Abstract

We derive the effects of dark matter (DM) decays and annihilations on structure formation. We consider moderately massive DM particles (sterile neutrinos and light DM), as they are expected to give the maximum contribution to heating and reionization. The energy injection from DM decays and annihilations produces both an enhancement in the abundance of coolants (H2 and HD) and an increase of gas temperature. We find that for all the considered DM models the critical halo mass for collapse, mcrit, is generally higher than in the unperturbed case. However, the variation of mcrit is small. In the most extreme cases, that is, considering light DM annihilations (decays) and haloes virializing at redshift zvir > 30 (zvir∼ 10), mcrit increases by a factor of ∼4 (∼2). In the case of annihilations the variations of mcrit are also sensitive to the assumed profile of the DM halo. Furthermore, we note that the fraction of gas which is retained inside the halo can be substantially reduced (to ≈40 per cent of the cosmic value), especially in the smallest haloes, as a consequence of the energy injection by DM decays and annihilations.

1 INTRODUCTION

One of the fundamental questions concerning the formation of first structures is the minimum halo mass (critical mass, mcrit) for collapse at a given redshift (Rees & Ostriker 1977; Silk 1977; White & Rees 1978; Couchman 1985; Couchman & Rees 1986; de Araujo & Opher 1988, 1991; Haiman, Thoul & Loeb 1996).

Tegmark et al. (1997; hereafter T97) thoroughly addressed such question, pointing out how mcrit crucially depends on the abundance of H2, the main coolant present in the metal free Universe. Subsequent studies (Abel et al. 1998; Galli & Palla 1998; Fuller & Couchman 2000; Galli & Palla 2002; Ripamonti 2007) refined the model of T97, accounting also for minor effects, such as the cooling induced by HD molecules.

The production of molecules and mcrit are sensitive to any physical process which can release energy in the intergalactic medium (IGM). In fact, the injection of energy in the IGM can either delay the collapse of first haloes (because of the increased gas temperature, or of photodissociation of molecules) or favour structure formation (because of the enhancement in the abundance of free electrons, which act as catalysts for the formation of molecules).

For this reason, it is crucial to understand the influence of reionization sources on structure formation. Many studies have shown that massive metal free stars are efficient in dissociating H2 molecules, quenching star formation in the first haloes (Haiman, Rees & Loeb 1997; Ciardi, Ferrara & Abel 2000a; Ciardi et al. 2000b; Haiman, Abel & Rees 2000; Ricotti, Gnedin & Shull 2002; Yoshida et al. 2003). Intermediate mass black holes, produced by the collapse of first stars, are thought to efficiently reheat the IGM, increasing mcrit and reducing star formation in the smaller mass haloes (Ricotti & Ostriker 2004; Ricotti, Ostriker & Gnedin 2005; Zaroubi et al. 2006).

Also particle decays and annihilations can be sources of partial reionization and heating (see Mapelli, Ferrara & Pierpaoli 2006 and references therein), and could influence structure formation. For example, Shchekinov & Vasiliev (2004) investigated the possible effect on mcrit due to ultrahigh energy cosmic rays (UHECRs) emitted by particles decaying in the early Universe. Biermann & Kusenko (2006) considered the impact on structure formation due to sterile neutrino decays. Both these studies found a substantial enhancement on the abundance of molecular coolants (H2 and/or HD). However, they neglected the possible increase of gas temperature due to UHECRs or decays, respectively.

More recently, Stasielak, Biermann & Kusenko (2006) evaluated the effect of sterile neutrino decays accounting also for the heating of the gas. However, their single-zone model is likely to oversimplify the crucial behaviour of gas density during the halo collapse.

In this paper, we consider the influence of dark matter (DM) decays and annihilations on structure formation, taking into account variations induced both in the chemical and in the thermal evolution of the IGM and of the gas inside haloes. Furthermore, we substitute the single-zone models, which are commonly adopted in previous papers (Haiman et al. 1996 is an important exception), with more sophisticated 1D simulations. We focus on relatively low-mass DM particles, such as sterile neutrinos and light DM (LDM), as their effect on the IGM is expected to be much more important than that of heavier (≳100 MeV) DM particles (Mapelli et al. 2006).

Sterile neutrinos are expected to decay into active neutrinos and keV-photons (Dolgov 2002), while LDM can either decay or annihilate producing electron–positron pairs (Boehm et al. 2004; Hooper & Wang 2004; Picciotto & Pospelov 2005; Ascasibar et al. 2006). keV-photons interact with the IGM both via Compton scattering and photoionization; instead, the electron–positron pairs undergo inverse Compton scattering, collisional ionizations, and positron annihilations (Zdziarski & Svensson 1989; Chen & Kamionkowski 2004; Ripamonti, Mapelli & Ferrara 2006, hereafter RMF06). RMF06 derived the fraction fabs(z) of energy emitted by sterile neutrino decays and LDM decays or annihilations which is effectively absorbed by the IGM through these processes. In this paper, we adopt the fits of fabs(z) given by RMF06.

In Section 2 we describe the hydrodynamical code used to derive mcrit and the DM models which we adopt. In Section 3 we discuss the effect of DM decays and annihilations on the chemical and thermal evolution of the IGM, giving an estimate of the Jeans mass. In Section 4 we describe the chemical and thermal evolution of the gas inside the haloes, deriving mcrit. In the discussion (Section 5) we address various points, such as the variations in the baryonic mass fraction inside the haloes induced by DM decays/annihilations and the influence of the concentration of the DM profile.

We adopt the best-fitting cosmological parameters after the 3-yr Wilkinson Microwave Anisotropy Probe results (Spergel et al. 2006), that is, Ωb= 0.042, ΩM= 0.24, ΩDM≡ΩM−Ωb= 0.198, ΩΛ= 0.76, h= 0.73, H0= 100 h km s−1 Mpc−1.

2 METHOD

2.1 The code

In order to estimate the effects of the energy injection from DM decays and annihilations, it is necessary to follow the chemical and thermal evolution of primordial gas. This can be done with single-zone models such as the one originally described in T97, or the adaptations by Shchekinov & Vasiliev (2004), and by Stasielak et al. (2006). However, this kind of models is bound to use some approximations which can be very crude. First of all, single-zone models cannot follow the density evolution of a virialized halo because of their lack of ‘resolution, so that it is usually assumed that after virialization the gas density in a halo is both uniform in space and constant in time. Even in the linear phase of the collapse, the top-hat model provides a reasonable description of the DM component, but it becomes a rough approximation when the baryonic component is considered, as hydrodynamical effects are likely to become important at scales below the Jeans or the filtering length (Peebles 1993; Gnedin 2000).

An exact description would require a 3D simulation (such as those of Abel, Bryan & Norman 2002, and of Bromm, Coppi & Larson 2002); but it is possible to capture the basic features of the collapse phenomenon by an intermediate, time-efficient approach, for example, by means of 1D simulation, as they are both more accurate than single-zone models, and much faster than 3D simulations (Haiman et al. 1996).

In this paper, we use the 1D Lagrangian, spherically symmetric code described by Ripamonti et al. (2002), as updated in Ripamonti (2006). Such a code includes the treatment of

• the gravitational and hydrodynamical evolution of the gas (by means of an artificial viscosity scheme);

• the chemical evolution of 12 species (H, H+, H, H2, H+2, D, D+, HD, He, He+, He++ and e; see table 1 of Ripamonti 2007 for a list of the considered reactions and of the adopted reaction rates);

• the cooling (or heating) due to a number of components, such as the compressional (adiabatic) heating, the emission and absorption of line radiation from H, H2 and HD (accounting for the effects of the cosmic microwave background, CMB), the heating (or cooling) from the Compton scattering of CMB photons off free electrons, and the heating (or cooling) due to chemical reactions (e.g. the formation or dissociation of H2 molecules);

• the gravitational effects of DM, according to a simple model which is based on the top-hat formalism up to the turn-around redshift, smoothly evolving into a concentrated profile after virialization (see Section 2.2 for more details).

Table 1

List of chemical reactions stimulated by the DM energy injection.

 Species Reaction Threshold (ev) H H → H++ e− 13.6 He He → He++ e− 24.6 He+ He+→ He+++ e− 54.4 D D → D++ e− 13.6 H2 H2→ H + H 4.48 HD HD → H + D 4.51 H+2 H+2→ H + H+ 2.65
 Species Reaction Threshold (ev) H H → H++ e− 13.6 He He → He++ e− 24.6 He+ He+→ He+++ e− 54.4 D D → D++ e− 13.6 H2 H2→ H + H 4.48 HD HD → H + D 4.51 H+2 H+2→ H + H+ 2.65

2.2 DM profiles

The code does not include a self-consistent treatment of DM. Instead, the function ρDM(r, z), describing the DM density profile and its redshift evolution, must be chosen a priori. Since our results might depend on this choice, we decided to study two quite different cases, which we call ‘isothermal’ and ‘NFW’ (from the profile described in Navarro, Frenk & White 1997) depending on the shape of the density profile after virialization.

In both cases the DM distribution was assumed to be spherically symmetric, and concentric with the simulated region, whose central part represents the halo which is being investigated. At any redshift a DM mass MDM=MhaloΩDMM is assumed to be within the truncation radius

1
where t(z) is the time corresponding to redshift z, and zvir, zta, Rvir and ρTH(z) are the halo virialization and turn-around redshifts, its virial radius, and the DM density inside the halo at z > zta (as derived from the evolution of a simple top-hat fluctuation; see, for example, Padmanabhan 1993, or T97), respectively. The exact definition of these quantities can be found in Ripamonti (2006).

The isothermal and NFW assumptions differ only after the halo virialization (z < zvir); both of them refer to a static DM profile. In the isothermal case

2
instead, in the NFW case the DM density profile is chosen to be
3
where ρ0≃ 1.88 × 10−29h2 g cm−3 is the critical density of the Universe at present. In both cases RcoreRvir, where ξ= 0.1 is a parameter (cf. Hernquist 1993; Burkert 1995 for the choice of its value; in the NFW case ξ the inverse of the more commonly used concentration parameter), and the densities ρcore and ρNFW can be found by requiring the DM mass within Rtr to be equal to MDM.

At z > zvir both the isothermal and NFW case assume the DM density profile described by equation (2). However, at such redshift the profile is not static, because the core radius is evolved with redshift

4
Such a choice combines the behaviour of a top-hat fluctuation (the density inside Rtr is assumed to be uniform until the turn-around) with a transition to the final density profiles.

We only considered the case ξ= 0.1 (i.e. concentration 10 for the NFW profile), instead of varying ξ, because the differences between the isothermal and NFW cases are quite relevant even with the same value of ξ. In fact, the NFW case is representative of concentrated haloes, whereas the isothermal case is representative of relatively shallow potentials.

However, it is important to note that the flat central profile of the isothermal case helps to ensure that the behaviour we observe near the centre is due to the self-gravity and hydrodynamics of the simulated gas, rather than to the assumed DM profile.

2.3 Treatment of the DM energy injection

The above code was modified in order to include the effects of the energy injection from DM decays/annihilations on both the chemical and thermal evolution of the gas.

The gas can be heated, excited and ionized by the energy input due to DM decays/annihilations. It is important to note that the fraction of the absorbed energy going into each one of these components is quite unrelated to how the energy was deposited in the IGM at the first step. For example, if a keV-photon ionizes an atom, the resulting electron will generate a cascade of collisions, and the energy of the photon will go not only into ionizations, but also into excitations and heating.

Thus, given the energy injection per baryon from DM decays and annihilations, ε(z) (described in Section 2.4), we split it into an heating and ionization component1

5

6
where x(z) is the ionization fraction. In the first equation we are using the fit to the results of Shull & van Steenberg (1985) which is provided in their paper (with and ), while in the second equation we are using the fit to the same results given by Chen & Kamionkowski (2004).

The heating component is simply added to the equations describing the thermal state of the gas. Instead, the ionization component is further split between H, He, He+, D, H2, HD and H+2, according to their number abundance:

7
where the indices i and j indicate chemical species, and if the species i is atomic, or as if the species i is molecular; Ni is the number density (per unit volume) of the chemical species i. In principle, the terms in the sum above should be weighted by the cross-section for each species. However, both the cross-section and the energy spectrum are too complex to be implemented in our simple calculations. In particular, the energy spectrum is expected to be the result of a cascade (Shull & van Steenberg 1985). The error in neglecting these factors is quite small, as mcrit is more sensitive to the temperature increase than to the chemistry (see Section 4).

The quantity εi approximates the energy which is absorbed by chemical reactions dissociating the species i, and is translated into a reaction rate (number of reactions per particle per unit time) through a division by the energy threshold Eth,i of the considered reaction. The list of the reactions and of the energy thresholds is given in Table 1.

We neglect the absorption of ionization energy by H+, He++, D+ and e, because these species cannot be ionized further. We also neglect the ionization energy absorption by H because the energy threshold for the transformation of H into H is negative, and cannot be treated with our simple formalism. However, the number abundance of H is always very small, and the number of dissociations induced by DM decays/annihilations is likely to be negligible.

2.4 DM models

We apply this formalism to two different DM candidates, that is, sterile neutrinos and LDM. Sterile neutrinos are one of the most popular warm DM (WDM) candidates (Colombi, Dodelson & Widrow 1996; Bode, Ostriker & Turok 2001; Sommer-Larsen & Dolgov 2001). They can decay via different channels (Dolgov 2002; Hansen & Haiman 2004). In this paper we are interested in the radiative decay, that is, the decay of a sterile neutrino into a photon and an active neutrino, because of its direct impact on the IGM (Mapelli & Ferrara 2005; Mapelli et al. 2006). The photon produced in the decay interacts with the IGM both via Compton scattering and photoionization (RMF06).

LDM particles have recently become of interest, because they provide a viable explanation for the detected 511-keV excess from the Galactic centre (Boehm et al. 2004; Knödlseder et al. 2005). If they are source of the 511-keV excess, then their maximum allowed mass mLDM should be 20 MeV, not to overproduce detectable gamma-rays via internal bremsstrahlung (Beacom, Bell & Bertone 2005). If we consider also the production of gamma-rays for inflight annihilation of the positrons, this upper limit might become ∼3 MeV (Beacom & Yüksel 2006).

LDM can either decay or annihilate, producing photons, neutrinos and pairs. In this paper we consider both LDM decays and annihilations, but we restrict their treatment to the case where the product particles are e+− e pairs. In fact, in the case of pair production the impact of LDM on the IGM is maximum (RMF06). The e+− e pairs are expected to interact with the IGM via inverse Compton scattering, collisional ionization and positron annihilation (RMF06).

2.4.1 The energy input from the ‘background’

We first consider the energy injected in the general IGM after cosmic DM decays/annihilations.

Both in the case of sterile neutrinos and of LDM, the rate of energy transfer (per baryon) to the IGM because of this ‘background’ contribution can be written as

8
where mDM is the mass of a DM particle and c is the speed of light. The energy absorbed fraction, fabs(z), has been derived in RMF06; is the decrease rate of the number of DM particles per baryon.

In the case of DM decays, is given by

9
where nDM,0 and τDM are the current number of DM particles per baryon and the lifetime of DM particles, respectively. τDM is assumed to be much longer than the present value of the Hubble time, as is the case for all the models we are considering.

For the annihilations:

10
where Nb(0) = 2.5 × 10−7 g cm−3 is the current baryon number density (Spergel et al. 2006), and 〈σv〉 is the thermally averaged DM annihilation cross-section.

Both for sterile neutrinos and LDM, the values of nDM,0, τDM and 〈σv〉 adopted in this paper are the same reported in RMF06; for convenience, they are listed in Table 2.

Table 2

List of the main characteristics of the considered DM models. In particular, from the leftmost to the rightmost column: DM model, nDM,0, τDM (if decaying particle), 〈σ v〉 (if annihilating), comoving free-streaming lengths (λFS,n and λFS,i) and the associated mass scale. λFS is taken to be the maximum between λFS,n and λFS,i.

 DM model nDM,0/103 τDM/(1027 s) 〈σ v〉/(10−29 cm3 s−1) λFS,n (pc) λFS,i (pc) MFS(M⊙) ν 4 keV (decaying) 1176 2.96 − 3 × 105 60 5 × 108 ν 15 keV (decaying) 314 1.98 − 8 × 104 35 1 × 107 ν 25 keV (decaying) 188 0.097 − 4.8 × 104 25 2 × 106 LDM 3 MeV (decaying) 1.49 1.2 − 2.2 <0.1 2 × 10−7 LDM 10 MeV (decaying) 0.446 4.0 − 0.5 <0.1 3 × 10−8 LDM 1 MeV (annihilating) 4.46 − 4 9.5 300 0.5 LDM 3 MeV (annihilating) 1.49 − 12 2.2 100 0.02 LDM 10 MeV (annihilating) 0.446 − 24 0.5 30 5 × 10−4
 DM model nDM,0/103 τDM/(1027 s) 〈σ v〉/(10−29 cm3 s−1) λFS,n (pc) λFS,i (pc) MFS(M⊙) ν 4 keV (decaying) 1176 2.96 − 3 × 105 60 5 × 108 ν 15 keV (decaying) 314 1.98 − 8 × 104 35 1 × 107 ν 25 keV (decaying) 188 0.097 − 4.8 × 104 25 2 × 106 LDM 3 MeV (decaying) 1.49 1.2 − 2.2 <0.1 2 × 10−7 LDM 10 MeV (decaying) 0.446 4.0 − 0.5 <0.1 3 × 10−8 LDM 1 MeV (annihilating) 4.46 − 4 9.5 300 0.5 LDM 3 MeV (annihilating) 1.49 − 12 2.2 100 0.02 LDM 10 MeV (annihilating) 0.446 − 24 0.5 30 5 × 10−4

2.4.2 The ‘local’ energy input

In addition to the background energy injection discussed above, the baryons inside a halo absorb extra energy from the additional decays/annihilations of overdense halo DM. In the case of decays the total excess energy ‘produced’ inside the halo is

11
and in the case of annihilations
12
where nDM(r, z) =ρDM(r, z)/[mDMNb(0)(1 +z)3] is the number of DM particles per baryon at redshift z and at a distance r from the centre of the halo.

Then, we compute the baryon column density Σb from the halo centre to the truncation radius, and find the fraction of the energy Eloc which is absorbed by such a column density, floc. If the DM produces photons of energy Eγ,

13
where σHe+H(E) is the photoionization cross-section (see Zdziarski & Svensson 1989; RMF06), σT is the Thomson cross-section, and the function g is defined in equation (4.9) of Zdziarski & Svensson (1989). Instead, if the DM produces electron–positron pairs with Lorentz factor γ (and energy Emec2),
14
where φe,ion(z, E) and φe,com(z, E) give the fraction of the energy of an electron/positron which is absorbed per unit time by the IGM, because of collisional ionizations and of Compton scattering of CMB photons, respectively. Such functions are given by equations (14)–(15), and (16)–(18) of RMF06.

We then assume that all the baryons inside the halo absorb the same amount of energy from this local contribution. So, the ‘local’ energy deposition in each baryon within the truncation radius Rtr(z) is

15
where Mgas(Rtr) is the mass of the gas inside the truncation radius, and mH is the mass of an H atom. Instead, for baryons at a distance larger than Rtr(z) from the centre of the halo we assume εloc(z, r) = 0.

The total energy input per baryon from DM decays/annihilations is then

16

2.5 The free-streaming/damping lengths

We need to take into account that small DM fluctuations might be washed out by damping mechanisms, and in particular by free-streaming (see e.g. Padmanabhan 1993; Sommer-Larsen & Dolgov 2001; Boehm et al. 2005).

The free-streaming scale depends on the considered particle, and on the strength of its interactions. If such interactions are negligible, the comoving free-streaming length is

17
where mDM is the mass of the considered DM particle, p and T are the modulus of the proper momentum and the temperature of the particle, respectively. The above expressions are derived from Abazajian, Fuller & Patel (2001), and from Boehm et al. (2005), respectively.

However, if the DM particle interacts at a non-negligible rate, the comoving free-streaming length is (Boehm & Schaeffer 2005; Boehm et al. 2005)

18
with
19
where ΓDM(zdec) is the DM interaction rate at its decoupling redshift zdec.

In the case of decaying particles, we assume that this rate is simply the inverse of the present lifetimes of DM particles; so, ΓDM(zdec) < 10−24 s−1 in all the cases we are considering (cf. Table 2). Since zdec≫ 103, equations (17) and (18) imply that λFS,i≪λFS,n; so, for decaying particles we will use a free-streaming scale λFSFS,n.

Instead, when annihilating particles are considered, the interaction rate is ΓDM(zdec) = (1/2) nDM,0Nb(0) 〈σvdec(1 +zdec)3, where 〈σvdec∼ 10−26 cm3 s−1 (see e.g. Ascasibar et al. 2006) is the thermally averaged annihilation cross-section at the epoch of decoupling.2 Then, in the case of annihilating particles, λFS,i≫λFS,n, and we must assume λFSFS,i. In Table 2 we give the detailed list of the free-streaming lengths for each DM model. The same table also lists the free-streaming mass scale

20

Damping erases fluctuations on scales smaller than λFS. Thus, objects of mass ≲MFS are unlikely to form, unless the small-scale power spectrum is regenerated by non-linear effects at low redshift (see Boehm et al. 2005; however, this regeneration appears to take plate at z∼ 2, which is much later than the epoch we are considering). To introduce corrections for this effect in our models is beyond the purposes of this paper.

Damping might also affect the density profile of haloes, erasing cusps on scales ≲λFS. The values of λFS and MFS listed in Table 2 show that this effect of damping can be important only for sterile neutrinos. In the case of isothermal density profiles, we account for it adopting the following correction. If the core radius Rcore(z) (equation 4) is smaller than λFS/2, we increase its value to Rcore, new(z) = min(Rtr(z), λFS/2). Instead, in the case of simulations with NFW density profile we do not introduce this correction, because these simulations are intended to explore the effects of high concentration (see Section 4.2).

2.6 The simulations

Our code was used to run a large number of simulations, in order to explore a wide range of the zvirMhalo parameter space: we considered halo masses in the range 104–107 M, and virialization redshifts between 10 and 100.

We actually simulate a mass which is 1000 times higher than that of the collapsing halo, in order to include in the simulation a mass which is larger than the cosmological filtering mass (Gnedin 2000); otherwise, our treatment of hydrodynamics might be incorrect. The simulated object is divided in 150 shells, whose spacing was chosen so that (i) the mass of the shells smoothly increases when moving outwards, (ii) the central shell always encloses a gas mass of ∼0.3 M and (iii) the central 100 shells initially enclose a mass ∼Mhalo (including the DM).

The simulations are started at z= 1000, when we assume that the gas density is uniformly equal to the cosmological value (whereas the DM density profile is not perfectly uniform; see Section 2.2), the gas temperature is equal to the CMB temperature (≃2.728 K), and adopt the chemical abundances listed in table 2 of Ripamonti 2007.3

They are stopped either when the gas density reaches the threshold ρ/mH= 105 cm−3, or after a time 2 t(zvir) has elapsed.

Each set of simulations was repeated for each different DM decay/annihilation model, and also for the ‘standard’ case without any energy injection from DM, which is used as a reference against which we compare our results.

3 IGM EVOLUTION

In order to look at the influence of DM decays and annihilations upon the IGM, we have used a simplified version of our code (where the density was assumed to evolve as the cosmological value).

3.1 Chemistry and temperature

Our code follows the chemical evolution of 12 chemical species (see previous section). Two of them, molecular hydrogen (H2) and HD are particularly important for our purposes, because they are the main coolant of the metal free gas. In Fig. 1 we show the fractional abundances of both H2 and HD together with the ionized fraction and the matter temperature as a function of redshift for the considered DM models.

Figure 1

Effects of decaying/annihilating DM on the IGM evolution. Left-hand axis: fractional abundances of free electrons (e), H2 and HD as a function of redshift. Right-hand axis: matter temperature as a function of redshift. Top panel: effect of decaying sterile neutrinos of mass 25 keV (dashed line). Central panel: decaying LDM of mass 10 MeV (dashed line). Bottom panel: annihilating LDM of mass 1 MeV (dashed line). In all the panels the dotted line is the CMB temperature and the solid line represents the thermal and chemical evolution without DM decays/annihilations.

Figure 1

Effects of decaying/annihilating DM on the IGM evolution. Left-hand axis: fractional abundances of free electrons (e), H2 and HD as a function of redshift. Right-hand axis: matter temperature as a function of redshift. Top panel: effect of decaying sterile neutrinos of mass 25 keV (dashed line). Central panel: decaying LDM of mass 10 MeV (dashed line). Bottom panel: annihilating LDM of mass 1 MeV (dashed line). In all the panels the dotted line is the CMB temperature and the solid line represents the thermal and chemical evolution without DM decays/annihilations.

In all the DM models, both the matter temperature and the abundance of H2, HD and free electrons are enhanced by DM decays/annihilations. This effect is smaller for sterile neutrinos than for LDM particles. The main difference between decays and annihilations is represented by the redshift range in which the influence of DM is important. For all the considered quantities (i.e. temperature and abundance of e, H2 and HD) the energy injection from DM decays starts to be significant at redshift lower than ∼100. Instead, the influence of annihilations is important already at redshift ∼900. The annihilations represent also the case where the abundance of the two coolants is most enhanced (a factor of ∼17 for the H2 and ∼90 for the HD). Furthermore, the annihilations keep the matter temperature close to the CMB temperature everywhere, up to z∼ 50–100. This fact can have important consequences for experiments searching for high redshift HI 21-cm line signals (Shchekinov & Vasiliev 2006).

3.2 Jeans mass

In order to establish the influence of DM decays or annihilations on the structure formation, the key point is the following. DM decays and annihilations increase both the matter temperature and the abundance of coolants. The former effect tends to delay the formation of structures, while the latter favours an early collapse of the haloes. Which of these two opposite effects is dominant? When looking at the average properties of the IGM, the most popular diagnostic is the cosmological Jeans mass, mJ (Peebles 1993):

21
where kB is the Boltzmann constant, G the gravitational constant and μ the mean molecular weight.

In Fig. 2 we show the evolution of mJ. For all the considered DM models mJ is considerably increased by the effect of decays and annihilations. This means that, when the IGM is considered, the increase of the matter temperature dominates over the enhancement of the coolant abundance. For example, at z= 10 mJ= 3.0 × 107 M in the case of 10-MeV LDM decays, a factor of ∼420 higher than in the unperturbed case (for which mJ= 7.2 × 104 M). The increase of mJ is less pronounced, but nevertheless significant in the case of sterile neutrino decays (mJ= 6.6 × 105 M for 25-keV sterile neutrinos) and LDM annihilations (mJ= 3.9 × 105 M for 1-MeV LDM).

Figure 2

Jeans mass as a function of redshift. Top panel: effect of decaying sterile neutrinos of mass 25 (dashed line), 15 (dotted line) and 4 keV (dot–dashed line). Central panel: decaying LDM of mass 10 (dashed line) and 3 MeV (dotted line). Bottom panel: annihilating LDM of mass 10 (dashed line), 3 (dotted line) and 1 MeV (dot–dashed line). In all the panels the solid line represents the thermal and chemical evolution without DM decays/annihilations. The horizontal dot–dashed line shows the behaviour of the Jeans mass calculated by assuming that the gas temperature is always equal to the CMB temperature.

Figure 2

Jeans mass as a function of redshift. Top panel: effect of decaying sterile neutrinos of mass 25 (dashed line), 15 (dotted line) and 4 keV (dot–dashed line). Central panel: decaying LDM of mass 10 (dashed line) and 3 MeV (dotted line). Bottom panel: annihilating LDM of mass 10 (dashed line), 3 (dotted line) and 1 MeV (dot–dashed line). In all the panels the solid line represents the thermal and chemical evolution without DM decays/annihilations. The horizontal dot–dashed line shows the behaviour of the Jeans mass calculated by assuming that the gas temperature is always equal to the CMB temperature.

From this fact one could naively infer that DM decays and annihilations strongly delay the formation of the first structures. However, mJ refers only to the global properties of the IGM, and does not account for the non-linear evolution of collapsing haloes.

4 EVOLUTION INSIDE HALOES

4.1 Chemistry and temperature

To assess the star forming ability of the first haloes, it is necessary to follow their hydrodynamical and chemical evolution. This has been done by using the code described in Section 2. Fig. 3 (Fig. 4) shows, as an example, the evolution of the most relevant properties of the gas at the centre of a 6 × 105 M (2 × 106 M) halo, virializing at zvir= 12. In these figures the effects of different DM decay/annihilation scenarios are compared with the unperturbed case. For the smaller halo (Fig. 3) LDM annihilations and sterile neutrino decays delay the collapse by ∼1 redshift units, and in the case of LDM decays this effect is even more significant: in the presence of 10-MeV LDM decays the halo has not collapsed yet after one Hubble time from virialization. Instead, for the larger halo (Fig. 4) the difference between the case with and without DM decays/annihilations is negligible.

Figure 3

Evolution of the central region of a 6 × 105 M isothermal halo virializing at zvir= 12. From left- to right-hand panels and from top to bottom: density, temperature, electron abundance and H2 abundance as function of redshift. The solid line represents the unperturbed case (i.e. without DM decays and annihilations). The dashed, dot–dashed and dotted lines account for the contribution of 25-keV sterile neutrino decays, 1-MeV LDM annihilations and 10-MeV LDM decays, respectively.

Figure 3

Evolution of the central region of a 6 × 105 M isothermal halo virializing at zvir= 12. From left- to right-hand panels and from top to bottom: density, temperature, electron abundance and H2 abundance as function of redshift. The solid line represents the unperturbed case (i.e. without DM decays and annihilations). The dashed, dot–dashed and dotted lines account for the contribution of 25-keV sterile neutrino decays, 1-MeV LDM annihilations and 10-MeV LDM decays, respectively.

Figure 4

Evolution of the central region of a 2 × 106 M isothermal halo virializing at zvir= 12. From left- to right-hand panels and from top to bottom: density, temperature, electron abundance and H2 abundance as function of redshift. The dashed, dot–dashed and dotted lines are the same as in Fig. 3.

Figure 4

Evolution of the central region of a 2 × 106 M isothermal halo virializing at zvir= 12. From left- to right-hand panels and from top to bottom: density, temperature, electron abundance and H2 abundance as function of redshift. The dashed, dot–dashed and dotted lines are the same as in Fig. 3.

It is worth discussing in detail the gas temperature (T) and number density (N=ρ/mH) evolution shown in Fig. 3. At the epoch of turn-around (z∼ 19–20) all the models have the same N; but in the unperturbed one T is 1.5–5 times lower than in the others. This is important, because the fast increase in T and N due to the virialization process is essentially adiabatic, as it happens on a time-scale much faster than that for cooling. In the adiabatic approximation, Γad=−PdVP [N−1− (NN)−1] (where P is the gas pressure), is proportional to the initial T for any given increase in N. T and P increase faster in models ‘pre-heated’ by DM decays/annihilations, so that the pressure gradient slowing and halting the collapse develops earlier. In fact, the phase of unimpeded collapse stops at N∼ 10 cm−3 for the unperturbed case, but only at N∼ 0.3–3 cm−3 for the other models. At these densities the cooling per unit mass is proportional both to the H2 fraction and to N, so that the higher central density of the model without DM decays/annihilations largely compensates its lower H2 abundance, that is, the unperturbed case is the fastest cooling one. However, it is interesting to note that the final T is much lower for the cases where DM energy input is included, because of the enhancement of the HD fraction, which provides an efficient cooling mechanism also at T≲ 200 K (Ripamonti 2007).

4.2 Critical mass

Each of our simulated haloes was classified as collapsing (or, equivalently, efficiently cooling) or non-collapsing (inefficiently cooling), depending on whether it reaches a maximum density larger than 105mH cm−3 in less than a Hubble time. We define the critical mass, mcrit, as the minimum mass of a collapsing halo at a given virialization redshift zvir.

The values of mcrit as a function of the virialization redshift are shown in Fig. 5 for isothermal haloes, and in Fig. 6 for NFW haloes. The main trend inferred from the behaviour of mJ (i.e. the delay of structure formation) is confirmed by mcrit. However, the influence of DM decays and annihilations on mcrit is much smaller than could be expected from mJ.

Figure 5

The critical mass as a function of the virialization redshift zvir for isothermal haloes. Top panel: effect of decaying sterile neutrinos. Central: decaying LDM. Bottom: annihilating LDM. The lines used in the three panels are the same as in Fig. 2. The thin solid line used in the top panel represents the case of non-decaying WDM (with mass 4 keV).

Figure 5

The critical mass as a function of the virialization redshift zvir for isothermal haloes. Top panel: effect of decaying sterile neutrinos. Central: decaying LDM. Bottom: annihilating LDM. The lines used in the three panels are the same as in Fig. 2. The thin solid line used in the top panel represents the case of non-decaying WDM (with mass 4 keV).

Figure 6

The critical mass as a function of the virialization redshift zvir for NFW haloes. The lines used in the three panels are the same as in Fig. 5.

Figure 6

The critical mass as a function of the virialization redshift zvir for NFW haloes. The lines used in the three panels are the same as in Fig. 5.

In the case of isothermal haloes, the effects of the DM decays on mcrit (top and central panel) are completely negligible at high redshift and become significant only at redshift less than 20. Even for zvir= 10 and 10-MeV decaying LDM (which has the strongest effect), mcrit increases only by a factor of 2.

Sterile neutrinos (especially with a mass of 4 keV) seem to significantly increase mcrit. However, this is not connected with sterile neutrinos decays; but it is mainly due to differences in the assumed DM density profile, as the free-streaming length for these WDM haloes is larger than the ‘standard’ value of the core radius (0.1Rvir) and the ‘damping’ correction described in Section 2.5 is important. In fact, mcrit is almost the same down to z∼ 20, if we consider (thick dot–dashed line in Fig. 5) or neglect (thin solid line) sterile neutrino decays, and the difference between these two cases is small even at lower redshift.

In the case of LDM annihilations in isothermal haloes, mcrit is higher than in the unperturbed case for every considered virialization redshift (zvir= 10–100), confirming that the annihilations play a role even at very high z. However, the difference with respect to the unperturbed mcrit is always less than a factor of ∼2.

If we assume a NFW profile for DM haloes (Fig. 6), the general effect is a substantial decrease of mcrit (independently of DM decays/annihilations), due to the higher central densities, and the consequent stronger gravitational pull. LDM decays tends to increase mcrit (at z < 20) with respect to the unperturbed case; the increase factor is only marginally larger than in the case of isothermal haloes. Sterile neutrino decays appear to slightly reduce mcrit with respect to the unperturbed case; but this result is unphysical, as it does not account for damping. It is worth to note that, even if we ignore damping effects and adopt the models with the highest possible concentration, the influence of sterile neutrino decays on structure formation remains very small.

Instead, in the case of LDM annihilations, mcrit generally increases more in NFW haloes than in isothermal haloes. This is due to the importance of the local contribution for annihilations (see Section 5.2).

5 DISCUSSION

5.1 Gas fraction

Why considering either mJ or mcrit leads to such widely different conclusions? As mentioned earlier, mJ is a rough tracer of the minimum halo mass for collapse; but it is not sensitive to the local properties of the halo. For example, to calculate mJ one has to assume that the density is uniform. On the contrary, mcrit depends on the evolution of the central region of the halo, where the density increases much more rapidly than in the outskirts, driving the collapse (Figs 3 and 4). When the density becomes sufficiently high, the molecular cooling largely overcomes the heating due to DM decays/annihilations (Figs 3 and 4). This is the reason why mcrit does not increase significantly in presence of DM decays/annihilations. Instead, mJ, which does not account for the density increase and for the consequent cooling enhancement, dramatically grows. Then, we can conclude that mJ does not provide a realistic estimate of the effects of DM decays/annihilations.

Is there any physical implication of the growth of mJ?mJ is directly related to the global hydrodynamical behaviour of the gas inside a halo: in haloes with mass below mJ the gas pressure prevents the development of a large gas overdensity, while in more massive haloes the gas accumulation should proceed almost unimpeded. If so, the mass fraction in baryons within the virial radius of a halo should be of the order of the cosmological average ΩbM when MhalomJ, but it should be lower when MhalomJ. To check this hypothesis, we derive the baryonic mass fraction (fgas) normalized to ΩbM, that is,

22
where Mgas(Rvir) is the mass of gas within the virial radius at the final stage of each simulation; instead Mhalo is the total mass of the halo at the beginning of the simulation.

In Fig. 7 we show fgas as a function of mass inside haloes virializing at redshift zvir= 10. The comparison between the various scenarios clearly shows that the energy injection from DM decays/annihilations can substantially reduce the gas fraction inside all haloes, especially the smallest ones. This is clearly related to the increase in mJ, as the largest variation in fgas occurs in the LDM decay scenario, where the increase of mJ is maximum (Fig. 2).

Figure 7

Halo baryonic mass fraction (see definition in equation 22) as a function of the halo mass for a fixed virialization redshift (zvir= 10) and for an isothermal DM profile. Top panel: effect of decaying sterile neutrinos. Central panel: decaying LDM. Bottom panel: annihilating LDM. The lines used in the three panels are the same as in Fig. 2. Thick (thin) lines indicate that the halo mass is larger (smaller) than mcrit.

Figure 7

Halo baryonic mass fraction (see definition in equation 22) as a function of the halo mass for a fixed virialization redshift (zvir= 10) and for an isothermal DM profile. Top panel: effect of decaying sterile neutrinos. Central panel: decaying LDM. Bottom panel: annihilating LDM. The lines used in the three panels are the same as in Fig. 2. Thick (thin) lines indicate that the halo mass is larger (smaller) than mcrit.

5.2 Concentration and local contribution

In this paper we basically considered two different DM profiles: isothermal and NFW haloes. These two models are comparable in terms of virial radius and main properties; but the NFW one is much more concentrated. From the comparison between Figs 5 and 6 we have already seen that this difference has important effects on the critical mass: mcrit is generally much lower for a NFW than for an isothermal halo, independently of DM decays and annihilations. For decaying particles, higher concentrations lead to marginally stronger effects in delaying the formation of structures, but the effect is minimal. For annihilations this effect of high concentrations is much larger (see Figs 5, 6 and 8).

Figure 8

Evolution of mcrit as a function of the virialization redshift in the case without DM (solid lines), with 3-MeV LDM decays (dotted lines) and with 1-MeV LDM annihilations (dot–dashed lines). Thin (thick) lines are with (without) the local contribution of the DM inside the halo. Note that the thin dotted line is completely superimposed to the thick dotted line, because the local contribution of decaying DM is always negligible. Top panel: isothermal haloes. Bottom panel: NFW haloes.

Figure 8

Evolution of mcrit as a function of the virialization redshift in the case without DM (solid lines), with 3-MeV LDM decays (dotted lines) and with 1-MeV LDM annihilations (dot–dashed lines). Thin (thick) lines are with (without) the local contribution of the DM inside the halo. Note that the thin dotted line is completely superimposed to the thick dotted line, because the local contribution of decaying DM is always negligible. Top panel: isothermal haloes. Bottom panel: NFW haloes.

Another significant characteristic of our model is the estimate of the local contribution due to DM decays/annihilations occurring inside the halo (see Section 2.4.2). Pointedly, the local contribution strongly depends on the DM profile. But what is the importance of the local contribution? In Fig. 8 we indicate the effect on mcrit of LDM decays (dotted line) and annihilations (dot–dashed line), by including (thin line) or not (thick line) the local contribution. In the case of DM decays the thin and thick dotted lines appear superimposed, both in the isothermal (top panel) and NFW (bottom) profile, indicating that the local contribution is always negligible for decays.

Instead, if we consider the annihilations, the case with (thin line) and without (thick line) the local contribution are very different, especially for the NFW profile. If we do not include the local contribution, mcrit is very close to the unperturbed value. If we switch on the local contribution in the isothermal profile, mcrit is substantially higher (a factor of ∼2) than in the unperturbed case, at least for high virialization redshifts (zvir > 30). Finally, if we account for the local contribution in a NFW halo, mcrit is always higher (a factor of ∼2–4) than in the unperturbed case (even at low virialization redshifts). This is consistent with the fact that the annihilation rate strongly depends on the local DM density.

6 CONCLUSIONS

In this paper we derived the effects of DM decays and annihilations on structure formation. We considered only moderately massive DM particles (sterile neutrinos and LDM), as they are expected to give the maximum contribution to heating and reionization (Mapelli et al. 2006). To describe the interaction between the IGM and the decay/annihilation products we followed the recipes recently derived by RMF06.

We accounted not only for the diffuse cosmological contribution to heating and ionization, but also for the local contribution due to DM decays and annihilations occurring in the halo itself. The local contribution results to be dominant in the case of DM annihilations especially for cuspy DM profiles.

The energy injection from DM decays/annihilations produces both an enhancement in the abundance of coolants (H2 and HD) and an increase of gas temperature. We found that for all the considered DM models (sterile neutrino decays, LDM decays and annihilations) the critical halo mass for collapse, mcrit is often higher than in the unperturbed case. This means that DM decays and annihilations tend to delay the formation of structures. However, the variation of mcrit is minimal. In the most extreme cases, that is, considering LDM annihilations (decays) and haloes virializing at redshift zvir > 30 (zvir∼ 10), mcrit increases by a factor of ∼4 (∼2). In the case of decays, the variations of mcrit are almost independent from the assumed concentration of the DM halo, although higher concentrations (corresponding to smaller values of mcrit) seem to be associated with slightly stronger effects of the DM energy injection. The dependence on concentration is more evident in the case of annihilating particles, where higher concentrations lead to substantially larger effects. This happens because the ‘local’ contribution is important.

In summary, the effects of DM decays and/or annihilations on structure formation are quite small, except in some extreme cases (e.g. very high concentration for annihilations). However, the energy injection from DM decays/annihilations has important consequences on the fraction of gas which is retained inside the halo. This fraction can be substantially reduced, especially in the smallest haloes (≲106 M).

Finally, we point out that our results are quite different from the conclusions of Biermann & Kusenko (2006) and Stasielak et al. (2006),4 who suggest that sterile neutrino decays can favour the formation of first objects. The discrepancy is likely due to our more complete treatment which includes the hydrodynamics of the collapsing structures. In fact, our hydrodynamical treatment allows us to describe the detailed gas density evolution during the collapse, resulting in markedly different temperature and chemical properties with respect to those found by a simple one-zone model.

1
The sum of these two components is less than ε(z), as a significant fraction of the injected energy goes in atomic/molecular excitations and does not affect the chemical or thermal state of the gas.
2
For annihilating particles, a third scalelength might be considered, that is, the mixed damping length λmd (Boehm et al. 2003). The upper limit of λmd is 6.7, 2.9 and 1.2 kpc for 1, 3 and 10 MeV LDM, respectively. These values are a factor of ∼20 larger than the corresponding λFS,i; but our conclusions about mcrit (see Section 4) are not significantly affected, because MFS is still smaller than the mass range we are considering.
3
The most important abundances listed there are the H ionization fraction (at z= 1000) N$H +$/NH= 0.0672, the helium abundance NHe/NH= 0.0833, and the deuterium abundance ND/NH= 2.5 × 10−5.
4
Biermann & Kusenko (2006) and Stasielak et al. (2006) do not calculate the critical mass mcrit. So, it is quite difficult to make a quantitative comparison between their results and ours.

We thank S. Zaroubi, C. Watson, A. Kusenko, P. Biermann and J. Stasielak for useful discussions. We also acknowledge the referee for a critical reading of the paper. The three authors thank both SISSA/ISAS and the Kapteyn Institute for mutual hospitality during the preparation of this paper. ER gratefully acknowledges support from the Netherlands Organization for Scientific Research (NWO) under project number 436016.

REFERENCES

Abazajian
K.
Fuller
G. M.
Patel
M.
,
2001
,
Phys. Rev. D
,
64b
,
3501
Abel
T.
Anninos
P.
Norman
M. L.
Zhang
Y.
,
1998
,
ApJ
,
508
,
518
Abel
T.
Bryan
G. L.
Norman
M. L.
,
2002
,
Sci
,
295
,
93
Ascasibar
Y.
Jean
P.
Boehm
C.
Knödlseder
J.
,
2006
,
MNRAS
,
368
,
1695
Beacom
J. F.
Yüksel
H.
,
2006
,
Phys. Rev. Lett.
,
97g
,
1102
Beacom
J. F.
Bell
N. F.
Bertone
G.
,
2005
,
Phys. Rev. Lett.
,
94q
,
1301
Biermann
P. L.
Kusenko
A.
,
2006
,
Phys. Rev. Lett.
,
96L
,
1301
Bode
P.
Ostriker
J. P.
Turok
N.
,
2001
,
ApJ
,
556
,
93
Boehm
C.
Schaeffer
R.
,
2005
,
A&A
,
438
,
419
Bromm
V.
Coppi
P. S.
Larson
R. B.
,
2002
,
ApJ
,
564
,
23
Boehm
C.
Mathis
H.
Devriendt
J.
Silk
J.
,
2003
, preprint (0309652)
Boehm
C.
Hooper
D.
Silk
J.
Casse
M.
Paul
J.
,
2004
,
Phys. Rev. Lett.
,
92j
,
1301
Boehm
C.
Mathis
H.
Devriendt
J.
Silk
J.
,
2005
,
MNRAS
,
360
,
282
Burkert
A.
,
1995
,
ApJ
,
447
,
L25
Chen
X.
Kamionkowski
M.
,
2004
,
Phys. Rev. D
,
70d
,
3502
Ciardi
B.
Ferrara
A.
Abel
T.
,
2000
,
ApJ
,
533
,
594
Ciardi
B.
Ferrara
A.
Governato
F.
Jenkins
A.
,
2000
,
MNRAS
,
314
,
611
Colombi
S.
Dodelson
S.
Widrow
L. M.
,
1996
,
ApJ
,
458
,
1
Couchman
H. M. P.
,
1985
,
MNRAS
,
214
,
137
Couchman
H. M. P.
Rees
M. J.
,
1986
,
MNRAS
,
221
,
53
De Araujo
J. C. N.
Opher
R.
,
1988
,
MNRAS
,
231
,
923
De Araujo
J. C. N.
Opher
R.
,
1991
,
ApJ
,
379
,
461
Dolgov
A. D.
,
2002
,
Phys. Rep.
,
370
,
333
Fuller
T. M.
Couchman
H. M. P.
,
2000
,
ApJ
,
544
,
6
Galli
D.
Palla
F.
,
1998
,
A&A
,
335
,
403
Galli
D.
Palla
F.
,
2002
,
Planetary Space Sci.
,
50
,
1197
Gnedin
N. Y.
,
2000
,
ApJ
,
542
,
535
Haiman
Z.
Thoul
A. A.
Loeb
A.
,
1996
,
ApJ
,
464
,
523
Haiman
Z.
Rees
M. J.
Loeb
A.
,
1997
,
ApJ
,
476
,
458
Haiman
Z.
Abel
T.
Rees
M. J.
,
2000
,
ApJ
,
534
,
11
Hansen
S. H.
Haiman
Z.
,
2004
,
ApJ
,
600
,
26
Hernquist
L.
,
1993
,
ApJS
,
86
,
389
Hooper
D.
Wang
L.-T.
,
2004
,
Phys. Rev. D
,
70f
,
3506
Knödlseder
J.
et al
,
2005
,
A&A
,
441
,
513
Mapelli
M.
Ferrara
A.
,
2005
,
MNRAS
,
364
,
2
Mapelli
M.
Ferrara
A.
Pierpaoli
E.
,
2006
,
MNRAS
,
369
,
1719
Navarro
J. F.
Frenk
C. S.
White
S. D. M.
,
1997
,
ApJ
,
490
,
493
T.
,
1993
,
Structure Formation in the Universe
.
Cambridge Univ. Press
, Cambridge
Peebles
P. J. E.
,
1993
,
Principles of Physical Cosmology
.
Princeton Univ. Press
, Princeton, NJ
Picciotto
C.
Pospelov
M.
,
2005
,
Phys. Lett. B
,
605
,
15
Rees
M. J.
Ostriker
J. P.
,
1977
,
MNRAS
,
179
,
541
Ricotti
M.
Ostriker
J. P.
,
2004
,
MNRAS
,
352
,
547
Ricotti
M.
Gnedin
N. Y.
Shull
J. M.
,
2002
,
ApJ
,
575
,
49
Ricotti
M.
Ostriker
J. P.
Gnedin
N. Y.
,
2005
,
MNRAS
,
357
,
207
Ripamonti
E.
2007
,
MNRAS
, in press (0701148)
Ripamonti
E.
Haardt
F.
Ferrara
A.
Colpi
M.
,
2002
,
MNRAS
,
334
,
401
Ripamonti
E.
Mapelli
M.
Ferrara
A.
,
2006
,
MNRAS
, in press (0606482) (RMF06)
Shchekinov
Y. A.
Vasiliev
E. O.
,
2004
,
A&A
,
419
,
19
Shchekinov
Y. A.
Vasiliev
E. O.
2006
,
MNRAS
, submitted (0604231)
Shull
J. M.
Van Steenberg
M. E.
,
1985
,
ApJ
,
298
,
268
Silk
J.
,
1977
,
ApJ
,
211
,
638
Sommer-Larsen
J.
Dolgov
A.
,
2001
,
ApJ
,
551
,
608
Spergel
D. N.
et al
2006
,
ApJ
, submitted (0603449)
Stasielak
J.
Biermann
P. L.
Kusenko
A.
,
2006
,
ApJ
, in press (0606435)
Tegmark
M.
Silk
J.
Rees
M. J.
Blanchard
A.
Abel
T.
Palla
F.
,
1997
,
ApJ
,
474
,
1
(T97)
White
S. D. M.
Rees
M. J.
,
1978
,
MNRAS
,
183
,
341
Yoshida
N.
Abel
T.
Hernquist
L.
Sugiyama
N.
,
2003
,
ApJ
,
592
,
645
Zaroubi
S.
Thomas
R. M.
Sugiyama
N.
Silk
J.
,
2006
,
MNRAS
, in press (0609151)
Zdziarski
A. A.
Svensson
R.
,
1989
,
ApJ
,
344
,
551