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Temperature measurements

\label{chp:Temperature measurements}

We find that the temperature of the cosmic gas traced by the Lyman-\(\alpha\) forest, \(T(\bar{\Delta})\), increases for increasing overdensity from \(T(\bar{\Delta})\sim 22670\) K to \(33740\) K in the redshift range \(z\sim 2.8-1.6\). Under the assumption of two reasonable values for the parameter \(\gamma\), that defines the slope of the IGM temperature–density relation, the temperature at the mean density (\(T_{0}\)) shows a tendency to flatten at \(z\lesssim 2.8\). In the case of \(\gamma\sim 1.5\), our results are consistent with previous ones which indicate a falling \(T_{0}\) for redshifts \(z\lesssim 2.8\). Finally, our \(T(\bar{\Delta})\) values show reasonable agreement with moderate blazar heating models.

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

The selection of the characteristic overdensities, \(\bar{\Delta}\), and the associated one-to-one function between temperature and curvature presented in Chapter 2 allows to infer information about the temperature of the gas traced by the Lyman-\(\alpha\) forest, \(T(\bar{\Delta})\). This measurement is independent of the choice of the parameter \(\gamma\) for the \(T\)\(\rho\) relation (Equation \ref{eq:TDrelation}) and for this reason will represent the main result of this project. We also translate our temperature measurements to values at the temperature at the mean density, \(T_{0}\), for reasonable values of \(\gamma\). In this Chapter we present our results and compare them with those of (citation not found: Becker11) at higher redshift. A broader discussion, taking into consideration theoretical predictions, along with the conclusions summarizing these measurements is also presented here.

Temperature at the characteristic overdensities

\label{sec:temperature}

The main results of this work are presented in Figure \ref{fig:TD} where we plot the IGM temperature at the characteristic overdensities traced by the Lyman-\(\alpha\) forest as a function of redshift. The 1\(\sigma\) errors are estimated from the propagation of the uncertainties in the curvature measurements. In fact, the uncertainties in the measured effective optical depth are reflected only in a small variation in the temperature measurements that falls well within the 1\(\sigma\) uncertainties due to the errors in the curvature measurements. Our temperature measurements show good agreement with the previous work of (citation not found: Becker11) at higher redshifts where they overlap. This accord is particularly significant because we analysed a completely independent set of quasar spectra, obtained from a different instrument and telescope.

The curvature method in fact demonstrates self-consistency: the lower overdensities recorded from our sample are in fact compensated by our higher values of observed curvature. In this way, interpolating at each redshift the \(T(\bar{\Delta})\)\(log\langle|\kappa|\rangle\) relationship in the simulations to the \(log\langle|\kappa|\rangle\) computed directly from the data, we obtained similar temperature values to the ones of (citation not found: Becker11) in the overlapping redshift range. Differences in the characteristic overdensities, \(\bar{\Delta}\), at a particular redshift between the two studies will cause variation in the derived temperature at the mean density (\(T_{0}\)) because we will infer \(T_{0}\) using the \(T\)\(\rho\) relation with the values of \(\bar{\Delta}\). However, this effect will be modest and will cause disparity at the level of the 1\(\sigma\) error bars of our values (see Section \ref{sec:T0} and Appendix A). For comparison, in Figure \ref{fig:TD} we show the \(z=2.4\) line-fitting result of (citation not found: Rudie12), with their \(T_{0}\) and \(\gamma\) values recalibrated and translated to a \(T(\bar{\Delta})\) value by (citation not found: Bolton13). Even if the line-fitting method is characterized by much larger 1\(\sigma\) error bars, it represents an independent technique and its agreement with our temperature values gives additional confidence in the results .

In general, the extension to lower redshifts (\(z\lesssim 1.9\)) that our new results provide in Figure \ref{fig:TD} do not show any large, sudden decrease or increase in \(T(\bar{\Delta})\) and can be considered broadly consistent with the trend of \(T(\bar{\Delta}\)) increasing towards lower redshift of (citation not found: Becker11). The increasing of \(T(\bar{\Delta}\)) with decreasing redshift is expected for a non-inverted temperature–density relation because, at lower \(z\), the Lyman-\(\alpha\) forest is tracing higher overdensities: denser regions are much more bounded against the cooling due to the adiabatic expansion and present higher recombination rates (and so more atoms for the photoheating process). We consider this expectation further in Section \ref{sec:discussion1} after converting our \(T(\bar{\Delta}\)) measurements to \(T_{0}\) ones, using a range of \(\gamma\) values.

\label{table:results}Numerical values for the results of this work: the mean redshift of each data bin is reported (column 1) with the associated characteristic overdensity (column 2). Also shown are the temperature measurement with the associated 1\(\sigma\) errors obtained for each data bin at the characteristic overdensity (column 3) and at the mean density under the assumption of two values of \(\gamma\) (column 4 & 5). Finally, the values of \(\gamma\), recovered from the fiducial simulation C15, are presented (column 4).
\(z_{mean}\) \(\bar{\Delta}\) \(T(\bar{\Delta})/10^{3}K\) \(\gamma\sim 1.5\) \(T_{0}^{\gamma\sim 1.5}/10^{3}K\) \(T_{0}^{\gamma=1.3}/10^{3}K\)
1.63 5.13 33.74\(\pm\)3.31 1.583 13.00\(\pm\)1.27 20.66\(\pm\)2.03
1.82 4.55 27.75\(\pm\)1.19 1.577 11.79\(\pm\)0.51 17.61\(\pm\)0.76
2.00 4.11 27.62\(\pm\)0.84 1.572 12.60\(\pm\)0.38 18.08\(\pm\)0.55
2.18 3.74 29.20\(\pm\)1.06 1.565 14.05\(\pm\)0.51 19.66\(\pm\)0.71
2.38 3.39 25.95\(\pm\)1.22 1.561 13.20\(\pm\)0.62 18.00\(\pm\)0.85
2.60 3.08 23.58\(\pm\)1.09 1.554 12.77\(\pm\)0.59 16.83\(\pm\)0.78
2.80 2.84 22.67\(\pm\)0.89 1.549 12.90\(\pm\)0.51 16.58\(\pm\)0.65

\label{fig:TD}IGM temperature at the characteristic overdensities, \(T(\bar{\Delta}\)), as a function of redshift for this work (green points), for the line-fitting analysis of (citation not found: Bolton13) (red triangle) and for the previous work of (citation not found: Becker11) (black circles). Vertical error bars are \(1\sigma\) for this work and \(2\sigma\) for Becker et al. and are estimated from statistical uncertainties in the curvature measurements.

Temperature at the mean density

\label{sec:T0}

Using the \(T\)\(\rho\) relation, characterized by different \(\gamma\) values, we translate our measurements of \(T(\bar{\Delta}\)) to values of temperature at the mean density (\(T_{0}\)). In Figure \ref{fig:slopes} we present \(T_{0}\) under two different assumptions for \(\gamma\): for \(\gamma\) values measured from fiducial simulations (A15–G15) in the top panel and for a constant \(\gamma\) =1.3 in the bottom panel. In both the plots our results are compared with those from (citation not found: Becker11).

In the first case we use a \(\gamma\) parameter that varies slightly with redshift, with \(\gamma\sim 1.5\) at \(z=3\) (the exact values are reported in Table \ref{table:results} and have been recovered from the fiducial simulation C15). These \(\gamma\) values are close to the maximum values expected in reality and therefore correspond to the minimum \(T_{0}\) case. Our results in the overlapping redshift range (\(2.1\lesssim z\lesssim 2.9\)) are broadly consistent with the previous ones but show slightly higher values, a difference that can be attributed to the variation with respect to the previous work in the values of the characteristic overdensities from which we derived our \(T_{0}\) values (see Section \ref{sec:characO}). The extension at lower redshifts suggests a tendency of flattening of the increase in \(T_{0}\) that can be interpreted as a footprint for the completion of the reheating of the IGM by He ii reionization. The least-square linear fit of our data presented in Figure \ref{fig:slopes} (top panel) for this particular choice of \(\gamma\) shows, in fact, quantitatively an inversion in the slope of the temperature evolution at the mean density: the general trend of the temperature is therefore an increase from \(z\sim 4\) to \(z\sim 2.8\) with a subsequent flattening of \(T_{0}\) around \(\sim 12000\)K at \(z\sim 2.8\). The evolution of the temperature for \(z\lesssim 2.8\) is generally consistent with a linear decrease of slope \(a=0.80\pm 0.81(1\sigma)\) generally in agreement with the decrease registered in (citation not found: Becker11) for the same choice of \(\gamma\).

The situation is similar in the second case for a constant \(\gamma=1.3\). This choice, which is motivated by the numerical simulations of (citation not found: McQuinn09), corresponds to a mild flattening of the temperature–density relation, as expected during an extended He ii reionization process. The trend of \(T_{0}\) again shows a strong increase in the temperature from \(z\sim 4\) to \(z\sim 2.8\) and then a tendency of flattening from \(z\sim 2.8\) towards lower redshift. However, the temperature obtained in this case is higher, fluctuating around \(\sim 17000\) K. The scatter between our data points and the ones from (citation not found: Becker11) is also smaller for this choice of \(\gamma\), even if ours are slightly higher on average. In this case the linear fit of our data points at \(z\lesssim 2.8\) suggests a change in the slope of the temperature evolution but, while in the previous case we register a positive slope, for this \(\gamma\) choice we see only a slowdown in the increasing temperature, with a slope that assumes the value \(a=-1.84\pm 1.06(1\sigma)\).

The exact redshift of the temperature maximum, reached by the IGM at the mean density approaching the tail-end of He ii reionization, is then still dependent on the choice of \(\gamma\), as already pointed out in (citation not found: Becker11). Nevertheless, the extension at lower redshift of our data points gives stronger evidence about the end of this event. In fact, while an increase in the temperature for \(z\sim 4-2.8\) has been recorded in the previous work, if \(\gamma\) remains roughly constant, a tendency to a temperature flattening at lower redshift is suggested for both the choice of \(\gamma\). A particularly important result is the suggestion of a decrease in \(T_{0}\) in the case of \(\gamma\sim 1.5\). In fact, according to recent analysis with the line fitting method ((citation not found: Rudie12); (citation not found: Bolton13)) at redshift \(z=2.4\) there is good evidence for \(\gamma=1.54\pm 0.11(1\sigma)\) and, because we expect that at the end of He ii reionization \(\gamma\) will tend to come back to the asymptotic value of 1.6, indicating equilibrium between photoionization and cooling due to the adiabatic expansion, the possibility to have \(\gamma\lesssim 1.5\) for \(z\lesssim 2.4\) seems to be not realistic. Even if in this work we did not attempt to constrain the temperature–density relation, the scenario in the top panel of Figure \ref{fig:slopes} seems likely to reproduce the trend in the evolution of the temperature, at least at low redshift, with our results reinforcing the picture of the reheating of the IGM due to He ii reionization being almost complete at \(z\sim 2.8\), with a subsequent tendency of a cooling, the rate of which will depend on the UV background.

\label{fig:slopes}Temperature at the mean density, \(T_{0}\), inferred from \(T(\bar{\Delta})\) in this work (green points/stars) and in the work of (citation not found: Becker11) (black point/stars) for different assumptions of the parameter \(\gamma\): \(T_{0}\) for \(\gamma\sim 1.5\) (see table 4 for the exact values) (top panel) and for \(\gamma=1.3\) (bottom panel). The linear least square fit of our data points (green line) are presented for both the choices of \(\gamma\) with the corresponding \(1\sigma\) error on the fit (shaded region). The fits show a change of slope in the temperature evolution: from the increase of \(T_{0}\) between \(z\sim 4\) and \(z\sim 2.8\) there is a tendency of a flattening for \(z\lesssim 2.8\) with a decrease in the temperature for \(\gamma\sim 1.5\) and a slowdown of the reheating for \(\gamma=1.3\).

The UV background at low redshifts

The tendency of our \(T_{0}\) results to flatten at \(z\lesssim 2.8\) seems to suggest that at these redshifts the reheating due to the He ii reionization has been slowed down, if not completely exhausted, marking the end of this cosmological event. In the absence of reionization’s heating effects the temperature at the mean density of the ionized plasma is expected to approach a thermal asymptote that represents the balance between photoionization heating and cooling due to the adiabatic expansion of the Universe. The harder the UV background (UVB) is, the higher the temperature will be because each photoionisation event deposits more energy into the IGM. In particular, under the assumption of a power-law ionizing spectrum, \(J_{\nu}\propto\nu^{-\alpha}\), and that He ii reionization no longer contributes any significant heating, the thermal asymptote can be generally described by ((citation not found: HuiGnedin1997);(citation not found: HuiHaiman03)):

\begin{equation} \label{eq:thermalA} \label{eq:thermalA}T_{0}=2.49\times 10^{4}K\times(2+\alpha)^{-{{1}\over{1.7}}}\biggl{(}{{1+z}\over{4.9}}\biggr{)}^{0.53},\\ \end{equation}

where the parameter \(\alpha\) is the spectral index of the ionizing source. The observational value of \(\alpha\) is still uncertain. From direct measurements of QSO rest-frame continua, this value has been found to range between 1.4 and 1.9 depending on the survey (e.g. (citation not found: Telfer02); (citation not found: Shull12)) whereas for galaxies the values commonly adopted range between 1 and 3 (e.g. (citation not found: Bolton07); (citation not found: Ouchi09); (citation not found: Kuhlen12)) even if, in the case of the emissivity of realistic galaxies, a single power law is likely be consider an oversimplification. .

Because our data at \(z\lesssim 2.8\) do not show any strong evidence for a rapid decrease or increase in the temperature, here we assume that this redshift regime already traces the thermal asymptote in Equation \ref{eq:thermalA}. Under this hypothesis we can then infer some suggestions about the expectation of a transition of the UV background from being dominated mainly by stars to being dominated mainly by quasars over the course of the He ii reionization (\(2\lesssim z\lesssim 5\)). In Figure \ref{fig:slopesUV} we show, as an illustrative example only, two models for the thermal asymptote: the first is the model of (citation not found: HuiHaiman03) for the expected cooling in the absence of He ii reionization, with \(\alpha\) scaled to 5.65 to match the flattening of the (citation not found: Becker11) data at \(z\sim 4\)–5, while in the second case \(\alpha\) was scaled to 0.17 to match our results (for \(\gamma\sim 1.5\)) at \(z\lesssim 2.8\). These values are at some variance with the quantitative expectations: in the first case our UVB spectrum is significantly softer compared to typical galaxies-dominated spectrum while, after He ii reionization, our value is somewhat harder than a typical quasar-dominated spectrum. We also emphasise that any such estimate of changes in the spectral index also involves the considerable uncertainties, already discussed, connected with the correct position of the peak in \(T_{0}\) and the choice of \(\gamma\), and so we cannot make firm or quantitative conclusions here. However, in general, the observed cooling at higher temperatures at \(z\lesssim 2.8\) seems to suggest that the shape of the UV background has changed, hardening with the increase in temperature during to the reionization event.

\label{fig:slopesUV}An example of thermal asymptotes before and after He II reionization with different UVB shapes. The evolution of the thermal asymptote for the model of (citation not found: HuiHaiman03) with \(\alpha\) (Eq. 4) scaled to match the high redshift (\(z\sim 4\)–5) data of (citation not found: Becker11) (black dashed line) is compared with the same model with \(\alpha\) scaled to match our results (assuming \(\gamma\sim 1.5\)) at \(z\lesssim 2.8\) (green dashed line). The significant change in \(\alpha\) required over the redshift range \(2.8\lesssim z\lesssim 4.5\) in this example suggests that the UV background has changed, hardening during the He ii reionization.

Discussion

\label{sec:discussion1}

The main contribution of our work is to add constraints on the thermal history of the IGM down to the lowest optically-accessible redshift, \(z\sim 1.5\). These are the first temperature measurements in this previously unexplored redshift range. In this section we discuss the possible implications of our results in terms of compatibility with theoretical models.

Measuring the low-redshift thermal history is important for confirming or ruling out the photo-heating model of He ii reionization and the new blazar heating models. According to many models of the former, the He ii reionization should have left a footprint in the thermal history of the IGM: during this event, considerable additional heat is expected to increase the temperature at the mean density of the cosmic gas (\(T_{0}\)) at \(z\lesssim 4\) (citation not found: HuiGnedin1997). The end of He ii reionization is then characterized by a cooling of the IGM due to the adiabatic expansion of the Universe with specifics that will depend on the characteristics of the UV background. However, even if some evidence has been found for an increase in the temperature at the mean density from \(z\sim 4\) down to \(z\sim 2.1\) (e.g. (citation not found: Becker11)), the subsequent change in the evolution of \(T_{0}\) expected after the end of the He ii reionization has not been clearly characterized yet and remains strongly degenerate with the imprecisely constrained slope of the temperature–density relation \(\gamma\) (see Equation \ref{eq:TDrelation}). This, combined with several results from PDF analysis which show possible evidence for an inverted temperature–density relation ((citation not found: Becker07); (citation not found: Bolton08); (citation not found: Viel09); (citation not found: Calura12); (citation not found: Garzilli12)), brought the development of a new idea of volumetric heating from blazar TeV emission ((citation not found: Chang12); (citation not found: Puchwein12)). These models, where the heating rate is independent of the density, seem to naturally explain an inverted \(T\)\(\rho\) relation at low redshift. Predicted to dominate the photo-heating for \(z\lesssim 3\), these processes would obscure the change in the temperature evolution trend due to the He ii reionization, preventing any constraint on this event from the thermal history measurements.

A main motivation for constraining the temperature at lower redshifts than \(z\sim 2.1\) is to confirm evidence for a flattening in the already detected trend of increasing temperature for \(z\lesssim 4\). A precise measurement of a change in the \(T_{0}(z)\) slope, in fact, could bring important information about the physics of the IGM at these redshifts and the end of the He ii reionization event. It is therefore interesting that our new temperature measurements in Figure \ref{fig:slopes}, which extend down to redshifts \(z\sim 1.5\), show some evidence for such a change in the evolution for \(z\lesssim 2.8\). However, in order to make a fair comparison with different heating models in terms of the temperature at the mean density, we must recognize the fact that we do not have yet strong constraints on the evolution of the \(T\)\(\rho\) relation slope as a function of redshift: assuming a particular choice of \(\gamma\) for the translation of the temperature values at the characteristic overdensities to those at the mean density, without considering the uncertainties in the slope itself, could result in an unfair comparison. Furthermore, the blazar heating models’ \(T\)\(\rho\) relation at each redshift can be parametrized with a power-law (of the form of Equation \ref{eq:TDrelation}) only for a certain range of overdensities that may not always cover the range in our characteristic overdensities ((citation not found: Chang12); (citation not found: Puchwein12)). Therefore, to compare our results with the blazar heating model predictions, we decided to use directly the \(T(\bar{\Delta})\) values probed by the forest.

In Figure \ref{fig:BB2} we compare the model without blazar heating contributions, and the weak, intermediate and strong blazar heating models of (citation not found: Puchwein12), with our new results for the temperature at the characteristic overdensities. The temperature values for all the models were obtained by computing the maximum of the temperature distribution function at the corresponding redshift-dependent characteristic overdensities [\(\bar{\Delta}\)] in Table \ref{table:results} by E. Puchwein (private communication). Each of the three blazar heating models has been developed using different heating rates, based on observations of 141 potential TeV blazars, under the assumption that their locally-observed distribution is representative of the average blazars distribution in the Universe. The variation in the heating rates is due to the tuning of a coefficient in the model that corrects for systematic uncertainties in the observations: the lower this multiplicative coefficient, the weaker the heating rate ((citation not found: Chang12); (citation not found: Puchwein12)). The observational constraints on the IGM thermal evolution in Fig. \ref{fig:BB2} seem to be in reasonable agreement with the intermediate blazar heating model, even if some fluctuations toward higher temperatures reach the range of values of the strong blazar model. This result in general reflects what was found by (citation not found: Puchwein12) (their figure 5) in their comparison with the temperature measurements of (citation not found: Becker11), even if in that case the models were tuned to different \(\bar{\Delta}\) values than ours. This general agreement could be explained by the fact that our \(T(\bar{\Delta})\) measurements closely match those of (citation not found: Becker11) in the common redshift range \(2.0\lesssim z\lesssim 2.6\) (see Fig. \ref{fig:TD}) and by the weak dependency on the density of the blazar heating mechanism.

\label{fig:BB2}Comparison of blazar heating models: the temperature values at the redshift-dependent characteristic overdensities, \(T(\bar{\Delta})\), inferred in this work (green points) are compared with the model without a blazar heating contribution (black dashed line) and the weak (green dashed line), intermediate (yellow dashed line) and strong (red dashed line) blazar heating models of (citation not found: Puchwein12). The blazar heating predictions were computed at the corresponding using the \(\bar{\Delta}(z)\) in Table 4 to order to allow a fair comparison with our \(T(\bar{\Delta})\) measurements. Our observational results seem to be in reasonable agreement with the intermediate blazar heating model. The vertical error bars represent the 1-\(\sigma\) errors on the temperature measurements.

According to Fig. \ref{fig:BB2}, the model from (citation not found: Puchwein12) without a blazar heating contribution, that is based on the UV background evolution of (citation not found: Faucher09), shows temperature values significantly lower than our (\(\bar{\Delta})\) measurements \(z\lesssim 3\). However, this model does not take into account the contribution of the diffuse hard X-ray background ((citation not found: Churazov07)). The excess energy of these ionizing photons could, in fact, contribute to the heating, shifting the range of temperatures towards higher values. Also, to definitely ruled out or confirm any of the different thermal histories, an interesting further test would be to compare the temperature at the mean density between observations and models.

That is, there is a strong need for model-independent measurements of \(T_{0}\) that would allow a straightforward comparison between different \(T\)\(\rho\) relations, and so different model predictions. In this context a promising prospect is the use of the He II Lyman-\(\alpha\) forest for the identification of the absorption features useful for an improved line-fitting constraint of \(\gamma\). The possibility to calculate directly the ratio between \(b\) parameters of the corresponding H I and He II lines would, in fact, make possible an easier and more precise selection of the lines that are dominated by thermal broadening (\(b_{HI}/b_{HeII}\simeq 2\)). Even if the low S/N of the UV spectra currently available make this identification difficult ((citation not found: Zheng04)), possible future, space-based telescopes UV with high resolution spectrographs (e.g. (citation not found: Postman09)) may offer the opportunity to improve the quality and the number of “clean” He II forests for future analyses (see also Chapter 5).

Conclusions

In this thesis so far we have utilized a sample of 60 VLT/UVES quasar spectra to make a new measurement of the IGM temperature evolution at low redshift, \(1.5\la z\la 2.8\), with the curvature method applied to the H i Lyman-\(\alpha\) forest. For the first time we have pushed the measurements down to the lowest optically-accessible redshifts, \(z\sim 1.5\). Our new measurements of the temperature at the characteristic overdensities traced by the Lyman-\(\alpha\) forest, \(T(\bar{\Delta}\)), are consistent with the previous results of Becker et al. (2011) in the overlapping redshift range, \(2.0<z<2.6\), despite the datasets being completely independent. They show the same increasing trend for \(T(\bar{\Delta}\)) towards lower redshifts while, in the newly-probed redshift interval \(1.5\la z\la 2.0\), the evolution of \(T(\bar{\Delta}\)) is broadly consistent with the extrapolated trend at higher redshifts.

The translation of the \(T(\bar{\Delta}\)) measurements into values of temperature at the mean density, \(T_{0}\), depends on the slope of the temperature–density relation, \(\gamma\), which we do not constrain in this first project (but see Chapter 4). However, for reasonable, roughly constant, assumptions of this parameter, we do observe some evidence for a change in the slope of the temperature evolution for redshifts \(z\lesssim 2.8\), with indications of at least a flattening, and possibly a reversal, of the increasing temperature towards lower redshifts seen in our results and those of (citation not found: Becker11) for \(2.8\lesssim z\lesssim 4\). In particular, for the minimum \(T_{0}\) case, with \(\gamma\sim 1.5\), the extension towards lower redshifts provided by this work adds to existing evidence for a decrease in the IGM temperature from \(z\sim 2.8\) down to the lowest redshifts probed here, \(z\sim 1.5\). This could be interpreted as the result of the completion of the reheating process connected with the He ii reionization.

Following the additional hypothesis that our low redshift temperature measurements are already tracing the thermal asymptote, the cooling of \(T_{0}\) inferred at \(z\lesssim 2.8\) (assuming \(\gamma\sim 1.5\)) may suggest that the UV background has changed, hardening during the He ii reionization epoch. However, the expectation for the evolution of \(T_{0}\) following He ii reionization will depend on the evolution in \(\gamma\) and on details of the reionization model.

We also compared our \(T(\bar{\Delta}\)) measurements with the expectations for the models of (citation not found: Puchwein12) with and without blazar heating contributions. To allow a fair comparison with our observed values, the model predictions were computed at the corresponding (redshift-dependent) characteristic overdensities (\(\bar{\Delta}\)). Our observational results seem to be in reasonable agreement with a moderate blazar heating scenario. However, to definitely confirm or rule out any specific thermal history it is necessary to obtain new, model-independent measurements of the temperature at the mean density.

With the IGM curvature now constrained from \(z\sim 4.8\) down to \(z\sim 1.5\), the main observational priority now is clearly to tightly constrain the slope of the temperature–density relation, \(\gamma\), and its evolution over the redshift range \(1.5\la z\la 4\). This is vital in order to fix the absolute values of the temperature at the mean density and to comprehensively rule out or confirm any particular heating scenarios. Therefore, in the second part of this thesis we will focus on the constraint of the parameter \(\gamma\), studying new applications of the curvature method that could allow a complete characterization of the IGM thermal state in this crucial redshift range.

Finally, we note that, even though our new measurements have extended down to \(z\sim 1.5\), there is still a dearth of quasar spectra with high enough S/N in the 3000–3300 Å spectral range to provide curvature information in our lowest redshift bin, \(1.5<z<1.7\). We have searched the archives of both the VLT/UVES and Keck/HIRES instruments for new spectra to contribute to this bin. However, the few additional spectra that we identified had relatively low S/N and, when included in our analysis, contributed negligibly to the final temperature constraints. Therefore, new observations of UV-bright quasars with emission redshifts \(1.5\la z_{\rm em}\la 1.9\) are required to improve the temperature constraint at \(1.5<z<1.7\) to a similar precision as those we have presented at \(z>1.7\).

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