What is Friedmann equations: Definition and 20 Discussions

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 density


{\displaystyle \rho }
and pressure


{\displaystyle p}
. The equations for negative spatial curvature were given by Friedmann in 1924.

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  1. T

    I "Energy is not conserved" vs. energy is conserved: Friedmann Equations

    First, "Energy is not conserved" as e.g. explained by Sean Carroll in https://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/ . Second, the Friedmann Equations are expressed in energy conservation, e.g. https://core.ac.uk/download/pdf/25318877.pdf equation (16). Do we...
  2. SH2372 General Relativity - Lecture 10

    SH2372 General Relativity - Lecture 10

    0:00 Cosmological redshift 18:08 Solving the Friedmann equations 34:34 Accelerated expansion and the cosmological constant 56:10 Cosmological inflation
  3. T

    I Is Minkowski spacetime a solution of the Friedmann Equations?

    The empty FRW-universe with curvature parameter ##k = -1## and expanding linearly is well known. Also that it is mathematically equivalent (after a coordinate transformation) with the Milne universe which also expands linearly. I wonder if the Friedmann Equations have another solution (I...
  4. Vick

    A Anisotropic Universe and Friedmann Equations

    The Friedman Equations is based on the cosmological principle, which states that the universe at sufficiently large scale is homogeneous and isotropic. But what if, as an hypothesis, the universe was anisotropic and the clustering of masses are aligned to an arbitrary axis (axial pole), how...
  5. T

    I Integrating the Friedmann Equations

    Hello, I was just wondering about the method for integrating Friedmann equation for k=1 and w=1/3 ∫(8πρ/a^(2)-1)^-1/2thank you
  6. D

    I Energy-momentum tensor and Friedmann Equations

    Hi everyone, I want to derive the Friedmann equations from Einstein Field Equations. However, I have a problem that stems from the energy-momentum tensor. I am also trying to keep track of ## c^2 ## terms. FRW Metric: $$ ds^2= -c^2dt^2 + a^2(t) \left( {\frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2...
  7. thecourtholio

    Angular diameter distance to surface of last scattering

    Homework Statement 1) Calculate the angular diameter distance to the last scattering surface in the following cosmological models: i) Open universe, ΩΛ= 0.65, Ωm = 0.30 ii) Closed universe, ΩΛ = 0.75, Ωm = 0.30 ii) Flat universe, ΩΛ = 0.75, Ωm = 0.25 Describe how the CMB power spectrum...
  8. thecourtholio

    Distance to surface of last scattering in +, -, and flat uni's

    Homework Statement 1) Calculate the angular diameter distance to the last scattering surface in the following cosmological models: i) Open universe, ΩΛ= 0.65, Ωm = 0.30 ii) Closed universe, ΩΛ = 0.75, Ωm = 0.30 ii) Flat universe, ΩΛ = 0.75, Ωm = 0.25 Describe how the CMB power spectrum...
  9. S

    F(R) gravity and the Hubble parameter

    Dear all, Considering Einstein Hilbert lagrangian, by using Einstein field equations one can get the form of Friedman equations and consequently the Hubble parameter. I know that in f(R) models, Einstein equations get modified. However, what happens to the Friedman equation and the Hubble...
  10. 1

    How Does General Relativity Predict an Expanding Universe

    Hello Everyone, Back when Einstein was formulating General Relativity his equations just could not predict a static universe. I have read that they actually predicted an expanding Universe. Later Friedmann derived an equation from GR that would explain how an Expanding Universe would evolve...
  11. J

    Definition of Energy in Friedmann equations?

    The first Friedmann equation for a flat Universe is given by: $$\bigg(\frac{\dot{a}(t)}{a(t)}\bigg)^2 = \frac{8 \pi G}{3} \rho(t)$$ The energy density ##\rho(t)## is given by: $$\rho(t) \propto \frac{E(t)}{a(t)^3}$$ where ##E(t)## is the energy of the cosmological fluid in a co-moving...
  12. Quarlep

    Calculating Surface Density of the Universe Using the Shell Model

    You know friedmann equations derived from kinetic energy and potantial energy conservation.I found these shell model for universe.Here I am curious about something. This shell is like a surface of sphere isn't it.I mean it has only surface and that surface mass is m.And we made our equations...
  13. binbagsss

    Solving Friedmann Equations with Cosmological Constant

    Hi, I'm looking at 'Lecture Notes on General Relativity' by Sean M.Carroll. I have a question about p. 227, solving for ##a(t)## in the dark energy case. So for dust and radiation cases it was Friedmann equations you solve. But in the case of a non-zero cosmological constant Eienstien equation...
  14. C

    Struggling with Friedmann equations

    Hi! I'm struggling to derive the accelerated Friedmann equation (shame on me!)... I'll tell you what I'm doing and maybe you can find where the mistake is. First of all, we know that: H^2 = \frac{\rho}{2M^2_p}+\frac{\Lambda}{3} Now, using the Einstein Equations and doing the trace of...
  15. M

    Help understanding the Friedmann equations?

    Hello! I'm a senior at San Francisco State University, and I'm currently enrolled in a cosmology class. It's a GWAR class, meaning General Writing Assessment Requirement- I didn't expect much math. In fact, the first half of the class was basically an anthropology course, which is more in line...
  16. sergiokapone

    About completeness of the Friedmann equations

    I have some misunderstanding about Fridman models of University. Friedman equation: ## \begin{array}{l} {\left( {\frac{{\dot a}}{a}} \right)^2} = \frac{{8\pi G}}{3}\rho - \frac{{k{c^2}}}{{{a^2}}}\\ \frac{1}{a}\frac{{{d^2}a}}{{d{t^2}}} = - \frac{{4\pi G}}{3}\left( {\rho + \frac{{3p}}{{{c^2}}}}...
  17. P

    Tips for deriving the friedmann equations

    Hi I am using the metric ds^{2}=c^{2}dt^{2}-a^{2}(t)[\frac{dr^{2}}{1-kr^{2}}+r^{2}(dθ^{2} + sin^2θd^{2}∅)] and am subsequently trying to derive the christoffel symbols: \Gamma^{\sigma}_{\mu\nu}=\frac{1}{2}g^{\sigma\rho}(\partial_{\nu}g_{\rho\mu} +\partial_{\mu}g_{\rho\nu}...
  18. $

    A question about Friedmann Equations

    Hello everyone, Its been 4 years now since I posted, Last time i was asking for school homework and now I'm about to finish my physics master degree with 1 exam left in cosmology. As I'm sure you are aware, the k term in the Friedmann equation represents the curvature term k = 0, -1 & +1...
  19. P

    Solutions of Friedmann Equations

    Hello everybody, could somebody tell me how to get the scale factor as a function of time [R(t)] from the Friedmann Equations for a simple dust, pressureless universe when k != 0, \Lambda = 0. Mathematica 5.2 doesn't want to give me any solution and wherever I searched the best thing I got...
  20. marcus

    Alternative forms of the Friedmann equations

    What is your preferred way of remembering and writing down the Friedmann equations and in what form do you include dark energy in them? Heres the way that seems best to me. It is a balance between making them uncluttered but not so simplified as to lose intelligibility. For one bit of...