Ok so let;s get into the cosmological fluid equation, the Friedmann equation. The most simplest form of the Friedmann equation is

[math]\frac{\ddot{R}}{R} = \frac{4 \pi G}{3c^2}\rho[/math]

One correction I made I applied in the past, inspired through the work of Motz and Kraft was that the equation need not be taken as a conservation equation - this is after all at odds with relativity in which energy is ill-defined under the Noether theorem, the distribution of a time variable indicates we get

[math]\frac{\dot{R}}{R} \frac{\ddot{R}}{R} = -\frac{4 \pi G}{3c^2}\dot{\rho}[/math]

Motz' quite rightly stated that the equation need not be conserved, just as this previous equation indicates, as he said it was an ''unfounded assumption by Friedmann to conserve the energy in such a way) - and general relativity hints at this since not only is energy ill-defined, it is ill-defined because time itself is ill-defined (as it states there is no cosmological constant) under the Wheeler de Witt equation, in which diffeomorphism invariance is not a true time evolution of the theory. Further more, the Wheeler de Witt equation is also often taken as a statement of a conserved universe in which no changes happen inside of it, the consequence of it having no time derivative. But logic should be indicating to us that many phase transitions have occurred and so the general statement should be taken with a pinch of salt.

Using the non-conserved form of the Friedmann equation, we can introduce the so-called continuity equation, but this is a misnomer if the derivative on the density term implies a change inside of the universe in a non-conserved way. The misnamed continuity equation takes the form:

[math]\dot{\rho}= -3H(\rho + P)[/math]

where we take all terms here as the energy density and pressure. [math]H[/math] is the usual Hubble constant, but we should read this that density is not fixed, but dynamic component. From previous posts I have shown that the total pressure would imply a correction to this relativity term:

[math]\dot{\rho}= -3H[\rho + (P_0 - P)] = -3H[\rho + q][/math]

You can plug this into this into the non-conserved form of the Friedmann equation as

[math]\frac{\dot{R}}{R} \frac{\ddot{R}}{R} = -\frac{4 \pi G}{c^2}H[\rho + (P_0 - P)][/math]

[math]= \frac{4 \pi G}{c^2}H[\rho - (P_0 - P)] = \kappa [\dot{\rho} + H(P_0 - P) = \kappa [\dot{\rho} + H(P_0 - P) = \kappa [\dot{\rho} + \dot{q}][/math]

[math]\kappa = \frac{4 \pi G}{3}[/math]

Also an important quantity, the density parameter is equivalent to

[math]\Omega \equiv \frac{P}{\rho} = \frac{8 \pi G \rho}{3H^2}[/math]

Later we will come back to this as it will play a role in the Bernoulli's principle. We must also consider for this following work the following work the presence of a second correction term for the total pressure of the fluid, as we had defined earlier as

[math]q = (P_0 - P)[/math]

Which is a correction term, for the ''continuity equation'' a misnomer if a third derivative actually implies a diabatic, non-conserved solution for the Friedmann equation:

[math]\dot{\rho}= -3H(\rho + (P_0 - P)) = 3H((\rho - (P_0 - P)) = - 3H((\rho - (P_0 - P)) = 3(\dot{\rho} - \dot{q})[/math]

the last set of terms in this equation has also featured in investigations into practically similar expanding and contracting models for sonoluminscence models - this is no accident of course since both those solutions are also derived from fluid mechanics.Plugging the main corrections into the non-conserved form of the Friedmann equation yields:

[math]\frac{\dot{R}}{R} \frac{\ddot{R}}{R} = \frac{4 \pi G H}{c^2}(\rho - (P_0 - P))[/math]

[math]= \frac{4 \pi G }{c^2}(\dot{\rho} - \dot{q})[/math]

I have made previous corrections terms, such as introduction a zero point energy term for a pre-big bang state that existded at very near absolute zero to attempt to formulate a semi-classical model. The Friedmann equation, even in principle, cannot predict why spacetime should be flat - let me quickly explain. The density parameter, [math]\Omega[/math], is defined as the ratio of the actual (or observed) density [math]\rho[/math] to the critical density[math]\rho_c[/math] of the Friedmann universe. The relation between the actual density and the critical density determines the overall geometry of the universe; when they are equal, the geometry of the universe is flat (Euclidean). Appealing to authority, Susskind himself and Lisi Garret have expressed their opinions, that the universe is not flat - a way to articulate why this is , is because the universe would have to be infinitely large and this isn't really how we think about the universe in a scientific way, because it would take the universe an infinite steps to do so - but steps involve numbers and infinities are not numbers by definition. At any moment in time, no matter what time, the universe could be measured being finite in value always.

In a similar notion, the only way a black hole can become flat, would that the radius

[math]R \rightarrow 0[/math]

Meaning the curvature of an expanding sphere must disappear

[math]K \rightarrow 0[/math]

But the universe can only approach these values, but never reach it perfectly.

As wiki states itself: ''In earlier models, which did not include a cosmological constant term, critical density was initially defined as the watershed point between an expanding and a contracting Universe. To date, the critical density is estimated to be approximately five atoms (of monatomic hydrogen) per cubic metre, whereas the average density of ordinary matter in the Universe is believed to be 0.2–0.25 atoms per cubic meter.''

I have become a hard disbeliever against dark matter, too much mounting evidence against it over the years - I could go into loads of reasons why, but I will spare the reader. I also believe that dark energy has been misunderstood, but we will get to that later. The entire basis of the density parameter to the actual observed ratio of matter in the universe, is too many magnitudes off, which should have been taken seriously that the Friedmann equation may itself be flawed fundamentally.

Next we will get into Bernoulli's principle and how the work above will come together to explain a source of the cosmological constant and how it applies to dark energy due to a weakening of gravity in the universe - the cosmological constant has often been believed that it is not really a constant, but there are some intriguing reasons why we may have to consider new alternatives to explain this in a more rationally-mechanical way due to pressure, perhaps thermodynamic in nature.This time around i plan to write the correct formula, in whcih we retain that dark energy remains constant, but the acceleration entertained by a weakening of gravity over galactic scales, only giving the impression dark energy is soley due to it - basically, the weakening of gravity only appears to make the constant change, when it is not. The gravitational binding of the universe would arise in my picture as an effect of gravitation become weaker as it gets larger and so only appears to accelerates. Anyway, back to this tomorrow.

**Edited by Dubbelosix, 18 June 2019 - 03:55 PM.**