The impact of spatial fluctuations in the ultra-violet background on intergalactic carbon and silicon

Spatial inhomogeneities in the spectral shape of the ultra-violet background (UVB) at the tail-end of HeII reionisation are thought to be the primary cause of the large fluctuations observed in the HeII to HI Ly-a forest optical depth ratio, tau_HeII/tau_HI, at z~2-3. These spectral hardness fluctuations will also influence the ionisation balance of intergalactic metals; we extract realistic quasar absorption spectra from a large hydrodynamical simulation to examine their impact on intergalactic SiIV and CIV absorbers. Using a variety of toy UVB models, we find that while the predicted spatial inhomogeneities in spectral hardness have a significant impact on tau_HeII/tau_HI, the longer mean free path for photons with frequencies above and below the HeII ionisation edge means these fluctuations have less effect on the SiIV and CIV ionisation balance. Furthermore, UVB models which produce the largest fluctuations in specific intensity at the HeII ionisation edge also have the softest ionising spectra, and thus result in photo-ionisation rates which are too low to produce significant fluctuations in the observed tau_SiIV/tau_CIV. Instead, we find spatial variations in the IGM metallicity will dominate any scatter in tau_SiIV/tau_CIV. Our results suggest that observational evidence for homogeneity in the observed tau_SiIV/tau_CIV distribution does not rule out the possibility of significant fluctuations in the UVB spectral shape at z~2-3. On the other hand, the scatter in metallicity inferred from observations of intergalactic CIV and SiIV absorption at z~2-3 using spatially uniform ionisation corrections is likely intrinsic, and therefore provides a valuable constraint on intergalactic metal enrichment scenarios at these redshifts.

A crucial ingredient in many of these studies are the ionisation corrections which convert the observed ionic abundances to metallicities. These corrections depend on the density and temperature of the IGM, as well as the intensity and spectral shape of the metagalactic ultra-violet background (UVB). The former may be modelled using cosmological hydrodynamical simulations which follow the enrichment history of the IGM (e.g. Theuns et al. 2002a;Cen et al. 2005;Oppenheimer & Davé 2006;Tescari et al. 2010). The latter is usually calculated using detailed models for the UVB intensity and spectral shape (Haardt & Madau 1996Fardal et al. 1998;Faucher-Giguère et al. 2009). Simulations of intergalactic metal enrichment typically assume the metagalactic ionising radiation field is spatially uniform. This is expected to be a reasonable approximation for hydrogen ionising photons at z ≃ 3, when the mean free path of ionising photons is much larger than the average separation between ionising sources (Meiksin & White 2004;Croft 2004).
However, the epoch of He II reionisation is thought to complete around z ≃ 3, when quasars are numerous enough to provide the hard photons (E > 4 Ryd) needed to doubly ionise helium (Furlanetto & Oh 2008;McQuinn et al. 2009). Indeed, observations of the H I and He II Lyα forest indicate the UVB spectral shape fluctuates significantly on scales of 4 − 10 Mpc at these redshifts, producing a wide range of values for the He II to H I column density ratio, η = NHeII/NHI ∼ 1 − 10 3 Fechner et al. 2006;Shull et al. 2004Shull et al. , 2010. These fluctuations are thought to arise from the small number of quasars lying within the short (He II ionising photon) mean free path expected at the tail-end of He II reionisation (Fardal et al. 1998;Bolton et al. 2006;Furlanetto 2009a), although small scale radiative transfer effects (Maselli & Ferrara 2005;Tittley & Meiksin 2007), collisionally ionised gas (Muzahid et al. 2010) or a significant number of thermally broadened lines ) may also produce column density ratios with η ≤ 10.
These spatial inhomgeneities in the UVB spectral shape should also have an impact on the ionisation balance of the metals in the IGM. 1 Intriguingly, independent of the spectral shape of the (spatially uniform) UVB used to make ionisation corrections, several studies find evidence for scatter in the IGM metallicity at fixed density (Rauch et al. 1997;Schaye et al. 2003;Simcoe et al. 2004). This scatter may be intrinsic, or it may instead be partially attributable to spatial fluctuations in the spectral shape of the UVB. On the other hand, Aguirre et al. (2004) find little evidence for inhomogeneity in the [Si/C] they infer from the observed optical depth ratio τSiIV/τCIV, implying that spatial variations in the UVB spectral shape are small. There is also some evidence from metal line analyses for a hardening of the UVB spectrum at z ≃ 3 (Savaglio et al. 1997;Songaila 1998Songaila , 2005Vladilo et al. 2003;Agafonova et al. 2005Agafonova et al. , 2007, perhaps due to the onset of He II reionisation. However, other studies are consistent with no evolution at the same redshifts (Kim et al. 2002;Boksenberg et al. 2003;Aguirre et al. 2004). Quantifying the effect of a spatially inhomogenous UVB on intergalactic metal absorption lines is thus desirable, both as a potential probe of He II reionisation and for examining the systematic bias, if any, spatially uniform ionisation corrections impart to constraints on the distribution of metals in the IGM.
Analytical arguments have previously highlighted the importance of small scale fluctuations in the hydrogen ionising radiation field for metals associated with high H I column density systems (Miralda-Escudé 2005;Schaye 2006). However, there has been no detailed examination of the impact of a spatially inhomogeneous UVB at E > 4 Ryd on metals 1 There is some evidence for a reduction in the number of C IV absorption systems in close proximity (≤ 0.3 Mpc) to quasars at 1.6 < z < 4 (Wild et al. 2008, see also Fox et al. 2008;Tytler et al. 2009). This line-of-sight proximity effect arises because of the 1/R 2 dependence of the ionisation radiation intensity around the quasar when the IGM is optically thin, and it will manifest itself even after He II reionisation has long completed. This effect is slightly different to the large scale fluctuations in the UVB spectral shape discussed here, which are expected to be important toward the tail-end of He II reionisation when the He II ionising photon mean free path is small. associated with lower H I column densities. A self-consistent treatment would require multi-frequency cosmological radiative transfer in a large ( > ∼ (200 Mpc) 3 ) volume coupled to a detailed chemodynamical model. In contrast, current stateof-the-art hydrodynamical simulations of IGM metal enrichment generally assume a spatially uniform UVB (e.g. Tescari et al. 2010;Shen et al. 2010;Cen & Chisari 2010;Wiersma et al. 2010, but see Oppenheimer et al. 2009) while large cosmological radiative transfer simulations of He II reionisation are still restricted to a treatment of hydrogen and helium only (Paschos et al. 2007;McQuinn et al. 2009).
In this paper we instead use a comparatively simple model which includes most of the relevant aspects of the spatially inhomogeneous UVB expected towards the tail-end of He II reionisation (e.g. Fardal et al. 1998;Bolton et al. 2006;Furlanetto 2009a). We combine this model with a large hydrodynamical simulation of the IGM to explore the impact of spatial fluctuations in the UVB spectral shape on intergalactic carbon and silicon at z = 3. In particular, we focus on the ratio of Si IV to C IV absorption, which is sensitive to the shape of the UVB either side of the He II ionisation edge Savaglio et al. 1997;Giroux & Shull 1997). We describe our hydrodynamical simulation and model for spatial fluctuations in the UVB spectral shape in section 2. Section 3 describes the four toy UVB models we use in this work. In section 4 we proceed to examine the impact of spatial fluctuations in the UVB spectral shape on synthetic absorption spectra constructed from our simulations. We compare our results to observational data in section 5, and conclude in section 6. All atomic data is taken from Morton (2003), solar abundances are from Asplund et al. (2009) and all distances are given in comoving units.

A simple model for spatial fluctuations in the UV background spectral shape
Our model for spatial fluctuations in the UVB spectral shape is similar to the approach described by Bolton et al. (2006) (see also Fardal et al. 1998;Furlanetto 2009a). The model applies to fluctuations in the UVB spectrum at the tail-end of He II reionisation only, following the overlap of He III regions when the He II ionising photon mean free path is set by the abundance of Lyman limit systems (Miralda-Escudé et al. 2000). There is good evidence to suggest that He II reioniation is indeed completing by z ≃ 3 (Shull et al. 2010;Becker et al. 2011), and this assumption should therefore be reasonable. Note, however, that the model will not apply to the patchy ionisation state of the pre-overlap IGM during the early stages of He II reionisation at z > 3, and does not include the effect of detailed, small scale radiative transfer effects on the IGM (Maselli & Ferrara 2005;Tittley & Meiksin 2007). Further modelling will thus be required to address the impact of spatial variations in the UVB spectral shape on intergalactic metals during the heart of He II reionisation. Firstly, to create a volume large enough to contain several He II ionising photon mean free paths, we stack the hydrodynamical simulation volume around the original 40h −1 Mpc box to create a cube 600h −1 Mpc on each side. We identify haloes in the volume using a friends-of-friends halo finder. Quasar luminosities are assigned to the haloes by Monte Carlo sampling the Hopkins et al. (2007) B-band quasar luminosity function at z = 3. The luminosities are uniquely mapped to the halo masses by rank ordering both quantities and assigning the brightest quasars to the most massive haloes. This approach enables us to distribute sources on large scales while following the ionisation state of gas in detail in the central 40h −1 Mpc box. However, it will only approximately model the clustering properties of the sources. Fortunately, the rarity of bright quasars means that Poisson fluctuations rather than clustering is expected to dominate the resulting UVB fluctuations (Furlanetto 2009a), so this limitation is unlikely to significantly affect our results.
We require the number of quasars in a volume, V, to satisfy where φ(LB) is the Hopkins et al. (2007) quasar luminosity function. The lower limit of the integral is Lmin = 10 43.5 erg s −1 , corresponding to MB = −20. The B-band luminosity of each quasar is converted into a luminosity below the H I Lyman limit using a broken power law spectrum (e.g. Madau et al. 1999) We consider two cases for the extreme UV (EUV) spectral index, αs. In the first case we assume a constant value of αs = 1.5, consistent with the mean obtained by Telfer et al. (2002) for radio quiet quasars, αs = 1.57 ± 0.17. However, a wide range of values are measured for the EUV spectral index (e.g. Zheng et al. 1997;Telfer et al. 2002;Scott et al. 2004). Consequently, we also consider a variable EUV spectral index. Scatter in αs is included by Monte-Carlo sampling a Gaussian with mean 1.5 and standard deviation 0.5 over the range 0 ≤ αs ≤ 3, similar to the dispersion measured by Telfer et al. (2002).
Once the luminosity and spectral energy distribution for each quasar is specified, the specific intensity of the UVB, J(r, ν) [erg s −1 cm −2 Hz −1 sr −1 ], at frequencies between the H I and He II ionisation edges, νHI < ν < νHeII, is given by The sum is made over all quasars in the 600h −1 Mpc volume, where |ri − r| is the distance of quasar i from r, which is always in the central 40h −1 Mpc box. Eq.
(3) assumes that the IGM is optically thin to H I ionising photons; this should be a reasonable approximation at z = 3. At frequencies above the He II ionisation edge, ν > νHeII, the specific intensity is instead where λHeII is the mean free path for photons at the He II ionisation edge and β is the power-law index describing the slope of the He II column density distribution, f (NHeII) ∝ N −β HeII . This expression assumes quasars are point sources surrounded by discrete, Poisson distributed absorbers ). We adopt β = 1.5, following the analogous distribution for H I absorbers (Petitjean et al. 1993;Kim et al. 2002). Note, however, the slope of the He II column density distribution is not constrained observationally and may deviate from the H I distribution for He II absorbers associated with high H I column densities, NHI > ∼ 10 16 cm −2 (Fardal et al. 1998;Faucher-Giguère et al. 2009). Finally, in this work we model the contribution to the UVB from quasars only. The non-thermal emission from quasars is thought to dominate the UVB at ν > ∼ νHeII, but star forming galaxies will make an increasingly significant contribution to the UVB at lower frequencies at z > 3 (Bianchi et al. 2001;Haehnelt et al. 2001;Bolton et al. 2005;Kirkman et al. 2005;Faucher-Giguère et al. 2008). However, the soft spectra of these sources are unlikely to significantly contribute to spatial fluctuations in the UVB spectral shape at E > 4 Ryd.

TOY UV BACKGROUND SPECTRA
The four toy UVB models used in this work are summarised in Table 1. The models are constructed by evaluating Eqs. (3) and (4) on a 14 3 grid within the central 40h −1 Mpc simulation box. The size of each grid cell, ∼ 4 Mpc, is chosen to match the scales of 4 − 10 Mpc on which the He II to H I optical depth ratio is observed to vary at z = 2.4 − 2.9 (Shull et al. 2010). This choice also represents the best compromise between resolution and efficiency; the construction of the photo-ionisation balance look-up tables (see section 4.1) becomes very time consuming if the number of grid cells much larger. However, note that the rarest, high amplitude fluctuations in the UVB spectral shape on small scales will not be fully captured in the simulation. In  Table 1. Models UVB1 (solid curve) and UVB4 (dot-dashed curve) are identical at E > 4 Ryd. Models UVB1 and UVB2 (dotted curve) are the same at E < 4 Ryd. Vertical tick marks correspond to the ionisation potentials of various ions. Right: The solid curve displays the spatially averaged specific intensity for model UVB1. The range in specific intensity, J(r, ν), encompassing 95 per cent of all fluctuations from the median is displayed by the dotted curves. For comparison, the dashed line shows the UVB background model of  for emission from galaxies and quasars at z = 3, renormalised to have specific intensity J(ν HI ) = 3.5 × 10 −22 erg s −1 cm −2 Hz −1 sr −1 . each grid cell the specific intensities are evaluated at 25 different photon energies spanning the range E = 1 − 100 Ryd. Lastly, all four models are renormalised by a factor of 0.74 to give identical mean specific intensities at the H I ionisation edge J(r, νHI) = 3.5 × 10 −22 erg s −1 cm −2 Hz −1 sr −1 . This renormalisation yields H I photo-ionisation rates of ΓHI ∼ 1.0 × 10 −12 s −1 (cf. ΓHI ∼ 1.3 × 10 −12 s −1 prior to the renormalisation), consistent with observational constraints derived from the Lyα forest opacity at z = 3 (Bolton et al. 2005;Faucher-Giguère et al. 2008).
Models UVB1 and UVB2 are constructed using a single EUV spectra index, αs = 1.5. However, the value of the mean free path at the He II ionisation edge, λHeII, is rather uncertain. In this work we adopt a fiducial value of λHeII = 30 Mpc at z = 3 (e.g. Fardal et al. 1998;Miralda-Escudé et al. 2000;Bolton et al. 2006), but the exact value will depend on the He II to H I column density ratio, η = NHeII/NHI, which exhibits significant fluctuations approaching z = 3 Shull et al. 2004Shull et al. , 2010. We thus also consider a smaller mean free path for model UVB2, λHeII = 15 Mpc, consistent with the constraints recently presented by Dixon & Furlanetto (2009) at z ≃ 3. Model UVB3 instead uses a variable EUV spectral index and λHeII = 30 Mpc. The final model, UVB4, is identical to model UVB1, aside from the spectral shape between 3 and 4 Ryd. Resonant absorption due to He II Lyman series lines will also produce significant spatial fluctuations in the UVB between He II Lyα at 3 Ryd and the He II ionisation edge at 4 Ryd (Madau & Haardt 2009). Model UVB4 mimics these fluctuations by assuming the UVB spectrum computed with Eqs. (3) and (4) follows a power law J(r, ν) = J(r, νLyα)(ν/νLyα) ξ for νLyα < ν < νHeII, where νLyα is the frequency of He II Lyα at 3 Ryd and ξ ≃ 8 log[J(r, νHeII)/J(r, νLyα)].
The spatially averaged spectra, J(r, ν) , for all four UVB models are displayed in the left hand panel of Fig. 1. The step at 4 Ryd in models UVB1, UVB2 and UVB3, and between 3 − 4 Ryd for model UVB4, is due to the attenuation of the spectrum by the He II opacity. The spectra recover their intrinsic shape at high energies, where the photon mean free path is long and the attenuation of the radiation by the intervening He II absorbers is negligible. Model UVB3, which assumes a variable EUV spectral index, is much harder than the other models at high energies. This model is dominated by the quasars with the hardest spectra, αs ∼ 0, in the simulation volume, even if these objects are not the brightest quasars at lower frequencies. The logarithm of the spatially averaged softness parameter for each model, defined as the ratio of the H I to He II photoionisation rates, S = ΓHI/ΓHeII, are listed in Table 1. Observational constraints on the softness parameter at z = 3 are dominated by the large uncertainty in the He II effective optical depth. Bolton et al. (2006) find 2.3 ≤ log S ≤ 3.2 at z = 3 by comparing hydrodynamical simulations of the H I and He II Lyα forest opacity to observational data. All four models are designed to be consistent with these constraints.
The right hand panel of Fig. 1 again displays model UVB1, but the dotted curves now show the range encompassing 95 per cent of all fluctuations around the median specific intensity. Fluctuations are largest close to the He II ionisation edge where the attenuation of the radiation is strongest, while towards higher frequencies the increasing mean free path produces much smaller departures from the median. For comparison, the dashed line shows the widely Figure 2. Left: The probability distribution for fluctuations in the specific intensity, J(r, ν HeII ), relative to the spatially averaged value in the simulation volume, J(r, ν HeII ) , for the four UVB models summarised in Table 1. The distributions are calculated using the specific intensity at the He II ionisation edge, where the fluctuations are largest. Note that the distributions for models UVB1 and UVB4 are identical at this frequency. Right: Cumulative probability distributions for fluctuations in the specific intensity of UVB2 at 4 Ryd. The four curves show the cumulative distributions obtained in pixel subsets corresponding to overdensities at log ∆ = −1, 0, 1 and 2.
used UVB model of  (HM01) for emission from galaxies and quasars at z = 3. The spectrum has been renormalised to match the specific intensity of UVB1 at 1 Ryd. The softness parameter of the HM01 spectrum is log S = 2.42, around 0.2 dex lower than UVB1. Although the features of the toy UVB spectra presented here are broadly similar to the more detailed model of HM01, there are some important differences. In particular, our toy models do not include more complex effects such as recombination emission (Haardt & Madau 1996;Fardal et al. 1998;Faucher-Giguère et al. 2009) or sawtooth absorption by the He II Lyman series (Madau & Haardt 2009) which are important to consider when performing detailed modelling of observational data.
The fluctuations in the specific intensity are quantified in more detail in Fig. 2 for all four UVB models. The left panel displays the probability distribution for the specific intensity relative to its spatially averaged value, log[J(r, νHeII)/ J(r, νHeII) ], at the He II ionisation edge. The distribution becomes broader for a smaller λHeII (UVB2) as the number of sources within one mean free path is reduced. Somewhat counter-intuitively, the variable EUV spectral index model (UVB3) produces fewer of the large, rare fluctuations from the mean than UVB1, which assumes a constant αs but is otherwise identical. This is because the three quasars which lie closest to the centre of our simulation volume (and hence dominate the rare, high amplitude fluctuations) have randomly assigned EUV spectral indices which conspire to reduce fluctuations in the specific intensity at 4 Ryd. Recall the quasar luminosities are normalised in the B-band, and in this instance the EUV spectra for the three closest quasars all have αs > 1.5. This produces lower specific intensities at the He II ionisation edge compared to model UVB1 with αs = 1.5, and hence fewer large fluctuations from the mean.
Lastly, the right panel of Fig. 2 displays cumulative distributions for log[J(r, νHeII)/ J(r, νHeII) ] obtained from model UVB2. The four separate curves show the cumulative distributions for subsets of pixels corresponding to regions in the IGM density field with overdensities at log ∆ = −1, 0, 1 and 2, respectively. The lack of a clear correlation with density 2 is mainly due the low number density of bright quasars, resulting in a radiation field which varies on scales much larger than the Jeans scale (see the discussions in e.g. Mc-Quinn et al. 2009;Furlanetto 2009b). Note, however, that fluctuations on scales < 4 Mpc (typically those associated with the radiation field in close proximity to the quasars) will be smoothed over, but these fluctuations are also rather rare. On the other hand, if additional physics such as small scale radiative transfer effects are important, or if He II ionising sources are far more numerous (e.g. galaxies), the ionising radiation field may be more closely correlated with the IGM density on small scales.
2 Bolton et al. (2006) found that the He II to H I column density ratio, η = N HeII /N HI , obtained from a Voigt profile analysis of synthetic absorption spectra was systematically higher in regions where τ HI > 0.05 compared to regions with lower H I optical depths (and hence densities). They suggested this implied the fluctuating UVB model they used was harder in higher density regions. However, this interpretation is in disagreement with our findings here. This earlier result is likely spurious, and probably results from the assumptions of pure turbulent broadening in the Voigt profile analysis ) and a spatially uniform H I photo-ionisation rate. The cumulative probability distribution for τ HeII /τ HI for all pixels in the sight-lines drawn from our hydrodynamical simulation. The curves correspond to models UVB1 (solid), UVB2 (dotted), UVB3 (dashed) and UVB4 (dot-dashed). Thick curves are obtained from spectra constructed using the spatially fluctuating UVB. Thin curves correspond to the optical depth ratios obtained using the spatially averaged UVB spectrum. Upper right: As for the upper left panel, but now for pixels with 0.02 < τ HI,HeII < 5 only. Models UVB1 and UVB4 are indistinguishable in both of these panels. Lower left: The cumulative probability distribution for τ SiIV /τ CIV for all pixels. Lower right: As in the lower left panel, but now for pixels with 0.02 < τ CIV,SiIV < 5 only. The thin and thick curves are almost indistinguishable in the lower panels. Note also the scales on the horizontal axes of the lower left and right hand panels are significantly different.

Construction of synthetic absorption spectra
We now turn to examine the impact of our spatially inhomogeneous UVB models on synthetic quasar absorption spectra. We use a simple a posteriori approach to including metals within our hydrodynamical simulation. We do not model the production and dispersal of metals in detail (see e.g. however, these metallicity constraints are derived from a statistical analysis of C IV and Si IV absorption lines which use an ionisation correction based on the HM01 UVB model. Different ionisation corrections obtained using significantly softer or harder UVB spectra will alter these constraints considerably. The equilibrium ionisation balance of the metals in our hydrodynamical simulation is calculated using the photoionisation code Cloudy (version 08.00), last described by Ferland et al. (1998). For each of our four spatially inhomogeneous UVB models, we create a five dimensional lookup table listing the ionisation fractions of hydrogen, helium, carbon and silicon as a function of gas temperature, T , hydrogen number density, nH, and position on the 14 3 UVB grid within our 40h −1 Mpc simulation box. Synthetic spectra are obtained from the hydrodynamical simulations by randomly selecting 1000 sight-lines from the simulation and performing an interpolation on the particle data weighted by the smoothing kernel (e.g. Theuns et al. 1998). Each line of sight is drawn parallel to the box boundaries and has 2048 pixels, each of which has an associated gas density, temperature, peculiar velocity and ionisation fraction. The ionisation fractions for H I, He II, C IV and Si IV are obtained by linearly interpolating the five dimensional look-up table. Absorption spectra are constructed for H I Lyα (λ1216), Lyβ (λ1026), Lyγ (λ973), Lyδ (λ950), He II Lyα (λ304), C IV (λλ1548, 1551) and Si IV (λλ1394, 1403).

Optical depth ratios
The ratio of the optical depths for two different species is sensitive to fluctuations in the spectral shape of the ionising background between their respective ionisation potentials. A ratio also has the advantage of being less susceptible to fluctuations in the IGM density field (Worseck et al. 2007;Furlanetto & Lidz 2010). We examine τHeII/τHI for the Lyα transitions and τSiIV/τCIV for λ1548 and λ1394 in our synthetic spectra. Both of these ratios have been widely used as probes of He II reionisation and the UVB spectral shape at z ≃ 3 (e.g. Songaila 1998;Schaye et al. 2003;Worseck et al. 2007;Shull et al. 2010). Note the latter will also be sensitive to fluctuations in the relative abundance of carbon and silicon.

The τHeII/τHI ratio
It is instructive to first revisit the effect of UVB fluctuations on the He II to H I optical depth ratio (but see also Fardal et al. 1998;Maselli & Ferrara 2005;Bolton et al. 2006;Tittley & Meiksin 2007;McQuinn et al. 2009;Furlanetto 2009a). The upper left panel of Fig. 3 shows the cumulative distribution for τHeII/τHI for all pixels in our synthetic spectra. This includes pixel optical depths that are in practice unobservable, either because of saturated absorption or noise. The thick curves correspond to the optical depth ratios for the four spatially fluctuating UVB models listed in Table 1. Models UVB1 and UVB4 are almost identical; neither H I or He II are significantly ionised by photons with E = 3−4 Ryd. Models with progressively harder spectra produce lower median values for the He II to H I ratio. The He II fraction decreases as the specific intensity at E > 4 Ryd increases; since the UVB is almost uniform at 1 Ryd in all our models (see the right panel in Fig. 1), fluctuations in the specific intensity at the H I ionisation edge will have little impact on τHeII/τHI. Note also that the smaller mean free path used for UVB2 produces a much wider range of τHeII/τHI values due to the wider range of fluctuations produced at the He II ionisation edge.
We may assess the extent to which τHeII/τHI is influenced by fluctuations in the UVB spectral shape by comparison to the ratio obtained for a spatially uniform UVB. The thin curves display the optical depth ratios obtained using the spatially averaged (i.e. spatially uniform) spectrum for each UVB model. The distributions for models including UVB fluctuations are indeed broader, especially toward higher values of τHeII/τHI. However, all four distributions still extend to low τHeII/τHI in the spatially uniform case. This behaviour is almost entirely due to variations in τHeII/τHI introduced by thermal broadening; the thermal width of an absorption line for species i is bi ∝ m −1/2 i , where mi is the mass of the particle. Fluctuations in the optical depth ratio therefore also originate from the wings of absorption lines which are predominantly thermally rather than turbulently broadened .
The upper right panel of Fig. 3 instead shows only the subset of pixels which have 0.02 < τHI,HeII < 5, roughly corresponding to range over which the optical depths may be reliably measured in observational data (e.g. Shull et al. 2010). These pixels correspond to 27.4, 4.1, 39.1 and 27.1 per cent of the total for models UVB1, UVB2, UVB3 and UVB4, respectively. The median τHeII/τHI for UVB2 is now significantly smaller. This is because the small number of optical depths with 0.02 < τHeII < 5 selected for UVB2 tend to probe regions where the He II fraction is lowest.
Recent observations of the He II Lyα opacity in the spectrum of the z ≃ 2.9 quasar HE2345−4342, made using the Cosmic Origins Spectrograph on the Hubble Space Telescope, reliably measure fluctuations in the optical depth ratio spanning the range τHeII/τHI ≃ 1.25−125 at 2.75 < z < 2.89 (Shull et al. 2010). In comparison, model UVB2 exhibits τHeII/τHI ≃ 0.8 − 248 (15 − 188 for 95 per cent around the median), while UVB1 and UVB4 have τHeII/τHI ≃ 0.4 − 77 (16 − 65) and UVB3 has τHeII/τHI ≃ 0.3 − 58 (13 − 46). We do not perform a more detailed comparison here, but as previously noted these results strongly suggest the observed τHeII/τHI fluctuations are largely attributable spatial variations in the He II ionising background. These fluctuations arise primarily from the small number of quasars within the short He II ionising photon mean free path, λHeII ∼ 15 − 30 Mpc, expected towards the tail-end of He II reionisation.

The τSiIV/τCIV ratio
The results for the τSiIV/τCIV ratio are shown in the lower panels of Fig. 3. The ionisation edges for Si IV and C IV are at 3.32 and 4.74 Ryd respectively; the ratio of these optical depths is thus sensitive to the shape of the UVB either side of the He II ionisation edge at 4 Ryd Giroux & Shull 1997). The left hand panel displays the distributions for all pixels in the synthetic spectra. Models with progressively harder spectra produce smaller median values for the ratio -the distribution for UVB3 in particular extends to significantly lower τSiIV/τCIV values. A harder UVB reduces both the Si IV and C IV ionisation fractions, but the Si IV fraction is lowered more rapidly as the break at 4 Ryd is reduced, leading to the smaller values for the τSiIV/τCIV ratio in UVB3. However, in contrast to the case for τHeII/τHI, the distributions for the fluctuating and uniform UVB models are almost identical. Only the UVB3 distribution displays a small difference for the lowest values of τSiIV/τCIV. This scatter in the τSiIV/τCIV ratio is instead dominated by the different dependences of the C IV and Si IV fractions on gas density, and to a much lesser extent variations in the absorption profiles in redshift space.
The lower right hand panel in Fig. 3 displays the subset of pixels with 0.02 < τSiIV,CIV < 5. These pixels represent a very small fraction of the total: 0.2, 0.3, 0.1 and 0.2 per cent for models UVB1, UVB2, UVB3 and UVB4, respectively, reflecting the much smaller volume filling factor of detectable C IV and Si IV absorption in the simulations. The medians of all four distributions are now significantly larger; most of the discarded pixels have C IV and Si IV optical depths which are too low to be observed. The distributions for models UVB1, UVB2 and UVB3 are now similar, suggesting that the τSiIV/τCIV ratio is a less sensitive probe of the UVB spectral shape for these pixels. Finally, all four distributions are again virtually indistinguishable from the distributions obtained with a spatially uniform UVB.

Gas temperature and density
We may gain insight into this behaviour by first examining the typical gas densities and temperatures which produce the absorption in our synthetic spectra. Contour plots of the optical depth weighted temperature against optical depth weighted overdensity are shown in Fig. 4 for spectra constructed using model UVB1. Clockwise from the upper left, each panel shows the temperature-density plane for H I, He II, Si IV and C IV optical depths with 0.02 < τ < 5. The number of pixels increases by an order of magnitude within successive contours. It is evident that H I and He II absorption primarily probes photo-ionised gas around mean density and below, with the He II absorption slightly more sensitive to gas in voids (Croft et al. 1997;McQuinn 2009). In contrast, C IV and especially Si IV tend to probe overdense regions in the IGM (e.g. Rauch et al. 1997). Nearly all the carbon and silicon absorption is furthermore associated with gas at T < 10 5 K.
The exact temperature and density of regions with measurable optical depths depends on the UVB model used. For example, for UVB3 (the hardest spectrum we consider) the median τHI weighted temperature and overdensity for pixels with 0.02 < τHI,HeII < 5 are Tτ = 7 100 K and log ∆τ = −0.6. For 0.02 < τCIV,SiIV < 5 the median τCIV weighted values are instead Tτ = 34 500 K and log ∆τ = 2.0. For UVB2 (the softest spectrum) the corresponding values are Tτ = 5 600 K, log ∆τ = −0.8 for 0.02 < τHI,HeII < 5 and Tτ = 43 400 K, log ∆τ = 1.6 for 0.02 < τCIV,SiIV < 5. Nevertheless, it is clear the τHI/τHeII and τSiIV/τCIV ratios probe rather different temperature and density regimes in the IGM. Furthermore, since the majority of the C IV and Si IV absorption in our models corresponds to gas with T < 10 5 K, collisional ionisation is unlikely to explain the similarity between the τSiIV/τCIV distribution for the spatially uniform and fluctuating cases shown in Fig. 3. Note, however, additional heating from feedback is not included in our hydrodynamical simulation, and we thus likely underpredict the amount of C IV and Si IV absorption due to collisionally ionised gas. However, because fluctuations in the photo-ionising background will be less important for the τSiIV/τCIV ratio if a larger fraction of these elements are in a hot, predominantly collisionally ionised phase with log ∆ > ∼ 2, this is unlikely to significantly alter our conclusions regarding the importance of inhomogeneities in the UVB spectral shape.

The UVB spectral shape
With the typical gas densities and temperatures of the absorption in hand, we may now examine the effect of varying the spectral shape of the UVB in more detail. It is instructive to simply rescale the spatially averaged UVB spectrum for each model by a constant factor above the He II ionisation edge: Figure 6. Individual ionisation fractions for several different ionisation states of carbon (upper panels) and silicon (lower panels for model UVB2 (left column) and UVB4 (right column). The ionisation fractions are plotted as a function of J(ν HI )/J(ν HeII ) for gas n H = 10 −3 cm −3 and T = 32 500 K. The horizontal extent of the shaded regions display the range encompassing 95 per cent of all fluctuations around the median J(ν HI )/J(ν HeII ) for the spatially inhomogeneous UVB models. J(ν) = J(r, ν) × 1 (ν < νHeII), where χ is a dimensionless constant. For model UVB4, this rescaling also modifies the spectral shape between E = 3 − 4 Ryd. Smaller (larger) values of χ thus produce softer (harder) UVB spectra and a stronger (weaker) break in the spectrum at 4 Ryd. Fig. 5 displays the Si IV to C IV ionisation fraction, fSiIV/fCIV, for the four UVB models considered in this work. Ignoring redshift space distortions, this is related to the optical depth ratio by where σi and ni are the absorption cross-section and number density for each species. The ratios are plotted as a function of the ratio of specific intensities at the H I and He II ionisation edges, J(νHI)/J(νHeII), following the above rescaling. The thick curves are computed using Cloudy for gas with T = 32 500 K, similar to the temperature of gas responsible for C IV and Si IV absorption in our simulated spectra. In each panel, the Si IV to C IV ratio is shown at four different gas densities, nH = 10 −2 , 10 −3 , 10 −4 and 10 −5 cm −3 . These have overdensities at z = 3 of log ∆ = 2.91, 1.91, 0.91 and −0.09, respectively. For comparison, the median C IV optical depth weighted overdensity probed by Si IV and C IV absorption with 0.02 < τCIV,SiIV < 5 for UVB1 is log ∆τ = 1.7.
The horizontal extent of the shaded regions in Fig. 5 correspond to the range encompassed by 95 per cent of all fluctuations from the median J(νHI)/J(νHeII) in the spatially fluctuating UVB models. This provides a rough guide to the range in fSiIV/fCIV present in the synthetic spectra as a function of density at fixed temperature. However, there are three important points to keep in mind. Firstly, variations in the gas temperature 3 will produce additional fluctuations in fSiIV/fCIV at fixed density. This is illustrated by the thin curves in Fig. 5, which are computed for gas with T = 65 000 K; this temperature lies toward the upper range of the scatter observed in Fig. 4. Secondly, only values of −1.6 < ∼ log(τSiIV/τCIV) < ∼ 1 (lower right panel, Fig. 3) are readily observable in our synthetic spectra, which from Eq. (6) corresponds to −1.8 < ∼ log(fSiIV/fCIV) < ∼ 0.8. Thirdly, the range of fSiIV/fCIV fluctuations will in practice be smaller than indicated by the range of J(νHI)/J(νHeII) shown by the shading in Fig. 5. Fluctuations in the UVB spectral shape are maximised at 4 Ryd where the mean free path is shortest (e.g. Fig. 1), whereas C IV and Si IV have ionisation potentials above and below 4 Ryd, respectively. Furthermore, the fluctuations do not have an equal probability of occurring in this range, and have a higher probability of lying close to the median (e.g. Fig. 2). The shaded regions thus correspond to an upper limit in the expected fSiIV/fCIV variation for each model.
With these points in mind, the fSiIV/fCIV ratio nevertheless shows only modest variation (< 1.0 dex) over the shaded range in Fig. 5 for nH = 10 −3 cm −3 , with the largest variations occurring for UVB2 and UVB4. Towards larger J(νHI)/J(νHeII) (i.e. softer spectra), fSiIV/fCIV noticeably flattens for all models, and becomes almost constant for UVB1, UVB2 and UVB3. Larger variations in fSiIV/fCIV do occur at lower densities, especially for model UVB3 where the Si IV to C IV ratio changes by ∼ 2.2 dex within the shaded range for nH = 10 −5 cm −3 . The Si IV to C IV optical depth ratio for all pixels in Fig. 3 is slightly broader than the uniform case for UVB3 at low values as a consequence. However, from Eq. (6) we infer that C IV and Si IV absorption from such low density gas is not detectable in the synthetic spectra.
In Fig. 6 the individual carbon (upper panels) and silicon (lower panels) ionisation fractions for several different ions are shown against J(νHI)/J(νHeII). The curves are again computed assuming densities and temperatures typical of the C IV and Si IV absorption in our simulated spectra, nH = 10 −3 cm −3 and T = 32 500 K. The left panels display the results for UVB2, while UVB4 is shown on the right. Models UVB1 and UVB3 display similar behaviour to UVB2 within the equivalent ranges for J(νHI)/J(νHeII). Note the C IV and Si IV fractions for model UVB2 are largely insensitive to the UVB spectral shape for log[J(νHI)/J(νHeII)] > ∼ 2 and 3, respectively. This is because the UVB is very soft, 3 We do not include the effect of large scale ( > ∼ 50 Mpc) spatial fluctuations in the temperature of the low density IGM expected during He II reionisation (Theuns et al. 2002b;McQuinn et al. 2009) resulting in photo-ionisation rates that are too low to significantly change the C IV and Si IV fractions in the higher density gas responsible for most of the absorption. On the other hand, the C IV fraction for model UVB4 is more sensitive to an increase in the strength of the 3 − 4 Ryd break. The break at 3 Ryd in this model produces a larger proportion of carbon in the form of C III relative to C IV for increasing J(νHI)/J(νHeII) (Agafonova et al. 2007;Madau & Haardt 2009;Vasiliev et al. 2010). However, Si IV again flattens towards larger values for J(νHI)/J(νHeII), moderating the change in the Si IV to C IV ratio. Furthermore, as the C IV fraction drops rapidly at log[J(νHI)/J(νHeII)] > 2 it will become progressively more difficult to detect the weakening C IV absorption.
Consequently, as the He II ionising photon mean free path becomes smaller toward higher redshift, the average UVB spectral shape softens and variations in the observed Si IV to C IV ratio do not become significantly more pronounced as a result of spatial fluctuations in the UVB spectral shape. The C IV and Si IV absorption systems typically originate from overdense regions with log ∆ ≃ 2 in our simulations, where the Si IV and C IV fractions change less rapidly with increasing J(νHI)/J(νHeII) compared to lower density. It is this behaviour, combined with the smaller fluctuations expected in the UVB spectral shape at frequencies above and below the He II ionisation edge, which explain the similarity between the Si IV to C IV optical depth ratios for the uniform and fluctuating models in Fig. 3. This is in striking contrast to the He II to H I ratio, where the same spatial variations in the UVB spectral shape significantly increase fluctuations in τHeII/τHI from lower density gas as the mean free path is lowered (Fardal et al. 1998;Bolton et al. 2006;Furlanetto 2009a). The absence of any observational evidence for fluctuations in the τSiIV/τCIV ratio (or the inferred silicon to carbon ratio) is thus not necessarily indicative of a spatially uniform UVB at z ≃ 3. On the other hand, slightly larger τSiIV/τCIV fluctuations are expected at lower densities, although the weaker absorption from these regions is much more difficult to detect. We now examine whether UVB fluctuations can be detected statistically in the observational data, and whether or not they contribute to the observed scatter in the IGM metallicity (Rauch et al. 1997;Schaye et al. 2003;Simcoe et al. 2004).

Method
The pixel optical depth (POD) procedure provides a powerful tool for statistically analysing metal absorption at low gas densities (e.g. Songaila 1998;Ellison et al. 2000;Pieri & Haehnelt 2004). Briefly, the procedure involves recovering optical depths for a base transition (e.g. H I Lyα absorption) on a pixel-by-pixel basis and pairing these with metal optical depths recovered at the same redshift. Given suitable calibration, typically achieved using a cosmological hydrodynamical simulation, information may then be derived on the abundance and distribution of metals in the IGM. The C IV POD analysis performed by Schaye et al. (2003) found evidence for scatter in [C/H] at fixed τHI, described by a normal distribution in [C/H] with standard deviation σ([C/H]) = 0.76 − 0.23(log ∆ − 0.5) at z = 3. Scannapieco et al. (2006) and Pieri et al. (2006) noted this scatter may be attributable to spatial variations in the metallicity of the IGM, although they did not rule out the possibility that spatial fluctuations in the UVB spectral shape (and hence the assumption of a uniform ionisation correction) may also play a role. On the other hand, Aguirre et al. (2004) performed a POD analysis for Si IV and found no evidence for additional scatter in the τSiIV/τCIV ratio beyond the aforementioned scatter in the metallicity. The study concluded that a uniform [Si/C] is favoured and that inhomogeneities in the spectral shape of the UVB are small. However, none of these studies modelled the effect of UVB fluctuations on the ionisation balance for carbon and silicon in detail.
We perform a POD analysis on our synthetic spectra using the procedure outlined by Aguirre et al. (2002). We consider the optical depths for H I Lyα through to Lyδ as well as C IV and Si IV absorption. For the POD analysis we construct absorption spectra which resemble high resolution, high signal-to-noise observational data (e.g. Schaye et al. 2003). The synthetic spectra are convolved with a Gaussian instrument profile of width 7 km s −1 , resampled onto pixels of width 3.1 km s −1 and Gaussian distributed noise with S/N = 100 is added. Continuum fitting errors are mimicked by performing an iterative continuum correction to the synthetic spectra. We compute the median transmitted flux in each synthetic line-of-sight and deselect all pixels below 1σ of this value, where σ is the rms noise amplitude in each pixel. The median flux is computed again for the remaining pixels, and the procedure is then continued until convergence is reached.

Comparison to observational data
The results of our POD analysis are presented in Fig. 7. The data points with 1σ error bars in the left panel correspond to the Schaye et al. (2003) observational measurements for Q1422+230. These data have a median absorption redshift of z = 3.225, at slightly higher redshift than our synthetic absorption spectra. From top to bottom, the three sets of data points correspond to the 84th, 69th and 50th percentiles of the recovered C IV optical depths; the 84th and 69th percentiles have been offset by +1.5 dex and +0.75 dex for clarity. The curves display the recovered C IV pixel optical depth against H I optical depth for spectra constructed using model UVB4, which we find has the largest impact on the recovered optical depths. The solid curves are obtained from the spatially inhomogeneous UVB model, while the dotted curves corresponds to the optical depths recovered from the spatially averaged spectrum. The curves are almost indistinguishable, indicating that the predicted spatial inhomogeneities in the UVB spectral shape approaching He II reionisation will not significantly impact on the ionisation correction for C IV.
The dashed and dot-dashed curves in Fig. 7 use the spatially uniform UVB model, but now also include different assumptions for the lognormal scatter in the IGM metallicity at fixed density. The dashed curves assume zero scatter, σ = 0, while the dot-dashed curves correspond to σ = 1.52. The impact on the recovered optical depths is especially prominent for the higher percentiles, suggesting that spatial variations in metallicity rather than the UVB spectral  Schaye et al. (2003) observational measurements for Q1422+230. From top to bottom, the data points correspond to the 84th, 69th and 50th percentiles of the recovered C IV optical depths. The 84th and 69th percentiles have been offset by +1.5 dex and +0.75 dex for clarity. The solid curves correspond to the pixel optical depths recovered using the spatially inhomogeneous UVB model, while the dotted curves shows the results for the spatially averaged (uniform) spectrum; these two curves are almost indistinguishable. The dashed curves assumes a spatially uniform spectrum, but include zero scatter in the metallicity at fixed density. The dot-dashed curves are for lognormal scatter drawn from a distribution with σ = 1.52, twice the fiducial value of σ = 0.76 . Right panel: The recovered Si IV optical depth as a function of C IV optical depth for UVB4. The simulations are compared to the observational data of Aguirre et al. (2004) for Q1422+230. The 84th and 69th percentiles are again offset by +1.5 dex and +0.75 dex for clarity, and the different curves are as described for the left hand panel.
shape will dominate any scatter. As noted by Schaye et al. (2003), the fiducial model with σ = 0.76 is in somewhat better agreement with the higher percentiles.
In the right hand panel we compare the same model to the POD measurements for Si IV and C IV presented by Aguirre et al. (2004) for the same quasar. Again, the recovered optical depths are very similar for the spatially inhomogeneous and uniform UVB models, and scatter in the metallicity has a much larger impact on the recovered optical depths. The fiducial model with σ = 0.76 is again in better agreement with the higher percentiles, although the agreement is poorer for the median. Note also, in contrast to the C IV and H I pixel optical depths, lognormal scatter in the metallicity at fixed density lowers each percentile for τSiIV as a function of τCIV. The explanation for this behaviour is that scatter in metallicity is added to both of the pixel optical depths, in contrast to the left hand panel of Fig. 7. Adding this scatter enables the detection of both C IV and Si IV absorption in more pixels at lower densities due to their higher metallicities; these pixels are otherwise hidden in the flat (noise and continuum error dominated) part of the correlation at log τCIV < −1.5. However, for this UVB model the Si IV fraction (and hence optical depth) decreases more rapidly with decreasing gas density relative to the C IV fraction. Consequently, a greater fraction of C IV optical depths with low τSiIV are now detectable, leading to the lowering of the τSiIV percentiles observed in Fig. 7.
Finally, we caution the reader not to take the differences between the observational data and simulations too seriously; the Schaye et al. (2003) and Aguirre et al. (2004) metallicities we have used to construct our spectra are derived under the assumption of a different UVB model. There is therefore no reason to expect our model to match the data exactly using these metallicities; the comparison here is instead largely illustrative and one may always fine tune the metallicity to enable a given UVB model to match the data.
The key point is that we clearly expect variations in the spatial distribution of metals to dominate any scatter in the C IV and Si IV optical depths at fixed density, even in the presence of the spatially inhomogeneous UVB expected at the tail-end of He II reionisation.

CONCLUSIONS
We use a large hydrodynamical simulation of the IGM combined with a toy model for spatial inhomogeneities in the UVB spectral shape to investigate the impact of spectral hardness fluctuations on the ionisation balance of intergalactic carbon and silicon at z ≃ 3. We construct synthetic quasar absorption spectra from the simulations, assuming the metallicity of the IGM traces the underlying gas density (Schaye et al. 2003;Aguirre et al. 2004). We carefully examine the impact of the UVB fluctuations on the Si IV and C IV optical depths. Four different spatially inhomogeneous UVB models which employ a variety of different assumptions for the UVB spectral shape are considered in our analysis. We reconfirm that fluctuations in the UVB spectral shape expected at the tail-end of He II reionisation have a significant impact on the He II to H I optical depth ratio (see also Fardal et al. 1998;Bolton et al. 2006;Furlanetto 2009a).
However, some of the lowest values for this ratio also result from lines which are predominantly thermally broadened . On the other hand, while the Si IV to C IV ratio is indeed sensitive to the average spectral shape of the UVB, we find the predicted fluctuations have little impact on τSiIV/τCIV measured in our synthetic spectra.
The majority of the detectable Si IV and C IV absorption in our synthetic spectra originates from regions with log ∆ ≃ 1.5−2 and T ≃ 35 000 K. These absorbers are predominantly photo-ionised in our simulations, and we may thus exclude collisional ionisation as a possible explanation for the small impact of UVB fluctuations on our simulated spectra. Instead, we find the ratio of observable Si IV to C IV optical depths varies relatively little considering the wide range of fluctuations in the UVB spectral shape. This is in part because of the longer mean free path for photons above and below the He II ionisation edge, which results in smaller fluctuations in the UVB spectral shape at these frequencies.
However, as fluctuations in the UVB spectral shape become larger as the He II opacity increases, the spatially averaged UVB spectral shape becomes softer; UVB models which produce the largest fluctuations in τHeII/τHI also have photoionisation rates which are too low to have a significant impact on the observed Si IV to C IV ratio. At lower gas densities, or for UVB models which predict a larger fraction of C III relative to C IV (e.g. for He II Lyman series absorption), the expected variation in the Si IV to C IV ratio can be slightly larger, but it is more difficult to detect due to the correspondingly smaller gas densities and/or fraction of triply ionised carbon produced.
Finally, we briefly examine the observational consequences for studies of the IGM metallicity using C IV and Si IV absorption at z ≃ 3. We perform a pixel optical depth analysis on our synthetic spectra, and find that the predicted UVB hardness fluctuations will have little impact on observations compared to spatial variations in the IGM metallicity. We conclude that the lack of any observed fluctuations in the τSiIV/τCIV ratio does not provide a stringent limit on the non-uniformity of the UVB spectral shape, and in particular does not preclude the possibility of He II reionisation completing around z ≃ 2 − 3. On the other hand, we confirm the observed scatter in the IGM metallicity inferred from C IV and Si IV absorption z ≃ 2 − 3 is likely to be intrinsic, reinforcing its potential as a powerful constraint on intergalactic metal enrichment scenarios.