# No Title Found

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## Resolving the curvature of the He ii Ly-$$\alpha$$ forest

While the curvature ratio, presented in Chapter 4, seems to represent a valid method to constrain the parameter $$\gamma$$ of the IGM temperature–density relation (Equation \ref{eq:TDrelation}), the contamination of the Ly-$$\beta$$ forest with lower redshift foreground Ly-$$\alpha$$ absorption may necessitate further refinement in the data analysis which may not be straightforward. In this Chapter we propose and explore the idea of using the He ii Ly-$$\alpha$$ forest, together with the corresponding H i Ly-$$\alpha$$ forest to measure $$\gamma$$. The aim of this Chapter is to understand the applicability of the curvature method to future He ii high-resolution forest spectra. In particular, we use hydrodynamical simulations to understand the specifics in terms of resolution and box size necessary to resolve the underdense gas traced by the He ii forest and statistically analyse the shape of its absorption features. The simulations used in Chapter 2, 3 and 4 are optimized in terms of resolution and box size ($$2\times 512^{3}$$ particles per $$10h^{-1}$$Mpc box) to accurately reproduce the absorption features derived from the typical overdensities probed by the H i Ly-$$\alpha$$ absorption, but so far have not been used to study the helium counterpart. We show here, through a convergence analysis, that the specific parameters of the simulations previously adopted are insufficient to apply the curvature statistic to the He ii forest.

## Motivation & proposal

The He ii Ly-$$\alpha$$ absorption ($$\lambda_{0}=303.78$$ Å) is accessible in the far UV (FUV) from space at $$z>2$$, due to the Galactic Lyman limit and, being free from H i Ly-$$\alpha$$ contamination, could represent a valid alternative to constrain $$\gamma$$ in the observable redshift range $$2\lesssim z\lesssim 3$$. The cross section for He ii Ly-$$\alpha$$ is 4 times smaller than for H i. Hence, from Equation \ref{eq:linecenter}, the ratio of optical depths can be define as:

$$\label{eq:HeHtau} \label{eq:HeHtau}\frac{\tau_{\text{He{\sc\,ii}}}}{\tau_{\text{H{\sc\,i}}}}=\frac{1}{4}\frac{b_{\text{H{\sc\,i}}}}{b_{\text{He{\sc\,ii}}}}\eta,\\$$

where $$\eta\equiv\frac{n_{\text{He{\sc\,ii}}}}{n_{\text{H{\sc\,i}}}}$$ is the ratio of He ii and H i number densities. For $$z\lesssim 2.8$$, after the end of the He ii reionization, where the absorption can be decomposed into distinct absorption features, $$\eta$$ can be approximated to the ratio of He ii to H i column densities (e.g. (citation not found: Heap00); (citation not found: Kriss01); (citation not found: Fechner07); (citation not found: Muzahid11)). The value of $$\eta$$ is connected with the hardness of the UVB and, if quasars dominate the ionizing background, at $$z\sim 3$$ it is expected to assume a value of $$\eta\sim 80-100$$ (e.g. (citation not found: Madau94); (citation not found: Haardt96)). Given the higher opacity of the He ii Ly-$$\alpha$$ transition with respect to the H i one, absorption systems which are optically thin in H i Ly-$$\alpha$$ will form saturated He ii Ly-$$\alpha$$ lines (e.g. (citation not found: Meiksin09)). That is, the He ii Ly-$$\alpha$$ forest at moderate optical depths is sensitive to lower gas densities.

In principle, by applying a curvature analysis to coeval H i and He ii Ly-$$\alpha$$ forests, it should be possible to trace the cosmic gas in different density regimes and obtain a measurement of $$\gamma$$ independent of $$T_{0}$$. Assuming a density-independent $$\gamma$$ and the same value of $$T_{0}$$ for the He and H in Equation \ref{eq:TDrelation}, at each redshift it is then possible to compute $$\gamma$$ from the following expression:

$$\frac{T(\bar{\Delta}_{\text{H}})}{T(\bar{\Delta}_{\text{He}})}=\left(\frac{\bar{\Delta}_{\text{H}}}{\bar{\Delta}_{\text{He}}}\right)^{\gamma-1},\\$$

where $$T(\bar{\Delta}_{\text{H}})$$ and $$T(\bar{\Delta}_{\text{He}})$$ are the temperatures at the characteristic densities traced by the H i and He ii Ly-$$\alpha$$ forests, respectively.

However, in practice, there is currently only a small sample of “clean” sightlines, i.e. those free from intervening higher redshift H i Lyman limit absorbers that would severely attenuate the quasar flux, making them unavailable for He ii Ly-$$\alpha$$ absorption studies. Moreover, the lower quality of the FUV spectra of the known He ii quasars ($$R\lesssim 20000$$ and $$S/N\lesssim 10$$ per resolution element; e.g. (citation not found: Zheng04); (citation not found: Syphers12); (citation not found: Worseck14)) currently does not allow this kind of constraint. Nevertheless, over the past 5 years great improvements have been possible in increasing the number of available He ii quasars, thanks to the discovery of new quasars at $$z_{\text{em}}>2.7$$ by the Sloan Digital Sky Survey, $$BOSS$$ and the FUV and near UV imaging of almost the entire extragalactic sky by $$GALEX$$ (e.g (citation not found: Syphers12); (citation not found: Worseck14)). Moreover, it is possible that future space-based UV telescopes with high resolution spectrographs may provide higher quality spectra for future analyses ((citation not found: Postman09)). Exploring the potentialities and limits of this new approach, based on the curvature statistic, could be particularly useful and may be also of some influence for the design of such future telescopes and spectrographs.

## The simulations

We performed new GADGET-3 hydrodynamical simulations with varied box size and mass resolution to extend the suite used in Appendix B for the curvature ratio convergence tests. The general parameters adopted for the cosmology and initial conditions are described in Section \ref{sec:sims} and the helium fraction (by mass) of the IGM is assumed to be $$Y=0.24$$ ((citation not found: Olive04)). All the runs assume the same thermal history (model C15 in Table \ref{table:simulations}) and are summarized in Table \ref{table:ConvkHe}.