Chapter 2 Baryons, Cosmology, Dark Matter and Energy. The potential energy of the test mass, as seen by an observer at the center of the sphere, is

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1 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 M(R) = 4 3 πr3 ρ (1) The potential energy of the test mass, as seen by an observer at the center of the sphere, is while its kinetic energy is U = G M(R)m R By Hubble s Law the expansion velocity is given by T = 1 2 mv2 = 1 ( ) 2 dr 2 m (3) dt (2) v = HR (4) where H = 1 dr is the Hubble constant. Although the While the size of H has been debated R dt in the past, recent determinations give a rather precise value of 71 ± 4 km/s/mpc. (One parsec = light years.) The total energy of the test particle is then E tot = T + U = 1 2 mr2 (H 2 8 πρg) (5) 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 ρ crit = 3H2 8πG h 2 g/cm 3 (6) where h 0.71 ± 0.04 is (today s) Hubble constant in units of 100 km/s/mpc. This means ρ < ρ crit continued expansion ρ > ρ crit 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 ρ crit? We can 1

2 certainly do one immediate calculation, for photons. You are probably aware that photons 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 ionization 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 n γ = 2 d 3 q 1 (2π) 3 exp(q/t γ ) 1 = 2ζ(3)T γ 3 /π 2 408/cm 3 (7) where ζ(3) is the Riemann zeta function and T γ the today s cosmic microwave background temperature, measured (with great accuracy) to be about 2.73 K. Similarly for the energy density in photons ρ γ = 2 d 3 q q (2π) 3 exp(q/t γ ) 1 = π2 Tγ 4 / g/cm 3 (8) It follows that photons contribute only 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 η BBN = (5.9 ± 0.8) η CMB = (6.14 ± 0.25) So these values are in great agreement. Using the CMB value, we then find n nucleons = η CMB n γ = /cm 3 and thus multiplying by the average nucleon mass (a detail but we know the n/p ratio is 1/7 for doing this average) ρ b = g/cm ρ crit That is, baryons provide only 4.4% of the closure mass. Clearly the electron contribution to ρ, ρ e (6m e /7m N )ρ b, is then neglible, about of ρ crit, comparable to the photon contribution. 2

3 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 ρ b 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 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 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 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 , or 24-50% of ρ b, makes up a warm/hot low-density intergalactic medium an estimate based both on large-scale numerical simulations of the intergalactic 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 intergalactic 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 of the closure density. So adding up the components, one gets totals in the range of of the closure density perhaps 50-75% of ρ b 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

4 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 ρ rot 20ρ vis 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 possibilities. But regardless of the origin of the dark matter, it appears that the matter/energy density of our universe is a lot closer to ρ crit than one would guess from our calculations of ρ b and ρ γ. 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., ν e and ν e ) contributes: It thus follows n ν = 2 d 3 q 1 (2π) 3 exp(q/t ν ) + 1 = 3ζ(3)T ν 3 /(2π 2 ) (9) n ν = 3 4 (T ν T γ ) 3 n γ (10) What about T ν? In the very early universe electrons, positrons, neutrinos, and photons would all be relativistic and in equilibrium, characterized by a single temperature. Then there is an epoch around 1 MeV when the neutrinos have decoupled, but the electrons are relativistic and in equilibrium with the photons. Let the temperature of this epoch be called T. Still sometime later the positrons and electrons annihilate into two γs. This will clearly heat the photons. The net effect can be estimate by recognizing that the annihilation occurs after weak decoupling. Thus energy is no longer being exchanged between the neutrinos and the electrons (or other matter) when the annihilation occurs. On the other 4

5 hand, equilibrium is maintained between the electrons and photons, as they are coupled by electromagnetic interactions. If energy is not being exchanged, then the entropy is constant. If we first study the neutrinos at times well before and after electron-positron annihilation call these times t b and t a, then S(t b ) = R(t b) 3 (ρ b T b ν + Pν b ) = 4 R(t b ) 3 ρ b 3 T b ν = 4 3 R(t a ) 3 ρ a ν (11) where ρ is the energy density and P the pressure. We have used P = ρ/3 for relativistic neutrinos and the constancy of the entropy. Now for the six neutrino flavors so we find ρ ν = 6 T a ν d 3 q q (2π) 3 exp(q/t ν ) + 1 = 6 7 ( π 2 T 4 ) ν 8 30 (12) R(t b )T b = R(t a )T a ν (13) We can do the same thing for the electrons, positrons, and photons, again starting when the electrons/positrons are relativistic and in equilibrium with the neutrinos, and ending after both weak decoupling and electron-positron annihilation. We find that is, 4 R(t b ) 3 ( ρ b 3 T b γ + ρ b e + ) 4 R(t a ) 3 ρb e = ρ a + γ (14) 3 ( ( )] 7 7 1/3 [ ] 4 1/3 R(t b )T [2 b = R(t a )Tγ 8) a [2] 1/3 R(t b )T b = R(t a )Tγ a (15) 8 11 Using Eq. (13) we immediately find T a γ Tν a Tγ a = ( 4 11 )1/3 (16) Thus, if todays CMB temperature is about 2.72 K, today s cosmic neutrino background temperature is about 1.92 K. Equation (10) then yields n ν = n γ = 3 11 n γ (17) where n ν is the number of neutrinos and antineutrinos of one flavor. Thus the total number of neutrinos (sum over flavors) is 9/11 the number of photons. So if there are 408 CMB photons/cm 3, there must be about 334 neutrinos/cm 3. Now consider today, when the temperatures are low. Nothing has occurred to change n ν. If neutrinos were massless, they would contribute very little to the mass energy, clearly. So lets assume they have a mass. We assume that mass is large compared to today s kinetic 5

6 energy that s the only way to make them important. On the other hand, they cannot be so massive to invalidate our assumptions of relativistic neutrinos on decoupling. (We know this is true experimentally.) It follows that their contribution to the mass/energy is their number density times their mass. Summing over three flavors ρ ν = 3 11 n γ m ν (i) = ρ crit h 2 i i m ν (i) 1 ev (18) We will see later that the maximum of the sum over neutrino masses, using only laboratory and neutrino oscillations, is 6.6 ev. And the minimum (from the neutrino mass difference measured with atmospheric neutrinos) is ev. Using h=0.71 we find < ρ ν /ρ crit < 0.14 (19) So two things are important about this. First, there is neutrino dark matter. Second, based on laboratory data only, it could be significant, though never more than 1/7 the closure density. We can also calculate, for the period in which neutrinos remain relativistic, the total energy density in relativistic species. This is ρ γ + ρ ν = ρ γ ( Tν T γ ) 4 ( = ρ γ ( 4 4/3 ) ( π 1.681ρ γ ) 2 Tγ 4 ) 15 ( π 2 T 4 ) ( γ nγ π 2 ) n 15 2ζ(3)Tγ 3 γ T γ (20) Let s equate this to the energy density in baryons, to determine the epoch when the energy in relativistic species matches that in baryons 4.54n γ T γ n nucleons M N = η CMB n γ M N ev n γ T γ 0.13 ev (21) Since the formation of atoms (recombination) occurred at about 0.35 ev, this time and the transition between a radiation-dominated universe and a matter dominated universe occurred at about the same time. The large-scale structure of our universe is sensitive to neutrino mass. Because neutrinos are relativistic as structure forms and decoupled from the matter (free-streaming) they retard the growth of structure on large scales. This in fact leads to tighter constraints on neutrino mass, m ν (i) < 0.7 ev (22) than so far obtained in the laboratory. This mass bound implies i < ρ ν /ρ crit <

7 That is, neutrinos could be about as important as the visible baryons in the universe s mass/energy budget, but not more. The mass bound also implies m ν (i) < 0.7/3 (23) as oscillation constraints require small mass splittings. This tells us that neutrinos remain relativistic until the time of recombination, which through precise temperatures maps of the CMB yields one of our best measures of structure formation at a precise time (380,000 y post Big Bang). It can be shown that neutrinos of mass m ν suppress the growth of structure for wave numbers larger than k free streaming m ν /0.05 ev Mpc 1. (24) Here 0.05 ev is used as a scale because at least one neutrino must be at least this massive, for compatibility with oscillation mass difference measurements. Thus we see that the effects of neutrinos in suppressing the growth of structure at large scales must start to diminish at distances > 250 Mpc. More complete analyses than we can attempt here can quantify the effects on a minimal neutrino mass, 0.05 ev, on the growth of structure as a function of both redshift and scale. One finds Z = power decrease 1.9% 1.0% 2.1% 3.5% for k > Mpc (25) Current large-scale surveys have achieved a sensitive to not dark matter (neutrinos) at the level of about ρ crit. To reach the level where neutrino effects must be seen, ρ crit, one needs an increase in sensitivity of about a factor of 10. The surveys are limited statistically, so this translates into surveys about a factor of 100 beyond those performed to date. Such improvements are expected over the next decade from a variety of surveys focused on the relevant scale of Mpc: higfh redshift galaxy surveys, the SDSIII BOSS 10 5 QSO survey, Planck CMB, weak lensing, 21-cm radio telescopes with large ( 0.1 km 2 ) collection areas, etc. There will of course be issues in combining data from different surveys and different techniques, as systematic effects could arise. 2.3 Dark matter We have gone through the calculation of the cosmological density of baryonic matter from the theory of BBN and the measurements of light element abundances in reasonable detail. We have also mentioned the CMB constraint on η, the baryon-to-photon ratio. This comes from an analysis of temperature fluctuations in the Cosmic Microwave Background blackbody 7

8 spectrum. Temperature anisotropies are found at the level of 1 part in 10 5 and involve a typical angular size of about one degree. This corresponds to distances on the order of 150 Mpc. One can understand the general physics relatively simply. First, the observers very carefully measure the temperature of the black body radiation as a function of solid angle, plotting the very small variations in this temperature as a function of multipolarity. In a Legendre expansion the l of the multipole maps into distance: the higher the l (for a peak in the power spectrum), the more rapid the variation with change of solid angle. The picture of structure formation is that dark matter seeds areas of higher density form the gravitational potential into which ordinary matter falls. This picture presumes that there is some spectrum of density fluctuations associated with early cosmology. Ordinary (or baryonic) matter acts differently from the dark matter because it not only responds to gravity, but also interacts with radiation. Gravity causes ordinary matter to flow into potential wells; radiation pressure increases in regions of higher density and thus acts to resist strong compression of ordinary matter. The result are acoustic oscillations of the ordinary matter that reflect the time scale the time matter has had to flow since the Big Bang. There are a couple of processes that connect temperature variations in the CMB to density fluctuations, and thus to the structure of the universe at recombination. The most important physics, at least on smaller scales of most interest to us, is the heating and cooling associated with the interactions between ordinary matter and radiation, as that matter is acoustically compressed or rarefied. If matter flows into a gravitational potential and achieves a higher density, some of that kinetic energy associated with the inflow will be converted into heating of the plasma. Thus a hot spot in the CMB at small scales indicates a high density region, while a cold spot indicates a rarefied region. There are a couple of other effects that can also alter the temperature. One, the Sachs- Wolfe, effect has to do with the gravitational red shift. If a photon comes out of a region of high density and thus from deeper in the gravitational well it will loose more energy opposite of the effect describe above. This effect becomes more effective on larger scales, as a large-scale overdensity generates a stronger gravitational potential and a larger gravitational red shift. Thus it has a different signature. Such temperature fluctuations, and their connections to density fluctuations, probe the dynamic processes that govern structure formation. The kinetic energy of inflow is transferred to the plasma by processes like Compton scattering. This provides a radiation pressure that resists matter flow, and can halt that flow. Likewise, if a flow is reversed, motion of matter outward in a gravitational well must lead to a cooling of the plasma, by energy conservation. The timescale for possible acoustic oscillations compression and rarefaction is governed by the age of the universe at recombination, 380,000 years. This limits the size scale of fluctuations: if the scale is too large, there is insufficient time for matter at that scale to 8

9 fully condense. It is relatively easy to appreciate intuitively that the largest structures that can be seen must correspond to the largest area that can condense over the lifetime of the universe. By condense here we mean reach the density where the radiation pressure just halts the flow. It is helpful to think of the process as an oscillator, with gravity working to compress the spring, and with radiation resisting the compression (and becoming more effective as the spring is compressed). When a spring oscillates, at the points of maximum compression and maximum rarefaction, the spring is at rest. If one were to sample the spring during its motion, therefore, the power would collect at these extremes. At recombination, of course, the sampling time is fixed at 380,000. What varies are the springs the variety of density fluctuations that presumably follow some characteristic spectrum. A special spring a special size scale are the fluctuations that, over 380,000 years, allow matter to reach the point of maximum compression. Power will collect in this mode. Another special mode corresponds to a size scale about half of this. There the matter has time to reach the point of maximum compression, be forced outward by the radiation pressure, and then again come to rest as gravity once again overcomes the diminishing radiation pressure. These are the π and 2π peaks in the power spectrum. One can continue, forming a second compression (3π) etc. Figure 2 shows with somewhat diminishing clarity the first (compression), second (rarefaction), and third (compression) power peaks in the CMB temperature fluctuations, as measure by WMAP and other CMB probes. The first peak corresponds to an l of about 200 an angular scale of about one degree. The second corresponds to an angular size of about half a degree. Peaks in the power spectrum are not seen for ls much smaller than 200 there has not been enough time for large-scale regions to compress to high density. As the mechanism for the rarefaction is the interaction of radiation with ordinary matter interactions, the peak structure must be sensitive to the baryon to photon ratio η that we introduced in our BBN discussion. The baryons act as a source of inertia in the compression and rarefaction. It should be clear that if one increases the number of baryons, then the amplitudes of the oscillations should increase: there is more inertia on infall that the radiation has to resist, overcome, and reverse. This is shown in the third figure. Low baryon density tends to reduce the radio of the first two peaks (corresponding to compression and rarefaction). The result is in good agreement with the BBN determination, as we noted earlier, favoring just slightly larger values of η. In terms of a closure density, it corresponds to a ρ baryons of about Both the CMB and the BBN calculations based on radically different physics governing the universe at very different times give similar results. We noted before this implies that about half of the baryonic matter is nonluminous. Among the possible hiding places are MACHOS massive compact halo objects being probed in gravitational microlensing 9

10 Figure 1: The NASA/WMAP plot of the temperature of the CMB radiation, showing variations on angular scales of about one degree. The bluer regions are slightly cooler, and the red slightly hotter. This reflects density fluctuations at the time of last scattering, which influence the time of recombination and alter the energy loss of radiation as photons emerge from regions of overdensity. 10

11 Figure 2: The measured power spectrum for CMB temperature fluctuations. From Wayne Hu s web page. 11

12 Figure 3: This shows how the CMB power spectrum is influenced by variations in the baryon density. From Wayne Hu s web site. 12

13 Figure 4: This shows how the CMB power spectrum is influenced by variations in the total matter density. Fits to the WMAP data require that there is about five times more cold dark matter than baryonic matter. From Wayne Hu s web site. 13

14 searches and matter hidden in nonluminous gas clouds. There are also a couple of reliable determinations of the total matter density. The height of the first acoustic peak in the CMB spectrum is quite sensitive to the matter density, as shown in the fourth figure. The position of this peak requires ρ M ± (fraction of the critical density). Red-shift surveys measurements of the shape of the power spectrum for large-scale matter inhomogeneities also probe this quantity, giving ρ M 0.40 ± These results are in reasonable agreement with each other, as well as with the values derived by combining the known baryon density with the baryon-to-total-mass density ratio in clusters. Distance Type Ia supernovae can be used as standard candles even at large distances and thus at past times to probe the Hubble expansion. These indicate that in addition to dark matter, space (the vacuum) is characterized by some dark energy. This dark energy is a sort of negative pressure working against gravity, causing the universe to expand more rapidly than it would due to matter along. They find ρ Λ ρ M 0.4. Again, this sensitivity is physically very plausible. Matter retards expansion, dark energy accelerates it. Thus the expansion rate should test the difference. Actually, one can appreciate that things are actually richer, as cold matter and dark energy evolve differently cosmologically. Since we know how the former evolves as the universe stretches, careful measurements can determine the equation of state of the dark energy. Distant supernovae are the tool for probing the condition of the universe at earlier times. Another possibility is to use the mean distance between neighboring regions of high (low) density in the CMB as a kind of ruler, defined in a statistical sense at the time of recombination. One can then watch how that ruler stretches if one has other measures of similar correlations at later times correlations that would map back into those of the CMB. Combined cosmological analyses also give ρ 1.0 ± 0.04 That is, the universe is close to critical density. Combined with the above results, one deduces that ρ M 0.27, again with of this being baryons (visible and otherwise) and the rest something beyond the standard model (like the lightest stable supersymmetric particle). The remainder is the dark energy, ρ Λ 0.73 whose nature is simply not understood. 14