Talk:Friedmann equations

message: In F.E., as given in the article now (2019 Aug 29), the k is the inverse of the comoving radius of curvature squared (positive or negative), i.e. +-1/R^2 and has dimensions of L^-2. However, in paragraph below k is used as a SIGN of k. It should either be a new variable (usually designated kappa), or it should say k>0, k<0. ...  — Preceding unsigned comment added by 2600:1702:1BD0:8CD0:855:E16C:9983:EF9F (talk) 07:41, 29 August 2019 (UTC)
thread_title: Inconsistent use of small k

message: The author of this article says that the Friedmann equations relate certain cosmological parameters in the context of general relavity.  I would like to modify this statement to say that these equations define certain cosmological models in general relativity, usually called the Friedmann dusts  (or matter dominated Friedmann models) and Friedmann radiation fluids (or radiation dominated Friedmann models).  The equations themselves arise in the course of deriving these models in a comoving coordinate chart.
:Isn't "relating practical real parameters within a theoretical context" the same as a "model"? All anyone needs to do is be careful when referring to the "FLRW metric", as derived from GR assuming the Cosmological Principle, or when referring to the "FLRW model", when certain additional assumptions are made in order to arrive at practical solutions within the metric.
Kentgen1 (talk) 13:10, 3 November 2010 (UTC)
thread_title: Context

message: There should be some more development of where the equations come from.  I may add some straightforward derivation.
Alejandr013 21:02, 4 August 2006 (UTC)
thread_title:  Development 

message: added a new section Friedmann equations#Useful solutions. The equation of state for the perfect fluid appeared to be the ideal gas law. But then I realized that the "constant" w is proportional to the absolute temperature. However, the temperature of a gas decreases when it expands, see Adiabatic cooling. So the equation should probably be corrected to account for the decrease in temperature. JRSpriggs (talk) 14:10, 23 April 2008 (UTC)
replies: text: Uh, no, w is not proportional to T. For simple models ("dust", "radiation", etc) w is constant and luckily these models are quite good approximations to the real universe for quite long periods of its history. Only when w is independent of T (or equivalently of a) do we get nice closed-form solutions. Of course the real universe is a mixture of dust, radiation, dark energy etc but usually one dominates. Only when there is a cross-over between cases does the effective value of w change significantly, which happens relatively quickly (especially on a log-log plot!). PaddyLeahy (talk) 12:12, 25 April 2008 (UTC)
replies: text: To Paddy: Thanks for clarifying that. JRSpriggs (talk) 13:14, 25 April 2008 (UTC)
thread_title:  Expansion lowers temperature 

message: The 2nd equation:
:\dot{H} + H^2 \equiv \frac{\ddot{a}}{a} =  -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3},
could be replaced by:
:\dot{\rho} = -3 H (\rho + \frac{p}{c^2}),
which eliminates \Lambda ,
The original 2nd equation can be recovered, when combined with the 1st equation.  Any objections to replacing the 2nd equation with this?
--Michael C. Price talk 10:13, 26 April 2008 (UTC)
replies: text: That is equivalent, so we might want to mention it in the article. However, the original second equation has historical significance; and it is named the "Friedman's acceleration equation". So I think we must retain it. Remember that the equations given emerge more or less directly from Einstein field equations. JRSpriggs (talk) 11:46, 26 April 2008 (UTC)
thread_title: Simpler 2nd equation

message: * "k \over a^2 is one-sixth of the spatial component of the scalar curvature"
The scalar curvature, R, is a scalar and as such it has only one component which cannot reasonably be called "spatial".
* "a and k have two possible equivalent definitions"
Even if the definitions which follow this are inter-convertible, they are not equivalent because they yield different values for k and a.
* It is not reasonable to call a the radius of curvature of the universe when k=0 because in that case the radius of curvature is either infinite or, more correctly, undefined. 
JRSpriggs (talk) 13:38, 1 June 2008 (UTC)
thread_title: MichaelCPrice's mistakes

message: Throughout the article, his name is spelled two different ways. I think it would be better if we settled on a single spelling; ideally, the same way he did. Apparently, he spelled it with a single 'n', "Friedman". I've hunted around on the web, a bit, and found what I think could be a good reference. It appears to be an image of the title page of one of his papers: http://www.springerlink.com/content/l23864w241673530/fulltext.pdf?page=1  Michael McGinnis (talk) 19:25, 15 March 2009 (UTC)
replies: text: Of his two famous papers, the earlier one (which you linked) spells it Friedman, and the later one () spells it Friedmann. The "Friedmann" spelling is overwhelmingly more common in modern physics papers, and it's also the spelling used in every online encyclopedia I checked except Wikipedia, so I think we should stick with "Friedmann", and I think the article on the man himself should be moved as well. -- BenRG (talk) 13:13, 16 March 2009 (UTC)
replies: text: Actually, his name is "Алекса́ндр Алекса́ндрович Фри́дман" in Russian. The final three characters "ман" transliterate as "man". However, we are supposed to use the name by which most English speaking readers would know him, according to Wikipedia:Naming conventions#Use the most easily recognized name. JRSpriggs (talk) 01:39, 17 March 2009 (UTC)
replies: text: Thank you, BenRG and JRSpriggs, for clearing this up for me. I do feel that a consistent spelling is needed, if only to reduce the possibility of confusion for people who might think that "Friedman" and "Friedmann" are two different people. I'm going to change all references to "Friedmann".Michael McGinnis (talk) 19:58, 19 March 2009 (UTC)
thread_title: Ambiguous Spelling: Friedmann or Friedman

message: Particles which have a proper motion in an expanding universe should decelerate relative to their immediate surroundings. To examine this effect, I use the formula for acceleration at User:JRSpriggs/Force in general relativity#Acceleration applied to a free-falling particle
: \ddot{x}^{\beta} = - \Gamma^{\beta}_{\alpha\mu} \dot{x}^{\alpha} \dot{x}^{\mu} + \Gamma^{0}_{\alpha\mu} \dot{x}^{\alpha} \dot{x}^{\mu} \dot{x}^{\beta} \,.
: \ddot{x}^{i} = -2 \Gamma^{i}_{0 j} \dot{x}^{j} - \Gamma^{i}_{j m} \dot{x}^{j} \dot{x}^{m} + \Gamma^{0}_{j m} \dot{x}^{j} \dot{x}^{m} \dot{x}^{i} \,.
: \ddot{x}^{i} \approx -2 a^{-1} \dot{a} \delta_{i j} \dot{x}^{j} - k x^i \delta_{j m} \dot{x}^{j} \dot{x}^{m} + c^{-2} a \dot{a} ( \delta_{j m} + k x^j x^m ) \dot{x}^{j} \dot{x}^{m} \dot{x}^{i} \,.
: \ddot{x}^{i} \approx -2 H \dot{x}^{i} - k x^i \dot{x}^{j} \dot{x}^{j} + c^{-2} a^2 H \dot{x}^{j} \dot{x}^{j} \dot{x}^{i} + c^{-2} a^2 H k x^j \dot{x}^{j} x^m \dot{x}^{m} \dot{x}^{i} \,.
JRSpriggs (talk) 11:44, 7 April 2013 (UTC)
thread_title:  Free fall in FLRW 

message: We can use  R = 6/a^2 (\ddot{a} a + \dot{a}^2 + k)  to make Lagrange function  L = \sqrt{-g}(R-2\Lambda)  to find Friedmann equations using variational principle. But we find only one equation, not both.95.55.17.46 (talk) 14:45, 30 January 2014 (UTC)
replies: text: See Einstein–Hilbert action for the derivation of the Einstein field equations. Your argument assumes that variations in the metric δgμν which violate the symmetry of the Friedmann–Lemaître–Robertson–Walker metric can be safely ignored. That is false. JRSpriggs (talk) 10:38, 31 January 2014 (UTC)
thread_title:  Friedmann qquations from R 

message: I can't find any other source for Omega-k "spatial curvature density". It seems intuitive that the definition should be Omega-k = Omega - 1. Which is correct? Further, how is Omega-k related to k? Asgrrr (talk) 12:27, 5 June 2016 (UTC)
thread_title:  Is Omega-k correctly defined? 

message: I am not sure what the meaning of the following sentence is:
If k = −1, then (loosely speaking) one can say that i· a is the radius of curvature of the universe.
I would suggest to remove the sentence if there is no easy way to supplement it with a simple clarification. --Falktan (talk) 10:34, 25 September 2016 (UTC)
replies: text: See Gaussian curvature. If the two principal curvatures have the same sign so that the Gaussian curvature is positive, then the radius of curvature, a, is the reciprocal of their geometric mean. What is meant here is that if the two principal curvatures have opposite signs (e.g. the Gaussian curvature is negative), then the reciprocal of the geometric mean of their absolute values is a. JRSpriggs (talk) 14:06, 25 September 2016 (UTC)
thread_title:  Imaginary radius of curvature 

message: The text following the last equation in the Density parameter section is confusing and misleading to a reader with little background knowledge. 
The problem is that the sign of Ω_0,k is backwards. That is the sign is positive for a universe with negative (hyperbolic) curvature, and negative for a universe with positive (hyperspherical) curvature. In particular, the variable k was previously introduced with a description that k=+1 corresponds to positive curvature, and k=-1 corresponds to negative curvature.
I hope someone with the appropriate skills will edit this text to remove this confusion.
BHBloom (talk) 18:01, 28 April 2017 (UTC)
thread_title: Confusing Text

message: ((((x GHz) * (3.71295774e-28 (m^2))) / Planck's constant) * (1 kg)) = 299792458^2 
((((160.390033967 GHz) * (3.71295774e-28 (m^2))) / Planck's constant) * (1 kg)) = c^2 
((((CMBR Hz) * (Friedmann Mass Density (m^2))) / Planck's constant) * (1 kg)) = c^2 
1/(160.390033967 GHz * 3.71295774e-28 kg   / Planck's constant  / c^2  )^0.5 = 1 m^2/s (square meter per second) 
https://en.wikipedia.org/wiki/Viscosity#Kinematic_viscosity 
The kinematic viscosity "momentum diffusivity" is the ratio of the dynamic viscosity μ to the density of the fluid ρ. It is usually denoted by the Greek letter nu (ν) and has units  m^2/s   — Preceding unsigned comment added by 2602:306:833C:5980:EF:7508:2AC:611B (talk) 01:24, 20 September 2017 (UTC)
thread_title:  Bulk Elastic Properties & Wave Speed & Friedmann_equations Density_parameter  

message: I suggest creating a separate article named Density of the universe because at this moment it redirects to this article (Friedmann equations), which is full of formulae. The people who want to read the article Density of the universe they actually just want to know how many atoms (or grams) of matter are in the average cubic meter of this Universe. —  Ark25  (talk) 00:36, 21 July 2017 (UTC)
replies: text: I modified the redirect to go to the section "Density parameter" which contains that information. JRSpriggs (talk) 10:59, 21 July 2017 (UTC)
thread_title:  Density of the universe 

message: \Omega_c=-\frac{kc^2}{H_0^2 a_0^2}\;
If k is meant to be K=1/R²=(c/H°)², then this turns to Ωc = -1/a² = -1. But what the heck is Ωc supposed to be? It is not the critical density, I guess it is supposed to be the curvature densitiy which is everywhere else called Ωk. On the other hand the value of Ωc ≈ -1 points to the critical value, but the minus sign is wrong then. So i guess it is a mixture and should be Ωc = 1+Ωk Ra-raisch (talk) 19:14, 18 June 2019 (UTC)
replies: text: As far as I understood now, there are two conventions: k using as curvature index (-1,0,1) or (usually K) as the curvature (1/m²) and also a using as the scaling factor or the scaling parameter (m) which should be A in order to distinguish them. Thus here it should be Ωk and not Ωc, leaving k and a as is, the ambiguity of which should be cleared and naming altered throughout the article. Ra-raisch (talk) 22:44, 23 June 2019 (UTC)
thread_title:  rescaled 

message: ... because a(t) wasn't invented until four years (1929) after his death (1925)!
A presentation of the original equation would be quite interesting.
Hilmer B (talk) 15:18, 19 August 2020 (UTC)
replies: text: Do you have any reliable source to back up that statement? JRSpriggs (talk) 19:34, 19 August 2020 (UTC)
replies: text: Do you have any reliable source to falsify that statement? Did really Friedmann invent and use "a(t)" as it is interpreted today?  Hilmer B (talk) 19:46, 20 August 2020 (UTC)
replies: text: To a mathematician using the radius of a hyper-sphere as a function of time when the hyper-sphere varies in size is obvious. If you need to see that Friedmann was aware of it, see his letter to Ehrenfest here where he says "a world, the space of which possesses a curvature radius varying with time;". JRSpriggs (talk) 20:44, 23 August 2020 (UTC)

text: Unfortunately, the original papers of Friedmann are behind a wall: you have to pay to see them and then they are in Russian or German. This is a common problem, forcing us to rely on secondary sources. But the mere fact that those papers are not available to us is not evidence that Friedmann was unaware of what is now purported to be their contents. JRSpriggs (talk) 19:30, 24 August 2020 (UTC)
thread_title:  Friedmanns original equation could not possibly have included the term "a(t)"! 

message: You properly derive the first equation as:
\frac{\dot a^2+k c^2}{a^2}=\frac{8\pi G \rho+\Lambda c^2}{3}
But then you go on to state that:
These equations are sometimes simplified by replacing
\rho \rightarrow\rho-\frac{\Lambda c^2}{8\pi G}
to give:
\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}
What is the justification for this substitution?  I might guess that you assume that the Cosmological Constant is so small with respect to the other terms that it is effectively zero, but you can't just drop a major variable without an explanation.
:It is a question of how the cosmological term is interpreted. Is it a complication in the law of gravity (as originally conceived)? Or is it a mysterious "dark energy" which should be included in the stress-energy tensor on the other side of the equation? Lately, people have been leaning to the latter interpretation. JRSpriggs (talk) 03:31, 21 February 2021 (UTC)
thread_title:  Justification for eliminating the Cosmological Constant from the First Equation