which may have a different spectral shape, we calculate the total H iionization rate, Γ =
Z ∞ ν0
4πJ(ν)σHI(ν)
hν dν . (5.4)
Here, ν0 is the frequency at the Lyman limit, J(ν) is the specific intensity at frequency ν, and σHI(ν) = 6.3×10−18(ν0/ν)3 cm2 is the H i absorption cross section. For the flux from a QSO, 4πJ(ν) = L(ν)/(4πR2), where R is the distance from the QSO. We adopt a power-law QSO continuum, fν ∝ν−α, with α= 0.5±0.5. For each QSO we compute the ionization rate at a distance of 10 proper Mpc, ΓQ10−12, where Γ−12 is in units of 10−12 s−1. These values are listed in Table 5.3. Errors include uncertainties in the observed flux as well as scatter in the power-law slope. We note that, due to the strong decrease in the absorption cross-section with frequency, ΓQ does not depend sensitively on the choice ofα within the range we assume.
fact that, although the mean transmitted flux does not correspond to the mean optical depth, the median transmitted flux should correspond to the median optical depth. Due to the different QSO redshifts and luminosities, however, we would not expect the size of the proximity region to be the same for all objects. The median is also very noisy, even for a large number of spectra. We will return to this approach, nevertheless, as a consistency check in§5.4.3.
5.3.1 Transmitted Flux Distribution
In order to make the best use of the optical depth information encoded in the Lyα forest, we will instead examine how the distribution of transmitted fluxes changes a a function of distance from a QSO. In analogy to the column density distribution used by other authors, we adopt the lognormalτ distribution measured by Becker et al. (2006a). A slowly-evolving lognormal Pτ(τ) was shown to correctly produce the observed distribution of transmitted fluxes over redshifts 1.7< z <5.8. The distribution depends on two parameters,µ=hlnτi, and σ= std dev(lnτ), as
Pτ(τ) = 1 τ σ√
2πexp
"
−(lnτ −µ)2 2σ2
#
. (5.6)
ForF =e−τ, this gives an expected distribution of transmitted fluxes,
PF(F) = 1
(−lnF)F σ√ 2πexp
"
−(ln (−lnF)−µ)2 2σ2
#
, (5.7)
for 0 ≤ F ≤ 1, 0 otherwise. The observed flux distribution is constructed by convolving PF(F) with a smoothing kernel that incorporates the noise properties of the spectrum (Becker et al. 2006a).
For a uniform UV background, an increase in Γ due to flux from the QSO will not changeσ, the logarithmic scatter in τ. However, µ will depend on Γ as eµ∝Γ−1. Hence, in the proximity region,
µ(z, R) =µbg(z)−ln
1 + R Req
!−2
, (5.8)
where µbg = hlnτbgi. For HIRES data, we use the redshift evolution of the lognormal
parameters measured by Becker et al. (2006a),
µHIRESbg (z) =−9.35 + 1.79(1 +z), (5.9)
σHIRESbg (z) = 4.19−0.46(1 +z). (5.10)
For ESI data, we have repeated the analysis of Becker et al. (2006a) after smoothing the HIRES data in that work to ESI resolution. The resulting parameters are
µESIbg (z) =−7.50 + 1.43(1 +z), (5.11)
σbgESI(z) = 2.72−0.28(1 +z). (5.12) Examples of the expected flux PDFs in the proximity region are shown in Figure 5.6. At z= 4−5, the values forµare very similar in the high- and moderate-resolution cases. This is not surprising since the median optical depth and, hence, the median transmitted flux, depends only on µ, and the median flux is not changed greatly by smoothing. However, σ is somewhat lower at moderate-resolution than at high-resolution, which produces slight differences in the flux PDFs. The lowersigmaarises from the fact that smoothing tends to remove pixels from the continuum and from saturated regions, which narrows the apparent distribution of optical depths. We will return to the issue of resolution in§5.6. As evidenced by the similarity of the expected HIRES and ESI flux PDFs in Figure 5.6, it has only a minimal impact on our final results.
5.3.2 Maximum Likelihood Method
We can estimate the size of a QSO’s proximity region by finding the value of Req that maximizes the probability that the observed fluxes near the QSO redshift are drawn from the expected flux distribution. This is analogous to the maximum likelihood method of Kulkarni & Fall (1993), which uses the distribution H i column densities. The expected transmitted flux PDF will evolve with distance from the QSO according to equations (5.7) and (5.8). For an individual QSO, we can construct a likelihood function of the form
L=Y
i
PF(Fi, zi, Ri, Req), (5.13)
Figure 5.6 Theoretical transmitted flux probability distribution functions in the proximity region of a QSO atz= 4.5. In each panel we plot the expected flux PDF, computed from equation (5.7), at the indicated distance from the QSO. The lognormal parametersµ(z, R) andσ(z) are computed from equations (5.8)–(5.12). Distances are expressed as a fraction of Req, the distance at which ΓQ= Γbg. The PDFs are smoothed to reflect a small amount of noise in the spectra (r.m.s= 3% of the continuum). Top panels are for HIRES resolution.
Bottom panels are for ESI resolution. As the distance to the quasar decreases, the flux distributions shift towards higher values. The PDFs at the two resolutions are similar at all distances, although the high-resolution PDFs tends to have more pixels atF ≈0.
wherei refers to the ith pixel in the proximity region. The proper distance from the QSO is calculated for small distances asR= ∆v/H(z), where ∆v is the velocity separation from the systemic redshift of the QSO, andH(z) is the Hubble parameter at redshiftz. For an ensemble of QSOs, the likelihood function becomes
L=Y
Q
Y
i
PF(FiQ, ziQ, RQi , RQeq), (5.14)
where Q refers to an individual QSO. The expected ReqQ for an individual QSO will scale with Γbg as RQeq = 10(ΓQ10/Γbg)1/2 Mpc. We can therefore estimate the UV background directly by maximizing L over a set of values for Γbg. Since the quantity we are actually measuring is a length scale, however, we instead compute L by considering a QSO with a fiducial ΓQ10 and allowReq for that QSO to vary. For each fiducialReq, the corresponding values ofRQeq for the QSOs in our sample are calculated according to their ionizing outputs, and our estimate of Γbg is the one that corresponds to the fiducial Req that maximizes L. This approach gives the same result as computing RQeq for each QSO directly over a range in Γbg. It makes it easier, however, to estimate non-symmetric errors in Γbg that arise from the non-linear dependence of Γbg on Req.