Density of the Universe

Contents

1.  The density of the Universe in the first (Newtonian) approach. Short introductory resume.  

     Flatness of the geometry of the Universe. Calculation of density of the Universe. Density dependence on 

     time.    

2. Critical density. Calculation of the critical density using

     simplified  Weinberg’s method. Calculation on the basis of

     Friedmann equation.

1. The density of the Universe in the first (Newtonian) approach                                         Those who will say to themselves: "Here it goes, the continuation of that naïve cosmology" surely won’t read it. The previous article they also left unread with a sense of superiority of an "initiate". And I go on. I envy those who are so sure of everything.

Short introductory resume                                                                                                         In the preceding article we have concluded that he mass of the Universe (meaning the Contractual Mass of Universe (CMU)) gradually increases, moreover it increases in coordination with the increase in its size. This makes me (not just me) ask the questions: What is its density? Does it remain constant? Of course not, because the volume increases faster than the mass, in proportion to the third power of the parameter defining the dimensions and it does not matter what is the actual topology of the Universe. However, so as to estimate (density), we should perform the appropriate calculations. For this purpose, we assume (tentatively) that the Universe is a sphere with a radius equal to the Schwarzschild radius. This can be done provided that the geometry of the Universe is flat*. This is, of course, an oversimplification, but let's not quarrel about the values of small deviations. It does not preclude the main goal. Is the geometry of the Universe really flat? According to the concept of this work, the flatness of space is an inherent feature of the Universe as a whole. The rationale for such a view is the assertion that "the building stuff of space is the relative motion of material objects." Motivational basis for this approach is also embodied in one of the (already stated) conclusions based on the cosmological principle, indicating that the resultant intensity of the global gravitational field is equal to zero. So this is not about the expansion of curved (by gravity) space - as it is thought in relation to the fact that today's cosmology is based on the field equations of general theory of relativity (GTR). It is interesting that with this, according to today's views, the expansion has not been caused by gravity, since it "only attracts". [And pressures? They were introduced to justify the possibility of repulsion by those unaware of the possibility of dual gravity. According to the hypothesis of inflation, at some point there was great pressure, which was equated with a negative gravity. In general, pressure is the second, besides attraction, element in the description of the Universe on the basis of GTR. For this reason, there are three options for the development of expansion (Friedmann’s models).] The Explosion itself together with inflation is a matter of quantum field theory (some mismatches, vacuum energy, the Higgs field, the inflation field bubble, etc.), in which gravity is neglected, even ignored. But after inflation, development of the object proceeded according to GTR equations and the theory of gravity. It ought to be admitted that its field equations contain all options of the beginnings, without going into the processes at the level of the structure of matter (...).

  While I persistently claim that the cause of expansion is in fact gravity. Dual Gravity. So there was no baton transfer as in a relay. This is clearly indicated in articles 4-7.          

     Currently, as I have noted above, the causes of the Explosion are attributed to the quantum microstructure of being, the vacuum energy recognized as an indisputable fact, and its fluctuation all of a sudden, which was to lead directly to the Big Bang. It was then that time allegedly began its existence (And what was before?). “First (right after the beginning) the events were caused by fluctuation of some inflation field, existing through the concept of inflation, then gravity started its work (when this something became flesh), which essence is solely attraction, although the Universe (exclusively) was fitted with the repulsion chip (pressure generated by the "dust"), and then, relatively recently, Einstein’s mistake was reactivated, that is the cosmological constant, and it was associates with the "invention" of dark energy..." Muddle? The less we understand nature the more complex it becomes.  

And exactly where this vacuum energy comes from? The hypothesis of its existence derives from calculations within the framework of the quantum field theory. So it is huge, given the scale of our perception. I am full of admiration for scholars who have come to this without in any way considering gravity (the one in sub-dimensions, gravity at the Planck scale). By the way, at great cost of time and work by the greatest minds. I had it a lot easier, thanks to them.             

Considering the elementary gravity (articles 5-8) we discover another world – let’s just remember the (absolute) maximum force of attraction between the two plankons, enormous even in astronomical scale. Let’s recall still greater (64X), maximum repulsive force of gravity – it is exactly this force which caused that event which we call the Big Bang. I think that the reader can already guess everything. Is it really about a mysterious vacuum energy, detectable only in the equations of quantum field theory (and not in experience); about nothingness masking potential enormity? The enormity of what? It borders on mysticism. Rather not gravity, I think, which is not recognized by the quantum mechanics. This (masking) is facilitated by the fact that in the atomic scale and in the scale of particles, as well as in our macroscopic scale, gravity is extremely weak. But we (and only we here) already know that the cause of this weakness is the almost total compensation of gravitational forces in the subatomic scale. Thanks to what? Thanks to dual gravity, thanks to the existence of a niche, gravitational potential energy - a place where particles can be created. So there are particles, atoms, there is the whole of our world. After all, if somewhere deep there was no repulsion, everything would disappear in an infinite black abyss. Could the matter at all come into existence? Nobody cares. "But we have the Pauli exclusion principle." And where does it come from? The job is performed by equations.  

Flatness of the geometry of the Universe.  
 And in the scale of our senses? At present the subject is based on the Friedmann equation (including the cosmological constant and dark energy) and on current knowledge of the microworld (quantum mechanics), while the unfortunate, " flatness of the Universe" is the result of observation. Is something wrong? Also the mathematical superstructure pretending to the role of the base, even absolute, and argumentation based on it plays here its prominent (if not fundamental) role. Is that how it should be? Rather not. The effect should not precede the cause. Nature does not need to adapt to the connotations resulting from equations devised to the extent of human's limited capabilities.  

 Flatness means that the density parameter (ratio of the average density of the Universe to the critical density) Ω = 1. As the most reliable calculations (of course based on the Friedmann equation - not necessarily correct) indicate, if in the first second from the beginning of the expansion this parameter was only slightly greater than unity, the Universe would have collapsed long time ago. If it were only slightly less than unity, the atoms could not be created (as a result of too rapid dissipation of matter). We just wouldn’t exist. And yet maybe even as much as 15 billion years have passed from the start of expansion. Robert Dicke in one of his lectures presented it in a very demonstrative way: „When the age of the universe was equal to one second, the value of the parameter Ω could not exceed the range of 1±0.00000000000000001", for the Universe to have current features of the structure of matter and the dynamics of development**. The flatness problem. Strange balancing of the Universe on the rope of a hair thickness. Is that the reason for the adoption of the anthropic principle? Personally, I reject this principle, seeing it even as a kind of mysticism (in a materialistic frame). Reason: below. By the way, earlier, already in the initial part of the previous article I stated that due to the inherent flatness of the space of the Universe, "criticality" is the only option. In this situation we can’t talk about criticality due to its semantics, and the density parameter loses its usefulness as only equal to one. Here we’ve got the base for a new cosmology.

   At present that is not seen this way. It is accepted that nowadays the density parameter is indeed (let's say "approximately") equal to unity, but perhaps in the future its value will be different, that’s why the symbol Ω is often accompanied by index zero. In addition, in accordance with today's modelling, its value results from a variety of factors independent of each other, which may mean that the unit value of omega is not all that obvious, it may be even accidental. The mind boggles at the observation made by Robert Dicke. According to the model which I present, this "randomness" is not possible (as randomness). According to the model presented here Ω does not change. The flatness of space is an objective feature of nature, and not an improbable state aimed at ensuring our existence. A little more and we’ll come to think that this is because everything that surrounds us is, for each of us individually, only an impression, as it was stated hundreds of years ago by certain Berkeley. Thus, the Universe was not created purposely for our existence, and its description based on basic and universal laws of nature is simpler, without any a hint of purposefulness... 

Calculation of the density of the Universe                                                                                    ...In any case, concerning the flatness of the Universe consensus omnium predominates, because this is indicated by observations. So let’s calculate the density of the Universe assuming (in this case perfectly digestible) simplification that the Universe is a spherical object. Thus: 

Here R –Schwarzschild (gravitational) radius: R = 2GM÷c^2. In view of the assumed flatness the Universe can be treated as a sphere of radius R. So we get:  

As you can see, judging from this formula, the density of the Universe (as well as the density of an object enclosed within the gravitational radius, called the black hole) is inversely proportional to the square of its mass, which, as we know, is increasing. I will refer to this formula time and again.

Density dependence on time 

So it is interesting how this density relates to time. For this purpose we use the postulated in the previous article equality of gravitational and Hubble radiuses. The Hubble law should be also taken into account.

However this non-linearity can be described differently, and thus avoid inconsistencies brought about by this hypothesis. Suffice to assume that the initial dimensions of the Universe were not zero. Then the non-linearity of increase in size is natural in view of the target speed c. The model corresponding to this thesis is described in the third part of the article entitled: "The first moments of the Big Bang", though quite a lot I blurted out earlier. The nonlinear expansion of my breeding, not based on "inflationary" assumptions I called: Urela (Ultra-Relativistic acceleration). In those very early times there wasn’t either the electromagnetic interaction, and so there was no limitation concerning the speed of expansion - as one might think.

So first we transform the formula for the Schwarzschild radius:                                             

and apply the Hubble’s law in relations to the horizon: c = HR. Thus we get: 

Here τ is the age of the Universe. Finally by joining formulas (1) and (2) we obtain the formula for the average density of the Universe:                

We see that the average density of the Universe depends explicitly of the value of constant H, and is proportional to its square (or inversely proportional to the square of the age of the Universe). [For the record, so defined age of the Universe is sometimes called the Hubble’s time. It is believed that to determine the (real) age of the Universe, the gravitational slowing down of the expansion (predicted by the Friedmann equation) must be taken into consideration as well as its acceleration due to dark energy. I see no justification for such an approach. I think that the "idealization" in the context of our discussion is in fact closer to the truth as it is freed from the imperfections of human investigative insight.] Let’s note also that in the last formula there is no constant c. So, according to this dependence the speed of light bears no relation to changes of density of the Universe. This does not mean that this speed is really constant against global time.  

     It is noteworthy that we deduced this formula starting from adopted in the previous article postulate of equality of gravitational and Hubble radiuses. We will get this formula following also a different, more agreed path. It is not a novelty. Currently this postulate surprises, in any case those to whom I present my thoughts. [In general, beforehand I invite them to supply themselves with a bottle of mineral water.] As it will (surprisingly) turns out further on, this postulate leads to the result concurrent with the calculation forecasting the development of the Universe in accordance with the critical model, constituting, as we know, one of the three options resulting from the Friedmann equation. Therefore, this "other way" of mine is not "so" meaningless in spite of the fact that it does not fully comply with the Friedman’s version, even in relation to "criticality". The point is that despite the (modelled here) flatness, the Universe, in relation to assumed periodicity of its physical and spatial characteristics, will not expand asymptotically toward infinity (as expected by equation Friedmann equation for the critical model). Yes, periodicity, not only postulated but simply justified on the basis of various criteria, to the extent that it "cannot be otherwise". Is it only my subjective conviction? Of course not. Many wiser than me are convinced about the periodicity of Nature. The problems which they encountered are essentially of doctrinal character, for example, "What to do with the increase of entropy?”. The question of entropy I will also address in due time. 

     The conviction of cyclicality can draw support from the ancient philosophical thought (not forgetting the intellectual heritage of the peoples of America before the arrival of the barbarians from Christian Europe. Now there are already other barbarians. Those have already invaded Europe.

They haven’t yet managed to learn humanity. They need for it hundreds of years and millions of victims of self-destruction under the imperative of hatred for someone completely different... Well, Homo Sapiens. How strange that they send their countrymen to virgins, instead of, for this godly purpose, blowing their own brains out. No, first they have to play some martyr’s jihad. This does not disturb political correctness, what matter is that they want to annihilate those whom Europe did not quite manage in its time. That was rather common in antiquity. Are humans really growing? The (negative) measure of growth is the potential of hatred. Ahead of us, humans, still a long way to go.

     An alternative to cyclicality in the form of a semi-infinity (because Everything had its beginning and stretches to infinity), though supported by the authorities of the church, does not seem serious. It simply gives an impression of an ontological botch. The Creator was not a bungler! Today, this strange alternative is accepted as if in the nature of things, and moreover, with "deep understanding".

     And (here) we have flatness of space, combined with cyclicality. Along with this, in a concept that I present, we have the development of the Universe: determined, occurring in one direction only. This is about the only possible process in the sense that there cannot be any alternative of various options (as it’s the case with three models in the Friedmann equation). Nature is one, and its description should therefore be unambiguous. And this is not about the content of my findings, not about "this particular model", but about the uniqueness of objective being. It’s about the idea of uniqueness. There is simply no more than one possibility. This means that even this particular, presented here model of the Universe is more credible with respect to the objectively existing nature than the traditional one. The Universe is simply not "this or that" (depending on the results of measurement). The theory not providing for a univocal development is only (and at most) a tentative model. I think that’s how we should treat Friedmann equation.

   Also, consideration of matters in the category of probability is here (I think) pointless. The Universe as such is, I emphasize again, determined. It cannot be described by some wave function. The Universe is simply a kind of an absolute and human perception does not have anything to do with. Epistemological superstructure should have no influence on the objective facts we learn and will learn in the future. And the anthropic principle? For me it’s a kind of oddity, which "emotional" premises I do not share. Excessive anthropocentrism? (Subconscious) recurrence to hierarchizing by Thomas Aquinas and geocentricism in the new edition? Relapses? Well, cyclicality does not bypass us either. This principle is most probably the result of the frustration of science at the crossroads. I encourage you to read Lee Smolin’s book***.

Purposefulness, a reversal of cause and effect. All of these arguments in support of the anthropic principle do not amount to any evidence. They lead astray. In 1600 Giordano Bruno was burned at the stake for having had the audacity to claim that there are other worlds where there is life, perhaps similar to our own. Since that time more than 400 years have passed, and mankind has not learned much despite technological and scientific achievements (in fact of only a handful of people). As a mass we are mentally far from these achievements. The evolution of life on Earth has been going on for more than 3 billion years. There was enough time. Unless someone thinks that the whole world began 5,776 years ago. In this case any questioning is unquestionably impossible. 

If the anthropic principle is a consequence of today's paradigms, it is a sign that we should discard some of them. When examining Nature and studying its objective laws one should be guided by simplicity and discard anthropocentrism.

     Another thing that research as such, observations, along with a variety of interpretations, the whole cognitive dynamics which has its main source in curiosity, is something extremely interesting and worthy of separate studies. It is for the pious. The search for truth of nature (and other objective truths), is also one of the essential elements of what should constitute the basis for humanistic thought, even if the search dissociates itself from the "anthropo-activism". How many declared humanists have that outright paranoid hatred (of course in relation to people)... This transpires in many Parlour (Salon in Polish) scraps and comments (Salon24.pl), not to mention the less intelligent portals. Among the real investigators of Nature (including human nature) hate is a rather rare phenomenon.

     Returning to equation (3) let’s note also that there is no mass in it, even though density is the invariant mass. In fact, this was to be expected given that the mass is an extensive value. Does this apply to the mass of the entire Universe? Here I would be careful.

This means unambiguity of being, you might say: absoluteness. In any case, the Creator knew what he was doing. Perhaps this way should lead the path to understanding Nature, even if the nature of the Creator is semantically undetermined (usually everything is limited to pagan idolatry, including kissing pictures and teaching of hatred in accordance with the priority of love of one’s neighbour).

What is the average density of the Universe? Anyone can calculate it for him/herself.

2. Critical density

Above we derived the density of the Universe based on the postulated equality of Hubble and gravitational radiuses of the Universe. Additionally I relied on the assumption of flatness of the geometry of the Universe, which I supported with perfectly acceptable rational premises. The flatness is also indicated by observation [even if it is (conceptually) surprising to some]. As for the flatness there is consensus omnium despite the fact that in my humble opinion it is an inherent, and not an "approximate" feature of the Universe.

Calculation of the critical density using simplified Weinberg’s method.

"Flatness", according to today's views, means that the Universe develops in accordance with the critical model, one of the three predicted by the Friedmann equation. It is therefore of critical density. [Almost, but we do not know which way. I omit here inflation, which eliminated this problem by smoothing everything to fit the order.] Let's calculate it using two different approaches. First one is actually a description of the method used by Steven Weinberg in his famous book “The First Three Minutes”, a description designed for lay persons. 

We choose at random a galaxy. Its mass is equal to m, and the radial velocity relative to us (in the cosmological sense) is equal to v. Its distance from us equals r. We constitute the origin of the coordinate system and, of course, the centre of the Universe. According to the cosmological principle each observer will tell you the same regardless of the galaxy it inhabits. Our chosen galaxy is located on the surface of a fictitious sphere of radius r, including certain number of galaxies, including ours (along with interstellar matter), which mass is M*. As a base we will use the Newton’s law of universal gravitation. It is known that the remaining Universe, apart for the chosen galaxy, bears no gravitational impact on the outcome of our deliberations. Just like a layer of any thickness above a person measuring his/her weight, and located at a certain depth below the surface. There, that person’s weight is determined solely by the mass of this part of the Globe, which lies below. In the centre the weight of each body is equal to zero. This can be proven by calculation. A more general description of this rule is expressed by Gauss' law, which is also true with respect to the electrostatic field. The potential energy of the galaxy (more precisely, the energy of its interaction with the rest located below) is equal to:

Its radial velocity, according to the Hubble’s law: v = Hr, so its kinetic energy:    

Thus, the total energy: 

After substituting in place of the mass:   

which can be done since we assume that space is flat, we get:

(We remember that, according to the cosmological principle, the local density, although of cosmological importance, is everywhere the same.)

The galaxy we have chosen can be on the very horizon, because we have not limited the distance at which it is located. Then the total mass "under" it:  M* → M is the mass of the whole Universe. The above equations remain, obviously, in force. Let us discuss the formula (4). We can immediately see that there are three possibilities. When E > 0, which means that the numerical value of the potential energy is less than the value of kinetic energy, and gravity is too weak to stop expansion – the opened model. When E<0, we have the reversed situation, gravity is strong enough to stop the expansion and cause in the aftermath of the collapse of the Universe. Of course, we are talking here about the closed model. When E = 0, the Universe develops according to the critical model. This is the case of our interest. From the formula (4), we obtain:   

Here: ρc – critical density of the Universe. As it comes out, following completely different path we have come to formula (3). We received the equation for the critical density identical to the formula for the density of the Universe based on the postulate of the equality of gravitational and Hubble radiuses. I think there is something in it, even if it's surprising. Here, however, it is about the critical density, and this is one of three possibilities, actually the improbable one, since it is the point border (see the previous chapter). However, in connection with a fairly strong indication of the flatness of the Universe, researchers’ attention has focused precisely on this option. The problem, however, lies in the fact that the observationally detected mass (actually the density parameter derived from the mass of the visible, and even the dark matter), is too small to ensure criticality.

In view of the above the search for additional mass (to achieve the critical density) is quite understandable. Or maybe the search for this additional mass is unnecessary? Is it certain that the density parameter is the proper indicator or that its measurement is correct? The question is justified not only for those who share my view that the GTR (say in the Friedmann version) is not the appropriate tool for cosmological determinations. And if we do not use the GTR and are not interested in the critical density, then the search for density parameter is beside the point. Heresy chases heresy. 

The observational determination of the average density is not an easy thing. Not all matter is visible, and it is not always is possible to determine it. An example of such efforts is the measurement of the amount of deuterium, which was created at the beginning of nucleosynthesis, in the early phase of BB, probably only then. The results, however, are far from expectations.

In the concept proposed in this work the problem of the value of density parameter does not exist, and "adding" mass equivalent to dark energy (apparently up to 70% of the mass of the Universe) is probably a big misunderstanding, it is simply a fiction, multiplication of entities beyond need. Perhaps today's concept of Ω measurement is at fault (and apart, also the treatment of this parameter as binding and mandatory). But this lack of mass had to be filled in some way. Hence the uncritical enthusiasm for the idea of dark energy. All en-masse herded this way and many were amply rewarded with PhDs and professorships. Even Mr. Nobel was seduced. [Wait, repulsion (dark energy) points to the negative mass, therefore these 70% should have been subtract (not added). We would get not 100% but 40% ... And here, as if out of spite, the space is flat.]

   My insolent approach is consistent with the previously expressed opinion that the Universe available to observation comprises its totality. Obviously this is not consistent with the currently applicable rules and presumptions (not to say prejudices). So as to appease the more incensed readers (fundamentalism is now in vogue) I admit that for the time being this opinion is not the final judgement banishing today's views. If someone was to be banished, then (as it can be seen in each case) it would have to be the writer of this work, despite the fact that he is basing it on rather rational assumptions, in any case no less rational than those which serve as basis for opposed views. It so happens that he was already exiled many years ago and for the completely different reasons****. But never mind.

     Coming back to the interrupted thought we should add that if in spite of everything, there is something beyond the horizon (as it is thought by the majority of those interested), the consideration of this would be something purely speculative, not much adding to the final vision due to its unprovability.

Here is the numerical value of the critical density corresponding to our assumed value of constant H = 20: 

It is of course today’s value. Let’s compare this value with the density determined on the basis of estimated in the previous article mass of the Universe and its corresponding Schwarzschild radius (here expressed in light years). Here is the calculation of density:                  

The results of these calculations are very close to each other. This probably testifies in favor of the concept presented in this article. [Anyone who read it all carefully knows that it is all clear of any "sly adjustments". I did not have to do that. And even if, then we still have a unique convergence given the same numbers, which we had at our disposal - huge. The probability of coincidence is virtually nil. Incidentally, the "sly adjustment" is used widely nowadays. Take the hypothesis of inflation. In that case "slyness" breaks all records.]

Calculation of critical density based on Friedman equation. 

Let us make the next step. Here's Friedman equation:              

where ȧ – the scale factor of the Universe (dot at the top means its derivative with respect to time), k – a constant value in time and in space, describes the geometry of the Universe, type of its curvature. 

k > 0 means spherical curvature of the closed Universe, k < 0 – hyperbolic curvature of the open Universe, and k = 0 – flat space in which the Universe evolves according to the critical model. Here we should add that the value c^2 (square of the speed of light) in general, especially in the professional writings, is ignored due to the assumption that it is equal to unity. This is justified not only by practical consideration (simplified calculation). But let’s not stray from the subject.

The scale factor (a) is a function of time and is directly related to the pace of expansion. If within certain time the factor, for example, triples, it means that the dimensions of the Universe have also tripled. This expansion, however, is not an "explosion of a grenade". It is the expansion of space which contains matter (according to today's understanding). This results in continued increase of distances between galaxies (in the cosmological scale), although it is not about their relative movement in the Newtonian sense. Can we therefore say that the movement is cosmological scale is not a kinematic quality in the Newtonian sense? This would result in a very convenient conclusion that relative "speed" of objects may exceed, even significantly, the speed of light in a vacuum. It would suffice that they were adequately distant from each other, far enough that it would not be possible to have coordination between them in less than the age of the Universe. This apparently happened in time of the so-called inflation, and the existence of these appropriately remote objects is its consequence.

So the whole expansion is a "personal" matter of  space-time, and galaxies remain, in fact, at rest relative to each other (not taking into account their insignificant (?) local movements), despite the resulting from it expansion and mutual distancing. [Would that mean that the actual movement is purely local, and cosmology is something else?] What is this constant distance between them (if not taking into account the Friedmann expansion)? – one could ask. Very interesting question, particularly in view of the adopted by some, even a priori, thesis that Everything started from (let say: almost) a point singularity. So are we talking here about the arbitrariness and activism of space against inertia and passivity of matter? Why not? What about De Sitter’s model? Yes, but the magnitude of curvature depends directly on the total mass. Also, on the mass of the smallest bodies, even the elementary particles... Well, yes, it is, after all, about the mutually moving coordinates.           

According to the concept presented in this work, which is, among other things, to serve as a checking test (by its alternativeness) maybe even for the sake of today's beliefs, it is after all about the actual movement, though in a closed (not Newtonian-infinite) space. Closed by the fact that it is contained within specific topological formation, which characteristics are indicated by  the features of the evolution of the Universe, suggested in the text and in different contexts. This formation makes the Universe a periodically variable creation.

Let us return to Friedman equation. Let’s consider the first part of its right side, and actually its dimension: 1/s^2 (the square of the inverse unit of time). The same dimension applies of course to the left side, which features the scale factor a. The dimension of the left side indicates that the scale factor has the dimension of length. So we have here the square of ratio of speed to distance. The meaning of distance is here, however, somewhat different than usually, because it is the magnitude associated with the expansion of space. There can be demonstrated the consistency of the thesis that the amount constituting the left side of the equation is equal to the square of the Hubble factor. [Let’s note that it is enough to change the letters in Hubble’s law, and represented the speed as a derivative of distance (r) with respect to time.] Therefore it seems to be perfectly valid to observe that constant H itself defines the rate of expansion. [The term "the rate of expansion" was introduced (actually derived) already in the second article.] Its decrease with time would mean the gradual reduction of this rate, which would correspond to slowing the movement of a body thrown upwards (and to diminishing curvature of space). My approach is different. For the record, the consistent application of the cosmological principle leads to the conclusion that global cosmological gravitational field intensity is equal to zero. So there is no question of delay (or acceleration). Thus, the speed at which the Universe expands is determined by the "expansion rate", that is by the upper bound of relative speeds - c, which, according to cosmological principle, is invariant. But let’s go back to our narrative. So we get:  

I noticed it already in the second article. Thus we can formulate the Friedman equation in a slightly modified form:

Using it we can calculate critical density. In this case the curvature (k) is equal to zero. Therefore:

Again, we get a familiar equation. As you can see, the postulated equality of gravitational and Hubble radiuses leads to a formula identical with that derived on the basis of the general theory of relativity. Thus this new path which I have proposed is consistent with the method based on this theory. This is well worth noting. Therefore the postulate of equality of gravitational and Hubble radiuses has some justification, even if it is a surprise. I have already drawn attention to it earlier. Let’s add to this, that this postulate indicates explicitly (and not "one way or the other") the character of expansion, it defines its course as consistent with the view that the space of the Universe is indeed flat, in addition, regardless of time. If so, we made some progress in understanding at least this cosmological issue.

*) Flat geometry is the Euclidean geometry.

**) Information on this subject can be found, among other things, in the book by  

       Alan H. Guth – The inflationary universe.

***) „Kłopoty z fizyką – The trouble with physics” (Prószyński i Ska 2008)

****) Martial law and emigration.