# No Title Found

AbstractNo Abstract Found

## Conclusions

\label{chp:Conclusions}

This thesis investigates the thermal state of the intergalactic medium in the redshift range $$z\simeq 1.5-3.8$$. This phase of the IGM evolution is particularly interesting because it is characterized by the second reionization event, the He ii reionization, that may have left detectable imprints in the evolution of the temperature at the mean gas density ($$T_{0}$$) and in the steepness of the power-law that defines the relation between the IGM temperature and density ($$\gamma$$). The picture drawn by previous observations of the thermal state of the IGM is still confusing, with some indication of the imprints of the He ii reionization but still much room for more definitive evidence. The findings presented in the previous Chapters can help to improve the understanding of the evolution of the IGM in this complex phase of its thermal history. While detailed descriptions of our measurements can be found in the Conclusions of Chapters 3, 4 and 5, in this final Chapter we summarise the main results, discuss them in the context of the status of the field, and describe new prospects for future follow-up work.

## Summary of the main results

The analyses presented in the previous Chapters aimed mainly to provide new constraints and methods to study the evolution of $$T_{0}$$ and $$\gamma$$ as a function of redshift. In particular, in Chapters 2 and 3 we provided new measurements of the IGM temperature at the characteristic gas overdensities probed by the H i Ly-$$\alpha$$ forest, $$T(\bar{\Delta})$$ (see Figure \ref{fig:TD}). With a sample of 60 high quality UVES spectra, for the first time we extended the curvature measurements (previously obtained for $$2\lesssim z\lesssim 4$$ by (citation not found: Becker11)) to the lowest optically-accessible redshifts, down to $$z\sim 1.5$$. We found that the values of $$T(\bar{\Delta})$$ increase for increasing overdensities probed by the forest towards lower redshifts, from $$T(\bar{\Delta})\sim 22670$$ K to 33740 K in the redshift range $$z\sim 2.8-1.6$$. These results are broadly consistent with the trend found by others at higher redshifts. Moreover, in the overlapping redshift range $$2.1\lesssim z\lesssim 2.9$$, our results (from a completely independent dataset) are in excellent agreement with the measurements of (citation not found: Becker11), demonstrating the self-consistency of the curvature method.

Using the temperature–density relation (Equation \ref{eq:TDrelation}) we translated the $$T(\bar{\Delta})$$ values to temperatures at the gas mean density $$T_{0}$$ to investigate the heating trend observed previously by others at redshifts $$2.8\lesssim z\lesssim 4$$. While, later in Chapter 4 we attempted to constrain the parameter $$\gamma$$, in Chapter 3 we assumed two reasonable values of $$\gamma$$ to obtain the measurements in $$T_{0}$$ (see Figure \ref{fig:slopes}). For both the choices of $$\gamma$$ we observed some evidence for a change in the tendency of the temperature evolution for $$z\lesssim 2.8$$, with indications of at least a flattening of the increasing temperature towards lower redshifts. In particular, for the case of a broadly constant $$\gamma\sim 1.5$$, the extension towards lower redshifts provided by our work shows the evidence for a decrease in the IGM temperature from redshifts $$z\sim 2.8$$ down to the lowest redshift $$z\sim 1.5$$. This supports tentative evidence for the same from (citation not found: Becker11). Such a reversal of the heating trend can be interpreted as the end of the IGM reheating process driven by He ii reionization, with a subsequent tendency of a cooling of a rate depending on the UV background.

In Chapters 4 and 5 we proposed two new possible applications of the curvature method to constrain the parameter $$\gamma$$. In particular, in Chapter 4 we presented and tested with hydrodynamical simulations the possibility of using the ratio of curvatures, $$\langle R_{\kappa}\rangle$$, in the Ly-$$\alpha$$ and $$\beta$$ forests to trace simultaneously two different gas density regimes and obtain measurements of the slope of the $$T$$$$\rho$$ relation. We applied this technique to a subsample of 27 UVES spectra selected for their high S/N ($$>24$$ per pixel) in both the Ly-$$\alpha$$ and $$\beta$$ regions and we analysed the strengths and weakness of this method. The technique, relatively fast and simple to compute, appears robust against observational uncertainties in the noise level, metal contamination and effective optical depth. The evolution of $$\langle R_{\kappa}\rangle$$, measured from our observational spectra in 3 redshift bins covering $$z=2.0-3.8$$, matches that derived from our nominal simulations with $$\gamma\sim 1.5$$. The statistical errors obtained from the 27 spectra are promising: in the absence of any systematic errors, the $$\approx$$6 per cent statistical uncertainties in our $$\langle R_{\kappa}\rangle$$ measurements would translate to a $$\lesssim$$10  per cent uncertainty in $$\gamma$$ in $$\Delta z\sim 0.6$$ bins. This would be competitive with the most precise results available in the literature derived using line decomposition ((citation not found: Rudie13); (citation not found: Bolton13)). However, we did not present $$\gamma$$ measurements with formal uncertainties because the nominal $$\gamma$$–log$$\langle R_{\kappa}\rangle$$ relation, used to translate $$\langle R_{\kappa}\rangle$$ measurements to $$\gamma$$ values, may also be sensitive to assumptions in the simulations about the evolution of the IGM thermal state that need to be addressed in detail.

Finally, in Chapter 5 we also proposed the possibility of using a double curvature constraint, from coeval H i and He ii Ly-$$\alpha$$ forests, to obtain measurements of the IGM temperature at the two different density regimes probed by these transitions and constrain $$\gamma$$, independently of $$T_{0}$$, in the redshift range $$2\lesssim z\lesssim 3$$. Currently, it is clear that any curvature analysis of the He ii forest is prevented by the low quality of the observed spectra; however, future instruments and space telescopes may improve dramatically the availability of high resolution and S/N spectra. We then analysed the available hydrodynamical simulations to understand if the specific parameters (i.e. box size and mass resolution), previously adopted to reproduce the H i Ly-$$\alpha$$ absorption features, could also be used to study the helium counterpart. Our convergence tests show that the underdense regions probed by the He ii Ly-$$\alpha$$ forest would require future refinements of the current suite of simulations.

Starting from the general question that opened the Introduction of this thesis, in the following Section we will explain the contribution of our results in addressing some more specific queries that have represented the key investigation subjects in this field for more than a decade.

## What is the thermal state of the IGM at low redshift?

The thermal state of the intergalactic medium is defined by its temperature–density relation that, in the simplest scenario, is represented by a power-law (Equation \ref{eq:TDrelation}). Understanding the evolution of the IGM thermal state therefore relies on measuring the evolution of the parameters $$T_{0}$$ and $$\gamma$$ of Equation \ref{eq:TDrelation}. For more than a decade, significant efforts have been invested in developing methods that would allow the two parameters to be determined independently of each other using quasar absorption lines. However, the necessity of using complex simulations and large statistical samples of high quality observational spectra, makes it particularly difficult. Nowadays, the main challenge is represented by the requirement to reduce the systematic and statistical uncertainties in the measurements to a level that would allow to clarify the confusing picture, drawn so far by previous estimations.

Including the constraints obtained in this work, Figure \ref{fig:finals} provides a summary of the state of the field to date. Panel (a) presents the observed evolution of $$T_{0}$$ with redshift while panel (b) shows the constraints on the evolution of $$\gamma$$ (reproduction of Figure \ref{fig:gamma}). Note that for clarity, in both the panels we could not include all the measurements available in the literature; rather, the comparison presented here is intended to provide a representative idea of the size of the error bars, obtained using different approaches, and of the dispersion of the values of temperature and $$\gamma$$ measured in the broad redshift range $$z\simeq 1.5-5$$.

\label{fig:finals}Summary of observational status of the IGM thermal evolution from this work and selected literature. Panel (a): Temperature at the mean density, $$T_{0}$$, inferred under the assumption of $$\gamma\simeq 1.5$$ in this work (green points) and in the work of (citation not found: Becker11) (black points) using the curvature method. The results of the line-profile analysis of (citation not found: Schaye00) (grey points) and wavelet analysis of (citation not found: Lidz10) (light green triangles ) are also presented for comparison. Note that the error bars quoted by different authors are either 1- or 2-$$\sigma$$ (shown in the symbol key). Panel (b): $$\gamma$$ measurements and statistical uncertainties obtained from the curvature ratio in this work (green points) compared with the line-fitting measurements of (citation not found: Schaye00) (grey points), (citation not found: Ricotti00) (azure triangles), (citation not found: McDonald01) (light blue squares) and (citation not found: Bolton13) (red circle). The non-equilibrium model of (citation not found: Puchwein14) (red dashed line) and the radiative transfer simulations of (citation not found: McQuinn09) with an early ($$z\gtrsim 6$$; blue dotted line) and late ($$z\sim 3$$; green dot-dashed line) He ii reionisation are also presented.

This summary figure allows us to put our results in context and address the following main questions.

## Is there any evidence for a peak in $$T_{0}$$ due to the He ii reionization?

The results of (citation not found: Schaye00), presented in Figure \ref{fig:finals} panel (a), were historically the first to show a large scatter, and so a possible variability, of the temperature at the mean gas density, measured through a Voigt profile decomposition of H i Ly-$$\alpha$$ absorption lines at redshifts $$z\simeq 2-4.5$$. However, the clearly large uncertainties characterizing these results do not allow any conclusion about the real presence of a temperature peak: a heating of the IGM during reionization followed by a gradual cooling as the temperature, at the end of the reionization, returns to a thermal asymptote. Similarly, large uncertainties characterized subsequent statistical analysis, such as the wavelength decomposition obtained by (citation not found: Lidz10). These large error bars are partially driven by the strong degeneracy between the effects of temperature and density on defining the shapes of the H i absorption lines.

In its attempt to reduce the error bars characterizing these measurements, the curvature method shows a considerable improvement. The first results obtained with this method by (citation not found: Becker11) in the redshift range $$z\simeq 2-5$$, present significantly smaller error bars that allow to trace the evolution of the temperature in a much clearer way. In particular, the curvature method measures with high precision (typically 1$$\sigma\lesssim 5$$ per cent) the temperature at the characteristic overdensity probed by the Ly-$$\alpha$$ forest, $$T(\bar{\Delta})$$ (see Figure \ref{fig:TD}), but it does not simultaneously constrain $$\gamma$$ because it only accesses a small range of densities at each redshift. The translation of the $$T(\bar{\Delta})$$ measurements into $$T_{0}$$ values therefore involves an assumption about the value of the parameter $$\gamma$$ which is itself independently very poorly constrained. In the work of (citation not found: Becker11), while strong evidence for an increase in $$T_{0}$$ between redshift $$z\sim 4$$ and $$z\sim 2.8$$ was found, the uncertainties in the value of $$\gamma$$ prevented an unambiguous identification of an actual peak: $$T_{0}$$ could continue to increase for $$z\lesssim 2.8$$ or could start to flatten, depending on the $$\gamma$$ assumed in the translation.

Our curvature analysis, presented in Chapters 2 and 3, investigated the temperature evolution of the IGM down to the lowest optically-accessible redshift ($$z\sim 1.5$$), providing further evidence for at least a flattening, and possibly a reversal, of the increasing IGM temperature for $$z\lesssim 2.8$$. In particular, for the $$T_{0}$$ case with $$\gamma\sim 1.5$$ (shown in Figure \ref{fig:finals}), our results provide evidence for a decrease in the IGM temperature that could be interpreted as an imprint of the completion of the reheating process connected with the He ii reionization. Note that, using an independent observational dataset, our results in the overlapping redshift range ($$2.1\lesssim z\lesssim 2.9$$) are broadly consistent with the previous curvature measurement of (citation not found: Becker11) but show slightly higher values; this difference can be attributed to the different optical depth calibration, as demonstrated in Appendix A. The curvature ratio statistic, developed in this work, shows preliminary results that are consistent with $$\gamma\sim 1.5$$ in the redshift range of interest. If this value is confirmed by future refinements of the method (see Section \ref{Rref}), the actual evolution of $$T_{0}$$ presented in Figure \ref{fig:finals} will be finally known to high precision.

## Is there any evidence for a flattening in the $$T$$–$$\rho$$ relation at low redshifts?

While a tight constraint on the evolution of the parameter $$\gamma$$ of the $$T$$$$\rho$$ relation would be of vital importance to measure the absolute value of $$T_{0}$$ and obtain the redshift of the peak with substantial precision, the actual observations presented in panel (b) of Figure \ref{fig:finals} show an even more confused scenario than for $$T_{0}$$. Large error bars dominate again the line-fitting measurements and, while in some cases some statistical evidence for a decrease in the value of $$\gamma$$ at $$z\sim 3$$ has been presented (e.g. (citation not found: Schaye00)), this scenario has not been supported by other analyses (e.g. (citation not found: McDonald01)). Moreover, several flux PDF analyses (not reported in Figure \ref{fig:finals}) suggested that the temperature–density relation could become inverted, with a $$\gamma<1$$ at $$z\lesssim 3$$ (e.g. (citation not found: Becker07)). A mild flattening, with a variation in $$\gamma$$ from its asymptotic value ($$\gamma\sim 1.6$$) of $$\Delta\gamma\lesssim 0.2$$ has been predicted by photo-heating models which include radiative transfer (e.g. (citation not found: McQuinn09)), but an inverted $$T$$$$\rho$$ relation is difficult to explain without assuming more exotic models such as blazar heating ((citation not found: Puchwein12)). An improvement in the statistical uncertainties has been recently demonstrated at $$z=2.4$$ by the measurement of (citation not found: Bolton13) (which was a recalibration of that by (citation not found: Rudie13)). However, that involved the profile fitting analysis of $$\sim 6000$$ individual H i absorbers, which is a large undertaking.

In Chapter 4 we tested a new approach to obtain statistically competitive $$\gamma$$ measurements over a large redshift range ($$z\simeq 2-3.8$$). This statistic, the curvature ratio, incorporates information from the two gas density regimes traced by the Ly-$$\alpha$$ and $$\beta$$ forest along the same line of sight to a quasar. Thus, if averaged over a large sample of spectra, it allows measurement of $$\gamma$$, this is largely independent of $$T_{0}$$ and different to all previous techniques. Relatively simple to compute, the curvature ratio appears robust against observational uncertainties in the noise level, effective optical depth and metal contamination.

Testing this application of the curvature statistic for the first time, a sample of 27 quasar spectra provided a $$\lesssim 10$$ per cent statistical uncertainty in $$\gamma$$ in $$\Delta z\sim 0.6$$ redshift bins. Such error bars are competitive with the recent results of (citation not found: Bolton13). However, we do not present here the $$\gamma$$ measurements with final error bars; the error bars presented in Figure \ref{fig:finals} do not incorporate yet the possible systematic uncertainties related to the sensitivity of our method to the assumptions about the thermal state of the IGM and its evolution. These assumptions have been necessarily adopted in the simulations used to calibrate our curvature ratio measurements and could be overcome with some future refinements of the analysis technique (see Section \ref{Rref}). Nevertheless, the evolution of the curvature ratio computed from the real spectra matches that derived from our nominal simulations which assume a $$\gamma\sim 1.5$$ with very little evolution in the redshift range $$z=2-3.8$$. That is, the same model used in Chapter 3 to translate the $$T(\bar{\Delta})$$ measurements into $$T_{0}$$ values (shown in Figure \ref{fig:finals}) seems to reproduce correctly the observational value of our new statistic, giving some further confidence in our temperature measurements.

Similar to the previous measurement of $$\gamma$$ by (citation not found: Bolton13), our preliminary $$\gamma$$ measurements do not support an inverted temperature–density relation. However, mild decreases in $$\gamma$$ at redshifts $$z\simeq 2-4$$ cannot be excluded at this stage. Distinguishing between models of the IGM thermal history (e.g. the late and early reionization models of (citation not found: McQuinn09)) could be improved in the future by using larger samples of high quality spectra. Future, refined curvature ratio analyses could then represent a valid companion statistic to the curvature method itself, allowing to combine $$T(\bar{\Delta})$$ and $$\gamma$$ measurements for a precise, first order characterization of the thermal evolution of the IGM.

## Is there any evidence for blazar heating mechanisms at low redshifts?

The evidence for a possible inverted $$T$$$$\rho$$ relation obtained in previous flux PDF observations (e.g. (citation not found: Becker07)), suggested the idea that other, more exotic heating mechanisms could dominate the photo-heating at $$z\lesssim 3$$ (e.g. (citation not found: Chang12); (citation not found: Puchwein12)). According to these models, blazar $$\gamma$$-ray emission would produce volumetric heating of the IGM. Because these models have a heating rate that can be considered independent of the density, if it dominates the photo-heating at $$z\lesssim$$3, it could explain naturally an inverted $$T$$$$\rho$$ relation at low-redshift and obscure the “imprint” of the He ii reionization in the IGM temperature evolution. The blazar heating models’ $$T$$$$\rho$$ relations, at each redshift, can be parametrized with a power-law only for a limited range of overdensities that may not always cover the range of characteristic overdensities probed by curvature measurement. Therefore, the only fair, model-independent comparison between our temperature results and the blazar heating model predictions is directly with the $$T(\bar{\Delta})$$ values at the same redshift-dependent characteristic overdensities. As presented in Figure \ref{fig:BB2}, from direct comparison, our $$T(\bar{\Delta})$$ measurements are in agreement with the intermediate blazar heating model of (citation not found: Puchwein12). However, in contrast, our preliminary measurements of $$\gamma$$ and the evolution as a function of redshift of the curvature ratio statistic are in considerable tension with possible scenarios of $$\gamma<1$$. Given these results, we cannot currently confirm or rule out any specific thermal history, as model-independent measurements of the temperature at the mean density would be necessary.

Nevertheless, more recently, new simulations that take into account the clustering of the sources, showed that the variability in the blazar heating is significant and leads to an “important scatter” in the temperature at $$z\gtrsim 2$$ ((citation not found: Lamberts15)). If this scatter is confirmed by future analyses of the Ly-$$\alpha$$ forest properties, these more sophisticated blazar heating models could take into account both the increase in the temperature observed in our results and a non-inverted $$T$$$$\rho$$ power-law.

## Future work

Despite the considerable improvements over the last decade, a precise characterization of the IGM thermal state at low (and high) redshift has still to be completed. Such a goal, if achieved, would be of great importance for understanding the physics of the IGM, the reionization epochs, their sources and in general the mechanisms of structure formation. In this section we present some short-term and long-term ideas to improve IGM thermal state measurements using the information coming from the shapes of quasar absorption lines.

### The curvature ratio statistic refinement

\label{Rref}

The curvature ratio statistic, $$\langle R_{\kappa}\rangle$$, was demonstrated in Chapter 4 to be a promising tool for obtaining independent constraints on $$\gamma$$. It is relatively easy to compute, can be applied to a vast range of redshifts ($$z\simeq 2-5$$) and is robust against the main observational uncertainties. However, due to the foreground Ly-$$\alpha$$ absorption at lower redshifts which contaminates the Ly-$$\beta$$ forest, the simulations used to calibrate and translate curvature ratio measurements in $$\gamma$$ values necessarily rely on assumptions about the IGM thermal evolution between the Ly-$$\beta$$ and the contaminant Ly-$$\alpha$$ redshifts. As demonstrated in Chapter 4, if the $$\gamma$$$$\log\langle R_{\kappa}\rangle$$ relation is sensitive to these assumptions, it could produce systematic uncertainties in the final $$\gamma$$ estimates.

Reliable sensitivity tests can only be conducted using new, self-consistent hydrodynamical simulations that assume different scenarios of $$T_{0}$$ and $$\gamma$$ evolution. Similar to the ‘T15fast’ and ‘T15slow’ simulations of (citation not found: Becker11), used in Section \ref{sec:SystErr} for self-consistently testing the effects of rapid changes in the evolution of $$T_{0}$$, the new models need to incorporate possible rapid variations of $$\gamma$$ between the Ly-$$\beta$$ and the foreground Ly-$$\alpha$$ redshifts. The effects on the $$\gamma$$$$\log\langle R_{\kappa}\rangle$$ relation of simultaneous rapid changes in the value of $$\gamma$$ and $$T_{0}$$ also need to be investigated. While the exploration of the entire possible parameter space would require a large undertaking, we estimate that a suite of $$\sim 10$$ new models would be sufficient to establish the magnitude of these effects in reasonable scenarios of $$\gamma$$ and $$T_{0}$$ evolution (with maximum variations of $$\Delta\gamma\simeq 0.4$$ and $$\Delta T_{0}\simeq 10000$$K in a $$\Delta z\simeq 0.6$$ redshift range). Once fully explored, the formal magnitude of possible systematics can be added to the statistical uncertainties for a final error budget, yielding reliable $$\gamma$$ measurements from this curvature ratio approach.

Alternatively, these systematics may be largely mitigated by identifying and masking out the contaminant foreground Ly-$$\alpha$$ absorption from the Ly-$$\beta$$ forest spectra. One possible approach to do so is to automatically fit the “pure” Ly-$$\alpha$$ forest and then the contaminant Ly-$$\alpha$$ lines in the Ly-$$\beta$$ forest after masking or dividing out the Ly-$$\beta$$ lines. Then, one could divide out these contaminant Ly-$$\alpha$$ profiles to allow a refined joint analysis of a relatively “clean” Ly-$$\beta$$ forest and its corresponding Ly-$$\alpha$$ forest. Some specific fitting problems have to be carefully taken into account (e.g. saturated Ly-$$\alpha$$ lines or metal contaminants) but this procedure, executed over large statistical sample, could help to reduce the possible systematic uncertainties analysed in Section \ref{sec:SystErr}.

Given the potential of the curvature ratio method in obtaining competitive statistical uncertainties on $$\gamma$$, its refinement represents a high priority in the short term.

### Global $$\gamma$$ constraint using a large sample of quasar spectra

After its refinement, the curvature ratio method could be applied to a larger sample of quasar spectra. Given the statistical uncertainties obtained for the sample of 27 UVES spectra used in this work, we estimate that a sample of a further $$\sim 100$$ quasar spectra, covering absorption redshifts $$z\sim 2.7-4$$, would improve the precision of the $$\gamma$$ measurements to $$1\sigma\simeq 0.05$$ in $$\Delta z\simeq 0.5$$ redshift bins. This precision would allow a $$\simeq 3\sigma$$ discrimination between the (citation not found: McQuinn09) models of Figure \ref{fig:finals}, finally clarifying much of the physics behind the evolution of $$\gamma$$ during the He ii reionization. Theoretically, the curvature ratio can be also used to measure $$\gamma$$ at higher redshift (up to $$z\sim 5$$). A constraint on the evolution of $$\gamma$$ up to $$z\sim 5$$ represents the highest redshift limit for this kind of measurement, due to the rapid increase of the H i Ly-$$\alpha$$ optical depth close to the end of the hydrogen reionization.

The large sample of high signal-to-noise ratio spectra, necessary for these measurements, is already available in the archives of the high resolution spectrographs VLT/UVES, Keck/HIRES and SUBARU/HDS, albeit mostly in un-reduced form. Moreover, further fully-reduced spectra can now be retrieved from the publicly available sample of 170 quasars at $$0.29\lesssim z_{em}\lesssim 5.29$$ of the KODIAQ Survey ((citation not found: OMeara15)). And, so far, high quality spectra from the High Dispersion Spectrograph (HDS) on the SUBARU Telescope have not been used for constraining the IGM thermal history. Furthermore, the new spectrograph, ESPRESSO (Echelle Spectrograph for Rocky Exoplanet and Stable Spectroscopic Observations) on the VLT will soon be available for new observations, providing further high-quality Ly-$$\alpha$$ and $$\beta$$ spectra at $$z\gtrsim 1.8$$. Finally, an entirely new sample of much higher S/N spectra will be accessible in future with the new generation of telescopes, such as the Giant Magellan Telescope (GMT), the Thirty Meter Telescope (TMT) and the European Extremely Large Telescope (E-ELT). However, even if the new echelle spectrographs mounted on these future telescopes, such as G-CLEF commissioned for GMT, will be able to provide high quality data in shorter exposure times, it will still take a considerable amount of time on such highly oversubscribed facilities to collect a sample of quasar spectra as large as that currently available from HIRES, UVES and HDS.

Clearly, the existing spectra also offer new $$T(\bar{\Delta})$$ measurements as well. Therefore, they can be combined with the new $$\gamma$$ constraints to provide more precise estimates of $$T_{0}$$’s evolution from $$z\sim 5$$ down to $$z\sim 1.5$$.

### Independent constraint on $$\gamma$$ from the He ii Ly-$$\alpha$$ curvature

In Chapter 5 we introduced an alternative possibility to measure $$\gamma$$ at redshifts $$z\sim 2-3$$ using the coeval H i and He ii Ly-$$\alpha$$ absorption. Even if the quality and the number of far UV spectra available today is not sufficient for the curvature analysis, new space telescopes, such as the Advanced Technology Large-Aperture Space Telescope (ATLAST) or the High Definition Space Telescope (HDST), with high resolution spectrographs (e.g. (citation not found: Postman09); (citation not found: Dalcanton15)) may be available in future, providing high quality spectra to be used for an independent constraint on the IGM thermal state. It is therefore important to explore the potential of such a constraint based on the curvature statistic, as the findings may be of some influence for the design of such future telescopes and spectrographs.

However, as described in Chapter 5, new hydrodynamical simulations are necessary to properly sample the underdense regions probed by the He ii Ly-$$\alpha$$ forest and obtain synthetic spectra usable for calibrating the curvature analysis. New simulations with a box size of at least $$20h^{-1}$$ Mpc and mass resolution $$M_{\text{gas}}\lesssim 9.2\times 10^{4}M_{\odot}$$ would be required, but with today’s technologies these are very time-consuming. However, it could be possible that, with advancements in supercomputing and new generations of supercomputers, this task will be feasible within the next $$\sim 5$$ years.