HEATING THE INTERGALACTIC MEDIUM BY X-RAYS FROM POPULATION III BINARIES IN HIGH-REDSHIFT GALAXIES

We further analyze the simulation in Xu et al. (2013) to study possible X-ray radiation from Pop III binaries in the high-density ∼138 comoving Mpc³ survey volume. The simulation is performed using the adaptive mesh refinement (AMR) cosmological hydrodynamics code Enzo (Bryan et al. 2014). The adaptive ray-tracing module Enzo+Moray (Wise & Abel 2011) is used for the radiation transfer of ionizing radiation, which is coupled to the hydrodynamics and chemistry in Enzo.

The initial conditions for the simulation is generated using Music (Hahn & Abel 2011) with second-order Lagrangian perturbations at z = 99. We use the cosmological parameters from the seven-year WMAP ΛCDM+SZ+LENS best fit (Komatsu et al. 2011): Ω_M = 0.266, Ω_Λ = 0.734, Ω_b = 0.0449, h = 0.71, σ₈ = 0.81, and n = 0.963. We use a hydrogen mass fraction X = 0.76. We simulate a comoving volume of (40 Mpc)³ that has a 512³ root grid resolution and three levels of static nested grids centered on a high-density region. We first run a 512³N-body-only simulation to z = 6. Then, we select the Lagrangian volume (a single rectangular box) around two ∼3 × 10¹⁰ M_☉ halos at z = 6 and re-initialize the simulation, having the Lagrangian volume at the center, with three more static nested grids to have an effective resolution of 4096³ and an effective dark matter mass resolution of 2.9 × 10⁴  M_☉ inside the highest nested grid, which just covers the Lagrangian volume, with a comoving volume of 5.2 × 7.0 × 8.3 Mpc³ (∼300 Mpc³). During the course of the simulation, we allow a maximum refinement level l = 12, resulting in a maximal resolution of 19 comoving pc. The refinement criteria employed are the same as in Wise et al. (2012b). The refinements higher than the static nested grids are only allowed in a sub-volume, which adjusts its size during the simulation to contain only the highest-resolution dark matter particles of the highest static nested grid. The highly refined region, covering the Lagrangian volume of the two massive halos at z = 6, has a comoving volume of 3.8 × 5.4 × 6.6 Mpc³ (∼138 Mpc³) at z = 15, which represents a 3.5σ density peak. We call this well-resolved volume, which is also our survey volume, the Rarepeak in this study and related papers (P. F. Chen et al. 2014, in preparation; Ahn et al. 2014). At this time, the simulation has more than 10,000 Pop III stars and remnants distributed over 3000 halos, most of them are more massive than 10⁷ M_☉. The simulation has 1.3 billion computational cells in the refined region and consumed more than 10 million CPU hours on the Kraken system at NICS and Blue Waters system at NCSA.

Both Pop II and Pop III stars are allowed to form inside the survey volume, and we distinguish them by the total metallicity of the densest star-forming cell. Pop III stars are formed if [Z/H] < −4, and Pop II stars are formed otherwise. We use the same star formation models and most of the parameters as in Wise et al. (2012b), as well as feedback models. For the initial mass of Pop III stars, we randomly sample from an IMF with a functional form:

$$\begin{equation} f(\log M)dM=M^{-1.3}\exp \left[-\left(\frac{M_{\rm char}}{M}\right)^{1.6}\right]\, dM, \end{equation} \tag{ 1 }$$

which behaves as a Salpeter IMF above the characteristic mass, M_char, but is exponentially cutoff below that mass (Chabrier 2003). Here, we use a characteristic mass of 40 M_☉ for the Pop III IMF, which agrees with the latest results of Pop III formation simulations (e.g., Turk et al. 2009; Greif et al. 2012). For the details of the star formation and stellar feedback schemes used, refer to Sections 2.2 and 2.3 of Wise et al. (2012b).

The simulation performs ray tracing to calculate the propagation of UV H i ionizing radiation, but to study the X-ray radiation transport and associated photoheating and photoionization effects from Pop III binaries, we post-process the data sets with Enzo+Morayto calculate the X-ray radiation transport, which also includes the ionization of He i and He ii. We consider secondary ionizations and heating by X-ray photons, using the fitting formula from Shull & van Steenberg (1985). Details of the implementation can be found in Wise & Abel (2011).

We first need to use our Pop III distribution from simulation to estimate the X-ray radiation from Pop III binaries. Since the occurrence fraction, properties, and evolution of Pop III binaries are not yet well constrained (e.g., Stacy & Bromm 2013), we build a simple model, which takes advantage of solid information on Pop III population and distribution and ignores the details of Pop III binary formation and evolution, to estimate the X-ray luminosities from Pop III binaries in Rarepeak. We ignore the Pop III star initial mass when we set the chance of a Pop III becoming binary and the evolution of the companion star. We assume that there is α (0 ⩽ α ⩽ 1) chance that a Pop III star becomes an XRB. We also assume that one companion directly collapses into a BH with a mass of tens of M_☉  without a ova (Fryer 1999), and the remaining mass exists in the companion star. In addition, the companion star is supposed to live as a normal star (e.g., no SN or BH) for a constant lifetime τ, no matter what its mass is. For simplicity, we set the initial M_{BH, 0} to 40 M_☉ if the Pop III star particle mass (M_⋆) is more massive than 40 M_☉, or 10 M_☉ if 10 M_☉ < M_⋆ < 40 M_☉. We do not consider XRB formation in Pop III stars smaller than 10 M_☉. The initial BH mass should be a free parameter in the model. However, considering that it introduces too much complication into the accretion and X-ray spectrum modeling, we simply fix them here. The BH then accretes matter from the companion star at the Eddington limit during the lifetime of the companion star τ, which we take to be a free parameter in our model or until all the mass of the companion star accretes onto the BH. We will discuss the effects of different τ on the X-ray luminosity in the next section. We do not consider the accretion from environment, so once the BH ceases accretion from the secondary star, its X-ray luminosity is zero. When electron scattering dominates opacity, the isotropic luminosity from accretion is limited to the Eddington luminosity,

$$\begin{equation} L_{\rm Edd} = 1.3 \times 10^{38} \left(\frac{M_{\rm BH}}{M_\odot }\right) \; \hbox{erg s}^{-1}. \end{equation} \tag{ 2 }$$

The radiation efficiency of accretion is = $L/\dot{M}c^2$, so the mass accretion rate at the Eddington limit is $\dot{M}_{\rm Edd}$ = L_Edd/c². Then, the mass evolution of the BH is

$$\begin{equation} M_{\rm BH} = M_{\rm BH,\,0} \exp \left(\frac{t}{t_{\rm Edd}}\right), \end{equation} \tag{ 3 }$$

where the Eddington time $t_{\rm Edd} = M_{\rm BH}/\dot{M}_{\rm Edd} = \epsilon c^2 M_{\odot } / 1.3 \times 10^{38} \; \hbox{erg s}^{-1} \sim 440 \epsilon \; \hbox{Myr}$ before the companion star runs out of matter.

Since we only update the X-ray luminosity of snapshots between large time steps (δz ∼ 0.5), which is ∼6 and 12  Myr at z = 20 and 15, respectively, we use the time averaged luminosity as the luminosity of each Pop III binary. The total BH accreted mass is

$$\begin{equation} M_{\rm acc} = \left\lbrace \begin{array}{@{}l@{\quad }l}M_{\rm BH,max} - M_{\rm BH,0} & \hbox{for}\quad M_{\star } > M_{\rm BH,max} \\ M_{\star } - M_{\rm BH,0} & \hbox{for}\quad M_{\star } \le M_{\rm BH,max}, \end{array} \right. \end{equation} \tag{ 4 }$$

where M_BH, max = M_BH, 0exp (τ/t_Edd) is the maximum mass of a BH after accreting at the Eddington limit for the lifetime τ.

The total radiation energy from the accretion process is M_accc², so the average luminosity of each binary is simply M_accc²/t_acc, where the accretion time t_acc is τ for M_⋆ > M_BH, max or ln (M_⋆/M_BH, 0)t_Edd for M_⋆ ⩽ M_BH, max. There is no delay between BH and Pop III formation considered; each Pop III binary luminous X-ray radiation is at this constant rate since the Pop III star forms in the simulation for its accretion time t_acc. Using this model, the X-ray energy output is simply determined by the three free parameters, lifetime τ, binary probability α, and radiation efficiency .

The propagation of X-ray radiation and the effects on the IGM are dependent on the photon energy because the H i cross-section approximately decreases rapidly as ∼ ν⁻³. Although, in post-processing, we consider a monochromatic spectrum and not a spectral energy distribution (SED); we can estimate the effects of different XRB SEDs by exploring different photon energies. We adapt a multi-color disk (MCD) blackbody (Mitsuda et al. 1984) plus a high-energy power-law model to model the radiation spectrum from Pop III binaries with BHs with masses of tens of M_☉. This model was also adopted for mini-quasars in a cosmological context in Kuhlen & Madau (2005), who considered the photoheating and photoionization effects from a 150  M_☉  BH that has a softer SED than the binaries presented in this work. Their model included a MCD component with luminosities equally divided between a MCD and a power-law component with a power index α, both with the same low-energy cutoff. In a MCD model, each annulus of a thin accretion disk radiates as a blackbody with a radius-dependent temperature, T(r)∝r^−3/4, and the temperature of the innermost portion of the disk decreases slowly with BH mass (Makishima et al. 2000) as T_in ∼ 1.2(M_BH/10 M_☉)^−1/4 keV. The inner disk temperature is about 1.2 keV and 0.8 keV for a 10 Ms and 40 M_☉ BH, respectively. We use a similar SED functional form as Alvarez et al. (2009), assuming L_ν∝ν for hν < 400 eV, L_ν∝ν⁻¹ for 400 eV < hν < 10 keV, and L_ν = 0 for hν > 10 keV, which has a mean photon energy of ∼770 eV. To study the effects of different photon energies, we consider several different monochromatic photon energies (E_ph) between 300 eV to 3 keV, covering most of the emissions from this model.

X-ray intensities are almost flat inside the Rarepeak and decrease externally because we do not consider star formation outside of the Rarepeak.

• 1.  
  Sub-kiloelectronvolt Photons. The 300 and 500 eV X-ray intensities are much weaker than kiloelectronvolt ones because the absorption is stronger at lower photon energies. Their intensities drop fast outside of the source region. Only small amounts of photons escape to the IGM. More specifically, at z = 15, there are about 15.8% and 50.5% of the total X-ray energy escaping the Rarepeak region at 200 kpc radius (3.2 comoving Mpc) for 300 eV and 500 eV photons, respectively. At 1 Mpc (16 comoving Mpc) radius, the escape fractions are only 0.11% and 13.1%, respectively.
• 2.  
  Kiloelectronvolt Photons. The attenuation of the kiloelectronvolt photons through the IGM is weak, resulting in their intensities dropping just slightly faster than 1/r² geometric dilution. This leads to a significant amount of radiation at kiloelectronvolt energy scales escaping into the IGM and possibly contributing to the X-ray background. At z = 15, there are about 73.9% and 81.0%, of the total X-ray energy escaping the Rarepeak region at 200 kpc radius (3.2 comoving Mpc) for 1 keV and 3 keV cases, respectively. At 1 Mpc (16 comoving Mpc) radius, the escape fractions are 44.3% and 53.2%, respectively.

The radial distribution of X-ray photoheating rates is more complicated than that of X-ray intensities. There is a large variance inside the Rarepeak because the heating rates are highest in star-forming halos, which have high electron fractions and thus have most of the photon energy deposited to thermal energy.

• 1.  
  300 eV Photons. Because the attenuation of low-energy photons through H i, He i, and He ii are much higher (recall that their cross-sections scale as ∼$1/E_{\rm ph}^3$), high heating rates are created by these low-energy photons, which extend to the boundary of the Rarepeak region. The X-ray photoheating can significantly change the thermal state of the gas inside the Rarepeak in just millions of years.
• 2.  
  500 and 1 keV Photons. Photons at this energy range heat the Rarepeak region more weakly, but can efficiently heat the IGM when compared to lower-energy photons. However, their overall rates are not high, and they need a long time to significantly increase the IGM temperature.
• 3.  
  3 keV Photons. Though the 3 keV photon intensity is strong, weak interactions between high-energy photons and gas result in very low heating rates.

We also plot the photoheating rate profile at z = 15 from the UV ionizing photons that were originally included in the simulation to illustrate the significance of X-ray heating. The spherically averaged UV heating rate is comparable to the X-ray photoheating inside the Rarepeak and drops to almost zero out of the star-forming region because the UV photons are all absorbed within a few kiloparsecs of their origins. Because the UV radiation can only heat the gas within these small-scale H ii regions, the thermal morphology from UV radiation is porous, whereas the X-ray radiation creates a smoothly varying component of the heated IGM.

The distribution of H i photoionization rates k_ph for different photon energies is similar to the photoheating rate Γ, but their profiles are much smoother inside the Rarepeak.

• 1.  
  300 eV Photons. The 300 eV photons have relatively high photoionizing rates inside Rarepeak and extend to the edge of the Rarepeak. However, at rates of ∼10⁻¹⁷ s⁻¹, they are insignificant compared to the UV photoionization for the star-forming region of Rarepeak.
• 2.  
  500 and 1 keV Photons. Similar to our photoheating results, radiations in the 0.5–1 keV band are more efficient at photoionizing the IGM, where the Rarepeak acts as a single source at large distances, still at very low rates <10⁻¹⁸ s⁻¹.
• 3.  
  3 keV Photons. Their photoionization rates are much lower compared to lower photon energy cases in both Rarepeak and the IGM.

We also plot the profile of the photoionization rate at z = 15 from the UV radiation. Contrary to the behavior in the heating rates, the photoionization by UV is much more efficient than the ionization by X-rays inside the Rarepeak, as a higher fraction of UV photon energies is used to ionize the IGM than to heat them. Outside of the Rarepeak region, the UV radiation is fully attenuated, and the photoionization rates drop to zero accordingly.

We first use a one-zone model to study the effects of X-rays of different photon energies and fluxes on the IGM. This model includes X-ray heating and ionization with secondary ionizations, collisional ionizations, recombinations (case B), and primordial atomic cooling. The changes of electron fraction (x_e) and thermal energy (E_th) are expressed as

$$\begin{equation} \frac{dx_e}{dt} = (1-x_e)(k_{\rm ph}+n_e k_1)-x_e n_e \alpha _B \end{equation} \tag{ 6 }$$

$$\begin{equation} \frac{dE_{\rm th}}{dt} = \Gamma _{\rm ph} - \Lambda . \end{equation} \tag{ 7 }$$

Here, the photoionization and photoheating rates k_ph and Γ_ph with the X-ray secondary ionizations use the results from Shull & van Steenberg (1985), the same as in our ray tracing calculation (Wise & Abel 2011). k₁ is the collisional ionization coefficient in Abel et al. (1997), using the fit in Janev et al. (1987). The case B recombination coefficient, α_B is 2.59 × 10⁻¹³ (T/10⁴ K)^−0.7 cm⁻³ s⁻¹ (Osterbrock 1989). The primordial atomic cooling rate Λ is computed using the table in Sutherland & Dopita (1993) when the gas temperature is over 10⁴ K.

The model considers a constant X-ray flux impacting an initial cold (T = 20 K) and neutral (x_e = 10⁻⁵) gas parcel with mean cosmic density. The calculation starts at z = 15 and evolves for 1 Gyr. The photon energy used in the calculation is not redshifted and is kept as a constant. Also, the decrease of temperature due to the expansion of the universe is not considered.

We first study the effects of different photon energies with an X-ray flux of 10⁻⁶ erg cm⁻² s⁻¹, which is close to the X-ray fluxes generated by Rarepeak in our ray-tracing simulations. We plot the evolution for the electron fraction and temperature in Figure 8 for this X-ray flux of different photon energies in a range from 300 eV to 6 keV.

• 1.  
  Sub-kiloelectronvolt Photons. X-rays below 1 keV can significantly heat and ionize the IGM. The 300 eV case heats the gas to 10⁴ K in ∼100 Myr, at which point atomic hydrogen cooling becomes efficient, resulting in an ionization boost of 10%, which is enough to contribute a non-negligible amount to the optical depth from Thomson scattering.
• 2.  
  Kiloelectronvolt Photons. The kiloelectronvolt photons can only moderately heat the gas and weakly ionize the IGM. For 1 keV photons, even for 1 billion years, the gas temperature only reaches slight over 600 K and the electron fraction is just 5 × 10⁻³, not high enough to impact the optical depth of the CMB from Thomson scattering. However, this electron fraction is high enough to stimulate the H₂ formation, and then, in turn, the Pop III star formation (Ricotti et al. 2001; Yoshida et al. 2004). For even higher photon energies, the heating and ionizing effects are negligible at this level of X-ray flux.

Zoom In Zoom Out Reset image size

For kiloelectronvolt radiation, stronger fluxes are needed to heat and ionize the IGM to a meaningful level. The evolution of the electron fraction and temperature is plotted in Figure 9 for 1 keV and 3 keV photon energies with various X-ray fluxes. X-ray radiation of 1 keV photons with a flux of 10⁻⁵ erg cm⁻² s⁻¹ can barely heat the gas to 10⁴ K and ionize it to about 4%. This shows that for increasing the X-ray flux, the heating is more significant than the associated ionization. For 3 keV photons, even this elevated X-ray flux is not sufficient to heat and ionize the gas within 1 Gyr.

Zoom In Zoom Out Reset image size

In a mean patch of the IGM, a small opacity for X-ray photons of E ≳ 1 keV renders the mean free path truly cosmological (∼100 comoving Mpc for 1 keV photons and several comoving gigaparsecs for 3 keV photons). Therefore, we need to consider the X-ray background caused by sources outside the simulation box, in addition to the local transfer of X-ray photons. This is supported by our results in Section 4, showing that large amounts of kiloelectronvolt X-rays easily escape out of their host halos, the Rarepeak region, and even the entire simulation box, which then contribute to the X-ray background. In this section, we describe our method that calculates the X-ray background from Pop III binaries.⁶

Our simulation only allow stars formed inside our refined Rarepeak region, which is only 0.22% of the entire survey volume, containing 0.37% of the total baryon mass. To compensate for these unresolved sources outside the Rarepeak region, we simply use the density distribution to populate X-ray sources out of the star-forming region in our simulation. We do not intend to establish a fully self-consistent correlation between X-ray luminosity from Pop III binaries and the underlying baryon density in such a coarse resolution, but our aim is to populate this region with X-ray sources in the most likely places. We first project the X-ray luminosity and baryon density to the root grid of 512³. For the cells with X-ray sources, we calculate the mean X-ray luminosity $\overline{L}_{X,0}$ and baryon density $\overline{\rho }_0$. Then, for each cell in the region external to the Rarepeak with a baryon density higher than the mean density $\overline{\rho }_0$, we assume there is an X-ray source with luminosity $\overline{L}_{X,0}$. We find that sources populated in this manner produce X-ray luminosities that are several times higher than in the Rarepeak region alone. The total X-ray luminosities of the entire simulated volume are 6.7, 3.1, and 8.5 times those of Rarepeak region at z = 24, 17.9, and 15, respectively.

In principle, the background itself can have spatial fluctuation due to the inhomogeneous distribution of radiation sources outside the simulation box. Nevertheless, simulating the structure formation on such a large scale, while resolving star-forming regions with the relevant local astrophysical processes in detail is almost impossible in practice at this time. Because inhomogeneity in the source distribution at large lookback times will be observed to be almost uniform inside the box, we simply treat the background sources to be uniformly distributed. We also assume that the entire simulation box is a good representation of the average universe and takes its luminosity as the mean, globally averaged luminosity.

The (proper) X-ray background intensity J_ν (erg s⁻¹ cm⁻²Hz⁻¹ sr⁻¹) at an observed frequency ν and redshift z_obs will then be given by the following:

$$\begin{equation} J_{\nu }(z_{{\rm obs}})=(1+z_{{\rm obs}})^{3}\int _{0}^{\infty }\frac{dr_{{\rm os}}}{1+z_{{\rm s}}}\,\bar{j}_{\nu _{{\rm s}}}(z_{{\rm s}})\,\exp [-\tau _{\nu _{{\rm obs}}}], \end{equation} \tag{ 8 }$$

where $\bar{j}_{\nu _{{\rm s}}}$ (erg s⁻¹ cm⁻³ Hz⁻¹ sr⁻¹) is the comoving emission coefficient given by

$$\begin{equation} \bar{j}_{\nu _{{\rm s}}}=\frac{1}{4\pi }\frac{L_{\nu _{{\rm s}},{\rm Box}}(z_{{\rm s}})}{V_{{\rm Box}}}, \end{equation} \tag{ 9 }$$

where $L_{\nu _{{\rm s}}}$ (erg s⁻¹ Hz⁻¹) is the proper, total luminosity inside the simulation box at source frequency ν_s and redshift z_s, and V_Box is the comoving volume of the simulation box. The optical depth originates from the absorption of photons by H i and He i:

$$\begin{eqnarray} \tau _{\nu _{{\rm obs}}} & = & \tau _{\nu _{{\rm obs}},{\rm HI}}+\tau _{\nu _{{\rm obs}},{\rm HeI}}\nonumber \\ & = & \int _{t(z_{{\rm s}})}^{t(z_{{\rm obs}})}c\, dt\, \nonumber \\ & & \times \lbrace \ n_{{\rm HI}}(z_{{\rm s}})\,\sigma _{{\rm HI}}(\nu _{{\rm s}},\, z_{{\rm s}})+n_{{\rm HeI}}(z_{{\rm s}})\,\sigma _{{\rm HeI}}(\nu _{{\rm s}},\, z_{{\rm s}})\rbrace \nonumber \\ & = & \int _{z_{{\rm s}}}^{z_{{\rm obs}}}\frac{c\, dz_{{\rm s}}\,(1+z_{{\rm s}})^{-5/2}}{H_{0}\sqrt{\Omega _{m}}} \nonumber \\ & & \times \lbrace n_{{\rm HI}}(z_{{\rm s}})\,\sigma _{{\rm HI}}(\nu _{{\rm s}},\, z_{{\rm s}})+n_{{\rm HeI}}(z_{{\rm s}})\,\sigma _{{\rm HeI}}(\nu _{{\rm s}},\, z_{{\rm s}})\rbrace, \nonumber \\ & & \end{eqnarray} \tag{ 10 }$$

where we neglect absorption by He ii because the ionization fraction of the IGM at z ⩾ 15 remains very low. r_os is the comoving line-of-sight distance that a photon has traveled, given by

$$\begin{equation} r_{{\rm os}}=\frac{2c}{H_{0}\sqrt{\Omega _{m}}}\lbrace (1+z_{{\rm obs}})^{-1/2}-(1+z_{{\rm s}})^{-1/2}\rbrace, \end{equation} \tag{ 11 }$$

and the redshifted frequency ν/ν_s = (1 + z_obs)/(1 + z_s). Over the frequency range of our interest, 100 eV ≲ hν ≲ 3 keV, absorption cross-sections are well approximated by power laws:

$$\begin{equation} \sigma _{{\rm HI}}(\nu _{s})=6.5\times 10^{-14}\,\left(\frac{h\nu _{s}}{{\rm eV}}\right)^{-3.25} \end{equation} \tag{ 12 }$$

and

$$\begin{equation} \sigma _{{\rm HeI}}(\nu _{s})=1.55\times 10^{-12}\,\left(\frac{h\nu _{s}}{{\rm eV}}\right)^{-3.22}. \end{equation} \tag{ 13 }$$

With these cross-sections, Equation (10) becomes

$$\begin{eqnarray} &&{ \tau _{\nu _{{\rm obs}}} = \left(\frac{\Omega _{b}}{0.044}\right)\left(\frac{h}{0.7}\right)\left(\frac{\Omega _{0}}{0.27}\right)^{-0.5}\left(\frac{1+z_{s}}{1+25}\right)^{1.5} } \nonumber \\ & & \times \left\lbrace 4.10863\left(\frac{X}{0.75}\right)\left(\frac{h\nu _{0}}{{\rm keV}}\right)^{-3.25}\left[\left(\frac{1+z_{s}}{1+z_{{\rm obs}}}\right)^{1.75}-1\right]\right.\nonumber \\ & & + \left. 10.13573\left(\frac{Y}{0.25}\right)\left(\frac{h\nu _{0}}{{\rm keV}}\right)^{-3.22}\left[\left(\frac{1+z_{s}}{1+z_{{\rm obs}}}\right)^{1.72}-1\right]\right\rbrace, \nonumber \\ & & \end{eqnarray} \tag{ 14 }$$

where X and Y are mass fractions of hydrogen and helium, respectively.

As we approximate the X-ray SED with a monochromatic frequency ν₀ and the bolometric X-ray luminosity L₀ ($\equiv \int _{\rm X-ray}L_{\nu _{{\rm s}}}d\nu _{{\rm s}}$), Equation (8) can be simplified as

$$\begin{eqnarray} L_{\nu _{{\rm s}},{\rm Box}}(z_{{\rm s}}) & = & L_{0,{\rm Box}}(z_{{\rm s}})\,\delta ^{D}(\nu _{{\rm s}}-\nu _{0})\nonumber \\ & = & L_{0,{\rm Box}}(z_{{\rm s}})\,\frac{1+z_{{\rm obs}}}{\nu }\delta ^{D} \left(1+z_{{\rm s}}-\frac{1+z_{{\rm obs}}}{\nu }\nu _{0}\right), \nonumber \\ & & \end{eqnarray} \tag{ 15 }$$

and thus using Equations (11) and (15),

$$\begin{equation} J_{\nu }(z_{{\rm obs}})=\left(\frac{\nu }{\nu _{0}}\right)^{3/2}\frac{c(1+z_{{\rm obs}})^{3/2}}{4\pi \nu _{0} H_{0}\sqrt{\Omega _{m}}}\,\frac{L_{0,{\rm Box}}(z_{{\rm s}})}{V_{{\rm Box}}}\,\exp (-\tau _{\nu _{{\rm obs}}}), \end{equation} \tag{ 16 }$$

where 1 + z_s = (ν₀/ν)(1 + z_obs) is implied.

In practice, Equation (16) should be integrated in piecewise frequency intervals due to the time-discrete nature of the simulation. Especially for the contribution from the most recent past to the present because spatial fluctuation will be non-negligible, we calculate the fluctuating 3D X-ray background by adopting a scheme by Ahn et al. (2009). We locate the 3D field of X-ray luminosities frozen at the most recent past in the full box periodically and calculate the contribution at every location of the box by summing over the full contribution from all the X-ray sources but within the corresponding lookback time. Since this includes an out-of-box contribution, it will differ from the X-ray intensity distribution calculated from sources only inside the box at the observed redshift.

We plot the X-ray background intensities of monochromatic 1 keV and 3 keV X-ray photons as functions of distance to the simulation box center in Figure 10. The X-ray background intensities from higher-redshift sources are represented by their redshifted photon energies, which are all uniform in space except for the most recent contribution. We also plot the local X-ray intensities from our ray-tracing calculation for comparison. We also perform the calculations for the sub-kiloelectronvolt X-rays. They show that the X-ray intensities from outside the simulation volume are negligible because most of their photons are absorbed locally.

Zoom In Zoom Out Reset image size

It is interesting to understand the relative importance of the X-ray effects between the local sources and the X-ray background in different regions. Inside the Rarepeak, the X-ray intensities from local sources are strong and are at the same level as the X-ray background from nearby sources, whereas outside the Rarepeak, the X-ray background dominates. The background outside has an equivalent X-ray intensity at the same order of magnitude as the Rarepeak region, which has a very high Pop III stellar density. The X-ray background intensities at ∼20 comoving Mpc away from the source region are about 10 and 50 times higher than those from the Rarepeak region alone at redshifts z = 17.9 and 15, respectively. Thus, for the X-ray feedback, the mean IGM should be mainly heated and ionized by the external X-ray background rather than by any local sources. In addition, as the external background resides in lower frequencies, its impact on heating and ionization of the IGM is boosted from the sheer value of X-ray intensity by a factor of ∼ (ν/ν₀)⁻³.

The X-ray background from 1 keV photons and 3 keV photons is indeed quite different due to its differing mean free paths. For the 1 keV photons, the background is dominated by cosmological nearby sources because the X-ray radiation from sources at a distance of dz > 2 is negligible. This is consistent with the fact that the mean free paths for the 1 keV X-ray photons are ∼100 comoving Mpc (more specifically, ∼4 × 10⁴/(1+z)² comoving Mpc). Therefore, the background from the 1 keV case is not building up from these early redshifts. At even higher energies, the mean free path for 3 keV photons is a few gigaparsecs in the mean IGM, resulting in a non-negligible fraction of the radiation from z ≲ 25 propagating to z = 15. However, since the total X-ray radiation energy density is increasing during this period, the contributions from distant (Δz ≫ 1) sources are still much weaker than those from the nearby sources.

We show the evolution of averaged intensities of the total X-ray background from all cases in Figure 11, including an additional monochromatic X-ray of 770 eV, which is the mean photon energy from the assumed SED in Section 2.3. At z = 15, while the total X-ray luminosity from all sources in the simulation box is 7.1 × 10⁴³ erg s⁻¹, the mean X-ray intensity is ranging from 5 × 10⁻⁹ to 3 × 10⁻⁷ erg cm⁻² s⁻¹ sr⁻¹ for low- to high-energy photons. The background of the 3 keV photon case is more than twice as high as the 1 keV case because of the large mean free path of the higher-energy photons, and the ratio of background at 3 keV and 1 keV shows little evolution from z = 25 to z = 15. The average intensities for the sub-kiloelectronvolt cases of the entire simulated box are much weaker because most of the radiation is confined inside the Rarepeak. The 300 eV background is about two orders of magnitude smaller than that of the 3 keV case. However, since their attenuation in the IGM is much stronger (as shown in Figure 8), their heating and ionizing effects may still be as important as the higher-energy background.

Zoom In Zoom Out Reset image size

At even lower redshifts, X-rays from Pop III binaries likely grow slowly or even start to drop due to Pop III star formation being suppressed by metal enrichment ⁷; the X-ray background of higher-energy photons, which survive from higher redshifts, should be more important at later times. However, since the heating and ionizing effects are much weaker for the high-energy photons, as shown in the previous subsection, the effects of their background are still expected to be weak. We will study the long-term effects of X-ray background in the next subsection.

We now estimate the heating and ionizing effects of our X-ray background model by applying our one-zone model to each cell in the 512³ base grid from the full simulation, starting at z = 24. At each snapshot that is separated by Δz ∼ 0.5, we fix the density distribution and allow the electron fraction x_e and temperature T to evolve according to the one-zone model and the X-ray background intensity at the given redshift z_i and frequency. The X-ray heating and ionization by the redshifted photons with different energies and intensities are calculated and summed at each timestep. We include the adiabatic expansion effect in calculating the temperature change. We then adjust the temperature and electron fraction in the next snapshot at redshifts z_i + Δz by ΔT and Δx_e, respectively. This process is repeated until we reach the final redshift z = 15 of the simulation. After this point, we fix the X-ray background and density distribution and continue the one-zone calculation for each cell to z = 6. We consider five cases with different photon energies of 300 eV, 500 eV, 770 eV, 1 keV, and 3 keV. As previously mentioned, the sub-kiloelectronvolt cases locally heat and ionize the gas instead of adding to the background intensity. Although all of these calculations consider a monochromatic X-ray spectrum, nevertheless, we can obtain more realistic synthetic results by averaging the effects weighed by the luminosities based on the assumed SED in Section 2.3.

We first show the temperature and electron fraction volume-weighted averaged radial profiles at z = 15 in Figure 12. We also plot the profiles from the original simulation with UV and other additional heating and cooling processes for comparison. Because the UV photoionization and photoheating are not considered in the one-zone model, the temperatures and ionization fractions at some radii inside Rarepeak are lower than the values from the original full simulation.

Zoom In Zoom Out Reset image size

There are very different heating and ionization patterns for different photon energies.

• 1.  
  X-ray photoheating
• 2.  
  X-ray photoionizationThe distribution of electron fraction is similar to the distribution of temperature. The electron fractions in all cases at ∼1 Mpc away from the center are smaller than 10⁻³. These ionization levels are not high enough to contribute to the optical depth to electron Thomson scattering of the CMB significantly (δτ ≪ 0.01).

  • (a)  
    Sub-kiloelectronvolt Photons. The highest electron fractions from the 300 eV case are just below 0.1 inside the Rarepeak. For the IGM outside of Rarepeak, the electron fraction of the 300 eV case is the highest at r ≲ 500 kpc, and then the 500 eV case has the highest electron fraction. The effect of the 770 eV photons is closer to the 1 keV case than to the 500 eV case.
  • (b)  
    Kiloelectronvolt Photons. The ionization effect of kiloelectronvolt photons is very weak (δx_e < 10⁻³), both inside and outside the Rarepeak. The photoionization of 1 keV photons decreases slowly with radius due to their high escape fraction and then close to those of sub-kiloelectronvolt cases at r > 1 Mpc.
  • (c)  
    SED. The synthetic ionization is close to the 770 eV case, inside and near the Rarepeak (inside ∼500 proper kpc or 8 comoving Mpc radius), and then is even weaker than those of the 1 keV cases at large distances.

  Figures 13 and 14 show the evolution of volume-weighted averaged temperature and electron fraction over the entire comoving (40 Mpc)³ simulated volume with results from the one-zone calculation of each cell, respectively. We exclude the central (8 comoving Mpc)³ region to obtain the mean values for a typical patch of the universe. For the temperature evolution, we also show the CMB temperature and the temperature from the hydrodynamical simulation. Star formation and feedback are not considered in the simulation after z = 15 for computational reasons.The results of this volumetric average are consistent with our one-zone calculation that uses a fixed initial conditions and X-ray flux, described in Section 5.1. This confirms that, at the same luminosity, the photon energy is the most important determinant in the heating and ionization history of the IGM.
• 3.  
  IGM temperature.Before z ∼ 16, the X-ray background is still weak, and the heating of the IGM is not enough to compensate the decrease of temperature due to the adiabatic expansion. After that, the X-ray background is strong enough to increase the mean IGM temperature, which (except for the 3 keV case) passes the CMB temperature at redshifts just below 15, passes 100 K by z = 10, and then gradually increases to a range of 400–900  K at z = 6. Some regions, especially for the 300 eV case, are heated to over 10⁴ K and then are cooled by atomic hydrogen, resulting in a slowing down temperature growth.

  • (a)  
    Sub-kiloelectronvolt Photons. Low-energy X-rays rapidly heat the IGM. The 300 eV photons significantly affect the regions near the sources, and the temperature growth slows approaching ∼640 K because of a combination of the weak heating of the IGM outside of the Rarepeak and efficient radiative cooling near the Rarepeak. On the other hand, 500 and 770 eV X-rays continue to heat all the IGM of the simulated volume more effectively to higher than 850 and 670 K, respectively.
  • (b)  
    Kiloelectronvolt Photons. The heating for the 1 keV radiation is much more pronounced than that of the 3 keV case, as its temperature reaches 450 K. For the 3 keV photons, though their intensity is more than twice stronger, the heating effect is so weak that the IGM temperature only increases by a few Kelvin by z = 6 when compared with the case without X-ray heating.
  • (c)  
    SED. The heating effect from the synthetic full spectrum is similar to that for the 1 keV case at large radii, and, accordingly, the mean temperature of the simulation volume is close to that of the 1 keV case. The mean IGM is heated to T ∼ 360 K at z = 6.
• 4.  
  IGM Electron FractionThe evolution of electron fraction is simpler than the evolution of temperature because the equilibrium between photoionization and case B recombination is not yet reached in most of the volume for all cases.

  • (a)  
    Sub-kiloelectronvolt Photons. The 500 eV X-rays are the most efficient in ionizing the entire volume because they can propagate farther away from their sources than lower-energy photons while having stronger interactions with the IGM than higher-energy photons. Our estimations show that at z = 6, the maximal change in the volume-averaged electron fraction is ∼0.01. The electron fractions are smaller for 300 and 770 eV photons at 0.8% and just <0.7%, respectively. Though unlikely, if all of the X-ray radiation exists at ∼500 eV, its ionizations might contribute to the optical depth to the Thomson scattering.
  • (b)  
    Kiloelectronvolt Photons. The X-ray background from 3 keV photons is inefficient in ionizing the IGM, only resulting in Δx_e ∼ 10⁻³ by z = 6. The 1 keV X-rays photoionize the IGM moderately to an electron fraction of x_e ∼ 5 × 10⁻⁴ at z ∼ 15 and ∼6 × 10⁻³ at z = 6.
  • (c)  
    SED. The mean IGM electron fraction from the synthetic full spectrum is only x_e ∼ 2 × 10⁻³ at z = 10 and x_e < 5 × 10⁻³ at z = 6, suggesting that ionizations from an X-ray background are a minor correction when calculating the optical depth due to Thomson scattering.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

In summary, the IGM can be heated and ionized by the X-ray background from Pop III binaries. The photoionization effect is likely unimportant in direct observations, at least to the optical depth to Thomson scattering of the CMB, as the electron fraction might not be elevated over 0.005. However, this electron fraction could have some really substantial positive effects on H₂ formation and cause a general uptick in Pop III SFR elsewhere. On the other hand, the temperature of a large volume of the IGM may exceed 100 K by redshift z = 10. These heated regions of the IGM might be barely detectable by 21 cm SKA observations. The details of the mock 21 cm observations of Rarepeak and the surrounding area will be presented in a forthcoming paper (Ahn et al. 2014).