# Introduction

\label{chp:Introduction}

In modern cosmology, the study of the properties of the intergalactic medium is one of the key goals for a more comprehensive understanding of galaxy evolution and formation. Starting from a very hot plasma made of electrons and protons after the Big Bang, to the gas that now fills the space between galaxies, the intergalactic medium (IGM) has been one of the main “recorders" of the different phases of evolution of the Universe. In fact, it is the thermodynamic state and chemical composition of this gas, the main reservoir of baryons in the Universe, that determined the conditions for the formation of the structures that we can observe today. In particular, the IGM thermal history can be an important source of information about the processes that injected vast amounts of energy into the intergalactic medium on relatively short cosmological timescales: the reionization events. Within the broader aim of a better understanding of the properties of the IGM and of the mechanisms governing its evolution, this thesis focuses on constraining the thermal state of the cosmic gas in the low-redshift Universe ($$1.5\lesssim z\lesssim 3.8$$), principally addressing the following open science questions: What is the thermal state of the IGM at low redshift and what are the possible heating processes that could explain it? and How is the temperature–density relation of the IGM evolving in the recent Universe and what are the possible phenomena that have determined it?

While a more detailed motivational context can be found at the beginning of each chapter, this Introduction gives a general background about the physical processes and the techniques relevant for a fuller understanding of the thesis.

## The origin of the IGM

According to the Standard cosmological model, after the Big Bang the Universe was filled with an hot plasma made mainly of electrons and protons in rapid thermal motion. For hundreds of thousands of years the radiation was coupled with matter, with visible and ultraviolet (UV) photons scattered inside this ionized medium. As the expansion of the Universe proceeded the temperature of the cosmic gas decreased until, at $$z\sim1100$$, the temperature dropped down to few thousand degrees Kelvin, low enough to allow the protons and electrons to recombine into neutral hydrogen (Hi), marking the cosmic recombination phase. As the particles recombined, the scattering of the photons became rarer and rarer and finally the radiation was let free to travel undisturbed, determining the origin of the Cosmic Microwave Background (CMB). After the recombination, the Universe entered the so-called Dark Ages: it was left filled by neutral gas mainly made of hydrogen and for a small quantity (i.e. $$\sim$$25per cent by mass) of helium. Its evolution was driven by the continuous gravitational collapse of overdense regions that, after hundreds of millions of years, allowed the formation of the first galaxies and their stars. These new structures and their UV radiation subsequently determined dramatic changes in the properties of the cosmic gas, including the IGM.

## The search for the IGM

The vast majority of all that is known about the properties and the structures of the IGM comes from the study of optical and UV spectra. While attempts to detect the IGM have started even before, the finding of new brilliant objects called Quasi-Stellar Objects or QSOs (e.g. (citation not found: Schmidt63)) allowed significant improvements in the measurements of the intergalactic Hi density. Studying the decrease in flux blueward the Lyman-$$\alpha$$ (Ly-$$\alpha$$) emission line of a QSO at $$z=2.01$$, it was suggested that the cosmic mass density of neutral hydrogen was much smaller than what expected from cosmological predictions, showing the first evidence that the IGM had been reionized((citation not found: GunnPe65)). In subsequent years, many absorption features were detected in higher resolution spectra, but until 1971 these were associated with intervening galaxy halos intercepted through the lines of sight. The real nature of these spectral imprints, as Ly-$$\alpha$$ absorption lines arising from discrete Hi clouds, was reported for the first time by (citation not found: Lynds71). Today, the sum of these discrete absorption features is called the Ly-$$\alpha$$ forest and after the disclosure of its intergalactic origin ((citation not found: Sargent80)) it has represented the best laboratory to study the IGM.

## The IGM absorption lines

The diffuse intergalactic gas can be studied detecting the absorption imprints that it leaves on the spectrum along the line of sight to a quasar. Figure \ref{fig:QSOspectrum} shows an example of flux density as a function of wavelength ($$\lambda$$) from the optical spectrum of a QSO at $$z_{em}=3.11$$. The densely distributed, apparently discrete absorption features that constitute the Ly-$$\alpha$$ forest spread bluewards of the Ly-$$\alpha$$ emission line ($$\lambda$$= 4996.40Å) down to the Lyman-$$\beta$$ (Ly-$$\beta$$) emission line ($$\lambda$$= 4215.71Å) shortwards of which the Ly-$$\alpha$$ absorption is accompanied by subsequent higher orders of Lyman transitions. Longwards of the Ly-$$\alpha$$ emission line, metal absorbers give rise to fewer, narrow absorption lines. The properties of all the absorption features produced in a spectrum are determined by the equation of radiative transfer that describes the propagation of the radiation emitted by a background source thorough a medium of interest (in this case the IGM).

\label{fig:QSOspectrum}

### The equation of radiative transfer

The equation of radiative transfer describes how the absorption features are produced by the intervening intergalactic gas in the spectrum of a background quasar and is defined as follows: $\frac{1}{c}\frac{\partial I_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})}{\partial t}+\hat{\mathbf{n}}\cdot\nabla I_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})=-\alpha_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})+j_{\nu}(\mathbf{r},t,\hat{\mathbf{n}}), \label{eq:radTrans}$ where $$I_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})$$ represents the specific intensity that characterizes the source of radiation, $$c$$ is the speed of light, $$\alpha_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})$$ is the attenuation coefficient of the intergalactic medium and $$j_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})$$ is the emission coefficient that describes the local specific luminosity per solid angle per unit volume emitted by the source. The specific intensity at any given time $$t$$ and position $$\mathbf{r}$$ represents the rate at which the energy, carried by photons of frequency $$\nu$$ in the direction $$\hat{\mathbf{n}}$$, crosses a unit area per unit solid angle per unit time ((citation not found: Meiksin09)).

In the case of a single background source, such as a QSO, $$j_{\nu}=0$$ and the solution of Equation \ref{eq:radTrans} will depend only on the way in which the incident radiation is attenuated by absorption and scattering of the photons due to the intervening gas. The attenuation coefficient is defined as follows: $\alpha_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})=\rho(\mathbf{r},t)\kappa_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})+n(\mathbf{r},t)\sigma_{\nu}(\mathbf{r},t,\hat{\mathbf{n}}), \label{eq:attenuation}$ where $$\rho(\mathbf{r},t)$$ is the mass density of the gas, $$\kappa_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})$$ is its opacity, $$n(\mathbf{r},t)$$ is the number density of scattering particles of a specific mean mass $$\bar{m}$$ and $$\sigma_{\nu}(\mathbf{r},t,\hat{\mathbf{n}})$$ is the scattering cross section ((citation not found: Meiksin09)).

The absorption features arise from the scattering of photons traveling from the background quasar through an intergalactic medium with number density $$n(\mathbf{r},t)$$. The resonance line scattering cross section will depend on the specific characteristics of the transition considered (rest-wavelength $$\lambda_{0}$$ or frequency $$\nu_{0}$$ and oscillator strength $$f$$) but also on the thermodynamic condition of the ensemble atoms. The IGM atoms are not at rest but they are generally affected by thermal motion and may have additional components due to peculiar velocity flows or turbulent motion if in shocked or collapsed regions. Taking into account the thermal motion of the particles, the resonance line scattering cross section is obtained using the Voigt profile that well incorporates the thermal broadening into the line profile: $\sigma_{\nu}=\left(\frac{\pi e^{2}}{m_{e } c}\right)\frac{1}{4\pi\epsilon_{0}} f\varphi_{V}(p,\nu), \label{eq:crossS}$ where $$\varphi_{V}(p,\nu)=\frac{1}{\pi^{1/2}\Delta\nu_{D}}H(p,x)$$ is the normalized Voigt profile defined by the Voigt function $$H(p,x)$$1 and the Doppler width $$\Delta\nu_{D}= \nu_{0} b/c$$. The Doppler parameter $$b$$ is generally defined with its thermal component $$b_{th}=\left(\frac{2k_{B}T}{m}\right)^{1/2}$$ where $$k_{B}$$ is Boltzmann’s constant and $$m$$ is the mass of the atoms, but a kinematic component ($$b_{kin}$$) can be added to take into account possible turbulence effects (e.g. (citation not found: Pradhan11)).

The optical depth to absorption at the frequency $$\nu$$, due to the presence of intervening gas along a line of sight (at the position $$s'$$ and time $$t'$$) from the source position ($$s_{0}$$) to a position $$s$$, is then given by the solution of Equation \ref{eq:radTrans} : $\tau_{\nu}=\int_{s_{0}}^{s}ds'n(s',t')\sigma_{\nu'} \label{eq:opticaldepthgen}$ where $$\nu'=\nu a(t)/a(t')$$ according to the frequency redshift due to the expansion factor $$\nu\varpropto a(t)^{-1}$$. From the optical depth the corresponding attenuation of the intensity of the background QSO can be obtained as $$e^{-\tau_{\nu}}$$.

While the above description is valid for any atomic transition of interest, the work presented in this thesis will mainly be focused on the Lyman-$$\alpha$$ transition of Hi (Chapters 2-3-4). Nevertheless, the Hi Lyman-$$\beta$$ transition will also be considered in Chapter 4 and the Heii Lyman-$$\alpha$$ transition will be used in Chapter 5.

### The Lyman-$$\alpha$$ forest

Since the first discovery, the sum of the absorption lines arising from Ly-$$\alpha$$ absorption in diffuse Hi gas along the line of sight to a quasar has proved to be the main laboratory to study the properties of the IGM at different redshifts. Observations have shown that the Ly-$$\alpha$$ absorption in high redshift quasar spectra creates a dense aggregation of discrete lines ((citation not found: Lynds71)). Such a distribution, called Ly-$$\alpha$$ forest ((citation not found: Weymann81)), suggested that these features arise in distinct localized regions of the IGM, implying the presence of an inhomogeneous density field ((citation not found: Sargent80)). To this day, the availability of new instruments and spectroscopic capabilities has led to a detailed analysis of the possible connection between galaxies and absorption lines, allowing a more detailed classification of the Ly-$$\alpha$$ absorption systems, based mainly (but not only) on the basis of their Hi content (e.g. (citation not found: Weymann81); (citation not found: Cowie95); (citation not found: Rauch98); (citation not found: Wolfe05); (citation not found: Prochaska05)). The properties of the Ly-$$\alpha$$ absorption systems and their classification are discussed below.

#### The Ly-$$\alpha$$ optical depth

As a first approximation, for $$z\lesssim4$$ it is possible to describe the Ly-$$\alpha$$ forest as a sum of discrete absorbers with a total optical depth ($$\tau_{\nu}$$) given by the sum of the individual optical depths, $$\tau_{\nu}(i)$$, corresponding to locally overdense regions ((citation not found: Meiksin09)): $\tau_{\nu}=\sum_{i}\tau_{\nu}(i)=\pi^{1/2}\tau_{0}\langle\phi_{V}(p,x)\rangle \label{eq:discreteOptical}$ where $$\phi_{V}=\frac{H(p,x)}{\pi^{1/2}}$$ is averaged over the line of sight, weighted by the density and $$\tau_{0}$$ is the optical depth at line centre defined as: $\tau_{0}=\frac{N\sigma\lambda}{\pi^{1/2}b}=\frac{e^{2}\pi^{1/2}}{m_{e}c}\left[\frac{1}{4\pi\epsilon_{0}}\right]\frac{N}{b}\lambda_{0}f \label{eq:linecenter}$ as a function of the column density $$N$$. For the Hi Ly$$\alpha$$ ($$\lambda_{0\alpha}=1215.67$$Å, $$f_{\alpha}=0.4164$$) Equation \ref{eq:linecenter} becomes: $\tau_{0}\sim0.38\left(\frac{N(\text{H{\sc \,i}})}{10^{13}\text{cm}^{-2}}\right)\left(\frac{b}{20 \text{km s}^{-1}}\right)^{-1} \label{eq:linecenterH}$

The Ly$$\alpha$$ absorption is highly sensitive to the presence of even small amounts of neutral hydrogen, for this reason at higher redshifts the distinction between individual lines becomes more difficult. The blending of absorption lines increases up to a redshift ($$z\gtrsim5.5$$) where the individual lines merge together forming the effect of a trough in the spectrum ((citation not found: GunnPe65)).

#### Equivalent width and curve of growth

For the purpose of determining abundances and to study blended absorption features, it is useful to define a quantity corresponding to the area between the line profile and the continuum, the equivalent width.

The equivalent width, $$W_{\lambda}$$ is usually defined in wavelength notation as: $W_{\lambda}=\frac{\lambda^{2}}{c}\int(1-e^{-\tau_{\nu}}) d\nu . \label{eq:EqW}$ For a Voigt profile, $$W_{\lambda}$$ depends on the different parameters that define the line shape, but particularly interesting is its correlation with temperature and column density, well represented through the curve of growth. In Figure \ref{fig:EqW} is presented a reproduction of the curve of growth for hydrogen Ly-$$\alpha$$; it is possible to distinguish three different segments, related to different line profiles ((citation not found: Pradhan11),(citation not found: Meiksin09)):

• The linear part: in optically thin regions ($$\tau\ll1$$) the equivalent width should increase linearly as the number of ions. The line deepens and broadens in direct proportion to the flux removed from the continuum by an increasing number of absorbers, i.e. $$W_{\lambda}\sim N$$(Hi).

• The logarithmic part: corresponds to saturated line profiles, when the density of ions is sufficient to absorb nearly all of the continuum photons at the line center wavelength. In this regime the increase in density results in a slow increase in $$W_{\lambda}\sim \sqrt{\text{ln}N\text{(H{\sc \,i})}}$$. For these features the equivalent width allows a precise measurement of the Doppler broadening.

• The square-root part: represents the case in which the line profile is dominated by the damping wings. The wings, on both the side of the line center are enhanced as the column density increases and $$W_{\lambda}\sim \sqrt{N\text{(H{\sc \,i})}}$$.

Reproduction of Figure 3 from (citation not found: Encyclopedia2000). The curve of growth for the Ly-$$\alpha$$ transition, relating the equivalent width $$W$$ of an absorption feature to its column density $$N$$(Hi). Different line-styles show the effect of different Doppler $$b$$ parameters in the logarithmic part. Different background colours mark the distinction, based on the column density ranges, among different types of absorbers: Ly$$\alpha$$ forest (blue), LLS (yellow), DLA(red).

\label{fig:EqW}

#### The classification of Ly-$$\alpha$$ absorption systems

The classification of the different absorption systems seen in quasar spectra is mainly due to their different physical origin. However, they can be broadly separated into three categories corresponding to their different column density ranges, as shown in Figure \ref{fig:EqW}. Historically, absorption systems with log$$N$$(Hi)$$\lesssim17.2$$ are called Ly-$$\alpha$$ forest, those with $$17.2\lesssim$$ log$$N$$(Hi)$$<20.3$$ Lyman limit systems (LLSs) and those with log$$N$$(Hi)$$\geq20.3$$ damped Ly$$\alpha$$ absorbers (DLAs) (e.g. (citation not found: Weymann81); (citation not found: Rauch98); ,(citation not found: Encyclopedia2000); (citation not found: Wolfe05); (citation not found: Prochaska05)).

The number of systems per unit of redshift has been found to decrease as their column density increases. That is, the Ly-$$\alpha$$ forest absorbers are the most common, they mainly contain ionised hydrogen and may be associated with metals absorbers (e.g. (citation not found: Songaila98); (citation not found: Tytler04)). The LLSs have similar characteristics but, due to their higher column density they are defined to be optically thick at the Lyman limit (912Å) (e.g. (citation not found: OMeara07)). The hydrogen contained in DLAs is, instead, mainly neutral and this makes them particularly interesting reservoirs of gas for star formation at high redshift (e.g.(citation not found: Wolfire03); (citation not found: Nagamine04); (citation not found: Wolfe05)). Moreover, DLAs are often associated with halos of intervening galaxies along the line of sight and the presence or the lack of significant amounts of associated metals have been used to obtain important information about galaxies formation and evolution (e.g. (citation not found: Wolfe05); (citation not found: Fumagalli11)). On the theoretical side, models to reproduce synthetic Ly-$$\alpha$$ absorption systems have progressed with the aim of a better understanding of their origin and their place within the context of the standard theory of structure formation. Using both semi-analytical and full hydrodynamical simulations, it was shown that the low density absorption features of the Ly-$$\alpha$$ forest arise naturally from the fluctuations of the continuous medium formed by the gravitational collapse of the initial density perturbations (e.g. (citation not found: GnedinHui98); (citation not found: Theuns98)). Moreover, the simulations show a variety of morphologies in the absorbing structures that correspond to different physical densities and Hi column densities in the Ly-$$\alpha$$ forest absorption systems (e.g. (citation not found: Zhang98)). According to these models, most of the volume of the Universe is occupied by low density gas with a $$\rho / \bar{\rho}\lesssim 10$$, where $$\bar{\rho}$$ is the mean baryon density of the IGM.

The broad aim of this work is to explore statistically the connection between the line shapes of the absorption features and the properties of the IGM. It is therefore clear that our objects of interest must mainly be the low column density Ly-$$\alpha$$ forest features, that incorporate information about the physics of the majority of the gas in which galaxies are embedded. While present along the lines of sight used in this work, higher column density systems – DLAs and LLSs – have been masked out of our analysis, as have the narrow metal lines which arise from these systems and which contaminate the forest region.

## The IGM Reionizations

Given the high sensitivity of the Ly-$$\alpha$$ absorption to the amount of neutral hydrogen, the lack of full absorption in quasar spectra implies that the IGM has been highly reionized after the recombination phase, and that it has been kept ionized from high redshift to the present. In Figure \ref{fig:IGMevol} is presented a schematic overview of the three identifiable epochs of reionization, one of hydrogen and two of helium. Given the similar ionization potential and recombination rate for both Hi and neutral helium (Hei), likely they have been reionized by the same radiation sources and, therefore, only the epochs of hydrogen reionization and full helium reionization to Heiii are generally thought of as distinct.

\label{fig:IGMevol}

### Hydrogen reionization

From the study of the cosmic microwave background anisotropy it seems that not later than $$z\simeq 11$$ ((citation not found: LL11)) the UV radiation emitted by the first objects, mostly stars ((citation not found: Haehnelt01)), was able to photoionize the cosmic neutral hydrogen (and Hei) but the nature of the sources that contributed to this massive injection of photons is not completely understood.

The end of this first Epoch of Reionization (EoR) is still not really well constrained: studies of the Hi Ly-$${\alpha}$$ absorption features of several quasars have shown that the intergalactic hydrogen was completely reionized by $$z\simeq6$$ (e.g. (citation not found: Becker01); (citation not found: Fan06); (citation not found: McGreer15)). This redshift corresponds to a rapid rise of the Hi Ly-$$\alpha$$ optical depth ($$\tau_{\text{H{\sc \,i}}}$$), from low to high redshifts, due to the increased amount of neutral Hi before the completion of the reionization. However, because only a small neutral fraction is adequate for providing a large $$\tau_{\text{H{\sc \,i}}}$$, this interpretation is not unique ((citation not found: Becker07)). Moreover, some evidence for the final stages of a patchy hydrogen reionization has been found recently in (citation not found: Becker15) at $$z\sim6$$, suggesting a later end of this process (at $$z\sim5$$). If for $$z\gtrsim6$$ the Ly-$$\alpha$$ forest becomes so thick that only lower limits on the optical depth and, therefore, on the end of the Hi reionization can be obtained, the second Reionization Epoch, that of Heii at $$z\sim3$$, is potentially much more accessible to direct observations.

### Heii reionization

Because the ionization potential of Heii (from Heii to Heiii) is 54.4 eV and fully ionized helium recombines more than 5 times faster than hydrogen, the second helium reionization event began later, when quasars started to dominate the UV background ((citation not found: MiraldaescudeQ00)). Theoretically, their much harder photons would have been able to fully ionize Heii at a redshift $$3\lesssim z\lesssim4.5$$, but these estimates change depending on assumptions about the abundance of QSOs and the hardness of their spectra ((citation not found: Meiksin05)).

#### Direct observation

The most direct evidence for Heii reionization derives from measurements of the Heii Lyman-$$\alpha$$ optical depth ($$\tau_{\text{He{\sc \,ii}}}$$). The evolution of $$\tau_{\text{He{\sc \,ii}}}$$ along a line of sight to a quasar traces the presence of intergalactic Heii ions: if, after the Heii reionization, these ions are located in discrete clumps they will produce discrete absorption lines; however, if they are diffused throughout the IGM (as expected before the reionization) they will smoothly depress the flux level of a quasar bluewards of its Heii emission line (e.g. (citation not found: Heap00); (citation not found: MiraldaescudeQ00)). The latter is known as the “Gunn–Peterson absorption" ((citation not found: GunnPe65)). There are some advantages in the physics of the Heii Gunn–Peterson effect that make its observation potentially more reliable with respect to the hydrogen one. In fact, due to the later redshift of its reionization, the lower abundance of helium versus hydrogen, the shorter wavelength of the Heii Lyman-$$\alpha$$ line (304 Å) and the density fluctuations in the IGM, the Heii Gunn–Peterson trough is sensitive to ion fractions $$x_{\text{He{\sc \,ii} }}\gtrsim 0.01$$, while the hydrogen one saturates at $$x_{\text{H{\sc \,i}}}\simeq 10^{-5}$$ ((citation not found: Fan06); (citation not found: McQuinn09)). In addition, luminous quasars create large fluctuations in the ionizing background, making the flux transmission possible even during the early stages of Heii reionization ((citation not found: Furlanetto09)).

The intergalactic Heii Lyman-$$\alpha$$ transition, at redshifts relevant for the reionization, is observed in the far UV, its detection in quasar spectra is particularly difficult because intervening low-redshift hydrogen absorption along the line of sight can severely attenuate the quasar flux. It is also observable from space only at $$z>2$$ due to the Galactic Hi Lyman limit. Most observations of the Heii Lyman-$$\alpha$$ forest have come from the $$Hubble$$ $$Space$$ $$Telescope$$ ($$HST$$), although some observations of a few brighter targets were possible with the $$Hopkins$$ $$Ultraviolet$$ $$Telescope$$ and the $$Far$$ $$Ultraviolet$$ $$Spectroscopic$$ $$Explorer$$. However, only a few percent of known $$z \simeq 3$$ quasar sightlines were found to be “clean” (free of intervening Hi Lyman limit systems) down to the Heii Lyman limit, and therefore usable for a possible detection of the Gunn–Peterson effect. It has been detected at $$z\gtrsim3$$ ((citation not found: Heap00); (citation not found: Zheng04)), whereas the absorption becomes patchy at $$z\lesssim3$$, reflecting an intermediate phase of reionization, and evolves into Heii Ly-$$\alpha$$ forest (discrete absorption lines) at $$z\lesssim2.7$$ (e.g. (citation not found: Shull10); (citation not found: Worseck11)). In recent years, the advent of the $$Galaxy$$ $$Evolution$$ $$Explorer$$ ($$GALEX$$) UV maps and the installation of the $$Cosmic$$ $$Origin$$ $$Spectrograph$$ ($$COS$$) on $$HST$$ have allowed very high quality re-observations of such known “Heii quasars” ((citation not found: Shull10)) and the discovery of 2 new ones at $$z<3$$ ((citation not found: Worseck11)). But even if in these recent works the end of the Heii reionization seems to be observed at $$z\simeq 2.7$$ (e.g. (citation not found: Worseck11); (citation not found: Syphers13)), there are strong variations between the sightlines and any current constraint on the physics of this phenomenon is limited by the cosmic variance among this small sample studied in detail.

Waiting for the promising cross-matching of new optical quasar catalogs (e.g. BOSS) with $$GALEX$$ UV catalogs, that has been very effective in finding new Heii quasars (e.g (citation not found: Syphers12)), and looking forward to new higher resolution observations with $$COS$$, other indirect methods have been developed to obtain a detailed characterization of the Heii reionization.

#### Indirect evidence

Various indirect methods of observing the Heii reionization have been used, either by examining metal systems or the Hi Ly-$$\alpha$$ forest. Variation of metal-line ratios (like Civ/Siiv; (citation not found: Songaila98)) above or below the Heii Lyman limit (228Å) can be used to determine changes in the ionizing UV background connected with changes in the helium opacity ((citation not found: Furlanetto09)). These measurements generally agree with a $$z\simeq3$$ reionization ((citation not found: Agafonova07)) but can be affected by metallicity variation that can complicate understanding them. The majority of the indirect evidence for the Heii reionization comes from the attempts to exploit the IGM heating associated with this epoch and its effect on the Hi Ly-$$\alpha$$ forest. In particular, one of the effects of an injection of a substantial amount of energy could have been a lower average Hi Ly-$$\alpha$$ opacity ((citation not found: Bernardi03); (citation not found: Faucher-Giguere08)). However, a small “dip” in the Hi Ly-$$\alpha$$ opacity observed at $$z\simeq3.2$$ ((citation not found: Faucher-Giguere08)) does not have a straightforward interpretation ((citation not found: Bolton09); (citation not found: McQuinn09)) and has not been confirmed in recent and refined measurements ((citation not found: Becker12)). On the other hand, the best indirect evidence of the reionization period so far seems to be provided by the study of the IGM temperature evolution.

## The IGM thermal state

\label{IntroThRE} One of the main impacts of the Epochs of Reionization is on the thermal state of the intergalactic medium: the IGM cooling time is long, so the low density gas retains some ‘memory’ of when and how it was reionized ((citation not found: HuiGnedin1997)). At different redshifts, the thermal state of the intergalactic medium can be described through its temperature–density ($$T$$$$\rho$$) relation. In the simplest scenario, for gas at overdensities $$\Delta\lesssim 10$$ ($$\Delta=\rho / \bar{\rho}$$, where \(\ba