[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 – 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).
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.
[5]: 408 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.
The equivalence principle applied to each galaxy means distance measurements can be made using special relativity locally.
[6]: 73 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.
[8]: 25.1.3 To apply the metric to cosmology and predict its time evolution via the scale factor
giving the Friedmann equations when the energy–momentum tensor is similarly assumed to be isotropic and homogeneous.
The resulting equations are:[9] 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.
As of 2003[update], 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.
, the spatial curvature index, serving as a constant of integration for the first equation.
The second equation states that both the energy density and the pressure cause the expansion rate 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 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.
In reduced-circumference polar coordinates the spatial metric has the form[12][13] k is a constant representing the curvature of the space.
There are two common unit conventions: 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.
FLRW space using Cartesian coordinates, the surviving components of the Ricci tensor are[14] and the Ricci scalar is In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are[15][failed verification] and the Ricci scalar is The Soviet mathematician Alexander Friedmann first derived the main results of the FLRW model in 1922 and 1924.
[16][17] Although the prestigious physics journal Zeitschrift für Physik published his work, it remained relatively unnoticed by his contemporaries.
In 1927, Georges Lemaître, 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 Société Scientifique de Bruxelles (Annals of the Scientific Society of Brussels).
[18][19] In the face of the observational evidence for the expansion of the universe obtained by Edwin Hubble in the late 1920s, Lemaître's results were noticed in particular by Arthur Eddington, and in 1930–31 Lemaître'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.
[20][21][22][23] 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.
By combining the observation data from some experiments such as WMAP and Planck with theoretical results of Ehlers–Geren–Sachs theorem and its generalization,[28] 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 [29] and quasars [30] 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,