Olivier, as you recall, we discussed this question previously elsewhere. Let me offer a hopefully more coherent summary of my previous comments for your consideration.

When we write down the field equations in a spatially homogeneous expanding (FLRW) universe, the terms that contribute to the curvature of spacetime can all be characterized by an equation of state w = p/rho, where p is the (effective) pressure, rho is the (energy) density associated with the contributing term.

If we treat the cosmological constant this way, we find that its effective equation of state is w = -1.

A classical potential associated with a field also has the same equation of state, w = -1.

The zero-point (vacuum) energy density of quantum fields would also be represented by w = -1.

There may be other, exotic forms of matter or fields with an equation of state w ~= -1.

Observational evidence (which includes the Type Ia supernova luminosity-distance relationship as well as the shape and position of peaks in the CMB angular spectrum) is consistent with the estimate that approximately 70% of curvature comes from a w ~= -1 term in the field equations. These observations cannot distinguish between the contributions of a fundamental constant, a classical potential, or vacuum energy.

The rephrasing that you propose, then, if I understand your intent correctly, amounts to treating the 30% (with equation of state w ~= 0, characterizing non-interacting "dust" and/or non-relativistic gas) as "stuff"; whereas the remaining 70% (with equation of state w ~= -1) as an intrinsic property of empty spacetime (vacuum, i.e., no "stuff").

Is this really a good idea? What if it turns out that this 70% (or most of it) is also "stuff" (that can be added, removed, or otherwise manipulated), and that whatever is left that can be called "vacuum" has less or no intrinsic curvature?

In other words, doesn't your rephrasing amount to deciding ahead of time that the 70% cannot be "stuff", even though we have no observational evidence that favors this view?

Lastly, I think it should also be emphasized for clarity that the curvature we discuss is spacetime curvature, distinct from the spatial curvature, usually represented with the symbol k in the Friedmann equations, which has an effective equation of state w = -1/3. (And as far as we know from observation, k = 0.)

Dear Viktor, thank you for your comment.

"The rephrasing that you propose, then, if I understand your intent correctly, amounts to treating the 30% (with equation of state w ~= 0, characterizing non-interacting "dust" and/or non-relativistic gas) as "stuff"; whereas the remaining 70% (with equation of state w ~= -1) as an intrinsic property of empty spacetime (vacuum, i.e., no "stuff")."

No, not necessarily. The rephrasing that I propose just gives the facts without deciding ahead what would be the explanation (at least as long as we are in the paradigm of general relativity): the acceleration of the expansion of the universe, as well as many other observations, suggest that the source of 70% of the space-time curvature at present epoch is unknown. Whether this curvature is due to a fundamental constant (that would not be "stuff"), to QFT zero energy fields (that would be "stuff" but from (mostly (or not)) known fields) or to some unknown fields that would spread everywhere (unknown "stuff"), remains unknown. No more than that.

In other words, I suggest that we explicitly say that the source of 70% of space-time's curvature is unknown instead of saying that 70% of the universe is made of dark energy.

From there, one can explain to lay people that a constant curvature of space-time (which may have several kind of source) can induce an accelerated expansion of the universe.

Saying that 70% of the universe is made of dark energy is not only potentially wrong, but also I believe really confusing for non-specialists.

Olivier, isn't "dark energy" just a synonym for "unknown with equation of state w = -1" just as "dark matter" (which, incidentally, is just as unknown) is a synonym for "unknown with equation of state w = 0"? The only other difference I see is that whereas the distribution of the w = -1 "stuff/no stuff" is homogeneous, the distribution of the w = 0 "stuff/no stuff" (darn, I feel like I am back in 1983 or so, playing the Infocom game The Hitch Hiker's Guide to the Galaxy on a TRS-80, trying to get both "tea and no tea" to prove my intelligence to an automatic door) is definitely clumpy. Is this why, perhaps, one may justifiably believe that "dark matter" is in fact "stuff" (as only "stuff" can be clumpy) and thus using the very similar term, "dark energy", is misleading as it implies that it, too, is "stuff", which is not necessarily true?

Olivier, isn't "dark energy" just a synonym for "unknown with equation of state w = -1"

Yes it is, as long as you know the problem well enough as you do. But not for lay people. Moreover it has the word "energy" in it, while the source of space-time's curvature might not have anything to do with a field.Therefore I suggest to rephrase it in a more transparent (and correct) way. I recall that I introduced the discussion by saying "In my opinion, the usual way to talk about the so-called dark energy problem can be misleading for non-specialists of the subject."

My suggestion is not intended to specialists. All theoretical physicists are well aware of the simple things that I say (I do not invent them, nor are they really complicated). For specialists, "dark energy" is indeed just a synonym for "source of constant curvature term -- in the late universe at least -- for space-time". But look at the popularization about dark energy (eg. forums, blogs etc.) and you will see that it is another story for non specialists (eg. journalists, non-cosmologists etc.).

My useless comment to this discussion is that I would support any proposal supporting the removal of the word "dark" in any expression like "dark energy/matter/whatever" in the current cosmological/astrophysical discussion.

Of course, they reach their "marketing" goal (lay people wondering what this "daaark" stuff is and headlines with "scientists look for daaaark whatever") but they introduce a lot of misundertandings between what people thing "dark energy/matter" are and what specilists think they are.

For example, when explaining to lay people what "dark matter" is, the first thing that I have to say it is that is NOT dark, it is ***transparent***. What it is mostly dark is the universe (Olbers paradox) , which is what we see ***through*** the dark matter.

Anyway, thanks Olivier for your proposal and good luck with it :-)

In my opinion, I would reconsider the demonstration made by Friedmann (and which uses 'an equation of state') and a 'whatever you call it' with negative pressure. As a matter of fact, the universe is 3d and curved, thus it may be virtually 4d (embedded in a 4d space). I wonder how an equation of state (made for a 3d space) might apply to a 4d space: a pressure is a force divided by a surface (2d)...

Article And if there was no need of dark energy to explain the accel...

Dear Nathalie: the concept of pressure is unambiguously defined in four-dimensional relativistic physics in the form of the stress-energy-momentum tensor. Components of this tensor correspond to three-dimensional (non-covariant) quantities such as energy density, stress (pressure) and shear. These quantities depend on the choice of coordinates of course. However, for an isotropic fluid in thermodynamic equilibrium, a comoving coordinate system can be uniquely defined, and in these coordinates, the stress-energy-momentum tensor becomes diagonal, with the timelike component representing energy density, and the three spacelike components, all being equal, representing pressure. And it is entirely legitimate (and generally covariant, i.e., fully relativistic and "4d") to use the dimensionless ratio of pressure (the spacelike components of the diagonalized, spatially isotropic stress-energy tensor) and energy density (the timelike component of the same) to characterize a constituent medium.

The Friedmann-Lemaitre-Robertson-Walker (FLRW) universe is not "3d and curved [...] embedded in a 4d space". In fact, we have observational evidence to suggest that our universe is spatially (that is, 3d) flat. An FLRW universe is 4d and curved, period, with the curvature being the intrinsic curvature (something that can be measured without referencing any embedding in a higher-dimensional manifold.)

Victor, I did not know that our universe is flat!

Nathalie, yes, at least in the Lambda-CDM concordance model (https://en.wikipedia.org/wiki/Shape_of_the_universe). To be clear, we are talking about spatial flatness (that is, 3d space is Euclidean; 4d spacetime, however, is not Minkowski spacetime, so there is still curvature, just not spatial curvature.)

Viktor, in your link they say that the universe is flat with a 0.4% error margin. This error margin is huge if you multiply it by cosmological lengths...

Nathalie, I don't think it is appropriate to consider the error margin that way. Every dimensionless error margin becomes huge if you multiply it into something large and dimensioned. This is why an error is best expressed instead in a dimensionless form as a relative error. What is important is that when you add the best observational fit for Omega_m (the dimensionless ratio of the matter content of the universe relative to the critical mass density required for the universe to be flat) and Omega_Lambda (as above for the dark energy/cosmological constant component) the result is 1 with a relative error of only 0.4%. Moreover, the recent history of the subject is that as better measurements became available, the outcome did not deviate from this result. Now of course this does not exclude alternative interpretations of the data, or the possibility that the observed small error may one day prove to be significant, which is why I phrased my comments cautiously, writing, e.g., "we have observational evidence to suggest that our universe is spatially flat".

Viktor, I hope you are cautious with saying the universe is flat with a relative error of 0.4%, because you could also say that the earth is flat with a relative error of 0.5%, depending on the distance you measure...(it increases with the distance)

Nathalie, there are two issues with your argument. First, the implied assumption that the flatness observation is based on some simple geometric measurement, like trying to measure the Earth's curvature with a yardstick. In principle, we could place three buoys on a quiet lake, measure their distances, verify the law of Pythagoras and conclude, for instance, that the Earth is flat with a 0.4% margin of error. Similarly, we could make a measurement in deep space using spacecraft and, say, laser beams, measure a triangle, and draw a similar conclusion.

But that is not what we are doing. What we are observing is the dynamics of the evolving physical universe, including not just its present-day geometry but its past evolution, as evidenced by structure formation, the microwave background, etc.

There is, however, another issue. If the relative contribution of spatial curvature to the total energy density today is less than 0.4%, it had to be much less than that in the past. The reason for this is simple. Given an equation of state expressed as the dimensionless ratio of pressure to density (customarily denoted by w) we can trivially calculate how the energy density changes as a function of the dimensionless scale factor of the universe, a. For pressureless matter, the energy density will be proportional to 1/a^3, as we would expect. For radiation, it is 1/a^4, which is why the early universe (when a was much smaller) was dominated by radiation. More generally, the energy density will be proportional to 1/a^(3+3w). When w = -1 (dark energy/cosmological constant) the energy density will remain constant. For spatial curvature, the effective equation of state is w = -1/3, so its contribution to the energy density will be proportional to 1/a^2.

To make a long story short, if at present, the contribution of spatial curvature to the total energy density is only 0.4%, it was much less than that in the early universe. And if you add inflation to the picture, in the pre-inflation universe the contribution of spatial curvature had to be of order 1e-60 or something. In other words, the statement "the universe is not spatially flat" is equivalent, in the inflationary scenario, to stating that "in the early universe, the sum of the contributions of matter, dark energy/cosmological constant and radiation was, Omega_m + Omega_Lambda + Omega_gamma = 0.999999999999999999999999999999999999999999999999999999999999..." ... but not 1.

That said, of course we can't exclude the possibility. The inflationary scenario may be wrong, for starters. And even if it was right, perhaps something else yet to be discovered is at work. Which, again, is why I used very careful language when I said, "we have observational evidence to suggest that our universe is spatially flat". It is not a certainty, only an assertion that best fits the data we currently have.

Great question!

Yes, I think also that dark energy can be seen as a constant negative curvature of the spacetime (so therefore one has a hyperbolic 4-manifold). Then one can find a foliation (or a splitting into space and time) so that the spatial curvature can be positive. But then the negative curvature goes along the time direction (with an exponential divergence of two neighbored geodesics, i.e. accelarated expansion).

But more importantly there is a very interesting mathematical property of hyperbolic n-manifolds (with n>2) called Mostow rigidity. According to this property, any diffeomorphism and any conformal transformation induces an isometry, i.e. geometrical properties like volume or curvature are toplogical invariants. So this property implies the constancy of the curvature. But this curvature (or better the scalar curvature) is the cosmological constant (according to Einsteins equation).

But then (because of Mostow rigidity) we obtain the state equation with w=-1 but not w=-1/3.

Dear Torsten, I have some questions: Isn't Mostow rigidity a property of finite-volume hyperbolic n-manifolds (n>2)? Furthermore, wouldn't this argument apply only to a universe that contains only dark energy (or is dominated by dark energy at all times), as otherwise, the manifold is not necessarily hyperbolic? (Our universe may be hyperbolic today, but that was not necessarily the case when the expansion was still decelerating, so there may exist neighborhoods that are not hyperbolic.) Also, I don't follow how any of this changes the effective equation of state, w = -1/3, for the spatial curvature term (not to be confused with the negative spacetime curvature induced by the w = -1 dark energy/cosmological constant term) in the Friedmann equations?

Dear Victor,

yes you are right, Mostow rigidity is only true for finite-volume hyperbolic n-manifolds for n>2 (I see you already know it). But I use it only to illustrate the fact that dark energy can be also explained geometrically.

In my work, I'm interested in exotic smoothness. The exotic 4-manifolds like an exotic S^3xR contains finite-volume hyperbolic submanifold but not for all "times". So matter and/or radiation dominated phases are also possible.

Now to the state equation, w=-1/3 is only true if the curvature scales as 1/a^2 but in case of the hyperbolic phase at least the hyperbolic submanifold do not scales so that the curvature is constant a^0.

Did you agree now? Best wishes, Torsten

Dear Torsten: I of course agree that the spacetime curvature (induced, e.g., by a cosmological constant) scales as a^0 and has equation of state w = -1. But this is distinct from the spatial curvature term in the Friedmann equations that scales as 1/a^2 with w = -1/3. Unless the universe is spatially flat, both terms are present in the Friedmann equations (and the effective equation of state will depend on which of the two dominates), don't you agree?

Dear Victor,

Thanks for the answer, now I understand. And I agree with you. The whole spatial curvature term scales like 1/a^2 and has state equation with w=-1/3. I also think that the universe is approximately flat (but has a very small positive curvature).

Furthermore, it is also possible that some part of the spatial curvature can be hyperbolic (you know of course the prime decomposition of 3-manifolds). I used this hyperbolic 3-manifold to describe the reason for the inflation.

Perhaps one way to rethink the dark energy problem is to consider possible solutions. I may have some to offer. They include the following. Dark energy is 'stuff,' possibly including peers of Standard-Model stuff and possibly including stuff based on spin-7/2 quark-like particles. The expansion of the universe and rates of changes thereof correlate with effects of a family of zero-mass bosons that includes photons, gravitons, and spin-3 and spin-4 particles. These 'solutions' come from a harmonic-oscillator based approach that may point to solutions to about 10 well-known physics problems, as well as reproduce the list of Standard Model particles.

For further details, please feel free to see the attachment. I welcome further discussion.

Dear Thomas, proposing a new specific source to the overall curvature of space-time is not the scope of this thread. Please read carefully the question that is asked before answering. Thanks.

Oliver: OK. Apologies. If you would like, please remove my (now) 2 comments. Or, if you would me to remove them, do you know how I can do so?

Dear All,

I think that the astrophysicists, as scientists of no other specialty, are responsible vis-a-vis the population for its naturalistic world outlook and by the believe that, in sciences, truth is always better than notorious invention.

The present astrophysics, including all its conclusions on the allegedly actual occurrence of such celestial objects as the dark matter and dark energy, black holes and different colored dwarfs, etc., on their sizes, ages, and distances between them, on the Universe age and size, and on the nature and transformations of the celestial objects and the Universe as a whole, is based on four assumptions which, being taken together, represent the basis for all subsequent calculations and conclusions. These assumptions are made about 90 years ago and, thus, are bases on the level of knowledge of those times and were intended for the explanations of the observations then available. The assumptions are as follows.

(i) Stars exist and transform as a result of three-dimensional fusion reactions in their interiors, and just these reactions are the source of the stellar luminosity, heat emission, and other forms of radiation (Eddington);

(ii) The parameters of the stellar state can be calculated on the basis of the ideal-gas laws (Eddington);

(iii) The gravitation coefficient value inherent in the Solar System is applicable to the entire Universe (Einstein, Special Relativity and General Relativity);

(iv) The speed of light is constant and identical everywhere over the Universe (Einstein, SR and GR).

None of these assumptions was proved.

The acceptance of the assumptions (i) and (ii) can be explained only by the absence in the 1920s of any alternative explanation for the solar luminosity and energy and neutrino emission. Indeed, it is unthinkable that a fusion reaction could proceed stationary over giant volumes of a pressed substance with liberation of so small specific energy amount as it is observable for the Sun and that the behavior of a hot highly-concentrated plasma might be assimilated to the ideal gas. Of course, “explanations” for these imaginary phenomena were invented, but they look like a square peg in a round hole. Just these assumptions were put in the basis of this so-called standard solar model which, in turn, represented first floor for the building of fantasies, including the structure and history of the stars, origination of chemical elements, etc.

Today, we have an alternative explanation for the solar luminosity and energy and neutrino emission, this explanation in its common form being apparently applicable to other stars independently of their age and sizes. I keep in mind the PFO-CFO Hypothesis of Solar System Formation that is available in ResearchGate. Isn’t it time to look critically at the fundamental assumptions of almost 100-year age and at the entire building constructed over them?

As for the assumption (iii), it is, as least, questionable, because it is quite unconditioned and nothing counts in its favor, although there are no proved objections against it.

The assumption (iv) is, in our opinion, completely unacceptable as a ground for any earnest conclusions on the Universe as a whole. However, this statement requires a rather detailed substantiation.

We see three factors that force us to doubt in the constancy of the light speed and direction (We think that it is clear that the variations in the direction are equipollent to variations in the speed) over the Universe. Moreover, we think that the speed of light should be dependent on the distance and, there is very likelihood, on the time by the following reasons:

(x) The gravitation coefficient might be different in different stellar systems or, all the more, in different galaxies. It is observationally shown that light deviates in gravitation field (just such an observation was taken as a confirmation of Einstein’s earlier theory). If the gravitation coefficient is different over the Universe and changes in an unknown manner, the light speed out of the Solar System is unpredictable.

(y) Even if the gravitation coefficient is the same over the Universe (We have grounds to think that this is improbable; see below), it is possible that light propagates through the inter-galaxy space, in preference, along the between-galaxy channels and through the intra-galaxy space, in preference, along the between-star channels , where gravitation to the adjacent objects is compensated; thus, it can propagate in a straight line along great but limited distances and, when propagating through multi-galaxy distances, choose the directions of minimum resistance; in this case, its direction is also unpredictable.

(z) In the 20th century, after development of the quantum field theory (1934), an opinion existed that gravitation is a property of each atom as such and that this property is caused by the existence of gravitons, i.e., massless spin-2 boson particles that represent carriers of this property. However, no gravitons were discovered within nuclei, in spite of their 80-year searches over the world. Eighty years is a rather long period; today, it is necessary to be rather obstinate optimists to wait the discovery of such particles and there are no scientific ground to believe that the gravitation constant is the same over the Universe; the recent LHC experiment is the source of optimism for some physicists, however no consensus exists even among its participants. It must not be ruled out (although we do not undertake to state) that gravitation is an integral feature of the space/celestial-body system but not the feature of each atom and that this feature depends on the size and, may be, age of the star that forms the stellar system.

We wrote earlier in ResearchGate about a wide criticism relative to the present calculations of the masses and distances for the celestial bodies. Eddington’s assumptions are arbitrary, like the assumption on the over-Universe constancy of the gravitation coefficient and on the light speed constancy over the Universe, and the miracles, to which this set of assumptions has led the astrophysics, show that these assumptions should be turned down. Therefore, we state that not only the Universe expansion is questionable. The opinions on the natures of the so-called standard solar model, all dark things, black holes, variegated dwarfs, etc. should be reconsidered. At the ResearchGate site, a number of researchers had tried to make objections against this conclusion but nobody of them presented scientific proofs counting in favor of these products of hypnotizing imagination. We are sure that the half-hearted hypotheses are scarcely acceptable. If a pseudo-science is build on sandy dunes, it is necessary to be ready that the sand may begin to move in any day. The today astrophysics includes the entire set of the four assumptions listed in the beginning of this letter; rejection of any one of them would lead to its full dismounting. Meanwhile, three first of them are not proved and the fourth one is practically disproved by the occurrence of the so-called Einstein’s rings and other figures obtained as a result of not-direct light incidence on an object. It is difficult to ignore that light is capable of changing at any boundary, where the space density varies.

The occurrence of the so-called Einstein’s rings in their ideal or deformed variants shows that the time of light propagation from any object to the Earth can’t be used to determine the object-Earth distance even if the gravitation coefficient is constant. (Really, there is quite likelihood that it is non-constant.) Einstein’s mistake who took in the SR and GR that the light speed is constant and identical everywhere over the Universe is explainable: he can’t know that, 50 years after his death, Hubble will discover on the vault of heaven a great number of rings and similar figures and so many “glowworms” that, even if 99.99% of them represent mirages, the real quantity of the stars is immeasurably greater than it was known in his time. May be, in the Universe, which was known to Einstein, it was possible to take approximately that light propagates in straight lines, but in the Universe, which is known to us after studies by Hubble, light propagates by more complicated routes.

Apparently, only one explanation of these phenomena occurs: if an illuminating object, the center of a half-way galaxy, and an observer are along the light stream (the conventional wisdom that these three objects should be along a straight line is not quite correct, because light deviates in gravitation fields) and if no other galaxies exist along this line, the observer sees a dark disk and a light ring around it (very random situation); if the light direction from the illuminating object is somewhat shifted from the center of the half-way galaxy, the observer sees any other more complicated figure instead of the ring (not so random situation). Thus, Einstein’s figures are observable as the result of rounding the half-way galaxies by light. However, it is clear that not the entire outer radiation rounds the half-way galaxy and the excluded light streams not necessarily interlock after it. Depending on the light pitch angle, on the galaxy size, and on other causes, some portions of the primary light stream pass through the galaxy and are being absorbed, reflected, and scattered by it. Apparently, just the scattered and reflected light is the cause of the observable space background radiation.

This means that a major portion of light doesn’t go through galaxies but chooses the way around them, where there is no gravity, and may go down and bifurcate. The way of each portion of a primary light stream from a radiating celestial object is much longer than the straight distance. The light from any remote star may round many galaxies. Therewith, it may separate, may walk over the Universe, and may come to the Earth even from a direction that is opposite to the real direction to the real remote star. Moreover, a portion of the light should be reflected and dissipated over the space.

The following concrete questions are waiting unambiguous answers to them.

(1) Is there any scientific proof that stars exist as a result of fusion reactions within them, or is this assumption taken only because there is no other idea to explain the stellar emission of light, heat, and neutrinos and the mechanism of formation of chemical elements?

(2) Is there any scientific proof that stars can be assimilated to balls of ideal gas to calculate their state independently of fulfillment the previous assumption, and is there the confidence in correctness of such calculations if the previous assumption is fulfilled and stars represent highly-pressed concentrate plasma?

(3) The independence of the gravitation coefficient over the Universe and time, can it really be obviously true and are there any scientific proofs that the gravitation coefficient is constant over the Universe?

(4) Are there any scientific proofs that the speed of light is constant over the Universe?

We contend that all these assumptions are artificial, wrong, and smell of the lamp, and their application had led to nothing but secondary fantasies, such as dark matter and energy, black holes, expansion of the Universe etc.; the masses, speeds, and distances of the celestial bodies are obtained just on the basis of these four fantastic assumptions supplemented with different other no more grounded fabrications. And I think that any attempts to save the widely distributed views on the Universe are today in the Internet epoch senseless.

The available notion of the way of formation of the chemical elements is too complicated and requires several meets of several objects and rather questionable processes, such as many-parsec collapses, their initiations, supernova, etc. Too many of unknown and questionable things are used to explain the incomprehensible things. Meanwhile, Nature is simple and doesn’t luxuriate in excesses. Such a complicated way of formation of the chemical elements is of low probability for one realization and is scarcely possible for billions realizations in any one galaxy. The explanations for such phenomena as the solar corona high temperature, high planetary moment, periodicity of the protuberances and of the variability of the magnetic moments, isotopic anomalies, cold Bok globules, and many others are obscure and can be easily subjected to criticism.

The following group of our questions to the members of the Community relates to the understanding of the Origin. In our opinion, when discussing Origin, scientists should keep in mind that Origin, according to the content of the term, should be absolute, i.e., the questions about the events and phenomena, which had been occurring before Origin, should not arise at all and that there are no grounds to ignore the physical laws of conservation and the laws of thermodynamics. Meanwhile, the notion of the so-called Big Bang fulfils none these conditions. In the philosophical plane, it cannot be considered as the Origin. The notion of it has no physical ground because the expansion of the Universe is no more than a myth, put new unreciprocated questions about the pre-explosion events, previous history of the "neutron egg" and “energetic field”, time and events before it, etc., and assumes its appearance outside physical laws. This notion solves no problems but initiates a number of new problems and has no scientific grounds. In this connection, I put the following question:

(5) Is there any definite opinion about the appearance of those materials that were the subject of the so-called Big Bang?

Meanwhile, apparently, there is a unique solution that leads to no additional problems. Only one solution is irreproachable from the scientific point of view, can be proved from the contrary and leaves no additional questions; it is given by Friedrich Engels (“Natural dialectics”, Papers, any edition: the last page of the Introduction) in his discussion with Eugen Dührings: “universe is eternal and infinite”. Not all agree to this, but, unfortunately, there is no absolutely closed question over the world. We put you one more question:

(6) Are there any reasonable objections against this Engels’s solution?

I use, may be, too undisguised expressions. I ask all readers of this issue to excuse me for the intransigence. The fact is that the many-decade scientific activity has taught me that science has no intermediate solutions and that efforts of accommodation of conflicting interests are non-productive.

The notions of fusion reactions within stars, gravitation coefficient and light speed constancy over the Universe, and stars as balls of ideal gas are the viruses of the terrible disease. No one of these assumptions is confirmed with independent measurements, and they led to ideas fix on the dark matter, dark energy, black holes, Universe extension, and Big Bang as the consequence of this entire devilry and just the shock therapy is the unique medicine for this disease. In any other science, any one of similar results would be sufficient to set aside the system of the initial assumptions and to start the construction of the scientific building on a new fundament. For 90 years, there was no new fundament. Today, such a fundament occurs. The rejection of the notion of fusion reactions is necessary because it was possible only at the dawn of discovery of such reactions to believe that they can be localized by Nature within giant volumes with no explosion, because the notion of fusion reactions gives no possibility of going to a realistic notion of the mechanism of formation of chemical elements, and because of other causes about which we wrote above and in our published works.

Until no new hypothesis was available, the criticism against the old one was non-productive because it could create a vacuum zone and because each vacuum zone in the system of knowledges is dangerous as different superstitions rush there.

The times have changed after formulation of the PFO-CFO Hypothesis by me together with Elena Kadyshevich, and its ignoring under Internet is senseless. Its first version was published in the Advances in Plasma Astrophysics, 2011; and, at present, we propose for discussion the works of 2013. The PFO-CFO Hypothesis is presented at the ResearchGate in its development including its finish state of 2013; it includes our version of the mechanism of formation of stars.

The main principles of the hypothesis are as follows.

The Universe is eternal and infinite, and the space consists of energy/mass of a low concentration and has infinite rotational moment. The stars are the knots in the energy/mass space. No fusion reactions proceed within stars. Just radiochemical decays over the stellar vicinities represent the unique alternative to fusion reactions. The energy/mass transformation into matter starts within stars and finishes in their surroundings. In each stellar system, all chemical elements originate over the star vicinity from radioactive and non-radioactive pico-drops of the stellar substance and just the radiochemical decays are the source of the stellar luminosity and heat-emission, the neutrinos resulting from the beta-decays of the pico-drops. The planetary systems around stars are formed on the basis of the stellar substance emitted by protuberances and as results of explosive destructions of the stellar radiation zones, after which stars return to their initial state. The hypothetical mechanisms of all processes are described in the available publications.

The point is that the conclusions on astrophysical problems are of great public importance; they propagate over the world by the mass media, writers, etc., get into the scholar books, and are taken by the population as the reality. Therefore, I think that, at least, the degree of validity of each statement should be strictly determined and the assumptions underlying these statements should be clearly outlined. It is an open secret that the results in the field of astrophysics are of ideological importance. The responsibility of each researcher who works in the field of ideology is too high vis-a-vis the population to be ignored.

I certainly commend the question's suggestion - 'dark' proxy elements have a way of shaping, at least the public's, consciousness. I have to question though how the problem can now be effectively rephrased in the public's consciousness?

I've always been concerned that the initial discoverers shaped expectations regarding the interpretation of their observations by evoking the preexisting cosmological constant (and diminishing the deceleration parameter in the then-standard cosmological models) to fit their model's results to their observations. As a casual observer, this appears to be a convenient 'tweaking' the model to produce desired results without sufficient theoretical justification. Almost by default, the cosmological constant has become the favored explanation for a reacceleration of universal expansion...

All that aside, I can't assess and do not wish to distract the discussion, but can't the universal curvature of spacetime be the product of temporally developing regional variations in its curvature? Specifically, I think it was proposed early on that we might be viewing the universe from within a void, thereby producing a 'favored' perspective - but that this was evaluated and I think dismissed.

However, if I understand correctly, it's thought that gravitation has condensed matter at large scales, producing filamentary structures stretching between large scale voids that have increased in size as the universe has expanded. If this is correct, couldn't the temporally increasing size of voided regions increase the curvature of universal spacetime?

Moreover, if spacetime is contracted by the presence of mass-energy and generally expands in its absence, might not the regionally increasing absence of mass-energy result in a temporally increasing rate of regional and universal expansion?

In this case, we would not be occupying a special location in the universe in relation to voided regions, just a location in the increasingly expanded more recent, observable universe...

James,

"I've always been concerned that the initial discoverers shaped expectations regarding the interpretation of their observations by evoking the preexisting cosmological constant (and diminishing the deceleration parameter in the then-standard cosmological models) to fit their model's results to their observations."

as a matter of fact, the cosmological constant is part of the most general equation that satisfies general relativity principles. It is not something added, it is simply there. Therefore, if one doesn't want to consider the cosmological constant (ie. set it to zero), he has to give a good reason for that. Within the paradigm of general relativity, as far as I know, no such a reason has been found. Now, since it turns out that the acceleration of the expansion of the universe is well explained by a cosmological constant and since we measure no deviation from the behavior of a simple cosmological constant, it says that for that matter, the paradigm of general relativity is still well suited to describe the evolution of the universe. Does that mean general relativity is the final answer? No, certainly not. But it means that its answer is good enough for now for this particular problem (ie. it fits observations well).

"Moreover, if spacetime is contracted by the presence of mass-energy and generally expands in its absence, might not the regionally increasing absence of mass-energy result in a temporally increasing rate of regional and universal expansion?"

Do the math, it simply doesn't work this way (at least within the paradigm of general relativity).

I mean, if you find a non-homogeneous solution of general relativity without cosmological constant that leads to an acceleration of the expansion of the Universe that mimics the behavior induced by a cosmological constant (that fits observations well), that would be a huge thing. But in any case, you have to work the math out. You simply cannot understand physics without working the math out.

Dear Olivier, Dear All,

I think, no rephrasing the dark energy problem will help.

We know nothing about the masses, rates, and distances of the celestial objects located out of the Solar System.

I wrote here about that earlier and an additional proof of this opinion is below.

Allow me to pass my opinion about the so-called Universe that Hubble sees on vault of heaven.

Let, you are inside a room with no light sources but several rather large spheres whose surfaces consist of a multitude of small luminous spheres with the gaps between them. Inside each large sphere, a multitude of small luminous spheres of different sizes are located. In the vicinities of a number of luminous spheres, non-luminous spheres are located. Let, each large sphere and each small sphere is fixed or rotates so slowly that you cannot notice its movement.

This is the model of the Universe, and the room with its walls, floor, and ceiling is the model of the vault of heaven. It is known that light has features of reflection, absorption, diffraction, interference, scattering, and polarization at the boundaries of the media with different densities and other properties and that the light waves are capable of interacting with each other.

On our model vault of heaven, we see the projection of the light picture that is born by our model Universe. Is any of you capable of believing that the picture that is seen on our model vault of heaven gives the real quantity of our model stars, their real luminosity, location in the space, wavelength of their radiation, or any other real parameters of each model star?

If not, you can know nothing either about our model Universe or about the Universe that is seen by Hubble.

It is necessary to take into account that we (and also you) know nothing about the values of the light speed and direction and about the values of the gravitation coefficient in different regions of the real Universe.

That’s why I say that we know nothing about the real state of the Universe and the population will know nothing about it at least for one or two centuries. That’s why I think that we should concentrate ourselves on the Solar System studies, in particular, on the studies of the Sun, because the existence of the population depends on the state of the Sun and because we know today (at least, before our PFO-CFO hypothesis and studies of the consequences from it) about the Sun almost nothing.

I would be grateful for your justified objections against our views.

Sincerely.

Victor,

I think you are neglecting too many evidences (many of them predicted before being observed) that are all pretty consistent. Therefore, I disagree with your statement. The current paradigm explains observations pretty well and I believe it gives a fairly good picture of our universe. Also, if one works out the literature, he'll see that so far, all attempts to modify our present paradigm -- even slightly -- have failed to fit observations better than the present standard model of cosmology.

It is your right to express your disbelief in the standard model, but as far as I am concerned, your comments are out of the scope of the question that I asked and therefore I would be grateful if you could not continue this debate in this specific thread.

Thanks.

Dear Dr. Olivier Minazzoli,

OK! I will write nothing in this page; I will never open this page. Be calm. But, before that, I should say the following.

Science has its own laws.

Either you should answer to the arguments or you should acknowledge them.

You answered to none of my arguments.

After my previous issue, neither you nor anybody other answered to it for 23 days.

You did not answer to my last issue also.

This means one of two things: ether you neglect the laws of scientific discussions and, thus, construct your own individual laws for them, or you don’t consider your passages about the Universe as the scientific ones and carry them into the field of imaginations and fantasies.

In both cases, your question has no scientific meaning.

This is the answer in the scope of your question.

You are young, and I wish you successes in your way.

Olivier,

"as a matter of fact, the cosmological constant is part of the most general equation that satisfies general relativity principles. It is not something added, it is simply there. Therefore, if one doesn't want to consider the cosmological constant (ie. set it to zero), he has to give a good reason for that. Within the paradigm of general relativity, as far as I know, no such a reason has been found. Now, since it turns out that the acceleration of the expansion of the universe is well explained by a cosmological constant and since we measure no deviation from the behavior of a simple cosmological constant, it says that for that matter, the paradigm of general relativity is still well suited to describe the evolution of the universe."

This version of the facts seem to conflict with the facts that I'm aware of. Please see http://en.wikipedia.org/wiki/Cosmological_constant#History - it (and its references) describes how Einstein included it in the field equations for general relativity in order to prevent the seemingly inevitable gravitational collapse of the universe. However, the universe that Einstein was then considering is now known to be the gravitationally bound Milky Way galaxy!

When Hubble identified that the Milky Way was just one of many billions of galaxies and that the previously unknown intergalactic spacetime was expanding, Einstein and others set the cosmological constant parameter value to 0. That was the generally accepted value used by all standard cosmological models from that time (in the 1930's) until 1998, when researchers attempting to more precisely constrain the then standard cosmological models' deceleration parameter. Please see http://arxiv.org/abs/astro-ph/9805201 - it states in section 6 - "Conclusions" that:

"2. The distances to the spectroscopic sample of SNe Ia measured by two methods are consistent with a currently accelerating expansion (q0¹0) at confidence levels from 99.5% (2.8 p) to more than 99.9% (3.9 p) for q04()M/2) [)" using the prior that )Mº0."

[see source text for correct formatting]

However, the preceding section concludes:

"Our detection of a cosmological constant is limited not by statistical errors but by systematic ones. Further intensive study of SNe Ia at low (z\0.1), intermediate (0.1¹z¹0.3), and high (z[0.3) redshifts is needed to uncover and quantify lingering systematic uncertainties in this striking result."

As far as I know, this systematic uncertainty has not been resolved...

IMO, if Einstein had not attempted to stabilize the cosmos (actually only the Milky Way) against the eventual effects of gravitational binding, it's unlikely that he would have included the cosmological constant fudge factor in his published field equations for GTR.

Moreover, I have to wonder how it is that vacuum energy density could possibly remain constant within expanding universal spacetime and contracting galactic spacetime! I suspect that vacuum energy density is a critical factor in both the universal expansion of spacetime and its localized gravitational contraction - but that it cannot be directly measured. However I do not believe it can be a constant or that it alone is physically producing any acceleration of spacetime expansion.

There are, after all, other potential explanations for the acceleration of universal expansion. For example, with the increasing development of large scale structure consisting of regionally voided spacetime and material filaments, the density of matter (and gravitational effects opposing expansion) is locally becoming increasingly disparate. If one modeled only regional voids I suspect their expansion rate should be increasing (accelerating) relative to that of the universe as a whole - accelerating in time. As voided regions increase in size the universal rate of expansion should also accelerate...

I hope you'll consider that there's more to understanding physics (or any other complex system) than just computational analysis. To be useful, the equations to be computed must properly represent the physical processes involved...

James,

"This version of the facts seem to conflict with the facts that I'm aware of. Please see http://en.wikipedia.org/wiki/Cosmological_constant#History - it (and its references) describes how Einstein included it in the field equations for general relativity in order to prevent the seemingly inevitable gravitational collapse of the universe."

I said "as a matter of fact, the cosmological constant is part of the most general equation that satisfies general relativity principles. It is not something added, it is simply there". This is a purely logical deduction,and it has nothing to do with historical considerations. You can do it yourself: take the principles of general relativity, then construct the most general Lagrangian from that and you'll naturally end up with a cosmological constant. That's pretty much all you can do about it from a physical point of view, and no matter how we got there from an historical point of view.

"However, the preceding section concludes:

"Our detection of a cosmological constant is limited not by statistical errors but by systematic ones. Further intensive study of SNe Ia at low (z\0.1), intermediate (0.1¹z¹0.3), and high (z[0.3) redshifts is needed to uncover and quantify lingering systematic uncertainties in this striking result."

As far as I know, this systematic uncertainty has not been resolved..."

Measuring the expansion rate through supernovae redshifts is not the only way to put a value on the cosmological constant. For instance, you can also use the measurement of temperature fluctuations in the cosmological background (http://arxiv.org/abs/1303.5076), or strong lensing phenomena (http://arxiv.org/pdf/1008.4802.pdf). Otherwise check out the references in the introduction of the following letter in order to see how things have improved since 1998: http://adsabs.harvard.edu/abs/2013PhRvL.111i1302F.

"There are, after all, other potential explanations for the acceleration of universal expansion. "

Yes there are, but as far as I know, none of them fit within the paradigm of general relativity, ie. none of them has a purely geometrical nature (due to inhomogeneity for instance) in an otherwise usual general relativity -- without cosmological constant.

You seem to be fond of inhomogeneous cosmology. A lot of of work has been done already in order to characterize inhomogeneous solutions of general relativity. I am no specialist of those solutions, but what I can say is that your suggestion (that one could explain the apparent acceleration of the expansion of the universe by cosmological inhomogeneities) is not new and has been thoroughly explored already. It doesn't mean that your suggestion cannot be correct, but it means that things are not as simple as you imagine them. In the end, if you want to sort that out by yourself (and I can only encourage you to do so), you'll have to do the math, there's no other option...

PS: You can find the "simplest" example of non-homogeneous solution to general relativity without a cosmological constant at the following location: http://www.scholarpedia.org/article/Exact_solutions_of_Einstein%27s_equations#Lema.C3.AEtre-Tolman-Bondi_.28LTB.29_solutions.

"For nearly 70 years standard cosmological models did not include a cosmological constant"

I am sorry to tell but history doesn't change what the laws of Nature are. I explicitly told you that from a logical point of view, one has to consider the cosmological constant, or explain why its value should be equal to zero. Within the paradigm of general relativity, no such a reason is known, regardless what people may have believed at some point (and certainly for historical reasons).

" In fact, all these 'measurements' are derived from exceedingly complex analytical interpretations of very indirect observational data."

Indeed, but also from very different phenomena. And yet, they are pretty consistent. We call it the "concordance" in Physics and it is what rules whether or not we consider that a theory may represent an accurate enough description of observations. For instance, if the apparent acceleration of the universe was due to inhomogeneities as you suggest, we shouldn't get the same value for the cosmological constant from the measurement of the CMB's temperature tiny fluctuations because, among other things, at that time the universe was really close to be homogeneous.

"There is at least one physicist whose analyses seems to agree with my independent, simplistic conception"

Thanks for the link. I now remember having been really excited by Wiltshire's talk at the 12th Marcel Grossmann meeting in Paris while I was still a phd student. The idea is indeed interesting but still faces the problem that we get the same value for the cosmological constant from Planck's data, while if inhomogeneities were responsible for the apparent acceleration of the expansion of the universe, we shouldn't.

I just found a review that I will read when I get a chance to do so: http://iopscience.iop.org/0034-4885/74/11/112901/

(Note however in the conclusion: "There are some scales where backreaction may be important—probably not the largest scales relevant to the cosmic acceleration, but others where precision cosmology is significant.")

Olivier,

"I explicitly told you that from a logical point of view, one has to consider the cosmological constant, or explain why its value should be equal to zero."

Sorry if I do not accept proclamations at face value. There has been no detected change in vacuum energy density - only the recent observations that can most conveniently be explained by adjusting the cosmological constant parameter. There has been no fundamental ('logical') justification for the sudden presence of the positive value for vacuum energy density - now it seems you're demanding a fundamental justification for its non-existence!

Thank you for the link to the very interesting review article, which after a very brief review seems to support at least serious consideration of my assertion.

"The idea is indeed interesting but still faces the problem that we get the same value for the cosmological constant from Planck's data, while if inhomogeneities were responsible for the apparent acceleration of the expansion of the universe, we shouldn't."

I don't understand your reasoning here - after all, the Plank satellite's perspective of the universe is essentially the same as our own. At very large scales, any minute small-scale variations should be imperceptible. As I understand, the expansion rate of the past billion years or so is indeterminable...

My assertion is that all the observed radiation has traversed temporally developing large scale structures - specifically voids, whose peculiar rate of expansion exceeds that of the universal average. As a result the temporally increasing contribution of the increasing expansion of voids to the universal average produces an accelerating universal average.

I'm no undergraduate student - merely a retired information systems analyst with no formal education or experience in physics or math, but I do have extensive experience identifying intractable problems in, even unfamiliar, very complex systems. While I have learned a little about physics, I will never become a qualified physicist or mathematician. However, please do not dismiss my suggestions on the basis of my lack of subject area expertise alone. Certainly I will never publish any formal journal articles. However, as has often been the case, I was completely unaware that qualified physicist had already been considering the solutions I independently suggest - from a very different perspective. IMO, that alone adds some credence to the proposal.

"There has been no fundamental ('logical') justification for the sudden presence of the positive value for vacuum energy density - now it seems you're demanding a fundamental justification for its non-existence!"

Ok, I'll repeat myself one last time: take the principles of general relativity (can you name them?), then construct the most general Lagrangian from them (do you understand what that means?). If you do so, you'll end up with a cosmological constant in your equations. Period.

Do it... Continuing this debate is useless if you don't understand my arguments. And the only way to challenge what I just said is to actually do the calculation and show that I'm wrong. Do that, and then we'll talk. Otherwise, I'm afraid I'll stop our discussion with the present post.

The value of the cosmological constant is not given by the theory. But if you want to consider a theory of general relativity for which this constant is set equal to zero, you must have a good reason for that. But there is not such a reason. Therefore, the cosmological constant cannot be set equal to zero a priori (ie. by logical deduction), within the paradigm of general relativity.

Now, if you are asking why it has its very specific value? (may it be positive or whatever). That's an excellent question! But it is as good as, for instance, why the constant of gravitation has its own very specific value too? Yet, you don't seem to be bothered by the latter, are you? Please read my blog post (http://bit.ly/1bR3QSn) and the references therein. If I mentioned it in my question, it is not for no reasons.

PS: see also http://bit.ly/17Qw5Mo

Again, as I understand for nearly 70 years (as recently as the late 1990's) the standard models of cosmology did in fact use the value 0 for the cosmological constant parameter. This issue is not central to my argument.

However, if the increasingly large scale structure of the universe alone could explain the apparent universal acceleration of expansion, would a return to the null cosmological constant still require justification?

I want to point out one other very serious logical inconsistency with the current model - the SNe search teams objective was actually to more precisely constrain the deceleration parameter. As I understand, this initial expansion of the universe would be expected to diminish as mass-energy was diminished by expansion - owing to increased entropy. That parameter seems to be excluded from the current model - are the effects of entropy in an expanding universe no longer a factor? Even if expansion is accelerating - the process that actually produces that expansion would have to overcome the decelerating effects of entropy - if the cosmological constant is in fact the cause of acceleration, its value would have to be much greater to overcome entropy. Perhaps that's why the fitted value is so unexpectedly small in comparison to the expectations of quantum theory!

As I understand, the gravitational constant has been confirmed, particularly by observation of two body interactions, in many galaxies. But then gravitational effects are, in general, consistently produced by the presence of localized mass-energy, while spacetime expansion is varyingly produced in the absence of localized mass-energy...

Sorry - I'll have to leave the mathematics to you and others - just as theoreticians must generally rely on astronomers for precise observations and, increasingly, their interpretation...

BTW, the Riess, et al. 1998 study fit the distance estimates for type Ia SNe host galaxies to their SN peak emission luminosity based distance estimates by not only specifying a positive cosmological constant parameter in their then standard models but by changing the deceleration parameter from positive to negative - specifying negative deceleration! This illustrates my assertion that those studies were simply fine tuning model parameters to fit with observations...

It is really regrettable that some laymen have the pretentiousness to challenge actual hard work from actual researchers with (merely) rough arguments. Do you really believe that your very incomplete knowledge of the subject might have led you to a proper understanding of the question? Seriously?

Enough of this. I already spent too much time answering your questions. Publish your stuff, if you are correct, time will tell.

Olivier,

I am willing to challenge Nobel Laureates on issues that I think are valid. I have in the past directly raised such issues to one contributing author of the Riess, et al. study who was kind enough to reply. I have often challenged subject area experts in an effort to resolve critical problems - and directly facilitated their solution as a result.

I presume that If you could respond to the deceleration parameter issues you would not be attempting to dismiss them - with my lay status and lack of subject area expertise as the only cause. What happened to entropy as a contributing factor in the temporal development of the expanding universe? No response?

Do you imagine that your or anyone else's knowledge of the subject is complete?

I do appreciate your referring me to Clarkson, et al.'s 2011 paper, available at http://arxiv.org/abs/1109.2314. It looks to be a very interesting read...

At least some of 'my' stuff has already been published by qualified researchers. You should know, however, that not even all published research receives appropriate consideration from the now enormous physics community...

"You should know, however, that not even all published research receives appropriate consideration from the now enormous physics community..."

Indeed, but as I said, time will tell.

Dear Oliver,

I think the problem with the cosmological constant is more related to global properties of spacetime. Therefore (because of he local charcter of QFT) it cannot determined by QFT (or one will get a very wrong value). With the advent of supersymmetry one was able to determine this constant to be zero in QFT.

My work is mainly occupied with the question: what are the global properties of the spacetime (and therefore also understanding the time line of our universe). There is one interesting global property (the smoothness) which is a natural (or the next) generalization of the metric structure. Unfortunately it carries the misleading term "exotic smoothness" (there is nothing exotic here). We (my coworkers and I) studied it for moer than 10 years. But currently we made progress to understand the cosmological constant from this global point of view. Here is the link to our recent paper:

https://www.researchgate.net/publication/259784427_Inflation_and_topological_phase_transition_driven_by_exotic_smoothness

At the end we calculate the cosmological constant which agreed very well with the experimental results. Maybe you have time for a look.

Best

Torsten

Article Inflation and Topological Phase Transition Driven by Exotic Smoothness

Dear Oliver,

I saw your blog post http://bit.ly/17Qw5Mo

We found a simple explaination of the constancy of the cosmological constant: exotic smoothness enforces a fnite volume part of the spacetime to be hyperbolic. But finite-volume, hyperbolic 4-manifolds are rigid (Mostow rigidity), i.e. every scaling or diffeomorphism is an isometry. Therefore the volume and the curvature of this hyperbolic 4-manifold are topological invariants (and therefore constant).

A mathematical property is the reason.

Best Torsten

Dear Torsten,

it sounds interesting. I hope I'll be able to follow the math in your articles as I know almost nothing about differential topology and strictly nothing about exotic smoothness, but I'll take a look anyway :)

Thanks.

Though, in the meantime, I have a simple question Torsten. Does your possible explanation through exotic smoothness have a chance to predict a specific value for the cosmological constant?

Dear Oliver,

yes we predict a value for the cosmological constant. The value depends on the Hubble constant (and we assume that the cosmos starts with a Planck size 3-sphere).

For H=74 km/(s MPc) we get 0.729 (in units of the critical density) in good agreement with the WMAP results. But Planck measured H=68 km/(s MPc) and then we got the wrong value 0.88 (in units of the critical density) but we have now the freedom to change the 3-sphere at the beginning. Using the area quantization of Loop quantum gravity we get 0.685 for the cosmological constant (again in units of the critical density). I'm not satisfied with the last explanation but think about a general correction coming from the path integral by the value exp(- chi(M)/4) where chi(M) is the Euler characteristics of M. In our case we have chi(M)=1 and so we obtain the true value 0.6853 again.

Best Torsten

PS: All details of the calculation can be found in the paper including the closed formula to calculate the cosmological constant.

Oliver,

In an attempt to directly address your question - my original comment stated:

"I certainly commend the question's suggestion - 'dark' proxy elements have a way of shaping, at least the public's, consciousness. I have to question though how the problem can now be effectively rephrased in the public's consciousness?"

I have two primary concerns:

1 - How can some consensus statement explaining the current understanding of the physical cause of accelerating expansion be effectively communicated to the general public?

2 - How can the established dark energy explanation that has already been widely publicized now be effectively replaced or superseded by some new explanation?

To the first concern, how do you expect the statement "... the acceleration of the expansion of the universe is mainly due to the curvature of space-time in vacuum" to be understood by the general public?

While I'm no physicist, I think that most people hearing the term 'curvature of spacetime' envision the rubber sheet with smaller balls rolling into a depression caused by the weight of a larger ball. I don't think that they can meaningfully relate that understanding to any acceleration of spacetime expansion.

Personally, I envision that any curvature of spacetime is an aspect of spacetime contraction resulting from the (for spherical masses, radially) mass-directed length contraction of space and dilation of the advancement of time.

In this context, the expansion of spacetime is the global expansion of spacetime in the absence of mass as opposed to the local contraction of spacetime focused on mass. But I don't know if that description would meet with any physicists' approval or could be successfully explained to the public.

I do not think that "the curvature of spacetime in [the] vacuum" will communicate anything meaningful to the public that can help explain the acceleration of spacetime expansion, much less anything about its physical cause.

Please consider your own quote of Feynman or, from the 1967 film "Cool Hand Luke", "What we've got here is (a) failure to communicate."

BTW, re.: "... Whether this curvature is due to a fundamental constant" IMO, numerical parameters do not cause physical phenomena - they only relate to physical processes that may actually produce them...

Torsten,

"yes we predict a value for the cosmological constant. The value depends on the Hubble constant".

hmmm, it's not really what I expected. When I asked whether or not your explanation could predict the value of the cosmological constant, I implicitly thought "and therefore the consequences of such a value". Hence, I was expecting that the value of the Hubble parameter would be a prediction rather than an input.

But I'll take a look at your papers as I don't quite follow what you are saying right now :) It may take some time before I've enough time to carefully read them though.

Oliver,

we calculate the absolute value of the cosmological constant but I presented the value in the usual units where the Hubble constant (determining the critical density) is used. So the absolute value is approximately 7.5 10^-50 1/m^2 or 5 10^-118 in Planck units. I have to think about it to determine the Hubble constant. It is rather amazing for me because this constant depends on the moment of the measurement (so the todays value is the time since the big bang). Maybe I miss something and you can correct me.

Torsten

Oh ok I see what you meant, you were simply talking of the so-called normalized cosmological constant. My bad, I should have seen that from the values that you gave to us. Never mind what I said :)

Olivier:

Permit me to provide some comments regarding your original question.

Recent observations suggest the large-scale universe may to flat (or, at least, have close to zero curvature).

Also, I think it is important to allow for separation of 2 uses of the term "dark energy."

* Stuff that contributes the majority of the density of the universe.

* Mechanisms that correlate with the observed rate (and changes to that rate) of expansion of the universe.

I think I offer theory that you might want to consider. One basis for it is correlations between solutions to isotropic harmonic oscillator equations and elementary particles (and their fields, some particle properties, and some interactions in which the particles participate). For each known elementary particle, I find a solution. Other solutions might correlate with yet-to-be-found elementary particles.

Some of the solutions could correlate with fermions and bosons that could be "dark energy stuff." Other solutions could correlate with zero-mass bosons (relatives of the photon and [hypothetical] graviton) that would drive and regulate the rate of expansion.

Other aspects of the work could correlate with an essentially flat universe.

While the above may not fulfill your desire to a simple explanation for non-specialists, I think we could develop such an explanation. (Of course, an issue is the extent to which my attempted research correlates with nature.) Permit me to attach an extract from a book describing my work. Sections 1.1, 1.2 1.3, and 2.1 discuss the above-mentioned family of zero-mass bosons, hint at the nature of dark energy, and discuss the possible mechanism governing the rate of expansion of the universe.