1 Introduction

Intensity Mapping of Lyman-alpha Emission During the Epoch of Reionization

Abstract

We calculate the absolute intensity and anisotropies of the Lyman- radiation field present during the epoch of reionization. We consider emission from both galaxies and the intergalactic medium (IGM) and take into account the main contributions to the production of Lyman- photons: recombinations, collisions, continuum emission from galaxies and scattering of Lyman-n photons in the IGM. We find that the emission from individual galaxies dominates over the IGM with a total Lyman- intensity (times frequency) of about at a redshift of 7. This intensity level is low so it is unlikely that the Lyman- background during reionization can be established by an experiment aiming at an absolute background light measurement. Instead we consider Lyman- intensity mapping with the aim of measuring the anisotropy power spectrum which has rms fluctuations at the level of at a few Mpc scales. These anisotropies could be measured with a spectrometer at near-IR wavelengths from 0.9 to 1.4 m with fields in the order of 0.5 to 1 sq. degrees. We recommend that existing ground-based programs using narrow band filters also pursue intensity fluctuations to study statistics on the spatial distribution of faint Lyman- emitters. We also discuss the cross-correlation signal with 21 cm experiments that probe HI in the IGM during reionization. A dedicated sub-orbital or space-based Lyman- intensity mapping experiment could provide a viable complimentary approach to probe reionization, when compared to 21 cm experiments, and is likely within experimental reach.

Subject headings:
cosmology: theory — large scale structure of Universe — diffuse radiation

1. Introduction

The epoch of reionization (EoR) is a crucial stage in the history of galaxy formation, signaling the birth of the first luminous objects, during which the universe went from completely neutral to almost completely ionized (Barkana & Loeb, 2001). This phase has been largely unexplored so far, although current observations suggest it was reasonably extended (Komatsu et al., 2011; Fan et al., 2006) and a wide variety of observational avenues are being explored to probe it. In particular the 21-cm line of neutral hydrogen is now understood to be a promising tool to study reionization and to understand the formation and evolution of galaxies during that epoch (see e.g. Furlanetto et al. 2006). It is also now becoming clear that we need complimentary data in order to obtain extra insight into the sources of reionization. Such complimentary data could also aid in the interpretation of the HI signal by allowing ways to pursue cross-correlations and providing ways to reduce systematics and foregrounds encountered in 21-cm observations.

Recently, intensity mapping of other atomic and molecular lines at high redshifts, in particular CO and CII (Gong et al., 2012, 2011; Lidz et al., 2011; Visbal & Loeb, 2010), has been proposed as a probe of reionization. In this work we study the viability of also using intensity mapping of the Lyman- (Ly-) line as an additional probe. For this study we include several Ly-emission mechanisms involving both individual sources of emission such as galaxies and the emission and scattering associated with the intergalactic medium (IGM).

We consider both the integrated intensity and anisotropies of the Ly-line and suggest the latter as a new probe of reionization. In particular we suggest that it will be possible to measure the amplitude of the Ly-intensity fluctuations with a narrow-band spectrometer either from the ground with a suppression of atmospheric lines or from the orbital/sub-orbital platform.

The Ly-line, corresponding to transitions between the second and first energy level of the hydrogen atom, has a rest wavelength of approximately . The signal present during reionization is observable in near-IR wavelengths today. Existing imaging observations made with narrow-band filters on 10m class telescopes focus on individual galaxy detections and are limited to a handful of narrow atmospheric windows at near-IR wavelengths. Given the strength of the line, it has now been seen in galaxies at (Iye et al., 2006), (Salvaterra et al., 2009) and (Lehnert et al., 2010), reaching well into the epoch of reionization.

Deep narrow-band surveys of high redshift Ly-emitters have led to detections of a sufficient number of galaxies at redshifts , , and to allow constraints on the bright-end of the Ly-luminosity function (LF) and its redshift evolution (e.g. Ouchi et al. 2008; Ota et al. 2010; Taniguchi et al. 2005; Iye et al. 2006; Shibuya et al. 2011). Observations of the Ly-LF indicate a decrease in the Ly-intensity from redshift to . This would require a strong evolution of the Ly-emitters population, which is not predicted by most recent galaxy evolution models (Ota et al., 2010; Shibuya et al., 2011), or could be explained as the result of an increase in the fraction of IGM neutral hydrogen that would absorb or scatter Ly-photons from the observed galaxies (Haiman et al., 2000; Ota et al., 2008).

The scattering of Ly-photons by neutral hydrogen in the ISM (interstellar medium) and the IGM is expected to disperse the photons in both frequency and direction (Santos, 2004). Such scattering could considerably decrease the Ly-intensity per frequency bin from an individual galaxy, making the detection of most of the high redshift galaxies impossible with current instruments. Exact calculations related to scattering are a difficult problem to solve analytically and in simulations the scattering problem requires ray tracing of photons through the neutral medium in a simulation box (Zheng et al., 2010). While scattering makes individual galaxies dimmer, intensity mapping of the Ly-line at high redshifts can be an improvement over the usual experiments that make detections of Ly-emission from point sources and are only sensible to the strongest Ly-emitters. These are likely to be some of the brightest star-forming galaxies, however, any dust that is present in such galaxies, especially during the late stages of reionization, is likely to suppress the Ly-line. An experiment targeting the integrated emission will be able to measure all the sources of Ly-photons in a large region and will be sensitive to the extended, low surface brightness Ly-emission that is now known to generally form around star-forming regions (e.g., Steidel et al. 2011; Zheng et al. 2011). The anisotropy power spectrum of Ly-intensity then would be a probe of the Ly-halos around star-forming galaxies present during reionization. The cross-correlation with the 21-cm data could provide a direct test on the presence of neutral hydrogen in the extended Ly-halo.

The paper is organized as follows: in the next section we estimate the contribution to the Ly-emission from galaxies. In section 3 we analyze the contributions to the Ly-emission from the IGM. In section 4 we calculate the intensity of the Ly-signal as well as its power spectrum using a modified version of the code SimFast21 (Santos et al., 2010, 2011). In section 5 we discuss the correlation of Ly-intensity maps with the 21 cm signal and finally in section 6 we comment on the experimental feasibility of measuring the Ly-intensity power spectrum.

2. Lyman- emission from Galaxies

The observed Ly-flux is mainly the result of line emission from hydrogen recombinations and collisional excitations in the interstellar clouds or in the IGM powered respectively by UV emission or UV and X-ray emission from galaxies. High energy photons emitted by stars ionize hydrogen that then recombines to emit a rich spectrum of lines including a Ly-photon (Gould & Weinberg, 1996; Fernandez & Komatsu, 2006). Moreover, the electron ejected during this ionization heats the ISM or the IGM, increasing the probability of Ly-photon emission caused by collisional excitation (Gould & Weinberg, 1996; Cantalupo et al., 2008). There is also a small contribution to the lyman alpha flux originated in the continuum emission from stars between the Ly-line and the Lyman-limit (Chuzhoy & Zheng, 2007; Barkana & Loeb, 2005) plus Ly-from continuum free-free or free-bound emission as well as 2-photon emission during recombinations. This continuum will also make contributions to a given observation from lower redshifts besides the ”Ly-” redshift (Cooray et al., 2012) which will confuse the Ly-signal. However, due to the smoothness of that continuum across frequency, we expect it should be possible to remove this contribution, for instance, by fitting a smooth polynomial in frequency for each pixel.

Another source of Ly-emission in the universe is cooling of gas that has suffered in-fall into a dark matter halo potential well. Several studies show that much of this cooling is made in the form of Ly-emission (Haiman et al., 2000; Fardal et al., 2001; Dijkstra et al., 2006a, b; Dayal et al., 2010; Latif et al., 2011). Cold gas is used by galaxies as fuel to form stars so there is a relation between the star formation rate (SFR) of a galaxy and the Ly-flux emitted as gas cools in that galaxy.

Since emission of Ly-radiation is closely connected with the star formation, the contribution from the several mechanisms by which Ly-radiation is emitted in galaxies and in the IGM can be related to the SFR of individual galaxies or galaxy samples. In order to calculate the emission of Ly-radiation from the IGM during the EoR we also need to know the ionized fraction of hydrogen as well as the temperature of the gas in the IGM. Unfortunately both these quantities are poorly constrained at (Larson et al., 2011; Ouchi et al., 2010; Zahn et al., 2011). Since hydrogen ionization should be a consequence of stellar ionization/X-ray emission, we can in principle estimate it by following the SFR history and making sure that the resulting evolution of hydrogen ionized fraction is consistent with current constraints on the CMB optical depth.

In order to obtain the SFR of galaxies at the high redshifts during the epoch of reionization we make use of parametrizations that reproduce a correct reionization history. Our parametrizations are non linear in a similar way to the relations found in the Guo et al. (2011) and in the De Lucia & Blaizot (2007) galaxies catalogs derived respectively from the high resolution Millennium II (Boylan-Kolchin et al., 2009) and Millennium I (Springel et al., 2005) simulations. Such relations, when available from observations, make an improvement on the models instead of relying purely on theoretical calculations and semi-numerical simulations to predict all of the observations (Mesinger & Furlanetto, 2007; Santos et al., 2010).

There are additional sources of radiation contributing to the Ly-emission, such as a strong non-local sources of ionizing photons as expected from quasars, which would emit a large amount of energy in X-ray photons that would be able to ionize several neutral atoms giving origin to a locally strong Ly-emission from recombinations. However, since the number of quasars is very small compared to the number of normal galaxies at the redshifts we are considering, we will neglect their contribution in the following calculations.

We encourage future works on Ly-intensity to see if the shape of the power spectrum and other statistics can be used to choose between reionization histories that involve both galaxies and quasars.

In the following sub-sections we discuss in more detail the four processes for Ly-emission from galaxies: recombinations, excitations/relaxations, gas cooling, and photon emission from continuum processes.

2.1. Lyman- emission from hydrogen recombinations

Assuming ionizing equilibrium, the number of recombinations in galaxies are expected to match the number of ionizing photons that are absorbed in the galaxy and does not escape into the IGM. Depending on the temperature and density of the gas, a fraction of the radiation due to these recombinations is emitted in the Ly-line.

In the interstellar gas, most of the neutral hydrogen is in dense clouds with column densities greater than cm. These clouds are optically thick to Ly-radiation and Lyman photons are scattered in the galaxy several times before escaping into the IGM. Such multiple scatterings increase the probability of absorption. Assuming that these clouds are spherical and that the gas temperature is of the order of K, Gould & Weinberg (1996) used atomic physics to study the probability of the Ly-emission per hydrogen recombination. They estimated that a fraction of the hydrogen recombinations would result in the emission of a Ly-photon and that most of the other recombinations would result in two-photon emission. These fractions should change with the temperature and the shape of the cloud, but such variations are expected to be small. Other calculations yield fractions between % and % according to the conditions in the cloud. In this paper we have chosen to use a value of since the overall uncertainty on this number is lower than the uncertainty on the number of hydrogen recombinations.

The absorption of Ly-photons by dust is difficult to estimate and changes from galaxy to galaxy, Gould & Weinberg (1996) estimated that for a cloud with a column density cm, the dust in the galaxy absorbs a fraction % of the emitted Ly-photons before they reach the galaxy virial radius however recent observations of high redshift galaxies indicate a much higher . In this study we will use a redshift parameterization for the fraction of Ly-photons that are not absorbed by dust that is double the value predicted by the study made by Hayes et al. (2011):

(1)

where and . The Hayes et al. (2011) parameterization was made so that gives the difference between observed Ly-luminosities and Ly-luminosities scaled from star formation rates assuming that the Ly-alpha photons emitted in galaxies are only originated in recombinations. The high redshift observations used to estimate are only of massive stars while the bulk of Ly-emission is originated in the low mass stars that cannot be detected by current surveys. According to several studies (Forero-Romero et al., 2011), decreases with halo mass, so it is possible that it is being underestimated in Hayes et al. (2011) which is why we decided to use a higher . Our results can however be easily scaled to other evolutions.

The number of Ly-photons emitted in a galaxy per second, , that reach its virial radius is therefore given by

(2)

where accounts for the fraction of photons that go into the ionization of helium ( is the mass fraction of helium), is the rate of ionizing photons emitted by the stars in the galaxy and is the fraction of ionizing photons that escape the galaxy into the IGM.

The ionizing photon escape fraction depends on conditions inside each galaxy and is difficult to estimate, especially at high redshifts. The precise determination of its value is one of the major goals of future observations of high redshift galaxies at . This parameter can be measured from deep imaging observations or can be estimated from the equivalent widths of the hydrogen and helium balmer lines. The ionizing photon escape fraction dependence with the galaxy mass and the star formation rate, as a function of redshift, has been estimated using simulations that make several assumptions about the intensity of this radiation and its absorption in the interstellar medium. However, for the halo virial mass range, to , and during the broad redshift range related to the epoch of reionization, there are no simulations that cover the full parameter space. Moreover the limited simulations that exist do not always agree with each other (Gnedin et al., 2008; Wise & Cen, 2009; Fernández-Soto et al., 2003; Siana et al., 2007; Haardt & Madau, 2012). Razoumov & Sommer-Larsen (2010) computed the escape fraction of UV radiation for the redshift interval to and for halos of masses from to M using a high-resolution set of galaxies. Their simulations cover most of the parameter space needed for reionization related calculations and their escape fraction parameterization is compatible with most of the current observational results. Thus, we use it for our calculations here.

According to Razoumov & Sommer-Larsen (2010) simulations, the escape fraction of ionizing radiation can be parameterized as:

(3)

where is the halo mass, and are functions of redshift (Table 1).

z
10.4
8.2
6.7
5.7
Table 1Fits to the escape fraction of UV radiation from galaxies as a function of redshift (based on Razoumov & Sommer-Larsen 2010).

The number of ionizing photons emitted by the stars in a galaxy depends on its star formation rate, metallicity and the stellar initial mass function (IMF). Making reasonable assumptions for these quantities we will now estimate . Since this UV emission is dominated by massive, short lived stars, we can assume that the intensity of ionizing photons emitted by a galaxy is proportional to its star formation rate. In terms of the star formation rate in one galaxy,

(4)

where is the average number of ionizing photons emitted per solar mass of star formation. This can be calculated through:

(5)

where is the stellar IMF, is a constant normalization factor and is the slope of the IMF. In our calculation we used a Salpeter IMF, with . is the star lifetime and its number of ionizing photons emitted per unit time. The values of and were calculated with the ionizing fluxes obtained by Schaerer (2002) using realistic models of stellar populations and non-LTE atmospheric models, appropriated for POP II stars with a metallicity.

Assuming that ionizing photons are only emitted by massive OB stars sets a low mass effective limit for the mass of stars contributing to the UV radiation field of a galaxy. This limit is a necessary condition for the star to be able to produce a significant number of ionizing photons. For the stellar population used for this work we take M (Schaerer, 2002; Shull et al., 2011). The integration upper limit is taken to be M. In this paper we calculated Q using the parameterization values published in Schaerer (2002). The number of ionizing photons per second emitted by a star as a function of its mass is given by:

(6)

where and the star’s lifetime in years is given by:

(7)

The use of these parameters results in M. In Shull et al. (2011) it has been suggested the use of a different model for stellar atmosphere and evolution (R. S. Sutherland J. M. Shull, unpublished) which yields M. This may imply that the stellar emissivity we calculated is an overestimation and that consequently our Ly-flux powered by stellar emission may be overestimated by about 35%. This is comparable to other large uncertainties, such as the ones in the parameters and . The Ly-luminosity is calculated assuming that the Ly-photons are emitted at the Ly-rest frequency, Hz with an energy of erg. To proceed, we will assume that the SFR for a given galaxy is only a function of redshift and the mass of the dark halo associated with that galaxy. The Ly-luminosity due to recombinations in the interstellar medium, , can then be parameterized as a function of halo mass and redshift as

(8)

2.2. Lyman- emission from excitations/relaxations

The kinetic energy of the electron ejected during the hydrogen ionization heats the gas and assuming thermal equilibrium this heat is emitted as radiation. Using atomic physics, Gould & Weinberg (1996) estimated that for a cloud with an hydrogen column density of cm, the energy emitted in the form of Ly-photons is about % for ionizing photons with energy and % for photons with energy , where eV is the Rydberg energy. The remaining of the energy is emitted in other lines.

Using the spectral energy distribution (SED) of galaxies with a metallicity from the code of Maraston (2005) we estimated that the average ionizing photon energy is eV and that more than 99% of the photons have an energy lower than . According to the Gould & Weinberg (1996) calculation, the fraction of energy of the UV photon that is emitted as Lyman alpha radiation due to the collisional excitations/relaxations is given by:

(9)

For a cloud with the properties considered here this yields an energy in Ly-per ionizing photon of eV or erg. This results in an average of Ly-photons per ionizing photon.

Finally, the Ly-luminosity due to excitations in the ISM, , is then:

(10)

where again it is assumed to be a function of the star formation rate.

2.3. Lyman- emission from gas cooling

During the formation of galaxies, gas from the IGM falls into potential wells composed mainly by dark matter which collapsed under its own gravity. The increase in the gas density leads to a high rate of atomic collisions that heats the gas to a high temperature. According to the study of Fardal et al. (2001) most of the gas in potential wells that collapses under its own gravity never reaches its virial temperature and so a large fraction of the potential energy is released by line emission induced by collisions and excitations from gas with temperatures K. At this temperature approximately of the energy is emitted in Ly-alone.

From Fardal et al. (2001) we can relate the luminosity at the Ly-frequency due to the cooling in galaxies to their baryonic cold mass, , using:

(11)

where both the luminosity and the mass are in solar units. To relate this baryonic cold mass to a quantity we can use in our models, we used the relation between cold baryonic mass and the halo mass from the galaxies in the Guo et al. (2011) catalog. From the equation above, we can then obtain an expression for the luminosity, which can be fitted by:

(12)

with in units of . The relation between the cold gas mass and the mass of the halo shows very little evolution with redshift during reionization. Thus we expect the relation in equation 12 to only depend on redshift due to the redshift evolution of .

2.4. Contributions from continuum emission

Continuum emission can also contribute to the Ly-observations. These include stellar emission, free-free emission, free-bound emission and two photon emission. Photons emitted with frequencies close to the Ly-lines should scatter within the ISM and eventually get re-emitted out of the galaxy as Ly-photons. Otherwise they will escape the ISM before redshifting into one of the Ly-lines and being reabsorbed by a hydrogen atom.

The fraction of photons that scatter in the galaxy can be estimated from the intrinsic width of the Lyman alpha line which has (Jensen et al., 2012). We calculated the stellar contribution assuming an emission spectrum for stars with a metallicity of estimated with the code from Maraston (2005) that can be approximated by the emission of a black body with a temperature of K for eV. The number of stellar origin Ly-photons per solar mass in star formation obtained with this method is:

(13)

We note that we are not accounting for the higher opacity at the center of the Ly-line which should push the photons out of the line center before exiting the star and so we may be overestimating the stellar Ly-photon emission.

Free-bound emission and free-free emission are respectively originated when free electrons scatter off ions with or without being captured. Following the approach of (Fernandez & Komatsu, 2006), the free-free and free-bound continuum luminosity can be obtained using:

(14)

where is the volume of the Strmgren sphere which can be roughly estimated using the ratio between the number of ionizing photons emitted and the number density of recombinations in the ionized volume,

(15)

is the total volume emissivity of free-free and free-bound emission, is the number density of protons (ionized atoms) and is the case A or case B recombination coefficient (see Furlanetto et al. (2006)).

The volume emissivity estimated by (Dopita & Sutherland, 2003) is given by:

(16)

where is the continuum emission coefficient including free-free and free-bound emission given in SI units by:

(17)

In here, ( is the Boltzmann constant, n is the level to which the electron recombines to and is the Rydberg unit of energy), and are the thermally average Gaunt factors for free-free and free-bound emission (Karzas & Latter, 1961, values from). The initial level is determined by the emitted photon frequency and satisfies the condition where is the Rydberg constant.

The continuum luminosity per frequency interval () is related to the Ly-luminosity emitted from the galaxies by: , where is the number of emitted lyman alpha photons per solar mass in star formation. We then obtain M for free-free emission and M for free-bound emission.

During recombination there is also the probability of two photon emission and although this photons have frequencies below the lyman alpha frequency there is a small fraction of them of that are emitted so close to the lyman alpha line that are included in the lyman alpha intrinsic width.

The number of lyman alpha photons that can be originated due to two photon emission during recombination is given by:

(18)

where P(y)dy is the normalized probability that in a two photon decay one of them is the range and is the probability of 2 photon emission during an hydrogen n=21 transition. The probability of two photon decay was fitted by Fernandez & Komatsu (2006) using Table 4 of Brown Mathews (1970) as:

(19)

Finally, the different contributions to the total Ly-luminosity from galaxies due to continuum emission, , are given by:

(20)

for stellar emission,

(21)

for free-free emission,

(22)

for free-bound emission and

(23)

for 2-photon emission.

Note that here we are only considering the part of the continuum emission from galaxies that could contribute to the same ”Ly-redshift”. There will be a continuum emission spectrum with frequencies below the Ly-line from the mechanisms above that will contribute to the same observation from lower redshifts and will generate a ”foreground” to the Ly-signal that needs to be removed. This should be possible due to the smoothness of this background across frequency, in the same manner as foregrounds of the 21-cm signal are removed (e.g. Wang et al. 2006).

2.5. Modeling the relation between star formation rate and halo mass

Simulations of galaxy formation and observations indicate that the star formation of a halo increases strongly for small halo masses but at high halo masses () it becomes almost constant (Conroy & Wechsler, 2009; Popesso et al., 2012).

In order to better estimate and constrain the SFR of a halo we used three non linear SFR versus Halo Mass parameterizations that are in good agreement with different observational constraints. In Sim1 we adjusted the SFR to reproduce a reasonable reionization history and a Ly-Luminosity Function evolution compatible with different observational constraints, in Sim2 we adjusted the SFR vs halo mass relation to the parameterizations from the Guo et al. (2011) galaxies catalogue (low halo masses) and the De Lucia & Blaizot (2007) galaxies catalogue (high halo masses). Sim2 results in an early reionization history with an optical depth to reionization compatible with the low bound of the current observational constraints. Finally Sim3 has the same halo mass dependence as Sim2 but evolves with redshift in a similar way to the De Lucia & Blaizot (2007) and to the Guo et al. (2011) galaxy catalogues.

We parametrized the relations between the SFR and halo mass as:

(24)

where , , , M and M for Sim1,

(25)

where , , , , M, M and M for Sim2 and

(26)

where , , , , M, M and M for Sim3.

Figure 1 shows these relations.

Figure 1.— Star formation rate versus halo mass. The dotted lines show the relations taken from the Guo et al. (2011) catalogue for low halo masses at (bottom dotted line) and (upper dotted line), the yellow crosses show the relation taken from the DeLucia catalogue for high halo masses at . The dash-dotted, solid and dashed lines show the parameterizations used in simulations Sim1, Sim2 and Sim3 respectively for .
Figure 2.— Star formation rate density evolution as a function of redshift. The blue solid line, the green dashed line and the black dashed dotted line were obtained from simulations made using the SimFast21 code (for informations about the code see section 4 and Santos et al. 2010) and the SFR vs. halo mass relations from equations 24, 25 and 26. The red dots are observational constraints derived from the UV luminosities corrected for dust extinction from Bouwens et al. (2011c). Please note that these observational values correspond to high mass galaxies while our results integrate over the halo mass function starting at solar masses (which at redshift 7 corresponds to star formation rates of s for Sim1, 7.83 s for Sim2 and 1.1 s for Sim3), so our star formation rate densities are expected to be higher.

In figure 2, the strong decline in the observational SFRD from to , imposed by the observational point at , was obtained with the observation of a single galaxy using the Hubble Deep Field 2009 two years data (Bouwens et al., 2011a; Oesch et al., 2012). It was argued in Bouwens et al. (2011b), based on an analytical calculation, that even with such low SFRD at high redshifts it was possible to obtain an optical depth to reionization compatible with the value obtained by WMAP () (Komatsu et al., 2011). However, this derivation would imply a high escape fraction of ionizing radiation and that reionization would end at which is hard to reconcile with the constraints from observations of quasars spectra (Mesinger & Haiman, 2007; Zaldarriaga et al., 2008). Our SFRDs are considerably higher than the current observational constraints, although the difference can be explained by a systematic underestimation of the SFR in observed galaxies. Moreover, current observations only probe the high mass end of the high redshift galaxies mass function which will underestimate the SFR density (also the obtained SFRs have very high error bars due to uncertainties in the correction due to dust extinction, the redshift and the galaxy type). In the following sections the results shown were obtained using Sim1 unless stated otherwise.

2.6. Total Lyman- luminosity: comparison with observations

In the previous sections we calculated the Ly-luminosity as a function of the SFR for several effects. The commonly used “empirical” relation between these two quantities is (Jiang et al., 2011)

(27)

and it is based on the relation between SFR and the H luminosity from Kennicutt (1998a) and in the line emission ratio of Ly-to H in case B recombinations calculated assuming a gas temperature of K. This empirical relation gives the Ly-luminosity without dust absorption (we have labeled it K98 for the remainder of the paper).

Our relation between luminosity and star formation is mass dependent (both from the escape fraction as well as due to the expression from the cooling mechanism), so in order to compare it with the result above, we calculate:

(28)

where the average of quantity is done over the halo mass function for the mass range considered. The results are presented in table 2 for a few redshifts.

A(z) A(z) A(z) A(z) A(z)
Table 2Average luminosity per star formation rate (in units ) averaged over the halo mass function for redshifts 10, 9, 8 and 7, from top to bottom.

Although our Ly-luminosities per SFR are slightly higher, at least for low redshifts, we point out that the ”empirical” relation is based on a theoretical calculation that only accounts for Ly-emission due to recombinations. Moreover the observational measurements of H and Ly-are primarily made at low redshifts, where the absorption of Ly-photons by dust in galaxies is expected to be high. Our relation has the advantage of evolving with redshift since it accounts for the evolution of the escape fraction of ionizing photons and for the evolution of the escape fraction of Ly-photons. This -dependence is not present in the standard empirical relation. This redshift evolution of the UV photons escape fraction is a consequence of the increase in the number of massive galaxies with more clumpy structure as the redshift decreases. The star forming regions of massive galaxies are embedded in clumps and therefore it becomes more difficult for the ionizing photons to escape from such dense regions (Razoumov & Sommer-Larsen, 2010; Yajima et al., 2011). The redshift evolution of the relation presented in equation 28 justifies why a theoretical calibration between Ly-luminosity and the SFR of a galaxy is useful for our work.

To check the consistency between our theoretical estimation of the Ly-luminosity and the existing observations during reionization we show in figure 3 the luminosity function (LF) using two of the star formation rate vs. halo mass parameterizations presented in section 2.5. This prediction is then compared to Ly-luminosity functions of photometric identified objects in Shimasaku et al. (2006) and in Kashikawa et al. (2006) near the end of the reionization epoch.

Figure 3.— Ly-luminosity functions obtained with our calculations are shown for redshifts (dashed lines) and (solid lines) for Sim1 (black thick lines) and Sim2 (blue thin lines). The green and red circles show the intrinsic (i.e. not affected by the IGM) Ly-LF from photometric identified objects in Shimasaku et al. (2006) and in Kashikawa et al. (2006) for and respectively.

Our luminosity functions were calculated assuming a minimum halo mass of which corresponds to a minimum luminosity of erg s for Sim1, erg s for Sim2 and erg s for Sim3. The agreement between our LFs and observations is reasonable for Sim1 however our Sim2 overpredicts the abundance of high luminosity Ly-emitters. This difference can be due to sample variance or a result of the high sensitivity of theoretical predictions to several parameters in our model. We point out that the luminosity range relevant for this comparison falls in a halo mass range outside the one for which the escape fraction of UV radiation we are using was estimated, so we could easily get a better fit between observations and Sim2 by reducing this escape fraction for high halo masses. This difference could also be related with the choice of halo mass function. Here we choose the Sheth-Tormen halo mass function (Sheth & Tormen, 1999) which has been shown to fit low-redshift simulations more accurately, but it is yet to be established the extent to which such a halo mass function can reproduce the halo distribution during reionization. Other possible explanation for this difference is the existence of a small amount of neutral gas in the IGM which would severely decrease the observed Ly-luminosity from galaxies. Also, we could have decreased the high luminosity end of our luminosity functions if we had use an Ly-escape fraction that decreased with halo mass such as the one used in (Forero-Romero et al., 2011). We do not consider a model fit to the data to optimize various parameters in our model given that the current constraints on the observed Ly-LFs have large overall uncertainties, especially considering variations from one survey to another.

2.7. Lyman- Average Intensity

In this section and the next one we will attempt to estimate the intensity and power spectrum of the Ly-signal using an analytical model. In section 4 we will improve the estimation by doing the same calculation using a semi-numerical simulation.

The total intensity of Ly-emission can be obtained from the combined luminosity of Ly-photons associated with different mechanisms described in the previous sub-sections, such that:

(29)

where is the halo mass function (Sheth & Tormen, 1999), is the halo mass, , , is the proper luminosity distance and the comoving angular diameter distance. Finally, , where is the comoving distance, is the observed frequency and m is the rest-frame wavelength of the Ly-line.

The evolution of the Ly-intensity predicted by this calculation is shown in figure 4 together with the scaling expected under the ”empirical” relation from Kennicutt (1998a) combined with an assumption related to the gas temperature.

Figure 4.— Ly-Intensity from galaxies in erg s cm sr as a function of redshift. The black dashed dotted line and the blue solid line were obtained using our theoretical calculation of the Ly-luminosity and the SFR halo mass relation from Sim1 and Sim2 respectively. The orange dotted line uses the Ly-luminosity SFR relation based on the relation between SFR and the H luminosity from Kennicutt (1998a) and the line emission ratio of Ly-to H in case B recombinations calculated assuming a gas temperature of K (labeled as the K98 relation). The K98 line is not corrected for dust absorption.

The intensities of Ly-emission from different sources are presented in table 3 for several redshifts.

Source of emission in I(z=7) I(z=8) I(z=10)
Recombinations
Excitations
Cooling
Continuum
Total
Table 3Surface brightness (in observed frequency times intensity) of Ly-emission from the different sources in galaxies at , and for Sim1.

These intensities can be extrapolated to other SFRDs, assuming that the only change is in the amplitude of the SFR halo mass relations presented in figure 1 by using the coefficients in table 4.

Redshift A(Sim1) A(Sim2)
10
9
8
7
Table 4Average Ly-Intensity from galaxies per SFRD (A) in units , calculated using the star formation rate halo mass relation from simulations Sim1 and Sim2.

The intensities from emission at , and are , and erg s cm sr, respectively. Such an intensity is substantially smaller than the background intensity of integrated emission from all galaxies (around erg s cm sr (Madau & Pozzetti, 2000), or from the total emission of galaxies during reionization, estimated to be at most erg scm sr (Yan et al. 2012).

2.8. Lyman- intensity power spectrum

The Ly-emission from galaxies will naturally trace the underlying cosmic matter density field so we can write the Ly-line intensity fluctuations due to galaxy clustering as

(30)

where is the mean intensity of the Ly-emission line, is the matter over-density at the location , and is the average galaxy bias weighted by the Ly-luminosity (see e.g. Gong et al. 2011).

Using one of the relations between the SFR and halo mass from section 2.5 we can calculate the luminosity and obtain the Lyman alpha bias following Visbal & Loeb (2010):

(31)

where is the halo bias and the halo mass function (Sheth & Tormen, 1999). We take and . The bias between dark matter fluctuation and the Ly-luminosity, as can be seen in figure 5, is dominated by the galaxies with low Ly-luminosity independently of the redshift.

Figure 5.— Bias between dark matter fluctuations and Ly-surface brightness from galaxies as a function of the galaxy Lyman alpha luminosity at redshifts , , and .

We can then obtain the clustering power spectrum of Ly-emission as

(32)

where is the matter power spectrum. The shot-noise power spectrum, due to discretization of the galaxies, is also considered here. It can be written as (Gong et al., 2011)

(33)

The resulting power spectrum of Ly-emission from galaxies is presented in figure 6. At all scales presented the Ly-intensity and fluctuations are dominated by the recombination emission from galaxies.

Figure 6.— Clustering power spectrum of the Ly-surface brightness from Galaxies at redshifts 7 to 10 (from top to bottom), from several sources: collisions and excitations, recombinations and continuum emission with frequencies inside the Ly-width. The power spectra from cooling emission is not shown since it is several orders of magnitude smaller than the contributions from the other sources. Also shown are the total power spectra (clustering (solid black line) and shot noise power spectra (dotted black line)) of the total contribution for Ly-emission in galaxies predicted by our theoretical calculation and total Ly-clustering power spectra predicted using the relation (orange solid line).

3. Lyman- emission from the IGM

The Ly-emission from the IGM is mostly originated in recombinations and collisions powered by the ionizing background. These processes are similar to the ones described inside the galaxies, although, since the physical conditions of the gas in the IGM are different from those in the ISM, the intensity of Ly-emission can no longer be connected to the ionizing photon intensity using the previous relations. The biggest challenge in doing these calculations is to connect the IGM ionizations and heating of the gas to the emission of ionizing radiation and the star formation rate assumed in the previous sections. Moreover, in the IGM, we also have to take into account the contribution of continuum radiation from stars between the Ly-and the Lyman limit which redshifts into the Ly-line.

In a schematic view, we have to take into account the following processes,

  1. The amount of energy in UV photons that escapes the galaxy

  2. This energy will then be distributed in the IGM into:

    1. ionizations

    2. direct excitations (followed by emission, partially into the Ly-line)

    3. heating of the gas

  3. Taking into account the state of the IGM in terms of temperature and ionization, we can then further determine how much it will radiate through the Ly-line from:

    1. Recombinations

    2. Radiative cooling (usually through excitations followed by decay in several lines including Ly-)

  4. The amount of Ly-photons that escape the galaxy, re-scattering in the IGM into Ly-photons

The proper calculation of all these processes will require simulations which we will address in section 4. In the following sub-sections we review the contributions through analytical calculations in order to get a better understanding of the dominating effects.

3.1. Lyman- emission from hydrogen recombinations

The UV radiation that escapes the interstellar medium into the intergalactic medium ionizes low density clouds of neutral gas. Part of the gas in these clouds then recombines giving rise to Ly-emission. The radiation emitted in the IGM is often referred to as fluorescence (Santos, 2004). The comoving number density of recombinations per second in a given region, , is given by:

(34)

where changes between the case A and the case B recombinations coefficient, is the ionized hydrogen comoving number density ( is the ionization fraction, the baryonic comoving number density). The free electron density can be approximated by .

The recombination coefficients are a function of the IGM temperature, . The case A comoving recombination coefficient is appropriate for the highly ionized low redshift Universe Furlanetto et al. (2006),

(35)

while the case B comoving recombination coefficient is appropriate for the high redshift Universe.

(36)

The use of a larger recombination coefficient when the process of hydrogen recombination is close to its end accounts for the fact that at this time, ionizations (and hence recombinations) take place in dense, partially neutral gas (Lyman-limit systems) and the photons produced after recombinations are consumed inside this systems so they do not help ionizing the IGM (see: eg. Furlanetto et al., 2006).

The fraction of Ly-photons emitted per hydrogen recombination, , is temperature dependent so we used the parameterization for made by Cantalupo et al. (2008) using a combination of fits tabulated by Pengelly (1964) and Martin (1988) for and respectively:

(37)

The luminosity density (per comoving volume) in Ly-from hydrogen recombinations in the IGM, , is then given by

(38)

3.2. Lyman- emission from excitations in the IGM

The UV radiation that escapes the galaxies without producing ionization ends up ionizing and exciting the neutral hydrogen in the IGM and heating the gas around the galaxies. The high energetic electron released after the first ionization spends its energy in collisions/excitations, ionizations and heating the IGM gas until it thermalizes (Shull & van Steenberg, 1985). We estimated the contribution of the direct collisions/excitations to the Ly-photon budget and concluded that it is negligible.

The Ly-luminosity density due to the collisional emission (radiative cooling in the IGM), , is given by:

(39)

where is the neutral hydrogen density, is the IGM ionized fraction and is the effective collisional excitation coefficient for Ly-emission which we calculated in the same way as Cantalupo et al. (2008), but using different values for the gas temperature and IGM ionization fraction.

Considering excitation processes up to the level that could eventually produce Ly-emission, the effective collisional excitation coefficient is given by:

(40)

The collisional excitation coefficient for the transition from the ground level () to the level () is given by

(41)

where is the temperature dependent effective collision strength, is the statistical weight of the ground state, is the energy difference between the ground and the level and is the Boltzmann constant.

3.3. Scattering of Lyman-n photons emitted from galaxies

Continuum emission of photons, by stars, from Ly-to the Lyman-limit travels until it reaches one of the Ly-lines where it gets scattered by neutral hydrogen. Most of this scattering will have as end result the production of Ly-photons which eventually redshift out of the line. Since a considerable fraction of this photons only reach a given Ly-frequency in the IGM this Ly-emission is formed as a flux that decays with around the star that emitted the continuum photons so it appears diluted in frequency in line observations of point sources (Chen & Miralda-Escudé, 2008). This continuum photons are much less likely to be absorbed by the dust in the ISM than photons originated in recombinations.

In intensity mapping the frequency band observed is much larger than in line observations so in principle all the continuum Ly-photons can be detected. Using the Spectral Energy Distribution (SED) made with the code from Maraston (2005) we estimated that the number of photons emitted by stars between the Ly-plus the lyman alpha equivalent width and the Lyman-limit is equivalent to . The higher frequency photons are absorbed by hydrogen atoms as they reach the Lyman beta frequency, reemitted and suffer multiple scattering until they reach the Ly-line. The fraction of the continuum photons emitted close to the Ly-line have already redshifted to lower frequencies before reaching the IGM so they will not be scattered by the neutral hydrogen in the IGM and will not contribute to the radiative coupling of the 21 cm signal (they are already included in the calculation of the Ly-emission from galaxies).

The intensity of this emission was calculated with a stellar emissivity that evolves with frequency as with and normalized to . The Ly-luminosity density originated from continuum stellar radiation and emitted in the IGM, , is then approximately given by:

(42)

where the SFRD is in units of per second. Note that in section 4, this calculation is done through a full simulation.

3.4. Lyman- Intensity

We calculated the intensities for the several Ly-sources in the IGM from their luminosity densities using: