The Friedmann–Lemaître–Robertson–Walker metric (FLRW;) is a metric based on an exact solution of the Einstein field equations of general relativity. The metric describes a homogeneous, isotropic, expanding (or otherwise, contracting) universe that is path-connected, but not necessarily simply connected.[1] [2] 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). This model is sometimes called the Standard Model of modern cosmology, 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.
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 that meets these conditions is
-c2d\tau2=-c2dt2+{a(t)}2d\Sigma2,
\Sigma
d\Sigma
In reduced-circumference polar coordinates the spatial metric has the form[3] [4]
d\Sigma2=
| dr2 | |
| 1-kr2 |
+r2d\Omega2, whered\Omega2=d\theta2+\sin2\thetad\phi2.
k is a constant representing the curvature of the space. There are two common unit conventions:
d\Sigma
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, i.e. a 3-sphere with opposite points identified.)
In hyperspherical or curvature-normalized coordinates the coordinate r is proportional to radial distance; this gives
d\Sigma2=dr2+
| 2 | |
| S | |
| k(r) |
d\Omega2
d\Omega
Sk(r)= \begin{cases} \sqrt{k}-1\sin(r\sqrt{k}),&k>0\\ r,&k=0\\ \sqrt{|k|}-1\sinh(r\sqrt{|k|}),&k<0. \end{cases}
As before, there are two common unit conventions:
d\Sigma
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
Sk(r)=
| infty | |
| \sum | |
| n=0 |
| (-1)nknr2n+1 | |
| (2n+1)! |
=r-
| kr3 | |
| 6 |
+
| k2r5 | |
| 120 |
- …
Sk(r)=r sinc(r\sqrt{k}),
\sqrt{k}
When k = 0 one may write simply
d\Sigma2=dx2+dy2+dz2.
x=r\cos\theta,
y=r\sin\theta\cos\phi
z=r\sin\theta\sin\phi,
In flat
(k=0)
Rtt=-3
| \ddot{a | |
R=6c-2\left(
| \ddot{a | |
| (t)}{a(t)} |
+
| |||||
| a2(t) |
\right).
In more general FLRW space using spherical coordinates (called "reduced-circumference polar coordinates" above), the surviving components of the Ricci tensor are[6]
Rtt=-3
| \ddot{a | |
Rrr=
| c-2(a(t)\ddot{a | |
| (t) |
+2
| a |
2(t))+2k}{1-kr2}
R\theta\theta=r2(c-2(a(t)\ddot{a}(t)+2
| a |
2(t))+2k)
R\phi\phi=r2(c-2(a(t)\ddot{a}(t)+2
| a |
2(t))+2k)\sin2(\theta)
R=6\left(
| \ddot{a | |
| (t)}{c |
2a(t)}+
| |||||
| c2a2(t) |
+
| k | |
| a2(t) |
\right).
See main article: Friedmann equations.
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
a(t)
\rho(t),
G\mu\nu+Λg\mu\nu=\kappaT\mu\nu
| \left( |
| ||
| a |
\right)2+
| kc2 | |
| a2 |
-
| Λc2 | |
| 3 |
=
| \kappac4 | |
| 3 |
\rho
| 2 | \ddota |
| a |
+\left(
| |||
| a |
\right)2+
| kc2 | |
| a2 |
-Λc2=-\kappac2p.
These equations are the basis of the standard Big Bang cosmological model including the current ΛCDM model.[7] 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 that 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 that follows the FLRW metric apart from primordial density fluctuations., 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 pair of equations given above is equivalent to the following pair of equations
| {\rho} |
=-3
| \left(\rho+ | |||
| a |
| p | |
| c2 |
\right)
| \ddota | |
| a |
=-
| \kappac4 | |
| 6 |
\left(\rho+
| 3p | |
| c2 |
\right)+
| Λc2 | |
| 3 |
k
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–Lemaître–Robertson–Walker metric).
The second equation states that both the energy density and the pressure cause the expansion rate of the universe
| {a} |
The cosmological constant term can be omitted if we make the following replacements
\rho → \rho-
| Λ | |
| \kappac2 |
p → p+
| Λ | |
| \kappa |
.
Therefore, the cosmological constant can be interpreted as arising from a form of energy that has negative pressure, equal in magnitude to its (positive) energy density:
p=-\rhoc2,
An attempt to generalize this to
p=w\rhoc2
In fact, in order to get a term that causes an acceleration of the universe expansion, it is enough to have a scalar field that satisfies
p<-
| \rhoc2 | |
| 3 |
.
This is due to McCrea and Milne,[8] although sometimes incorrectly ascribed to Friedmann. The Friedmann equations are equivalent to this pair of equations:
-a3{
| \rho} |
=3a2{
| a} |
\rho+
| |||||
| |||
| 2}{2} |
-
| \kappac4a3\rho | |
| 6a |
=-
| kc2 | |
| 2 |
.
The first equation says that the decrease in the mass contained in a fixed cube (whose side is momentarily a) is the amount that 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–energy (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.
The Soviet mathematician Alexander Friedmann first derived the main results of the FLRW model in 1922 and 1924.[9] Although the prestigious physics journal Zeitschrift für Physik published his work, it remained relatively unnoticed by his contemporaries. Friedmann was in direct communication with Albert Einstein, who, on behalf of Zeitschrift für 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 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).[10] 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. 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.
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
| a |
=\ddot{a}=0
RE=c/\sqrt{4\piG\rho},
c
G
\rho
See also: Shape of the universe. 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–Geren–Sachs theorem and its generalization,[11] 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 [12] and quasars [13] 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,
H0