2. POPULAR CANDIDATES FOR NON-BARYONIC DARK MATTER

A new kind of elementary particle has been the dominant (exclusive?) candidate for non-baryonic dark matter.

A major classification of non-baryonic dark matter is based on its temperature at the time of galaxy formation, which occurs at a photon temperature of about 1 keV. Hot dark matter was relativistic at the time of galaxy formation, and as a consequence hindered the formation of the smallest objects by streaming out of the forming structures. An example of a hot dark matter particle is a light neutrino, much lighter than ~keV. Cold dark matter was non-relativistic when galaxies formed, and thus was able to collapse effectively under the action of gravity because of its negligible pressure. Examples of cold dark matter particles are neutralinos, axions, WIMPZILLAs, solitons (B-balls and Q-balls), etc. Warm dark matter was semi-relativistic at the time of galaxy formation, and is therefore an intermediate case between hot and cold dark matter. Two examples of warm dark matter are keV-mass sterile neutrinos and gravitinos.

Another important classification of particle dark matter rests upon its production mechanism. Particles that were in thermal equilibrium in the early Universe, like neutrinos, neutralinos, and most other WIMPs (weakly interacting massive particles), are called thermal relics. Particles which were produced by a non-thermal mechanism and that never had the chance of reaching thermal equilibrium in the early Universe are called non-thermal relics. There are several examples of non-thermal relics: axions emitted by cosmic strings, solitons produced in phase transitions, WIMPZILLAs produced gravitationally at the end of inflation, etc.

For the sake of presentation, we find still another classification useful. We will divide candidates for particle dark matter into three categories: Type Ia, Type Ib, and Type II (following a common practice in superconductors and supernovas). Type Ia candidates are those known to exist, foremost among them are the neutrinos. Type Ib candidates are candidates which are still undiscovered but are `well-motivated.' By this we mean that (1) they have been proposed to solve genuine particle physics problems, a priori unrelated to dark matter, and (2) they have interactions and masses specified within a well-defined (and consistent) particle physics model. We are aware of the arbitrariness of this classification, and reserve the honor of belonging to the Type Ib category only to a sterile neutrino, the axion, and the lightest supersymmetric particle (which may be a neutralino, a gravitino, or a sneutrino) Finally, Type II candidates are all other candidates, some of which are examples of maybe fruitful ideas, such as WIMPZILLAs, solitons (B-balls, Q-balls), dark matter from extra-dimensions, self-interacting dark matter, string-inspired dark matter, string-perspired dark matter, etc. It goes without saying that a candidate may move up from Type II to Type Ib and even to Type Ia as our understanding of particle physics models progresses.

We now examine some of the current candidates.

2.1. Type Ia: candidates that exist

Dark matter candidates that are known to exist in Nature have an obvious advantage over candidates that have not been detected. The chief particles in this category are the neutrinos.

There are three known `flavors' of neutrinos: the electron neutrino _e, the muon neutrino _µ, and the tau neutrino . They are so named because they are produced or destroyed in concomitance with the electron, the muon, and the tau lepton, respectively.

"If neutrinos had a mass, they would be a good candidate for the dark matter," said Steven Hawking. We now know that neutrinos, or at least some of the neutrinos, do have a mass. This was discovered indirectly through the observation of neutrino flavor oscillations, i.e. the spontaneous conversion of one neutrino flavor into another as a neutrino propagates from point to point. The connection between flavor oscillations and neutrino masses can be seen as follows.

Consider for simplicity two flavors of neutrinos instead of three, _e and _µ, say. Weak interactions produce the flavor eigenstates |_e> and |_µ>, which are associated with their respective charged leptons. However, these flavor eigenstates are not energy eigenstates. Let |₁> and |₂> denote the two energy eigenstates for the two-flavor system, with energies E₁ and E₂ respectively. Then the flavor and the energy eigenstates are connected by a unitary transformation,

(5)

Imagine that at time t = 0 we produce a _e, so that the initial wave function is

(6)

After a time t, the wave function evolves according to the system Hamiltonian as (we use natural units = c = 1)

(7)

To see the evolution explicitly, we expand |_e> into energy eigenstates, and obtain

(8)

We can now ask what is the probability of observing the neutrino in the state |_µ> after a time t, i.e. of observing a neutrino with flavor _µ instead of the initial _e. According to standard rules of quantum mechanics, this probability is

(9) (10) (11) (12)

where we have used Eq. (5) and <₁ | ₁> = <₂ | ₂> = 1, <₁ | ₂> = 0. For free relativistic neutrinos, with momentum p much larger than their mass m, we have

(13)

and

(14)

Hence

(15)

This equation shows that the probability of conversion from flavor _e to flavor _µ oscillates in time with a frequency proportional to the difference of the squares of the neutrino masses m₁₂² = m₂² - m₁². The observation of neutrino flavor oscillations therefore implies that neutrino masses differ from each other, and in particular that at least one of them is different from zero.

Neutrino oscillations have up to now been detected in two systems. Atmospheric muon neutrinos, which originate from the collision of cosmic rays with the Earth atmosphere, have been observed to oscillate into tau neutrinos (Fukuda et al., 1998),

(16)

Solar neutrinos, produced in the nuclear reactions that make the Sun shine, also show oscillations (Ahmad et al., 2002),

(17)

These results can be used to set a lower limit on the mass of the heaviest neutrino. Indeed, the mass of the heaviest neutrino must be greater than or equal to the square root of the largest mass-squared difference (just take the mass of the other neutrino to vanish). This gives the lower limit

(18)

Upper limits on neutrino masses come from laboratory experiments, such as tritium decay and high-energy accelerator experiments, and are (see Review of Particle Physics, Hagiwara et al., 2002)

(19)

However, the small mass differences implied by Eqs. (16) and (17) imply that the smallest of the three upper limits applies to all three active neutrino masses. Thus we have

(20)

It follows from this mass constraint that reactions such as _{e e} e⁺e^- in the hot early universe were able to keep standard-model neutrinos in thermal equilibrium. The neutrino density then follows from a computation of the neutrino number density (see Section 2.2). The result is

(21)

where g_i = 1 for a neutrino which is its own antiparticle (Majorana neutrino) and g_i = 2 for a neutrino which is not its own antiparticle (Dirac neutrino).

We already mentioned in Section 1 that cosmology provides an upper limit on the neutrino density h². This translates into a cosmological upper limit on the neutrino mass using Eq. (21). The cosmological limit is strictly speaking on the mass density in relativistic particles at the time of galaxy formation. An excessive amount of relativistic particles when galaxies form, i.e. of particles with mass m << keV, would erase too much structure at the smallest scales. A combination of cosmic microwave background measurements, galaxy clustering measurements, and observations of the Lyman- forest gives the upper limit quoted before (Spergel et al., 2003)

(22)

Eq. (21) then gives

(23)

On the other hand, Eq. (21) can be used in conjunction with inequality (18) to obtain a lower bound on the cosmological density in neutrinos. Taking only one massive Majorana flavor,

(24)

Thus neutrinos are definitely a form of dark matter, although perhaps a minor component of it.

The results for the known neutrinos as dark matter can be summarized by the constraints

(25) (26)

where the constraint in Eq. (23) has somewhat been relaxed by taking g_i = 1.

The upper limit on h² forbids currently known neutrinos from being the major constituents of dark matter. Moreover, since they are light and relativistic at the time of galaxy formation, the three neutrinos known to exist are hot, not cold, dark matter.

The three active neutrinos are our only known particle candidates for non-baryonic dark matter. Since they fail to be cold dark matter, we are lead to consider hypothetical particles.

2.2. Type Ib: `well-motivated' candidates

We will discuss three cold dark matter candidates which are `well-motivated', i.e. that have been proposed to solve problems in principle unrelated to dark matter and whose properties can be computed within a well-defined particle physics model. The three candidates we discuss are: (1) a heavy active neutrino with standard model interactions, (2) the neutralino in the minimal supersymmetric standard model, and (3) the axion. Examples of other candidates that can be included in this category are a sterile neutrino (see, e.g., Abazajian, Fuller, & Patel, 2001) and other supersymmetric particles such as the gravitino (see, e.g., Ellis et al., 1984) and the sneutrino (see, e.g., Hall, Moroi, & Murayama, 1998).

The first two candidates we discuss belong to a general class called weakly interacting massive particles (WIMPs). ² WIMPs that were in thermal equilibrium in the early universe (thermal WIMPs) are particularly interesting. Their cosmological density is naturally of the right order of magnitude when their interaction cross section is of the order of a weak cross section. This also makes them detectable in the laboratory, as we will see later. In the early Universe, annihilation reactions that convert WIMPs into standard model particles were initially in equilibrium with their opposite reactions. As the universe expanded, and the temperature became smaller than the WIMP mass, the gas of WIMPs, still in equilibrium, diluted faster than the gas of standard model particles. This occurred because the equilibrium number density of non-relativistic particles is suppressed by a Boltzmann factor e^-m/T with respect to the number density of relativistic particles. After a while, WIMPs became so rare that the WIMP annihilation reactions could no longer occur (chemical decoupling), and from then on the number density of WIMPs decreased inversely with volume (or in other words, the number of WIMPs per comoving volume remained constant). Chemical decoupling occurs approximately when the WIMP annihilation rate _ann = <_ann v>n became smaller than the universe expansion rate H. Here _ann is the WIMP annihilation cross section, v is the relative velocity of the annihilating WIMPs, n is the WIMP number density, and the angle brackets denote an average over the WIMP thermal distribution. Using Friedmann's equation to find the expansion rate H gives

(27)

for the relic density of a thermal WIMP. An important property of this equation is that smaller annihilation cross sections correspond to larger relic densities ("The weakest wins.") This can be understood from the fact that WIMPs with stronger interactions remain in chemical equilibrium for a longer time, and hence decouple when the universe is colder, wherefore their density is further suppressed by a smaller Boltzmann factor. Figure 2 illustrates this relationship.

Figure 2. Evolution of a typical WIMP number density in the early universe. The number of WIMPs in a volume expanding with the universe (comoving density) first decreases exponentially due the Boltzmann factor e-m/T and then `freezes out' to a constant value when the WIMP annihilation reactions cannot maintain chemical equilibrium between WIMPs and standard model particles. In the figure, < v> is the thermally averaged annihilation cross section times relative velocity. WIMPs with larger annihilation cross section end up with smaller densities.

Heavy neutrino

The WIMP par excellence is a heavy neutrino. The example we consider is a thermal Dirac neutrino of the fourth generation with Standard Model interactions and no lepton asymmetry. Figure 3 summarizes its relic density as a function of mass. Also shown in the Figure are the current constraints from accelerator experiments and dark matter searches.

Figure 3. Neutrinos as dark matter. Relic density of a thermal Dirac neutrino with standard-model interactions, together with current constraints from cosmology, accelerators (LEP), and dark matter searches. See text for explanations. (The `dark matter' band is quite generous in light of the WMAP measurements.)

A neutrino heavier than ~ 1 MeV decouples while non-relativistic. Its relic density is determined by its annihilation cross section, as for a general WIMP (see Eq. (27)). The shape of the relic density curve in Figure 3 is a reflection of the behavior of the annihilation cross section. The latter is dominated by the Z-boson resonance at m m_Z / 2. This resonant annihilation gives the characteristic V shape to the relic density curve. Above m ~ 100 GeV, new annihilation channels open up, namely the annihilation of two neutrinos into two Z- or W-bosons. The new channels increase the annihilation cross section and thus lower the neutrino relic density. Soon, however, the perturbative expansion of the cross section in powers of the (Yukawa) coupling constant becomes untrustworthy (the question mark in Figure 3). An alternative unitarity argument limits the Dirac neutrino relic density to the dashed curve on the right in the Figure. Neutrinos heavier than 10 TeV `overclose' the universe, i.e. have a relic density that corresponds to a universe which is too young.

The `dark matter' band in Figure 3 indicates where the neutrino is a good dark matter candidate (the band is actually quite generous in light of the most recent measurements of h²). A thermal Dirac neutrino is a good dark matter candidate when its mass is around few eV, a few GeV or possibly a TeV. For masses smaller than about an eV and between ~ 10 GeV and ~ 100 GeV, it is an underabundant relic from the Big Bang, too dilute to be a major component of the dark matter but nevertheless a cosmological relic. For other masses, it is cosmologically excluded.

Dark matter neutrinos with a mass around 1 eV would be relativistic at the time of galaxy formation (~ keV), and would thus be part of hot dark matter. From the bounds on hot dark matter in the preceding Section, however, they cannot be a major component of the dark matter in the Universe.

Neutrinos can be cold dark matter if their masses are around few GeV or a TeV. However, fourth-generation heavy neutrinos lighter than 45 GeV are excluded by the measurement of the Z-boson decay width at the Large Electron-Positron collider at CERN. Moreover, direct searches for WIMP dark matter in our galaxy exclude Dirac neutrinos heavier than ~ 0.5 GeV as the dominant component of the galactic dark halo (see Figure 3). Thus although heavy Dirac neutrinos could still be a tiny part of the halo dark matter, they cannot solve the cold dark matter problem.

We need another non-baryonic candidate for cold dark matter.

Neutralino

The WIMP par default is the lightest neutralino ₁⁰, or sometimes simply , which is often the lightest supersymmetric particle in supersymmetric extensions of the Standard Model of particle physics. Supersymmetry is a new symmetry of space-time that has been discovered in the process of unifying the fundamental forces of nature (electroweak, strong, and gravitational). Supersymmetry also helps in stabilizing the masses of fundamental scalar particles in the theory, such as the Higgs boson, a problem, called the hierarchy problem, which basically consists in explaining why gravity is so much weaker than the other forces.

Of importance for cosmology is the fact that supersymmetry requires the existence of a new particle for each particle in the Standard Model. These supersymmetric partners differ by half a unit of spin, and come under the names of sleptons (partners of the leptons), squarks (partners of the quarks), gauginos (partners of the gauge bosons) and higgsinos (partners of the Higgs bosons). Sleptons and squarks have spin 0, and gauginos and higgsinos have spin 1/2.

If supersymmetry would be an explicit symmetry of nature, superpartners would have the same mass as their corresponding Standard Model particle. However, no Standard Model particle has a superpartner of the same mass. It is therefore assumed that supersymmetry, much as the weak symmetry, is broken. Superpartners can then be much heavier than their normal counterparts, explaining why they have not been detected so far. However, the mechanism of supersymmetry breaking is not completely understood, and in practice it is implemented in the model by a set of supersymmetry-breaking parameters that govern the values of the superpartners masses (the superpartners couplings are fixed by supersymmetry).

The scenario with the minimum number of particles is called the minimal supersymmetric standard model or MSSM. The MSSM has 106 parameters beyond those in the Standard Model: 102 supersymmetry-breaking parameters, 1 complex supersymmetric parameter µ, and 1 complex electroweak symmetry-breaking parameter tan (see, e.g., the article by Haber in Review of Particle Physics, Hagiwara et al., 2002). Since it is cumbersome to work with so many parameters, in practice phenomenological studies consider simplified scenarios with a drastically reduced number of parameters. The most studied case (not necessarily the one Nature has chosen) is minimal supergravity, which reduces the number of parameters to five: three real mass parameters at the Grand Unification scale (the scalar mass m₀, the scalar trilinear coupling A₀, and the gaugino mass m_1/2) and two real parameters at the weak scale (the ratio of Higgs expectation values tan and the sign of the µ parameter). Other scenarios are possible and are considered in the literature. Of relevance to dark matter studies is, for example, a class of models with seven parameters specified at the weak scale: µ, tan, the gaugino mass parameter M₂, the mass m_A of the CP-odd Higgs boson, the sfermion mass parameter , the bottom and top quark trilinear couplings A_b and A_t. See the reviews by Jungman, Kamionkowski & Griest (1996) and Bergström (2000) for more details.

It was realized long ago by Goldberg (1983) and Ellis et al. (1984) that the lightest superposition of the neutral gauginos and the neutral higgsinos (which having the same quantum numbers mix together) is an excellent dark matter candidate. It is often the lightest supersymmetric particle, it is stable under the requirement that superpartners are only produced or destroyed in pairs (called R-parity conservation), it is weakly interacting, as dictated by supersymmetry, and it is massive. It is therefore a genuine WIMP, and it is among the most studied of the dark matter candidates. Its name is the lightest neutralino.

Several calculations exist of the density of the lightest neutralino. An example is given in Figure 4, which reproduces a figure from Edsjö & Gondolo (1997), updated with the WMAP value of the cold dark matter density. This figure was obtained in a scenario with seven supersymmetric parameters at the weak scale. The relic density is not fixed once the neutralino mass is given, as was the case for a Dirac neutrino, because the neutralino annihilation cross section depends on the masses and composition of many other supersymmetric particles, thus ultimately on all supersymmetric parameters. Therefore the density in Figure 4 is not a single-valued function of the neutralino mass, and the plot must be obtained through an extended computer scan in the seven-dimensional parameter space. ³ It is clear from Figure 4 that it is possible to choose the values of the supersymmetric parameters in a way that the neutralino relic density satisfies the current determination of h². Although it may seem ridiculous to claim that the neutralino is naturally a good dark matter candidate, let us notice that the neutralino relic density in Figure 4 and the neutrino relic density in Figure 3 have a similar range of variation. It is the precision of the cosmological measurements that make us think otherwise.

Figure 4. Relic density of the lightest neutralino as a function of its mass. For each mass, several density values are possible depending on the other supersymmetric parameters (seven in total in the scenario plotted). The color code shows the neutralino composition (gaugino, higgsino or mixed). The gray horizontal line is the current error band in the WMAP measurement of the cosmological cold dark matter density. (Figure adapted from Edsjö & Gondolo, 1997.)

Reversing the argument, the precision of the cosmological measurements can be used to select the regions of supersymmetric parameter space where the lightest neutralino is cold dark matter. With the current precision of cosmological measurements, these are very thin regions in supersymmetric parameter space. For this approach to be carried out properly, the theoretical calculation of the neutralino relic density should match the precision of the cosmological data. The latter is currently 7%, as can be gathered from Eq. (4), and is expected to improve to about 1% before the end of the decade with the launch of the Planck mission. A calculation of the neutralino relic density good to 1% now exists, and is available in a computer package called DarkSUSY (Gondolo et al., 2002, 2000). The 1% precision refers to the calculation of the relic density starting from the supersymmetric parameters at the weak scale. The connection with the parameters at the Grand Unification scale, which is vital for minimal supergravity, introduces instead large errors in important regions of parameter space, errors that some authors estimate to be as big as 50% (Allanach, Kraml, & Porod, 2003).

Given the importance of this calculation, we give now a rapid survey of the ingredients needed to achieve a precision of 1%. Firstly, the equation governing the evolution of the neutralino number density n,

(28)

has to be solved numerically. The equation being `stiff' (i.e. its difference equation being stable only for unreasonably small stepsizes), a special numerical method should be used. The thermal average of the annihilation cross section <_ann v> at temperature T should be computed relativistically, since the typical speed of neutralinos at decoupling is of the order of the speed of light, v ~ c / 3. For this purpose, we can use the expression in Gondolo & Gelmini (1991):

(29)

where W(p) is the annihilation rate per unit volume and unit time (a relativistic invariant), s = 4(m² + p²) is the center-of-mass energy squared, and K₁, K₂ are modified Bessel functions. In passing, let us remark that the common method of expanding in powers of v², _ann v = a + bv² + ^... , and then taking the thermal average to give <_ann v> = a + b(3T / 2m) + ^... is unreliable, since it gives rise to negative <_ann v>, and thus negative h², near resonances and thresholds. These are nowadays the most important regions of parameter space. Finally, an essential ingredient in the calculation of the neutralino relic density is the inclusion of coannihilation processes. These are processes that deplete the number of neutralinos through a chain of reactions, and occur when another supersymmetric particle is close in mass to the lightest neutralino (m ~ T). In this case, scattering of the neutralino off a particle in the thermal `soup' can convert the neutralino into the other supersymmetric particle close in mass, given that the energy barrier that would otherwise have prevented it (i.e. the mass difference) is easily overcome. The supersymmetric particle participating in the coannihilation may then decay and/or react with other particles and eventually effect the disappearance of neutralinos. We give two examples. Coannihilation with charginos ^± (partners of the charged gauge and Higgs bosons) may proceed via, for instance,

(30)

(subscripts on superpartner names indicate particles with different masses). Coannihilation with tau sleptons may instead involve the processes

(31)

Coannihilations were first included in the study of near-degenerate heavy neutrinos by Binetruy, Girardi, & Salati (1984) and were brought to general attention by Griest & Seckel (1991). The current state-of-the-art treatment of neutralino coannihilations, which involves several thousands of processes, is contained in the work by Edsjö & Gondolo (1997) and Edsjö, Schelke, Ullio, & Gondolo (2003).

To illustrate the power of the cosmological precision measurements in selecting regions of supersymmetric parameter space, Figure 5 shows the WMAP constraint in one of the figures in Edsjö, Schelke, Ullio, & Gondolo (2003). The constraint is as a very thin line that approximately follows the edges of the allowed region in this slice of parameter space (the m₀-m_1/2 plane with µ < 0, tan = 30 and A₀ = 0).

Figure 5. Illustration of the power of WMAP constraints on the minimal supergravity parameter space. The figure shows a slice in m0 and m1/2 with µ < 0 at tan = 30 and A0 = 0. The WMAP constraint is a very thin (grey) line t hat approximately follows the edges of the allowed region. (Figure adapted from Edsjö, Schelke, Ullio, & Gondolo, 2003.)

Axion

Our third and last example of a `well-motivated' cold dark matter candidate is the axion.

Axions were suggested by Peccei & Quinn (1977) to solve the so-called "strong CP problem". Out of the vacuum structure of Quantum Chromodynamics there arises a large CP-violating phase, which is at variance with stringent measurements of the electric dipole moment of the neutron, for example. A possible solution to this problem is that the CP-violating phase is the vacuum expectation value of a new field, the axion, which relaxes dynamically to a very small value. The original axion model of Peccei and Quinn is today experimentally ruled out, but other axion models based on the same idea have been proposed. Among them are the invisible axions of Kim (1979) and Shifman, Vainshtein, & Zakharov (1980) (KSVZ axion) and of Dine, Fischler, & Srednicki (1981) and Zhitnitsky (1980) (DFSZ). They differ in the strength of the axion couplings to matter and radiation.

In a cosmological context, axions, contrary to neutrinos and neutralinos, are generally produced non-thermally (although thermal axion production is sometimes considered, as are non-thermal neutrino and neutralino productions). The two main mechanisms for non-thermal axion production are vacuum alignment and emission from cosmic strings. In the vacuum alignment mechanism, a potential is generated for the axion field at the chiral symmetry breaking, and the axion field, which can in principle be at any point in this potential, starts moving toward the minimum of the potential and then oscillates around it. Quantum-mechanically, the field oscillations correspond to the generation of axion particles. In the other main non-thermal mechanism for axion production, axions are emitted in the wiggling or decay of cosmic strings. In both cases, axions are produced with small momentum, << keV, and thus they are cold dark matter despite having tiny masses, between 1 µeV and 1 meV. This is in fact the range of masses in which axions are good dark matter candidates. Figure 6 shows the current constraints on the axion mass from laboratory, astrophysical, and cosmological data.

Figure 6. Laboratory, astrophysical, and cosmological constraints on the axion mass mA. The inflation scenario and the string scenario are referred in the text as the vacuum alignment scenario and the string emission scenario, respectively. fA is the axion decay constant, which is inversely related to mA. The axion is a good dark matter candidate for 1 µeV mA 1 meV. (Figure from Raffelt in Review of Particle Physics, Hagiwara et al., 2002.)

Searches for axions as galactic dark matter rely on the coupling of axions to two photons. An incoming galactic axion can become a photon in the magnetic field in a resonant cavity. For this to happen, the characteristic frequency of the cavity has to match the axion mass. Since the latter is unknown, searches for galactic axions use tunable cavities, and scan over the cavity frequency, a time-consuming process. The U.S. axion search at Livermore is currently exploring a wide range of interesting axion masses, and has put some constraint on the KSVZ axion as a dominant component of the galactic halo (Asztalos et al., 2004). Figure 7 shows the constraints on the local galactic density in axions as a function of the axion mass. KSVZ axions with mass in the range 1.91-3.34 µeV cannot be the main component of galactic dark matter. The Livermore search is still continuing to a larger range of axion masses. It is fair to say that axion dark matter is either about to be detected or about to be ruled out.

Figure 7. Experimental constraints on the density of axions in the galactic halo near the Sun as a function of the axion mass (upper scale) and cavity frequency (lower scale). The regions above the curves marked `DFSZ' and `KSVZ' are excluded fro the respective axion models. The currently accepted value for the local dark halo density is 0.45 GeV/cm3, which is approximately the extension of the excluded region for the KSVZ axion. (Figure from Asztalos et al., 2004.)

2.3. Type II: other candidates

In the Type II category we put all hypothetical cold dark matter candidates that are neither Type Ia nor Type Ib. Some of these candidates have been proposed for no other reason than to solve the dark matter problem. Others are examples of beautiful ideas and clever mechanisms that can provide good possibilities for non-baryonic dark matter, but in some way or another lack the completeness of the theoretical particle physics models of Types Ia and Ib. Although Type II candidates are not studied as deeply as others, it may well be that eventually the question of the nature of cold dark matter might find its answer among them.

Below we present the idea of self-interacting dark matter and gravitationally-produced WIMPZILLAs. Other interesting candidates have been proposed recently in models with extra dimensions, such as Kaluza-Klein dark matter (Cheng, Feng, & Matchev, 2002) and branons (Cembranos, Dobado, & Maroto, 2003).

There are several ways in which one may be able to come up with an ad hoc candidate for non-baryonic cold dark matter. A humorous flowchart on how to do this was put together around 1986 by a group of graduate students at Princeton (Lauer, Statler, Ryden, & Weinberg, 1986). The flowchart involves multiple options, and one possibility runs as follows. "A new particle is envisioned which is cooked up just to make everything OK but violates federal law ...still, it doesn't prevent a paper being written by Spergel ..." We have now the proper setting to introduce a candidate suggested by Spergel & Steinhardt (2000), self-interacting dark matter.

Self-interacting dark matter

The idea behind the introduction of self-interacting dark matter is to find a solution to the cusp and satellite problems of standard cold dark matter scenarios. These two problems are in effect discrepancies between observations and results of simulations of structure formation on the galactic scale. Namely, in the cusp problem, numerical simulations predict a dark matter density profile which increases toward the center of a galaxy like a power law r^- with ~ 1 or higher. This sharp density increase is called a cusp. On the other hand, kinematical and dynamical determinations of the dark matter profile in the central regions of galaxies, especially of low surface brightness galaxies, tend not to show such a sharp increase but rather a constant density core. The observational situation is still rather confused, with some galaxies profiles being compatible with a cusp and others with a core. The theoretical situation is also not fully delineated, with higher resolution simulations showing a dependence of the slope on the mass of the galaxy. Although many ideas have been proposed for the resolution of the cusp problem, it is still not completely resolved.

Perhaps connected with the cusp problem is the satellite problem, which is a mismatch between the observed and the simulated numbers of satellites in a galaxy halo. Too many satellites are predicted by the simulations. In reality, observations can detect only the luminous satellites while simulations contain all satellites, including the dark ones. It may be that many dark satellites do not shine, thus solving the satellite problem, but how and which satellites become luminous is not understood yet.

Spergel & Steinhardt (2000) suggested another way to solve both problems. They realized that if dark matter particles would interact with each other with a mean free path of the order of the size of galactic cores, the dark matter interactions would efficiently thermalize the system and avoid the formation of both a central cusp and too many satellites. The requirement on the mean free path is roughly ~ 10 kpc. We can figure out the necessary cross section for dark matter self-interactions by recalling that the mean free path is related to the cross section and the number density n, or matter density = mn, through the relationship = 1 / (n) = m / ( ). Taking a typical ~ 0.3 GeV/cm³ gives / m ~ 60 cm²/g. So Spergel & Steinhardt (2000) suggested a new self-interacting dark matter particle with / m in the range

(32)

To understand the magnitude of this number, it is useful to compare it with the geometric cross section of a proton, which is one of the largest known cross sections for elementary particles. We have

(33)

Thus the desired / m seems rather big. It is therefore not surprising that astrophysical constraints on self-interacting dark matter are rather stringent. Gnedin & Ostriker (2001) considered the evaporation of halos inside clusters and set the constraint / m < 0.3-1 cm²/g. Yoshida, Springel, White, & Tormen (2000) considered the shape of cluster cores, which is rounder for self-interacting dark matter than for standard cold dark matter, and concluded that / m must be < 10 cm²/g. This bound was later strengthened by Miralda-Escudé (2002) to / m < 0.02 cm²/g. Markevitch et al. (2003) discovered a gas bullet lagging behind dark matter in the merging galaxy cluster 1E0657-56. They combined Chandra X-ray maps of the hot gas in the cluster with weak lensing maps of its mass distribution. From estimates of the column mass densities and of the distance between the gas and the dark matter, Markevitch et al. were able to set the upper limit / m < 10 cm²/g with direct observations of the dark matter distribution. All these bounds leave little room, if any, to self-interacting dark matter a la Spergel & Steinhardt.

WIMPZILLAs

Our last example of cold dark matter candidates illustrates a fascinating idea for generating matter in the expanding universe: the gravitational creation of matter in an accelerated expansion. This mechanism is analogous to the production of Hawking radiation around a black hole, and of Unruh radiation in an accelerated reference frame.

WIMPZILLAs (Chung, Kolb, & Riotto, 1998, 1999; Kuzmin & Tkachev, 1998) are very massive relics from the Big Bang, which can be the dark matter in the universe if their mass is 10¹³ GeV. They were produced at the end of inflation through a variety of possible mechanisms: gravitationally, during preheating, during reheating, in bubble collisions. It is possible that their relic abundance does not depend on their interaction strength but only on their mass, giving great freedom in their phenomenology. To be the dark matter today, they are assumed to be stable or to have a lifetime of the order of the age of the universe. In the latter case, their decay products may give rise to the highest energy cosmic rays, and solve the problem of cosmic rays beyond the GZK cutoff.

Gravitational production of particles is an important phenomenon that is worth describing here. Consider a scalar field (particle) X of mass M_X in the expanding universe. Let be the conformal time and a() the time dependence of the expansion scale factor. Assume for simplicity that the universe is flat. The scalar field X can be expanded in spatial Fourier modes as

(34)

Here a_k and a_k^† are creation and annihilation operators, and h_k() are mode functions that satisfy (a) the normalization condition h_k h_k'^* - h'_k h_k^* = i (a prime indicates a derivative with respect to conformal time), and (b) the mode equation

(35)

where

(36)

The parameter is = 0 for a minimally-coupled field and = 1/6 for a conformally-coupled field. The mode equation, Eq. (35), is formally the same as the equation of motion of a harmonic oscillator with time-varying frequency _k(). For a given positive-frequency solution h_k(), the vacuum |0_h> of the field X, i.e. the state with no X particles, is defined as the state that satisfies a_k| 0_h> = 0 for all k. Since Eq. (35) is a second order equation and the frequency depends on time, the normalization condition is in general not sufficient to specify the positive-frequency modes uniquely, contrary to the case of constant frequency ₀ for which h_k⁰() = e^-i₀ / (2₀)^1/2. Different boundary conditions for the solutions h_k() define in general different creation and annihilation operators a_k and a_k†, and thus in general different vacua. ⁴ For example, solutions which satisfy the condition of having only positive-frequencies in the distant past,

(37)

contain both positive and negative frequencies in the distant future,

(38)

Here _k^± = lim _{± k}(). As a consequence, an initial vacuum state is no longer a vacuum state at later times, i.e. particles are created. The number density of particles is given in terms of the Bogolubov coefficient _k in Eq. (38) by

(39)

These ideas have been applied to gravitational particle creation at the end of inflation by Chung, Kolb, & Riotto (1998) and Kuzmin & Tkachev (1998). Particles with masses M_X of the order of the Hubble parameter at the end of inflation, H_I 10^-6 M_Pl 10¹³ GeV, may have been created with a density which today may be comparable to the critical density. Figure 8 shows the relic density h² of these WIMPZILLAs as a function of their mass M_X in units of H_I. Curves are shown for inflation models that have a smooth transition to a radiation dominated epoch (dashed line) and a matter dominated epoch (solid line). The third curve (dotted line) shows the thermal particle density at temperature T = H_I / 2. Also shown in the figure is the region where WIMPZILLAs are thermal relics. It is clear that it is possible for dark matter to be in the form of heavy WIMPZILLAs generated gravitationally at the end of inflation.

Figure 8. Relic density of gravitationally-produced WIMPZILLAs as a function of their mass MX. HI is the Hubble parameter at the end of inflation, Trh is the reheating temperature, and Mpl 3 × 1019 GeV is the Planck mass. The dashed and solid lines correspond to inflationary models that smoothly end into a radiation or matter dominated epoch, respectively. The dotted line is a thermal distribution at the temperature indicated. Outside the `thermalization region' WIMPZILLAs cannot reach thermal equilibrium. (Figure from Chung, Kolb, & Riotto, 1998.)

² Notice that according to the Merriam-Webster Dictionary of the English Language, a wimp is a weak, cowardly, or ineffectual person. Back.

³ We may ask if there can be points in the empty regions, or in more general terms, what is the meaning of the density of points in Figure 4, and in similar figures in Section 3. For a discussion on this, see Bergström & Gondolo (1996). Back.

⁴ The precise definition of a vacuum in a curved space-time is still subject to some ambiguities. We refer the interested reader to Fulling (1989); Wald (1994); Fulling (1979); Birrell & Davis (1982) and to the discussion in Chung, Notari, & Riotto (2003) and references therein. Back.