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The curvature ratio

\label{chp:The curvature ratio}

The post-reionization thermal state of the intergalactic medium is characterized by a power-law relationship between temperature and density, with a slope determined by the parameter \(\gamma\). We describe a new method to measure \(\gamma\) using the ratio of flux curvature in the Lyman-\(\alpha\) and \(\beta\) forests. At a given redshift, this curvature ratio incorporates information from the different gas densities traced by Lyman-\(\alpha\) and \(\beta\) absorption, thereby breaking the degeneracy between \(\gamma\) and the temperature inferred at the gas mean density. It is relatively simple and fast to compute and appears robust against relevant sources of observational uncertainty. We apply this technique to a sample of 27 high-resolution quasar spectra from the Very Large Telescope, finding preliminary results broadly consistent with \(\gamma\sim 1.5\) over the redshift range \(z\sim 2.0\)\(3.5\). However, while promising statistical errors appear to be achievable in these measurements, uncertainties in the assumptions about the thermal state of the gas and its evolution may complicate this picture.

Boera, E., et al. 2015, MNRAS Letters, submitted

The results presented in Chapter 3 highlight the necessity of a precise constraint of the parameter \(\gamma\) of Equation \ref{eq:TDrelation} to completely characterize the IGM thermal evolution shown in Figure \ref{fig:slopes} in a model-independent way. Therefore, the second part of this thesis will be dedicated to this aim.

During and immediately after the H i and He ii reionisations (citation not found: McGreer15) (citation not found: Syphers14) (citation not found: Worseck14) cosmological simulations predict that \(\gamma\) may vary, becoming multi-valued and spatially-dependent according to the dynamics, heating and radiative transfer mechanisms involved (citation not found: Bolton04) (citation not found: McQuinn09) (citation not found: Meiksin12) (citation not found: Compostella13) (citation not found: Puchwein15). Despite considerable recent improvements, accurately simulating the effect of reionisation events on the IGM remains an open challenge. Furthermore, current measurements offer a somewhat confusing observational picture.

The main laboratory to detect variations in the \(T\)\(\rho\) relation has been the H i Lyman-\(\alpha\) forest in quasar spectra. Efforts to infer the thermal state of the IGM and search for signals of reionisation have used either line-profile decomposition to measure gas temperature as a function of column density (citation not found: Schaye00) (citation not found: Ricotti00) (citation not found: McDonald01) (citation not found: Rudie13) (citation not found: Bolton13) and a variety of statistical approaches which are valuable at higher redshifts, \(z>3\), where line fitting is problematic (citation not found: Theuns02) (citation not found: Becker07) (citation not found: Bolton08) (citation not found: Lidz10) (citation not found: Becker11) (citation not found: Boera14). While these methods probe wide redshift and density ranges (\(z\approx 1.6\)–5, \(\Delta\approx 0.3\)–8), large uncertainties in the measurements of \(T_{0}\) and \(\gamma\) may be caused by strong degeneracies between the effects of temperature and density on Ly-\(\alpha\) forest absorption.

One way to reduce these degeneracies is to constrain the \(T\)\(\rho\) relation by comparing Ly-\(\alpha\) and higher-order Lyman-series transitions, such as Ly-\(\beta\). Ly-\(\beta\) lines of moderate optical depth (\(\tau\sim 0.1\)–1.0) arise from higher overdensities at which Ly-\(\alpha\) lines are saturated. That is, the Ly-\(\alpha\)-to-\(\beta\) optical depth ratio is \(f_{\alpha}\lambda_{\alpha}/f_{\beta}\lambda_{\beta}=6.24\) (proportional to the ratios of oscillator strengths and rest wavelengths). Statistically comparing Ly-\(\alpha\) and \(\beta\) absorption is therefore a promising approach for measuring \(\gamma\). Indeed, using the Ly-\(\beta\) forest in IGM temperature measurements has been suggested in several theoretical works (citation not found: Dijkstra04) (citation not found: Furlanetto09) (citation not found: Irsic14). However, so far no practical attempt has been made to directly measure \(\gamma\) from a joint Ly-\(\alpha\) and \(\beta\) forest analysis. One challenge is that the Ly-\(\beta\) forest (the region between the Ly-\(\beta\) and Ly-\(\gamma\) emission lines) is entangled with absorption from Ly-\(\alpha\) at lower redshifts and it is difficult to separate the two. Hereafter we will refer to the total Ly-\(\beta\) plus the foreground lower redshift Ly-\(\alpha\) absorption as the Ly-\(\beta\)\(+\)\(\alpha\) region. However, assuming that these Ly-\(\beta\) and \(\alpha\) lines arise from physically uncorrelated IGM structures, a possible strategy to overcome this problem is to statistically compare the properties of the Ly-\(\beta\)\(+\)\(\alpha\) and corresponding Ly-\(\alpha\) regions.

In this Chapter we present a new method to constrain the slope of the \(T\)\(\rho\) relation using the two forest regions (Ly-\(\alpha\) and Ly-\(\beta\)\(+\)\(\alpha\)) in 27 high resolution quasar spectra. We use a statistic based on the flux curvature analysis of (citation not found: Becker11) and Chapters 2 & 3. These previous works demonstrated that the curvature method can measure the temperatures at the (redshift dependent) characteristic densities probed by the Ly-\(\alpha\) forest. However, as only a narrow density range is constrained, it has not yet been used to measure the slope of the \(T\)\(\rho\) relation (but see (citation not found: Padmanabhan15) for a recent theoretical analysis using the Ly-\(\alpha\) forest). Using hydrodynamical simulations, we show that, at each redshift, the ratio between the curvatures of corresponding Ly-\(\alpha\) and Ly-\(\beta\)\(+\)\(\alpha\) forest regions (where in the Ly-\(\beta\)\(+\)\(\alpha\) region the redshift always refers to the Ly-\(\beta\) absorption) is sensitive to differences in the IGM thermal state between the two density regimes. Averaged over many lines of sight, this curvature ratio appears to allow \(\gamma\) to be measured with little sensitivity to \(T_{0}\). We demonstrate the potential for this technique using 27 quasar spectra spanning the He ii reionisation redshift range. That potential is currently limited by assumptions underpinning the available simulation suite; we propose how these limitations can be overcome with future simulations and a refined data analysis approach.

The Chapter is organized as follows. In Section LABEL:sec:data is presented the selection of the subsample of quasar spectra used for this analysis from the larger sample of Table \ref{table:datatable}. The production of synthetic forest spectra from hydrodynamical simulations is explained in Section \ref{sec:sim}. The curvature ratio statistic and the main steps of the analysis are introduced in Section \ref{sec:method}. Possible systematic uncertainties in the evolution of the IGM thermal state are extensively discussed in Section \ref{sec:SystErr}. Our preliminary measurements of \(\gamma\) are presented in Section \ref{sec:gamma}. Finally, the results are summarized and interpreted in Section \ref{sec:conclusions2}.

The observational data

\label{sec:data}

The 27 quasar spectra were originally retrieved from the archive of the Ultraviolet and Visual Echelle Spectrograph (UVES) on the Very Large Telescope (VLT). They were selected on the basis of quasar redshift, wavelength coverage and signal-to-noise ratio from the sample of 60 spectra used in Chapters 2 and 3 for measuring the \(z=1.5\)–3 Ly-\(\alpha\) forest curvature. They have resolving power \(R\sim 50000\) and continuum-to-noise ratio \(\geq\)24 pix\({}^{-1}\) in the Ly-\(\alpha\) forest region (see Section \ref{sec:obs} for details). This level of spectral quality is necessary so that the curvature measurement is not dominated by noise and unidentified metal lines. Because our new method compares the curvature of the Ly-\(\alpha\) and \(\beta\)\(+\)\(\alpha\) forest regions, we extended these same criteria to the Ly-\(\beta\)\(+\)\(\alpha\) region at \(z=2.0\)–3.5, reducing the sample from 60 to 27 spectra. The quasar sample details are provided in Table \ref{table:subsample}.