What is Friedmann equation: Definition and 39 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. D

    A Mass density of radiation in Friedmann equation?

    \frac{1}{R}\frac{{{d^2}R}}{{d{t^2}}} = - \frac{{4\pi G}}{3}\left[ {{\rho _m} + {\rho _{rad}} + {\rho _\Lambda } + \frac{{3({P_m} + {P_{rad}} + {P_\Lambda })}}{{{c^2}}}} \right] In the Friedmann equation, ρ is the mass density. === https://en.wikipedia.org/wiki/Friedmann_equations They were...
  2. atyy

    How Are Tsinghua Students Using the Friedmann Equation to Protest Lockdowns?

    "Tsinghua students, true to form, protesting lockdowns with the Friedmann Equation: the basic reality of the universe is constant, eternal expansion, or put another way, opening up." [Moderator's note: chinese twitter link removed.]
  3. Buzz Bloom

    I Is there a Friedmann equation form with inflation?

    If it is known that such an equation exists, I would much appreciate seeing a link to a reference. If it is known that such an equation does not exist, I would much appreciate seeing a reference with an explanation of implications regarding the compatibility of general relativity with a...
  4. hnnhcmmngs

    Modern Physics: Friedmann equation

    I was shown that adot^2/a^2 = c/a^3, adot = c / √(a), then da/dt = c / √(a) . Then I was told that I have to integrate this, but I don't understand where to go from there or how this will show me that the scale factor grows as t^(2/3).
  5. Buzz Bloom

    Find the Metastatic Conditions for Omegas in the Friedmann Equation

    Required Equations Note: Includes inequalities. [1] H(s) = 0, and [2] H’(s) = 0? [3] [4] H = a’/a [5] a(t0) = a0 =1 [6] H0 = H(t0) [7] ΩR = 0 [8] Ωk<0, Ωk = -K, K > 0 [9] ΩM>1, ΩM = M, M > 1 [10] ΩΛ = 1-Ωk- ΩM = 1+K-M > 0 → M-1 < K Solution Note: I have assigned this problem to myself...
  6. J

    I Does the Friedmann vacuum equation have a linear solution?

    Does the Friedmann vacuum equation have a linear solution rather than an exponential one? Using natural units one can write Friedmann's equation for the vacuum as $$ \begin{eqnarray*} \left(\frac{\dot a}{a}\right)^2 &=& \frac{8\pi G}{3}\rho_{vac}\\\tag{1} &=& L^2 \left(\frac{\rho_0}{L^4}\right)...
  7. A

    B The k term in the Friedmann equation

    One simple question whether the k in the friedmann equation H(t)2=∑8πGε(t)/3c2-k/a2 is something related to curvature or is simply constt.?? If related to curvature whether it is 1/R where R is the radius of the 3-sphere.??
  8. T

    Rearranging the Friedmann Equation

    Homework Statement Assume that the universe today is flat with both matter and a cosmological constant, the latter with energy density which remains constant with time. Integrate Eq. (1.2) to find the present age of the universe. That is, rewrite Eq. (1.2) as: dt = H0-1 da / a [ΩΛ + (1 - ΩΛ )...
  9. Arman777

    Insights A Journey Into the Cosmos - FLRW Metric and The Friedmann Equation - Comments

    Greg Bernhardt submitted a new PF Insights post This article is part of our student writer series. The writer Arman777, is an undergraduate physics student at METU A Journey Into the Cosmos - FLRW Metric and The Friedmann Equation Continue reading the Original PF Insights Post.
  10. Arman777

    I Friedmann Equation - Positive Curvature

    Friedmann Equation without the cosmological constant can be written as; $$H^2=\frac {8πρG} {3}-\frac {1} {a^2(t)}$$ for ##Ω>1## (which in this case we get a positive curvature universe), and for a matter dominant universe; $$ρ=\frac {1} {a^3(t)}$$ so in the simplest form the Friedmann equation...
  11. Arman777

    Insights A Journey Into the Cosmos - The Friedmann Equation - Comments

    Greg Bernhardt submitted a new PF Insights post A Journey Into the Cosmos - The Friedmann Equation Continue reading the Original PF Insights Post.
  12. D

    I Understanding the 2nd Term of the Friedmann Equation: Replacing U with -kc^2

    while deriving the friedmann equation using Newtonian Mechanics the 2nd term of the r.h.s is coming to be 2^U/(r^2*a^2) where U is a constt,but it is replaced by -kc^2/(r^2*a^2)?
  13. binbagsss

    Friedmann equation - show Big Bang happened given conditions

    Homework Statement Use Friedmann equations to show that if ##\dot{a} > 0##, ##k<0## and ##\rho>0## then there exists a ##t*## in the past where ##a(t*)=0## Homework Equations [/B] Friedmann : ##(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{k}{a^2}##The Attempt at a Solution [/B]...
  14. S

    I Friedmann Equation Analysis, expansion of the universe?

    Friedmann's Eq can be viewed here https://ned.ipac.caltech.edu/level5/March08/Frieman/Equations/paper1x.gif What I don't get is that all the texts/analyses of Friedmann's equation say that if the right hand side is negative it means that the universe will expand reach a critical point and then...
  15. Buzz Bloom

    Question regarding Friedmann equation

    It has occurred to me that the Friedmann equation allows for a solution H = H0 = 0 . This seems to say that independently of the value of the scale factor a, and the various Ωs, a static universe is possible. I am guessing that this solution is spurious and is a side effect of the...
  16. RyanH42

    Newtonian Friedmann Equation, Referance frame, Homogeneity

    Hi all I want to ask a question about NFE(Newtonian Friedmann Equation).I know that NFE is not usefull to describe universe.But we can have a general idea about universe to use that formula. I know that the only spatial coordinate system is CMB referance frame and NFE is derived from...
  17. Quarlep

    Referance Frame Calculating Newtonian Friedmann Equation

    In friedmann equation we start to make a model kinetic energy-potantial energy=U kinetic energy depends observer and referance frame so what's the referance frame and observer in this calculation. Thanks
  18. K

    Solving expansion rate for a variant of the Friedmann equation

    Homework Statement For the equation H^2 = \frac{8 \pi G \rho_m}{3} + \frac{H}{r_c} how do I find the value of H for scale factor a \rightarrow \infty , and show that H acts as though dominated by \Lambda (cosmological constant) ? Homework Equations \rho_m \propto \frac{1}{a^3} H > 0...
  19. Greg Bernhardt

    What is the Friedmann Equation

    Definition/Summary The Friedmann equation is a dynamical equation that describes the expansion of the universe. Equations H^2 = \left( \frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3} \rho - \frac{kc^2}{a^2} Extended explanation The Friedmann equation is derived from the 0-0...
  20. K

    Questions on Hubble's law and Friedmann equation

    The Hubble's Law v=H0D LHS of The Friedmann equation H2/H02 I am a bit confused with the following: 1: The Hubble constant H0 is the current universe velocity/distance ratio, it is basically a constant, so galaxies in the universe with a further distance, move away from each other at a...
  21. S

    Rewriting the Friedmann Equation

    Homework Statement Consider a universe with Cold Matter, Radiation, and Dark Energy satisfying an equation of state p = w\rho. Variables: p is the pressure of the universe, w is a constant, and \rho is the density of the universe. a) Show that the Friedmann equation can be rewritten as...
  22. B

    Energy and the Friedmann Equation

    Homework Statement To relate E to the total energy of the expanding sphere, we need to integrate over the sphere to determine its total energy. These integrals are most easily carried out by dividing the sphere into shells of radius r, and thickness dr, so that each shell has a volume dV...
  23. 4

    Integrating Friedmann Equation of Multi-component universe respect to a and t

    I am having a trouble finding relationship between 'a' and 't' by integrating friedmann equation in a multi-component universe. It would be very helpful if you can help me with just matter-curvature only universe and matter-lambda only universe. The two integrals looks like following...
  24. D

    First Friedmann Equation forms

    Homework Statement the first friedmann equation is: (\frac{\dot{a}}{a})^2=\frac{8\pi G\rho}{3}-\frac{kc^2}{a^2}+\frac{\Lambda}{3} In the case of a closed Universe (k > 0) containing only non-relativistic matter and no cosmological constant, write the Friedmann equation in terms of, H(a), H0...
  25. P

    Does Friedmann equation allow for complex scale factor?

    Looking at the Friedmann equation H^2=\left[\frac{\dot{a}}{a}\right]^2=\frac{8\pi G\rho}{3}-\frac{kc^2}{a^2} and considering positive curvature, then for the limit where the second term dominates, we're left with \left[\frac{\dot{a}}{a}\right]^2=-\frac{kc^2}{a^2} This implies a complex...
  26. K

    Scalefactor calculations for special cases using the Friedmann Equation

    (i) Obtain the scale factor a(t) and redshift z when the energy density of matter and radiation were equal. (ii) Next use the a(t) relation for a matter-only universe to estimate the time of matter-radiation equality. (iii) Repeat (ii) but using the a(t) relation for a radiation-only universe...
  27. F

    What exactly is the relation between LQG and the Friedmann equation?

    I'm just beginning to look into loop quantum cosmology, and I've recently run into some fundamental confusion concerning how exactly the Friedmann equation fits in. Is it just that the application of loop quantum gravity yields a modified/effective Friedmann equation, or is it the other way...
  28. G

    Friedmann Equation demonstration - Liddle

    Are you confortable with the demonstration in page 19 of Introduction to Modern Cosmology - A. Liddle ? My problem is in particular eq. (3.4) when he writes the potential energy of a test particle. I think there is a factor of 2 missing in the denominator.
  29. R

    Density in the friedmann equation

    I'm a little confused about the density \rho in the equation: H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3}\rho - \frac{kc^2}{a^2} Measuring \rho at a single instant in time seems easy. But \rho changes with time. The time dependence of \rho is given as...
  30. A

    Cosmology-Application of the friedmann equation

    Homework Statement The universe is expanding as shown by a Hubble constant H of 75 Km s^-1 Mpc^-1, but a constant density of matter of 10^-26 kg m^-3 is maintained by the continuous creation of hydrogen atoms as the universe expands. Find the rate of creation of hydrogen atoms per unit...
  31. P

    Arriving at Friedmann Equation with Newton's Mechanics

    Hey! I'd like to see an easy way to arrive to Friedmann equation by using Newton's Mechanics. I've seen many ways but they skip many steps so I don't understand the whole. THANKS!
  32. nicksauce

    Solution of Friedmann equation w/ Cosmological Constant

    Homework Statement Solve for the age of the universe as a function of scale factor in a cosmology with \Omega_{M}=0.3, \Omega_{\Lambda}=0.7.Homework Equations The Attempt at a Solution (\frac{\dot{a}}{a})^2=H_0^2(0.3a^{-3}+0.7) \frac{da}{a\sqrt{0.3a^{-3}+0.7}}=H_0dt At this point, Maple seems...
  33. S

    Help a mature student Friedmann equation

    Anybody like to help a physics student who did their BSc six years ago and has forgotten all their maths! Its a problem from Andrew Liddle's Introduction to cosmology 5.5 The Friedmann eqn is (a'/a)2 = 8TTG/3 P - k/a2 Consider the case k>o, with the universe containing matter only, so...
  34. A

    The Friedmann equation in a lambda-dominated universe

    Homework Statement Show that a lambda-dominated euclidean universe entails an exponential expansion. Homework Equations Equation of state is P=-E (E=epsilon) There is no curvature (i.e. a flat universe) The Friedmann equation is then ((a(dot)/a)^2)=((8piG)/3c^2)E + lambda/3...
  35. L

    Solving Friedmann Equation w/ Friedmann Eqn Hmwk Stmt

    Homework Statement By substituting in \left( \frac{\dot{a}}{a_0} \right)^2 = H^2_0 \left(\Omega_0 \frac{a_0}{a} + 1 - \Omega_0 \right) show that the parametric open solution given by a(\psi)=a_0 \frac{\Omega_0}{2(1-\Omega_0)}(\cosh{\psi} - 1) and t(\psi)=\frac{1}{2H_0} \frac{\Omega_0}{(1 -...
  36. M

    Newtonian derivation of Friedmann equation

    I'm trying to derive Friedmann's equation for cosmology using Newtonian physics. I've got the force equation F=ma for the case without a cosmological constant. But now I'm trying to incorporate the cosmological constant into this force equation. But I'm having trouble seeing how the...
  37. B

    Deriving Friedmann Equation from RW Metric - Blumfeld

    Hi does anyone know a website or paper where the friedmann equation is derived from the robertson walker metric? it should have the calculation of atleast a few of the christoffel symbols etc thanks blumfeld
  38. D

    Friedmann Equation and the age of empty universe

    Got stuck in this question, the question asked to use friedmann equation to show that the age of an empty universe (average density = 0) is τ=1/H₀ where H₀ is the present day value of Hubble's Constant. Well, i got the friedmann equation by put the value ρ(average)=0 but the trouble is...
  39. wolram

    Failure of Friedmann equation?

    arXiv:astro-ph/0503715 v2 31 Mar 2005 Late time failure of Friedmann equation Alessio Notari* Physics Department, McGill University, 3600 University Road, Montr´eal, QC, H3A 2T8, Canada (Dated: March 31, 2005) It is widely believed that the assumption of homogeneity is a good zeroth...