Calculation of the Age of the Universe

Transcription

1 Calclation of te Age of te Univese Clade Mecie eng., Apil 9 t, Rev. Octobe 7 t, 5 clade.mecie@gctda.com og is obsevations, Edwin Powell Hbble discoveed in 99 tat te nivese is expanding. He noticed tat te galaxies, independently of tei own movements, move away fom eac ote at speeds even geate tan wen tey wee moe distant fom eac ote []. He dedced a law and a paamete tat e called H (bette nown as te Hbble constant). Based on tis fact, many astopysicists appoximate te extapolated age of te nivese sing te eqation /H, especially if te density of te nivese is low. is is te metod sed by NASA to calclate a nivese age of.75 billion yeas (see te eslts of te sevent yea of te WMAP poject []). Howeve, te Fiedman-Lemaîte-Robetson-Wale model [4,5,6,7,8,9] pedicts tat te age of te nivese wold ate be aond /(H ). Using te paametes c (te speed of ligt in te vacm), G (te nivesal gavitational constant) and H (te Hbble constant), it is possible to calclate pecisely te age of te nivese tog an integal. is new metod as te meit of econciling te two metods mentioned above by maing a fndamental distinction between te "eal" and te "appaent" age of te nivese. KEY WORDS: Age of te nivese, ligt, big bang, expansion of te nivese. INRODUCION e NASA cently estimates te age of te nivese by sing te invese of te Hbble constant [], tat is to say /H. e model pesented by te pysicists Alexande Fiedman, Monsigno Geoge Lemaîte, Howad P. Robetson and At G. Wale pedicts tat fo a flat nivese dominated by te pesence of matte [4,5,6,7,8,9], te te age of te nivese wold ate be aond /(H ). Einstein was consideing te nivese as being static [8], wit constant spacetime dimensions. Wen e noticed tat is geneal elativity teoy was leading to a nivese in expansion o in contaction, e added a cosmological constant to is eqations to podce is model of te static nivese [8]. Let s mention tat in is teoy of elativity, Einstein was taing fo ganted tat te speed of ligt in te vacm was constant []. It was entiely consistent wit is view of te nivese. A static nivese leads to a constant speed of ligt, except, of cose, on te otsits of lage masses [,].

2 C. Mecie In 99, Hbble fond tat te nivese was expanding []. Wen Einstein became awae of Hbble s obsevations, e was foced to admit tat adding a cosmological constant to is model of te nivese to mae it static was te biggest blnde e as made in is life [8]. It loos lie e did not see, at tat moment, tat te acceleation of ligt ove time was a diect conseqence of an expanding nivese. Howeve, wit ecent wo, we sowed tat it is possible tat te speed of ligt as neve been constant ove time []. Accoding to Einstein, te gavitational potential (conseqently, te mass) canges te index of efaction of te vacm and slows te ligt down. e nivese is expanding [] and we move away fom its cente of mass [6]. eefoe, te index of efaction diminises and ligt acceleates ove time []. Initially, we will fist sow ow te appoximate age of te nivese is calclated. In a second step, we will se some eslts fom a wo we ave done ecently [] to estimate te age of te nivese by pefoming te integal of te invese of te expansion velocity of te mateial nivese accoding to te cvate adis of te nivese. Finally, we will appoximate te age of te nivese. We will sow tat /H actally epesents a good appoximation of te appaent age of te nivese and tat /(H ) epesents te eal pat of te age of te nivese. We can ten compae te eslts and comment.. DEVELOPMEN.. Cent Metods fo te Calclation of te Age of te Univese In 99, Edwin Powell Hbble fond tat galaxies distance temselves fom one anote at a speed popotional to te distance between tem []. He dedced a law involving a constant tat e called H. It epesents te aveage ecession velocity v of galaxies pe nit of distance. v () H v wee t e vale of H is pobably between 7.4 [] and 76.9 m/(smpasec) [7]. In tis pape, we will se te vale of 7.4 m/(smpasec), since it comes fom te most ecent eslts of te WMAP poject of NASA []. Fom te eqation (), we get: His fist pape on te sbject dates fom 9 []. Howeve, sing te geneal elativity bases [], e ad to mltiply te index of efaction vaiation by a facto [4,5]. His teoy is confimed by te gavitational lenses [5,6].

3 Calclation of te Age of te Univese () t H It is wit tis eqation tat NASA is cently evalating te age of te nivese in te appoximation of.75 billion yeas []. is way of calclating te age of te nivese assmes tat te expansion ate of te latte is constant. Accoding to te model of Fiedman-Lemaîte-Robetson-Wale [4,5,6,7,8,9], te te age of te nivese wold ate be aond: 9 () t 9. yeas o t H H.. Calclation of te Age of te Univese A ecent wo tat we made leads s to believe tat te speed of te ligt and te speed of expansion of te mateial nivese ave not been constant ove time []. We will se some eslts fom tis wo to solve te integal of te invese of te speed of expansion of te mateial nivese as a fnction of te adis of cvate of te nivese. In tis way, we will calclate te age of te nivese. Some Einstein wos sowed tat te pesence of a massive body canges te sape of te space-time [] and inceases te index of efaction of te vacm aond te body []. By moving away fom te body, te gavitational inflence is being edced and te speed of ligt tends towad c. o smmaize o pevios wo [], we can say tat te nivese itself is te biggest existing mass. Since te nivese is expanding [], we move away fom te cente of mass of te latte. is cases te index of efaction to diminis ove time. e speed of ligt is teefoe inceasing ove time to eventally tend towad a constant tat we named (wen te appaent adis of cvate of te nivese tends towad te infinity). e cent speed of ligt in te vacm is c. Accoding to o pevios wo [], te speed of ligt inceases by m/s evey 5.4 yeas (te acceleation of ligt is abot a L ch / m/s ). Recent wos allow s to evalate pecisely te vale of te Hbble constant to ±.46 m/(smpasec) [9]. Please efe to Appendix A fo moe details.

4 4 C. Mecie We smmaize ee te vales of te vaios paametes of te nivese wic wee calclated in te wo cited in []. ese paametes wee based only on c (te speed of ligt in vacm), G (te nivesal gavitational constant) and H (te cent Hbble constant). e atio between te cent speed v m of te expansion of te mateial nivese and te cent speed of ligt c is []: 5.76 (4) e asymptotical speed of ligt (wen te appaent adis of cvate of te nivese will tend towads infinity) is []: c 8 5 c 6 m / s (5) e appaent mass m of te nivese is [,] : (6) c 5 m.8 g G H e appaent adis of cvate of te nivese is []: c 6 (7). m H As we mentioned peviosly, Einstein sowed tat te speed of ligt is inflenced by te gavitational potential. Accoding Scwazscild s calclations based on te geneal elativity, it is possible to calclate te speed of ligt v c as a fnction of [5,6]. c ω (8) G m vc ( ) wee n () and ω n( ) ω c v c () New speed of ligt as a fnction of te distance R Distance between te cente of mass m and wee v c () is evalated M Mass ceating te gavitational field G x - m /(gs ) Univesal gavitational constant C x 8 m/s Speed of ligt in vacm is eqation is valid fo te calclation of te speed of ligt aond a blac ole, a sta o a galaxy. Anote way of expessing tis: tis eqation is only valid fo te pesent time. Howeve, as soon as we ty to calclate te speed of ligt fo te time in te past o in te fte, we mst tae into accont te fact tat te speed of ligt canges

5 Calclation of te Age of te Univese 5 as a fnction of te appaent adis of cvate of te nivese []. We mst teefoe cange c in te eqation (8) to (see eqation (5)). e speed of ligt as a fnction of te appaent adis of cvate becomes: y G m (9) v L ( ) wee n( ) and y n( ) y In tis eqation, tee is a adis of cvate fo wic te speed of ligt v L ( ). is position is called te oizon of te nivese. is is te position fo wic te denominato of te sqae oot of te eqation (9) becomes zeo. In a simila way, in a blac ole, te adis of cvate of te oizon is obtained by te Scwazcild s adis wee we eplace c by : G m 5 () 6. m It is te same pinciple as tat of a blac ole []. In fact, te nivese is te biggest existing blac ole since it as te biggest mass. Howeve, nlie a blac ole, ate tan aving te mass located witin te limits of te oizon, a big pat of te blac ole mass of te nivese lies otside te bondaies te oizon. In fact, te blac ole of te nivese is located aond te cente of te mass of te nivese. e speed of te expansion of te nivese is cently te speed of ligt []. Based on te pinciples of Einstein's elativity, matte cannot move at te speed of ligt witot aving an infinite enegy. Conseqently, te pevios assetion can be te only fo ligt. e mateial nivese (containing te galaxies, integalactic dst clods, etc.) is expanding at a speed eqal to c. e facto mst necessaily be less tan. Moeove, accoding to o eqation (4), its vale wold be aond.76. Wen te nivese began its expansion fom, te expansion stated wit a speed eqal to zeo (since te speed of ligt is zeo fo tis adis of cvate). Fo a adis of cvate geate tan, te speed of te expanding mateial nivese v m () is always times te speed of ligt v L (): y G m () v m ( ) v L ( ) wee n( ) and y n( ) y Pefoming te integal of te invese of te expansion velocity of te mateial nivese v m () elative to te adis of cvate, it is possible to calclate te age of te nivese moe pecisely tan by sing a single tangential pojection.

6 6 C. Mecie Let s find te age of te nivese by pefoming te following integal between te adis of cvate of te oizon and te appaent adis of cvate of te nivese : () d d d v m ( ) v m ( ) v m ( ) Pefoming te integal calclation, we obtain: ( z( ) G m ln( [ z( ) ] ) () d v ( ) m 4 wee z( ) 4 G m Conseqently, te vale of becomes: 9 (4) ( i) yeas wee i is eslt is of a complex type. In te eqation (4), te fist pat of te integal is of a eal type (between and ). Howeve, te second pat of tis latte one is of an imaginay type (between and ). Fo now, no one is able to obseve wat appens inside te limits of te oizon of a blac ole (between and ). In tis pape, we mae no conjecte in egad to wat goes on inside a blac ole. e pysical intepetation of a complex type of time is left to posteity fo analysis. Wat is cetain is tat only te potion of te time elapsed (aond 9.7 billion yeas) between te adis of cvate of te oizon (wee te speed of ligt becomes zeo) and te appaent adis of cvate of te nivese is of a eal type and as good and well elapsed. Wen we wis to conside te time elapsed between te position of te big bang and te adis of cvate of te oizon, we mst calclate te modle of te time elapsed. We define tis vale as te appaent age of te nivese becase it does not necessaily epesent te te age of te nivese. is nmbe epesents only an appaent age in te liely event tat te big bang existed. 9 (5) 4.48 yeas We see tat te obtained vale is only 4. % ove te estimated vale of ().. APPROXIMAION OF HE AGE OF HE UNIVERSE As in te calclation of te powe fo an electical moto (wit te eal powe, te indctive powe and te appaent powe), te age of te nivese may be seen as follows: te «eal» pat of te age of te nivese, te «imaginay» pat of te

7 Calclation of te Age of te Univese 7 age of te nivese and te «appaent» age of te nivese. e modle of te two components (eal and imaginay) can be calclated sing te Pytagoean teoem by finding te sqae oot of te sm of te sqaes of te eal pat of te age of te nivese and te imaginay pat of te age of te nivese. e calclation of te appoximation of te age of te nivese will be made in tee pats: te appoximation of te eal pat of te age of te nivese, te appoximation of te imaginay pat of te age of te nivese and te calclation of te modle of te appaent age of te nivese. Imaginay pat of te age of te nivese Real pat of te age of te nivese Appaent age of te nivese Fige Hee, te modle of te appaent age is decomposed into two vectos: its eal pat and its imaginay pat... Appoximation of te Real Pat of te Age of te Univese Let s pefom te appoximation of te eal pat of te age of te nivese. Fo a adis of cvate of te oizon, te following sqae oot eqals zeo: 4 (6) z ( ) 4 G m So, accoding to te eqations () and (), we obtain: (7) z( ) G m ln ( [ z( )]) G m ln( ) 4 wee z( ) 4 G m Using te eqation () and by doing a few simplifications, we obtain:

8 8 C. Mecie ln (8) is same eqation cold be ewitten in tis manne witot canging anyting: (9) c c ln As sown, te content of te bacet is appoximately eqal to. By doing tis appoximation and sing te eqation (7), te eqation (9) becomes: H () We will se tis eqation to eventally pefom te calclation of te appaent age of te nivese. Let s sow tat tis eqation can be fte appoximated to obtain an eqation commonly sed by some astonomes to calclate te actal age of te nivese. Accoding to eqation (4), te vale of.76. Let s se tis appoximation to ewite te eqation (). Afte a few simplifications, we get: 69 H H () is eqation can be dedced fom te model of Fiedman-Lemaîte-Robetson- Wale [4,5,6,7,8,9]. eefoe, te eqations () and () epesent good appoximations of te eal pat of te age of te nivese... Appoximation of te Imaginay Pat of te Age of te Univese Now, let s find te appoximated vale of te imaginay pat of te age of te nivese. Fom te eqation (), () and (6), we get:

9 Calclation of te Age of te Univese 9 () 4 ln 4 ln m G m G m G m G Using te eqation () and pefoming a few simplifications, we get: ( ) [ ] i i ln wee i () Using te following elation: ( ) i i ln π (4) e eqation (), wic is te eslt of a pely imaginay type, can be ewitten: i π (5) We can ewite te eqation (5) in te following manne witot canging anyting: i c c π (6) As sown, te content of te paenteses is appoximately eqal to. e eqation (9) ten becomes: i H π (7).. Appoximation of te Appaent Age of te Univese Let s calclate te appaent age of te nivese by sing te eqation (5) wit te eqations () and (7). Afte a few simplifications and a development of /H, we obtain:

10 C. Mecie π H (8) e content of te paentesis of te eqation (8) is appoximately eqal to. Accoding to eqation (7), te eqation (8) ten becomes appoximately eqal to eqation (): (9) H Conseqently, we ave sown tat te integal of te eqation () can be appoximated by te eqation (). Accoding to s, based on te appoximation calclation made at te eqation (9), te eqation () epesents only an appaent age of te nivese. In fact, it comes fom te calclation of te modle of a complex sm of te eal pat and te imaginay pat of te age of te nivese. 4. CONCLUSION In tis wo, we limited oselves to calclate te age of te nivese. We did not mae any conjecte in egad to wat appens witin te limits of te oizon of te nivese. We ave sown tat te vale /H is a valid appoximation of te appaent age of te nivese. Assming tat tis vale coesponds to te eal pat of te age of te nivese, pysicists assme incoectly, we believe, tat te speed of ligt in vacm was constant ove time. We ave sown tat te vale of /(H ) calclated by some pysicists coesponds to an appoximation of te actal age of te nivese. It also epesents te best estimate of te te age of te nivese. e oiginality of o wo is tat we ave calclated te age of te nivese sing te integal of te invese of te speed of ligt as a fnction of te appaent adis of cvate of te nivese. e only eqied paametes to pefom tis calclation ae c, G and H. is calclation wold not ave been possible

11 Calclation of te Age of te Univese witot te ypotesis fomlated in te wo cited in []. We sowed in te latte, among otes, tat te speed of ligt as neve been constant ove time. e second oiginal point of o wo was to conciliate te fact tat cetain pysicists calclated te age of te nivese by sing te eqation /H and tat otes sed te eqation /(H ). is conciliation is allowed only if we become awae tat te age of te nivese is in fact of a complex type. e modle of te appaent age can be well and tly appoximated by /H wile te eal pat of te age of te nivese can be appoximated by /(H ). O wo aises qestions tat will eqie deepe stdy: Wy is te potion of time between te big bang and te adis of te oizon of an imaginay type? Is te nivese eally bon fom a singla point? Wat is appening witin te limits of te oizon of te nivese? How does te nivese manage to mae te beatog on te oizon since te speed of ligt tee was zeo? Despite te adical cange in concept tat we made, we find tat o calclations ae flly consistent wit te vales of te age of te nivese commonly sed by astopysicists. is fact einfoces te possibility tat o assmptions made in o pevios wo ae coect []. 5. REFERENCES [] Hbble, E. and Hmason, M. L., "e Velocity-Distance Relation among Exta-Galactic Neblae", e Astopysical Jonal, v. 74, p.4. [] Macleod, Alasdai, "Evidence fo a Univese Expending at te Speed of Ligt", Univesity of Higlands and Islands pysics, Scotland, UK, Apil 4. [] Jaosi, N. and al., Seven-Yea Wilinson Micowave Anisotopy Pobe (WMAP) Obsevations: Sy Maps, Systematic Eos, and Basic Reslts, e Astopysical Jonal Spplement Seies, v. 9, no, Febay, pp. -5. [4] Fiedmann, Alexande, "Übe die Kümmng des Rames ", Zeitscift fü Pysi, v., 9, pp (Fo a positive spatial cvate) [5] Fiedmann, Alexande, "Übe die Mögliceit eine Welt mit onstante negative Kümmng des Rames ", Zeitscift fü Pysi, v., 94, pp. 6-. (Fo a negative spatial cvate) [6] Lemaîte, Geoge, "Un nives omogène de masse constante et de ayon coissant endant compte de la vitesse adiale des nébleses extagalactiqes", Annales de la société scientifiqe de Bxelles, v. A47, 97, pp

12 C. Mecie [7] Robetson, Howad P., "On te Fondations of Relativistic Cosmology", Poceedings of te National Academy of Sciences of te United States of Ameica, v. 5, 99, pp (Unified witing of te Fiedman s eqations fo evey vale of spatial cvate) [8] Robetson, Howad P., "Relativistic Cosmology", Review of Moden Pysics, v. 5, 9, pp [9] Wale, At G., "On Milne s teoy of wold s stcte", Poceedings of te London Matematical Society, v. 4, 96, pp (Identical eslts to tose fond independently by HP Robetson) [] Cavalo, Joel C., "Deivation of te Mass of te Obsevable Univese", Intenational Jonal of eoetical Pysics, v. 4, no, Decembe 995, p. 57. [] Mecie, Clade, "e Speed of Ligt May Not Be Constant", Pagtec, Baie-Comea, Qébec, Canada, Octobe 8,, pape available on Intenet at: [] Einstein, Albet, "On te Inflence of Gavitation on te Popagation of Ligt", e Pinciple of Relativity (Dove Boos on Pysics), New Yo, Dove Pblications, 95 (oiginal pape of 9), pp [] Einstein, Albet, "e Fondation of te Geneal eoy of Relativity", e Pinciple of Relativity (Dove Boos on Pysics), New Yo, Dove Pblications, 95 (oiginal pape of 96), pp [4] Binney, James and Micael Meifield, Galactic astonomy, Pinceton Univesity Pess, 998, p. 7, fom te eqation A. [5] Manegetti, Massimo, "Intodction to Gavitational Lensing, Lecte scipts", Institt fü eoetisce Astopysi, Bologna, Italy, 6, p. 7, fom te eqation.9, Web. <ttp:// [6] Matzne, Ricad A., "Dictionay of Geopysics, Astopysics, and Astonomy (Compeensive Dictionay of Pysics)", Boca Raton: CRC,. [7] Bonamente, Massimiliano and al, "Detemination of te Cosmic Distance Scale fom Snyaev- Zel dovic Effect and Canda X-ay Measements of Hig Redsift Galaxy Clstes", vesion, Apil 6, p., axiv:asto-p/549v, Web. ttp://axiv.og/ps_cace/asto-p/pdf/5/549v.pdf [8] Weinbeg, Steven, "Einstein s Mistaes", Pysics oday, Novembe 5, pp. -5. [9] Mecie, Clade, "Calclation of te Gavitational Constant G", Pagtec, Baie-Comea, Qebec, Canada, Mac,, aticle available on Intenet at: [] "Latest () Vales of te Constants", NIS Standad Refeence Database, latest pdate: Apil, Intenet aticle: ttp://pysics.nist.gov/c/constants/index.tml [] Wang, Xiaofeng and al., Detemination of te Hbble Constant, te Intinsic Scatte of Lminosities of ype Ia SNe, and Evidence fo Non-Standad Dst in Ote Galaxies, Mac, pp. -4, axiv:asto-p/69v

13 Calclation of te Age of te Univese 6. APPENDIX A (Added Octobe t 4, evised Octobe 7 t 5) Witot canging te entie docment tat pecedes te appendix, we want to bing a few pecisions on te Hbble constant H. is "constant", wic isn t one, vaies slowly ove time. Since its discovey by Hbble, many ways ave been sed to mease it sing te diffeent obsevation made by telescope. In spite of all effots, tis paamete, wic is essential to descibe o nivese, sffes a ge lac of pecision. Effectively, diffeent measements made by diffeent teams sow tat tis paamete is pobably between 69 and 77 m/(smpasec). All obseves popose incetitde magins wic do not matc. Howeve, witot aving a measement tat joins eveybody togete and as long as someting bette is being poposed, a cetain consenss is made aond a vale of 7 m/(smpasec). Most seios astopysics boos se tis vale. Since new calclations on te nivesal gavitational constant G we ave been able to mease pecisely te Hbble constant [9]: 9 / () c α H ±,46 m/(s MPasec) e e following vales come fom te CODAA []: e speed of ligt in vacm c m/s e fine stcte constant α e classical adis of te electon e m e constant epesents te atio between te expansion speed of te mateial nivese and te speed of ligt (see eqation (4)). e vale sown in eqation () is confimed by measements made by te Xiaofeng Wang team [] tat obtained H 7. ±.9 m/(smpasec). e appaent age of te nivese is given by: H e MPasec is a nit of distance defined as following: MPasec m () ()

14 4 C. Mecie e appaent age of te nivese, in yeas, is given by te following eqation: () s MPasec m m yea day min m MPasec m 65.5 days 4 6 min 6s e vale of te appaent age of te nivese is teefoe: (4).5646 yeas.5646billion yeas Let s emind tat, accoding to o model of te nivese, te te age of te nivese is of complex type.