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Dark Matter

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astro-ph/0301505 UMN–TH–2127/03 TPI–MINN–03/02 January 2003

arXiv:astro-ph/0301505v2 25 Jan 2003

TASI LECTURES ON DARK MATTER∗

KEITH A. OLIVE† William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 USA E-mail: olive@umn.edu
Observational evidence and theoretical motivation for dark matter are presented and connections to the CMB and BBN are made. Problems for baryonic and neutrino dark matter are summarized. Emphasis is placed on the prospects for supersymmetric dark matter.

1. Lecture 1 The nature and identity of the dark matter of the Universe is one of the most challenging problems facing modern cosmology. The problem is a long-standing one, going back to early observations of mass-to-light ratios by Zwicky1 . Given the distribution (by number) of galaxies with total luminosity L, φ(L), one can compute the mean luminosity density of galaxies L= which is determined to be2 L ≃ 2 ± 0.2 × 108 ho L⊙ M pc−3 (2) Lφ(L)dL (1)

where L⊙ = 3.8 × 1033 erg s−1 is the solar luminosity. In the absence of a cosmological constant, one can define a critical energy density, ρc = 3H 2 /8πGN = 1.88 × 10−29 ho 2 g cm−3 , such that ρ = ρc for three-space curvature k = 0, where the present value of the Hubble parameter has been defined by Ho = 100ho km Mpc−1 s−1 . We can now define a critical mass-to-light ratio is given by (M/L)c = ρc /L ≃ 1390ho(M⊙ /L⊙ ) (3)

∗ Summary of lectures given at the Theoretical Advanced Study Institute in Elementary Particle Physics at the University of Colorado at Boulder - June 2-28, 2002. † This work was supported in part by DOE grant DE-FG02-94ER40823 at Minnesota.

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which can be used to determine the cosmological density parameter Ωm = ρ = (M/L)/(M/L)c ρc (4)

Mass-to-light ratios are, however, strongly dependent on the distance scale on which they are determined3 . In the solar neighborhood M/L ≃ 2 ± 1 (in solar units), yielding values of Ωm of only ∼ .001. In the bright central parts of galaxies, M/L ≃ (10 − 20)ho so that Ωm ∼ 0.01. On larger scales, that of binaries and small groups of galaxies, M/L ∼ (60 − 180)ho and Ωm ≃ 0.1. On even larger scales, that of clusters, M/L may be as large as (200 − 500)ho giving Ωm ≃ 0.3. This progression in M/L seems to have halted, as even on the largest scales observed today, mass-to-light ratios imply values of Ωm < 0.3 − 0.4. Thus when one considers the scale ∼ of galaxies (and their halos) and larger, the presence of dark matter (and as we shall see, non-baryonic dark matter) is required. Direct observational evidence for dark matter is found from a variety of sources. On the scale of galactic halos, the observed flatness of the rotation curves of spiral galaxies is a clear indicator for dark matter. There is also evidence for dark matter in elliptical galaxies, as well as clusters of galaxies coming from the X-ray observations of these objects. Also, direct evidence has been obtained through the study of gravitational lenses. On the theoretical side, we predict the presence of dark matter (or dark energy) because 1) it is a strong prediction of most inflation models (and there is at present no good alternative to inflation) and 2) our current understanding of galaxy formation requires substantial amounts of dark matter to account for the growth of density fluctuations. One can also make a strong case for the existence of non-baryonic dark matter in particular. The recurrent problem with baryonic dark matter is that not only is it very difficult to hide baryons, but given the amount of dark matter required on large scales, there is a direct conflict with primordial nucleosynthesis if all of the dark matter is baryonic. In this first lecture, I will review the observational and theoretical evidence supporting the existence of dark matter. 1.1. Observational Evidence Assuming that galaxies are in virial equilibrium, one expects that one can relate the mass at a given distance r, from the center of a galaxy to its rotational velocity by M (r) ∝ v 2 r/GN (5)

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Figure 1. Synthetic rotation curve5 for galaxies with M = −21.2. The dotted curve shows the disk contribution, whereas the dashed curve shows the halo contribution.

The rotational velocity, v, is measured3,4 by observing 21 cm emission lines in HI regions (neutral hydrogen) beyond the point where most of the light in the galaxy ceases. A subset of a compilation5 of nearly 1000 rotation curves of spiral galaxies is shown in Fig. 1. The subset shown is restricted to a narrow range in brightness, but is characteristic for a wide range of spiral galaxies. Shown is the rotational velocity as a function of r in units of the optical radius. If the bulk of the mass is associated with light, then beyond the point where most of the light stops, M would be constant and v 2 ∝ 1/r. This is not the case, as the rotation curves appear to be flat, i.e., v ∼ constant outside the core of the galaxy. This implies that M ∝ r beyond the point where the light stops. This is one of the strongest pieces of evidence for the existence of dark matter. Velocity measurements indicate dark matter in elliptical galaxies as well6 . Galactic rotation curves are not the only observational indication for the existence of dark matter. X-ray emitting hot gas in elliptical galaxies also provides an important piece of evidence for dark matter. A particularly striking example is that of the large elliptical M87. Detailed profiles of the temperature and density of the hot X-ray emitting gas have been mapped out7 . Assuming hydrostatic equilibrium, these measurements allow one to determine the overall mass distribution in the galaxy necessary to bind the hot gas. Based on an isothermal model with temperature kT = 3keV (which leads to a conservative estimate of the total mass), Fabricant and Gorenstein7 predicted that the total mass out to a radial distance of 392 kpc is 5.7 × 1013 M⊙ , whereas the mass in the hot gas is only 2.8 × 1012 M⊙

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or only 5% of the total. The visible mass is expected to contribute only 1% of the total. M87 is not the only example of an elliptical galaxy in which X-ray emitting hot gas is observed to indicate the presence of dark matter. X-ray observations have shown that the total mass associated with elliptical galaxies is considerably larger than the luminous component in many examples of varying morphological types8 . Mass-to-light ratios for these systems vary, with most being larger than 30h0 and some ranging as high as ∼ 200h0. In addition, similar inferences pertaining to the existence of dark matter can be made from the X-ray emission from small groups of galaxies9. On these scales, mass-to-light ratios are typically > 100h0 and detailed studies ∼ have shown that the baryon faction in these systems is rather small. Furthermore, it was argued10 that cluster baryon fractions should not differ from the Universal value given by ΩB /Ωm . Using an estimate of ΩB = 0.04 from big bang nucleosynthesis (BBN, see below), and baryon fractions ranging from 0.1 to 0.3, one would obtain an estimate for the total matter density of 0.13 – 0.4. Another piece of evidence on large scales is available from gravitational lensing11 . The systematic lensing of the roughly 150,000 galaxies per deg2 at redshifts between z = 1 − 3 into arcs and arclets allow one to trace the matter distribution in a foreground cluster. Lensing observations can be categorized as either strong (multiple images) or weak (single images). Both require the presence of a dominant dark matter component. Strong lensing is particularly adept in testing the overall geometry of the Universe12,13 . While a cluster which provides multiple lenses of a single background galaxy (at known redshift) is useful for determining the total cluster mass, when several background galaxies are lensed, it is possible to constrain the values of Ωm and ΩΛ 14 . The recent results of 13 show a degeneracy in the Ωm – ΩΛ plane. Nevertheless, the allowed region is offset from similar types of degeneracies found in supernovae searches and the CMB (see below). Indeed, these lensing results are much more constraining in Ωm than the other techniques, though a residual uncertainty of about 30 % persists. While in principle, these results find that any value (from 0 to 1) is possible for ΩΛ , Ωm < 0.5 for low values of ΩΛ and Ωm < 0.4 for higher values of ΩΛ (> 0.6). ∼ Weak lensing of galaxies by galaxies can (on a statistical basis) also probe the nature of galactic halos. Recent studies based on weak lensing data indicate that galactic halos may be far more extended than previously thought15 (radii larger than 200 h−1 kpc). These results also imply a sub0

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stantial contribution to Ωm (of order 0.1-0.2) on this scale. On larger scales, using many cluster lenses enables one to estimate Ω ≃ 0.3 16 . Another use of weak lensing statistics is to determine the evolution of cosmic shear and hence an estimate of Ωm 17 . Finally, there exist a number of examples of dark clusters, ie., lenses with no observable counterpart18. The contribution of these objects (if they are robust) to Ωm is not clear. For a recent review of weak lensing see 19 . Finally, on very large scales, it is possible to get an estimate of Ωm from the distribution of peculiar velocities of galaxies and clusters. On scales, λ, where perturbations, δ, are still small, peculiar velocities can be expressed20 as v ∼ HλδΩ0.6 . On these scales, older measurements of the m peculiar velocity field from the IRAS galaxy catalogue indicate that indeed Ω is close to unity21 . Some of the new data indicates a lower value in the range22 0.2 – 0.5, but does not conclusively exclude Ωm = 1 23 . The above discussion of observational evidence for dark matter pertains largely to the overall matter density of the Universe Ωm . (For a comprehensive review of determinations of the matter density, see ref. 24 . ) However, the matter density and the overall curvature are not related one-to-one. The expansion rate of the Universe in the standard FRW model is expressed by the Friedmann equation H2 = ˙ 8πGN ρ Λ k R2 = − 2+ 2 R 3 R 3 (6)

where R(t) is the cosmological scale factor, k is the three-space curvature constant (k = 0, +1, −1 for a spatially flat, closed or open Universe), and Λ is the cosmological constant. The Friedmann equation can be rewritten as (Ω − 1)H 2 = k R2 (7)

so that k = 0, +1, −1 corresponds to Ω = 1, Ω > 1 and Ω < 1. However, the value of Ω appearing in Eq. (7) represents the sum Ω = Ωm + ΩΛ of contributions from the matter density (Ωm ) and the cosmological constant (ΩΛ = Λ/3H 2). There has been a great deal of progress in the last several years concerning the determination of both Ωm and ΩΛ . Cosmic Microwave Background (CMB) anisotropy experiments have been able to determine the curvature (i.e. the sum of Ωm and ΩΛ ) to with in about 10%, while observations of type Ia supernovae at high redshift provide information on a (nearly) orthogonal combination of the two density parameters.

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The CMB is of course deeply rooted in the development and verification of the big bang model25 . Indeed, it was the formulation of BBN that led to the prediction of the microwave background. The argument is rather simple. BBN requires temperatures greater than 100 keV, which according √ 2 to the standard model time-temperature relation, ts TMeV = 2.4/ N , where N is the number of relativistic degrees of freedom at temperature T , and corresponds to timescales less than about 200 s. The typical cross section for the first link in the nucleosynthetic chain is σv(p + n → D + γ) ≃ 5 × 10−20 cm3 /s This implies that it was necessary to achieve a density 1 ∼ 1017 cm−3 (9) σvt for nucleosynthesis to begin. The density in baryons today is known approximately from the density of visible matter to be nB o ∼ 10−7 cm−3 and since we know that that the density n scales as R−3 ∼ T 3 , the temperature today must be n∼ To = (nB o /n)1/3 TBBN ∼ 10K (10) (8)

thus linking two of the most important tests of the Big Bang theory. Microwave background anisotropy measurements have made tremendous advances in the last few years. The power spectrum26,27,28 has been measured relatively accurately out to multipole moments corresponding to ℓ ∼ 1000. The details of this spectrum enable one to make accurate predictions of a large number of fundamental cosmological parameters27,29,30,31,32,33. An example of these results as found by a recent frequentist analysis34 is shown in Fig. 2. Of particular interest to us here is the CMB determination of the total density, Ωtot , as well as the matter density Ωm . The results of recent CMB anisotropy measurements are summarized in Table 1. As one can see, there is strong evidence that the Universe is flat or very close to it. Furthermore, the matter density is very consistent with the observational determinations discussed above and the baryon density, as we will see below, is consistent with the BBN production of D/H and its abundance in quasar absorption systems. The discrepancy between the CMB value of Ωm and ΩB is sign that non-baryonic matter (dark matter) is required. Furthermore, the apparent discrepancy between the CMB value of Ωtot and Ωm , though not conclusive on its own, is a sign that a contribution from the vacuum energy density or

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Figure 2. ∆χ2 calculated with the MAXIMA-1 and COBE data as a function of parameter value. Solid blue circles show grid points in parameter space, and the green lines were obtained by interpolating between grid points. The parameter values where the green line intercepts the red dashed (dotted) line corresponds to the 68% (95%) frequentist confidence region34 .

Table 1.

Results from recent CMB anisotropy measurements. Ωtot Ωm h 2 0.12 ± 0.05 0.17+0.08 −0.04 0.25+0.07 −0.09 0.14 ± 0.04 0.17+0.08 −0.06 0.13+0.08 −0.05 – ΩB h 2 0.021+0.004 −0.003 0.0325 ± 0.0125 0.026+0.010 −0.006 0.022+0.004 −0.003 0.022+0.015 −0.009 0.029 ± 0.009 0.019+0.006 −0.007

BOOMERanG27 MAXIMA29 MAXIMA (freq.)34 DASI30 CBI31 VSA32 Archeops33

1.03 ± 0.06 0.9+0.09 −0.08 0.89+0.13 −0.10 1.04 ± 0.06 0.99 ± 0.12 1.03 ± 0.012 1.16+0.24 −0.20

cosmological constant, is also required. The preferred region in the Ωm −ΩΛ plane as determined by a frequentist analysis of MAXIMA data is shown in Fig. 3.

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Figure 3. Two-dimensional frequentist confidence regions in the (ΩM , ΩΛ ) plane. The red, orange and yellow regions correspond to the 68%, 95%, and 99% confidence regions respectively. The dashed black line corresponds to a flat universe, Ω = Ωm + ΩΛ = 1.

The presence or absence of a cosmological constant is a long standing problem in cosmology. To theorists, it is particularly offensive due to the necessary smallness of the constant. We know that the cosmological term is at most a factor of a few times larger than the current mass density. Thus from Eq. (6), we see that the dimensionless combination, GN Λ < 10−121 . ∼ Nevertheless, even a small non-zero value for Λ could greatly affect the future history of the Universe: allowing open Universes to recollapse (if Λ < 0), or closed Universes to expand forever (if Λ > 0 and sufficiently large). Another exciting development has been the use of type Ia supernovae, which now allow measurement of relative distances with 5% precision. In combination with Cepheid data from the HST key project on the distance scale, SNe results are the dominant contributor to the best modern value for H0 : 72 km s−1 Mpc−1 ± 10% 35 . Better still, the analysis of high-z SNe has allowed the first meaningful test of cosmological geometry to be carried out, as shown in Fig. 4. These results can be contrasted with those from the CMB anisotropy measurements as in Fig. 5. We are led to a seemingly conclusive picture. The Universe is nearly flat with Ωtot ≃ 1. However the density in matter makes up only 20-50% of this total, with the remainder in a cosmological constant or some other form of dark energy.

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Figure 4. The type Ia supernova Hubble diagram36 taken from 37 . The first panel shows that, for z ≪ 1, the large-scale Hubble flow is indeed linear and uniform; the second panel shows an expanded scale, with the linear trend divided out, and with the redshift range extended to show how the Hubble law becomes nonlinear. Comparison with the prediction of Friedmann-Lemaˆ models appears to favor a vacuum-dominated itre universe.

1.2. Theory Theoretically, there is no lack of support for the dark matter hypothesis. The standard big bang model including inflation almost requires Ωtot = 1 38 . This can be seen from the following simple solution to the curvature problem. The simple and unfortunate fact that at present we do not even

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Figure 5. Likelihood-based confidence contours27 over the plane ΩΛ (i.e. Ωv assuming w = −1 vs Ωm . The SNe Ia results very nearly constrain Ωv −Ωm , whereas the results of CMB anisotropies (from the Boomerang 98 data) favor a flat model with Ωv + Ωm ≃ 1. The intersection of these constraints is the most direct (but far from the only) piece of evidence favoring a flat model with Ωm ≃ 0.3.

know whether Ω is larger or smaller than one, indicates that we do not know the sign of the curvature term further implying that it is subdominant in Eq. (6) 8πG k < ρ (11) R2 3 In an adiabatically expanding Universe, R ∼ T −1 where T is the temperature of the thermal photon background. Therefore the quantity ˆ k= k 8πG < < 2 × 10−58 2 R2 T 2 3To (12)

is dimensionless and constant in the standard model. This is known as the curvature problem and can be resolved by a period of inflation. Before inflation, let us write R = Ri , T = Ti and R ∼ T −1 . During inflation, R ∼ T −1 ∼ eHt , where H is constant. After inflation, R = Rf ≫ Ri but T = Tf = TR < Ti where TR is the temperature to which the Universe ∼ ˆ reheats. Thus R ∼ T and k → 0 is not constant. But from Eqs. (7) and

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ˆ (12) if k → 0 then Ω → 1, and since typical inflationary models contain much more expansion than is necessary, Ω becomes exponentially close to one. The inflationary prediction of Ω = 1 is remarkably consistent with the CMB measurements discussed above. Furthermore, we know two things: Dark matter exists, since we don’t see Ω = 1 in luminous objects, and most (about 90%) of the dark matter is not baryonic. The latter conclusion is a result of our forthcoming discussion on BBN which constrains the baryonto-photon ratio and hence ΩB . Thus 1 − ΩB is not only dark but also nonbaryonic. Furthermore, the matter density is surely composed of several contributions: Ωm = ΩB + Ων + Ωχ where the latter represents the dark matter contribution. Another important piece of theoretical evidence for dark matter comes from the simple fact that we are living in a galaxy. The type of perturbations produced by inflation39 are, in most models, adiabatic perturbations (δρ/ρ ∝ δT /T ), and I will restrict my attention to these. Indeed, the perturbations produced by inflation also have the very nearly scale-free spectrum described by Harrison and Zeldovich40 . When produced, scalefree perturbations fall off as δρ ∝ l−2 (increase as the square of the wave ρ number). At early times δρ/ρ grows as t until the time when the horizon scale (which is proportional to the age of the Universe) is comparable to l. At later times, the growth halts (the mass contained within the volume l3 has become smaller than the Jean’s mass) and δρ = δ (roughly) indeρ pendent of the scale l. When the Universe becomes matter dominated, the Jean’s mass drops dramatically and growth continues as δρ ∝ R ∼ 1/T . ρ The transition to matter dominance is determined by setting the energy densities in radiation (photons and any massless neutrinos) equal to the energy density in matter (baryons and any dark matter). For three massless neutrinos and baryons (no dark matter), matter dominance begins at Tm = 0.22mB η (13)

and for η < 7 × 10−10 , this corresponds to Tm < 0.14 eV. Because we are considering adiabatic perturbations, there will be anisotropies produced in the microwave background radiation on the order of δT /T ∼ δ. The value of δ, the amplitude of the density fluctuations at horizon crossing, has been determined by COBE41, δ = (5.7 ± 0.4) × 10−6 . Without the existence of dark matter, δρ/ρ in baryons could then achieve a maximum value of only δρ/ρ ∼ Aλ δ(Tm /To ) < 2 × 10−3 Aλ , where ∼ To = 2.35 × 10−4 eV is the present temperature of the microwave back-

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ground and Aλ ∼ 1 − 10 is a scale dependent growth factor. The overall growth in δρ/ρ is too small to argue that growth has entered a nonlinear regime needed to explain the large value (105 ) of δρ/ρ in galaxies. Dark matter easily remedies this dilemma in the following way. The transition to matter dominance is determined by setting equal to each other the energy densities in radiation (photons and any massless neutrinos) and matter (baryons and any dark matter). While if we suppose that there exists dark matter with an abundance Yχ = nχ /nγ (the ratio of the number density of χ’s to photons) then Tm = 0.22mχ Yχ (14)

Since we can write mχ Yχ /GeV = Ωχ h2 /(4 × 107 ), we have Tm /To = 2.4 × 104 Ωχ h2 which is to be compared with 600 in the case of baryons alone. The baryons, although still bound to the radiation until decoupling, now see deep potential wells formed by the dark matter perturbations to fall into and are no longer required to grow at the rate δρ/ρ ∝ R. With regard to dark matter and galaxy formation, all forms of dark matter are not equal. They can be distinguished by their relative temperature at Tm 42 . Particles which are still largely relativistic at Tm (like neutrinos or other particles with mχ < 100 eV) have the property43 that (due to free streaming) they erase perturbations out to very large scales given by the Jean’s mass MJ = 3 × 1018 M⊙ mχ 2 (eV ) (15)

Thus, very large scale structures form first and galaxies are expected to fragment out later. Particles with this property are termed hot dark matter particles. Cold particles (mχ > 1 MeV) have the opposite behavior. Small scale structure forms first aggregating to form larger structures later. It is now well known that pure HDM cosmologies can not reproduce the observed large scale structure of the Universe. In contrast, CDM does much better. Current attention is focused on so-called ΛCDM cosmologies based on the ΩΛ − Ωm contribution to the curvature discussed above. 2. Lecture 2 2.1. Big Bang Nucleosynthesis The standard model44 of big bang nucleosynthesis (BBN) is based on the relatively simple idea of including an extended nuclear network into a homogeneous and isotropic cosmology. Apart from the input nuclear cross

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sections, the theory contains only a single parameter, namely the baryonto-photon ratio, η. Other factors, such as the uncertainties in reaction rates, and the neutron mean-life can be treated by standard statistical and Monte Carlo techniques45,46 . The theory then allows one to make predictions (with well-defined uncertainties) of the abundances of the light elements, D, 3 He, 4 He, and 7 Li. 2.1.1. Theory Conditions for the synthesis of the light elements were attained in the early Universe at temperatures T > 1 MeV. In the early Universe, the energy ∼ density was dominated by radiation with ρ= 7 7 π2 (2 + + Nν )T 4 30 2 4 (16)

from the contributions of photons, electrons and positrons, and Nν neutrino flavors (at higher temperatures, other particle degrees of freedom should be included as well). At these temperatures, weak interaction rates were in equilibrium. In particular, the processes n + e+ ↔ p + νe ¯

n + νe ↔ p + e−

n ↔ p + e− + νe ¯

(17)

fix the ratio of number densities of neutrons to protons. At T ≫ 1 MeV, (n/p) ≃ 1. The weak interactions do not remain in equilibrium at lower temperatures. Freeze-out occurs when the weak interaction rate, Γwk ∼ G2 T 5 F falls below the expansion rate which is given by the Hubble parameter, √ √ H ∼ GN ρ ∼ T 2 /MP , where MP = 1/ GN ≃ 1.2 × 1019 GeV. The β-interactions in eq. (17) freeze-out at about 0.8 MeV. As the temperature falls and approaches the point where the weak interaction rates are no longer fast enough to maintain equilibrium, the neutron to proton ratio is given approximately by the Boltzmann factor, (n/p) ≃ e−∆m/T ∼ 1/6, where ∆m is the neutron-proton mass difference. After freeze-out, free neutron decays drop the ratio slightly to about 1/7 before nucleosynthesis begins. The nucleosynthesis chain begins with the formation of deuterium by the process, p + n → D + γ. However, because of the large number of photons relative to nucleons, η −1 = nγ /nB ∼ 1010 , deuterium production

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exponential tail of the photon energy distribution with energies E > EB ¯ despite the fact that the temperature or Eγ is less than EB . The degree to which deuterium production is delayed can be found by comparing the qualitative expressions for the deuterium production and destruction rates, Γp ≈ nB σv (18)
−EB /T

is delayed past the point where the temperature has fallen below the deuterium binding energy, EB = 2.2 MeV (the average photon energy in a ¯ blackbody is Eγ ≃ 2.7T ). This is because there are many photons in the

Γd ≈ nγ σve

When the quantity η −1 exp(−EB /T ) ∼ 1, the rate for deuterium destruction (D + γ → p + n) finally falls below the deuterium production rate and the nuclear chain begins at a temperature T ∼ 0.1M eV . The dominant product of big bang nucleosynthesis is 4 He and its abundance is very sensitive to the (n/p) ratio 2(n/p) ≈ 0.25 [1 + (n/p)]

Yp =

(19)

i.e., an abundance of close to 25% by mass. Lesser amounts of the other light elements are produced: D and 3 He at the level of about 10−5 by number, and 7 Li at the level of 10−10 by number. The resulting abundances of the light elements46 are shown in Figure 6, over the range in η10 = 1010 η between 1 and 10. The left plot shows the abundance of 4 He by mass, Y , and the abundances of the other three isotopes by number. The curves indicate the central predictions from BBN, while the bands correspond to the uncertainty in the predicted abundances based primarily the uncertainty in the input nuclear reactions as computed by Monte Carlo in ref. 46 . This theoretical uncertainty is shown explicitly in the right panel as a function of η10 . The dark shaded boxes correspond to the observed abundances of 4 He and 7 Li and will be discussed below. The dashed boxes correspond to the ranges of the elements consistent with the systematic uncertainties in the observations. The broad band shows a liberal range for η10 consistent with the observations. At present, there is a general concordance between the theoretical predictions and the observational data.

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Figure 6. η10 .

The light element abundances from big bang nucleosynthesis as a function of

2.1.2. Abundances In addition to it BBN production, 4 He is made in stars, and thus coproduced with heavy elements. Hence the best sites for determining the primordial 4 He abundance are in metal-poor regions of hot, ionized gas in nearby external galaxies (extragalactic HII regions). Helium indeed shows a linear correlation with metallicity in these systems, and the extrapolation to zero metallicity gives the primordial abundance (baryonic mass fraction)47 Yp = 0.238 ± 0.002 ± 0.005. (20)

Here, the first error is statistical and reflects the large sample of systems, whilst the second error is systematic and dominates. The systematic uncertainties in these observations have not been thoroughly explored to date48 . In particular, there may be reason to suspect that the above primordial abundance will be increased due to effects such as underlying stellar absorption in the HII regions. We note that other analyses give similar results: Yp = 0.244 ± 0.002 ± 0.005 49 and 0.239

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±0.002 50 . The primordial 7 Li abundance comes from measurements in the atmospheres of primitive (Population II) stars in the stellar halo of our Galaxy. The 7 Li/H abundance is found to be constant for stars with low metallicity, indicating a primordial component, and a recent determination gives
7

Li = (1.23 ± 0.06+0.68 )×10−10 (95% CL), −0.32 H p

(21)

where the small statistical error is overshadowed by systematic uncertainties51. The range (21) may, however, be underestimated, as a recent determination52 uses a different procedure to determine stellar atmosphere parameters, and gives 7 Li/Hp = (2.19 ± 0.28) × 10−10 . At this stage, it is not possible to determine which method of analysis is more accurate, indicating the likelihood that the upper systematic uncertainty in (21) has been underestimated. Thus, in order to obtain a conservative bound from 7 Li, we take the lower bound from (21) and the upper bound from52 , giving 9.0×10−11 <
7

Li < 2.8×10−10 . H p

(22)

Deuterium is measured in high-redshift QSO absorption line systems via its isotopic shift from hydrogen. In several absorbers of moderate column density (Lyman-limit systems), D has been observed in multiple Lyman transitions53,54 . Restricting our attention to the three most reliable regions53, we find a weighted mean of D = (2.9 ± 0.3) × 10−5 . Hp (23)

It should be noted, however, that the χ2 per degree of freedom is rather poor ( ∼ 3.4), and that the unweighted dispersion of these data is ∼ 0.6 × 10−5 . This already points to the dominance of systematic effects. Observation of D in systems with higher column density (damped systems) find lower D/H 55 , at a level inconsistent with (23), further suggesting that systematic effects dominate the error budget56 . If all five available observations are used, we would find D/H = (2.6 ± 0.3) × 10−5 with an even worse χ2 per degree of freedom (∼ 4.3) and an unweighted dispersion of 0.8. Because there are no known astrophysical sites for the production of deuterium, all observed D is assumed to be primordial57. As a result, any firm determination of a deuterium abundance establishes an upper bound on η which is robust. Thus, the recent measurements of D/H 53 at least

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provide a lower bound on D/H, D/H > 2.1 × 10−5 (2σ) and hence provide an upper bound to η, η10 < 7.3 and ΩB h2 < 0.027. Helium-3 can be measured through its hyperfine emission in the radio band, and has been observed in HII regions in our Galaxy. These observations find58 that there are no obvious trends in 3 He with metallicity and location in the Galaxy. There is, however, considerable scatter in the data by a factor ∼ 2, some of which may be real. Unfortunately, the stellar and Galactic evolution of 3 He is not yet sufficiently well understood to confirm whether 3 He is increasing or decreasing from its primordial value59 . Consequently, it is unclear whether the observed 3 He abundance represents an upper or lower limit to the primordial value. Therefore, we can not use 3 He abundance as a constraint. By combining the predictions of BBN calculations with the abundances of D, 4 He, and 7 Li discussed above one can determine the the 95% CL region 4.9 < η10 < 6.4, with the peak value occurring at η10 = 5.6. This range corresponds to values of ΩB between 0.018 < ΩB h2 < 0.023
2

(24)

with a central value of ΩB h = 0.020. If we were to use only the deuterium abundance from Eq. 23, one obtains the 95% CL range 5.3 < η10 < 7.3, with the peak value occurring at η10 = 5.9. This range corresponds to values of ΩB between 0.019 < ΩB h2 < 0.027 (25) with a central value of ΩB h2 = 0.021. As one can see from a comparison with Table 1, these values are in excellent agreement with determinations from the CMB. 2.2. Candidates 2.2.1. Baryons Accepting the dark matter hypothesis, the first choice for a candidate should be something we know to exist, baryons. Though baryonic dark matter can not be the whole story if Ωm > 0.1, the identity of the dark matter in galactic halos, which appear to contribute at the level of Ω ∼ 0.05, remains an important question needing to be resolved. A baryon density of this magnitude is not excluded by nucleosynthesis. Indeed we know some of the baryons are dark since Ω < 0.01 in the disk of the galaxy. ∼ It is interesting to note that until recently, there seemed to be some difficulty in reconciling the baryon budget of the Universe. By counting the

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visible contribution to Ω in stellar populations and the X-ray producing hot gas, Persic and Salucci60 found only Ωvis ≃ 0.003. A subsequent accounting by Fukugita, Hogan and Peebles61 found slightly more (Ω ∼ 0.02) by including the contribution from plasmas in groups and clusters. At high redshift on the other hand, all of the baryons can be accounted for. The observed opacity of the Ly α forest in QSO absorption spectra requires a large baryon density consistent with the determinations by the CMB and BBN62 . In galactic halos, however, it is quite difficult to hide large amounts of baryonic matter63. Sites for halo baryons that have been discussed include Hydrogen (frozen, cold or hot gas), low mass stars/Jupiters, remnants of massive stars such as white dwarfs, neutron stars or black holes. In almost every case, a serious theoretical or observational problem is encountered. 2.2.1.1 Hydrogen A halo predominately made of hydrogen (with a primordial admixture of 4 He) is perhaps the simplest possibility. Hydrogen may however be present in a condensed snow-ball like state or in the form of gas. Aside from the obvious question of how do these snowballs get made, it is possible to show that their existence today requires them to be so large as to be gravitationally bound63 . Assuming that these objects are electrostatically bound, the average density of solid hydrogen is ρs = 0.07 g cm−3 and the binding energy per molecule is about 1 eV. Given that the average velocity of a snowball is v ∼ 250 kms−1 , corresponding to a kinetic energy Ek ∼ 600 eV, snowballs must be collisionless in order to survive. Requiring that the collision rate Γc = ns σv be less than tu −1 (tu is the age of the halo ≃ age of the Universe) with ns = ρH /ms and σ = πrs 2 one finds rs > 2 cm and ms > 1 g, assuming ∼ ∼ a halo density ρH = 1.7 × 10−26 g cm−3 (corresponding to 1012 M⊙ in a radius of 100 kpc). However, collisionless snowballs also require that their formation occur when the overall density ρ = ρH . In this case, snowballs could not have formed later than a redshift (1 + z) = 3.5 or when the microwave background temperature was 9.5K. At this temperature, there is no equilibrium between the gaseous and condensed state and the snowballs would sublimate. For a snowball to survive, rs > 1016 cm is required making this no longer an electrostatically bound object. If snowballs sublimate, then we can consider the possibility of a halo composed of cold hydrogen gas. Because the collapse time-scale for the halo (< 109 yrs) is much less than the age of the galaxy, the gas must be

19

in hydrostatic equilibrium. Combining the equation of state P (r) = (2ρ(r)/mp )kT where mp is the proton mass, with GM (r)ρ(r) r2 one can solve for the equilibrium temperature dP (r)/dr = − T = (27) (26)

Gmp M (r) ≃ 1.3 × 106 K (28) 4kr This is hot gas. As discussed earlier, hot gas is observed through X-ray emission. It is easy to show that an entire halo of hot gas would conflict severely with observations. Cooling of course may occur, but at the expense of star formation. 2.2.1.2 Jupiter-like objects A very popular candidate for baryonic dark matter is a very low mass star or JLO. These are objects with a mass m < mo = 0.08M⊙, the mass necessary to commence nuclear burning. Presumably there is a minimum mass64 m > mmin = (0.004 − 0.007)M⊙ based on fragmentation, but the exact value is very uncertain. The contribution of these objects to the dark matter in the halo depends on how much mass can one put between mmin and mo and depends on the initial mass function (IMF) in the halo. An IMF is the number of stars formed per unit volume per unit mass and can be parameterized as φ = Am−(1+x) (29)

In this parametrization, the Salpeter mass function corresponds to x = 1.35. Because stars are not observed with m < mo , some assumptions about the IMF for low masses must be made. An example of the observed65 IMF in the solar neighborhood is shown in Fig. 7. It is possible to use infrared observations66 to place a lower limit on the slope, x, of the IMF in the halo of galaxies by comparing a mass-to-light ratio defined by Q = (ρm /ρL ) L⊙ /M⊙ where the total mass density in JLO’s and low mass stars is given by mG (30)

ρm = mmin mφdm

(31)

20

1000 100 10

φ

1 0.1 0.01 0.001 0.0001 0.1 1.0 10.0 100.0

Μ/ΜΟ .
Figure 7. The IMF in the solar neighborhood65 .

where mG = 0.75M⊙ is the mass of a giant. The luminosity density given by such a stellar distribution is mG ρL = mo Lφdm + ρG

(32)

where L(m) is the luminosity of a star of mass m. ρG is the contribution to the luminosity density due to giant stars. The observed66 lower limits on Q translate to a limit63 on x x > 1.7 (33)

with a weak dependence of mmin . Unfortunately, one can not use Eq. (33) to exclude JLO’s since we do not observe an IMF in the halo and it may be different from that in the disk. One can however make a comparison with existing observations, none of which show such a steep slope at low masses. Indeed, most observations leading to a determination of the IMF (such as the one shown in Fig. 7) show a turn over (or negative slope). To fully answer the questions regarding JLO’s in the halo, one needs a better understanding of star formation and the IMF. For now, postulating the existence of a large fraction of JLO’s in halo is rather ad-hoc.

21

Despite the theoretical arguments against them, JLO’s or massive compact halo objects (MACHOs) are candidates which are testable by the gravitational microlensing of stars in a neighboring galaxy such as the LMC67 . By observing millions of stars and examining their intensity as a function of time, it is possible to determine the presence of dark objects in our halo. It is expected that during a lensing event, a star in the LMC will have its intensity rise in an achromatic fashion over a period δt ∼ 3 M/.001M⊙ days. Indeed, microlensing candidates have been found68 . For low mass objects, those with M < 0.1M⊙, it appears however that the halo fraction of MACHOs is very small. The relative amount of machos in the halo is typically expressed in terms of an optical depth. A halo consisting 100% of machos would have an optical depth of τ ∼ 5 × 10−7 . The most recent results of the MACHO collaboration69 indicate that τ = 12+4 × 10−8 , cor−3 responding to a macho halo fraction of about 20% with a 95% CL range of 8 – 50% based on 13 – 17 events. They also exclude a 100% macho halo at the 95% CL. The typical macho mass falls in the range 0.15 – 0.9 M⊙ . The EROS collaboration has set even stronger limits having observed 5 events toward the LMC and 4 toward the SMC70 . The observed optical depth from EROS1 is τ = 4+10 × 10−8 and from EROS2 τ = 6+5 × 10−8 . They −4 −3 have excluded low mass objects (M < 0.1M⊙ ) to make up less than 10% of the halo and objects with 2 × 10−7 M⊙ < M < 1M⊙ to be less than 25% of the halo at the 95% CL. 2.2.1.3 Remnants of Massive Stars Next one should possibility that the halo is made up of the dead stellar remnants of stars whose initial masses were M > 1M⊙ . Briefly, the problem which arises in this context is that since at least 40% of the stars initial mass is ejected, and most of this mass is in the form of heavy elements, a large population of these objects would contaminate the disk and prevent the existence of extremely low metallicity objects (Z ∼ 10−5 ) which have been observed. Thus either dust (from ejecta) or dead remnants would be expected to produce too large a metallicity63,71 . Clearly star formation is a very inefficient mechanism for producing dark matter. Many generations of stars would would be required to cycle through their lifetimes to continually trap more matter in remnants. This question has been studied in more detail72 . By allowing a variable star formation rate, and allowing a great deal of flexibility in the IMF, a search for a consistent set of parameters so that the halo could be described primarily in terms of dead remnants (in this case white dwarfs) was per-

22

formed. While a consistent set of chemical evolution parameters can be found, there is no sensible theory to support this choice. In such a model however, the dark matter is in the form of white dwarfs and the remaining gas is is heavily contaminated by Carbon and Nitrogen73. Though it is not excluded, it is hard to understand this corner of parameter space as being realistic. 2.2.1.4 Black Holes There are several possibilities for black holes as the dark matter in halos. If the black holes are primordial74 which have presumably formed before nucleosynthesis, they should not be counted as baryonic dark matter and therefore do not enter into the present discussion. If the black holes are formed as the final stage of star’s history and its formation was preceded by mass loss or a supernovae, then the previous discussion on remnants of massive stars applies here as well. However, it is also possible that the black halos were formed directly from very massive stars (m > 100M⊙ ?) through gravitational instability with no mass loss75. Though there are limits due to overheating the disk76 and stellar systems77 . In this case I know of no argument preventing a sufficiently large population of massive black holes as baryonic dark matter (Of course, now the IMF must have mmin > 100M⊙ .) ∼ 2.2.2. Neutrinos Light neutrinos (m ≤ 30eV ) are a long-time standard when it comes to nonbaryonic dark matter78 . Light neutrinos produce structure on large scales, and the natural (minimal) scale for structure clustering is given in Eq. (15). Hence neutrinos offer the natural possibility for large scale structures79,80 including filaments and voids. Light neutrinos are, however, ruled out as a dominant form of dark matter because they produce too much large scale structure81 . Because the smallest non-linear structures have mass scale MJ and the typical galactic mass scale is ≃ 1012 M⊙ , galaxies must fragment out of the larger pancake-like objects. The problem with such a scenario is that galaxies form late80,82 (z ≤ 1) whereas quasars and galaxies are seen out to redshifts z > 6. ∼ In the standard model, the absence of a right-handed neutrino state precludes the existence of a neutrino mass (unless one includes nonrenormalizable lepton number violating interactions such HHLL). By adding a right-handed state νR , it is possible to generate a Dirac mass for

23

√ the neutrino, mν = hν v/ 2, as is the case for the charged lepton masses, where hν is the neutrino Yukawa coupling constant, and v is the Higgs expectation value. It is also possible to generate a Majorana mass for the neutrino when in addition to the Dirac mass term, mν ν¯ νL , a term M νR νR R is included. If M ≫ mν , the see-saw mechanism produces two mass eigenstates given by mν1 ∼ m2 /M which is very light, and mν2 ∼ M which is ν heavy. The state ν1 is a potential hot dark matter candidate as ν2 is in general not stable. The simplicity of the standard big bang model allows one to compute in a straightforward manner the relic density of any stable particle if that particle was once in thermal equilibrium with the thermal radiation bath. At early times, neutrinos were kept in thermal equilibrium by their weak interactions with electrons and positrons. As we saw in the case of the β-interaction used in BBN, one can estimate the thermally averaged lowenergy weak interaction scattering cross section σv ∼ g 4 T 2 /m4 W (34)

for T ≪ mW . Recalling that the number density scales as n ∝ T 3 , we can compare the weak interaction rate Γ ∼ n σv , with the expansion rate given by eqs. (6) with ρ=
B

gB +

7 8

gF
F

π2 4 π2 T ≡ N (T ) T 4 30 30

(35)

Neutrinos will be in equilibrium when Γwk > H or T3 >
−1/2

8π 3 N/90 m4 /MP W

(36)

= 1.22 × 1019 GeV is the Planck mass. For N = 43/4 where MP = GN (accounting for photons, electrons, positrons and three neutrino flavors) we see that equilibrium is maintained at temperatures greater than O(1) MeV (for a more accurate calculation see 83 ). The decoupling scale of O(1) MeV has an important consequence on the final relic density of massive neutrinos. Neutrinos more massive than 1 MeV will begin to annihilate prior to decoupling, and while in equilibrium, their number density will become exponentially suppressed. Lighter neutrinos decouple as radiation on the other hand, and hence do not experience the suppression due to annihilation. Therefore, the calculations of the number density of light (mν < 1 MeV) and heavy (mν > 1 MeV) neutrinos differ ∼ ∼ substantially.

24

The number of density of light neutrinos with mν < 1 MeV can be ∼ expressed at late times as ρν = mν Yν nγ (37)

where Yν = nν /nγ is the density of ν’s relative to the density of photons, which today is 411 photons per cm3 . It is easy to show that in an adiabatically expanding universe Yν = 3/11. This suppression is a result of the e+ e− annihilation which occurs after neutrino decoupling and heats the photon bath relative to the neutrinos. In order to obtain an age of the Universe, t > 12 Gyr, one requires that the matter component is constrained by Ωh2 ≤ 0.3. (38)

From this one finds the strong constraint (upper bound) on Majorana neutrino masses84: mtot = ν mν < 28eV. ∼

(39)

where the sum runs over neutrino mass eigenstates. The limit for Dirac neutrinos depends on the interactions of the right-handed states (see discussion below). Given the discussion of the CMB results in the previous section, one could make a case that the limit on Ωh2 should be reduced by a factor of 2, which would translate in to a limit of 14 eV on the sum of the light neutrino masses. As one can see, even very small neutrino masses of order 1 eV, may contribute substantially to the overall relic density. The limit (39) and the corresponding initial rise in Ων h2 as a function of mν is displayed in the Figure 8 (the low mass end with mν < 1 MeV). ∼ Combining the rapidly improving data on key cosmological parameters with the better statistics from large redshift surveys has made it possible to go a step forward along this path. It is now possible to set stringent limits on the light neutrino mass density Ων h2 , and hence on the neutrino mass based on the power spectrum of the Ly α forest86 , mtot < 5.5 eV, and the limit is even stronger if the total matter density, Ωm is less than 0.5. Adding additional observation constraints from the CMB and galaxy clusters drops this limit87 to 2.4 eV. This limit has recently been improved by the 2dF Galaxy redshift88 survey by comparing the derived power spectrum of fluctuations with structure formation models. Focussing on the the presently favoured ΛCDM model, the neutrino mass bound becomes mtot < 1.8 eV for Ωm < 0.5. When even more constraints such as HST

25

4 2 ην = 0 /

Log(Ω ν h2 )

0 −2 −4 −6 KeV MeV GeV TeV



Figure 8. Summary plot85 of the relic density of Dirac neutrinos (solid) including a possible neutrino asymmetry of ην = 5 × 10−11 (dotted).

Key project data, supernovae type Ia data, and BBN are included89 the limit can be pushed to mtot < 0.3 eV. The calculation of the relic density for neutrinos more massive than ∼ 1 MeV, is substantially more involved. The relic density is now determined by the freeze-out of neutrino annihilations which occur at T < mν , after ∼ annihilations have begun to seriously reduce their number density90. The annihilation rate is given by Γann = σv ann nν



m2 ν (mν T )3/2 e−mν /T m4 Z

(40)

where we have assumed, for example, that the annihilation cross section ¯ is dominated by ν ν → f f via Z-boson exchangea and σv ann ∼ m2 /m4 . ¯ ν Z When the annihilation rate becomes slower than the expansion rate of the Universe the annihilations freeze out and the relative abundance of neutrinos becomes fixed. For particles which annihilate through approximate weak scale interactions, this occurs when T ∼ mχ /20. The number density this is approximately true for Dirac neutrinos, the annihilation cross section of Majorana neutrinos is p-wave suppressed and is proportional of the final state fermion masses rather than mν . a While

26

of neutrinos is tracked by a Boltzmann-like equation, ˙ R dn = −3 n − σv (n2 − n2 ) (41) 0 dt R where n0 is the equilibrium number density of neutralinos. By defining the quantity f = n/T 3 , we can rewrite this equation in terms of x, as df = mν dx 8π 3 GN N 90
1/2 2 (f 2 − f0 )

(42)

The solution to this equation at late times (small x) yields a constant value of f , so that n ∝ T 3 . Roughly, the solution to the Boltzmann equation goes as Yν ∼ f ∼ (m σv ann )−1 and hence Ων h2 ∼ σv ann −1 , so that parametrically Ων h2 ∼ 2 1/mν . As a result, the constraint (38) now leads to a lower bound90,91,92 on the neutrino mass, of about mν > 3 − 7 GeV, depending on whether ∼ it is a Dirac or Majorana neutrino. This bound and the corresponding downward trend Ων h2 ∼ 1/m2 can again be seen in Figure 8. The result ν of a more detailed calculation is shown in Figure 9 92 for the case of a Dirac neutrino. The two curves show the slight sensitivity on the temperature scale associated with the quark-hadron transition. The result for a Majorana mass neutrino is qualitatively similar. Indeed, any particle with roughly weak scale cross-sections will tend to give an interesting value of Ωh2 ∼ 1. The deep drop in Ων h2 , visible in Figure 8 at around mν = MZ /2, is due to a very strong annihilation cross section at Z-boson pole. For yet higher neutrino masses the Z-annihilation channel cross section drops as ∼ 1/m2 , leading to a brief period of an increasing trend in Ων h2 . However, ν for mν > mW the cross section regains its parametric form σv ann ∼ m2 ν ∼ due to the opening up of a new annihilation channel to W -boson pairs93, and the density drops again as Ων h2 ∼ 1/m2 . The tree level W -channel ν cross section breaks the unitarity at around O(few) TeV 94 however, and the full cross section must be bound by the unitarity limit95 . This behaves 2 again as 1/mν , whereby Ων h2 has to start increasing again, until it becomes too large again at 200-400 TeV 95,94 (or perhaps somewhat earlier as the weak interactions become strong at the unitarity breaking scale). If neutrinos are Dirac particles, and have a nonzero asymmetry the relic density could be governed by the asymmetry rather than by the annihilation cross section. Indeed, it is easy to see that the neutrino mass density corresponding to the asymmetry ην ≡ (nν − nν )/nγ is given by96 ¯ ρ = mν ην nγ , (43)

27

1

0
Log (Ων h2)
400

-1

150

-2

-3 0 5 10 mν 15
(GeV)

20

25

30

Figure 9. The relic density of heavy Dirac neutrinos due to annihilations92 . The curves are labeled by the assumed quark-hadron phase transition temperature in MeV.

which implies Ων h2 ≃ 0.004 ην10 (mν /GeV). (44)

where ην10 ≡ 1010 ην . The behaviour of the energy density of neutrinos with an asymmetry is shown by the dotted line in the Figure 8. At low mν , the mass density is dominated by the symmetric, relic abundance of both neutrinos and antineutrinos which have already frozen out. At higher values of mν , the annihilations suppress the symmetric part of the relic density until Ων h2 eventually becomes dominated by the linearly increasing asymmetric contribution. In the figure, we have assumed an asymmetry of ην ∼ 5 × 10−11 for neutrinos with standard weak interaction strength. In this case, Ων h2 begins to rise when mν > 20 GeV. Obviously, the bound ∼ (38) is saturated for mν = 75 GeV/ην10 . Based on the leptonic and invisible width of the Z boson, experiments at LEP have determined that the number of neutrinos is Nν = 2.9841 ± 0.0083 97 . Conversely, any new physics must fit within these brackets, and thus LEP excludes additional neutrinos (with standard weak interactions) with masses mν < 45 GeV. Combined with the limits dis∼ played in Figures 8 and 9, we see that the mass density of ordinary heavy neutrinos is bound to be very small, Ων h2 < 0.001 for masses mν > 45 GeV up to mν ∼ O(100) TeV. Lab constraints for Dirac neutrinos are

28

available98, excluding neutrinos with masses between 10 GeV and 4.7 TeV. This is significant, since it precludes the possibility of neutrino dark matter based on an asymmetry between ν and ν 96 . Majorana neutrinos are ex¯ cluded as dark matter since Ων ho 2 < 0.001 for mν > 45 GeV and are thus cosmologically uninteresting. A bound on neutrino masses even stronger than Eqn. (39) can be obtained from the recent observations of active-active mixing in both solar- and atmospheric neutrino experiments. The inferred evidence for νµ − ντ and νe − νµ,τ mixings are on the scales m2 ∼ 1 − 10 × 10−5 and ν m2 ∼ 2 − 5 × 10−3. When combined with the upper bound on the electronν like neutrino mass mν < 2.8 eV 99 , and the LEP-limit on the number of neutrino species, one finds the constraint on the sum of neutrino masses: 0.05 eV < mtot < 8.4 eV. ∼ ∼ (45)

Conversely, the experimental and observational data then implies that the cosmological energy density of all light, weakly interacting neutrinos can be restricted to the range 0.0005 < Ων h2 < 0.09. ∼ ∼ (46)

Interestingly there is now also a lower bound due to the fact that at least one of the neutrino masses has to be larger than the scale m2 ∼ 10−3 eV2 set by the atmospheric neutrino data. Combined with the results on relic mass density of neutrinos and the LEP limits, the bound (46) implies that the ordinary weakly interacting neutrinos, once the standard dark matter candidate78 , can be ruled out completely as a dominant component of the dark matter. If instead, we consider right-handed neutrinos, we have new possibilities. Right-handed interactions are necessarily weaker than standard left-handed interactions implying that right-handed neutrinos decouple early and today are at a reduced temperature relative to νL 100 ( 43 Tχ 3 ) = Tγ 4N (Td ) (47)

As such, for Td R ≫ 1 MeV, nνR /nνL = (TνR /TνL )3 ≪ 1. Thus the abundance of right-handed neutrinos can be written as YνR = 3 Tν 3 nνR = ( )( R )3 ≪ nγ 11 TνL 11 (48)

In this case, the previous bound (39) on neutrino masses is weakened. For a suitably large scale for the right-handed interactions, right-handed neutrino

29

masses may be as large as a few keV 101 . Such neutrinos make excellent warm dark matter candidates, albeit the viable mass range for galaxy formation is quite restricted102 . 2.2.3. Axions Due to space limitations, the discussion of this candidate will be very brief. Axions are pseudo-Goldstone bosons which arise in solving the strong CP problem103,104 via a global U(1) Peccei-Quinn symmetry. The invisible axion104 is associated with the flat direction of the spontaneously broken PQ symmetry. Because the PQ symmetry is also explicitly broken (the CP violating θF F coupling is not PQ invariant) the axion picks up a small mass similar to pion picking up a mass when chiral symmetry is broken. We can expect that ma ∼ mπ fπ /fa where fa , the axion decay constant, is the vacuum expectation value of the PQ current and can be taken to be quite large. If we write the axion field as a = fa θ, near the minimum, the 2 potential produced by QCD instanton effects looks like V ∼ m2 θ2 fa . The a axion equations of motion lead to a relatively stable oscillating solution. The energy density stored in the oscillations exceeds the critical density105 unless fa < 1012 GeV. ∼ Axions may also be emitted stars and supernova106. In supernovae, axions are produced via nucleon-nucleon bremsstrahlung with a coupling gA N ∝ mN /fa . As was noted above the cosmological density limit requires fa < 1012 GeV. Axion emission from red giants imply107 fa > 1010 GeV ∼ ∼ (though this limit depends on an adjustable axion-electron coupling), the supernova limit requires108 fa > 2 × 1011 GeV for naive quark model ∼ couplings of the axion to nucleons. Thus only a narrow window exists for the axion as a viable dark matter candidate. 3. Lecture 3: Supersymmetric Dark Matter Although there are many reasons for considering supersymmetry as a candidate extension to the standard model of strong, weak and electromagnetic interactions109, one of the most compelling is its role in understanding the hierarchy problem110 namely, why/how is mW ≪ MP . One might think naively that it would be sufficient to set mW ≪ MP by hand. However, radiative corrections tend to destroy this hierarchy. For example, one-loop diagrams generate α Λ 2 ≫ m2 (49) δm2 = O W W π

30

where Λ is a cut-off representing the appearance of new physics, and the inequality in (49) applies if Λ ∼ 103 TeV, and even more so if Λ ∼ mGUT ∼ 1016 GeV or ∼ MP ∼ 1019 GeV. If the radiative corrections to a physical quantity are much larger than its measured values, obtaining the latter requires strong cancellations, which in general require fine tuning of the bare input parameters. However, the necessary cancellations are natural in supersymmetry, where one has equal numbers of bosons and fermions with equal couplings, so that (49) is replaced by δm2 = O W α π |m2 − m2 | . B F (50)

The residual radiative correction is naturally small if |m2 − m2 | < 1 TeV2 . B F ∼ In order to justify the absence of interactions which can be responsible for extremely rapid proton decay, it is common in the minimal supersymmetric standard model (MSSM) to assume the conservation of R-parity. If R-parity, which distinguishes between “normal” matter and the supersymmetric partners and can be defined in terms of baryon, lepton and spin as R = (−1)3B+L+2S , is unbroken, there is at least one supersymmetric particle (the lightest supersymmetric particle or LSP) which must be stable. Thus, the minimal model contains the fewest number of new particles and interactions necessary to make a consistent theory. There are very strong constraints, however, forbidding the existence of stable or long lived particles which are not color and electrically neutral111. Strong and electromagnetically interacting LSPs would become bound with normal matter forming anomalously heavy isotopes. Indeed, there are very strong upper limits on the abundances, relative to hydrogen, of nuclear isotopes112, n/nH < 10−15 to 10−29 for 1 GeV < m < 1 TeV. A strongly ∼ ∼ ∼ interacting stable relic is expected to have an abundance n/nH < 10−10 ∼ with a higher abundance for charged particles. There are relatively few supersymmetric candidates which are not colored and are electrically neutral. The sneutrino113 is one possibility, but in the MSSM, it has been excluded as a dark matter candidate by direct98 and indirect114 searches. In fact, one can set an accelerator based limit on the sneutrino mass from neutrino counting, mν > 44.7 GeV 115 . In this ˜ ∼ case, the direct relic searches in underground low-background experiments require mν > 20 TeV 98 . Another possibility is the gravitino which is ˜ ∼ probably the most difficult to exclude. I will concentrate on the remaining possibility in the MSSM, namely the neutralinos.

31

3.1. Parameters The most general version of the MSSM, despite its minimality in particles and interactions contains well over a hundred new parameters. The study of such a model would be untenable were it not for some (well motivated) assumptions. These have to do with the parameters associated with supersymmetry breaking. It is often assumed that, at some unification scale, all of the gaugino masses receive a common mass, m1/2 . The gaugino masses at the weak scale are determined by running a set of renormalization group equations. Similarly, one often assumes that all scalars receive a common mass, m0 , at the GUT scale. These too are run down to the weak scale. The remaining parameters of importance involve the Higgs sector. There is the Higgs mixing mass parameter, µ, and since there are two Higgs doublets in the MSSM, there are two vacuum expectation values. One combination of these is related to the Z mass, and therefore is not a free parameter, while the other combination, the ratio of the two vevs, tan β, is free. If the supersymmetry breaking Higgs soft masses are also unified at the GUT scale (and take the common value m0 ), then µ and the physical Higgs masses at the weak scale are determined by electroweak vacuum conditions (µ is determined up to a sign). This scenario is often referred to as the constrained MSSM or CMSSM. Once these parameters are set, the entire spectrum of sparticle masses at the weak scale can be calculatedb . In Fig. 10, an example of the running of the mass parameters in the CMSSM is shown. Here, we have chosen m1/2 = 250 GeV, m0 = 100 GeV, tan β = 3, A0 = 0, and µ < 0. Indeed, it is rather amazing that from so few input parameters, all of the masses of the supersymmetric particles can be determined. The characteristic features that one sees in the figure, are for example, that the colored sparticles are typically the heaviest in the spectrum. This is due to the large positive correction to the masses due to α3 in the RGE’s. Also, one finds that the B (the partner of the U (1)Y gauge boson), is typically the lightest sparticle. But most importantly, notice that one of the Higgs mass2 , goes negative triggering electroweak symmetry breaking116. (The negative sign in the figure refers to the sign of the mass2 , even though it is the mass of the sparticles which are depicted.)

b There

are in fact, additional parameters: the supersymmetry-breaking tri-linear masses A (also assumed to be unified at the GUT scale) as well as two CP violating phases θµ and θA .

32

Figure 10.

RG evolution of the mass parameters in the CMSSM.

3.2. Neutralinos There are four neutralinos, each of which is a linear combination of the R = ˜ −1 neutral fermions111: the wino W 3 , the partner of the 3rd component ˜ of the SU (2)L gauge boson; the bino, B; and the two neutral Higgsinos, ˜ 1 and H2 . Assuming gaugino mass universality at the GUT scale, the ˜ H identity and mass of the LSP are determined by the gaugino mass m1/2 , µ, and tan β. In general, neutralinos can be expressed as a linear combination ˜ ˜ ˜ ˜ χ = αB + β W 3 + γ H1 + δ H2 (51)

The solution for the coefficients α, β, γ and δ for neutralinos that make up the LSP can be found by diagonalizing the mass matrix   2 √v M2 0 −g22 1 g√v2 ˜  W3 2  0 1 √v M1 g√v1 −g12 2   B   ˜  2 ˜ ˜ ˜0 ˜0  (W 3 , B, H1 , H2 )  −g2 v1 g1 v1 (52)  ˜0 √ √ 0 −µ   H1   2 2 g√v2 −g1 v2 ˜ 2 H0 √ −µ 0
2 2 2

33

where M1 (M2 ) is a soft supersymmetry breaking term giving mass to the U(1) (SU(2)) gaugino(s). In a unified theory M1 = M2 = m1/2 at the unification scale (at the weak scale, M1 ≃ 5 α1 M2 ). As one can see, the 3 α2 coefficients α, β, γ, and δ depend only on m1/2 , µ, and tan β. In Figure 11 117 , regions in the M2 , µ plane with tan β = 2 are shown in which the LSP is one of several nearly pure states, the photino, γ , the ˜ ˜ a symmetric combination of the Higgsinos, H(12) , or the Higgsino, ˜ bino, B, ˜ ˜ ˜ S = sin β H1 +cos β H2 . The dashed lines show the LSP mass contours. The ˜ ˜ cross hatched regions correspond to parameters giving a chargino (W ± , H ± ) 118. This state with mass mχ ≤ 45GeV and as such are excluded by LEP ˜ constraint has been extended by LEP119 and is shown by the light shaded region and corresponds to regions where the chargino mass is < 103.5 GeV. ∼ The newer limit does not extend deep into the Higgsino region because of the degeneracy between the chargino and neutralino. Notice that the ˜ ˜ parameter space is dominated by the B or H12 pure states and that the photino only occupies a small fraction of the parameter space, as does ˜ the Higgsino combination S. Both of these light states are experimentally excluded.

3.3. The Relic Density The relic abundance of LSP’s is determined by solving the Boltzmann equation for the LSP number density in an expanding Universe. The technique92 used is similar to that for computing the relic abundance of massive neutrinos90. The relic density depends on additional parameters in the MSSM beyond m1/2 , µ, and tan β. These include the sfermion masses, mf and the Higgs pseudo-scalar mass, mA , derived from m0 (and m1/2 ). To ˜ determine the relic density it is necessary to obtain the general annihilation cross-section for neutralinos. In much of the parameter space of interest, the LSP is a bino and the annihilation proceeds mainly through sfermion exchange. Because of the p-wave suppression associated with Majorana fermions, the s-wave part of the annihilation cross-section is suppressed by the outgoing fermion masses. This means that it is necessary to expand the cross-section to include p-wave corrections which can be expressed as a term proportional to the temperature if neutralinos are in equilibrium. Unless the neutralino mass happens to lie near near a pole, such as mχ ≃ mZ /2 or mh /2, in which case there are large contributions to the annihilation through direct s-channel resonance exchange, the dominant contribution ˜˜ to the B B annihilation cross section comes from crossed t-channel sfermion

34

−µ
Figure 11. Mass contours and composition of nearly pure LSP states in the MSSM 117 .

exchange. Annihilations in the early Universe continue until the annihilation rate Γ ≃ σvnχ drops below the expansion rate. The calculation of the neutralino relic density proceeds in much the same way as discussed above for neutrinos with the appropriate substitution of the cross section. The final neutralino relic density expressed as a fraction of the critical energy density can be written as111 Ωχ h2 ≃ 1.9 × 10−11 Tχ Tγ
3

Nf

1/2

GeV axf + 1 bx2 f 2

(53)

where (Tχ /Tγ )3 accounts for the subsequent reheating of the photon temperature with respect to χ, due to the annihilations of particles with mass m < xf mχ 100 . The subscript f refers to values at freeze-out, i.e., when annihilations cease. The coefficients a and b are related to the partial wave expansion of the cross-section, σv = a + bx + . . .. Eq. (53 ) results in a very good approximation to the relic density expect near s-channel annihilation

35

poles, thresholds and in regions where the LSP is nearly degenerate with the next lightest supersymmetric particle120 . When there are several particle species i, which are nearly degenerate in mass, co-annihilations are important. In this case120 , the rate equation (41) still applies, provided n is interpreted as the total number density, n≡ ni , i (54)

n0 as the total equilibrium number density, n0 ≡ n0,i , i (55)

and the effective annihilation cross section as n0,i n0,j σeff vrel ≡ σij vrel . n2 0 ij

(56)

In eq. (42), mχ is now understood to be the mass of the lightest sparticle under consideration. Note that this implies that the ratio of relic densities computed with and without coannihilations is, roughly, R≡ Ω0 ≈ Ω σeff ˆ σ0 ˆ xf x0 f , (57)

where σ ≡ a + bx/2 and sub- and superscripts 0 denote quantities comˆ puted ignoring coannihilations. The ratio x0 /xf ≈ 1 + x0 ln(geff σeff /g1 σ0 ), f f where geff ≡ i gi (mi /m1 )3/2 e−(mi −m1 )/T . For the case of three degenerate slepton NLSPs 121 , geff = i gi = 8 and x0 /xf ≈ 1.2. The effects of f co-annihilations are discussed below. 3.4. Phenomenological and Cosmological Constraints For the cosmological limits on the relic density I will assume 0.1 ≤ Ωχ h2 ≤ 0.3. (58)

The upper limit being a conservative bound based only on the lower limit to the age of the Universe of 12 Gyr. Indeed, most analyses indicate that Ωmatter < 0.4 − 0.5 and thus it is very likely that Ωχ h2 < 0.2 (cf. the ∼ CMB results in Table 1). One should note that smaller values of Ωχ h2 are allowed, since it is quite possible that some of the cold dark matter might not consist of LSPs.

36

The calculated relic density is found to have a relevant cosmological density over a wide range of susy parameters. For all values of tan β, there is a ‘bulk’ region with relatively low values of m1/2 and m0 where 0.1 < Ωχ h2 < 0.3. However there are a number of regions at large values of m1/2 and/or m0 where the relic density is still compatible with the cosmological constraints. At large values of m1/2 , the lighter stau, becomes nearly degenerate with the neutralino and co-annihilations between these particles must be taken into account121,122 . For non-zero values of A0 , ˜ there are new regions for which χ − t coannihilations are important123 . At large tan β, as one increases m1/2 , the pseudo-scalar mass, mA begins to drop so that there is a wide funnel-like region (at all values of m0 ) such that 2mχ ≈ mA and s-channel annihilations become important124,125 . Finally, there is a region at very high m0 where the value of µ begins to fall and the LSP becomes more Higgsino-like. This is known as the ‘focus point’ region126. As an aid to the assessment of the prospects for detecting sparticles at different accelerators, benchmark sets of supersymmetric parameters have often been found useful, since they provide a focus for concentrated discussion. A set of proposed post-LEP benchmark scenarios127 in the CMSSM are illustrated schematically in Fig. 12. Five of the chosen points are in the ‘bulk’ region at small m1/2 and m0 , four are spread along the coannihilation ‘tail’ at larger m1/2 for various values of tan β. This tail runs along the shaded region in the lower right corner where the stau is the LSP and is therefore excluded by the constraints against charged dark matter. Two points are in rapid-annihilation ‘funnels’ at large m1/2 and m0 . At large values of m0 , the focus-point region runs along the boundary where electroweak symmetry no longer occurs (shown in Fig. 12 as the shaded region in the upper left corner). Two points were chosen in the focus-point region at large m0 . The proposed points range over the allowed values of tan β between 5 and 50. The light shaded region corresponds to the portion of parameter space where the relic density Ωχ h2 is between 0.1 and 0.3. The effect of coannihilations is to create an allowed band about 25-50 GeV wide in m0 for m1/2 < 1400 GeV, which tracks above the mτ = mχ ∼ contour. Along the line mτ = mχ , R ≈ 10, from (57) 121 . As m0 increases, the mass difference increases and the slepton contribution to σeff falls, and ˆ the relic density rises abruptly. This effect is seen in Fig. 13. The light shaded region corresponds to 0.1 < Ωh2 < 0.3. The dark shaded region has mτ < mχ and is excluded. The light dashed contours indicate the corresponding region in Ωh2 if one ignores the effect of coannihilations.

37

m0

mh, b→sγ

g-2

m1/2
Figure 12. Schematic overview of the CMSSM benchmark points proposed in 127 . The points are intended to illustrate the range of available possibilities. The labels correspond to the approximate positions of the benchmark points in the (m1/2 , m0 ) plane. They also span values of tan β from 5 to 50 and include points with µ < 0.

Neglecting coannihilations, one would find an upper bound of ∼ 450 GeV on m1/2 , corresponding to an upper bound of roughly 200 GeV on mB . ˜ Instead, values of m1/2 up to ∼ 1400 GeV are allowed corresponding to an upper bound of ∼ 600 GeV on mB . ˜ The most important phenomenological constraints are also shown schematically in Figure 12. These include the constraint provided by the LEP lower limit on the Higgs mass: mH > 114.4 GeV 128 . This holds in the Standard Model, for the lightest Higgs boson h in the general MSSM for tan β < 8, and almost always in the CMSSM for all tan β. Since mh ∼ is sensitive to sparticle masses, particularly mt , via loop corrections, the ˜ Higgs limit also imposes important constraints on the CMSSM parameters, principally m1/2 as seen by the dashed curve in Fig. 12.

38

110

m ± =104 χ m0

100

~ e

114

200

tan β =10, µ > 0

m∼ < m ∼ τ1 χ

100

200

300

m 1/2

400

500

600

Figure 13. The light-shaded ‘bulk’ area is the cosmologically preferred region with 0.1 ≤ Ωh2 ≤ 0.3. The light dashed lines show the location of the cosmologically preferred region if one ignores coannihilations with the light sleptons. In the dark shaded region in the bottom right, the LSP is the τ1 , leading to an unacceptable abundance ˜ of charged dark matter. Also shown is the isomass contour mχ± = 104 GeV and mh = 110, 114 GeV, as well as an indication of the slepton bound from LEP.

The constraint imposed by measurements of b → sγ 129 also exclude small values of m1/2 . These measurements agree with the Standard Model, and therefore provide bounds on MSSM particles, such as the chargino and charged Higgs masses, in particular. Typically, the b → sγ constraint is more important for µ < 0, but it is also relevant for µ > 0, particularly when tan β is large. The BNL E821 experiment reported last year a new measurement of aµ ≡ 1 (gµ − 2) which deviated by 2.6 standard deviations from 2 the best Standard Model prediction available at that time130 . However, it had been realized that the sign of the most important pseudoscalar-meson pole part of the light-by-light scattering contribution131 to the Standard Model prediction should be reversed, which reduces the apparent experimental discrepancy to about 1.6 standard deviations (δaµ ×1010 = 26±16). The largest contribution to the errors in the comparison with theory was

39

thought to be the statistical error of the experiment, which has been significantly reduced just recently132. The world average of aµ ≡ 1 (gµ − 2) now 2 deviates by (33.9 ± 11.2) × 10−10 from the Standard Model calculation of Davier et al.133 using e+ e− data, and by (17 ± 11) × 10−10 from the Standard Model calculation of Davier et al.133 based on τ decay data. Other recent analyses of the e+ e− data yield similar results. On the subsequent plots, the formal 2-σ range 11.5 × 10−10 < δaµ < 56.3 × 10−10 is displayed.

800 700 600 500 400 300 200 100 0 100

tan β = 10 , µ > 0 mh = 114 GeV

m0 (GeV)

mχ± = 103.5 GeV

200

300

400

500

600

700

800

900

1000

m1/2 (GeV)
Figure 14. Compilation of phenomenological constraints on the CMSSM for tan β = 10, µ > 0, assuming A0 = 0, mt = 175 GeV and mb (mb )M S = 4.25 GeV. The nearSM vertical lines are the LEP limits mχ± = 103.5 GeV (dashed and black)119 , and mh = 114.1 GeV (dotted and red)128 . Also, in the lower left corner we show the me = 99 GeV ˜ contour135 . In the dark (brick red) shaded regions, the LSP is the charged τ1 , so this ˜ region is excluded. The light(turquoise) shaded areas are the cosmologically preferred regions with 0.1 ≤ Ωh2 ≤ 0.3 125 . The medium (dark green) shaded regions are excluded by b → sγ 129 . The shaded (pink) region in the upper right delineates the 2 σ range of gµ − 2. The dashed curves within this region correspond to the 1 − σ bounds.

40

Following a previous analysis125,134, in Figure 14 the m1/2 −m0 parameter space is shown for tan β = 10. The dark shaded region (in the lower right) corresponds to the parameters where the LSP is not a neutralino but rather a τR . The cosmologically interesting region at the left of the figure ˜ is due to the appearance of pole effects. There, the LSP can annihilate through s-channel Z and h (the light Higgs) exchange, thereby allowing a very large value of m0 . However, this region is excluded by phenomenological constraints. Here one can see clearly the coannihilation tail which extends towards large values of m1/2 . In addition to the phenomenological constraints discussed above, Figure 14 also shows the current experimental constraints on the CMSSM parameter space due to the limit mχ± > 103.5 ∼ GeV provided by chargino searches at LEP 119 . LEP has also provided lower limits on slepton masses, of which the strongest is me > 99 GeV ˜ ∼ 135 . This is shown by dot-dashed curve in the lower left corner of Fig. 14. Similar results have been found by other analyses136. As one can see, one of the most important phenomenological constraint at this value of tan β is due to the Higgs mass (shown by the nearly vertical dot-dashed curve). The theoretical Higgs masses were evaluated using FeynHiggs137, which is estimated to have a residual uncertainty of a couple of GeV in mh . The region excluded by the b → sγ constraint is the dark shaded (green) region to the left of the plot. As many authors have pointed out138 , a discrepancy between theory and the BNL experiment could well be explained by supersymmetry. As seen in Fig. 14, this is particularly easy if µ > 0. The medium (pink) shaded region in the figure corresponds to the overall allowed region (2σ) by the new experimental result. As discussed above, another mechanism for extending the allowed CMSSM region to large mχ is rapid annihilation via a direct-channel pole 1 when mχ ∼ 2 mA 124,125. This may yield a ‘funnel’ extending to large m1/2 and m0 at large tan β, as seen in Fig. 15. In principle, the true input parameters in the CMSSM are: µ, m1 , m2 , and B, where m1 and m2 are the Higgs soft masses (in the CMSSM m1 = m2 = m0 and B is the susy breaking bilinear mass term). In this case, the electroweak symmetry breaking conditions lead to a prediction of MZ , tan β ,and mA . Since we are not really interested in predicting MZ , it is more useful to assume instead the following CMSSM input parameters: MZ , m1 , m2 , and tan β again with m1 = m2 = m0 . In this case, one predicts µ, B, and mA . However, one can generalize the CMSSM case to include non-universal Higgs masses139,140 (NUHM), in which case the

41

1500

tan β = 50 , µ > 0

m0 (GeV)

1000

mh = 114 GeV

0 100

1000

2000

3000

m1/2 (GeV)
Figure 15. As in Fig. 14 for tan β = 50.

input parameters become:MZ , µ, mA , and tan β and one predicts m1 , m2 , and B. The NUHM parameter space was recently analyzed140 and a sample of the results found is shown in Fig. 16. While much of the cosmologically preferred area with µ < 0 is excluded, there is a significant enhancement in the allowed parameter space for µ > 0. 3.5. Detection Because the LSP as dark matter is present locally, there are many avenues for pursuing dark matter detection. Direct detection techniques rely on an ample neutralino-nucleon scattering cross-section. The effective fourfermion lagrangian can be written as L = χγ µ γ 5 χqi γµ (α1i + α2i γ 5 )qi ¯ ¯

42

tan β = 10 , m1/2 = 300 , m0 = 100
1000 900 800 700 600 500 400 300 200 100 -2000

mA (GeV)

-1000

µ (GeV)

0

1000

2000

Figure 16. Compilations of phenomenological constraints on the MSSM with NUHM in the (µ, mA ) plane for tan β = 10 and m0 = 100 GeV, m1/2 = 300 GeV, assuming A0 = 0, mt = 175 GeV and mb (mb )M S = 4.25 GeV. The shading is as described in SM Fig. 14. The (blue) solid line is the contour mχ = mA /2, near which rapid direct-channel annihilation suppresses the relic density. The dark (black) dot-dashed line indicates when one or another Higgs mass-squared becomes negative at the GUT scale: only lower |µ| and larger mA values are allowed. The crosses denote the values of µ and mA found in the CMSSM.

+ α3i χχqi qi + α4i χγ 5 χqi γ 5 qi ¯ ¯ ¯ ¯ + α5i χχqi γ 5 qi + α6i χγ 5 χqi qi ¯ ¯ ¯ ¯ (59)

However, the terms involving α1i , α4i , α5i , and α6i lead to velocity dependent elastic cross sections. The remaining terms are: the spin dependent coefficient, α2i and the scalar coefficient α3i . Contributions to α2i are predominantly through light squark exchange. This is the dominant channel for binos. Scattering also occurs through Z exchange but this channel requires a strong Higgsino component. Contributions to α3i are also dominated by light squark exchange but Higgs exchange is non-negligible in

43

most cases. tan β = 10 , µ < 0 mh = 114.1 GeV
700 600 500 400 300 200 100 0 100

800

800 700 600 500 400 300 200 100 0 100

tan β = 10 , µ > 0

mh = 114.1 GeV

m0 (GeV)

m0 (GeV)

σSI = 10-12 pb

σSI = 10-9 pb

200

300

400

500

600

700

800

900

1000

200

300

400

500

600

700

800

900

1000

m1/2 (GeV)

m1/2 (GeV)

Figure 17. Spin-independent cross sections in the (m1/2 , m0 ) planes for (a) tan β = 10, µ < 0, (b) tan β = 10, µ > 0. The double dot-dashed (orange) curves are contours of the spin-independent cross section, differing by factors of 10 (bolder) and interpolating factors of 3 (finer - when shown). For example, in (b), the curves to the right of the one marked 10−9 pb correspond to 3 × 10−10 pb and 10−10 pb.

Fig. 17 displays contours of the spin-independent cross section for the elastic scattering of the LSP χ on protons in the m1/2 , m0 planes for (a) tan β = 10, µ < 0, (b) tan β = 10, µ > 0 141 . The double dot-dashed (orange) lines are contours of the spin-independent cross section, and the contours σSI = 10−9 pb in panel (a) and σSI = 10−12 pb in panel (b) are indicated. The LEP lower limits on mh and mχ± , as well as the experimental measurement of b → sγ for µ < 0, tend to bound the cross sections from above, as discussed in more detail below. Generally speaking, the spin-independent cross section is relatively large in the ‘bulk’ region, but falls off in the coannihilation ‘tail’. Also, we note also that there is a strong cancellation in the spin-independent cross section when µ < 0 142,143 , as seen along strips in panel (a) of Fig. 17 where m1/2 ∼ 500 GeV. In the cancellation region, the cross section drops lower than 10−14 pb. All these possibilities for suppressed spin-independent cross sections are disfavoured by the data on gµ − 2, which favour values of m1/2 and m0 that are not very large, as well as µ > 0, as seen in panel (b) of Fig. 17. Thus gµ − 2 tends to provide a lower bound on the spin-independent cross section. Fig. 18(a) illustrates the effect on the cross sections of each of the princi-

44
10-7 tan β = 10, µ > 0

10-7 µ>0 10-8
Higgs b to s γ

10-8

Relic Density b to s γ

σSI (pb)

g-2 Higgs

σSI (pb)

10-9

All

10-9 g-2 10-10

10-10 relic density

10-11 0 100 200 300 400 500 600 700

10-11
0 10 20 30 40 50 60

m χ (GeV)

tan β

Figure 18. Allowed ranges of the cross sections for tan β = 10 (a) µ > 0 for spinindependent elastic scattering. The solid (blue) lines indicate the relic density constraint, the dashed (black) lines the b → sγ constraint, the dot-dashed (green) lines the mh constraint, and the dotted (red) lines the gµ − 2 constraint. The shaded (pale blue) region is allowed by all the constraints. (b) The allowed ranges of the spin-independent cross section for µ > 0. The darker solid (black) lines show the upper limits on the cross sections obtained from mh and b → sγ, and (where applicable) the lighter solid (red) lines show the lower limits suggested by gµ − 2 and the dotted (green) lines the lower limits from the relic density.

pal phenomenological constraints, for the particular case tan β = 10 µ > 0. The solid (blue) lines mark the bounds on the cross sections allowed by the relic-density constraint 0.1 < Ωχ h2 < 0.3 alone. For any given value of m1/2 , only a restricted range of m0 is allowed. Therefore, only a limited range of m0 , and hence only a limited range for the cross section, is allowed for any given value of mχ . The thicknesses of the allowed regions are due in part to the assumed uncertainties in the nuclear inputs. These have been discussed at length in 143,142. On the other hand, a broad range of mχ is allowed, when one takes into account the coannihilation ‘tail’ region at each tan β and the rapid-annihilation ‘funnel’ regions for tan β = 35, 50. The dashed (black) line displays the range allowed by the b → sγ constraint alone. In this case, a broader range of m0 and hence the spin-independent cross section is possible for any given value of mχ . The impact of the constraint due to mh is shown by the dot-dashed (green) line. Comparing with the previous constraints, we see that a region at low mχ is excluded by mh , strengthening significantly the previous upper limit on the spin-independent cross section. Finally, the dotted (red) lines in Fig. 18 show the impact of the gµ −2 constraint. This imposes an upper bound on m1/2 and hence mχ , and correspondingly a lower limit on the spin-independent cross section.

45

10 10 10 10 10 10

-40

-41

σ χ-p [cm2]

-42

-43

-44

-45

10

1

10 mχ [GeV]

2

10

3

Figure 19. Limits from the CDMS145 and Edelweiss146 experiments on the neutralinoproton elastic scattering cross section as a function of the neutralino mass. The Edelweiss limit is stronger at higher mχ . These results nearly exclude the shaded region observed by DAMA147 . The theoretical predictions lie at lower values of the cross section.

This analysis is extended in panel (b) of Fig. 18 to all the values 8 < tan β ≤ 55 and we find overall that 141 2 × 10−10 pb < σSI ∼ 2 × 10−7 pb < σSD ∼ < 6 × 10−8 pb, ∼ < 10−5 pb, ∼ (60) (61)

for µ > 0. (σSD is the spin-dependent cross-section not shown in the figures presented here.) As we see in panel (b) of Fig. 18, mh provides the most important upper limit on the cross sections for tan β < 23, and b → sγ for larger tan β, with gµ − 2 always providing a more stringent lower limit than the relic-density constraint. The relic density constraint shown is evaluated at the endpoint of the coannihilation region. At large tan β, the Higgs funnels or the focus-point regions have not been considered, as their locations are very sensitive to input parameters and calculational details144. The results from a CMSSM and MSSM analysis142,143 for tan β = 3

46

and 10 are compared with the most recent CDMS145 and Edelweiss146 bounds in Fig. 19. These results have nearly entirely excluded the region purported by the DAMA147 experiment. The CMSSM prediction142 is shown by the dark shaded region, while the NUHM case143 is shown by the larger lighter shaded region. Other CMSSM results148 are also available. I conclude by showing the prospects for direct detection for the benchmark points discussed above149 . Fig. 20 shows rates for the elastic spinindependent scattering of supersymmetric relics, including the projected sensitivities for CDMS II150 and CRESST151 (solid) and GENIUS152 (dashed). Also shown are the cross sections calculated in the proposed benchmark scenarios discussed in the previous section, which are considerably below the DAMA147 range (10−5 − 10−6 pb). Indirect searches for supersymmetric dark matter via the products of annihilations in the galactic halo or inside the Sun also have prospects in some of the benchmark scenarios149.

Figure 20. Elastic spin-independent scattering of supersymmetric relics on protons calculated in benchmark scenarios149 , compared with the projected sensitivities for CDMS II 150 and CRESST151 (solid) and GENIUS152 (dashed). The predictions of our code (blue crosses) and Neutdriver153 (red circles) for neutralino-nucleon scattering are compared. The labels A, B, ...,L correspond to the benchmark points as shown in Fig. 12.

47

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