Friedmann equations
The Friedmann equations build on three assumptions:[3]: 22.1.3 The metric in turn starts with the simplifying assumption that the universe is spatially homogeneous and isotropic, that is, the cosmological principle; empirically, this is justified on scales larger than the order of 100 Mpc.
Inserting this metric into Einstein's field equations relate the evolution of this scale factor to the pressure and energy of the matter in the universe.
using the present day value The Friedmann equations can be written in terms of this dimensionless scale factor:
The winner depends upon the k value in the total energy: if k is +1, gravity eventually causes the universe to contract.
, then it is called "spatially closed": in this simple approximation the universe would eventually contract.
[4]: 74 The critical density is equivalent to approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic metre.
However, the largest part comes from so-called dark energy, which accounts for the cosmological constant term.
An expression for the critical density is found by assuming Λ to be zero (as it is for all basic Friedmann universes) and setting the normalised spatial curvature, k, equal to zero.
However, one can also subsume the spatial curvature and vacuum energy terms into a more general expression for Ω in which case this density parameter equals exactly unity.
We see that in the Friedmann equations, a(t) does not depend on which coordinate system we chose for spatial slices.
[4]: 73 Direct observation of stars has shown their velocities to be dominated by radial recession, validating these assumptions for cosmological models.
[3]: 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.
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 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.
where p is the pressure, ρ is the mass density of the fluid in the comoving frame and w is some constant.
For example, w = 0 describes a matter-dominated universe, where the pressure is negligible with respect to the mass density.
From the generic solution one easily sees that in a matter-dominated universe the scale factor goes as
radiation-dominated Note that this solution is not valid for domination of the cosmological constant, which corresponds to an w = −1.
In this case the energy density is constant and the scale factor grows exponentially.
He adopted the same homogeneity and isotropy assumptions used by Albert Einstein and by Willem de Sitter in their papers, both published in 1917.
Einstein postulated an additional term to his equations of general relativity to ensure this stability.
[14]: 157 Friedmann's second paper, "On the possibility of a world with constant negative curvature," published in 1924 explored more complex geometrical ideas.
However once Edwin Hubble published astronomical evidence that the universe was expanding, Einstein became convinced.
Unfortunately for Friedmann, Georges Lemaître discovered some aspects of the same solutions and wrote persuasively about the concept of a universe born from a "primordial atom".
[15] Several students at Tsinghua University (CCP leader Xi Jinping's alma mater) participating in the 2022 COVID-19 protests in China carried placards with Friedmann equations scrawled on them, interpreted by some as a play on the words "Free man".
