Topological Dark Matter

Kibble mechanism drastically underestimates the production of topological defects, as confirmed recently in atomic and condensed matter systems. If non-thermally produced, they can be cosmological dark matter of mass 1-10 PeV. If thermalized, skyrmion of mass 1-10 TeV is also a viable dark matter candidate, whose decay may explain $e^\pm$ spectra in cosmic rays recently measured by PAMELA, FERMI, and HESS. Models that produce magnetic monopoles below the inflation scale, such as Pati-Salam unification, are excluded.

Topological defects are of common interest to condensed matter physics, atomic physics, astrophysics and cosmology, as well as algebraic topology [1]. When the symmetry group G spontaneously breaks down to its subgroup H, there are continuously connected ground states parametrized by the coset space G/H. The homotopy groups of the coset space then tell us what kinds of topological effects are possible. In most cases, non-trivial π d (G/H) implies the existence of (2 − d)-dimensional topological defect. If the coset space has disconnected pieces (π 0 (G/H) = 0), we expect domain walls. For multiply-connected space (π 1 (G/H) = 0), there are strings (vortices). If the boundary of space can map nontrivially to the coset space (π 2 (G/H) = 0), we expect pointlike defects such as magnetic monopoles. An exception to the rule is when the whole space is mapped non-trivially to the coset space (π 3 (G/H) = 0), where skyrmions are stabilized by non-renormalizable terms in the low-energy effective theory [3]. In this case, it is not the boundary condition that is topologically non-trivial, but the configuration in the bulk.
To estimate the initial abundance of defects produced by a phase transition in early universe, Kibble pointed out that the correlation length diverges at the critical temperature while the causality does not permit exchange of information beyond the horizon scale [4]. He therefore came up with a lower bound on the amount of defects, namely approximately one per horizon, called Kibble mechanism. Most of the literature uses this lower bound as the estimate of the abundance of topological defects from phase transitions in early universe. For point-like topological defects, one finds n T D /s ∼ (T c /M P l ) 3 . Therefore, only phase transitions close to the grand-unification scale produce abundance of topological defects worthy of consideration.
We point out that the Kibble-Zurek mechanism provides a substantially larger abundance of topological defects from phase transitions in early universe than the original estimate by Kibble. Therefore even phase transitions just above the TeV energy scale may produce interesting (or dangerous) amount of topological defects.
In particular, we discuss the possibility that point-like topological defects may be the cosmological dark matter, which is arguably one of the most pressing mysteries in cosmology, astrophysics, and particle physics [2] [39]. The dominant paradigm to explain the dark matter is the thermal relic of yet-undiscovered particle. Within this paradigm, we consider dark matter candidates below approximately 100 TeV in mass because of the unitarity bound [16]. Our main result in this Letter is that the natural range for topological dark matter, if non-thermally produced by a second-order phase transition, is O(1 ∼ 10) PeV, which obviously violate the unitarity limit. Note that a symmetry breaking at this energy scale in the hidden sector is of great interest in many attempts to understand the origin of hierarchy between the Planck and electroweak scales such as dynamical supersymmetry breaking, and extra dimensions. In addition, we also point out that skyrmions at the order 10 TeV, once thermalized, are also interesting dark matter candidates that are often ignored in the literature [17]. The existence of skyrmion solution is quite generic in models where Higgs serves as a pseudo Nambu Goldstone boson, which opens the new possibility to connect the origin of electroweak symmetry breaking and dark matter.
If the dark matter particles are produced thermally at temperatures higher than their mass, their initial abundance is the same as any other relativistic particle species. Then the final abundance is determined by their annihilation cross section, freeze-out temperature, and we used g * ≈ 100 and ℓ = 0 (S-wave). Assuming that only one partial wave J would contribute, the annihilation cross section is limited from above by [16] Combining Eqs. (1,2), we find m < 110 TeV assuming Swave annihilation and J = 0.
On the other hand, the Kibble-Zurek mechanism predicts a very different abundance of point-like topological defects. Throughout this paper, we assume second-order phase transition. The correlation length ξ and relaxation time τ diverge near the critical temperature which can be parametrized using the critical exponents ν and µ respectively where ǫ ≡ (T c − T )/T c characterizes the proximity to the critical temperature T c . The system is quenched when it passes through the critical temperature with a finite speed. It is characterized by the quenching rate τ Q ≡ (t − t c )/ǫ to the linear order around time t c when T = T c . During the quenching, there exists a particular time t * when the time remaining before the transition equals the equilibrium relaxation time |t * − t c | = τ (t * ). Beyond this point the system can no longer adjust fast enough to follow the changing temperature of the bath, and at time t * the fluctuation becomes frozen until a time |t * − t c | after the critical temperature is reached. It is easy to see that |ǫ(t * )| = (τ Q /τ 0 ) −1/(1+µ) . Therefore, the fluctuation does not get smoothed out beyond the correlation length [40] In radiation dominated universe, Assuming the free energy of the Landau-Ginzburg form In radiation dominated universe and hence the correlation is shorter than the horizon size by a factor ∼ (T c /M P l ) 2/3 , leading to a far larger number of defects than the original Kibble's estimate. For point-like defects (PD), we expect approximately one per ξ 3 . Assuming (g * ≈ 10 2 − 10 3 , λ ≈ 0.3-1), we find If we consider the quantum corrections to the system, the critical exponents µ and ν could be different from 1/2. Generally speaking, it is related with the anomalous dimension of the leading relevant operators in the of the Lagrangian of the scalars that triggers the symmetry breaking. Causality ξ ≤ cτ dictates ν ≤ µ and we will assume ν = µ below as the Hubble friction term for scalar field φ could be ignored in the vicinity of critical temperature indicated in footnote [40]. For typical quantum systems based on O(N )-symmetric φ 4 theory in three dimensions [41], the critical exponents are ν = 0.625 in binary liquid system (N = 1), ν = 0.672 in superfluid 4 He experiment (N = 2), and ν = 0.70 in EuO, EuS system (N = 3). As we can see, ν is quite close to 2/3 and does not vary very much for different N . By plugging in the same numbers as the classical case, we obtain We see that the magnetic monopoles are produced orders of magnitude more than the original Kibble's estimate and hence even models with phase transitions down to TeV scale are subject to serious constraints. For instance, Pati-Salam model [36] assumes the symmetry breaking SU (4) C × SU (2) L × SU (2) R → SU (3) C × SU (2) L × U (1) Y , and hence predicts magnetic monopoles. Once produced, the monopoles are stable and their number can only be reduced by annihilation of M −M pairs. The magnetic (hidden) monopoles will stay in kinetic equilibrium with the (hidden) thermal plasma of electrically charged particles. The long-range forces between M and light charged particles will dissipate the energy of M drifting towards a nearbyM , allowing capture and annihilation to occur. Preskill [19] found the annihilation was negligible assuming the initial abundance given by the Kibble mechanism. With the Kibble-Zurek mechanism, however, the annihilation must be considered for magnetic monopoles, and we must use Eq. (5) in [19], where B = (3/4π 2 )ζ(3) i (hq i /4π) 2 which sums over all spin states of relativistic charged particles and h = 2π/q is the magnetic coupling.
In Fig. 1, combing Eq. (8) and (9), we show how magnetic monopole density to entropy ratio depends on the critical exponent ν in 2nd order phase transition for different critical temperatures. It is clearly that the Parker limit [37] excludes the such monopoles assuming the phase transition below the unification scale unless the critical exponent ν is significantly above 1. However, this is not the case normally considered in relativistic field theories for phase transitions.
On the other hand, the point-like defects may be magnetic monopole under a U (1) gauge theory unrelated to electromagnetism ("hidden U (1)"). We assume that there is no corresponding hidden plasma to dissipate the energy of M and we ignore the annihilation. We can also ignore the annihilation for point-like defects based on global symmetries because there is no long-range force among them. For nonthermal production of topological defects to dominate, they have to be heavy enough so that they never stay in chemical equilibrium once produced. At the critical temperature, when x c ≡ m P D /T c > x f , which ranges from 20 to 30 for differ-   10) if it is non-thermally produced during a second order phase transition. The yellow horizontal band denotes the relic density 0.094 < Ωmh 2 < 0.129 preferred by WMAP data. We assume xc = mP D /Tc = 50. ent monopole coupling strengths and phase transition temperatures, the relics density could be derived from Eq. (8) as If we take x c = 50, the relic density is a function of T c , as shown in Fig. 2. In order to account for the cold dark matter abundance, we need T c ∼ O(1) PeV to in the classical case and T c ∼ O(10) PeV in the typical quantum cases [42]. What kind of model can lead to realistic topological dark matter? One obvious possibility is that there is a hidden non- abelian gauge theory whose breaking to U (1) produces magnetic monopoles. As long as the U (1) does not mix with QED, strong bounds such as Parker's limit [37] do not apply. Their annihilation cross section in the plasma is negligible. As an example, the vector-like model of dynamical supersymmetry breaking by Izawa-Yanagida [23] and Intriligator-Thomas [24] has SO(6) global symmetry. Gauging SO(3) subgroup embedded diagonally into SO(3) × SO(3) ⊂ SO (6), one can see that it breaks to SO(2) and produces magnetic monopoles. It is not clear if skyrmions can be created by the same Kibble-Zurek mechanism, as they are topologically nontrivial configurations in the bulk rather than the boundary conditions. However, skyrmions are baryonic composites in the underlying gauge theory and hence may be thermalized independent of the production mechanism. Note that many composite Higgs models (e.g., little Higgs theories) proposed in the literature can have skyrmions as topological solitons (see Table I [43]). Their masses are expected in the 10 TeV region because f π ≈ 1 TeV in these theories from the naturalness argument. Once thermally produced, the correct abundance of topological dark matter could be obtained with a relatively strong coupling g P D ∼ 3. Since the global symmetry G in those models in Table I is approximate, we may ask whether the skyrmion is metastable. Gauging a subgroup of G may induce the skyrmion to decay through instanton effects [27]. However, the enormous suppression factor proportional to exp(−8π 2 /g 2 ) will make its life time much longer than the age of our universe [28] [44].
Let us now comment on the consequence of topological dark matter on cosmic ray signals. For the case of skyrmion dark matter, we can imagine that skyrmions will decay through some higher dimension operators analogous to proton decay in Grand Unified Theory (GUT). The most economical way is to consider GUT-suppressed dimension 6 operators, with its lifetime [35] τ ∼ 8π The final decay products of the skyrmions would be some meson states with extra fundamental fermions, for instance charged leptons. We can imagine that the main branching ratio of the skyrmion decay is the one into a light meson state below GeV. The light meson mixes with the Higgs boson, so its coupling to the SM fermions is proportional to their masses and will dominantly decays into µ pairs. As long as the mass of the skyrmion is multi-TeV, the muon dominated leptonic final state will naturally explain [29,30] the PAMELA excess in e + [31] while no excess inp [32] as well as slow decline E −3.0 in the e − + e + spectrum as reported by FERMI [33] which steepens at about 1 TeV as measured by H.E.S.S. [34].
In summary, we have considered the possibilities that pointlike topological defects, such as monopoles and skyrmions, as the viable dark matter candidates. We apply the Kibble-Zurek mechanism to the non-thermal production of monopoles by a second order phase transition, and find that the abundance is much larger than the one originally estimated by Kibble. Depending on critical exponent in the correlation length, the hidden monopoles could account for the correct relics density for the mass range of approximately 1-10 PeV. The thermally produced skyrmion of mass 1-10 TeV can also provide the correct relics density of cold dark matter, whose decay may account for the flux and spectral shape of electrons and positrons recent observed by PAMELA and FERMI satellites and HESS. A similar consideration excludes any models that produce magnetic monopoles above TeV scale but below the inflation scale, such as Pati-Salam unification.