APP-IV Introduction to Astro-Particle Physics. Maarten de Jong

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1 APP-IV Introduction to Astro-Particl Physics Maartn d Jong 1

2 cosmology in a nut shll Hubbl s law cosmic microwav background radiation abundancs of light lmnts (H, H, )

3 Hubbl s law (199) 1000 vlocity [km/s] pc.6 ly distanc [Mpc]

4 Dopplr ffct Cphid T v L 1+ β λ' = λ0 1 β v c (1 + v ) c λ 0 rd-shift t L = f ( T) Standard candl Vlocity Distanc 4

5 Hubbl s law (II) v= H d Hubbl constant 5

6 xpanding Univrs a(t 1 ) tim a(t ) a(t) = univrsal scal paramtr 6

7 univrsal scal paramtr v= H d H() t = at () a a at () t 7

8 gnralisation of rd-shift obsrvd (t 1 ) mittd (t 0 ) λ' = at ( 1) λ at ( ) 0 λ (1 + z) rd-shift 0 z < 8

9 small rd-shift 1+ z = at ( ) 1 at ( ) = at ( ) at ( ) 1 0 at ( ) 0 at ( ) ( t t ) at ( ) at ( 0) ( t1 t0) c = 1+ at ( ) c 0 d = 1+ H = 1+ c v c z 1 Taylor xpansion Dopplr ffct 9

10 what valu has H today? 1 1 H = h 100 km s Mpc h = obsrvation 10

11 Gravitational forc a m Nwton: M 4 = ρ πa ma = G mm a mmg 1 mmg da ma + = ma = a a constant 1 Kc m 11

12 mmg 1 1 ma = a Kc m ma 1 mmg 1 ( ma ) ( Kc m) a ma = ma a 8πGM Kc 8πGρ Kc = = a π a a a 4 1

13 Fridrich-Lmaîtr-Robrtson-Walkr Enrgy dnsity curvd spac historical 0? H a 8πGρ Kc Λ = = + a a gnral solution of th Einstin quations Mass Enrgy (dividd by c ) homognous and isotropic Univrs 1

14 critical dnsity H 8πGρ Kc Λ = + a K =Λ=0 ρ c = H 8π G usd to normalis particl dnsitis (Ω) in th Univrs 14

15 fat of th Univrs nglct cosmological constant Λ = 0 assum mattr domination Mc non-rlativistic ρ = V mattr consrvd M = constant 15

16 Expansion rat: 8π GMc Kc = 4 π a a a a a GMc = a Kc K = 1 a c K = 0 a 0 K =+ 1 a 0 opn Univrs flat Univrs closd Univrs K = 1, 0, + 1 gnral mtric of homognous, isotropic spac 16

17 fat of th Univrs (Λ = 0) K = 1, opn now K = 0, flat a(t) K = +1, closd t 17

18 Ag of th Univrs K = Λ 0 H = 8π G ρ non-rlativistic mattr = 8π G Mc 4 π a Volum 18

19 a a = 8π G Mc 4 π a try: at () = αt β β 1 αβt = β αt G Mc β α t β α β t = G Mc t β = α = 9G Mc 19

20 Ag of th Univrs (II) a H = = a t t = = 8 11Gyr Gyr H obsrvd 0

21 arly Univrs H 8 = πg ρ 1

22 radiation dominatd volum xpansion rd-shift rlativistic gas: 1 ρ at () at () ρ ρ = a 4 4H a =

23 ρ 8πGρ 4 ρ = try: ρ() t = α t β αβt αt β 1 β βt 8π G = 4 8π G β / 1 β / = 4 α t α t β = α = π G ρ() t t Big Bang!

24 particl dnsity N ( E) dp = π 1 / h p dp E kt g ± 1 numbr of spin stats Numbr of particls pr unit volum in th momntum rang p p + dp Bos-Einstin ± photons (and all othr bosons, i.. spin 0, 1,, ) Frmi-Dirac ± + frmions (spin 1/, /, 5/, ) 4

25 rlativistic gas N ( E) dp = g 1 p dp E / kt π h ± 1 E = pc = hv g hν ( hν) ρν ( ) dν = dhν h / kt π ( hc) ν ± 1 Enrgy pr unit volum in th frquncy rang ν ν + dν 5

26 g hν ( hν) ρν ( ) dν = dhν h / kt π ( hc) ν ± 1 bosons 4 x π dx = x dx frmions 4 x 7 π = x ρ = π 4 ( kt ) 0 ( hc) g ff all particls ar intracting with ach othr 6

27 rlativistic gas xpansion of arly Univrs ρ = π π ( kt ) h c g ff ρ = π G t kt 1 MV t[ s] 7

28 Enrgy dnsity Tmpratur ρ 1 t T 1 t hot Big Bang 8

29 Hubbl constant H 8 πg ( kt ) = ρ 1.66 gff M c h Pl G hc ( M c ) Pl c 4 M Pl c Planck mass GV 9

30 xpanding Univrs At som momnt, photons dcoupld intraction rat xpansion rat Thn, numbr of photons consrvd but Univrs kps xpanding dnsity volum N 1 ( ) d dv ν ν ν = d ν dv = h / kt π ( hc) ν 1 constant 0

31 N ( ν ') dν ' dv ' = N ( ν) dνdv Bos-Einstin distribution T ' = π 1 ν dν dv ( hc) 1 hν / kt 1 (1 + z) ν ' = (1 + z) dν ' dv h(1 + z) ν '/ kt π ( hc) 1 1 ν ' = dν ' (1 + z) dv hν '/ kt ' π ( hc) 1 T = cooling 1 + z dv ' 1

32 cosmic microwav background radiation wavlngth [cm] intnsity photon dnsity distribution T 0 =.7 K frquncy [GHz]

33 Transparncy of th Univrs chargd particls nutral atoms 1 p H γ + + larg photon-mattr cross sction small photon-mattr cross sction Univrs opaqu Univrs transparnt

34 quilibrium non-rlativistic N N, N! γ p 1 + p H + γ chmical potntial µ H = µ p + µ ( µ γ = 0) binding nrgy Q m + m m 1.6 V p H baryon numbr dnsity nb np + nh 4

35 n = g 1 p dp E / kt h 0 π ± 1 kt Q m c i g π 1 h 0 E / kt p dp E mc + p m x p mkt mc / kt x g ( mkt ) = x dx π h 0 5

36 x x x x x dx = x 1 x dx = 1 x 1 dx v ' u = = = = π x ( x ) r dx + y dr rdφ dxdy 6

37 non-rlativistic particl dnsity n = = g ( mkt ) mc / kt π π h 4 mc kt g π ( hc) mc / kt 7

38 diffrnt particl spcis chmical potntial mi c kt ni = g i π ( hc) µ mc i i kt 8

39 photon dnsity g 1 ν N( ν) dν = dν h / kt π ( hc) ν 1 0 dx x x 1 =.404 n γ.404 kt = π hc 9

40 mh c kt nh = g H π ( hc) = µ m c H H kt gh mc mpc kt nn p ggp π ( hc) m c H n n n n µ ( m m m ) c H µ µ p+ + p H kt 0 p p Q Univrs is nutral n p n = g H g g p n p m c kt π ( hc) Q kt 40

41 Ionisation fraction barion / photon ratio X np np = n n + n B p H η B n B n γ n + 1 X nh p H nh = n γ = η B X np nγ np n n γ Saha quation 1 X.404 kt = ηb X π mc Q kt 41

42 limiting cass lim 0 kt X 1 X X 0 Univrs transparnt lim kt Q 1 X X 0 X 1 Univrs opaqu X ( kt ) 0.5? η B ( kt ) numrical calculation 4

43 Ionisation fraction Ionisation fraction transparnt opaqu rd shift 4

44 Ionisation tmpratur T0.7 K z 100 kt = kt (1 + z) 0. V Q = 1 V 0 ηb 1! kt Q / 40 44

45 cross-chck 0.5 = 1 X X.404 ln 0.5 ln ln ( ) B ln kt = + η + + π mc Q kt!

46 Nuclosynthsis 46

47 kt 1 MV t T nutrinos and nuclons in quilibrium T 1 Hlium formation T t 1 t t 47

48 T 1 T T 1 > T < T1 ν ν + n + p + + p + n n p+ + ν nutron dcay 48

49 Q ( M M m ) c 1.9 MV n p kt Q M c p nuclons ar non-rlativistic E Mc + p M m n m p n n n p = Q / kt 49

50 quilibrium intraction rat xpansion rat Γ H dcoupl 50

51 intraction rat pr nuclon dnsity cross sction rlativ vlocity Γ= n σ v 51

52 intraction rat (II) dnsity lctrons & nutrinos ar rlativistic n ν, 1.0 kt = 4 π hc g cross-sction m c s np, σ α ( hc) p p 4 ( M c ) W ν, np, rlativ vlocity lctrons & nutrinos ar rlativistic v c 5

53 intraction rat (III) my guss pν, pnp, kt M Nc +,, ν, ν rlativistic p, n at rst 5

54 quilibrium (II) α M N c 4 ( kt ) ( kt ) 4 W Pl ( M c ) h M c h kt1 0.8 MV m c 511 kv M c 940 MV N 54

55 nutron-proton ratio at t t 1 n ( t) = n (0) n n t / τ r n n Q / kt = 1 n p 0. t / τ nn r () t = np 1+ r r t / τ τ nutron liftim t / τ n ( t) = n (0) + n (0)(1 ) p p n 55

56 T T T > T < T n + p H + γ H + H H + n H + H H + p 4 H + H H + p 4 H + H H + n formation of stabl H nutrons ar savd 56

57 T Q. MV rmindr η B ffctiv tmpratur kt Q / 40 = 0.05 MV 57

58 nutron-proton ratio at t 1MV 1MV t = kt kt1 400 s τ = 896 ± 10 s nutron liftim nn rt ( 1) rt ( ) ( t) = 0.1 np t / τ 1 + rt ( ) rt ( ) 1 1 t / τ 58

59 H mass ratio Y = MH M nn 4 n + n p n 4n H n r = ± 0.01 obsrvd! 1+ r H cosmological origin 59