HHypothesis on the nature and origin of cold darkmatter
Roman Schnabel
Institut f¨ur Laserphysik & Zentrum f¨ur Optische Quantentechnologien, Universit¨atHamburg, Luruper Chaussee 149, 22761 Hamburg, GermanyE-mail: [email protected]
June 6, 2020
Abstract.
One of the greatest mysteries in astrophysics and cosmology is the natureand the origin of cold dark matter, which represents more than 84% of the mass inthe universe. Dark matter reacts on and produces gravitational forces and governs thedynamics of stars around galactic centres, however, does not absorb or emit any kindof electromagnetic radiation. So far, any relation to known types of matter has notbeen conclusive, and proposed new particles have not been found. Here, I propose anddiscuss how dark matter evolved from ultra-light fermionic particles that decoupledfrom the rest of the universe shortly after the Big Bang. My description explicitlyconsiders their interference, and reveals the emergence of entanglement between twosuch particles, as well as their transformation to massive dark-matter quantum fieldsof cosmic sizes. Furthermore, I argue that dark matter and supermassive black holeshave the same origin and evolved simultaneously. If the particles’ decoupling time wasabout half a second after the Big Bang, my hypothesis predicts a minimum mass forsupermassive black holes that fits well to the smallest known such object of 5 · solar masses. It seems very much likely that the ultra-light fermionic particle was theneutrino.
1. The dark-matter mystery
The existence of dark matter has become evident from a large number of astronomicalobservations that call for much more gravitating matter compared to what can be seenwith any kind of telescope. Already in the 1930s it was inferred from the motionsof galaxies inside galactic clusters that most matter in the universe might be dark[1, 2]. More recent observations of the rotational properties of galaxies confirmed this[3, 4, 5, 6, 7], cf. Fig. 1, and suggested the dark-matter contents of galaxies to be aboutten times higher than the stellar components in the galactic disk and bulk [8]. Theexistence of dark matter has further been supported by the observation of gravitationallensing of background electromagnetic radiation by galaxy clusters [1, 9, 10]. Itsexistence is also necessary to explain today’s structure of the spatial distribution of a r X i v : . [ phy s i c s . g e n - ph ] J un ypothesis on the nature and origin of cold dark matter Figure 1.
Photograph of a spiral galaxy (here: NGC3672). Stars in spiral arms atdistances from galactic centres between one third and full radius have about the samerotational velocities, as observed by V. Rubin [6]. The fact that velocities do not reducewith higher distances can be explained by largely extended halos of an invisible massof unknown nature, called cold dark matter . Credit: Adam Block/Mount LemmonSkyCenter/University of Arizona. stars, galaxies, and galaxy clusters as well as the anisotropy of the cosmic micro-wavebackground (CMB).The CMB originates from 380.000 years after the Big Bang, when protons, otherlight nuclei and electrons combined to neutral atoms. Only then, light could freelypropagate because it was not anymore continuously scattered by the highly chargedmatter. In turn, the repelling radiation pressure forces between baryonic particlesgot strongly reduced and atoms could start clumping together to form seeds for stars.Computer models have shown, however, that the time since the release of the CMB wastoo short to allow the growth from baryonic seeds to the cosmic structure composedof galaxies and galaxy clusters we observe. And indeed, the anisotropy of the cosmicmicrowave background as observed by the satellite missions
COBE [11],
WMAP [12],and
Planck [13] proves that matter had already formed seeds much earlier than 380.000years after the Big Bang, calling for (cold) dark matter that is completely invisible,i.e. does not emit, absorb or reflect any kind of electromagnetic radiation.A highly successful model that reproduces the evolution of the universe inconsistency with astronomical observations is the ‘Lambda Cold Dark Matter’ model(ΛCDM model) [14]. In addition to a number of fixed parameters, the model usessix free parameters, whose values are deliberately set to achieve consistency. Amongstthese are the age of the universe with a value of about 13.8 billion years, the amountof cold dark matter (DM), and the amount of ordinary, baryonic matter [13], see Fig. 2.Although successful, the model does not give any further hints concerning the nature ofdark matter.Candidates for the cold dark matter particle should have non-zero rest mass and beneutral, non-baryonic, and stable over cosmic time-scales. Furthermore, there is anargument why candidates should not be very light fermions: According to the Pauli ypothesis on the nature and origin of cold dark matter Figure 2.
Energy densities according to the ΛCDM model and the experimental dataof the Planck mission [13] as they contribute to the total energy density of the universe.Cold dark matter (‘CDM’) contributes 26.8%, ordinary (baryonic) matter just 4.9%.CDM thus represents 84.5% of the gravitating energy density. The remaining 68.3%are attributed to an unknown energy that causes the observed accelerated expansionof today’s universe and is called ‘dark energy’, described by the symbol ‘Λ’. Photons,ultra-relativistic neutrinos, black holes as well as gravitational waves do not contributesignificantly. exclusion principle [15], identical fermions cannot occupy the same space. Each of themoccupies a volume that corresponds to the full-width position uncertainty cubed. Thefermionic character thus limits their number density and demands a minimum mass perfermion to be able to reach the observed dark matter mass density [16]. There is anotherargument why candidates, at least if they were in thermal equilibrium with the primevalplasma at some point shortly after the Big Bang, should not have too low rest masses.Their average velocities would be very high even at moderate temperatures, such thatgravitational structure would be washed out immediately. These arguments basicallyexclude all known particles as DM candidates, including the neutrino [16, 17]. Searchesfor new kinds of particles have started many years ago, but were not successful so far[18, 19, 20, 21, 22].
2. Proposition for solving the dark-matter puzzle
In this article, I present the hypothesis that ultra-light fermions that were in thermalequilibrium with the hot primeval plasma cannot be excluded as cold DM candidates.I argue that such fermions are in fact a strong candidate for cold dark matter.First, I outline the mechanism that naturally transforms ultra-light fermions that haveinteracted with the hot plasma in the past into macroscopic cold quantum fields oncethey stop interacting (or – in other words – ‘decouple’). Second, I argue that two suchfermions combine to form pairs of zero spin and zero momentum, similarly to electronpairs in superconductors. Third, I show that the excitation of these quantum fieldsnaturally reach occupation numbers at which they must transform to (supermassive)black holes, around which remaining quantum fields should aggregate as cold DM halos. ypothesis on the nature and origin of cold dark matter
The purpose of this subsection is to recapitulate well-known facts about the neutrinoswhen they still interacted with the primeval plasma and when they started todecouple. In brief, shortly before decoupling, the neutrino position uncertainties almostoverlapped. From this I conclude that the neutrino gas was at the transition to adegenerate Fermi gas.The event of neutrino decoupling is a well-established fact. It took place abouthalf a second after the Big Bang ( t ν d = 0 . T ν d ≈ · K [23]. The term ‘decoupling’ describes a rather instantaneousprocess, after that the interaction rate between a neutrino and any other particle wasbasically zero. Neutrino decoupling was caused by the expansion of the universe and theassociated reduction of energy density and temperature [24]. At decoupling temperature T ν d , the (weak-force) interaction rate of the neutrinos ( ∝ T ) dropped below the relativeone-dimensional expansion rate of the universe ( ∝ T ), which meant that the intervalbetween two momentum-changing interactions started to exceed the age of the universe[24]. Before decoupling, neutrinos were mainly coupled to the primeval plasma viathe annihilation of charged leptons. For instance, neutrinos with electron flavor ( ν e )were coupled via the annihilation and creation of electron (e − ) / positron (e + ) pairs [24]according to ν e + ¯ ν e ↔ e − + e + ↔ γ + γ , (1)where ¯ ν e is the electron anti-neutrino. (Note, neutrino and anti-neutrino might beidentical, however, this is currently unknown [25]). Electrons and positrons, in turn,were in equilibrium with the annihilation and creation of photons ( γ ). Before neutrinodecoupling, energy was equally distributed over the degrees of freedom in the chainabove. Neutrino pairs, electron/positron pairs, and photon pairs were just differentcoexisting forms of energy.The neutrinos’ average thermal kinetic energy was much higher than the energyrelated to their tiny rest mass. Neutrinos were ultra-relativistic. With the approxima-tions that neutrinos behaved similar to a relativistic ideal fermionic gas, the expressionfor the relativistic kinetic energy and the equipartition theorem yield for the averageabsolute momentum in three dimensions (cid:104)| (cid:126)p |(cid:105) th ≈ k B T /c , where k B is the Boltzmannconstant and c the speed of light. As a well-known fact, the same quantity is also agood approximation for the full width (fw) of the Fermi-Dirac momentum distribution, ypothesis on the nature and origin of cold dark matter (cid:104)| (cid:126)p |(cid:105) th ( T ) ≈ ∆ fwth | (cid:126)p | ( T ), see for instance Fig. 1 in Ref. [26].When the ultra-relativistic neutrinos approached their decoupling, interactions be-tween individual neutrinos became negligible. In this case, position and momentumuncertainties (or ‘indeterminacies’) of the neutrinos can be considered in the approxi-mate limit of massless particles. A quantification of the neutrino position uncertaintyis given by the (one-dimensional) thermal De Broglie wavelength λ th ≈ h/ (cid:104)| p x |(cid:105) th ≈ hc/ ( k B T ) ≈ h/ ∆ fwth p x , where h is the Planck constant and (cid:104)| p x |(cid:105) th the one-dimensionalaverage momentum. Interestingly, the product λ th · ∆ fwth p x ≈ h does not dependent ontemperature. This fact suggests that the product represents minimal-uncertainty (pure)states in the phase-space spanned by position and momentum, because any contributionon top of minimal uncertainty should reveal itself by a monotonically increasing functionof temperature. The same fact suggests that the entire momentum spectrum is givenby quantum uncertainty, meaning that there are no sub-classes of different momentumvalues and all relativistic particles have the same momentum of (cid:104) ˆ p x (cid:105) ≈
0. A looseconnection to the standard deviations in Heisenberg’s uncertainty relation is given by∆ˆ x ≈ λ th / (2 π ) and ∆ˆ p x ≈ ∆ fwth p x /
2. Here, the factors of 1 / /π takes roughly into account that the energy distribution within a ‘wavelength’ has twomaxima, whereas valid is just the width of one maximum. One finds the lower boundof Heisenberg’s uncertainty relation ∆ˆ x ∆ˆ p x ≥ h/ (4 π ), which is known to certify anensemble of pure states.In the following I contrast the neutrino position uncertainty with their averagedistance at times shortly before decoupling and find that they are approximately equal.The average neutrino number density per flavor, spin and cubic meter as a function ofthe temperature of the universe as derived by the (fully classical) Boltzmann equationreads [24, 27]. n ν = 6 π · ζ (3) (cid:18) k B Thc (cid:19) , (2)where ζ (3) ≈ .
2. The average one-dimensional distance between two identical neutrinosis thus given by d ≈ (cid:115) π · ζ (3) · hck B T ≈ . · hck B T . (3)At decoupling temperature, one gets d ν d ≈ . · − m. For comparison, the full spreadof the neutrino position uncertainty around decoupling was2∆ˆ x ≈ λ th π = 1 π · hck B T ≈ . · hck B T , for T ≈ T ν d . (4)At decoupling temperature, one gets 2∆ˆ x ν d ≈ . · − m. 2∆ˆ x and d turn out to beapproximately identical, keeping in mind the rough estimations Eq. (4) is based on. ypothesis on the nature and origin of cold dark matter (cid:104) ˆ p x (cid:105) ≈ (cid:104) ˆ p y (cid:105) ≈ (cid:104) ˆ p z (cid:105) ≈
0. Since(local) hidden variables do not exist [28], the conception of neutrinos having a trajec-tory, i.e. moving like classical particles, was not adequate at decoupling time. Instead,a valid description of neutrinos at and after decoupling has to include interference ofposition uncertainties and the Pauli exclusion principle.
Without any coupling to the primeval plasma, the neutrinos had an undisturbed(‘free’) evolution in the continuously expanding universe. In relevant literature, theevolution of neutrinos is usually described as a free, particle-like propagation of mutuallyindependent neutrinos. Accordingly, the expansion of the universe red-shifted theirmomentum spectrum in the same way as the frequency spectrum of radiation. Whileat decoupling time, the neutrinos constituted a considerable fraction of the universe’stotal energy, today the have lost most of their relativistic energy. As a matter of fact,the cosmological redshift does not remove the thermal appearance of neither momentumnor frequency spectra. For this reason, ‘temperatures’ are conveniently used to describethe relic neutrinos as well as the cosmic micro-wave background. Based on the view ofparticle-like propagation of independent neutrinos, the temperature of today’s cosmicneutrino back-ground (C ν B) is derived to T C ν B = 1 .
945 K. This value is below thetemperature of the CMB of T CMB = 2 .
725 K, because the electro-magnetic radiationfield was reheated when the electrons and positrons of the primeval plasma annihilatedshortly after neutrino decoupling [24, 27]. In this view, it seems to be justified to expandthe validity range of Eq. (2) to today’s low temperatures. By doing so, T turns into ameasure of the expansion of the universe. It is standard in relevant literature to usethe temperature value of the C ν B and Eq. (2) to derive the well-known value of today’saverage relic neutrino number density of n ν, ≈
56 cm − for each of the six types ofneutrinos (two spin values × three flavors) [24, 27].To conclude this subsection, I note that the prevailing particle-like description ofneutrinos after their decoupling is not conform with the results derived in the previoussubsection, since ‘free evolution’ needs to be understood as a continuously increasing po-sition uncertainty in three dimensions. In the next subsections, I present my argumenta-tion why the overlapping of neutrino wave functions eventually resulted in a much higheraverage kinetic energy of today’s C ν B than the one associated with T C ν B = 1 .
945 K. ypothesis on the nature and origin of cold dark matter According to quantum physics, as it is mathematically described by quantum fieldtheory, a ‘neutrino’ is the quantized excitation of a Fourier-limited mode of the neutrinofield. The ‘mode’ is understood as a physical object and represented by its wave function.In case of fermionic modes, occupation numbers cannot exceed one.Before decoupling, the neutrinos belonged to modes that were distinguishable due to their different locations in space-time. Energy quanta in terms of neutrinosgot frequently transferred from one mode to another. After decoupling, the modes’occupation numbers did not change any more. The free evolution of the modes, however,was fundamentally different from the classical concept of free particle-like propagation.The free evolution of a neutrino mode was governed by the momentum uncertainties∆ˆ p x (likewise ∆ˆ p y and ∆ˆ p z ), which resulted in a continuous ultra-relativistic increaseof the spatial mode size:∆ˆ x ( t ) = ∆ˆ y ( t ) = ∆ˆ z ( t ) ≈ ∆ˆ x ν d + c ( t − t ν d ) . (5)The consequences were mode overlapping and the emergence of indistinguishability. Asan example, consider two neutrino modes in the very instance when any kind of energyexchange with the rest of the universe stopped. Due to their interactions in the pastthey be highly localized to a level of about 10 − m and accidentally have an occupationnumber of one. The modes have spherical symmetry with Gaussian energy distributionand are fully identical, except for a spatial separation of 1 . · − m. Due to thehigh localization and the low neutrino rest mass, their momentum uncertainty is ultra-relativistic. The radii of the modes’ position uncertainties have to grow with almostthe speed of light, while their distance hardly changes. After 1 µ s, the modes haveradii of about 300 m and they fully overlap. Their occupation number in terms of twoneutrinos in total does not change. In the course of overlapping the two modes becomeindistinguishable, because (i) their separation becomes negligible compared with theirposition uncertainty, (ii) the expectation values of their momenta remain identical (andsmall), and (iii) the momentum uncertainties also remain identical. The neutrinos arethus also indistinguishable. Their position uncertainty corresponds to the size of thejoined (degenerate) mode. There is no other way how interaction-free physical systemscan evolve.The question arises, why this result is not in conflict with the Pauli exclusionprinciple. The answer is given by the internal structure of the new mode, which isa consequence of interference. The two neutrinos are excitation quanta of the same (new) mode, and as such indeed indistinguishable. The internal structure of the mode,however, guarantees that the two neutrinos would always localize with a separationof 1 . · − m. Here, I use the word ’would’ to point out the fact that localizationrequires an interactions, which, as requested for this example, has zero probability. ypothesis on the nature and origin of cold dark matter x (1 µ s) = 300 m radii. Due to particle number conservation, each was occupiedby up to N ≈ indistinguishable neutrinos. As in the example above, each modeevolved without violating the Pauli exclusion principle. The emerged internal quantumanti-correlation had to have the property of a three-dimensional standing wave, sinceit was produced from spherical waves from all directions. The high neutrino densityenforced a wavelength that was (almost) as short as the thermal De Broglie wavelengthat decoupling time ( λ = λ th ,νd ). In the nodes, the probability of finding a neutrinowas zero. The locations of the nodes, however, were not determined with respect tothe primeval plasma. As in the example above, these quantum correlations guaranteedspatial separation of the neutrinos in case they got localized due to the unlikely case ofinteractions.While the neutrino uncertainties grew, inhomogeneities of the neutrino energydensity on scales smaller than the neutrino fields got smeared out. But inhomogeneitieson larger scales endured for a while. Space-time was not flat, in particular also due tothe mass contribution of the neutrino fields. Obviously, neutrino modes experienced theattractive potentials due to their own masses. Since all field modes continuously evolvedfurther to larger sizes and larger neutrino numbers N , while the universe continuouslyexpanded, it seems plausible that the neutrino fields started to localize due to the space-time curvature produced by themselves. By ‘self-localization’ I do not mean any gravitational collapse of the neutrino mode,but just a stable energy density that is positioned around its own centre. Increasingthe density of neutrino fields was not possible because it would have violated the Pauliexclusion principle. Taking this fact into account, neutrino-field self-localization simplycorresponded to the formation of space-time curvature. The crucial question is whetherthe thermodynamic temperature of the neutrino field was low enough for gravitationalself-localization.A reasonable estimation for T ν is not obvious. T ν ( t ) certainly is a decreasingfunction of time, but it is not related to the ‘temperature’ of the C ν B. The latteris not the temperature of the neutrinos but the (red-shifted) representation of thetemperature of the primeval plasma at the time when neutrinos decoupled. (Similarly,the temperature of the cosmic microwave background is not the temperature of themicrowave field, but a representation of the temperature of the primeval plasma atthe time when electromagnetic radiation decoupled. A radiation field can only have a ypothesis on the nature and origin of cold dark matter T ν ( t ) is supposed to be the thermodynamic neutrino temperature, as it is given bythe average interaction due to neutrino collisions per time interval. Indeed, k B T ν ( t )quantifies the action per second for one neutrino at temperature T ν ( t ), where thesmallest unit of ‘action’ is Planck’s quantum of action h [30]. Before decoupling, theaverage weak-force collision (scattering) rate per neutrino was in thermal equilibriumand f ν, s = ( T ν /T ν d ) /t ν d with f ν, d , s = 2 Hz at decoupling temperature [24]. The actionper collision as an integer multiple of Planck’s ‘quantum of action’ ( h ) was given by l ν, s ( T ν ) h = S ν, s ( T ν ) = k B T ν f ν, s = k B T ν d T ν t ν d , for 1 ≤ S ν, s h ≤ k B T ν d h t ν d . (6)This equation is in full agreement the prevailing description of neutrino decoupling.The neutrinos decoupled from the primeval plasma because even a collision of maximalenergy transfer per neutrino ( k B T ν /
2) could not provide the required action S ν, s , sincethe collision rate was too low. Remarkably, Eq. (6) sets a relation between neutrino de-coupling temperature T ν d and an upper temperature, beyond which neutrino collisionsare unphysical since S ν, s < h . Taking the value T ν d = 3 · K, the highest possibleweak-force collision rate is reached at T ew ≈ · K. Here, I use subscript ‘ew’, because T ew should correspond to the temperature above that the weak force and the electro-magnetic force were unified by the electroweak force. My conclusion is well justifiedbecause the impossibility of weak-force collisions indicate the absence of the weak forceper se. One might further argue that neutrinos (or neutrino modes) as non-degeneratesystems only existed at temperatures below T ew .Eq. (6) describes the reason of decoupling, which is 1 /f ν, s ( t ) > t for t > t ν d . Theupper bound of Eq. (6) suggest that after decoupling not the collision frequency but theage of the universe t was relevant to quantify the temperature of the neutrinos. Sincethe size of the neutrino position uncertainty is a perfect measure of the time that hadpassed since the last interaction, I propose the following expression for the temperatureof the neutrino field k B T ν ( t ) = k B T ν d ∆ x ν d ∆ x ν d + c ( t − t ν d ) = (cid:126) c ∆ x ν d + c ( t − t ν d ) ≈ (cid:126) ( t − t ν d ) , for t − t ν d (cid:29) ∆ x ν d c . (7)The neutrino field is thus much colder than the ‘temperature’ assigned to the cosmicneutrino background. For instance, 0.4 s after decoupling the neutrino field temperaturewas T ν (0 . ≈ .
02 nK, and accordingly k B T ν (0 . ≈ . · − eV. The neutrinotemperature according to Eq. (7) provides a very reasonable value for a system that isdecoupled from the rest of the universe. Another such system is a black hole. In fact,Eq. (7) claims that an ultra-relativistic neutrino field with radius r = c ( t − t ν d ) has thesame temperature as a black hole with Schwarzschild radius r s = GM/c of the samesize: ypothesis on the nature and origin of cold dark matter T ν ( r ) = (cid:126) ck B r = r s r · (cid:126) c k B GM , (8)where the right term corresponds to the well-known Hawking temperature [31]. G isthe gravitational constant. This finding strongly supports my proposal for the neutrinofield temperature in Eq. (7).Having found the expression for the thermodynamic temperature of the neutrinofield after decoupling (Eq. (7)) it is possible to estimate whether gravitational self-localization of neutrino fields was possible or even self-evident. As said at the beginningof this subsection, self-localization must have happened without an increase of neutrinonumber density. In this case, gravitational self-localization can be approximated by thewell-known statistics of indistinguishable non-interacting bosonic particles:For simplification, I consider just two energy levels for a total number of N neutrinos.The lower one corresponds to the quanta’s ground state in the trapping potential ofgravitational self-localization. The second level corresponds to an elevated neutrinoenergy of flat space-time. The potential difference be | E | . The probability of having anupper level population of K < N/ N → ∞ is given by P ( K ) ≈ e − K | E | / ( k B T ν ) . Itcan be shown that the total upper level population is given by K ≈ e −| E | / ( k B T ν ) / (1 − e −| E | / ( k B T ν ) ), which is independent of N . For any arbitrarily small value of | E | > K becomes negligible as N approaches infinity. In practice, neutrino field modes reachedextremely large, but nevertheless finite values of N . For this reason, the trappingpotential | E | needs to be quantified.My quantitative estimation of the trapping potential | E | considers the potentialof a black hole. In section 3, I show that 0.9 s after the Big Bang the first primordialblack holes emerged. For the time being, I refer to the fact that galaxies not only havedark matter halos but also supermassive black holes at their centres. For a massiveneutrino field that accumulates around a black hole, the trapping potential is | E | = mc ,with m the neutrino rest mass. There are in fact three different neutrino rest masses.Whereas the three neutrino flavors are the eigenvalues of the weak interaction, thethree neutrino rest masses are eigenvalues of neutrino propagation. The observation ofneutrino oscillations showed that all these masses are non-zero. The individual values,however, are not known, just sums and differences of their squares [32, 33].Using the lower bound for the average neutrino rest mass of m ≈ .
033 eV /c as derivedfrom the observation of neutrino oscillations [32], | E | is thirteen orders of magnitudegreater than k B T ν (0 . K is zero and not a single neutrino is not localizedaround the black hole. The lowest neutrino mass, however, might be much lower than theaverage neutrino mass. In Sec. 4, I present a very conservative estimation of the value forthe lowest neutrino mass and find a lower bound of about 10 − eV/ c . Consequently, | E | is at most eleven orders of magnitude lower than k B T ν (0 . K ≈ . Toconclude, even in case of an extremely low neutrino rest mass, K is negligible comparedto the typical numbers of N .In summary, already a fraction of a second after decoupling, significant parts of the ypothesis on the nature and origin of cold dark matter A Fermi sea is unstable against pairing of its quanta if an arbitrarily weak attractiveforce exists between them. This is the claim of the Bardeen, Cooper and Schrieffer (BCS)theory [34], which was formulated to describe the formation of Cooper pairs from initiallydistinguishable electrons [35] and superconductivity in metals at low temperatures. Myproposition is that the neutrinos in Fermi seas paired as well, in fact due to the tinygravitational attraction between them.The physical description of pairing of initially independent quanta starts fromthree preconditions. First, the quanta are indistinguishable, from which follows thatdecoherence is negligible. Second, there exists a mutually attractive force. And third,the first two preconditions last for an expanded period of time in order to take intoaccount the spatial spread of the quanta as well as to counterbalance the finite natureof the attractive force. All three preconditions are fulfilled for a neutrino Fermi sea.Indistinguishability guarantees that all quanta have maximal spatial overlap of theirposition uncertainties, simply because they excite the same spatial mode. Maximalspatial overlap is relevant since it maximizes the efficacy of the attractive force. Theattractive force itself is relevant because it produces an attractive potential.The formation of pairs is only manifested in case of interactions with theenvironment that lead to localization. Neutrinos of a Fermi sea do not have suchinteractions, however, we can consider such interactions hypothetically. Since wepostulate indistinguishability, we have to conclude the following. If an interactionlocalizes a neutrino ‘ A ’ precisely at position (cid:126)x A then there is another neutrino ‘ B ’ atprecisely the same position with certainty, i.e. 100% probability (regardless whether thesecond neutrino is detected or not). This is enforced because of the vanishing quantumuncertainty in the differential position (cid:126)x A − (cid:126)x B = 0 and the attractive force. If the sameinteraction additionally determines the spin value, the second neutrino has oppositespin, again with certainty. This is the anti-correlation enforced by the Pauli principle,which leads to bosons of zero spin. Alternatively, the interaction may determine aneutrino with a precise momentum value (cid:126)p A with respect to the centre of mass ofthe neutrino field. In this case, the interaction must realize a second neutrino that hasprecisely the same momentum but with opposite sign. This is due to indistinguishabilitycombined with momentum conservation. In the concept of pairing, the interaction withthe environment affects exactly two neutrinos while all others remain unaffected andthus indistinguishable. The total momentum of all remaining N − ypothesis on the nature and origin of cold dark matter N neutrinos beforethe interaction. In conclusion, there is no uncertainty in the sum of the momenta (cid:126)p A + (cid:126)p B = 0. A simultaneous spin measurement would again show opposite spins.The zero-spin bosons thus have precisely zero momentum. Using the concept of pairingprovides a complementary view on a neutrino Fermi sea, which is that of a Bose-Einsteincondensate of zero-spin particles.Finally, I consider a large number of such neutrino measurements with perfectquantum efficiency. The measurements correspond to an ensemble measurement ofidentical pairs. Regardless whether (cid:126)x A or (cid:126)p A is measured, it allows to predict thecorresponding value for the corresponding second neutrino without any uncertainty.This is correctly described by quantum theory, since the relevant quantities commute([ x A − x B , p x,A + p x,B ] = 0 for all three dimensions). The measurements demonstrateEinstein-Podolsky-Rosen (EPR) entanglement [36]. The physics that describes theevolution to pairs from initially individual systems thus describes the emergence ofEPR entanglement. (For comparison I refer to [37], which investigates the physics ofthe emergence of EPR entanglement between two massive mirrors with respect to theirmotional degree of freedom in one dimension.) It is well-known that the rest mass of all neutrinos in the visible universe is much lessthan the mass of the inferred cold DM. The prevailing view is that also their today’ssum of rest mass and relativistic mass is too low for constituting a major fraction ofDM, because the neutrinos are supposed to have lost most of their relativistic mass dueto the expansion of the universe. In the foreground of gravitational self-attraction ofneutrino fields, this assumption should be incorrect. My argumentation is the following.First, dense neutrino fields, as they spread over sizes as large as galaxy clusters andbeyond, resembled a strong gravitational force that locally acted against the expansionof the universe. Second, there were also space-time volumes containing less neutrinofield energy. At these space-time locations the dilution of energy density was faster dueto higher expansion rate. In other words, the initial thermal density fluctuations gotamplified and the expansion rate of the universe became more and more inhomogeneousas time progressed.From this point of view, the well-known cosmic voids [38] seem to be a naturalconsequence. My argumentation also fits well to the observation that the cosmicDM distribution forms a gravitational scaffold [39]. While electromagnetic radiationexperienced a redshift due to the average expansion of the universe, the neutrinofields were predominantly located in space time regions of reduced expansion. Theyexperienced a reduced redshift that is given by the average of local expansions weightedby the neutrino field density distribution.In the following I estimate the average rest-mass density of the decoupled neutrinosand from that the cumulated factor by which the expansion of the universe was reduced ypothesis on the nature and origin of cold dark matter . · m − , taking into account two spin and three mass values [24, 27],see also Eq. (2) for T = T C ν B = 1 .
945 K. Using the upper bound for the average neutrinorest mass of 0 .
04 eV /c ( ≈ · − kg) [40] the average rest-mass density is less than2 . · − kg/m . Using the lower bound for the average neutrino rest mass of 0 .
033 eV /c ( ≈ · − kg) as derived from the observation of neutrino oscillations [32], the totalrest mass of all neutrinos is at least 2 · − kg/m . The average DM mass density,however, is more than a hundred times higher, namely about 2 . · − kg/m . Thisvalue is given by the data shown in Fig. 2 and the total energy density of the universe.Due to the WMAP mission [41] it is known that the latter almost corresponds to the(mass equivalent) critical density of a flat universe of ρ crit ≈ · − kg/m .If the decoupled neutrinos constitute all cold dark matter, their relativistic energytoday needs to be about a hundred times higher than their average rest mass. In otherwords, the effective expansion of the ‘neutrino-field universe’ must have reduced theaverage relativistic energy in three dimensions from initially 3 × . × . ≈ × .
04 eV), which corresponds to an average neutrino field red-shiftof z νν d = 2 . / . ≈ · instead of the literature value of ˜ z νν d = T ν, d /T C ν B ≈ . · .The summary of my hypothesis on the nature and origin of cold dark matter is thefollowing. Cold DM as observed today are neutrino fields of zero temperature. Its totalmass is dominated by relativistic mass. The quantized excitation of the neutrino fieldcan be understood as (degenerate) neutrino pairs. Their position uncertainty spreadsover the connected parts of the cold DM scaffold described in [39]. Neutrino pairs arenot strictly localized to a single galactic halo but are excitations of connected parts ofthe entire scaffold. This subsection complements the physics of neutrino decoupling by the quantum fieldview. As outlined in subsection 2.3, neutrinos are the excitation quanta of neutrinofield modes. At any temperature T of the early universe, these modes expanded withalmost the speed of light. The only way of avoiding the overlap of excited modes, whichwould have violated Pauli’s exclusion principle, was a high rate of their depopulation.According to Eq. (3), relevant overlapping would have occurred already after a timeinterval of about (3 · − T ν d /T ) s. During such a short period of time, one out of twopopulated modes had to be depopulated. This fact requested a depopulation frequencyof slightly more than (10 T /T ν d ) Hz per neutrino. The collision frequency of individualneutrinos of T / ( T ν d t ν d ) was too low. The highest possible interaction frequency of arelativistic gas in thermal equilibrium, however, was sufficiently high:Γ ≈ k B Th ≈ · TT ν d , for T ≥ T ν d . (9) ypothesis on the nature and origin of cold dark matter (c) (d)(a) (b) ! < ! ! ≈ ! ! ≳ ! ! > ! Figure 3.
Hypothesis of neutrino-to-dark-matter evolution starting with neutrinodecoupling at t ν d ≈ . I propose that the above equation represents the frequency of neutrino-pair annihilationaccording to Eq. (1), balanced by an equally high creation rate. Twice the valuecorresponds to the depopulation rate of individual neutrino modes, which meets thevalue requested above. My physical justification of this statement is the following. SinceEq. (1) describes coexisting forms of energy, indeed, a single quantum h is sufficient tocomplete a process in one or the other direction. In fact, Eq. (9) nicely represents thequantization of action, as given by Planck’s constant h : All degrees of freedom of arelativistic gas in thermal equilibrium are excited by k B T , because the tiny quant ofaction ‘ h ’ is applied with frequency Γ.When the depopulation of modes via collisions became less than one per age of theuniverse, depopulation of modes could only happen via annihilation. The total rate wasjust too low to maintain thermal equilibrium. The neutrino modes overlapped and thetransition to a degenerate Fermi gas occurred. I thus propose the following complemen-tary description of neutrino decoupling. The neutrino decoupling temperature T ν d is theone, at which the depopulation rate of effectively half the neutrino modes dropped belowypothesis on the nature and origin of cold dark matter the relative expansion rate of the neutrinos’ position uncertainty. It is not surprising that the annihilation of electron-positron pairs and thus thecreation of neutrino pairs happened at a high rate as given in Eq. (9). The electro-magnetic attraction of electrons and positrons supported their annihilation. It is lessobvious, what mechanism resulted in an equally high rate for neutrino-pair annihilation.My answer to that question is the following. The parity of Eqs. (3) and (4) shows that theneutrino position/momentum phase space before decoupling was almost packed. Here,I slightly refine this statement. In fact, balancing between creation and annihilation,requested 50% of all modes that were addressed by the neutrino pair creation were notoccupied. In this situation, 50% of all neutrino-pair creation processes indeed resultedin the population of field modes, but in 50% of the neutrino pair ‘creation’ processes anoccupied modes were addressed and the Pauli exclusion principle enforced destructiveinterference. In other words, in 50% of the creation processes, the net effect was neutrinoannihilation. This way, creation and annihilation were balanced even when the collisionrate of individual neutrinos was small.This changed when the neutrino collision rate got reduced to the inverse of the ageof the universe. The overall depopulation rate of neutrino modes reached the criticalvalue and neutrino modes overlapped and became degenerate. The newly evolveddegenerate modes did not have sufficient spatial overlap with the modes that wereaddressed by neutrino-pair creation. Destructive interference and annihilation stopped.Consequently, the creation of neutrino pairs also stopped. The model outlined hereexplains why neutrinos decoupled although the temperature of the universe was stillhigh enough for neutrino-pair production at high rate. (This eventually changed afterreheating).
3. Primordial formation of super-massive black holes
The direct and necessary consequence of large-scale neutrino Fermi seas was theprimordial formation of super-massive black holes. Their formation did not require anyagglomerations of mass via the gravitational force, i.e. no gravitational collapses, sincethe mass/energy density of the neutrino field was already sufficient for the emergenceof event horizons. As a matter of fact, the mass/energy density in the early universe,at any time, was that high that the integration over finite volumes resulted in valuesbeyond the critical value for the emergence of event horizons. In this section, I presentmy argumentation why black holes did rarely emerge before neutrino decoupling, butnecessarily emerged right after neutrino decoupling.The well-known necessary condition for the emergence of a black hole can beformulated as follows. For a black hole without spin and charge (a ‘Schwarzschild blackhole’), the total mass within a volume of radius r s (the ‘Schwarzschild radius’) has tobe above the threshold ypothesis on the nature and origin of cold dark matter M BH = r s, BH c / (2 G ) ≈ r s, BH M (cid:12) /r s , (cid:12) . (10)where G is the gravitational constant, M (cid:12) ≈ · kg is the mass of our sun and r s , (cid:12) ≈ sufficient for BHformation. My answer is the following. The energy in finite volumes of the earlyuniverse fluctuated by too much and too quickly. The continuous redistribution ofmass/energy density in the early universe frustrated the necessary condition for BHformation to remain fulfilled for a sufficiently long time. The emergence of event horizonscompeted with the emergence of those of partially overlapping volumes. An importantissue was the finite speed of light. A direct consequence of Einstein’s special theoryof relativity [42] is the fact that any kind of information can maximally travel at thespeed of light, because otherwise the principle of cause and effect (‘causality’) wouldbe violated. The above condition becomes only necessary and sufficient, if it persistsfor the same unambiguously defined volume over a time period that light requires topropagate over the distance that corresponds to the Schwarzschild radius. In the earlyuniverse, the sufficient condition for BH formation was not met. (Due to the strongthermal fluctuations it can not be fully excluded that a small number of black holesemerged during all phases of the early universe.) But the situation changed completely,once the neutrinos had decoupled. Neutrino fields of large dimension had a well-defined centre of mass location in spacetime and showed only small and slow internalenergy redistributions; neutrino fields in their ground states did not show any energyredistributions, not even due to quantum fluctuations.In the following, I quantitatively consider the sufficient condition of the primordialformation of BHs and find a lower bound to their mass. Required is the expression forthe total neutrino mass density at and after decoupling time when the neutrinos wereultra-relativistic. Neglecting the neutrino rest mass, Eq. (2) yields for all six neutrinospecies and all three degrees of freedom at decoupling time t ν d Ω ν ( t ν d ) ≈ n ν d (cid:104)| (cid:126)p |(cid:105) ν d c ≈ π · ζ (3) ( k B T ν d ) h c ≈ . · kgm . (11)After decoupling, the above mass density got reduced due to the expansion of theuniverse. The one-dimensional expansion of the universe is given by the scale factor a = 1 / (1 + z ). Generally, relativistic energy densities are inversely proportional to theincrease in the volume and to the red shift, i.e. they are proportional to a − . Neutrinodecoupling as well as subsequent times are part of the radiation-dominated era, where a ( t ) ∝ t / , with t again being the age of the universe [24]. In subsection 2.6, I argue thatthe expansion of the universe after neutrino decoupling became the more inhomogeneousthe more time passed. Right after neutrino decoupling this effect can be neglected. Themass-equivalent relativistic neutrino energy density at least for several seconds afterdecoupling can thus be approximated to ypothesis on the nature and origin of cold dark matter ν ( t ) ≈ π · ζ (3) ( k B T ν d ) h c (cid:18) t ν d t (cid:19) ≈ . · (cid:18) t ν d t (cid:19) kgm . (12)Also required is an estimation for the total mass-equivalent energy density of the restof the universe, since it contributed to the formation of black holes when event horizonsformed. The energy density of the primeval plasma Ω pp was mainly given by the degreesof freedom of the other relativistic species, i.e. electrons, positrons and photons. Its timedependence is thus equivalent to the one of Ω ν ( t ), at least for the relevant few secondsafter decoupling.Using Eqs. (10) and (12), it is evident that the necessary condition for the formationof an event horizon at time t (cid:38) t ν d was fulfilled for a sphere with radius r s ( t ) = (cid:115) M (cid:12) πr s , (cid:12) · (Ω ν ( t ν d ) + Ω pp ( t ν d )) · tt ν d . (13)The event horizon had to evolve if in addition the sphere was filled with a neutrinofield having negligible density fluctuations. In space-time volumes of initially largeneutrino field energy densities, such fields had grown since decoupling with almost thespeed of light, i.e. r s ( t ) = c ( t − t ν d ) , (14)which obeyed the principle of cause and effect.The identity of the right sides of Eqs. (13) and (14) yield the time when the firstprimordial black holes formed t minpBH = ct ν d ct ν d − (cid:112) M (cid:12) / [4 πr s , (cid:12) · (Ω ν ( t ν d ) + Ω pp ( t ν d ))] ≈ . , (15)where I use the approximation Ω pp ( t ν d ) ≈ Ω ν ( t ν d ). The first primordial black holesformed about 0.4 s after neutrino decoupling. Inserting Eq. (15) into Eq. (13) yields thecorresponding Schwarzschild radius r mins , pBH = (cid:32)(cid:115) π r s , (cid:12) (Ω pp + Ω ν d )3 M (cid:12) − c t ν d (cid:33) − ≈ . · m , (16)and the corresponding black hole mass M minpBH ≈ M (cid:12) · r mins , pBH /r s , (cid:12) ≈ · M (cid:12) , (17)again with the approximation Ω pp ( t ν d ) ≈ Ω ν ( t ν d ).Eqs. (15) to (17) are just lower bounds. Smaller primordial black holes at earliertimes could not form, however, larger black holes at later times could. The reason isthe afore discussed self-amplifying inhomogeneity of the expansion of the universe. At ypothesis on the nature and origin of cold dark matter · solar masses certainly correspond to a supermassive black hole. This result fits in an excellent way to the mass of the smallestcurrently known supermassive black hole of 5 · M (cid:12) [43].I now consider two examples well above the lower bound. The supermassive blackhole in the centre of our galaxy has a mass of 4 · M (cid:12) . Its Schwarzschild radius is1 . · m. From the perspective of flat space time, the light travel time over the radiusis 40 s, which means that a black hole of this size formed at the earliest about 40 s afterdecoupling. It is more likely, however, that the Fermi sea that formed such a large blackhole did not grow with the speed of light. If one deliberately assumes a formation after88 s, the relativistic energy density since decoupling got reduced by (88 / . . Eq. (12)together with the approximation Ω pp ( t ν d ) ≈ Ω ν ( t ν d ) yields an energy density of about10 kg / m . Integrated over a sphere with a radius of 1 . · m yields the consistentvalue for the black hole mass of 4 · M (cid:12) .An even larger supermassive black hole is located in our neighboring galaxy M87[44]. It has a mass of 6 . · M (cid:12) and its Schwarzschild radius is 2 · m, whichcorresponds to a light travel time of 18.5 h. If one deliberately assumes a formationafter 41 h, the relativistic energy density got reduced by 8 . · . Eq. (12) togetherwith the approximation Ω pp ( t ν d ) ≈ Ω ν ( t ν d ) yields an energy density of about 0 . / m .Integrated over a sphere with a radius of 2 · m yields a mass of about 6 . · M (cid:12) .According to my rationale, regions of lower neutrino energy density produced Fermiseas at later times, but with larger masses. The consequences were larger supermassiveblack holes. Multiplying Eq. (12) by the volume of a sphere with the Schwarzschildradius r s, pBH = r s, (cid:12) M pBH /M (cid:12) shows that the mass of a supermassive primordial blackhole is proportional to the time when it emerged, given by t pBH = t ν d · M pBH M (cid:12) · (cid:115) (Ω pp + Ω ν d ) 4 π r s, (cid:12) M (cid:12) ≈ · − s · M pBH M (cid:12) . (18)The emergence of supermassive black holes without gravitational collapse is aninteresting physical scenario. In the instant when the event horizon is physically defined,the enclosed space-time is of the same archetype as the outside. From then on, however,the space-time beyond the horizon needs to be seen as an independent universe withrather low total energy, which follows its own independent evolution. ypothesis on the nature and origin of cold dark matter
4. Dark matter halos and the neutrino rest mass
Astronomical observations suggest that every spiral galaxy contains a supermassiveblack hole surrounded by a DM halo. The mass of the latter usually is orders ofmagnitude larger than that of the black hole.If my hypothesis is correct, supermassive BHs and their DM halos were producedin the same process. Neutrino fields that never reached the threshold for blackhole formation (with respect to our space time) aggregated around the supermassiveBHs. The latter mainly acted as gravitational seeds in expanding space-time.Black holes that got in contact with initially independent (supermassive) neutrinofields radiated away kinetic energy in terms of gravitational waves [45] and formedsupermassive BH/DM halo systems. Black holes that emerged within larger neutrinofields had rather low differential speed vectors and produced little gravitational-waveemission.The formation of a supermassive BH/DM halo system can be understood similarlyto the formation of a hydrogen atom from a proton and a largely uncertain (free)electron. The localization of the electron in the vicinity of the proton enforces anincrease of the electrons momentum uncertainty. The increased kinetic energy balancesthe reduced potential of the electron. The electron remains in a pure, zero temperaturestate. No electro-magnetic energy is radiated away, if the process of electron localizationis spherical. A supermassive black hole together with a dark matter halo can be seenas a giant gravitationally-bound atom. Different from true atoms, whose positivelycharged core is (partially) compensated by a trapped electron, the mass of the blackhole is not compensated by a trapped neutrino field. The neutrino field rather expandsthe attractive 1 /r -potential to a larger and less steep one.Similar to electron ground-state orbitals in hydrogen atoms, DM halo cores should haveclose to rotationally symmetric shapes. The DM halo of our galaxy has a triaxial shape[46] that is not far off a sphere. Deformations might be explainable by a significant spinof the central black hole, by the partial overlap with nearby DM halos, as well as thedistribution of other masses such as smaller black holes and the ordinary baryonic mass.Following my hypothesis, the energy density of the DM halo of our galaxycorresponds to the product of the average galactic neutrino number density and theneutrino energy. According to [47] its energy density accounts to 1 . / cm in thehalo core. To provide this energy density in terms of a neutrino field, the effectiveneutrino red shift in the Milky Way according to Eq. (11) is z ν MW ν d ≈ · . If theneutrinos of the galaxy’s DM halo core indeed experienced a red shift smaller than thatof photons, their number density scales up accordingly with respect to the literaturevalue of today’s neutrino density of 3 . · / cm [24, 27]. It translates to a neutrinonumber density in the Milky Way halo core of (˜ z νν d /z ν MW ν d ) × . · / cm ≈ / cm and to an average (relativistic) neutrino energy of about 3 · . / (5 · ) ≈ .
15 eV.Wellenlaenge fordert die Knoten! Nicht die Fermionen.If galactic DM halos are neutrino-pair fields that are gravitationally allocated ypothesis on the nature and origin of cold dark matter x = (cid:114) (cid:126) m ω , (19)where ω = E p / (cid:126) is the trapping frequency and E p the trapping potential for the particle. E p corresponds to the energy that is added to the halo if a single particle that haszero average momentum with respect to the BH at infinite distance and a quantumdelocalization identical to that in the halo falls into the BH, i.e. E p = mc , yielding∆ˆ x = (cid:126) √ m c . (20)The above equation shows that the neutrino-pair mass m could be calculated if the lowerbound of DM halo radii was known. For completeness, the momentum uncertainty isgiven by ∆ˆ p = (cid:114) (cid:126) ωm mc √ , (21)fulfilling Heisenberg’s uncertainty principle. The expectation values of the particle’sposition (cid:104) x (cid:105) and momentum (cid:104) p (cid:105) , as defined in the BH’s inertial frame and with respectto the BH’s event horizon, are zero. Finally, the zero point energy of the (empty) DMhalo is given by E zpe = (cid:126) ω p m + mω ∆ x mc , (22)where ∆ denotes the variance of the observable’s quantum uncertainty.In reality, masses of galactic DM halos are several orders of magnitude larger thanthe masses of the central black holes and their core radii should be several orders ofmagnitude larger than ∆ˆ x in Eq. (20) because an extended mass distribution allows fora larger ∆ˆ x without increasing the potential energy. Heisenberg’s uncertainty relationthus offers a smaller value for ∆ˆ p , resulting in reduced kinetic energy ∝ ∆ p and totalenergy.Not only spiral galaxies have a dark matter halo and a supermassive black hole, butalso dwarf galaxies [48]. An example is the Ursa Minor spheroidal dwarf galaxy, whichis a satellite galaxy of the Milky Way and which has a dark matter halo core of 0.1 kpcradius [49]. It is a DM dominated galaxy, with a halo mass several orders of magnitudelarger than the mass of its black hole. The smallness of its halo, nevertheless, providesan interesting, albeit very conservative lower bound for the lightest neutrino mass m .Applying ∆ˆ x DM = 0 . ≈ · m to Eq. (20) yields m ≫ (cid:126) √ · c · ∆ˆ x DM ≈ · − kg ≈ · − eV /c . ypothesis on the nature and origin of cold dark matter m certainly is several orders of magnitude larger than this lowerbound. The standard model of particle physics makes the (wrong) prediction thatthe neutrino has zero rest mass. Based on this, it seems not unlikely that one ofthe three neutrino rest masses is much lower than the other two, i.e. m ≈ . m ≈ . m ≪ m .
5. Galaxy clusters and structure formation
Galaxy clusters contain hundreds or even thousands of galaxies. A specific feature ofsuch clusters is that they do not expand with the general expansion of space time, similarto individual galaxies. In contrast, some ‘super clusters’ do expand with the universe[50]. So far, it is assumed that clusters are held together by solely the gravitational force.Based on the nature of dark matter proposed here, I conclude that cluster formation aswell as cluster internal dynamics have been strongly influenced by a quantum mechanicaleffect – the quantum mechanical binding energy of galaxies due to joint DM halos. Thisbinding energy results in a so-called covalent bond, whose fundamentals are known.Consider two neighboring DM-halo/BH-systems whose halos partially overlap. Sucha joint DM halo represents a joint orbital excited by a joint neutrino field. The sizeof the joint orbital is elongated along the symmetry axis by a factor of order two. Incomparison to two DM-halo/BH-systems at large distance, the position uncertainty ofthe dark matter ∆ˆ x is increased (without an increase in potential energy). Consequently,Heisenberg’s uncertainty principle allows for the reduction of the momentum uncertainty∆ˆ p x . The neutrino field’s kinetic and total energy are reduced as well, which establishthe covalent bond. The proposed covalent bond is similar to the covalent bond inmolecules, such as in hydrogen or oxygen molecules, as outlined in Refs. [51, 52, 53].(Unfortunately, the physical description of the covalent bond on the basis of quantumuncertainties ist still not standard in relevant textbooks.) From this point of view,galaxy clusters could be seen as cosmological molecules. Different from molecules, thereis no repelling force between the nuclei (the supermassive black holes), since gravitationis always attractive. The repelling mechanism, against which the bond balances, isthe expansion of the universe. Similar to molecules, ‘gravito-covalent’ bonds stabilizeagainst merging of galaxies, because merging would reduce ∆ˆ x of the joint orbital,which requires energy that is not available. This mechanism is identical to the one thatprevents galactic dark matter halos falling into the central black hole. The quantummechanical binding energy between galaxies may correspond to a significant fraction ofthe total mass of joint DM halos. It should then significantly contribute to the stabilityof galaxy clusters against both, the expansion of the universe as well as merging ofsupermassive BHs. This fits well to the observation that DM distributions act likescaffolding [39].My hypothesis might shine new light on the analysis of collisions of galaxy clusters. ypothesis on the nature and origin of cold dark matter
6. Summary
Based on the position uncertainty of the neutrino wave-packets, their quantummechanical evolution, as well as interference, this work elaborates on the mechanismthrough which neutrinos lost interaction with the rest of the universe about half a secondafter the Big Bang. This event of neutrino ‘decoupling’ is well known, but prevailingdescriptions convey a physical picture of neutrinos having trajectories, i.e. moving likeclassical particles. I argue that such a particle-like picture is incorrect, even at timesbefore decoupling.After decoupling, according to my rationale, the neutrino wave-functions expanded withalmost the speed of light, overlapped, and evolved into macroscopic and massive neutrinofields. I provide an expression for the temperature of a neutrino field, according to whicha field of radius r has the same temperature as a Schwarzschild black hole of the sameradius. I show that the neutrino fields evolved into their quantum mechanical groundstates, so-called Fermi seas. Following the BSC theory, I argue that the neutrinosin a Fermi sea paired to bosons of zero spin, similar to Cooper pairing of electronsin superconducting metals. I show that the neutrinos of a pair are Einstein-Podolsky-Rosen entangled. Furthermore, the nonclassical feature of the Fermi seas prohibited anykind of density fluctuation, whose direct consequence was the emergence of primordialsupermassive black holes without gravitational collapse. My hypothesis suggests a directlink between neutrino decoupling time and the mass of the smallest supermassive blackholes. For the actual decoupling time of half a second after the Big Bang, my hypothesispredicts a lower bound of about 4 · solar masses. This value is in excellent agreementwith the smallest known supermassive black hole (5 · M (cid:12) ) [43]. I show that the smallexemplars evolved as early as one second after the Big Bang. Larger ones emerged inregions of smaller quantum field density, but over larger volumes and at later times.Subsequently, supermassive black holes and remaining dark-matter fields formed thedark component of todays galaxies and galaxy clusters.I argue that dark-matter fields around supermassive black holes experienced a muchlower average expansion of space-time than electromagnetic radiation, since theirgravitational attraction locally resisted against the expansion. According to myestimation, neutrino fields can explain all cold dark matter in the universe if the red- ypothesis on the nature and origin of cold dark matter z νν d ≈ · instead of ˜ z νν d ≈ . · ,where the latter is the red-shift of relativistic cosmic-background neutrinos if they werehomogeneously distributed in space-time.My ‘neutrino-field hypothesis’ answers the long-standing question why the centres ofdark matter halos have the shapes of cores rather than ‘cusps’ [54] in the following way.Similar to the electron probability distribution in a hydrogen atom in ground state,quantum position uncertainties generally have a rather flat shape at its centre. Myhypothesis also answers the question why DM halos are stable since their formation.Again it is useful to compare a DM halo around a supermassive black hole with theelectron ground-state orbital around a proton. The electron does not fall into theproton because a more precisely determined position uncertainty would request a largermomentum uncertainty and thus a larger kinetic energy. My hypothesis also providesa solution for the so far unknown mechanism of primordial formation of super-massiveblack holes. And finally it suggests a gravito-covalent bond due to joint DM orbitals asa significant contribution to galaxy-cluster formation and stability. Acknowledgements
This work was funded by the European Research Council (ERC) project ‘MassQ’(Grant No. 339897). The author acknowledges Wilfried Buchm¨uller, Ludwig Mathey,Henning Moritz for helpful discussions, and Mikhail Korobko and Christian Rembefor valuable comments on the manuscript. The author further acknowledges usefuldiscussion within ‘Quantum Universe’ (Grant No. 390833306), which is financed bythe Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) underGermany’s Excellence Strategy - EXC 2121.
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