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## The Curvature method at low redshifts

\label{chp:The Curvature method}

According to the photo-heating model of the intergalactic medium, He ii Ireionization is expected to affect its thermal evolution. Evidence for additional energy injection into the IGM has been found at $$3\lesssim z\lesssim 4$$, though the evidence for the subsequent fall-off below $$z\sim 2.8$$ is weaker and depends on the slope of the temperature–density relation, $$\gamma$$. Here we present, for the first time, an extension of the IGM temperature measurements down to the atmospheric cut-off of the H i Lyman-$$\alpha$$ forest at $$z\simeq 1.5$$. Applying the curvature method on a sample of 60 UVES spectra we investigated the thermal history of the IGM at $$z<3$$ with precision comparable to the higher redshift results.

Boera, E., et al. 2014, MNRAS, 441, 1916

As discussed in Section \ref{IntroThRE} the IGM thermal history can be an important source of information about reionizing processes that injected vast amounts of energy into this gas on relatively short timescales and, in particular, about the He ii reionization. While the direct observation, through the detection of the “Gunn–Peterson effect”, recently suggests the end of the He ii reionization at $$z\sim 2.7$$ (e.g. (citation not found: Shull10); (citation not found: Worseck11); (citation not found: Syphers13); (citation not found: Syphers14)), any current constraint on the physics of this phenomenon is limited by the cosmic variance among the small sample of “clean” lines of sight, those along which the He ii Ly-$$\alpha$$ transition is not blocked by higher-redshift H i Lyman limit absorption. For this reason indirect methods have been developed to obtain a detailed characterization of the He ii reionization.

The reionization event is expected to reheat the intergalactic gas leaving the characteristic signature of a peak, followed by a gradual cooling, in the temperature evolution at the mean gas density (e.g. (citation not found: McQuinn09)). In the last decade, the search for this feature and the study of the thermal history of the IGM as a function of redshift have been the objectives of different efforts, not only to verify this basic theoretical prediction and constrain the timing of He ii reionization, but also to obtain information on the nature of the ionizing sources and the physics of the related ionizing mechanisms.

To obtain measurements of the temperature of the IGM, studying the absorption features of the H i Ly-$$\alpha$$ forest has proven to be a useful method so far. The widths of the Ly-$$\alpha$$ lines are sensitive to the thermal broadening, among the other effects,. Therefore, using the comparison with cosmological simulations, different approaches have been able to extract from them information about the “instantaneous” temperature of the gas at the moment of absorption. However, as summarized in Section \ref{PreviousMethods}, the observational picture drawn by the results of previous efforts does not have a straightforward interpretation.

Recently, (citation not found: Becker11) developed a statistical approach based on the flux curvature. This work constrained the temperature over $$2\lesssim z\lesssim 4.8$$ of an “optimal” or “characteristic” overdensity, which evolves with redshift. The error bars were considerably reduced compared to previous studies, partially at the expense of determining the temperature at a single density only, rather than attempting to constrain the temperature–density relation. Some evidence was found for a gradual reheating of the IGM over $$3\lesssim z\lesssim 4$$ but with no clear evidence for a temperature peak. Given these uncertainties, the mark of the He ii reionization still needs a clear confirmation. Nevertheless, the curvature method is promising because it is relatively robust to continuum placement errors: the curvature of the flux is sensitive to the shape of the absorption lines and not strongly dependent on the flux normalization. Furthermore, because it incorporates the temperature information from the entire Lyman-$$\alpha$$ forest, this statistic has the advantage of using more of the available information, as opposed to the line-fitting method which relies on selecting lines that are dominated by thermal broadening.

Moreover, an injection of substantial amounts of thermal energy may also result in a change in the temperature–density relation (Equation \ref{eq:TDrelation}). The detailed study of this process has to take into consideration the effects of the IGM inhomogeneities driven by the diffusion and percolation of the ionized bubbles around single sources, and currently constitutes an important object of investigation through hydrodynamical simulations (e.g. (citation not found: Compostella13)). Some analyses of the flux PDF have indicated that the $$T$$$$\rho$$ relation may even become inverted (e.g. (citation not found: Becker07); (citation not found: Bolton08); (citation not found: Viel09); (citation not found: Calura12); (citation not found: Garzilli12)). However, the observational uncertainties in this measurement are considerable (see discussion in (citation not found: Bolton13)). A possible explanation was suggested by considering radiative transfer effects ((citation not found: Bolton08)). Although it appears difficult to produce this result considering only He ii photo-heating by quasars ((citation not found: McQuinn09); (citation not found: Bolton09)), a new idea of volumetric heating from blazar TeV emission predicts an inverted temperature–density relation at low redshift and at low densities. According to these models, heating by blazar $$\gamma$$-ray emission would start to dominate at $$z\simeq 3$$, obscuring the “imprint” of He ii reionization ((citation not found: Chang12); (citation not found: Puchwein12)). In the most recent analysis, with the line-fitting method ((citation not found: Rudie13); (citation not found: Bolton13)), the inversion in the temperature–density relation has not been confirmed, but a general lack of knowledge about the behavior of the $$T$$$$\rho$$ relation at low redshift ($$z<3$$) still emerges, accompanied with no clear evidence for the He ii reionization peak. A further investigation of the temperature evolution in this redshift regime therefore assumes some importance for obtaining constraints on the physics of the He ii reionization and the temperature–density relation of the IGM.

The purpose of the work presented in this Chapter and in Chapter 3 is to apply the curvature method to obtain new, robust temperature measurements at redshift $$z<3$$, extending the previous results, for the first time, down to the optical limit for the Lyman-$$\alpha$$ forest at $$z\simeq 1.5$$. By pushing the measurement down to such a low redshift, we attempt to better constrain the thermal history in this regime, comparing the results with the theoretical predictions for the different heating processes. Furthermore, the exploration of this new redshift regime allows to search for the end of the heating seen previously in (citation not found: Becker11), such an evidence is necessary to help bolster the interpretation as being due to the He ii reionization. We infer temperature measurements by computing the curvature on a new set of quasar spectra at high resolution obtained from the archive of the Ultraviolet and Visual Echelle Spectrograph (UVES) on the Very Large Telescope (VLT). Synthetic spectra, obtained from hydrodynamical simulations used in the analysis of (citation not found: Becker11), and extended down to the new redshift regime, are used for the comparison with the observational data. Similar to (citation not found: Becker11), we constrain the temperature of the IGM at a characteristic overdensity, $$\bar{\Delta}$$, traced by the Lyman-$$\alpha$$ forest, which evolves with redshift. We do not attempt to constrain the T–$$\rho$$ relation, but we use fiducial values of the parameter $$\gamma$$ in Equation \ref{eq:TDrelation} to present results for the temperature at the mean density, $$T_{0}$$.

While the actual temperature measurements and their discussion will be presented in Chapter 3, this Chapter intends to explain the curvature analysis procedure and the necessary preparation of the observational and synthetic spectra; it is organised as follows. In Section \ref{sec:obs} we present the observational data sample obtained from the VLT archive, while the simulations used to interpret the measurements are introduced in Section \ref{sec:sims}. In Section \ref{sec:curvature} the curvature method and our analysis procedure are summarized. In Section \ref{sec:analysis} we present the data analysis and we discuss the strategies applied to reduce the systematic uncertainties. Finally, the calibration and the analysis of the simulations from which we obtain the characteristic overdensitieis is described in Section \ref{sec:SimAnalysis}.