A cosmological view on Milgrom s acceleration constant

Transcription

1 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v A osmologil view on Milgrom s elerion onsn Engel Roz Sripperwei, 555 ST Vlkenswrd, The Neherlnds Emil: engel.roz@onsbrbnne.nl Summry In his rile wo-prmeer model is developed for he universe. The wo prmeers re he ge of he universe nd he vlue of Einsein s Cosmologil Consn. I is shown h hese re suffiien o lule he mouns of mer nd drk energy in he universe, s well s he onribuions of drk mer nd bryoni mer in he mer pr. All his, no only for presen ime, bu lso s funion of osmologil ime. Moreover, he model llows esblishing he numeril vlue for Milgrom s elerion prmeer for presen ime. The developed heory gives n deque explnion for he phenomen of he elered sling of he universe nd he nomly of he sellr roion urves in glxies. The numeril resuls re in greemen wih hose of he md-cdm model. Keywords: Cosmologil Consn; MOND; drk mer; drk energy; md-cdm model. Inroduion This rile is men s n exension o previous sudy []. I is my im o give some view on he osmologil spe, in whih I will pply he oneps h I hve developed o give n explnion for he negive pressure exeued by he spil fluid h mus be presen in vuum o jusify posiive vlue of he Cosmologil Consn in Einsein s Field Equion. Suh vlue is required o remove he nomly of priulr osmologil phenomen, like he roion urves of srs in glxies nd he elered expnsion of he universe. In my previous rile, i hs srighforwrdly been derived h in grviionl sysem wih enrl mss M in vuum, he Cosmologil Consn, while independen of spe-ime oordines, mouns o / 5MG, ( where ( - m/s is Milgrom s elerion onsn [,,] nd G he grviionl onsn. I is my im o show h his relionship beween Milgrom s elerion onsn nd he Cosmologil Consn no only pplies o glxies, bu holds for he universe s whole s well, in spie of he obvious diffiuly o idenify enrl mss. The sudy is promped by he wish o find mens o ssess he numeril vlue of Milgrom s elerion onsn by heory, for whih no lue ould be found wihin he sope of glxies. Sisfying Einsein s Field equion in vuum under bsene of mssive soure under he ondiion of posiive Cosmologil Consn requires he presene of bkground energy [5]. Under inlusion of he mssive soure, he bkground energy will of ourse sill be here nd shows up s polrized dipoles [,], whih explins he / M dependeny of he Cosmologil Consn. I my seem h grviionl dipole onep, in he sense of 7 by he uhor(s. Disribued under Creive Commons CC BY liense.

2 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v bond beween posiive mss nd negive mss, violes physis, beuse negive mss is no vible onep. I showed up, however, in model h explins he wek limi soluion of Einsein s equion wih posiive Cosmologil Consn. As is well known, suh soluion fores viewing he vuum s fluidl spe wih negive pressure (orresponding wih posiive bkground mss densiy. A lose inspeion will revel (see ppendix in [] h his soluion is nohing else bu he resul of modulion of he bkground energy densiy, due o disurbne used by he inserion of enrl mss M. This llows o oneive he disurbne s grviionl dipoles on he pedesl of he bkground energy densiy. Wh my seem s negive mss in he grviionl dipole is dip in he bkground energy. The dipoles onsiue grins in he fluidl vuum wih negive pressure h neurlizes he grviionl fore beween he poles. This mkes he grviionl dipole vlid onep. The vuum fluid is n idel one. This implies h is sress-energy ensor onins digonl elemens only nd h in spe-ime ( ( i, x, y, z wih (+,+,+,+ meri hey ll hve he sme vlue. The pressure pv is due o he bkground energy of he fluidl spe, expressed by, p v ρ, ( 8 π G where is he ligh veloiy in vuum nd where ρ is he fluidl mss densiy. I mus be presen for giving soluion of Einsein s Field Equion wih for he vuum wihou ny bryoni soures. Assuming he orreness of (, he fluidl pressure would rise o infiniy under he bsene of bryoni soure. Bryoni soures e from he fluid, hereby gining mss nd reduing he energy level of he fluid. This model is pplible o osmologil objes wih mssive kernel, like glxies. Ineresingly, hough, i n be evolved o model h fis o he osmos where mss is disribued. For resons o be explined ler, semni differene is mde beween somehing denoed s osmos nd somehing else denoed s universe.. Cosmologil model e us model he osmos s sphere wih disribued mer nd some rdius. This disribued mer is mixure of fluidl mer M D s men by ( nd bryoni mer M B s men by (. Effeively, he ler is he ol of enpsuled mss. ene, from ( nd (, M D πr dr. ( 5M G B e ΔM D he differene beween he fluidl mer in sphere + Δ nd he fluidl mer in sphere. I follows redily h

3 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v ΔM D M B G Δ. ( Beuse of he influene of he bryoni mss on he energy level of he fluidl mss, i is fir o se h, M β, (5 B M D where β is dimensionless quniy. ene, by inegring (,. ( β G M D The ol grviionl mssive energy M G M G (fluidl plus bryoni, herefore, is M D + M B ( + β. (7 5βG ene, he mssive energy densiy ρ G in he sphere wih rdius is given by ρg π ( + β π ( + β. (8 5βG 5βG This unifiion of bryoni mss nd fluidl mss llows us o ompre his mssive energy wih he one s obined in he presen osmologil model, known s he md-cdm model (CDM Cold Drk Mss, md Einsein s Cosmologil Consn, [7] The CDM-md model hs been evolved from he Einsein-de Sier model h hs been he preferred one for he universe up o he 98s. I hs been refined o he presen one o ope wih erin osmologil phenomen, like for insne he disovery of he elering universe in 998, [8,9].. The md-cdm model The model evolves from he soluion of Einsein s Field Equion under he onsrin of priulr meri. Gμν + g μν T μν wih Gμν Rμν Rg μν where soure(s nd where, (9 Tμν is he sress-energy ensor, whih desribes he energy nd he momen of he Rμν nd R re respeively he so-lled Rii ensor nd he Rii

4 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v slr, whih n be luled if he meri ensor omponens g μν re known [,,]. The doped meri, known s he Friedmnn-emirre-Roberson-Wlker (FRW meri [], is dr ds dq + ( ( + r sin ϑdϕ + r dϑ, ( kr where q i is he normlized ime oordine ( i, nd where k is mesure for he urving of spe-ime. The sle for ( expresses he ime-dependene of he size of he universe. The rio (, ( is known s he ubble for. I is he min observble of he universe, beuse is numeril vlue n be esblished from red shif observions on osmologil objes ( d / d. By moving he erm g μν o he righ side of (9, i n be oneived s n ddiionl onribuion o he energy densiy, ρ( ρ( + ρ ; ρ ; κ κ. ( p( p( ρ. (b The soluion of (9 under onsrin of he meri ( re he wo Friedmnn equions [], k ( + 8π G ρ ( k 8 G π + ( + p. (b Under he onsrin k (fl universe, nd king ino onsiderion (,b, he firs Friedmnn equion evolves s, ( ( ρ + ρ ( ρ ; ρ + ρ ρ. ( The seond Friedmnn equion reds s, ρ π 8 α. (b ρ G(ρ ρ πg(ρ + ρ ρ πgρ ( α;

5 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v Differeniing he mss densiy ρ in ( gives, d ρ ( [ { ( }] d 8 { πg( α } ρ {( α } ρ α ρ dρ ρ d dρ d ρ α ρ ρ ρ d ρ d d d dρ dρ d + ρ. d d d (5 Beuse he bkground mssive densiy spe-ime oordines, (5 is sisfied if, ρ is ime-independen ( is independen of dρ d ρ ρ m α nd ρ, (,b d d where m, nd re onsns. The quniy is he ubble prmeer / (. I is emping o believe h m nd re, respeively, he relive moun of bryoni mss ρ / ρ nd he relive moun of bkground mss ρ / ρ (. This, however, is no neessrily be rue, beuse (wihou furher onsrins he differenil equion ( is sisfied for ny disribuion beween m nd s long s m +. Applying (,b on he firs Friedmnn equion (, resuls ino, ( m +. (7 This equion represens he md-cdm model in is mos simple form (ully, more erms re heurisilly dded under he squre roo operor o model empiril evidene from erin osmologil phenomen. Eq. (7 n be nlyilly solved s [5], ( m. (8 / / ( sinh ( / ; A presen ime P, he sle for equls uniy ( nd he ubble prmeer is he observble. Equing presen ime wih ubble ime P is jusified if ( would hve shown liner inrese over ime up o now, under onsn re of sy, beuse in h se ( nd (. This is ubble s empiril lw. Equing P in (8 s n xiomi ssumpion, indeed resuls in behvior of he sle urve h, up o presen ime, is prey lose o ubble s empiril lw. ene, from (7, 5

6 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v ( ; ( +. (9 ene, from (9 nd (8, m ( / sinh / ( /.77; m.. ( These vlues re only slighly differen from hose in he six-prmeer md-cdm model. The differene is due o he simple form (7. Figure demonsres he vibiliy of he xiomi ssumpion o eque presen ime wih ubble ime h Fig.: The sling for ( s funion of osmologil ime. The lower urve represens ubble s lw. The upper urve shows he urve of elered sling due o Einsein s Cosmologil Consn.. rmonizing he view on he universe wih he view on he osmos e us proeed by rying o hrmonize he view on he osmos s disussed in he seond prgrph wih he view on he universe s disussed in he hird prgrph. This will be done by ompring he moun of mssive energy in he universe h is presenly observble wih he mssive energy h would be observble on he bsis of he osmologil model. Noe h we wish only king re of observbiliy, hereby ignoring possible mssive energy in non-observble rnges. The presen mssive densiy ρ (, known s riil mss ρ in he md-cdm model observed wihin he ubble horizon, n be esblished from ( s, ρ. ( Is mssive energy is equivlen wih he mssive energy s defined by (7. ene, ( + β. ( π 5βG

7 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v e us elimine nd of elerion (m/s, from ( by inroduing quniy wih he dimensionliy, ( ene, from nd (9,. ( is no free vrible, bu known quniy h n be I hs o be emphsized h luled from ( s 7x - m/s. Expression ( evolves s, ( 8 ( + β π ( + β { } 5βG 5β 8G ( + β 8x5 β x ( + β 5βG. (5 This expression esblishes relionship beween he presen rio of bryoni mer over fluidl mer nd Milgrom s elerion onsn. Tking Milgrom s elerion onsn for presen ime nd he ge of he universe s independen prmeers, llows o esblish he rio of he vlues for bryoni mer nd drk mer, whih in he md-cdm model re onsidered s primry prmeers. From.8 Gyer, we hve /(.9 m/s. For.7 m/s we ge from (5 he resul β.85, whih is he rio bryoni mer/drk mer in he md-cdm model. The orrespondene of he luled vlue for Milgrom s elerion prmeer in presen ime wih evidene of observions gives srong suppor for he vibiliy of he heory developed in his rile. 5. Drk mer nd drk energy The issue o be resolved sill is he problem how o rele he heoreilly esblished quniies m nd for mer nd drk energy, whih hve heir origin in Einsein s Cosmologil Consn s well, wih bryoni mer nd drk mer. While under expnsion of he universe he mer densiy dereses proporionlly wih he size of he universe, he bryoni pr of sill oninues eing from he drk mer pr. The densiy of he drk energy, however, remins invrin under expnsion, possibly fer n iniil loss of mer. Th mens h he mer pr is mde up by bryoni mer nd drk mer in rio h sles under expnsion of he universe. The sling behviour n be esblished from he following onsiderions. For he mssive energy of he universe, we hve from (, 7

8 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v M r dr π G. ( From ( nd (, he Cosmologil Consn for he universe (whih is differen from he of glxies n be esblished s G. (7 5 MG 5 G (Noe h, beuse of he ime-independeny of nd he sling of Milgroms elerion onsn sles wih α /, ene, from (7 nd (5, 5 8x5 5 x ( β + β onsn (8 β ( + β onsn. ene, k ( / β k ( β ( + β k ( / ± k ( /, (9 where he onsn k n be esblished from he known vlue of β.85 presen ime. Figure shows he rios β nd β s funion of osmologil ime. Beuse he bryoni onen es from he fluidl mss onen, he bryoni pr nno exeed he drk mer pr. Therefore β, suh s shown in he grph. rio h Fig.. The upper urve shows he relive moun of drk mer s funion of osmologil ime. The lower urve shows he relive moun of bryoni mer. 8

9 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v. Conlusion Puing everyhing ogeher, we rrive very simple model for he universe, whih neverheless explins phenomen s drk mer, drk energy, he nomly of he sellr roion urves in he glxies nd he elered expnsion of he universe. I gives n explnion s well for he empiril lws of ubble nd Milgrom. Nex o Newon s grviionl onsn G, only wo oher prmeers re needed. These re he ge of he universe nd Einsein s Cosmologil Consn for he universe. Everyhing else follows from Einsein s Field Equion in onjunion wih he FRW-meri. e me summrize he bsi equions nd heir impliions. The ime behviour of he sling for of he universe is soluion of Einsein s Field Equion under he FRW-meri, ( ( sinh ( / m / / ;. ( As onsequene of (, he relive vlues for mer densiy nd drk energy re esblished s, m. nd.77. ( The relionship beween he Cosmologil Consn of he universe nd Milgrom s elerion prmeer is esblished s, 5 ; ;. ( Aeping he life ime of he universe.8 Gyer nd 7.9 x -5 m - s primry independen quniies, we ge.7 m/s. The relionship beween Milgrom s elerion prmeer nd he rio β of bryoni mer over drk mer is esblished s, 9 8x5 β. ( x ( + β The resuling vlue for he presen vlue of his rio mouns o β.85. The evoluion over ime is shown in figure. Beuse he bryoni mer nno exeed he iniil moun of drk mer, he rio β hs he upper limi β. All his hs been derived srighforwrdly by his wo-prmeer heory. The luled quniies orrespond niely wih hose of he six-prmeer md-cdm model, whih is lrgely empiril. The benefi of inluding more prmeers is he modelling of osmologil effes beyond he sope of he simple model. The benefi of he simple model is is srengh o show he relionship beween drk energy nd drk mer s well s he relionship

10 Preprins ( NOT PEER-REVIEWED Posed: Deember 7 doi:.9/preprins7.77.v beween Milgrom s empiril MOND heory [] nd he md-cdm model. Moreover, he wo prmeer heory srenghens he prediion mde before h lrge osmologil disne grviy urns on nd-off ino nigrviy wih some spil periodiiy []. 7. Disussion The previous prgrph onins he onlusions of his work. rder hn formule is no possible. And beer proof hn he mh wih known empiril evidene nno be given. This leves he problem of inerpreion. This will be open for disussion nd opinions migh diverge. In he piure of he uhor, whih he wns o give free for beer one, he universe seems ppering s bubble in he osmos. The osmos is se of fluidl energy. Oherwise Einsein s Cosmologil Consn would be zero (n empy universe does no llow vible soluion of Einsein s Field equion for. The universe is reed from he subrion of mer bubble from his se. The rio of he fluidl energy nd he mer energy is found from Einsein s Field Equion nd he xiom h he universe is fl one. The mer subrion is he even h mrks he birh of he universe. The mer bubble migh be quid pro quo for sponneously reed ubiquius ones. The mer bubble onsiss of drk mer, whih is grdully onvered ino bryoni mer, in re h is deermined by he vlue of Einsein s Cosmologil Consn for he universe, whih is nurl onsn nex o he Grviionl Consn. The onversion re is reled wih he expnsion re of he universe, whih is elered beuse of he non zero-vlue of he Cosmologil Consn, hene beuse of he energei fluid in he osmos. As shown in he previous sudy [], in whih he relionship beween Milgrom s elerion onsn nd he Cosmologil Consn hs been derived, he se of fluidl energy onins grins wih grviionl dipole. The negive pole is dip in he se of energy. Th mens h he grins re ripples in he se nd h he grviionl dipole hs pedesl. In glxy, he grins influene he grviionl fore, beuse he field from he enrl bryoni mss polrizes heir grviionl dipole momen, hereby reing he grviionl equivlen of displemen hrge h dds o he bryoni one. Effeively, i mens h mer from ouside is pushed in wihin he osmologil horizon. This displed mer is he drk mer. This drk mer model is less ler for he universe, where bryoni mer hs uniform disribuion, hereby using n overll rndom disribuion of he grviionl dipole momens of he grins. Neverheless, he universe is n ssembly of glxies s well, whih ll show he mehnism of ring mss from beyond heir osmologil horizons. I migh well be h ll heir onribuions sum up o he drk mer of our universe. The developed heory predis he onversion of drk mer ino bryoni mer. owever, in spie of n deque desripion of he proess, s onsequene of Einsein s Cosmologil Consn, i does no revel he very physil sruure of he onversion. This onversion seems o be n irreversible proess, kin o osmosis, in whih, for some reson, sble bryoni sruures re reed from he enropi behviour of he drk mer kernels. Furher reserh is required o gin more insigh in his.

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