ЯДЕРНАЯ ФИЗИКА, 2010, том 73, № 1, с. 183-194
= ЭЛЕМЕНТАРНЫЕ ЧАСТИЦЫ И ПОЛЯ SMALL-SCALE CLUMPS IN THE GALACTIC HALO
© 2010 V. S. Berezinsky1),2), V. I. Dokuchaev2)*, Yu. N. Eroshenko2)
Received October 24, 2008
A mass function of small-scale dark matter clumps is calculated. We take into account the tidal destruction of clumps at early stages of structure formation starting from a time of clump detachment from the Universe expansion. Only a small fraction of these clumps, ^0.1%, in each logarithmic mass interval AlogM ~ 1 survives the stage of hierarchical clustering. We calculate the probability of surviving of the remnants of dark matter clumps in the Galaxy by modelling the tidal destruction of the small-scale clumps by disk and stars. It is demonstrated that a substantial fraction of clump remnants may survive through the tidal destruction during the lifetime of the Galaxy if a radius of core is rather small. The resulting mass spectrum of survived clumps is extended down to the mass of the core of the cosmologically produced clumps with a minimal mass. The survived dense remnants of tidally destructed clumps provides a large contribution to the annihilation signal in the Galaxy. We describe the anisotropy of dark matter clump distribution caused by tidal destruction of clumps in the Galactic disk. A corresponding annihilation of dark matter particles in small-scale clumps produces the anisotropic gamma-ray signal with respect to the Galactic disk.
1. INTRODUCTION
A primordial power-law spectrum of density fluctuations in the Dark Matter (DM) ranges from the largest scales above the scales of superclusters of galaxies to the smallest sub-stellar scales according to prediction of inflation models. This permits to predict the properties of smallest DM structures from the known CMB fluctuations at large scales.
The cosmological formation and evolution of small-scale DM clumps have been studied in numerous works [1 — 12]. The minimum mass of clumps (the cutoff of the mass spectrum), Mmin is determined by the collision and collisionless damping processes (see, e.g., [5] and references therein). Additionally, the cutoff of mass spectrum is influenced by the acoustic absorption [13] at the time of kinetic decoupling of DM particles [14] and also by the horizon-scale perturbation modes [15]. The low-mass cutoff of the clump mass spectrum accompanies the process of decoupling. It starts when DM particles couple strongly with surrounding plasma in the growing density fluctuations. The smearing of the small-scale fluctuations is due to the collision damping occurring just before decoupling, in analogy with the Silk damping [16]. It occurs due to diffusion of DM particles from a growing fluctuation, and only the small-scale fluctuations can be destroyed by this
!)infn, Laboratori Nazionali del Gran Sasso, Assergi (AQ), Italy.
2)Institute for Nuclear Research, Russian Academy of Sciences, Moscow.
E-mail: dokuchaev@lngs.infn.it
process. The corresponding diffusive cutoff M^ff is very small. As coupling becomes weaker, the larger fluctuations are destroyed and Mmin increases. One may expect that the largest value of Mmin is related to a free-streaming regime. However, as recent calculations show [9], the largest Mmin is related to some friction between DM particles and cosmic plasma similar to the Silk damping. The predicted minimal clump masses range from very low values, Mmin ~ 10-12M© [17], produced by diffusive escape of DM particles, up to Mmin ~ 10-4M©, caused by acoustics oscillations [15] and quasi-free-streaming with limited friction [9].
In the case of the Harrison—Zeldovich spectrum of primordial fluctuations with CMB normalization the first Earth-mass small-scale DM clumps are formed at redshift z ~ 60 (for 2a fluctuations) with a mean density 7 x 10-22 g cm-3, virial radius 6 x x 10-3 pc and internal velocity dispersion 80 cm s-1, respectively. Only very small fraction of these clumps survives the early stage of tidal destruction during the hierarchial clustering [3]. Nevertheless, these survived clumps may provide the major contribution to the annihilation signal in the Galaxy [3, 6, 18— 20]. At a high redshift neutralinos considered as DM particles (see, e.g., [21]), may cause the efficient heating of the diffuse gas [22] due to annihilation in the dense clumps.
One of the unresolved problem of DM clumps is a value of the central density or core radius. Numerical simulations give a nearly power density profile of DM clumps. Both the Navarro—Frenk—White (NFW)
and Moore profiles give formally a divergent density in the clump center. A theoretical modelling of the clump formation [23] predicts a power-law profile of the internal density of clumps
Pint(r) =
3-p_fr\-в
(1)
where p and R are the mean internal density and a radius of clump, respectively, ft — 1.8—2 and pint(r) = 0 at r > R. A near isothermal power-law profile (1) with ft — 2 has been recently obtained in numerical simulations of small-scale clump formation [24].
It must be noted that density profiles of small-scale DM clumps and large-scale DM halos may be different. The galactic halos are well approximated by the NFW profile outside the central core where dynamical resolution of numerical simulations becomes insufficient. Different physical mechanisms are engaged for formation of a central core during the formation and evolution of clumps. A theoretical estimation of the relative core radius of DM clump xc = Rc/R was obtained in [23] from energy criterion, xc = Rc/R — S3q, where Seq is a value of density fluctuation at the beginning of matter-dominated stage. A similar estimate for DM clumps with the minimal mass ~10-6M© originated from 2a fluctuation peaks gives Seq — 0.013 and Rc/R — 1.8 x 10-5, respectively. In [3] the core radius xc — 0.3v-2 has been obtained, where v is a relative height of the fluctuation density peak in units of dispersion at the time of energy—matter equality. This value is a result of the influence of tidal forces on the motion of DM particles in the clump at stage of formation. This estimate may be considered as an upper limit for the core radius or as the break-scale in the density profile, e.g., a characteristic scale in the NFW profile. It could be that a real core radius, where the density ceases to grow, is determined by the relaxation of small-scale perturbations inside the forming clump [25]. Another mechanism for core formation arises in the "meta-cold dark matter model" due to late decay of cold thermal relics into lighter nonrelativistic particles with low phase-space density [26, 27].
Nowadays numerical simulations have a rather low space resolution in the central region of clumps to determine the core radius. The only example with some indication to presence of a core with radius xc — — 10-2 is numerical simulation of small-scale clump formation [24]. In this work we consider the relative core radius xc = Rc/R of DM clumps as a free parameter in the range 0.001—0.1.
Correct approach includes a gradual mass loss of a system [28—31], in particular, by small-scale DM clumps [4, 32]. In this work we will describe a gradual
mass loss of small-scale DM clumps assuming that only the outer layers of clumps are involved and influenced by the tidal stripping. Additionally, we assume that inner layers of a clump are not affected by tidal forces. In this approximation we calculate a continuous diminishing of the clump mass and radius during the successive galactic disk crossings and encounters with the stars. We accept now for criterium of clump destruction the diminishing of radius of tidally stripped clump down to the core radius. Small remnants of clumps may survive in the Galaxy. These remnants would be an additional source of amplification of the DM annihilation signal in the Galaxy.
2. FORMATION AND DESTRUCTION OF CLUMPS IN HIERARCHIAL CLUSTERING
The process of hierarchical clustering and tidal destruction of DM clumps can be outlined in the following way. The DM clumps of minimal mass are formed first in the expanding Universe. The clumps of larger mass, which host the smaller ones, are formed later and so on. Some part of clumps are destroyed tidally in the gravitational field of their host clumps. In this section we study the destruction of DM clumps in the process of hierarchical structuring long before the final galaxy formation. At small-mass scales the hierarchial clustering is a fast and rather complicated nonlinear process. We use a simplified model which nevertheless takes into account the most important features of hierarchial clustering.
To describe the formation of clumps we will use the model of spherical collapse [33] in flat cosmology without the A term. This assumption is well justified at early times of clumps formation when the A term is negligible in comparison with the matter density. In this model a formation time of clump with an internal density p is t = (Kpeq/p)1/2teq, where k = 18^2 and peq = po(1 + zeq)3 is a cosmological density at the time of matter-radiation equality teq, 1 + zeq = 2.35 x x 104ttmh2 and po = 1.9 x 10-29ttmh2 gcm-3. The index "eq" here and throughout below refers to quantities at the time of matter-radiation equality teq.
The DM clumps of mass M can be formed from density fluctuations of different peak height v = Seq/aeq(M), where aeq(M) is the fluctuation dispersion on a mass scale M at the time teq. A mean internal density of clump p is fixed at the time of clump formation and according to [33] equals p = Kpeq[vaeq(M)/5c]3, where Sc = 3(12^)2/3/20 — ~ 1.686.
The tidal destruction of clumps is most effective at the early epochs of the Galactic halo formation, when the host density profiles are not finally established. The tidal interaction of clumps is a complicated process and depends on many factors: a clump formation history, host density profile, an existence of different substructures inside the host, orbital parameters of individua
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.