Sound waves before recombination

Transcription

1 9 Feb Feb Feb Feb Feb 2012 Sound waves before recombinaion è Ouline è Is sound carried by radiaion or by maer? è Sound speed è Calculaion of he angular scale for a universe dominaed by radiaion è Precise calculaion of he angular scale ü Physical condiions a recombinaion ü A recombinaion, which has he greaer mass densiy, pressureless maer or radiaion? W m0 = 0.26 pressureless maer, mosly dark maer, maer ha does no inerac wih ligh W b0 = baryons, ordinary maer W r0 = 1.2μ10-5 radiaion r b =r b0 a -3 r r = r r0 a -4 r m r r = r m0 r r0 a A a eq = (z = 3600), he mass-energy densiy of baryonic maer and radiaion are equal. Q: A recombinaion (a=0.0009), which has greaer mass densiy, pressureless maer or radiaion? We will discuss sound waves, which has o do wih radiaion and maer. Q: Does dark maer paricipae in he sound waves? Q: A recombinaion, are elecrons pressureless? The energy of a CBR phoon is 2.3μ10-4 ev. ü Which has greaer number densiy, baryonic maer or radiaion? A he presen ime, he mass of baryonic maer is 938MeV. The mass of a phoon is 2.73 K K ev = 2.3μ10-4 ev. The number densiy n r n b = μ938 MeV 2.3μ10-4 ev = 1.1μ10 9. More precisely, because phoons have differen energies, I need o inegrae he Planck number specrum.

2 2 cosmology.nb n r = 0.41μ10 9 phoon m -3 T K 3 n b = 0.25 nucleon m -3 W b0.043 H 0 72 km s Mpc 2 n r n b = 1.64μ10 9. The number of phoons and baryons do no change. As he universe expands, he number of baryons in a coexpanding box does no change. The number of baryons enering mus equal he number exiing, because of homogeneiy. Same argumen is rue for phoons. How can you say n r n b = 1.64μ10 9 in a dramaic way? ü Is sound carried by phoons or by maer? The composiion of he gas is ordinary maer and phoons. ü Does maer or radiaion provide more pressure? Pressure P = np x v x, where n is number densiy, p and v are momenum and speed. For maer, P = n m mv 2 x = 1 n 3 m mv 2. For radiaion, P = 1 n 3 r E (from earlier class). Equipariion: In hermal equilibrium, he average energy of each paricle is he same. (More precisely, he energy of each degree of freedom is 1 kt. In QM, degrees of freedom may be frozen wih 0 energy.) 2 For maer P m = 2 n 3 m 3 kt= n 2 m kt. For radiaion: P r º 1 n 3 r 1 kt. More accuraely, 2 P r = 0.90 n r kt. There are 10 9 phoons for every baryon. Therefore he pressure of radiaion is 10 9 imes he pressure of maer. ü Sound speed is P r 1 2 The emperaure a recombinaion is 3000K There are many phoons for every baryon or elecron. n r n b = 1.64μ10 9 A a eq = (z = 3600), he mass-energy densiy of baryonic maer and radiaion are equal. The speed of sound v s = P r 1 2 where he derivaive is for adiabaic changes.

3 cosmology.nb 3 Proof: Newon's 2nd law F = ma deermines he movemen of a sound wave, The force is due o an excess pressure. The mass is due o an excess densiy. Consider a slab of gas beween x and x + x. Because of he presence of he disurbance, x moves o x +c, x. The ma erm becomes r 0 x 2 c. 2 The force comes from he difference in pressure. The force erm is - P x. x I need o relae pressure o mass densiy: P =- P r c r 0 x Collec all; cancel r 0 and x: 2 c 2 = P r 2 c x 2 The speed of sound is v s = P r 1 2 The derivaive is aken wih no hea flow, if he wavelengh is large compared o he mean-free pah. ü Sound speed for a perfec gas For adiabaic changes P = cons r g. For a monoonic gas, g= 5. 3 Take derivaive o ge v 2 s =gp r =gpv r V =gkt m Equipariion fl 3 kt= 1 mv avg Therefore v s = g v avg The sound speed is approximaely he average speed of he gas paricles. ü Values Calculaing he sound speed Consider a box of gas wih a fixed number of paricles. The box expands or shrinks because of he sound wave. 1. du= dq-pdv=-pdv Because here is no hea flow, dq= 0. du=-pdv. Recall u = a B T 4 and P = 1 a 3 B T 4 d uv = Vdu+ udv =-PdV du=- u + P dv V 4 T 3 dt=- T T4 dv V 3 dt T =-dv V

4 4 cosmology.nb 2. dp= 4 a 3 B T 3 dt. 3. d r=d r b + d r r d r b =-r b dv V, since mass of he baryons in he box (r b V) is unchanged. d r b = 3 r b dt T d r r = 4 a B T 3 dt 4. Gaher all: v 2 s = P = 4 a r 3 B T 3 dt 3 r b dt T + 4 a B T 3 dt -1 = r b -1 r r v s = R -1 2, where R = 3 r b 4 r r Q: If R ` 1, how fas do sound waves ravel? Q: Why do baryons slow he speed of sound? Recall v s = dp d r 1 2. Number densiy of phoons: Inegrae x 2 x 1, x, 0, 2 Zea 3 Energy densiy: Inegrae x 3 x 1, x, 0, 4 15 Average energy: N E is Calculaing he horizon How far does a sound wave ravels from he big bang (=0) o he ime of recombinaion? e a be he expansion parameer a las scaering (recombinaion) a E be he expansion parameer a epoch when r r =r b.

5 cosmology.nb 5 The disance of he horizon is d = v s d. v s d is how far he sound wave moves. As he wave is moving, he ending poin is moving oo. Calculaion is wrong. Beer posed: Wha is he comoving coordinae r of a sound wave ha ravels from he big bang (=0) o he epoch of recombinaion? v s d= adr r = 0 v s a -1 d. Given r, how do you ge he disance of he horizon? d = a r d = a 0 v s a -1 d. The sound speed v s = R -1 2 depends on R = 3 4 r b r r = 3 4 a a E. ü To gain undersanding, consider his simplified case where maer is negligible: R ` 1. Then d = a v s 0 a -1 d Use Friedman's equaion a H 0 2 = W k0 +W m0 a -1 +W r0 a -2 +W v0 a 2 ØW r0 a -2 d = a v s 0 a -1 d = H a 0 a v s W r0 0 da = H -1 0 a v s W r0 Apply F's eqn a a H a = H 0 W r0 a o ge he ransparen resul d = H -1 a v s Q: Inerpre he formula for d. A dense region produces a sound wave ha goes in all direcions o cover a lengh 2 d. The angle subended is q= 2 d ra = 2 H -1 a v s a r a Q: Inerpre he formula for q. ü Resuls for bes cosmological values d = a 0 a -1 v s a d Change = H -1 a -1 a, and inegrae o ge (Weinberg 2008, Cosmology, p. 145) d = 2 H -1 0 a R W m0-1 2 ln 1 + R R E + R R E For W m0 = 0.26, W v0 = 0.74, W b0 = 0.043, R = 0.62 R E = 0.21 d = 1.16 H a

6 6 cosmology.nb q= 2 d r a a = 1 48 = 1.2 ü Plo