Abstract
We use a set of Monte Carlo simulations to follow the cascade produced by a primary electron of energy E_{in} in the intergalactic medium. We choose E_{in}= 3–10 keV as expected from the decay of one of the most popular warm dark matter (WDM) candidates, sterile neutrinos. Our simulation takes into account processes previously neglected such as free—free interactions with ions and recombinations, and uses the best available crosssections for collisional ionizations and excitations with H and He and for electron—electron collisions. We precisely derive the fraction of the primary electron energy that heats the gas, ionizes atoms and produces line and continuum photons as a function of the ionization fraction. Handy fitting formulae for all the above energy depositions are provided. By keeping track of the individual photons, we can distinguish between photons in the Lyα resonance and those with energy E < 10.2 eV that do not interact further with gas. This separation is important because a Lyα background can heat or cool the gas depending on the nature of the photons, and can have effects on the 21cm radiation emitted by neutral H, which will probably become detectable at z > 6 in the near future by the next generation radio interferometers.
Introduction
The determination of the gas temperature and ionized fraction in the intergalactic medium (IGM) and interstellar medium (ISM) is fundamental for a number of astrophysical studies. In particular, it is essential for the investigation of the nearly uniform, dark, neutral state of the Universe known as Dark Ages, which has become one of the most studied topics in cosmology in the last decade. The cosmic phase between hydrogen recombination at z∼ 1000 and the socalled Epoch of Reionization (EoR) at z∼ 9 can be directly detected by the 21cm hyperfine tripletsinglet level transition of the ground state of neutral hydrogen. A new generation of lowfrequency radio interferometers such as the LOw Frequency ARray (LOFAR), the 21 Centimeter Array (21CMA), the Mileura Widefield Array (MWA) and the Square Kilometer Array (SKA), is expected to reach the sensitivity required to map the H i distribution at angular resolution of the order of a few arcmin (e.g. Kassim et al. 2004; Pen, Wu & Peterson 2005; Wyithe, Loeb & Barnes 2005; Bowman, Morales & Hewitt 2006). If a few issues such as foreground removal, interferences from humangenerated signals and ionospheric corrections will be successfully taken care of these large radio arrays will be able to perform a tomography of the Universe before and during the EoR.
It is therefore crucial to be able to predict the magnitude of the signal. The standard scenario predicts the gas temperature, T_{K}, to decouple from the cosmic microwave background (CMB) temperature, T_{CMB}, at z∼ 300. T_{K} then starts decreasing adiabatically until the first sources of light heat again the gas above T_{CMB} at z∼ 20. In this case, the 21cm radiation would be seen in absorption against the CMB in the redshift interval 20 < z < 300 as discussed in detail by many recent studies (e.g. Loeb & Zaldarriaga 2004).
A number of more complex theoretical scenarios have been investigated recently. In particular, if there was a mechanism to heat the gas at z= 10, thus producing a thermal history different from the standard one, a direct imprint would be left on the 21cm radiation from the neutral gas, which could then be observable in emission rather than in absorption.
First stars (Ciardi & Salvaterra 2007), intermediatemass black holes (Zaroubi et al. 2006; Ripamonti, private communication) and decaying or annihilating warm and cold dark matter (CDM) particles (Furlanetto, Oh & Pierpaoli 2006; Shchekinov & Vasiliev 2007; Valdés et al. 2007) could be capable of leaving a significant trace in the evolution of the neutral intergalactic gas. Future 21cm observation will possibly rule out or confirm these theoretical predictions, opening a new exciting frontier in cosmology and possibly unveiling the nature of dark matter particles.
All the most popular dark matter particle candidates inject energy into the IGM, either via decays or annihilations, initiating an energy cascade from energetic primary photons or electrons. The energy deposition depends on the large number of interactions taking place during the propagation of the cascade particles through the IGM. It is therefore very important for achieving correct results to follow in detail these cascades and to know exactly how much of the energy injected ionizes the gas, produces radiation by collisional excitations and heats surrounding medium, respectively.
This fundamental problem has received attention in the past (e.g. Bergeron & CollinSouffrin 1973), and the results achieved were corrected by an extensive study by Shull & van Steenberg (1985), which we will denote hereafter as SVS85, which was a development of a previous work by Shull (1979), hereafter S79.
In this work, we present the results of a Monte Carlo calculation in the spirit of the one from S79 and SVS85; in comparison, we use more recent and accurate crosssections for collisional ionization and excitations from electron impacts, for electronelectron collisions, freefree interactions and recombinations. In addition to this, we follow in detail the radiation produced by the excitations and we are able to predict precisely how much of the energy goes into photons that do not further interact with the gas (E = 10.2 eV) and how much of it contributes to the Lyα background, which directly affects the physics of the 21cm line radiation by the socalled WouthuysenField effect (e.g. Wouthuysen 1952; Field 1959; Hirata 2006) and which can heat or cool the gas depending on the nature of the Lyα photons (Chen & MiraldaEscudé 2004; Chuzhoy & Shapiro 2007).
Method
Description of the calculation
We calculate the effects produced by an Xphoton of ∼keV energy injected into the IGM with T≪ 10^{4} K. At these energies, the dominant interaction is photoionization (see e.g. Zdziarski & Svensson 1989) and the Xphoton ionizes an H or He atom producing an energetic primary electron. We follow the subsequent secondary cascade products.
For our calculations, we choose two specific energies for the primary electron, E_{in}= 3 and 10 keV. This energy range is of great interest because sterile neutrinos, one of the most promising WDM candidates (Dodelson & Widrow 1994; Dolgov & Hansen 2002), are expected to emit line radiation at an energy between 3 and 25 keV. A large effort has been done recently by several authors to constrain the mass and lifetime of radiatively decaying sterile neutrinos from Xray observations (Abazajian, Fuller & Tucker 2001; Mapelli & Ferrara 2005; Abazajian 2006; Abazajian & Koushiappas 2006; Boyarsky et al. 2006a,b; Watson et al. 2006).
Once the primary electron is injected into the IGM the code calculates the crosssections relative to a list of possible processes: H, He, He i ionization; H, He excitation; collisions with thermal electrons; freefree interactions with ionized atoms; recombinations. We then estimate the mean free paths and derive the probability for a single electron to have any of the aforementioned interactions. We assume, in agreement with the 3yr Wilkinson Microwave Anisotropy Probe (WMAP) data analysis, that the helium fraction by mass is f_{He}= 0.248 (Spergel et al. 2007), and we follow the fate of a primary electron and its secondary products for different values of the ionized fraction x_{e} of the gas, which, as showed by SVS85 and S79, is the free parameter which affects the results. We assume, similarly, that x_{e}=n(H^{+})/n(H) =n(He^{+})/n(He).
For each primary energy E_{in} and for each assumed value of the gasionized fraction x_{e}, we performed 1000 Monte Carlo realizations, a number sufficient to produce consistent results not biased by the random nature of the computation. We will return on this aspect at the end of this section.
If the primary electron (or an energetic secondary electron) collisionally ionizes an H or He atom then the resulting two electrons have to be followed separately as they interact further with the gas.
Once the electron energy has degraded below 10.2 eV, we assume that its entire energy is deposited as heat. This is a simplification as the electron would actually thermalize with the gas and as a consequence the precise heating should be calculated by taking into account the gas temperature. If the temperature is of the order of 10^{4} K electrons with energies lower than 1 eV could even cool the gas.
Collisional excitations of H ad He produce photons that escape freely in the surrounding medium if their energy is lower than 10.2 eV or that can further interact with the gas if they have higher energy.
The study from S79 and SVS85 derives the amount of energy which is deposited in excitations but does not give details about the individual photons. We want to estimate instead the amount of energy that goes into Lyα photons: this radiation can in fact interact further with the gas by the WouthuysenField effect and is crucial to correctly estimate the 21cm signal. Furthermore, Lyα photons by scattering resonantly off neutral hydrogen can cool or heat the gas, depending on whether they enter the resonance from its red or blue wing, respectively, as we will explain more in detail later in this Letter.
An additional novel feature of our model is the inclusion of the previously neglected twophoton forbidden transition 2s → 1s. We included both the direct collisional excitation crosssection to the 2s level and the probability that a collisional excitation to a level n= 3 results in a cascade through the 2s level rather than through 2p. As noted recently (Hirata 2005; Chuzhoy & Shapiro 2007), this effect is not negligible, in general, and constitutes the most probable decay channel for H atoms excited to a level n= 3 or higher. The result is the emission of two photons below the Lyα energy that do not further interact with the gas.
With our code, we are therefore able to separate the Lyα radiation from the less energetic photons produced by cascade from the 2s level. All Lyα photons that result from collisional excitations (or injected Lyα photons, see Chen & MiraldaEscudé 2004) have a cooling effect on the gas.
We also included in our calculation processes that can produce continuum photons, such as recombinations and Bremsstrahlung freefree interactions of electrons with ionized atoms. This effects are negligible as we will see and there is virtually no production of photons between the Lyα and the Lyβ resonances. This radiation would have redshifted into continuum Lyα photons (Chen & MiraldaEscudé 2004) and by entering the Lyα resonance from its blue wing would have heated the gas instead.
Electronelectron collisions between secondary and thermal electrons were implemented as in SVS85 and S79, and we treated similarly the energy distribution of secondary electrons following collisional ionization of H and He, so we refer the reader to those works for a detailed explanation.
As we mentioned earlier, we found that 1000 Monte Carlo realizations gave consistent results. Going from 50 to 1000 realizations changed the averaged values by less than 5 per cent and the respective s by less than 10 per cent. We deemed 1000 to be a sufficient number of realizations for a stable result. The 1s values of the energy deposition fractions are included in Table 1 and they vary from 2.9 to 0.04 per cent of the total amount of energy.
Crosssections
For our purposes, it is important to use the best available crosssections for the several interaction channels considered in our calculation. We will give here the references rather than entering in the description of physical details that are not the scope of this Letter. The crosssections (s_{i}) for collisional ionization of H, He, He^{+} were taken from Kim & Rudd (1994), Shah, Elliot & Gilbody (1987) and Shah et al. (1988). A simple functional fit is
where x=E_{kin}/E_{B} is the ratio between the incoming electron energy and the binding energy of the atomic electron.The collisional excitation crosssections of H and He are from Stone, Kim & Desclaux (2002), while for the excitation to the 2s level of H we used the work from Bransden & Noble (1976). The crosssection for Coulomb collisions between electrons is from Spitzer & Scott (1969) as in S79, while the freefree crosssection (s_{ff}) for electrons interacting with protons is given by the BetheHeitler quantummechanical Born approximated result (see e.g. Haug 1997),
where m_{e} the electron mass, r_{0} is the classical electron radius, a is the fine structure constant, k=E/mc^{2} is the photon energy in units of m_{e}c^{2}, and p_{i}, p_{f} are the momenta of the incident and scattered electron, respectively, again in units of m_{e}c^{2}.As mentioned before, we also included the recombination crosssection s_{r}, to take into account a process which could produce continuum photons. Neglecting helium recombinations, we have that for hydrogen:
where g_{fb} is the Gaunt factor of (1), v is the emitted radiation frequency, v_{e} is the electron velocity and n is the level at which the electron recombines.Results
Our results are summarized in Table 1 and in Fig. 1. In the table, we report the fraction of the energy of a 10 keV primary electron which, for different values of the gasionized fraction x_{e}, is deposited into heat, Lyα excitations, ionizations and photons with E < 10.2 eV. The errors correspond the standard deviation from the mean calculated over 1000 Monte Carlo realizations of the experiment.
The energy fraction that goes into heating grows rapidly as the gas ionization fraction value becomes higher. The physical reason of this behaviour is that electronelectron interactions become dominant for high values of x_{e}.
We performed our calculations for 3 and 10 keV to account for the theoretically and observationally inferred range of the sterile neutrino mass and found that the energy fraction distributed among the different processes remains similar in both cases (within 2 per cent). This is in agreement with the results from S79 and SVS85 that find that for primary electrons with energies higher than 100 eV, the fractional energy depositions converge rapidly to a common behaviour.
We gave a convenient functional form to our results by fitting the data in Table 1 with an accuracy greater than 3.5 per cent.

(i)

(ii)
Fraction f_{a} of the primary energy converted into Lyα radiation:

(iii)
Fraction f_{i} of the primary energy deposited into ionizations:

(iv)
Fraction f_{c} of the primary energy deposited into continuum radiation:
Our results differ sensibly from those of the past studies, with differences as high as 30 per cent for some values of x_{e}. The reason for this is a combination of the better and more modern crosssections which we used and of the additional processes included in the calculation such as the transitions through the 2s level of H. We compare in detail our results with those obtained by SVS85 in Fig. 2. While for small values of x_{e}, the curves are similar it is evident that the differences become substantial for x_{e}≳ 0.1.
Discussion and Conclusions
We have presented an updated calculation of the energy cascade arising from a primary electron with energy in the range (3–10 keV) predicted for one of the most popular dark matter particle candidates, i.e. sterile neutrinos. We have computed the fractional energy deposition into ionizations, excitations and heating to a new level of detail and followed the fate of individual photons to be able to distinguish between Lyαinjected radiation and continuum photons with energies under 10.2 eV.
As mentioned previously in this work, Lyα radiation affects the gas in several ways. It has a thermal effect on the matter, with a heating or a cooling effect depending on the nature of the photons. Line or injected Lyα photons cool the gas, while photons between the Lyα and Lyβ resonances that redshift to 10.2 eV (continuum Lyα photons) act as a heat source for the gas.
A Lyα background is also responsible for the WouthuysenField process, which directly affects the physics of the 21cm line radiation from neutral H and can make it visible in emission or absorption against the CMB. This aspect is important because 21 cm observations of the highredshift Universe will be performed in the near future by next generation lowfrequency radio interferometers such as LOFAR.
We included in our calculations the only mechanisms that could produce continuum Lyα radiation, recombinations and freefree interactions with ions. Both processes are more probable as the ionized fraction increases, but at the same time the crosssection for electronelectron collisions becomes dominant, so we found that both these channels are practically negligible and also found that electrons injected in a highly ionized gas are thermalized before they can produce continuum photons.
We expect that the same calculations performed for relativistic electrons could produce interesting results in this sense, also taking into account that processes such as inverseCompton on CMB photons would become important and produce continuum radiation. This could be a useful extension to this work and could be applied to study the effects of light dark matter decays/annihilations in the energy range around ∼10 MeV.