A NOVEL APPROACH TO CONSTRAIN THE ESCAPE FRACTION AND DUST CONTENT AT HIGH REDSHIFT USING THE COSMIC INFRARED BACKGROUND FRACTIONAL ANISOTROPY

Modern cosmology is now able to constrain details about the era of reionization. Observations show that the reionization of the universe occurred early and was extended in time, with an equivalent of an instantaneous reionization at z ∼ 11 (Komatsu et al. 2009, 2011). Stars are likely candidates for being responsible for the majority of reionization because they are efficient producers of ultraviolet photons. Thanks to a wave of modern, sensitive telescopes, we can begin to observe and understand the frontier of reionization, along with these high-redshift stellar populations (z > 6). For example, we can observe these galaxies directly via high-redshift surveys, which can now routinely identify a population of bright galaxies up to a redshift of about z ∼ 8. However, these surveys can only locate those galaxies that are both above the limiting magnitude and common enough to be present in the survey field. It is now thought (Bouwens et al. 2010; Robertson et al. 2010; Fernandez & Shull 2011) that reionization needed a large population of smaller galaxies below the current detection limits. Because we cannot yet observe these galaxies directly, we can instead look for their cumulative light, which should exist as background radiation. Because reionization is said to have occurred around z ∼ 11, the photons responsible for reionization should be present in the Cosmic Infrared Background (CIB). The spectral peak of this radiation is around the Lyα line, which will be redshifted to 1–4 μm. However, continuum emission will create an extended tail at longer wavelengths.

Here, we discuss the CIB from about 1 to 200 μm. The majority of the CIB will be emission from sources below z ∼ 6, such as our Galaxy, foreground galaxies, and other sources of infrared light, such as zodiacal light. If these sources can be subtracted away to a high precision, it is possible that the remainder could be from the era of reionization and, if so, could tell us about the properties of these high-redshift stars.

There have been many attempts to theoretically model the high-redshift component of the CIB, especially in the near-infrared, from the mean (Santos et al. 2002; Magliocchetti et al. 2003; Salvaterra & Ferrara 2003; Cooray & Yoshida 2004; Madau & Silk 2005; Fernandez & Komatsu 2006) to fluctuations (Kashlinsky et al. 2002, 2004, 2005, 2007c, 2012; Kashlinsky 2005; Magliocchetti et al. 2003; Cooray et al. 2004; Thompson et al. 2007a, 2007b; Fernandez et al. 2010, 2012). In this paper, we examine another way to analyze the CIB, the fractional anisotropy, which is the ratio of the fluctuations to the mean. By looking at the fractional anisotropy, many free parameters are removed and more information about this elusive stellar population can be extracted. Specifically, we discuss using the CIB as a probe for the escape fraction of ionizing photons. Finding the escape fraction is important for understanding reionization and its duration. There have been several attempts to measure the escape fraction through analytical models, simulations, and observations (see Fernandez & Shull 2011 and references therein). These papers have shown that the escape fraction appears to vary greatly from galaxy to galaxy. Therefore, instead of trying to measure the escape fraction of an individual galaxy, here we discuss the average escape fraction of all galaxies, which will give more of a global view of reionization. In addition, the fractional anisotropy can reveal information about the dust content of galaxies, which is mostly unknown at high redshifts.

We describe our simulations in Section 2 and our models in Section 3. In Section 4, we discuss our method for finding the mean CIB, the fluctuations of the CIB, and the fractional anisotropy. In Section 5, we discuss our results of the fractional anisotropy for various bands. In Section 6, we discuss the most recent observations. We conclude in Section 7. Throughout this paper, we use the cosmological parameters (Ω_m, Ω_Λ, Ω_b, h) = (0.27, 0.73, 0.044, 0.7), consistent with the simulations from Iliev et al. (2012), which are based on the Wilkinson Microwave Anisotropy Probe five-year results and other available constraints (Komatsu et al. 2009).

In order to predict the angular power spectrum of the CIB, we used simulations from Iliev et al. (2012), which are N-body simulations combined with radiative transfer, which allow us to see how sources are affected by the reionization process. The high resolution of these simulations (with a minimum mass of 10⁸ M_☉) allows us to also include Jeans-mass filtering on low-mass halos. This effectively allows suppression of star formation within small halos (10⁸–10⁹ M_☉) because of elevated gas temperatures that could be caused by the proximity to other star-forming galaxies. These simulations have a box size of either 114h⁻¹ Mpc (for cases with suppression of small sources) or 37h⁻¹ Mpc (with no suppression, where M_min = 10⁸ M_☉, or complete suppression, where M_min = 10⁹ M_☉). These simulations are summarized in Table 1 (Iliev et al. 2012; Fernandez et al. 2012).

To describe the stellar populations within these halos and their relationship with their environment, we define a parameter f_γ, which describes the number of ionizing photons produced per stellar atom that can escape the galaxy and reionize the intergalactic medium (IGM). f_γ is defined as a product of the star formation efficiency, or the fraction of baryons that are in stars (f_*), the escape fraction (f_esc), and the number of ionizing photons per stellar atom (N_i):

$$\begin{equation} f_\gamma = f_{{\rm esc}}f_*N_i. \end{equation} \tag{ 1 }$$

We allow f_γ to have different values, dependent on the mass of the halo, assuring it is consistent with reionization.

True first-generation stars are metal free (Population III stars). As time goes on, stars die and enrich the universe, and eventually these Population III stars give way to stars with metals (Population II stars). It is unclear when this happens, and this process is probably very inhomogeneous. Therefore, we assume two limiting cases—all of the stars from 6 < z < 30 are either Population III (Z = 0) or Population II (Z = 1/50 Z_☉) stars.

In addition to the metallicity, there is uncertainty in the mass spectrum of these stars. These stars could be very large, or they could be similar in size to what we see today. To model these two extremes, we choose either a heavy, Larson mass function (Larson 1998)

$$\begin{equation} f(m)\propto m^{-1}\left(1+\frac{m}{m_c}\right)^{-1.35}, \end{equation} \tag{ 2 }$$

with mass limits of m₁ = 3 M_☉, m₂ = 500 M_☉, and m_c = 250 M_☉ to model a population of large stars, or a Salpeter mass function (Salpeter 1955)

$$\begin{equation} f(m) \propto m^{-2.35}, \end{equation} \tag{ 3 }$$

with mass limits of m₁ = 3 M_☉ and m₂ = 150 M_☉ to simulate a mass spectrum similar to what we see in the local universe.

If we combine our limiting cases for both mass and metallicity, we can establish our two limiting stellar models: Population III stars with a Larson mass spectrum, and Population II stars with a Salpeter mass spectrum. In reality, these stellar limits are extreme. In addition, we expect stellar properties to be inhomogeneous throughout redshift. However, these examples were chosen as limiting cases: which represent a population with the smallest and largest amplitude for the angular power spectrum of a large range of models, studied in detail in Fernandez et al. (2010). We would expect the actual amplitude for the angular power spectrum to lie between these extremes. These populations are summarized in Table 2.

4.1. The Mean Cosmic Infrared Background

Now that we have our stellar and galactic models, we are in a position to calculate the fractional anisotropy, δI/I. To do this, we must compute both the mean intensity and the angular power spectrum of the CIB. The mean CIB is a combination of emission from the star, which is modeled as a stellar blackbody, and emission from the nebula, which is a combination of the Lyα line, two-photon, free–free and free–bound emission, and with the possibility of emission from dust (see Section 4.2). This nebular emission is either produced within the halo itself, or within the IGM if some fraction of the ionizing radiation (f_esc) escapes the halo. However, the mean CIB does not depend on f_esc, since the nebular emission is the same, regardless of whether it is from the halo or the IGM.

Each emission process (stellar, free–free, free–bound, two-photon, and Lyα) was modeled analytically (for details, see Fernandez & Komatsu 2006) and integrated over a range of redshifts from 6 < z < 30. The total intensity (I) is then

$$\begin{eqnarray} \nonumber I &=& \frac{c}{4\pi }\left(f_*\frac{\Omega _b}{\Omega _m}\right)\int \frac{dz}{H(z)(1+z)} \bar{\rho }_M^{{\rm halo}}(z)\\ & &\times \left[\bar{l}^*(z) +\bar{l}^{ff}(z)+\bar{l}^{fb}(z)+\bar{l}^{2\gamma }(z)+\bar{l}^{\rm Ly\alpha }(z)\right]. \end{eqnarray} \tag{ 4 }$$

Here, $\bar{\rho }_M^{\rm halo}(z)$ is the mean mass density collapsed into halos from the simulation. The luminosity per stellar mass, $\bar{l}(z)$, is given for each component of the luminosity, * for stellar, ff for free–free, fb for free–bound, 2γ for two-photon, and Lyα for Lyα emission. The luminosity of any component "α" can be written as³

$$\begin{equation} l^\alpha _\nu (z) = \frac{d\ln \rho _*(z)}{dt} \frac{\int ^{m_2}_{m_1} dm f(m)L^\alpha _{\nu }(m)\tau (m)}{\int ^{m_2}_{m_1} dm f(m) m}. \end{equation} \tag{ 5 }$$

The luminosity of each emission component (L^α_ν) and the stellar lifetime (τ(m)) are integrated over a mass spectrum of stars. The first part of this expression is the inverse of the star formation timescale, t_SF(z) = [dln ρ_*(z)/dt]⁻¹. This star formation timescale is unknown, but we assume a value of 11.5 Myr, consistent with the value from the simulations of Iliev et al. (2012). This expression then reduces to

$$\begin{equation} l^\alpha _\nu (z) = \frac{1}{t_{{\rm SF}}(z)} \frac{\int ^{m_2}_{m_1} dm f(m)L^\alpha _{\nu }(m)\tau (m)}{\int ^{m_2}_{m_1} dm f(m) m} \end{equation} \tag{ 6 }$$

(Fernandez et al. 2010).⁴

4.2. Dust

4.3. Fluctuations in the Cosmic Infrared Background

The next step is to compute the angular power spectrum. These fluctuations will arise from both the emitting halos and their surrounding H ii regions within the IGM. As shown in Fernandez et al. (2010), the fluctuations from the IGM are probably quite small (from two to seven orders of magnitude smaller than that of the halos themselves) so can safely be ignored.

The angular power spectrum C_l can then be written as

$$\begin{eqnarray} \nonumber C_l &=& \frac{c}{(4\pi)^2}\left(f_*\frac{\Omega _b}{\Omega _m}\right)^2\int \frac{dz}{H(z)r^2(z)(1+z)^4}\\ \nonumber & &\times \left[\bar{\rho }_M^{{\rm halo}}(z) \left\lbrace \bar{l}^*(z) + (1-f_{\rm esc})\bar{L}(z)\right\rbrace \right]^2\\ & &\times b^2_{{\rm eff}}\left(k=\frac{l}{r(z)},z\right)P_{\rm lin}\left(k=\frac{l}{r(z)},z\right). \end{eqnarray} \tag{ 7 }$$

Here, b_eff is the effective bias, P_lin is the linear matter power spectrum, r(z) = c∫^z₀dz'/H(z') is the comoving distance, and the luminosity is

$$\begin{equation} \bar{L}(z)=\bar{l}^{ff}(z)+\bar{l}^{fb}(z)+\bar{l}^{2\gamma }(z)+\bar{l}^{\rm Ly\alpha }(z). \end{equation} \tag{ 8 }$$

The simulations provide both the halo bias and the linear matter density fluctuations. Note that the angular power spectrum depends on f_esc. The angular power spectrum of these simulations was computed in Fernandez et al. (2012).

4.4. The Fractional Anisotropy

The fractional anisotropy of the CIB is obtained by dividing the angular power spectrum C_l (shown in Equation (7)) by the mean intensity I (Equation (4)):

$$\begin{equation} \delta I/I\equiv \sqrt{l(l+1)C_l/(2 \pi I^2)}. \end{equation} \tag{ 9 }$$

Most of the free parameters then cancel out, including the star formation efficiency f_*. The luminosity $\bar{l}^\alpha$, however, will only cancel out when f_esc = 0. Therefore, the fractional anisotropy serves as a test to constrain f_esc.

At large values of l, the minimum mass of the star-forming halos and suppression history will change the shape of the angular power spectrum due to nonlinear bias effects (Fernandez et al. 2012). Since the minimum mass of these star-forming halos is unknown, we compute the fractional anisotropy for l = 3000, avoiding any flattening or steepening of the angular power spectrum that could occur at larger l.

The fractional anisotropy at l = 3000 is shown in Figures 1 and 2 as a function of observed wavelength. The shaded regions are bounded by our two fiducial models (Population II stars with a Salpeter mass function will give the upper limit of the shaded region, while Population III stars with a Larson mass function will give the lower limit). Other reasonable models, varying the mass or metallicity of the stars, will lie between these two limiting cases, since the amplitude of the angular power spectrum will lie between these cases (Fernandez et al. 2010). We also show a range of f_esc, from f_esc = 1 (where all the ionizing photons escape from the halo into the IGM) to f_esc = 0.1. Results are shown for a case where reionization progresses with a high efficiency, the minimum halo mass is 10⁸ M_☉, and small halos can be suppressed. However, these assumptions do not greatly affect the results.

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As seen in Figure 1, the escape fraction has a large effect on δI/I. The mean level of the CIB, given in Equation (4), has no dependence on the escape fraction. The angular power spectrum, given by Equation (7), has a factor of (1 − f_esc). Therefore, when f_esc rises, the level of the nebular contribution to the angular power spectrum will fall. This causes the overall level of δI/I to fall.

This drop-off of δI/I for larger values of the escape fraction is more pronounced at longer wavelengths. To see why this occurs, we can look at the mean spectrum of starlight and nebular emission for a high-redshift galaxy in Figure 3. (The definitions of the bands shown are given in Table 3.) In the near-infrared bands (λ < 4 μm), there is always a large contribution from the stellar blackbody emission. At longer wavelengths, the stellar emission drops off very quickly while, if the escape fraction is low, the nebular emission remains relatively high. If the escape fraction is high, however, the nebular emission component of the angular power spectrum would be diminished. This is particularly noticeable at longer wavelengths.

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For small values of f_esc, the range of allowed values for δI/I is quite narrow. This range widens if f_esc = 1. This is a consequence of the spectral shape for our two stellar models. For Population III stars, the blackbody spectrum of the star is steeper and there are more ionizing photons to be processed into nebular emission. The stellar spectrum is almost always equal to or less than that of the nebular component. On the other hand, the stellar blackbody is greater than the nebular component at short wavelengths for metal-poor Population II stars. As the wavelength increases, the emission from the stellar component will drop below the level of emission from the nebula. If f_esc is small, the total emission of the halos for Population III and Population II stars is similar, so the range of δI/I is narrow. If f_esc is large, the amplitude of the angular power spectrum of large Population III stars will be more affected than smaller Population II stars, widening the range of allowed values of δI/I.

In Figure 2, we see the fractional anisotropy for the case when galaxies contain dust. Here, we no longer see a decrease in δI/I at long wavelengths. This is because dust will reprocess the stellar and nebular emission, re-emitting the light at longer wavelengths. (An illustration of the SED with dust included is also shown in Figure 3.) Unlike nebular emission, the dust component will not fall to zero when f_esc = 1. In fact, the change between the dust emission from f_esc = 0 to 1 is only slight. Therefore, while the nebular component will disappear if f_esc = 1, the dust component will still be present, causing the angular power spectrum at long wavelengths to remain high, raising δI/I. This is a direct result from the fact that nebular emission only results from ionizing photons, while photons of lower energies can be converted into dust emission.

Measurements of the CIB are notoriously hard to perform. Finding an accurate mean is particularly difficult because precise foreground subtraction is needed. However, observations, such as with Ciber, Akari, and Herschel, continue to improve, resolving foregrounds in more detail and obtaining more reliable observations for the CIB.

Many observations have been made to try to understand the contribution of high redshifts to the CIB from 1 to 4 μm. In order to uncover any residual emission in the mean or fluctuation observations, one must carefully take into account all of the foreground components. Zodiacal light is a major contaminant, and because it is very difficult to model, it is not straightforward to subtract from the CIB. In addition, foreground galaxies at lower redshifts must be taken into account. Despite the difficulty, there have been many attempts to measure the mean level of the CIB in the near-infrared due to high-redshift stars (Dwek & Arendt 1998; Gorjian et al. 2000; Kashlinsky & Odenwald 2000; Kashlinsky 2005; Wright & Reese 2000; Wright 2001; Cambrésy et al. 2001; Totani et al. 2001; Kashlinsky et al. 2002, 2004, 2007a, 2007b, 2012; Magliocchetti et al. 2003; Odenwald et al. 2003; Cooray et al. 2004; Matsumoto et al. 2005; Thompson et al. 2007a, 2007b).

Fluctuation observations are, in theory, easier to perform, since they do not need an accurate zero point, and instead rely on variations from one region of the sky to another. However, these observations still rely on careful and complete subtraction of foreground sources, and also remain controversial (Kashlinsky et al. 2005, 2007b, 2012; Cooray et al. 2007; Thompson et al. 2007a; Matsumoto et al. 2011).

Observations are even more difficult in the mid- and far-infrared. One problem is that foregrounds that were present in the near-infrared are even more prevalent in the mid- and far-infrared. Zodiacal light peaks at about 20 μm, which washes out most detections of the CIB in this range. In addition, Galactic cirrus is a main contaminant. However, observations are possible in clean regions of the sky. Finally, as wavelength increases, foreground galaxies become more difficult to resolve. All of these problems lead to only a fraction of the CIB in the mid- and far-infrared being resolved into low-redshift galaxies. It is likely that only a very small (and currently unknown) percentage of this excess is from z > 6, so care must be taken in interpreting observations.

There has been a great push to understand the CIB at longer wavelengths. At 100 and 160 μm, Pénin et al. (2011) measured the mean and fluctuation power of galaxies at all redshifts using observations from IRIS/IRAS and Spitzer/MIPS, respectively. Fluctuations of the cumulative CIB have been taken in the mid-infrared to submillimeter wavelengths (Kashlinsky & Odenwald 2000; Lagache & Puget 2000; Miville-Deschênes et al. 2002; Grossan & Smoot 2007; Lagache et al. 2007; Amblard et al. 2011; Matsuura et al. 2011; Planck Collaboration et al. 2011; Pyo et al. 2012). The cumulative mean level of the CIB from galaxies at all redshifts has been measured as well (Fixsen et al. 1998; Hauser et al. 1998; Lagache et al. 2000; Wright 2004; Odegard et al. 2007; Matsuura et al. 2011). The mean CIB has been measured as a function of redshift (Berta et al. 2011; Jauzac et al. 2011; Béthermin et al. 2012), while measurements from BLAST and Planck from 250 to 1400 μm (Planck Collaboration et al. 2011) could indicate that galaxies at a higher redshift (here, z > 1.2–2) could contribute more to the CIB as the wavelength increases. Currently, the best measurements (Planck Collaboration et al. 2011; Viero et al. 2009) show that the fractional anisotropy is of the order of 15%; however, these measurements include galaxies at all redshifts.

One way to subtract unresolved low-redshift galaxies in the mid- to far-infrared to a more complete level is to use a stacking algorithm. This typically involves using the locations of known galaxies at a shorter wavelength, stacking these locations of a longer wavelength image, and utilizing this stack to calculate the CIB accounted for from these galaxies. If stacking is relied upon, more of the CIB at long wavelengths can be resolved into lower redshift galaxies. For example, Marsden et al. (2009) used stacking to resolve 100% of the CIB as detected with FIRAS (Fixsen et al. 1998) at 250, 350, and 500 μm using BLAST. Dole et al. (2006) used 24 μm sources from Spitzer/MIPS data to stack images at 70 and 160 μm. They were able to resolve 79%, 92%, and 69% of the CIB at 24, 70, and 160 μm, respectively. Berta et al. (2010) resolved 45% and 52% (without stacking) and 50% and 75% (with stacking) of the CIB at 100 and 160 μm using Herschel/PACS data. At longer wavelengths, Greve et al. (2010) resolved 16.5% of the CIB at 870 μm using stacking. While it is possible that some of the remaining flux is from low-redshift galaxies, it is also possible that some of this unresolved CIB could be due to high-redshift galaxies. (See, e.g., Matsuura et al. 2011.)

We compare some of these observations to our models for the fractional anisotropy at high redshifts. While precise measurements of the mean are challenging, measurements of the fluctuation power are becoming more reliable. Because of this, in Figure 4 we show the fractional anisotropy predicted using various recent observations of the fluctuation power, assuming an upper limit of the mean CIB due to high-redshift stars is either 10 or 1 nW m⁻² sr⁻¹. Because it is unlikely that a z > 6 component of the mean CIB will be much higher than this, very low values of δI/I, and thus very high values of the escape fraction from a dust-free population, can be ruled out. More definitive conclusions can be reached as observations continue to improve and our understanding of foreground emission grows.

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It is important to remember that there is still low-redshift contamination contributing to both the mean and fluctuations, especially at longer wavelengths. Therefore, these results should be interpreted with this in mind. As observations improve, these results will become more reliable.

We have shown the observable signatures of high-redshift populations with different values of the escape fraction f_esc and dust content in the CIB. It is possible to distinguish these populations through observations of the fractional anisotropy of the CIB. The global escape fraction of high-redshift galaxies is a main variable that can be probed in this way, since the angular power spectrum is dependent on it, while the mean is not. In addition, dust will transform the SED of the galaxy, thus leaving an imprint on the fractional anisotropy. Therefore, low values of the fractional anisotropy will be indicative of a population of stars with a high escape fraction and little dust. This will be more noticeable at longer wavelengths. While observations are still difficult, improved observations could be able to distinguish between these populations.

We thank Masami Ouchi and Kristian Finlator for helpful discussions. In addition, we also thank Melanie Koehler and Laurent Verstraete for help with DustEM. This work was supported by the Science and Technology Facilities Council (grant Nos ST/F002858/1 and ST/I000976/1), the ANR program ANR-09-BLAN-0224-02, and The Southeast Physics Network (SEPNet). The authors acknowledge the Texas Advanced Computing Center (TACC) at The University of Texas at Austin for providing HPC resources that have contributed to the research results reported within this paper (http://www.tacc.utexas.edu). This research was supported in part by the National Science Foundation through TeraGrid resources provided by TACC and NICS.