aa r X i v : . [ phy s i c s . g e n - ph ] J a n The Dark Energy Universe
B.G. SidharthInternational Institute for Applicable Mathematics & Information SciencesHyderabad (India) & Udine (Italy)B.M. Birla Science Centre, Adarsh Nagar, Hyderabad - 500 063 (India)
Abstract
Some seventy five years ago, the concept of dark matter was intro-duced by Zwicky to explain the anomaly of galactic rotation curves,though there is no clue to its identity or existence to date. In 1997,the author had introduced a model of the universe which went dia-metrically opposite to the existing paradigm which was a dark matterassisted decelarating universe. The new model introduces a dark en-ergy driven accelarating universe though with a small cosmologicalconstant. The very next year this new picture was confirmed by theSupernova observations of Perlmutter, Riess and Schmidt. These as-tronomers got the 2011 Nobel Prize for this dramatic observation.All this is discussed briefly, including the fact that dark energy mayobviate the need for dark matter.
By the end of the last century, the Big Bang Model had been worked out.It contained a huge amount of unobserved, hypothesized ”matter” of a newkind - dark matter. This was postulated as long back as the 1930s to ex-plain the fact that the velocity curves of the stars in the galaxies did not falloff, as they should. Instead they flattened out, suggesting that the galax-ies contained some undetected and therefore non-luminous or dark matter .The identity of this dark matter has been a matter of guess work, though.It could consist of Weakly Interacting Massive Particles (WIMPS) or SuperSymmetric partners of existing particles. Or heavy neutrinos or monopoles1r unobserved brown dwarf stars and so on.In fact Prof. Abdus Salam speculated some two decades ago [1] ”And nowwe come upon the question of dark matter which is one of the open prob-lems of cosmology. This is a problem which was speculated upon by Zwickyfifty years ago. He showed that visible matter of the mass of the galaxies inthe Coma cluster was inadequate to keep the galactic cluster bound. Oortclaimed that the mass necessary to keep our own galaxy together was atleast three times that concentrated into observable stars. And this in turnhas emerged as a central problem of cosmology.”You see there is the matter which we see in our galaxy. This is what wesuspect from the spiral character of the galaxy keeping it together. And thereis dark matter which is not seen at all by any means whatsoever. Now thequestion is what does the dark matter consist of? This is what we suspectshould be there to keep the galaxy bound. And so three times the mass ofthe matter here in our galaxy should be around in the form of the invisiblematter. This is one of the speculations.”The universe in this picture, contained enough of the mysterious dark matterto halt the expansion and eventually trigger the next collapse. It must bementioned that the latest WMAP survey [2], in a model dependent resultindicates that as much as twenty three percent of the Universe is made upof dark matter, though there is no definite observational confirmation of itsexistence.That is, the Universe would expand up to a point and then collapse.There still were several subtler problems to be addressed. One was the fa-mous horizon problem . To put it simply, the Big Bang was an uncontrolledor random event and so, different parts of the Universe in different directionswere disconnected at the very earliest stage and even today, light would nothave had enough time to connect them. So they need not be the same. Ob-servation however shows that the Universe is by and large uniform, rather likepeople in different countries showing the same habits or dress. That wouldnot be possible without some form of faster than light intercommunicationwhich would violate Einstein’s Special Theory of Relativity.The next problem was that according to Einstein, due to the material con-tent in the Universe, space should be curved whereas the Universe appearsto be flat .There were other problems as well. For example astronomers predicted thatthere should be monopoles that is, simply put, either only North magneticpoles or only South magnetic poles, unlike the North South combined mag-2etic poles we encounter. Such monopoles have failed to show up even afterseventy five years.Some of these problems as we noted, were sought to be explained by whathas been called inflationary cosmology whereby, early on, just after theBig Bang the explosion was super fast [3, 4].What would happen in this case is, that different parts of the Universe, whichcould not be accessible by light, would now get connected. At the same time,the super fast expansion in the initial stages would smoothen out any dis-tortion or curvature effects in space, leading to a flat Universe and in theprocess also eliminate the monopoles.Nevertheless, inflation theory has its problems. It does not seem to explainthe cosmological constant observed since. Further, this theory seems to im-ply that the fluctuations it produces should continue to indefinite distances.Observation seems to imply the contrary.One other feature that has been studied in detail over the past few decadesis that of structure formation in the Universe. To put it simply, why isthe Universe not a uniform spread of matter and radiation? On the contraryit is very lumpy with planets, stars, galaxies and so on, with a lot of spaceseparating these objects. This has been explained in terms of fluctuationsin density, that is, accidentally more matter being present in a given region.Gravitation would then draw in even more matter and so on. These fluctua-tions would also cause the cosmic background radiation to be non uniform oranisotropic. Such anisotropies are in fact being observed. But this is not theend of the story. The galaxies seem to be arranged along two dimensionalstructures and filaments with huge separating voids.From 1997, the conventional wisdom of cosmology that had concretized fromthe mid sixties onwards, began to be challenged. It had been believed thatthe density of the Universe is near its critical value, separating eternal expan-sion and ultimate contraction, while the nuances of the dark matter theorieswere being fine tuned. But that year, the author proposed a contra view,which we will examine.
To proceed, as there are N ∼ such particles in the Universe, we get,consistently, N m = M (1)3here M is the mass of the Universe. It must be remembered that the energyof gravitational interaction between the particles is very much insignificantcompared to the above electromagnetic considerations.In the following we will use N as the sole cosmological parameter.We next invoke the well known relation [5, 6, 7] R ≈ GMc (2)where M can be obtained from (1). We can arrive at (2) in different ways.For example, in a uniformly expanding Friedman Universe, we have˙ R = 8 πGρR / R = c at R , the radius of the universe, we get(2). Another proof will be given later in Section 3.10.We now use the fact that given N particles, the (Gaussian)fluctuation in theparticle number is of the order √ N [7, 8, 9, 10, 11, 12], while a typical timeinterval for the fluctuations is ∼ ¯ h/mc , the Compton time, the fuzzy intervalwithin which there is no meaningful physics as argued by Dirac and in greaterdetail by Wigner and Salecker. So particles are created and destroyed - butthe ultimate result is that √ N particles are created just as this is the nettdisplacement in a random walk of unit step. So we have, dNdt = √ Nτ (3)whence on integration we get, (remembering that we are almost in the con-tinuum region that is, τ ∼ − sec ≈ T = ¯ hmc √ N (4)We can easily verify that the equation (4) is indeed satisfied where T is theage of the Universe. Next by differentiating (2) with respect to t we get dRdt ≈ HR (5)where H in (5) can be identified with the Hubble Constant, and using (2) isgiven by, H = Gm c ¯ h (6)4quation (1), (2) and (4) show that in this formulation, the correct mass,radius, Hubble constant and age of the Universe can be deduced given N ,the number of particles, as the sole cosmological or large scale parameter.We observe that at this stage we are not invoking any particular dynamics- the expansion is due to the random creation of particles from the ZPFbackground. Equation (6) can be written as m ≈ H ¯ h Gc ! (7)Equation (7) has been empirically known as an ”accidental” or ”mysterious”relation. As observed by Weinberg [13], this is unexplained: it relates a sin-gle cosmological parameter H to constants from microphysics. We will touchupon this micro-macro nexus again. In our formulation, equation (7) is nolonger a mysterious coincidence but rather a consequence of the theory.As (6) and (5) are not exact equations but rather, order of magnitude rela-tions, it follows, on differentiating (5) that a small cosmological constant ∧ is allowed such that ∧ ≤ H )This is consistent with observation and shows that ∧ is very small −− thishas been a puzzle, the so called cosmological constant problem alluded to,because in conventional theory, it turns out to be huge [14]. But it poses noproblem in this formulation. This is because of the characterization of theZPF as independent and primary in our formulation this being the mysteri-ous dark energy. Otherwise we would encounter the cosmological constantproblem of Weinberg: a ∧ that is some 10 orders of magnitude of observ-able values!To proceed we observe that because of the fluctuation of ∼ √ N (due to theZPF), there is an excess electrical potential energy of the electron, which infact we identify as its inertial energy. That is [9, 7], √ N e /R ≈ mc . On using (2) in the above, we recover the well known Gravitation-Electromagnetismratio viz., e /Gm ∼ √ N ≈ (8)or without using (2), we get, instead, the well known so called Weyl-Eddingtonformula, R = √ N l (9)5It appears that (9) was first noticed by H. Weyl [15]). Infact (9) is thespatial counterpart of (4). If we combine (9) and (2), we get,
Gmlc = 1 √ N ∝ T − (10)where in (10), we have used (4). Following Dirac (cf.also [16]) we treat G asthe variable, rather than the quantities m, l, c and ¯ h which we will call microphysical constants because of their central role in atomic (and sub atomic)physics.Next if we use G from (10) in (6), we can see that H = cl √ N (11)Thus apart from the fact that H has the same inverse time dependance on T as G , (11) shows that given the microphysical constants, and N , we candeduce the Hubble Constant also, as from (11) or (6).Using (1) and (2), we can now deduce that ρ ≈ ml √ N (12)Next (9) and (4) give, R = cT (13)Equations (12) and (13) are consistent with observation.Finally, we observe that using M, G and H from the above, we get M = c GH This relation is required in the Friedman model of the expanding Universe(and the Steady State model too). In fact if we use in this relation, theexpression, H = c/R which follows from (11) and (9), then we recover (2). We will be repeatedlyusing these relations in the sequel.As we saw the above model predicts a dark energy driven ever expanding andaccelerating Universe with a small cosmological constant while the densitykeeps decreasing. Moreover mysterious large number relations like (6), (12)6r (9) which were considered to be miraculous accidents now follow from theunderlying theory. This seemed to go against the accepted idea that thedensity of the Universe equalled the critical density required for closure andthat aided by dark matter, the Universe was decelerating.However, as noted, from 1998 onwards, following the work of Perlmutter , Schmidt and
Riess , these otherwise apparently heretic conclusions havebeen vindicated.It may be mentioned that the observational evidence for an acceleratingUniverse was the American Association for Advancement of Science’s Break-through of the Year, 1998 while the evidence for nearly seventy five per-cent of the Universe being Dark Energy, based on the Wilkinson MicrowaveAnisotropy Probe (WMAP) and the Sloan Sky Digital Survey was the Break-through of the Year, 2003 [17, 2]. The trio got the 2011 Nobel for Physics.The zero point field or dark energy has another signature. It causes for exam-ple the Lamb Shift, which for a Hydrogen atom is 1
M Hz and is ubiquitous.This gives us an approximate idea of the strength of the signal. So we shouldexpect a uniform Cosmic Radio Background of roughly 1
M Hz to about 1metre, remembering the various dissipative processes that exist in space.NASA’s ARCADE, balloon borne experiment detected a mysterious isotropicradio radiation (or hiss) six times as powerful as expected, but not from anyspecific radio sources. This is consistent with a power law and is precisely inthe wavelength region of 10 cm to 1 metre. This otherwise inexplicable radiowave background could well be the above signature.
1. We observe that in the above scheme if the Compton time τ → τ P , werecover the Prigogine Cosmology [18, 19]. In this case there is a phasetransition in the background ZPF or Quantum Vacuum or Dark Energyand Planck scale particles are produced.On the other hand if τ → | ∆ x | = < ∆ x > = ν · ∆ tν = ¯ h/m, ν ≈ lv (14)This way we could explain a process similar to the formation of Benard cells[20, 18] – there would be sudden formation of the “cells” from the backgrounddark energy, each at the Planck Scale, which is the smallest physical scale.These in turn would be the underpinning for spacetime.We could consider an array of N such Planckian cells [21]. This would bedescribed by r = √ N ∆ x (15) ka ≡ k ∆ x = 12 k B T (16)where k B is the Boltzmann constant, T the temperature, r the extent and k is the spring constant given by ω = km (17) ω = km a ! r = ω ar (18)We now identify the particles or cells with Planck masses and set ∆ x ≡ a = l P , the Planck length. It may be immediately observed that use of (17)and (16) gives k B T ∼ m P c , which ofcourse agrees with the temperature ofa black hole of Planck mass. Indeed, Rosen [22] had shown that a Planckmass particle at the Planck scale can be considered to be a Universe in itselfwith a Schwarzchild radius equalling the Planck length. We also use the factalluded to that a typical elementary particle like the pion can be consideredto be the result of n ∼ Planck masses.Using this in (15), we get r ∼ l , the pion Compton wavelength as required.Whence the pion mass is given by m = m P / √ n which of course is correct, with the choice of n . This can be described by l = √ nl P , τ = √ nτ P , (19)8 P = ¯ hm P τ P The last equation is the analogue of the diffusion process seen, which is infact the underpinning for particles, except that this time we have the sameBrownian process operating from the Planck scale to the Compton scale (Cf.also [23, 24]).We now use the well known result alluded to that the individual minimaloscillators are black holes or mini Universes as shown by Rosen [22]. Sousing the Beckenstein temperature formula for these primordial black holes[25], that is kT = ¯ hc πGm we can show that Gm ∼ ¯ hc (20)We can easily verify that (20) leads to the value m ∼ − gms . In deducing(20) we have used the typical expressions for the frequency as the inverse ofthe time - the Compton time in this case and similarly the expression for theCompton length. However it must be reiterated that no specific values for l or m were considered in the deduction of (20).We now make two interesting comments. Cercignani and co-workers haveshown [26, 27] that when the gravitational energy becomes of the order ofthe electromagnetic energy in the case of the Zero Point oscillators, that is G ¯ h ω c ∼ ¯ hω (21)then this defines a threshold frequency ω max above which the oscillationsbecome chaotic. In other words, for meaningful physics we require that ω ≤ ω max . Secondly as we can see from the parallel but unrelated theory of phonons[8, 28], which are also bosonic oscillators, we deduce a maximal frequencygiven by ω max = c l (22)In (22) c is, in the particular case of phonons, the velocity of propagation,that is the velocity of sound, whereas in our case this velocity is that of light.9requencies greater than ω max in (22) are again meaningless. We can easilyverify that using (21) in (22) gives back (20).In other words, gravitation shows up as the residual energy from the forma-tion of the particles in the universe via Planck scales (Benard like) cells.3. It has been mentioned that despite nearly 75 years of search, Dark Matterhas not been found. More recently there is evidence against the existenceof Dark Matter or its previous models. The latest LHC results for exampleseem to rule out SUSY.On the other hand our formulation obviates the need for Dark Matter. Thisfollows from an equation like (10) which shows a gravitational constant de-creasing with time. Starting from here it is possible to deduce not just theanomalous rotation curves of galaxies which was the starting point for DarkMatter; but also we could deduce all the known standard results of GeneralRelativity like the precession of the perihelion of mercury, the bending oflight, the progressive shortening of the time period of binary pulsars and soon (Cf.ref.[20]).4. Epilogue : The idea of a perfect vacuum began to get frayed in the 19thcentury. In the 20th century with the advent of Quantum Theory the conceptof a Quantum Vacuum came into being. This Quantum Vacuum is seethingwith energy and activity, and it is there everywhere. With this backgroundwe can see the following:Around 1997 I had put forward a radically different model. In this, therewasn’t any Big Bang, with matter and energy being created instantaneously.Rather the universe is permeated by an energy field of a kind familiar tomodern physicists. The point is, that according to Quantum theory which isundoubtedly one of the great intellectual triumphs of the twentieth century,all our measurements, and that includes measurements of energy, are at bestapproximate. There is always a residual error. This leads to what physicistscall a ubiquitous Zero Point Field or Quantum Vacuum. We will return tothis “Dark Energy” soon. Out of such a ghost background or all pervadingenergy field, particles are created in a totally random manner, a process thatkeeps continuing. However, much of the matter was created in a fractionof a second. There is no “Big Bang” singularity, though, which had posedWheeler’s greatest problem of physics. The contents of this paper went dia-metrically opposite to accepted ideas, that the universe, dominated by darkmatter was actually decelarating. Rather, driven by dark energy, the uni-verse would be expanding and accelerating, though slowly. I was quite surethat this paper would be rejected outright by any reputable scientific jour-10al. So I presented these ideas at the prestigious Marcell Grossmann meet inJerusalem and another International Conference on Quantum Physics. But,not giving into pessimism, I shot off the paper to a standard Internationaljournal, anyway. To my great surprise, it was accepted immediately!There is a further cosmic foot print of this model: a residual miniscule en-ergy in the Cosmic Microwave Background, less than a billion billion billionbillionth of the energy of an electron. Latest data has confirmed the presenceof such an energy. All this is in the spirit of the manifest universe springingout of an unmanifest background, as described in the Bhagvad Gita. Thereare several interesting consequences.Firstly it is possible to theoretically estimate the size and age of the universeand also deduce a number of very interesting interrelationships between sev-eral physical quantities like the charge of the electron, the mass of elementaryparticles, the gravitational constant, the number of particles in the universeand so on. One such, connecting the gravitational constant and the massof an elementary particle with the expansion of the universe was dubbed asinexplicable by Nobel Laureate Steven Weinberg. But on the whole theseintriguing interrelationships have been considered by most scientists to bemiraculous coincidences.With one exception. The well known Nobel Prize winning physicist PaulDirac sought to find an underlying reason to explain what would otherwisepass off as a series of inexplicable accidents. In this model, there is a depar-ture from previous theories including the fact that some supposedly constantquantities like the universal constant of gravitation are actually varying veryslowly with time. Interestingly latest observations seem to point the fingerin this direction.However my model is somewhat different and deduces these mysterious re-lations. Further, it sticks its neck out in predicting that the universe is notonly expanding, but also accelerating as it does so. This went against allknown wisdom. Shortly thereafter from 1998 astronomers like Perlmutterand Kirshner began to publish observations which confirmed exactly such abehavior. These shocking results have since been reconfirmed. The universehad taken a U Turn.When questioned several astronomers in 1998 confided to me that the ob-servations were wrong! After the expansion was reconfirmed, some becamecautious. Let us wait and see. At the same time, some rushed back totheir desks and tried to rework their calculations. The other matter was,what force could cause the accelerated expansion? The answer would be,11ome new and inexplicable form of energy, as suggested by me. Dark Energy.Later the presence of dark energy was confirmed by the Wilkinson MicrowaveProbe (WMAP) and the Sloane Digital Sky Survey. Both these findings weredeclared by the prestigious journal Science as breakthroughs of the respectiveyears.The accelerated expansion of the universe and the possibility that supposedlyeternally constant quantities could vary, has been the new paradigm giftedto science, a parting gift by the departing millennium.A 2000 article in the Scientific American observed, “In recent years the fieldof cosmology has gone through a radical upheaval. New discoveries havechallenged long held theories about the evolution of the Universe... Nowthat observers have made a strong case for cosmic acceleration, theoristsmust explain it.... If the recent turmoil is anything to go by, we had betterkeep our options open.”On the other hand, an article in Physics World in the same year noted , “Arevolution is taking place in cosmology. New ideas are usurping traditionalnotions about the composition of the Universe, the relationship between ge-ometry and destiny, and Einstein’s greatest blunder.”It is this greatest blunder of Einstein which got the Nobel Prize for Physics in2011 for three US astronomers, Perlmutter, Reiss and Schmidt who observedthe accelerated expansion of the universe in 1988.
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