Effect of Density Components in the Evolution of the Universe

Transcription

1 Effect of Desity Compoets i the Evolutio of the Uiverse Kara Farsworth, Christia Kig, Rya Price Uiversity of Arizoa The formatio, expasio ad evolutio of the structure of the uiverse are determied by the ratio of varyig desities of the compoets that comprise it. Usig the Friedma Equatio to describe the evolutio of the uiverse, we ivestigated the properties of the each compoet of the uiverse to determie the role played by each i the evolutio of the uiverse. Usig umerical methods, we solved the Friedma Equatio usig various parameters for the desities. From the computatios, we foud that the age of the uiverse is largely depedet upo the dark eergy ad matter compoets of the uiverse while the radiatio desity has less of a effect. Itroductio Curretly, most scietists believe that the uiverse is approximately 13.4 billio years old. There are multiple ways to approach the measuremet the age of the uiverse. Oe method is to use the rate of expasio of the elemets i the uiverse to ifer the evolutio of it. Edwi Hubble was oe of the first scietists to discover that the uiverse was expadig. The discovery that distat galaxies are all recedig from us with velocities that are proportioal to their distaces shows that all the galaxies came from oe sigle poit ad expaded at the same time 1. By measurig the rate of expasio, assumig that it is costat, ad how far apart the objects are today, scietists ca estimate the amout of time eeded for these astroomical bodies to become as far apart as they are curretly 3. Usig this method the uiverse is roughly 14 billio years old. Aother approach is the observatio of cosmic microwave backgroud or CMB. By measurig the CMB scietists have studied the radiatio of the uiverse as far back i time as the Big Bag 1. The uiverse is so large that whe scietists observe far away materials ad radiatio, they are i fact lookig back i time. This is due to the time take for light to travel from these places to Earth 3. Therefore, by usig the amout of radiatio of the material to fid the distace to these sources, scietists ca estimate the age of the uiverse. Similarly, measurig specific microwaves from the begiig of the uiverse are used like the CMB to calculate the uiverse s age. The 1 Va De Bergh, Sidey. "Size ad Age of the Uiverse." Gould, Adam. "IPS Offical Statemet o the Aciet Age of the Earth ad Uiverse." 3 Barkaa, Rea. "The First Stars i the Uiverse ad Cosmic Reioizatio."

2 Farsworth, Kig, Price detectio of the 3 K microwave backgroud by Pezias ad Wilso, which represets the red shifted radiatio of the primeval fireball provides evidece of a sigle startig poit ad also evidece for the Big Bag, which may scietists believe is how the uiverse started 4. Despite how the uiverse started, most agree that it has a defiite begiig. There is also other evidece that makes scietists thik that there is a sigle begiig to the uiverse. The uiverse could also be ope, i which there would be o begiig ad it would be idefiitely old. However, there is ot as much evidece supportig this argumet. Though these methods give a approximate age of the uiverse, the most exact method is solvig Friedma s equatio. Aleksadr Friedma was a Russia physicist ad mathematicia who, after studyig Eistei s Friedma derived his equatio usig the Eistei field equatios, assumig a homogeeous ad isotropic uiverse 5 ad wrote the paper O the curvature of Space to Zeitschrift für Physik. I this paper Friedma proposed that the radius of curvature of the uiverse ca be either a icreasig or a periodic fuctio of time 6 ad derived his famous equatio. Assumptios ad Equatios While comparig the red-shifts ad the distaces of differet objects i the uiverse, Edwi Hubble realized there was a relatioship betwee the distace of the object ad its velocity away from us. From this iformatio, Hubble cocluded the uiverse is expadig. He expressed this expasio usig his Hubble costat. v = Hd (1) Where H is the Hubble costat. With further observatio, scietists cocluded that the uiverse is ot oly expadig, but also acceleratig. This meas that the Hubble costat is ot actually costat. If the distace betwee objects i the uiverse is represeted as R(t), ad the distace betwee objects curretly is represeted as R, the the ratio betwee them,α, is R( t)!( t ) = () R 4 Va De Bergh, Sidey. "Size ad Age of the Uiverse." 5 Weisstei, Eric W. "Friedma-Lemaitre Cosmological Model." 6 "Aleksadr Aleksadrovich Friedma."

3 Farsworth, Kig, Price 3 Where α () = 1. The the Hubble costat ca be writte i terms of α. 1 H = ' & $ % d' #! dt " (3) By takig the iverse of the preset Hubble costat at time =, H, a rough estimate of the age of the uiverse ca be foud. 1 $ = t " & 71# 1m H % 1km # 31536s # 1yr 1Mpc ' 3*1 ) m( *1 =1.34 *1 1 yrs (4) Or the uiverse is approximately 13.4 billio years old. But sice the Hubble costat chages with time, this umber is oly a approximatio. The real age of the uiverse is better calculated usig the Friedma equatio. The Friedma equatio icludes the Hubble costat to give a much more accurate age of the uiverse as well as icorporatio the acceleratio of the uiverse. 1 * d! ' ( % dt 8# G = 3 (" r + " m + " d ) $! ) & R kc! (5) I this equatio, G is the gravitatioal costat = 6.67*1-11 Nm /kg, ρr is the radiatio desity of the uiverse, ρm is the matter desity of the uiverse, ad ρd is the dark eergy desity of the uiverse. The variable k is the curvature parameter of the uiverse ad ca be either -1,, or 1 for a closed, flat, ad ope uiverse respectively. R as stated before is the separatio distace betwee objects i the uiverse presetly, ad c is the speed of light = 3*1 8 m/s. The various desities of the uiverse also vary with time. " r (t) = " r # 4 (6) " m (t) = " m # 3 (7) " d (t) = " d (8) These desities are usually expressed i terms of the critical desity, ρc. 3H " c = (9) 8! G

4 Farsworth, Kig, Price 4 I order to make equatio 5 dimesioless, the various desities are better expressed as a ratio of the critical desity. The ratios of the desity of the uiverse vs. the critical desity are expressed as Ω. " r (t) = # (t) r = # r # c # c $ (1) 4 " m (t) = # m (t) # c = # m # c $ 3 (11) " d (t) = # d (t) # c = # d # c (1) " = " r + " m + " d (13) I order for the Friedma equatio to work, there must be a relatioship betwee k (the curvature parameter) ad Ω. Whe Ω < 1, k = -1. Whe Ω = 1, k =. Whe Ω > 1, k =1. Takig these parameters ito accout, the age of the uiverse ca be foud for a ope, flat or closed uiverse. Substitutig equatios 9, 1, 11 ad 1 ito equatio 5 gives: d! = H! dt " 4 " 3 kc (# r! + # m! + # d )" H R! (14) At time =, = 1, H = H, ad =. The iitial coditios were used to evaluate equatio 14 to fid the solutio for the value of R. The resultig relatioship becomes: kc 1 = "! (15) H R kc Equatio 15 ca be rearraged to be: 1"! =, which is equal to the H R last term i equatio 14. By substitutig these values, a dimesioless form of the Friedma Equatio is give by:! 4! 3! (" # + " # + " )! ( 1! " ) d# = H# r m d # dt (16) We put equatio 16 ito a program ad solved for α(t) usig the Ruge-Kutta method.

5 Farsworth, Kig, Price 5 Depedig o the type of material icluded i our uiverse, the uiverse will have differet predicted ages. Research has foud that for the curret uiverse Hubble s costat is about 71 km/s/mpc, r=3x1-5, m=.7, ad d=.73. Usig umerical solutios, we ivestigated the age of the uiverse for differet values of the cosmological parameters. The equatios we used i our research caot be solved aalytically for all solutios. However, usig umerical methods, the Friedma equatio, equatio 16, was solved for various desity parameters ad scearios. To solve the Friedma equatio umerically, we used the 4 th -order Ruge- Kutta method. The Ruge-Kutta method propagate(s) a solutio over a iterval by combiig the iformatio from several Euler-style steps (each ivolvig oe evaluatio of the right-had f s), ad the usig the iformatio obtaied to match a Taylor series expasio up to some higher order. 7 The Ruge-Kutta method works by usig iitial values say x ad y ad watig to fid y+1. We used the followig algorithm 6 to umerically solve the Friedma equatio: k k k k y = h! f ' = h! f % x & ' = h! f % x & = h! f + 1 = y ( x, y ) h, y k + ( x + h, y + k ) + h +, y k1 $ + " # k1 k k3 k Where h is the step size ad f(x,y) is the differetial equatio: the Friedma equatio. Usig a give set of iitial parameters, the algorithm was able to produce a plot of the value of versus the time t. For each sceario that is umerically solved ad plotted, we iferred the age of the uiverse from the poit where equals zero o the t axis. 3 $ " # Verificatio of umerical method The simplest way to verify the umerical method to solve the Friedma Equatio is to look at specific scearios, formulate their outcomes, ad compare them to the umerical results with similar parameters. To do this, we started with a variatio of the Friedma Equatio: 7 Flaery, Bria P., William H. Press, Saul A. Teukolsky, ad William T. Vetterlig. Numerical Recipes i C: the Art of Scietific Computig.

6 Farsworth, Kig, Price 6 H = H & # ' 4 ' 3 kc $! ( r a + ( ma + ( d ' (17) % H R a " Where H is the Hubble parameter described i equatio 3. Usig this form, there are three specific cases that ca be solved aalytically. These cases are the matter domiated uiverse, the radiatio domiated uiverse ad the dark eergy domiated uiverse. For each case, the desity parameter was assumed to equal 1, makig k =, i order to simplify the result. I the case of a matter-domiated uiverse, the uiverse is affected most by the -3 term of equatio 17. By this, Friedma s equatio becomes: H ( 3 )! ( 3 / ) = H a! Substitutig equatio 3 ito equatio 18 gives: H = H a (18) ( a ) da = ( H ) dt (19) Ad upo itegratig the terms ad usig the predefied value of () = 1: / 3 & 3H # ( t) = t! " ( ) a $ () % We used this method of substitutio ad itegratio also with the dark eergy ad radiatio domiated uiverses. I the case of dark eergy, (t) is give by: ( t) = exp H t (1) ( ) a Ad for a radiatio domiated uiverse: ( H ) 1 / a ( t) = t () Equatios, 1, each provide isight ito how each costituet of the uiverse affects the fial plot, acceleratio, ad, most importatly, the age of the uiverse. Each of the equatios is plotted i Figure 1 as a example of each variatio of the evolutio of the uiverse:

7 Farsworth, Kig, Price 7 Figure 1 - Plot of Aalytical Solutios Usig these equatios, we ca check the effect of the umerical method agaist these scearios. To verify the solutio, we used the parameters for the umerical method, set to the same values as the assumptios used for the aalytical solutios. Whe we used the umerical method to solve for the sigle domiatig factor i the uiverse, the outcome is closely modeled by the aalytical solutios, which verify the accuracy of the umerical method employed. The plot of the umerical method usig the scearios described above is show i figure : Figure - Plot of Numerical Solutios By comparig the two plots we saw that both the aalytical ad umerical plots display the same behavior ad thus verified the legitimacy of the umerical method.

8 Farsworth, Kig, Price 8 Results For each compoet of desity of the uiverse, multiple values were measured agaist the calculated age of the uiverse for each sceario. By doig such, a tred occurs that shows to what extet each plays i the evolutio ad more importatly the age of the uiverse. I the followig tables ad plots are the results of varyig the desity parameter of a sigle compoet while holdig the other two costat at the experimetally measured values. For example, for a varyig radiatio desity, the desities of matter ad dark eergy were held at.66 ad.73 respectively. This allowed us to sigle out the effects of each compoet ad determie its impact o the age of the uiverse. A plot of the actual uiverse was also calculated ad plotted i Figure 6 as a referece. Ωr t Actual Time (1 9 years) Table 1 - Radiatio Desity Time Scale Age of Uiverse (1^9 Years) Age of Uiverse vs. Radiatio Desity Radatio Desity Figure 3 - Radiatio Desity Time Scale Ωm t Actual Time (1 9 years) Table - Matter Desity Time Scale Age of Uiverse (1^9 Years) Matter Desity Timescale Matter Desity Figure 4 Matter Desity Time Scale

9 Farsworth, Kig, Price 9 Ωd t Actual Time (1 9 years) Table 3 - Radiatio Desity Time Scale Age of Uiverse (1^9 Years) Dark Matter Desity Timescale Dark Eergy Desity Figure 5 Dark Eergy Desity Time Scale Figure 6 Expasio of the Uiverse for Predicted Values of Desity Parameters Coclusio We ca see the effects of various desities of the uiverse from the data produced. From the results, we determied the relatioships betwee each of the compoets of the uiverse, its desity, ad the age of the uiverse. The age of the uiverse has differet relatioships amog each compoet suggestig that a variatio i oe parameter would ot have the same effect as aother parameter. May of the results that occurred from solvig the Friedma equatio umerically could have bee iferred from the aalytical solutios. The depedecies upo the value of (t) for each desity parameter describe how each compoet of matter effects the evolutio of the uiverse. The a -4 term

10 Farsworth, Kig, Price 1 describes that the radiatio desity affects the evolutio at small values of t, or the early uiverse. The radiatio at this poit causes rapid expasio. As the effect of the radiatio decreases o the uiverse, the ifluece of the matter with the a -3 depedecy icreases. At larger values of t, the dark eergy becomes domiat causig a acceleratig expasio. Because at the curret time both the matter desity ad dark eergy desities are playig the largest role i the evolutio ad acceleratio of the uiverse, they also have much more of a role i age of the uiverse. As the Friedma equatio is solved backwards i time, the large expasio caused by the two compoets will also have a importat role i how quickly the uiverse will cotract back ito a sigularity. The results verify this hypothesis as the chages i the matter desity ad the dark eergy desity ca create a wide rage of ages for the uiverse while the radiatio desity has a much more limited rage i which it ca effect the age of the uiverse (see Tables 1-3 ad Figures 3-5).

11 Farsworth, Kig, Price 11 Refereces "Aleksadr Aleksadrovich Friedma." Dec Nov. 6 < Barkaa, Rea. "The First Stars i the Uiverse ad Cosmic Reioizatio." Sciece 313 (6): Nov. 6 < Flaery, Bria P., William H. Press, Saul A. Teukolsky, ad William T. Vetterlig. Numerical Recipes i C: the Art of Scietific Computig. d ed. Cambridge: Cambridge UP, 199. < Gould, Adam. "IPS Offical Statemet o the Aciet Age of the Earth ad Uiverse." 1 July 6. Iteratioal Plaetarium Society. 9 Nov. 6 < Va De Bergh, Sidey. "Size ad Age of the Uiverse." Sciece 13 (1981): Nov. 6 < Weisstei, Eric W. "Friedma-Lemaitre Cosmological Model." 6. 9 Nov. 6 < LemaitreCosmologicalModel.html>.

12 Farsworth, Kig, Price 1 #iclude<stdio.h> #iclude<math.h> Code float d(float t, float a) { float k,c,h,omegar,omegam,omegad,f; c=3e8; /*speed of light*/ H=.3e18; /* Hubble costat*/ k=.; /*curvature parameter*/ omegar=3e-5; /*radiatio desity*/ omegam=.66; omegad=.73; /*matter desity*/ /*dark matter desity*/ /*curret desity values omegar=3e-5 omegam=.66 omegad=.73*/ if(t<=-1.) { f=a*sqrt(omegar/(a*a*a*a)+omegam/(a*a*a)+omegad-(1- (omegam+omegar+omegad))/(a*a)); } else /*for egative square root*/ { f=(-1.)*a*sqrt(omegar/(a*a*a*a)+omegam/(a*a*a) +omegad-(1-(omegam+omegar+omegad))/(a*a)); } retur f; } it mai(void) { float m1,m,m3,m4,f; float t,h,ub,iter; it i,n,j,a,k; N=1; /*Number of steps*/ f=1.; t=.; a=1.; h=.1; for(i=; i<=n; i=i+1) /*Ruge-Kutta method*/ { m1=h*d(t,f); m=h*d(t+.5*h,f+m1/.); m3=h*d(t+.5*h,f+m/.); m4=h*d(t+h,f+m3);

13 Farsworth, Kig, Price 13 } f=f+m1/6.+m/3.+m3/3.+m4/6.; pritf("%e %e\",-t,f); } retur;