arXiv:astro-ph/0610306v1 11 Oct 2006

evtoso ihrdhf usr.Te ocueta the that conclude They quasars. redshift high of servations o msinrt oniigwt h nw S popula- QSO known the with at tion coinciding pho- the the rate of compute emission reionisation then the ton They maintain (IGM). to medium possible inter-galactic be not would it h ugto oiigpooswas photons ionising of budget the L)cmue from computed (LF) mtee l 02,i sotntogtta tr dominated stars that at thought flux often UV is it the 2002), al. et Smette ihatrsodpoo msinrt at rate estab- emission MHR99) have photon hereafter threshold era results a (1999, inconsistent reionisation lish al. to the et led Madau of and example, For end fronts several the on to proceeded quasars of bution a ta.(01 opt h usrcnrbto othe to contribution quasar at the rate compute emission photon (2001) al. et Fan oteinsn akrudby background by ionising made the equal make contributions contribute stars to relative and evidence quasars the Although observational sources. estimate these of to stars lack difficult theoretical first However, it a the 2001). and by reioni- Loeb produced uncertainties & that (2006); (Barkana photons believed quasars UV Fan commonly & to and is Gnedin due by It was F05); sation (2003)]. completed hereafter al. was et (2005, White hydrogen al. reion- et cosmic the [Fan that of imply quasars isation redshift high of Observations INTRODUCTION 1 o.Nt .Ato.Soc. Astron. R. Not. Mon. rf Version Draft Wyithe J.S.B. & Srbinovsky J.A. Reionisation Hydrogen the Cosmic to of Contribution Quasar the Constraining c colo hsc,Uiest fMlore akil,Vic Parkville, Melbourne, of University Physics, of School 06RAS 2006 tepst aeqatttv siae ftecontri- the of estimates quantitative make to Attempts z ∼ n ocueta h usrcnrbto to contribution quasar the that conclude and 5 z & eg a ta.2001). al. et Fan (e.g. 6 la iia k Survey Sky Digital Sloan z 000 ∼ sn h uioiyfunction luminosity the using 6 1– , ?? n 14 and e words: Key luminosity. bopinsetao ihrdhf usr ugs httere the that near suggest complete quasars was redshift drogen high of spectra Absorption ABSTRACT tv usrcnrbto (at 2005) al. contribution b et quasar made (Fan ative observations luminosity contributions reported t relative quasar with has minimum the comparison through model the constrain overall we and Our parameters, efficiency function. these formation luminosity star bac m quasar clumpy the ionising a the ters, overlap in for post reionisation model the of descriptions a of semi-analytic evolution t combining quasars by the by and made tion compute stars contributions to relative be is the to of approach thought estimate quantitative generally a are make reionisation this for z 20)Pitd2 uut21 M L (MN 2018 August 22 Printed (2006) ∼ ∼ Kise l 2001; al. et (Kriss 3 . 10 % h ag fucranyi oiae yteukonmnmmqu minimum unknown the by dominated is uncertainty of range The 5%. z − eo which below 5 = 2 Analogously, . ( omlg:ter aais omto neglci medium intergalactic - formation galaxies: - theory cosmology: SDSS z ob- ) ∼ oi,AsrlaEal [email protected] Email: Australia toria, ly 6 . z ∼ z ee htwudb otiue yapplto fquasars of population a by could contributed that the be beneath would is that background com- X-ray level unresolved soft the observed that the taken show in who was ponent (2004), approach al. different et A Dijkstra by galaxies. forming star from nucett rdc eoiainat reionisation produce to insufficient rahsue odsrb h oiigsucsadmore and t sources assumed rates ionising photo-emission the different describe the significantly, to used proaches flux. ionizing missing this tribute otiuint h oiigbcgon rmsa forming the star and in from identified background ionising galaxies the to contribution 98 al hr fterqie u orins h univers the reionise al. to et flux (Boyle required LF the the at of of form short statis- law falls a power 1988) by double the described to population fit quasar tical Meiksin a Alternatively, that 2006). finds be (2005) could Loeb & it followin (Wyithe or suppressed reionisation quasars, were which by galaxies dwarf generated by generated be could flux ionising ySivlie l 20)i upr fteve httema- at the background that view ionising the the noted of to and support contribution in (2004) jor (2004) Windhorst al. & et con- Yan Stiavelli Similar by by LF. drawn the are in clusions evolution luminosity negative large usrcnrbto oteinsn akrudat background also ionising the to contribution quasar otiuo finsn htn at photons ionising of contributor .Tedmnn ore finsn htn responsible photons ionising of sources dominant The 6. 5 = z hsrneo ocuin eet h ieigap- differing the reflects conclusions of range This nteohrhn ukre l 20)so htthe that show (2004) al. et Bunker hand other the On ra bevtre rgn epSre (GOODS) Survey Deep Origins Observatories Great ∼ ∼ . )t h oiigbcgon a ewe 1 between was background ionising the to 7) 10 nyb atro 2 of factor a by only 6 − 2 fully a n ugs htqaascudntb major a be not could quasars that suggest and T E tl l v2.2) file style X oieteuies at universe the ionise ubeUtaDe il (UDF) Field Ultra-Deep Hubble − n ol hrfr con- therefore could and 3 oiaino omchy- cosmic of ionisation z ∼ efidta h rel- the that find We . tr n quasars and stars y z ihu noiga invoking without 6 z nti ae we paper this In . ∼ eesucs Our sources. hese ∼ ofe parame- free wo 6. gon radia- kground yadjusting By . .Temissing The 6. z du with edium ∼ z comes 6 was 6 = asar . 4% is g o e 2 Srbinovsky & Wyithe be necessary for reionisation. In the aforementioned analy- model ∆crit is fixed prior to overlap, after which it evolves ses, reionisation in an inhomogeneous (clumpy) universe was with redshift [see equation (6) and relevant discussion in treated using a clumping factor, C = hn2i/hni2 (where n is WL03]. In this formalism it is possible to compute the ef- the hydrogen number density), to account for the increased fective clumping factor in regions where ∆i < ∆crit such recombination rate. Larger values of C increase the emis- that C = C[z, ∆crit(z)]. For example, assuming overlap to sion rate necessary to achieve reionisation. MHR99 consider occur at z ∼ 6 and ∆crit= 20 prior to overlap (following a value of C = 30 corresponding to the numerical simu- WL03), C(z = 6) ∼ 2.3. Thus, the effective clumping fac- lation of Gnedin & Ostriker (1997), however they note the tor in ionised regions is considerably smaller than the value considerable uncertainty in the applicability of this value. often assumed, which is over the whole IGM. Fan et al. (2001), Yan & Windhorst (2004), Stiavelli et al. The sources of ionising photons in this reionisation (2004) and Bunker et al. (2004) use this same value for C. model were assumed to be quasars, in addition to popu- Dijkstra et al. (2004) use a value of C = 10 and Meiksin lation II (popII) and population III stars. In the current (2005) assumes values of 1 matter halo over a large range of luminosities at z & 3. hosts containing star forming galaxies (the additional flux Our model includes two free parameters. First, the min- required to keep this overdense region ionised is accounted imum luminosity for quasars Lmin, which is not predicted ∗ for by the value of escape fraction of ionising photons). The by the model. Second, the parameter fesc represents the volume over which the clumping factor should be computed product of star formation efficiency and escape fraction, and also excludes those overdense regions in the IGM that form characterises the contributions made by stars. Our approach Lyman limit systems. These overdense clouds set the mean- is to compute a reionisation history given a particular set of ∗ free-path (MFP) for ionising photons and the recombina- Lmin and fesc. With this history in place we then compute tion rate should be computed only up to densities separat- the evolution of the background radiation field due to these ing the ionised IGM and these clouds. A framework within same sources. Post overlap, ionising photons will experience which the clumping factor appropriate for reionisation in attenuation due to residual overdense pockets of HI gas. We an inhomogeneous IGM may be computed was described in use the reionisation model of WL03 to describe the ionis- Miralda-Escud´eet al. (2000, hereafter M-E00). We follow ing photon MFP, and subsequently derive the attenuation the formalism developed in that work throughout this paper. of ionising photons. We then compute the flux at the Lyman 15 We begin by describing our models for the reionisation limit (νL = 3.29 × 10 Hz) in the IGM due to sources imme- history and quasar LF in § 2. We then review our proce- diate to each epoch, in addition to redshifted contributions dures for calculation of the ionising background using these from earlier epochs. models in § 3 and present results from our analysis in § 4, We note that HI Lyman limit photons (13.6 eV ) are before discussing our conclusions in § 5. Throughout the incapable of ionising helium (He) [24.6 eV (HeII), 54.4 eV paper we assume values for cosmological parameters based (HeIII)]. We therefore neglect He when computing the in- on WMAP3 results (Spergel et al. 2006) but obtained by tensity of the ionising background. However, the presence of averaging over the observational data sets from WMAP3, He is fully incorporated in the reionisation model of WL03 SDSS, HST, SN Astier and BAO. The resulting parameters which we use to describe the history of reionisation. are ΩΛ = .743, ΩM = .257, ΩB= .0437 and h = .718 (Renyue Cen, private communication). In computation of the mass function we assume a primordial power spectrum defined by 3 EVALUATING FLUX AT THE LYMAN a power law with index n = .95, an exact transfer function LIMIT given by Bardeen et al. (1986) and rms mass density fluctu- −1 ations with a sphere of radius R8 = 8h Mpc of σ8 = 0.765. In this section we review the process for calculating the ion- ising background flux at a particular redshift. We assume the background to be generated by a combination of stars and quasars. We first define z0 to be the redshift at which the 2 SEMI-ANALYTIC MODELS FOR flux is to be evaluated. The flux at the Lyman limit is nor- REIONISATION AND THE QUASAR −21 2 malised in physical units of J21 (10 ergs/sec/Hz/cm /sr), LUMINOSITY FUNCTION and at redshift z0 is related to the energy density by The density field in an inhomogeneous universe can be de- 2 tot scribed by the overdensity, ∆i = ρi/ρ. M-E00 show how c d E (z0) 1 J (z )= νL (1 + z )3, (1) the effective recombination rate in an inhomogeneous uni- 21 0 4π dV dν 10−21 0 verse may be determined dynamically by consideration of the maximum overdensity (∆crit) penetrated by ionising 1 This model is less successful at reproducing the number of very photons within HII regions (i.e. the volume of the IGM to luminous quasars at z ∼ 6 using the WMAP third year cosmology, be maintained in an ionised state). Wyithe & Loeb (2003a, indicating the necessity of revision of its physical basis. However hereafter WL03) employed this prescription for the recombi- we maintain the LF described in Wyithe & Loeb (2003b) as an nation rate into a semi-analytic model of reionisation. In this empirically successful description.

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2 tot d E (z0) νL highly uncertain. We therefore treat Lmin as a free parame- where c is the speed of light and dV dν is the energy per unit frequency interval, per co-moving volume, at frequency ter in our description of quasar emissivity. To convert from L to the luminosity at the Lyman limit (L ) we use the νL, and z0. Contributions to the radiation field at νL (and B νL relation described by Schirber & Bullock (2003), z0) from sources at redshift z were emitted at frequency 1+z z L z ν = 1+z0 ν . Note that ν remains below the He Lyman 1+z limit (24.6 eV ) whilst < 2. The semi-analytic model of LνL 18.05 −1 −1 −1 1+z0 ∼ ηνL = = 10 ergs s Hz LB . (5) the reionisation history predicts an average redshift of over- LB lap, which we refer to as zol. While the observed redshift of In order to compute the Lyman limit flux at z0, we require overlap is subject to some scatter among different lines-of- the contribution from intrinsically higher frequency photons sight, we make the approximation that the universe becomes from higher redshifts. We assume a powerlaw spectrum Lν ∝ −α transparent to Lyα photons at zol along all lines-of-sight. ν , whence we obtain The ionising background flux contains contributions −α φ(νz) 1+ z from sources at z0 in addition to redshifted flux from sources ηνz = ηνL = ηνL . (6) at higher redshift. To compute the ionising background φ(νL)  1+ z0  flux we consider contributions from sources with redshifts Telfer et al. (2002) find α = 1.57 in the wavelength range z0 galaxy [note however that the Lyα halos photon frequency is cancelled by the redshifting of the fre- 3 around some QSOs indicate reprocessing of this ionising flux d Eνz (z) quency interval. The term dV dνdt represents the frequency (Francis & McDonnell 2006)]. Finally, using equation (2), dependent (co-moving) energy density per unit time gener- we find the energy density due to quasars at frequency νL ated by ionising sources. Finally, the relation between proper and redshift z0, time and redshift is 2 tot α − z0 tot dt 3 1 d EνL (z0) dLB (z) 1+ z0 = [H0(1 + z) (1 + z) ΩM + ΩΛ] , (3) = η dz dz νL p dV dν Zzol dV  1+ z  where H0 is the current value of the Hubble parameter. − dt × e τ(z,z0) . (7) Equations (1) & (2) describe the ionising flux at z0 given dz contributions of ionising sources at higher redshifts. In the This energy density may be converted to a flux and ex- next two sub-sections we describe the calculation of the pressed in units of J21 using equation 1. emissivity of quasars and stars, which provide the sources of ionising radiation. 3.2 Lyman limit photons from Stars

3.1 Lyman limit photons from Quasars We next describe the contribution made by popII stars. To begin, we describe the spectral energy distribution (SED) Our estimate of the emissivity of quasars is based on the B- of popII star forming galaxies using the model presented in 3 d Eν band quasar LF [Φ(LB,z)], i.e. the number density per co- Leitherer et al. (1999). The SED has units of , where dνdtdM˙ moving volume per solar B-band luminosity LB. We use the M˙ is expressed in units of (baryonic) solar masses per year. theoretically derived LF of Wyithe & Loeb (2003b), which In the instance that ionising photons are produced primarily successfully reproduces known properties of the observed in starbursts, with lifetimes much shorter than the Hubble quasar LF at both the bright and faint ends at z & 2.5, time, we may express the star formation rate per unit time including the isolated observations discussed in § 4.3. This as model is based on the physics of galaxy mergers, accretion ˙ powered luminosity and the self-regulated growth of super- dM ∗ dF (z) (z)= f ρb, (8) tot massive black holes. The total B-band luminosity [LB (z)] dV dtyear per volume is calculated directly from Φ(LB,z) where ρb is the co-moving baryonic mass density, and F (z) tot ∞ is the collapsed fraction of mass in halos above a critical dLB (z) ∗ = dLB Φ(LB,z) × LB. (4) mass at z. The factor f (star formation efficiency) de- dV Z Lmin scribes the fraction of collapsed matter that participates in While this integral converges at high luminosities, the shal- star formation. This fraction is largely unknown, however ∗ low slope of the LF at low luminosities means that the value Wyithe & Loeb (2006) constrain f to be ∼ 10 − 15% by of the integral is sensitive to the lower limit Lmin, which is fitting to the galaxy LF at high redshift.

c 2006 RAS, MNRAS 000, 1–?? 4 Srbinovsky & Wyithe

To compute F (z) we use the Press-Schechter (1976) the MFP is set by residual HI in ionisation equilibrium with mass function with the modification of Sheth & Tormen the background flux as well as Lyman limit systems. Our (1999) to describe the evolution of the collapsed fraction estimate of the MFP therefore provides an upper limit. Fur- [F(z)] within halos above a critical virial temperature. In thermore, our reionisation model assumes ionising photons a cold neutral IGM beyond the redshift of reionisation, to be instantly absorbed, an assumption which becomes less this critical virial temperature is set by the temperature applicable post overlap. 4 (TN ∼ 10 K) above which efficient atomic hydrogen cool- Finally the MFP as a function of redshift may be used ing promotes star formation. Following the reionisation of a to compute the attenuation of ionising photons between red- region, the Jeans mass in the heated IGM limits accretion shifts z and z0. The resulting optical depth is 5 to halos above TI ∼ 2.5 × 10 K (Thoul & Weinberg 1996). The corresponding virialised mass can be determined using z0 cdt 1 equation (26) in Barkana & Loeb (2001). τ(z,z0)= dz . (13) Z dz λi We may therefore write the time derivative of the col- z lapsed fraction

dF dF (z,TI) dF (z,TN) (z)= Qm(z) + [1 − Qm(z)] 4 RESULTS dtyear  dz dz  dz The model described in the previous section has two free × , (9) ∗ dtyear parameters Lmin and fesc. For any combination of these pa- rameters we are able to compute a reionisation history, the here Qm is the ionised baryonic mass fraction in the universe. evolution of the MFP, the evolution of the ionising back- To describe the ionising flux from stars we require one ground, and the relative contributions of stars and quasars further parameter. We have assumed that all ionising pho- to the ionising background. In this section we first sum- tons produced by quasars escape the host galaxy (§ 3.1). marise the observations to which we compare our model be- However, due to the softer spectra of galaxies it is likely fore describing the results of this comparison. that only a fraction of ionising photons produced by stars enter the IGM. Therefore, an additional factor of fesc (the escape fraction) must be included when computing the emis- 4.1 Observational estimates of the ionising sivity due to stars. Ciardi & Ferrara (2005) include a review background and mean-free-path of existing constraints on fesc which suggests that its value is < 15%. The star formation efficiency and escape fraction F05 have analyzed the absorption spectra of 19 high redshift ∼ ∗ may be combined into a single free parameter (fesc) to de- quasars. Based on this analysis F05 present estimates of the scribe the contribution of stars in our model. neutral fraction at a range of redshifts near the end of the Finally the energy density due to popII stars at z0 may reionisation era, as well as the background ionising flux and 3 d Eνz (z) MFP. In this paper it is our aim to constrain the contribu- be computed using equation 2 with dV dνdt given by tions of quasars and stars to the ionising background based 3 3 d Eνz (z) d Eνz (z) dF (z) ∗ on these observations. = ρbfesc. (10) dV dνdt dνdtdM˙ dtyear The observed estimates of the ionising background are This energy density may then be converted to a flux and quoted as photoionisation rates (F05), derived from the ob- combined with the contribution due to quasars. served (Lyα) optical depth. These photoionisation rates may be related to ionising background fluxes using

3.3 Attenuation of ionising photons ∞ Jν Γ = 4π σν dν, (14) Given the luminosity density we may find the ionising back- ZνL hpν ground using equation (2), following calculation of the op- where σν is the HI photoionisation cross-section and hp is tical depth (τ) and thereby the attenuation of ionising pho- Planck’s constant. Using equation (14) and the spectra de- tons. Following overlap, we use the approach of M-E00 to scribing our sources, we find Γ12 = 1.69J21 (where Γ12 is the − − estimate the MFP (λi) of ionising photons as a function of ionising rate in units of 10 12s 1) for a background in which redshift (zi). stars are the dominant source. Similarly, if the background radiation is dominated by quasars we obtain Γ12 = 2.64J21 . −2/3 λi = λ0(1 − Fv) . (11) In comparing our model to observation we normalise the ob- served flux using a weighted sum of the instantaneous flux Here we use the reionisation model of WL03 to compute the contributions from each of the stellar and quasar sources. redshift evolution of the the volume filling factor in ionised F05 also estimate the ionising photon MFP. They com- regions. pute MFPs using the frequency averaged cross-section (hσν i)

∆crit(z) to Lyman limit absorption assuming an ionising background −α Fv = d∆P (∆), (12) spectrum of the form Jν ∝ ν . They compute hσν i in the Z 0 frequency range between the HI and the HeI Lyman limits, where [P (∆)] is the volume weighted probability distri- and assume α = 5 which describes an ionising background bution of the overdensity (M-E00). The product λ0H = dominated by stars (Barkana & Loeb 2001). In § 3.2, we out- − 60kms 1 was obtained from comparison to the scales of Lyα lined our calculation of stellar emissivity using the spectrum forest structures in simulations at z = 3 (M-E00). In reality, of Leitherer et al. (1999). We therefore use this spectrum to

c 2006 RAS, MNRAS 000, 1–?? The Quasar Contribution to the Reionisation 5

∗ 12.5 Figure 1. The best fit model (see text) corresponding to fesc= .00815, Lmin= 10 LB and ∆crit=20. (a) The reionisation history. Solid, dotted and dashed lines represent the evolution of the volume and mass filling fractions [Fv(HII),Fm(HII),Fv(HeIII)]. (b) The fractional contribution of quasars to the instantaneous production rate of ionising photons (solid line). The fractional contribution of quasars to the cumulative number of ionising photons produced by z (dashed line). (c) The evolution of the MFP. Our model (solid line) and observational data points from F05. (d) The redshift evolution of J21. The model combined flux (solid line) and contributions from stellar sources (dotted line) and quasars (dashed line). The data points are from F05.

adjust the values of hσν i from F05 (assuming stellar domi- tons. The dashed line shows the fractional contribution of nance) and present data points using this modified value. quasars to the cumulative number of ionising photons pro- duced by redshift z. Ionising photons above the HeI Lyman limit also contribute to the reionisation of HI (WL03). We 4.2 Comparison with observation therefore consider the contribution of ionising photons made In this section we show results for a particular model in by quasars and stars at all frequencies above the HI Lyman 12.5 limit. The contribution of quasars to the instantaneous pro- which we assume Lmin = 10 LB. In this case we find that the model best fits the ionising background observations for duction rate rises towards low redshift and becomes ∼ 40% ∗ that of stars by z ∼ 2.5, in broad agreement with obser- the value of fesc= .00815. For these parameters we find that model fit to observations of the ionising background yields a vation (Kriss et al. 2001; Smette et al. 2002). Cumulatively reduced χ2 value of 1.3. In a subsequent section, we explore the stellar contribution remains dominant even at low red- shifts. the range of values for Lmin permitted by observation. For our representative model, Figure 1 shows the red- In Figure 1(c) we show the evolution of the MFP (solid shift evolution of four quantities computed. Here we have line) with comparison to the points from F05. It is clear from assumed a critical overdensity of ∆crit= 20 prior to overlap, this plot that the models abilty to predict the MFP dimin- which occurs in this case at zol ∼ 6. Figure 1(a) shows the ishes towards lower redshift. In Figure 1(d), we plot J21 as a full reionisation history. The volume fraction of HII (solid function of redshift. The evolution of the total (model) flux line), the mass fraction of HII (dotted line) and the vol- due to quasars and stars is shown as a solid line. The con- ume filling fraction of doubly ionised HeIII (dashed line) tribution to the ionising background due to quasars (stars) are shown. Note that overlap is marked by the point when alone is shown as a dashed line (dotted line). In Figures (c) the volume filling fraction of HII reaches unity. Figure 1(b) and (d), excepting the two highest redshift bins the data shows the resulting fractional contribution of quasars to the points show the mean value for J21 with 1 − σ error bars. instantaneous production rate (solid line) of ionising pho- In the two highest redshift bins optical depth measurements

c 2006 RAS, MNRAS 000, 1–?? 6 Srbinovsky & Wyithe

11 11.75 12.5 Figure 2. The a posteriori probability (P) assuming values for Lmin(LB) = 10 (solid), 10 (dashed), 10 (dot-dashed). (a) ∗ The cumulative distribution as a function of fesc. (b) dP/dF where F is the fractional quasar contribution to the instantaneous photon production rate at z = 5.7. were not always possible due to complete Gunn-Peterson In Figure 2(a) we show the resulting cumulative a poste- ∗ troughs along some lines of sight (see F05). In these cases riori conditional probability distribution for fesc. The mod- the mean and standard deviation were calculated using lower els which best fit the ionising background observations, for ∗ limits on the optical depth. Hence these data points repre- each given Lmin, have corresponding values for fesc in the ∗ sent upper limits on J21. narrow interval .0074 . fesc . .008. Furthermore, for each particular Lmin the distribution is restricted to a narrow in- ∗ terval in fesc space. This is a reflection of the sensivity of the 4.3 Constraining f ∗ and the quasar flux ∗ esc goodness of fit to fesc. The best fits for each Lmin correspond contribution to the IGM to overlap redshifts suggested by the reionisation model at In the previous sub-section we discussed the results from a z ∼ 6. This is slightly lower than the recent numerical anal- 12.5 ysis of Gnedin & Fan (2006) which suggested overlap occurs particular model which assumed a value for Lmin= 10 LB. ∗ between 6.1 . zol . 6.3. Better fits are obtained for larger In this sub-section we investigate values of fesc, given various minimum quasar luminosities. In particular, for a range of values of Lmin. However, this reflects systematic uncertainty in the model rather than a real constraint. In Figure 2(b) we Lmin we compute the a posteriori probability of our model ∗ ∗ dP dP dfesc show the corresponding distribution = ∗ , where flux, as a function of the free parameter fesc, given the like- dF dfesc dF lihoods of our model fits. We include an a priori probability F represents the fractional contribution of quasars to the distribution which is flat in the logarithm and consider a instantaneous photon production rate at z = 5.7. For the ∗ range for fesc ∈ [.006, .01]. values of Lmin assumed in Figure 2 the most probable (frac- < At low redshift (z ∼ .5) quasars are observed at lumi- tional) quasar contributions to the ionising background are 10.7 nosities corresponding to ∼ 10 LB (Boyle et al. 2000). F ∼ .145, .061, .014. This range covers the systematic uncer- Barger et al. (2003) have inferred from X-ray observations tainty introduced by Lmin. 11.3 a quasar with luminosity of ∼ 10 LB at z = 5.7. At The models presented thus far have assumed a value of z = 5.85 observations have revealed the presence of quasars ∆crit = 20 beyond zol. We have repeated our analysis for 12.8 at luminosities corresponding to ∼ 10 LB (Cool et al. different choices of ∆crit and find that while the results are 2006). This suggests an upper limit to Lmin as even at similar, the fit to J21 improves as ∆crit is increased between high redshift we can see these quasars directly. At z ∼ 4.5 5 and 20. In particular, for higher ∆crit the flux immedi- ately following overlap rises more sharply due to both the Dijkstra & Wyithe (2006) find a downturn in the density ∗ 11 of AGN with L< 10 LB, based on the absence of AGN in increased fesc that is required to match observation and also Lyα surveys. However, current data is insufficient to strongly the higher value of Fm at zol. constrain the lower limit of QSO activity. Therefore, within this range we construct conditional probability distributions 11 11.75 12.5 4.4 Additional Uncertainties at Lmin(LB) = 10 , 10 , 10 . In each case probability distributions are extracted from the aforementioned likeli- Our model contains several additional uncertainties. These hood analysis. For display we normalise the probability den- include, but are not limited to, the following. First, the sity functions such that the maximum likelihood is set to model LF describes the density of high redshift quasars. The one. uncertainty in the overall level of quasar flux at z ∼ 6 (where

c 2006 RAS, MNRAS 000, 1–?? The Quasar Contribution to the Reionisation 7 there is only 1 point) is at least a factor of 2. This uncer- Barger A. J., Cowie L. L., Capak P., Alexander D. M., tainty needs to be added to the systematic uncertainty in Bauer F. E., Brandt W. N., Garmire G. P., Hornschemeier Lmin. Second, the MFP computed by our model is an upper A. E., 2003, Astroph. J. L., 584, L61 limit. Indeed comparison of our model with the results of Barkana R., Loeb A., 2001, Astroph. J. Suppl. Ser., 349, Fan et al. (2005) suggest an overestimate by a factor of ∼ 2 125 at z . 5.5. This overestimate results in an underestimate Bernardi M., Sheth R. K., SubbaRao M., Richards G. T., of the attenuation in our model. We may therefore under- Burles S., Connolly A. J., Frieman J., Nichol R., Schaye ∗ estimate the quantity fesc required for a good description J., Schneider D. P., Vanden Berk D. E., York D. G., of the ionising background, and therefore overestimate the Brinkmann J., Lamb D. Q., 2003, Astron. J., 125, 32 fractional contribution to quasars. Thirdly, our model does Boyle B. J., Shanks T., Croom S. M., Smith R. J., Miller L., not consider the possible evolution of Lmin with redshift. Loaring N., Heymans C., 2000, Month. Notic. R. Astron. Soc., 317, 1014 Boyle B. J., Shanks T., Peterson B. A., 1988, Month. Notic. R. Astron. Soc., 235, 935 5 CONCLUSIONS Bunker A. J., Stanway E. R., Ellis R. S., McMahon R. G., In this paper we have estimated the relative contributions 2004, Month. Notic. R. Astron. Soc., 355, 374 made by quasars and stars to the ionising background at Ciardi B., Ferrara A., 2005, Space Science Reviews, 116, the end of the reionisation epoch. We have characterised 625 these contributions by two free parameters. Lmin which de- Cool R. J., Kochanek C. S., Eisenstein D. J., Stern D., fines the minimum luminosity of contributing quasars and Brand K., Brown M. J. I., Dey A., Eisenhardt P., Fan ∗ fesc which describes the product of star formation efficiency X., Gonzalez R. F., Jannuzi B. T., McKenzie E. H., Rieke and escape fraction of ionising photons from galaxies. We M., Soifer B. T., Spinrad H., Elston R. J., 2006, astro- compute the relative quasar contribution for the range of ph/0605030 11 11.75 12.5 values, Lmin(LB) = 10 , 10 , 10 . We find the relative Dijkstra M., Haiman Z., Loeb A., 2004, Astroph. J., 613, contributions made by quasars to the ionising background 646 at z = 5.7 are F ∼ .145, .061, .014, respectively. The value of Dijkstra M., Wyithe J. S. B., 2006, astro-ph/0606334 Lmin provides the greatest uncertainty. Elitzur M., Schlosman I., 2006, astro-ph/0605686 We are able to quantitatively constrain the contribu- Fan X., Narayanan V. K., Lupton R. H., Strauss M. A., tion of quasars to the ionising background because our model 2001, Astron. J., 122, 2833 (following M-E00) computes the clumping factor in the IGM Fan X., Strauss M. A., Becker R. H., White R. L., Gunn which has represented a major uncertainty in previous stud- J. E., Knapp G. R., Richards G. T., Schneider D. P., ies. There are a few independent checks which enhance our Brinkmann J., Fukugita M., 2005, astro-ph/0512082 confidence in the results. Firstly, our quasar LF accurately Francis P. J., McDonnell S., 2006, Monthly Notices of the reproduces observations at z > 2.5. This quasar popula- Royal Astronomical Society, 370, 1372 tion was responsible for the double reionisation of helium, Gnedin N. Y., Fan X., 2006, astro-ph/0603794 11 which for values of Lmin(LB) ∼ 10 is predicted by our Gnedin N. Y., Ostriker J. P., 1997, Astroph. J., 486, 581 reionisation model to occur at 3 . z . 3.5, in agreement Kriss G. A., Shull J. M., Oegerle W., Zheng W., Davidsen with observation (Bernardi et al. 2003; Theuns et al. 2002). A. F., Songaila A., Tumlinson J., Cowie L. 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