Новиков Евгений Алексеевич: другие произведения.

Age of the universe and interface between dark and ordinary matter [возраст Вселенной и взаимодействие тёмной и обычной материи]

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   Evgeny A. Novikov
   University of California - San Diego, BioCircuits Institute,
   La Jolla, CA 92093 -0328; E-mail: enovikov@ucsd.edu; http://evgenyn.blogspot.com
   This work is based on modification of the general relativity, which includes effects of production /absorption of matter by the vacuum. The theory (without fitting parameters) is in good quantitative agreement with cosmological observations (SnIa, SDSS-BAO and reduction of acceleration of the expanding universe). In this theory, at least in the dust approximation and in a flat case with pressure, there is no singularity and no critical density of the universe. It is shown that an effective age of the universe is about 328 billion years. Production of dark matter particles have started 43 billion years later. From the theory it also follows that an interface exist between dark and ordinary matter (IDOM). Manifestation of that interface is considered.
   Key words: cosmology; age of the universe; dark matter; interface between dark and ordinary matter; subjectivity.
   1. Introduction
   The standard theory (ST) in conventional cosmology is based on three major assumptions: Big Bang (BB), Cosmological Constant (CC) and Inflation (INF). Huge and useful work have been done in frames of ST. But, doubts about the basic assumptions are remaining.
   BB corresponds to a particular Friedmann solution [1] of the classical equations of general relativity (GR). But, is it natural and physical? I do not think so [2-4] and I am far from been alone. There is growing evidence that many stars are older than 13.8 billion years (assumed for BB) and age of other cosmic objects are hardly compatible with ST( see, for example, recent collection of such data in [5]). These observations are not at the level to proof that BB did not took place, but, at least, they give some warning. Note, that Friedmann solution create controversial critical density of the universe, which in turn create controversial dark energy.
   As concern to CC, it is known, that inventor of CC Einstein was unhappy about it, especially after his friend Gödel gave him, as a birthday present, unphysical GR solution with CC [6]. It was a long before it turns out, that in order to be consistent with global scale observation, CC should be unphysically small. Numerical solutions of GR with such CC contradict observations on the scale of galaxies [7].
   INF appeared as a rescue mission, when it was found that BB+CC contradict observations. Recent hopes to support INF by observation [8] turn sour [9]. Again, the data did not proof that there was no INF, but create some doubts.
   In my opinion, such singularities, as BB and INF can be temporary used as some mathematical asymptotics, but not as real basis for physical theory. But, my primary motivation for this work was unphysically small CC with unclear physical sense.
   Below we consider a different theory, supported by observations, which dismisses all three major assumption of ST. In order to better explain such fundamental change, let us briefly describe how this theory came about. I became interested in gravitation in late 1960-th. Because of my experience in fluid dynamics, two things surprised me at the time: the Lagrangian description of gravity (LDG) were not used and situations with spatial dimension less than 3 were not considered (a taboo?). So, I decided to do both and obtained Lagrangian invariant (relative acceleration of particles) and exact general analytical solution for (1+1)-dimensional Newtonian gravitation [10]. This is an example of trivialisation, which I always enjoy (see below). Before publication, this paper was discussed with Ya. B. Zel'dovich, who express great enthusiasm and a few months later told me that he and his collaborators have a continuation of ideas presented in my paper. The editor E. M. Lifshitz was surprised, but did not object publication, even did not object the remark in the paper: <>[10]. Than came Zel'dovich approximation [11], "pancakes" and further development in this direction [12].
   I returned to fluid dynamics for a long time until acceleration of the universe was observed [13,14]. The acceleration was explained by using CC, which is hundred orders smaller than can be predicted in the frames of classical GR. That was to much for me to accept. GR has to be modified. But how? Major player in GR is the spacetime curvature, which is balanced by the energy. But global curvature is close to zero in our universe. So, what else can play a role compatible to curvature? I had no desire to deal with new unknown fields and , in accord with trivialisation (Occam's razor), was looking for something very simple. And here, again, came help from fluid dynamics: divergency (stretching) of velocity field, which is related to creation/absorption of particles by the vacuum. So, to begin with, I have invented a new type of fluid, namely, dynamics of distributed sources/sinks (DoDSS), in which divergency is Lagrangian invariant [15]. With constant initial distribution of divergency, it gives effect similar to CC [15]. The next step was relativistic DoDSS [16] with the covariant divergency (2). Finally, came modified general relativity (MGR) [2], which is described below. I think, Einstein will be happy with such modification. This alternative to CC did not occurred to him, probably, because he came to GR from electricity, so to speak.
   2. Modified general relativity
   Now, from words we are coming to equations of MGR [2]:
   R_{i}^{k}-(1/2)δ_{i}^{k}R=8πG_{∗}T_{i}^{k}+λ_{N}δ_{i}^{k},T_{i}^{k}=wu_{i}u^{k}-δ_{i}^{k}p,w=ε+p, #1
   λ_{N}=λ₀+β((dσ)/(ds))+γσ²,σ=((∂u^{k})/(∂x^{k}))+(1/(2g))((dg)/(ds)),(d/(ds))=u^{k}(∂/(∂x^{k})) #2
  Here R_{i}^{k} is the curvature tensor, p, ε and w are pressure, energy density and heat function, respectively, G_{∗}=Gc⁻⁴(G- gravitational constant, c- speed of light), u^{k} - components of velocity (summation over repeated indexes is assumed from 0 to 3, x⁰=τ=ct), λ₀ is CC (which we will put zero), σ is the covariant divergency, β and γ are nondimensional constants (which we will put β=2γ=2/3) and g is the determinant of the metric tensor. With β=γ=0 we recover the classical equation of GR. Let us note that curvature terms in (1), dσ/ds and σ² all contain second order (or square of first order) derivatives of metric tensor, which make these terms compatible. The importance of σ also follows from the fact that it is the only dynamic characteristic of media, which enters into the balance of the proper number density of particles n: dn/ds+σn=q, where q is the rate of particle production (or absorption) by the vacuum. So, if n is constant (see the exact analytical solution (5) below) or changing slowly, than the σ-effect is, certainly, very important in quantum cosmology.
   Some exact analytical solutions of equations (1,2) where obtained in Ref. 2. On the basis of these solutions, it was concluded that the effect of spacetime stretching (σ) explains the accelerated expansion of the universe and for negative σ (collapse) the same effect can prevent formation of singularity. Equations (1,2) reproduce Newtonian gravitation in the nonrelativistic asymptotic, but gravitational waves can propagate with speed, which is not necessary equal to speed of light [3]. In the case β=2γ equations (1,2) can be derived from the variational principle by simply replacing the cosmological constant λ₀ (in the Lagrangian) by λ=λ₀-γσ²[3].
   The natural next step was quantitative comparison with cosmological data and choice of nondimensional constants β and γ. Let us consider equations for the scale factor a(τ) in homogeneous isotropic universe (Eq. (8,9) in Ref. 2):
   (2-3β)(a/a)+(1+3β-9γ)((a/a))²+(k/(a²))-λ₀=-8πG_{∗}p, #3
   -β(a/a)+(1+β-3γ)((a/a))²+(k/(a²))-((λ₀)/3)=((8π)/3)G_{∗}ε. #4
  Here points indicate differentiation over τ, the discrete curvature parameter k=0,+1,-1 corresponds to flat, closed and open universe, respectively.
   With indicated in [2] unique choice β=2γ=2/3, these equations take simple form:
   (k/(a²))=λ₀-8πG_{∗}p, #3'
   H=((3k)/(2a²))-((λ₀)/2)-4πG_{∗}ε, H≡(a/a) #4'
   From (3') with λ₀=0, we see that sign of curvature is opposite to sign of pressure. From observations we know that global curvature is close to zero. So, the dust approximation (p=0 ) is natural for this theory with λ₀=0 and β=2γ=2/3.
   In the dust approximation with λ₀=0,k=0, two special cases for system (3-4) have been indicated [2]: 1) for β=2/3 and γ≠1/3 stationary solution exist; 2) for β=2γ the global energy is conserved, except for β=2γ=2/3. The choice β=2γ=2/3 is really exceptional and in the dust approximation with λ₀=0,k=0, equation (3') is identity and from (4') we have exact analytical Gaussian solution:
   a(τ)=a₀exp[H₀τ-2π(τ/L_{∗})²],L_{∗}=(G_{∗}ε₀)^{-1/2} #5
   Here subscript 0 indicate present epoch (τ=0) and H₀ is the Hubble constant. In the solution, obtained in [3], instead of ε₀ was w₀=ε₀+λ₀/8πG_{∗}, for generality.
   Solution (5) corresponds to continuous and metric-affecting production of dark matter (DM) particles out of vacuum, with its density ρ₀=ε₀c⁻² being retain constant during the expansion of spatially flat universe. In this solution there is no critical density of the universe, which is a kind of relief.
   The solution (5) is shown [3] to be stable in the regime of cosmological expansion until t_{max} about 34 billion years from now. After that time, the solution becomes unstable and characterizes the inverse process of dark matter particle absorption by the vacuum in the regime of contraction of the universe. This can imply the need for considering the change of regime (5) at t>t_{max} to a different evolutionary regime, possibly, with a different value of the parameter γ or with the more general model (2) from [2].
   In this context, it is tempting to consider equations (1,2) without curvature terms in (1). In the dust approximation (with λ₀=0,k=0), equations (3,4) give not only stationary regime with ε=0, but also dynamical solution:
   a(τ)=a₀(1+θ₁H₀τ)^{1/θ₁},θ₁=3γ/β #6
   With H₀>0,θ₁>0, from (6) we get: a=0 at τ=-1/H₀θ₁, a≈a₀(θ₁H₀τ)^{1/θ₁} for τ≫1/θ₁H₀ - power-law expansion. With H₀>0,θ₁<0, (6) gives: a→0 at τ→-∞, a→∞ at τ→1/|θ₁|H₀ - blowup at finite time. With θ₁=0: H=H₀, a(τ)=a₀exp{H₀τ}.
   In order to solve equations (3,4) in more general case, we choose the simplest equation of state, which does not introduce a dimensional constant: p=ϰε, where ϰ is nondimensional constant. Particularly, with ϰ=0 we return to the dust approximation, ϰ=1/3 corresponds to ultrarelativistic matter. From (3,4) with λ₀=0, we obtain invariant:
   I=(a²+μ)(((a₀)/a))^{2(1-θ)}=(a₀²+μ), μ=((k(1+ϰ))/(1+3ϰ+3(1+ϰ)(β-3γ))), θ=((3(1+ϰ)(1-3γ))/(2-3β(1+ϰ))) #7
   This is a generalization of invariants, introduced first in [2] and than used in [3] for more special cases. Without stretching effects (β=γ=0), we have μ=k(1+ϰ)/(1+3ϰ), θ=3(1+ϰ)/2.For k>0 and ϰ>-1/3, we get μ>0, θ>1. With such parameters, gravitational collapse (a₀<0), according to (7), will lead to singularity (a→0,a→-∞). Stretching effects can prevent singularity. With θ<1,μ>0, from (7) it follows that gravitational collapse will lead to a finite core:
   a→a_{∗}=a₀[(μ/((a₀)²+μ))]^{(1/(2(1-θ)))} #8
   Expression for θ in (7) indicate special case with γ=1/3, as before, and β(1+ϰ)=2/3. Assuming, as above, λ₀=0, from (3) -(4) we get ka⁻²(1+3ϰ)=0. For k=0, from (4) we get solution similar to (5) with substitution ε₀ by ε₀(1+ϰ). For ϰ=-1/3, the solution in quadrature is more complicated and deserves detailed consideration in future.
   Mass m₀ of dark matter particles have been estimated [4] by comparing characteristic scale L_{∗} from (5) with the relativistic uncertainty of particle position [17] (or Compton wavelength) ħ/m₀c, where ħ is the Plank constant. We have:
   m₀∼ħ(G_{∗}ρ₀)^{1/2}∼0,5 10⁻⁶⁶gram. #9
   Here, according to WMAP data, we use ρ₀≈0.26⋅10⁻²⁹gcm⁻³, which includes dark and ordinary matter, but not dark energy. In this theory, flatness of the universe is supported by the divergency terms in (1-2).
   Estimate, similar to (9), we got before [3] from more complicated consideration, which involves solution of a model equation for a quantum field, so, this is also an example of trivialisation. According to (9), DM particles are ultralight and their uncertainty of position L_{∗}≈76 billion light years (bly) is of the same order as size of the visible universe a₀≈46,5 bly. So, we can say, that mass of primary dark matter particles (PDMP) m₀ predetermined the size of universe (see also next section). It also means that universe has a halo of DM particles. This halo potentially can influence the visible part of universe, producing effects similar to influence of hypothetical multiverse. The same effect (large uncertainty of DM particle position) can explain halo of a galaxy, which is more easy to observe (see, for example, paper [18] and references therein). Formula (5) does not have any fitting parameters and shows good quantitative agreement with cosmological observations (SnIa, SDSS-BAO and reduction of acceleration of the expanding Universe [19]) [3,4].
   In retrospect, some early theoretical papers are relevant to our work, particularly, [20-23]. These and others relevant papers are discussed in [3]. The physical nature of the ultralight dark matter particles is also discussed in [3] and arguments in favor of scalar massive photon pairs are presented there. So, the dark matter, which penetrate our visible universe and beyond (halo), could be light, packed into photon pairs. Irrespective of this particular interpretation, the quantity m₀ defined in (9) can also serve as a basis for subsequent reconsideration of the problem of divergence in quantum field theory [24,25].
  3. Age of the universe
   According to (5), our universe was born in infinite past from small fluctuation. But, physically speaking, we can choose some initial scale for an effective beginning of the universe. From (5) we get:
   T=h₀+(h₀²+s)^{1/2}, T=τ/L_{∗}, h₀=H₀L_{∗}/4π, s=(1/(2π))ln(((a₀)/(a(τ)))) #10
  For τ<0 we have s>0 and in formula for T the sign is minus. It seems natural to choose Planck length l_{P}=(G_{∗}cħ)^{1/2} as an initial scale, at which we can expect beginning of a smooth metric. With a(τ)=l_{P} and h₀≈0.45 (H₀c≈2.4⋅10⁻¹⁸s⁻¹), from (10) we get corresponding time t₁≈-327 billion years. So, at the effective beginning of the universe there was a spec of matter, which we will call Premote, with size l_{P} and mass M₁=ρ₀l_{P}³≈10⁻¹²⁸gram. The uncertainty of position for Premote is L₁=ħ/M₁c≈10⁹⁰cm≈10⁶³bly. So, the probability of finding Premote can be estimated by (l_{P}/L₁)³∼4⋅10⁻³⁶⁹.
   The next step is when universe is ready to accommodate production of PDMP with mass (9). Solution (5) corresponds to constant mass density ρ₀ with concentration of particles n and characteristic scale l:
   n=ρ₀/m₀∼0.5⋅10³⁷cm⁻³, l=n^{-1/3}∼0,27⋅10⁻¹²cm. #11
  With that scale from (10) we get t₂≈-284 billion years. So, it took about 43 billion years to accommodate universe for production of PDMP. The mass of the universe at time t₂ was M₂=ρ₀l³∼m₀. As was said above, the uncertainty of position L_{∗} predetermined the size of the visible universe and, according to (5), a_{max}/a₀≈3.56.
   4. Dark matter
   According to cosmic observations, DM interacts with ordinary matter (OM) only gravitationally. So far, in frames of our theory, we know the mass (9) of PDMP, which is very small, and averaged concentration (11), which is not only enormous, but also constant. It means, that these particles somehow communicate with each other (perhaps, even superluminally) and polarize vacuum in order to maintain averaged distance l (11). Remember, that we are dealing with unusual fluid [15]. In the areas of gravitational condensation (future galaxies) the density was even much higher. With certain critical density, we can expect multiple collisions with formation of new particles in some sort of "natural selection". During the steady and stable expansion of the universe, the OM was synthesized in this way, probably, starting with light particles. This process was accompanied by radiation, which is reflected in CMB. The eqilibrium character of CMB and the global condition R≈0 are naturally explained by the large amount of time available for the evolution. Some peculiarities of CMB can be associated with synthesis of various particles in expanding universe. Particularly, the observed anisotropy of CMB can be connected with nonsynchronous processes in galaxies. In context of the type of evolution, which is described by exact solution (5), what we call ordinary matter is, in fact, an exotic matter, which was synthesized from PDMP and, so far, constitute only small fraction of the total mass of the universe (about 4%). The theory of elementary particles should be modified by considering DM as primary basis for all particles. Moreover, we can not be sure that DM obeys all the rules of the conventional quantum theory. It is possible, that DM produces some quantum effects for "ordinary" matter (see new interpretation of quantum theory [26]). The presented cosmology is more "quiet"(at least, in the beginning) than the usually accepted "Big Bang + Inflation" scenario. Father development of such theory and corresponding experimental investigation, in our opinion, will greatly benefit humankind (see, particularly, one human aspect of the theory in the next section).
   However, this is not work for one person. The short list of what we need to do is: 1) based on equations (1,2) without fitting parameters (λ₀=0, β=2γ=2/3), or in more general case (λ₀=0,β(1+ϰ)=2/3,γ=1/3), calculate formation of galaxies and compare results with Sloan Digital Sky Survey and Canada-France-Hawaii Telescope Legacy Survey; 2) using the same equations (1, 2), for simplicity in spherically-symmetric case, calculate gravitational collapse and look what modification of the classical singular Black Holes we got in this theory; 3) develop detailed model for PDMP interaction and synthesis of OM particles; 4) calculate temperature and polarization anisotropies of CMB and compare with measurements ( WMAP and Plank missions). I will be happy if cosmologists, with experience in corresponding work in frames of ST, can contribute in this development. Below we consider another important aspect of the theory, which can help in the project 3) in the above list.
   5. Interface between dark and ordinary matter (IDOM)
   Description of DM (sections 3, 4) leads us to very old big mystery in science, namely, the physical nature of our subjective experiences. The qualia (subjectivity) required for its description enormous number of degrees of freedom (NDF) and was historically considered as otherworldly. DM is otherworldly in the indicated above sense and, according to (11), has huge NDF. It is generally accepted, that qualia is not matter, but some sort of information. At the same time, qualia is imbedded in our body, which is made from OM. And here is a catch. Are we sure, that our body does not have a little bit (mass) of DM? The described above MGR, supported by cosmic observations, shows that DM is omnipresent and continuously produced everywhere. If we accept that, than qualia can be connected with DM. How? By been something in between two different types of matter, say, an interface. Indeed, if we, the people, have some DM in our body, than Mother Nature had plenty of time to make use of it by creating special conditions in our neural system in favor of some form of interaction with DM. This special form of interaction may not be easily detectable in cosmic data or in the supercollider. So, our neural system could be the natural detector for a new form of interaction between two different types of matter. Qualia seems to be a manifestation of this interaction. In what follows, we will unveil some details. Taking into account that problem of subjective experiences is of great interest and importance for general public, the main text in this section does not include explicit mathematics. However, for experts, in cited papers there is plenty of mathematical modeling. Some equations and technical remarks are presented in the Appendix (A1-A3).
   There is huge literature on modeling of consciousness ( see an interesting collection of papers [27] and references there). For the purpose of this paper, we will need only specific aspect of such modeling. The phenomena of consciousness can be considered as hierarchy of observations and control [28]. Hierarchical structures appear naturally in systems with big NDF. Typical signatures of such hierarchy are so-called similarity laws. Particularly, in turbulence the concept of scale-similarity was developed and was associated with the infinitely-divisible distributions [29, A1]. The activity of the human brain also revealed the regime of scale-similarity, which was discovered by using the multi-channel MEG (magnetoencephalogram) [30,31] and EEG (electroencephalogram) [32] (see also [33,34]). Hundreds of billions of interconnected neurons and surrounding sells (particularly, astroglia), apparently, is favorable playground for hierarchical structures in the brain. The electrochemical brain activity is taking place in wet and warm surroundings. To reproduce such activity in artificial systems, even approximately, seems impossible. However, modeling of the effects of consciousness [35-37] can be used to enhance performance of artificial stochastic systems [28]. In the modeling [36,37], the subjective experiences were divided into three major groups: sensations (S), emotions (E) and reflections (R). Note, that subjective S should be distinguished from the automatic sensory input into the neuron system of the brain [38]. Consider so called quaternion (generalization of complex number, see A2), which in our case has real component (the electric current density perpendicular to the cortical surface) and three imaginary components representing the indicated above (S, E, R, or simply SER) - effects. Corresponding imaginary units satisfy conditions: 1) square of each of them is equal to -1; 2) product of two different imaginary units is antisymmetric (changes sign with transposition) and is equal to the third unit with sign determined by the cyclic order ( say, product of the first and second units is equal to the third unit with sign plus, while product of the third and second units gives the first unit with sign minus). The quaternion is a function of time and space coordinates on the surface of the cortex. The model equation for this quaternion [36,37] is a nonlinear partial differential equation, which contains the linear wave terms (with the second order time and space derivatives), linear relaxation term and a nonlinear term representing the sigmoidal firing rate of neurons [A2]. If we omit the (SER) -effects, than equation will be similar in spirit to equation used for interpretation of EEG and MEG spatial patterns (see [39] and references therein). Note, that without (SER)-effects the system behaves robot-like, while with (SER)-effects it is more flexible.
   The essential point of (SER) - modeling is that imaginary fields produce real effects (testability) because of the nonlinear firing rate of neurons. Note, that complex fields have been used [25] to eliminate classical electromagnetic divergencies, namely, the infinite self-energy of electrons and the paradoxical self-acceleration of electron. The same (algebraic) approach works for the quantum interaction of charges. In new interpretation of quantum theory [26] imaginary trajectory and corresponding momentum play an important role. Such broad usefulness of imaginary field is indicative of a new form of interaction in Nature (see above).
   The (SER) - modeling is designed for description of the effects of consciousness on the electric currents in the human brain. In order to advance in the problem of qualia (subjectivity) we now turn to MGR (described in section 2). Note, that gravitation is resisting quantization, unlike the other three interactions (EM, strong and week). In a sense, interaction of DM with OM can be presented in the form of indicated above quaternion with gravitation as real component and other three components imaginary (see also [36]). If we continue with this analogy, than (apart from gravitation) some indirect form of interaction, similar to nonlinear firing rate of neurons, can exist between dark and ordinary matter. Indeed (see above), OM was synthesized from DM as a result of multiple hierarchical collisions. In this sense, dark matter is working similarly to neural system.
   From what was described in this paper, it seems natural to suggest that qualia is inherited from DM. The indicated above ultralight particles (PDMP) are constantly produced by the vacuum everywhere, including our body and our brain. Perhaps, so called biophotons (see [40] and references therein) are related to production of PDMP. Inside neurons and in surrounding sells we may have special conditions, which can facilitates interaction with PDMP. Every living creature may have inside the body and in a halo an enormous number of PDMP without noticeable gravitational effect. At the same time, hierarchical processes in such system with huge number of PDMP can be associated with qualia. In this way, some macroscopic "objective" degrees of freedom are effectively transforming into structures with internal ("subjective") degrees of freedom. In this sense, qualia is manifestation of an interface between dark and ordinary matter (IDOM) [A3]. An analogue of such interface are the ocean waves. More relevant analogue is the scale-similar intermittency [29] with viscous dissipation on very small scales (for turbulence it is Kolmogorov microscale with intermittency correction [29], for qualia it will be Plank scale with possible intermittency correction).
   The best way to investigate these effects is, probably, during events of extremal qualia, such as pain or orgasm (preferable). Orgasm has many definitions [41], none of them totally satisfactory. Generally, orgasm has different feeling depending of sources of stimulation (including mental stimulation) and corresponding nerves. Combinations of sources in simultaneous stimulation produce so-called blended orgasms, which are, generally, more powerful (particularly, in women). The physical nature of orgasm is a total mystery. The electrochemical signals repeatedly reach brain and than something happens, which reminds lightning, but in a "mental world". Another case of extremal qualia is improvisational dance (spontaneously creating movements).
   The modeling of the effects of consciousness suggests existence of a particle or a group of particles - mediators between dark and ordinary matter (MeDOM), which may have a superluminal component, related to imaginary field in modeling [A2]. PDMP can produce MeDOM spontaneously, or, more likely, during collisions. MeDOM in turn produce additional ordinary photons during the nonlinear process of neuro-firing. So, the one thing, which can be tested during orgasm (or improvisational dance) is enhanced radiation with a peculiar spectrum (power law with possible log-periodic modulation [A1]).
   Similar scheme can be applied to cosmic events. Collisions of PDMP produce MeDOM - sparks of dark matter. In nonlinear process of hierarchical collisions, the "plasma" of PDMP and MeDOM produces particles of ordinary matter, including ordinary photons. Note, that only small fraction of PDMP collisions produces ordinary matter. Cosmological observations (for example, [42]) indicate that more substantial portion of such interactions produce some lumps and clouds of dark matter.
   Of course, this is only an outline of future theory. Particularly, MeDOM with possible connection to Premote (section 3) should be worked out in detail. But the major conclusion that qualia manifests IDOM seems to be insensitive to many details of the theory. Indeed, DM is the background, from which emerged OM and than emerged Qualia (A3). It is argued above, that Qualia remains dependent on the background. So, Qualia (information with huge NDF) depends on two different types of matter. Such connection can be considered as an interface [43].
   Do dark matter, which we now observe only by the gravitational effect, has some sort of qualia (perhaps, connected to MeDOM)? If so, are they similar to indicated above SER-qualia, which we possess? And, finally, can we (perhaps, with a proper equipment) communicate with dark matter? The positive answer to this question can lead to revolution in the history of humankind. Particularly, humans can get access to enormous energy resources and computational power.
   The idea of omnipresent substance is, actually, very old and some useful medical recommendations are based on it. The progress is that we now understand that it is a special kind of matter, invisible for us directly, but gravitating.
   The main conclusion of this section is that such seemingly divorced phenomena as consciousness and dark matter, in fact, could be closely connected. These two very important areas of research can greatly benefit each other from their proper coordination.
   A1. We should distinguish between discrete and continuous self-similarity. In the discrete case there is a preferable scale factor leading to the logarithmically periodic modulations [29].
   A2. Consider quaternion:
   q=α+i_{p}ψ_{p} #12
   Here α(t) is the average (spatially uniform) current density perpendicular to the cortical surface, ψ_{p}(t) represent the indicated above (S, E, R) - effects and summation is assumed on repeated subscripts from 1 to 3. The imaginary units i_{p} satisfy condition:
   i_{p}i_{s}=ε_{psr}i_{r}-δ_{ps}, where ε_{psr} is the unit antisymmetric tensor and δ_{ps} is the unit tensor. It is a compact form of conditions: i₁²=i₂²=i₃²=-1, i₁i₂=-i₂i₁=i₃, i₂i₃=-i₃i₂=i₁, i₃i₁=-i₁i₃=i₂.
   The model equation for the quaternion q has the form [36,37]:
   ((∂q)/(∂t))+kq=f(q+σ)+φ, σ=s+i_{p}ϕ_{p} #13
  (Notations in this appendix should not be confused with notations in sections 2 and 3). Here k is the relaxation coefficient, f represents the sigmoidal firing rate of neurons [for example, f(α)=tanh(α)], φ represents the external electromagnetic (EM) excitations. The quaternion σ is the averaged sensory input, which has real component s and imaginary components ϕ_{p} (which can be associated with the influence of DM).
   For the case of spatially nonuniform q(t,x), σ(t,x) and φ(t,x), we can use more general equation, which include typical propagation velocity of signals in the neuron system of the cortex v. Time differentiation of (13), simple algebra and addition a term with the two-dimensional spatial Laplacian Δ gives [36,37]:
   ((∂²q)/(∂t²))+(k+m)((∂q)/(∂t))+(km-v²Δ)q=(m+(∂/(∂t)))f(q+σ)+((∂φ)/(∂t)) #14
  where m is an arbitrary parameter (see below). Real and imaginary projections of (14) give a system of four partial differential equations for α and ψ_{p}. If we put ψ_{p}=0 and φ=0, than equation for α will be similar in spirit to equation used for interpretation of EEG an MEG spatial patterns (see [39] and references therein). In this context we have parameters: k∼m∼v/l_{c} , where l_{c} is the connectivity scale.
   A3. Interface between Dark and Ordinary matter (IDOM) with presence of Qualia can be described as part of general scheme:
   Pr emote→DM→MeDOM↑→OM→Qualia↑ #15
   where Premote and MeDOM are explained in sections 3 and 5 respectively. This simple scheme can have loops (indicated by vertical arrows) for potential communication of human with DM.
   I thank S. G. Chefranov for useful comments.
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