Impact of primordial ultracompact minihaloes on the intergalactic medium and first structure formation

Monthly Notices of the Royal Astronomical Society, Dec 2011

The effects of dark matter annihilation on the evolution of the intergalactic medium (IGM) in the early Universe will be more important if the dark matter structure is more concentrated. Ultracompact minihaloes (UCMHs), which formed through dark matter accretion on to primordial black holes (PBHs) or an initial dark matter overdensity produced by a primordial density perturbation, provide a new type of compact dark matter structure to ionize and heat the IGM after matter–radiation equality zeq, which is much earlier than the formation of the first cosmological dark halo structure and later the first stars. We show that the dark matter annihilation density contributed by UCMHs can completely dominate over the homogeneous dark matter annihilation background, even for a tiny UCMH fraction fUCMH=ΩUCMH(zeq)/ΩDM≥ 10−15(1 +z)2(mχc2/100 GeV)−2/3 with a standard thermal-relic dark matter annihilation cross-section, and can provide a new gamma-ray background in the early Universe. UCMH annihilation becomes important to IGM evolution for approximately fUCMH > 10−6(mχc2/100 GeV). The IGM ionization fraction xion and gas temperature Tm can be increased from the recombination residual xion∼ 10−4 and adiabatically cooling Tm∝ (1 +z)2 in the absence of energy injection, to a maximum value of xion∼ 0.1 and Tm∼ 5000 K at z≥ 10 for the upper bound on UCMH abundance constrained by the cosmic microwave background optical depth. A small fraction of UCMHs are seeded by PBHs. The X-ray emission from gas accretion on to PBHs may totally dominate over dark matter annihilation, and may become the main cosmic ionization source for a PBH abundance fPBH=ΩPBH/ΩDM≫ 10−11 (10−12) with PBH mass MPBH∼ 10−6 M⊙ (102 M⊙). However, the constraints on gas accretion rate and X-ray absorption by baryon accumulation within UCMHs, together with accretion feedback, show that X-ray emission can only be a promising source much later than UCMH annihilation at z < zm≪ 1000, where zm depends on the PBH masses, their host UCMHs and the dark matter particles. Also, UCMH radiation including both annihilation and X-ray emission can significantly suppress the low-mass first baryonic structure formation. The effects of UCMH radiation on baryonic structure evolution are quite small as regards the gas temperature after virialization, but more significant in enhancing gas chemical quantities such as the ionization fraction and molecular hydrogen abundance in baryonic objects.

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Impact of primordial ultracompact minihaloes on the intergalactic medium and first structure formation

Impact of primordial ultracompact minihaloes on the intergalactic medium and first structure formation Dong Zhang 0 0 Department of Astronomy, Ohio State University , 140 W. 18th Avenue, Columbus, OH 43210 , USA A B S T R A C T The effects of dark matter annihilation on the evolution of the intergalactic medium (IGM) in the early Universe will be more important if the dark matter structure is more concentrated. Ultracompact minihaloes (UCMHs), which formed through dark matter accretion on to primordial black holes (PBHs) or an initial dark matter overdensity produced by a primordial density perturbation, provide a new type of compact dark matter structure to ionize and heat the IGM after matter-radiation equality zeq, which is much earlier than the formation of the first cosmological dark halo structure and later the first stars. We show that the dark matter annihilation density contributed by UCMHs can completely dominate over the homogeneous dark matter annihilation background, even for a tiny UCMH fraction f UCMH = UCMH(zeq)/ DM ≥ 10−15(1 + z)2(mχ c2/100 GeV)−2/3 with a standard thermal-relic dark matter annihilation cross-section, and can provide a new gamma-ray background in the early Universe. UCMH annihilation becomes important to IGM evolution for approximately f UCMH > 10−6(mχ c2/100 GeV). The IGM ionization fraction xion and gas temperature T m can be increased from the recombination residual xion ∼ 10−4 and adiabatically cooling T m ∝ (1 + z)2 in the absence of energy injection, to a maximum value of xion ∼ 0.1 and T m ∼ 5000 K at z ≥ 10 for the upper bound on UCMH abundance constrained by the cosmic microwave background optical depth. A small fraction of UCMHs are seeded by PBHs. The X-ray emission from gas accretion on to PBHs may totally dominate over dark matter annihilation, and may become the main cosmic ionization source for a PBH abundance f PBH = PBH/ DM 10−11 (10−12) with PBH mass MPBH ∼ 10−6 M (102 M ). However, the constraints on gas accretion rate and X-ray absorption by baryon accumulation within UCMHs, together with accretion feedback, show that X-ray emission can only be a promising source much later than UCMH annihilation at z < zm 1000, where zm depends on the PBH masses, their host UCMHs and the dark matter particles. Also, UCMH radiation including both annihilation and X-ray emission can significantly suppress the low-mass first baryonic structure formation. The effects of UCMH radiation on baryonic structure evolution are quite small as regards the gas temperature after virialization, but more significant in enhancing gas chemical quantities such as the ionization fraction and molecular hydrogen abundance in baryonic objects. intergalactic medium - galaxies; structure - cosmology; theory - dark matter - early Universe 1 I N T R O D U C T I O N Ultracompact minihaloes (UCMHs) are primordial dark matter structures that formed by dark matter accreting on to primordial black holes (PBHs) after matter–radiation equality, zeq ∼ 3100, or directly by collapse on to an initial dark matter overdensity produced by a small density perturbation before zeq, e.g. in several Universe phase-transition epochs (Mack, Ostriker & Ricotti 2007; Ricotti & Gould 2009). If the density perturbation in the early Universe exceeds a critical value δc = (δρ/ρ)c ∼ 1/3, this region becomes gravitationally unstable and collapses directly to form a PBH (Hawking 1971; Carr & Hawking 1974; see Khlopov 2010 for a review, and the references therein). PBHs that form with a sufficiently high mass ≥1016 g do not evaporate but begin to grow by accreting the surrounding dark matter and form a compact dark matter halo, which will grow by two orders of magnitude in mass during the matter-dominated era (Mack et al. 2007). These haloes are the so-called ultracompact minihaloes (UCMHs), or alternatively ‘primordially laid ultracompact minihaloes’ (PLUMs). On the other hand, a small density perturbation in the early Universe 10−3 < δ < δc will form a compact dark matter overdensity instead of a PBH. Such an overdense cloud can also seed the formation of UCMHs (Ricotti & Gould 2009; Scott & Sivertsson 2009; Josan & Green 2010). Note that the initial density perturbations from inflation were just δ ∼ 10−4–10−5; it is proposed that a far more viable formation method for UCMHs is by accretion on to a dark matter overdensity, which requires a much lower perturbation threshold than do PBHs. Also, the UCMHs seeded by primordial overdensities have a different profile from those seeded by PBHs (Bertschinger 1985; Mack et al. 2007). UCMHs have recently been proposed as a new type of nonbaryonic massive compact gravitational object (MACHO: Ricotti & Gould 2009) as well as gamma-ray and neutrino sources (Scott & Sivertsson 2009). UCMHs could produce a microlensing light curve, which can be distinguished from that of a ‘point-like’ object such as a star or brown dwarf, thus becoming a promising new target for microlensing searches. Moreover, the abundance of UCMHs can be constrained by the observation of the Milky Way gammaray flux and the extragalactic gamma-ray background, although this constraint is still very uncertain based on today’s data (Josan & Green 2010; Lacki & Beacom 2010; Saito & Shirai 2011). Since we know the growth of an isolated UCMH as a function of redshift (Mack et al. 2007), we can natively trace the fraction of UCMHs back to very high redshift without considering mergers and tidal destruction. Until now, most work on UCMHs has focused on the properties of nearby UCMHs at z < 1. Another important question that has barely been discussed is the consequences of UCMH radiation at very high redshift, since UCMHs are ‘remnants’ originally from the early Universe. As sources of heating and ionization before the first structures, stars and galaxies, sufficient UCMHs might play an important role in changing the chemical and thermal history of the early Universe. Our main purpose in this paper is to investigate the impacts of UCMH emission on the intergalactic medium (IGM) in the Universe’s reionization era and the first baryon structure formation and evolution that followed. The process of reionization of all hydrogen atoms in the IGM would have been completed at redshift z ≈ 6 (Becker et al. 2001; Fan et al. 2002). However, much earlier ionization at z > 6 is implied by Wilkinson Microwave Anisotropy Probe (WMAP) observations (Dunkley et al. 2009; Komatsu et al. 2009). It is commonly suggested that possible contributions to high-redshift reionization in range approximately 6 < z < 20 are the first baryonic objects to produce significant ultraviolet light, early (Pop III and Pop II) stars and old quasars (Barkana & Loeb 2007; Wise & Abel 2008; Meiksin 2009; Volonteri & Gnedin 2009). However, it is still unclear whether quasars and first stars were sufficiently efficient to reionize the Universe. Dark matter, on the other hand, is suggested as an exotic source of ionization and heating at high redshift due to its self-annihilation or decay. It is usually proposed that weakly interacting massive particles (WIMPs) provide a compelling solution to identify the dark matter component. The mass of the dark matter particles, mχ , and the average annihilation cross-section, σ v , are the two crucial parameters that affect the ionizing and heating processes. Under the thermal-relic assumption that the cross-section σ v ≈ 3 × 10−26 cm3 s−1 to match the observed DMh2 ≈ 0.110, most previous studies showed that the effects of homogeneous dark matter background annihilation or decay on the high-redshift IGM are expected to be important only for light dark matter, mχ c2 ≤ 1 GeV, or sterile neutrinos (Hansen & Haiman 2004; Pierpaoli 2004; Belotsky et al. 2005; Mapelli & Ferrara 2005; Mapelli, Ferrara & Pierpaoli 2006; Zhang et al. 2006; Ripamonti, Mapelli & Ferrara 2007a,b; Chluba 2010). The annihilation flux would be enhanced only after the formation of the first dark objects at z < 60, as dark matter becomes more clumpy (Chuzhoy 2008; Natarajan & Schwarz 2008, 2009, 2010; Belikov & Hooper 2009, 2010). However in our case dark matter is more concentrated in UCMHs, which are significantly denser than the homogeneous dark matter background, so WIMP dark matter annihilation within UCMHs may produce powerful gamma-ray sources that dominate over the homogeneous background annihilation, even though UCMHs are very rare. A small fraction of UCMHs are seeded by PBHs (Mack et al. 2007). In this paper we call these UCMHs ‘PBH-host UCMHs’. Since PBH abundance is still uncertain for a broad range of PBH mass (Josan & Green 2009; Carr et al. 2010) we only give a qualitative estimate that the abundance of PBH-host UCMHs should be much less than that of other UCMHs. For PBH-host UMCHs, the Xray emission from the accreting baryonic gas that flows on to PBHs may totally dominate over dark matter annihilation within the host UCMH, since the Eddington luminosity is several orders of magnitude brighter than that of annihilation from the host UCMH and the photoionization cross-section for hydrogen or helium is much larger than the Klein–Nishina or pair-production cross-section for energetic gamma-rays. It is very difficult for a ‘naked’ PBH to reach a sufficiently high accretion rate in the IGM environment (Barrow & Silk 1979; Carr 1981; Gnedin, Ostriker & Rees 1995; Miller & Ostriker 2001; Ricotti 2007; Mack & Wesley 2008; Ricotti, Ostriker & Mack 2008), but the situation will be quite different when PBHs are surrounded by UCMHs. The accretion rate and X-ray luminosity of baryons can change significantly when the effects of a growth UCMH are involved (Ricotti 2007; Ricotti et al. 2008). However, it is possible that the gas is heated and piles up around the PBH if the host UCMH is sufficiently massive. Also, accretion feedback such as outflows or radiation pressure prevents gas from being totally eaten by the PBH immediately, if the gas accretion rate significantly exceeds the Eddington limit. As a consequence, the gas density and temperature within the UCMH may be significantly higher than the cosmic universal gas density and the X-ray emission is totally absorbed by the UCMH but reradiated basically in the infrared band. In this paper we will give criteria for X-ray emission escaping from the host UCMH to ionize the IGM. We will also compare the importance of X-ray emission from PBH-host UCMHs and dark matter annihilation from total UCMHs in the early Universe, depending on the abundance of both total UCMHs and PBH-host UCMHs. Another topic related to UCMH radiation is that the formation and evolution history of the first baryonic structure can be changed by UCMH radiation. Previous studies showed that the annihilation or decay of extended distributed dark matter in the first structures changes both the gas temperature and the chemical properties such as the abundance of molecular coolants like H2 and HD (Biermann & Kusenko 2006; Stasielak, Biermann & Kusenko 2007; Ripamonti et al. 2007b). Higher coolant abundances help to decrease the gas temperature and favour an early collapse of the baryon gas inside the halo, but dark matter energy injection delays this collapsing process. It is still under debate whether dark matter annihilation or decay inside the dark halo will promote or suppress the first structure formation. Nevertheless, it is concluded that the promotion or suppression effect is quite small for most dark matter models, as the change of gas temperature in a virialized halo for various dark Lann Lhalo UCMH mass at redshift z UCMH (density) profile at z extent radius of UCMH DM particle mass DM average anni cross-section anni lum of a single UCMH f UCMH(zeq) = UCMH(zeq)/ DM UCMH anni lum per volume homogeneous DM anni background X-ray rad density from PBH-host UCMHs PBH/ DM Bondi accretion radius amplification factor of the IGM Tm dimensional accretion rate baryonic fraction in a UCMH charac. redshift for gas accretion reionized baryon fraction at z energy deposition rate per volume at z photon energy from anni DM charac. energy for X-ray emission IGM gas temperature molecular hydrogen fraction in IGM gas total anni lum from a dark halo anni lum from UCMHs inside a halo anni lum from extended DM in a halo energy deposited by isothermal DM halo anni energy deposited by UCMHs anni inside a halo energy deposited by X-rays inside a halo zm for PBH-host UCMHs inside/outside a halo energy deposition rate density by δ-func SED ratio of rad of differently distributed UCMHs Section 2.1, equation (1) Section 2.1, equation (2) Section 2.1, equation (3) Section 2.1, equation (4) Section 2.1, equation (4) Section 2.1, equation (5) Section 2.1, equation (7) Section 2.1, equation (10) Section 2.1, equation (11) Section 2.2, equation (13) Section 2.2, equation (13) Section 2.2, equation (19) Section 2.2, equation (19) Section 2.2, equation (23) Section 2.2, equation (24) Section 2.2, equation (28) Section 3.1, equation (35) Section 3.1, equation (39) Section 3.1, equation (41) Section 3.1, —— Section 3.1, equation (43) Section 3.1, equation (47) Section 4.1, —— Section 4.1, —— Section 4.1, —— Section 4.1, equation (48) Section 4.1, equation (49) Section 4.2, equation (50) Section 4.2, —— Section 5.3, equation (52) Section 5.4, equation (64) matter models is small. If the first large-scale dark haloes contain UCMHs, these UCMHs can inject more annihilation energy into the halo than the first dark haloes and can potentially play a more important role in changing the properties of the first haloes than the extended distributed dark matter in haloes. Moreover, X-ray emission that comes from PBHs also suppresses the formation of the first baryonic objects. Therefore it is also worthwhile to study the effects of UCMH radiation on the first structure formation and evolution. This paper is organized as follows. In Section 2, we calculate the dark matter annihilation luminosity from UCMHs and the X-ray emission from PBH gas accretion. We emphasize the importance of UCMH annihilation compared with homogeneous dark matter background annihilation, and focus on the physical reasons as to whether and when the X-ray emission from PBHs becomes more important than UCMH annihilation in the early Universe. In Section 3 we discuss the gas heating and ionization process arising from the two types of UCMH radiation from z ∼ 1000–10 and investigate the impact of UCMH radiation on IGM evolution. Next, in Section 4 we show the influence of UCMH radiation on the first baryonic structure formation and evolution. The main results of this paper are given in Sections 3 and 4. In Section 5 we discuss the importance of UCMHs in the reionization era and the effects of a single massive UCMH on the first baryonic structure as well as other secondary effects. The reader could skip this section and go directly to Section 6, which presents the conclusions. In this paper we do not consider dark matter decay, which should have similar consequences to the annihilation process. Also, we fix the annihilation cross-section to be the thermal-relic σ v = 3 × 10−26 cm3 s−1, although much larger cross-sections σ v = 3 × 10−24 cm3 s−1– 10−20 cm3 s−1 are proposed in the hope of explaining the reported Galactic cosmic ray anomalies as the results of dark matter annihilation (Aharonian et al. 2008; Chang et al. 2008; Abdo et al. 2009). A larger cross-section with the same dark matter particle mass mχ can have higher luminosity and more significant influence on ionization and heating of the early Universe. Table 1 gives the notation and definition of some quantities in this paper. 2 R A D I AT I O N F R O M U C M H S In this section we discuss two types of energy emission from UCMHs in the early Universe: dark matter annihilation and X-ray emission from the accreting baryonic gas on to PBHs. As mentioned in Section 1, the second type of emission is related to a small fraction of UCMHs that host PBHs. Generally we still call the second type of energy emission ‘UCMH radiation’; this is because a PBH is always located in the centre of its host UCMH and belongs to a PBH–UCMH system. where the factor (3 − α)/4π in equation (2) is obtained by normalizing the total mass inside the radius of maximum halo extent Rh as δm, and Rh is calculated by Rh(z) ≈ 0.019 pc where K = (3−α)(4.66×108)α−3(1+zeq)α/3(4π)−1 in cgs units and nχ (r) = ρχ (r)/mχ is the dark matter particle number density. The UCMH density profile can change from a steep slope α = 3 (Mack et al. 2007) in the outer region to α = 1.5 (Bertschinger 1985) in the inner region, if there is a PBH in the centre of the UCMH. In particular, radial infall on to a central extended overdensity shows a profile ρ ∝ r−9/4, which is more widely used as the typical density profile for most regions in UCMHs (Ricotti & Gould 2009; Scott & Sivertsson 2009; Josan & Green 2010). Taking α = 9/4, we have Lann = 36.3 L where σ v s = σ v /3 × 10−26 cm3 s−1 and mχ,100 = mχ c2/100 GeV. According to equation (6), the annihilation luminosity decreases with the evolution of the Universe, basically because the annihilation flattens the inner density profile as shown in equation (4). The left panel of Fig. 1 gives UCMH annihilation luminosity for different halo profiles α and dark matter particle masses mχ . Note that the annihilation luminosity can be much brighter for lighter dark matter particles, and a shallower density profile reduces Lann significantly.1 In the limiting case α 1.5 or 3, we have Lann ∝ σ v /mχ or Lann, in order to be independent of mχ and σ v . The abundance of UCMH as a function of redshift is still uncertain today. It can be presented by a parameter fUCMH(z) = UCMH(z)/ DM, where UCMH and DM are the comoving abundances of UCMH and total dark matter with UCMH(z) = UCMH(zeq)(1 + zeq)/(1 + z) and DM(z) = DM(z = 0). We have f UCMH(z = 0) ∼ f UCMH(zeq)(1 + zeq)/(1 + 10) ∼ 3 × 102f UCMH(zeq), which shows that the UCMH mass grows by up to two order of magnitude from zeq to z ∼ 10. From now on we take f UCMH as f UCMH(zeq), to show the initial abundance of UCMHs at matter–radiation equality, and at the current stage f UCMH is taken as a parameter for simplicity. 1 In Fig. 1 the annihilation luminosities following the density profile equa tion (2) are somewhat overestimated for a steep profile α ∼ 3, because in this case the halo mass within the truncated radius rcut can no longer be neglected. Thus the normalization factor of the density profile is ∝ [ln (Rh/rcut)]−1, which is different from the factor (3 − α)/(4π) in equation (2). However, as we show that a steeper UCMH profile leads to a Lann several orders of magnitude higher than that with α = 2.25, the conclusion that a steeper profile gives a brighter annihilation will not be changed too much even in the limiting case α = 3. More details of the UCMH profile are discussed in Section 5.2. 2.1 Dark matter annihilation The dark matter annihilation luminosity of nearby UCMHs (z = 0) has been calculated recently (Scott & Sivertsson 2009; Josan & Green 2010; Lacki & Beacom 2010; Saito & Shirai 2011). We assume that UCMHs stop growing at z ≈ 10 when structure formation has progressed deeply enough to prevent dark matter from accreting further. Now we calculate the annihilation luminosity as a function of redshift in the early Universe before z ≈ 10 and compare the result with homogeneous background annihilation. The mass of the UCMHs accreted by dark matter radial infall is given by (Mack et al. 2007; Ricotti & Gould 2009; Scott & Sivertsson 2009; Josan & Green 2010) where zeq ≈ 3100 is the redshift of matter–radiation equality and δm is the mass of the initial dark matter overdensity. The density profile in a UCMH ∝ r−α can be written as where t ≈ 32 (1 + z)−3/2( m,0)−1/2H0−1 is the age of the Universe at a certain redshift z and ti ≈ 77 kyr is the initial age at zeq. Thus the total dark matter annihilation luminosity within the UCMH can be calculated as Dark matter annihilation reduces the density in the inner region of a UCMH and makes the density in this region flat. Following Ullio et al. (2002), the UCMH power-law density distribution is truncated at the maximum density K3/α σ v (3−α)/α(1 + z)(9−4α)/α ×δm(t − ti)(3−2α)/αm(α−3)/α, χ Lann = The mean free path of gamma-ray photons from an UCMH with energy Eγ is written as 1 λUCMH = nA(z)σ (Eγ ) ≈ 3 × 103 pc where nA(z) = nA(1 + z)3 is the atomic number density at redshift z. The average distance betwen UCMHs is estimated as dUCMH = nUCMH(z) ∼ 7 pc fU−C1M/3H,−4 We have the mean free path exceed the inter-UCMH distance λUCMH dUCMH, except for extremely small f UCMH 10−12. Therefore cosmic UCMH annihilation also gives a uniform gamma-ray background radiation field as well as that produced by homogeneous dark matter. UCMH annihilation luminosity per volume is given by lann = = 1.4 × 10−28 erg cm−3 s−1 × σ v s1/3mχ−,11/030fUCMH(1 + z)4. On the other hand, the energy injection rate by the self-annihilation of homogeneous dark matter background per volume is = 3.2 × 10−43 erg cm−3 s−1 σ v smχ−,1100(1 + z)6. We compare the radiation between UCMH and normal dark matter annihilation: = 4.5 × 1012 σ v s−2/3m2χ/,3100 If f UCMH ≥ 2.2 × 10−15 σ v s2/3mχ−,21/030(1 + z)2, the gamma-ray background due to dark matter annihilation is dominated by UCMH annihilation. More details depending on the density profile α and dark matter mχ can be seen in the right panel of Fig. 1. 2.2 Gas accretion on to PBHs The abundance of PBHs PBH as a fraction of total dark matter DM at z < zeq can be parametrized as f PBH = PBH/ DM. We ignore the PBH growth and take f PBH as a constant in the matter-dominated Universe for two reasons. The first reason is that, as the accretion processes had been significantly suppressed before z ∼ 10 due to the relative motion between PBHs and baryon gas, the PBH growth time-scale, tgrowth ∼ tSalp 5 × 108 yr, just reaches – or is longer than – the age of the Universe at t(z ∼ 10) ∼ 5 × 108 yr. The second reason is that low-mass PBHs have a lower accretion rate while high-mass PBHs are inclined to produce outflows, which further increase the accretion time-scale and make PBH growth negligible compared with its host UCMH growth. A very similar statement to keep f PBH constant was also proposed in Ricotti et al. (2008). Keep in mind that the PBH abundance f PBH is different from the UCMH initial abundance f UCMH at zeq as mentioned in Section 2.1, because a large proportion of UCMH seeds at zeq should be initial primordial dark matter overdensity but not PBHs. According to density primordial perturbation theory, generally we have the relation f PBH f UCMH, which will be discussed in detail in Section 2.2.1. Much work has been done to show the effects of radiation from PBH or early black hole accretion on the thermal and ionization history of the early Universe (Barrow & Silk 1979; Gendin et al. 1995; Miller & Ostriker 2001; Ricotti 2007; Ricotti et al. 2008; Ripamonti, Mapelli & Zaroubi 2008). Our goal in this section is to focus on the importance of PBH gas accretion radiation compared with the overall UCMH dark matter annihilation. The X-ray emission from accreting PBHs may lead to a very different heating and ionization history of the early Universe, compared with dark matter annihilation. The X-ray luminosity from an individual PBH with mass MPBH can be written as ηLEdd = 4πηGmpMPBH/σTc 3.3 × 103η−1 L (MPBH/M ), with LEdd and η−1 = η/0.1 being the Eddington luminosity and average radiation efficiency of all PBHs respectively. This X-ray luminosity is much higher than the dark matter annihilation luminosity in equation (6). Thus the X-ray radiation density in the early Universe z > 10 can be written as lacc = ηfPBHρDM(z = 0)(1 + z)3 1.3 × 10−26 erg cm−3 s−1 η−1fPBH(1 + z)3. Combing equations (10) and (13), the ratio between PBH accretion luminosity and UCMH dark matter annihilation luminosity is lann fUCMH 1 + z Since the IGM heating rate due to energy injection by PBH X-ray emission or UCMH dark matter annihilation is proportional to both the energy injection rate and the IGM cross-section for all interactions suffered by the UCMH-emitted photons in the X-ray band (EX) or annihilation emitted by gamma-ray photons (Eγ ), the importance of IGM gas heating by X-ray emission and UCMH dark matter annihilation can be estimated through the ratio laccσ tot(EX)/lannσ tot(Eγ ), with σ tot labelling the total cross-sections in different photon energy ranges. If we approximate the X-ray and IGM interaction crosssection σ tot(EX) as the Thomson cross-section and the high-energy photon interaction σ tot(Eγ ) as the Klein–Nishina cross-section (see Section 3.1 for more accurate calculations), the energy deposition in the IGM due to gas accretion acc and annihilation ann is estimated as 3.2 × 105 σ v s−1/3m4χ/,3100 ann fUCMH 1 + z which gives the first conclusion that X-ray heating may become totally dominant over dark matter annihilation in the early Universe if the PBH abundance exceeds a critical value, ≥ 3.1 × 10−7 σ v s1/3mχ−,41/030 Theoretically the value of ηf PBH/f UCMH includes many uncertainties. In general, there are at least three reasons to have a low value ηf PBH/f UCMH 1: density perturbation scenarios prefer a low initial value of f PBH/f UCMH; inefficient radiation η 1 is favoured by low-mass PBHs while accretion feedback decreases η for high-mass PBHs or PBHs with high-mass UCMHs; and X-ray emission from PBHs can be trapped inside the surrounding host UCMHs, which accumulate baryons. 2.2.1 PBH abundance Either PBHs or UCMH overdensity seeds are produced by density perturbations in the very early Universe during some special epochs such as inflation or phase transitions. The cosmological abundance of UCMHs can be estimated by integrating from the overdensity seed threshold ∼10−3 to the PBH formation threshold δc ∼ 1/3 (Ricotti & Gould 2009; Scott & Sivertsson 2009). Similarly, the PBH abundance is estimated by integrating the perturbation above δc ∼ 1/3 (Green & Liddle 1997; Green, Liddle & Riotto 1997). Assuming a Gaussian perturbation at a formation redshift zf zeq produces both PBHs and UCMH seeds, the ratio f PBH/f UCMH at matter–radiation equality can be directly traced back to formation time zf (Carr et al. 2010; Khlopov 2010). As a result, the relative abundance of PBHs to UCMH overdensity seeds formed at redshift zf can be written as The perturbation variance at zf is roughly given by σ (zf ) 9.5 × 10−5[Mhor(zf )/1056 g](1−n)/4 (Green & Liddle 1997), with Mhor(zf ) and n being the horizon mass and mass spectrum index at zf . Taking n ≤ 1.3 (Lidsey, Carr & Gilbert 1995), the ratio f PBH/f UCMH from a Gaussian perturbation is a function of horizon mass: ≤ exp − (n−1)/2 which means that the value of f PBH/f UCMH becomes 1 for Mhor(zf ) 5.5 × 1010 g, not to mention the fact that the masses of dark matter overdensity seeds or PBHs are even lower than the horizon mass, δm Mhor(zf ) and MPBH < Mhor(zf ). Combing equations (15) and (18), X-ray emission from gas accretion hardly becomes the dominant heating source in the early Universe, except for low-mass PBHs MPBH Mhor(zf ) < 3.7 × 1018 g in the Gaussian perturbation scenario. However, PBHs in this mass range should either have disappeared within a Hubble time due to Hawking evaporation or be too small to accrete IGM gas. As a result, the initially Gaussian density perturbation at a certain epoch is not able to generate sufficiently abundant PBHs to dominate over the total UCMH dark matter annihilation emission, basically because the large-amplitude part of the Gaussian distribution is highly suppressed. On the other hand, a non-Gaussian perturbation may give an even lower probability of PBH formation, as large fluctuations can be suppressed in the non-Gaussian distribution, further decreasing the ratio of f PBH/f UCMH (Bullock & Primack 1997). However other mechanisms, such as different formation epochs for UCMHs and PBHs, different early inflationary potential, double inflation models, various phase transitions and cosmic-string collapse may enhance high-amplitude perturbations and increase the PBH abundance (see Khlopov 2010 and references therein). Also, it is still arguable whether all δ > 10−3 perturbations could produce dark matter overdensity in the radiation-dominant era. For example, Ricotti & Gould (2009) require the host UCMHs around PBHs to have similar initial perturbation amplitude to PBHs, while Scott & Sivertsson (2009) have a less strict requirement of δ > 10−3 to form the initial dark matter overdensity. There are more physical uncertainties involved in estimating the abundance of PBHs and UCMHs produced by other mechanisms than a simple Gaussian distribution assumption. Therefore we still take f PBH as a free parameter satisfying f PBH f UCMH to describe the relative abundances between PBHs and UCMHs. 2.2.2 Inefficient radiation Another effect through which to constrain the X-ray luminosity density in the early Universe by gas accretion on to PBHs is the low radiation efficiency resulting from the low accretion rate on to lowmass PBHs, or the significant radiative feedback, thermal outflow and suppressed accretion rate from accretion on to high-mass PBHs or PBHs with high-mass host UCMHs. In principle the mass distribution of PBHs is broad enough to cover the range from the Planck mass ∼10−5 g to thousands of solar masses, 105 M (e.g. Carr et al. 2010). As mentioned in Section 2.2.1, if PBHs are formed from a Gaussian perturbation with variation σ ∝ M−(n−1)/4 and index n > 1, low-mass PBHs should be more abundant because of the higher density perturbation variance σ for lower mass M. Also, phase-transition models give a PBH mass or UCMH seed of less than 1 M (Scott & Sivertsson 2009). On the other hand, we should keep in mind that in an IGM environment a ‘naked’ PBH without a host UCMH can never reach the Eddington accretion rate M˙ EPdBdH = LEdd/c2 1.4×1017(MPBH/M ) g s−1, unless its mass is MPBH ≥ 360 M [1000/(1 + z)]3/2. The surrounding host UCMH increases the accretion rate if the PBH mass is MPBH > 100 M (Ricotti et al. 2008, their fig. 4). Note that an ideal case is η min{0.1 m˙, 1} after the accretion becomes super-Eddington, while the typical accretion efficiency for quasars or microquasar discs is η ∼ 0.15. For low-mass PBHs with m˙ 1 the radiation efficiency is estimated as η 0.01 m˙2 for the spherical case (Shapiro 1973a,b), which gives a much lower efficiency than the high accretion rate for η 0.1. If high-mass PBHs (100 M < MPBH < 105 M ) successfully form with an appreciable abundance compared with low-mass PBHs, as discussed by some previous authors (Mack et al. 2007; Saito, Yokoyama & Nagata 2008; Frampton et al. 2010), or the host UCMH seeds are more massive than the PBHs, δm MPBH (Ricotti & Gould 2009, for more details see Section 2.2.3), the Bondi accretion rates on to these PBHs with their host UCHMs can significantly exceed the Eddington limit after some critical redshift (Ricotti et al. 2008). However, spherical super-Eddington accretion is generally unstable and inclined to drive high mass-loss rates for thermal outflows (e.g. Smith & Owocki 2006). Recent simulations show that radiative feedback may become important to reduce or even quench the accretion process periodically (Milosavljevic´, Couch & Bromm 2009a; Milosavljevic´ et al. 2009b; Park & Ricotti 2011). Also, thermal heating by the outflow energy or radiative feedback will increase the temperature of the gas around PBHs and decrease the Bondi radius and accretion rate on to PBHs. Besides the spherical accretion case, the falling gas angular momentum will become important for m˙ 1 and will form an accretion disc around PBHs. However, the physics of super-Eddington accretion discs is still not clearly known. Various types of super-Eddington accretion-disc models have been proposed, such as optically thick advection-dominated accretion flow (ADAF: Narayan & Yi 1994; Narayan, Mahadevan & Quataert 1998), adiabatic inflow–outflow (ADIO: Blandford & Begelman 1999), convection-dominated accretion flow (CDAF: Narayan, Igumenshchev & Abramowicz 2000), the ‘Polish doughnuts’ torus (Abramowicz, Jaroszyn´ski & Sikora 1978) and thick slim disc (Abramowicz et al. 1988). In most cases the super-Eddington accretion disc advects most of its heating energy inward into the black hole without emission, and has a low radiation efficiency η for a high accretion rate η < 1 (Abramowicz et al. 1988; Narayan et al. 1998; see Abramowicz & Fragile 2011 for a review). In a brief summary, low average radiation efficiency η in equations (13)–(15) is favoured because of the low accretion rate on to low-mass PBHs and radiative or viscous feedback and outflows of accretion on to high-mass PBHs or PBHs with high-mass host UCMHs, which also leads to a low value of ηf PBH/f UCMH and suppresses the importance of PBH X-ray radiation from PBHs compared with the overall UCMH dark matter annihilation. From now on we consider that the X-ray emission is mainly contributed by accreting PBHs with m˙ 1. 2.2.3 Radiation trapping in host UCMHs Some previous works discussed the fact that the accretion flow around PBHs is Compton-thin in most cases, since in the subEddington accretion case the spherical flow is transparent near the PBH, while in the super-Eddington accretion case accretion flows are inclined to form an accretion disc (e.g. Ricotti et al. 2008). However, sufficiently high-mass UCMHs can accrete and thermalize baryons from the ambient IGM, even when there are no PBHs in the centre of these UCMHs. As the gravity potential at the outer edge of the host UCMH is mainly contributed by the UCMH mass, but the accretion on to the centre PBHs is according to the PBH mass, the accretion rate on to host UCMHs is not necessarily equal to the accretion rate on to centre PBHs. In other words, baryons can be first accumulated and virialized inside the host UCMH during accretion from the IGM to the inner UCMH region, followed by a secondary accretion on to the centre PBH and feedback (outflow) from the accreting PBHs. Based on this consideration, the baryons inside the UMCH can be divided into two components: the piled-up baryons inside the UCMH and the accretion spherical flow or disc around the centre PBH. Although the optical depth of the accretion gas or disc, which is mainly contributed by the depth around the inner horizon region r ∼ RSch, is transparent to X-ray photons, the Xray emission can still be trapped and absorbed by piled-up baryons inside the host UCMH and can reradiate photons with much longer wavelength into the outer IGM environment. Quantitative analysis is given as follows. Part of the treatment is similar to an analogous discussion on dark matter structure formation and the baryon-filling process (Hoeft et al. 2006; Okamoto, Gao & Theuns 2008). If UCMH dark matter annihilation does not change the IGM temperature evolution, the IGM temperature is approximately coupled with the cosmic microwave background (CMB) temperature before the decoupling time zdec ∼ 100, and the IGM sound speed before zdec is cs 5.7 km s−1[(1 + z)/1000]1/2. In general, UCMH annihilation heating and PBH emission without trapping increase the IGM temperature. We introduce an amplification factor A such that Tm = ATCMB at z > zdec, where Tm and TCMB are the temperature of the IGM and CMB respectively and A depends on the UMCH profile and annihilation properties, as we will calculate in Section 3. The sound speed cs ∝ T1/2 becomes cs 5.7 km s−1A1/2 [(1 + z)/1000]1/2 and the Bondi accretion radius (i.e., the accretion sonic sphere) of a PBH–UCMH system at z > rB ≈ −1 Equation (19) is derived under the assumption that the Bondi radius is larger than the UCMH size rB > Rh. Furthermore, if rB > 2Rh, the virial temperature of the host UCMH, is greater than the temperature of the ambient IGM gas, Tvir > Tm. According to the general virial theorem, the thermal pressure of the gas due to virialized heating is weak compared with the gravity of the UCMH. In this case we consider that IGM baryons should fall into the UCMH unimpeded, regardless of the centre PBH mass (Hoeft et al. 2006; Okamoto et al. 2008). The criterion rB > 2Rh at z > zdec gives Note that higher IGM temperature around a UCMH, i.e. higher A, gives a higher minimum UCMH mass to attract baryons. A similar result can be derived for the case after decoupling, z < zdec, where the IGM gas temperature decouples from the CMB temperature and drops adiabatically as Tad ∝ (1 + z)2 without any heating sources. We still take the factor A ≥ 1 to measure the IGM temperature increase due to annihilation, Tm = ATad. Then, using the criterion rB > 2Rh, we find that baryons fall into UCMHs unimpeded at z < As a result, if the UCMH initial overdensity seed is δm > 1600A3/2 M for zdec < z < 1000, or δm > 160A3/2 M for z < zdec, the IGM gas can always fill the UCMH no matter whether it includes a PBH or not. Otherwise, for a lower δm, the critical redshift zc below which the UCMH accretes is (1 + zc) < 0.63A−3/2(δm/M ) for z > zdec and (1 + zc) < 13A−3/5(δm/M )2/5 for z < zdec. If the UCMH hosts a PBH in the centre, baryons are still able to pile up and become thermalized in the host UCMH due to gas virialization. The lower bound of the gas accretion rate into the UCMH can be estimated as M˙ UCMH = 4πrB2 csρgas(z) > 4πRh2vff (Rh)mbnb(z) ∼ 8.2 × 1016 g cm−3(1 + z)1/2 where vff (Rh) is the free-fall velocity at Rh. If all the gas in the UCMH is totally accreted on to the centre PBH, the dimensionless accretion rate of the PBH m˙ = M˙ g/M˙ EPdBdH is m˙ > 18.0 with M˙ EPdBdH being the Eddington limit accretion rate on to the central PBH. Note that the ideal m˙ can be even higher if the initial host UCMH is more massive than the centre PBH, δm MPBH, as discussed in Ricotti & Gould (2009). However, the real accretion rate should be lower than the value in equation (23) for two reasons. First, baryons can be heated and virialized during the accretion process in the UCMH and have a temperature ∼Tvir, warmer than the IGM Tm, to increase the gas pressure and decrease the accretion rate on to the PBH. Additionally, the gas temperature is further increased, Tvir, near the PBH due to PBH emission and ionization. Also, super-Eddington accretion discs are also inclined to drive outflows. The positive Bernoulli parameter over most of the ADAFs due to small radiation loss may trigger strong outflows or jets (Narayan & Yi 1994) and produce ADIOs in which outflow carries away most of the flow mass and energy (Blandford & Begelman 1999). Also, CDAFs may produce a ‘convective envelope’ with no accretion on to the black hole (Narayan et al. 2000). In general, accretion discs with super-Eddington accretion rate are inevitably accompanied by outflows and winds, which significantly decrease the final accretion rate on to the black hole. In the PBH case, these outflows should be injected back to the host UCMH environment. The upper bound of the baryonic fraction in the UCMH is the universal fraction b/ m. However, as the UCMH grows following equation (1), we adopt a more conservative method to estimate the lower bound of baryon fraction f b inside the UCMH. We estimate the baryonic fraction in a UCMH, f b (the mass ratio between gas and dark matter), as (M˙ UCMH − M˙ PBH)(t − ti) ∼ mh(z)fb, where we take M˙ UCMH M˙ PBH, i.e. most of the gas accreted into the UCMH is piled up without being immediately eaten by the PBH. Combining equations (22) and (24), we have a lower bound on the baryon fraction of f b ≥ 7.6 × 10−3, which is a constant independent of the redshift. The optical depth of the piled-up gas in the UCMH due to Compton scattering is Since f b from equation (24) is a constant, and UCMH growth does not change the steep region ρDM ∝ r−α with α = 9/4 but only increases Rh and flattens the region r < rcut (see equation [4]), we take the baryon fraction to be uniformly distributed in the UCMH, in both steep and flat regions. Therefore the column density of the baryon gas inside the UCMH depends on the UCMH profile, which depends on the dark matter properties ( σ v , mχ ) given by equation (4). Actually the baryon profile can be steeper in the flat region of the halo, r < rcut, since the dark matter annihilation flattens the inner halo profile, thus giving an even larger optical depth. Furthermore, we consider that the baryon gas is ionized, xe ∼ 1, inside the UCMH, at least in the flat region r < rcut. We check that the Stro¨mgren radius of the PBH emission rS satisfies rcut < rS < Rh, as the heated gas near the PBH can reach a temperature as high as the Compton temperature, ∼10 keV in the ionized region. The hot ionized gas around the PBH produces a small sonic sphere in the dense baryon region near the PBH and decreases the accretion rate on to the PBH, giving M˙ UCMH M˙ PBH as mentioned in equation (24). Note that there should be two distinct sonic spheres, the sphere for the host UCMH outside Rh and that for the centre PBH inside the UCMH. This scenario is similar to that of Wang, Chen & Hu (2006) in which an accreting BH has two Bondi spheres, a smaller inner sphere in the hot gas region and a larger one in the outer cooler region. Therefore we take xe ∼ 1 in equation (25). The optical depth is written as ≥ 10 Hereafter we take σ v s = 1. Combining equations (20) and (26), we conclude that before the decoupling z > zdec the gas is always Compton-thick to X-ray emission from the centre PBH accretion when the host UCMH itself accretes baryons. After decoupling, z < zdec, the redshift range in which X-ray emission escapes is −2/5 A−3/5, From equation (27) there is a maximum zm in the case of rB > 2Rh: zm(rB>2Rh) = 29mχ−,11/030A−3/10 − 1. If z > zm, X-ray photons will be totally trapped. Note that zm insensitively decreases with the increase of IGM temperature factor A. A larger optical depth due to a deeper baryon profile at r < rcut gives an even lower zm. Also, the range of δm applied in equation (27) is In other words, in the rB > 2Rh case no X-rays can escape the host UCMH if δm ≥ 80 M mχ−,51/030. More massive PBHs find it easier to reach the Eddington accretion rate, but more difficult to produce a transparent baryon environment in the host UCMHs. On the other hand, if rB < 2Rh (i.e. Tvir(Rh) < Tm), most of the UCMH gravity potential well is not deep enough to compress the gas and overcome the pressure barrier of gas virialization heating. In this case the UCMH itself cannot accrete and thermalize baryons except for the region within radius rB, which satisfies [Gmh(r ≤ rB)/(2rB)] = cs2. At z > zdec, the critical radius rB for a pure UCMH profile ρχ ∝ r−α is rB = Rh where we apply α = 9/4 from equation (5). The part of the UCMH inside rB can accrete and heat baryons. Similarly to equations (22) to (26), the accretion rate in region r ≤ rB in units of Eddington accretion rate of the centre PBH at z > zdec is m˙ = ≈ 3.9 × 10−11A−15/2 where the factor 2π is due to the suppressed accretion at r > rB in the UCMH; thus the baryon density is half of the ambient gas density. Assuming δm = MPBH, equation (31) shows that only highmass PBHs (MPBH > 100 M ) are able to produce super-Eddington accretion if there is no accretion feedback. This is basically consistent with the results in Ricotti et al. (2008). However, we should mention two points. The first point is that, if δm > MPBH, the ideal accretion rate in equation (31) also increases. The second point, which is similar to the analysis below equation (23), is that the real accretion rate on to the PBH is lower than the ideal m˙ due to higher gas temperature and accretion feedback. As a result, we find that baryons can be accumulated and virialized inside the UCMH region r ≤ rB with a baryon fraction of approximately fb ∼ 10−3A−9/2(1 + z)−3(δm/M )3. Using the condition m˙ > 1 in equation (31) at z > zdec for sufficient radiation efficiency, the baryon optical depth inside radius rB is τ ∼ 1.1 × 10−2A−9/2(1 + z)−13/6m5χ/,9100 where we also take the baryon profile inside the UCMH as proportional to the dark matter profile for simplicity, and xe ∼ 1. According to equation (32), we consider that the UCMH is optically thick to X-ray emission in the case of rB < 2Rh and z > zdec, unless δm MPBH or dark matter particle mass mχ,100 1. After decoupling, z < zdec, we find that baryons can be accumulated and virialized inside a radius 8 3 rB ≈ 0.012 pc A−4 with upper bound zm, such that 32A−3/8 below which (z < zm) the gas is Compton-thin inside radius rB. A higher ratio δm/MPBH 1 or lighter dark matter mχ,100 1 increases zm, which also decreases slightly if the annihilation effect is included to heat the IGM gas (A > 1). Not only can baryons inside UCMHs trap X-ray photons, but also outflows driven by PBH accretion feedback absorb X-ray emission. As mentioned before, the super-Eddington accretion rate on to the PBH is unstable to trigger strong outflows in both spherical and disc cases. An optically thick ‘outflow envelope’, in both polar and equatorial regions around the PBHs, forms to cover the PBH and totally or mostly absorbs X-ray emission from the inner accretion flows (Igumenshchev, Narayan & Abramowicz 2003; Kohri, Narayan & Piran 2005; Poutanen et al. 2007; Abolmasov, Karpov & Kotani 2009). In this case, the Compton heating in the outflow region should be important, in order to increase the gas pressure and temperature, balance the gravity well, reemit thermalized photons from outflows and regulate the accretion rate on to the PBH (Wang et al. 2006). It is likely to have a steady-state or periodically changing outflow envelope covering the whole PBH, but the details are still an open question beyond the purpose of this paper. What we want to show is that, even though the accretion disc itself around a PBH is optically thin to X-ray radiation, X-ray emission can still be absorbed in the outflow envelope due to accretion feedback and disc instability in the super-Eddington case, not to mention optically or geometrically thick discs, which absorb X-ray emission by themselves. We give a summary of Section 2.2. Equation (13) computes the Xray radiation density due to baryon gas accretion on to PBHs. The ratio ηf PBH/f UCMH is parametrized in this paper. The much lower probability of PBH formation compared with UCMH formation and inefficient radiation due to low-mass PBHs (δm 100 M ) or accretion feedback from high-mass PBHs or PBHs with massive UCMHs (δm MPBH) give ηf PBH/f UCMH 1. Moreover, we simply introduce a critical redshift zm below which (z < zm) X-ray emission from super-Eddington accretion PBHs ( m˙ 1) becomes important to heat and ionize the early Universe. A hotter IGM heated by other energy sources (e.g. annihilation) slightly decreases zm. Ricotti et al. (2008) showed that the accretion becomes m˙ 1 at zm ∼ 20 (100) for MPBH = 102 (300) M , and m˙ is always m˙ 1 ( m˙ 1) for MPBH > 103 M (MPBH 10 M ). However, as we discussed in Section 2.2.3, the stage of super-Eddington accretion heating the IGM can be delayed, because X-ray photons are trapped inside the total or inner region of the UCMHs due to the accumulation and virialization of the accretion gas. Outflows can also (partly) absorb X-rays. We take the critical redshift zm as shown in equations (28) and (34); for z < zm, X-ray photons from super-Eddington accretion PBHs could escape their host UCMHs. If the PBH abundance is much less than that of UCMHs, but still satisfies equation (16), XIf we take the monochromatic dark matter annihilation emission for simplicity, i.e. photons produced by dark matter annihilation have the rest energy of the dark matter particle mχ c2, the photon flux spectral density then can be calculated using the δ-function ray emission should dominate over early dark matter annihilation at z < zm. 3 R E I O N I Z AT I O N A N D H E AT I N G O F T H E I G M 3.1 Basic equations The evolution of baryon ionization fraction xion(z) is given by the differential equation (e.g. Cirelli, Iocco & Panci 2009) nA(1 + z)3 dxion(z) = I (z) − R(z), (35) dt where I(z) and R(z) are the ionization and recombination rates per volume respectively and nA is the atomic number density today. We use the rate R(z) from Natarajan & Schwarz (2008). The ionization rate per volume due to dark matter annihilation or X-ray emission from gas accretion is given by I (z) = where Eχ = mχ c2 is the maximum energy of the emitted photon, Eeq Eχ (1 + z)/(1 + zeq) and the differential term dn(z)/dEγ is the photon spectral number density at redshift z. We follow Cirelli et al. (2009, see also Belikov & Hooper 2009; Natarajan & Schwarz 2008, 2009) to calculate the probability of primary ionizations per second P(Eγ , z), and the number of final ionizations generated by a single photon of energy Eγ , Nion(Eγ ). Note that Nion(Eγ ) is proportional to the ionization factor ηion(xion) ≈ (1 − xion)/3 (Shull & van Steenberg 1985; Chen & Kamionkowski 2004), which means that approximately one-third of emitted energy goes into the reionization of atoms if xion 1. First of all, we consider the ionization is due to UCMH dark matter annihilation. The spectral number density is obtained as ∞ where lann is the annihilation luminosity as mentioned in Section 2.1. Eγ (z ) = Eγ (1 + z )/(1 + z). The optical depth τ is with Eγ = Eγ (1 + z )/(1 + z ). The total cross-section σ (tot) for the DM annihilation photon to interact with electrons in the IGM mainly includes the Klein–Nishina cross-section for Compton scattering (Rybicki & Lightman 2004) and the photonionization cross-section for H and He, σH+He (Zdziarski & Svensson 1989). Pair production on matter becomes important for mχ > 1 GeV. CMB photons also contribute to the total cross-section for mχ > 10 TeV, which can be neglected in our case. The total energy deposition per second per volume at redshift z is given by (z) = Thus we have an energy deposition (z) of (z) = where z0 satisfies and τ is calculated from z0 to z. Keep in mind that in the above formula (41) the dark matter annihilation products are simplified as gamma-ray photons with sole energy Eχ = mχ c2. A more realistic annihilation spectrum is model-dependent. For example, dn/dEγ can be chosen following the model in Bergstr o¨m, Ullio & Buckley (1998) and Feng, Matchev & Wilczek (2001). For X-ray photons from an accreting PBH, equation (41) will still be available if we choose σ tot as the X-ray total cross-section, σtot ≈ σH+He + σT, and change Eχ to be the X-ray characteristic energy EX as EX 3 keV (MPBH/M )−1/4 (Salvaterra, Haardt & Ferrara 2005). As the real accreting PBH spectral energy distribution is very model-dependent (Shakura & Sunyaev 1973; Sazonov, Ostriker & Sunyaev 2004; Salvaterra et al. 2005; Ripamonti et al. 2008), in the very first calculation we simplify the X-ray emission as singlefrequency emission at a characteristic energy EX, which can mostly be considered as the peaked energy in the real spectral energy distribution. We will also discuss a more realistic PBH spectral energy distribution in Section 5.3. Now we list the heating and cooling processes in the IGM. The heating of the IGM by UCMH annihilation or X-ray emission can be written as dTm 2 ηheat(xion) (z), (43) dt ann = 3kB nA(1 + z)3(1 + fHe + xion) where the heating fraction ηheat(xion), which shows the portion of energy (z) that goes into heating the IGM, is adopted as ηheat = C[1 − (1 − xiaon)]b, with C = 0.9971, a = 0.2663 and b = 1.3163 (Shull & van Steenberg 1985). We can approximate the He fraction in the IGM as f He 0.073. Moreover, CMB photons can be treated as another heating source for the IGM if the IGM gas is colder than the CMB (Tm < TCMB), otherwise the IGM gas would transfer energy into the CMB environment. The coupling between IGM gas and CMB photons can be important when the difference between Tm and TCMB is significant (Weynmann 1965; Tegmark et al. 1997; Seager, Sasselov & Scott 2000): 4 ≈ kcompTCMBxion(TCMB − Tm), where the coupling rate coefficient kcomp 5.0 × 10−22 s−1. Other IGM cooling terms are dominated by the adiabatic cooling during the expansion of the Universe as H2 (1 − xion − 2fH2 )fH2 [nA(1 + z)3]2, chemical cooling processes such as bremsstrahlung, helium-line cooling, H2 line cooling and hydrogen three-body reaction. HD cooling is important for T < 200 K and low densities (Yoshida et al. 2006), but the gas adiabatic cooling will be dominant in this case. As we mainly pay attention to the evolution of the ionized fraction xion and IGM temperature Tm, we only include the evolution of hydrogen (H, H−, H+, H2) and electron gas (e−) as the main species for ionization. More detailed simulation including other species and cooling processes is beyond the purpose of this paper. The evolution of the H2 fraction is adopted from the semi-analytic model in Tegmark et al. (1997): 3.2 Solutions of IGM evolution Since no direct evidence related to the UCMH radiation has been confirmed until now, the UCMH abundance is still uncertain. In this paper we take the UCMH fraction f UCMH as a free parameter. In Fig. 2 we show the ionization fraction xion(z) and the IGM temperature Tm for f UCMH = 10−4 and 10−6, which corresponds to today’s expected abundance ∼1 per cent and 10−4 respectively. Moreover, general discussion on the UCMH abundance will be given later. The initial xion at z = 1000 is taken as 0.01, and Tm as the CMB temperature (Galli & Palla 1998; Ripamonti 2007; Ripamonti et al. 2007a,b). The first basic conclusion, which is similar to the previous work, is that lighter dark matter particles or higher UCMH abundance give larger xion and higher Tm. An extremely bright UCMH annihilation background (e.g. mχ c2 ≤ 1 GeV for f UCMH 10−4 or mχ c2 ≤ 100 MeV for f UCMH 10−6) even gives a monotonic increase in xion and Tm ≥ 104 K without a standard reionization epoch in the early Universe. We compare the UCMH annihilation results with the homogeneous background annihilation; note that the homogeneous dark matter annihilation background only produces noticeable effects for light dark matter particles mχ c2 < 1 GeV or sterile neutrinos. UCMHs, which provide a new dominant dark matter annihilation gamma-ray background as shown in Section 2.1, play a more important role in ionizing and heating the early Universe. Furthermore, as the cosmological Jeans mass mJ can be taken as an indicator of IGM structure evolution, the left panel of Fig. 3 gives the evolution of Jeans mass in the cosmic UCMH annihilation background. The Jeans mass mJ ∝ T3m/2ρ−1/2 should be a constant if the gas temperature is always equal to the CMB temperature Tm = TCMB. In this paper we call this constant ‘CMB mass’. In the left panel of Fig. 3, mJ with various Tm is generally normalized in units of ‘CMB mass’. The remaining Thomson scattering optical depth contributed by UCMH annihilation is shown in the right panel of Fig. 3. For 6 ≤ z < 30, the remaining CMB optical depth is estimated as δτ 0.046 ± 0.016 by the WMAP five-year measurement.2 We assume a linear increase of xion from z = 10 to the full ionization 2 The WMAP five-year measurements give the CMB Thomson scattering optical depth τ 0.084 ± 0.016 (Komatsu et al. 2009), which is mostly due to ionization at late times z < 30 (Ricotti et al. 2008; Natarajan & Schwarz 2010). If we subtract the optical depth contributed by the totally ionized gas τ (z ≤ 6) = 0.038, the remaining depth is δτ 0.046 ± 0.016 ≤ 0.062 (Cirelli et al. 2009). time z = 6; thus the upper bound contribution of UCMH annihilation to the measurable CMB optical depth is δτ 0.028 ± 0.016 ≤ 0.044. Based on this consideration, in our examples only one extreme case, mχ c2 = 100 MeV with f UCMH = 10−4, is ruled out by the CMB remaining optical depth in Fig. 3. UCMH annihilation can significantly increase the Thomson optical depth in the early Universe, z 100, up to δτ ∼ 0.5 without stringent constraints from the CMB optical depth measurement at z < 30. After the last scattering epoch, the ionization fraction xion can change from xion ∼ 10−4 (without dark matter annihilation) to an upper bound xion ∼ 0.1 (e.g, mχ c2 = 1 GeV and f UCMH = 10−4). Also, the IGM can be heated from a temperature Tm of adiabatically cooling Tm ∝ (1 + z)2 in the absence of a heating source to the upper bound Tm > 103 K with a sufficient amount of heating contributed by UCMH annihilation. A Jeans mass much higher than the ‘CMB mass’, an increase of ∼2–3 orders of magnitude, can be obtained due to the hotter IGM temperature. Therefore, we can natively estimate that the formation of small baryonic objects can be strongly suppressed, although more investigations need to be carried out in Section 4. In general, we find that the impact of UCMH annihilation on IGM evolution can be empirically estimated by the factor mχ−,1100f UCMH, while the threshold of UCMH abundance affecting IGM evolution is approximately given by mχ−,1100f UCMH > 10−6, with an upper bound constrained by the CMB optical depth at late times z < 30 as xion ∼ 0.1 and Tm ∼ 5000 K at mχ−,1100f UCMH ∼ 10−2. The CMB optical depth enhancement at early times z > 30 can be more dramatic than that at late times due to the higher annihilation luminosity at early redshifts (equation 6). This is different from the PBH radiation, which has higher luminosity at late times (Section 2.2.3; Ricotti et al. 2008). Further phenomenological constraints should be made by CMB polarization anisotropies; this is left for a future investigation. Keep in mind that another channel to concentrate dark matter rather than primordial density perturbations is the formation of the first dark objects, which should affect the IGM evolution much later (z < 100) than UCMHs. Therefore UCMH annihilation definitely has a much earlier and more important impact on IGM evolution from last scattering to structure formation time. So far we have given results only for UCMH dark matter annihilation. Whether the X-ray emission from the PBH-host UCMHs will significantly change the above results depends mainly on the fraction of PBHs f PBH, the average inflow radiation efficiency η and the critical redshift zm, as given in Section 2.2. In our paper we combine the product ηf PBH as one parameter. Remember the results in Section 2.2: when X-rays from the centre PBH region successfully passes through the transparent baryon medium in the host UCMH at z < zm, the much brighter X-ray luminosity and much larger interacting cross-section σ tot(EX) compared with the annihilation luminosity and σ tot(Eγ ) usually guarantee that X-ray emission dominates over UCMH annihilation (equations 15), except for a much lower PBH fraction f PBH below the value in equation (16). We will give a lower limit of f PBH above which X-rays have obvious impact on the IGM evolution at z < zm. In the following calculation we assume that equation (16) is always satisfied and do not distinguish between the redshift dividing the UCMH radiation into annihilation-dominated or X-ray-dominated and the redshift giving a transparent baryon environment in UCMHs, but simply use a single parameter zm. In Fig. 4 we take zm and ηf PBH as parameters with characteristic emission frequency EX = 1, 10 and 100 keV, which corresponds to typical PBH masses of 102 M , 10−2 M and 10−6 M respectively. Higher zm means a higher ratio δm/MPBH or lighter dark matter particles. Only X-ray emission as an energy source is calculated in this figure.3 As shown in Fig. 4, the final properties of the IGM at z ∼ 10 with the same ηf PBH and EX are more or less close to each other regardless of the value of zm, which means that lower zm gives more dramatic thermal and chemical change at z < 3 The more realistic case is that both zm and EX are functions of the mass of PBH and its host UCMH, thus in principle neither zm nor EX are single free parameters. The real X-ray radiation background due to PBH accretion happens at a certain zm as shown in equations (28) and (34), and changes its spectral energy distribution as a function of redshift (Ripamonti et al. 2008). Technically one may calculate the X-ray emission variation by introducing the initial mass function of PBHs and the host UCMHs. However, both of these initial mass functions are poorly known. More elaborate models only make our model more complicated and uncertain. What we focus on in our calculations is the key differences between X-ray radiation and the much earlier occurring UCMH annihilation, so we use only the characteristic energy EX and the ratio ηf PBH/f UCMH to show the importance of the X-ray radiation. More discussion of PBH spectral energy distribution can be seen in Section 5.3. zm. According to equation (34), lower zm corresponds to lighter mχ , which also increases the UCMH annihilation. On the other hand, xion and Tm vary by more than three orders of magnitude from EX ∼ 100 keV (10−6 M ) to EX ∼ 1 keV (102 M ) with the same ηf PBH, which means that massive PBHs favour IGM ionization. Also, the X-ray radiation effect can be neglected when ηf PBH ≤ 10−11 for EX ∼ 100 keV, but a smaller limit ηf PBH ≤ 10−12 is applied for EX ∼ 1 keV. Below the lower limit, the PBHs are not expected to have any promising effects on reionization. The estimate in Fig. 5 shows that no strict constraints are made for ηf UCMH ≤ 10−7 by the remaining CMB depth δτ , which allows a dramatically increased Jeans mass due to the hot IGM gas ∼104 K. As a result, X-ray emission from PBHs gives a more promising impact on the IGM evolution if ηf PBH 10−11 (10−12) for MPBH ∼ 10−6 M (102 M ), or empirically ηf PBH 1.8 × 10−12(M/M )−1/8. As we assume f UCMH ≤ 10−4, we expect that the UCMH dark matter annihilation only played its role on IGM evolution at very high redshift zm < z < 1000, but X-ray emission changes the Universe reionization history dramatically at relatively lower redshift z < zm. Considering η ∼ 0.1, the upper-bound value f PBH ≤ 10−6 is two orders of magnitude higher than the upper PBH abundance ≤10−8 in Ricotti et al. (2008) for MPBH > 103 M , but much lower than the low-mass PBH abundance constraint in Ricotti et al. (2008). However, the massive PBH abundance constraint in Ricotti et al. (2008) is made by the Compton y-parameter estimate at zrec < z < zeq without local UCMH trapping, while we consider two-step accretion, first by host UCMHs and then by centre PBHs, as mentioned in Section 2.2.3. X-rays can locally heat the accreted gas inside UCMHs but not the entire cosmic gas at high redshift. The abundance of IGM molecular hydrogen fH2 in various UCMH radiation models is shown in Fig. 6 in this section. We see that when UCMH energy injection can be neglected, this fraction returns to fH2 ∼ 10−6, which is consistent with the standard result (e.g. Galli & Palla 1998 and references therein). The upper bound of enhanced fH2 is fH2 ∼ 10−3, due either to the allowed UCMH dark matter annihilation constrained by the CMB optical depth or the X-ray emission at zm ≤ 100. 4 F I R S T S T R U C T U R E S In the hierarchical cold dark matter (CDM) scenario, the first cosmological objects are dark matter haloes, which are formed by gravitational instability from the scale-free density fluctuations (Diemand, Moore & Stadel 2005; Green, Hofmann & Schwarz 2005; Yoshida 2009). The formation of the first baryonic objects depends on the detailed gas dynamical processes. The first baryonic objects can successfully collapse and form inside dark haloes when the cooling time-scale for dissipating the kinetic energy is much shorter than the Hubble time. Tegmark et al. (1997) showed that the formation of baryonic structure crucially depends on the abundance of molecular hydrogen fH2 . Biermann & Kusenko (2006) considered that photons emitted by dark matter annihilation or decay inside a halo can boost the production of H2, and may favour the formation of the first structure. On the other hand a different conclusion, that dark matter annihilation or decay can slightly delay the first baryonic structure formation, was given by Ripamonti et al. (2007b). As they discussed, the higher central density in the baryonic cloud without dark matter energy injection could compensate for the lower abundance of H2 and still lead to the fastest cooling. Nevertheless, regardless of the promotion or suppression caused by halo extended dark matter annihilation or decay, such effects are very small. UCMHs can also be captured by the first dark matter objects; in this case, the UCMH radiation can be much brighter than that from the extended large dark matter halo, even if the fraction of UCMHs in the dark halo is tiny. In this section we focus on dark halo structures with mass ∼106 M (or 107 M in the PBH heating case), as they favour later first star formation (Broom et al. 2009; Yoshida 2009). For a typical 106-M halo in the early Universe, ∼1020 (1018) or ∼108 (106) UCMHs within this halo can be expected for an initial UCMH fraction f UCMH ∼ 10−4 (10−6), if the seeds of UCMHs are generated in electroweak or quantum chromodynamic (QCD) phase transitions respectively (Scott & Sivertsson 2009). In these cases, UCMH emission can also provide a new type of radiation background in the dark halo. However, the uniform UCMH distribution treatment will break down if the number of massive UCMHs in a halo is less than ∼10. This happens for the massive UCMH case f UCMHMDM/mh ≤ 10. In this section we study the effects of UCMH radiation on the first structure formation and evolution with f UCMHMDM mh; the case of ‘only several luminous UCMHs’ will be discussed separately in Section 5.4 as a supplement. As the virial temperature of ∼106 M haloes is less than the threshold for atomic hydrogen line cooling, these haloes are often referred as ‘minihaloes’ in the literature. However, for clarity, in this paper we call the first dark matter structures (cosmological) dark matter haloes or dark haloes, which should not be confused with UCMHs. 4.1 Dark matter annihilation The profiles of cosmological dark matter haloes are chosen before our calculation. The equations for the halo profile are listed in the Appendix. Before the formation of the first stars, the energy injection inside a large dark matter halo is mainly contributed by local emission from UCMHs in the halo, local annihilation or decay of the extended dark matter within the halo, and the outside radiation background that is injected into the halo. We check the total energy produced by dark matter annihilation with a dark halo, i.e. Lhalo = LUCMH + Lext, with LUCMH and Lext being the annihilation luminosity from UCMHs and the extended dark matter in this halo. The typical ratio LUCMH/Lext is demonstrated in Fig. 7, where we adopt an isothermal dark halo model and f UCMH = 10−6. We find that both LUCMH and Lext are proportional to the total halo mass MDM, so LUCMH/Lext is independent of MDM. In Fig. 7, LUCMH is mostly dominant in the halo, except for large zvir with light dark matter particles (e.g. <10 GeV for f UCMH = 10−6 and zvir = 100; much lighter dark matter particles are required for more abundant UCMHs or smaller zvir). Therefore, similarly to the IGM environment, the energy injection mechanism inside a dark halo that contains UCMHs can be very different from the no-UCMH case. We focus on the ionization and heating inside the dark matter halo. Also, we mention that the results for LUCMH/Lext in Navarro, Frenk & White (1996) haloes are very similar to the isothermal haloes in Fig. 7. The simple criterion of baryonic matter filling in a dark halo is approximately taken as the IGM gas temperature being cooler than the virial temperature of the halo, Tm < Tvir, otherwise the gas pressure prevents the gas from collapsing. We do not include the temperature cooling-slope criterion as in Tegmark et al. (1997) to study further baryon cooling and collapse in the dark halo. Fig. 8 gives the dark matter halo mass for the critical case Tm = Tvir with UCMH annihilation heating the ambient IGM gas. Generally, more massive dark halo mass is needed for gas filling into dark haloes and forming baryonic structure in the haloes. If Tm = TCMB, the minimum halo mass for gas infilling is 6.1 × 103 M , while the minimum halo mass increases significantly for Tm TCMB. In this sense, the formation of the first baryonic objects will obviously be suppressed in small dark haloes located in host IGM gas. However, in the pure UCMH annihilation case without PBH radiation, we expect that ∼106 M dark haloes attract baryons at z > 10 in most cases, expect for f UCMHmχ−,1100 ≥ 10−2. Figure 8. The minimal dark halo mass for Tm = Tvir for an IGM heated by UCMHs. The shaded area is for the case in which the minimal halo mass for gas collapse becomes MDM < 6.1 × 103 M , due to a lower ambient gas temperature compared with the CMB temperature. The lines from mχ c2 = 100 MeV–500 GeV are as in Fig. 3. The annihilation energy deposited in a dark halo can be linearly divided into two parts: the energy from the background where cosmic UCMH annihilation occurs, bgd(z), and that within the local dark halo, loc(z). The term bgd(z) is obtained by equation (39). The local energy loc(z) is a function of position inside the halo. We focus on energy deposition at the centre of the halo. The contribution by the extended dark matter in an isothermal halo is loc,iso(z) = 6 Rc6ore 5 Rcore − 5Rt5r where f χ = DM/ M ≈ 0.833. On the other hand, the local energy deposited by UCMH annihilation depends on the UCMH distribution inside the halo. If we assume that the UCMH number density is uniformly distributed depending on the halo mass density, i.e. dnUCMH(r)/dMDM(r) ∝ constant (for a relevant distribution simulation see Sandick et al. 2011), we have 4 Rc4ore 3 Rcore − 3Rt3r In the following calculation we adopt equation (49) for UCMH annihilation and also include extended dark matter annihilation within the halo. Fig. 9 shows the gas evolution at the centre of a 106 M isothermal halo virializing at zvir = 20 or zvir = 100. We also show the protohalo stage at z > zvir. More energetic annihilation due to larger f UCMH or lower mχ gives higher xion and fH2 . The UCMH annihilation gives a significant impact on Tm before virialization in the protohalo stage; this is because the cooling and heating mechanisms are different before and after virialization. The change of Tm before virialization is mainly due to heating by background UCMH annihilation, which is completely dominant over extended dark matter annihilation; thus, brighter UCMH annihilation luminosity gives a higher gas temperature at z > zvir. However, after a dramatic temperature increase during the virializing process, z ∼ zvir, H2 cooling becomes the main process to cool the denser gas at z < zvir. The peak temperature during virializing is around ∼1000–2000 K. We find that higher H2 abundance, which is caused by brighter UCMH annihilation, gives a lower gas temperature after virialization for small zvir (∼20) but a higher temperature for large zvir (∼100). This result is between those in Biermann & Kusenko (2006), who considered that the effects of sterile neutrino decay can favour structure formation, and Ripamonti et al. (2007b), who showed that dark matter annihilation will slightly delay structure formation. The main reason for our difference from Ripamonti et al. (2007b) for zvir 100 is that we take the baryon gas density to be proportional to the halo density nb ∝ ρ as in Tegmark et al. (1997), therefore more molecular gas due to stronger heating just means more efficient cooling. A more elaborate result can be made by adding more detailed gas dynamics and energy transfer including the UCMH radiation within the halo (Tegmark et al. 1997; Ripamonti et al. 2007b). However, such a new calculation should not change the fact that UCMH radiation, as well as extended dark halo annihilation, cannot change the gas temperature in the halo in an obvious way after virialization. Figure 10. The minimal dark halo mass at Tm = Tvir for PBH radiation with ηf PBH = 10−7 (dash–dotted lines), 10−8 (solid lines), 10−9 (dashed lines), 10−10 (dotted lines) and zm = 100 (dark lines), 50 (red lines) and 20 (blue lines). The characteristic PBH radiation is EX = 10 keV. The shaded area is for the case in which the minimal halo mass for gas collapse becomes MDM < 6.1 × 103 M due to a lower ambient gas temperature compared with the CMB temperature. Note that the temperature Tm in the halo only changes by a factor of ∼3 for a several orders of magnitude change to the UCMH annihilation luminosity inside the halo. Therefore, we cannot expect that the first baryonic structure formation can be obviously promoted or suppressed. After virialization, the effects of dark matter annihilation are always secondary compared to the H2 cooling mechanism. On the other hand, gas chemical properties such as xion and fH2 can be changed significantly, so that higher xion and fH2 are produced by brighter dark matter annihilation. 4.2 Gas accretion on to PBHs and X-ray emission If we consider gas accretion on to PBHs, and take the PBH fraction as ηf PBH > 10−11 (10−12) for MPBH = 10−6 M (102 M ), the first baryonic structure formation will be different. Fig. 10 shows the minimal dark matter mass for the critical case Tm = Tvir with different ηf PBH and characteristic radiation EX = 10 keV. In contrast to the dark matter annihilation heating case (Fig. 8), the minimal dark halo mass increases dramatically after zm. For zvir = 20, the minimal dark haloes increase to >106 M for ηf PBH ≥ 10−8. Moreover, if we combine the annihilation before zm with the X-ray emission after zm, the minimal dark halo mass for Tm = Tvir can be even larger. Moreover, we expect that gas accretion on to PBHs in the dark halo environment above the critical mass in Fig. 10 will be slightly different from that in the ambient IGM, because the baryon gas is denser within a dark halo than in the ambient IGM, which leads to a different accretion rate and baryon fraction inside UCMHs compared with IGM-located UCMHs. The accretion rate M˙ UCMH and baryon fraction f b inside a PBH-host UCMH should be higher than those outside the halo. Therefore it is more difficult for X-rays from PBHs to pass through the host UCMH without absorption. The critical redshift zhalo for X-rays escaping from the UCMH baryonic m environment should be slightly delayed inside the halo compared with the background zhalo < zmbkgd. It is possible that in a period m of time the X-ray energy injection and deposition within a halo are mainly from the background, even after virialization. The evolution of the baryonic structure inside a 107-M isothermal dark halo, as an example, is shown in Fig. 11. We consider two models: the number density of PBH-host UCMHs being uniformly distributed per halo mass, as mentioned in Section 4.1, and also uniformly distributed UCMHs per halo volume inside the halo, i.e. dnUCMH/dVhalo ∝ constant. Different UCMH distributions inside the halo will give different baryonic evolution. The local energy deposition for the UCMHs with uniform distribution per halo volume is written as where Lacc is the total X-ray luminosity due to gas accretion on to PBHs inside the first baryonic objects. The main results in Fig. 11 are very similar to those of dark matter annihilation in Fig. 9. For low virialization redshift, zvir = 20, the gas temperature is cooler for higher X-ray luminosity but Tm inside the halo only changes by a factor of 2–4 for a four orders of magnitude change in X-ray luminosity. A more obvious change of chemical quantities (xion, fH2 ) than the temperature change occurs for different X-ray luminosity within the halo. This means that the effects of X-ray emission from PBHs on the gas evolution inside a halo are very small. PBHs that are uniformly distributed per halo volume give a slighter cooler gas within the halo, as well as lower xion and fH2 than uniformly distributed UCMHs per halo mass. That is because the latter model makes the average distance of UCMHs closer to the halo centre and this yields stronger effects to change the gas properties at the centre. In summary, UCMH radiation including both annihilation and PBH gas accretion enhances the baryon chemical quantities such as xion and fH2 inside dark matter haloes above the minimal halo mass for Tm = Tvir, but the impact of UCMH radiation on the temperature of the first baryonic objects is small (by a factor of several), which shows that the change in first baryonic structure formation due to UCMH radiation is less important than the H2 cooling and dark halo virialization time. However, the new chemical conditions provided by UCMH radiation can be more important in affecting later gas collapse and the first star formation after first baryonic object formation, because more abundant H2 and electrons acting as the cooling agents can cool the gas more efficiently during the gas collapse process and provide a lower fragmentation mass scale and first star mass (e.g. Stacy & Bromm 2007). As we mainly focus on the first baryonic structure formation and evolution, the calculation of the first star formation due to the changed gas chemical components should be investigated in more detail in future. 5 D I S C U S S I O N 5.1 Status of UCMH radiation in reionization, other sources A variety of cosmological sources can reionize and heat the IGM at different redshifts before z 6. So far we have shown that UCMHs, even if they merely occupy a tiny fraction of the total dark matter mass, provide a new gamma-ray background for gas heating and ionization. Also, the X-ray emission from accreting PBHs could change the IGM gas evolution history dramatically after zm 1000, where the value of zm depends on the masses of PBH, host UCMHs and dark matter particles. Furthermore, we investigate how both dark matter annihilation and X-ray emission from UCMHs can dominate over the annihilation of extended dark matter haloes. Therefore UCMHs are also an important energy source in dark matter haloes before the first star formation. In this section we briefly review all candidate energy sources during the Universe’s reionization era, 10 ≤ z < zeq. In particular, we emphasize the importance of UCMH radiation among all these sources at different times. In this paper we focus on heating and ionization processes after the last scattering epoch z ∼ 1000, but some interesting effects can be produced by primordial energy sources at earlier times z > 1000. For example, cosmic gas heating and CMB spectral distortion at zrec < z < zeq produced by PBH gas accretion can be used to constrain the PBH abundance, so that f PBH ≤ 10−8 for MPBH ≥ 103 M in the absence of UCMH annihilation (Ricotti et al. 2008). However, as shown in Section 2.2, UCMH annihilation can be more important than PBH gas accretion in the very early Universe even though UCMHs are just beginning to grow at that time. Hotter cosmic gas heated by dark matter annihilation suppresses the PBH gas accretion to become the dominant source to distort the CMB, and even changes the cosmic recombination process, as shown in Fig. 2. The Compton y parameter is likely to be used to constrain the UCMH abundance based on the annihilation scenario in future work. The influence of dark matter annihilation or decay at z ≤ 1000 on the IGM during the reionization era has been discussed by many authors (Chen & Kamionkowski 2004; Hansen & Haiman 2004; Pierpaoli 2004; Mapelli & Ferrara 2005; Padmanabhan & Finkbeiner 2005; Zhang et al. 2006; Mapelli et al. 2006; Ripamonti 2007; Chluba 2010; Yuan et al. 2010). Some authors also use observational data to constrain the cross-section of dark matter interaction (Cirelli et al. 2009; Galli et al. 2009; Slatyer, Padmanabhan & Finkbeiner 2009; Kanzaki, Kawasaki & Nakayama 2010). The basic assumption is that the dark matter distribution is smooth and homogeneous at z ≥ 100. However, the annihilation power can be strongly increased by UCMHs in the early Universe, as discussed in this paper. The next commonly suggested source of ionization and heating is the first dark objects (dark haloes), which formed approximately at z ≤ 100. Dark matter haloes enhanced the overall cosmic dark matter annihilation density due to the dark matter concentration in haloes (Iliev, Scannapieco & Shapiro 2005; Oda, Totani & Nagashima 2005; Ciardi et al. 2006; Chuzhoy 2008; Myers & Nusser 2008; Natarajan & Schwarz 2008; Belikov & Hooper 2009; Natarajan & Schwarz 2009; Natarajan & Schwarz 2010). The mass distribution of dark haloes varies from very low mass (∼10−6 M ) to high mass (∼1012 M ) (Green et al. 2005; Diemand et al. 2005; Hooper et al. 2007), depending on the different damping scales due to different dark matter models (Abazajian, Fuller & Patel 2001; Boehm et al. 2005) as well as the mass–halo function (Press & Schechter 1974; Sheth & Tormen 1999). If UCMHs collapse together with homogeneous dark matter, the UCMH annihilation flux could still dominate over the total annihilation flux within dark matter haloes, at least in massive dark haloes with f UCMHMDM mh. However, remember that small dark matter haloes contribute a significant part of the total annihilation rate after structure formation. The profiles of earth-mass dark matter haloes and the gamma-ray flux due to annihilation have also been studied recently (Diemand et al. 2005; Ishiyama, Makino & Ebisuzaki 2010). Similarly to UCMH emission, a large enhancement in the annihilation signal is also expected due to emission from dark matter subhaloes as the remnant of structure formation at z < 60. The issue of whether UCMHs or small haloes are more important for ionization and heat after z ∼ 60 should be investigated in future. The next ionization sources are accreting PBHs, which are located in their host UCMHs, as mentioned in Section 2.2. PBHs with host UCMHs lead to a faster accretion than naked PBHs, but also absorb the X-ray emission due to baryon accumulation within the UCMH. The PBH accretion can only be more important than UCMH annihilation at z ≤ zm < 1000 with sufficient abundance and radiation efficiency. Keep in mind that the X-ray emission here is from PBHs, or perhaps PBH–UCMH systems, which play an earlier role than the so-called accreting ‘first black holes (BHs)’, which are the remnants of the first stars at z ∼ 15. As mentioned in Section 4, dark matter annihilation or X-ray emission affects baryonic structure formation and evolution. Also, it affects the process of the first star formation. The standard first star formation is carried out at z ∼ 20 (Abel, Bryan & Norman 2002; Broom et al. 2009), but the first star-forming history can be affected by primordial magnetic fields (Tashiro & Sugiyama 2006) or by extended dark matter annihilation in the halo (so-called ‘dark stars’: see Spolyar, Freese & Gondolo 2008; Spolyar, Freese & Gondolo 2009). Previously it was said that the first stars gave the first light to end the cosmic ‘dark age’, which cannot be true if exotic sources such as dark matter annihilation and accreting PBHs are included. The next ionization sources are more familiar to us. The first stars emitted UV light and produced ‘ionized bubbles’, which could partially ionize the Universe at z < 20 directly or could affect the forthcoming formation of next-generation stars and later galaxy formation (e.g. Haiman & Loeb 1997; Wyithe & Loeb 2003; Shull & Venkatesan 2008; Whalen, Hueckstaedt & McConkie 2010). The death of the first stars produced ‘first-generation BHs’, which emitted X-rays and ionized the Universe at z ∼ 15 or even closer (Cen 2003; Ricotti & Ostriker 2004; Madau et al. 2004; Ricotti et al. 2005; Ripamonti 2007; Thomas & Zaroubi 2008). The reionization process was completed after galaxy formation, as galaxies are generally considered to be the main candidates for reionization of the Universe at z ∼ 6 (Meiksin 2009, and references therein). Future work that can be carried out includes studying the heating and ionization processes at z > 1000 due to annihilation, comparing the total annihilation rate from small dark haloes (10−6 M < MDM < f U−C1MHmh, as mentioned above) with that from UCMHs and distinguishing the impacts of different ionization sources using CMB polarization anisotropies and observational constraints from 21-cm spectra. Actually, CMB polarization and the hydrogen 21cm line are powerful potential probes of the era of reionization with which to constrain early energy sources. The high multipoles of polarization anisotropies may be able to distinguish between UCMHs and small dark structures formed at z < 100 and to constrain UCMH and PBH abundances further. 5.2 Different UCMH profiles Remember that in Section 2.1, although several UCMH annihilation rates due to various profiles were given in Fig. 1, we chose the UCMH profile as ρ ∝ r−9/4 with a cut-off at ρmax ∝ (t − ti)−1. Such a profile is based on the analytical solution of radial infall on to a central overdensity (Bertschinger 1985). A shallower density profile ρ(r) ∝ r−1.5, which is given if the central accretor is a black hole, or a steeper profile ρ(r) ∝ r−3, simulated by Mack et al. (2007), will change the total annihilation luminosity of a UCMH significantly. Fig. 12 gives an example of the different annihilation luminosities due to different density profiles in an entire UCMH. A more concentrated dark matter distribution in a steeper profile leads to a much higher total annihilation rate within the UCMH, because the centre region of a UCMH contributes a major part to the total annihilation rate. However, we conclude that the overall cosmic annihilation luminosity density equation (10) will not be changed too much, for two reasons. First of all, the ρ ∝ r−3 profile usually appears in the outer region of a UCMH, but the change in UCMH density profile at the outer region r rcut will not dramatically change the total annihilation rate, as the density distribution in the central region is crucial to determine the total annihilation rate. Secondly, the contribution of PBH-host UCMHs (with profile ρ(r) ∝ r−1.5 near the centre) to the overall cosmic annihilation should be much less important than the initial overdensity-seeded UCMHs (ρ(r) ∝ r−2.25), due to both their shallower inner profile and their much lower abundance f PBH f UCMH. The annihilation luminosity should still be taken into account if the dark matter particle inner trajectory is highly eccentric with a much closer pericentre than the cut-off radius, as in equation (4), but it is still lower than the luminosity of overdensity-seeded UCMHs as shown in Section 2.1 (Lacki & Beacom 2010). 5.3 Extended radiation spectral energy distribution In Section 3 the products of annihilation of two dark matter particles are assumed to be two gamma-ray photons, both with energy mχ c2, and X-ray photons emitted from PBH-host UCMHs are characterized by a single energy reflecting the centre PBH mass, EX 3 keV (MPBH/M )−1/4. This simplified treatment with a monochromatic (i.e. δ-function) spectrum of local UCMH emissivity is a good approach to demonstrate crucial UCMH effects on the IGM evolution dependent on the most crucial parameters such as UCMH and PBH abundances f UCMH and f PBH, as well as dark matter particle mass mχ . In this section we will experiment with other extended photon spectral energy distributions (SEDs) for UCMH radiation, rather than a δ-function, and will study the dependence of UCMH density can be written as In this case the energy density can be increased compared with the monochromatic spectrum if E1 > Ei, or decreased for E1 < Ei. However, the logarithmic enhancement ln (E1/Ei) in the X-ray emission case is less significant than in the annihilation case. For a more general power-law spectrum, F(E, z) ∝ E−a (E1 ≤ E ≤ E2) with a > 1, the number-density spectrum and energy deposition are calculated as (1 + z)−3/2 emissivity on the SED. We will see that more elaborate considerations can quantitatively change the ionization results, but will not qualitatively change the basic conclusions regarding UCMH radiation. We first adopt a power-law spectrum F(E, z) ∝ E−a and then discuss another spectrum F(E, z) ∝ Ebexp ( − dE); these are the two most commonly used SEDs for annihilation and BH X-ray emission. First of all we give an analytic calculation for UCMH radiation with a locally monochromatic spectrum, based on an approximation that the optical depth described in equation (38) can be neglected, τ < 1. This condition is applied to the spectrum Eγ (EX) 10 keV. Under the approximation τ 1, equation (37) in Section 3 can be simplified as where the emitted photon number n and luminosity l(z) can be applied to both annihilation [nann, lann(z)] and X-ray luminosity [nX, lacc(z)], E0 for the characteristic energy Eχ or EX respectively, and l(z) ∝ (1 + z)ζ with ζ ≈ 4 for annihilation and ζ ≈ 3 for X-ray emission. We take the interaction cross-section between photons and the IGM gas as σ tot(E) ∝ E−k. The total energy deposition per second per volume (equation 40) is integrated as −1 where the subscript δ is marked for a monochromatic SED as in equation (40). For high-energy gamma-ray photons we can take k ≈ 2 for the Klein–Nishina cross-section, but for X-ray photons we take k ≈ 0 for EX 1 keV. Another thing we mention is that ionization rate I(z) ∝ (z) for xion 1, so we only track (z) based on different SEDs. The first extended SED is F(E, z) ∝ E−1 for E1 ≤ E ≤ E2, which gives a spectral number density for τ 0 of where = E2/E1. Note that the number spectrum equation (53) is different from equation (51) in the monochromatic SED case. If we consider dark matter annihilation, k 2, the total energy deposition for UCMH annihilation is written as Compared with equation (52), the most obvious difference is the additional two factors (E0/E1)k and (1 + zeq)k/(1 + z)k, which significantly increase the annihilation energy density if E0 E1. However this energy amplification for a power-law SED might be overestimated, as we take the lower limit of the energy (40) as Eeq = E1(1 + z)/(1 + zeq), i.e, the Universe at redshift z can receive emission from the matter–radiation equality era. A more general expression (E1/Ei)k can be used instead of (1 + zeq)k/(1 + z)k, with Ei being the threshold-energy photons from UCMH radiation; the amplified factor compared with equation (52) can then be written as (E0/Ei)k 1, i.e. no direct relation betweenE1 and E2. On the other hand, for X-ray emission from a PBH-host UCMH, since k ≈ 0 and the cross-section is more or less the Thomson cross-section, the energy a+k−1 If we do not focus on the linear changed factors such as (a − 1)/[a + (11/2) − ζ ], in most cases the power-law spectrum will significantly increase the energy deposition (z) as well as the ionization rate I(z) (even a dramatic increase in some cases) by a factor of (E0/Ek)k(E1/Ei)a+k−1. A blackbody-like or multicomponent spectral distribution F(E, z) ∝ Ebexp ( − dE) with a peak E0 and c > 0 can be approximately written as F(E, z) ∝ xb for x < 1 and F(E, z) ∝ exp ( − dx) for x > 1, where x = E/E0. In this case we find the number-density spectrum as + (f (d) − 1) where A is an algebraic factor A = [(1 + b)−1 + d−1]−1 and 0 < f (d) < 1 can be obtained numerically, which is not important for the following discussion. The terms in the number-density spectrum are proportional to Ebγ or E11/2−ζ . According to equation (58), the γ integrated energy density (z) is expected to be similar to equation (52) for b > k − 1. Otherwise the enhancement should be (z) ∝ δ(z)(E0/E1)k−b−1. A steep xc with c > 1 for annihilation k ∼ 2 and all the blackbody-like SEDs for X-rays k 0 are more or less similar to the δ-function SED at E0. Now we study the case of EX 10 keV for X-ray emission from very massive PBHs. Photon absorption in the IGM is important for EX ∼ 1 keV. The mean free path of X-ray photons undergoing redshift change z can be written as or we have z ∼ 0.40(1 + z)−1/2 (1 + z)−5/2nA(1 + z)3σtot(1 keV) −k where k 3.3 for the photonionization cross-section. Similarly to the former calculation, the energy density for a δ-function spectrum is lacc(z)nA m Note that we now have a shallower density evolution δ(z) ∝ (1 + z)3 compared with the transparent propagation for the case of equation (54), δ(z) ∝ (1 + z)ζ +3/2. For a power-law SED, F(E, z) ∝ E−a (E1 ≤ E ≤ E2), we obtain the energy density (z) ∼ a−1 a−1 Therefore the energy density is enhanced for E1 > Ei, but the enhancement is less significant compared with that of equation (57) for the same spectral index a. In brief summary, the strength of UCMH radiation depends on its extended spectral energy distribution, which will increase the properties of IGM ionization I(z) and heating (z). Compared with the basic results with a monochromatic spectrum Eχ for annihilation or EX for X-ray emission, a power-law spectrum ∝ E−a with a > 0 can increase the energy density effectively, but a blackbody-like spectrum is more like the monochromatic SED case. For local heating of EX ∼ 1 keV, the heating increases are less significant than for the transparent propagation case τ 1. Also, a power-law spectrum changes (z) and I(z) more significantly for annihilation than for X-ray radiation. 5.4 More massive UCMHs inside the first dark haloes In Section 4 we assume that the number of UCMHs inside a dark halo is so huge that UCMHs in the halo are uniformly distributed per halo mass (dnUCMH/dMDM = const) or per volume (dnUCMH/dVDM = const). This assumption can be invalid if the mass seed of a single UCMH mh is comparable to f UCMHMDM. In this case, the position of each UCMH is important to determine the energy deposition within the halo. Usually X-ray emission can be neglected in a massive UCMH due to photon trapping (Section 2.2.3), so we only focus on UCMH annihilation. Equation (49) in Section 4.2 is invalid for massive UCMHs, mh ∼ f UCMHMDM. In this case loc,iso in equation (49) should be calculated as the summation of individual UCMHs: loc,UCMH(z) = where LUCMH,i and ri are the annihilation luminosity and position of the ith UCMH within the halo. An extreme case is the one in which only one massive UCMH is inside a dark halo; the energy deposited in the gas at the centre of the halo can then be obtained by equation (63) in Section 4.1. Remember that for uniformly distributed UCMHs with the same total mass as the single UCMH inside a halo we use equation (49). The ratio factor between (63) and (49) is − 2 4 − 3 −1 factor = where r0 is the location of the single massive UCMH from the halo centre. If we take Rcore = ξ Rtr with ξ 1, the factor can Figure 13. Ionization fraction xion (thick lines) and molecular hydrogen fraction H2 (thin lines) in a 106-M isothermal halo with a single massive UCMH located at Rcore (dashed lines), 2Rcore (dotted lines) and 10Rcore (dash–dotted lines) from the centre of the halo. The solid lines show the results given by many small UCMHs with number density proportional to the halo density. For both the single UCMH and many small UCMHs, a total UCMH fraction of f UCMH = 10−4, zvir = 100 and mχ c2 = 10 GeV are adopted. be approximately written as (3/4ξ )(Rcore/r0)4. For a single UCMH located close to the isothermal case r0 Rcore, the factor (64) is 3/(4ξ ). For a single UCMH located at the turnaround radius r0 Rtr, equation (64) becomes 3ξ 3/4. Taking a typical value ξ ∼ 0.1, the energy deposition loc contributed by a single UCMH varies from ∼10 higher to ∼10−3 lower compared with that contributed by the same total mass but uniformly distributed small UCMHs. Fig. 13 gives the gas xion and fH2 inside a 106-M isothermal halo with a single UCMH located in different locations. UCMH annihilation at the halo centre will be less important for UCMHs located further away, >2Rcore. Uniformly distributed UCMHs are more or less equivalent to a singe UCMH located at approximately ∼1.5Rcore. As an example in this figure, we take the UCMH fraction as f UCMH = 10−4 and mχ c2 = 10 GeV. More massive dark matter particles or a lower UCMH fraction will give a faster decrease in energy deposition, with the single UCMH moving outward from the halo. We also check the importance of a single UCMH’s position on the gas temperature inside a halo, but we see that the effect on temperature can be neglected. On the other hand, the energy deposition by a single UCMH compared with a volume of uniformly distributed small UCMHs with the same total mass can be amplified by a factor 1 − which gives an enhancement factor of about ∼1/(6ξ 3) ∼ 1/ξ 2 for r0 = Rcore and a weaker factor ξ /6 for r0 = Rtr. These results show that a massive single UCMH is always more important for energy deposition compared with a volume of uniformly distributed UCMHs. Therefore we conclude that different UCMH distributions with the same total mass but different individual mass can change the energy deposition and structure evolution inside a halo. A more concentrated distribution towards the halo centre or a more closely located single halo near the centre gives a more significant effect on gas ionization and heating at the halo centre. 6 C O N C L U S I O N S Ultracompact minihaloes (UCMHs) have been proposed as primordial dark matter structures that formed by dark matter accreting on to an initial overdensity or primordial black holes (PBHs) after matter–radiation equality, zeq 3100 (Mack et al. 2007; Ricotti & Gould 2009). The key difference between UCMHs and the first dark halo structures is that UCMHs are seeded by primordial density perturbations produced in the very early Universe, such as phasetransition epochs (10−3 ≤ δ ≤ 0.3 for the initial overdensity or δ > 0.3 for PBHs), so they can grow shortly after z ∼ zeq. The radiation from UCMHs in the early Universe includes dark matter annihilation from all UCMHs and X-ray emission from gas accretion on to PBHs. In this paper we investigate the influence of UCMH radiation on the early ionization and thermal history of the IGM and the subsequent evolution of the first massive baryonic objects. Our conclusions are as follows. (1) UCMH annihilation can completely dominate over homogeneous dark matter background annihilation and can provide a new gamma-ray background even for a tiny UCMH fraction, f UCMH = (zeq)/ DM ∼ 2.2 × 10−15mχ−,21/030(1 + z)2 with mχ,100 = mχ c2/100 GeV. We conclude that the influence of dark matter annihilation on IGM evolution can be significantly enhanced when we include UCMHs in addition to the homogeneous dark matter background. In most cases, UCMH annihilation has been the dominant source of ionization and gas heating since the matter–radiation equality epoch, until X-ray emission from PBHs or large-scale structure formation became important at z ≤ 100. (2) The impact of UCMH annihilation on the IGM can be approximately estimated by the quantity mχ−,1100f UCMH. The threshold for UCMH abundance f UCMH to affect IGM evolution by dark matter annihilation is mχ−,1100f UCMH > 10−6. After the matter–radiation equality epoch, the IGM ionization fraction xion can be increased from xion ∼ 10−4 in the absence of any energy injection to an upper bound xion ∼ 0.1, and the IGM temperature from the adiabatic cooling Tm ∝ (1 + z)2 to a maximum value Tm ∼ 5000 K for the upper-bound case mχ−,1100f UCMH ∼ 10−2, which is constrained using the CMB optical depth at late times z < 30. UCMH annihilation is able to increase significantly the Thomson optical depth τ ≥ 0.1 in the early Universe z 30, which is unrelated to the measured CMB optical depth at z < 30. The UCMH annihilation luminosity is based on the UCMH profiles, where we take ρ ∝ r−2.25 from the literature (Bertschinger 1985). Steeper (shallower) profiles decrease (increase) the allowed upper limit of f UCMH, but the variations in the overall IGM chemical and thermal quantities should not be changed too much, because the fraction of UCMHs with a profile ρ ∝ r−1.5 as PBH hosts is very small, and ρ ∝ r−3 occurs only in the halo outskirts r rcut. (3) Each PBH is located in its host UCMH (Mack et al. 2007). We emphasize that the impact of X-ray emission from PBH-host UCMH systems is limited by the low abundance of PBHs (f PBH f UCMH), the average inefficient radiation (η 1), the photon-trapping effect by accreted baryons in host UCMHs and outflows produced by rapid accretion feedback. Sufficiently massive host UCMHs can accrete and thermalize infalling baryons, which are accumulated inside UCMHs with a mass fraction f b > 10−3, and trap X-rays from accreting PBHs until a critical redshift zm ∼ 32(δm/MPBH)1/2mχ−,51/0102, below which X-rays from super-Eddington accretion flows on to PBHs could escape the surrounding baryon environment in host UCMHs. Although the PBH abundance is f PBH f UCMH due to a much higher perturbation threshold for PBH formation, X-ray emission could dominate over UCMH annihilation and become a more promising cosmic energy source of IGM ionization and heating at z < zm if the PBH abundance is above a threshold ηf PBH/f UCMH ∼ 3.1 × 10−8mχ−,41/030(1 + z), which is only allowed beyond the standard Gaussian density perturbation scenario. (4) As UCMHs are expected to exist in our Galaxy, we expect that UCMHs collapse with the homogeneous dark matter background to form the first large-scale dark matter objects (dark haloes). If this is the case, the dark matter annihilation from UCMHs inside the first dark haloes still dominates over the extended dark matter annihilation background inside the haloes even after halo virialization. UCMH radiation, including both dark matter annihilation and accretion emission, can dramatically suppress the formation of lowmass first baryonic structure, since UCMH radiation heats the IGM and provides a hot ambient gas environment up to Tm ∼ 104 K. The UCMH radiation enhances baryon chemical quantities such as xion and fH2 by orders of magnitude from xion ∼ 10−6 and fH2 ∼ 10−4 to the upper bound of xion ∼ 10−4 and fH2 ∼ 5 × 10−3. However, the impact of UCMH radiation on the baryon temperature of the first baryonic objects is very small, which shows that the the influence of UCMH radiation on the temperature of the first baryonic objects is small compared with the molecular hydrogen cooling and virialization time zvir. However, the more abundant xion and fH2 provided by UCMH radiation decrease the gas temperature in the later gas collapse phase and can produce a lower fragmentation mass scale and lower mass first stars. Also, we point out that different spectral energy distributions of UCMH radiation also affect the processes of ionizing radiation and heating gas. A more concentrated UCMH distribution within a dark matter halo provides a more promising ionization phenomenon for the gas in the first dark haloes. UCMHs should be distinguished from small dark structures that formed during the structure formation epoch after z ∼ 100. Future work needs to be done to investigate the importance of small dark matter structure down to Earth mass, compared with UCMH radiation in the early Universe at z ∼ 60. Also, the CMB polarization anisotropies, 21-cm spectrum and Compton y parameter affected by UCMH radiation also need to be studied further for a better constraint on UCMH abundance. AC K N OW L E D G M E N T S The author is grateful to John Beacom and Brain Lacki for stimulating discussions and useful comments on the manuscript and also thanks David Weinberg and Alexander Belikov for helpful discussions on the baryonic fraction within dark haloes and dark halo structure formation history. Furthermore, Todd Thompson and Zi-Gao Dai are acknowledged. R E F E R E N C E S A P P E N D I X A : P R O F I L E S O F F I R S T D A R K S T R U C T U R E The dark matter profile we use in this paper are as follows. Most of the equations listed below can be found in Tegmark et al. (1997), Ripamonti (2007) and Ripamonti et al. (2007a). A dark matter halo with a mass MDM is assumed to be distributed inside a truncation radius Rtr, which is given by ⎧ Rtr(z, zvir) = ⎪⎪⎪⎪⎪⎨⎪ t (z) − t (zta) Rvir 2 − t (zvir) − t (zta) z ≥ zta zvir ≤ z ≤ zta, z ≤ zvir. Here zvir is the redshift of halo virialization and zta 1.5(1 + zvir) − 1 is the turnaround redshift. The dark matter mass inside the halo MDW is given as a parameter here. Dark matter within the halo is assumed to be distributed uniformly at z > zta with a density evolution ρth(z, zvir) = ρDM(z)e1.9A/(1−0.75A2), For an isothermal halo profile with a core radius Rcore inside Rtr, the dark matter density ρ(z) is a constant ρcore for r ≤ Rcore, ρ(z) ∝ r−2 for Rcore ≤ r ≤ Rtr. The density tends to the background dark matter density for r > Rtr. Here the core radius Rcore is obtained as Rcore(z, zvir) = Rvir 2 − (2 − ξ ) t(tz(vzi)r−)−t(tz(tzat)a) , where the coefficient ξ is introduced as a parameter and ρcore can be obtained by integrating the halo mass within Rtr as MDM. The total luminosity produced by the annihilation of dark matter particles distributed within the isothermal halo is Lext,iso = 2πmρχc2ore σ v c2Rc2ore Rtr − 23 Rcore . (A5) On the other hand, the widely used NFW dark matter profile is ρ(r) ∝ r−1(1 + r/Rcore)−1 for r ≤ Rtr. Similarly, we can write the complete formula for the density distribution as a function of redshift and the luminosity produced by dark matter annihilation. The virial temperature of a halo Tvir is calculated as Tvir = 2kB Rvir where sometimes the total mass Mhalo is used instead of MDMf χ . However, as we consider whether gas from the background can collapse into the halo, we adopt the pure dark halo mass in order to calculate the initial virial temperature. Therefore we obtain a virial temperature of Tvir = 8.2 × 10−3 K (1 + zvir)(MDM/M )2/3, or Tvir = 380 K (MDM/104 M )2/3(1 + zvir)/100. Baryon-falling can occur when the IGM temperature Tm around the halo is Tm < Tvir, otherwise the gas pressure will impede gas from falling into the halo and collapsing to form smaller structures. Moreover, the condition TCMB = Tvir with TCMB = 2.73 K (1 + zvir) gives a critical halo mass MDM = 6.1 × 103 M . This paper has been typeset from a TEX/LATEX file prepared by the author. Abazajian K. , Fuller G. M. , Patel M. , 2001 , Phys. Rev . D, 64 , 023501 Abdo A. A. et al., 2009 , Phys. Rev. Lett. , 102 , 181101 Abel T. , Bryan G. L. , Norman M. L. , 2002 , Sci, 295 , 93 Abolmasov P. , Karpov S. , Kotani T. , 2009 , PASJ, 61 , 213 Abramowicz M. A. , Fragile P. C. , 2011 , preprint (arXiv:1104.5499) Abramowicz M. A. , Jaroszyn´ski M. , Sikora M. , 1978 , A&A, 63 , 221 Abramowicz M. A. , Czerny B. , Lasota J. P. , Szuszkiewicz E. , 1988 , ApJ, 332 , 646 Aharonian F. et al., 2008 , Phys. Rev. Lett. , 101 , 261104 Barkana R. , Loeb A. , 2007 , Rep. Prog. Phys., 70 , 627 Barrow J. D. , Silk J. , 1979 , Gen. Relativ. Gravitation, 10 , 633 Becker R. H . et al., 2001 , AJ, 122 , 2850 Belikov A. V. , Hooper D. , 2009 , Phys. Rev . D, 80 , 035007 Belikov A. V. , Hooper D. , 2010 , Phys. Rev . D. , 81 , 043505 Belotsky K. M. , Khlopov M. Y. , Legonkov S. V. , Shibaev K. I. , 2005 , Gravitation and Cosmology, 11 , 27 Bergstro¨m L., Ullio P. , Buckley H. , 1998 , Astropart. Phys., 9 , 137 Bertschinger E. , 1985 , ApJS, 58 , 39 Biermann P. L. , Kusenko A. , 2006 , Phys. Rev. Lett. , 96 , 1301 Blandford R. D. , Begelman M. C. , 1999 , MNRAS, 303 , L1 Boehm C. , Mathis H. , Devriendt J. , Silk J. , 2005 , MNRAS, 360 , 282 Bromm V. , Yoshida N. , Hernquist L. , McKee C. F. , 2009 , Nat, 459 , 49 Bullock J. S. , Primack J. R. , 1997 , Phys. Rev . D, 55 , 7423 Carr B. J. , 1981 , MNRAS, 194 , 639 Carr B. J. , Hawking S. W. , 1974 , MNRAS, 168 , 399 Carr B. J. , Kohri K. , Sendouda Y. , Yokoyama J. , 2010 , Phys. Rev . D, 81 , 104019 Cen R. , 2003 , ApJ, 591 , 12 Chang J. et al., 2008 , Nat, 456 , 362 Chen X.-L. , Kamionkowski M. , 2004 , Phys. Rev . D, 70 , 043502 Chluba J. , 2010 , MNRAS, 402 , 1195 Chuzhoy L. , 2008 , ApJ, 679 , L65 Ciardi B. , Scannapieco E. , Stoehr F. , Ferrara A. , Illiev I. T. , Shapiro P. R. , 2006 , MNRAS, 366 , 689 Cirelli M. , Iocco F. , Panci P. , 2009 , J. Cosmol. Astropart. Phys., 10 , 9 Diemand J. , Moore B. , Stadel J. , 2005 , MNRAS, 364 , 665 Dunkley J. et al., 2009 , ApJS, 180 , 306 Fan X. , Narayanan V. K. , Strauss M. A. , White R. L. , Becker R . H., Pentericci L. , Rix A.-W. , 2002 , ApJ, 123 , 1247 Feng J. L. , Matchev K. T. , Wilczek F. , 2001 , Phys. Rev . D, 63 , 045024 Frampton P. H. , Kawasaki M. , Takahashi F. , Yanagida T. T. , 2010 , J. Cosmol. Astropart. Phys., 4 , 23 Galli D. , Palla F. , 1998 , A&A, 335 , 403 Galli S. , Iocco F. , Bertone G. , Melchiorri A. , 2009 , Phys. Rev . D, 80 , 3505 Gnedin N. Y. , Ostriker J. P. , Rees M. J. , 1995 , ApJ, 438 , 40 Green A. M. , Liddle A. R. , 1997 , Phys. Rev . D, 56 , 6166 Green A. M. , Liddle A. R ., Riotto A. , 1997 , Phys. Rev . D, 56 , 7559 Green A. M. , Hofmann S. , Schwarz D. J. , 2005 , J. Cosmol. Astropart. Phys., 8 , 3 Haiman Z. , Loeb A. , 1997 , ApJ, 483 , 21 Hansen S. H. , Haiman Z. , 2004 , ApJ, 600 , 26 Hawking S. , 1971 , MNRAS, 152 , 75 Hoeft M. , Yepes G. , Gottlo¨ber S., Springel V. , 2006 , MNRAS, 371 , 401 Hollenbach D. , McKee C. F. , 1979 , ApJS, 41 , 555 Hooper D. , Kaplinghat M. , Strigari L. E. , Zurek K. M. , 2007 , Phys. Rev . D, 76 , 3515 Igumenshchev I. V. , Narayan R. , Abramowicz M. A. , 2003 , ApJ, 592 , 1042 Iliev I. T. , Scannapieco E. , Shapiro P. R. , 2005 , ApJ, 624 , 491 Ishiyama T. , Makino J. , Ebisuzaki E. , 2010 , ApJ, 723 , L195 Josan A. S. , Green A. M. , 2009 , Phys. Rev . D, 79 , 103520 Josan A. S. , Green A. M. , 2010 , Phys. Rev . D, 82 , 083527 Kanzaki T. , Kawasaki M. , Nakayama K. , 2010 , Prog. Theor. Phys., 123 , 853 Khlopov M. Y. , 2010 , Res. Astron. Astrophys., 10 , 495 Kohri K. , Narayan R. , Piran T. , 2005 , ApJ, 629 , 341 Komatsu E. et al., 2009 , ApJS, 180 , 330 Lacki B. C. , Beacom J. F. , 2010 , ApJ, 720 , L67 Lidsey J. E. , Carr B. J. , Gilbert J. H. , 1995 , Nucl. Phys. Proc. Suppl. , 43 , 75 Mack K. J. , Wesley D. H. , 2008 , preprint (arXiv:0805.1531) Mack K. J. , Ostriker J. P. , Ricotti M. , 2007 , ApJ, 665 , 1277 Madau P. , Rees M. J. , Volonteri M. , Haardt F. , Oh S. P. , 2004 , ApJ, 604 , 484 Mapelli M. , Ferrara A. , 2005 , MNRAS, 364 , 2 Mapelli M. , Ferrara A. , Pierpaoli E. , 2006 , MNRAS, 375 , 1399 Meiksin A. A. , 2009 , Rev. Mod. Phys. , 81 , 1405 Miller M. C. , Ostriker E. C. , 2001 , ApJ, 561 , 496 Milosavljevic ´ M. , Bromm V. , Couch S. M. , Oh S. P. , 2009 , ApJ, 698 , 766 Milosavljevic ´ M. , Couch S. M. , Bromm V. , 2009 , ApJ, 696 , L146 Myers Z. , Nusser A. , 2008 , MNRAS, 384 , 727 Narayan R. , Yi I. , 1994 , ApJ, 428 , L13 Narayan R. , Mahadevan R. , Quataert E. , 1998 , in Abramowicz M. A., Bjornsson G. , Pringle J. E., eds, Theory of Black Hole Accretion Disks . Cambridge Univ. Press, Cambridge, p. 148 Narayan R. , Igumenshchev I. V. , Abramowicz M. A. , 2000 , ApJ, 539 , 798 Natarajan A. , Schwarz D. J. , 2008 , Phys. Rev . D, 78 , 103524 Natarajan A. , Schwarz D. J. , 2009 , Phys. Rev . D, 80 , 043529 Natarajan A. , Schwarz D. J. , 2010 , Phys. Rev . D, 81 , 123510 Navarro J. F. , Frenk C. S. , White S. D. M. , 1996 , ApJ, 462 , 563 Oda T. , Totani T. , Nagashima M. , 2005 , ApJ, 633 , L65 Okamoto T. , Gao L. , Theuns T. , 2008 , MNRAS, 390 , 920 Padmanabhan N. , Finkbeiner D. P. , 2005 , Phys. Rev . D, 72 , 023508 Park K. H. , Ricotti M. , 2011 , ApJ, 739 , 2 Pierpaoli E. , 2004 , Phys. Rev. Lett. , 92 , 031301 Poutanen J. , Lipunova G. , Fabrika S. , Butkevich A. G. , Abolmasov P. , 2007 , MNRAS, 377 , 1187 Press W. H. , Schechter P. , 1974 , ApJ, 187 , 425 Ricotti M. , 2007 , ApJ, 662 , 53 Ricotti M. , Gould A. , 2009 , ApJ, 707 , 979 Ricotti M. , Ostriker J. P. , 2004 , MNRAS, 350 , 539 Ricotti M. , Ostriker J. P. , 2004 , MNRAS, 352 , 547 Ricotti M. , Ostriker J. P. , Gnedin N. Y. , 2005 , MNRAS, 357 , 207 Ricotti M. , Ostriker J. P. , Mack K. J. , 2008 , ApJ, 680 , 829 Ripamonti E. , 2007 , MNRAS, 376 , 709 Ripamonti E. , Mapelli M. , Ferrara A. , 2007a , MNRAS, 374 , 1067 Ripamonti E. , Mapelli M. , Ferrara A. , 2007b , MNRAS, 375 , 1399 Ripamonti E. , Mapelli M. , Zaroubi S. , 2008 , MNRAS, 387 , 158 Rybicki G. B. , Lightman A. P. , 2004 , Radiative Processes in Astrophysics. Wiley, New York Saito R. , Shirai S. , 2011 , Phys. Lett . B. , 697 , 95 Saito R. , Yokoyama J. , Nagata R. , 2008 , J. Cosmol. Astropart. Phys., 6 , 24 Salvaterra R. , Haardt F. , Ferrara A. , 2005 , MNRAS, 362 , L50 Sandick P. , Diemand J. , Freese K. , Spolyar D. , 2011 , J. Cosmol. Astropart. Phys., 1 , 18 Sazonov S. Y. , Ostriker J. P. , Sunyaev R. A. , 2004 , MNRAS, 347 , 144 Scott P. , Sivertsson S. , 2009 , Phys. Rev. Lett. , 103 , 211301 Seager S. , Sasselov D. D. , Scott D. , 2000 , ApJ, 128 , 407 Shakura N. I. , Sunyaev R. A. , 1973 , A&A, 24 , 337 Shapiro S. L. , 1973a , ApJ, 180 , 531 Shapiro S. L. , 1973b , ApJ, 185 , 69 Sheth R. K. , Tormen G. , 1999 , MNRAS, 308 , 119 Shull J. M. , van Steenberg M. E. , 1985 , ApJ, 298 , 268 Shull J. M. , Venkatesan A. , 2008 , ApJ, 685 , 1 Slatyer T. R. , Padmanabhan N. , Finkbeiner D. P. , 2009 , Phys. Rev . D, 80 , 3526 Smith N. , Owocki S. P. , 2006 , ApJ, 645 , L45 Spolyar D. , Freese K. , Gondolo P. , 2008 , Phys. Rev. Lett. , 100 , 051101 Spolyar D. , Freese K. , Gondolo P. , 2009 , ApJ, 705 , 1031 Stacy A. , Bromm V. , 2007 , MNRAS, 382 , 229 Stasielak J. , Biermann P. L. , Kusenko A. , 2007 , ApJ, 654 , 290 Tashiro H. , Sugiyama N. , 2006 , MNRAS, 368 , 965 Tegmark M. , Silk J. , Rees M. J. , Blanchard A. , Abel T. , Palla F. , 1997 , ApJ, 474 , 1 Thomas R. M. , Zaroubi S. , 2008 , MNRAS, 384 , 1080 Ullio P. , Bergstro¨m J., Edsjo ¨ J., Lacey C. , 2002 , Phys. Rev . D, 66 , 123502 Volonteri M. , Gnedin N. , 2009 , ApJ, 703 , 2113 Wang J.-M. , Chen Y.-M. , Hu C. , 2006 , ApJ, 637 , L85 Weynmann R. , 1965 , Phys. Fluids, 8 , 2112 Whalen D. , Hueckstaedt R. M. , McConkie T. O. , 2010 , ApJ, 712 , 101 Wise J. H. , Abel T. , 2008 , ApJ, 684 , 1 Wyithe S. , Loeb A. , 2003 , ApJ, 588 , L69 Yoshida N. , 2009 , preprint (arXiv:0906.4372) Yoshida N. , Omukai K. , Hernquist L. , Abel T. , 2006 , ApJ, 652 , 6 Yuan Q. , Yue B. , Bi X.-J. , Chen X. , Zhang X. , 2010 , J. Cosmol. Astropart. Phys., 1010 , 23 Zdziarski A. , Svensson R. , 1989 , ApJ, 344 , 551 Zhang L. , Chen X.-L. , Lei Y .-A., Si Z.-G. , 2006 , Phys. Rev . D, 74 , 103519


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Dong Zhang. Impact of primordial ultracompact minihaloes on the intergalactic medium and first structure formation, Monthly Notices of the Royal Astronomical Society, 2011, 1850-1872, DOI: 10.1111/j.1365-2966.2011.19602.x