I find another aspect highly misleading. In a homogeneous and isotropic fluid there is no pressure gradient at any slice of time. Your correlations to positive and pressure matter suggests incorrectly that expansion is due to pressure.
In particular this section
"This all was about “normal” matter, with positive pressure. With “negative pressure matter” the effect is opposite – this was already described in the previous paragraph – that’s “dark energy”."
Reference
https://www.physicsforums.com/insights/brief-expansion-universe/
In my opinion taking a lesson from Andrew Liddle's Introductory to Cosmology"
You should consider defining expansion in terms of potential vs kinetic energy.
There is some rewording here but this set of equations demonstrate the section
One of the key aspects to understand expansion is to understand the potential and kinetic energy aspects with conservation of energy.
For this we can detail using Newtons laws.
F=\frac{GMm}{r^2}
Mass density we will use \rho which is the mass per unit volume.
Now assume a field of test particles. Motion and mass currently unimportant.
One of the aspects of the shell theorem in Newtons laws is the test particle will only notice a force from the center of mass.
In a homogeneous and isotropic distribution any test particle or CoM can be used.
As we're dealing with test particles we just need the mass relation.
M=\frac{4\pi\rho^3}{3}
So
E_p=-\frac{GMm}{r^2}=-\frac{4\pi G\rho^3 m}{3}
Kinetic energy is E_k=1/2m\dot{r}^2
U=E_k+E_p
U is just a dimensionless constant to equate total energy must be set as a constant value.
So the above translates to
U=\frac{1}{2}m\dot{r}^2-\frac{GMm}{r^2}=-\frac{4\pi G\rho^3 m}{3}
Now with the vector relation of the radius to length we can denote the scale factor.
\overrightarrow{r}=a(t)\overrightarrow{x}
Where a is a function of time. This leads toU=\frac{1}{2}m\dot{a}^2x^2-\frac{4\pi}{3}G\rho a^2x^2 m
Multiply each side by 2/ma^2x^2
Leads to
(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}
kc^2=-2U/mx^2
k=-2U/mc^2x^2
K is the curvature constant.
Andrew Liddle covers this in better detail to the Einstein field equations later in the book.
Key note you don't require the cosmological constant for a matter only universe to expand.
A good book showing expansion of a matter only universe is Barbers Rydens "Introductory to Cosmology"
I prefer the solution above for the potential vs kinetic energy aspects.
The end result equation doesn't include the Cosmological constant.
With the cosmological constant the equation becomes.
(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\rho-\frac{kc^2}{a^2}+\frac{\Lambda}{3}
This above better reflects the FLRW metric prior to the discovery of the cosmological constant. It shows you don't require the cosmological constant to have an expanding universe during any era
