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.
These three possibilities correspond to parameter k of (0) flat space, (+1) a sphere of constant positive curvature or (-1) a hyperbolic space with constant negative curvature.
Here the radial position has been decomposed into a time-dependent scale factor,
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.
With the stress–energy tensor for a perfect fluid, results in the equations are described below.
[4]: 73 There are two independent Friedmann equations for modelling a homogeneous, isotropic universe.
We see that in the Friedmann equations, a(t) does not depend on which coordinate system we chose for spatial slices.
In earlier models, which did not include a cosmological constant term, critical density was initially defined as the watershed point between an expanding and a contracting Universe.
[citation needed] To date,[citation needed] the critical density is estimated to be 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.
Assuming a zero vacuum energy density, if Ω is larger than unity, the space sections of the universe are closed; the universe will eventually stop expanding, then collapse.
If Ω is less than unity, they are open; and the universe expands forever.
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]
where p is the pressure, ρ is the mass density of the fluid in the comoving frame and w is some constant.
In spatially flat case (k = 0), the solution for the scale factor is
This family of solutions labelled by w is extremely important for cosmology.
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.
If the matter is a mixture of two or more non-interacting fluids each with such an equation of state, then
To make the solutions more explicit, we can derive the full relationships from the first Friedmann equation:
Solutions for the dependence of the scale factor with respect to time for universes dominated by each component can be found.
The Λ-dominated universe solution is of particular interest because the second derivative with respect to time is positive, non-zero; in other words implying an accelerating expansion of the universe, making ρΛ a candidate for dark energy:
Where by construction ai > 0, our assumptions were Ω0,Λ ≈ 1, and H0 has been measured to be positive, forcing the acceleration to be greater than zero.
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".
