The Friedmann model of the universe forms the basis of the Standard Model. Yet, it is usually considered not as a pair of evolution equations, but, rather, a condition where the Hubble parameter and average density of the universe are determined empirically and the sum of all the densities, the mass density, the curvature term, and the vacuum energy density (cosmological term), add to unity.
What is the difference between the Friedmann equations and those resulting from the Robertson-Walker (RW) metric? The Friedmann equations supposedly determine the time evolution of the spatial curvature, based on the half-space model of the hyperbolic plane where the half-plane can be viewed as the boundary at infinity. It therefore makes no sense in making the scalar factor in what was the constant negative curvature a function of time.
As a matter of fact, the Friedmann equations are identical to the equations for the RW metric for the "cosmic scale factor", which introduces an additional element not present in the Friedmann model: a spatial scalar curvature constant, k, which can assume one of three values. One of these values corresponds to hyperbolic space, but k can magically vanish, when a "critical density" is reached. That's equivalent to changing the entire model, which is like changing the card game once the hands have been dealt.
According to the Riemann metric k=const implies a metric of constant curvature. However, Friedmann has the curvature evolving in time in a "non-stationary world". Friedmann does not address the question of why should the scalar curvature satisfy Einstein's field equations, but just assumes that it does.
How can the density of the universe remain constant in an evolving universe? Contraction of the Einstein equations with respect to a timelike unit vector equates the density to the spatial scalar (Ricci) curvature, which a part from a sign is the constant Gaussian curvature, K. (This supposedly introduces an additional three categories of curvature, K, (=0.0), and the relation that the energy density is equal to a "negative" pressure. Unlike its garden variety, "negative" pressure causes an explosion rather than an implosion.) This is consistent with the Friedmann equations, the lambda term in his equations appears in both the energy density and pressure equations, but with opposite signs.
Moreover, it additionally constrains the Hubble parameter in that it becomes inversely proportional to the spatial curvature of our world". So once we determine the scalar curvature R, both the Hubble parameter and average energy density of the universe are fixed, and there is nothing to solve!
According to Robertson, "the constant curvature models can be rescued by relaxing either the assumption of the field equations of general relativity or the hydrodynamic interpretation of the cosmological pressure." The second option is hardly feasible since pressure is pressure.
Friedmann supposes the scalar curvature is a function of time, which by the contraction of the Einstein equations, makes the average density also a function of time. How can the dark matter density supposedly make up a substantial percentage of the critical density when the latter is evolving in time? The Einstein equations then make the Hubble parameter also a function of time. Does this constitute a viable description of the temporal evolution of the universe?
This is a side comment direct towards the people who would care for a discussion on the Friedmann-Lemaitre Model. They would be the target people for this question
Why would the Cosmological Standard Model be based on a 4D Spacetime???? Considering it makes extremely difficult to explain Cosmology.
Any 3D Explosion would normally have a radial symmetry. Cosmologists have to jump hoops, create Rube Goldberg Machines to explain something that is trivially explained by a Hyperspherical Lightspeed Expanding Universe.
Why would Physics be driven by such non-physical considerations:
"the constant curvature models can be rescued by relaxing either the assumption of the field equations of general relativity or the hydrodynamic interpretation of the cosmological pressure."
This is what a Mathematician would say. A Physicist would say.. What the Hell..:) This has to be wrong. The Universe fate (its curvature) cannot be determined by the amount of mass the Universe contains. That leads to hypercritical dependences on the mass...
Below are the d(z) (distance versus redshift) derived using simple trigonometry in HU and its predictions of 650 Supernovae SN1a explosions distances.
This is accomplished without a single parameter.
Is there any reason to persist using the Friedmann-Lemaitre Model in Cosmology or a 4D Spacetime for that matter?
https://issuu.com/marcopereira11/docs/huarticle
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