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en.wikipedia.org/wiki/Friedmann_equations

The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity.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 ...

diposit.ub.edu/dspace/bitstream/2445/59759/1/TFG-Arnau-Romeu-Joan.pdf

Derivation of Friedman equations Joan Arnau Romeu points of the universe. The scale factor is de ned to be 1 in the present time. From now on the time dependence of the scale factor can be implicit, so a(t) a. K 2 is directly related to the curvature radius of the spatial hypersurface.

astronomy.nmsu.edu/aklypin/COSMOLOGY/SimpleDerivation.pdf

Friedmann Equation: Newtonian derivation Consider a sphere, which expands in a homogeneous Universe. For non-relativistic particles the mass inside the sphere is constant. We need to find how the radius of the sphere changes with time. Later we will add corrections due to effects of GR. Use the fact that mass inside comoving radius is preserved:

uncw.edu/phy/documents/thefriedmannequations.pdf

Friedmann equation with values of either (1, -1, 0) Cosmic Scale Factor a(t) • Represents relative expansion of the universe • Dimensionless function of time • Also called the Robertson-Walker scale factor • Relates proper distance between a pair of objects undergoing

www.sciencedirect.com/topics/mathematics/friedmann-equation

The Friedmann equation 9 relates the scale factor S (t), curvature constant k, and the effective total energy density μ(t), which is defined by this equation whatever dynamics may be involved (multiple scalar fields, higher order gravity, higher dimensional theories leading to effective 4-dimensional theories, etc.). 18 The Raychaudhuri ...

www.physicsforums.com/insights/journey-cosmos-friedmann-equation

Chapter 1- Derivation of The Friedmann Equation. In this part of the journey we will take a general look at the evolution and eventual fate of the Universe. The Friedmann Equation is the heart of these concepts. It explains the evolution of the universe as a function of time for different type of models.

hyperphysics.phy-astr.gsu.edu/hbase/Astro/fried.html

The Friedmann Equation Alexander Friedmann of Russia is credited with developing a dynamic equation for the expanding universe in the 1920s. This was a time when Einstein, Willem de Sitter of the Netherlands, and Georges Lemaitre of Belgium were also working on equations to model the universe. Friedmann developed it as a relativistic equation in the framework of general relativity, but the ...

hep.itp.tuwien.ac.at/~wrasetm/files/2017S-FriedmannFromGR.pdf

Deriving the Friedmann equations from general relativity The FRW metric in Cartesian coordinates is ds2 = g dx dx = 2dt2 + g ijdx idxj = dt + a(t) 2 dx i + K x2 i dx 2 i 1 Kx2 i ... Dividing by -2 leads to equation (17) in the lecture notes and the second Friedmann equation a(t) a(t) 1 3 =

www.astro.caltech.edu/~george/ay21/Ay21_Lec02.pdf

The Friedmann Equation A more complete derivation, including the cosmological constant term, gives: The Friedmann Eqn. is effectively the equation of motion for a relativistic, homogeneous, isotropic universe. In order to derive cosmological models from it, we also need to specify the equation of state of the “cosmological