Friedmannien equations


 Percival Warner
 4 years ago
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2 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions nd obtins eqution Fiedmnnien equtions which descibe the evolution of ou Univese.
3 nd thee Fiedmnnien equtions e: 4 kc G c p G 4 stte the eqution of nd one moe is ) ( c p H dt d c q p
4 Motion of test pticle Let conside the motion of test pticle lunched fom the Eth. If the peiod of cceletion is shot enough, one cn conside the motion of the pticle s motion in the Eth s gvittionl field only. 4
5 The eqution of enegy consevtion povides us with eqution like v GM const Const= R Const> Const<. Const > is infinite motion. Const= is infinite motion with escping velocity. Const< is finite motion 5
6 A plnet moves ound the Sun in elliptic tjectoy. The Sun is in focus of the ellipse. Hee is ttction body (the Sun) nd plnet which is moving the gvittionl field of the Sun (in fct, in sum of the two 6 fields: the Sun nd the plnet).
7 Ou Univese is filled with mtte medi. The mtte in the Univese is distibuted homogeneous nd isotopic. Homogeneity is independence of the min physicl mtte chcteistics of position in spce, while isotopy is independence of tht in diffeent diections. So, the motion of the mtte is not the sme s motion of plnet ound the Sun. of 4G ( t ) How one cn solve it in the cse homogeneou s density? 7
8 The motion of the Univese (filled homogeneously by mtte) is simil, but not exctly the sme s keplein motion of celestil bodies in ou sol system. We hve to conside the motion of mtte medi in gvittionl field of the medi. The solution in odiny integl diveges. 8
9 F F GM n GM n n n 9
10 M A M A A d A d F dn F dn F F d n n ( )
11 (t)=(t)x hee is distnce to test pticle nd x is lgngin coodinte of this pticle, (t) is clled scle fcto. x
12 Eqution of enegy consevtion d( t) ( ) dt GM ( t) const M V 4 ( ) t M V 4 ( t) x 4 G x Fist Fiedmnnien eqution
13 Themodynmicl Reltions (equtions) Hee we will use equtions of the specil eltivity (SR) nd themodynmicl equtions. The eqution of SR is E Mc enegy mss eqution de pdv entopy consevtion
14 E Mc Vc diffeentil of both sides of the eqution is de Vd c dvc using the entopy consevtion eqution nd substitute it one cn get eqution fo density evolution o thid Fiedmnnien eqution pdv Vd c dvc o d dt c p ( ) dv Vdt 4
15 if one emind the eqution fo volume V 4 ( t) x volume diffeentil ove volume is dv Vdt d dt Let intoduce the definition d dt H nd now d dt p H ( c ) 5
16 How one cn obtin the eqution of ocket motion? v GM const R One cn pply the diffeentition with espect to time to both sides of this eqution d dt d GM v dt Diffeentition povides us with eqution of motion 6
17 The fist tem eds d dt v vv v d dt nd the second is d dt GM GM v hee M const is constnt with espect to time 7
18 velocity is cncelled in both sides, so one emins with the eqution of motion G M minus sign designte ttction The sme pocedue cn be pplid to obtin cosmologicl eqution of motion 8
19 d dt d GM dt d dt The second tem is bit difficult. The mss is no moe constnt. The M is vible, it is function of time. d dt GM GM GM 9
20 To clculte deivtive of mss with espect to time we hve emembe the themodynmicl nd SR equtions. E Mc de pdv fom these follow p dm c dv
21 fom tht follows GM G dv dt c p the deivtive of volume with espect to time one cn do s follows dv dt 4 nd fte this pocedue one cn cncel velocity nd obtin the following eqution
22 GM p 4G c the cceletion of test pticle on the sufce of cut sphee is then 4G ( c p c ) in tems of density nd pessue
23 One cn substitute eulein coodinte (t) by lgngin coodinte x ccoding to eqution (t) = (t) x nd obtin the second Fiedmnnien eqution: 4G p c In fct, eulein coodinte is chnged duing the Univese evolution nd lgngin coodinte does not chnged. So, we cn conclude tht sptil coodinte x foms comoving coodinte system in the sense tht typicl glxy hs constnt lgngin coodinte.
24 The dust dominted Univese p= 4 c G Negtive sign = ttction The dition dominted Univese p=e/ 8 c G Negtive sign = ttction. The mixtue of these types of mtte poduces ttction. 4
25 Let us conside negtive eltivistic pessue p c in this cse p c nd gvittionl ttction is exchnged by gvittionl epulsion 8 c G 5
26 The nlysis of solution of Fiedmnnien equtions The metic intevl is: ds c dt ( t) d f ( ) d sin d 6
27 7 nd thee Fiedmnnien equtions e: 4 kc G c p G 4 ) ( c p H dt d c q p
28 The dust dominted Univese p= nd ou eqution become 4 kc G 4G d dt H 8
29 The thid eqution cn be ewitten s follows d dt nd the solution is ( t) ( t) The physicl sense of the eqution is ( t) mn( t) m const N 4 n( t) V ( t) x ( t) V ( t) 9
30 o const t t n ) ( ) ( 4 G kc 8 4 H H G G kc 8 H G cit
31 kc 4 G cit If k> then cit nd ou Univese is closed nd finite in volume If k= then cit nd ou Univese is flt nd infinite in volume If k< then cit nd ou Univese is open nd infinite in volume
32
33 If k= the eqution becomes GM dt d whee 4 M nd solution is / ) ( in g t t R t nd ) ( t t t
34 The W pmete One cn intoduce the W pmete which is moe convenient in mny cses cit the closed Univese the flt Univese the open Univese 4
35 kc H is fo thepesent density pmete If we hve Univese filled with diffeent types of mtte Ω pmete is sum of sevel contibution. m... 5
36 The dition dominted Univese p=c / nd k= Fiedmnnien equtions become d dt 4 4G G 4H 6
37 The thid eqution cn be ewitten s follows d dt 4 nd the solution is 4 ( t) 4 ( t) The physicl sense of the eqution is ( t) mn( t) m / ( t) nd n( t) / ( t) photon s mss ed shifted duing expnsion. Theefoe, this is vlid fo eltivistic pticles 7
38 If k= the eqution becomes d dt GM whee M 4 nd solution is ( t) t t 8
39 9
40 The vcuum dominted Univese p c nd k in this cse Fiedmnnien equtions e : the fist is : 4 G nd thesecond is : 8G esults ep ulsion insted of ttction! 4
41 the density of this medi is const nt d const dt Let intoduce definit ion 8G H nd cll it thehubble pmete 4
42 In this cse the solution of Fiedmnnien equtions is ( t) e Ht 4
43 The Stndd Cosmologicl Model The Stndd Cosmologicl Model is: the model of expnding Univese with flt hypesufce which is filled by diffeent types of mtte: smll mount of eltivistic mtte (photons), byonic mtte nd dk mtte which lso obey dust like eqution of stte, nd dk enegy o quintessense which obeys vcuum dominted eqution of stte. 4
44 WMAP 44
45 45, nd components uns independently medi the diffeent the evolution of c p c p p p p p p m m m
46 46 nd thee Fiedmnnien equtions e: ) ( 4 m G G m 4 ) 4 ( ) ( m m H dt d
47 47 const z t z t m m m m nd ) ( nd ) ( s nd ewite these equtions edshift functions of s densities One cn put
48 Accoding to WMAP the totl density of ou Univese is: W totl = nd contibution of diffeent type of mtte in density is: W m =.7 nd W q =.7 Theefoe, the fist Fiedmnnien eqution is: H m( z) q ( z ) 4 hee W is pesent density of the CMBR with espect to citicl density 48
49 nd eqution becomes nd H m( z) we hve two egimes z z z.7 thesecond Fiedmnnien m m / / is is descelet ion stge cceletion stge ( z) 4 49
50 RED SHIFT The ed shift is most known cosmologicl phenomen. v c v is velocity of n emitte which is moving fom obseve e o o e z 5
51 The genel desciption of the phenomen is s follows: spectl line fom nothe glxy i emitted with the sme fequency s in lbotoy. But obseved fequency is diffeent. Let conside the motion of light ys in the expnding Univese. The light y is moving long stight line ccoding to the eqution ds=. This eqution is postulte of the Specil Reltivity Which is vlid in genel Reltivity too. In the expnding Univese metic hs fom (flt hypesufce): ds c dt ( t) d d sin d 5
52 Let ssume tht n obseve is in the cente of spheicl coodinte system nd the light ys move long the dil coodinte. So, we cn put dq=, df=. In this cse metic eqution is educed to fom dt ( t) o dt ( t) One cn solve this eqution s t o dt ( t) t e o e 5
53 In this eqution t is physicl time. One cn intoduce new vible which is clled confoml time. The eqution fo this time is d=dt/(t). In this cse the solution of bove eqution is vey simple e o e o Now one cn clculte the intevl of n event in emitte nd in obseve. Suppose tht the lgngin distnce ( ) between the emitte nd the obseve is constnt. In this cse intevl of n event t emitte position ( D ) is equl to intevl t obseve position: e o 5
54 One cn ewite this eqution in tems of physicl time s dte dto ( te ) ( to ) Suppose tht the event is one cycle of dition. One cn ewite bove eqution in tems of fequency o ( t ( t o e ( t ( t e o ) ) e ) z ) o s genel definition of ed shift 54
55 55 o e o o o e t t t e o e l t dt t d t H ) ( ) ( e o t t z z c Hl e o c Hl c Hl c Hl z e e e
56 fo Hl e c one obtin z Hl c e wht is Hubble lw, in the cse z Hl one obtin e c 56
57 The definition of cosmic distnce is: e o t o t e dt ( t) One cn ewite the definition fo ed shift in fom: o ( t) z( t) nd ewite the cosmic distnce in tems of ed shift e z ( z) dt( z) 57
58 Also one cn ewite the eqution fo ed shift in the fom ( t) o z( t) nd obtin the Hubble pmete s function of ed shift H( t) dt dz z nd one cn substitute these equtions into definition of cosmic distnce nd obtin the eqution which detemines the distnce to object s function of its ed shift: 58
59 e c H z mo ( z) dz q o ( z) 4 59
60 6
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