Derivation of the Friedmann Equations[1] from the Newtonian
First Derivative of the Scale Factor
a is the scale factor, not acceleration. Notice it is not expressed as a vector. The scale factor is scalar; It has no direction, only magnitude. It describes the instantaneous relative space between all galaxies, and relative points, in the universe simultaneously. It is like we imposed a three-dimensional grid or matrix on the universe, and whatever size the boxes were when we did that, we called it “1”.
a (dot) is the time derivative of the scale factor. It describes how the space between all points in the universe expands or contracts over time. Mathematically, it is the instantaneous rate of “inflation” between all points in the universe at any point in time.
Inflation here is described both as in the wider context of the physical process of inflation, regardless of speed, as in the case of a balloon inflating; not the initial (rapid) “inflationary” expansion of the early universe.
Further, in the mathematical context we regard both “positive” and “negative” inflation as simply “inflation,” just as (positive) “acceleration” equally means “retardation” in the mathematical sense.
Thus, both physically and mathematically, “inflation” means only the rate of change of the scale factor, without regard to direction nor magnitude.
The ratio 
a (dot) divided by a is the ratio of the rate of change of the scale factor at any point in time, to the value of scale factor at that point in time. It is also known as the Hubble Parameter.
8 pi divided by 3 is derived from the volume of a sphere, which is
G is the gravitational constant throughout the universe, and has the extremely small currently measured value of 
N means Newtons, and it is simply a unit of force.
m in this case stands for meters. So this is, how far in meters can a force of this many Newtons move an object of so many kilograms (kg).
is the energy (mass) density of the universe.
We can further define the density by saying that is the mass per unit volume of the universe as defined by the cube of the scale factor. So essentially this is saying how much mass is in any box of a certain size.
Thus, 
v (nu
) is a constant and simply the mass per unit coordinate volume. By changing your coordinates you can change the amount of mass that’s in a coordinate volume, so is largely irrelevant in interpretation of the equation’s effects.
is the cube of the scale factor. This gives us the scale factor in three dimensions, i.e. 
. It essentially is a box – a cube – of volume a * a * a.
Substituting 
into the original equation 
, we get 
.
Everything in 
is made up of constants, so setting that all to 
for the sake of simplicity, we can think of the “moving parts” of this equation as 
.
So we can describe the motion as almost entirely based on the value of the scale factor and its derivatives. It is a differential equation – which is one in which a function and its derivatives behave in certain ways relative to each other. As 
changes, 
is changing in a very specific way, such that the equation always remains true.
Looking at the expression 
, we can see that the right hand side is always positive, and never goes to 
. No matter how large the three-dimensional scale factor becomes, and thus how small becomes, it remains positive. As gets very big, becomes smaller and smaller.
This tells us that 
– the Hubble Parameter – never goes to , which means it can’t change sign. But as increases, the Hubble Parameter () gets smaller and smaller over time.
Since Hubble’s Parameter is the fractional rate of change of the scale factor, it is as if the universe just gets tired of expanding and expands slower and slower, but it is never tired enough to stop.
This type of equation is central to all of cosmology, and this is what we thought the universe did, until we discovered dark energy.
The preceding derivation is taken almost directly from Leonard Susskind’s Stanford Lecture:
Second Derivative of the Scale Factor
is the time derivative of inflation. It describes how that instantaneous rate of change between all points in the universe fluctuates over time, i.e. the acceleration of the rate of inflation.
Mathematically, acceleration may be positive or negative with respect to the object under acceleration. Likewise, “acceleration” in the context of the scale factor is analogous. The rate of inflation (deflation) may increase (decrease), and transition between positive and negative inflation (contraction) and the reverse.
These are key components of Friedman’s[2] equations.
| Scale Factor & Derivatives | Mathematical Description | Physical Description | Analogs (Not Equivalents) in the Derivatives of Motion | Description |
| Instantaneous distance between any two galaxies. | The amount of “spacetime[3]” in the universe, between all points. |  | Position |
  | First derivative of the scale factor with respect to time. | “Inflation” or “deflation;” i.e. change in the amount of “space” in the universe. |  | Velocity |
,  | Second derivative of the scale factor with respect to time. | Change in rate of inflation. i.e change in direction and/or magnitude of inflation. i.e. speeding up or slowing down the rate of inflation or contraction. |  | Acceleration |
Not addressed here but later addressed is the third derivative of the scale factor, which is equivalent to the third derivative of position in motion, the jerk.
| Scale Factor Derivatives | Mathematical Description | Physical Description | Analogs (Not Equivalents) in the Derivatives of Motion | Description |
 ,  | Third derivative of the scale factor with respect to time. | The rate of change of the acceleration of inflation. |  | Jerk |
Simplified Friedmann Equation
Here we have the relation of the second derivative of the scale factor to the scale factor itself. This describes the speeding up or slowing down the inflation or contraction relative to the size of the scale factor at that same time.
is the volume of a sphere.
is again the gravitational constant, which has a value of .
The minus sign means this force is directed “inward”.
is the density. The density of the universe does not depend on position. The formula doesn’t care where you are.
One of the consequences of this equation is that the universe can not be static unless its density is . In a universe with any density, it must be moving. It is impossible to have a static universe unless it’s empty. And we know ours is not empty.
We also know that in terms of whether the universe will expand forever or recontract, the universe can be above the escape velocity, at the escape velocity, or below the escape velocity. The escape velocity is where energy is equal to zero and is the point of no return in the expansion.
According to this equation, the universe can in fact turn around and re-contract. So this equation doesn’t tell us whether the universe is expanding or contracting, but it tells us that the second derivative of the scale factor is negative. That is, even if the universe is expanding, it is tending to slow down. And if it’s contracting it tends to speed up the contraction.
But it is in fact expanding and not slowing down, and herein lies the origin of dark energy. In 1998 we discovered that it is accelerating positively in its expansion of space, so there must be something missing in this equation.
The Need for Friedmann’s Equations to Correct the Newtonian
In Newtonian theory, space is flat. A Newtonian universe would have been infinite, spatially flat, and expanding or contracting in a constant manner. It would have been entirely Newtonian; i.e. formulated or behaving according to the principles of classical physics.
In Newton’s universe, the scale factor increases as a ⅔ power of the time such that 
, and thus is completely “clockwork,” i.e. it keeps a steady rate throughout.
But our universe does not act that way. Dark energy makes this increase accelerate, and so we need to go back to the original Friedmann Equations, which included Einstein’s Cosmological Constant term 
– originally included to counteract the gravitational force in the equation to keep the universe static – but now known to in fact accelerate the inflation in the latter third of the universe’s existence.
is always the speed of light.
is the 
, or more precisely, a term of unknown origin that counteracts 
, i.e. 
. We know it should be in the equations, but we don’t know its source[4].
So the entire question of current cosmology comes down to what it is that is making the inflation accelerate once again, violating the power law of the Newtonian. This is the fundamental question that we have been trying to answer in our work on the Identity Principle and Blast Wave models.
And as we can see the entire equation really comes down to the relationship between density
(), energy (), and pressure (
), so we can see why the blast wave makes such an appealing candidate for a mathematical solution to the dark energy problem; as the blast wave by nature deals with the exact same variables.
[1] The Friedmann Equations are an exact solution to General Relativity using a perfect fluid.
[2] Alexander Friedman’s name is properly spelled with one “n.” Einstein once mistakenly referred to Friedman – who is Russian, as Friedmann – which would be his own common German spelling. Friedman’s colleagues suggested that since Einstein used the German spelling, he should keep it for future publications or risk being unrecognized.
[3] Spacetime can be thought of as a single dimension. However, Einstein’s later use of the term “timespace” leads more quickly to a proper interpretation; Time (T) giving the total volume of the universe, and time (t) giving the distance between any two points.
[4] Of course, we have postulated the blast wind phase of the blast wave as its source.