THE SIGNATURES OF PARTICLE DECAY IN 21 cm ABSORPTION FROM THE FIRST MINIHALOS

DM profiles are described as suggested by Ripamonti (2007): the DM of M_DM = Ω_DMM_halo/Ω_M is a truncated isothermal sphere with a truncation radius R_tr evolving as in Tegmark et al. (1997), and a flat core of radius R_core; the ratio η = R_core/R_vir is assumed to be 0.1 throughout the simulations, as commonly used to mimic the evolution of a simple top-hat fluctuation (e.g., Padmanabhan 1993). The dynamics of the baryons is described by a one-dimensional Lagrangian scheme similar to that proposed by Thoul & Weinberg (1995); a reasonable convergence is found at a resolution of 1000 zones over the computational domain.

The chemical and ionization composition includes a standard set of species: H, H⁺, H⁻, He, He⁺, He⁺⁺, H₂, H$_2^+$, D, D⁺, D⁻, HD, HD⁺, and e, with the corresponding reaction rates from Galli & Palla (1998) and Stancil et al. (1998). The energy equation includes radiative losses typical of a primordial plasma: Compton cooling, recombination and bremsstrahlung radiation, and collisional excitation of H i (Cen 1992), H₂ (Galli & Palla 1998), and HD (Flower 2000; Lipovka et al. 2005).

Calculations start at a redshift of z = 100. The initial parameters (gas temperature, chemical composition, and other quantities) are taken from simple one-zone calculations beginning at z = 1000 with typical values taken at the end of recombination: T_gas = T_CMB, x[H] = 0.9328, x[H⁺] = 0.0672, x[D] = 2.3 × 10⁻⁵, and  x[D⁺] = 1.68 × 10⁻⁶ (see references and details in Ripamonti 2007, Table 2).

Decaying DM contributes to the ionization and heating of baryons (see the recent discussion in Bertone 2010). The nature of decaying DM particles as well as the decay products is still under debate. Several candidates have been proposed: superheavy particles (Bertone et al. 2005), axions (Boehm & Fayet 2004; Boehm et al. 2004), sterile neutrinos (Dodelson & Widrow 1994; Hansen & Haiman 2004), and weakly interacting massive particles (e.g., Jungman et al. 1996). The expected mass regime may range from the very light (a few keV; sterile neutrinos) to the relatively light (a few GeV; e.g., Chen & Kamionkowski 2004) to the very heavy (several TeV, as recently discussed by Valdés et al. 2012). Depending on the nature of the decaying particle, the decay products may include γ photons, electrons, positrons, and other more exotic particles. For example, detection of positrons of tens of GeV by PAMELA (Adriani et al. 2009) can be explained by the injection from annihilation and/or the decay of DM (He 2009; Nardi et al. 2009). It is clear, however, that through the formation of particle showers, the decays end when weakly interacting stable particles escape, while the other particles couple to baryons via electromagnetic interactions, ionize them, and deposit energy.

The most generic form of the ionization rate from decaying DM particles is proposed by Chen & Kamionkowski (2004):

$$\begin{equation} I_e(z) = \chi _i f_x \Gamma _X { m_p c^2 \over h \nu _c}, \end{equation} \tag{ 1 }$$

where χ_i is the energy fraction deposited into ionization (Shull & van Steenberg 1985), m_p is the proton mass, f_x = Ω_X(z)/Ω_b(z), Ω_b(z) is the baryon density parameter, Ω_X(z) is the fractional abundance of decaying particles, Γ_X is the decay rate, and hν_c is the energy of Lyman continuum photons. It is obvious that the contribution from decaying DM is determined by the product ξ_i = χ_if_xΓ_X, which masks the nature of DM particles.

The corresponding heating rate can be written in the form (Chen & Kamionkowski 2004)

$$\begin{equation} K = \xi _h m_p c^2, \end{equation} \tag{ 2 }$$

where ξ_h = χ_hf_xΓ_X and χ_h is the energy fraction deposited into heating. Based on an order of magnitude approximation, χ_i ∼ χ_h ∼ 1/3 for the conditions we are interested in (Shull & van Steenberg 1985).

Using CMB datasets, Zhang et al. (2007) have constrained the ionization rate associated with radiatively decaying DM to be ξ ≲ 1.7 × 10⁻²⁵ s⁻¹. An extended analysis of the data of Type Ia Supernovae, the Lyα forest, large-scale structure, and weak lensing observations have lead DeLope Amigo et al. (2009) to provide a stronger constraint: ξ ≲ ξ_L = 0.59 × 10⁻²⁵ s⁻¹. It is worth noting that all datasets favor long-living decaying DM particles with a lifetime $\Gamma _X^{-1} \gtrsim 100$ Gyr (DeLope Amigo et al. 2009). Further improvement is expected from the Planck satellite. In this paper, we consider models with an ionization rate within this limitation ξ ⩽ ξ_L. As will be seen below, such a constraint does not affect our understanding of what happens outside the limit ξ > ξ_L.

Equations (1) and (2) were written under the assumption of a uniform production of ionizing photons by decaying DM with a density Ω_X, without accounting for its concentration in minihalos. Virialized minihalos are transparent for sufficiently energetic photons for which τ(E) = r_virn_virσ(E) < 1, where r_vir and n_vir are the radius and the mean density of a virialized halo, and σ(E) is the photoionization cross section. The corresponding lower energy of escaping photons is thus

$$\begin{equation} E_l \gtrsim 300 {\rm \,eV} \left({M\over 10^8 M_\odot }\right)^{1/9} \left({1+z\over 10}\right)^{2/3}. \end{equation} \tag{ 3 }$$

In this sense, our estimates correspond to a lower limit of the ionization and heating rates.

The upper energy limit for the photons being able to heat and ionize the IGM is determined by the assumption that they are absorbed in the IGM within a Hubble time; this limit can be readily estimated as E_u ≲ 30 keV (for a detailed description of the "transparency window," see Chen & Kamionkowski 2004). In the energy range E_l < E < E_u, photons are supposed to fill the IGM homogeneously except for relatively small circumgalactic regions. Note in this connection that even though each minihalo is a source of ionizing photons, their influence on the circumhalo baryons is obviously negligible because of a very low optical depth τ(E) at the energies of interest.

Contrary to decaying DM, stellar and quasi-stellar sources of ionizing radiation emerge only at redshifts z < 20. In addition, they heat nearby surrounding gas, which then emits at 21 cm with a patchy, spot-like distribution on the sky; the imprints from non-stellar and stellar sources can therefore be discriminated (see discussion below in Section 4).

The spin temperature of the H i 21 cm line is determined by atomic collisions and the scattering of UV photons (Field 1958; Wouthuysen 1952). In our calculations, we used collisional coefficients from Kuhlen et al. (2006) and Liszt (2001). In general, decaying DM is deposited into the budget of Lyα photons due to recombinations. However, in our present calculations, we neglect UV Lyα pumping because it is small compared with collisional processes for suitable parameters of the decaying DM and the redshifts of interest (see the discussion in Shchekinov & Vasiliev 2007). Moreover, the contribution from the stellar Lyα background is small due to the scarcity of stellar objects at redshifts z = 10–15.

The optical depth along a line of sight at a frequency ν is

$$\begin{eqnarray} \tau _\nu &=& {3h_p c^3 A_{10}\over 32 \pi k \nu _0^2} \int _{-\infty }^{\infty } dx {n_{{\rm H}\,{\scriptsize{I}}}(r) \over \sqrt{\pi } b^2(r) T_s(r)} \nonumber\\ && \times {\rm exp}\left[{-{[v(\nu) - v_l(r)]^2\over b^2(r)}}\right], \end{eqnarray} \tag{ 4 }$$

where r² = (αr_vir)² + x², α = r_⊥/r_vir is the dimensionless impact parameter, v(ν) = c(ν − ν₀)/ν₀, v_l(r) is the infall velocity projected on the line of sight, and b² = 2kT_k(r)/m_p is the Doppler parameter.

The observed line equivalent width is determined by $W_\nu ^{{\rm obs}} =W_\nu / (1+z)$, where the intrinsic equivalent width is

$$\begin{equation} {W_\nu \over 2} = \int _{\nu _0}^{\infty }{(1-e^{-\tau _\nu })d\nu } - \int _{\nu _0}^{\infty }{(1-e^{-\tau _{{\rm IGM}}})d\nu }, \end{equation} \tag{ 5 }$$

where τ_IGM is the optical depth of the background neutral IGM.

We consider here only evolving minihalos with masses in the range M = 10⁵–10⁷ M_☉ virialized at z_v = 10. At lower redshifts, reionization by stellar and quasi-stellar sources comes in to play and contaminates the effects from the decaying DM. On the other hand, at higher redshifts, the number of bright background radio sources decreases. The choice of the halo mass range is motivated by the fact that halos with M = 10⁵ M_☉ lie below the limit where the baryons can efficiently cool and form stars. Only inside minihalos with M ≳ 2 × 10⁶ M_☉ can star formation potentially occur in the sense that these objects collapse at z_vir = 10, i.e., the gas density in the inner-most shell reaches the critical value 10⁸ cm⁻³ when three-body reactions turn on to form H₂ (Ripamonti 2007). On the other hand, minihalos of M = 10⁷ M_☉ do eventually collapse and form stars and thus represent the opposite dynamical regime. The results we present here illustrate therefore the typical features of 21 cm absorptions produced by non-star-forming (M = 10⁵ M_☉), marginally star-forming (M = 10⁶ M_☉), and star-forming (M = 10⁷ M_☉) minihalos.

Heating from decaying DM (DDM) weakens the accretion rate of baryons onto minihalos. Therefore, in comparison with the evolution of minihalos in the standard scenario, one can expect smaller baryon masses and higher temperatures within the virial radius of a minihalo. The influence of decaying particles depends on the minihalo mass and the rate of the decay energy deposited in the gas, ξ.

Figure 1 shows radial profiles of density (upper panels), temperature (upper middle panels), velocity (lower middle panels), and the ionized hydrogen fraction (lower panels) in minihalos of M = 10⁵, 10⁶, and 10⁷ M_☉, virialized at z_vir = 10 (from left to right); 21 cm absorption profiles from collapsing minihalos in the standard recombination model can be found in Vasiliev & Shchekinov (2012). It is clearly seen that the IGM temperature under DDM heating can approach or even exceed the virial temperature of minihalos: DDM with ξ ≳ 0.1ξ_L heats the IGM up to T ≃ 250 K, which is only half of the virial temperature of a M = 10⁵ M_☉ halo, while ξ ≳ 0.3ξ_L elevates the IGM temperature close to the virial temperature of M = 10⁶ M_☉ minihalos. DDM heating results in a weakening of baryon accretion. Thus, the most obvious and important difference between evolving minihalos in the standard recombination regime and the presence of DDM is the absence of an accretion shock wave in minihalos with M ≲ 10⁶ M_☉, as obviously seen on the lower panels of Figure 1.

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The radial density, temperature, and velocity profiles in low-mass (M ≲ 10⁶ M_☉) minihalos in the presence of DDM heating look similar. The profiles² in massive halos (M = 10⁷ M_☉) resemble to those in the standard recombination scenario (see the dashed lines in Figure 1), in spite of strong DDM heating. Thus, contrary to the halos of lower mass where gravitation is too weak to overcome additional heating, massive halos, M ⩾ 10⁷ M_☉, instead remain able to support the accretion rate on a practically unchanged level. It is obvious that within the limit ξ ⩽ ξ_L, minihalos with M = 10⁷ M_☉ represent the least massive halos sensitive to an additional heating from DDM. However, as readily seen, further increasing ξ results in a practically proportional growth of the IGM temperature with T, $T_{_{{\rm IGM}}}\propto \xi$, and consequently suppresses the accretion rate approximately as $T_{_{{\rm IGM}}}^{-3/2}$, which in turn increases the lower limit of halo mass insensitive to DDM heating.

Ionization from the DDM is considerable: it is seen from Figure 1 that in the low-mass range (M ⩽ 10⁶ M_☉), the fractional ionization within minihalos and outside them is nearly invariant in models with ξ ≳ 0.3ξ_L. The ionization rate of ∼10ξ_L can affect the evolution of minihalos with masses of classical dwarf galaxies (M ∼ 10⁹ M_☉).

New evolutionary features of minihalos introduced by DDM can be manifested in the optical depths and equivalent widths of the 21 cm line. Figure 2 shows the maps of optical depth in the "impact parameter–frequency" plane in the standard recombination model (left panels) and in the presence of DDM with ξ/ξ_L = 0.1 (right panels). The analysis of optical depth in the 21 cm line from minihalos in the standard recombination scenario can be found in Xu et al. (2011), Meiksin (2011), and Vasiliev & Shchekinov (2012), so here we will discuss only the models with DDM. In addition, we will discuss only models with a single value of ξ because the other models are either similar or have no remarkable features and can be readily scaled.

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An obvious result is that DDM heavily suppresses optical depth. In the low-mass end, M = 10⁵–10⁶ M_☉, the center of line optical depth (ν − ν₀ ≲ 10 kHz) for small impact factors (α ≲ 0.5) at ξ = 0.1ξ_L is less than half compared with the standard case. A small peak in optical depth at ν − ν₀ ≃ 40 kHz for M = 10⁵ M_☉ disappears (the origin of this peak is discussed in Vasiliev & Shchekinov 2012). At higher impact factors α ≳ 1, the optical depth remains insensitive to ξ ≲ 0.1ξ_L. However, higher ξ can suppress optical depth down to the background value τ ∼ 10⁻³ for all α: for M = 10⁵ M_☉, this process occurs when ξ ≳ 0.3ξ_L, and for M = 10⁶ M_☉, it can be met when ξ ≳ ξ_L. Thus, DDM with ξ ≳ 0.3ξ_L practically erases 21 cm absorption features from low-mass minihalos.

Massive minihalos, M = 10⁷ M_☉, demonstrate a more complex frequency dependence on optical depth: it does not peak in the center of line; the maximum is instead at ν − ν₀ ∼ 45–50 kHz. Such a horn-like dependence originates from a strong accretion in massive halos (Xu et al. 2011; Vasiliev & Shchekinov 2012). In the presence of DDM, the range of impact factors with a horn-like profile decreases strongly even for ξ ∼ 0.1ξ_L. The center of line optical depth (ν − ν₀ ≲ 30 kHz) at small α ≲ 0.1 becomes flat and insensitive to ξ: it stays at a level ∼0.05 even for ∼ξ_L. At higher α, the optical depth merges with the background value.

It is clear that both the heating and the ionization from the DDM contribute to the decrease of optical depth. In order to evaluate their relative contributions, we performed calculations of two sets of models: in the first set, we turned off the ionization from the DDM I_e = 0, leaving the heating unchanged, while in the second set, the ionization remains unchanged, while the heating is turned off K = 0. The results are shown in Table 1, where the first column shows the line center optical depths τ₀ in the standard ionization scenario, and τ₀ is shown in the second column for the model with the DDM particles treated self-consistently. The second and the third columns correspond to the models with I_e = 0 and K unchanged and I_e unchanged and K = 0, respectively. An obvious trend of increased τ₀ in the shortened models with either I_e = 0 or K = 0 is caused by a reduction in the DDM to the total contribution. It is also clearly seen that the influence of the ionization I_e is not mainly due to a decrease in the fraction of H i in virialized halos, which is negligibly small: $\Delta x({\rm H\,\scriptsize{I}})=-x({\rm H\,\scriptsize{II}})\sim -10^{-3}$. The influence of the ionization stems predominantly from an enhanced Compton heating beginning from the turnaround point. Therefore, both the ionization and the heating decrease the optical depth via the additional heating. Separate contributions from these processes depend on minihalo mass: for instance, for 10⁵ M_☉ halos, the contributions are approximately 1/3 for the ionization and 2/3 for the heating, although their contributions are not independent and their combined influence is not simply a sum of the two separate components; the additional heating from these sources changes the temperature, density, and velocity profiles that are involved in the optical depth (Equation (4)) nonlinearly.

Figure 3 shows the evolution of the radial profiles of the equivalent widths measured in the observer's restframe. As expected, the equivalent widths decrease significantly with ξ except for halos with M = 10⁷ M_☉: for instance, low-mass halos (M = 10⁵ M_☉) at z = 10 (solid lines) show a decrease in $W_\nu ^{\rm obs}$ from ∼0.2 kHz for the standard recombination to a negligible value ≲ 0.02 kHz for ξ ⩾ 0.3ξ_L within r_⊥/r_vir ≲ 0.5. At z = 15.5 and 12 (dashed and dot-dashed lines), $W_\nu ^{\rm obs}$ can hardly be resolved on future telescopes and may only contribute to the total signal in broadband observations (Xu et al. 2011).

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More massive halos (M ≳ 10⁶ M_☉) show strong equivalent widths $W_\nu ^{\rm obs} \gtrsim 0.2$ kHz near virialization (z = 10) even at a relatively high ionization ξ ≲ 0.3ξ_L within the entire halo α ≲ 1. At $\alpha \lesssim 0.5, W_\nu ^{\rm obs}$ can reach even ∼0.5–0.9 kHz. At a higher redshift range, z = 15.5 and 12, M ≳ 10⁶ M_☉ halos can produce such strong equivalent widths only in the standard recombination scenario. More massive halos, M = 10⁷ M_☉, show strong absorption ($W_\nu ^{\rm obs}\gtrsim 0.5$ kHz) even for high ionization rates ξ ∼ ξ_L, although only in a narrow range of impact factors α ≲ 0.1. As a consequence, the DDM leads to a considerable decrease in the number of strong absorption lines: for ξ ≳ 0.3ξ_L, sufficiently strong lines form only in high-mass (M ⩾ 10⁷ M_☉) halos nearly along the diameter with a small probability of contributing absorption. A similar effect stems from the action of the stellar X-ray background at lower redshifts (Furlanetto & Loeb 2002; Xu et al. 2011); see also Section 4.

The sensitivity of future telescopes is at least an order of magnitude lower than the flux limit needed to separate spectral lines from individual halos, i.e., to observe with a spectral resolution of Δν ∼ 1 kHz. Therefore, Xu et al. (2011) have proposed broadband observations with lower resolution. In this case, the measured quantity is the average signal from different halos lying along a line of sight. Accordingly, we introduce a radially averaged equivalent width

$$\begin{equation} {\left\langle W_\nu ^{{\rm obs}} \right\rangle = {2\over 9} \int _0^{3r_{{\rm vir}}}{W_\nu ^{{\rm obs}}(r_\perp)r_\perp dr_\perp \over r_{{\rm vir}}^2}. } \end{equation} \tag{ 6 }$$

Figure 4 shows the dependence of $\langle W_\nu ^{{\rm obs}}\rangle$ on ξ/ξ_L. The increase of $\langle W_\nu ^{{\rm obs}}\rangle$ for M = 10⁵ M_☉ at z = 10 and ξ/ξ_L ⩽ 0.1 is explained by a widening of $W_\nu ^{{\rm obs}}$ in the peripheric region r_⊥/r_vir ≲ 2 (top panels in Figure 3). For higher masses and redshifts, the radially averaged $W_\nu ^{{\rm obs}}$ decreases with ξ/ξ_L by factor of ≲ 2 and apparently remains sufficient to distinguish effects from the ionization caused by DDM particles.

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In previous studies, the theoretical spectrum of the 21 cm forest was simulated under the assumption of steady-state minihalos with fixed profiles corresponding to virialization (Furlanetto & Loeb 2002; Xu et al. 2011). When dynamics is taken into account, minihalos lying on a given line of sight are, in general, at different evolutionary stages and have different density, velocity, and temperature profiles. Moreover, a Press–Schechter formalism cannot be used anymore for a description of their mass function as soon as the minihalos stray too far from virialization. It is therefore problematic to describe correctly the number density of minihalos at each evolutionary stage. An additional complication stems from continuous structure formation. While major mergers involving merging minihalos with M ≳ 0.5 M_h occur on times twice as long as a Hubble time at z = 10, less destructive mergers with M ≲ 0.25 M_h proceed within ≲ 0.7 of a Hubble time (Lacey & Cole 1993). This time is shorter than the free-fall time in halos with M ∼ 2 × 10⁶ M_☉, thus making estimates based on the assumption of the virial equilibrium inapplicable to less massive halos. The possible formation of stars in massive evolved minihalos also introduces additional complications. Overall, the analysis of a theoretical spectrum within a statistical simulation becomes exceedingly cumbersome. A more relevant picture can be obtained from high-resolution cosmological gas dynamic simulations, although the current resolution with ΔM ∼ 10⁶M_☉ does not seem to be sufficient.

However, with upcoming low-frequency interferometers (LOFAR and SKA), such modeling is apparently excessive. Indeed, the absorption line profiles and the spectral features from separate minihalos located at z = 10 are of 1–5 kHz in width—close to the spectral resolution limit 1 kHz. As discussed in Xu et al. (2011), the corresponding sensitivity requires enormously bright background sources. In order to resolve spectral lines of Δν ∼ 1 kHz with a typical for the SKA telescope ratio of the aperture-to-system temperature A_eff/T_sys = 5000 m² K⁻¹ and a signal-to-noise ratio S/N = 5, an integration time of ∼30 days and a minimum flux density of about 500 μJy are needed (Xu et al. 2011). This flux density is four times lower than the flux density of an object similar to the bright radio-loud galaxy Cygnus A (Carilli et al. 2002), but an order of magnitude higher than typical gamma-ray burst (GRB) afterglow fluxes (Xu et al. 2011). Note, in addition, that there are no confirmed quasi-stellar objects at z > 7.5; moreover, the mass of a black hole in the Cygnus A galaxy is about 2.5 × 10⁹ M_☉ (Tadhunter et al. 2003), which should apparently be a very rare object at z ≳ 10. GRB afterglows seem to be more promising background sources for 21 cm absorption studies. In this case, the integration time is restricted by the duration of the bright phase of the afterglow, ∼100 days (Frail 2003). Thus, low resolution or broadband observations may be a better alternative.

In broadband observations, suppression of optical depth at 21 cm caused by decaying particles manifests itself as a factor of 2–4 decrease in the absorption in the frequency range ν < 140 MHz, where the contribution from low-mass minihalos at z ⩾ 10 dominates. Indeed, the number of halos in the low-mass range 10⁵–10⁷ M_☉ at z = 10 scales as n(M, z) ∼ M⁻¹. The probability of minihalos intersecting a line of sight is proportional to n(M, z) × (αr_vir(M))², resulting in a decrease in the number of strong absorption lines with $W_\nu ^{\rm obs} \gtrsim 0.3$ kHz by a factor of at least 2.5 for ξ/ξ_L = 0.3 and more than 4.5 for ξ/ξ_L = 1. Such a decrease inevitably nullifies the average 21 cm forest signal from low-mass halos. This decrease also results in a several times decrease in the mean flux decrement D_A, and correspondingly the detectability limit in broadband observations. A strong suppression of the optical depth at ξ/ξ_L ≳ 0.3 would require time integration scales longer than the duration of the bright phase of GRB afterglows. Low-mass (M ∼ 10⁵ M_☉) minihalos produce weak absorptions, $W_\nu ^{\rm obs} \lesssim 0.2$ kHz, even for the standard ionization scenario.

Thus, as far as the characteristic scales corresponding to such broadband observations are of the order of hundreds of kiloparsecs, an immediate consequence is that the advantages expected initially from the 21 cm forest observations are not that obvious. However, once observations have been made covering a wide range of frequencies corresponding to redshifts 10 to 15, these data would in principle provide information about how the influence from DDM varies with redshift. More explicitly, a non-detection of absorption from 21 cm in the frequency range ν < 140 MHz can in principle serve to bound the ionization rate from DDM from below: ξ/ξ_L ≳ 0.3, unless contributions from stellar sources compete to ionize low-mass halos.