THE HE II PROXIMITY EFFECT AND THE LIFETIME OF QUASARS

To study the impact of quasar radiation on the intergalactic medium we post-process the outputs of a smooth particle hydrodynamics (SPH) simulation using a one-dimensional radiative transfer algorithm. For the cosmological hydrodynamical simulation, we use the Gadget-3 code (Springel 2005) with 2 × 512³ particles and a box size of 25h⁻¹ cMpc. We extract 1000 sightlines, which we will refer to as skewers, from the SPH simulation output at a snapshot corresponding to z = 3.1. In Section 5.2 we also consider skewers at a higher redshift of z = 3.9. We identify quasars in the simulation as dark matter halos with masses $M\gt 5\times {10}^{11}{M}_{\odot }$, by running a friends-of-friends algorithm (Davis et al. 1985) on the dark matter particle distribution. While this mass threshold is 1 dex lower than the halo mass inferred from quasar clustering measurements (White et al. 2012; Conroy & White 2013), our simulation cube does not capture enough such massive halos. However, because the He ii proximity effect extends many correlation lengths (${r}_{0}\approx 10{h}^{-1}$ Mpc, White et al. 2012), the use of the less clustered halos will not make a difference at most radii we consider.

Starting from the location of the quasars, we create skewers by casting rays through the simulation volume at random angles, and traversing the box multiple times. We use the periodic boundary conditions to wrap a skewer through the box along the chosen direction. This procedure results in skewers from the quasar location through the IGM that are 318 cMpc long, significantly larger than the length of our simulation box and the He ii proximity region. Our skewers have 27,150 pixels which corresponds to a spatial interval of dR = 0.012 cMpc or a velocity interval dv = 0.893 km s⁻¹. This is sufficient to resolve all the features in the He ii Lyα forest.

We extract the one-dimensional density, velocity, and temperature distributions along these skewers and use them as input to our post-processing radiative transfer algorithm⁶ , which is based on C²-Ray algorithm (Mellema et al. 2006). Because this algorithm is explicitly photon conserving, it enables great freedom in the size of the grid cells and the time step. Our one-dimensional radiative transfer algorithm is extremely fast, with one skewer calculation taking ∼5 s on a desktop machine, allowing us to run many realizations. In this section, we describe the salient features of our radiative transfer algorithm, and we refer the reader to the original Mellema et al. (2006) paper on the C²Ray code for additional details.

We put a single source of radiation, the quasar, at the beginning of each sightline, and trace the change in the ionization state and temperature of the IGM for a finite time, t, as the radiation is propagated.⁷ The spectral energy distribution (SED) of the source is modeled as a power-law, such that the quasar photon production rate at any frequency $\nu \geqslant {\nu }_{\mathrm{th}}$ is

$$\begin{eqnarray}&&{N}_{\nu }=\displaystyle \frac{\alpha {Q}_{4\mathrm{Ry}}}{{\nu }_{{th}}}{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{th}}}\right)}^{-\alpha -1},\end{eqnarray} \tag{ 1 }$$

where Q_4Ry is the photon production rate above the He ii ionization threshold of 4 Ry or a corresponding threshold frequency ${\nu }_{\mathrm{th}}=1.316\times {10}^{16}\ {\rm{Hz}}$ corresponding to this threshold. We assume a spectral index of α = 1.5, ${f}_{\nu }\sim {\nu }^{-\alpha }$, consistent with the measured values in stacked UV spectra of quasars (Telfer et al. 2002; Shull et al. 2012; Lusso et al. 2015).

The r-band magnitudes of He ii quasars in the HST/COS archive range from r = 16.0–19.4, with a median value of r ≃ 18.25. Following the procedure described in Hennawi et al. (2006), we model the quasar SED using a composite quasar spectrum which has been corrected for IGM absorption (Lusso et al. 2015). By integrating the Lusso et al. (2015) composite redshifted to z = 3.1 against the SDSS filter, we can relate the r-band magnitude of a quasar to the photon production rate at the H i Lyman limit Q_1Ry, which is then related to Q_4Ry by ${Q}_{4\mathrm{Ry}}={Q}_{1\mathrm{Ry}}\times {4}^{-\alpha }$, assuming the same power law SED governs the quasar spectrum for frequencies blueward of 1 Ry. This procedure implies that quasars in the range r = 16.0–19.4 have Q_4Ry = 10^56.0–57.4 s⁻¹, with median r ≃ 18.25 and Q_4Ry =10^56.5 s⁻¹.

The quasar photoinization rate in every cell is given by

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{QSO}}={\int }_{{\nu }_{\mathrm{th}}}^{\infty }\displaystyle \frac{{N}_{\nu }{e}^{-\langle {\tau }_{\nu }\rangle }}{{h}_{{\rm{P}}}\nu }\displaystyle \frac{1-{e}^{-\langle \delta {\tau }_{\nu }\rangle }}{\langle {n}_{\mathrm{He}{\rm{II}}}\rangle {V}_{{\rm{cell}}}}d\nu ,\end{eqnarray} \tag{ 2 }$$

where $\langle {\tau }_{\nu }\rangle$ is the optical depth along the skewer from the source to the current cell, $\langle \delta {\tau }_{\nu }\rangle$ is the optical depth inside the cell, $\langle {n}_{\mathrm{He}{\rm{II}}}\rangle$ is the average number density of He ii in this cell, V_cell is the volume of the cell, and h_p is Planck's constant. Here the angular brackets indicate time averages over the discrete time step δt.

Combining with Equation (1), we can then rewrite Equation (2) as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{QSO}}=\displaystyle \frac{\alpha {Q}_{4\mathrm{Ry}}}{{n}_{\mathrm{He}{\rm{II}}}{V}_{\mathrm{cell}}{\nu }_{\mathrm{th}}}{\int }_{{\nu }_{\mathrm{th}}}^{\infty }{\left(\displaystyle \frac{\nu }{{\nu }_{\mathrm{th}}}\right)}^{-(\alpha +1)}{e}^{-\langle {\tau }_{\nu }\rangle }(1-{e}^{-\langle \delta {\tau }_{\nu }\rangle })d\nu .\end{eqnarray} \tag{ 3 }$$

Although this equation for the photoionization rate is an integral over frequency, in practice the code is not tracking multiple frequencies. For the power-law quasar SED that we have assumed, and He ii ionizing photon absorption cross section ${\sigma }_{\nu }\approx {\sigma }_{\mathrm{th}}{(\nu /{\nu }_{\mathrm{th}})}^{-3}$, Equation (3) has an analytic solution that depends on $\langle {\tau }_{\mathrm{th}}\rangle$ and $\langle \delta {\tau }_{{\rm{th}}}\rangle$, the time-averaged optical depth to the cell and inside the cell, respectively, evaluated at the He ii ionization threshold. Thus, given the values for these optical depths evaluated at the single edge frequency, we can compute the frequency-integrated photoionization rate.

The foregoing has considered the case of a single source of radiation, namely the quasar. This scenario is likely appropriate early on in He ii reionization, when each quasar photoionizes its own He iii zone. However, at later times, the IGM is filled with He ii ionizing photons emitted by many sources. In order to properly model the radiative transfer along our skewer, we need to account for additional ionizations caused by this intergalactic ionizing background. We approximate this background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ as being a constant in space and time, and simply add it into each pixel of our sightline to model the presence of these other sources

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{\mathrm{tot}}={{\rm{\Gamma }}}_{\mathrm{QSO}}+{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}.\end{eqnarray} \tag{ 4 }$$

As such, this procedure does the full one-dimensional radiative transfer to compute the photoionization rate Γ_QSO produced by the quasar, but it would treat the background as an unattenuated and homogeneous radiation field present at every location. However, the densest regions of the IGM will be capable of self-shielding against ionizing radiation, thus giving rise to He ii Lyman-limit systems (He ii-LLSs). Within these systems our 1D radiative transfer algorithm always properly attenuates the quasar photoionization rate Γ_QSO, however, if we take the the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ to be constant in every pixel along the skewers, this would be effectively treating the He ii-LLSs as optically thin, thus underestimating the He ii fraction in the densest regions. In order to more accurately model the attenuation of the He ii background by the He ii-LLSs, we implement an algorithm described in McQuinn & Switzer (2010). In Appendix B we summarize this approach and show that the effect of self-shielding of He ii-LLSs to the He ii background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ has a negligible effect on the structure of the He ii proximity zones. We nevertheless treat the He ii LLSs with this technique for all of the results described in this paper. In Section 5.2 we also discuss the impact of spatial fluctuations in the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ on our results (showing that the average transmission profile is largely unaffected by such fluctuations).

Having arrived at an expression for the photoionization rate in each cell, the next step is to determine the evolution of the He ii fraction x_{He ii} in response to this time-dependent ${{\rm{\Gamma }}}_{\mathrm{tot}}(t)$. The time evolution of the He ii fraction is given by

$$\begin{eqnarray}&&\displaystyle \frac{{{dx}}_{\mathrm{He}{\rm{II}}}}{{dt}}=-{{\rm{\Gamma }}}_{\mathrm{tot}}(t)\;{x}_{\mathrm{He}{\rm{II}}}+{\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}(1-{x}_{\mathrm{He}{\rm{II}}}),\end{eqnarray} \tag{ 5 }$$

which if n_e is independent of x_{He ii} (which holds at the 6% level) has the solution of

$$\begin{eqnarray}&&{x}_{\mathrm{He}{\rm{II}}}(t)={x}_{\mathrm{He}{\rm{II}},0}{e}^{-{\int }_{0}^{t}{dt}^{\prime} {t}_{\mathrm{eq}}{(t^{\prime} )}^{-1}}+{\int }_{0}^{t}{dt}^{\prime} {\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}{e}^{-{\int }_{0}^{t^{\prime} }{dt}^{\prime\prime} {t}_{\mathrm{eq}}{(t^{\prime\prime} )}^{-1}}.\end{eqnarray} \tag{ 6 }$$

where ${x}_{\mathrm{He}{\rm{II}},0}$ is the initial singly ionized fraction and t_eq is the equilibration timescale given by

$$\begin{eqnarray}&&{t}_{\mathrm{eq}}={({{\rm{\Gamma }}}_{\mathrm{tot}}+{n}_{{\rm{e}}}{\alpha }_{{\rm{A}}})}^{-1},\end{eqnarray} \tag{ 7 }$$

n_e is the electron density, and α_A is the Case A recombination coefficient. In the limit when Γ_tot(t) does not change over t (and also the same for α_A n_e which is less of an approximation) Equation (6) simplifies considerably to

$$\begin{eqnarray}&&{x}_{\mathrm{He}{\rm{II}}}(t)={x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}+({x}_{\mathrm{He}{\rm{II}},0}-{x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}){e}^{-t/{t}_{\mathrm{eq}}},\end{eqnarray} \tag{ 8 }$$

where we have defined the equilibrium He ii fraction as

$$\begin{eqnarray}&&{x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}={\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}{t}_{\mathrm{eq}}.\end{eqnarray} \tag{ 9 }$$

We discuss our treatment of He ii ionizing photons produced by recombinations and the justification for adopting Case A in Section 2.5 below. We have ignored collisional ionizations which contribute negligibly in proximity zones, where gas is relatively cool $(T\sim {10}^{4}{\rm{K}})$, and are even more highly suppressed for He ii relative to H i. Given that the He ii fraction starts at an initial value x_{He ii,0}, Equation (8) gives the He ii fraction at a later time t, provided that Γ_tot, n_e, and α_A are constant over this interval. As such, this equation is only exact for infinitesimal time intervals.

Following Mellema et al. (2006), we can compute the time averaged He ii fraction inside a cell by averaging Equation (8) over the discrete time-step δt, yielding

$$\begin{eqnarray}&&\langle {x}_{\mathrm{He}{\rm{II}}}\rangle ={x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}+({x}_{\mathrm{He}{\rm{II}},t}-{x}_{\mathrm{He}{\rm{II}},\mathrm{eq}})\left(1-{e}^{-\tfrac{\delta t}{{t}_{\mathrm{eq}}}}\right)\displaystyle \frac{{t}_{\mathrm{eq}}}{\delta t},\end{eqnarray} \tag{ 10 }$$

where x_{He ii, t} is the He ii fraction at the previous timestep t. We then use Equation (10) to calculate the time-averaged optical depth in the cell at the He ii edge

$$\begin{eqnarray}&&\langle \delta {\tau }_{\mathrm{th}}\rangle =\langle {x}_{\mathrm{He}{\rm{II}}}\rangle {n}_{\mathrm{He}}{\sigma }_{\mathrm{th}}{\rm{\Delta }}r,\end{eqnarray} \tag{ 11 }$$

where Δr is the size of the cell. Likewise, the time-averaged optical depth to the cell $\langle {\tau }_{\mathrm{th}}\rangle$ is computed by adding, in causal order, all the $\langle \delta {\tau }_{\mathrm{th}}\rangle$ of the cells lying between the source and the cell under consideration. The electron density $\langle {n}_{{\rm{e}}}\rangle$ and number density of He ii atoms $\langle {n}_{\mathrm{He}{\rm{II}}}\rangle$ are similarly computed using Equation (10).

The iterative process that we employ to find the new ionization state in each cell (see the flow-chart in Figure 2 of Mellema et al. 2006) can be described as follows. Starting with the cell nearest the source and moving outward, we:

• 1.  
  Set the mean He ii fraction ${\langle x}_{\mathrm{He}{\rm{II}}}\rangle$ to that given by the previous time-step (or the initial conditions).
• 2.  
  Evaluate the optical depth between the source and the cell $\langle {\tau }_{\mathrm{th}}\rangle$ by summing the δ τ_th from the previous time-step.
• 3.  
  Iterate the following until convergence of the He ii fraction $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle$ is achieved

  • (i)  
    compute the time-averaged optical depth $\langle \delta {\tau }_{\mathrm{th}}\rangle$ within the cell (Equation (11));
  • (ii)  
    compute the photoionization rate Γ_tot (Equations (3) and (4));
  • (iii)  
    compute the mean electron number density based on the current mean ionization state;
  • (iv)  
    calculate the new He ii fraction ${x}_{\mathrm{He}{\rm{II}}}(t)$ at this time step (Equation (8)), as well as its time-step averaged value $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle$ (Equation (10))
  • (v)  
    check for convergence
• 4.  
  Once convergence is reached, advance to the next cell.

Following this procedure for every time-step, we thus integrate the time-evolution over time interval t_Q, which denotes the quasar lifetime, yielding the He ii fraction x_{He ii} at each location in space in the proximity zone.

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We assume that the excess from photoionization of helium heats the surrounding gas. In reality, instead of heating, the electron produced during the photoionization, can become a source of secondary (collisional) ionization, but they are unimportant for the highly ionized IGM, which will clearly be the case for He ii proximity zones. The amount of heat injected per time interval is given by (Abel & Haehnelt 1999)

$$\begin{eqnarray}&&\displaystyle \frac{{dT}}{{dt}}\approx \displaystyle \frac{2{Y}_{\mathrm{He}}}{3{k}_{{\rm{B}}}(8-5{Y}_{\mathrm{He}})}\langle E\rangle \displaystyle \frac{{{dx}}_{\mathrm{He}{\rm{II}}}}{{dt}}\end{eqnarray} \tag{ 12 }$$

where ${{dx}}_{\mathrm{He}{\rm{II}}}/{dt}$ is the change in the He ii fraction over the time step and $\langle E\rangle$ is the average excess energy of a photon above He ii ionization threshold, which is given by

$$\begin{eqnarray}&&\langle E\rangle =\left[\displaystyle \frac{{\int }_{{\nu }_{\mathrm{th}}}^{\infty }{N}_{\nu }{\sigma }_{\nu }{e}^{-\langle {\tau }_{\nu }\rangle }(1-{e}^{-\langle \delta {\tau }_{\nu }\rangle })({h}_{{\rm{P}}}\nu -{h}_{{\rm{P}}}{\nu }_{\mathrm{th}})d\nu }{{\int }_{{\nu }_{\mathrm{th}}}^{\infty }{N}_{\nu }{\sigma }_{\nu }{e}^{-\langle {\tau }_{\nu }\rangle }(1-{e}^{-\langle \delta {\tau }_{\nu }\rangle })d\nu }\right]\end{eqnarray} \tag{ 13 }$$

For a power law SED, this frequency integral can be solved analytically, and yields a function that depends on the optical depth between the source and the current cell $\langle {\tau }_{\nu }\rangle$ and optical depth of the cell $\langle \delta {\tau }_{\nu }\rangle$. This function is evaluated at each time step and the resulting change in the temperature added to each cell. We ignore the heating caused by the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, since this is accounted for via the photoionization heating in the hydrodynamical simulations (and it does not depend on the normalization of the photoionizing background, which we freely adjust). We also do not include cooling in our simulations because the gas cooling time at mean density is comparable to the Hubble time which is much longer than the expected lifetime of the quasar (i.e., ${t}_{\mathrm{cool}}\gg {t}_{{\rm{Q}}}$) and thus can be safely ignored.

Our code works in the infinite speed of light limit, similar to other one-dimensional radiative transfer codes which attempt to model observations along the line of sight (White et al. 2003; Shapiro et al. 2006; Bolton & Haehnelt 2007a; Lidz et al. 2007; Davies et al. 2014). Thus, the results do not depend on the speed of light, which may seem problematic because the light travel times across the proximity region can be tens of Myr. However, in Appendix A, we show that because absorption observations also occur along the light-cone, the infinite speed of light assumption exactly describes the ionization state of the gas probed by line of sight observations.

We use the Case A recombination coefficient throughout the paper. Quasar proximity zones represent highly ionized media which typically have ${x}_{\mathrm{He}{\rm{II}}}\lesssim {10}^{-2}$, making most cells optically thin to ionizing photons produced by recombinations directly to the ground state. This recombination radiation is an additional source of ionizations, but we do not include it in our computations. We show in Appendix C that neglecting secondary ionizations from recombination radiation is justified as it is negligible in comparison to the quasar radiation.

We calculate He ii spectra along each of the sightlines taken from the SPH simulations following the procedure described in Theuns et al. (1998), which we outline in Appendix D. Figure 2 shows various physical properties along an example line of sight. The quasar is located on the right side of the plot at R = 0. These physical properties are drawn from a model that has the quasar lifetime set to t_Q = 10 Myr, the photon production rate ${Q}_{4\mathrm{Ry}}={10}^{56.1}\;{{\rm{s}}}^{-1}$, and the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, which is our preferred value following the discussion in Section 3.1.

The uppermost panel of Figure 2 shows the gas density along the skewer in units of the cosmic mean density, illustrating the level of density fluctuations present in the z = 3.1 IGM. The second panel shows the H i transmitted flux, which exhibits the familiar absorption signatures characteristic of the Lyα forest. A weak H i proximity effect (Carswell et al. 1982; Bajtlik et al. 1988) is noticeable by eye near the quasar for R ≲ 20 cMpc. This weak H i proximity effect is expected: given the high value of the H i ionizing background, the region where the quasar radiation dominates over the background is relatively small. Furthermore, the low H i Lyα optical depth at z = 3.1 reduces the contrast between the proximity zone and regions far from the quasar. On the contrary, the He ii transmission clearly indicates the large and prominent (≃50 cMpc) He ii proximity zone around the quasar. At larger distances R > 50 cMpc the transmission drops to nearly zero, giving rise to long troughs of Gunn-Peterson (GP; Gunn & Peterson 1965) absorption, as is commonly observed in the He ii transmission spectra of quasars observed with HST (Worseck et al. 2011; Syphers & Shull 2014). This GP absorption results from the large He ii optical depth in the ambient IGM, which is in turn set by our choice of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$. The He ii transmission follows the radial trend set by the He ii fraction x_{He ii}. As expected, close to the quasar R ≲ 20 cMpc, helium is highly ionized (x_{He ii} < 10⁻³) by the intense quasar radiation. At larger radii the quasar photoionization rate weakens, dropping approximately as R⁻², as indicated in the bottom panel. Eventually, at large distances R ≳ 70 cMpc the quasar radiation no longer dominates over the the background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, and x_{He ii} gradually asymptotes to a value ${x}_{\mathrm{He}{\rm{II}},0}\simeq {10}^{-2}$, set by the chosen He ii ionizing background.

In order to model the He ii proximity regions, we need to make some assumptions about the background radiation field in the IGM. At z ≳ 2.8 this background may not be spatially uniform, as it has been argued that He ii reionization is occurring (McQuinn 2009; Shull et al. 2010). He ii reionization is thought to be driven by quasars turning on and emitting the hard photons required to doubly ionize helium (Madau & Meiksin 1994; Miralda-Escudé et al. 2000; McQuinn et al. 2009; Haardt & Madau 2012; Compostella et al. 2013). At z = 3.1, redshifts characteristic of much of the COS data, the IGM is still likely to consist of mostly reionized He iii regions, but there may be some quasars that turn on in He ii regions. The latter case should become increasingly more likely with increasing redshift. In this paper we model the full range of possibilities, but let us first get a sense for what currently available data implies about typical regions of the IGM.

Recent measurements of the He ii effective optical depth τ_eff from Worseck et al. (2014) & Worseck et al. (2016, in preparation), albeit with large scatter, imply $\langle {\tau }_{\mathrm{eff}}^{\mathrm{He}\;{\rm{II}}}\rangle \simeq 4\mbox{--}5$ on scales Δz = 0.04 (ΔR ≃ 40 cMpc) at z ∼ 3.1 (see the upper panel of Figure 3). We use our 1D radiative transfer algorithm to try to understand how these observational results constrain the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$. Similarly to Worseck et al. (2014), we exclude the proximity zone from our calculations by turning the quasar off. The resulting transmission through the IGM is solely due to the He ii background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$. We then vary ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, and calculate the effective optical depth defined by ${\tau }_{\mathrm{eff}}\equiv -\mathrm{ln}\langle F\rangle$, where we take the average of the simulated transmission F in 5 regions along 100 skewers, each region with size Δz = 0.04 equal to the size of the bin in the observations. We also calculate the average value of the He ii fraction $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle$ in the same bins.

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The results are shown in Figure 3. The solid red line in the middle panel shows the values of our modeled He ii effective optical depth as a function of the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$. The corresponding He ii fraction $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle$ is shown in the bottom panel of Figure 3. We find that the effective optical depth τ_eff ≃ 4–5 implies a characteristic He ii ionizing background of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}\simeq {10}^{-14.9}\;{{\rm{s}}}^{-1}$, which will refer to henceforth as our fiducial value for z = 3.1. This He ii background corresponds to an average He ii fraction $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle \simeq 0.02$.⁸ While uniformity is likely not a good assumption during He ii reionization, which makes the above number highly approximate (and probably an underestimate for x_{He ii}), it is a good assumption thereafter, i.e., z ≃ 2.5, where fluctuations of He ii ionizing background are on the order of unity (McQuinn & Worseck 2014).

We also preformed the same set of calculations at higher redshift of z = 3.9 and plot the results as blue curves in Figure 3. Note, that at higher redshifts the gas in the IGM is becoming more dense, and the steep redshift dependence of the GP optical depth ${\tau }_{\mathrm{GP}}\propto {(1+z)}^{3/2}$ gives rise to significantly higher optical depths at z = 3.9. Thus at these high redshifts even He ii fractions of x_{He ii} ∼ 0.01 give rise to large effective optical depths τ_eff ≳ 5. While we mostly consider z = 3.1, Section 5.2 considers z ≃ 3.9.

Consider the case of a quasar emitting radiation for time t_Q into a homogeneous IGM with He ii fraction x_{He ii}. The time evolution of the He iii ionization front is governed by the equation (Haiman & Cen 2001; Bolton & Haehnelt 2007a)

$$\begin{eqnarray}&&\displaystyle \frac{{{dR}}_{\mathrm{IF}}}{{dt}}=\displaystyle \frac{{Q}_{4\mathrm{Ry}}-\tfrac{4}{3}{R}_{\mathrm{IF}}^{3}{\alpha }_{\mathrm{He}{\rm{III}}}{n}_{\mathrm{He}}^{2}}{4\pi {R}_{\mathrm{IF}}^{2}{x}_{\mathrm{He}{\rm{II}}}{n}_{\mathrm{He}}},\end{eqnarray} \tag{ 14 }$$

which has the solution

$$\begin{eqnarray}&&{R}_{\mathrm{IF}}={R}_{S}{\left[1-{\rm{exp}}\left(-\displaystyle \frac{{t}_{{\rm{Q}}}}{{x}_{\mathrm{He}{\rm{II}}}{t}_{\mathrm{rec}}}\right)\right]}^{1/3},\end{eqnarray} \tag{ 15 }$$

where ${t}_{\mathrm{rec}}=1/{n}_{{\rm{e}}}{\alpha }_{{\rm{A}}}$ is the recombination timescale and R_S is the classical Strömgren radius ${R}_{{\rm{S}}}={\left(\tfrac{3{Q}_{4\mathrm{Ry}}}{4\pi {\alpha }_{{\rm{A}}}{n}_{\mathrm{He}{\rm{III}}}{n}_{{\rm{e}}}}\right)}^{1/3}$, which is the radius of the sphere around a source of radiation, within which ionizations are exactly balanced by recombinations.

Previous observational studies of the H i Lyα proximity zones around z ≃ 6 quasars have primarily focused on measuring proximity zone sizes (Cen & Haiman 2000; Madau & Rees 2000; Mesinger & Haiman 2004; Fan et al. 2006; Carilli et al. 2010), guided by the faulty intuition that, for assuming a highly neutral IGM x_{H i} ∼ 1, the location where the transmission profile goes to zero can be identified with the location of the ionization front R_IF in Equation (15). However, the transmission profile for a H ii region expanding into a significantly neutral IGM can be difficult to distinguish from that of a "classical" proximity zone embedded in an already highly ionized IGM (Bolton & Haehnelt 2007b; Lidz et al. 2007; Maselli et al. 2007). This degenerate situation arises, because even very small residual neutral fractions in the proximity zone ${x}_{{\rm{H}}{\rm{I}}}\gtrsim {10}^{-5}$ are sufficient to saturate H i Lyα, which may occur well before the location of the ionization front is reached (Bolton & Haehnelt 2007a). Thus naively identifying the size of the proximity zone with the location of the ionization front R_IF, can lead to erroneous conclusions about the parameters governing Equation (15), e.g., the quasar lifetime and the ionization state of the IGM.

A similar degeneracy also exists for He ii proximity zones, which is exemplified in Figure 2 where the He ii transmission saturates at R ≃ 50 cMpc, whereas the ionization front is located much further from the quasar R_IF ≃ 67 cMpc (dashed vertical line). However, all previous work analyzing the structure of He ii proximity zones has been based on the assumption that the edge of the observed proximity zone can be identified with R_IF (Hogan et al. 1997; Anderson et al. 1999; Syphers & Shull 2014; Zheng et al. 2015). In addition, the majority of studies have assumed that helium is completely singly ionized x_{He ii} = 1 for quasars at 3.2 < z < 3.5 (Hogan et al. 1997; Anderson et al. 1999; Zheng et al. 2015), whereas our discussion in the previous section (see Figure 3) indicates that at z = 3.1 observations of the effective optical depth suggest that ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ implying x_{He ii} ≃ 0.01 (although the average x_{He ii} could be much larger if some regions are predominantly He ii at z > 2.8, Compostella et al. 2013; Worseck et al. 2014). Hence, in many regions one is actually in the classical proximity zone regime, where radiation from the quasar increases the ionization level of nearby material which was already highly ionized to begin with and the location of the ionizaton front is irrelevant. Furthermore, as we describe below, the background level determines the equilibration timescale ${t}_{\mathrm{eq}}\approx 1/{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ (see Equation (7)), which is the characteristic time on which the He ii ionization state of IGM gas responds to the changes in the radiation field. For our fiducial value of the background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, ${t}_{\mathrm{eq}}\simeq 2.5\times {10}^{7}\;{\rm{years}}$ is comparable to the Salpeter time. This suggests that the approach of x_{He ii} to equilibrium is the most important physical effect in He ii proximity zones, and in what follows we introduce a simple analytical equation for understanding this time evolution.

The full solution to the time evolution of ${x}_{\mathrm{He}{\rm{II}}}$ is given by Equation (6), which involves a nontrivial integral because Γ_tot, n_e and α_A are all functions of time. Indeed, this is exactly the equation that is solved at every grid cell in our 1D radiative transfer calculation, by integrating over infinitesimal timesteps (see Equation (8)). However, in the limit of a highly ionized IGM ${x}_{{\rm{H}}{\rm{I}}}\ll 1$, ${n}_{{\rm{e}}}\propto {(1+z)}^{3}$ which is approximately constant over the quasar lifetimes we consider. Furthermore, as we will demonstrate later, the low singly ionized fraction ${x}_{\mathrm{He}{\rm{II}}}\ll 1$ implies that the attenuation is small in most of the proximity zone and, hence, the photoionization rate Γ_tot is also approximately constant in time. Similarly, α_A varies only weakly with temperature (i.e., $\propto {T}^{-0.7}$), and given that the temperature will also not vary significantly with time if the He ii already has been reionized, α_A can also be approximated as constant. In this regime where Γ_tot, n_e and α_A are constant in time, Equation (8) reduces to the simpler expression evaluated at t = t_Q, the quasar lifetime:

$$\begin{eqnarray}&&{x}_{\mathrm{He}{\rm{II}}}({t}_{{\rm{Q}}})={x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}+({x}_{\mathrm{He}{\rm{II}},0}-{x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}){e}^{-{t}_{{\rm{Q}}}/{t}_{\mathrm{eq}}},\end{eqnarray} \tag{ 16 }$$

Given that the recombination timescale ${t}_{\mathrm{rec}}\equiv 1/{\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}\;\simeq {10}^{9}\;{\rm{years}}$ is very long compared to the longest ionization timescales $1/{{\rm{\Gamma }}}_{\mathrm{HeiI}}^{{\rm{bkg}}}$, we can write ${t}_{\mathrm{eq}}\approx 1/{{\rm{\Gamma }}}_{\mathrm{tot}}$, ${x}_{\mathrm{He}{\rm{II}},0}\;\approx {\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}/{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, and ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\approx {\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}{t}_{\mathrm{eq}}$.

In the left panel of Figure 4, the solid black curve shows the time evolution of ${x}_{\mathrm{He}{\rm{II}}}(t,R)$ at a single location in the proximity zone R = 25 cMpc, produced from our radiative transfer solution for the quasar photon production rate ${Q}_{4\mathrm{Ry}}={10}^{56.1}\;{{\rm{s}}}^{-1}$ and finite background case ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ corresponding to an initial He ii fraction x_{He ii,0} = 0.02. The dashed blue curve is the analytical solution Equation (16). where x_{He ii,eq} and t_eq have been evaluated from the code outputs, using the total photoionization rate Γ_tot for a fully equilibrated IGM. In other words, we evaluate ${{\rm{\Gamma }}}_{{\rm{tot}}}(t=\infty )={{\rm{\Gamma }}}_{\mathrm{QSO}}(t=\infty )+{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ where $t=\infty$ is taken to be our last output at t = 100 Myr. It is clear that for the case ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ and thus an initially highly ionized IGM ${x}_{\mathrm{He}{\rm{II}}}\simeq {10}^{-2}$, Equation (16) provides an excellent match to the result of the full radiative transfer calculation.

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Due to the patchy and inhomogeneous nature of He ii reionization some regions on the IGM might, however, have a very high He ii fraction. Therefore, it is important to check if our analytical approximation also holds in this case. The middle and right panels of Figure 4 show the time evolution of the He ii fraction at the same location and value of Q_4Ry, but now for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-16.3}\;{{\rm{s}}}^{-1}$ and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$, which correspond to x_{He ii,0} ≃ 0.3 and x_{He ii,0} = 1 respectively. The analytical approximation (blue squares and curve) clearly fails to reproduce the time evolution. Specifically, it predicts too rapid a response to the quasar ionization relative to the true evolution, and this discrepancy is largest for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ where the true evolution to equilibrium is delayed by ∼5.5 Myr.

Recall that, because we observe on the light cone, the speed of light is effectively infinite in our code. Thus, at the location R = 25 cMpc we expect no delay in the response of the proximity zone to the quasar radiation caused by finite light travel time effects. For the x_{He ii,0} ≃ 0.02 (${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$) case the ionization front travels at nearly the speed of light and, because we observe on the light cone, there is thus no noticeable delay between the evolution of the He ii fraction and Equation (16). However, if the ionization front does not travel at the speed of light, which will be the case for the lower backgrounds ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-16.3}\;{{\rm{s}}}^{-1}$ and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ and correspondingly higher He ii fractions (x_{He ii,0} ≃ 0.3 and x_{He ii,0} = 1), then the time that it takes the ionization front to propagate to the location R = 25 cMpc is no longer negligible relative to the equilibration timescale, and the response of the He ii fraction will be delayed.

We can estimate this time delay by noting that the location of the ionization front (see Equation (15)) is given by

$$\begin{eqnarray}&&{R}_{\mathrm{IF}}\approx {\left[\displaystyle \frac{3{Q}_{4\mathrm{Ry}}{t}_{\mathrm{IF}}}{4\pi {n}_{\mathrm{He}}{x}_{\mathrm{He}{\rm{II}}}}\right]}^{1/3},\end{eqnarray} \tag{ 17 }$$

assuming that ${t}_{{\rm{Q}}}\ll {x}_{\mathrm{He}{\rm{II}}}{t}_{\mathrm{rec}}=({x}_{\mathrm{He}{\rm{II}}}/1.0)1.16\times {10}^{9}\;{\rm{years}}$, valid for x_{He ii} ∼ 0.3–1.0 and the quasar lifetimes we consider. In this regime the ionization front is simply the radius of the ionized volume around the quasar. Inverting this equation, we obtain that at a location R, the time delay between the quasar turning on and the arrival of the first ionizing photons is⁹

$$\begin{eqnarray}&&{t}_{\mathrm{IF}}(R)=6.7\ \left(\displaystyle \frac{{x}_{\mathrm{He}{\rm{II}}}}{1.0}\right){\left(\displaystyle \frac{{Q}_{4\mathrm{Ry}}}{{10}^{56}{{\rm{s}}}^{-1}}\right)}^{-1}{\left(\displaystyle \frac{R}{25{\rm{cMpc}}}\right)}^{3}{\rm{Myr}}\end{eqnarray} \tag{ 18 }$$

For R = 25 cMpc this delay is very nearly the delay seen in the middle and right panels of Figure 4, suggesting a simple physical interpretation for the behavior of the He ii fraction in the proximity zone for the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-16.3}\;{{\rm{s}}}^{-1}$ and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ cases. Namely, Equation (16) still describes the equilibration of the He ii fraction, but it must be modified to account for the delay in the arrival of the ionization front, only after which equilibration begins to occur. We thus write

$$\begin{eqnarray}&&{x}_{\mathrm{He}{\rm{II}}}({t}_{Q},r)={x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}+(1-{x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}){e}^{\tfrac{-({t}_{{\rm{Q}}}-{t}_{\mathrm{IF}})}{{t}_{\mathrm{eq}}}}\end{eqnarray} \tag{ 19 }$$

The green curves in the middle and right panels of Figure 4 illustrate that the simple equilibration time picture, but now modified to account for a delay in the arrival of the ionization front, provides a good description of the time evolution of x_{He ii} in the proximity zone for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-16.3}\;{{\rm{s}}}^{-1}$ and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ cases.

To summarize, we have shown that the He ii fraction in quasar proximity zones is governed by a simple analytical equation (Equation (16)), which describes the exponential time evolution from an initial pre-quasar ionization state x_{He ii,0} to an equilibrium value x_{He ii,eq}. The enhanced photoionization rate near the quasar Γ_tot sets both the timescale of the exponential evolution ${t}_{\mathrm{eq}}=1/{{\rm{\Gamma }}}_{\mathrm{tot}}$, and the equilibrium value attained ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\approx {\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}{t}_{\mathrm{eq}}$. For very high He ii fractions x_{He ii,0} ≃ 1, this exponential equilibration is delayed by the time it takes the sub-luminal ionization front to arrive to a given location.

Previous work on H i proximity zones at z ∼ 6 have pointed out that the quasar lifetime and ionization state of the IGM (or equivalently the ionizing background) are degenerate in determining the location of the ionization front R_IF (Bolton & Haehnelt 2007a, 2007b; Lidz et al. 2007; Bolton et al. 2012), which is readily apparent from the exponent in Equation (15). Although, many studies simply assume a fixed value for the quasar lifetime of ${t}_{Q}={10}^{7}\;{\rm{years}}$ when making inferences about the ionization state of the IGM (but see Bolton et al. 2012, for a more careful treatment). This degeneracy between lifetime and ionizing background is complicated by the fact that, at z ∼ 6 only lower limits on the hydrogen neutral fraction x_{H i} (upper limits on the background photoionization rate ${{\rm{\Gamma }}}_{{\rm{H}}\;{\rm{I}}}^{{\rm{bkg}}}$) can be obtained from lower limits on the GP absorption optical depth. The situation is further exacerbated if there are significant spatial fluctuations in the background caused by foreground galaxies that may have "pre-ionized" the IGM (Bolton & Haehnelt 2007b; Lidz et al. 2007; Wyithe et al. 2008).

An analogous degeneracy exists between t_Q and x_{He ii} for He ii proximity zones, as we illustrate in Figure 5. The upper panel shows three example transmission spectra for the same skewer and value of He ii background, but different values of quasar lifetime, which are clearly distinguishable. The situation changes if we also allow the He ii background to vary, which is shown in the bottom panel of Figure 5 where transmission spectra are plotted for the same skewer, but several distinct combinations of lifetime and background. The nearly identical resulting spectra indicate that the same degeneracy exists at z ≃ 3.1 between the quasar lifetime t_Q and the value of the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$. In what follows, we discuss this degeneracy in detail, aided by our analytical model for the time evolution of the He ii fraction from the previous section.

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We can understand this degeneracy by rearranging Equation (16)

$$\begin{eqnarray}&&{x}_{\mathrm{He}{\rm{II}}}({t}_{{\rm{Q}}},R)={x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\left[1+\displaystyle \frac{{{\rm{\Gamma }}}_{\mathrm{QSO}}}{{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}}{e}^{-\tfrac{{t}_{{\rm{Q}}}}{{t}_{\mathrm{eq}}(R)}}\right]\end{eqnarray} \tag{ 20 }$$

where for simplicity we have focused on the finite background case where the ionization front time delay can be ignored. There are two regimes that are relevant to this degeneracy. First, very near the quasar the attenuation of Γ_QSO is negligible, implying that ${t}_{\mathrm{eq}}\propto {{\rm{\Gamma }}}_{\mathrm{QSO}}^{-1}\propto {R}^{2}$ and given by

$$\begin{eqnarray}&&{t}_{\mathrm{eq}}=0.08{\left(\displaystyle \frac{{Q}_{4\mathrm{Ry}}}{{10}^{56}{{\rm{s}}}^{-1}}\right)}^{-1}{\left(\displaystyle \frac{R}{5{\rm{cMpc}}}\right)}^{2}\;{\rm{Myr}}.\end{eqnarray} \tag{ 21 }$$

At small distances (R ≲ 15 cMpc in Figure 5) in the highly ionized "core" of the proximity zone, ${t}_{\mathrm{eq}}\ll {t}_{{\rm{Q}}}$ for the quasar lifetimes we consider, and Equation (20) indicates that the proximity zone structure depends only on the luminosity of the quasar, which determines ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\approx {\alpha }_{{\rm{A}}}{n}_{{\rm{e}}}/{{\rm{\Gamma }}}_{\mathrm{QSO}}$, but there is no sensitivity to either ${t}_{{\rm{Q}}}$ or ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$.

Second, at larger distances the equilibration time grows as ${t}_{{\rm{eq}}}\propto {R}^{2}$ and will eventually be comparable to the quasar lifetime. We define the equilbration distance R_eq to be the location where ${t}_{\mathrm{eq}}({R}_{\mathrm{eq}})\equiv {t}_{{\rm{Q}}}$, which gives

$$\begin{eqnarray}&&{R}_{\mathrm{eq}}=17{\left(\displaystyle \frac{{Q}_{4\mathrm{Ry}}}{{10}^{56}{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{{t}_{{\rm{Q}}}}{1{\rm{Myr}}}\right)}^{1/2}\;{\rm{cMpc}},\end{eqnarray} \tag{ 22 }$$

where for simplicity we neglect the impact of attenuation of Γ_QSO. At distances comparable to the equilibration distance R ∼ R_eq, Equation (20) indicates that the He ii fraction will be sensitive to t_Q. Note that at R ∼ R_eq the quasar still dominates over the background ${{\rm{\Gamma }}}_{\mathrm{QSO}}\gg {{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, such that x_{He ii,eq} is still independent of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$. One then sees from Equation (20) that for any change in quasar lifetime t_Q one can always make a corresponding change to the value of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ to yield the same value of x_{He ii}. But note that this degeneracy holds only at a single radius R, because ${t}_{\mathrm{eq}}(R)$ is a function of R, whereas our ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ is assumed to be spatially constant. Therefore, there is no way to choose a constant ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ such that the ${x}_{\mathrm{He}{\rm{II}}}({t}_{{\rm{Q}}},R)$ matches different values of t_Q at all R. In reality, ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ will fluctuate spatially, but it will not have the required dependence on R to counteract the lifetime dependence.

Similar arguments also apply when ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$, and the x_{He ii}(t, R) evolution is governed by Equation (19). In this case, the time evolution depends only on the quasar lifetime t_Q and the ionizing photon production rate of the quasar Q_4Ry. Very close to the quasar (R ≲ 15 cMpc), there is no sensitivity to quasar lifetime provided that ${t}_{{\rm{Q}}}-{t}_{{\rm{IF}}}(R)\gg {t}_{\mathrm{eq}}$, which is the case for the long quasar lifetime model t_Q = 63 Myr shown in Figure 5 with ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ (dashed blue curve), which is indistinguishable from the finite background models at small radii. Although note that for much shorter quasar lifetimes ${t}_{{\rm{Q}}}\sim {t}_{\mathrm{IF}}(R)$ comparable to the ionization front travel time, Equation (19) indicates one may retain sensitivity to the quasar lifetime even in the core of the zone. At larger distances R ≳ 15 Mpc where ${t}_{{\rm{Q}}}-{t}_{\mathrm{IF}}(R)\sim {t}_{\mathrm{eq}}$, the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ case becomes sensitive to quasar lifetime according to Equation (19), but Figure 5 still indicates that the transmission profile is remarkably similar to the finite background case. In principle Q_4Ry could be varied to produce a curve that appears even more degenerate, but we do not explore that here (but see the discussion in Section 5.2).

Finally, an obvious difference between the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ and finite background case is of course the transmission level far from the quasar, which is zero for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$, but corresponds to a finite value of τ_eff for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}\ne 0$ (see Figure 3). At low redshifts z ≃ 3 where the effective optical depth can be measured, this provides an independent constraint on ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ which rules out a ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ model. As discussed in Section 3.1, the enhanced sensitivity of τ_eff measurements to the He ii background level for He ii GP absorption at z ∼ 3, as compared to H i GP absorption at z ∼ 6–7 (where only upper limits on the background are available), constitutes an important difference between He ii and H i proximity zones, which can be leveraged to break the degeneracy between the quasar lifetime t_Q and He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, as we will elaborate on in the next section.

In Figure 7 we show stacks of 1000 skewers for a sequence of proximity zone models with different quasar lifetimes in the range t_Q = 1–500 Myr, indicated by the colored curves, but with other parameters (${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ and Q_4Ry) fixed. The panels show, from top to bottom, the stacked transmission profiles in He ii Lyα region, the He ii fraction x_{He ii}, and the total photoionization rate ${{\rm{\Gamma }}}_{\mathrm{tot}}={{\rm{\Gamma }}}_{\mathrm{QSO}}+{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ together with the unattenuated photoionization rate Γ_unatt.. The panels on the left show the models with fixed values of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, whereas the right side illustrates the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ case. The photon production rate has been set to fiducial value ${Q}_{4\mathrm{Ry}}={10}^{56.1}\;{{\rm{s}}}^{-1}$ throughout.

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Several qualitative trends are readily apparent from Figure 7. First, as was also mentioned in Section 3.3, at the smallest radii R ≲ 5 cMpc, there is a "core" of the proximity zone, which is insensitive to the changes of the quasar lifetime such that all transmission profiles overlap. Second, it is clear that both with (${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$) and without (${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$) a He ii ionizing background, increasing the quasar lifetime t_Q results in larger proximity zones, reflecting the longer time that the nearby IGM has been exposed to the radiation from the quasar. Third, in the presence of a He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, the transmission profile shape loses sensitivity to quasar lifetime for models with t_Q > 25 Myr, whereas for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ case, the proximity zone continues to grow with increasing quasar lifetime up to large values of t_Q = 500 Myr.

We can gain a better physical understanding of the origin of these trends from the equation that describes the time evolution of the He ii fraction ${x}_{\mathrm{He}{\rm{II}}}(t)$ given by Equation (16) and discussed in Sections 3.2 and 3.3. In what follows we focus on the specific example ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, but our discussion also applies to the case of zero ionizing background, provided that the equation for the time evolution is modified to account for the time-delay associated with the arrival of the ionization front (see Equation (19) in Section 3.2). At z ∼ 3, observations of the effective optical depth strongly favor a finite background (see Section 3.1) ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}\simeq {10}^{-14.9}\;{{\rm{s}}}^{-1}$ (${x}_{\mathrm{He}{\rm{II}}}\simeq {10}^{-2}$), although most of the previous interpretations of observed He ii proximity regions have concentrated on the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ case (Shull et al. 2010; Syphers & Shull 2014; Zheng et al. 2015).

The three panels in Figure 8 show the time evolution of the He ii fraction at three different distances from the quasar $R=[3,25,50]$ cMpc, labeled A, B, and C, respectively. The black curves show the average ${x}_{\mathrm{He}{\rm{II}}}(t)$ computed from 100 skewers, where the quasar was on for the entire t_Q = 100 Myr that is shown. The dashed blue curves show the time evolution from Equation (16), where for the input parameters we have averaged the outputs from the radiative transfer code. Specifically, to compute the blue curves we take an average value of photoionization rate $\langle {{\rm{\Gamma }}}_{\mathrm{tot}}\rangle$ which is a mean of 100 skewers and we do the same for the equilibrium ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}=\langle {\alpha }_{{\rm{A}}}({\rm{T}}){n}_{{\rm{e}}}\rangle /\langle {{\rm{\Gamma }}}_{\mathrm{tot}}\rangle$, and initial neutral fractions ${x}_{\mathrm{He}{\rm{II}},0}=\langle {\alpha }_{{\rm{A}}}({\rm{T}}){n}_{{\rm{e}}}\rangle /{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$.¹⁰

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This procedure excellently reproduces the evolution given by the solid black curves, computed from a full time integration of the radiative transfer. Whereas we previously saw that this analytical approximation provides a good fit to the time evolution of the He ii fraction of a single skewer (see Figure 4), it is somewhat surprising that it also works so well for the stacked spectra using these averaged quantities.

As Figure 8 shows the full time evolution of He ii fraction over 100 Myr, observing a quasar with a given quasar lifetime t_Q is equivalent to evaluating ${x}_{\mathrm{He}{\rm{II}}}(t)$ at the time t = t_Q. The green, yellow, and magenta vertical lines indicate three possible quasar lifetimes of t_Q = 1, 4, and 25 Myr, respectively. The spatial profile of x_{He ii} and Lyα transmission are shown in the middle and bottom panels of Figure 9 for the same three quasar lifetime models (with the same line colors as in Figure 8). The black curves in these two panels show the "equilibrium" $t=\infty$ profiles for x_{He ii} and the transmission, which we define to correspond to t_Q = 100 Myr, at which time x_{He ii} has fully equlibrated. The vertical dashed lines labeled A, B, and C in Figure 9 indicate the three distances from the quasar $R=[3,25,50]$ cMpc for which the time evolution is shown in Figure 8.

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First, consider location A in the inner "core" of the proximity zone at a distance of R = 3 cMpc, at which the He ii fraction and transmission profiles in Figure 9 are all identical for the quasar lifetimes we consider. The uppermost panel of Figure 9 shows the equilibration time as a function of distance from the quasar, which indicates that at R = 3 cMpc t_eq = 0.025 Myr. The reason why all quasar lifetimes are indistinguishable at this distance can be easily understood from the ${x}_{\mathrm{He}{\rm{II}}}(t)$ evolution in the left panel of Figure 8. Irrespective of whether the quasar has been emitting for t_Q = 1 Myr (green), t_Q = 4 Myr (yellow) or t_Q = 16 Myr (magenta), because at this distance the equilibration time (red vertical dashed line) ${t}_{\mathrm{eq}}\ll {t}_{{\rm{Q}}}$, the IGM has already equilibrated ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\simeq 2.5\times {10}^{-5}$, and there is thus no sensitivity to quasar lifetime (see also the discussion in Section 3.3 and Equation (20)).

Note however that the ${t}_{\mathrm{eq}}\sim {R}^{2}$ dependence of equilibration time shown in the top panel of Figure 9 indicates that at greater distances, the equilibration time is larger and becomes comparable to the lifetimes we consider. This manifests as significant differences in the stacked x_{He ii} and transmission profiles for different lifetimes in Figure 9, which can again be understood from the time evolution in Figure 8. For example, consider location B (middle panel of Figure 8) at R = 25 cMpc, at which the equilibration time is ${t}_{\mathrm{eq}}=1.7\;{\rm{Myr}}$ (red vertical dashed line). For the shortest quasar lifetime of 1 Myr (green line), the IGM has not been illuminated long enough to equilibrate, i.e., ${t}_{{\rm{Q}}}\lt {t}_{\mathrm{eq}}$, and thus still reflects the He ii fraction x_{He ii,0} consistent with the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ that prevailed before the quasar turned on. This ${x}_{\mathrm{He}{\rm{II}},0}\simeq 2.2\times {10}^{-2}$, much larger than the equilibrium value ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\simeq 1.5\times {10}^{-3}$, explains why the corresponding transmission at location B in Figure 9 (green curve) lies far below the fully equilibrated model t_Q = 100 Myr (black curve). Likewise, the t_Q = 4 Myr model (yellow) is still in the process of equilibriating, whereas the t_Q = 25 Myr model (magenta) has fully equilibrated, explaining the respective values of the x_{He ii} and transmission for these models at location B in Figure 9.

Because equilibration time increases with distance from the quasar, the stacked transmission profile becomes sensitive to larger quasar lifetimes at larger radii. This is evident from the transmission profiles in Figures 7 and 9, where models with progressively larger lifetimes peel off from the equilibrium model (t_Q = 100 Myr) at progressively larger radii. However, eventually far from the quasar, this sensitivity saturates, as t_eq approaches its asymptotic value ${t}_{\mathrm{eq}}\sim 1/{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ (see upper panel of Figure 9), which for our fiducial ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ corresponds to t_eq = 25 Myr. This saturation effect is illustrated by the time evolution at location C, R = 50 cMpc from the quasar, in the right panel of Figure 8. If the quasar has been illuminating the IGM for t_Q = 25 Myr (magenta curve), the He ii fraction has nearly reached equilibrium x_{He ii,eq}, and thus at location C Figure 9 exhibits only a small but still noticeable difference between the x_{He ii} at t_Q = 25 Myr and the equilibrium model (black curve), and consequently the transmission profiles (lower panel) hardly differ at all. It would clearly be extremely challenging to distinguish between different models with t_Q > 25 Myr. Finally, at the largest radii R > R_bkg =70 cMpc, where R_bkg is defined to be the location where ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={{\rm{\Gamma }}}_{\mathrm{QSO}}$, the quasar no longer dominates over the background, and all the transmission profiles converge to the mean transmission set by the He ii background. Finally, we again note that if ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$, there is no such asymptote in the equilibration time, and the transmission profile continues to be sensitive to values of quasar lifetime as large as t_Q = 500 Myr as shown in Figure 7. However, in practice for very long quasar lifetimes and therefore very large proximity zones, one might eventually encounter locations in the universe where the background is no longer zero.

In Figure 10 we illustrate the impact of varying the He ii ionizing background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ on the stacked transmission profile, with t_Q and Q_4Ry held fixed. The left panels show a quasar lifetime of t_Q = 1.5 Myr, whereas the right show t_Q = 10 Myr. Four different values for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ are plotted, including the ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ case.

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From Figure 10, we see that in the inner core R < 5 cMpc of the proximity zone, the transmission profile is independent of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, analogous to the behavior in Figure 7, where we saw that the core is also independent of t_Q. As discussed in Section 3.3 (see also the previous section), this insensitivity to ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ and t_Q can be understood from Equation (20) governing the time evolution of x_{He ii}. For ${t}_{\mathrm{eq}}\ll {t}_{{\rm{Q}}}$, the IGM has already equilibrated and ${x}_{\mathrm{He}{\rm{II}}}({t}_{{\rm{Q}}})\approx {x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}$. At small distances $R\ll {R}_{\mathrm{bkg}}$ the quasar dominates over the background ${{\rm{\Gamma }}}_{\mathrm{QSO}}\gg {{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ and attenuation is negligible, hence the equilibrium He ii fraction ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\propto {{\rm{\Gamma }}}_{\mathrm{QSO}}^{-1}$ is determined by the quasar photon production rate alone, and is independent of the background and quasar lifetime.

In the previous section we argued that the equilibration time picture explains why the transmission profiles for progressively larger lifetimes peel off from the equilibrium model (t_Q = 100 Myr) at progressively larger radii (see Figure 7). The curves in Figure 10 illustrate that varying the ionizing background has a different effect, namely to change the slope of the transmission profile about this peel off point, as well as to determine the transmission level far from quasar. The different response of the transmission profile to these two parameters, t_Q and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, results from the functional form of Equation (20). The stronger peel off behavior with t_Q is due to the exponential dependence on the quasar lifetime t_Q in Equation (20), whereas the milder variation of the slope with ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ arises because of the inverse proportionality of x_{He ii,eq} on ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$. Furthermore, at large distances from the quasar where ${{\rm{\Gamma }}}_{\mathrm{tot}}\simeq {{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$, x_{He ii,eq} approaches ${x}_{\mathrm{He}{\rm{II}},0}\propto 1/{{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ and the background sets the absorption level as expected. As we also noted in Section 3.3, the different dependence of the transmission profile on t_Q and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ suggests that the degeneracy between these parameters could be broken by the shape of the transmission profile, which we discuss further in Section 5.2.

Figure 11 shows the effect of the photon production rate on the structure of the proximity zone. The quasar lifetime has been set to the value t_Q = 10 Myr, and the background is ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ on the left and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ on the right. One can see that the impact of the photon production rate Q_4Ry on the resulting transmission profile is twofold. First, as expected, more luminous quasars produce larger proximity zones, i.e., increasing the photon production rate by 0.5 dex expands the characteristic size of the proximity zone by a factor of ∼2. Second, besides increasing the overall size, the slope of the stacked transmission profile becomes shallower when Q_4Ry is increased.

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The increase in proximity zone size with Q_4Ry can be easily understood in the equilibration time picture. From Equation (22), we see that the equilibration distance R_eq, which sets the location where the transmission profile becomes sensitive to t_Q and approaches the mean transmission level of the IGM (see Figure 7), scales as ${R}_{\mathrm{eq}}\propto {Q}_{4\mathrm{Ry}}^{1/2}$. Hence an increase (decrease) in photon production rate Q_4Ry results in a larger (smaller) characteristic size of the proximity zone.

To understand the change in transmission profile slope with Q_4Ry, consider that in the core of the proximity zone where ${t}_{{\rm{eq}}}\ll {t}_{{\rm{Q}}}$, the He ii fraction has equilibrated and is given by ${x}_{\mathrm{He}{\rm{II}}}(t)={x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}$, where ${x}_{\mathrm{He}{\rm{II}},\mathrm{eq}}\propto {Q}_{4\mathrm{Ry}}^{-1}$. As the transmitted flux is just the exponential of a constant times x_{He ii,eq}, an increase (decrease) in Q_4Ry makes this exponent smaller (larger) and thus the slope of the transmission profile becomes shallower (steeper).

As we illustrated in Section 2.2, the total quasar photon production rate Q_4Ry at frequencies above the He ii ionization threshold depends on the assumed slope of the quasar SED α blueward of 4 Ry. Throughout the paper we have assumed that α = 1.5 is constant at all frequencies blueward of 1 Ry and simply varied the value of Q_4Ry to illustrate its effect on the stacked transmission profiles (see Section 4.3). However, the spectral slope regulates the number of hard photons with long mean free path which can affect the transmission profiles at large distances from the quasar. Therefore, it is not obvious whether Q_4Ry and α impact the structure of the proximity zone in the same way.

We thus run a set of radiative transfer simulations to further explore this question. In order to disentangle the effects of ${Q}_{{\rm{4Ry}}}$ and α on the transmission profiles we chose to fix the quasar specific luminosity N_4Ry at 4 Ry (see Equation (1)), which can be deduced from the observable quasar specific luminosity at the H i ionization threshold of 1 Ry by scaling it down to He ii ionization threshold with constant spectral slope ${\alpha }_{1\mathrm{Ry}\to 4\mathrm{Ry}}=1.5$. We then freely vary the spectral slop ${\alpha }_{4\mathrm{Ry}\to \infty }$ within the range ${\alpha }_{4\mathrm{Ry}\to \infty }=1.1\mbox{--}2.0$. Although, the actual slope of quasar SED at $\lambda \leqslant 912\;\mathring{\rm A}$ is not currently well known, the chosen range of ${\alpha }_{4\mathrm{Ry}\to \infty }$ values is motivated by the constraints on the power-law index from Lusso et al. (2015), who found α_ν = 1.70 ± 0.61 at $\lambda \leqslant 912\;\mathring{\rm A}$.

Figure 12 illustrates the effect of the spectral slope on the structure of the proximity zone. The quasar lifetime and He ii background are fixed at ${t}_{{\rm{Q}}}=10\;{\rm{Myr}}$ and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, respectively. Three curves show stacked transmission profiles for our fiducial value ${\alpha }_{4\mathrm{Ry}\to \infty }=1.5$ (black), harder slope ${\alpha }_{4\mathrm{Ry}\to \infty }=1.1$ (blue), and softer slope ${\alpha }_{4\mathrm{Ry}\to \infty }=2.0$ (red). It is apparent from Figure 12 that the effect of the spectral slope on the structure of the proximity zone is negligible in comparison to the effect of the overall photon production rate Q_4Ry (see Figure 11).

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Consider that when specific luminosity N_4Ry is fixed at 4 Ry, the quasar photon production rate Q_4Ry scales with ${\alpha }_{4\mathrm{Ry}\to \infty }$ as ${Q}_{{\rm{4Ry}}}\propto {\alpha }_{4\mathrm{Ry}\to \infty }^{-1}$ (see Equation (1)). Thus, varying the value of ${\alpha }_{4\mathrm{Ry}\to \infty }$, results in 25%–45% changes in the photon production rate Q_4Ry (which is the ratio of assumed spectral slopes ${\alpha }_{4\mathrm{Ry}\to \infty }$ in different models). Thus, one would expect a significant difference in the resulting transmission according to Figure 11. However, the more important quantity here is the cross-section weighted quasar He ii photoionization rate Γ_QSO given by Equation (3), which regulates the time-evolution of the He ii fraction (and, consequently, the transmission profile). Since the ionization cross-section scales as ${\sigma }_{\nu }\propto {(\nu /{\nu }_{{\rm{th}}})}^{-3}$, the resulting dependence of the He ii photoionization rate in the unattenuated limit on the spectral slope is ${{\rm{\Gamma }}}_{{\rm{QSO}}}\propto {({\alpha }_{4\mathrm{Ry}\to \infty }+3)}^{-1}$. Hence, the difference between the photoionization rates of the fiducial model ${\alpha }_{4\mathrm{Ry}\to \infty }=1.5$ and the two models in Figure 12 with ${\alpha }_{4\mathrm{Ry}\to \infty }=1.1$ and ${\alpha }_{4\mathrm{Ry}\to \infty }=2.0$ is only ${\rm{\Delta }}{\rm{\Gamma }}\approx 10 \%$. Therefore, there is no significant effect in the resulting transmission in He ii proximity zone even for relatively large variations in the assumed spectral slope of quasar SED blueward of 4 Ry.

In what follows we study the impact of the distribution of quasar lifetimes and He ii backgrounds on the shape of the stacked transmission profiles in He ii Lyα regions at z = 3.1 using the same stacking technique described in Section 4.

First, we consider the distribution of quasar lifetimes only, while keeping He ii background and quasar photon production rate fixed to our fiducial values ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ and ${Q}_{4\mathrm{Ry}}\;={10}^{56.1}\;{{\rm{s}}}^{-1}$, respectively. The upper panel of Figure 13 shows the comparison between the stacked transmission profiles of our fiducial model with single quasar lifetime t_Q = 10 Myr and three models with different distributions of t_Q. We model the distribution of quasar lifetimes as a uniform sampling from 5 discrete t_Q values centered on ${\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=1.00$ spanning a total range of $0\leqslant {\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})\leqslant 2$, but with different widths ${\rm{\Delta }}{\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=[0.5,1.0,2.0]$. This is done by constructing stacks of 1000 skewers, where each skewer is randomly chosen from one of the single lifetime models over the range specified by the width. The blue curve in the upper panel of Figure 13 represents the stack of skewers taken from models with ${\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=[0.75,0.875,1.00,1.125,1.25]$, red is a stack of skewers with ${\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=[0.50,0.75,1.00,1.25,1.50]$ and green is ${\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=[0.00,0.50,1.00,1.50,2.00]$. The numbers in square brackets represent the values of t_Q or ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ (see below) used in the models that contributed to the stacks of the distribution models.

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The upper panel of Figure 13 clearly shows that in comparison to the single lifetime model, there appears to be a reduction in the transmission in the range R ≃ 10–40 cMpc in the stacked spectra of models with the distribution of quasar lifetimes. This is due to the skewers with lower values of t_Q. In these models the IGM did not have enough time to respond to the changes in the radiation field caused by the quasar and still reflects a higher He ii fraction set by the He ii ionizing background, resulting in decreased transmission. Furthermore, the transmission becomes more depressed as the width of the quasar lifetime distribution increases, indicating that stacked transmission profiles are also sensitive to the width of the quasar lifetime distribution.

Similarly, we now investigate how the distribution of He ii backgrounds affects the stacked transmission profiles. As we discussed in Section 3.1, our fiducial value of He ii background ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$ is derived from measurements of the He ii effective optical depth τ_eff which give τ_eff ≃ 4.5 at z = 3.1 (see Figure 3). Therefore, the distribution of He ii ionizing backgrounds should also correspond to the same observed mean effective optical depth of τ_eff ≃ 4.5. Analogously to Section 3.1, we run our radiative transfer calculation with the quasar turned off for 100 skewers chosen from several models with different values of the He ii background. We focus on two uniform distributions for ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ which yield the same effective optical depth τ_eff ≃ 4.5: ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}\;=[{10}^{-14.7},{10}^{-14.8},{10}^{-14.9},{10}^{-15.0},{10}^{-15.1}]\;{{\rm{s}}}^{-1}$ and ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=[{10}^{-14.6},{10}^{-14.8},{10}^{-15.0},{10}^{-15.2},{10}^{-15.4}]\;{{\rm{s}}}^{-1}$. We then run our 1D radiative transfer algorithm with the quasar on for t_Q = 10 Myr and calculate 10 different models with the above mentioned values of He ii background. Similar to the distribution of quasar lifetimes, we calculate two stacked transmission profiles using 1000 skewers randomly chosen from these models.

The results of this exercise are shown in the middle panel of Figure 13, where the black curve corresponds to the model with a single value of He ii background fixed at our fiducial value of ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}={10}^{-14.9}\;{{\rm{s}}}^{-1}$, the blue curve shows the model with a background distribution that spans a range of 0.4 dex ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=[{10}^{-14.7}\mbox{--}{10}^{-15.1}]\;{{\rm{s}}}^{-1}$, and the red curve is for the model spanning 0.8 dex ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=[{10}^{-14.6}\mbox{--}{10}^{-15.4}]\;{{\rm{s}}}^{-1}$. We also include a model in which 20% of the IGM at z ≃ 3 is still represented by the regions with x_{He ii} = 1, or, equivalently, ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$, which is shown in the middle panel of Figure 13 by the green curve. It is apparent that for the distributions we consider here, varying the He ii background such that the mean effective optical depth is fixed, has only a small effect on the resulting transmission profile. However, if measurements of the He ii effective optical depth cannot rule out the existence of the regions of IGM with high fractions of singly ionized helium, including them into the distribution of backgrounds can produce a stronger effect on the stacked transmission profile. Nevertheless, comparing to the upper panel of Figure 13 we can conclude that the distribution of quasar lifetimes has a dominant effect on the stacked transmission profile and future attempts to model stacked proximity zone should take this into account. This is illustrated in the bottom panel of Figure 13 where we combine these effects and model both the distributions of quasar lifetimes t_Q and He ii backgrounds ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}$ simultaneously, by combining skewers drawn from different models in both parameters. Note, that even in the model where 20% of IGM has ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ (x_{He ii} = 1), when convolved with a broad distribution of quasar lifetimes, the impact of these regions with x_{He ii} = 1 is not very significant.

In Section 4 we showed that variations in the quasar lifetime and He ii background impact stacked transmission profiles in distinct ways. Our analysis here shows that the width of the distribution of quasar lifetimes, can significantly change the shape of the stacked transmission profile, and should be included in any attempt to model real observations. Nevertheless, we argue that that at z ≃ 3.1 one should be able to put interesting constraints on the quasar lifetime given that the average He ii background can be determined from the level of transmission in the IGM as quantified by effective optical depth measurements (see Figure 3 in Section 3.1), and due to the fact that the stacked transmission profile is relatively insensitive to fluctuations in the He ii background.

Recall Figure 3, where the blue curve shows the evolution of the mean He ii effective optical depth τ_eff and the mean He ii fraction at z = 3.9 in our simulations. At this higher redshift, the ${\tau }_{\mathrm{GP}}\propto {(1+z)}^{3/2}$ dependence of the optical depth, implies that even backgrounds which correspond to a IGM with on average relatively low He ii fraction of only ≃0.03–0.04, correspond to very large effective optical depth is τ_eff ≃ 8–9. Thus, unlike the situation at z ≃ 3, it would be extremely challenging to measure an optical depth that high with HST/COS, making it virtually impossible to measure the He ii background and thus distinguish between IGM with low He ii fraction (x_{He ii} < 0.05) and the high He ii fraction (x_{He ii} = 1) at z ≃ 4. It is, therefore, interesting to explore whether the shape of the transmission profile of He ii proximity zones can be used to independently probe both the quasar lifetime and the He ii background at z ∼ 4, where the background cannot be independently constrained.

We begin by applying our method to skewers drawn from the same hydrodynamical simulation at redshift z = 3.9. Following our approach in the previous subsection, we choose a finite background model with uniform distribution ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}\;=[{10}^{-14.5},{10}^{-14.7},{10}^{-14.9},{10}^{-15.1},{10}^{-15.3}]\;{{\rm{s}}}^{-1}$. For the distribution of lifetimes we adopt ${\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=[0.50,0.75,1.00,1.25,1.50]$, and fix the photon production rate to ${Q}_{4\mathrm{Ry}}\;={10}^{56.1}\;{{\rm{s}}}^{-1}$. The resulting stacked transmission profile is shown by the blue curve in Figure 14. Despite the finite background and relatively small average He ii fraction $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle \simeq 0.05$, far from the quasar R > 50 cMpc, the average transmission is very nearly zero reflecting the large effective optical depth τ_eff ≃ 8–9 for this model. The dashed black curve in Figure 14 shows the stacked transmission profile for a model with the same distribution of quasar lifetimes, but with ${{\rm{\Gamma }}}_{\mathrm{He}\;{\rm{II}}}^{{\rm{bkg}}}=0$ and hence an IGM with $\langle {x}_{\mathrm{He}{\rm{II}}}\rangle =1.0$, and ${Q}_{4\mathrm{Ry}}$ fixed to the same value. In the absence of the He ii background, one sees that the proximity zone is smaller, and the transmission approaches zero at smaller distance from the quasar than for the finite background model. This significant difference in the transmission profile naively suggests that proximity zones can be used to determine the value of the background.

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However, clearly one way of compensating for this difference between the transmission profiles is to consider: (1) a distribution with longer quasar lifetimes and, (2) a change in the quasar photon production rate Q_4Ry, for the zero background model. Both of these parameter variations change the size of the proximity zone, which could make the two models look more similar. In principle the photon production rate Q_4Ry should be determined by our knowledge of the quasar magnitudes, however in practice the average quasar SED is not well constrained at energies above 4 Ry, giving rise to at least ∼50% relative uncertainty in Q_4Ry (see e.g., Lusso et al. 2015). To illustrate these parameter degeneracies, we increase the values of t_Q in our distribution for the zero background model by 0.50 dex to ${\rm{log}}\ ({t}_{{\rm{Q}}}/{\rm{Myr}})=[1.00,1.25,1.50,1.75,2.00]$, and simultaneously slightly reduce the value of the photon production rate by 0.1 dex to ${Q}_{4\mathrm{Ry}}={10}^{56.0}\;{{\rm{s}}}^{-1}$. The result of this exercise is shown by the solid red curve in Figure 14, which shows that the the transmission profiles for a highly doubly ionized helium ($\langle {x}_{\mathrm{He}{\rm{II}}}\rangle =0.05$) and a completely singly ionized helium ($\langle {x}_{\mathrm{He}{\rm{II}}}\rangle =1.0$) are essentially indistinguishable.

In conclusion we see, that at z ∼ 4 the limited sensitivity of HST/COS combined with the steep rise of effective optical depth with redshift (see Figure 3), implies that one will likely only be able to place lower limits on the the average effective optical depth τ_eff, and hence upper limits on the value of the He ii background. The two very different models that we considered for the He ii background in Figure 14 are expected to be consistent with such limits. Without an independent measurement on the He ii background, and given our poor knowledge of the SEDs of quasars above 4 Ryd, the similarity of the models in Figure 14 (blue and red curves) illustrate that it will be extremely challenging to break the degeneracies between quasar lifetime, ionizing background, and photon production rate, given existing data or even data that might be collected in the future with HST/COS. Thus, in contrast with z ∼ 3, where an independent determination of the the He ii background from effective optical depth measurements allows one to infer the quasar lifetime from proximity zones, the proximity zones of z ∼ 4 quasars alone cannot independently constrain the quasar lifetime and ionization state of the IGM. Nevertheless, a degenerate combination of these parameters would still be extremely informative, and could be combined with other measurements to yield tighter constraints.

Previous studies of H i proximity zones at z ≃ 6 have concentrated on the location of the "edge" of the ionized regions in order to infer the unknown parameters governing proximity zones. In this section we adopt a similar technique to investigate if it is a better diagnostic tool than stacking, considered in the previous section, for constraining the properties of the IGM and quasars at z = 3.9.

We follow previous conventions (Fan et al. 2006) and define the size of the He ii proximity zone to be the location where the appropriately smoothed transmission profile crosses the threshold value F = 0.1 for the first time. For the choice of smoothing we follow conventions that have been adopted in the study of H i proximity zones at z ∼ 6 (Fan et al. 2006; Bolton & Haehnelt 2007a; Lidz et al. 2007; Carilli et al. 2010). Specifically, following the work of Fan et al. (2006), these studies smooth the spectra by a Gaussian filter with $\mathrm{FWHM}=20\;\mathring{\rm A}$ in the observed frame, which corresponds to a velocity interval Δv ∼ 700 km s⁻¹ or proper distance R_prop ≃ 0.97 Mpc. We adopt the same value of the smoothing scale in proper units R_prop^z=3.9 = 0.97 Mpc, which corresponds to a comoving scale of R_com = 4.75 cMpc at z = 3.9, or a velocity interval Δv ∼ 410 km s⁻¹. This is approximately twice the FWHM of HST/COS (for G140L grating).

Figure 15 shows the distribution of the proximity zone sizes determined in this way measured from a set of 1000 skewers for the two models shown in Figure 14 whose stacked spectra were degenerate. Given the large degree of overlap between the histograms for these two models, and the relatively small number $\;\simeq 8$ of z ≳ 3.5 quasars with HST/COS spectra it is clear that it will be extremely challenging to measure the value of quasar lifetime or He ii background using this definition of the proximity zone size. In Section 4 we discussed how density fluctuations also introduce scatter in the distribution of the proximity zone sizes and thus complicate our ability to infer parameters (see Figure 6). We conclude that this statistical approach of measuring the sizes of He ii proximity zones does not result in higher sensitivity to the properties of quasars or the IGM parameters at high redshift z ∼ 4.

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