Chapter 2Baryons, Cosmology, Dark Matter and Energy2.1 Hubble expansionWe are all aware that at the present time the universe is expanding. However, what willbe its ultimate fate? Will it continue to expand forever, or will the expansion slow andfinally 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 answeringthis 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 universehaving radius R. If the mean energy density of the universe is ρ, then the mass containedinside the spherical volume isM(R) =43πR3ρ (1)The potential energy of the test mass, as seen by an observer at the center of the sphere, isU = −GM(R)mR(2)while its kinetic energy isT =12mv2=12m dRdt!2(3)By Hubble’s Law the expansion velocity is given byv = HR (4)where H =1RdRdtis the Hubble constant. Although the While the size of H has been debatedin the past, recent determinations give a rather precise value of 71 ± 4 km/s/Mpc. (Oneparsec = 3.262 light years.) The total energy of the test particle is thenEtot= T + U =12mR2(H2−83πρG) (5)and the fate of the universe depends on the sign of this number, or equivalently with therelation of the density to a critical valueρcrit=3H28πG∼ 1.88 × 10−29h2g/cm3(6)where h ∼ 0.71 ± 0.04 is (today’s) Hubble constant in units of 100 km/s/Mpc. This meansρ∼<ρcrit⇒ continued expansionρ∼>ρcrit⇒ ultimate contraction2.2 Photon, baryon, and neutrino contributions to mass/energy densitySo how does the measured mass/energy density of the universe match up to ρcrit? We can1certainly 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 protonsand electrons. But just as we calculated the n + p ↔ d + γ equilibrium, we can evaluatethe 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 thatphotocapture can no longer efficiently break up newly formed atoms. One can show thiscorresponds to a temperature of about 1 eV and to a time about 380,000 years after theBig Bang. After this point, the photons decouple from the matter as they no longer see freecharges to scatter off. This decoupled background of photons is now redshifted to microwaveenergies.For the photon number densitynγ= 2Zd3q(2π)31exp(q/Tγ) − 1= 2ζ(3)T3γ/π2∼ 408/cm3(7)where ζ(3) ∼ 1.20206 is the Riemann zeta function and Tγthe today’s cosmic microwavebackground temperature, measured (with great accuracy) to be about 2.73 K. Similarly forthe energy density in photonsργ= 2Zd3q(2π)3qexp(q/Tγ) − 1= π2T4γ/15 ∼ 4.6 × 10−34g/cm3(8)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) contributionto the ρ as well. The baryon to photon number density is η, which either BBN or cosmicmicrowave background studies finds to beηBBN= (5.9 ± 0.8) × 10−10ηCM B= (6.14 ± 0.25) × 10−10So these values are in great agreement. Using the CMB value, we then findnnucleons= ηCM Bnγ= 2.51 × 10−7/cm3and thus multiplying by the average nucleon mass (a detail – but we know the n/p ratio is1/7 for doing this average)ρb= 4.19 × 10−31g/cm3∼ 0.0442ρcritThat is, baryons provide only 4.4% of the closure mass. Clearly the electron contributionto ρ, ρe∼ (6me/7mN)ρb, is then neglible, about 2 ×10−5of ρcrit, comparable to the photoncontribution.2One can count the “visible” nucleons, by integrating over all of the luminous matter in starsand gas clouds, and by making some model assumptions that take into account simulationsof the behavior of the interstellar medium, etc. It should be appreciated that our two testsof ρbare from the first three minutes and from the time of recombination, say 400,000 y postBBN. So it is quite an interesting question to ask where those baryons are now, more than10 b.y. later.There is a nice summary of this problem by Joseph Silk, which will be posted on the website. It deals with dark matter in general, but recounts the baryon number inventory as partof 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 thegalaxy, including globular clusters and isolated binary stars, with low metallicity oftenof 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 oldto 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, asbeing 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 intergalacticmedium – an estimate based both on large-scale numerical simulations of the inter-galactic medium and by observations of excess soft x-ray emission. The mechanismwarming 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 normallywould form stars or fall into the gravitational wells of galaxies, at least. So this wouldbe the residuals remaining in the ISM. Theory suggests this contributes no more that0.008 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 ρbcan be accounted for, plausibly. Most of the above inventory is inthe intergalactic medium.Roughly half of the baryons are not visible – though whether there is a baryon inventoryproblem 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 predictionfor η is now confirmed by CMB