Friedmann equations
They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density ρ and pressure p.[1] The equations for negative spatial curvature were given by Friedmann in 1924.
[2] 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:
[5]: 3 There are two independent Friedmann equations for modelling a homogeneous, isotropic universe The first is:[3]
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.
The spatial geometry of the universe has been measured by the WMAP spacecraft to be nearly flat.
The first Friedmann equation is often seen in terms of the present values of the density parameters, that is[9]
We see that in the Friedmann equations, a(t) does not depend on which coordinate system we chose for spatial slices.
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.
Both of the earlier works also assumed the universe was static, eternally unchanging.
Einstein postulated an additional term to his equations of general relativity to ensure this stability.
[11]: 152 The idea of static universe was a fundamental assumption of philosophy and science.
Starting with Einstein's 10 equations of relativity, Friedmann applies the symmetry of an isotropic universe and a simple model for mass-energy density to derive a relationship between that density and the curvature of spacetime.
[11]: 157 Friedmann's second paper, "On the possibility of a world with constant negative curvature," published in 1924 explored more complex geometrical ideas.
Friedmann used two concepts of a three dimensional sphere as analogy: a trip at constant latitude could return to the starting point or the sphere might have an infinite number of sheets and the trip never repeats.
[11]: 167 Friedmann's paper were largely ignored except – initially – by Einstein who actively dismissed them.
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".
[12] 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".
