Published as: Thomas, R.M., et. al., – “Cooperative reionization: Quasars and Stars”2008, MNRAS, submitted
Chapter 1
Cooperative Reionization & Heating by Miniqsos and Stars
But when you combine the two – thats special
Anonymous The potential of PopIII stars and power-law sources like quasars to heat and ionize the Universe has been studied extensively by several authors, although their combined influence on the inter-galactic medium (IGM) as they co-evolve has sel- dom been addressed. In reality, at least in the observable Universe, stars and quasars do co-exist. In this chapter, we model the effect in ionization and heating of the IGM by populations of both stars and quasars. BEARS has been used in the previous chapter to model the ionization around different types of sources. Here, we will use the temperature profile around sources calculated by the 1-D radiative transfer code and embed this into the 3-D box, in a fashion similar to that in the previous chapter.
A simple prescription is used to convert a dark matter halo into a stellar and quasar component. Results are presented for the heating, first due to stars and quasars in- dividually, and then when they are combined. Also, the spin temperature evolution has been followed self-consistently, relaxing the assumption that Tspin ≫ TCMB. Lyα coupling was introduced, one by assuming a background flux and the other by calculating the Lyα flux due to “secondary” electrons. The radiative transfer was performed on a number of snapshots spanning redshifts from ≈ 12 to 6. These snap- shots were then merged together to make a contiguous cube of reionization histories in each of the three cases. As a final step, the images, as a function of frequency, were subjected to convolution with the point spread function of the LOFAR telescope to obtain sets of mock observations.
1.1 Introduction
P
hysical processes that occur during reionization are numerous and complex.Nevertheless, ionization of neutral gas (hydrogen & helium) and heating of the intergalactic medium can be considered the two primary influences of radiating ob-
jects during reionization. Although there has been extensive research and develop- ment in 3-D cosmological radiative transfer (RT) codes, only a few attempt couple the evolution of temperature with the ionization of the IGM. And all of these RT codes that couple temperature to the rate equations of ionization do so for relatively low energy photons and not for hard (X-ray) photons.
Most studies of reionization have focussed on stars as being the primary source [Mellema et al., 2006, Abel et al., 2000, 2002, Bromm et al., 2002, Yoshida et al., 2003].
Due to the deficiency of hard photons in the spectral energy distributions (SEDs) of these “first stars”, heating due to these objects are limited in extent [Thomas and Zaroubi, 2008]. On the other hand, miniquasars (miniqsos), characterized by central black hole masses < 106 M⊙, have also been considered as an important contributor to
reionization [Madau et al., 1997, Ricotti and Ostriker, 2004a,b, Nusser, 2005, Furlanetto et al., 2004, Wyithe and Loeb, 2004, Furlanetto and Loeb, 2002, Thomas and Zaroubi, 2008].
Ionization aspect of the miniquasar radiation has been explored by several authors [Madau et al., 1997, Ricotti and Ostriker, 2004a,b, Madau et al., 2004, Zaroubi and Silk, 2005, Thomas and Zaroubi, 2008, Thomas et al., 2008]. In Thomas and Zaroubi [2008]
it was shown that although the ionization due to miniqsos is similar to that of stellar- type sources, the heating due to the presence of hard photons in miniqsos is very different.
Miniqsos heat the surrounding inter-galactic medium (IGM) well beyond their ionization front [Thomas and Zaroubi, 2008, Chuzhoy et al., 2006]. In Nusser [2005]
the importance of heating the IGM with respect to the observability of the redshifted 21 cm radiation in either emission or absorption was explained. The observed quan- tity, i.e., the brightness temperature (δTb), is a function of the so-called 21 cm spin temperature, Tspin, defined according to the equation nu/nl = 3 exp(−T∗/Tspin).
Here, nu and nl are the number of electrons in the triplet and singlet states of the hyperfine levels of the ground state of the neutral hydrogen atom, and T∗ = 0.0681 K is the temperature corresponding to the 21 cm wavelength. In order for the 21 cm radiation to be observed relative to the CMB background, the spin tem- perature must be decoupled from the CMB [Wouthuysen, 1952, Field, 1958, 1959, Hogan and Rees, 1979]. The decoupling is achieved through either Lyα radiation or collisional excitations and heating. Collisional excitations due to X-ray pho- tons result in a “secondary” Lyα pumping [Shull and van Steenberg, 1985] which dominates the spin temperature and CMB temperature decoupling in some regions around the miniquasar; this effect has been recently pointed out by Chuzhoy et al.
[2006]. We in this paper have primarily considered this “secondary” Lyα flux and not the Lyα intrinsic to the spectrum of the ionizing source. Several other authours have also discussed heating due to X-rays from starbursts in a semi analytical man- ner [Chen and Miralda-Escude, 2006, Pritchard and Furlanetto, 2006].
1.1. Introduction 3
We have seen that unlike stars, quasars have an additional property of heat- ing the IGM to a large extent and through secondary Lyα radiation making the neutral IGM visible to a 21-cm experiment. However, Dijkstra et al. [2004] and Salvaterra et al. [2005] argue that miniquasars alone can not reionize the Universe as they will produce far more soft X-ray background radiation than currently observed [Moretti et al., 2003, Sołtan, 2003] while simultaneously satisfying the WMAP3 po- larisation results [Page et al., 2007, Spergel et al., 2007]. It should be noted, however, that the Dijkstra et al. [2004] & Salvaterra et al. [2005] calculations have been carried out assuming specific models for the evolution of black hole mass density and spec- tral energy distributions of UV/X-ray radiation of the miniquasars and for some other models the discrepancy is not so severe [Zaroubi et al., 2007, Ripamonti et al., 2008]. Add to that a strong evidence for stars being the primary source of reioniza- tion from the excess IR background radiation [Kashlinsky et al., 2005]. This too has been subject to controversy because of the sensitivity of the result to the subtraction of the Zodiacal light within the same waveband [Cooray et al., 2007]. The fact that the contribution of the Zodiacal light is model-dependent is also a matter of concern.
Although uncertainity looms about the sources that resided during the dark ages, it is conceivable from observations of our Universe up to redshifts of 6.5, that sources of reionization could have been a mixture of both stellar and quasar sources.
As obvious as this might seem, implementing radiative transfer that include both ionizing and hard X-ray photons has been difficult and as a result most 3-D radiative transfer schemes restrict themselves to ionization due to stars [Gnedin and Abel, 2001, Ciardi et al., 2001, Ritzerveld et al., 2003, Susa, 2006, Razoumov and Cardall,
2005, Nakamoto et al., 2001, Whalen and Norman, 2006, Rijkhorst et al., 2006, Mellema et al., 2006, Zahn et al., 2007, Mesinger and Furlanetto, 2007, Pawlik and Schaye, 2008]. In
Ricotti and Ostriker [2004a], a “semi” hybrid model of stars and quasars like the one hinted at above, has been used, albeit in sequential order instead of a simultaneous implementation. That is, pre-ionization due to quasars has been invoked between 7 ≤ z ≤ 20, after which stars reionize the Universe at redshift 7. We, in this chap- ter, will attempt modeling the co-evolution of quasars and stars, and contrast these results with reionization being done by one of these sources alone.
For the purpose of modelling the effect of heating by the first sources, we ex- tended the formalism used in BEARS [Thomas et al., 2008] to include X-ray heat- ing. To summarize, BEARS uses N -body simulations and the halo locations within them to embed spherically symmetric ionization profiles, depending on a model to convert the dark matter halo masses to ionizing radiation. Fig 1.1 shows the ioniza- tion maps thus obtained for four different redshifts for the scenario in which stars were the source of reionization.
The focus of this chapter is therefore to introduce the algorithm that is used to
Figure 1.1:Ionized fraction: the BEARS algorithm was used to calculate the ionization frac- tion with stars as the source, for four different redshifts. One slice from each of the four boxes at redshifts 10, 8, 7 and 6 is plotted ( panels left to right and top to bottom, respectively). The ionized fraction is expressed as a grey-scale from completely ionized (black) to completely neutral (white). As expected, at redshift of 10, there are only a small number of ionized bubbles and as we progress in time (lower redshifts) many more bubbles appear that start overlaping, eventually ionizing the entire Universe by a redshift of 6.
implement X-ray heating of IGM in BEARS and use this new version to study the effect of heating due to quasars and stars, both as a hybrid population and individ- ually. Also a unique feature of the code would be to follow the evolution of the Lyα flux produced by secondary electrons and hence calculate self-consistently the spin temperature as a function of redshift.
The chapter is organized as follows: in §1.2 we describe the procedure adapted to include kinetic temperature in BEARS, followed by the calculation of the bright-
1.2. Including X-ray heating in BEARS 5
ness temperature within the simulation box. BEARS is then applied to three differ- ent scenarios of reionization, viz., the primary source of reionization being stars in
§1.3, quasars in §1.4 and a hybrid population of stars and quasars in §1.5. Series of simulations for the three scenarios are run and observational cubes of the brightness temperature (δTb) are generated and discussed in §1.6. Conclusions and discussions of the results are presented in §1.7, along with a mention of a few topics that can be addressed using the data set simulated in this chapter.
1.2 Including X-ray heating in BEARS
In chapter ?? we introduced BEARS, a special-purpose 3-D radiative transfer scheme used in simulating the cosmological EoR signal and detailed the philosophy and im- plementation of the code for the purpose of calculating the reionization history of the Universe for different plausible reionizing sources. In this section, we extend the features of BEARS by including the effect of heating within the same framework.
Our final aim is to model the brightness temperature δTb as a function of red- shift or frequency. From Eq. 1.5 we see that δTb is a function of neutral fraction, density of the IGM and the spin temperature (the rest depend on the cosmology).
Using BEARS as in chapter ??, we obtain the neutral fraction. The density is de- rived from the N -body simulations. The spin temperature is the quantity that needs to be estimated. Thus far we have assumed Tspin≫ TCMB. In this chapter we relax this assumption and model the evolution of the spin temperature depending on the source of reionization. The spin temperature is in turn a function of the kinetic and CMB temperature, and depending on the physical conditions of the IGM interpo- lates between them.
In order to calculate the correct spin temperature evolution we need to estimate the kinetic temperature and the coupling strengths of the various processes at every point in the simulation box. We start by calculating the kinetic temperature and the coupling coefficients at every location and subsequently estimate the spin tempera- ture.
The algorithm to implement heating begins in a manner similar to that for ion- ization. The kinetic temperature profile from the 1-D radiative transfer code in chap- ter ?? is used to embed a spherically symmetric “temperature bubble” at the loca- tions of the dark matter haloes. The luminosity of the source is a function of the halo mass. One of the major problems embedding a temperature profile in the simulation box is that, the radius of the bubble being so large (> 5 Mpc), it results in extensive overlap. Treating overlapped zones directly in terms of temperature is difficult in a simulation box. This is primarily due to the fact that a pre-ionized zone is heated
less than an initially neutral zone. And ionization in turn depends on the proximity of the zone to the ionizing source and the density profile around it, which affects the ionization-front velocity, and so on. The next section outlines an approach which basically utilizes energy conservation to counter this problem.
1.2.1 Treating overlaps: Energy conservation
In the case of ionization, as we approach the late stages (z < 7) of reionization, bubbles of ionized material overlap significantly. To correct for this we conserved the number of photons used for ionization, i.e., the “excess” photons used were calculated from the volume of the overlapped region and redistributed around each source involved in the overlapped zone [Thomas et al., 2008]. In the case of heating however, large volumes of hot ionized and non-ionized gas overlap, well before a redshift of seven. In a quasar dominated part of the Universe this is particularly severe because of the large extent of heating caused by them.
There are various aspects to treating the overlaps of these large “temperature bubbles”. As alluded to before, the problem of estimating the temperature in a re- gion of overlap is complex. A good approximation to this problem is to consider the energy available at the overlapped region. Because the total energy needs to be conserved, we add the contributions to the energy budget at a given location from all sources. This total energy is then redistributed equally to all the contributing sources, i.e., the energy output of each source is modified to account for the over- lapping.
For example, consider there are N sources that overlap at a particular location and the total energy estimated at that location is Etot. The fraction of “excess” en- ergy attributed to each of the N sources is δE = Etot/N . This energy δE is then added to the total energy from the source to estimate a new normalization constant for the SED of the source. As an illustration, consider Eq. 1.10, which is used to estimate the normalization constant for power-law-type sources. On the right hand side of this equation Etotalis to be replaced by Etotal+ δE. Having now obtained the kinetic temperature at every location in the box the next section explains the calculations involved in estimating the brightness temperature δTb.
1.2.2 Calculating δT
bin the volume
In the previous subsection we have seen how to deal with overlap in the case of heating and thus obtain maps of the kinetic temperature of the gas (Tk). In this section, we detail the calculations to obtain the spin temperature of the medium and consequently the observable of an interferometric experiment, the brightness
1.2. Including X-ray heating in BEARS 7
temperature, δTb.
Spin temperature (Tspin) is a shorthand to represent the level population of the hyperfine states of the ground level of a hydrogen atom, i.e., n2/n1= 3×exp(−T⋆/Tspin).
Thus, if Tspin≫ T⋆= 0.068 K this implies that most atoms of neutral hydrogen have their electrons in the triplet hyperfine state. Depending on the physical processes and background radiations that dominate a medium, the spin temperature is either coupled to the background CMB temperature or to the kinetic temperature of the hydrogen gas in the medium. Formally Field [1958] derived Tspin as a weighted sum of the kinetic temperature Tkand the CMB temperature at a particular redshift
’z’ ,TCMB, as;
Tspin=TCMB(z) + yαTk+ ycoll
1 + yα+ ycoll
, (1.1)
where yαand ycollare parameters that reflect the coupling of kinetic temperature to the spin temperature via Lyα-coupling and collisions respectively. The efficiency of Lyα-coupling is far higher than that of collisions, especially further away from the source [Thomas and Zaroubi, 2008, Chuzhoy et al., 2006]. But in our treatment of calculating Tspin, we include both the coupling parameters.
The coefficient of collisional coupling, ycollis a function of the gas temperature Tk and the ionized fraction of the medium XHII. On the other hand, accounting for yα requires the calculation of the flux of secondary Lyα photons available at a particular location in space and time. During the simulations of the 1-D profiles we do calculate the Lyα coupling term as:
yα= 16π2T⋆e2f12Jo
27A10Temec . (1.2)
Here, Jois the Lyα flux density. For the miniqsos high energy photons Lyα coupling is mainly caused by collisional excitation due to secondary electrons [Chuzhoy et al., 2006]. This process is accounted for by the following integral,
Jo(r) = φαc
4πH(z)ναnHI(r)
∞
Z
Eo
σ(E)N (E; r)dE, (1.3)
where N (E; r; t) is the radiation flux for an energy ’E’ at radius ’r’ and time ’t’, as in Zaroubi and Silk [2005]. The cross-section of neutral hydrogen in the ground state for energy ’E’ is given by σ(E); f12= 0.416 is the oscillator strength of the Lyα tran- sition; e and meare the electron’s charge and mass, respectively. φαis the fraction of the absorbed photon energy that goes into excitation [Shull and van Steenberg, 1985]. The contribution of this term is important close to the miniquasar.
Now, instead of embedding a sphere of the spin temperature, as in the case of the kinetic temperature, we embed in the 3-D simulation box a bubble whose radial profile is the secondary Lyα flux, Jo(r). Since Jo(r) is basically the number of Lyα photons at a given location, the overlap of two “Jobubbles” implies that the photons and hence the Joflux has to be added, i.e., at a given spatial location, ~x, ~y, ~z and time t, the total Lyα flux is given by;
Jotot(~x, ~y, ~z, t) =
N
X
i=0
Joi(~x, ~y, ~z, t), (1.4) where Joi(~x, ~y, ~z, t) is the secondary Lyα flux contributed by ith source at the spatial location, ~x, ~y, ~z and time t. Equipped with the kinetic temperature, Tk, and Lyα flux density, Jo, at each point on the simulation, we can calculate yα(Eq. 1.2) and subsequently the spin temperature through Eq. 1.1. Now all the terms required for the calculation of δTb, as in the equation below, are obtained.
δTb= (20 mK) XHI
h
1 − TCMB
Tspin
× Ωbh2 0.0223
1 + z 10
0.24 Ωm
1/2
. (1.5)
1.3 Heating due to stars
As a first application of the extension of BEARS to include heating, we embed the dark matter halo locations in the simulation box with stars. In this section we de- scribe the model used for stars and the prescription adopted to embed these stars into haloes of dark matter identified in the N -body simulation.
1.3.1 Modeling stellar radiation in BEARS
Stars approximately behave as a blackbody of a given temperature, although details of the SED of the star depend on more complex physical processes, and on the age, metallicity and mass of the star. This blackbody nature of the source leaves a char- acteristic signature in the manner the IGM is heated and ionized. From Schaerer [2002], we know that the temperature of the star only weakly depends on its mass.
Thus, we fix the blackbody temperature of the stars to 5 × 104K and simulate 1-D radiative transfer using the code in chapter ?? for masses between 10 and 1000 M⊙. The total luminosity for a given stellar mass is calculated from Table 3 of Schaerer [2002].
For blackbodies with temperatures around 105K, as we have assumed above, the spectrum peaks in between ≈ 20 eV and ≈ 24 eV (Wien’s displacement law). Thus,
1.3. Heating due to stars 9
Figure 1.2: Different quantities as a function of the radial distance away from the source in kpc is shown. The source is a miniqso powered by a 106M⊙black hole at redshift 10. The solid line is the neutral fraction of hydrogen between 0 and 1. The ionized region extend to about 200 kpc. The dotted line plots the Lyα flux (ergs s−1 cm2). From Eq. 1.3 we see that Lyα flux is proportional to neutral fraction and thus we see low values of flux close to the source. The kinetic temperature (K) is indicated by the dashed line. Within the ionized zone the kinetic temperature is in the order of 106 K or more. Although the temperature drops sharply where the IGM is neutral, it is still well above the background CMB (30 K). The dash dot line represents the spin temperature (K). Initially (< 200 kpc) the spin temperature is lower than the kinetic temperature because the Lyα flux is very low, but at the transition into the neutral regime the Lyα flux increases sharply, coupling the kinetic temperature to the spin temperature. Finally the dash dot dot line plots the brightness temperature (mK) calculated according to Eq. 1.5. δTbis obviously zero inside the ionized region and then rises sharply because the spin temperature, being coupled to the kinetic temperature, is very high and hence completely decoupled from the background CMB. This is when δTbis positive and the 21-cm signal is visible in emission. Further away (> 1 M pc) the kinetic temperature falls below the CMB, but since the spin temperature is still coupled to the kinetic temperature δTb becomes negative and the signal observed in absorption. At > 5 M pc, the spin temperature no longer couples to the kinetic temperature, instead returning to that of the CMB, causing δTbto climb back to zero.
from the form of the blackbody spectrum, we can a priori expect in the case of stars that the ionization induced by these objects will be high, but that the heating they cause will not be substantial given the exponential cutoff of the radiation towards higher frequencies.
Following the prescription in §?? we associate stellar spectra with dark matter haloes using the following procedure. The global star formation rate density ( ˙ρ⋆(z)) as a function of redshift was calculated using the empirical fit
˙
ρ⋆(z) = ˙ρm
β exp [α(z − zm)]
β − α + α exp [β(z − zm)] [M⊙yr−1Mpc−3], (1.6) where α = 3/5, β = 14/15, zm= 5.4 marks a break redshift, and ˙ρm= 0.15 M⊙yr−1Mpc−3 fixes the overall normalisation [Springel and Hernquist, 2003]. Now, if δt is the time interval between two outputs in years, the total mass density of stars formed is
ρ⋆(z) ≈ ˙ρ⋆(z)δt [M⊙Mpc−3]. (1.7) Notice that this approximation is valid only if the typical lifetime of the star is much smaller than δt, which is the case in our model because we assume 100M ⊙ stars as the source that have a lifetime of about few Myrs [Schaerer, 2002].
Therefore, the total mass in stars in the box is M⋆(box) ≈ L3boxρ⋆ [M⊙]. This mass in stars is then distributed among the haloes according to the mass of the halo as,
m⋆(halo) = mhalo
Mhalo(tot)M⋆(box), (1.8) where m⋆(halo) is the mass of stars in a “halo”, mhalois the mass of the halo and Mhalo(tot) the total mass of haloes in the box.
We then assume that all of the mass in stars is distributed in stars of 100 solar masses, which implies that the number of stars in the halo is N100= 10−2×m⋆(halo).
The luminosity of a 100 M⊙star is obtained from Fig. 1 of Schaerer [2002], assuming zero metallicity. The luminosity of a 100 M⊙star thus derived is in the range 106— 107L⊙ and this value is multiplied by N100to get the total luminosity emanating from the “halo”. The radiative transfer is then done by normalizing the blackbody spectrum at 5 × 105K to this value. The escape fraction of ionizing photons from early galaxies is assumed to be 10 per cent.
1.3.2 Results: Stellar sources
In this section we discuss the results of the evolution of the kinetic, spin and bright- ness temperatures of the IGM, when the sources of reionization consist only of stars.
Figs 1.3, 1.4 and 1.5 show the kinetic, spin and brightness temperatures.
1.3. Heating due to stars 11
Figure 1.3:Kinetic temperature for stellar sources: The slices corresponds to those of Fig. 1.1.
The evolution of the kinetic temperature is plotted. We see that for stellar sources the extent of heating, both in amplitude (maximum around ≈ 105 K towards the center) and spatial extent (< 100 kpc), is extremely small. Basically, only the central part (ionized region) is at a high temperature and there is a sharp fall at the transition into the neutral IGM.
The blackbody type stellar spectra do not have sufficient high energy photons to heat the IGM substantially far away from the source. Thus we see compact regions (bubbles) of high temperatures in the immediate vicinity of the sources. In the inner parts where the ionized fraction is high (XHII > 0.95) the temperatures are on the order of ≈ 105K and drop sharply in the transition zone to neutral IGM. The ionized region is restricted though, to about 100 kpc in physical coordinates.
Spin temperature (Tspin), which reflects the efficiency of driving the brightness temperature (δTb) away from zero, depends on the coupling terms, collisional (ycoll) and Lyα (yα). Although collisional coupling is important close to the radiating
Figure 1.4: Spin temperature for stellar sources: The slices corresponds to those of Fig. 1.1.
The Lyα flux is high enough to couple the kinetic temperature to the spin temperature, but the small extent of the region of high kinetic temperature implies that the spin temperature does not decouple from the CMB temperature beyond the ionized region. We see that there is sufficient Lyα flux to couple the kinetic temperature to the spin temperature for about 50 kpc beyond the ionized region, where the kinetic temperature is lower than the CMB temperature.
source (< 200 kpc), the reason for Tspinfollowing the kinetic temperature Tkis pri- marily the “secondary flux” of Lyα photons, Jo. It has be emphasized here that the Lyα photons are produced only due to secondary processes and their abundance is
calculated on the basis of the Monte Carlo simulations performed by Shull and van Steenberg [1985]. It is known that stellar sources do have copious amounts of “intrinsic” Lyα
flux from the spectra. Although small at the initial stages of reionization, it quickly builds up a significant background of Lyα flux, sufficient to couple Tspinto Tk. As an example, in Fig 1.14 we show the reionization history (δTb) assuming a high back-
1.3. Heating due to stars 13
Figure 1.5: Brightness temperature for stellar sources: The slices corresponds to that of Fig. 1.1. δTbfor stellar sources ranges between a few degree Kelvin just outside the ionized bubbles around the source to about -300 mK for small extent beyond that. Because the values we are dealing with spans a few dynamic ranges and goes from negative to positive, we con- tour plot the absolute values of the brightness temperature. We see that by and large the δTb values are not very high (< 1 mK). Early on, (top-left) at redshift around 10, there are only a few sources and the heating or the Lyα flux is not high enough to render the Universe visible.
At later times, the ionized bubbles overlap significantly driving the brightness temperature to zero.
ground Lyα flux.
For stellar sources, δTbranges from a couple of degrees above zero Kelvin, just beyond the ionizing front of the source, to well below 300 mK below zero further away from the ionizing front. The spin temperature is not decoupled from that of the CMB at extended regions from the source because of the lack of Lyα flux. In
Fig 1.5 δTb is shown at the same four redshifts as in the figure 1.1. The values in the case of stars span a few dynamic ranges and are mostly negative. Therefore, we contour plot the absolute values of the brightness temperature in log scale. We see that by and large the δTbvalues are not very high (< 1 mK) at redshifts < 7. Early on, (top-left) at redshift around 10, there are only a few sources and the heating of the IGM or the secondary Lyα flux is not high enough to render the Universe visible.
And, at later times, the ionized bubbles overlap significantly driving the brightness temperature to zero.
1.4 Heating due to miniqsos
In §1.3 we considered the heating due to stars, and because of the SED considered for stars (blackbody) there weren’t enough hard X-ray photons to cause extensive heating of the IGM. In this section we will consider the heating due to high energy X-ray photons emanating from power-law type sources. Because of the large mean free paths involved in transporting them, it has been difficult to incorporate the effect of heating by quasars self-consistently in a 3-D radiative transfer simulation.
1.4.1 Modeling quasars in BEARS
Studies have shown that energy spectrum of quasars typically follow a power- law of the form E−α[Vanden Berk et al., 2001, Vignali et al., 2003, Laor et al., 1997, Elvis et al., 1994]. Specifically, we explore the power law of the form:
I(E) = Ag × E−α 10.4 eV < E < 104eV, (1.9) where the normalization (Ag) is done according to;
Ag = Etotal
R
Erange
I(E)dE, (1.10)
where Etotalis the total energy output of the quasar-type objects within the energy range (Erange), and τ (E; r; t) is the optical depth. For the simulations performed using quasars, in this paper we consider the value of α to be unity for most of the study. More complex multi-slope spectral templates as in Sazonov et al. [2004] could also be adopted, but this is not done in this study. The variety of SEDs that can be considered is numerous and serves as a reminder of the extent of unexplored
“parameter space”, even while considering only the case of quasars, and argues for the need of an extremely quick RT code like BEARS.
1.4. Heating due to miniqsos 15
The miniquasars are assumed to accrete at a constant fraction ǫ (normally 10%) of the Eddington rate. Therefore, the luminosity of a miniqso, with a central black hole of mass M , is given by:
L = ǫ Ledd(M) (1.11)
= 1.38 × 1037 ǫ 0.1
M M⊙
erg s−1, (1.12)
Where Leddis the Eddington luminosity.
The luminosity derived from the equation above is used to normalize the relation in equation 1.9 according to equation 1.10. The normalization takes place across the energy range of 10.4 eV to 10 keV. Simulations were carried out for a range of masses between 105and 109 M⊙. Although the number of photons at different energies is a function of the total luminosity and spectral index, if we assume that all photons are at the hydrogen ionization threshold, then the number of ionizing photons thus obtained for the mass range given above is of the order of 1050to 1055. These number are of the same order of magnitude as the number of ionizing photons being employed for simulations by various authors likr Mellema et al. [2006] and Kuhlen and Madau [2005], for example.
Now, in order to embed the quasars into the output of the N -body simulations we follow the prescription in §??. dark matter haloes were identified using the friends-of-friends algorithm [Davis et al., 1985] and masses of black holes assigned according to Zaroubi et al. [2007],
MBH= 10−4× Ωb
Ωm
Mhalo, (1.13)
where the factor 10−4reflects the Magorrian relation between the halo mass (Mhalo) and black hole mass (MBH), and ΩΩmb gives the baryon ratio [Ferrarese, 2002]. The template spectrum is assumed like in Eq. 1.9.
1.4.2 Results: Power-law sources
Contour plots of the kinetic, spin and brightness temperatures at four different red- shifts are plotted in Figs 1.6, 1.7 and 1.8, respectively. The high energies of the X-rays implies that they influence the photoionization and thermal energy balance of the IGM surrounding them to a large extent. This is in turn reflected in the brightness temperature as a strong emission signal from the redshifted 21-cm line.
The marked difference between power-law type and stellar sources, in the extent to which they heat the IGM, is evident in Fig 1.6. In the inner parts of the “temper- ature bubble” the amplitude of the kinetic temperature is of the order of ≈ 106.5K.
Figure 1.6: Kinetic temperature for power-law sources: The slices corresponds to those of Fig. 1.1. The evolution of the kinetic temperature is plotted. Here we see the striking dif- ference between a quasar and a stellar source in the the extent of heating. The amplitude is
≈ 106.5K towards the centre and the spatial exent of the heating is about a few Mpc. Already at redshift 10 (top-left) the average temperature is of the order 103K.
The photoelectrons created by high energy X-ray radiation thermalize to these high temperature values. Also contributing to extremely high temperatures in the very inner parts of the bubble is Compton heating [Thomas et al., 2008]. The mean free paths of X-ray electrons are extremely large resulting in a spatial exent of heating of the IGM that reaches a few Mpc. Thus, we see that already at redshift 10 (top-left) the average temperature of the IGM is in the order of 103K.
Owing to a high secondary Lyα flux produced due X-ray photons, the spin tem- perature (Tspin) is coupled to the kinetic temperature of the IGM for the most part.
The snapshots of the Tspinevolution (Fig. 1.7) shows a peculiar ring-like behaviour.
1.4. Heating due to miniqsos 17
Figure 1.7: Spin temperature for power-law sources: The slices corresponds to those of Fig. 1.1. Spin temperatures show an interesting ring-like behaviour (easily visible in the top- left panel). The reason for this is that towards the central part the IGM is ionized and hence from Eq. 1.3 we see that the Lyα flux is very low. And because the Lyα flux progressively gets lower away from the source, regions at >5Mpc also receive low Lyα flux. This implies that there is a region between these two extremes where the IGM is not ionized and the Lyα flux is high enough to decouple the spin temperature from the CMB.
The explanation of this lies in the fact that towards the central parts of the bubble around the source, the IGM is highly ionized and we see from Eq. 1.3 that the Lyα flux, being proportional to the neutral fraction, attains an extremely low value. On the other hand the Lyα flux gets progressively lower away from the source (regions
> 5 Mpc). Therefore there is a relatively narrow zone (a few hundred kpc) in be- tween the ionizing front and regions further out where the Lyα flux is high enough to decouple the spin temperature from the CMB [see also Thomas and Zaroubi [2008]].
Figure 1.8: Brigtness temperature for power-law sources: The slices corresponds to those of Fig. 1.1. Contours plotted are the observable of an interferometric experiment: the brightness temperature. Apart from being the quantity of interest for mapping the neutral hydrogen content of the Universe, the profile of the brightness temperature, especially at earlier times, reflects the nature of the radiation-emitting source. We see again from the top-left panel that although the number of sources in the field are few, they are efficient in both increasing the kinetic temperature dramatically and providing sufficient Lyα flux to light up the Universe around them. The spheres in the top-left panel have δTbof zero because they are highly ionized.
Fig 1.8 shows the brightness temperature of the IGM for redshifts 10, 9, 8 and 7 in panels top-left to bottom-right, respectively. Brightness temperatures plotted in this figure are high enough to be detectable by upcoming telescopes like LOFAR and MWA, whose sensitivities can detectδTb2 1/2
> 5mK. Apart from being within the observable range of LOFAR to map the evolution of the neutral hydrogen con-
1.5. Modeling co-evolution of stars & quasars 19
tent of the Universe, the profiles of the δTbaround the sources, especially at earlier times, reflect the nature of radiation-emitting sources. We see from from the top-left panel of the figure that even though the number of sources in the field is small, they are efficient in both increasing the kinetic temperature dramatically and providing sufficient Lyα flux to light up the Universe around them. The δTbinside the spheres is zero because they are highly ionized (XHII> 0.99).
1.5 Modeling co-evolution of stars & quasars
In §1.3 and §1.4 we have seen the result of ionization and heating caused sepa- rately by stars and quasars, respectively. From observations of our Universe we know stars and quasars do co-exist. Although there are indications of the quasar number-density peaking around redshift two [Fan, Nusser and Silk, 1993] and de- clining thereafter, there are also measurements by Fan et al. [2006] of high-mass (> 108M⊙) quasars at redshifts around six. This allows us to envisage a scenario of reionization in which stars and quasars contributed to the ionization and heating of the IGM. In this section we present our model of a combined star-quasar spectral energy distribution (SED) and examine the results of ionization and heating caused by this model SED.
1.5.1 The modeled “Hybrid” SED
The uncertainty regarding the properties and distributions of objects during the dark ages allows us to come up with strikingly different ways of incorporating the co-existence of stars and quasars. For example, consider a volume of the Universe like the simulations we are using (100 h−1 Mpc3, comoving) and suppose that we have identified N different haloes. One approach would be to suppose that each of these haloes host both a quasar and a stellar population, whose mass/luminosity is derived according to §??and §??, respectively. The other approach would be to place one or two massive quasars, while within the black hole mass density predictions of Volonteri et al. [2008], while populating the rest of the haloes with stars. For this analysis we have chosen to adopt the former approach.
In our approach, given the halo mass we determine the black hole as previously discussed and further assume a stellar component that is 10−3times the black hole mass. Radiative transfer is performed using this hybrid SED, i.e., a superposition of power-law type quasar SED and black body spectrum. The results for ionization and heating are presented below.
Figure 1.9: Ionization due to the hybrid model: The slices corresponds to those of Fig. 1.1.
We see here that for the hybrid model we have assumed the ionization proceeds quickly and the Universe is ionized by redshift 6 (see Fig. 1.13 for the reionization history.
1.5.2 Results: Hybrid model
We present results of the impact of X-ray and stellar sources on the IGM. High- energy X-ray photons from early quasars influence the temperature and ionization of the IGM prior to full reionization, before the fully ionized bubbles associated with individual sources have overlapped. X-rays have large mean free paths relative to UV photons, and their photoelectrons can have significant effects on the thermal and ionization balance. We find that hydrogen ionization is dominated by the X-ray photoionization of neutral helium and the resulting secondary electrons. Thus, the IGM may have been warm and weakly ionized prior to full reionization. Fig 1.9 shows the ionization due to the hybrid model we have considered here.
1.6. Reionization histories: A comparison 21
In this model we have assumed we have scaled down the mass of the black hole by an order of magnitude compared to that assumed in the model described in §1.4.
The stellar mass is then assumed to be 10−3of the black hole mass, motivated by the well known Magorrian relationship. We see that in this case the number of UV photons was not high enough to ionize the Universe completely by redshift 6. Full reionization thus could be achieved either by increasing the mass of the black hole and/or by increasing the factor 10−3 assumed between the black hole and stellar mass.
This hybrid model was used in BEARS to calculate the evolution of kinetic (Fig. 1.10), spin (Fig. 1.11) and brightness temperatures (Fig. 1.12) of the IGM at various redshifts.
The hybrid model produces heating patterns that are qualitatively in between those of the former two models (Fig 1.10). The power-law components take over the role of heating an extended region and the stellar component maintains a very high temperature in the central parts.
The spin temperature (Fig 1.11) again shows the characteristic ring-like struc- ture around the source. The amount of secondary Lyα radiation produced due to quasars enable the efficient coupling of the spin temperature to the kinetic tempera- ture. Towards the end of reionzation (z < 7) the Lyα is high enough at all locations in the simulation box that the kinetic and spin temperatures are coupled.
The brightness temperature for the hybrid model is plotted in Fig 1.12. We see that the secondary Lyα flux produced due to the power-law type sources efficiently couples the high ambient kinetic temperature of the IGM to the spin temperature and thus enables a large fraction of the Universe to be visible in the brightness tem- perature, albeit with a few sources (top-left panel of the figure). The brightness tem- perature does not go to zero at redshift six because, in the model we have assumed, the Universe is not entirely ionized by redshift six.
1.6 Reionization histories: A comparison
Radiative transfer including X-ray heating was performed for three distinct scenar- ios in the previous sections. In this section we will make a comparative study of their ionization histories. To create the continous frequency cube of observations, we adopt the procedure described in Thomas et al. [2008]. This data cube is then convolved with the point spread funtion (PSF) of the LOFAR telescope to produce the mock data cube of the redshifted 21-cm signal as seen by LOFAR.
As expected the signatures (both visual and in the RMS) of the three scenarios (power-law objects only, stellar objects only and the hybrid; panels from top respec-
Figure 1.10: Kinetic temperature for a hybrid model: The slices corresponds to those of Fig. 1.1. The heating pattern is peculiar in that the temperature shows a much “cuspier”
behaviour towards the center. The reason is that the stellar component of the source ionizes the very central part and the X-ray radiation can thus heat this central part to a higher tem- perature. On the other hand the large extent of the heating is attributed to the power-law component of the source.
tively in Fig 1.13) are markedly different. In the power-law only scenario, reion- ization proceeds extremely quickly and the Universe is almost completely (XHII>
0.95) reionized by around redshift 7. The case in which stars are the only source of photons sees reionization complete at a redshift of 6. In this case, compared to the previous one, reionization proceeds in a rather gradual manner. The hybrid model we have described above reflects a different outcome from the two mentioned above, the reionization being completed in a redshift interval that is in between the previous two scenarios.
1.6. Reionization histories: A comparison 23
Figure 1.11: Spin temperature for the hybrid model: The slices corresponds to those of Fig. 1.1. Spin temperatures for the case of the hybrid model again shows the ring like be- haviour as in the case of power-law sources (discussed in Fig. 1.6.
The brightness temperature (δTb) itself, as seen in the figure displays striking differences. We remind the reader that the δTb in Fig 1.13 is calculated based on the effectivness of secondary Lyα flux, produced by the source, in decoupling the CMB temperature (TCMB) from the spin temperature (Tspin). This flux, both in spa- tial extent and amplitude, is obviously much large in the case of power-law sources compared to the stars. This is the reason for the much larger brightness temper- atures in both the power-law only and the hybrid model compared to that of the stars. However, we know that stars themselves produce Lyα radiation in their spec- trum. Apart from providing sufficient Lyα flux to their immediate surroundings, this radiation builds up, as the Universe evolves, into a strong background Lyα flux [Ciardi and Madau, 2003]. Thus, we plot in Fig 1.14 the same set of reionization his-
Figure 1.12:Brightness temperature for the hybrid model: The slices corresponds to those of Fig. 1.1. The large extent of heating and sufficient Lyα flux ensures that the IGM well beyond the location of the source is rendered visible in the 21-cm emission.
tories, now assuming that there is enough background Lyα flux to couple Tspin to the kinetic temperature (Tkin) of the IGM.
It has to be noted that the results we are discussing here are extremely model de- pendent and any changes to the parameters can influence the results significantly.
This, on the other hand, is a demonstration of the capability and the need for a
“BEARS like” algorithm: spanning the enormous parameter space of the astrophys- ical unknowns.
1.6. Reionization histories: A comparison 25
Figure 1.13: Contrasting reionization histories: Starting from the top, reionization histo- ries (brightness temperatures) are plotted for power-law, stellar and hybrid sources, respec- tively. The bottom panel plots the RMS of the brightness temperature as a function of red- shift/frequency.
1.6.1 Looking through LOFAR
To distinguish the observational signatures of the various sources of reionization we need to filter the cosmological 21-cm maps generated in §1.6 using the LOFAR telescope response. The point spread function (PSF) of the LOFAR array was con- structed according to the latest configuration of the antenna layouts for the LOFAR- EoR experiment.
The latest configuration of the LOFAR telescope will consist of up to 48 stations of which approximately 24 will be located in the core region (see Fig. ??), in The Netherlands. The core marks an area of 1.7 × 2.3 kilometres. Each High Band An-
Figure 1.14:Reionization histories (Tspin≈ TCMB): Reionization histories (brightness temper- atures), are plotted as in Fig. 1.13, except now with the assumption that the spin temperature (Tspin) has been completely decoupled from that of the CMB (TCMB).
tenna (HBA; 110-240 MHz) station in the core is further split into two “half-stations”
of half the collecting area (35-metre diameter), separated by ≈ 130 metres. This split further improves the uv-coverage. The central region of the core consists of six closely-packed stations, to ensure improved coverage of the shortest baselines nec- essary to map out the largest scales on the sky, such as the Milky Way. For details on the antenna layout refer to Labropoulos et al, in preparation.
In its current configuration, the resolution of the LOFAR core is expected to be around 3 arcmin. This implies that, for example, at redshift 10 all scales below ≈ 800 kpc will be filtered out. Fig 1.15 shows this effect for the reionization histories corresponding to Fig 1.13. The corresponding changes in the RMS of the brightness temperature are also plotted.
1.7. Conclusions & Outlook 27
Figure 1.15:Convolution with LOFAR - PSF: Brightness temperatures after convolution with a point spread function corresponding to LOFAR are plotted in the same order as in Fig. 1.13.
1.7 Conclusions & Outlook
The focus of this chapter was two-fold. First, to introduce the algorithm to incorpo- rate heating (including X-ray heating) in BEARS and discuss its application to two cases of reionization and heating, i.e., for blackbody (stellar) and power-law (quasar) type sources. Second, to use this algorithm to study the influence of a hybrid popu- lation of stars and quasars as reionizing sources. Also, an important addition to the code is the self-consistent calculation of the spin temperature evolution as a function of redshift.
In order to incorporate heating into BEARS, we followed the procedure of em- bedding “temperature bubbles””, much like the algorithm used to obtain the ion- ized fraction. Overlaps were treated by considerations of the conservation of en-
ergy. The BEARS algorithm implemented here is extremely quick, in that it takes
∼ 5 hours to perform the radiative transfer (including heating) on about 25 differ- ent boxes of 5123 particle, 100 h−1comoving Mpc in size, and then to interpolate between redshifts to produce a contiguous data cube running from redshift 6 to 12, the observational range of the LOFAR-EoR experiment. This implies that various scenarios of reionization can be implemented and tested for observational signa- tures in the redshifted 21-cm emission. Apart from predicting the nature of the underlying cosmological signal, these simulations can be used in conjunction with simulations of others probes of reionization to enhance detectability and/or con- strain new parameters concerning reionization. In a forthcoming paper we (Jeli´c et al., in preparation) will cross-correlate the data set simulated here with that of the CMB from the Planck satellite.
It is clear from the simulations that quasar type sources are not only very efficient in increasing the kinetic temperature of the IGM, the secondary Lyα flux produced by them is sufficient to drive the spin temperature Tspinaway from the CMB temper- ature hence rendering the environment around the source visible in the brightness temperature. Therefore, if a quasar with a black hole in the range > 106M⊙is within the observing window of LOFAR, the brightness temperature produced by it would be high enough to be visible in the experiment. This is not the case for stars because the sizes of the regions and the extent of heating are both much smaller than for quasars.
Stars and quasars were combined in a hybrid model of reionization. Every dark matter halo was embedded with both a quasar and a stellar component. The relation between the two was set according to the Magorrian relation we observe today.
The quasar masses were limited such as not to violate the black hole mass densities estimated by Volonteri et al. [2008].
Results from all these simulations, i.e. stars, quasars and the hybrid, were used to create reionization histories and the observational cube in each of the cases. In creating the histories for each of these cases, we calculated the evolution of the spin temperature as a function of redshift for two scenarios. First, with only the Lyα flux produced due to secondary electrons and second, assuming a very high Lyα background. Stark differences can be found between these simulations. Statistical studies of the differences in various models are deferred to another paper.
Also in the future these calibration errors/residual will be folded into these sim- ulation boxes along with the effects of the ionosphere and expected radio frequency interference (RFI). Subsequently Galactic and extra-galactic foregrounds modelled as in Jeli´c et al. [2008] will be merged with these simulations to create the final “re- alistic” data cube. These data cubes will be processed using the signal extraction and calibration schemes being developed to retrieve the underlying cosmological
1.7. Conclusions & Outlook 29
signal in preparation for the actual experiment scheduled to begin towards the end of 2009.
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