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Chapter 2 Baryons, Cosmology, Dark Matter and Energy 2.1 Hubble

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Chapter 2 Baryons, Cosmology, Dark Matter and Energy 2.1 Hubble Chapter 2 Baryons, Cosmology, Dark Matter and Energy 2.1 Hubble expansion We are all aware that at the present time the universe is expanding. However, what will be its ultimate fate? Will it continue to expand forever, or will the expansion slow and finally reverse? In order to see what role the constituent matter and energy { baryons, photons, neutrinos, and other stuff not yet identified { of our universe may play in answering this question, we explore their effects in an expanding homogeneous and isotropic universe. Consider a small test mass m which sits on the surface of a spherical chunk of this universe having radius R. If the mean energy density of the universe is ρ, then the mass contained inside the spherical volume is 4 M(R) = πR3ρ (1) 3 The potential energy of the test mass, as seen by an observer at the center of the sphere, is M(R)m U = −G (2) R while its kinetic energy is 1 1 dR!2 T = mv2 = m (3) 2 2 dt By Hubble's Law the expansion velocity is given by v = HR (4) 1 dR where H = R dt is the Hubble constant. Although the While the size of H has been debated in the past, recent determinations give a rather precise value of 71 ± 4 km/s/Mpc. (One parsec = 3.262 light years.) The total energy of the test particle is then 1 8 E = T + U = mR2(H2 − πρG) (5) tot 2 3 and the fate of the universe depends on the sign of this number, or equivalently with the relation of the density to a critical value 3H2 ρ = ∼ 1:88 × 10−29h2g=cm3 (6) crit 8πG where h ∼ 0:71 ± 0:04 is (today's) Hubble constant in units of 100 km/s/Mpc. This means ρ <∼ ρcrit ) continued expansion ρ >∼ ρcrit ) ultimate contraction 2.2 Photon, baryon, and neutrino contributions to mass/energy density So how does the measured mass/energy density of the universe match up to ρcrit? We can 1 certainly do one immediate calculation, for photons. You are probably aware that pho- tons remained in thermal equilibrium with the matter as long as there were free protons and electrons. But just as we calculated the n + p $ d + γ equilibrium, we can evaluate the p + e− $ H + γ equilibrium, where H denotes the hydrogen atom. Given the ioniza- tion potential of H of 13.6 eV, one can calculate when the photons cool to the point that photocapture can no longer efficiently break up newly formed atoms. One can show this corresponds to a temperature of about 1 eV and to a time about 380,000 years after the Big Bang. After this point, the photons decouple from the matter as they no longer see free charges to scatter off. This decoupled background of photons is now redshifted to microwave energies. For the photon number density Z 3 d q 1 3 2 3 nγ = 2 3 = 2ζ(3)Tγ /π ∼ 408=cm (7) (2π) exp(q=Tγ) − 1 where ζ(3) ∼ 1:20206 is the Riemann zeta function and Tγ the today's cosmic microwave background temperature, measured (with great accuracy) to be about 2.73 K. Similarly for the energy density in photons Z 3 d q q 2 4 −34 3 ργ = 2 3 = π Tγ =15 ∼ 4:6 × 10 g=cm (8) (2π) exp(q=Tγ) − 1 It follows that photons contribute only 0.0000485 of the closure density. Now what we did in BBN allows us to estimate the baryonic (or nucleonic) contribution to the ρ as well. The baryon to photon number density is η, which either BBN or cosmic microwave background studies finds to be −10 ηBBN = (5:9 ± 0:8) × 10 −10 ηCMB = (6:14 ± 0:25) × 10 So these values are in great agreement. Using the CMB value, we then find −7 3 nnucleons = ηCMBnγ = 2:51 × 10 =cm and thus multiplying by the average nucleon mass (a detail { but we know the n/p ratio is 1/7 for doing this average) −31 3 ρb = 4:19 × 10 g=cm ∼ 0:0442ρcrit That is, baryons provide only 4.4% of the closure mass. Clearly the electron contribution −5 to ρ, ρe ∼ (6me=7mN )ρb, is then neglible, about 2 ×10 of ρcrit, comparable to the photon contribution. 2 One can count the \visible" nucleons, by integrating over all of the luminous matter in stars and gas clouds, and by making some model assumptions that take into account simulations of the behavior of the interstellar medium, etc. It should be appreciated that our two tests of ρb are from the first three minutes and from the time of recombination, say 400,000 y post BBN. So it is quite an interesting question to ask where those baryons are now, more than 10 b.y. later. There is a nice summary of this problem by Joseph Silk, which will be posted on the web site. It deals with dark matter in general, but recounts the baryon number inventory as part of the survey. Joe's estimates are (all in units of the closure density): • About 0.0026 is in the spheroid stars { those in the galactic bulge and surround halo { or Population II stars (these are old stars, found especially in the stellar halo of the galaxy, including globular clusters and isolated binary stars, with low metallicity often of the order of 1/100-1/1000 of solar, e.g., typically with compositions by mass of 75% hydrogen, 24.99% He, and 0.01% metals). • About 0.0015 in disk stars { the disk is the flat, wispy spiral structure of our galaxy (and others) { and cold gas. These are the Population I stars, which range from old to young, with roughly solar metallicity, e.g., 70% hydrogen, 28% He, and 2% metals. • About 0.0026 in intracluster gas in rich clusters, which can be mapped in x-rays, as being in a cluster with star activities keeps the gas warm. • About 0.01-0.015, or 24-50% of ρb, makes up a warm/hot low-density intergalactic medium { an estimate based both on large-scale numerical simulations of the inter- galactic medium and by observations of excess soft x-ray emission. The mechanism warming the gas is gravitational, due to shock waves that propagate into the inter- galactic medium from the peripheries of galaxies, galaxy groups, and galaxy clusters. • Simulations also suggest there may be some cold intergalactic gas { this stuff normally would form stars or fall into the gravitational wells of galaxies, at least. So this would be the residuals remaining in the ISM. Theory suggests this contributes no more that 0.008 of the closure density. So adding up the components, one gets totals in the range of 0.02-0.03 of the closure density { perhaps 50-75% of ρb can be accounted for, plausibly. Most of the above inventory is in the intergalactic medium. Roughly half of the baryons are not visible { though whether there is a baryon inventory problem is a matter of whom one talks with. Presumably these nucleons are some place { perhaps nonluminous gas clouds { because we believe BBN, and because the BBN prediction for η is now confirmed by CMB results. This problem is sometimes called the dark baryons problem { though there are even more intriguing \dark" problems. 3 A second dark problem has to do with large-scale gravitational interactions of galaxies, galaxy clusters, etc. For some time it has been clear that the total ρ is much larger than that coming from photons and baryons (and electrons). For example, Doppler studies of the rotation rates of spiral galaxies indicate that these systems are much more massive than their luminosities seem to suggest ρrot ∼ 20ρvis This is too large a discrepancy to attribute just to the dark baryons. The origin of the \dark matter" responsible for this discrepancy is a matter of current study: there are several pos- sibilities. But regardless of the origin of the dark matter, it appears that the matter/energy density of our universe is a lot closer to ρcrit than one would guess from our calculations of ρb and ργ. Just as we have a CMB, there will be a relic neutrino spectrum left over from the big bang. These neutrinos would have decoupled when temperatures were slightly above 1 MeV. Since that first second of the big bang, no further interactions have occurred. If we had some means to detect these neutrinos, they would tell us about conditions at that very early time, e.g., their temperature fluctuations (probably exceeding tiny!) over the sky would tell us about the structure of the universe at 1 sec. We do the calculation of the neutrino contribution to ρ making two assumptions. First is the assumption that we have three flavors (thus 6 neutrinos in all), as the standard model tells us, all of which are light. We will see that this is know from both cosmology, and from a combination of tritium β decay and recent discoveries of neutrino oscillations. The upper bound on the masses of the light neutrinos is about 1 eV. With this assumption neutrinos are relativistic when they decoupled. It follows that each neutrino flavor (e.g., νe andν ¯e) contributes: Z 3 d q 1 3 2 nν = 2 3 = 3ζ(3)Tν =(2π ) (9) (2π) exp(q=Tν) + 1 It thus follows 3 Tν 3 nν = ( ) nγ (10) 4 Tγ What about Tν? In the very early universe electrons, positrons, neutrinos, and photons would all be relativistic and in equilibrium, characterized by a single temperature.
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