Model of the Universe based on the Repulsive Dark Matter

Transcription

1 Model of te Universe based on te Repulsive Dark Matter Jaroslav Hynecek 1 Abstract Tis article describes te new model of te Universe tat is an alternative for te wellknown Big Bang (BB) model. Te recently publised paper [1], were te autors ave presented data on te oldest star in te Milky Way galaxy alo, callenges te validity of te BB model claim about te age of te Universe. It is tus apparent tat tere is a need to develop an alternative model for te Universe tat would not ave tis problem and provide a better agreement wit observations. Te model presented in tis paper offers suc a new alternative by assuming tat te Universe is not expanding and is filled wit a static gravitating "dark matter" (DM) tat is transparent and terefore does not absorb ligt. Tis matter provides a framework in wic te visible matter moves similarly as defects or vacancies move in a crystal floating from te bulk to te surface. It is furter assumed tat te visible matter may ave been created from tis dark transparent matter by an unspecified process sometime in te past, or is being constantly created wit a smaller rate. After aggregation to stars and galaxies te visible matter is driven out to te edge of te Universe were it disintegrates and generates te immense Gamma Ray Bursts (GRB). Tis radiation ten may contribute to te generation of new matter trougout te Universe similarly as te assumption tat te matter is being constantly created in te Fred Hoyle's model of te Universe []. Te DM model provides equations for te observed recession velocities of distant galaxies, and for many remaining parameters tat follow directly from te Hubble constant suc as: te size and te mass of te Universe, te maximum observable luminosity modulus, te maximum observable Z sift, te maximum galaxy recession velocity, te size of te average galaxy, etc.. An important relation, also derived from te model, is te relation between te Hubble constant and te temperature of te cosmic microwave background radiation (CMBR). Tis relation allows a precise calculation of Hubble constant from tis temperature. Te developed teory is compared wit te available data of te GRBs, te Supernova Cosmology project, and te BATSE catalog, and a very good agreement is obtained. 1 jynecek@netscape.net, ISETEX, Inc. 16 1

2 Key words: Repulsive DM Model of te Universe, Hubble Constant, Hubble mass, Galaxy maximum recession velocity, Galaxy maximum age, Gamma Ray Bursts, CMBR temperature, Relation of Hubble constant to CMBR temperature. 1. Introduction "Astropysicists are always wrong, but never in doubt", R. P. Kirsner. Te model of te Universe tat as attained te large popularity is te so called Big Bang (BB) model. Tis model was developed by a simplistic extrapolation from te famous Hubble discovery and from many later measurements tat are clearly sowing most of te observed galaxies receding from Eart wit an increasing velocity in a linear proportion to teir distance from Eart. However, during te process of model refinement and furter development several problems ave surfaced and are discussed in many publications []. An increasingly complex modifications and improvements of te model, usually by adding new tunable parameters, are tus necessary to explain tese problems away. Tere are also well known critics of te BB teory tat are bringing forward strong arguments against its validity [], wic is in agreement wit te recent Hubble telescope observations of vast regions of young star formations. However, a new difficulty for te BB model as appeared following te data publised in a paper in 7 [1] were te autors describe te discovery of a star in our Milky Way galaxy alo tat is 1. Gyr old. Te autors commented tat tis age is witin te current BB model age limit of 1.7 Gyr, so tis sould not be a problem. However, it now seems tat te model does not give enoug time for te ydrogen to condense into stars, burn troug te two star generations, and ten form te galaxies wit alos. Tere sould be many galaxies identical to our own wit te same age as a consequence of te ad oc assumption of te initial rapid superluminal inflation postulated in te BB teory. Tese galaxies are located far away and must now ave te same alos and old stars in tem as well. Since we see tese galaxies almost fully developed and te ligt from tem as traveled to us peraps 1.1 Gyr, tis results in an unreasonably sort galaxy formation times of.5~.6 Gyr since te BB. Finally, as will become clear later, te galaxies explode near te edge of te Universe generating GRBs. Terefore, in order to travel tere yet anoter time needs to be added to te average galaxy age. It is tus more reasonable to assume tat te

3 average galaxy, wic includes our Milky Way, is at least ~4 Gyr old. Tis certainly does not fit te BB model and a new model of te Universe needs to be developed. Te work described in tis paper is an extension of te previously publised work were te uniform dark matter density was assumed. Tis assumption is now removed and a muc better agreement wit te available data of te GRBs and te Supernova Cosmology project is obtained [4]. Te teory provides equations for te duration of te GRBs and te frequency of te GRB occurrences tat correlates well wit te temperature of te cosmic microwave background radiation CMBR. Te new model also allows to calculate te Hubble constant from te CMBR temperature, supplies values for te size and te mass of te Universe, te dark matter pressure at te origin of te Universe, te limit for te maximum observable Z sift, te maximum value for te luminosity modulus tat can possibly be observed, te maximum galaxy recession velocity, te size of te average galaxy, and te time to te Milky Way galaxy destruction. None of tese interesting parameters tat uniquely follow from te Hubble constant and te current recession velocity of te Milky Way galaxy are available from te standard Big Bang model of te Universe.. Model assumptions Te key assumption of te new Universe model is tat te visible matter represents only a small portion of te total matter of te Universe and can be essentially neglected in its long range gravitational effects. Te most of te matter of te Universe is terefore dark. Tis dark (transparent) matter (DM) will be considered attractive to itself everywere but causing a repulsive force to visible matter. In te previously publised paper te autor as assumed tat te DM as a uniform density [5]. Tis assumption is now relaxed and te DM is considered compressible aving te density m and te pressure P at te origin. More details about te pysical nature of te DM mass density, about te origin of its repulsive force, and about its relation to te CMBR temperature are given in te last section of tis paper. Te propagation speed of te pressure disturbances in te DM abstract and flat pysical space-time will be considered constant, equal to te speed of ligt, and satisfying te well-known formula for te speed of sound: c = P / m. Tis is consistent wit te relation P = mc, wen P is identified wit te energy density. It will also be considered tat te radiation tat is moving at te speed of ligt relative to te

4 DM reference frame does not ave any gravitational mass [6]. Te visible matter natural space-time description will be based on te Riemann metric ypotesis of curved spacetime and te DM space will be considered aving a finite size. Te constants suc as te speed of ligt, te Hubble constant, te gravitational constant, te Milky Way recession velocity relative to te CMBR, etc., are te results of measurements and are not te part of assumptions.. Matematical background of te model Tis section describes te logical consequences tat follow directly from te assumptions stated above. Te validity of assumptions can be, of course, justified only by comparing te derived consequences wit observations and measurements. Since te long range gravitational effects of visible radiating mater and all of te radiation can be neglected, te space-time metric can be considered static, sperically symmetric, and described by te following differential metric line element [7] : ds ( cdt) g dr g Ω = gtt rr r ttd (1) were: d dϑ sin ϑ dϕ Ω = +, gtt exp( ϕ v ) =, g = 1, and c is te local intergalactic speed of ligt. Te cosmological Newton gravitational potential for te visible matter, ϕ v, normalized to tt g rr c is calculated using te well-known equation: ( ) 4πκ ϕv ( ) = m( ) ( ) d c ( ) () were κ is te Newton gravitational constant. Due to te deformation of te observed natural radius r by te DM gravity, te pysical radius r (r) must be used in te formula and tis parameter is found from te differential equation tat follows from te metric: ϕv dr = grr dr = e dr () Because any particular galaxy now represents only a small test body in tis Universe, te well-known and many times verified Lagrange formalism will be used to describe te motion of suc galaxies. Te Lagrangian is terefore as follows [7] : ϕ cdτ ϕ d ϕ dω v v v (4) L = e e e dτ dτ dτ For a purely radial motion te Lagrangian can be simplified and te first integrals of te 4

5 corresponding Euler-Lagrange equations easily found using te initial condition at te origin were te recession velocity is zero and were: Eliminating dτ dt d dr = c dτ from tese equations, because dτ te formula for te recession velocity: dr dt t d t = dt. Te results are: ϕv = e (5) c e ϕv (6) is not an observable parameter, leads to ϕ v ϕv = 1 (7) c e For te relatively near objects, were te cosmological gravitational potential ϕ v is still e small, it olds tat: r ( r) r, and m ( ) = m. Tis simplifies Eq.7 as follows: dr dt c 8 ϕ v = π κ m r = H r (8) From tis result it is ten clear tat te recession velocity is linearly proportional to te natural coordinate distance r of suc objects from te origin and tat te Hubble constant H is related to te DM density m at te origin according to te following equation: H 8 = π κ m (9) Te recession velocity and te Hubble constant are referenced to te DM coordinate system, so te value of te Hubble constant sould be corrected and referenced to te Eart's centered coordinate system were it is actually measured. However, te correction is very small and it will be neglected. Eart and its Milky Way galaxy are located relatively near te center of te Universe in comparison to its immense size. In order to proceed furter in te model development it is necessary to find te relation for te DM density m () as a function of te pysical radius. Tis is obtained by adapting te well-known approac described, for example, by Zel'dovic [8] were te DM pressure gradient can be expressed as a function of te pysical radial distance: 4π κ m( ) = m( ) dp d d (1) After substituting for te DM pressure from te relation: c = P / m, and defining te 5

6 normalized mass density function: m n ( ) = m( ) / m, Eq.1 can be rearranged wit te elp of te Green's function as: were A is a constant equal to: ( ) m n ( ) = eξp A mn ( ξ ) ξ ξ / dξ (11) A πκm = 4 / c. Tere is no known analytic closed form solution for tis equation, so it is necessary to use te numerical iterative approac or find an approximating function. Te approximating function approac was selected for te next steps to avoid very long computing times during iterations. Te selected function, owever, underestimates te true value of te DM mass density at large, but te error as only a small overall effect. Te first two iterations and te approximating function: ) = exp (1) m a ( 1 are sown in a grap in Fig.1. More details of te approximating function derivation are given in te Appendix. Te introduced parameter defined as: = c / H, is called te 6 9 Hubble distance or te Hubble pysical radius: =.7 1 m = Ly..8 ma( x).6 m1( x) m( x) x Fig.1: Te first two iterations (dased and dot-dased traces) and te approximating function describing te dark matter mass density as a function of te pysical radius were: x = /. Anoter advantage of using te approximating function is tat te DM concentration tail extending past te maximum radial distance can be easily cut off by suitably truncating te power series expansion in te exponent. Tis feature is advantageous if it is considered tat te visible matter debris from explosions of galaxies are accumulating at 6

7 te edge of te Universe and are forming a loosely bound sell tere. Of course, it is possible to add more terms tan sown in Eq.1; owever, tis will not be pursued any furter in tis paper, because te accuracy of te approximation was found reasonable as will be discussed later by comparison wit observations. Once te mass density function is known it is easy to find te normalized gravitational potential for te visible matter using te formula in Eq., and for te dark matter using te Green's function formula derived also from te Gauss law as follows: ξ dξ m ( ζ ) ζ dζ. = A m ( ξ )( ξ ξ / ) ξ a a ϕd ( ) = A d. (1) ξ Bot potentials are plotted in te graps sown in Fig ϕv( s) 1.5 ϕd( s) r( s) Ly Fig.: Te dependencies of normalized gravitational potentials for te visible matter (solid trace) and for te dark matter as functions of te pysical radius. Te integration constants were adjusted suc tat te potentials at infinity are zero. In te next step of te model development it is necessary to find te formula for te Z sift, since tis is te parameter tat is directly measured by astronomers. Te Z sift typically consists of tree components: te star gravity induced red sift, te cosmological potential induced red sift, and te Doppler red sift resulting from te recession velocity. Te star gravity red sift does not ave to be considered ere, because after te star or te galaxy explosion as occurred most of te principal source of te gravitational field as been converted to radiation and radiated away and only te remnants or te afterglow produce te ligt tat is observed. Te cosmological potential induced red sift does not ave to be considered eiter, since in tis model te galaxies 7

8 are in a radial free fall and tis compensates for te sift [5]. Te only remaining Z sift component is tus te Doppler red sift resulting from te radial recession velocity v r. Te Doppler red sift observed on Eart is: 1 vr / cr gtt exp( ϕv ) Z( r ) = 1 = 1 = 1 (14) 1 v / c 1 1 g 1 1 exp(ϕ ) r r were c r indicates te ligt speed at te galaxy location in reference to Eart. Te grap of te Z dependency on te natural coordinate radius r is sown in Fig.. 1 tt v 1 Z( s) r( s) Ly Fig.: Te dependency of Z sifts on te natural coordinate radial distance. Te maximum Z sift tat can be observed is Z =1. 5. Te visible matter does not exist at larger distances tan: mx 9 r mx.11 1 = Ly, since it disintegrates a te Universe's edge. Te radial distance r, also called in tis paper te natural radial distance, wic is te observable parameter, is calculated according to Eq. as follows: ( ) = exp( ϕv ) d (15) It is also convenient to introduce te average Universe radius a, wic is te radius corresponding to a spere wit te constant mass density m tat as te same total dark mass as te Universe, and te maximum Universe radius mx tat corresponds to te radial distance were te visible matter potential as its minimum. Te numerically computed values for tese parameters, including also te Hubble distance for a comparison, all expressed in ligt years (Ly), are: 8

9 9 9 = Ly, r = Ly (16) 9 9 = Ly, r a =.4 1 Ly (17) a 9 9 = Ly, r mx =.11 1 Ly (18) mx Te radius of te observable Universe, were te DM ends and only witin wic te visible matter can exists is terefore: =.11 1 Ly. Te interesting parameter is te r mx 9 DM pressure at te center of te Universe, terefore, near our Eart. Te pressure is equal to: 1 P = Pa. Tis is an extremely low value but neverteless te gradient of tis pressure is causing galaxies and all te visible matter to float to te edge of te Universe were tey disintegrate. Te recession velocity following Eq.7 is sown in a grap in Fig.4 as a function of te natural radial distance. Te graps also include te current velocity of te Milky Way galaxy and te velocity at its formation 4 Gyr ago v( s) 1 vg 1 vo 1 cr( s) r( s) Ly Fig.4: Numerically computed galaxy recession velocity in km/sec as a function of te natural coordinate radius in ligt years measured from te center of te Universe (purple trace), te limiting vacuum speed of ligt c r (green trace), te current Milky Way recession velocity: 55 km/sec (dased trace), and te Milky Way galaxy recession velocity: 4.18 km/sec during its formation 4 Gyr ago (dotted trace). Te galaxies disintegrate at te distance of:.11 bly. For completeness and before making comparisons wit observations, it is interesting to find te values of te remaining parameters tat directly follow from te single Hubble galaxy recession velocity constant. Tis is in contrast to te Big Bang model tat needs at 9

10 least tree to six adjustable parameters and a postulate of a sudden superluminal inflation tat as no known pysical cause to obtain an agreement wit observations. Te total dark mass of te Universe calculated from te field is: M du c ϕ d( ) = d = m π k m ( ) ϕ ( ) d = Tis value is peraps more accurate tan te direct integration of te mass approximating function: M dua 54 a d 54 kg (19) = kg, since tis satisfies te energy-mass equivalence rule. Bot results are larger tan te Hubble mass defined as: 5 M = 4c / k H = kg. An agreement between M du and M dua is obtained wen it is considered tat about 19% of dark matter is sielding te visible matter, predominantly at te edge of te Universe. Te definitions of Hubble parameters and an example of teir use in calculations are given in section 5 and in te Appendix. Te minimum normalized potential of te visible matter at te Universe's edge is: ϕ min = 1.746, resulting in te maximum for te observable Z sift: Z mx = Tis v value surprisingly agrees wit te re-ionization Z sift: Z = 1.4 ± 1. of te BB model. Te maximum time to te Milky Way galaxy explosion observed by te observers on ri Eart tat may be travelling wit it is: τ max = 1. Gyr. Tis is, of course, muc longer tan te Hubble time: τ = 1/ H = Gyr. Te time to destruction is calculated using te integral following from Eq. and Eq.6: were mx 1 d τ max = c () exp( ϕ ( )) 1 g g is te current pysical distance of our galaxy from te center of te Universe calculated based on te measured Milky Way recession velocity relative to te cosmic microwave background radiation reference frame: v v g = 55km / sec. Tere is no equation in tis model to find te age of te DM Universe, in particular te age is not related to te Hubble time. Te age as to be deduced from some oter observations and considerations as is, for example, discussed in te already referenced paper [1] and as commented on in te introduction. Neverteless, it is possible to estimate 1

11 tat te maximum galaxy lifetime is approximately equal to: τ glife = 14 Gyr. Tis is te maximum time limit for any intelligence to develop an intergalactic travel capability. Te lifetime of any particular galaxy is, of course, dependent on te place of its creation, wic would substantially reduce tis limit. 4. Comparison wit observations Te astronomers typically evaluate teir observations in terms of te apparent and intrinsic stellar magnitudes by introducing te luminosity modulus: m = m M and plotting it as a function of te Z sift, wic can be precisely measured. Te modulus can be expressed in terms of te luminosity distance, wic is defined as: = ( r) ( Z + 1) were r (r) is te pysical distance. Te resulting equation is ten as follows: ( Z 1) s sa si d L r, r( r) + µ s ( r) = 5log1 (1) 1 pc were pc is te distance of one parsec. It is ten not too difficult to find te teoretical prediction of luminosity modulus as a function of te Z sift. It is only necessary to invert te formula in Eq.14 and find te pysical radius as a function of te Z sift, wic is ten substituted into te formula in Eq.1. Te teoretical dependency of te luminosity modulus on Z sift is sown in Fig. 5 togeter wit te measured values publised by Kowalski [4] and Scaefer [9]. It is also possible to plot te luminosity modulus directly as a function of te natural radial distance as sown in Fig.6. For completeness te plots also include te dependency of te galaxy recession velocity and te speed of ligt on te natural radial distance. Te agreement of teoretical predictions wit te measurement is, considering te simplicity of te teory, stunning. Te agreement is noticeably better in comparison wit te uniform DM density model introduced in te previous publication. Tis result tus unquestionably confirms te correctness of te developed model, validates te model assumptions, and at te same time raises new doubts about te validity of te BB model. It is also wort mentioning tat te upward bending of te luminosity modulus curve in te range of ~ 1 1 Ly, or similarly a sligt upward bending of te luminosity modulus curve in Fig.5 in te range of:.5 < Z < 1 is not caused by te accelerated Universe expansion, wic te main stream BB astropysicists claim tat exists. 11

12 51 51 MEASURED SN AND GRB MODULUS THEORETICAL MODULUS MEASURED SN AND GRB Z SHIFT Fig.5: Measured 4 Supernova (squares) and 69 GRB (circles) data points of luminance modulus µ s plotted togeter wit te corresponding teoretical values of modulus (black dots) as functions of te Z sift. Tis diagram is sometimes also called te Hubble diagram and it is te direct comparison of observations wit te teory. SN, GRB, AND THEORETICAL MODULUS NATURAL COORDINATE DISTANCE [ligt years] RECESSION VELOCITY AND LIGHT SPEED Fig.6: Measured 4 Supernova (squares) and 69 GRB (circles) data points of modulus plotted togeter wit te corresponding teoretical values of modulus (black dots) as functions of te natural radial distance. Te recession velocity is also sown on te same grap (plus signs) wit te speed of ligt (triangles). Te DM density at te origin used in calculations was: = kgm, as derived from te Hubble constant: H = 68. km s Mpc. m o µ s 1

13 It is interesting, owever, to observe in te plots in Fig.6 tat most of te GRB explosions occur in te region were te galaxies decelerate, just before te maximum Universe natural radial distance of: r mx 9 =.11 1 Ly is reaced. If te GRBs were caused by te galaxy collisions, as is typically claimed, te probability of GRB occurrences trougout te Universe's volume would be approximately uniform or more likely skewed towards te nearby older galaxies and some GRBs would occur relatively near Eart. It is possible to estimate, based on te current observed rate of explosions, tat during te entire Eart's existence at least 4 GRBs witin te radius of 1 million ligt years would ave occurred. Te GRB explosions in suc a near proximity to Eart would certainly destroy all te life every time and sterilize Eart forever. Again, te fit of te modulus data to te teory seems very good wit a perfect matc in many cases. Tis result, terefore, provides te experimental support for te teory and for te correctness of te metric used in Eq.1. Te maximum recession velocity of: 9 v mx =. 85c occurs at te distance of: r vmx = Ly. Te maximum luminosity modulus at te Universe s edge tat can ever be observed is: m (max) = It is fascinating to see tat tis value as already almost been reaced. Tis particular GRB signal must ave, terefore, come to us from te region tat is very close to te edge of te Universe. Te compressibility of te DM allows for te propagation of DM disturbances in te pysical space-time wit te speed of ligt c. Te corresponding radial speed of ligt in te natural space time, as observed from Eart tat is currently positioned relatively close to te center of te Universe, is, of course, equal to: c = c g = c exp( ϕ ). Te reason for te GRB explosions as not been establised wit te certainty yet, owever, one of te contributing factors is te reduction of gravitating mass of stars in te galaxy wen tey approac te edge of te Universe. Te gravitating mass depends on te cosmological potential for te visible matter as was explained in more detail in te autor's previous paper [5]. Te most likely possibility, owever, is tat te visible matter aggregates at te boundary of te Universe forming tere a sell from debris, or a sell from clouds of ydrogen plasma, or even a sell from clouds of elementary particles suc as neutrons, since te visible matter gravitational potential as its minimum tere. Te r tt s v 1

14 neutrons could ten be te predominant source of te cosmic neutrinos detected ere on Eart. Tis structure as a temperature of te CMBR and te new galaxy arrivals collide wit it converting most of teir mass into radiation. Finally, it is possible to consider tat te reduction of te DM pressure at te edge of te Universe cannot keep te visible matter compacted togeter any longer and causes it to disintegrate. However, te extremely low value of te DM pressure does not seem to make tis possibility very likely. Neverteless, it migt be interesting to investigate if te DM pressure, or te deep negative DM potential, could someow be dynamically created ere on Eart and induce te controlled nuclear fission of any visible matter as a source of energy. Te correct understanding of te pysics of gravity and te Universe is tus very important for te advancement of tecnology. Te frequency of te GRB explosions per day can rougly be estimated assuming tat tey are caused by te anniilation of te Milky Way size galaxy central masses equal approximately to: Suns. From te CMBR temperature of: T b. 75K =, using te Stepan-Boltzmann law, it is simple to calculate te total eat energy radiated from te interface back into te Universe considering tat tere is approximately in te average galaxy. Te mass of te galaxy is terefore: 11 1 Suns 11 M G = 1 M. Te efficiency of te mass conversion to eat is assumed: η = 1.%, since most of te energy from te explosions is converted to gamma rays. Te efficiency of te GRB detection is assumed: ξ = 5% due to te Eart sielding effect. Te observation time of one day S expressed in seconds is: t d = sec. Te equation for te count of te GRB explosions per day is ten as follows: N GRB / day 4 x 8 6 kbt mx gtttd g b tt = π () 4 15 g tt c gtt ( M G / gtt ) were k B is te Boltzmann constant. Tere is no cosmological potential effect on k B, similarly as tere is none on te Planck constant [5]. Te effect on te remaining parameters is included in te formula as indicated. Te surface area of te interface was calculated using te metric given in Eq.1. After simplification and substitution of values for te parameters, te GRB count per day is: 14

15 x 8 6 kbt mx t b d N GRB / day = π =. 88 () 15 c gtt M G Te obtained result is in a reasonable agreement wit te observed frequency of te long duration GRB pulse occurrences. Te detected CMBR is terefore te image of te boundary region of te Universe and not te remnant of te Big Bang. Te supporting observational evidence for tis conclusion is obtained from te angular dependence of te CMBR power spectrum ripples [1]. Te arriving galaxy explosion disturbances propagate along te surface of te Universe's boundary sell forming te well-known circles on te CMBR background, wic are detected as sown in Fig.7. 4 Fig.7: NASA WMAP data of angular dependence of te CMBR power spectrum ripples caused by te galaxy explosions at te edge of te Universe [1]. Te calculation of te gamma ray flux measured on Eart from te conversion of te galaxy central masses to energy is less reliable, since te amount of absorption of te gamma rays on teir way to Eart is not known. It is assumed tat te absorption is considerable since it may be contributing to te generation of new visible matter. Te lengt of te long duration GRB pulses, on te oter and, can be readily found in te current DM model. Te calculation is best performed in te pysical space-time were it is simple to find te minimum pysical radius to wic any large mass can be compacted to and divide te result by te speed of ligt. After te pysical time lengt of te explosion is found it is ten only necessary to add te cosmological time dilation factor to it to obtain te Eart's observed duration. Te minimum radius of te galaxy 15

16 central mass is: min = Rs / 4, as was derived previously elsewere [7], wit R s being te Scwarzscild radius of te galaxy central mass. Te resulting equation for te GRB pulse duration observed on Eart is tus as follows: 6 r κ M 4. 1 min S t grb = = = 56.4 sec (4) c g c g Te value of g tt substituted into Eq.4 is te value at te Universe's edge: tt tt.487 g tt = e. Tis result agrees well wit te statistic obtained from te BATSE catalog [11] as is sown in Fig.8, wic te standard BB model does not predict. Tis also experimentally disproves te existence of Black Holes, wic are now replaced by a very compact masses witout event orizons [5,7], and can, terefore, explode. Te peak of te sort GRB pulse durations corresponds to te explosions of Quasars tat did not ave enoug time yet to develop into te full size galaxies, or corresponds to te first generation of massive stars. Te sort pulse duration agrees again well wit te prediction obtained from Eq.4 wen te average mass of Quasars is substituted into te formula. Tis provides once more a good experimental support for te presented model of te Universe. Fig.8: Statistical distribution of te long and sort GRB pulse durations as publised in te BATSE 4B catalog [11]. Te described Repulsive Dark Matter teory as only one adjustable parameter to fit te data, as already mentioned, te Hubble constant: H 1 1 = 68. km s Mpc. Tis is in stark contrast to te existing Big Bang model, wic requires at least tree or more parameters 16

17 tat need to be adjusted and an ad oc postulate of a sudden superluminal inflation tat is difficult to justify by te well-establised laws of pysics. Te more adjustable parameters tere are, te less reasonable te model becomes, since wit a large number of adjustable parameters it is always possible to fit any data. A significant confirmation of validity of te presented DM model, owever, would come from te prediction of te galaxy rotation curves. In order to accomplis tis task it is necessary to consider tat te DM is partially depleted wit its concentration reduced in te vicinity of te galaxy central mass. Te depletion of te dark matter is due to te introduction of te DM compressibility and results from te mutual repulsive force between tese two types of matter. Tis penomenon was difficult to justify in te previous model of te uniform DM density, but arises naturally in te deformable DM model concept. Te gravitational force tat is generated from te depleted region, wic is depleted relative to te locally approximately uniform background, can tus be viewed as additional attractive force acting on te visible matter. A significant depletion, owever, occurs only in te very near vicinity of te compact masses, but a small fraction of depletion extends to larger distances. Equation for te DM concentration in te neigborood of te mass derivation of Eq.1. M S can be derived following te same concept as in te dm( ) 4πκ κm S = m( ) d + (5) m( ) d c c In te case of a galaxy, te non-uniform distribution of visible matter in te galaxy's arms, te mutual gravitational star interaction, and te compensating centrifugal forces due to te arms' rotation including te local rotation or counter rotation of DM itself tat may be dragged along by te arms, unfortunately complicate te calculations of te DM distribution furter and as a result te complete and accurate model of te galaxy rotation is not expected to be easy to develop. Tis work is deferred to future publications. Despite of tese complications, owever, te DM model of te Universe offers an interesting calculation tat is related to te spere of te gravitational influence of eac star or te entire galaxy. At a certain radius te DM mass density will again return to te local background level and at tat point te derivative of te density will be equal to zero. From Eq.5 ten follows tat at tat radius it will old: 17

18 M S i = 4 π m( ) d (6) Since te cange of te DM density over te local background in most of te volume of te spere of gravitational influence is very small, it is reasonable to consider tat m( ) m. Tis simplification ten yields te equation for te radius. Also, for te visible matter potential at te edge of te spere we ave from te Gauss law: dϕ = 4 i v κm s πκ m( ) d = (7) d It is terefore clear tat at te edge of te spere of gravitational influence te gravitational field is completely compensated and sielded by te DM. For te star of te mass equal to te mass of our Sun te diameter of te spere of gravitational influence is tus simply determined as follows: d is M S = = 8. Ly. (8) c κ If we now consider tat te arms of te Milky Way galaxy on average consist of stars similar to our Sun it is clear tat te Milky Way disc tickness sould be approximately equal to or sligtly larger tan te diameter of te spere of te gravitational influence of individual stars considering only a moderate spere overlaps. It is estimated tat te Milky Way galaxy tickness is approximately equal to: d ig = 1 Ly. Tis value is reasonably close to te value calculated from te formula in Eq.8, considering tat any vertical star distribution in te galaxy's arms was neglected and tat te average mass of te stars in te galaxy may be greater tan te mass of our Sun. Similarly, if it is considered tat te Milky Way galaxy center arbors a compact star wit te mass equal to: 6 1 M, te diameter of te galaxy's spere of te gravitational influence is found S from te same formula as in Eq.8 and is equal to: d ig = 6 κ M S 1 5 = Ly. (9) c Te astronomers estimate tat te Milky Way galaxy diameter is: 1, Ly. Again, tis value is in a reasonable agreement wit observations. However, te interaction of te DM mass wit te visible matter is not linear and te mutual interaction of te moving stars in 18

19 te galaxy's arms is also complicated. It is, terefore, interesting tat despite te large number of stars in te galaxy, estimated to be approximately: 11 1, no stable structure is formed past te radius of te gravitational influence of te central body. If all te stars were concentrated in one central mass ten te diameter of te gravitational influence would be 4.5 times larger. Terefore, it appears tat te motion of stars in te galaxy's arms and teir large aggregate mass is disruptive enoug tat wen te stabilizing effect of te central body is sielded by te DM mass no stable structure except te galaxy alo can develop at larger distances. It is tus also clear tat most of te galaxies on a large scale are not bound among temselves by te gravitational forces and will not evolve into a single giant Universe galaxy during teir long lifetime as it migt be oterwise expected. Te galaxies can form clusters or strings only if teir mutual distances do not exceed te 4 times teir diameter. Te most galaxies tus move independently in a free fall to te edge of te Universe, wic is consistent wit te original assumption of te repulsive DM model. Te DM sielding effect tus seems to be te necessary requirement for te existence of individual galaxies in te first place. Te gravitational field of te visible matter terefore does not extend to infinity. It is also interesting tat te mass of te galaxy central star, te Hubble distance, and te galaxy diameter are all related by te simple formula introduced in Eq.9. It would be interesting to confirm te validity of tis formula by investigating if te size of te galaxies depends on teir distance from Eart, since te DM density is reduced at larger distances. As a last topic of tis section it is also necessary to mention te gravitational waves. It is peraps witout any doubt tat te gravitational waves in te tensor form or in a simplest case in te scalar form must exist and are produced. However, te repulsive interaction between te visible matter and te dark matter may significantly affect teir long range propagation and detection due to te various screening and matter depletion effects. So, until a more sopisticated dynamic model of tis interaction is developed it is difficult to claim wit certainty tat te gravitational waves can be detected by te currently proposed metods [1], in particular from te extragalactic sources. To tis date only two gravitational wave events were detected: ttp://vixra.org/abs/166.. In conclusion of tis section it is once more necessary to empasize tat te DM frequently described in te literature, were it is used to explain te gravitational lensing, 19

20 were it is also claimed tat it causes additional forces needed to keep te galaxies togeter, and were it is needed to obtain an agreement wit te measurements of te galaxy rotating curves, is different tan te DM described in tis paper. In tis paper te DM is causing te sielding effect for te standard gravitational forces and terefore te DM measurements publised in te literature are most likely describing te missing or depleted repulsive DM and oter forces resulting from te galaxy arms motion. 5. Te origin of te DM repulsive force Tis section provides explanation for te possible origin of te DM repulsive force, derives te specific value for te DM mass quantum, and derives an interesting relation between te Hubble constant and te CMBR temperature. Te Hubble constant introduced in Eq.9 is related to te Universe's DM mass density at te origin as follows: H 8 = π κ m () From te simple dimensional analysis and te distance comparisons in Eqs.16,17,18, also follows tat te Hubble radius or te Hubble distance is correctly defined as: = c / H (1) and not as is sometimes claimed in te mainstream literature equal to: = c / H. Similarly, te Hubble DM mass of te Universe can be calculated as: and finally, for te Hubble time it is customary to write: 4π 4c M = m = () κ H τ = 1 / H () Te Hubble constant can, terefore, be considered also as a nature's lowest frequency and used for te frequency normalization. Te interesting parameter is te Planck-Hubble mass quantum: m q 68 = H / c = kg = ev (4) and its dimensionless ratio to te Hubble DM mass of te Universe: N p = M m q 5 4c = κ H = (5)

21 Tis very large number can be establised as te fundamental constant of te Universe. Anoter and muc more interesting fundamental DM mass quantum can be obtained by assuming tat te DM is almost a mass-less crystal-like structure consisting of cells wit te standing wave vibrations. An example of suc vibrations is te trapped potons tat, as is well known, exert a repulsive force on te cell walls. Tis is consistent wit te idea of repulsive DM force acting on te visible matter. Te cell mass is ten derived from tese vibrations and any oter mass tat could be te constituent of te cells will be neglected. Te cells may ave random sapes and sizes, but in teir simplest representation may be considered on te average as cubes. Te formula for te cell mass can ten be written as: m λ = = kg (6) 8 c m q 8 Tis formula is derived from te consideration tat eac cell as an average volume: ( λ / ) V = ξ qλ and contains te energy E qλ = m c equal to: qλ λ c E q λ = mc ξ = λ Te parameter ξ = represents te treefold spatial degeneracy corresponding to te tree ortogonal cell vibrations wit te same frequency. Wen tis mass is compared wit te mass of neutrinos, te particular neutrino would ave to ave its mass equal to: (7) m ni = mev (8) Suc a low neutrino mass as already been predicted elsewere [1]. However, wen te mass m qλ is converted to energy and te energy to temperature at te edge of te Universe, were te visible and te dark matter are in a termal equilibrium due to teir strong mutual gravitational interaction tere, te result is surprisingly close to te temperature of te CMBR observed on Eart: Tb =.755 ±. 6 o K [14]. Te Eart observed temperature T bλ corresponding to qλ m is calculated according to te formula: were for g tt it was again substituted: mqλc gtt o Tbλ = =. 77 K (9) k g tt B.487 = e. Tis result, terefore, clearly sows 1

22 tat te Hubble constant and te CMBR temperature are not te mutually independent parameters. Te close agreement of tis result wit te measurement provides once more a clear experimental support for te correctness of te repulsive DM model of te Universe. Te Hubble constant can tus be directly and precisely calculated from te measured CMBR temperature according to te following formula: 8 p k c k BTb 1 1 = 67. km s c gtt H b = 9 Mpc and te mass quantum from te formula: mq λ = kbtb / c gtt = meV (41) wit te corresponding resonant DM frequency: (4) kbtb ν b = =1. 856THz (4) g tt wic te standard BB model of te Universe cannot offer. All te Hubble parameters are, terefore, determined by te CMBR temperature. Tese relations are very interesting and may be important for te evaluation of various models of te Universe were te precise value of Hubble constant may elp to distinguis between te correct and incorrect teories tat are attempting to describe te reality. 6. Conclusions In tis paper te previously developed uniform DM density model of te Universe was generalized to a new model were te DM is deformable tus permitting te propagation of disturbances wit te speed of ligt. Te model provides equations for te recession velocity of galaxies and a number of oter parameters suc as te total DM mass of te Universe, te size of te Universe, te maximum galaxy recession velocity, te maximum observable luminosity modulus, te maximum observable red sift, and te relation between te CMBR temperature and te Hubble constant. All of tese parameters are determined by only tree constants: te Hubble constant, te gravitational constant, and te speed of ligt. Te recession of galaxies in tis model resembles te motion of defects or vacancies in a solid matter tat seem to float from te bulk to te surface, te Universe's edge, were tey disintegrate. It is terefore reasonable to conclude tat te disintegration of te galaxy centers is te cause for te long duration GRB pulses. Te

23 sort duration GRB pulses seem to result from te disintegration of Quasars. Te pulse duration was calculated from te mass and te size of te typical galaxy central bodies and a good agreement wit observations was obtained. It was also concluded tat te galaxy central masses cannot be te Black Holes but are very compact massive objects witout te event orizons [5,7]. Te GRB radiation back to te Universe's bulk may be te cause for te creation of new matter troug te Universe, wic ten condenses to new stars and new galaxies, repeating endlessly te cycle of destruction and creation. Te CMBR temperature seems to also correlate well wit te number of te destroyed galaxies per day. Te detected CMBR radiation pattern is tus te image of te Universe's edge region and not te remnant of te BB. Te developed alternative model provides values for te luminosity modulus as function of te radial distance from te center of te Universe. Tis function was compared wit te extensive data available from te GRBs, and te Supernova Cosmology project, and an excellent agreement between te teory and observations was obtained. Finally, te model also provided a roug estimate for te size of te average galaxy and determined its approximate relation to te Hubble distance and te mass of te galaxy's central body. Te agreement of teory wit observations tus suggests tat te model is correct. It is, terefore, clear tat te new model presents a good alternative for and a considerable callenge to te main stream BB teory. Te new model also avoids te number of implausible and very strange assumptions, wic te BB model must ave; te sudden creation of all te Universe's visible matter from a single point singularity, a sudden space inflation but only between te galaxies not witin te individual atoms of matter, te endless Universe expansion wit galaxies accelerating witout a force acting on tem again only in places were it seems to fit te narrative, and finally te Universe's mass disappearance to notingness after its final conversion to radiation. All of tese strange assumptions including few oters tat are also well known are discussed elsewere [] in te publised literature. Appendix Tis section illustrates te use of te Hubble radius and te average Hubble radius defined previously tat naturally followed from te Hubble constant to simplify te derived formulas by transforming tem into te dimension-less forms. For example, by a

24 introducing te dimension-less radius: x = /, and ten substituting for te mass: m ( ) = m exp( ψ ( )) into te formula in Eq.11 te result, after differentiation, becomes: ( ψ ( )) xψ ( x)' ' + ψ ( x)' = 6x exp x (A1) were te prime denotes te derivative wit respect to x. Similarly, using te substitution: m( ) = m exp( ψ ( ) κm / c ), Eq.5 is transformed into: S ( ψ ( x) x) x ψ ( x)' ' + ψ ( x)' = 6x exp α / (A) were te constant α is equal to: α = κm / c. Eq.A1 is well known as te Emden's S equation [8,15], and its solution was approximated ere by expanding te function ψ (x) into a power series and comparing te coefficients wit te same powers of x on bot sides of te equation. Te result tat was used in Eq.1 is: ψ ( x) = x + x x + x x... (A) Te calculation of te total DM mass of te Universe is also simplified, as is sown below, but sown only for te case of te uniform DM mass density for te sake of simplicity. Assuming tat te uniform DM density extends to te distance for te total mass calculated from te field te following relation: were te potential as been normalized to calculated from te formula derived from te Gauss law: a, we ave c ϕ d( ) M du = d (A4) κ c. Te potential inside of te DM region is πκm ϕ di = + const = x + const (A5) c and outside of te DM region according to te formula tat follows from te Newton gravitational potential: ϕ do 4 a = πκm = (A6) x wic satisfies te zero potential condition at infinity. Similarly for te visible matter te potential inside of te DM region it is: a ϕ (A7) vi = x 4

25 Because at te edge of te DM region te potential must be a continuous function te constant can be determined and te DM potential inside of te DM region written as: From Eq.A4 ten follows for te DM mass of te Universe: ϕ 4 c a di = x H (A8) H a 1 d + d = M + 1 c 8κ c 5 H M du = (A9) 8κ A significant portion of te DM mass is, terefore, derived from te field energy inside of te DM region, wic was in te previous paper neglected [5]. Peraps tis sould not be neglected wen a more accuracy is needed as it was done in te formula derived in Eq.19. Te field energy is equal to te DM mass in tis model following te relation: E = Mc. Once all te DM mass of te Universe is known te Universe's average radius a can be found: 1/ ( 1+ 1/ 5) = (A1) a = Finally, we can also determine te minimum DM potential at te origin were x = : / ( + 1/ 5) = ϕ () = 1 (A11) d Peraps tis is te absolute minimum of te DM potential tat can ever exist in te Universe. Te value found for te potential in Eq.1 did not exceed tis limit. References 1. Frebel, N. Cristlieb, J. E. Norris, C. Tom, T. C. Beers, and J. Ree, arxiv:astro-p/7414v1, 7.. F. Hoyle, G. Burbidge, and J.V. Narlikar, A Different Approac to Cosmology, Cambridge University Press, Cambridge, U.K.,.. Proceedings of te First Crisis in Cosmology Conference (CCC-I), Monaco, Portugal, -5 June 5, ttp:// 4. M. Kowalski et al., ApJ. arxiv.org/abs/asrto-p/84.414, J. Hynecek, Pysics Essays, v, No., July 9, p L. B. Okun, K. G. Selivanov, and V. L. Telegdi, Pys. Usp. 4, 1, 145 (1999). 7. J. Hynecek, Pysics Essays, v 4, No., June 11, p Ya. B. Zel'dovic, I. D. Novikov, Stars and Relativity, Dover Publications, Inc. Mineola, New York, July 1, B. E. Scaefer, arxiv.org/abs/astro-p/6185v1, 11 Dec ttp://wmap.gsfc.nasa.gov/results/ 11. ttp:// 1. ttp:// 1. W. Królikowski, Acta Pysica Polonica B, vol. 6, No.6 (5), p D. J. Fixsen, arxiv: v, [astro-p.co] 1 Nov E. Kamke, Differentialgleicungen, Leipzig 1959, eq