17 Lecture 17: Recombination and Dark Matter Production

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1 PYS 652: Astrophysics Lecture 17: Recombination and Dark Matter Production New ideas pass through three periods: It can t be done. It probaby can be done, but it s not worth doing. I knew it was a good idea a aong! Arthur C. Carke The Big Picture: Today we continue discussing the recombination epoch in the eary Universe. We aso extend the Botzmann formaism to the production of dark matter partices. Recombination continued Just as the neutron-nuceon ratio n is important to the abundance of ight eements, the abundance of free eectrons e is of great significance to the observationa cosmoogy. Recombination, which takes pace around z 1000 directy eads to decouping of photons from matter. Decouping means that the photons stopped scattering off eectrons, which become bound to neutra atoms during this epoch. The mean-free paths of photons become on the order of the size of the Universe, meaning that the Universe has become opaque. The resuting CMB radiation represents a snapshot of the Universe at the time of the ast scatter. Roughy speaking, decouping occurs when the rate of Compton scattering of photons off eectrons becomes smaer than the expansion rate of the Universe. The scattering rate is n e σ T e n b σ T, 337 where σ T cm 2 is the Thomson cross-section, and we continue to ignore contribution of 4 e, by approximating n e + n n b. The ratio of the baryon density to the critica density is Ω b ρ b m pn b ρ cr0 ρ cr0 ρ cr h 2 g cm 3 Ω b Ω b0 a 3 so that the eq. 337 the becomes From eq. 73, we have 0 n b ρ cr0 Ω b0 a g cm 3 m p g h 2 Ω b0 a 3 n b h 2 Ω b0 a 3 cm 3, 338 n e σ T cm 1 e Ω b0 h 2 a h years 1 year s s 1 h, h s 0, 340 so that the eq. 339 can be rewritten as n e σ T cm 1 e Ω b0 ha s s cm 1 e Ω b0 ha

2 PYS 652: Astrophysics 89 In order to get a dimensioness equation, we mutipy the eq. 341 by c/ but in the equation we sti omit c: n e σ T s cm cm s 1 e Ω b0 ha e Ω b0 ha During the eary epochs, the Universe is either radiation- or matter-dominated, which means that the ratio 0 / can be soved from the first Friedmann s equation eq. 101a: Ω T 0 2 Ωm0 a 3 + Ω r0 a 4 Ω m0 a 3 + Ω r0 a 4 1/2 1/2 Ω m0 a 3/2 1 + Ω 1/2 r0 a 1 0 Ω m0 Ω 1/2 m0 a 3/2 1 + a eq 1/2, a where we have used the resuts from Appendix to Lecture 9 or eqs in the textbook: a eq Ω r Ω m0 Ω m0 h Finay, we can rewrite eq. 342 in terms of z reca a 1/1 + z: n e σ T 1/ e Ω b0 ha 3 Ω 1/2 m0 a 3/2 1 + a eq a e Ω b0 h1 + z 3/2 Ω 1/ m z 113 e Ωb0 h Ω m0 h 2 1/2 1 + z 1000 Ω m0 h 2 3/ z / /2 Ω m0 h 2, 345 where the constants have been normaized to the best-fit vaues obtained from observations. When the free eectron fraction e drops beow 10 2, photons decoupe from matter. This happens before the recombination is over, i.e., before the eectron fraction e eves off beow Even if the Universe remained ionized throughout its history, at some point photons woud decoupe from baryons. This can be easiy seen from the eq. 345, if we set e 1 i.e, a eectrons are free. Then, after some agebra, we arrive at /3 Ωm0 h 2 1/3 1 + z decoupe 43 Ω b0 h 2, which, if the terms in parenthesis are taken to be equa to one, corresponds to z decoupe 42, i.e., t 60 miion years. Recombination timeframe. We can compute when the recombination took pace, by computing how od the Universe was at z 1000 see Tabe 5: tz 1 0 z d z 347 Ωm0 1 + z 5 + Ω r0 1 + z 6 + Ω de0 1 + z 2, which gives t ,000 years for h 0.72, Ω m0 0.28, Ω r h 2, Ω de

3 PYS 652: Astrophysics 90 Figure 28: Free eectron fraction e as a function of redshift. Recombination takes pace abrupty at about z 1000, which corresponds to T 0.25eV. The Saha approximation in eq. 330 is a correct description during equiibrium and accuratey identifies the onset of recombination, but not the ong-term behavior, for which the fu Botzmann equation is necessary. ere Ω b0 0.06, Ω m0 1, h 0.5. Earier Appendix to Lecture 9 or eq in the textbook, we have derived that the Universe made a transition from radiation- to matter-dominated at about z eq Ω m0 h , which corresponds to when the Universe was about 50,000 years od. This means that the recombination happened during the matter-dominated epoch. Structure formation. Recombination was foowed by the dark ages during which the baryonic matter was neutra. It is during this time that the first structures in the Universe started to form. Structure formation in the Big Bang mode proceeds hierarchicay, with smaer structures forming before arger ones. The first structures to form are quasars, which are thought to be bright, eary active gaaxies, and popuation III stars. Before this epoch, the evoution of the Universe coud be understood through inear cosmoogica perturbation theory a structures coud be understood as sma deviations from a perfect homogeneous Universe. This is computationay reativey easy to study. At this point noninear structures begin to form, and the computationa probem becomes much more difficut, invoving, for exampe, N-body simuations with biions of partices. Reionization. Reionization took pace when the first objects started to form in the eary Universe energetic enough to ionize neutra hydrogen. As these first objects formed and radiated energy, the Universe went from being neutra back to being an ionized pasma, between 150 miion and one biion years after the Big Bang at a redshift 6 < z < 20. When protons and eectrons are separate, they cannot capture energy in the form of photons. Photons may be scattered, but scattering interactions are infrequent if the density of the pasma is ow. Thus, a Universe fu of ow density ionized hydrogen wi be reativey transucent, as is the case today. 90

4 PYS 652: Astrophysics 91 Dark Matter Earier, in Lectures 10 and 11, we discussed the evidence for nonbaryonic matter in the Universe, and came to the genera concusion that the tota contribution of the such a matter to the energy density is Ω dm 0.3. We aso estabished WIMPs as the eading candidates for the nonbaryonic dark matter. Even though we do not know yet what these partices are, we do know that if such partices exist, they were at some point in equiibrium with the rest of the cosmic pasma at high temperatures of the eary Universe. At some point, they experienced freeze-out as the temperature of the Universe dropped beow the WIMP s mass. ad it not been for faing out of the equiibrium freeze-out, the abundance of the dark matter partices woud decay as e m/t, which woud ead to their extinction. owever, they do freeze out at some point, which is why we use the Botzmann equation instead of its equiibrium version, the Saha equation to determine when they froze-out and quantify their reic abundance. The idea is to use the concusions from observations and the earier epochs of the Big Bang the BBN, such as Ω dm 0.3, to constrain the properties of the unknown WIMPs: their mass and cross-section. Putting such constraints on the WIMPs woud be usefu in the experimenta attempts at their direct detection. We now consider a generic scenario, in which two heavy WIMPs denoted as annihiate and produce two ight essentiay massess partices. The ight partices are assumed to be in compete equiibrium to the cosmic pasma, which means n n 0. This means that in the reaction + + 1, 2, 3, 4, there is ony one unknown n, the abundance of the WIMPs. Again, we use the Botzmann equation eq. 280: n a 3 n 0 n0 σv n a 3 σv n 0 n n n 0 n 0 n n n 0 n0 2 n As we did before, we continue to massage the Botzmann equation into something mathematicay more eucidating. After recaing that the temperature scaes as T a 1, we can rewrite the RS of the eq. 348 above as: n a 3 n a 3 a 3 a 3 d dt n d n dt. 349 After defining the quantity Y as Y n, 350 we can rewrite the eq. 348 above as dy dt σv T 6 n 0 2 n 2, dt σv Y 2 EQ Y 2, 351 where Y EQ n 0 /. It is, again, beneficia to introduce a new time variabe: x m T, 352 where m is the mass of the WIMP. Again, very high temperatures correspond to x 1, which is when the reactions proceed so rapidy to maintain equiibrium Y Y EQ. Since the WIMPs 91

5 PYS 652: Astrophysics 92 are reativistic at that time, their equiibrium abundance is given by the m T portion of the eq. 276, so Y Y EQ n0 gt3 π 2 g π2 For x 1, the exponent e x dominates and suppresses the equiibrium abundance Y EQ. Eventuay, the WIMPs become so rare due to this suppression that they no onger can find each other fast enough to maintain the equiibrium abundance. This is when the freeze-out begins. We rewrite the Botzmann equation in terms of the new integration variabe x: dy dt dy dt dy dy 1 xx x 2 T T x dy x ka 2ȧ ka 1 dy x ȧ dy x 1 σv YEQ 2 Y 2 x x x 1 σv YEQ 2 Y 2 m3 σv x 1 m 3 a x Y 2 EQ Y 2 dy x λ x 2 Y 2 Y 2 EQ, 354 where the ratio of annihiation rate to the expansion rate is given by λ m3 σv x In most theories, λ is a constant. Some theories, however, have a temperature-dependent thermayaveraged cross-section, which eads to a variabe λ. This changes the quantitative resuts sighty, whie the quaitative soutions remain the same. 92