While it is accepted that the Friedmann equations are a solution to Einstein’s field equations of general relativity, a more classical derivation seems to also provide some possibly interesting insights to the underlying physics. Therefore, it might be worth highlighting the following points about the classical derivations in comparison to the GR based derivations:
- Both lead to Friedmann's equation set, i.e. there are no correction
- Although the Newtonian premise is based on gravitational energy
rather than the curvature of space, the results are still the same.
- Although the Newtonian derivation is based on mass-density, the equations can be converted to an energy-density using E=mc2.
It may also be highlighted that while general relativity and Newtonian mechanics describes the effects of gravity in a fundamentally different ways, i.e. spatial curvature as opposed to a gravitational force, in relatively weak gravitational fields, the results of both approaches must converge. One other point of convergence is that a gravitational centre is still required by both approaches, i.e. gravitation curvature or a gravitational force. However, in order to reconcile the Friedmann equation to the ΛCDM model, we also need to consider the density [ρ] and pressure [P] implications that arise out of the Fluid equation, which were based on the classical idea of the conservation of energy and tied to the 1st law of thermodynamics.
While it is accepted that speculation has a valid role in science, there appears to be a growing preference is to ‘market’ what may be little more than speculation as a theory or hypothesis, especially when it appears to be underpinned by some degree of mathematical logic. However, given that many hypotheses are eventually proven to be unfounded, it might also be recognised that mathematics, in isolation, does not necessarily lead to an accurate description of the physical universe. So let us try to put the scope of this speculation into some overall perspective, based on the starting assumption that some form of energy-density model is thought to underpin the basic premise of the Big Bang concept. In this context, we might wish to highlight that dark energy must still be labelled as hypothesis, at least, at this point in time. However, within the assumed energy-density model, it can be shown that dark energy does not have a sufficient presence in the universe, until about +7 billion years, when its density rises to a sufficient proportion to be a ‘possible’ explanation to the ‘apparent’ observation of the accelerated expansion of the universe. However, it is unclear how dark energy maintains a constant energy density under expansion, as this would ‘seem’ to contradict the conservation of energy, at least, within an isolated thermodynamic system. Of course, ‘maybe’ the universe is not all there is, ‘maybe’ there is some form of quantum energy in the vacuum of space, as both may be valid speculations, but not ‘necessarily’ verified fact. However, dark energy does not ‘seem’ to explain how the universe initially expanded or why it kept expanding in the absence of any obvious energy-density for the 1st 7 billion years. At this point, we might ‘speculate’ further regarding scalar fields of the inflationary model and on the 'idea' of the momentum of expanding space. Of course, others may wish to alternatively ‘speculate’ that space is infinite and that the expansion of our local universe took place within existing space. In essence, all that is being highlighted in this commentary is that there is plenty of scope for speculation, such that we might wish to table a question at the start of this section of the discussion for further consideration:
Can any of these ‘speculations’ be ruled out by the mathematics of GR or is it just as susceptible to the adage ‘garbage in, garbage out’ as any other branch of science?
While those who already have an in-depth understanding of cosmology may possibly be able to debunk the following interpretation of the Friedmann equations, as either wrong or no longer relevant to modern cosmology, such arguments can only be presented when found and understood. Unfortunately, many technical references either appear to avoid, trivialize or suggest that certain issues are so complex that they can only be explained based on a prerequisite and deep understanding of general relativity and its mathematics. If true, it might seem to suggest that any substantive understanding of cosmology may be beyond the educational reach of most people.
Is this an acceptable position?
While possibly it is not, what may have to be accepted is that it does actually represent the current situation, which then returns to the issues as to whether the ‘accepted worldview’ is one of consensus of the majority or simply the imposition of a minority, albeit in this case of an educated minority. However, at this point another adage is possibly apt:
In the land of the blind, the one eyed man is king.
Anyway, leaving these issues aside, we may wish to pursue the questioning of cosmology based on more widely understood principles of physics, even though they may only approximate the solutions within general relativity. As such, we shall continue the discussion of the Friedmann equations in order to try to determine where problems may appear to arise in terms of the classical approximations made so far, i.e. the conservation of energy, 1st law of thermodynamics, centres of gravity and inertia of expansion. As such, we need to return to the issue as to whether there is anything in the Friedman equations that actually explains why the universe is expanding, as the inference of Hubble’s law [H=v/d] is often directly interpreted in terms of an expansion velocity. The following equations in  below correspond to the previously derived Friedmann equation set, although the form of each has now been modified to show the component energy densities, which have been subsequently introduced as requirements of the current ΛCDM model.
- matter energy density [ρM≈4%]
- radiation energy density [ρR⇒0%]
- dark matter energy density [ρD≈23%]
- space curvature energy density [ρK=0%]
- dark energy density [ρΛ≈73%]
The values associated with the respective energy-densities above correspond to the present era, which will change as a function of time due to expansion - see Cosmic Calculator for details. The numeric weightings associated with each energy density in  below are determined by the ‘equation of state’ associated with each 'energy-density’. Both these issues will be explained in more detail later within this section of the discussion.
The equations in  corresponds to the Friedmann, Fluid and Acceleration equations, where pressure [P] has been normalised to just energy-density using the associated equation of state. They effectively represent the rate of change of the scale-factor [a], i.e. the normalised radius [r] of the universe, and the rate of change of the energy-density [ρ].
With reference to the first equation in , the effect of each energy-density might be said to be unambiguous, at least, in the sense that the sign of each quantity is assumed to be positive. As such, we may combine the component densities back into a summed total [ρT] and revert [da/dt] back to the velocity [v]. It might also be useful to remind ourselves of the conservation of energy that underpinned the classical derivation of the Friedmann equation:
In terms of Newtonian physics, what  appears to be telling us is that the velocity [v] would normally be interpreted as the a free-fall velocity towards the centre of gravity associated with [M]. Of course, what this interpretation would clearly fail to explain is how the velocity in  is then interpreted as the expansion velocity of the physical universe. However, the scope of this issue is picked up in the discussion of Energy Density, while the assumption concerning zero energy will be further questioned in the discussion entitled 'Model Assumptions'. Of course, if pressure [P] can be converted to an equivalent energy-density, as in the following discussions of the Fluid and Acceleration equation, might we reverse the process for the Friedmann equation, which normally makes no reference to pressure?
So, based on the same arguments associated with the equation of state for each component energy-density, baryon and cold dark matter would exert no pressure. Radiation pressure would appear to be expansive, which seems to contradict the implications of its energy-density, but either way, would have only been effective in the very early universe. The curvature pressure would oppose the radiation pressure, irrespective of the direction of the radiation pressure when finally resolved, although generally this component is ignored in a spatially flat universe, i.e. when [k=0]. This leaves dark energy, which also opposes radiation, but its energy-density only becomes effective after about 7 billion years along the timeline of expansion, as predicted by the acceleration equation.
The second equation in  corresponds to the Fluid equation and reflects the rate of change of the energy density components with time. At this point, the key interpretation to highlight is the zero numeric weighting for dark energy, which implies that this energy-density does not change with expansion. As such, dark energy must become an increasingly dominant factor as the universe expands, although we might need to again question what this means to any assumptions relating to the conservation of energy, which underpins the classical derivation of both the Friedmann and Fluid equations. However, it might also be suggested that the Fluid equation, in isolation, does not explain the cause of expansion or contraction, only that should either occur, then there is a defined relationship between the rate of change of density [dρ/dt] and velocity [v] of expansion or contraction.
First, it might be worth highlighting that the negative sign in the acceleration equation is a direct result of the substitution of the Fluid equation in its derivation. However, it might be useful to simplify the form of the acceleration equation, as shown in , by setting all constants to unity and subsitituting in the relative energy-density of each component in the present era, where [a=1]:
What we can now see more clearly is that the acceleration equation carries a negative sign, by default, which would appear to reflect gravitational collapse, not expansion. However, the collective value of each component density, in the present era, now seems to suggest a positive accelerating expansion, which is solely based on the negative pressure of dark energy. At this point, we might initially try to interpret the implication of this acceleration in terms of Newton’s 2nd Law, i.e. F=ma. At face value,  would then suggest that prior to +7 billion years, the default negative acceleration would result in an ‘inwards’ force, i.e. gravitational slow-down, and only after this point would the positive acceleration suggest an ‘outward’ force of accelerated expansion. Of course, this description is problematic on a number of levels:
- Irrespective of the direction of the force implied by the acceleration,
it is assumed that the universe has always been expanding in accordance
to the usual interpretation of the Friedmann’s equation. However,
in both cases, it is unclear what is thought to be driving the physical
expansion, although it would appear that dark energy cannot account
for this expansion.
- Within the general concept of expanding space, any mass [m]
considered to be stationary with respect to the CMB radiation would
have no velocity or acceleration and therefore experience no obvious
force. However, according to the acceleration equation, any two
points in space must now be subject to accelerated expansion.
- Again, we must list the problem of gravitational slow-down, which appears to operate without the acceptance of any obvious centre of gravity.
As such, we either have to accept that the expansion of space continued based on some notion of inertial momentum or consider that some, as yet, undefined process continued to drive the expansion of space after the process of inflation completed.