# Interpreting Friedmann

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
factors.
- 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=mc
^{2}.

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 1^{st} 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 ‘*apparen*t’ 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 ‘*necessaril*y’ 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 [1] 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 [ρ
_{M}≈4%]_{ } - radiation energy density [ρ
_{R}⇒0%]_{ } - dark matter energy density [ρ
_{D}≈23%] - space curvature energy density [ρ
_{K}=0%]_{ } - dark energy density [ρ
_{Λ}≈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 [1] 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.

[1]

The equations in [1] 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].

## Friedmann Equation:

With reference to the first equation in [1], 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 [ρ_{T}] 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:

[2]

In terms of Newtonian physics, what [2] 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 [2] 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?

[3]

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.

## Fluid Equation:

The second equation in [1] 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.

## Acceleration Equation:

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 [1], by setting all constants to unity and subsitituting in the relative energy-density of each component in the present era, where [a=1]:

[4]

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 2^{nd} Law, i.e. F=ma.
At face value, [4] 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.