Journal of Modern Physics, 11,, 1331-1347 doi:1.436/jp.11.11165 Published Online Noveber 11 (http://www.scirp.org/journal/jp) Transforation of the Angular Power Spectru of the Cosic Microwave Background (CMB) Radiation into Reciprocal Spaces and Consequences of This Approach Abstract Ladislav Červinka Institute of Physics, Acadey of Sciences of the Czech Republic, Prague, Czech Republic E-ail: L.Cervinka@icaris.cz Received July 3, 11; revised Septeber 8, 11; accepted Septeber, 11 A foralis of solid state physics has been applied to provide an additional tool for the research of cosological probles. It is deonstrated how this new approach could be useful in the analysis of the Cosic Microwave Background (CMB) data. After a transforation of the anisotropy spectru of relict radiation into a special two-fold reciprocal space it was possible to propose a siple and general description of the interaction of relict photons with the atter by a relict radiation factor. This factor enabled us to process the transfored CMB anisotropy spectru by a Fourier transfor and thus arrive to a radial electron density distribution function (RDF) in a reciprocal space. As a consequence it was possible to estiate distances between Objects of the order of ~1 [] and the density of the ordinary atter ~1 [kg 3 ]. Another analysis based on a direct calculation of the CMB radiation spectru after its transforation into a siple reciprocal space and cobined with appropriate structure odelling confired the cluster structure. The internal structure of Objects ay be fored by Clusters distant ~1 [c], whereas the internal structure of a Cluster consisted of particles distant ~.3 [n]. The work points in favour of clustering processes and to a cluster-like structure of the atter and thus contributes to the understanding of the structure of density fluctuations. As a consequence it ay shed ore light on the structure of the universe in the oent when the universe becae transparent for photons. On the basis of our quantitative considerations it was possible to derive the nuber of particles (protons, heliu nuclei, electrons and other particles) in Objects and Clusters and the nuber of Clusters in an Object. Keywords: CMB Radiation, Analysis of CMB Spectru, Radial Distribution Function of Objects, Early Universe Cluster Structure, Density of Ordinary Matter 1. Introduction The angular power spectru (anisotropy spectru) of the Cosic Microwave Background (CMB) radiation ([1,]) shows incredible siilarity with X-ray or neutron scattering easured on non-crystalline aterials ([3], [4]), see Figures 1 and. Astronoers ascribe to various peaks of the anisotropy spectru of the CMB radiation different processes [5]. It is the Sachs-Wolf effect, Doppler effect, Silk daping, Rees-Sciaa effect, Sunyaev-Zeldovich effect, etc. In this connection it should be stated that all theoretical predictions of the standard cosological odel are in very good agreeent with the course of the anisotropy spectru of CMB radiation. However, the foral siilarity in the for of both figures initiates the tepting idea if an analysis of the anisotropy spectru of relict radiation using an analogous approach as is coon in solid state physics, i.e. in the structural analysis of disordered aterials, would bring ore inforation on the structure of the early universe. The inspiration for this approach we found further in the nowadays situation: Although the individual disciplines in physics are highly specialized, nevertheless their ethods and results are shared in areas that at the first sight ay see to be far apart. An exaple of this is the already established use of eleentary particle physics
133 L. ČERVINKA Intensity [Arbitrary units] 6 5 4 3 1 5 1 15 5 3 Multipole oent L Figure 1. Anisotropy spectru of the CMB radiation [1]. The figure describes the dependence of the agnitude of the intensity of icrowave background on the ultipole oent L = 18 /α, where α is the angle between two points in which the teperature fluctuations are copared to an overall ediu teperature. The description of the Y-axis is for siplicity described in [Arbitrary units]. The original description was given as L(L + 1)C L /π in [μk ] units, where L is the ultipole oent, C L is a function reflecting the width of the window easuring the teperature fluctuations. Intensity [Arbitrary units] 1 8 6 4 4 6 8 1 1 14 16 Reciprocal space scattering vector s [n -1 ] Figure. X-ray scattering diagra taken on a saple of a chalcogenide glass of a coposition (Ge.19 Ag.5 Se.5 ) using the MoK α radiation, see [4] for detail. The reciprocal space scattering vector s is defined in Equation (A5). in cosology. Siilarly, we hope that it ay be tie now to apply the foralis of solid state physics to soe special cosological probles and in this way to provide an additional tool for their research. First of all our new approach ay be useful in the analysis of the CMB data. We will show how after the transforation of the anisotropy spectru of relict radiation into a special two-fold reciprocal space we will be able to process the transfored CMB anisotropy spec- tru by a Fourier transfor and thus calculate a radial distribution function (RDF) of the atter in a reciprocal space. Because the CMB radiation reflects the fluctuations in the density of the atter, we hope that in this way our study will contribute to the understanding of the structure of these density fluctuations. Siultaneously, as a consequence, it ay shed ore light on the structure of the universe in the oent when the universe becae transparent for photons (see Subsection 5.1.). Moreover, in contrast to solid state physics where the atoic (coherent) and Copton (incoherent) scattering factors are describing theoretically the interaction of X-rays (or neutrons) with all kinds of atos, this new foralis will present a general description of the interaction of relict radiation with the atter by a single relict radiation factor, which should unify all processes realized during the interaction of relict radiation with various kinds of particles foring the priordial atter, see Subsections.3.3. and 5... Construction of the and Relict Reciprocal Space In solid state physics the principal atheatical ethod during the structure analysis of the atter is the Fourier transfor of the intensity of X-rays (or neutrons) scattered by atos building the aterial. The experiental data are collected in the reciprocal space and their Fourier transfor brings the required inforation on the distribution of atos in the real space. In this contribution we will try to apply this approach to the CMB spectru (see Figure 1) and siultaneously point out the coplications we have to overcoe in this direction. The necessary basic atheatical apparatus is suarized in the Appendix A, the ost iportant basic equations for the analysis of scattered radiation and leading to the radial density distribution function (RDF) are Equations (A1) and (A). The essential difference in the use of ters scattering and interaction of photons will be elucidated in the next Subsection.1..1. The Relict Radiation Factor During a conventional structure analysis with X-rays or neutrons, the X-ray or neutron atoic scattering factors are a precise picture of the interaction of radiation with the atter and are known precisely [6]. They enter into the calculation of the RDF in correspondence with the coposition of the studied aterial; see Equations (A6), (A7) and (A1). Generally, for coherent scattering, the atoic scattering factor f is the ratio of the aplitude of X-rays scattered by a given ato E a and that scattered according to the classical theory by one single electron
L. ČERVINKA 1333 E e, i.e. f Ea Ee (f Z ), where Z is the nuber of electrons in the ato. Moreover, there are scattering factors not only for the coherent but also for the incoherent (Copton) type of scattering, see e.g. later on Figure 6. In our study, however, the basic obstacle is that with CMB photons we have not a classic scattering process of photons on atos; i.e. a process described in equations of the Appendix A. There are not atos, there are particles only (e.g. baryons, electrons, etc.), which participate in the foration of the structure of density fluctuations. Therefore we will speak throughout this article about an interaction instead of scattering in all cases when instead of the classic atoic scattering factor the new relict radiation factor will be used. It is true that a part of the interaction of photons with electrons before the recobination ay be realised as Thoson scattering (elastic scattering of electroagnetic radiation by a free charged particle, as described by classical electroagnetis) 1, but the coplex picture of physical processes describing the interaction of relict photons with the non unifor atter coposed of various particles (electrons, ions, etc.) is not known to such an extent in order to enable a theoretical calculation of this interaction (on the basis of scattering factors). It is therefore evident that it will not be possible to use the conventional atoic scattering factors and that a new special factor reflecting the coplexity of interaction processes of photons with the priordial atter has to be constructed. We only point out that the description of these interactions is possible only in a special two-fold reciprocal space into which the CMB spectru is transfored. This new factor will be called the relict radiation factor and substitutes all coplicated processes which participate in the foration of the angular power spectru of CMB radiation. The construction of the relict radiation factor is presented in Subsection.3.3... The Wavelength of Radiation The wavelength of radiation is a quantity of highest iportance, too. It follows fro Equation (A5), that the greater the wavelength the saller is the axial possible value s ax of the reciprocal space vector. At the sae tie the upper liit of the integral in Equation (A) strongly influences the quality of the Fourier transfor. Although there is a broad distribution of wavelengths of photons (see later on the discussion in Subsection 5.3.) the calculation will be undertaken for the wavelength 1 It is just the low-energy liit of Copton scattering: the particle kinetic energy and photon frequency are the sae before and after the scattering, however this liit is valid as long as the photon energy is uch less than the ass energy of the particle. corresponding to the axiu of the wavelength distribution which corresponds to the teperature.75 K of the Universe today (see later on Figure 18), i.e. for the wavelength λ = 1.9 []. That this wavelength is rational is based on three arguents. First of all photons with this wavelength bring us the inforation on their last several interactions with particles today, in the second place the CMB radiation spectru is the sae for all wavelengths and in the third place the wavelength corresponding to the axiu of the wavelength distribution secures the highest probability of the interaction process of photons with the atter..3. Preparatory Calculations.3.1. The Reciprocal Space During a classic scattering experient we easure the intensity of the scattered radiation (e.g. X-rays) as a function of the scattering angle θ. This scattering angle describes in real space the angle between the incident and scattered radiation. Its relation with the scattering vector in reciprocal space was described in Equation (A5). On the other hand the angle α in the anisotropy spectru of relict radiation (see already Figure 1) is not a scattering angle. It is an angle characterizing the distance between an arbitrary point to another in those different points the teperature fluctuation is easured and copared with the overall ediu one. In order to overcoe the incoparableness between the angles α and θ, we will construct an angle dependent reciprocal space to the angle α. The basic quantity deterining this reciprocal space will be the scattering angle θ. We will suppose that the axiu possible value of ax the classic scattering angle 9, corresponds to the axiu value of the ultipole oent L ax = 3. As a consequence we receive a transforation coefficient Q ax ax QL, (1) (its value in this case is Q =.3). We are then able to calculate the whole set of θ angles ax ax LQ L L () and because L = 18/α, then i.e. ax ax 1 18 L, (3) 1 P, (4)
1334 L. ČERVINKA where P Q 18 deg is a coefficient enabling the transition between the space α and the space θ and where the angular space θ is reciprocal to the angular space α. According Equation (A5) we are now able to construct the whole set of scattering vectors s (5) s 4π sin, (6) where λ is the wavelength of the relict radiation. It should be noted that the quantities s and α are in an indirect relation. The space of the vector s will be further on called a reciprocal space. It should be pointed out that in this construction (see Equation (6)) the scattering vector s is defined in the reciprocal space (1/λ) and that this space is now dipped into the reciprocal space (1/α), see Equations (), (4) and (6). For this dipping we will use further on the expression that the space s is a -fold reciprocal space to the space α. The recalculation of the original data presented in Figure 1 using Equations (4) and (6) is shown in Figure 3. This new intensity dependence is labelled I ( s )..3.. The Relict Reciprocal Space There is a possibility to construct another reciprocal space which will be based directly on the angle α. For a better coparison and lucidity we will use now for the angle α the labelling θ Relict. i.e. hence Relict, (7) L = 18 /α = 18 /θ Relict. (8) In close analogy with Equation (A5) we now transfor the anisotropy spectru of CMB (relict) radiation into a reciprocal space (1/λ) described by the paraeter S Relict Relict S 4π sin, (9) Relict where λ is the wavelength of the relict radiation. The space of the vector S Relict will be further on called the Relict reciprocal space. It should be noted that quantities S Relict and Relict are in a direct relation. The anisotropy spectru of the CMB radiation rescaled on the basis of Equation (9) is here labelled I Relict (S Relict ) and is shown in Figure 4..3.3. Construction of the Relict Radiation Factor Generally, a correct scattering factor has to fulfil three criterions: Intensity [Arbitrary units] 6 5 4 3 1 I =.7169 [n] 4 6 8 1 1 14 16 18 s [n -1 ] Figure 3. Anisotropy spectru of the relict radiation shown in Figure 1 is recalculated as a function of s, i.e. after a rescaling of the angular oent L and is labelled I (s ). The rescaling of the angular oent L is realized on the basis of Equations (), (4) and (6) and using the MoK α radiation wavelength λ =.7169 [n]. Intensity [Arbitrary units] 6 5 4 3 1 I Relict =.7169 [n] 4 6 8 1 1 14 16 18 S Relict = 4(sin(18/L))/ [n -1 ] Figure 4. Anisotropy spectru of the relict radiation shown in Figure 1 is after a rescaling of the angular oent L, recalculated as a function of the relict reciprocal space vector S Relict and labelled I Relict (S Relict ). The rescaling of the angular oent L is realized on the basis of Equations (8), (9) and (11) using the MoK α radiation wavelength λ =.7169 [n]. The dashed line represents a soothed curve. 1) the I nor (s) curve should oscillate along the I gas (s) curve and as a consequence according Equation (A9); ) the curve I distr (s) should oscillate along the zero value of the intensity axis; 3) the resulting RDF ust not be containated by parasitic fluctuations due to bad scaling (see Section A.) as a consequence of a bad course of the scattering factor. The utual relation between quantities I nor (s), I gas (s) and I distr (s) is explained in the Appendix A, see equations (A9), (A1) and (B1) with (B). In Figure 5 the calculation of the crucial curve I gas is
L. ČERVINKA 1335 undertaken for the relict radiation factor f Relict. The for of this factor was deterined by the trial and error ethod and is shown in Figure 6. In this figure is the X factor f Relict copared with the coherent ( f coh ) and inco- X herent ( f incoh ) atoic scattering factor for X-rays corresponding to the Hydrogen ato (according the International Tables for Crystallography [6]). Siilarly as for X-rays we have set the relict radiation factor f Relict frelict 1 for s (1) and further, we have set in Equation (A7) Z = 1 and = 1, hence in Equation (A6) is K = 1. Fro this point of view our construction of the relict radiation factor f Relict should forally correspond to a hydrogen-like particle. Further we have to point out that in connection with the presentation of the quantity I gas (s) in Equation (A1) its course in Figure 5 is given now by the relation gas I s f. (11) Relict In Figure 5 we see that the function I nor (s) is properly oscillating along the function I gas (s) and therefore the function I distr (s) is properly oscillating along the zero line. The consequence is that we will obtain a proper radial distribution function, i.e. without any parasitic axia, see the Subsection 3.1..3.4. Relation between the and Relict Distribution of distances We rewrite now the basic Equation (A) using the scat- Intensity [Arbitrary units] 1,5. 1,.,5.,. -,5. I distr I gas I nor =.7169 [n] -1,. 4 6 8 1 1 14 16 18 s [n -1 ] Figure 5. Calculation of quantities I nor (s) full line, I gas (s) dashed line (see Equation (11)) and of I distr (s) dashed dotted line, according Equations (A9), (A1) and (B1), (B) using the artificial relict radiation factor f Relict for the wavelength λ =.7169 [n]. Oscillations of the curve I distr (s) are along the x-axis; hence the criterions set at the beginning of this section are fulfilled. See text for details. Atoic factors f Relict and f X [e] 1,.,8.,6.,4.,.,. f X coh f X incoh f Relict =.7169 [n] 5 1 15 s and s X classic [n-1 ] Figure 6. Behaviour of the relict radiation factor f Relict is shown. For coparison the courses of the classic coherent X and incoherent X-ray atoic scattering factors f coh and f X for Hydrogen are included. The paraeter s incoh is X defined in Equation (6), the paraeter s is described X X in Equation (A5). Data for f coh and f incoh are taken fro [6]. The calculation is deonstrated for the wavelength λ =.7169 [n]. tering vector in the classic reciprocal space s, see Equation (6). Mediu Fourier r[n*] r r, I s, (1) Mediu where r is the eber which is not Fourierdependent and describes the structure-less total disorder depending on the density of the atter. The paraeter r is easured in [n*] in order to ephasize that the calculation of the RDF ρ(r) is realized on the basis of the paraeter s, which is dipped in a -fold reciprocal space (see Subsection.3.1.). In other words: the calculation of the RDF ρ(r) is realized in the reciprocal space of classic distances, which have the diension [n*]. Here we again point out the fact, that classic distances are distances between Objects calculated on the basis of the function I (s ), see Figure 3, which we analyze using Equation (A) or (1). In order to receive now the inforation in the real space of classic distances (characterized by the paraeter R) we ust calculate the reciprocal value of the paraeter r, hence the relation between r and R is 1 r[n*] R[n]. (13) It would be now possible to rewrite quite forally Equation (A) using the scattering vector in the relict reciprocal space S Relict, see Equation (9). Siilarly as for Equation (1) we would receive Mediu Fourier R R R, ISR elict. (14)
1336 L. ČERVINKA Quite hypothetically the RDF ρ(r) would then bring us inforation on the real space of relict distances, which have the diension [n]. Actually, however, a RDF will not be calculated in this case, because the distribution I(S Relict ), see Figure 4, is not convenient for a Fourier transfor. The calculation of relict distances in the real space, i.e. of distances between coplex Objects (big clusters) will be done on the basis of a theoretical calculation of the function I(S Relict )) using the Debye forula (18) calculated for appropriate odels, see later on Section 4. 3. Calculations in the Reciprocal Space s 3.1. Calculation of RDFs In our first exaple we calculate in Figure 7 the RDF of Objects corresponding to the Fourier transfor of inten- Fourier sities r, Is for the wavelength λ =.7169 [n] which is a frequently used wavelength (λ MoKα ) in e.g. structure analysis, see Equation (A) andor (1). The scaling of intensities has been already deonstrated in Figure 5 on the basis of the relict radiation factor f Relict constructed in Figure 6. The calculated RDF shows a for typical for RDFs obtained for disordered aterials. It turns out that in the region fro.1 to.4 [n*] we observe peaks r and 1 r. Moreover there is a iniu in r which separates this region fro a structure-less course starting at the position r. Such behaviour indicates the existence of ordering in the atter. In other words, there is a distinctive separation of the atter ending its ordering at 1 r =.31 [n*] fro the residual structure-less phase which starts at r =.395 [n*]. For these reasons we will consider as a boarder between the ordered and disordered state the gap at in r =.348 [n*]. In the sae way we calculated RDFs for four ore typical wavelengths, i.e..11674 (λ SeKα ),.154178 (λ CuKα ),.5466 (λ VKα ) and.537334 [n] (λ SKα ). Fro these calculations it follows that, as expected, the dependence of the agnitude of corresponding coordination spheres on the wavelength λ is linear, see Figure 8, oreover, all RDFs had the sae appearance. In this connection we have to point out, that the distances are easured in reciprocal space distances [n*] and that, with respect to Equation (13), these distances have to be recalculated to real space distances, e.g. in [k]. This recalculation is realized in Table 1 only for the ost iportant distance in r =.348 [n*]. Siultaneously we review this paraeter for all wavelengths (Figure 8) and siultaneously extrapolate this distance to the wavelength of relict radiation photons λ = 1.9 Radial distribution function [e /n] 4 35 3 5 15 1 5-5 -1 Based on I(s ) r =.17 1 r =.31 r =.395 in r =.348 =.7169 [n] D = 18.6 [kg. -3 ] -15.1..3.4.5.6.7.8.9 1. r [n*] Figure 7. Calculation of the radial distribution function (RDF) according equations (A) and-or (1) for the wavelength λ =.7169 [n]. The dashed-dotted line corresponds to the second eber in equation (1), the dashed line is the first eber in this equation (dependent on density) and full line is the su of both coponents, see text for details. Value of the density D necessary to shift the iniu at.348 [n*] to positive values of the RDF is indicated in the upper right corner. Positions of spheres 1 r, in r, r [n*] 3..5. 1.5 1..5.,154.111.71.5..1..3.4.5 Wavelength [n] r in r 1 r.537 Figure 8. Dependences of ost iportant distances, i.e. of coordination spheres 1 r (squares), r (circles) separated by the iniu in r (down triangles) on the wavelength λ in the reciprocal space [n*] according Figure 7 and fro analogical calculations for wavelengths.11674,.154178,.5466 and.537334 [n]. For an easier survey error bars are inserted only for the sphere 1 r. []. Real space distances in R calculated in Table 1 are visualized in Figure 9. The extrapolation to the wavelength of relict photons 1.9 [] indicates that for this wavelength the shortest in R distances are of the order 1 eters. Later on (see Subsections 4. and 5.1.) the distance in R will be ascribed to the distance between Objects.
L. ČERVINKA 1337 in R, i.e. distance between "Objects" [k] 1E7 1 1 1 1 1 1 1.1.1 Calculated values Extrapolated values 18 []) = 1 9 [n] = 1.9 [].1 1 1 1 1 1 1 1 Wavelength [n] Figure 9. Dependence of the real space distances in R (full circles) on the wavelength λ (see Table 1 for details). Siultaneously an extrapolation to a distance corresponding to the wavelength of CMB (relict) photons 1.9 [] is visualized (epty circles). Later on (Sections 4. and 5.1.) the quantity in R will describe the distance between Objects. Table 1. Review of the ost iportant distance in r characterizing the separation of the ordered region fro the structure-less one on the wavelength λ (see Figure 8). Recalculation to the real space distances in R [k] is included. Extrapolation of this distance to the wavelength of relict radiation photons 1.9 [] is coputed together with an estiate of final errors. Review of the reciprocal space distance in r in [n*] on the basis of results presented in Figures 8 and 9 λ [n] in r [n*] Recalculation of the reciprocal space distance in r in [n*] into the real space distance in R [k] in R [k] = 1/ in r [n* 1 ].7169.348,873,563.11674.54 1,845,18 3.. Calculation of the Density The calculation presented in Figure 7 and repeated for four additional wavelengths enabled us to estiate the density of the atter, i.e. the iportant paraeter effecting the first eber ρ Mediu (r) in Equation (1). We siply supposed that the fluctuations of the RDF should not be negative. In order to shift in Figure 7 the iniu at in r =.348 [n*] to positive values we had to set the density to a value D = 18.6 [kg 3 ]. In the sae way we have deterined densities for the reaining four wavelengths. The results are suarized in Figure 1 and Table. In the log-scale is the dependence of density on the wavelength nearly linear and therefore enables again an extrapolation to higher wavelengths. This extrapolation is presented in Table and visualized in Figure 11. It follows fro Table and Figure 11 that the ost probable ediu density of density fluctuations of the atter with which CMB (relict) photons realized their last interaction is ~9 1 3 [kg 3 ]. Taking in account the liits of our calculation then the density can be for- 3 ally written as D 1 1 [kg 3 ], see also Figure 11 and Table. 4. Modelling in the Relict Reciprocal Space S Relict In the case when Figure 4 should be an X-ray scattering picture of a disordered aterial (e.g. of a glass) then such record would represent a picture typical for a aterial with well developed clusters. Their utual distance should then characterize the position of the first assive peak. It follows fro theory and experience that it is.154178.75 1,39,787.5466 1.1 819,1.537334.618 381,971 Extrapolation to higher wavelengths λ λ [n] in r [n*] in R [k] 1 4.87 5,31 1 49,53 1 487 5 Log (Density D) [kg. -3 ] 1 1 1.71.111.154.5.537 5 436 41 1 4 873 5 1 4 87 561.5 1 9 9 57 865.18 = 1.9 [] = 9.3 ±.1 [*] = 18 ± [].7.1..3.4.5.6 Log (Wavelength ) [n] Figure 1. Dependence of acroscopic densities on short wavelengths. In the log-scale this dependence is nearly linear. Nuerical values are given in Table. Nubers indicate wavelengths, for which the corresponding RDF was calculated.
1338 L. ČERVINKA Table. Review of nuerical values of densities according Figure 1 is presented. Extrapolation of the sequence of densities to higher wavelengths, especially to the wavelength of relict radiation photons 1.9 [] is shown. First five densities D were calculated following the description in Subsection 3.. Possible final error is estiated and the values of the critical density according [7] and [8] are given. Log (Density D) [kg. -3 ] Wavelength λ [n] Macroscopic density D [kg 3 ].7169 18.6.11674 4.84.154178 17.18.5466 4.39.537334.46 Extrapolation to higher wavelengths λ λ [n] D [kg 3 ] 1 9.E 1 6.E 5 1 4.E 8 1.E 11 1 1.E 1 1 9 = 1.9 [] 9.E 3 ± E 3 Critical density according [7,8]: D critical = 5. to 7. E 7 [kg 3 ] 1.1 1E-6 1E-1 1E-14 1E-18 1E- Calculated Extrapolated values Extrapolation liits Density = 9E-3 [kg. -3 ] = 1 9 [n] = = 1.9 [] radiation on the basis of the Debye forula n n Relict i j sin ij Relict ij Relict I S f f d S d S. (18) i1 j1 Here f i and f j are the scattering factors of n input particles and d ij are the distances in real space between all available particles in the odel and S Relict is the scattering vector in the relict reciprocal space defined in equation (9). It should be pointed out that as scattering factors f i and f j we have used now the relict radiation factor f Relict calculated in Subsection.3.3. The suation is over all n particles in the odel. This forula gives the average scattered intensity for an array of particles (or atos in solid state physics) with a copletely rando orientation in space to the incident radiation. Our odel was quite siple: For the wavelength λ =.7169 [n] the Cluster was a tetrahedron (5 particles) with an inter-particle distance.63 [n] i.e. located in a cube with an edge.67 [n]. In order to find the best fit with the scattering curve according Equation (18), the distance between Clusters (tetrahedrons) had to be d = 3 [n], i.e. the tetrahedrons were located in positions of the basic skeleton, see Figure 1, characterized now by a side a = 6.93 [n]. This odel had 5 particles, i.e. a total of 11 particles. This calculation is shown in Figure 13. For all other wavelengths (λ.11674 [n]) we had to increase the diensions of the Cluster. The Cluster had then the for of the skeleton shown in Figure 1 with an edge.67 [n] and consisted of particles (again with an inter-particle distance.63 [n]) ebedded in 8 edge-bound tetrahedrons. Only this Cluster occupied then the positions of the cubic basic skeleton shown in Figure 1 foring now an Object. (A ore 1E-6 1E-3 1 1 1 1 Log (Wavelength ) [n] Figure 11. Extrapolation of the dependence of densities on wavelengths λ to the wavelength of relict (CMB) photons λ = 1.9 []. Epty circles represent values shown already in Figure 1. Full circles are extrapolated values. Dashed lines show the liits of possible extrapolations. not possible to get fro this peak inforation on the internal structure of Clusters, only on their agnitude and utual distance. The ethod which has to be used for an analysis of this type of scattering is a direct calculation of scattered Figure 1. The basic skeleton (and-or a part of a Cluster structure) consists of positions fored by 8 edgebound tetrahedrons. All positions are identical, for a better graphic representation are the centres of tetrahedrons drawn white. The picture has been constructed using progras [9] and [1].
L. ČERVINKA 1339 instructive scheatic presentation of an Object is shown in Figure 17 where Clusters are presented as sall darker circles filled with particles.) When changing the diension of this basic skeleton, we siultaneously changed again the distance d between Clusters. In order to reach for λ =.11674 [n] the correct position of the assive peak at 1.6 [n 1 ] an inter-cluster distance d = 4.65 [n] had to be used, i.e. the diension of the skeleton was characterized by the side a = 1.74 [n]. This odel had then particles, i.e. a total of 484 particles and siulated a part of the Object structure. The Intensity [Arbitrary units] 6 5 4 3 1 =.7169 [n] 1 3 4 5 6 7 8 9 1 11 1 13 14 15 16 17 18 19 S Relict = 4(sin(18/L)) / Figure 13. Calculation of the profile of the recalculated anisotropy spectru for λ =.7169 [n] based on a set of Clusters with a utual distance d = 3 [n]. The Cluster was fored by a tetrahedron (5 particles); hence there were 11 particles in a odel, see text for details. Full line - experient, epty circles - calculated scaled and soothed curve, dashed line - calculated scaled scattering. Intensity [Arbitrary units] 6 5 4 3 1 =.11674 [n] 1 3 4 5 6 7 8 9 1 S Relict = 4(sin(18/L)) / [n -1 ] Figure 14. Calculation of the profile of the recalculated anisotropy spectru for λ =.11674 [n] and for a set of Clusters with a utual distance d = 4.65 [n]. The Cluster consisted of particles, hence there were 484 particles in the odel, see text for details. Full line - experient, epty circles calculated scaled and soothed curve, dashed line calculated scaled scattering. calculation is shown in Figure 14. Calculations of Cluster distances for additional wavelengths (.154178,.5466 and.537334 [n]) have shown (see Figure 15) that the dependence of Cluster distances on the corresponding wavelength is linear. This fact enabled an extrapolation of the Cluster distance d to the wavelength of relict photons λ = 1.9 [n], see Table 3. This extrapolated distance is drelict 1 1 [c]. The extrapolation is visualized in Figure 16. It should be noted that the recalculated anisotropy spectru depends in this case directly on the angle θ Relict which is equal to the angle α (see Equation (8)) and therefore a recalculation of the inter-cluster distance d into real space distances is not necessary because the Debye forula analyzes the relict reciprocal space represented by the vector S Relict directly in real space distances, see the quantity d ij in Equation (18). 5. Discussion In the following discussion we will concentrate on several iportant ideas which ay arise when reading this paper. In particular this contribution should deonstrate how the foralis iported fro solid state physics could be useful in solving specific cosological probles: It ay shed soe new light on the physical processes taking place in the priordial plasa. First of all, according our opinion, this work points in favour of clustering processes and consequently to a cluster-like structure of the atter in the oent when Distance d between Clusters [n] 35 3 5 15 1 5.111.71.154.5.534..1..3.4.5.6 Wavelength [n] Figure 15. Dependence of distances d between Clusters on the wavelength λ (nuerical values are shown). With exception of the.71 case odels consisted of Clusters with particles in each Cluster, i.e. a odel included 484 particles. Inter-Cluster distances characterize the position of the assive peak, see Figures 13 and 14 and the text for details.
134 L. ČERVINKA Table 3. Extrapolation of distances d between Clusters to the wavelength of relict photons λ = 1.9 [n]. These distances influence the position of the assive peak, see Figures 14, 15 and 16. The estiate of the final error is based on errors given in Figure 15. Wavelength λ [n] Distance between Clusters d [n].7169 3. ± 1.5.11674 4.65 ± 1..154178 7. ± 1..5466 13. ± 1..537334 3. ± 1. λ [n] Extrapolation to higher wavelengths λ d [n] 1 6.81 1 68 1 681 5 3,44 1 6,88 radiation with the atter and ay help in an iproveent of the theoretical predictions of the CMB pattern (see Subsection 5..). Finally this new approach ay be useful in the analysis of the CMB data. We have shown that the transforation of the anisotropy spectru of relict radiation into a special two-fold reciprocal space and into a siple reciprocal space was able to bring quantitative data in real space. Probles with the transforation into reciprocal spaces, ainly with the use of the proper wavelength of relict photons will be discussed in Subsection 5.3. 5.1. The Cluster-Like Structure of the Priordial Matter The ost iportant consequence of our quantitative results obtained in Sections 3 and 4 is, according our opinion, the idea of clustering processes taking part in the foration of the priordial atter. There we have arrived to three distances, which we interpret in a following way: The first distance ~1 [], (Table 1) should indicate the distance between Objects (big clusters), the second one ~1 1 [], (Table 3) should indicate the distance between saller Clusters, while the internal struc- 1 6,8,79 1 6,87,919 1 9 115,535,46 = 1.9 [] d Relict = 1 ± 1 [c] Distance d between Clusters [n] 1E8 1E7 1 1 1 1 1 1 1 1.15535 E8 [n] ~ 1 [c] = 1.9 E6 [n] = 1.9 [].1 1 1 1 1 1 1 1 Wavelength [n] Figure 16. Extrapolation of distances d between Clusters to the wavelength λ = 1.9 [] full squares; calculated values epty squares (see Figure 15 and Table 3). the universe becae transparent for photons (see Subsection 5.1.). In the second place the new foralis enabled us a siple and general description of the interaction of relict Figure 17. A scheatic arrangeent of Clusters (darker regions) with particles (sall white points) in an Object (white region). In our odel the distance between Objects is ~1 [], see Table 1. A detailed structure of a Cluster and of an Object in this odel is presented in Figure 1. The ost probable odel distance between Clusters is ~1 1 [], see Table 3. The distance between particles in the odel is.6 [n]. We estiate that there are ~1 11 particles in one Object and ~1 particles in one Cluster (see Appendix C1 and C) and ~1 9 Clusters in one Object, see Appendix C3.
L. ČERVINKA 1341 ture of a single Cluster was fored in the odel by particles with a ediu particle distance.6 [n], see Section 4. In Figure 17 we show a scheatic picture of this cluster odel. The big circle represents an Object. An Object is a clup of Clusters, where only a part of this clup was siulated in the odel (the odel of an Object had Clusters each consisting of particles, i.e. it consisted of a total of 484 particles). Although this odel gave a sufficiently well agreeent with the width of the assive peak, as deonstrated in Figure 14, our estiates (see Appendix C) show that the nuber of Clusters as well as the nuber of particles in one Cluster is greater., i.e. that there ay be as far as 1 11 particles in one Object and 1 particles in one Cluster. That the density plays an iportant role in these calculations will be discussed in Subsection 5.4. Even when the cluster odel gave a good profile of the assive peak at e.g. 1.6 [n], than such a odel of internal structure of Clusters cannot be a unique one, because the calculation of the profile is not sensitive to the internal cluster structure, nevertheless the cluster-like character of the odelling process has to be aintained. 5.. The Relict Radiation Factor We have already pointed out in Subsection.1. why during the analysis of the CMB spectru it has not been possible to apply conventional atoic scattering factors used in solid state physics and why a new special factor reflecting the coplexity of interaction processes of photons with the priordial atter has to be constructed. It is iportant to have on ind that the description of these interactions is possible only in a special two-fold reciprocal space into which the CMB spectru was transfored. We have called this new factor the relict radiation factor and it had to substitute all coplicated processes which participate in the foration of the angular power spectru of CMB radiation. Because relict photons realize their interaction with various kinds of particles and we have generated only one radiation factor, this factor represents, as a atter of fact, a ediu fro all possible individual relict radiation factors. In this way this new foralis offers a general description of the interaction of relict radiation with the atter and siultaneously reflects the coplexity of processes which influence the anisotropy spectru of CMB radiation fro the cosological point of view [5]. During our study we have concentrated on three iportant facts which ay justify the attept to interpret the anisotropy spectru of CMB radiation as a conesquence of the interaction of photons with density fluctuations which characterize the distribution of particles be- fore the recobination process. The first fact is that teperature fluctuations in the CMB spectru are related to fluctuations in the density of atter in the early universe and thus carry inforation about the initial conditions for the foration of cosic structures such as galaxies, clusters or voids [11]. Secondly, it is the fact that the inforation on these density fluctuations in the distribution of particles (electrons, ions, etc.) has been brought by photons. Photons which we observe fro the icrowave background have travelled freely since the atter was highly ionized and they realized their last Thoson scattering (see already Subsection.1.). If there has been no significant early heat input fro galaxy foration then this happened when the Universe becae cool enough for the protons to capture electrons, i.e. when the recobination process started [1]. The third fact is that the anisotropy spectru is angular dependent, see Figure 1. Although we know that the anisotropy spectru of CMB radiation, as presented in Figure 1, has no direct connection with a scattering process of photons, it was the transforation of the CMB spectru into a two-fold reciprocal space, which enabled us to interpret the anisotropy spectru of CMB radiation as a result of an interaction process of photons with density fluctuations of the atter represented by electrons, ions or other particles. This approach enabled us to reach an advantageous approxiation of this process. The process consisted of two steps: First of all we have constructed in Subsection.3.1. an angular reciprocal space characterized by the scattering angle θ, see Equations () and (4). This space is reciprocal to the space characterized by the angle α (α is the angle between two points in which the teperature fluctuations of CMB radiation are copared to an overall ediu teperature). Then, we have constructed an additional classic reciprocal space (1/λ) into which the first one (the θ space) was dipped, by defining in this new two-fold reciprocal space the classic scattering vector s, see Equation (6). Only after these transforations we treated in this new classic reciprocal space the transfored anisotropy CMB spectru as a scattering picture of relict photons. It was only this space in which we siulated (in Subsection.3.3.) the interaction of CMB (relict) photons with density fluctuations by the relict radiation factor f Relict. The criterion for the trial and error construction of the relict radiation factor f Relict has been that this factor had to fulfil the three requireents set at the beginning of Subsection.3.3. Only then it was secured that after the Fou-
134 L. ČERVINKA rier transfor, according Equations (A) and-or (1), there will not be any (or at least sall) parasitic fluctuations on the curve ρ(r) and-or ρ Fourier. That we have achieved these deands is docuented in Figure 7 where we do not see any parasitic fluctuations on the curve ρ Fourier and as a consequence on the curve ρ(r). To suarize: It is true that in our foral analogy between scattering of e.g. short-wave radiation on disordered atter (Figure ) and scattering of CMB photons on electrons, ions and other particles (Figure 1) is an essential difference, because the physical processes are copletely different, e.g. the scattering process itself, length scales involved, etc., however, the difference between physical processes is reflected and siultaneously eliinated by the special relict radiation factor f Relict (Subsection.3.3.), which we have included into all calculations based on the classic two-fold reciprocal space (see Subsection.1.). Moreover, additional calculations in the relict reciprocal space (see Subsection 4.) based on the relict radiation factor were done directly for the transfored angular power spectru of relict radiation (see I Relict (S Relict ) in Figure 4) and thus present an inforation on distance relations between Clusters (fored by particles) in real space. 5.3. The Wavelength Proble The proble is to which wavelength of relict photons we have to relate our calculations. One possibility ay be to refer this wavelength to that tie when 379. years after the Big Bang the Universe cooled down to 3 K and the ionization of atos decreased already only to 1%. Then according Wien s law bt (16) ax where λ ax is the peak wavelength, T is the absolute teperature of the blackbody, and b is a constant of proportionality called Wien s displaceent constant, b.89781 3 [K], we obtain for the teperature 3 K a wavelength value ax 966 [n], see [13]. However, siultaneously we ust be aware of the fact that we are analyzing CMB photons now when the teperature of the universe, due to its expansion, is.75 K. Then the wavelength of photons according the Wien s law should be ~1 []. On the other hand the COsic Background Explorer (COBE) easured with the Far Infrared Absolute Spectrophotoeter (FIRAS) the frequency spectru of the CMB, which is very close to a blackbody with a teperature.75 K [11,14]. The results are shown in Figure 18 in units of intensity (see the text to Figure 18). It follows that the wavelength corresponding to the axiu is 1.9 []. Intensity [MJy/sr] 4 FIRAS data: Blackbody teperature.75 K 3 1 Wavelength [] 1.67.5 5 1 15 [c -1 ] Figure 18. Dependence of the intensity of the CMB radiation on frequency as easured by the COBE Far InfraRed Absolute Spectrophotoeter (FIRAS) ([11,14]). The thick curve is the experiental result; the points are theoretically calculated for an absolute black body with a teperature of.75 [K]. The x axis variable is the frequency in [c 1 ]. The y-axis variable is the power per unit area per unit frequency per unit solid angle in MegaJanskies per steradian [sr], (1 [Jansky] is a unit of easureent of flux density used in radioastronoy, abbreviated Jy (1 [Jansky] is 1 6 [W Hz 1 ]). After all we have decided to relate our results to the wavelength of CMB photons λ = 1.9 [] which corresponds to the axiu of the intensity distribution. Because the distribution of the spectru covers a relatively broad interval of wavelengths, see Figure 18, calculations based on the wavelength 1.9 [] should then represent the ost probable estiate. Moreover, this consideration is supported by the fact that the angular distribution of CMB radiation is the sae for all wavelengths. However, on the basis of graphs in Figures 1, 1 and 16 an easy recalculation of distances and-or of the density would be possible when another CMB photons wavelength would be considered as ore appropriate. 5.4. The Density of the Mass and Distances between Objects, Clusters and Particles The way how we arrived to nubers characterizing the density of the atter was described in Subsection 3.. In a conventional X-ray analysis the density is the acroscopic density of the aterial under study. Therefore we suppose that also in this case the density which influences the parabolic shape of the curve of total disorder (see the first eber on the right side of equation (A) and-or (1) and Figure 8) should be understood as a real ediu density of density fluctuations.
L. ČERVINKA 1343 The dependence of the density on the wavelength as deonstrated in Figures 11 and 1 is not perfectly linear; therefore we have arked in Figure 11 the extent of possible linear dependences. This result can be forally written as D This ediu value is about 1 5 ties higher than the critical density D critical ~ (5 to 7) 1 7 [kg 3 ], see Table. Further, we should have in ind that the local density in a Cluster or in an Object has to be greater. We are able to docuent this fact on the basis of our Cluster odel. Based on particle distances d particles =.63 [n], we have siulated a part of the Cluster structure by a cube with an edge a Cluster =.67 [n]. There were particles in this cube which can be closed in a sphere with a radius R Cluster = d particles =.63 [n] =.53 [n]. The volue of this sphere is V Cluster =.6 [n 3 ] =.6 1 7 [ 3 ]. Supposing that particles are represented ac- 1:1:1 cording expression (C6) by their ediu ass part.771 7 [kg], we obtain for the density in the Cluster the value 3 1 1 [kg 3 ]. (17) 1:1:1 DCluster part VCluster 6.94.6 98 [kg 3 ], (18) i.e. a value approaching density values known fro solid state physics (i.e. values lying between the densities of gases and liquids). At the sae tie we have to take in account that the estiates concerning the density of atter are really coplicated. The icrowave light seen by the Wilkinson Microwave Anisotropy Probe (WMAP), suggests that fully 7% of the atter density in the universe appears to be in the for of dark energy [15] and 3% is dark atter. Only 4.6% is ordinary atter. So less than 1 part in is ade out of atter we have observed experienttally or described in the standard odel of particle physics. Of the other 96%, apart fro the properties just entioned, we know absolutely nothing [16]. In this connection we consider the density value we have received (9 1 3 [kg 3 ]) as the density of the ordinary atter. Last reark should be given to the probability of Object interactions in the case of their apparently large utual distances (~1 []). It follows fro the Maxwell speed distribution that the root ean square particle velocity v corresponding to the teperature T = 3 [K], is 3kT, (19) where k is the Boltzann constant (k = 1.38 1 3 [Joule K 1 ]) and is the ass of the particle, which ay be here for exaple the ass of the proton = 1.67 1 7 [kg]. Then we obtain 3 3 3 1.38 1 3 1 1.67 1 7 1.41 1.671 7 3 8.6 1 [ s 1 ] This is already a velocity, which should ake possible an intensive interaction of Objects fored by Clusters consisting of particles. 6. Conclusions A foralis of solid state physics has been applied to provide an additional tool for the research of cosological probles. It was deonstrated how this new approach could be useful in the analysis of the CMB data. After a transforation of the anisotropy spectru of relict radiation into a special two-fold reciprocal space it was possible to propose a siple and general description of the interaction of relict photons with the atter 38. years after the Big-Bang by a relict radiation factor. This factor, which ay help in an iproveent of the theoretical predictions of the CMB pattern, enabled us to process the transfored CMB anisotropy spectru by a Fourier transfor and thus arrive to a radial electron density distribution function (RDF) in a reciprocal space. As a consequence it was possible to estiate distances between Objects of the order of ~1 [] and the density of the ordinary atter ~1 [kg 3 ]. Another analysis based on a direct calculation of the CMB radiation spectru after its transforation into a siple reciprocal space and cobined with appropriate structure odelling confired the cluster structure. It indicated that the internal structure of Objects ay be fored by Clusters distant ~1 1 [], whereas the internal structure of a Cluster consisted of particles distant ~.3 [n]. In this way the work points in favour of clustering processes and to a cluster-like structure of the atter and thus ay contribute to the understanding of the structure of density fluctuations and hence to a refineent of paraeters describing the Standard Model of Cosology [17]. Siultaneously, the work sheds ore light on the structure of the universe in the oent when the universe becae transparent for photons. On the basis of quantitative considerations it was possible to estiate the nuber of particles (protons, heliu nuclei, electrons and other particles) in Objects and Clusters and the nuber of Clusters in an Object. 7. Acknowledgeents My thanks are due to Prof. Richard Gerber (University of Salford, Manchester) for discussion and proposals di-