): 184001. arXiv: 2105.09790. Bibcode: 2021CQGra..38r4001K. doi: 10.1088/1361-6382/ac1a81. ISSN  0264-9381. S2CID  234790314.
  24. ^ See pp. 351ff. in Hawking, Stephen W.; Ellis, George F. R. (1973), The large scale structure of space-time, Cambridge University Press, ISBN  978-0-521-09906-6. 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 Stoeger, W. R.; Maartens, R; Ellis, George (2007), "Proving Almost-Homogeneity of the Universe: An Almost Ehlers-Geren-Sachs Theorem", Astrophys. J., 39: 1鈥5, Bibcode: 1995ApJ...443....1S, doi: 10.1086/175496.
  25. ^ See Siewert et al. for a recent summary of results Siewert, Thilo M.; Schmidt-Rubart, Matthias; Schwarz, Dominik J. (2021). "Cosmic radio dipole: Estimators and frequency dependence". Astronomy & Astrophysics. 653: A9. arXiv: 2010.08366. Bibcode: 2021A&A...653A...9S. doi: 10.1051/0004-6361/202039840. S2CID  223953708.
  26. ^ Secrest, Nathan J.; Hausegger, Sebastian von; Rameez, Mohamed; Mohayaee, Roya; Sarkar, Subir; Colin, Jacques (2021-02-25). "A Test of the Cosmological Principle with Quasars". The Astrophysical Journal. 908 (2): L51. arXiv: 2009.14826. Bibcode: 2021ApJ...908L..51S. doi: 10.3847/2041-8213/abdd40. S2CID  222066749.
  27. ^ Krishnan, Chethan; Mohayaee, Roya; 脫 Colg谩in, Eoin; Sheikh-Jabbari, M. M.; Yin, Lu (2021-05-25). "Does Hubble tension signal a breakdown in FLRW cosmology?". Classical and Quantum Gravity. 38 (18): 184001. arXiv: 2105.09790. Bibcode: 2021CQGra..38r4001K. doi: 10.1088/1361-6382/ac1a81. S2CID  234790314.

Further reading