The evolution of the helium-ionizing background at z ~ 2-3
aa r X i v : . [ a s t r o - ph . C O ] J un Draft version November 8, 2018
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THE EVOLUTION OF THE HELIUM-IONIZING BACKGROUND AT Z ∼ Keri L. Dixon & Steven R. Furlanetto
Department of Physics and Astronomy, University of California, Los Angeles, CA 90095, USA; email: [email protected]
Draft version November 8, 2018
ABSTRACTRecent observations suggest that helium became fully ionized around redshift z ∼
3. The He II optical depth derived from the Lyman- α forest decreases substantially from this period to z ∼ z ∼ . . z . .
2. This model includes aninhomogeneous metagalactic radiation background, which is expected during and after helium reion-ization. We find that assuming a uniform background underestimates the required photoionizationrate by up to a factor ∼
2. When averaged over the (few) available lines of sight, the effective opticaldepth exhibits a discontinuity near z ≈ .
8, but the measurement uncertainties are sizable. Thisfeature translates into a jump in the photoionization rate and, provided the quasar emissivity evolvessmoothly, in the effective attenuation length, perhaps signaling the helium reionization era. We thencompute the evolution of the effective optical depth for a variety of simple helium reionization models,in which the measured quasar luminosity function and the attenuation length, as well as the evolvingHe
III fraction, are inputs. A model with reionization ending around redshift z ≈ . Subject headings: intergalactic medium – diffuse radiation – quasars: absorption lines INTRODUCTION
In the standard reionization history of the Universe,the metagalactic ionizing background evolved relativelyslowly except for the reionization of hydrogen ( z & z ∼ II has an ionization potential of54.4 eV, bright quasars with hard spectra are requiredfor its ionization. As such, the distribution and intrinsiccharacter of quasars, in addition to the properties of theIGM, determine the radiation background at these highenergies.These quasars are quite rare, implying a strongly fluc-tuating background even after reionization (Fardal et al.1998; Bolton et al. 2006; Meiksin 2007; Furlanetto 2008).Direct evidence for these source-induced variations hasbeen seen in the “transverse proximity effect” of thehardness ratio through comparisons of the H I andHe II Lyman- α (Ly α ) forests with surveys for nearbyquasars (Jakobsen et al. 2003; Worseck & Wisotzki 2006;Worseck et al. 2007). These variations are exag-gerated by the strong attenuation from the IGM(Haardt & Madau 1996; Faucher-Gigu`ere et al. 2008a;Furlanetto & Oh 2008a). Furthermore, radiative trans-fer through the clumpy IGM can induce additionalfluctuations (Maselli & Ferrara 2005; Tittley & Meiksin2007). During reionization, fluctuations in the back-ground are even greater, because some regions receivestrong ionizing radiation while others remain singly-ionized with no local illumination.Recent observations indicate that helium reionizationoccurs at z ∼
3. The strongest evidence comes from far-ultraviolet spectra of the He II Ly α forest along the lines of sight to bright quasars at z ∼
3. These observationsof the He II Ly α transition ( λ rest = 304 ˚A) are difficult,because bright quasars with sufficient far-UV flux and nointervening Lyman-limit systems are required. To date,six such lines of sight have yielded opacity measurements:PKS 1935-692 (Tytler et al. 1995; Anderson et al.1999)), HS 1700+64 (Davidsen et al. 1996; Fechner et al.2006), HE 2347-4342 (Reimers et al. 1997; Kriss et al.2001; Smette et al. 2002; Shull et al. 2004; Zheng et al.2004b), SDSS J2346-0016 (Zheng et al. 2004a, 2008),Q0302-003 (Jakobsen et al. 1994; Hogan et al. 1997;Heap et al. 2000; Jakobsen et al. 2003) and HS 1157-3143 (Reimers et al. 2005). The effective helium opticaldepth from these studies decreases rapidly at z ≈ . Hubble SpaceTelescope should add to the current pool of data.Several indirect methods attempt to probe the im-pact of helium reionization on the the IGM. One ex-pected effect of helium reionization is an increase inthe IGM temperature (Hui & Gnedin 1997; Gleser et al.2005; Furlanetto & Oh 2008b; McQuinn et al. 2009).Schaye et al. (2000) detected a sudden temperature in-crease at z ∼ . I Ly α forest lines (see also Schaye et al.1999; Theuns et al. 2002b). Around the same time,the IGM temperature-density relation appears to be-come nearly isothermal, another indication of recenthelium reionization (Schaye et al. 2000; Ricotti et al.2000). However, not all studies agree (McDonald et al.2001), and temperature measurements via the Ly α for-est flux power spectrum show no evidence for any sud-den change (Zaldarriaga et al. 2001; Viel et al. 2004;McDonald et al. 2006). Furthermore, this temperatureincrease should decrease the recombination rate of hy-drogen, decreasing the H I opacity (Theuns et al. 2002a).Three studies with differing methods have measureda narrow dip at z ∼ . I effective opti-cal depth (Bernardi et al. 2003; Faucher-Gigu`ere et al.2008b; Dall’Aglio et al. 2008). While initially attributedto helium reionization (Theuns et al. 2002a), more recentstudies find that reproducing this feature with heliumreionization is extremely difficult (Bolton et al. 2009;Bolton et al. 2009; McQuinn et al. 2009).The (average) metagalactic ionizing backgroundshould also harden as helium is reionized, because theIGM would become increasingly transparent to high-energy photons. Observations of the He II /H I ratioare qualitatively consistent with reionization occurringat z ∼ IV to Si IV at z ∼
3. Modeling of the ionizing backgroundfrom optically thin and optically thick metal line systemsalso shows a significant hardening at z ∼ II /H I ratio even after helium reionization iscomplete (Shull et al. 2004).In this paper, we focus on interpreting the He II Ly α forest and the significance of the jump in the opacity at z ≈ .
8. After averaging the effective optical depth overall sightlines, we calculate the expected photoionizationrate given some simple assumptions. In particular, weinvestigate the impact of a fluctuating radiation back-ground, comparing it to the common uniform assump-tion. We interpret our results in terms of an evolving at-tenuation length for helium-ionizing photons R as wellas state-of-the-art models of inhomogeneous reionization.We use a semi-analytic model, outlined in §
2, to in-fer the helium photoionization rate from the He II Ly α forest. The helium opacity measurements in the redshiftrange 2 . . z . .
2, which serve as the foundation forour calculations, are compiled from the literature in § §
4, given theaverage measured opacity. Motivated by these results,we examine some fiducial reionization histories in §
5. Weconclude in § m =0 . , Ω Λ = 0 . , Ω b = 0 . , H = h (100 km s − Mpc − )(with h = 0.74), n = 0 .
95, and σ = 0 .
8, consistentwith the most recent measurements (Dunkley et al. 2009;Komatsu et al. 2009). METHOD
The helium Ly α forest observed in the spectra ofquasars originates from singly-ionized helium gas in theIGM. Quantitative measurement of this absorption istypically quoted as the transmitted flux ratio F , definedas the ratio of observed and intrinsic fluxes, or the relatedeffective optical depth τ eff ≡ − ln h F i . (1) We have F = e − τ eff = h e − τ i > e −h τ i , thus τ eff < h τ i .From the current opacity measurements, we wish to inferthe He II photoionization rate. This connection dependson the details of the IGM, including the temperature,density distribution, and ionized helium fraction. Fluctuating Gunn-Peterson approximation
The Gunn-Peterson (1965) optical depth for He II Ly α photons is τ GP = πe m e c f α λ α H − ( z ) n HeII . (2)Here, the oscillator strength f α = 0 . λ α = 304 ˚A, and n HeII is the density of singly-ionized helium in the IGM.For simplicity, we approximate the Hubble constant as H ( z ) ≈ H Ω / m (1 + z ) / . Since the Ly α forest probesthe low-density, ionized IGM, most of the hydrogen (massfraction X = 0 .
76) and helium ( Y = 0 .
24) are in the formof H II and He III , respectively, after reionization. Underthese assumptions, photoionization equilibrium requiresΓ n HeII = n He n e α A , (3)where Γ is the He II photoionization rateand the case-A recombination coefficient is α A = 3 . × − ( T / K) − . cm s − accord-ing to Storey & Hummer (1995). For a clumpy universe,most photons emitted by recombinations are producedand subsequently reabsorbed in dense, mostly neutralsystems. These ionizing photons, therefore, cannotescape to the low density regions of interest for theforest, so we use case-A (Miralda-Escud´e 2003).The Ly α forest, and therefore the optical depth, tracethe local overdensity ∆ of the IGM, where ∆ ≡ ρ/ ¯ ρ and ¯ ρ is the mean mass density. Since n e ∝ n He in ahighly ionized IGM, equation (3) implies that n HeII ∝ n ∝ ∆ . The optical depth is proportional to n HeII (seeeq. 2), which introduces a ∆ factor. Additionally, thetemperature of the IGM, which affects the recombinationrate, is typically described by a power law of the form T = T ∆ γ − (Hui & Gnedin 1997), where T and γ aretaken as constants. Including the above equations andcosmological factors, τ GP ≃ − (cid:18) T K (cid:19) − . (cid:18) Ω b h . (cid:19) (cid:18) Ω m h . (cid:19) − / × (cid:18) z (cid:19) / ∆ − . γ − , (4)where Γ = 10 − Γ − s − .The fluctuating Gunn-Peterson approximation(FGPA) (e.g., Weinberg et al. 1999), equation (4),relates the effective optical depth, or the continuumnormalized flux, to the local overdensity ∆ and thephotoionization rate Γ − . This approximation showsthe relationship between the opacity and the IGM, butit ignores the effects of peculiar velocities and thermalbroadening on the Ly α lines. In practice, when compar-ing to observations an overall proportionality constant, In actuality, during and after helium reionization both T and γ likely become redshift- and density-dependent. We will considersuch effects in § § κ , is introduced to the right hand side of equation (4)to compensate for these factors, as described in § T , Ω b , Ω m , and h . A semi-analytic model for Ly α absorption To fully describe helium reionization and computethe detailed features of the Ly α forest, complex hy-drodynamical simulations of the IGM, including radia-tive transfer effects and an inhomogeneous background,are required. Recent simulations (Sokasian et al. 2002;Paschos et al. 2007; McQuinn et al. 2009) have madegreat advances to incorporate the relevant physics and toincrease in scale. The simulations remain computation-ally intensive, and they cannot simultaneously resolvethe ∼
100 Mpc scales required to adequately study in-homogeneous helium reionization and the much smallerscales required to self-consistently study the Ly α forest,necessitating some sort of semi-analytic prescription todescribe baryonic matter on small scales. On the otherhand, the semi-analytic approach taken here, includingfluctuations in the ionizing background, should broadlyreproduce the observed optical depth, especially consid-ering the uncertainties in the measurements and the lim-ited availability of suitable quasar lines of sight. Tooutline, the model has four basic inputs: the IGM den-sity distribution p (∆), the temperature-density relation T (∆), the radiation background distribution f ( J ), andthe mean helium ionized fraction ¯ x HeIII .Miralda-Escud´e et al. (2000) suggest the volume-weighted density distribution function p (∆) = A ∆ − β exp " − (cid:0) ∆ − / − C (cid:1) δ / , (5)where δ = 7 . / (1 + z ) and β (for a few redshifts) aregiven in Table 1 of their paper. Intermediate β valueswere found using polynomial interpolation. The remain-ing constants, A and C , were calculated by normaliz-ing the total volume and mass to unity at each redshift.The distribution matches cosmological simulations rea-sonably well for the redshifts of interest, i.e. z = 2 − § T = T ∆ γ − in theFGPA. Generally, T ∼ − × K and 1 ≤ γ ≤ . T = 2 × K More recent simulations by Pawlik et al. (2009) andBolton & Becker (2009) basically agree with the above p (∆) forlow densities, which the Ly α forest primarily probes. and γ = 1 throughout our calculations. The isothermalassumption also suppresses the temperature, and there-fore density, dependence of the recombination coefficient α A .A uniform radiation background has been a com-mon assumption in previous studies, but the sources(quasars) for these photons are rare and bright. There-fore, random variations in the quasar distribution cre-ate substantial variations in the high-frequency radia-tion background (Meiksin 2007; Furlanetto & Oh 2008a;McQuinn et al. 2009). Furthermore, the 1 /r intensityprofiles of these sources induce strong small-scale fluctu-ations (Furlanetto 2009), which may in turn significantlyaffect the overall optical depth of the Ly α forest. For theprobability distribution f ( J ) of the angle-averaged spe-cific intensity of the radiation background J , we followthe model presented in Furlanetto (2008). In the post-reionization limit, the probability distribution can becomputed exactly for a given quasar luminosity functionand attenuation length, assuming that the sources arerandomly distributed (following Poisson statistics). Thisdistribution can be derived either via Markov’s method(Zuo 1992) or via the method of characteristic functions(Meiksin & White 2003).During reionization, the local He III bubble radius,i.e. the horizon within which ionizing sources are visible,varies across the IGM and so becomes another impor-tant parameter. Due to the rarity of sources, with typi-cally only a few visible per He
III region, a Monte Carlotreatment best serves this regime (Furlanetto 2008). Fora given bubble of a specified size, we randomly choosethe number of quasars inside each bubble according toa Poisson distribution. Each quasar is then randomlyassigned a location within the bubble as well as a lu-minosity (via the measured luminosity function). Next,we sum the specific intensity from each quasar. After10 Monte Carlo trials, this procedure provides f ( J ) fora given bubble size; the solution converges to the post-reionization scenario for an infinite bubble radius. Thefinal ingredient is the size distribution of discrete ion-ized bubbles, which is found using the excursion set ap-proach of Furlanetto & Oh (2008a) (based on the hydro-gen reionization equivalent from Furlanetto et al. 2004).After integrating over all possible bubble sizes, f ( J ) de-pends on the parameters z, R , and ¯ x HeIII , in additionto the specified luminosity function.In the following calculations, we scale J (which is eval-uated at a single frequency) to Γ (which integrates overall frequencies) simply using the ratio of the mean pho-toionization rate to the mean radiation background. Thisis not strictly correct, because higher frequency pho-tons have larger attenuation lengths and so more uniformbackgrounds; however, it is a reasonable prescription be-cause the ionization cross section falls rapidly with pho-ton frequency. However, it does mean that we ignore thelarge range in spectral indices of the ionizing sources (seebelow), which modulate the shape of the local ionizingbackground and lead to an additional source of fluctu-ations in Γ relative to J that we do not model. Thelargest problem occurs during helium reionization, whenthe highest energy photons can travel between He III re-gions (a process we ignore); however, they have small ion-ization cross sections and so do not significantly changeour results, except very near the end of that process(Furlanetto 2008).For the majority of the paper, we consider the IGMto be fully-ionized, i.e. ¯ x HeIII = 1 .
0. In this post-reionization regime, R , the attenuation length of theionizing photons, determines the shape of f (Γ), giventhe redshift and other model assumptions. We showseveral example distributions in the left panel Figure 1.From widest to narrowest, the curves correspond to R = 5 , , , and 100 Mpc ( z = 2 . R , making the peak photoionization ratemore likely, i.e. the curve is narrower. A uniform back-ground corresponds to R → ∞ . Although low Γ valuesare more likely for low R , the high-Γ tail is nearly inde-pendent of R . This is because a large Γ occurs withinthe “proximity zone” of a single quasar, making it rel-atively independent of contributions from much largerscales (unless R is much smaller than the proximity zoneitself). During reionization, as in §
5, the ionized fraction and the mean free path affect the distribution function,as shown in the right panel of Figure 1. Here, we take z = 2 . R = 35 Mpc, and ¯ x HeIII = 0 . , . , . , . , and1.0 (from lowest to highest at the peak). A broader distri-bution of photoionization rates is expected, because thelarge spread in He III bubble sizes restricts the sourcehorizon inhomogeneously across the Universe.To estimate τ eff (or F ), we integrate over all densitiesand photoionization rates: F = e − τ eff = Z ∞ d Γ f (Γ) Z ∞ d ∆ e − τ (∆ | Γ) p (∆) . (6)This integral is valid if Γ and ∆ are uncorrelated. Sincerelatively rare quasars ionize He II , random fluctua-tions in the number of sources (as opposed to theirspatial clustering) dominate the ionization morphology(McQuinn et al. 2009), justifying our assumption of anuncorrelated IGM density and the photoionization rate. The UV background from quasars
Since quasars ionize the He II in the IGM, the UVmetagalactic background can, in principle, be calcu-lated directly from distribution and intrinsic propertiesof quasars. Currently, these details, i.e. the quasar lu-minosity function and attenuation length, are uncertain.The following method for estimating Γ is used as a ref-erence for the semi-analytic model. The He II ionizationrate is Γ = 4 π Z ∞ ν HeII J ν hν σ ν dν, (7)where σ ν = 1 . × − ( ν/ν HeII ) − cm is the photoion-ization cross section for He II and ν HeII is the photon fre-quency needed to fully ionize helium. For the radiationbackground at frequency ν , J ν , we assume a simplifiedform, the absorption limited case in Meiksin & White(2003): J ν = 14 π ǫ ν ( z ) R ( z ) , (8)where ǫ ν is the quasar emissivity and R is the attenua-tion length. We begin with the B -band emissivity ǫ B , derived fromthe quasar luminosity function in Hopkins et al. (2007,hereafter HRH07). To convert this to the extreme-UV(EUV) frequencies of interest, we follow a broken power-law spectral energy distribution (Madau et al. 1999): L ( ν ) ∝ ν − . < λ < ν − . < λ < ν − α λ < . (9)The EUV spectral index α is a source of debate and isnot the same for all quasars. Telfer et al. (2002) find awide range of values for individual quasars, e.g. α = -0.56for HE 2347-4342 and 5.29 for TON 34. Most quasarslie closer to the mean, but HE 2347-4342 is a He II Ly α line of sight in §
3. Unless otherwise noted, we use themean value h α i ≈ . α . Zheng et al. (1997) found h α i ≈ . α affects the amplitude, not the shape, ofthe Γ curve derived from the QLF (see Fig. 2). Sinceour semi-analytic calculations are normalized to a singlepoint on the Γ curve, this uncertainty translates into anamplitude shift in our results, i.e. changes κ .The next ingredient for J ν is the attenuation length ofhelium-ionizing photons R . As described above, R de-pends on the photon frequency; for example, high-energyphotons can propagate larger distances. For simplicity,we use a single frequency-averaged attenuation lengthand focus only on the redshift evolution (in any case, theabsolute amplitude can be subsumed into our normal-ization factor κ below). For concreteness, we apply the comoving form found in Bolton et al. (2006): R = 30 (cid:18) z (cid:19) − Mpc , (10)which assumes the number of Lyman limit sys-tems per unit redshift is proportional to (1 + z ) . (Storrie-Lombardi et al. 1994) and uses the normaliza-tion based on the model of Miralda-Escud´e et al. (2000).An alternate approach, probably more appropriate dur-ing reionization itself, is to estimate the attenuationlength around individual quasars, as in Furlanetto & Oh(2008a). This method gives a similar value at z = 3but with a slower redshift evolution, which would onlystrengthen our conclusions. This empirical equation forthe mean free path (with the above quasar emissivity)provides the photoionization rate Γ, shown in Figure 2.The photoionization rate from the Madau et al. (1999)QLF (with α = 1 . § α forest data requires an uncertain correctionto the τ − ∆ relation in equation 4. For this purpose, weassume the above emissivity and attenuation length tobe accurate. Then, the semi-analytic model is adjustedso that the predicted τ eff matches the measured value ata particular redshift. To do so, we insert a prefactor, κ , to the right hand side of equation (4). This factorcompensates for line blending and other detailed physicsignored by the FGPA, but it also includes any uncertain- Fig. 1.—
Distribution of the photoionization rate Γ relative to its mean value in a fully-ionized IGM, h Γ i , at redshift z = 2 . Left panel:
The curves assume R = 5 , , , and 100 Mpc, from widest to narrowest, in a post-reionization universe. Right panel:
The curves take¯ x HeIII = 0 . , . , . , . , and 1.0, from lowest to highest at peak, for R = 35 Mpc. Fig. 2.—
Evolution of the mean photoionization rate (in units of10 − s − ) with redshift. The solid curve represents the inferredionization rate from the HRH07 quasar luminosity function with α = 1 .
6. The dotted (dashed) curve follows Madau et al. (1999)with extreme-UV spectral index α = 1 . . z = 2 . ties in the underlying cosmological or IGM parameters(such as T ). A suitable normalization redshift shouldbe after reionization and have data from more than oneline of sight (see the right panel of Fig. 3). Throughoutthis work, we take z = 2 .
45 as our fiducial point.If κ is a constant with redshift, our choice of refer-ence point mainly affects the overall amplitude of thephotoionization rate or attenuation length, not the red-shift evolution. However, since κ depends on the IGMproperties, it may change with redshift, density, and/ortemperature. For example, an increase in temperature broadens the widths of the absorption lines in the Ly α forest, decreasing the importance of saturation but in-creasing the likelihood of line blending. Reionization, adrastic change to the IGM, should also affect κ . Here,we take κ to be independent of z . The precise value liessomewhere between 0.1 and 0.5, depending on the spe-cific model. Furlanetto & Oh (2009) find κ = 0 . α effective optical depth,which is similar to our values. In any case, we emphasizethat our method cannot be used to estimate the absolutevalue of the ionizing background – for which detailed sim-ulations are necessary – but we hope that it can addressthe redshift evolution of Γ. EVOLUTION OF THE HE II EFFECTIVE OPTICALDEPTH
Measurements of the He II effective optical depth arechallenging. Suitable lines of sight require a brightquasar with sufficient far-UV flux and no interveningLyman-limit systems. Currently, only five quasar spectrahave provided He II opacity measurements appropriatefor our analysis, displayed in the left panel of Figure 3with the averaged values. Zheng et al. (2004a) measurea lower limit on the optical depth at z ∼ . §
5. Due to the limited scope of thedata, the observed opacities may not be representativeof the IGM as a whole.
HE 2347-4342:
This quasar ( z em = 2 . . < z < .
9, including Ly α and Ly β . Zheng et al.(2004b) and Shull et al. (2004) (2 . < z < . α only) utilized high-resolution spectra from the Far Ul-traviolet Spectroscopic Explorer (FUSE; R ∼ , R ∼ , . < z < . z = 2 . Fig. 3.—
Evolution of the He II effective optical depth based on the observations of the Ly α forest for five quasar spectra: HE 2347-4342,HS 1700+64, Q0302-003, HS 1157+314, and PKS 1935-692. The squares are the opacity measurements averaged over redshift bins ∆ z = 0 . Left panel:
Data and uncertainties as quoted in the literature are plotted.
Right panel:
The average valuesfor each line of sight are displayed, elucidating the origin of the uncertainties in the average opacities used throughout the paper. The smallredshift offsets within each bin are for illustrative purposes only. the effective helium optical depth evolves smoothly. Athigher redshifts, the opacity exhibits a patchy structurewith very low and very high absorption, often describedrespectively as voids and filaments in the literature.
HS 1700+64:
The Ly α forest of this quasar ( z em =2 .
72) has been resolved with FUSE over the redshiftrange 2 . . z . .
75 (Fechner et al. 2006). An olderstudy using the Hopkins Ultraviolet Telescope (HUT)is consistent with the newer, higher resolution results(Davidsen et al. 1996). The helium opacity evolvessmoothly and exhibits no indication of reionization.
Q0302-003:
The spectrum of this quasar ( z em = 3 . . . z . .
22, excluding a void at z ∼ .
05 due to a nearby ionizing source (Bajtlik et al.1988; Zheng & Davidsen 1995; Giroux et al. 1995). Thedata were averaged over redshift bins of ∆ z ≃ . II optical depth near z ∼ . HS 1157+314:
Reimers et al. (2005) obtained low res-olution HST/STIS spectra of the He II Ly α forest to-ward this quasar ( z em ∼ . ≤ z ≤ .
97) of the study a patchy structure, simi-lar to HE 2347-4342, is present. The given optical depthwas averaged over a redshift bin of ∆ z ≃ . PKS 1935-692:
The HST/STIS spectrum for thisquasar ( z em = 3 .
18) was analyzed by Anderson et al.(1999). Only one optical depth was quoted, but the spec-trum exhibited the usual fluctuations.To merge these data sets, we initially binned them inredshift intervals of 0.1, starting with z = 2 .
0. Each binwas assigned the median redshift value, e.g. z = 2 . . ≤ z < .
4. To objectively combine the data, the transmission flux ratios for each data set were aver-aged (weighted by redshift coverage) in the redshift bins.Then, the values for each quasar were averaged, sincemultiple data sets may cover the same line of sight. Fi-nally, if more than one line of sight contributes to a bin,the fluxes are averaged once again. The process trans-lates the left panel to the right panel of Figure 3. Theuncertainties along each line of sight in the right panel aresimply the errors from each separate point in the litera-ture, added in quadrature, without regard to systematicerrors; our averaged values then take errors equal to therange spanned by these separate lines of sight. For theremainder of the paper, the error bars result from assum-ing F ± dF for each redshift bin, where F are the squaresand ± dF are the upper/lower error bars on the squaresin the figure. The small number of well-studied lines ofsight limits the amount of truly quantitative statementsthat can be made. RESULTS
Mean He II photionization rate We now apply our semi-analytic model to the observedHe II opacity found in the right panel of Figure 3. Foreach redshift bin, the photoionization rate is calculatedby iteratively solving equation (6), i.e. varying h Γ i (themean photoionization rate for a fully-ionized IGM) until F matches the measurements. As noted in § f (Γ) for thefluctuating background, but the uniform case has no suchrequirement except insofar as it affects h Γ i .First, to provide some intuition, we fix κ to be the samefor both cases (here, κ = 0 . § τ eff is significantly smaller for a uniform back-ground, especially at higher redshifts (probably due to Fig. 4.—
The He II effective optical depth, assuming the HRH07QLF and eq. (10), as a function of redshift. The normalizationfor all curves is constant, κ = 0 . γ = 1 . . the shrinking attenuation length assumed in the fluctuat-ing model). This is because most points in the IGM haveΓ < h Γ i for the fluctuating background, so most of theIGM has a higher opacity; the “proximity zones” aroundeach quasar are not sufficient to compensate for this ef-fect. The measured opacities are included in the figurefor reference, showing that the shapes of both (normal-ized) models are consistent with observations for z < . τ eff is the measured quantity, from whichwe try to infer Γ. We find that, for the fiducial at-tenuation lengths, including a realistic fluctuating back-ground increases the required h Γ i by about a factor oftwo – a nontrivial effect that is important for reconcil-ing quasar observations with the forest. The magnitudeof the required adjustment is comparable to that foundby Bolton et al. (2006), who used numerical simulations.However, as we have emphasized above, we cannot useour model to estimate the absolute value of the ionizingbackground because the FGPA does not fully describethe Ly α forest; instead we need a renormalization factor κ . For the remainder of this paper, we therefore fix thephotoionization rate at the z = 2 .
45 HRH07 value (to-gether with our fiducial R ), as described in § κ = 0 .
457 and 0.291 for the uniform andfluctuating background, respectively. We strongly cau-tion the reader that the remainder of our quoted resultswill therefore mask the overall amplitude disparity be-tween these two cases, and we will focus on the redshiftevolution of Γ instead.Figure 2 displays the mean photoionization rate for auniform (triangles) and fluctuating (squares) UV back-ground (normalized to the fiducial point). The curvesrepresent the photoionization rate inferred from vari-ous quasar luminosity functions, as described in § z = 2 .
45 reference point was chosen arbitrarily,the overall amplitude should not be considered reliable, which is emphasized by the spread in the QLF curves.The difference between the fluctuating and uniform UVbackground is small and certainly within the uncertain-ties. The effect of including fluctuations on the photoion-ization rate is not straightforward. Generally, at lower z ,the fluctuating Γ is smaller than the uniform result, andthe opposite is true for higher redshift. The redshift ofthe crossover between the two behaviors depends on theamplitude of the measured transmission ratio; a higher F decreases this crossover redshift.We therefore attribute this effect to changing the char-acteristic overdensity of regions with high transmission,which is larger at lower redshifts because of the Uni-verse’s expansion. In this regime, where the underlyingdensity field itself has a relatively broad distribution, thefluctuating ionizing field makes less of a difference to therequired h Γ i . Remember, however, that these relativelysmall changes are always swamped by the differing κ ’s,and a fluctuating background always requires a larger h Γ i than the uniform case.For z < .
7, the normalized points lie near the HRH07curve. The averaged opacity at z = 2 .
25, which is signif-icantly lower, relies on a single line of sight, HE 2347-4342, and differs significantly from the trend seen inFigure 3. As expected, the inferred Γ fluctuates con-siderably over this redshift range. In part, these vari-ations are due to the limited amount of data, both inthe number and redshift coverage of usable quasar sight-lines. But the UV background fluctuates considerably,especially during reionization (see the f (Γ) discussion in § z > .
8, the calculated photoionization rateconsistently undershoots the model prediction, possiblyindicating the end of helium reionization around thattime: there is much more He II than can be accommo-dated by a smoothly varying emissivity or attenuationlength. Evolution of the attenuation length
Because the measured quasar emissivity evolvessmoothly with redshift, the most natural interpretationof this discontinuity is in terms of the attenuation length,which intuitively evolves rapidly at the end of reioniza-tion when He
III regions merge together and sharplyincrease the horizon to which ionizing sources are visi-ble. Following the prescription for the UV backgroundin § R , given theHRH07 QLF and the Γ − from the previous section.This procedure amounts to varying the solid curve, via R , to match the points in Figure 2. The redshift evo-lution of the attenuation length for uniform (triangles)and fluctuating (squares) radiation backgrounds is plot-ted in Figure 5, with equation (10) as a reference. Thenormalization remains the same as the previous section,i.e. z = 2 .
45 is the fiducial point.Similarly to the inferred photoionization rate, thepoints vary about the reference curve for z < . z > .
8. The uncertainties, whichare shown only for the uniform UV background (but arecomparable for the other case), are again quite large. In-corporating fluctuations reduces the severity of, but doesnot eliminate, the jump in the evolution of the attenua-tion length. Once again the results lie consistently belowthe curve for higher redshifts. From the viewpoint of Γor R , there appears to be a systematic change in be- Fig. 5.—
Evolution of the helium-ionizing attenuation lengthwith redshift. The calculated mean free path generally increaseswith a discontinuity around z = 2 .
8. The fluctuating (squares)background appears to smooth the evolution as compared to theuniform (triangles) background. The results are matched to theBolton et al. (2006) attenuation length (solid curve) at z = 2 . havior above z ≈ .
8. The marked decrease in R thatis required, by at least a factor of two from the fidu-cial model, indicates an important change in the state ofthe IGM. However, as we have described above, a singleattenuation length is no longer appropriate during reion-ization, so in § The IGM Temperature-Density Relation
As discussed in § γ = 1in T = T ∆ γ − . In reality, the temperature may dependon the density of the IGM. A further complication arisesduring (and shortly after) helium reionization when theIGM is inhomogeneously reheated and subsequently re-laxes to a power law (Gleser et al. 2005; Furlanetto & Oh2008b; McQuinn et al. 2009).To partially address the former issue, we repeat the cal-culation of Γ for a homogeneous radiation background,but now with γ = 1 .
6, shown in Figure 6. The differencebetween the two cases is . − near redshift z ≈ . κ γ =1 . = 0 .
457 and κ γ =1 . = 0 .
277 (see Fig. 4). Inother words, a model with a higher γ requires a higher h Γ i to achieve the same optical depth. This is becausea steeper temperature-density relation makes the low-density IGM, which dominates the transmission, colderand hence more neutral. Overall, then, the temperature-density relation and fluctuating ionizing background leadto a systematic uncertainty of nearly a factor of four inthe mean photoionization rate inferred from the He II forest. Fig. 6.—
Comparison of the inferred photoionization rate (inunits of 10 − s − ) for two temperature-density relations, givena homogeneous radiation background. The triangular points arederived using an isothermal model. The square points assume asteeper temperature-density relation, γ = 1 .
6. The points have asmall redshift offset for illustrative purposes. The photoionizationrate computed from the HRH07 quasar luminosity function is plot-ted for reference. Both models are normalized to the observationsat z = 2 . MODELS FOR HE II REIONIZATION
We have seen that the observations appear to requirea genuine discontinuity in the properties of the IGM at z ≈ .
8, although large statistical errors stemming fromthe small number of lines of sight prevent any strong con-clusions. This change is often attributed to reionization;here we investigate this claim quantitatively with several“toy” models for the evolution of the helium ionized frac-tion ¯ x HeIII . This fraction determines f (Γ) as describedin § we calculate the effectiveoptical depth via equation (6). The fiducial point fornormalizing κ remains at z = 2 . τ eff and ion-ized fraction ¯ x HeIII as a function of redshift for five reion-ization models. Each scenario is characterized by theredshift, z He , at which ¯ x HeIII reaches 1.0. From leftto right in the figure, the curves correspond to z He =2 . , . , . , . z He > .
8, i.e. post-reionization forthe entire redshift range in question. The rate of ioniza-tion varies slightly between the models. The measuredopacities are plotted for reference, including the recentlydiscovered SDSS J2346-0016 at z = 3 .
45 (Zheng et al.2004a, 2008). Note that we do not estimate any cosmicvariance uncertainty.The effective optical depth evolves smoothly in thepost-reionization regime, which seems compatible with Again, we note that during helium reionization the atten-uation length should be evaluated with reference to individualquasars; in that case, it does not evolve strongly with redshift(Furlanetto & Oh 2008a). This consideration will only strengthenour conclusions. The unevenness in the curves arises because generatingour Monte Carlo distributions is relatively expensive computa-tionally, so we only generated a limited number at ¯ x HeIII =(0 . , . , . , . , and 1.0) for each redshift. Fig. 7.—
The effective helium optical depth and He
III frac-tion for five toy reionization models. From left to right, the curvescorrespond to helium fully-ionized by z He = 2 . , . , . , . z He > . III fraction evolution isvaried slightly between models. The measured opacities are in-cluded for reference, including the lower limit at z = 3 .
45 fromSDSS J2346-0016 (Zheng et al. 2004a). the data below z ≈ .
8. The z He = 2 . z He > z & DISCUSSION
We have applied a semi-analyic model to the inter-pretation of the He II Ly α forest, one of the few directobservational probes of the epoch of helium reionization.Using simple assumptions about the IGM, the ionizationbackground, and our empirical knowledge of quasars, wehave inferred the evolution of the helium phoionizationrate and the attenuation length from the He II effectiveoptical depth. We averaged the opacity measurementsover 5 sightlines, which show an overall decrease in τ eff with decreasing redshift and a sharp jump at z ≈ . h Γ i above z ≈ .
8. Although the uncertainties are large, these re-sults suggest a rapid change in the IGM around thattime.Our semi-analytic model is based the quasar luminos-ity function and the helium-ionizing photon attenuationlength, which are determined empirically. The uncer-tainty in these quantities significantly affects our results.In particular, the plausible range of the mean EUV spec-tral index, 1 . . α . .
8, shifts the amplitude of ourresults by a factor of about two. Our model takes onlythe mean value and does not account for the variationin α from different quasars. Furthermore, our treatmentof the attenuation length ignores any frequency depen- dence. However, these factors likely only affect the am-plitude, not the evolution, of the inferred photoionizationrate, especially the the jump at z ≈ .
8. A steeper red-shift evolution for the attenuation length would decreasethe severity of the feature around z ≈ .
8, but a simplepower law cannot eliminate it, given our method.In calculating Γ and R , we compared uniform andfluctuating backgrounds. Although helium reionizationis thought to be inhomogeneous (see Furlanetto & Oh2008a), the assumption of a uniform background hasbeen common. We found that the uniform case pro-duces an effective optical depth approximately a factorof two smaller for a fixed h Γ i . Thus, properly incorporat-ing the fluctuating background is crucial for interpretingthe He II forest in terms of the ionizing sources. Further-more, we find that the inclusion of background variationsslightly smoothes, but does not remove, the jump in theattenuation length at z ≈ .
8. A clear change in the IGMdoes appear to occur around this redshift.The discontinuous behavior in Γ and R led us to in-clude helium reionization in our model through the dis-tribution f (Γ). During reionization, the ionized heliumfraction determines this distribution, and we studied sev-eral toy models for the redshift evolution of ¯ x HeIII . Thesemodels suggest z He ≈ . τ eff are large. We donot account for cosmic variance and only consider themean effective optical depth. Our method also makesassumptions that are not valid during reionization, e.g.a power-law temperature-density relation, but this doesnot appear to affect the discontinuity significantly.In fact, the most important caveat to our model isthe use of the fluctuating Gunn-Peterson approximation,which is a simplified treatment of the Ly α absorption.The approach ignores the wings of absorption lines, pecu-liar velocities, and line blending; overall, these effects re-quire us to add an unknown renormalization factor (of or-der ∼ . .
5) when translating from Γ to optical depthand compromises attempts to measure the absolute valueof h Γ i . One danger is the possible redshift evolution ofthis factor: we have assumed that it does not evolve, butin reality the line structure and temperature of the for-est do evolve, especially at the end of reionization. Moredetailed numerical simulations that incorporate both thebaryonic physics of the Ly α forest and the large-scaleinhomogeneities of helium reionization are required toexplore this fully.Interestingly, if our interpretation is correct then it ap-pears that helium reionization completes at z He . after theepoch suggested by indirect probes of the H I Ly α for-est. Specifically, some measurements of the temperatureevolution of the forest show a sharp jump at z ≈ . z ≈ . I Ly α forest opacity in terms of helium reionization (see alsoBolton et al. 2009). Another indirect constraint is con-0sistent, however, with our picture: reconstruction of theionizing background from optically thin metal systemsfinds an effective optical depth in He II Ly α photonsslightly higher than the direct measurements, but with asimilar redshift evolution (Agafonova et al. 2005, 2007).The most significant limitation in the data is the rela-tively small number of lines of sight, producing large vari-ations in the measured transmission, especially at z & HubbleSpace Telescope adds a new instrument to our arsenal.Although the nominal wavelength range of COS limits itto z & .
8, this is precisely the most interesting range forstudying reionization. Our models show that τ eff . x HeIII & . III and H I reionization: the near-uniformity of the ionizing background at the end of H I reionization means that very little residual transmissioncan be expected at z & α forest relatively useless for studying reionization.In contrast, the large variance intrinsic to the He II -ionizing background produces much stronger fluctuationsand makes the epoch of reionization itself accessible withthe He II Ly α forest.Another interesting difference between helium and hy-drogen is the effect of including fluctuations on the pho-toionization rate inferred from the Ly α forest. We find that assuming a uniform ionizing background underesti-mates Γ by up to a factor of two, while during hydrogenreionization the effect is much smaller – only a few per-cent (Bolton & Haehnelt 2007; Mesinger & Furlanetto2009). During and after helium reionization, the fluc-tuations are much more pronounced than the hydrogenequivalent, leading to a much broader f ( J ), so that moreof the Universe lies significantly below the mean. For hy-drogen reionization, the distributions are much narrower,favoring Γ near the mean. In addition, after hydrogenreionization the density distribution is much wider than f (Γ), so that the latter provides only a small perturba-tion; the opposite is true in our case.Our general approach is very similar to Fan et al.(2002) and Fan et al. (2006), who also interpreted theH I Ly α forest data at z ∼ z ∼ . I reionization. But during thisearlier epoch, that inference is less clear because of thenear saturation of the forest and the unknown attenua-tion length (whose evolution really determines the overallionizing background, but which may evolve rapidly evenafter reionization; Furlanetto & Mesinger 2009). 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